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High Reynolds Number Simulation and Drag Reduction Techniques

A Thesis Presented by

J in X u

toDivision of Applied Mathematics

in partial fulfillment of requirements for the degree of

Doctor of Philosophy in the field of

Applied Mathematics

Brown University Providence, Rhode Island

March 2005


UMI Number: 3174698

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This dissertation by Jin Xu is accepted in its present form by the Division of Applied Mathematics as satisfying the dissertation requirement

for the degree of Doctor of Philosophy.

Date

Date

Date

Date

G.E. Karniadakis, Director

Recommended to the Graduate Council

4-3n-o5 I( P C u

(CoT)iM.R. Maxey(Co-Director), Reader

G. Tryggvason, Reader

Approved by the Graduate Council

Karen Newman Dean of the Graduate School.

111


Abstract

Turbulence remains one of the most challenging problems in Physics. The com

plexity of solving turbulence in theory is beyond people’s ability now. The develop

ment of the supercomputer has made it possible to explore the dynamics of turbulent

flows not just through experiments and theoretical analysis as in the past, but also

through Direct Numerical Simulation (DNS).

Since Kim, Moin and Moser published their paper on DNS of turbulent flow

in a channel in 1987 [68], DNS has become an important tool for investigating the

dynamics of wall-bounded, turbulent shear flows. The advantages of using DNS for

turbulence research are: (1) no turbulence closure models are required, as for example

as used in Reynolds Average Equations, and the full Navier-Stokes equations can

be solved without approximation; (2) much more detailed and accurate information

about the flow quantities, which are quite difficult and expensive to measure through

experiment, can be obtained. At the early age of this research, experimental data

was used to check DNS results. Now in some cases this situation has been reversed,

i.e. DNS data has been used to verify the accuracy of experimental results at low

Reynolds number. Since the number of grid points needed to resolve the flow in

DNS is proportional to Re9/4, this approach is presently limited to low Reynolds

number. As supercomputers become faster, simulations at higher Reynolds number

will be feasible. This has great impact in fundamental developments of turbulence

modelling, including industrial complexity application.

iv


The first part of this thesis is an attempt to perform high Reynolds number

DNS and large Eddy Simulation (LES). In order to conduct DNS at high Reynolds

number, we have implemented and benchmarked several different parallel models.

Domain decomposition has been done in either stream-wise only or in both stream-

wise and span-wise directions, based on MPI or OpenMP to parallelize the code. In

order to overcome in part the limitations of Reynolds number on DNS, LES has been

implemented whereby the large scale dynamics of flow are accurately simulated, but

the small scale features are parameterized by an approximate model. LES requires

fewer grid points and less computer time for a simulation. We have conducted rela

tively high Reynolds number DNS, and LES at Re* = 600,1000 using the Spectral

Vanishing Viscosity (SVV) method. SVV has been combined with standard and

dynamic Smagorinsky models. In order to improve the numerical stability, SVV has

been implemented implicitly. The LES results have been compared with DNS, and

show that SVV has potential to be a good approach for LES. We also investigated

the effects of applying SVV to a 2d spectral element discretization, which is more

difficult than before. We added SVV terms in all three directions, and the viscosity

has been added only to high Fourier and soectral polynomial modes. Comparison

of channel turbulence simulation has been performed, and good results have been

achieved. Besides the channel flow, we also investigated DNS and LES for flow past

a cylinder. DNS at Re=3900 has been done. LES at Re=10,000 have also been

conducted.

In the second half of thesis, we have investigated several turbulent drag reduction

techniques using DNS. Turbulence drag reduction by adding micro-bubbles into a

turbulent boundary layer has been well established in experiments. However, it has

been difficult until now to capture such effects in numerical simulations due to a

lack of an accurate interaction model between turbulence and micro-bubbles. In

this thesis, a series of DNS of small bubbles seeded in turbulent channel flow at

average volume fractions of up to 13% have been carried out. The results show that

about 10% drag reduction is reached. This is consistent with low speed and low

v


void fraction experiments, but significantly less than 70% to 80% reductions in skin

friction reported in some experiments. A nondeformable, spherical bubble shape has

been assumed, and the Force Coupling Method (FCM) has been used to simulate

bubbles in turbulent flow. It is found that the bubble size should be small enough to

produce a sustained level of drag reduction over time. Drag reduction effects have

been investigated in detail at different Reynolds numbers.

Inspired by the mechanisms of micro-bubble drag reduction, we have investigated

the effect of a constant streamwise force distribution acting against the flow direc

tion. This produces drag reduction as much as 30% at low Reynolds number and

70% at high Reynolds number. In order to realize such effects, the length scale of the

forcing should be small enough and should be localized near the wall. In addition,

the excessive shear stresses observed during the laminar to turbulence transition can

be substantially reduced. These results provide insights into the dynamics of turbu

lence and drag reduction, and have potential for applications requiring turbulence

suppression.

Finally, motivated by the apparantly lower levels of drag reduction found in the

numerical simulation as compared to many experiments, we consider the possibility

of other physical process not captured so far. One such process is a partial slip flow

condition at the wall. A slip boundary condition can arise from effects of hydrophobic

surfaces or the formation of a thin gas film on the wall. The simulation results

show that a large level of drag reduction can be achieved by applying slip boundary

condition. The effect of combining slip boundary conditions with micro-bubbles has

also been investigated, and a detailed analysis has been carried out.

vi


Acknowledgments

In the past five years, many people have provided invaluable advices and sub

stantial supports, without these my thesis is not be possible to finish. Many things

happened during these 5 years, and many people showed their performances. Some

are really excellent, which benefit me for long time. Some made me more clear of

this society and people.

Among all these people, firstly I would like to express my gratitude to Professor

George Em Karniadakis, my Ph.D advisor in Applied Mathematics. Throughout my

five years at Brown, his advices and suggestions benefit me in many fields. Several

research directions he suggested have been proved later to be correct. His energitic

way of doing research and broad view of knowledge in computational mathematics

impressed me a lot. I have learned a lot of things during these years. I would also

like to acknowledge his great support in providing supercomputer resources which

made it possible to have lots of good results in last 5 years. I am looking forward to

continue our professional relationship, and continue to collaborate with him.

Secondly, I would like to thank my co-advisor, Prof. Martin R. Maxey. Half of

my thesis work has been directed in detail by him. His regorous and attentive way

of research is respectable and that made my research work much easier. Without his

base work on bubbles, I can not publish my first important paper in JFM. Definitely

I would like to keep close connection with him, and collaborate more on fluids and

bubbles researches later.

vii


Among the faculties in Applied Mathematics, I would specially like to thank

professors in scientific computing group. They are all famous and respected. Prof.

Chi-Wang Shu, Prof. David Gottlieb, and Prof. Jan Hesthaven, either their classes

or qualification exam made me progress forward.

CRUNCH group has produced many excellent researchers, like Dr. Ron D. Hen

derson, Dr. Spencer Sherwin, Dr. Tim Warburton, Dr. Mike Kirby, et al, and that

attract me to Brown. I am very glad to have opportunities to work and discuss with

these former students. And I hope to have a chance of collaborating with them in

the future.

I would also like to thank Dr. Steve Dong and Ma Xia, especially Steve, for their

collaboation and helps in my research works. Nearly half of my thesis works have

been cooperated with Steve , his diligent and concentration impressed me a lot. The

cooperation between us is long and helpful. And I wish to have opportunities to

continue later.

I came to CRUNCH group at the same time with Vasileios, and we went through

all these 5 years. The cooperation and experience in research and class are cheerful.

I would also like to acknowledge some of the younger member in the group, Guang

Lin, Xiaoliang Wan, Leopold Grinberg and Jasmine Foo. They are fresh flood in

this group, and axe all clever guys.

I would like also to give many thanks to the system administrators: Dave John

son, Jie Zhang, George Loriot, Sam Fulcomer, Melih Bitim. I have caused so many

troubles either local or outside during these 5 years, which have been recovered by

their great works. Without their support, I can not complete my work successfully.

Same thankfulness should also give to the secretaries in Applied Math., Made

line Brewster, Janice D’Amico, Jean Radican, Laura Leddy, Roselyn Winterbottom.

They gave me much help in daily life , and their work created a nice working envi-

viii


ronment in Applied Math.

At last, I would like to thank my wife (Iris) Bing Jing, her support is extremly

important to me during these five years. Without her, I can not imagine what will

be now. This thesis is also a comfort for her great support and effort. I will and am

confident to have more success to share with her in the future.

The computation work have been done in Pittsburgh Supercomputer center

(PSC), Arctic Supercomputer Center (ARSC), The Naval Oceanographic Office Ma

jor Shared Resource Center (NAVOCEANO MSRC), National Center for Supercom

puting Applications (NCSA), and National Partnership for Advanced Computational

Infrastructure (NPACI).

This work was supported by following grant:

DARPA: SO 4-15

AFOSR: F49620-03-4-0218

AFOSR: F49620-01-1-0035

DOE: DE-FG02-95ER25239

DARPA/ONR: N00014-01-1-0177

DARPA: MDA972-01-C-0024

ix


Contents

1 Introduction 1

1.1 Turbulent Flow S im u la tio n .................................................................. 2

1.1.1 Reynolds-Averaged Equations (RANS)........................................ 2

1.1.2 Large Eddy Simulation (L ES)..................................................... 2

1.1.3 Direct numerical Simulation (DNS)............................................ 3

1.2 Turbulent Drag Reduction Techniques................................................ 3

1.2.1 Micro bubble Drag Reduction..................................................... 4

1.2.2 Constant forcing con tro l.............................................................. 6

1.2.3 Slip boundary condition.............................................................. 6

1.3 Motivation .............................................................................................. 6

1.4 Objectives................................................................................................. 7

1.4.1 DNS and LES................................................................................. 8

x


1.4.2 Drag reduction techniques........................................................... 8

1.5 Outline of T h es is ...................................................................................... 9

2 Spectral Element M ethod for Channel Flow 10

2.1 Numerical Method and Parallel Models .............................................. 10

2.1.1 Numerical M eth o d ....................................................................... 10

2.1.2 Direct and Iterative Solver ................................................... . 13

2.1.3 Parallel Im plem entation.............................................................. 14

2.2 Validation/Verification.............................................. 18

2.2.1 Kovasznay Flow .......................................................................... 19

2.2.2 3D Accurate solution.................................................................... 19

2.2.3 Turbulence Statistics.................................................................... 20

2.3 Parallel Benchm arks................................................................................ 21

2.3.1 Comparison on different models................................................... 21

2.3.2 Comparison on different p la tfo rm s............................................ 23

2.3.3 Comparison of Model B in x and z direction............................. 23

2.3.4 Comparison of Model C in (x,z) p la n e ...................................... 25

2.3.5 Comparison of Model B for Re* = 400 ...................................... 25

xi


2.4 High Reynolds DNS r e s u l t s ................................................................... 27

2.4.1 Re* = 400 ..................................................................................... 28

2.4.2 Re* = 600 ..................................................................................... 30

2.4.3 Re* = 1000 .................................................................................. 33

2.5 S u m m ary ................................................................................................. 33

3 High Reynolds number Large Eddy Simulation 39

3.1 Background.............................................................................................. 39

3.2 LES methods and im plem entation....................................................... 42

3.2.1 Basic Assumption and C o n c e p ts .............................................. 42

3.2.2 Filtering Techniques and Implementations.............................. 43

3.2.3 Filtered Navier-Stokes E quations.............................................. 45

3.2.4 Energy Balance E q u a tio n .......................................................... 47

3.3 Subfilter Scale M o d e ls ............................................................................ 49

3.3.1 Smagorinsky Eddy-Viscocity Model ........................................ 49

3.3.2 Spectral Vanishing Viscosity .................................................... 51

3.3.3 Implicit Spectral Vanishing Viscosity Implementation . . . . 53

3.3.4 Filtering in Orthogonal B a s is .................................................... 54

xii


3.3.5 Dynamic LES M odel.................................................................... 56

3.4 LES R esults............................................................................................... 57

3.4.1 Comparison to standard Smagorinsky M o d e l .......................... 57

3.4.2 LES results at low Reynolds num b er......................................... 62

3.4.3 LES Results at high Reynolds n u m b er..................................... 70

3.5 S u m m ary ................................................................................................... 78

4 Large Eddy Simulation in Complex Geometry 80

4.1 Background............................................................................................... 80

4.2 LES methods and im plem entation........................................................ 81

4.2.1 Standard SVV m ethod................................................................ 81

4.2.2 Smagorinsky Model ................................................................... 82

4.3 Channel F lo w ............................................................................................ 84

4.4 Cylinder F low ............................................................................................ 88

4.4.1 Verification of DNS at Re=3900 ............................................... 88

4.4.2 DNS and LES at Re=10,000 ..................................................... 90

4.5 S u m m ary ................................................................................................... 102

5 Simulation M ethods for Bubbles/Particles in Channel Flow 103

xiii


5.1 Background Review ................................................................................ 103

5.2 Simulation M ethod ................................................................................... 105

5.3 Validation and Verification .................................................................... 107

5.3.1 Validation...................................................................................... 107

5.3.2 Verification................................................................................... 116

5.4 Flow Analysis............................................................................................ 118

5.5 Collision M odel......................................................................................... 120

5.6 Parallel Implementation and B enchm arks........................................... 122

5.6.1 Parallel Im plem entation.............................................................. 122

5.6.2 Benchmarks................................................................................... 123

5.7 S u m m ary .................................................................................................. 124

6 Microbubble Drag Reduction 126

6.1 Low Reynolds Number Flow .................................................................... 129

6.1.1 Fluid characteristics.................................................................... 130

6.1.2 Bubble characteristics................................................................. 139

6.1.3 Bubbles versus P artic les .............................................................. 143

6.1.4 V isualization................................................................................ 145

xiv


6.2 High Reynolds Number F lo w ................................................................. 148

6.2.1 Fluid characteristics.................................................................... 148

6.2.2 Bubble characteristics................................................................. 153

6.2.3 Dispersion....................................................................................... 159

6.2.4 V isualization................................................................................. 164

6.3 S u m m ary ................................................................................................... 164

7 Drag Reduction by Constant Forcing 168

7.1 Near-wall forcing and simulation m ethod............................................... 171

7.2 Results on drag red u c tio n ....................................................................... 173

7.3 Turbulence m odification.......................................................................... 181

7.3.1 Mean velocity profile.................................................................... 181

7.3.2 Reynolds stresses........................................................................... 183

7.3.3 Vorticity fluctuations.................................................................... 186

7.3.4 Influence of force am plitude........................................................ 190

7.4 Flow V isualization................................................................................... 194

7.5 Summary and D iscussion ....................................................................... 202

8 Slip Flow DNS 207

xv


8.1 Background.............................................................................................. 207

8.2 Slip boundary cond ition ......................................................................... 208

8.2.1 Verification................................................................................... 209

8.3 Drag Reduction Under Different Slip Length b (Constant Slip Length) 210

8.4 Combined Slip with Bubbles................................................................... 216

8.5 V isualization ........................................................................................... 219

8.6 S u m m ary ................................................................................................. 225

9 Summary and Conclusion 226

xvi


List of Tables

2.1 KMM’s and MKM’s DNS runs................................................................. 27

2.2 Current DNS runs...................................................................................... 28

2.3 DNS runs at Re* = 1000............................................................................ 33

3.1 Simulation parameters for DNS................................................................ 58

3.2 LES runs at different Reynolds number.................................................. 62

4.1 Simulation parameters for flow pass cylinder at Re=10,000.................. 94

5.1 Experimental values for a single particle rising in an inclined channel. 109

5.2 Computational parameters for the single particle rising in an inclined

channel. The characteristic length L = — 10mm and the charac

teristic velocity U — lOmm/s 110

6.1 Simulation parameters and drag reduction............................................. 129

xvii


6.2 Balance check at Re* = 380 for t=20, Term I for the no bubble case

is 14.55e-4..................................................................................................

6.3 Dispersion data at Re* = 380 ................................................................

7.1 Simulation parameters: Reynolds numbers; domain size; numerical

resolution including the number of elements and the spectral order;

and mean pressure gradient, scaled by pU^/h, of the base flow. . . .

7.2 Percentage change in long term drag at Re* = 135 for various A and

I. Results to within ±2%.........................................................................

7.3 Long term, percentage reduction in turbulent drag from simulations

at Re* = 135,192,380,633, for various values of I and A. Results for

drag to within ±2 points...........................................................................

xviii


List of Figures

2.1 Sketch for channel dom ain....................................................................... 11

2.2 Spectral accuracy of iterative solver, N is the grid p o in ts .................... 14

2.3 Sketch for Model A parallelization ........................................................ 15

2.4 Sketch for Model B parallelization ........................................................ 16

2.5 Sketch of the first way for Fourier transform in Model BI ................ 17

2.6 Sketch of the second way for Fourier transform in Model BII . . . . 17

2.7 Sketch for Model C parallelization ........................................................ 18

2.8 Spectral convergence of solving 2d Kovasznay flow p ro b le m ............. 19

2.9 Spectral convergence solving 3d problem ............................................... 20

2.10 Re* = 180, Solid: KMM; Dash: SEM solver. Mean Velocity (left);

Reynold Stress ( r ig h t ) ............................................................................ 21

2.11 Re* = 180, Solid: KMM; Dash: SEM solver. Turbulence fluctuation

(left); turbulence vorticity fluctuation (right) ..................................... 22

xix


2.12 Model C: Speed comparison of different models on S G I.................... 22

2.13 Speed on different platforms: Model A (left), Model B (right) . . . . 23

2.14 Model B: Speed comparison for MPI in x and z directions: SGI(left),

SP4(right).................................................................................................. 24

2.15 Model B: Speed comparison in (x,z) plane: SGI(left), SP4(right) . . 24

2.16 Model C: Speed comparison in x(MPI) and z(OpenMP) directions . 25

2.17 Model C: Speed comparison in x(MPI) and z(OpenMP) directions . 26

2.18 Model B: scaling on large number of processors, IBM SP4, ARSC . . 27

2.19 Re* = 380, Solid: KMM; Dash: SEM solver. Mean Velocity normal

ized by (left); Reynold Stress normalized by u*2(right)................... 28


normalized by u*(left); turbulence vorticity fluctuation normalized by

u*2/ ^(rig h t)............................................................................................... 29

2.21 Streaks of Re* — 380 at y+ = 5.............................................................. 29

2.22 Vortices at Re* = 380............................................................................... 30

2.23 Re* — 600, Solid: KMM; Dash: SEM solver. Mean Velocity normal

ized by it* (left); Reynold Stress normalized by it*2(right)................... 31


normalized by u*(left); turbulence vorticity fluctuation normalized by

u*2/^ ( r ig h t) ............................................................................................... 31

2.25 Streaks of Re* = 633 at y+ = 5.............................................................. 32

xx


2.26 Vortices at R e* = 633 32

2.27 Re* = 1000. Mean Velocity normalized by u*(upper), Solid: KMM;

Dash: SEM solver; Reynold Stress normalized by w*2(low )................. 34


normalized by u*(upper); turbulence vorticity fluctuation normalized

by u*2/v (\o v f) ............................................................................................ 35

2.29 Streaks of Re* = 1000 at y+ = 5............................................................. 36

2.30 Q contour at Re* = 1000......................................................................... 37

3.1 SVV in Fourier Space ............................................................................ 54

3.2 Gaussian Filter (left); exponentail Filter (r ig h t)................................... 57

3.3 Mean Velocity, 40 x 41 x 40. Smagorinsky SVV, Solid line: DNS on

128 x 130 x 128; dash line: coarse DNS, one element; dot line: LES

using SVV, C8 = 0.005; dashdot line: standard Smagorinsky model,

Cs = 0.005 (upper); Dynamic Smagorinsky SVV, Solid line: DNS on

128 x 130 x 128; dash line: coarse DNS, one element; dot line: LES

using SVV (C=0.125); dashdot line: standard Dynamic Smagorinsky

model (below)............................................................................................ 59

3.4 Reynold Stress. Smagorinsky model, solid line: DNS on 128 x 130 x

128; dash line: coarse DNS, one element; dot line: LES using SVV,

Cs = 0.005; dashdot line: standard Cs = 0.005 (left), Dynamic

Smagorinsky model, solid line: DNS on 128 x 130 x 128; dash line:

coarse DNS, one element; dot line: LES using SVV (C=0.125); dash

dot line: standard (0.125) ( r ig h t) .......................................................... 60

xxi


3.5 Turbulence fluctuation. Smagorinsky model, solid line: DNS on 128 x

130 x 128; dash line: coarse DNS, one element; dot line: LES using

SVV, Cs = 0.005; dashdot line: standard Cs = 0.005 (upper), Dy

namic Smagorinsky model, solid line: DNS on 128 x 130 x 128; dash

line: coarse DNS, one element; dot line: LES using SVV (C=0.125);

dashdot line: standard (0.125) (below )................................................. 61

3.6 RE* = 180. Mean Velocity. 40 x 65 x 40, Smagorinsky SVV, Cs =

0.005. solid line: DNS on 128 x 130 x 128 dashed line: LES using

SVV, Cutoff=0; dotted line: LES using SVV, Cutoff=l; dash-dot

line: LES using SVV, Cutoff=2 (upper); Dynamic Smagorinsky SVV,

C = 0.05. solid line: DNS on 128 x 130 x 128; dashed line: coarse

DNS; dotted line: LES using SVV, cutoff=0; dash-dot line: LES using

SVV, Cutoff=l; long dashed line: LES using SVV, cutoff=2; dash-dot

dotted: LES using SVV, cutoff=3 (below )........................................... 63

3.7 RE* = 180. Reynold Stress. Smagorinsky (40 x 65 x 40, Cs = 0.005),

solid line: DNS on 128 x 130 x 128; dashed line: LES using SVV,

cutoff=0; dotted line: LES using SVV, cutoff=l; dash-dot line: LES

using SVV, cutoff=2 (left); Dynamic Smagorinsky (40 x 65 x 40, C =

0.05), solid line: DNS on 128 x 130 x 128; dashed line: coarse DNS;

dotted line: LES using SVV, cutoff=0; dash-dot line: LES using SVV,

cutoff=l; long dashed line: LES using SVV, cutoff=2; dash-dot dotted

line: LES using SVV, cutoff=3 ( r ig h t) ................................................. 64

xxii


3.8 RE* = 180. Turbulent fluctuation. 40 x 65 x 40, Smagorinsky SVV,

Cs = 0.005, solid line: DNS on 128 x 130 x 128; dashed line: LES

using SVV, cut off =0; dotted line: LES using SVV, cutoff=l; dash-dot

line: LES using SVV, cutoff=2 (upper); Dynamic Smagorinsky SVV,

C — 0.05, solid line: DNS on 128 x 130 x 128; dashed line: coarse


SVV, cutoff=l; long dashed line: LES using SVV, cutoff=2; dash-dot

dotted: LES using SVV, cutoff=3 (below )........................................... 65

3.9 RE* = 180. SVV viscosity (40 x 65 x 40, one element): Smagorinsky,

Cs = 0.005 (left); Dynamic Smagorinsky, C = 0.05 ( r ig h t) ............... 66

3.10 Mean Velocity. 40 x 65 x 40, two elements, Smagorinsky SVV, Cs =

0.01. solid line: DNS on 128 x 130 x 128; dashed line: coarse DNS;

dotted line: LES using SVV, cutoff=l; dash-dot line: LES using SVV,

cutoff=2; long dashed line: LES using SVV, cutoff=3; dash-dot dotted

line: LES using SVV, cutoff=4 (upper); Dynamic Smagorinsky SVV,

C — 0.075. solid line: DNS on 128 x 130 x 128; dashed line: coarse


SVV, cutoff=l; long dashed line: LES using SVV, cutoff=2; dash-dot

dotted line: LES using SVV, cutoff=3 (below)..................................... 67

3.11 Reynold Stress. Smagorinsky (40 x 65 x 40, two elements, Cs = 0.01),

solid line: DNS on 128 x 130 x 128; dashed line: coarse DNS; dotted

line: LES using SVV, cutoff=l; dash-dot line: LES using SVV, cut-

off=2; long dashed line: LES using SVV, cutoff=3; dash-dot dotted:

LES using SVV, cutoff=4 (left); Dynamic Smagorinsky (40 x 65 x 40,

two elements, C = 0.075), solid line: DNS on 128 x 130 x 128; dashed

line: coarse DNS; dotted line: LES using SVV, cutoff=0; dash-dot

line: LES using SVV, cutoff=l; long dashed line: LES using SVV,

cutoff=2; dash-dot dotted line: LES using SVV, cutoff=3 (right) . . 68

xxiii


3.12 Reynold Stress. 40 x 65 x 40, two elements, Smagorinsky SVV, Cs =

0.01, solid line: DNS on 128 x 130 x 128; dashed line: coarse DNS;

dotted line: LES using SVV, cutoff=l; dash-dot line: LES using

SVV, cutoff=2; long dashed: LES using SVV, cutoff=3; dashdotdot

line: LES using SVV, cutofF=4 (upper); Dynamic Smagorinsky SVV,

C=0.075, solid line: DNS on 128 x 130 x 128; dashed line: coarse DNS;

dotted line: LES using SVV, cutoff=0; dash-dot line: LES using SVV,

cutoff=l; long dashed line: LES using SVV, cutoff=2; dash-dot dotted

line: LES using SVV, cutoff=3 (below)................................................ 69

3.13 SVV viscosity. Smagorinsky (left); Dynamic Smagorinsky (right) . . 70

3.14 Mean velocity. 80 x 129 x 80, two elements, Cs = 0.005. Smagorinsky

SVV, solid line: DNS on 384*361*384; dashed line: coarse DNS; dot

ted line: LES using SVV, cutoff=0; dash-dot line: LES using SVV,

cutoff=2; dash-dot dotted line: LES using SVV, cutoff=4 (upper); Dy

namic Smagorinsky SVV, C = 0.02, solid line: DNS on 384 x 361 x 384;

dashed line: coarse DNS; dotted line: LES using SVV, cutoff=0; dash-

dot line: LES using SVV, cutoff=2; long dashed line: LES using SVV,

cutoff=4 (b e lo w )...................................................................................... 71

3.15 Reynold Stress. 80 x 129 x 80, two elements, Smagorinsky SVV,

Cs = 0.005, solid line: DNS on 384 x 361 x 384; dashed line: coarse

DNS; dotted line: LES using SVV, cutoff=0; dash-dot line: LES

using SVV, cutoffs2; dashdotdot line: LES using SVV, cutoff=4

(upper); Dynamic Smagorinsky SVV C = 0.02, solid line: DNS on

384 x 361 x 384; dashed line: coarse DNS; dotted line: LES using

SVV, cutoff=0; dash-dot line: LES using SVV, cutoff=2; long dashed

line: LES using SVV, cutoff=4 (below)................................................ 72

xxiv


3.16 Turbulent fluctuation. 80 x 129 x 80, two elements, Smagorinsky SVV,

Cs = 0.005, solid line: DNS on 384 x 361 x 384; dashed line: coarse


SVV, cutoff=2; dashdotdot line: LES using SVV, cutoff=4(upper);

Dynamic Smagorinsky SVV, C = 0.02, solid line: DNS on 384 x

361 x 384; dashed line: coarse DNS; dotted line: LES using SVV,

cutoff=0; dash-dot line: LES using SVV, cutoff=2; long dashed line:

LES using SVV, cutoff=4 (below).......................................................... 73

3.17 SVV Viscosity. Smagorinsky, Cs = 0.005 (left); Dynamic Smagorin

sky, C=0.02 (rig h t)................................................................................... 74

3.18 Mean velocity. 128 x 193 x 128, 2 elements, Cs = 0.012. Smagorin

sky SVV, solid line: DNS on 768 x 521 x 768; dashed line: coarse

DNS; dotted line: CutofF=0; dash-dot line: Cutofl=2; long dashed

line: Cutoff=4; dash-dot dotted line: Cutoff=6 (upper); Dynamic

Smagorinsky SVV, solid line: DNS on 768 x 521 x 768; dashed line:

coarse DNS; dotted line: Cutoff=0; dash-dot line: Cutoff=2; long

dashed line: Cutoff=4; DashDotDot line: Cutoff=6 (b e lo w ) 75

3.19 Reynold Stress. Smagorinsky(128 x 193 x 128,2 elements,Cs = 0.012),

solid line: DNS on 768 x 521 x 768; dashed line: coarse DNS; dotted

line: LES using SVV, Cutoff=0; dash-dot line: LES using SVV, Cut

o ff^ ; long dashed line: LES using SVV, Cutoff=4; dashdotdot line:

LES using SVV, CutofL=6 (upper); Dynamic Smagorinsky(128 x 193 x

128,2 elements,C = 0.028), solid line: DNS on 768 x 521 x 768; dashed

line: coarse DNS; dotted line: LES using SVV, CutofF=0; dash-dot

line: LES using SVV, Cutoff=2; long dashed line: LES using SVV,

Cutoff=4 (below )...................................................................................... 76

xxv


3.20 Turbulent fluctuation. Smagorinsky(128 x 193 x 128,2 elem ents,^ =

0.012), solid line: DNS on 768 x 521 x 768; dashed line: coarse DNS;

dotted line: LES using SVV, Cutoff=0; dash-dot line: LES using SVV,

Cutoff=2; long dashed line: LES using SVV, Cutoff=4; dashdotdot

line: LES using SVV, Cutoff=6 (upper); Dynamic Smagorinsky(128 x

193 x 128,2 elements,C = 0.028), solid line: DNS on 768 x 521 x 768;

dashed line: coarse DNS; dotted line: LES using SVV, Cutoff=0;

dash-dot line: LES using SVV, Cutoff=2; long dashed line: LES using

SVV, Cutoff=4 (below) ......................................................................... 77

3.21 SVV Viscosity. Smagorinsky,Cs = 0.012(left); Dynamic Smagorinsky,

C=0.028 ( r ig h t) ................................................................................ 78

4.1 Mesh for Re* = 180 (x,y) p la n e ..................................................... 83

4.2 Viscosity profile at Re* = 180, Smagorinsky model, Cs = 0.005, Chan

nel code, Solid line: No filter; Dash line: Filtering all polynomial

coefficients of Smagorinsky viscosity larger than 4 to z e r o ................. 84

4.3 Viscosity profile at Re* = 180, LES Smagorinsky model, Cs = 0.005,

n e k ta r F ..................................................................................................... 85

4.4 Coarse DNS turbulent fluctuations at Re* = 180, Solid line: Channel

code; Dash line: NektarF code ............................................................. 86

4.5 Turbulent fluctuation at Re* = 180 using SVV, e = 0.5, MN=0,

MNF=0. Solid:Channel code; Dash:nektarF c o d e ............................... 86

4.6 Turbulent fluctuation at Re* = 180 using SVV, e = 1.0, MN=3,

MNF=3. Solid line: Channel code; Dash line: NektarF code 87

xxvi


4.7 Turbulent fluctuation at Re* = 180 using Smagorinsky model, Cs =

0.005, MN=0, MNF=0. Solid line: Channel code; Dash line:nektarF

c o d e ........................................................................................................... 87

4.8 Mesh for Re=3900 Full m e s h ................................................... 88

4.9 Mesh for Re=3900 Local mesh around cy lin d e r.................... 89

4.10 Drag and lift coefficients for Re=3900 ................................................. 89

4.11 Mean and flucuation velocity profile at x = l.06,1.54,2.02 and 3 for

Re=3900. Solid line:George Karamanos’s results; Dash line: our results 91

4.12 Mean and flucuation velocity profile at x=4,7 and 10 for Re=3900.

Solid line:George Karamanos’s results; Dash line: our results . . . . 92

4.13 Mesh for Re= 10,000 Full mesh obtained from Steve D o n g . 93

4.14 Mesh for R,e=10,000 Local mesh around cy lin d e r..... 93

4.15 Viscosity contour for Smagorinsky Model at Re=10,000 ................. 94

4.16 Experiment Reynolds Stress from A. Ekmekci[36].............................. 95

4.17 Case I. Reynolds Stress (upper); Drag and lift coefficient history (lower) 96

4.18 Case VI. Reynolds Stress (upper); Drag and lift coefficient history

(lower)........................................................................................................ 97

4.19 Case VII. Reynolds Stress (upper); Drag and lift coefficient history

(lower)........................................................................................................ 98

4.20 Averaged streamwise velocity profile at x=2,5,10 for Re=10,000. . . 99

xxvii


4.21 Fluctuation U velocity profile at x=2,5,10 for Re=10,000 100

4.22 Fluctuation V velocity profile at x=2,5,10 for Re=10,000.................... 101

5.1 Experimental Setup ................................................................................ 107

5.2 Comparision of experimental and computed particle trajectory (a)

and velocities ((b) upward and (c) lateral) in an inclined channel for

Re'pMX = 0.0145. (•) Experiment, (-) FCM with the monopole term,

(— —) S. Lomholt’s monopole only. The line (— • —) indicate the

direction of gravity in the frame of the channel. The particle positions

are given in mm and the velocities are in mm/s in the frame of the

experimental setup..................................................................................... I l l



Re™ax = 0.044. (•) Experiment, (-) FCM with the monopole term,



are given in mm and the velocities axe in mm/s in the frame of the

experimental setup..................................................................................... 113



Re™ax — 0.84. (•) Experiment, (-) FCM with the monopole term,




experimental setup..................................................................................... 114

xxviii




Re™ax = 7.9. (•) Experiment, (-) FCM with the monopole term,




experimental setup......................................................................... 115

5.6 Configuration(left), Bubble pressure(right)................................ 117

5.7 Bubble velocity: u(left), v ( r ig h t) ............................................... 117

5.8 Configuration(left), Bubble pressure(right)................................ 118

5.9 Bubble velocity: u(left), v ( r ig h t) .................................................. 119

5.10 Collision M odel................................................................................. 122

5.11 Bubble Parallelization.................................................................. 123

6.1 Normalized drag force against time t at Re* = 135 for: 1, no bubbles;

2, 242 bubbles and a+ = 20; 3, 800 bubbles and a+ = 13.5; 4, 2450

bubbles and a+ = 10...................................................................... 130

6.2 Normalized drag force against time t at Re* = 135 comparing random

to near-wall seeding of 800 bubbles of size o+ = 13.5: (solid), no

bubbles; (dots), random seeding; (dash), near-wall seeding...... 131

6.3 Balance history at Re* — 135, a+ = 13.5 (left); a+ = 20 (right).

1-Wall friction; 2-Pressure drop; 3-Bubble acceleration; 4-Residue . 133

6.4 Balance history for a+ = 40.5 at Re* = 135................................... 133

xx ix


6.5 Mean and conditional mean velocity profiles for a+ — 13.5 averaged

over t = 50 — 80.......................................................................................... 134

6.6 Reynolds stress profile for a+ = 13.5 averaged over t = 50 — 80. Solid

line is turbulent flow without bubbles ................................................. 135

6.7 RMS profiles compared for a+ = 10,13.5. a+ = 13.5 has been aver

aged over t=50-80, and a+ = 10 has been averaged over t=50-65 . . 136

6.8 Reynolds stress profiles averaged over different intervals for case (Ilia):

(1) No bubbles; (2) t = 0-10; (3) = 10-20; (4) t = 20-30; (5) t = 30-40. 136

6.9 Reynolds stress profiles averaged over different intervals for case (V):

(1) No bubbles; (2) t = 0-10; (3) t = 10-20; (4) t = 20-30; (5) t =

30-40............................................................................................................. 137

6.10 RMS vorticity profiles averaged for t=50-80 for a+ = 13.5.................. 137

6.11 RMS vorticity profiles at t=59 for a+ = 10............................................ 138

6.12 Void fraction profiles for a+ = 13.5 (top) and a+ = 10 (bottom)

at:(solid), t = 20; (dash), t = 40; (dots), t = 60; (dash-dots), t = 80. . 138

6.13 Void fraction profiles for random seeding of 800 bubbles, a+ — 13.5:

left, t = 20,40,60,80; right, long term average compared to theory. . 139

6.14 Force profile for a+ = 10 at Re* = 135 averaged over t = 50 — 65.

Solid line is < / >, Dash line are from Eqn:(6.8) and (6 .9 )............... 141

6.15 Force profile for a+ = 13.5 at Re* = 135 averaged over t = 50 — 80.

Solid line is < / >, Dash line are from Eqn:(6.8) and (6 .9 )............... 141

6.16 Drag history of adding bubbles and particles......................................... 144

xxx

6.17 Force profile of adding partic les............................................................ 144

6.18 Comparation of Rms by adding particles or n o t ................................. 145

6.19 Re* = 135 at y+ = 5 ............................................................................... 146

6.20 Re* = 135 at y+ = 5, 800 bubbles o+ = 13.5....................................... 146

6.21 Q contour at Re* = 135. Threshold is 0 .1 6 1 8 .................................... 147

6.22 Q contour at Re* = 135 with 800 a+ = 13.5 bubbles. Threshold is

0 .1 6 1 8 147

6.23 Drag history for each wall at Re* = 380: 9100 bubbles a+ = 13.5

(left); 3200 bubbles a+ = 19 (right). Solid line is the mean drag for

no bubble f l o w ......................................................................................... 149

6.24 Drag history at Re* = 380: 6400 bubbles a+ = 19 (left); 9600 bubbles

a+ = 19 (right). Solid line is the mean drag for no bubble flow . . . 149

6.25 Balance at Re* = 380, 1-Wall friction; 2-Pressure drop; 3-Bubble

acceleration; 4-Residue. 9100 bubbles a+ = 13.5 (left); 3200 bubbles

a+ = 19 (right)........................................................................................... 150

6.26 Balance at Re* = 380, 1-Wall friction; 2-Pressure drop; 3-Bubble

acceleration; 4-Residue. 6400 bubbles a+ = 19 (left); 9600 bubbles

a+ = 19 (right)........................................................................................... 150

6.27 Turbulent fluctuation at Re* = 380, averaged for t=15-20: 9100 bub

bles a+ = 13.5 (left); 3200 bubbles a+ = 19 ( r ig h t) ............................ 151

6.28 Turbulent fluctuation at Re* = 380, averaged for t=15-20: 6348 bub

bles a+ = 19 (left); 9600 bubbles a+ = 19 ( r ig h t) ............................... 151

xxxi


6.29 Reynolds Stress at Re* = 380: 9100 bubbles a+ = 13.5, averaged for

t=10-12 (left); 3200 bubbles a+ = 19, averaged for t=15-20 (right) . 152

6.30 Reynold Stress at Re* = 380, averaged for t=15-20: 6348 bubbles

a+ = 19 (left); 9600 bubbles a+ = 19 (righ t)........................................ 152

6.31 Concentration profile at Re* = 380: 9100 bubbles a+ = 13.5 (left);

3200 bubbles a+ = 19 ( r ig h t ) ................................................................ 153

6.32 Concentration profile at Re* = 380: 6400 bubbles o+ = 19 (left); 9600

bubbles a+ = 19 ( r ig h t ) ............................................................... 154

6.33 Force profile at t=20 for Re* = 380: 9100 bubbles a+ = 13.5 (left);

3200 bubbles a+ = 19 ( r ig h t ) ................................................................ 154

6.34 Force profile at t=20 for Re* = 380: 6400 bubbles a+ — 19 (left);

9600 bubbles a+ — 19 ( r ig h t ) ................................................................ 155

6.35 Variations of f ^ h UiX2 2 dx2 for 9600 (a+ = 19) bubbles at Re* = 380 157

6.36 Bubble Reynold stress at Re* = 380, 9600 bubbles a+ = 19: < Cv2 >

(left); < Cw2 > (righ t)............................................................................. 158

6.37 Bubble Reynold stress at Re* = 380, 9100 bubbles a+ = 13.5: <

Cv2 > (left); < Cw 2 > (right)................................................................ 159

6.38 Dispersion for 9600 a+ = 19 bubbles at Re* = 380 ............................ 159

6.39 Re* = 380, 3200 a=0.05 bubbles. t=5 (upper left); t=10 (upper

right); t=20 (low left); Dispersion relation (low right) ...................... 161



xxxii



6.42 Re* = 380, 9600 a=0.05 bubbles............................................................. 164

6.43 Q contour at Re* = 380 for base flow (upper); for 9100 bubbles

a=0.0355 (middle); and 9600 bubbles a=0.05 bubbles (below), all at

t= 2 0 ........................................................................................................... 165

6.44 Re* = 400, 14400 a=0.05 bubbles.......................................................... 166

7.1 Profile of the gradient of the Reynolds shear stress, scaled by v%/u, against

distance from the wall y+. Results at Re* — 135,200,380,633 ................... 169

7.2 Profiles of the average streamwise force density for bubbles and particles in

a channel flow at Re* = 135,200.................................................................. 170

7.3 Idealization of the excitation force and notation; see equation (??)............. 173

7.4 Time history of the skin friction at Re* = 135, group AA, with I = 0.02,

for: (1) No forcing; (2) A+ = 13.5; (3) A+ = 18.9; (4) A+ = 35.1.................. 174

7.5 Time history of normalized skin friction for A+ = 18.9 and different ampli

tudes I at Re* = 135, group AA: (1) No forcing; (2) I = 0.01; (3) I — 0.02;

(4) I = 0.03................................................................................................... 175

7.6 Time history of the skin friction at Re* = 633 with I = 0.4 for A+ =

7.0,12.0,13.3................................................................................................. 177

7.7 Percentage of drag reduction as a function of the Reynolds number............ 180

xxxiii


7.8 Mean velocity profiles, normalized with the friction velocity of the base flow:

(A) A+ = 13.5,1+ = 0.073 at Re* = 135; (B) case B3; (C) case C3; (D) case

D3. The solid line (S) represents the no-control case at Re* = 380............. 182

7.9 Profiles of normalized mean velocity gradient in wall variables for Re* = 135

and A+ = 13.5, I+ = 0.109; case B3; case C3; and case D3. Solid line is the

no-control result at Re* = 380...................................................................... 182

7.10 Profiles of rms velocity fluctuations at Re* = 380. Solid lines denote the

no-control results; dashed lines denote results for case C4............................ 184

7.11 Profiles of Reynolds shear stress at (a) Re* = 135, case AA; (b) Re* = 380,

case C4. Solid lines denote the no-control results......................................... 184

7.12 Profiles of Reynolds stress at Re* = 380 for cases C3, A+ = 8.7; C4, A+ =

13.7; all for 1=0.2.......................................................................................... 185

7.13 Profiles of Reynolds stress at Re* = 633 for cases D2, A+ = 10.8; D3,

A+ = 12.0; D4, A+ = 13.3; all for 1=0.4........................................................ 186

7.14 Profiles of rms vorticity fluctuations at Re* = 380: uj[. Results for A+ =

12.9, case C3; A+ = 13.7, case C4; no-control................................................ 187

7.15 Profiles of rms vorticity fluctuations at Re* = 380: w'2. Results for A+ =


7.16 Profiles of rms vorticity fluctuations at Re* = 380: uj'3. Results for A+ =


7.17 Turbulent charge at Re* = 380. Solid lines: no-control; dash lines:

I = 0.2, A = 0.034....................................................................................... 189

xxxiv


7.18 Profiles of rms velocity fluctuations and Reynolds stress at Re* = 380 for

case CIA....................................................................................................... 191

7.19 Profiles of rms vorticity fluctuations at Re* = 380 for case C2.................... 192

7.20 Profiles of Reynolds stress gradient, vortex stretch and vortex transport for

case C3, case CIA and the base flow............................................................ 193

7.21 Contours of Q at Re* = 380, no control (upper) and case C4 (lower). . . . 195

7.22 Contours of Q at Re* = 380 for case C5...................................................... 195

7.23 Contours of streamwise velocity in the x\ — X3 plane at Re* = 380, no control

(left) and case C4 (right): (a) y+ = 5; (b) y+ = 10; (c) y+ = 30................... 197

7.24 No control Q detection at Re* = 633........................................................... 198

7.25 Control case: A = 0.019,1=0.4 at Re* = 633 ............................................. 198

7.26 Contours of streamwise velocity in the x\ — X3 plane at y+ = 5, Re* = 380,

no control (left) and case C3 (r ig h t).......................................................... 199

7.27 Contours of streamwise velocity in the xi - X3 plane at y+ = 10, Re* = 380,

no control (left) and case C3 (r ig h t) .......................................................... 200

7.28 Contours of streamwise velocity in the Xi — X3 plane at y+ = 30, Re* = 380,

no control (left) and case C3 (r ig h t).......................................................... 201

7.29 Contour in y-z plane at Re* = 380......................................................... 202

7.30 Control in y-z plane at Re* — 380, A = 0.034, 1=0.2............................ 203

xxxv


7.31 (a) Time history of pressure drop during transition in channel flow at Re* =

135. All values are normalized with the value of the uncontrolled flow in the

turbulent state, (b) Streamwise elocity profile in the laminar state at time

indicated by A.............................................................................................. 204

7.32 Time history of normalized drag force with Lorentz forcing in channel flow

at Re* = 135................................................................................................. 205

8.1 Solution of slip t e s t .................................................................................... 210

8.2 Drag Reduction and Pressure drop of channel flow with different slip

length at Re* = 135 ................................................................................ 211

8.3 Drag Reduction and Pressure drop of channel flow with different slip

length at Re* = 400 ................................................................................ 212

8.4 Drag Reduction vs. b+ .............................................................................. 212

8.5 Mean velocity at Re* = 135....................................................................... 213

8.6 Statistics at Re* = 135. Turbulence fluctuation (left); Reynolds stress

(r ig h t) ........................................................................................................ 214

8.7 Statistics at Re* = 135. Mean vorticity (left); Vorticity fluctuation

(r ig h t) ........................................................................................................ 215

8.8 Statistics at Re* = 135. Mean vorticity (left); turbulence fluctuation

(r ig h t) ........................................................................................................ 216

8.9 Re* = 135, 800 a+ = 13.5 bubbles, 6+ = 1.35. Drag history (left);

Pressure drop (rig h t)................................................................................ 216

xxxvi


8.10 Re* = 380, 7200 a+ = 20 bubbles, b+ = 1.35. Drag history (left);

Pressure drop (r ig h t)................................................................... 217

8.11 Re* = 135, b+ = 1.35. Turbulence fluctuation (upper); Reynolds

stress ( lo w ) ................................................................................... 218

8.12 Streaks at y+ = 5 of Re* = 135................................................... 219

8.13 Streaks at y+ = 5 of Re* = 135, b+ = 2.7.................................... 220

8.14 Q contour without slip BC at Re*.= 135...................................... 220

8.15 Q contour with slip BC at Re* = 135, b+ = 1.35................................... 221

8.16 Q contour with slip BC at Re* = 135, = 2.7..................................... 221

8.17 Q contour with slip BC at Re* — 135, b+ = 1.35 and 800 a+ = 13.5

bubbles............................................................................................ 222

8.18 Q contour without slip BC at Re* = 380.................................... 223

8.19 Q contour with slip BC at Re* = 380, b+ — 1.35................................... 223

8.20 Q contour with slip BC at Re* = 380, b+ = 2.7..................................... 224

8.21 Q contour with slip BC at Re* = 380, b+ = 2.7 and 6348 a+ = 19

bubbles............................................................................................ 224

9.1 Shape of modal expansion modes for a polynomial order of polynomial

order P = 5..................................................................................... 234

9.2 Schematic of direct stiffness summation of local matrices to form the

global matrix A ........................................................................................ 235

xxxvii


Chapter 1

Introduction

Turbulence belongs to some of the most difficult problems in Physics. After many

years of research, it still remains an open challenging to the fluid mechanics research

community. Many of the difficulties in understanding turbulence stem from

* Time dependence.

* Three dimensional motion.

* Contain coherent structure in the near wall region.

* Highly non-linear process.

As it is almost impossible to solve turbulence problems theoritically, people are

turning increasingly towards simulations. A typical approach for turbulence research

was to use turbulence model based on Reynolds Average Equations. Until 1960s,

turbulence was studied solely through wind-tunnel experiments. In 1973, Steven

Orzag performed the first DNS on a 32s mesh at NCAR on a CDC7600 computer with

only 50 Mbytes memeory. Then in 1987, Kim, Moin and Moser (KMM) published

their first paper on Direct Numerical Simulation (DNS) of channel turbulence. After

1


that, interests on DNS greatly increased. Because DNS does not use any ad hoc

turbulence model, and it can provide detailed information about the turbulence

fluctuations. Besides, with the advent of supercomputers, turbulence modelling and

simulation have become important tools to study turbulence. DNS data has been

used as database to verify turbulent models and theories.

1.1 Turbulent Flow Simulation

1.1.1 R eynolds-Averaged Equations (R A N S)

The Navier-Stokes equation is averaged over time. By averaging, the non-linearity of

the Navier-Stokes equations gives rise to terms that can only be determined through

ad hoc models, in order for the system of equations to be closed. RANS has been

extensively used in industry to provide flow statistics. With careful calibration it

may produce accurate results, though it cannot provide any detailed time-dependent

information. Furthermore, the complexity of turbulence makes it unlikely that any

single model can represent all turbulent flows. This limits its usage to simple engi

neering applications.

1.1.2 Large Eddy Sim ulation (LES)

Through careful investigation of turbulent flow structure, there are many different

scales in turbulence, and the large scale motions have much more energy than small

scales. Small scale motions play a less important role in the process of transport

of mass, energy and other scalar properties. Therefore, if we treat the large ed

dies more accurately than the small ones, we can obtain an effective approach to

turbulent simulation with less computational cost. The LES approach separates ve

locity into the large scale and small scale components. The large scale is obtained

2


by filtering, and small scales are represented by a turbulence model, called subfilter

model. There are a wide variety of models, and the most popular and simple one is

the Smagorinsky eddy-viscosity model. In this thesis, a modified approach different

from eddy-viscosity modeling for LES is presented.

1.1.3 D irect numerical Sim ulation (D N S)

The most advanced approach for turbulence simulation is DNS, which can solve

the Navier-Stokes equations without any model. In order to use DNS, the mesh

size should be small enough to be comparable with the Kolmogorov length scale of

the turbulent flow. At low Reynolds number, this is a powerful research tool as it

can provide detailed description of turbulent flow. Corrsin (1961) [24] pointed out

that the number of grid points required for DNS of fully developed turbulent flow

increases as i?e9/4 per time step, and a full simulation requires a large number of

time steps proportional to Re3/4. Thus the total cost scales like Re3. As Reynolds

number increases, the computation requirement become prohibative, and it makes

DNS inapplicable to solve practical industrial problems with current computer ca

pabilities.

1.2 Turbulent Drag Reduction Techniques

Turbulence control is very important in industry, and drag reduction has great eco

nomical and military value. There are two control methods considers energy expen

diture and the control loop involved: active control and passive control. Passive

control requiring no auxiliary power, while active control requiring energy expendi

ture. Active control can be further divided into predetermined or reactive control.

Predetermined control includes the application of steady or unsteady energy input

without regard to the particular state of the flow. The control loop in this case is

3


open and no sensors are required. Reactive control is a special class of active control

where the control input is continuously adjusted based on measurements of some

kind. The control loop in this case can either be an open, feedforward one or a

closed, feedback loop. Classical control theory deals, for the most part, with reactive

control.

Usually active control is more effective than passive control. Loren tz force belongs

to active control. Adding polymer or micro-bubbles and slip boundary condition

belong to passive control.

It has been found that adding polymer or microbubbles can lead to 70% to 80%

drag reduction, and Lorentz force acting in spanwise direction can reach about 30%

drag reduction. In this work, several techniques, such as adding micro-bubbles,

constant forcing, and slip boundary have been investigated in detail.

DNS has been used to investigate all above techniques, as we outline below.

1.2.1 Micro bubble Drag R eduction

This phenomenon was first demonstrated in [93] and subsequentially verified in a se

ries of experiments by [81, 82] and [95]. More recent experiments have been reported

in [69] and [49].

Although it has been reported that 80% drag reduction can possibly be achieved

in experiment, it is very hard to reach this level in numerical simulation. The reason

is that it is very difficult to simulate the interaction between bubbles and turbulent

fluids.

There are basically two methods to investigate turbulent multiphase flow. The

first one is a Lagrangian description, in which the equation for each bubble motion

needs to be solved, and the second one is Eulerian description, which treats the

4


bubbles as a continuous medium and modified Navier-Stokes equation are used to

describe both fluid and bubble phases.

The Arbitary Lagrangian-Eulerian (ALE) method is a Lagrangian method. It

fully resolves particles or bubbles, adaptive mesh has been updated as particle moves.

Forces on particle/bubble are calculated directly from flow variables and used to

move the particles. The Front Tracking [127] and Distributed Lagrage Multiplier

(DLM) [45] use static mesh, fictitious forces are used in flow to simulate presence

of particle/bubble. They allow for deformation of bubble shape. Force coupling

method (FCM) [88, 89] is also a Lagrangian approach, and the grids are all fixed in

this method. The Front Tracking technique captures the bubble boundary at each

time step, so it requires a lot of grid points inside the bubble. For the liquid-liquid

or gas-liquid flow, there are many methods similar to Front Tracking method, such

as marker and cell (MAC) method, volume of fluid (VOF) method [115], level set

method [104, 121], constrained interpolation profile method [137], and the phase field

method [58, 59].

The FCM method is introduced by Maxey (1997), and was studied in the PhD

thesis of Patel (2001) and Lomholt (2002). In this method, each bubble is represented

by a finite force monopole that generates a body force distribution which

transmits the resultant force of the bubble to the fluid. The dynamics of bubbles

and fluid are considered as one system, where fluid drag on the bubbles, added-mass

effects and bouyancy forces are treated as internal to the system. The equations

of fluid motion are applied to the whole domain, including the volume occupied by

bubbles. In this way, the body forces induce a fluid motion equivalent to that of the

bubble motion.

With Distributed Lagrangian Method (DLM) [45], the problem on a time-dependent

geometrically complex domain is extended to a stationary, larger, but simpler do

main so that a fixed mesh can be used. The no-slip boundary conditions between

the bubbles and the fluid are satisfied through the constraints of rigid-body motion

5


of the fluid in the volume of bubbles. These constraints are enforced through DLM,

which represents the additional body force required to maintain the rigid-body mo

tion inside the bubbles. The DLM method has been applied to study sedimentation,

fluidization, and viscoelastic bubble flows with the number of bubbles reaching the

order of 1000-10,000 in 2D and 100-1000 in 3D.

1.2.2 Constant forcing control

Inspired by the force of micro-bubbles acting on the turbulence, a sine wave form

constant force has been found to reduce drag dramatically. Different Reynolds num

ber were investigated, and it has been found that for larger Reynolds number, a

larger amount drag reduction can be achieved. However, there exists an upper-limit

for maximum drag reduction at each Reynolds number.

1.2.3 Slip boundary condition

Careful observation on real condition of bubble motion in the near wall region sug

gests that it is a common phenomenon that bubbles attach to the wall and may form

a thin film, next to wall. This motivated us to investigate more realistic boundary

condition, and slip boundary condition has been implemented to investigate the

hypothesis.

1.3 Motivation

With the rapid development of supercomputer, the possiblities of high Reynolds

number DNS become feasible. DNS turbulence data is invaluable in many fields. For

example, besides the turbulent data in plane channel flow, people are also interested

in turbulent data in complex geometries. Spectral/hp method provides a powerful

6


tool to do such simulations.

LES has also been investigated by using different models. In the frame of spec

tral/hp method, few work has been done. Due to the advantages of spectral/hp

methods, it is meaningful to do LES based on this high-order accuracy numerical

scheme. It incorporates both multi-domain spectral methods based on the ideas of

A. T. Patera, and high-order finite element methods based on the ideas of B. A.

Szabo. Polynomial spectral methods were first introduced by Gottlieb and Orszag

(1977) and have been covered in Canuto (1987) and Boyd (1989). Hp finite ele

ments and spectral elements have been discussed in Szabo and Babuska (1991), and

Bernardi and Maday(1992) respectively. They have been extended to their current

unstructured form by Sherwin and Karniadakis (1995). The concept is simple: the

solution is approximated using a series of polynomials, the order of which can be

chosen arbitrarily. However, to apply this concept is by no means easy. Despite

the complex scheme which is necessary for its application, unstructured spectral el

ements have some very useful properties. They exhibit convergence reached either

by increasing the number of elements or by increasing the polynomial order of the

expansion bases. The high accuracy at high order ensure it as an effective method

to do long-time integration of turbulence simulation.

Using unstructured spectral element method to do LES has some difficulties,

because filter width is related to the size of the structured grid size or volume. For

unstructured mesh, we need to define a new filter length. This may influence the

final simulation results of LES. A new method has been developed in the current

thesis to do LES using spectral element/hp method.

7


1.4 Objectives

The aim of the thesis is focused on two parts. The first one is to do high Reynolds

number DNS and LES by Spectral Vanishing Viscosity (SVV) method using Fourier

Spectral Element Method (FSEM). In order to reach high Reynolds number, several

parallel models were implemented and compared. The optimized channel turbulence

solver was used to do high Reynolds number DNS. DNS results at low Reynolds num

ber were compared with Moser’s result. Based on the results at different Reynolds

number, a database has been created. We also investigated using Spectral Vanish

ing Viscosity (SVV) method to do LES, and compared their results with our DNS

results. The second part is to investigate several drag reduction techniques using

DNS.

1.4.1 D N S and LES

Firstly, several parallel models were used to implement a channel solver using MPI

or OpenMP. Their parallel efficiency was thoroughly investigated, and the best model

was chosen for the parallel production runs. Secondly, DNS databases at different

Reynolds number were created and we compared all the statistics with results of

KMM [68] and MKM [99]. Specifically we did DNS at Re* = 180,400,600,1000.

Thirdly, we tried to explore whether SVV can be an efficient way to do LES. We

did our LES-SVV research at different Reynolds numbers, i.e. Re* = 180,600,1000.

Those results were also compared with standard LES and DNS data.

1.4.2 Drag reduction techniques

Using our DNS code, we investigated several drag reduction techniques as follows:

8


* Micro-bubble drag reduction.

* Constant forcing.

* Slip boundary condition.

The constant forcing method was inspired by micro-bubble drag reduction, and

the wave length must be small enough to have drag reduction. The tradional nu

merical methods usually do not satisfy such fine mesh, therefore it has never been

detected before. DNS has fine mesh in the wall region, so it can be used as a powerful

tool to investigate turbulent flow.

1.5 Outline of Thesis

The thesis is divided into two parts. The first part is high Reynold number DNS and

LES, including chapter 2-4. Chapter 2 presents different parallel implementation of

channle turbulence DNS, and gives DNS results at different Reynolds number. After

that, discussion on channel turbulence LES is presented in Chapter 3 and 4. The

second part is about drag reduction techniques, including chapters 5-6. Chapter 5

explains the FCM method, which has been used to simulate turbulent bubble flow

and its parallelization. Chapter 6 discuss the simulation results of micro-bubble drag

reduction at low and high Reynolds number. Chapter 7 discusses the constant forcing

control technique which is inspired from micro-bubble drag reduction. Chapter 8

investigates the drag reduction effect by using the slip boundary condition. Summary

and conclusion are given in Chapter 9.

9


Chapter 2

Spectral Element Method for

Channel Flow

The domain shown in figure (2.1) is a plane channel with periodic boundary

conditions in streamwise and spanwise directions.

2.1 Numerical Method and Parallel Models

2.1.1 Num erical M ethod

The discretization is similar to that of Kim, Moin and Moser [68]. The difference

is in the wall normal direction, where we use a spectral elements expansion instead

of the Chebychev polynomials. In the streamwise and spanwise directions, we use a

Fourier expansion as the expansion basis, which is the same as KMM’s. Under this

10


Figure 2.1: Sketch for channel domain

framework, the velocity can be expressed in the following form within an element:

M / 2 - 1 N / 2 - l p

u{x,y,z,t)= y Y fi(miP’n't')e~iamXe~l0nzpp(y)’ (2 -1)m = —M / 2 n = —N / 2 0

where Pp(y) are the Legendre Polynomials (P$’°(y)).

Using this method, we can choose the position of an element boundary, so that we

can control the number of points in the near wall region. Usually the DNS simulation

needs at least 13 points in the first 10 wall units, so that it is enough to resolve the

smallest turbulent structure in the near wall region.

The velocity field V (x, t) of incompressible flow satisfies

-V p + jA72V (2 .2)

(2.3)V -V 0

where v is the fluid viscosity and p is the pressure.

11


We use the high-order time splitting method of Karniadakis, Israeli, Orszag

(1991) to do the time integration.

1. Nonlinear step:

fs+i/3 - £ aqV s~<i J*~l AtS = = J2 f}qN{Vs-i) (2.4)

9= 0

2. Pressure step:

U s + 2 /3 _ y s + l / S

Xt = -V IT +1 (2.5)

V . y-s+2/3 = o (2.6)O T T J e ~ 1 J e ~ 1

- ^ = r H £ / y V ( V - « ) + ^ ^ ( - V x ( V x n ) ] (2.7)9 = 0 9 = 0

3. Viscous step:

1VP+1 — ('f/V+2/3At = i/V2(V)fl+1 (2.8)

In order to eliminate the aliasing error generated in nonlinear step, we perform the

nonlinear step in physical space using 3/2 rule. This means that we expand the mesh

by 3/2 times larger in both streamwise and spanwise directions. After evaluating the

nonlinear terms, we transform back to the normal mesh. The code uses FFTW, which

can optimize its performance on different platforms to do Fast Fourier Transform.

The high performance numerical libraries such as BLAS, LAPACK have been used

in the code. Since high Reynolds number DNS usually needs to run on parallel

supercomputers, the code has been parallized using MPI. Its parallel efficiency is

relatively high on various platforms, such as SGI, AIX, HP, LINUX Cluster, etc.

12

The performance is quite good, and we will benchmark the code in details later in

this chapter.

2.1.2 D irect and Iterative Solver

We have tried both direct and iterative solvers in our code for the pressure and

viscous steps, and compared their efficiency. For a direct solver, we use LU decompo

sition. Static condensation technique has been implemented, and high performance

has been reached. At low Reynolds number DNS, the advantage of static condensa

tion technique may not be obvious, because the time need to solve a global matrix

is relatively small. While at high Reynolds number DNS, the advantage is promi

nent. Matrices obtained from LU decomposition take a large mount of memory, the

speed without static condensation is relatively slow. This indicates that the static

condensation technique is necessary to do DNS at high Reynolds number. The work

of decoupling boundary points from internal points can be compensated by saving

memory and high efficiency by solving much smaller matrices.

For an iterative solver, we use the conjugate gradient method with a diagonal

preconditioning. Fig. (2.2) shows convergence of iterative solvers. The test probelm

is 3d acurate solution shown in equation(2.11) to equation(2.13). The speed of the

iterative solver is approximately the same as the direct solver. As the mesh becomes

larger, most time of each time step was spent in the nonlinear step, and we adopted

a direct solver in our DNS simulations.

13


_L5

_i_10N

Figure 2.2: Spectral accuracy of iterative solver, N is the grid points

2.1.3 Parallel Im plem entation

Because DNS is time and memory consuming at high Reynolds number, the code

has been parallelized using MPI. In order to obtain maximum optimization, we

have implemented several different parallel models. Their performances have been

compared and the best model has been chosen from them. The first model of the

code uses domain decomposition in the stream-wise direction (model A), and it is

the easiest implement. However, this model has a potential limitation on the number

of processors that can be used since we need to have at least two planes located in

each processor in order to do de-aliasing in nonlinear step. In order to increase the

number of processors that can be used, we have also implemented another parallel

model, which decomposed the domain in both stream-wise and span-wise directions

(model B). Using this parallel model, we can use thousands of processors at high

Reynolds number DNS. According to our computation experience, we found that

as the mesh becomes larger, most of the time in each time step is spent in the

nonlinear step (approximately 75%). Since we need to expand the mesh 9/4 times

larger than before, we need to exchange large amounts of data between different

processors in a high Reynolds number simulation. This is the bottleneck of the

14


Figure 2.3: Sketch for Model A parallelization

code speed. In order to speed-up nonlinear steps, we also tried different ways to do

de-aliasing. The best way has been used in the high Reynolds number simulation

later. We will give detailed explanation in the following. As a comparison, we also

implemented a hybrid parallel model using MPI and OpenMP. MPI has been used

in the streamwise direction, while OpenMP has been used in spanwise direction

(Model C). The benchmark of this model will give us a clear comparison of MPI and

OpenMP.

1. Model A: MPI in x direction

Figure (2.3) is the sketch of model A; the domain has been decomposed only in

the stream-wise direction.

In the nonlinear step, the data on each processor has been shifted N times (N is

the total number of processors been used) to the “following” processor. That means

0 —> 1 -> 2 —> N — 1 —>• 0. During this process, each processor can hold the

15


ZL

1 I 1 1 1 I

1 '1 '

1' 1

; 1 ' i i 1 1 1 i 1 1 1 i 1 1 1

1 1 1 1 I ' ' 1 [ ' ' I

' 1 1

11 I 1 1 I 1

' 1 1 1 1 1 1 1

_ — _!

! 1 ' i 1 11 1 1

- - 1

1 1 ! i 1 1 1

1 11i L L i .

1 1 7 - - U - 1- 1 yl1 h ---- -1 ± - J J t - —I -j -fc .___ A / ___ J f ___ - Y -----

/ / / / 7 77

12 13 14 15

8 9 10 11

4 5 6 7

0 1 2 3

Figure 2.4: Sketch for Model B parallelization

data which should belong to it after the 3/2 expansion.

2. Model B: MPI in x and z direction

Figure (2.4) is the sketch of model B; the domain has been decomposed in both

stream-wise and span-wise directions.

In the nonlinear step, we have implemented the Fast Fourier Transform in two

different ways. The first approach (referenced as model Bl) is to form two seperate

communication groups, one is composed of processors that have the same x planes,

and the other is composed of processors which have same z planes as shown in figure

(2.5). In the model, the FFT has been done seperately in x and z direction, one

after another. The second approach (refeneced as model B2) is to collect data from

all processors at first, and then put data on one or several planes to one processor

as shown in figure (2.6). Then a two-dimensional FFTW has been used, and after

transformation, the data will be distributed back to original processors.

16


I

II

I_ i __

Com m unication G roup X Comm unication G roup Z

Figure 2.5: Sketch of the first way for Fourier transform in Model BI

1 ! '

^ ' 7

11 :

/ 1 i

M >:

- T z| K , 1ii

/

11' !" i 1

/ k/" 1

“*>“ ' /

77

V

/ I / I

/

ii;"iJ1 '-H- I - / £ 1 l7/ J/ _/ r-K- • / 17 _/

/7

C P U I

--hr. 1

C P U II

Figure 2.6: Sketch of the second way for Fourier transform in Model BIT

17


Figure 2.7: Sketch for Model C parallelization

The second approach (B2) shows better performance, and it has been used in

our high Reynolds number DNS simulation.

3. Model C: MPI in x and OpenMP in z direction

Figure (2.7) is the sketch of model C. It is same as Model A, using MPI in the

stream-wise direction, besides, OpenMP has also been used in span-wise direction.

2.2 Validation/Verification

At first, we validate our code by two cases, and show spectral convergence as

it’s supposed to be. The first case is a two-dimensional Kovasznay flow, and the

second case is three-dimensional accurate solution. At last we verify our turbulence

statistics at Re* = 180 with standard KMM’s DNS results.

18


Figure 2.8: Spectral convergence of solving 2d Kovasznay flow problem

2.2.1 Kovasznay Flow

The Kovasznay flow is a model for laminar flow behind a two-dimensional grid,

the exact solution is given by Kovasznay (1948). The solution is a function of the

Reynolds number Re, and is of the form

where A = Re2/ 2 — {Re2/4 + 47r2)0,5 at Re=40. All the boundary conditions are

Dirichlet conditions, defined by the above exact solution. Figure (2.8) shows the

convergence of spectral accuracy, it proved the correctness of our code.

2.2.2 3D A ccurate solution

The second case we used is a three-dimensional flow problem, which satisfies

periodic boundary condition in both stream-wise and span-wise directions. This

u

v

1 - eXxcos{2-Ky)

—-eXx sin{2'Ky)

(2.9)

(2 .10)

19


Figure 2.9: Spectral convergence solving 3d problem

problem is designed to satisfy the divergence-free condition.

w

u

V

cos(x)sin(y)sin(z)

—2sin(x)cos(y)sin(z)

sin(x)sin(y)cos(z)

(2 .11 )

(2 .12)

(2.13)

Figure (2.9) shows the convergence of spectral accuracy, and again demonstrates

that our code can solve 3D problem accurately.

2.2.3 Turbulence Statistics

After validating our code with accurate solutions, we compare the turbulent

statistics obtained at Re* = 180 with KMM’s DNS results. Since its publication in

1987, KMM’s DNS results at Re* = 180 have been used as a standard benchmark

to verify other DNS results. Our statistics show good agreement with their data.

20


>3

y101

Figure 2.10: Re* = 180, Solid: KMM; Dash: SEM solver. Mean Velocity (left);

Reynold Stress (right)

It proves that our code can produce correct results at low Reynolds number. Our

turbulent fluctuation is a little bit larger than KMM’s in the center of channel,

because we use three elements, and the middle one has the largest length of 1.0 ,

from -0.5 to 0.5. So the spacing points in the center of channel is relatively larger

compared to the spacing points in the near wall region. Later in this Chapter, we

will also compare statistics at high Reynolds number; they also match KMM’s [68]

and MKM’s [99] result quite well.

2.3 Parallel Benchmarks

2.3.1 Comparison on different m odels

First we compare different models on an SGI computer as shown in Figure (2.12).

We can see all of them decreased nearly linearly, and Model B is the best model

among them. It shows that the Model B has the highest parallel efficiency.

21


(AEcc

y

Figure 2.11: Re* = 180, Solid: KMM; Dash: SEM solver. Turbulence fluctuation

(left); turbulence vorticity fluctuation (right)

180

6 4 X 6 5 X 6 4 6 5 p o in ts /1 e l e m e n t

b M odel A_ — M odel B

M odel C

160

140

120

Number of Processors

Figure 2.12: Model C: Speed comparison of different models on SGI

22


7065 i polnts/460

55

50

45

40

E 35 H 30

25

20

15

10

5

0N um ber of P ro c e sso rs

70

60

50

i-30

20

10

0.

N um ber of P ro c e s so rs

Figure 2.13: Speed on different platforms: Model A (left), Model B (right)

2.3.2 Comparison on different platforms

Next, we compare the speed of Model A and Model B on different machines.

Figure (2.13) shows that they have similar performance on different platforms. This

means the parallel efficiency of our code is independent of the platforms, so it can

be run on various supercomputers for high Reynolds number DNS.

2.3.3 Comparison of M odel B in x and z direction

Since in model B, we decompose the domain in both streamwise and spanwise

directions, we can increase the number of processor in either direction. We wish to

know the performance by increasing the number of processors in these directions.

As shown in figures(2.14) and (2.15), the parallel efficiency in x and z directions are

basically the same as they both use MPI. Figure (2.15) showed the performance in

a 2D contour plane. Because of the symmetry of the channel domain, the parallel

efficiency should be also symmetric and similar in the two directions.

23


4)EI -

N um ber of P ro c e sso rs

oEi—

N um ber of P ro c e sso rs

Figure 2.14: Model B: Speed comparison for MPI in x and z directions: SGI(left),

SP4(right)

Models on SP4 126X145X128 36 polnts/4 elements

Figure 2.15: Model B: Speed comparison in (x,z) plane: SGI(left), SP4(right)

24


200

175 M o d e l C o n S O I 1 2 8 X 1 4 5 X 1 2 8 3 6 p o in ts /4 e l e m e n t s

150 MPI in x O penM P in z

125

Number of Processors

Figure 2.16: Model C: Speed comparison in x(MPI) and z(OpenMP) directions

2.3.4 Comparison o f M odel C in (x,z) plane

Since we adopt different parallel models in streamwise and spanwise directions,

figures (2.16) and (2.17) show the different performance in x and z directions. The

parallel efficiency is higher using MPI than using OpenMP. The time decreased lin

early using MPI, but saturated quickly using OpenMP. OpenMP has some advantage

when the machine has some large block of shared memory, and with small number of

processors. It is not so efficient with a large number of processors, here the number

of processors is small (4-16). This suggests that MPI is more suitable to do large

scale computation than OpenMP as for high Reynolds number DNS.

2.3.5 Comparison of M odel B for R e * = 400

We have shown Model B2 is the most efficient model for parallelization, and we

will benchmark this model on a large number of processors. The grid is 256 x 241 x

25


M o d e l C o n S G I 1 2 8 X 1 4 5 X 1 2 8 3 6 p o in ts /4 e l e m e n t s

Figure 2.17: Model C: Speed comparison in x(MPI) and z(OpenMP) directions

256. In the wall normal direction, it has 8 elements with 31 points in each element.

The elements are distributed unevenly, so that we have more than 13 points in the

first 10 wall units in the near wall region.

Figure (2.18) show scaling of our code on large number processors, and the paral

lel efficiency is good even using a large number of processors. We also benchmarked

our code on larger mesh of 384 x 361 x 384, each time step takes about 4 seconds on

256 processors using the IBM SP4 machine, Iceberg at Arctic Supercomputer Center

(ARSC). This means that our code can be used to do long time integration of high

Reynolds number DNS. Based on DNS databases obtained, various fluid problems

occured in turbulent flow can be investigated thoroughly, such as turbulent control,

microbubbles motion, etc.

26


1514

R e .s 4 0 0 a t P S C 2 5 6 X 2 4 1 X 2 5 6 3 1 p o in ts /8 e l e m e n t s

1312

11

10

98

76

5432

1

0 400 600Number

800Number of Processors

1000

Figure 2.18: Model B: scaling on large number of processors, IBM SP4, ARSC

N x x Ny x Nz Norm Re* Real Re* Ele Lx Lz 5x+ Sy+ Sz+

128 x 129 x 128 180 178.1 1 47T 47r/3 17.7 4.4 5.9

256 x 193 x 192 395 392.2 1 2tt 7r 10.0 6.5 6.5

384 x 257 x 384 595 587.2 1 27T 7r 9.7 7.2 4.8

Table 2.1: KMM’s and MKM’s DNS runs.

2.4 High Reynolds DNS results

Firstly, we summarize the DNS runs we have performed in table(2.2). Table(2.1)

shows the parameters of KMM’s and MKM’s DNS, and the meshes we use are

similar to or more than KMM’s. Re* = 5x+ = 8y+ = 8z+ =

Secondly, we give turbulent statistics and visualization at Re* = 400 and Re* —

600 in the following.

27


N x x Ny x N z Norm Re* Real Re* Ele Lx Lz 8x+ Sy+ Sz+

128 x 130 x 128 180 178 3 2 tt 2 tt 8.98 6.6 8.98

256 x 241 x 256 400 380 8 27T 7T 9.31 10.3 4.66

384 x 361 x 384 600 633 10 2 tt 7r 9.81 12.9 4.91

Table 2.2: Current DNS runs.

> -0.4

y

Figure 2.19: Re* = 380. Solid: KMM; Dash: SEM solver. Mean Velocity normalized

by u*(left); Reynold Stress normalized by u*2(right)

2.4.1 Re* = 400

All the statistics at Re* = 400 match MKM’s results quite well, except that the

Reynolds stress show some difference. The correct value should be a straight line,

which is located between our result and MKM’s. Both results needs to be improved

a little bit.

Figure (2.21) shows the pattern of streamwise velocity contours or low-speed

streaks at y+ = 5 plane for Re* = 380. The domain is 2-k x 2 x 7r, corresponding

to 2388 x 760 x 1194 in wall units. Clear evidence of streaks can be seen, and the

average distance between them is approximately 100 wall units. Figure(2.22) shows

the vorticity contour in a box region of 3 x 0.5 x 1, which is 1140 x 190 x 380 in wall

units.

28


Ecc

y

0.35

0.3

0.25

g 0.2«EIT

0.15

0.1

0.05

0


normalized by w,*(left); turbulence vorticity fluctuation normalized by «,*2/T(right)

V : 0.0100 0.1292 0.2484 0.3676 0.4868

Figure 2.21: Streaks of Re* = 380 at y+ — 5.

29


vortices

150100

1000

800x600

300 400200

200100

Figure 2.22: Vortices at Re* = 380.

2.4.2 R e * = 600

Our statistics at Re* = 633 also match MKM’s results quite well. We have averaged

our results for about 20 time units, there is some small discrepency due to this

short time average. Figure (2.25) shows the streaks at y+ = 5 for Re* = 633,

the streaks becomes thinner and denser than those at Re* = 380. The observed

feature are consistent still with an average spacing of 100 wall units. Figure (2.26)

shows the vorticity contour, the sub-domain is the same as for figure (2 .22), which

is 1899 x 316.5 x 633 in wall units. For the fixed channel dimensions and flow region

shown, the vortices appear more dense and smaller than those at Re* = 380. As

Reynolds number increases, the apparent turbulent scale decrease in physical units,

but in terms of wall units there is little change. More mesh points are needed to

resolve turbulence at Re* = 633.

30


3

y

£3

Figure 2.23: Re* = 600, Solid: KMM; Dash: SEM solver. Mean Velocity normalized

by (left); Reynold Stress normalized by u*2(right)

co£cc

y

“sr 0.2


normalized by ?/* (left); turbulence vorticity fluctuation normalized by u*2/u (right)

31


Figure 2.26: Vortices at Re* = 633.

32


Case N x x Ny x N z Re* Ele Lx Lz 8x+ 5y+ Sz+

Moser 768 x 769 x 768 1901 1 7T 7t/ 2 7.8 7.8 3.9

Moser 3072 x 385 x 2304 934 1 87T 37r 7.6 3.8 7.6

Iwamoto 1152 x 513 x 1024 1160 1 67T 2 tt 19 - 7.1

Jin 768 x 521 x 768 934 20 67T 1.57T 22.9 7.66 5.7

Table 2.3: DNS runs at Re* = 1000.

2.4.3 Re* = 1000

We have also obtained results at Re* = 1000, and we give turbulent statistics and

visualization. Table(2.3) shows the parameters used by several researchers. The

mesh is 768 x 521 x 768 with 20 elements in wall normal direction, domain size is

67T X 2 X 1.57T.

2.5 Summary

In this chapter, we have developed and benchmarked Fourier Spectral Element (FSE)

code. The code has been validated by two dimensional Kovasznay flow problem

and three dimensional accurate solution. Spectral convergence has been satisfied.

We also verified the code by comparing turbulent statistics at Re* = 180 with

KMM’s standard results. The agreement is excellent, and the code has been used

to perform DNS at high Reynolds number. In order to optimize the code, parallel

benchmarks have been done, several parallel models have been investigated in detail,

and an optimized model has been used to implement the production code. Detailed

statistics at high Reynolds number of Re* = 380, Re* = 633 and Re* = 933 have

been obtained and compared with MKM’s results, and good agreement has been

reached for all of them. Streaks at y+ = 5 and vortices have been shown at these

33


R e * = 9 3 4S o l id -------D a s h -------

— M o s e r , 3 0 7 2 * 3 8 5 * 2 3 0 4 — J in , 7 6 8 * 5 2 1 * 7 6 8

BD

0.5

+A’>3V

-0.5

500 1000 1500Y+

Figure 2.27: Re* — 1000. Mean Velocity normalized by w*(upper), Solid: KMM;

Dash: SEM solver; Reynold Stress normalized by w*2(low)

34


2.5•M o se r, 3 0 7 2 * 3 8 5 * 2 3 0 4 ■Jin, 7 6 8 * 5 2 1 * 7 6 8

2

.5

1

0.5

00 200 400 600 800

20 40y+


normalized by w*(upper); turbulence vorticity fluctuation normalized by u*2/v( low)

35


Figure 2.29: Streaks of Re* = 1000 at y+ = 5.


vortex detection: Q

Figure 2.30: Q contour at Re* — 1000.

37


Reynolds number, and these high Reynolds databases are reliable and valuable for

research related to turbulent flow. Our work presented in the later chapters is based

upon these data.

38


Chapter 3

High Reynolds number Large

Eddy Simulation

3.1 Background

After many decades of intense research on large-eddy simulations (LES) of turbulent

flows, the results show that standard LES methods are still subject to some funda

mental limitations. Different implementations to improve the LES results are under

current investigation.

An interesting approach is the scale-similarity model, first proposed by Bardina

[4], and its subsequent variants. It assumes that the subfilter stress is proportional

to the so-called Leonard stresses, which are expressed in terms of the filtered velocity

gradients. Preliminary results with mixed models have shown significant improve

ment. However, such mixed models are typically computationally more expensive.

If the LES discretization lacks entropy dissipation, then Gibbs oscillations are

produced and eventually render the solution unstable. In high Reynolds number

39


flows, the situation is analogous. The conflict between monotonicity and accuracy

was analyzed by Godunov [46] and Tadmor [123]. Specifically, Tadmor introduced

artificial dissipation via Spectral Vanishing Viscosity (SVV), which is sufficiently

large to suppress oscillasions, yet small enough not to affect the solution’s accuracy.

The SVV approach guarantees an essentially nonoscillatory behavior, although

some small oscillations of bounded amplitude may be present in the solution. This

theory is based on three key components:

1. a vanishing viscosity amplitude which decreases with the mode number;

2 . a viscosity-free spectrum for the lower, most energetic modes;

3. an appropriate viscosity kernel for the high wavenumbers.

It is worth pointing out an important distinction between the classical LES for

mulation and the currently proposed SVV formulation. In standard large eddy for

mulations, the small scale dynamics is coupled to the dynamics of the large scale

dynamics with explicit contributions from the subgrid scales, whereas in comparison,

the SVV approach ignores this coupling.

Many of the applications of SVV method so far have dealt with one-dimensional

conservation laws. Andreassen et al[ 1] used SVV for two-dimensional simulations of

waves in a stratified atmosphere. Standard Fourier or Legendre discretization was

employed by Tadmor and his colleagues. Karamanos [64], Kirby and Karniadakis

[70] have done some pioneering work on using SVV for LES of channel turbulence.

Based on their efforts, our current work combined the concepts of standard LES with

SVV method.

In the standard Smagorinsky model, the subgrid stress term (SGS) is computed

from filtered quantities, and the coefficient is related to grid size and constant Cs.

In the Fourier Spectral Elements solver, these SGS terms are added to all modes

40


in the Fourier space for standard implementation. However, based on assumption,

these terms are not accurate. Adding these terms in low modes may not be correct

according to the concept of LES. SVV method tries to avoid such situation. By

incorporating Dynamic Smagorinsky model into SVV, the viscosity e can be calcu

lated from local flow information. This usually gives better results than standard

Smagorinsky model. We will present the results of SVV incorporated with standard

and Dynamic Smagorinsky models in the following sections.

Since high modes correspond to the small scale motions in the flow, we intend to

increase the small scale motion, which means to increase the momentum transport

in the near wall region. This is different from before, since no low order mode has

been changed, and the improvement of mean flow quantities is due to the change of

small scale turbulent structure, which is introduced by adding SGS terms in high

modes.

In this chapter, we present the method to do large eddy simulation using SVV.

In particular, we applied a Jacobi-based spectral element discretization along the

inhomogeneous direction [67] and Fourier collocation along the other two homoge

neous directions. We implemented this methods implicitly, and this improves the

stability. Detailed statistics at low Reynolds number are compared with DNS results

with single and multiple elements in wall normal direction. We also performed high

Reynolds number LES and compared the results with those from DNS.

41


3.2 LES methods and implementation

3.2.1 Basic A ssum ption and Concepts

As we have introduced briefly in Chapter 1, the basic idea of LES is to model

the small scale motion while simulating the large scale motion. This is based on the

following assumptions:

* Most of transport of mass, momentum and energy is due to the large eddies.

The small eddies dissipate the energy of large scale motion, but affect the mean

properties only slightly.

* Large scale motion is strongly dependent on the geometry of flow, and it is

anisotropic.

* Small scale motion is much more universal.

The LES equation is obtained by averaging the Navier-Stokes equation in space.

The method is different according to different definitions of averaging. Once the

filtered equation of motion is specified, the effect of the small scales on the large

ones requires modelling. The most common assumption in the turbulence modelling

is that production and dissipation terms dominate the turbulence budget and as a

first approximation, we asssume they are equal when other terms are ignored. This is

acceptable if the Navier-Stokes equation is filtered in the inertial subrange, with the

eddies carrying most of the energy being resolved by the filtered equation of motion.

In the inertial subrange, there is an absence of sources and sinks at each wave number

because the energy transfer dominates both production and dissipation. This makes

Production = Dissipation an acceptable approxiamtion in physical space. This

model holds in homogeneous turbulent flows and in free shear flows, althougth in

42


wall bounded flows, the structures responsible for much of the momentum transport

maybe quite small, especially close to the solid boundary. Special care is necessary

in these cases.

3.2.2 Filtering Techniques and Im plem entations

Incompressible Navier-Stokes equations and the continuity equation are:

—Vp + i/V2V (3.1)

(3.2)V - V 0

where 7 is the fluid momentum viscosity and p is the pressure.

The velocity field V, can be decomposed as

V = V + V'

where v is the part of variable separated by filtering and therefore called the subfilter

or small scale quantity. V is the part of the variable remaining after the filtering

operation and it represents the filtered or large scales motion. A subgrid motion

is one which is not captured by the numerical discretization, while a subfilter scale

(SFS) motion is one which is not captured by a filter. If the filter width is equal

to the mesh width, the subfilter and subgrid scales are the same. It follows that

a resolved motion is one resolved by the numerical discretization, while a filtered

motion is the motion subtracting the resolved motions from original ones.

The filtering operation is a convolution of the filter function G, with the variable

V, and produces the filtered variable:

G{—x) = G(x), I G(x)dx = 1, G(oo) —> 0, I G(x)x”dx < 00 (3.4)«/ — OO J — 00

43

(3.3)

The filter should satisfy following constraints:

This is normally evaluated in physical space using finite difference or finite volume

discretization. The integration is relatively computationally expensive, since the

integration envelope usually contains several points in each direction. For example,

if each direction has 3 points, then overall the integration envelope contains 27

points in three directions. One time integration requires 27 multiplication step per

grid point. Besides, when the domain is decomposed in streamwise or spanwise

direction, the integration in physical space must be done by each processor at first,

and then a summation needs to be done in order to get final result. This may

complicate the implementation. In order to simplify the work, one usually tries

to do this computation in coefficient space. However, there is some difficulty to

connect filter width in physical space with mode number in coefficient space using a

spectral element method. In our research, we tried both cutoff and Gaussian filters

in coefficient space to do filtering.

There are different filters defined as following:

Box Filter:

where A is the filter width. These are normally associated with explicit filtering

where the filter function G is usually convolved with the velocity field in Fourier

(3.5)

Sharp Cutoff Filter:

0(1) = (3.6)

Gaussian Filter:

(3.7)

Differential Filter:

(3.8)

44


space to produce the filtered velocity field.

3.2.3 Filtered Navier-Stokes Equations

Filtering the Navier-Stokes equations with a constant filter width, while neglect

ing body forces, density fluctuations and assuming that the differentiation operator

commutes with the filtering operation, i.e.

d u dud x d x 5

the equation of motion for an incompressible large eddy simulation is:

d(puj) djpufuj) _ dp d2Uj dt dxj dxi V dxj2'

Define = p{ujju,j — UiUj), the equation can be rewritten as

d{puj) d(pujUj) __ dp d'2Uj drjjdt dxj dxi dxj2 dxj '

In order to solve equation (3.10), we need to choose the filter width. As filter size

decreases, the subfilter models smaller and smaller scales. Thus, the effect of the

small scales on the larger ones is reduced, transforming the LES computation into a

direct numerical simulation. There are some tradeoff between obtaining an accurate

simulation and keeping the computational cost as low as possible. Basically the filter

cutoff should be in the inertial subrange, and the computational domain needs to be

large enough to have the largest turbulent scales of the flow.

Writing Ui = fq + , the term r,>; can be decomposed to

U i U j = U p l j -I- U i U j + U j U j + U jU •,

45


where L.j} = utUj — u,u,j are the Leonard terms, Cij = U{Uj + u^Uj are the cross

terms and Rij = u^u- are the subfilter Reynolds stresses. While the subfilter stresses

Tij are invariant with respect to a Galilean transformation, neither the Leonard nor

the cross terms are invariant. So the SFS stress should not be decomposed, but

modelled as a whole to retain Galilean invariance. The term Tij represents the SFS

scales, and has to be approximated using a subfilter model. In order to solve equation

(3.10), we need to choose proper filter width, as filter size decreases, the subfilter

model represents smaller and smaller scales. Thus the effect of the small scales on

the larger ones is reduced, transforming the LES computation into a direct numerical

simulation. A balance has to be found that allows for an accurate simulation, while

keeping the computational cost as low as possible. Basically the filter cut-off should

be in the inertial subrange, and the computational domain has to be large enough

to have capture the largest turbulent scales of the flow.

Hartel &; Kleiser (1998) analysed a DNS database of turbulent channel flow at

Re* = 115, Re* = 220 and Re* = 300. They argued that a filter width should be

as large as possible in order to minimize the computational needs of the simulation,

but larger filter widths gives rise to a more complex SFS turbulence, thus placing

greater demands on the subfilter scale models. If the filter cut-off is incorrect, then

a considerable fraction of the kinetic energy could reside in the subfilter scales. The

subfilter turbulence will then contain structures which play an important role in

the momentum transport but which are not resolved. To avoid such a situation

in low to moderate Reynolds numbers, some criterion is necessary for the spatial

resolution which will ensure that important features such as the evolution of near

wall streaks or bursts can be captured by the numerical grid and the filter. For

channel flow, Zang (1991) suggested grid spacings of about Ax+ = 80 wall units in

the streamwise direction and Az+ = 30 wall units in the spanwise direction. In the

wall-normal direction, grid spacings vary from Ay+ = 0.04 to Ay+ = 5 wall units.

Such restrictive resolutions have been used at low-Reynolds number flows, but are

extremely difficult to achieve at moderate and high Reynolds number flows, due to

46


computational expense. In most moderate Reynolds number simulations, the grid

spacings are of order of Ax+ = 120, Az+ = 70, 0.5 < Ay+ < 70. At high Reynolds

numbers, it is impractical to resolve the near wall region accurately. ’’Off the surface”

boundary conditions are then required, in order to reduce the computational cost.

3.2.4 Energy Balance Equation

In order to develop SFS models, it is useful to understand the physical phenomena

which the models should represent. The most important effect of the SFS scales on

the filtered ones is the energy exchange that results from the interaction between

filtered and subfilter scales. To understand this interaction, consider the transport

equation for the kinetic energy of the resolved field. For the total filtered energy

q2 = \uiUi,

dq2dt + d 2- \ 9 .__ . d dq2

I d , , du; du;dxj dxj + Tij S,ij 1 3 , (3.11)

or

da2 + (Advection of q2) =

-(Pressure Diffusion of q2) + (Viscous Diffusion of q2) -

(SFS Diffusion ) — (Viscous Dissipation of q2) + (SFS Dissipation).

A detailed explaination of this equation may be found in Hinze(1975). It should

be noticed that the advection and diffusion terms do not create nor destroy filtered

energy, but redistribute it spatially. The viscous dissipation and SFS dissipation

represent, the filtered energy lost by viscous dissipation at the filtered-scale level,

and the net energy exchanged between the filtered and subfilter scales. The subfilter

47

dissipation can be positive or negative, although on average energy flows from the

large scale to the small scales. Energy flowing from the small scales to the large ones

is called backscatter. For the SFS kinetic energy, q ^ s = Tkk/2. it may be shown

(Hinze 1975) that

dvifsdt

1 d d+ d^'fafsVj) = - 9 7^:(u^ ui - uiuiui) - 7ET(Puj - P“j )2 dx

+d . dqafa

— — [y----------- —

O X j d X j

Ti j S i j ,

d _ dui dui) + - fa .V iW ~ dxj dxj

dxi duj duj , dxj dxj '

(3.12)

or

dq2— — I- (Advection of qffj =(Jv

— (Turbulence Transport) - (Pressure Diffusion of d sfs) +

(Viscous Diffusion of q fs) + (SFS Diffusion) —

(Viscous Dissipation of qgfs) — (SFS Dissipation).

The energy exchange terms for the subfilter transport equation are similar to

those of total filtered energy transport equation. The advection and diffusion terms

are again redistribution terms. The energy lost by the filtered scales to the subfilter

ones appears now as a source term in equation (3.12), while the viscous dissipation

represents the SFS energy dissipated by the viscous forces and is modelled by the

subfilter scale model. The SFS diffusion and dissipation have opposite signs in the

transport equations for the filtered (3.11) (3.12).

48


3.3 Subfilter Scale Models

There are many different LES models. To test the quality of the various subfilter

scale models, there are two methods, a posteriori testing and a priori testing. A

posteriori testing involves the comparison of LES results with experiments or DNS

databases. It has the drawback that it is not always easy to pinpoint that cause

a model to fail, nor seperate the SFS models from other elements that affect the

results, such as the numerics. A priori testing offers a direct comparation of subfilter

variables to the exact small scale variables of fully resolved field. It is more strict

than a posteriori testing, so even a SFS model fails in a priori testing, it may still

be good in a posteriori testing. In our work, we will do the posteriori testing.

3.3.1 Smagorinsky E ddy-V iscocity M odel

Most subfilter scale models in use are eddy-viscosity models of the form

S' ■T ij - ~ ^ T kk = - 2v T S i j , (3.13)

which relates the subfilter scale stresses,Tij, to the filtered strain rate

The eddy viscosity ut is, by dimensional analysis, the product of a length scale,

la, and a velocity scale, usf s. Since the most active of the subfilter scales are those

closest to the cut-off, the natural length scale in LES modelling is the filter width,

which determines the size of the smallest resolved scales in the flow. The velocity

scale is usually taken to be the square-root of the trace of the SFS stress tensor, i.e.

u s f s = \ J Q ^ f s ~ y / T k k -

49


The equilibrium assumption is based on the assumption that the small scales

of motion have shorter time scales, and recover equilibrium nearly instantaneously.

Under this ’’local equilibrium” assumption, the transport equation for q ^ s simplifies

significantly, since all terms drop out, except the production term, esf s = Tij Si j, and

the viscous dissipation of SFS energy ev, to yield

This assumption implies the absence of transport effects, i.e. no source and

sink exist at each wave number. This equivalent to a pure transfer in the inertial

subrange, i.e. energy is generated at the filter scale level, and transmitted to smaller

scales, where the viscous dissipation takes place. All eddy-viscosity models are based

on the assumption of Production — Dissipation with Smagorinsky model being the

simplest one. Assuming the viscous dissipation is modelled as ev = u ^ s/ls, and

vt — h usfsi so from (3.13) « 2vr§ij. And from (3.15) we have

where ls is a length scale, called Smagorinsky length scale or subfilter length

scale. It is equal to l3 = c,A, where cs is a constant, called Smagorinsky constant,

and A is the filter width.

For the definition of the filter width A, Deardorf(1970) suggested

(3.15)

(3.16)

A = (AaqAo^AaJs)1/ 3 (3.17)

50


with A;ci, Ax2, Aas3 the filter width in each direction. Bardina(1983) argued• I AX' 1 /\-t*~that in homogeneous turbulent flow with anisotropic filters, A = y —1— ^ a is

better.

Near the wall, the size of the eddies decreases, and therefore the subfilter model

will dissipate more energy than expected unless some modification has been made.

In the near wall region, the Reynolds shear stress asymptotes to zero as y3 (Panton

1997), but the Smagorinsky eddy-viscosity model does not behave in such a manner.

As a result a damping function is needed, such as the van Driest damping function

(van Driest 1956). This is the most popular one used but it does not give the

correct behaviour close to the wall, instead it follows a y4 law. Phanton (1997) [106]

suggested another damping function, which follows a y3 law.

g = - a r c t a n ( ^ —)[ 1 - e x p ( -y - ) ] 2. (3.18)7T 7T G _r

The LES equations in essence describe a non-Newtonian Smagorinsky fluid, called

’’Smagorinsky fluid” or an ”LES-fluid”, in which the viscosity is proportional to a

deformation tensor amplitude, i.e. vies = I2 \S\.

3.3.2 Spectral Vanishing V iscosity

Not all LES models are based on eddy viscosity assumption. The spectral vanishing

viscosity is another approach to do LES. Tadmor first introduced the concept of SVV

using the ID inviscid Burgers equation

9 , 9 ,u2(x, t) .+ 2— ) = ° (3' 19>

subject to given initial and boundary conditions. The distinct feature of solutions

51


to this problem is that spontaneous jump discontinuities (shock waves) may develop,

and hence a class of weak solutions can be admitted. Within this class, there are

many possible solutions, and in order to single out the physically relevant one an

additional entropy condition is applied, of the form

du2(x,t) d u3(x,t).+ s 0 <3-20>

Directly solving the inviscid Burgers equation will lead to a difficulty due to Gibbs

oscillations. Tadmor introduced the spectral vanishing viscosity method, which adds

a small amount of controlled dissipation that satisfies the entropy condition, yet

retains spectral accuracy. It is based on viscosity solutions of nonlinear Hamilton-

Jacobi equations, which have been studied systematically in [25]. Specifically, the

viscosity solution for the Burgers equation has the form

where e —>• 0 is a viscosity amplitude and Qf is a viscosity kernel, which may be

nonlinear and, in general, a function of x. Convergence may then be established by

compensated compactness estimates combined with entropy dissipation arguments.

Equation (3.21) can be written in discrete form as

d , d ,v?(x,t).^ d _ OumIxA)^g-t nN(x,t) + - [ P N{— ^ - ) ] = e- [ Q N * ], (3.22)

where the star * denotes convolution and is a projection operator. Q,\r is a

viscosity kernel, which is only activated for high wave numbers. In Fourier space,

this kind of spectral viscosity can be efficiently implemented as multiplication of the

52

Fourier coefficients of w,,v with the Fourier coefficients of the kernel Qn ■ i.e.,

e ^2 k2Qk{t)iik{t)e‘ M<\k\<N

( 3 . 2 3 )

where k is the wave number, N is the number of Fourier modes, and M is the cutoff

wavenumber above which the spectral vanishing viscosity is activated. Originally,

Tadmor used

In subsequent work, a smooth kernel was used, and e & N 1, activated for modes

k > M « 5y/N, with

Karamanos et al[64] made the first extension of the spectral vanishing viscosity

concept to spectral/hp element methods.

Our SVV has been added to the 4 corners in figure(3.1).

3.3.3 Im plicit Spectral Vanishing V iscosity Im plem entation

First we would like to analyze the terms in SVV formulation.

. 1 0 , 1*1 < M

1, \k\ > M(3.24)

( k - N ) 2

(3.25)

53

Figure 3.1: SVV in Fourier Space

3.3.4 Filtering in Orthogonal Basis

In this chapter, we will do the filtering operation on an orthogonal basis which

needs a transform from hierarchical C° basis. We will explain the formulation in

following. Burgers equation written in a strong form is given by:

du 1 du2 ddt 2 dx e dx dx ’ (3.26)

If we examine the weak form of SVV term only, ignore boundary terms, we have

the following basic form of the SVV operator

<£■«£> (3.27)

where v is a test function from ij>k, and u = Y^k=i ^k^k = ^kVk- 4>k are basis

functions used for continuous Galerkin formulation, and ipk are orthogonal basis

functions span the same space as <j>k. i>k are test functions for (j>k-

54

Let B be the matrix which transforms the model coefficients u for the basis

functions {(j)/.} to u in {(pk} space. Let F be a diagonal matrix which acts as a

filtering function(the diagonal entries of which are given by equation (3.25).

In the notation above, we have that u = Bu. We want to filter the coefficients

u instead of filtering the coefficients u. Hence we will transform to orthogonal space

and filter there, then transform back. This is accomplished as follows:

h = B _1FBu. (3.28)

We can rewrite this as expression u = 0 u where 0 = B XFB. The formula

(3.27) can be expressed as:

St B~1FB M -1 Su. (3.29)

where Sy = (</);, and M,;j = (</;,;, Since B _1 = M _1Br , the formula

(3.30) can be written as:

St M “1Bt F B M -1Su. (3.30)

This operator is a symmetric matrix, because

(s t M '1b t f b m ' 1s )t = s t m - t b t f t b m - t s

= St M “1Bt FBM “ 1S

using the fact that FT = F,M~~T = M~l . Also this oprator is positive semi-

definite, as

nr STM “ 1BTFB M “1Su

(B M '1Su)t F(BM “1Su)

e n , ((, = BM _1Su)

(3.31)

55

Since F is a diagonal matrix in which all the diagonal elements are greater than

or equal to zero, so £r F£ > 0 for all £ of the form BM_1Su. Therefore the operator

of SVV term in weak form is positive semi-definite.

3.3.5 Dynam ic LES M odel

The dynamic model was first presented by Germano (1991) who proposed a

method for computing the coefficients of subfilter scale eddy-viscosity models as a

function of space and time. Detailed of Dynamic Smagorinsky method can be found

in appendix A.

Although the dynamic model is popular now, there are some problems associated

with the model. For Dynamic Smagorinsky model, cs can be negative, and this cor

responding to backscatter of turbulent energy, which means an energy transfer back

from small scales to the large ones. This may cause instability during computation.

For our dynamic SVV, we compute e at each time step according to the Smagorin

sky eddy viscosity model. In Fourier space, we average the viscosity y-profile over

all Fourier modes, and then apply it in all Fourier modes.

Vsmagjy)^ v i s

csA2|5(y)|1

Re*

56

e(l/) =

v sraag { y ) —

Vvis —


0 5 10 1 5 20 25 30 35 0 5 10 15 20 25 30 35

Figure 3.2: Gaussian Filter (left); exponentail Filter (right)

3.4 LES Results

There are several different filters for LES computation. First is Gaussian filter

defined as

_ . . - C 2 *m2Q(m) = e 24 (3 .32)

shown on left of figure(3.2), and Exponential filter is defined as

— m3Q(m) = e c * N a (3.33)

shown on right of figure(3.2).

3.4.1 Comparison to standard Smagorinsky M odel

First we compare our results with LES results using standard Smagorinsky and

Dynamic Smagorinsky model. Reynolds number is Re* = 180, and the mesh is

57


Re* Re Domain DNS mesh LES mesh Mesh ratio

1 8 0 4 3 0 0 27r x 2 x 2n 1 2 8 x 1 3 0 x 128 4 0 x 6 5 x 4 0 21

6 0 0 1 8 0 0 0 27t x 2 x 7r 3 8 4 x 3 6 1 x 3 8 4 8 0 x 1 2 9 x 80 6 5

1 0 0 0 2 7 5 0 0 67T X 2 X 1.57T 7 6 8 x 5 2 1 x 7 6 8 1 2 8 x 1 9 3 x 1 2 8 9 8

Table 3.1: Simulation parameters for DNS.

40*41*40, with one element in the wall normal direction.

Figure(3.3) show that our SVV implementation has similar effect as standard

one, the mean velocity and fluctuations both improved. The difference is that we

add SGS stresses implicitly, and the viscosity has been averaged over all Fourier

modes. On each Fourier mode, the viscosity is same as averaged value.

Figure(3.4) and (3.5) show that we achieved similar or better results than stan

dard LES method in Reynolds stress and turbulent fluctuation. For Smagorinsky

model, the Cs = 0.005, and for Dynamic Smagorinsky model, we chose C = 0.05. We

observed that the mean velocity we obtained was similar to the standard implemen

tation, but since we added same viscosity in each Fourier mode, makes the difference

from standard models. We observed that the turbulent fluctuation increased, and

the Reynolds stress profile was a little bit larger than standard implementation. This

means we add more disturbation at high Fourier modes, which increase the energy at

these modes. Even the Reynolds stress becomes larger, we found that our turbulent

fluctuation are better than the standard implementation. These comparison suggest

that our implementation has a similar effect as standard method, and we will use

this code to do more investigation on LES using SVV.

Table 3.1 shows the simulation parameters for DNS and LES at different Reynolds

numbers. The mesh ratio show how much larger of DNS mesh to LES mesh at each

Reynolds number. The value of Re corresponds to \ jv in the present scalings.

58


b

b

Figure 3.3: Mean Velocity, 40 x 41 x 40. Smagorinsky SVV, Solid line: DNS on

128 x 130 x 128; dash line: coarse DNS, one element; dot line: LES using SVV, Cs =

0.005; dashdot line: standard Smagorinsky model, Cs = 0.005 (upper); Dynamic

Smagorinsky SVV, Solid line: DNS on 128 x 130 x 128; dash line: coarse DNS,

one element; dot line: LES using SVV (C=0.125); dashdot line: standard Dynamic

Smagorinsky model (below)

59


■0.1

- 0.2

-0.5

-0.(

■0.7

0 50 100 150

- 0.2

> -0.4

- 0.6

50 100 150

Figure 3.4: Reynold Stress. Smagorinsky model, solid line: DNS on 128 x 130 x 128;

dash line: coarse DNS, one element; dot line: LES using SVV, Cs = 0.005; dashdot

line: standard Cs — 0.005 (left), Dynamic Smagorinsky model, solid line: DNS on

128 x 130 x 128; dash line: coarse DNS, one element; dot line: LES using SVV

(C=0.125); dashdot line: standard (0.125) (right)

60


2.5

Eoc

0.5

150100y+

2.6

2.4

2.2

+<flEcc

0.4

0.2

50 150100

Figure 3.5: Turbulence fluctuation. Smagorinsky model, solid line: DNS on 128 x

130 x 128; dash line: coarse DNS, one element; dot line: LES using SVV, Cs — 0.005;

dashdot line: standard Cs = 0.005 (upper), Dynamic Smagorinsky model, solid line:

DNS on 128 x 130 x 128; dash line: coarse DNS, one element; dot line: LES using

SVV (C=0.125); dashdot line: standard (0.125) (below)

61

Re* Domain LES mesh Elements in y Smagorinsky Dynamic Smagorinsky

180 2 tt x 2 x 2 tt 40 x 65 x 40 1 Cs = 0.005 C=0.05

180 2 tt x 2 x 2 tt 40 x 65 x 40 2 Cs = 0 .0 1 C=0.075

600 27t x 2 x 7r 80 x 129 x 80 2 Ca = 0.005 C=0.02

1 0 0 0 37t x 2 x 1.57r 128 x 193 x 128 2 Cs = 0.012 C=0.028

Table 3.2: LES runs at different Reynolds number.

Table 3.2 shows the LES runs at different Reynolds numbers. The detailed statis

tics will be reported in following.

3.4.2 LES results at low Reynolds number

Next we investigated the LES by SVV results on single element and multiple elements

in wall normal direction. The mesh is 40 x 65 x 40 with single or two elements in the

wall normal direction. Since DNS uses 128 x 130 x 128 mesh points, the LES mesh

has about 1/20 the number of points as compared to the mesh of DNS.

1. Single element in the wall normal direction

Similar to what we did before, we present the statistics of incorporating standard

and Dynamic Smagorinsky with SVV methods.

Figure(3.6) to (3.8) show that the SVV effects are quite similar in mean velocity

and Reynolds stress profile. Adding SGS stresses in all polynomial modes seem to

give better results in turbulent fluctuation profile, but this is not always true. As

before, we found that Reynold stress increased a little bit, which is consistent with

62


b

Figure 3.6: RE* = 180. Mean Velocity. 40 x 65 x 40, Smagorinsky SVV, Cs = 0.005.

solid line: DNS on 128 x 130 x 128 dashed line: LES using SVV, Cutoff=0; dotted

line: LES using SVV, Cutoff=l; dash-dot line: LES using SVV, Cutoff=2 (upper);

Dynamic Smagorinsky SVV, C = 0.05. solid line: DNS on 128 x 130 x 128; dashed

line: coarse DNS; dotted line: LES using SVV, cutoff=0; dash-dot line: LES using

SVV, Cutoff=l; long dashed line: LES using SVV, cutoff=2; dash-dot dotted: LES

using SVV, cutoff=3 (below)

63


- 0.1

- 0.2

V -0.4

-0.5

-0.750 100 150

- 0.1

-0.2

-0.3

= -0.4

-0.5

- 0.1

-0.7

0 50 100 150

Figure 3.7: R E * = 180. Reynold Stress. Smagorinsky (40 x 65 x 40, Cs — 0.005),

solid line: DNS on 128 x 130 x 128; dashed line: LES using SVV, cutoff=0; dotted

line: LES using SVV, cutoff=l; dash-dot line: LES using SVV, cutoff=2 (left);

Dynamic Smagorinsky (40 x 65 x 40, C — 0.05), solid line: DNS on 128 x 130 x 128;

dashed line: coarse DNS; dotted line: LES using SVV, cutoff=0; dash-dot line: LES

using SVV, cutoff—1; long dashed line: LES using SVV, cutoff=2; dash-dot dotted

line: LES using SVV, cutoff=3 (right)

64


2.5

+«£cc

0.5

100 150y+

2.5

+<fl£oc

0.5

50 100 150y+

Figure 3.8: R E * — 180. Turbulent fluctuation. 40 x 65 x 40, Smagorinsky SVV,

Cs =0.005, solid line: DNS on 128x130x128; dashed line: LES using SVV, cutoff=0;

dotted line: LES using SVV, cutoff=l; dash-dot line: LES using SVV, cutofF=2

(upper); Dynamic Smagorinsky SVV, C = 0.05, solid line: DNS on 128 x 130 x 128;

dashed line: coarse DNS; dotted line: LES using SVV, cutoff=0; dash-dot line: LES

using SVV, cutoff=l; long dashed line: LES using SVV, cutoff=2; dash-dot dotted:

LES using SVV, cutoff=3 (below)

65

IV)>

y y

Figure 3.9: RE* = 180. SVV viscosity (40 x 65 x 40, one element): Smagorinsky,

Cs = 0.005 (left); Dynamic Smagorinsky, C = 0.05 (right)

adding SVV terms at high modes. Corresponding viscosity of Smagorinsky SVV

and Dynamic Smagorinsky SVV are shown in Figure(3.9). The SVV viscosity is

normalized with fluid viscosity at Re* = 180, which is 1/4300. The peak value of

normalized SVV viscosity is only about 0.5, which means that compared to the fluid

viscosity, the SVV viscosity is not too big. The statistics though has been improved

quite a lot. In this case, we found that when adding SVV terms in more polynomial

modes, the statistics becomes much better.

2. Multiple elements in the wall normal direction

Figure(3.10) to (3.12) show that SVV methods achieved better statistics of tur

bulence than adding SGS terms in all modes. The best cutoff number is 2 and 3,

for standard and Dynamic Smagorinsky models, respectively. This proved that SVV

has some advantages over standard implementation of adding SGS stresses. The

SVV viscosity added in two elements is larger than that in one element, and this

makes the Reynolds stress a little bit larger. This is consistent with previous results.

Figure(3.12) shows the SVV viscosity with two elements in wall normal direction.

66


20

Figure 3.10: Mean Velocity. 40x65x40, two elements, Smagorinsky SVV, Cs = 0.01.

solid line: DNS on 128 x 130 x 128; dashed line: coarse DNS; dotted line: LES using

SVV, cutoff=l; dash-dot line: LES using SVV, cutoff=2; long dashed line: LES using

SVV, cutoff=3; dash-dot dotted line: LES using SVV, cutoff=4 (upper); Dynamic

Smagorinsky SVV, C = 0.075. solid line: DNS on 128 x 130 x 128; dashed line:

coarse DNS; dotted line: LES using SVV, cutoff=0; dash-dot line: LES using SVV,

cutoff=l; long dashed line: LES using SVV, cutoff=2; dash-dot dotted line: LES

using SVV, cutoff=3 (below)

67


- 0.2

3 -0-4

- 0.6

0 50 100 150

- 0.2

*A

£ -0.4 V

500 100 150

Figure 3.11: Reynold Stress. Smagorinsky (40 x 65 x 40, two elements, Cs = 0.01),


SVV, cutoff=l; dash-dot line: LES using SVV, cutoff=2; long dashed line: LES

using SVV, cutoff=3; dash-dot dotted: LES using SVV, cutoff=4 (left); Dynamic

Smagorinsky (40 x 65 x 40, two elements, C = 0.075), solid line: DNS on 128 x 130 x

128; dashed line: coarse DNS; dotted line: LES using SVV, cutoff=0; dash-dot line:

LES using SVV, cutoff—1; long dashed line: LES using SVV, cutoff=2; dash-dot

dotted line: LES using SVV, cutoff=3 (right)

68


2.42.2

+o>EOC

0.80.6

0.40.2

10050 150

3.2

2.62.42.2

+toEoc

0.60.40.2

100 150

Figure 3.12: Reynold Stress. 40 x 65 x 40, two elements, Smagorinsky SVV,

Cs = 0.01, solid line: DNS on 128 x 130 x 128; dashed line: coarse DNS; dot

ted line: LES using SVV, cutoff=l; dash-dot line: LES using SVV, cutoff=2; long

dashed: LES using SVV, cutoff=3; dashdotdot line: LES using SVV, cutoff=4 (up

per); Dynamic Smagorinsky SVV, C=0.075, solid line: DNS on 128 x 130 x 128;

dashed line: coarse DNS; dotted line: LES using SVV, cutoff—0; dash-dot line: LES

using SVV, cutoff=l; long dashed line: LES using SVV, cutoff=2; dash-dot dotted

line: LES using SVV, cutoff=3 (below)

69


JrU>

y

&Ms«>

y

Figure 3.13: SVV viscosity. Smagorinsky (left); Dynamic Smagorinsky (right)

3.4.3 LES R esults at high Reynolds number

Finally we investigate the SVV method at high Reynolds number, Re* = 600 and

Re* = 1000. The mesh are 80 x 129 x 80 and 128 x 193 x 128 respectively, with

two elements used in wall normal direction. Same as before, standard and Dynamic

Smagorinsky models have been incorporated into SVV method.

(1). Re* = 600

Figure(3.14) to (3.16) show that the SVV method gives better results. For the

standard Smagorinsky model, the best cutoff number is 6; while for the Dynamic

Smagorinsky model, the best cutoff number is 4. DNS has been done on 384 x 361 x

384 mesh, which is only about 65 times larger than the LES mesh. Our LES results

are very similar to DNS results. This indicates that our SVV implementation of LES

can be used at high Reynolds number LES simulation. The SVV viscosity is shown

in Figure(3.17).

70


20

y*

20

Figure 3.14: Mean velocity. 80 x 129 x 80, two elements, Cs = 0.005. Smagorinsky

SVV, solid line: DNS on 384*361*384; dashed line: coarse DNS; dotted line: LES

using SVV, cutoff=0; dash-dot line: LES using SVV, cutoff=2; dash-dot dotted line:

LES using SVV, cutoff=4 (upper); Dynamic Smagorinsky SVV, C = 0.02, solid line:

DNS on 384 x 361 x 384; dashed line: coarse DNS; dotted line: LES using SVV,

cutoff=0; dash-dot line: LES using SVV, cutoff=2; long dashed line: LES using

SVV, cutoff=4 (below)

71


- 0.2

> -0 4

100 200 300+ 500400 600

-0.2

+A>3V

-0.6

1000 200 300 500400 600

Figure 3.15: Reynold Stress. 80 x 129 x 80, two elements, Smagorinsky SVV, Cs =

0.005, solid line: DNS on 384 x 361 x 384; dashed line: coarse DNS; dotted line: LES

using SVV, cutoff=0; dash-dot line: LES using SVV, cutoff=2; dashdotdot line: LES

using SVV, cutoff=4 (upper); Dynamic Smagorinsky SVV C = 0.02, solid line: DNS

on 384 x 361 x 384; dashed line: coarse DNS; dotted line: LES using SVV, cutoff=0;

dash-dot line: LES using SVV, cutoff=2; long dashed line: LES using SVV, cutoff=4

(below)

72


2.4

2.2

+coEOC

0.6

0.4

0.2

200 400 600100 300 500y+

2.6

2.4

2.2

+MEoc

0.6

0.4

0.2

200 400 600100 300y+

500

Figure 3.16: Turbulent fluctuation. 80 x 129 x 80, two elements, Smagorinsky SVV,

C8 = 0.005, solid line: DNS on 384 x 361 x 384; dashed line: coarse DNS; dotted line:

LES using SVV, cutoff=0; dash-dot line: LES using SVV, cutoff=2; dashdotdot line:

LES using SVV, cutoff=4(upper); Dynamic Smagorinsky SVV, C — 0.02, solid line:

DNS on 384 x 361 x 384; dashed line: coarse DNS; dotted line: LES using SVV,

cutoff=0; dash-dot line: LES using SVV, cutoff=2; long dashed line: LES using

SVV, cutoff=4 (below)

73


£ 07> 0.5

y

&§

y

Figure 3.17: SVV Viscosity. Smagorinsky, Cs = 0.005 (left); Dynamic Smagorinsky,

C=0.02 (right)

(2). Re* = 1000

We will also give the results at Re* = 1000. The mesh is 128 x 193 x 128, with

two elements used in wall normal direction. Same as before, standard and Dynamic

Smagorinsky models have been incorporated into SVV method.

Figure(3.18) to (3.20) show that the SVV method gives better results. It is

very obvious that all LES simultation give better results than coarse DNS. For the

standard Smagorinsky model, the best cutoff number is 6; while for the Dynamic

Smagorinsky model, the best cutoff number is 4. DNS has been done on 768 x

521 x 768 mesh, which is only about 97 times larger than the LES mesh. Our LES

results are very similar to DNS results. The SVV viscosity is shown in Figure(3.21).

Although the maximum value of viscosity is about the same, in Smagorinsky model

the viscosity has a sharp peak in the near wall region.

74


20

b

20

Figure 3.18: Mean velocity. 128 x 193 x 128, 2 elements, Cs — 0.012. Smagorinsky

SVV, solid line: DNS on 768 x 521 x 768; dashed line: coarse DNS; dotted line:

Cutoff=0; dash-dot line: Cutoff=2; long dashed line: Cutoff=4; dash-dot dotted line:

Cutoff=6 (upper); Dynamic Smagorinsky SVV, solid line: DNS on 768 x 521 x 768;

dashed line: coarse DNS; dotted line: Cutoff=0; dash-dot line: Cutoff=2; long

dashed line: Cutoff=4; DashDotDot line: Cutoff=6 (below)

75


-0.2

+A>3V

- 0.6

200 400 600 800

-0.25

> -0.5

-0.75

200 400 600 800y+

Figure 3.19: Reynold Stress. Smagorinsky(128 x 193 x 128,2 elements,Cs = 0.012),


SVV, Cutoff=0; dash-dot line: LES using SVV, Cutoff—2; long dashed line: LES

using SVV, Cutoff=4; dashdotdot line: LES using SVV, Cutoff=6 (upper); Dynamic

Smagorinsky(128 x 193 x 128,2 elements,C = 0.028), solid line: DNS on 768 x 521 x

768; dashed line: coarse DNS; dotted line: LES using SVV, Cutoff=0; dash-dot line:

LES using SVV, Cutoff=2; long dashed line: LES using SVV, CutofF=4 (below)

76


2.62.42.2

wEoc

0.60.40.2

200 400 800600y+

2.75

2.5

2.25

1.75+</>Eoc

1.25

0.75

0.5

0.25

400 800200 600

Figure 3.20: Turbulent fluctuation. Smagorinsky(128 x 193 x 128,2 elements,Cs =

0.012), solid line: DNS on 768 x 521 x 768; dashed line: coarse DNS; dotted line:

LES using SVV, Cutoff—0; dash-dot line: LES using SVV, Cutoff=2; long dashed

line: LES using SVV, Cutoff=4; dashdotdot line: LES using SVV, Cutoff=6 (up

per); Dynamic Smagorinsky(128 x 193 x 128,2 elements,C = 0.028), solid line: DNS

on 768 x 521 x 768; dashed line: coarse DNS; dotted line: LES using SVV, Cut-

off=0; dash-dot line: LES using SVV, Cutoff=2; long dashed line: LES using SVV,

Cutoff=4 (below)

77

&'(A8M>

y

w>

y

Figure 3.21: SVV Viscosity. Smagorinsky,^ = 0.012(left); Dynamic Smagorinsky,

C=0.028(right)

3.5 Summary

We have investigated a new approach for LES simulation using Spectral Vanish

ing Viscosity (SVV) Method. Fourier Id Spectral Element method has been used

for discretization. In order to increase stability, the SVV terms have been imple-

mentated implicitly, and in corporated with standard Smagorinsky and Dynamic

Smagorinsky models. Using standard Smagorinsky or Dynamic Smagorinsky model,

the viscosity added is variable across the channel, which is different from standard

SVV method. Through this way, the viscosity can be determined according to local

flow information, which is more reasonnable than the constant value need before.

Comparing to standard LES models, SVV approach has the SGS terms only

added to high modes in coefficient space, no low modes have been touched. But

through change of small scales of channel turbulence, the mean statistical quantities

have also been improved indirectly.

Detail statistics have been compared with DNS results and standard LES results

at both low and high Reynolds number. At low Reynolds number, we found that the

78


advantage of SVV is more obvious, while at high Reynolds number SVV produce

similar results as standard implementation. The results are encouraging, and it

shows that SVV method is an efficient way to do LES. More work need to be done

in this direction.

79


Chapter 4

Large Eddy Simulation in

Complex Geometry

4.1 Background

Most tests of Large Eddy Simulation so far have been done for channel flow, but the

most interesting applications involve flows in complex geometries. The finite volume

and finite difference methods have been used in this area, as they are relatively

easier to implement in complex geometry. Spectral and Spectral element method

have achieved great success in Direct Numerical Simulation of channel turbulence

because of its high accuracy. The LES work reported in the earlier chapter was done

in channel flow using a spectral element method in one direction. Few LES works

have been done using 2d or 3d spectral element methods.

Unstructured spectral/hp finite elements is a high-order scheme with the capa

bility of efficient discretization of complex geometries. It incorporates both multi

domain spectral methods based on the ideas of A.T. Patera and B.A. Szabo. Poly

nomial spectral methods were first introduced by Gottlieb & Orszag (1977) and have

80


been covered in Canuto et al. (1987) and Boyd (1989). They have been extended to

their present unstructured form by Sherwin & Karniadakis (1995). The concept is

simple, the solution is approximated by a series of polynomials, the order of which

can be chosen arbitrarily. Implementation of this method is complicated, but it

does show very useful properties. They reach convergence either by increasing the

elements or by increasing the order of polynomial. This method provides spectral

convergence, which mean that the error decrease faster than exponential rate. Yet

this method has not been used in LES simulation too much so far. As mentioned

before, my work continue the efforts of previous LES efforts, see [70, 64, 8].

4.2 LES methods and implementation

The method is similar to LES using Fourier Spectral Element method in Chapter 3.

The difference is the NS equation been solved in (m,y,n) space, but now it is been

solved in (x,y,n) space. m,n are Fourier modes, and y is in physical space.

4.2.1 Standard SV V m ethod

Our work is different from Mike Kirby[70] in following several points:

1. SVV term in z direction has been only added to high polynomial modes;

2. Using wall damping function in channel flow;

3. SVV terms in (x,y) plane have been only added to high Fourier modes.

In order to use SVV, we have to filter high polynomial modes. Simlar to what

we have done in the earlier chapter, we transform the velocities to an orthogonal

basis, then do the filtering there, and finally transform back to the original basis. In

81


our implementation, this only needs to be done in the first several time steps for the

SVV method. Because the viscosity term added in the SVV method is constant, and

it does not change with time. But for Smagorinsky model, since the viscosity added

is a viariable both in space and time, this transformation between orthogonal basis

and original one needs to be done at each time step in theory. As the time step we

use is quite small, and the flow structure will not change too much over a few time

steps, we have done the transformation every 20 time steps in our simulation. This

will speed up the code.

4.2.2 Smagorinsky M odel

The idea is similar to that in the earlier chapter. Instead of adding constant viscoity,

we evaluate it by Smagorinsky model, which is based on the local stress in the flow.

This locally adds a larger viscosity in the high shear rate region, and less viscosity

in the other regions.

There is an additional difficulty for LES using the Spectral Element method.

All the concepts and techniques of LES have been developed under a structured-

grid formulation. In finite volume, finite difference or spectral method codes, the

definition of the subfilter model relies on length scales that are based on the grid

spacing. Using unstructured spectral elements, the model has to be modified in order

to take account of the order of accuracy of the calculation and the triangular shape

of the element. In this chapter, only the Smagorinsky model had been used to do

LES.

The choice and application of a subfilter model is very important. The subfilter

length scale ls = c,s A needs to be defined in a manner that is consistent with the

LES formulation, and incorporates the properties of spectral elements efficiently. A

filter width is proposed of the form

82


0.5

-0.5

0 0.5 1 1.5 2

Figure 4.1: Mesh for Re* = 180 (x,y) plane

A = (A (^ )2A z) ^ (4.1)

where P is the polynomial number of the expansion basis used and A is the

area of the triangle. Equation(4.1) is proportional to the ’volume’ of the prismatic

shaped element used in nektarF. The triangular plane is resolved using spectral

elements, while the third direction is resolved using Fourier series. The length of the

prismatic, Az, is the distance between the Fourier planes, while the height and base

of the triangle, used in calculating the area, A, of the triangle, are converted by the

term tt/P , according to equation(4.1) of Gottlieb & Orszag (1977).

The computed viscosity has been averaged over Fourier planes in z direction,

which is different from the channel case, where we average the viscosity over x and

z directions.

83


R e’=180,48*49*48

Smagorinsky, Cs=0.005Solid-----------No filterD ash-----------Filtering a tP - 4

0.4

0o(0>0.2

-0.5 0.5y

Figure 4.2: Viscosity profile at Re* = 180, Smagorinsky model, Cs = 0.005, Chan

nel code, Solid line: No filter; Dash line: Filtering all polynomial coefficients of

Smagorinsky viscosity larger than 4 to zero

4.3 Channel Flow

We use 4 elements in (x,y) plane, each element has 25 points. This makes total mesh

49 x 49 x 48. In order the verify the correctness of nektarF with SVV, we also run

the channel code with 48 x 49 x 48 mesh, and compare the results.

First we plot the viscosity profile using same Smagorinsky constant Cs = 0.005.

In figure (4.2), the dash line corresponds to the viscosity profile with filtering at

polynomial order 4. This means setting the coefficients of polynomial order larger

than 4 to be zero. In figure (4.3), the two curves correspond to the viscosity before

and after applying wall damping. Without damping, the viscosity has a large value

on the wall, but it decreases to zero after damping. This is very important to avoid

instability when doing the LES simulation.

Next, we want to check if doing DNS on these similar mesh will give similar

84


1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 50 100 150 200 250 300 350 400

Figure 4.3: Viscosity profile at Re* = 180, LES Smagorinsky model, Cs — 0.005,

nektarF

statistics.

Then we use Smagorinsky model to compute viscosity across the channel in both

codes, and compare the statistics results. Figure (4.5) show the comparison of them.

We can see they are close to each other, which confirm that our implementation is

correct. We can see streamwise fluctuation increase while spanwise and wall normal

fluctuations decrease, whichmake them closer to the DNS results.

Based on this result, we continue to test adding viscosity only at high polynomial

modes, and compare the results. Figure (4.6) gives the results, and it also shows the

effects are quite similar on both codes.

At last we compare the LES results using Smagorinsky model. We apply a

Phanton function as wall function in each code, and constant C=12.0. Cs = 0.005.

We can see both of them have been improved using Smagorinsky model.

85


i/i£cc

0.5

100 150

Figure 4.4: Coarse DNS turbulent fluctuations at Re* = 180, Solid line: Channel

code; Dash line: NektarF code

2.5

2

1.5

1

0.5

00 50 100 150

Figure 4.5: Turbulent fluctuation at Re* = 180 using SVV, e = 0.5, MN=0, MNF=0.

Solid:Channel code; Dash:nektarF code

86


2.5

</>£cc

0.5

100 150

Figure 4.6: Turbulent fluctuation at iie* = 180 using SVV, e = 1.0, MN=3, MNF=3.

Solid line: Channel code; Dash line: NektarF code

2.5

0.5

100 150y+

Figure 4.7: Turbulent fluctuation at ife* = 180 using Smagorinsky model, Cs =

0.005, MN=0, MNF—0. Solid line: Channel code; Dash line:nektarF code

87

5

0

■5

-10 0 10 20

Figure 4.8: Mesh for Re=3900 Full mesh

4.4 Cylinder Flow

We do not use wall function in the computation of cylinder flow.

4.4.1 Verification of D N S at R e=3900

Figure(4.8) shows the mesh we use for Re=3900, and figure(4.9) shows the local

mesh around the cylinder.

Figure(4.10) show the drag and life coefficients at Re=3900. C,i = 1.0 and

Cl = 0.1, St=0.2045 which match the results of X. Ma et al [92].

We also compare mean and fluctuation U velocity profile with results from [64].

88


Figure 4.9: Mesh for Re=3900 Local mesh around cylinder

0.8

0.6

0.2

- 0.2

3925 3930 3935 3940 3945t

Figure 4.10: Drag and lift coefficients for Re=3900

89


Prom these, our results match his data quite well.

Prom all results above, our simulation produce similar results as before. We will

investigate flow at Re=10,000 below.

4.4.2 D N S and LES at R e=10,000

Figure(4.13) shows the mesh we use for Re= 10,000, and figure(4.14) shows the local

mesh around the cylinder.

Figure(4.15) shows the viscosity computed from Smagorinsky model, with Cs =

0.005 at R,e=l 0.000 using the mesh of 6272 elements.

Table4.1 shows the simulation parameters in each simulation. As the speed of

NektarF is not very fast right now, the results we obtained so far only averaged over

short time period, and they are still not converge yet.

We show the drag and lift coefficent history, Reynolds stress contour for case I,VI

and VII in following figures. They are similar to each other, and LES results show

some improvements.

We also compare the mean streamwise velocity and velocity fluctuation profile of

u and v at x=2,5,10 with DNS data which use 9272 elements. At some position, LES

results do show improvement than DNS result on 6272 Elements. But as the time

average period is not too long right now, so we still need more careful investigation

on LES simultaion.

90


7.5

5.5

4.5

o

2.5 x -1 .54

0.5

-0.5

y

x-2 .02

= 0.4

0.2

-4 ■2 0 2 4y

Figure 4.11: Mean and flucuation velocity profile at x—1.06,1.54,2.02 and 3 for

Re=3900. Solid line:George Karamanos’s results; Dash line: our results

91


X=1Q

X“ 7

x=4

0.5

y

0.3

0.25

0.2

§0.15

x - 7

0.1

0.05

x=4

-0.05

y

Figure 4.12: Mean and flucuation velocity profile at x=4,7 and 10 for R,e=3900.

Solid line:George Karamanos’s results; Dash line: our results

92


Y

P X

20

10

0

10

20

-20 0 20 40

Figure 4.13: Mesh for Re=10,000 Full mesh obtained from Steve Dong

1

V :. . ■ !'■ ' '■ ■ i

P ' 7 ' ~r-~.v.. ■ ■■

H 'v yy : / 1

■'

; ..I-

Figure 4.14: Mesh for Re= 10,000 Local mesh around cylinder

93


vis: 0.665901 1.85823 5.18548 14.4703 40.3802

X

Figure 4.15: Viscosity contour for Smagorinsky Model at Re=10,000

Case K P M L z / ( ttD ) Cs or e c d - c p St CLNor berg - - - - - - - 0.202 0.394

Williamson - - - - - - 1.112 - -

Gopalkrishnan - - - - - 1.186 - 0.193 0.384

I 6272 5 64 1.0

ooII 1.06 1.114 0.211 0.5

II 6272 5 64 1.0 e = l /P 1.16 1.22 0.207 0.51

III 6272 5 64 1.0 e = 1 1.3 1.52 0.211 0.53

IV 6272 5 64 1.0 Cs = 0.005 1.24 1.27 0.2 0.53

V 6272 5 64 1.0 Cs = 0.05 1.2 1.0 0.248 0.45

VI 6272 5 64 1.0 e = l /P

MN=MNF=2

1.15 1.23 0.213 0.5

VII 6272 5 64 1.0 Ca = 0.05

MN=MNF=2

1.25 1.1 0.203 0.55

Table 4.1: Simulation parameters for flow pass cylinder at Re=10,000.


Figure 4.16: Experiment Reynolds Stress from A. Ekmekci[36]

95


Level 1 41 81 121 161 201p: -1.1335 -0.7335 -0.3335 0.1265 0.5265 0.9265

0.6

0.4

0.2

- 0.2

- 0.4

-0.6

■ 4 ■ L. i t I I I I I-0.80.4 0.6 0.8 1.2 1.4

X2.2

0.8

0.6

° 0.4 T3O

0.2

730 740 750t

Figure 4.17: Case I. Reynolds Stress (upper); Drag and lift coefficient history (lower)

96


Level 1 41 81 121 161 201p: -1.1335 -0.7335 -0.3335 0.1265 0.5265 0.9265

0.6

0.4

0.2

>.

- 0.2

- 0.4

- 0.6

-0.8 0.4 0.6 0.8 1.4 1.6 2.2X

1.2

1

0.8

0.6

0.2

0

- 0.2

- 0.4

730 740 750t

Figure 4.18: Case VI. Reynolds Stress (upper); Drag and lift coefficient history

(lower)

97


- 0.5 A)18?

j L.0.5

_i I I Ll_

1.5

1.4

1.2

1

0.8

0.6

° 0.4TS° 0.2

0

-0.2

- 0.4

-0.6

725 730 735 740 745 750t

Figure 4.19: Case VII. Reynolds Stress (upper); Drag and lift coefficient history

(lower)

98


0.9

0.8

0.7

"=> 0.6

0.5

- D N S, 6 2 7 2 E, C a s e I— L ES, 6 2 7 2 E, C a s e VI

L ES, 6 2 7 2 E, C a s e VII D N S, 9 2 7 2 E

0.4

0 .3

0.2

y

0.9

0.8

0.7

- D N S, 6 2 7 2 E, C a s e I' " LE S , 6 2 7 2 E, C a s e VI

LES, 6 2 7 2 E, C a s e VII D NS, 9 2 7 2 E

0.6

0.5

y

0.9

0.8ED x=10

0.7

D N S, 6 2 7 2 E, C a s e I L ES, 6 272 E, C a s e VI L ES, 6 272 E, C a s e VII D N S, 9 2 7 2 E

0.6

0.5

y

Figure 4.20: Averaged streamwise velocity profile at x=2,5,10 for Re=10,000.

99


0.6- D NS, 6 2 7 2 E, C a s e I

LES, 6 2 7 2 E, C a s e VI LES, 6 2 7 2 E, C a s e VII D NS, 9 2 7 2 E

0.5

0.4

x=2£ 0.33

0.2

0.1

y

0.4DNS, 6 2 7 2 E, C a s e I LES, 6 2 7 2 E, C a s e VI LES, 6 2 7 2 E, C a s e VII DNS, 9 2 7 2 E

0.3

x X\v>E 0.2

0.1

y

0.3D N S, 6 2 7 2 E, C a s e I LES, 6 2 7 2 E, C a s e VI LES, 6 2 7 2 E, C a s e VII D N S, 9 2 7 2 E

0.2 x=10

ViE

0.1

y

Figure 4.21: Fluctuation U velocity profile at x=2,5,10 for Re—10,000.

100


- D N S, 6 2 7 2 E , C a s e I L E S , 6 2 7 2 E , C a s e VI

L E S , 6 2 7 2 E , C a s e VII D N S, 9 2 7 2 E

0.8

0.7

0.6

0.5

(A

§ 0.4>0.3

0.2

0.1

y

- D N S, 6 2 7 2 E, C a s e I L E S , 6 2 7 2 E, C a s e Vi

L E S , 6 2 7 2 E , C a s e VII D N S, 9 2 7 2 E

0.6

0.5

0.4

§ 0.3 x=5

0.2

0.1

y

0.5D N S, 6 2 7 2 E, C a s e I LE S , 6 2 7 2 E, C a s e VI LE S , 6 2 7 2 E, C a s e VII D N S, 9 2 7 2 E0.4

0 .3 -(0E>

0.2

V /0.1

y

Figure 4.22: Fluctuation V velocity profile at x—2,5,10 for Re—10,000.

101


4.5 Summary

We have extended our LES works on Triangle spectral element discretization. Both

channel turbulence and flow pass cylinder have been investigated, and encouraging

results have been obtained. At first we compare the LES results using SVV at Re* =

180, similar trends have been gotten. With same Smagorinsky constant Cs, similar

viscosity profile has been obtained. Both standard SVV and Smagorinsky model

have been implemented in two codes, and results have been compared with each

other. In channel flow, using same Smagorinsky constant Cs, they both decreased

vrms and wrms, but increased urms. In cylinder flow, adding small e or Cs will

not change the flow too much, but large values do make change of the flow. Using

Smagorinsky model, the viscosity has been added only in the wake region behind

the cylinder, where the flow has much more variation. In the region away from the

cylinder, the added viscosity is nearly zero. This performance is more reasonable

than standard SVV method to do LES simulation. Due to the speed of NektarF

code, the result we obtained for flow past cylinder are still not converged yet, and

more investigation need to be done in this direction.

102


Chapter 5

Simulation Methods for

Bubbles/Particles in Channel

Flow

In the following chapters we investigate the effects of microbubbles on the dynamics

of turbulence in a channel flow. The first step in this chapter is to set out the

simulation procedures used to include the bubble phase in the turbulence dynamics.

The presence of the bubbles and their influence on the flow is represented by the

force-coupling method (FCM) introduced by [89] and developed by [88] and [78].

5.1 Background Review

Several research groups have been doing experiments on micro-bubble drag re

duction since the 1970’s [93, 81]. Up to the present date, 70% drag reduction has

been reported. However, it is hard to reach such result in numerical simulation,

103


because it is quite difficult to model the interaction between bubbles and turbu

lence. We were the first to report up to 10% drag reduction by a DNS method

[134]. Since then, other research groups have reported numerical simulations of

micro-bubble drag reduction. The group at Worcester Polytechnic Institute (WPI)

are using front-tracking methods to investigate large deformable bubbles. With this

detailed resolution of individual bubbles they focus on dynamics of a small number of

bubbles, at relatively low Reynolds number [80]. Ferrante and Elghobashi [38] from

University of Carlifornia at Irvine are interested in turbulent boundary flow using

a mixed Lagrangian-continuum model for the bubbles. The bubble phase is limited

to low void fractions. Professor Ceccio’s group at the University of Michigan have

recently conducted a series of high Reynolds number flow experiments with micro

bubble injection in a flat-plate boundary flow, at flow speeds up to 18 m/sec. Their

results are consistent with earlier results and a preliminary report of their results are

given in [114]. One specific feature that they note is that the near wall concentration

is very important to drag reduction and that as bubbles disperse away from the wall

drag reduction is lost. This is the issue of persistence of drag reduction. We have

also been collaborating with the Applied Research Laboratory at Pennsylvania State

University where turbulence simulations based on Reynolds-Averaged Navier Stokes

(RANS) models of microbubble flows have been developed and tested [75].

Our goal for the simulations is to investigate small, finite size bubbles at mean

ingful void fractions. Bubble radius a is in the range a+ = 10 — 40, and we assume the

bubbles to be spheres without deformation. This is an appropriate assumption when

the bubble size is small. The following sections describe FCM method, simulation

procedures and provide test results for verification/validation.

104


5.2 Simulation Method

As in [134, 135] and [91], the coupled two-phase flow dynamics of the micro-bubbles

and the turbulence is simulated by the force-coupling method (FCM), described in

[88 , 78] and [77]. Fluid is assumed to fill the whole flow domain, including the

volume occupied by the bubbles. The presence of each bubble is then represented by

a finite force monopole (and optionally a force dipole) that generates a body force

distribution f (x, t) on the fluid. This transmits the resultant force of the bubbles on

the flow to the fluid. The volumetric velocity field u(x, t) is incompressible

V • u = 0 (5.1)

and satisfies

P j ^ = -V p + pV2u + f(x,f), (5.2)

where /i is the fluid viscosity and p is the pressure.

The body force due to the presence of Ng bubbles isn b

f(x,l) = ^ F W A (x - Y W (l) ) (5.3)n ~ 1

where is the position of the n,th spherical bubble and is the force the nth

bubble exerts on the fluid. The force monopole for each bubble is determined by the

function A(x) which is specified as a Gaussian function

A(x) = (27rcr2)_3,/2 exp(—x2/2ct2) (5.4)

and the length scale a is set in terms of the bubble radius a as a/a = y/rr. The velocity

of each bubble (t) is found by forming a local average of the fluid velocity over

the region occupied by the bubble as

V (n)(i) = | u (x ,t)A (x -Y w (t))d3x (5.5)

The dynamics of the bubbles and the fluid are considered as one system where

fluid drag on the bubbles, added-mass effects and lift forces are internal to the

105


system. The equations of fluid motion are applied to the whole domain, including

the volume nominally occupied by the bubbles. In this way the body forces induce

a fluid motion equivalent to that of the bubbles. If mp and mp denote the mass of

a bubble and the mass of displaced fluid, the force of the bubble acting on the fluid

is

Fw = ( m B - m f ) ( g — ) (5 . 6)

This force is the sum of the net external force due to buoyancy of the bubble and the

excess inertia of the bubble over the corresponding volume of displaced fluid. For

the present study we exclude the effects of buoyancy and the mass of the bubble is

neglected.

In addition to the forces specified a short-range, conservative force barrier maybe

imposed to represent collisions between bubbles and prevent overlap. A similar

barrier force maybe imposed, normal to the wall, to represent collisions between a

bubble and a rigid wall [27].

A spectral/hp element method [67] has been used to solve for the primitive vari

ables u, p in the Navier-Stokes equations for the channel with rigid walls at x -2 = ±h.

Periodic boundary conditions are applied in the other two directions with dimensions

L\ x L3. A uniform mean pressure gradient —dP/dx\ is applied in the streamwise

direction and adjusted continuously to ensure that a constant volume flow rate is

maintained.

These FCM equations can be solved analytically for conditions of Stokes flow,

where they give good results for the motion of isolated particles, particle pairs, and

suspensions of particles at void fractions of less than 20% as shown by Maxey & Patel

(2001) [88]. The results are also reliable for unsteady flow conditions, matching those

obtained from particle-tracking equations such as [90]. The results have been tested

at finite particle Reynolds numbers, up to 40, by doing comparison with full direct

numerical simulations. [30] and [78] provided comparisons with experiments at low

to moderate Reynolds numbers for systems of one to three particles again with good

106


Mirror

10 mm

150 mm

Figure 5.1: Experimental Setup

general agreement. The method does not resolve flow details near the surface of a

bubble or particle, and indeed the no-slip condition is not satisfied on surface. Only

the constraint (5.5) that the bubble moves with velocity of the surrounding fluid is

imposed. At distances of about half a particle radius from the surface the flow is

well represented.

5.3 Validation and Verification

5.3.1 Validation

We validated FCM method and the present implementation by comparing simu

lation results with experiments in Lomholt Ph.D. thesis [79]. Details of the experi-

107


ments may be found in [78]. Briefly the setup for these experiments is shown in figure

(5.1). The setup consists of a rectangular channel made in transparent PVC with

height Li = 150mm, width L 2 = 10mm, and depth L3 = 100mm. Thus, the aspect

ratio of the channel is L3 /L 2 = 10. The numerical model described above is therefore

a good approximation for particles moving in the center part (40mm < x% < 60mm)

of the channel. The fluid was a mixture of glycerol and water in order to keep the

viscosity high enough to obtain low Reynolds numbers (Rep < 10). The particles

were polyamid spheres with a radius of a = 1mm, thus the ratio of particle radius

to channel width was a/L 2 = 0.1. The particles were introduced into the channel

through five small holes in the bottom, and since the density of the polymer particles

was smaller than the fluid density, the particles moved upwards toward the top of

the channel.

The motion of the particles was recorded by a standard CCD camera placed in

front of the channel. Therefore the motion in the x \x 2 plane is captured immediately.

The motion in the x\x?, plane is captured using a mirror placed on the left side of

the channel at an angle of 45°. The single video camera captures the motion in

both directions with 25 whole frames per second. The movie is saved either on video

tape or directly into a hard-disk. The particle trajectories are determined afterwards

using the tracking software Diglmage (Dalziel (1992)). The particle positions were

determined as the weighted average of the intensities from the bright particle. In this

way the positions were determined with an accuracy better than 0.1mm. Diglmage

also computes the particle velocities from the time dependent trajectories. The error

on the results for the particle position are 1 0 .1mm and for the particle velocities it

is ± 0 .1mm/s.

Based on experiment results, we will verify simulation results of a single sphere

falling or rising due to buoyancy in the channel. The experiment is a sphere rising

in an inclined channel, where the computed trajectories and velocities are compared

with the above experimental results.

108

The investigations presented in this section were initiated in order to see if the

force coupling method was able to reproduce trajectories and velocities of a real

particle. Furthermore, the inclination of the channel made it possible to study the

effect of the wall, since the sphere will move across the channel toward one of the

walls. The experiments were performed with the channel tilted an angle 9 from

vertical (the top of the channel is moved to the left). The experimental data are

given in table 5.1.

The Reynolds number Reptokes based on the Stokes settling velocity W is deter

mined as

Re,Stokes 2aW 2 a v

2 a29 - ( * - « ) 9

4a39^2

Pp

Pf- 1 9 (5.7)

where g = 9.82m/s2 is the absolute value of the gravitational acceleration. The

particle Reynolds number Re™ax is based on the maximum velocity of the sphere in

the stream-wise direction, i.e.

2a\V{aax\R,eT'x = (5.8)

Values of some of the important parameters used for the simulations are given in

table 5.2. The characteristic length was set to L = = 10mm and the characteristic

velocity to U = lOmm/s. The nondimensional radius of the particle was a = 0.1.

Exp. no. Pf (g/cm3) Pp (g/cm3) 9(°) v (mm2/s) j^gStokes Re™ax

1 1.237 1.081 11.15 172.7 0.019 0.015

2 1.222 1.081 8.08 95.24 0.056 0.044

3 1.180 1.081 8.08 18.52 1.07 0.84

4 1.115 1.081 8.23 3.125 13.6 7.9

Table 5.1: Experimental values for a single particle rising in an inclined channel.

109


Exp. no. L\ x L2 x L3 Ni X n 2 X n 3 St R e = ^ L 11

1 10 x 1 x 8 64 X 65 X 64 0.10 0.58 982.0

2 5 x 1 x 4 64 X 65 X 64 0.02 1.05 982.0

3 5 x 1 x 4 64 X 65 X 64 0.02 5.40 982.0

4 5 x 1 x 4 64 X 65 X 64 0.01 32.0 982.0

Table 5.2: Computational parameters for the single particle rising in an inclined

channel. The characteristic length L = L2 = 10mm and the characteristic velocity

U = lOmm/s.

For all the results presented in this section, the figures will have the following

common format. The particle trajectory is shown in subfigure (a), while the stream-

wise and the wall normal particle velocities are shown in subfigures (b) and (c),

respectively. The positions and velocities are given in the frame of the experimental

channel in mm and mm/s. The full drawn line shows the computational results with

the monopole term, while the dashed line denotes prior results from S. Lomholt. The

experimental results are shown as (•). The straight upward line drawn at x2 = 4

indicates the wall, i.e. when the center of the sphere is at x2 = 4 the particle touches

the wall. The straight line shown as dash-dot in the figures corresponds to the

direction of gravity in the frame of the channel.

In figure (5.2) the results for Re"'ax = 0.0145 are presented. The agreement

between the computed and the experimental trajectories is good, and simulation

results using only force monopole term also matches Lomholt’s computation quite

well. Closer to the wall (x2 > 3) the computed trajectories deviated from the ex

perimental, and Lomholt’s computational spheres moves through the wall (x2 > 4).

The reason is the lack of a collision model. When the sphere approaches the wall,

a collison force or velocity builds up between the sphere and the wall. During this

build-up, the collision force or velocity slows down the sphere and ultimately the

lateral motion of the sphere is stopped (see also the section 5.5). A comparison

110


I

/

0.8

/0.6

0.4

0.2

t n n l i i n l i i i i l ' i i i l i i i i l i m t

1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5

(b)

0.3

0.2

>

0.1. ‘i

55

11'*1'11111'1,111 '*'''' *-1 0 1 2

y3 4

Figure 5.2: Comparision of experimental and computed particle trajectory (a) and

velocities ((b) upward and (c) lateral) in an inclined channel for Re™ax = 0.0145.

(•) Experiment, (-) FCM with the monopole term, (— —) S. Lomholt’s monopole

only. The line (— ■ —) indicate the direction of gravity in the frame of the channel.

The particle positions are given in mm and the velocities are in mm/s in the frame

of the experimental setup.

I l l


between the computed velocities and the experimental velocities is more difficult,

because of the scatter in the experimental data. The scatter is due to experimental

uncertainties, since the velocities are small and therefore relatively difficult to mea

sure. For example the normal velocity component figure(5.2c) is of the order of the

limit for the experimental accuracy of about O.lmm/s. Nevertheless, the agreement

is reasonably good. The initial steep increase in the computed velocities, is due to

the initial velocities of the computational spheres being zero. Since we compute the

particle velocity from the fluid velocity field, resulting from the force the particle

exerts on the fluid, it is not possible to specify an initial velocity of the particle.

When the computational sphere is introduced, it will almost immediately attain the

velocity of a sedimenting sphere. This is a result of neglecting the particle inertia.

Therefore the curves in figure (5.2b) and (5.2c) initially appears as a step function,

but they are continuous.

Figure (5.3) shows a comparision at Re™ax = 0.044 and essentially the computed

trajectory and velocities agree with the experimental and Lomholt’s results. Again

there is same scatter in the velocity measurements. This is observed from the figures

of the upwards velocity figure in (5.2b) and (5.3b), and for the lateral velocity in

(5.2c) and (5.3c).

The two examples presented above were both at Reynolds number small enough

to be considered as approximations to Stokes flows. The results are consistent with

the Stokes flow results. Namely, that for distances larger than the sphere radius,

the force coupling method performs very well, and it is able to reproduce the par

ticle trajectories and velocities both qualitatively and quantitatively. For distances

smaller than the sphere radius the collision forces are not negligible and as a result

the discrepancy increases.

In the next two examples the Reynolds number is increased in order to examine

the effect of the convective inertial terms.

112


■1 0 1 2 3 4 5

2.5

0.3

0.2

0.5

■ 1 1 4 5 ■10 1 2 3 4

y y y


velocities ((b) upward and (c) lateral) in an inclined channel for R.e™ax = 0.044. (•)

Experiment, (-) FCM with the monopole term, (---- ) S. Lomholt’s monopole only.

The line (— • —) indicate the direction of gravity in the frame of the channel. The

particle positions are given in mm and the velocities are in mm/s in the frame of the

experimental setup.

113


/

-J5

y y

1.4

1.2

(c)

0.8

>0.6

0.4

0.2

iiiliiiiliiiiliiiilini\


velocities ((b) upward and (c) lateral) in an inclined channel for Re™ax = 0.84. (•)




experimental setup.

The first higher Reynolds number example with Re™ax = 0.84 is shown in

figure(5.4). The first thing to notice is the very good agreement of the computed

trajectory with the experimental trajectory. The trajectories are almost identical

until the particle collides with the wall, where the computational particle continues

its lateral motion through the wall. Comparing the normal particle velocities in fig

ure (5.4(c)) with those in figure (5.2(c)) and (5.3(c)) the profile of the velocity curve

has sharpened, because of the larger maximum in the center of the channel.

The final example in this section is shown in figure (5.5). The particle Reynolds

number for this case is R.e"IMX = 7.9. Again the computed trajectories and velocities

114


>25

1.75

/.25

3 7

).75

0.5

).25

J5

iJ5

y y


velocities ((b) upward and (c) lateral) in an inclined channel for R,e"MX = 7.9. (•)




experimental setup.

agree well with those observed experimentally and Lomholt’s results. Furthermore,

the difference between the two computations is larger than in the three previous

examples. The velocities from the two computations differs not only near the wall,

but also in the middle of the channel. When the particle Reynolds number becomes

larger, the nonlinear effects comes into flow, and it is more difficult to predict the

bubble motion and velocity.

115


5.3.2 Verification

Next we compare simulation results with some theoritical solutions, these cases

can be found in Lomholt and Maxey [77, 41] and [42], Two configurations will be

tested, they are bubbles moving parallel to the wall and perpendicular to the wall.

1. Parallel to the wall

We simulate a sphere in 3D channel, which has dimensions of 30 x 10 x 30 in

figure(5.6), and sphere radius is 1. The sphere falls down at the center of channel

because of gravity, the velocity increases with time, until it reaches a constant value.

This value is the Stokes velocity of a bubble falling in a static fluid if there is no wall

nearby, but in our cases it is smaller than the Stokes velocity.

In this simulation, we have V\jWs = 0.8214949019, A = F/(6irfj,aVi)) = 1.2173.

The particle does not rotate, and the local volume-averaged rate of strain is:

! —5.825686455e - 9 -1.976289423e - 10 7.807779895e - 11 ^

—1.976289423e - 10 6.457138716e - 09 1.171208893e - 08

7.807779895e - 11 1.171208893e - 08 5.266018229e - 09 ;

Here the non-zero rate of strain represents a relatively small error. This may be

corrected by including a force dipole term.

Increasing the resolution doesn’t affect these results, which match those obtained

previously.

The pressure and velocity profiles at different location nearby are shown in figure

(5.6) and figure (5.7).

116


Pressure profiles

I

Figure 5.6: Configuration (left), Bubble pressure(right)

Profiles parallel to wall Wall normal profiles

Y-2aY-a

_Y

1

Figure 5.7: Bubble velocity: u(left), v(right)

117


Pressure profiles

Figure 5.8: Configuration (left), Bubble pressure(right)

2. Perpendicular to the wall

Next we simulate a sphere in 3D channel, falling down towards the wall of the

channel due to the gravity. The velocity will become zero, until it reaches a constant

value. The stokes velocity of a bubble falling in a static fluid.

In this simulation, we have V\/Ws = 0.4906803882, A = F/(6ir(j,aVi)) = 2.037986.

Particle does not rotate, The rate of strain eij is:

( 0.00143011315 1.278958745e - 08 1.236229578e - 08 ^

1.278958745e - 08 -0.0007139948814 3.056193238e - 09

1.236229578e - 08 3.056193238e - 09 -0.0007275386888 y

5.4 Flow Analysis

The flow simulations are based on the volumetric velocity field u(x,f), which is

incompressible (5.1) and so ensures that the volume occupied by each phase, and

the mass of each phase, is conserved. Outside a bubble, in the liquid phase, u(x, t)

118


Profiles parallel to wall Wall normal profiles

Y-2aY-a

1

Figure 5.9: Bubble velocity: u(left), v(right)

computed with FCM is a good approximation to the liquid phase velocity. Inside a

bubble the volume-average of u(x, t) matches the bubble velocity. For each bubble,

one conventially defines an indicator function x(x -1)- x(x? t) = 1 if the point is inside

the bubble and x(x, t) — 0 if the point is outside the bubble.

For each bubble the function

-m FA(x - YW) (5.9)

serves as an indicator function. From these quantities, the more familiar continuum

field variables of two-phase flow can be constructed for any single instantaneous

realization of the flow. The instantaneous bubble concentration isN bI

c(x,i) = - 5 ] m FA (x -Y W ).P 71=1

(5.10)

The ensemble-averaged profile for the bubble void fraction is defined as

NbC(x2,t) = / - Y1 ™fa (x - y W ) \

\ P 77=1 /(5.11)

and here the angle brackets indicates a spatial average over the homogenous direc

tions Xi,X%.

The instantaneous mixture momentum density, equal to the momentum density

119


of the liquid phase,Ng

pui(x, t) - Y2 m Fv}n A(x - Y w ) (5.12)n ~ 1

is determined by the momentum deficit associated with the bubble phase. We can

form a conditional, bubble-phase velocity field v(x, t) as

i n b

cvi(x,t) = - ^ 2 mFVin^A(x - Y ^ ) (5.13)P n = 1

where 0. It is straightforward to verify from these definitions the bubble-phase

conservation lawdc— + V • cv = 0 (5-14)at

The liquid-phase density is p(l — c(x, t)) and the conditional, liquid-phase velocity

field w(x,t) is defined from (5.12) and (5.13) as

p(l - c(x, t))w(x, t) = pu(x, t) - pcv(x., t) (5.15)

provided the liquid-phase density is not zero. Again it is straightforward to verify

the usual liquid-phase mass conservation law. In contrast to the volumetric velocity

field u(x, t), both v(x. t) and w(x. t) are compressible.

5.5 Collision Model

In order to keep bubbles away from each other and wall, we need to use either a

force collision model or a velocity collision model, see [27].

In the turbulent flows we found that the force-barrier method was slow to respond

adequately to prevent overlapping bubbles and become numerically stiff. Instead we

adopted a velocity barrier. The motion of each bubble is computed as

= V (") (t) + + V $ (5.16)

120


The additional term vffi is the bubble-collision velocity and represents the effect of

short-term contact forces between bubbles and similarly represents the effect

of bubble contact with a wall. It is important to include these collision or contact

effects in order to maintain the bubble void fraction. Details of these short-range

interactions are given by [27].

The bubble collision velocity for a pair of bubbles is defined in terms of the

seperation vector between the centers of the two bubbles, i tij = and

rij = the force barrier has following form:

t ‘l = ljp ~,2* ref T ref ij2a l&ref-4a2

2^ . . ,ri K. Ri

, otherwise(5.17)

And the velocity barrier has similar form:

Vref 2 a ref •4a2 irij < Rref

, otherwise(5.18)

\^lJ is the collision velocity for bubble i from bubble j. The collision velocity Vc''

is the sum of all the terms from other bubbles.

From the wall collisions, is directed away from the wall and is set by the

distance from the wall in a manner similar to (5.18).

Figure (5.10) shows the collision velocity at different bubble separation distances.

In the simulations Rref= 1.2*2a is used and Vref = u*. Tests were made for other

values, but there was no significant difference.

121

12

10

8

6

4

2

00 0.5 1.51 2 2.5

Figure 5.10: Collision Model

5.6 Parallel Implementation and Benchmarks

A spectral/ h/p element method [67] has been used to solve for the primitive variables

u ,p in the Navier-Stokes equations for the channel with rigid walls at x? = ±/t.

Periodic boundary conditions are applied in the other two directions with dimensions

L\ x L?l. A uniform mean pressure gradient —dP/dx\ is applied in the streamwise

direction and adjusted continuously to ensure that a constant volume flow rate is

maintained. The bulk velocity is kept constant, Ub = 2/3. At Re* = 135, the

domain size is 27T x 2 x 2ir with a numerical resolution of 64 x 65 x 64. Dealiassing

schemes are used to evaluate the nonlinear terms. At Re* = 380 the domain size is

27r x 2 x 7r and the resolution is 256 x 241 x 256.

5.6.1 Parallel Im plem entation

Since we will simulate high Reynolds number turbulent bubble flow later, the

code must be parallelized. For the fluid simulation, we decomposed the domain in

the streamwise direction. For the simplicity of bubble parallelization, on each CPU,

we kept information of all bubbles. When we need to evaluate the bubble velocity

122


and compute the bubble force exerted on the fluid, we just compute them in the

respective region on each CPU. For the bubble velocity, a global summation needs

to be done, in order to get the correct bubble velocity. Figure (5.11) shows the

sketch of bubble parallelization. The velocity of the bubble in CPU 2 needs to be

calculated by a summation of the local velocity in CPU 1 and CPU 2.

o ' — - *

UJ

;1111

*

O______ i

C P U 1 C P U 2 C P U 3 C P U 4

Figure 5.11: Bubble Parallelization

5.6.2 Benchmarks

We benchmark the code at Re* = 135 with 800 bubbles and at Re* = 380 with

9100 bubbles. The machine we use is Iceberg, which is an IBM SP4 located at Arctic

Region Supercomputer Center (ARSC).

At Re* = 135, without bubbles, each time step took 1.05 seconds on 8 CPUs.

The mesh is 64 x 65 x 64, and the domain size is 27r x 2 x 2ir. When adding 800

bubbles of radius a+ = 13.5, each time step took 1.47 seconds on 8 CPUs. The time

spent on bubble computation is 0.42 seconds, which is about 40% of the no-bubble

123


simulation for one time step. Of the time for the bubble simulation, the force steps

which evaluates the force the bubbles exerts on the fluid takes about 0.21 seconds,

and the other processing steps takes about 0.2 seconds. The other processing includes

evaluating bubble velocity, bubble statistics and balance calculation. The interesting

thing is the time for bubble collisions is not too much, only 0.0005 seconds, which is

about 0.05%.

At Re* = 380, without bubbles, each time step took 16.486 seconds on 128

CPUs. The mesh is 256 x 241 x 256, and the domain size is 2?r x 2 x 7r. When

adding 9100 bubbles of radius a+ = 13.5, each time step took 22.986 seconds on 128

CPUs. The time spent on the bubble computation is 6.5 seconds, which is about

39.4% of the time for no-bubble simulation for one time step. Of the time for the

bubble simulation, the force step takes about 4.12 seconds, and the other processing

steps for the of bubbles takes about 2.68 seconds. The interesting thing is that the

time for bubble collisions is not large, provided we split the domain into more small

boxes to check for collisions.

5.7 Summary

Prom the above verification and validation, we know that FCM can be used correctly

to prescribe the bubble motion in both Stokes flow and finite Reynolds number flows.

In the present work we represent the bubbles with just the force monopole term

although there is the option of including higher-order dipole terms. These additional

terms would improve the accuracy of the results for the motion of individual bubbles

or particles. There are several reasons why we choose not to do this at this stage.

Firstly, we wish to investigate the so-called “density effect” of the bubbles on the

flow separately from other factors. The dipole terms would give the additional effect

of an enhanced suspension viscosity. Further, the dipole terms enforce a stronger

constraint of a rigid spherical bubble as opposed to the effect of the monopole term

124


alone. Some degree of bubble deformation, even if small, is likely. Finally, we are

interested in the bulk response of the turbulence to the presence of the bubbles and

less in the dynamics of individual bubbles. The force monopole simulatios involve

lower computational costs and allow us to investigate separate dynamical processes

in a manner that would not be directly possible in full-scale experiments..

125


Chapter 6

Microbubble Drag Reduction

The quest for ‘taming’ turbulence with the objective of reducing the skin-friction on

air- and sea-vehicles as well as in gas- and liquid-carrying pipes has been pursued for

more than a century. This has led to many proposals for drag reduction techniques

including microgrooves, polymer and microbubble injection, and electromagnetic

and acoustic excitation, see [40, 13]. The use of microgrooves or riblets mounted on

the wall surface has proved to be effective in partially suppressing turbulence and

reducing skin friction by about 5% to 10%, see [7, 21]. Transverse oscillations and

traveling waves, induced mechanically or electromagnetically, can lead to turbulent

drag reduction of about 50% as reported by [62, 35]. Injection of high molecular

weight polymer solutions or gas bubbles in the near-wall region of a liquid boundary

layer can result in turbulent drag reduction of more than 70%, see [110, 32, 108].

Closed-loop, active control of turbulence has also been successful in reducing

turbulent drag but requires a system sensors and actuators for implementation. A

theoretical example is provided by the technique of ‘opposition control’ proposed

by [20, 52], Opposition control employs a sensor plane at y+ = 10 — 25, where

the wall normal velocity is measured and opposing inflow/outflow velocity boundary

conditions are imposed at the wall so as to give an effective zero wall normal velocity

126


at each point in the sensing plane. The level of drag reduction is about 26% at

Re* = 100 but drops to 19% at Re* = 720, as reported by [17]. The level of drag

reduction is also sensitive to the location of the control plane.

Recently, there has been a renewed interest in the polymers and microbubbles

fueled by the new emphasis in the USA and Japan in developing fast ships with

cruising speeds of 50 to 100 knots. It is estimated that this goal could possibly

be accomplished if at least 50% skin-friction reduction can be sustained. Practical

application of these techniques, however, has hinged on two still-unresolved issues.

The first one is the loss of persistence of drag reduction downstream of the injection

port observed by [82, 114]. The second is related to the large storage and especially

preparation of the injectant, which is particularly problematic for polymers. It seems

that between the two, microbubble drag reduction may have the greater potential,

and it has been used already in sea tests in Japan by [132, 73].

Among the most successful and robust methods for drag reduction in a turbulent

boundary layer has been the injection of gas micro-bubbles into the liquid flow.

This phenomenon was first demonstrated by [93] and subsequentially verified in a

series of experiments by [81, 82]; see also the review article of [95]. More recent

experiments have been reported by [69] and [49]. In most of the experiments, gas is

injected through a porous plate mounted on the wall of an open boundary layer flow

or one side of a channel flow. The injection of the micro-bubbles leads to significant

drag reduction, as measured by the frictional wall stress, over a significant distance

downstream of the injection site. Reductions of 20-30% are readily achieved, and

there is a nearly linear increase in the drag reduction, up to 50 or 60%, as the relative

gas flow rate is increased.

There are some preliminary results on bubble simulations by [63]. They have

reported numerical simulations of bubbly, turbulent channel flow where the flow is

seeded with 27 bubbles of diameter 0.16 in a unit cube flow domain and average void

fraction of 6%; however, their results are inconclusive.

127


Many factors undoubtedly contribute to the observed dynamics of micro-bubble

drag reduction. These may include void fraction levels, bubble size, bubble deforma

tion, bubble splitting or coalescence, buoyancy and correlations between the bubble

motion and that of the turbulent fluctuations not to mention the way in which the

bubbles are injected into the flow. In this chapter we seek to focus on a simpler,

limited set of issues most relevant to the influence of the smaller bubbles. Specif

ically for small micro-bubbles, surface tension has a strong effect and the bubbles

remain essentially spherical. This is characterized by the Weber number, which for a

bubble of radius a in a turbulent channel flow may be defined in terms of the friction

velocity scale u* and fluid density p as We = pu*2aj7 , where 7 is the coefficient

of surface tension. The value of We is small for micro-bubbles under a range of

operating conditions. Additionally, small bubbles in water, especially in seawater,

tend rapidly to become coated with surface contaminants. As a result, it is often

observed that a small bubble will respond approximately as a rigid body, see [83, 23]

and [31]. The effect of surfactants though on bubble dynamics is still an open issue

as illustrated by the recent work by [140].

The current work is motivated by the experimental work on microbubble drag

reduction over the last thirty years, see [32] and references therein. Even though

there is a substantial body of experimental data, many fundamental questions remain

about the underlying mechanisms for drag reduction. Various models have been

proposed by [76, 96, 85] that focus on the change in effective density and viscosity

of the bubble-liquid mixture. These results are instructive but shed little light on

the dynamics or the interactions between the bubbles and the turbulence. Most

importantly, they do not address the question of lack of persistence in drag reduction.

In this chapter we consider the effects of bubble seeding levels, bubble size and in

teractions with the turbulent flow. We present results of direct numerical simulations

of a turbulent channel flow that is seeded with small, rigid, spherical bubbles up to

an average void fraction of 10%. The monodisperse bubbles are initially distributed

128


Simulation I II Ilia Illb IIIc IV V

a/h 0.05 0.075 0.1 0.1 0.1 0.15 0.3

a+ — au*/v 6.75 10 13.5 13.5 13.5 20 40.5

Nb 7200 2450 800 800 1600 242 60

Average void fraction 4.8% 5.5% 4.24% 4.24% 8.5% 4.3% 8.6%

Bubbles per layer 3600 1225 400 400 400 121 30

Layers near X2 /h = — 1 1 1 1 2 2 1 1

Layers near x^/h = 1 1 1 1 0 2 1 1

First layer position, y^ 8.1 10.8 20 20 20 25 50

Reduction in mean drag 5% 10% 6 .2% 3.8% 5.5% 2.5% 0.5%

Table 6.1: Simulation parameters and drag reduction.

in layers near each wall and then dispersed in the channel under the action of the

turbulence. The drag force on each wall, corresponding changes in mean pressure

gradient, and void fraction profiles are calculated. The results provide clear evidence

of drag reduction produced by the smaller bubbles, based on these assumptions.

6.1 Low Reynolds Number Flow

The simulations cover a range of bubble sizes, with radius a/h=0.05,0.075,0.1,0.15

and 0.3, and with average void fractions range from 4% to 8%. We explored both the

effects of bubble seeding positions and bubble size. The parameters of the different

simulations and result summary are listed in Table 6.1. In the third set of simulations

(Illa-IIIc), the effect of the initial seeding level was tested for the smaller bubbles.

The bubbles were introduced in layers of 400 bubbles each, with the centers of the

bubbles in the first layer at a distance of yf} = 20 from the wall. In simulations

Illb and IIIc, a second layer was placed adjacent to the first, at a distance y+ = 54

from the wall. The mean drag force is computed by integrating the mean viscous

129


shear stress over each wall. The mean drag force averaged over the time interval

t = 10 - 40, or t+ = 61 - 243, is compared to the mean drag in the flow without

bubbles and the results are listed in Table 6.1. The results for simulations Ilia and

IIIc both show a significant degree of drag reduction but the additional second layer,

and the higher average void fraction, does not enhance the effect. Seeding bubbles

near to just one wall, as in Illb, yields about half the overall level of drag reduction.

There was no significant change in the mean wall stress on the opposite wall. Based

on these results we focus on seeding the flow with a single layer adjacent to each

wall.

6.1.1 Fluid characteristics

1.05

0.95

100t

Figure 6.1: Normalized drag force against time t at Re* = 135 for: 1, no bubbles;

2, 242 bubbles and a+ = 20; 3, 800 bubbles and a+ = 13.5; 4, 2450 bubbles and

a+ = 10 .

By comparing the results of simulations II, Ilia and IV with the corresponding

results for the flow without bubbles, we can see the effects of using different size

of bubbles. The larger bubbles are supposed to have a more limited response to

130


1.05

o>

0 .9 5

0 .9 100 15 0t

Figure 6.2: Normalized drag force against time t at Re* = 135 comparing random

to near-wall seeding of 800 bubbles of size a+ = 13.5: (solid), no bubbles; (dots),

random seeding; (dash), near-wall seeding.

the turbulence. Firstly, there is the effect of spatial scale as the bubble velocity is

influenced most by turbulent fluctuations on a scale larger than the bubble diameter.

Secondly, the inertial response time of a bubble increases with bubble size, roughly

as a2. The integrated, viscous drag force on the two walls is shown in figure (6.1),

where the results are normalized by the mean drag force in the flow without bubbles.

For the two smaller sizes, II and Ilia, the drag force shows a sustained reduction

following an initial transient stage, with the bubbles of radius a+ = 13.5 clearly being

more effective for the same average void fraction. The larger bubbles (IV) show a

short term decrease in the drag force before it increases to a larger than ambient

level.

We can see that initially adding bubbles will lead to drag reduction. The smaller

size of bubble, the more drag reduction can be reached. However, after about 100

time units, drag reduction will be lost and the drag returns to the no-bubbles level.

When bubbles are seeded in the near wall region they disperse under the action

131


of the turbulence. The number of bubbles close to each wall decreases and the

number of bubbles in the central region increase, eventually tending to uniformly

dispersed distribution of the bubbles. We compare the drag history for 800 bubbles

of radius a /h= 0.1 that are initially dispersed uniformly throughout the channel to

the drag history for bubbles seeded in concentrated layers near each wall, case Ilia.

The results are shown in figure (6.2). The random seeding leads to a brief transient

reduction in drag, but in the long term there is no significant reduction in drag.

These results show that there must be sufficient bubbles in the near-wall region to

achieve drag reduction.

In two-phase flow the mean pressure gradient is influenced by two factors. One

is the mean viscous drag force on each wall, while the other is the acceleration of the

bubble phase. The total volume-integrated flow rate in the channel

is held constant throughout the simulation, Qq = 167r2/ 3 . The volume-integrated

flow rate associated with the bubble phase is

where Qb is the volume of one bubble and pQs is the momentum deficit in the flow

domain due to the presence of the bubbles. The corresponding flow rate for the

liquid phase Ql is (Qo — Qn)- As the bubbles disperse from their initial locations,

under the action of the turbulent flow, they migrate from the relatively slow flow

regions near the walls to the faster flow in the core of the channel. This produces an

increase in Qb at least for some initial period as shown in figure (6.4).

A control-volume integral of equation (5.1) for the flow domain (L x 2 h x L) gives

the balance for the mean pressure gradient

(6 . 1)

£ a Bv}n) = Qb (6 .2 )n = 1

+ r w > - ^ , (6.3)

132


where t {—K),t {K) are the mean viscous shear stresses at each wall, averaged over

the plane of the wall. The pressure gradient required to sustain the flow is lower if

either the mean drag is reduced or the bubble phase is accelerating.

0.14

0.12

0.1

S.

0.04

0.02

0

t

2

0.14

0.12

0.1

S<JS

•0.08

0.04

0.02

o

t

Figure 6.3: Balance history at Re* = 135, a+ — 13.5 (left); a+ = 20 (right). 1-Wall

friction; 2-Pressure drop; 3-Bubble acceleration; 4-Residue

0.18

0.14

0.12

0.1

0.08

0.06

0.04

0.02

Figure 6.4: Balance history for a+ — 40.5 at Re* — 135.

Further information is provided by considering the time variations of the mean

pressure gradient, shown in figure (6.3)(6.4) , for the bubbles of different size. While

the mean pressure gradient in the turbulent flow without bubbles fluctuates in time,

it remains stationary. The average value for the smallest bubbles, after an initial

transient, is about 12-15% lower. The mean pressure gradient for the largest bubbles

(V) is reduced for an interval before returning to the ambient level, or higher.

133

The largest bubbles, (V) a+ = 40.5, show a somewhat different trend. Left figure

(6.4) shows the various components of the force balance. Initially, the pressure

gradient decreases in response to the net acceleration of the bubble phase where

dQ [j / dt > 0. Over the second half of the simulation, Qb decreases while the pressure

drop returns back to the no-bubble level. While the drag curve seems only to decrease

slightly, and does not show such variation.

Figure (6.5) shows the mean and conditional mean bubble velocity profiles for

a+ = 13.5 averaged over t = 50 — 80. The conditional mean velocity is not zero at

the wall, which is different from the mean velocity. Over most of the channel, the

average streamwise velocity of the bubble lags behind that of the total flow. This

time-interval average was chosen for case Ilia to give better statistical resolution

and because the drag history remained relatively stable. The bubble concentrations

(shown later) also show small changes.

0 .7

Ao.v* 0.6V A3V

— - /<c> , with b u b b les

, n o b u b b les

0 .5

0 .4-0.6 -0 .3 0 .3 0.6

Figure 6.5: Mean and conditional mean velocity profiles for a+ = 13.5 averaged over

t = 50 - 80.

Figure (6 .6) shows the corresponding time-average of the Reynolds stress, which

134

decreases by adding micro-bubbles. This is consistent with drag reduction achieved

at this Reynolds number. Figure (6.7) also shows the same trend as the Reynolds

stress, turbulent fluctuations decrease by adding bubbles. It is also found that for

smaller sized of the bubbles where a larger decrease of turbulence fluctuation will be

achieved.

0.001

0 .0 0 0 5

0

-0 .0 0 0 5

- 0.001

■1 -0 .5 0 0 .5 1

Figure 6 .6: Reynolds stress profile for a = 13.5 averaged over t = 50 — 80. Solid

line is turbulent flow without bubbles

The results so far indicate that the initial seeding of the bubbles in the flow has

an important effect on the pressure gradient and viscous drag force. The large bub

bles give a transient reduction in drag history while the smaller bubbles give a more

sustained reduction in drag. A further indication of how the flow is modified by the

bubble phase is provided by the short-time averaged profiles of the unconditional

Reynolds shear stress, computed from the flow field u(x, t). These are plotted in

figure (6 .8) and (6.9) for bubbles of radius a+ = 13.5 (Ilia) and a+ = 41 (V) respec

tively. The Reynolds stresses, averaged over the four time intervals, for (Ilia) are in

general lower than the corresponding stresses for the turbulence without bubbles. In

contrast the Reynolds stresses for the larger bubbles (V) are in general higher than

the single phase turbulence.

135


2.5

EDC

0.5

-0 .5 0 .5

Figure 6.7: RMS profiles compared for a+ = 10,13.5. a+ = 13.5 has been averaged

over t=50-80, and a+ = 10 has been averaged over t=50-65

0.001

0 .0 0 0 5

-0 .0 0 0 5

■0.001

-0 .5 0 .5y

Figure 6 .8 : Reynolds stress profiles averaged over different intervals for case (Ilia):

(1) No bubbles; (2) t = 0-10; (3) = 10-20; (4) t = 20-30; (5) t = 30-40.

136


0.001

0 .0 0 0 5

■0.0005

■0.001

-0 .5 0.5y

Figure 6.9: Reynolds stress profiles averaged over different intervals for case (V): (1)

No bubbles; (2) t = 0-10; (3) t = 10-20; (4) t = 20-30; (5) t = 30-40.

Figure (6.10) and (6.11) show the Rms fluctuations of vorticity, which have the

same trend as fluctuations of velocity.

§«£QC

3Vs<0E

QC

y

>t roa>EQC

y

Figure 6.10: RMS vorticity profiles averaged for t=50-80 for a+ = 13.5.

137

•N o b u b b le s- 2 4 5 0 a = 0 .0 7 5 b u b b le s , T = 5 9D a s h -

1.5

O,>'5f£(E

0 .5

-0 .5 0 .5y

Figure 6.11: RMS vorticity profiles at t=59 for a+ = 10.

Figure 6.12: Void fraction profiles for a+ = 13.5 (top) and a+ = 10 (bottom)

at:(solid), t = 20; (dash), t = 40; (dots), t = 60; (dash-dots), t = 80.

138


6.1.2 Bubble characteristics

Specific information on the bubble distribution across the turbulent channel is

given by the void fraction profile C{x2 ,t), evaluated by (5.11).

Figure (6.12) gives the mean concentration profile of the bubbles for a+ = 13.5

and a+ = 10, as the distribution evolves over time. Initial layers gives peak con

centration of cmax = 30.0% for 400 bubbles of a=0.1, and cmax = 51.7% for 1225

bubbles of a=0.075.

The figure indicates that the smaller bubbles disperse faster than larger bubbles.

This is consistent with the shorter inertial response time of the smaller bubbles

and the more limited spatial averaging to obtain the bubble velocity. There are

strong peaks of bubble concentration near each wall that reflect the initial seeding.

These steadily decrease as the bubbles disperse across the channel, tending to a more

uniform average distribution. The larger bubbles, (IV) a+ = 20, show very similar

trends. For both, there is still a noticeably higher void fraction nearer to each wall

even at t = 40. The value of Qb increases steadily over this time interval and through

the force balance (6.3) contributes to the reduction in the mean pressure gradient.

cot>Eu.■oo>

x2 x2

Figure 6.13: Void fraction profiles for random seeding of 800 bubbles, a+ = 13.5:

left, t = 20,40,60,80; right, long term average compared to theory.

139


When the bubbles are seeded randomly in the channel, the distribution of the

bubbles remain approximately uniform. Figure (6.13) shows the bubble concentra

tion profile at different times and average concentration profile compared to theoreti

cal value. The long time average matches theoretical value quite well. The constraint

of the wall means that the bubble center must lie between —0.9 < Y Nb * 4tt * a3/ 3 1 [°-9

N b * a2 1/u.y

*/2

37T 2 h J - 0.9 y/2/•u.y i (x9- Y 9y j/ _ 0~ dv>

J - o . i(6.4)

The presence of the bubbles and their effect in modifying the volumetric flow

field is captured by the body force term f(x,t) in (5.3). This force represents the

excess of fluid inertia in the volume occupied by the bubble and corresponds to a

“density effect” of the bubbles. The force has an average effect on the mean flow,

Nb d V ^ = < £ m F - ^ - A ( x - Y ( B>(f))> (6.5)

n —l

which will be non-zero in both streamwise and wall normal directions. Profiles for

the mean body force are shown in figure (6.14) and figure (6.15) for the two cases

a+ = 10 and a r = 13.5 averaged over t=50-65 and t=50-80 respectively. The

streamwise components of these force < f \ > is characteristically negative close to

each wall and positive in the core of the channel. The integrated value of 

across the channel is propotional to dQs/dt, discussed previously in figure (6.4). It

is typically positive as bubbles disperse from the wall and accelerate with the mean

flow. Once the bubbles are uniformly dispered, Qb is constant and the integrated

value is zero. The wall normal component < > is directed away from the walls

and on average is an odd functions of X2 -

140

An initial estimate for < fi > is obtained by assuming statistical independence

of the variations in the bubble acceleration and the bubble position, so that

-d '0002

X 2

Figure 6.14: Force profile for a+ = 10 at Re* = 135 averaged over t = 50 - 65. Solid

line is < / >, Dash line is (6.8 and 6.9)

-SE-05

Figure 6.15: Force profile for a+ = 13.5 at Re* = 135 averaged over t = 50 — 80.

Solid line is < f >, Dash line is (6.8 and 6.9)

fix/. Nb< f i> ~ < - ^ >< ^ m jp A (x - Y (ra)(t)) >

n = 1

- < ¥ > „ < C > (6.6)

with both terms evaluated at corresponding positions. If the bubble is small enough it

responds almost as a Lagrangian tracer particle in the flow. The average acceleration

141

< > may then be estimated from the local fluid acceleration to give

dV, Du;< ~ d t> ~ < ~ D t >

= < UiU■* >

in a stationary turbulent flow. The combination of these estimates (6 .6) and (6.7)

suggests then that

< fi > ~ ~P < C > < - U \ U 2 > (6.8)A

< h > ~ p < C > — (6.9)

The Reynolds shear stress < —u\U2 > increases from zero at wall to a maximum

value at about y+ = 30 (for Re* = 135) and then decreases. Within this distance the

gradient is positive and (6.8) would predict a negative value of the force component

< f i >. In figure (6.14) and figure (6.15) the measured values of < f i > are compared

to the estimate (6 .8) showing a reasonable consistency of the results. The value of

< f i > in the core region are larger than (6.8) in part due to the net acceleration of

the bubble phase. Near the wall the values of are lower in magnitude than

predicted by (6 .8) and may reflect a limited response of the bubble to the turbulence.

Between the two bubble sizes the estimated value in (6.8) is larger at the wall for the

smaller bubble due to the larger local concentration, 9% versus 5.5%, and the closer

proximity of the smaller bubbles to the wall where the gradient of the Reynolds

stress is larger.

The wall normal component < fa > shows a good general correspondence to

the estimate (6.9) for both bubbles. The force is directed away from the wall as

the fluctuation levels < w, > increase from zero at the wall and reach a maximum

away from the wall. The location of this peak is further from the wall than for

142

the shear stress. The action of this force on average is balanced by a simple mean

pressure gradient and within a periodic channel will not modify the mean flow. It

may produc a weak “turbuphoretic effect” on the bubble motion, analogous to the

effect described by Brook et al (1994) [12].

6.1.3 Bubbles versus Particles

In order to further understand the influence of the “density effect” of the bubbles on

the turbulence, we compare the results to the effect of solid particles with a density

twice of the fluid. The force monople term for FCM is given by (5.3) and in the

absence of buoyancy,

F (n)(x,f) = - ( m p - m F) ^ (6.10)

With a dense particle, mp = 2m p, the force monopole term is equal in magnitude

to that of a corresponding bubble but of opposite sign. This reverses then the body

force distribution. Figure (6.16) compares drag history, normalized by the mean

drag for the base flow for the particles and bubbles. Both have radius a/h=0.1 and

are seeded initially with the same conditions at an average void fraction of 4.2%, as

in case Ilia. There is a clear indication of an increase in drag of about 3% for the

particles as opposed to the 6-7% drag reduction for the bubbles. The profiles of the

particle force < / > are shown in figure (6.17) for =20. At this stage the particles

are still concentrated close to the wall and this evident in the strong positive and

negative peaks of < f \ >. The basic trends are for a reversal in the profile of 

with a positive peak near the wall and an overall negative value that contributes to

an increase in the mean pressure gradient needed to sustain this flow. The action of

the force < /2 > is reversed also with the force now directed towards each wall, the

components of the rms fluctuations of velocity, given in figure (6.18) are based on a

time average over t=20-40 and show a small increase for all the components.

143

S o lid - ___________D a s h — 6 0 0 a = 0 .1 b u b b le sD a s h D o t- — 8 0 0 a = 0 .1 p a r t i c le s

-N o b u b b le

1.05

0.95

0 10 20 30 40T im e

Figure 6.16: Drag history of adding bubbles and particles.

0.001

0.0005

-0.0005

- 0.001

-0.0015

•1 -0.5 0 0.5 1y

Figure 6.17: Force profile of adding particles

144


In summary we see that for particles as opposed to bubbles, the reversal of the

force profiles is linked to the increase in drag and the increase in the turbulence

levels.

S o l id — '- N o p a r t i c le s D a s h — 8 0 0 a = 0 .1 p a r t i c le s

0.1

0 .0 9

0 .0 8

0 .0 7

0 .0 4

0 .0 3

0.02

0.01

-0 .5 0 .5y

Figure 6.18: Comparation of Rms by adding particles or not

6.1.4 V isualization

Figure (6.19) and (6.20) show the streaks of turbulence and turbulence with bub

bles. The blue region, which corresponds to the low speed streaks, becomes larger

by adding 800 a+ = 13.5 bubbles. This is due to the interaction of bubble and tur

bulence, the turbulent speed in the near wall region decreased due to the presence

of small bubbles.

We will use Q contour to visualize vorticity distribution. The second invariant Q

for the velocity gradient is defined as, Q = (dui/dxj)(duj/dxi). Figure (6.22) shows

the Q contour plane at Re* = 135, compared to figure (6.21), the vorticity decrease,

opposite to the increase of streaks. Near wall vorticity is connected with energy

production and transfer, decrease of vorticity will lead to the turbulence decrease,

then it cause drag reduced. This is consistent with drag reduction. Since there only

145


Figure 6.19: Re* = 135 at y+ = 5

z

0 1 2 3 4 5 6

Figure 6.20: Re* = 135 at y+ = 5, 800 bubbles a+ = 13.5

146


Q detection

0 0

Figure 6.21: Q contour at Re* — 135. Threshold is 0.1618

Q d e tec tio n

0 0

Figure 6.22: Q contour at Re* = 135 with 800 a+ = 13.5 bubbles. Threshold is 0.1618

147


is 10% drag reduction, so the difference is not so obvious.

6.2 High Reynolds Number Flow

In the following sections we consider the effects of microbubbles seeding at higher

Reynolds number. This provides a greater degree of scale seperation between the

near-wall and buffer region dynamics and that of the core region of the channel. We

are able to simulate small size bubbles.

The mesh used is 256 x 241 x 256, the domain size is 2n x 2 x n. Reynolds number

is Re* = 380, based on friction velocity (u* = 0.038). 5t = 1/800. The initial flow

data comes from fully developed turbulent channel flow at this Reynolds number.

Drag history has been compared with no bubble case in the following.

The cases we have run are listed as follows:

1. 9100 a~ = 13.5 bubbles, seeding initially in the near wall region;

2. 3200 a+ = 19 bubbles, seeding initially in the near wall region;

3. 6400 a+ = 19 bubbles, seeding initially in the near wall region;

4. 9600 a+ = 19 bubbles, uniform seeding initially in whole channel;

6.2.1 Fluid characteristics

Figure (6.23) to (6.24) show the drag history of these 4 cases compared with no

bubble cases. It is clear that seeding bubbles initially near wall can reach about 10%

148


drag reduction. This effect is similar to what we found at Re* = 135. The uniform

seeding case can only have 3-4% drag reduction due to the lower concentration in

the near wall region.

o>5o

t

O) 13

t

Figure 6.23: Drag history for each wall at Re* = 380: 9100 bubbles or = 13.5 (left);

3200 bubbles a+ = 19 (right). Solid line is the mean drag for no bubble flow

Figure 6.24: Drag history at Re* — 380: 6400 bubbles a+ = 19 (left); 9600 bubbles

a+ = 19 (right). Solid line is the mean drag for no bubble flow

Figure (6.25) to (6.26) show the pressure balance for these cases. We can see that

the residual term is around zero, which verifies that the pressure drop term balances

the wall drag and bubble acceleration terms. The acceleration of the bubbles as they

disperse from the wall reduced the overall pressure gradient. With uniform seeding

149


Bal

ance

this effect is very small. For the no bubble case, based on the wall friction velocity

the equilibrium value is L\ x L3 x 2 x « 2 = 2-ir x tt x 2 x (0.038)2 = 0.05702.

$ 0 .0 3

-0.02L

t

0.02

t

Figure 6.25: Balance at Re* = 380, 1-Wall friction; 2-Pressure drop; 3-Bubble ac

celeration; 4-Residue. 9100 bubbles o+ = 13.5 (left); 3200 bubbles a+ = 19 (right).

t

0.06

«j0.03

t

Figure 6.26: Balance at Re* = 380, 1-Wall friction; 2-Pressure drop; 3-Bubble ac

celeration; 4-Residue. 6400 bubbles a+ = 19 (left); 9600 bubbles a+ = 19 (right).

Since adding bubbles changes turbulent structures in the near wall region, tur

bulent statistics must also be changed. Figure (6.27) and (6.28) show the turbulent

fluctuations. The rms fluctuations are all reduced at higher void fractions, and the

most significant changes are in the core of the channel.

150


Rm

s+ R

ms*

2.5

2

1.5

1

0.5

0200 6000 400y

2.5

2

1.5(AEIT

1

0.5

0200 4000 600y+

Figure 6.27: Turbulent fluctuation at Re* = 380, averaged for t=15-20: 9100 bubbles

a+ = 13.5 (left); 3200 bubbles a+ = 19 (right)

2.5

2

1.5(AEDC

1

0.5

00 200 400 600y

2.5

2

1.5

1

0.5

0,0 200 400 600y

Figure 6.28: Turbulent fluctuation at Re* = 380, averaged for t=15-20: 6348 bubbles

a+ = 19 (left); 9600 bubbles a ' = 19 (right)

151


Figures (6.27) and (6.30) show a comparison of the turbulent Reynolds stress.

They also show similar trends as the turbulent fluctuations. All these are consistent

with a drag reduction effect.

A>3V

-0.4

0 200 400 600y

0.5

A> 03V

-0.5

Figure 6.29: Reynolds Stress at Re* = 380: 9100 bubbles a+ = 13.5, averaged for

t=10-12 (left); 3200 bubbles a ' = 19, averaged for t=15-20 (right)

0.8

0.6

0.4

0.2

A> 0 3V

-0.2

-0.4

-0.6

-0.8 0 200 400 600y

0.5

+A > 0 3 V

-0.5

0 200 400 600y

Figure 6.30: Reynold Stress at Re* — 380, averaged for t=15-20: 6348 bubbles

a 1 = 19 (left); 9600 bubbles fth = 19 (right)

152


6.2.2 Bubble characteristics

Figure (6.31) and (6.32) show the concentration history, t=0-20. For case 1 and 2,

bubbles are distributed randomly in near wall region at the beginning, the concen

tration profile has a flat shape in the near wall region. With the time going, they

dispersed away from the wall. In case 3, bubbles are distributed initially in three

layers near each wall, and in case 4, bubbles have a uniform distribution in channel

from the beginning.

Figure 6.31: Concentration profile at Re* = 380: 9100 bubbles a+ = 13.5 (left);

3200 bubbles a+ = 19 (right)

As for Re* = 135, there is a negative region in bubble force profile close to each

wall, which might contribute to the drag reduction effect. Here, force profiles are

obtained for t=20. Figure (6.33) and (6.34) show that the negative region is narrower

than that of Re* = 135, and the magnitude has decreased over time. Since the flow

is homogeneous in spanwise direction, < > is much smaller than and

< /2 >i about 10 times smaller.

We will check the balance of terms in more detail. In the streamwise direction,

after averaging in the streamwise and spanwise directions, the Navier-Stokes equation

R e * » 3 8 0 ,256*241*256 9 1 0 0 a = 0 .0 3 5 5 b u b b le s------------ t=0----------------t= 5.................. t=10----------------1=15-------------- ts2 0

R e ’= 3 8 0 , 256*241*256 3 2 0 0 a = 0 .0 5 b u b b le s ---------------- t= 0

y y'

is:

153

<F

x>,<

Fy>

,<F

z>

0 .1 4

0.13

0.12

0.11

0.1

0.09

0.06

0.05

0.04

0.03

0.02

0.01-1 -0.5 0y 0.5 1

O 0.2

Figure 6.32: Concentration profile at Re* = 380: 6400 bubbles a+ = 19 (left); 9600

bubbles a+ = 19 (right)

y

lE -0.0004

-0.0012-1 -0.5 0y 0.5 1

Figure 6.33: Force profile at t=20 for Re* — 380: 9100 bubbles a+ = 13.5 (left);

3200 bubbles a+ = 19 (right)

154

0.0004

a n 0.0002

V -0.0002

-0.0008

-0 .0 0 1 2-0.00125

Figure 6.34: Force profile at t=20 for Re* = 380: 6400 bubbles a+ = 19 (left); 9600

bubbles a+ = 19 (right)

dm dP d2m d , , ,<’-» = + + a ^ < ■ p “ , “ 2 > + > ( 6 - n )

where U\ is the mean streamwise velocity, and

Nb (!Vnfi{x,t) = '£j m F—± - A { x - Y n) (6 .12)

n = 1

Consider the momentum f ^ h X2 2 Equation(6 .ll)dx 2 , so we have the following

balance: Lhs = Rhsl+Rhs2+Rhs3+Rhs4

Lhs f k> / xJ-h

2 dUl i~ ^ r d x 2 dt

= "I (6.13)

dP rhRhs 1 = — —— / x<2 'dx2dx i J-h

155

/ .h d2 U\X2 2^ odx 2

-h OX 2h dx 2 2= - h 2 [t ( / j ) + r ( —/ i ) ] + A ^ K U s (6.15)

r h q 2

Rhs3 = / X2 2-= o < -pU\U2 > dX2J - h OX2 1

r h

= - 2 x 2 < -pu\U 2 > dx2 (6.16)J —h

rhRhs4

l - h

r h

= / X 2 < f i > d x 2 (6.17)J —h

then we have following equation,

d f h „ o , <9P 2o 7 oP 2J U\X2 dx2 = - - ^ - h 3 - /i2[t(/i) + t(-/j)] + 4///jPb

/ h r h

2 x 2 < ~(ni\U2 > efcc2 + / Z22 £/x2(6.18)- / i J —h

Since we have

? 1 i fh7 = Zh{Tl~h) + T(,l)1 “ 2h l - h < h > i x 2 (6' 19)

dP 1dx

so we have

2 dx21 3 3 f h 3 Ch d-[r(h) + r(-h )] = - / /P B ~ 2 h? J h X 2 < ~Puiuz > dx2 “ J ■

+ ( ^ p / P22 < / l > dx2 - ^ / ck2) (6 .20)

156

No. Term Case 1 Case 2 Case 3 Case 4

I i[r(h) + r(-h)} 13.9e-4 14.3e-4 13.56e-4 13.75e-4

II 2e-4 2e-4 2e-4 2e-4

III ~ ,l '\ X'l < ~PulU2 > dx2 13e-4 12.1e-4 11.6e-4 11.7e-4

IV 4h J-h X ? 2 < f l > d x 2

~ \S -h < /l > d x 2

-0.16e-4 -0.2e-4 -0.46e-4 -0.19e-4

I-(II+III+IV) -0.94e-4 0.4e-4 0.42e-4 0.24e-4

Table 6.2: Balance check at Re* = 380 for t=20, Term I for the no bubble case is

14.55e-4

The term ^ - X 2 2 dx, 2 usually is very small, especially where the bubbles

have reached equilibrium. We compute J^h U\X2 2 dx2 versus time in figure (6.35) for

9600 uniform seeding bubbles (Case 4). It nearly is a straight line after t= 10 for

random seeding case, so the time derivative is nearly zero.

0 .4 j -

R e '= 3 8 0 , 2 5 6 * 2 4 1 * 2 5 6 9 6 0 0 R a n d o m s e e d in g a s 0 .0 5

0 .3 9 -

0 .3 8 -ADV*v

0 .3 7 -

0 .3 6

10t

15 20

Figure 6.35: Variations of f ^ h U\X2 1 dx2 for 9600 (a+ = 19) bubbles at Re* = 380

A comparison of the various terms in (6.20) is given in table(6.2). These values

157

are calculated at t=20 for each of the four different simulations. The darg (I) in

each case is lower than the average value for the no-bubble flow. The reduction is

linked mainly to the reduced values of Reynolds stress in (III). The direct effect of

 in (IV) is small, and the residual term reflects errors and the effects of the

flow acceleration dU — 1/dt. The residual term is 5% or less.

An underlying question is the extent to which the bubbles respond to the tur

bulent fluctuations. Near the walls, the length scale for the turbulence is smaller

and comparable to or smaller than the bubble size. The bubble velocity is set by an

average over the bubble and this will reduce the response. In order to avoid compli

cations from the nonuniform distribution of the bubbles, we compare < Cv2 > and

< Cw2 > to the corresponding terms < C >< v2 > and < C >< w2 > for 9600

a=0.05 and 9100 a=0.0355 bubbles. They have similar shape, and bubble fluctuation

are lower than turbulent fluctuation, which is due to the finite bubble size.

1 ,256*241*256

'a 0.00015

y

aO.0001

y

Figure 6.36: Bubble Reynold stress at Re* = 380, 9600 bubbles a+ = 19: < Cv2 >

(left); < Cw2 > (right)

158

-0.5y

30 .0 0 0 1 5

A„ 0.0001

5E-05

y

Figure 6.37: Bubble Reynold stress at Re* — 380, 9100 bubbles a+ = 13.5: < Cv2 >

(left); < Cw2 > (right)

6.2.3 Dispersion

Figure(6.38) shows that the moments for the bubble positions < K>2 > and < >

change very slowly after t—10 for random seeding bubbles, in case 4.

0 .3 2

0.3

0 .2 8

0 .2 6

0 .2 4

0.22

A 0 2V0.18

VA 0 .1 6

N> 0 . 1 4

0.12

R e*=400, 256*241*256 960 0 uniform se e d in g a= 0 .05

0 .0 8

0 .0 6

0 .0 4

0.02

2015t

Figure 6.38: Dispersion for 9600 a+ = 19 bubbles at Re* = 380

Case Time Max < C > Max < Cvw > < Cvb > at x+ = 300 Max < Cv >

I 5 0.15 0.0275 0.007 0.037

I 10 0.13 0.0175 0.005 0.024

I 20 0.09 0.0095 0.003 0.018

II 5 0.14 0.024 0.01 0.044

II 10 0.10 0.024 0.006 0.024

II 20 0.09 0.0095 0.003 0.013

III 5 0.28 0.072 0.028 0.044

III 10 0.23 0.056 0.017 0.03

III 20 0.17 0.034 0.01 0.024

Table 6.3: Dispersion data at Re* = 380

Bubbles with near-wall seeding are dispersed away from the wall by the turbu

lence. The bubble flux term < Cv > is non-zero and there is a mean drift of bubbles

from each wall. Dispersion leads to an eventual loss of drag reduction, which explains

the lack of persistence seen in experiments. Regardless of the dynamics contributing

to drag reduction the rate of bubble dispersion is critical to estimating persistence.

The following figures show < Cv >, < Cvf, > and < Cvw > profiles for case I,

II and III. From these figures, we can see that in the near wall region the bubble-

bubble collision term < Cvj > balanced the bubble-wall collision term < Cvw >.

The turbulent flux term < Cv > dominates in the center of channel, and decreases

as bubbles dispersed away from the wall. We also notice that there is a small peak of

bubble-bubble collision term < Cv^ > at y+ = 300. We plot the maximum value of

< Cvw >, < Cv > and < Cvw > at y+ = 300 against maximum value of < Cv >. It

shows that all these collison terms are closely related to peak near wall concentration

values. Specific results for the bubble-wall and bubble-bubble collisions are given in

table (6.3)

160

sD 01

-300 -200 •100 0+ 100 200 300

> 0 .0 3

00.02

M a x < C >

0.01

<cv„>A*>ov

0>oVa">*ov

-0 .01

Figure 6.39: Re* = 380, 3200 a=0.05 bubbles. t=5 (upper left); t=10 (upper right)

t=20 (low left); Dispersion relation (low right)

161

Re*=3S0, 256*241'256 6346 asO.OS bubbles, ts10 --------------- <CVv>

<cvw>

> 0.02

£-0.02

-200 -100

M a x <CV^> < C V W> a t y* = 3 0 0

O 0.045

V 0.03

M a x < C >

<CVv>

<cv>0.02

>ov,A

o° 0v

>ov

-0.02

-300 -200 -100 100 200 300

Figure 6.40: Re* = 380, 6400 a=0.05 bubbles. t=5 (upper left); t=10 (upper right)

t=20 (low left); Dispersion relation (low right)

162

V-0.005A> * -0.01

V -0 .015

-300 -200 -100 0+ 100 200 300

<cv >A 0.01

o 0.005

\/\

O -0.01

-300 -200 -100 0 100 200 300

0.03

V 0.015

0.125M ax<C>

q 0.005

Figure 6.41: Re* = 380, 9100 a=0.0355 bubbles. t=5 (upper left); t=10 (upper

right); t=20 (low left); Dispersion relation (low right)

163

In the random seeding, < Cv > is small compare to other cases, as shown in

figure(6.42).

0.0004 Ef11 i

0.0005R e * = 3 8 0 , 2 5 6 * 2 4 1 * 2 5 6 9 6 0 0 a = 0 . 0 5 r a n d o m s e e d i n g

-0.0003

-0.0006

-0.0005<cv>.

-1 -0.5 0y 0.5

Figure 6.42: Re* = 380, 9600 a=0.05 bubbles

6.2.4 V isualization

The second invariant Q for the velocity gradient, Q = (d'Ui/ dxj) (chi,j/ dxj) has been

used to show vortex structure in the flow. A contour of plot of the regions where

Q+ < —3 highlight the vortex structures in the flow. Figure(6.43) shows the Q

contour for 9100 a=0.0355 and 9600 a=0.05 bubbles. It clearly shows that adding

bubbles will decrease the vortices, and 9100 bubbles has less vorticies than 9600

bubbles. This is due to larger near wall bubble concentration for 9100 a=0.0355

bubbles.

6.3 Summary

The results presented give a clear indication of the reduction in drag force that

164

stream w ise vortices

vortices, Q d e tec tion

vortices, Q d e tec tion

Figure 6.43: Q contour at Re* = 380 for base flow (upper); for 9100 bubbles a=0.0355

(middle); and 9600 bubbles a=0.05 bubbles (below), all at t=20165


Figure 6.44: Re* = 400, 14400 a=0.05 bubbles

comes from seeding micro-bubbles in a turbulent shear flow. The strongest, sustained

reduction in drag is achieved for the small bubbles, a+ = 13.5. For this small size,

the assumption of spherical shape is reasonable and acceptable. The results for the

larger bubbles point to the limited response of these bubbles to the turbulence due

to their increased size and time scales. They also illustrate the importance of the

evolving void fraction distributions and the initial bubble seeding. As the bubbles

disperse away from the walls and as the bubble flow rate Qb increases, there is a

displacement of liquid towards the walls from the center region of the channel. These

results point to at least three mechanisms involved: one linked to the initial seeding

of the bubbles, the second associated with density effects, where the bubbles reduce

the turbulent momentum transfer, and the third governed by specific correlations

between the bubbles and the turbulence.

With respect to the modification of turbulent structures in the near-wall region

we observe a very different picture as compared to other turbulent drag reduction

techniques, e.g. riblets, traveling waves or polymers, see [35]. Typically, when drag

is reduced, there is a modification of the sublayer and lifted streaks, which in most

cases become more coherent while the spacing of the streaks increases. The Reynolds

number for the present simulations is low and the viscous stresses of the flow is still

important throughout the flow. This will limit the degree of drag reduction that can

be achieved under these conditions.

166


We have also performed simulations at higher Reynolds numbers with different

size of bubbles. Relatively similar levels of drag reduction has been obtained and

sustained. Although this is not consistent with experiment results, it suggests that

there may exist other mechanism which cause the large amount drag reduction effect.

We will report other techniques in the following chapters.

If the bubbles are treated as rigid spherical inclusions due to the effects of surface

tension and surfactant contamination, then a symmetric dipole term Gij should be

considered too. For a laminar flow this additional stresslet term leads to an enhanced

viscous dissipation and increase in the effective viscosity of a random suspension.

With both monopole and dipole terms included, there is an increase in the overall

skin friction of about 8%. There is a small increase in the Reynolds shear stress but

mostly the stresslet term (force dipole) has its strongest effect in the near-wall region

where the viscous shear stresses are large. Whether the dipole terms, or suspension

viscosity effect, increases or reduces drag depends on the size and location of the

bubbles as well as the Reynolds number [33].

167


Chapter 7

Drag Reduction by Constant

Forcing

In Chapter 6 , we simulated the dynamics of microbubbles in a turbulent channel

flow. Spectral//;,/) elements (see [67]) were used for spatial discretization, and the

force coupling method developed by [89] was employed to represent the bubbles.

At Re* = u*H/v = 135 {y is the kinematic viscosity and H is the channel half

width), the realizable sustained drag reduction in such DNS is of the order of 10%

in agreement with experimental data in this low-speed regime, see [82, 16]. In more

recent simulations at Re* — 200 and 380, we have observed similar levels of drag

reduction. Drag reduction of up to 20% has been reported more recently by [38] who

represented the microbubbles as point forces.

In this chapter we consider the effects of a localized, constant stream-wise forcing

applied to the flow. The forcing F is applied in the near-wall region and acts to resist

the flow. The forcing is uniform in directions parallel to the wall. One motivation

for this study is a consideration of the effect of the Reynolds shear stress on the

mean momentum distribution. For turbulent Poiseuille flow in a channel, the mean

168


.06

.05

.04

.03

.02

.01

0

.01 0 10 20 30 40 50y+

Figure 7.1: Profile of the gradient of the Reynolds shear stress, scaled by «*/v, against

distance from the wall y+. Results at Re* — 135,200,380,633

momentum equation is

am dp d f ___ d u n „p~ w = “ S 7 + & ; { “ '“ 1" 2 + ' ‘ 8 ^ } + F i (7A>

and under conditions of steady flow, without forcing, this gives the usual linear shear

stress profile, where the sum of the Reynolds stress and viscous stress varies linearly

across the channel. The Reynolds stress is zero at the wall and attains a maximum

value at a distance from the wall that in terms of wall variables scales with Re*1/2,

see [119]. The acceleration of the mean flow, or the response to an applied force

density, though depends on the gradient of the shear stress. The gradient of the

Reynolds stress is positive near the wall and then has a smaller negative value in the

core of the channel. This is illustrated by the simulation results described below and

shown in figure (7.1) where the profile of the gradient of the Reynolds shear stress

is plotted, scaled in wall variables, for different Reynolds numbers Re*. In all cases

the gradient has a maximum at y+ = 8 , while the location at which the gradient is

zero is greater as the Reynolds number is increased.

The effect of the constant stream-wise forcing proposed here counteracts the

influence of the gradient of the Reynolds shear stress. One context in which a

corresponding feature has been observed is in the simulation of a turbulent channel

169


Re.=200, bubbles

Re.=135, bubbles

0.5

-0.5

Re.=135, particles

0.2 0.3 0.4 0.5y/H

Figure 7.2: Profiles of the average streamwise force density for bubbles and particles in a

channel flow at Re* = 135,200.

flow seeded with microbubbles, [134, 91]. Here the bubbles displace the liquid and

one of the dynamic effects is to reduce the local inertia of the flow. This may be

represented by a body force density acting on the flow that is proportional to the local

concentration of the bubbles and the local acceleration of the fluid. On average, the

streamwise component of the force opposes the flow close to the wall and enhances

the flow further away from the wall. This is illustrated in figure (7.2) which shows

sample profiles for the averaged streamwise component of this effective force density.

The orientation of the force near the wall is linked to that of the Reynolds stress

gradient. For a flow seeded with denser solid particles, the force density is positive

close to the wall. It is observed too that the bubbles reduce the effectiveness of

momentum transfer by the Reynolds shear stress and there is a reduction in the skin

friction, while for denser particles there is an increase in skin friction. While the

dynamics of drag reduction by microbubbles is complex and involves many factors,

the results provide a motivation for the present study and suggest that the effect of

streamwise forcing is worth investigating as an independent issue.

As we will show, the imposition of a streamwise forcing modifies the turbulence,

170


which in turn influences the appropriate parameters to use for the forcing. In the

following sections we summarize the simulation procedures and give results for the

change in skin friction as the parameters of the forcing are varied. Following this,

the influence of the forcing on turbulence structure is discussed.

7.1 Near-wall forcing and simulation method

Simulations of turbulent Poiseuille flow in a channel are performed by numerical

integration of the momentum equations for incompressible flow

^ + U ' V u = — Vp + zA^u + F (7.2)

where the fluid density and the kinematic viscosity are p. v respectively. In the

channel, the fluid velocity u(x, t) is determined for 0 < sq < L\ and 0 < x% < L%

with the rigid planar walls located at X2 = ±/i. No-slip boundary conditions are

applied at the walls with periodic boundary conditions in both the stream-wise, x\

and span-wise, x% directions. In these simulations variables are scaled by h and a fluid

velocity scale Uq, with the fluid density p = I. The bulk velocity Ub is maintained

at a constant value of 2C/q/3 by a control procedure that adjusts the mean pressure

gradient —dP/dxi to ensure that the Reynolds number Res = UBh/v is fixed for

any simulation.

The equations 7.2 are solved in terms of primitive variables using a spectral/hp

element scheme, [67]. Fourier pseudo-spectral representations are used in the two

periodic directions together with de-aliassing procedures for nonlinear terms. Spec

tral elements are chosen in the wall-normal direction to ensure good resolution in

both the near-wall regions and in the core of the channel. A third-order, stiffly stable

scheme is used for integration in time, see [67]. Table 7.1 summarizes the different

simulation conditions, listing the domain size and numerical resolution, including

171

Group Re* Res L i/h Lz/h Afi n 2 Nz d P dx 1

A 135 2000 27r 27r 64 1 x 65 64 0.00203

AA 135 2000 47T 2 tt 128 1 x 65 64 0.00203

B 192 3000 2 n 2 ir 128 4 x 37 128 0.001805

C 380 6,667 2 ir 7T 256 8 x 31 256 0.001455

D 633 12,000 27T 7T 384 10 x 37 384 0.001234

Table 7.1: Simulation parameters: Reynolds numbers; domain size; numerical reso

lution including the number of elements and the spectral order; and mean pressure

gradient, scaled by pU^/h, of the base flow.

the number of spectral elements used at each Reynolds numbers. The Reynolds

number Re* = u*h/v, based on the friction velocity v*, is varied between 135 — 630.

The results for the turbulence statistics of the base flows, without any stream-wise

forcing, are consistent with previously published results such as [99].

The stream-wise forcing is specified by the force density = (F]. 0,0) and

has the form shown in the sketch of figure(7.3). In particular, the force density is

given by

OjrFi(y) = —p7 sin(— y), 0 < y < A (7.3)

F\ (:y) = 0, y > A.

where y = h ,± x2 is the distance from the nearest planar boundary. This forcing

acts to decelerate the flow close to the wall, 0 < y < A/2, and then accelerate the

flow in the adjacent region A/2 < y < A. Overall the total force, integrated across

the channel, is zero and the mean pressure gradient balances the skin friction to

within the limits of numerical resolution. We parameterize the force density with

two parameters I and A, representing the amplitude and the spatial region of the

172

Flow

v 10-

Figure 7.3: Idealization of the excitation force and notation; see equation (7.3).

excitation, respectively. The amplitude of the force density I may be scaled in

convective flow variables by U^/h or in terms of wall variables as I + — Ii>/u*3,

using the value of u* for the unforced flow. Similarly A may be given in terms of

wall variables as A+ or as X/h.

In the following, we will present DNS results with the above force included in

the governing flow equations as the controlling mechanism. In particular, we have

simulated turbulent channel flow at Re* = 135,192,380 and 633. The discretization

involved resolutions of 64 x 65 x 64 at low Re* = 135, 128 x 145 x 128 at Re* = 192,

256 x 241 x 256 at Re* = 380, 384 x 361 x 384 at Re* = 633. Domian size is 27r x 2 x 27t

at Re* — 135,192 and 2tt x 2 x 7r at Re* = 380,633.

7.2 Results on drag reduction

In figure (7.4) we first show representative results at Re* = 135 for the effect of

varying A+ on the turbulent drag over time. The value of the forcing amplitude is

kept constant at 7 = 0.02 (in convective units). We see that for the two lower values

of A+ a persistent, long term drag reduction is achieved, whereas for the two larger

173


1 — No control2 — X-=13.53 — r=18.94 — X+=35.10 .4

0.25 0 7 0

t

Figure 7.4: Time history of the skin friction at f?e* = 135, group AA, with I = 0.02, for:

(1) No forcing; (2) A+ = 13.5; (3) A+ = 18.9; (4) A+ = 35.1.

values of A+ there is only a transient drag reduction. Later this transient reduction

gives way to either no drag reduction or an increase in drag. A similar transient

reduction in drag, followed by a drag increase was found for A+ = 59.4, while for

A = 27 there was a sustained reduction in drag of about 12%.

If the length scale of the forcing is fixed at A+ = 19 and the amplitude I is varied,

then sustained drag reduction is achieved as shown in figure(7.5). The level of drag

reduction increases initially as the amplitude is increased but for I > 0.02 there is no

additional reduction in the long term drag, and for I = 0.03 we again see a transient

response.

These results indicate that there is a preferred range of values for I and A to

achieve a sustained reduction in drag. The optimum values at Re* = 135, for the

results shown in figures(7.4) and 7.5, are approximately I = 0.02 and A = 0.14, or

A+ = 18.9. Additional results on the long term change in drag at Re* = 135 are

given table 7.2. The largest reduction in drag observed was for A+ = 13.5 and a

174


g 0 . 6 1 — No control2 — 1= 0.013 — 1=0.024 — 1=0.03

0 .4

0.220 4 0 5 0 6 0

t

Figure 7.5: Time history of normalized skin friction for A+ = 18.9 and different amplitudes

I at i?e* = 135, group AA: (1) No forcing; (2) I = 0.01; (3) I = 0.02; (4) I = 0.03.

stronger forcing I = 0.03. As the value of A+ is increased, the optimum level of

drag reduction is achieved with weaker forcing I. For A+ > 30 — 35 there is a switch

from sustained drag reduction to a transient response followed by a small net drag

increase.

At Re* = 633 the general features are the same but the level of sustained drag

reduction is significantly higher, up to 70%. Figure(7.6) shows the time history of the

normalized turbulent drag. For A+ = 12.0 there is a transient period with up to 90%

drag reduction followed by sustained drag reduction of 70%, while for A+ = 13.3 there

is a larger transient but a slightly lower level of the final sustained drag reduction for

the same level of the force amplitude. The time scale for the transient adjustment

period following the application of the forcing is roughly t = 5.5 — 6.5 at Re* = 633

and t = 20 — 25 at Re* = 135, in terms of convective time units. These both

correspond to an interval t+ = 125 — 150 and are consistent with a viscous wall-

variable scaling for this process.

175


I = 0.01 7 = 0.015 7 = 0.02 7 = 0.025 7 = 0.03

A A+ 7+ = 0.0365 7+ = 0.0547 7+ = 0.073 7+ = 0.0913 7+ = 0.109

0.1 13.5 -13% -12% -21% -28% -33%

0.14 18.9 -16% -23% -24% -24% -25%

0.2 27 -10% -13% -12% -11% -11%

0.26 35.1 +4% +4% 0% +3% +2%

0.44 59.4 +8%

Table 7.2: Percentage change in long term drag at Re* = 135 for various A and 7.

Results to within ±2%.

A summary of results for a range of Reynolds numbers are given in table 7.3,

covering Re* = 135—633 for selected values of the forcing parameters. A skin-friction

reduction of about 33% is achieved at the lowest Reynolds number, Re* = 135, that

increases up to about 40% at Re* = 192 for case B3 and 70% at Re* = 380, case C4,

and beyond. We note, however, that these results were obtained at increasing values

of the amplitude parameter 7; specifically I = 0.03 (case A); 0.08 (case B3); 0.20

(case C4), and 0.4 (case D3). These values were initially selected based on several

coarser simulations for each Reynolds number.

The parameters used for the present results may be compared with those asso

ciated with the variation in the Reynolds stress gradient for the base flows, shown

in figure(7.1). At the lowest Reynolds number the preferred value of A : = 13.5 and

as the Reynolds number is increased the preferred values of A+ remains fixed, or

decreases slightly, with A+ ~ 12 at Re* = 633. By comparison, the location of the

maximum gradient of the Reynolds stress occurs at ~ 8. An attempt to use the

forcing to simply cancel the effect of the Reynolds stress gradient would suggest then

a value of A+ = 4 The results at Re* = 135 indicate that at this value of A+ ~ 30

there would be at best only a slight reduction in drag. These values may be com

pared with the location of the peak Reynolds stress, yp . which as noted previously,

176


Figure 7.6: Time history of the skin friction at Re* = 633 with I = 0.4 for A+ =

7.0,12.0,13.3.

varies with oc \/R.e*. according to [119]. At Re* = 380 we have « 40 and this

increases further with increasing Reynolds number.

The amplitude of the forcing may also be compared. At Re* = 135 the value of

I = 0.02 corresponds to I + = 0.073, as given in table 7.2. This is comparable to the

peak values of the Reynolds stress gradient shown in figure(7.1), which has the same

dimensions as 7. At higher Re* the values of I + are significantly larger and increase

with Reynolds number, as given in table 7.3. The integrated negative contribution

of the force density retarding the near-wall flow is I +\ + /ir in terms of wall variables,

see (7.3), and this shows less variation with Reynolds number. Dimensionally IX/n

scales with u*2 and may be compared with the wall shear stress of the base flow.

At Re* = 135, with A+ = 13.5 and I + = 0.109, this gives a value corresponding to

0.44u*2. This is also equal to the value for A+ = 18.9 and I + — 0.073, where the

drag reduction is 24%. The preferred combinations of I and A at other Reynolds

numbers give values of I +\ +/ir corresponding to 0.91 w,*2 for case B3, 1.5u*2 for case

C3, and 2.0u* 2 for case D3.

177


These observations show that the profile of the Reynolds stress gradient may

provide an initial guide to suitable values of A+ and I +. The required amplitude

of the forcing required to achieve the largest levels of drag reduction increases with

Reynolds number. The magnitude of the forcing is then comparable to or slightly

larger than the wall shear stress of the base flow. It is possible that similar levels of

drag reduction could be achieved with slightly lower amplitudes, as indicated by the

results in table 7.3.

Results for other Reynolds numbers are given in table 7.3, covering Re* =

200 — 633 for selected values of the forcing parameters. In figure(7.7), we show

the levels of drag reduction that can be achieved at the different Reynolds number

flows simulated. A skin-friction reduction of about 25% is reached at the lowest

Reynolds number, that increases up to about 70% at Re* = 380 and beyond. We

note, however, that these results were obtained at increasing values of the amplitude

parameter I; specifically for the four DNS results in the plot I = 0.02 (case A);

0.08 (case B); 0.20 (case C), and 0.4 (case D) in increasing order of Re*. These

values were initially selected based on several coarser simulations for each Reynolds

number, which indicated that drag reduction is achieved above a threshold value of

1.

The present results may be compared with the variation in the Reynolds stress

gradient for the base flows, shown in figure(7.1). At the lowest Reynolds number the

preferred value of A ~ 18 is roughly twice yp ~ 8, the location of the maximum

gradient of the Reynolds stress. As the Reynolds number is increased the preferred

values of A+ remains fixed, or decreases slightly, with A+ ~ 12. At Re* = 633, the

optimal value of A+ is more tightly defined. By contrast, the location of the peak

Reynolds stress, yp\ as noted previously, varies with yp oc \ /Re* according to [119].

At Re* = 380 we have yp « 40 and this increases further with Reynolds number.

The amplitude of the forcing may also be compared. At Re* = 135, the value of

I = 0.02 corresponds to I + = 0.073, as given in table 7.2, and this is comparable to

178


Index i?e* I 7+ A A+ IX/nul — ADrag %

A1 135 0.02 0.073 0.1 13.5 0.31 21

A2 135 0.03 0.109 0.1 13.5 0.47 33

AA 135 0.02 0.073 0.14 18.9 0.44 24

B1 192 0.04 0.116 0.064 12.3 0.45 19

B2 192 0.06 0.174 0.064 12.3 0.68 30

B3 192 0.08 0.232 0.064 12.3 0.91 39

Cl 380 0.1 0.18 0.034 12.9 0.74 34

C2 380 0.15 0.27 0.034 12.9 1.11 47

C3 380 0.2 0.36 0.023 8.7 1.00 33

C4 380 0.2 0.36 0.034 12.9 1.48 70

C5 380 0.2 0.36 0.036 13.7 1.57 63

C6 380 0.2 0.36 0.04 15.2 1.74 52

C7 380 0.2 0.36 0.05 19 2.18 35

D1 633 0.4 0.513 0.011 7.0 1.14 35

D2 633 0.4 0.513 0.017 10.8 1.76 68

D3 633 0.4 0.513 0.019 12.0 1.96 71

D4 633 0.4 0.513 0.021 13.3 2.17 65

Table 7.3: Long term, percentage reduction in turbulent drag from simulations at

i?e* = 135,192,380,633, for various values of I and A. Results for drag to within

±2 points.


100

<0SouaO)&a

20

200100 300R e .

400 500 600

Figure 7.7: Percentage of drag reduction as a function of the Reynolds number.

the peak values of the Reynolds stress gradient shown in figure(7.1). The integrated

positive contribution of the force density, given by (7.3), is I +X 1 /V in terms of wall

variables and in this instance gives a value about 0.44w,*2. At higher Re* the values

of I + are larger. The preferred combination of I and A give values of I +\ +/n equal

to 0.91u*2 for case B3, 1.5w*2 for case C4, and 2.On*2 for case D3.

These observations suggest that the profile of the Reynolds stress gradient may

provide an initial guide to suitable values of A+ and J+. The required amplitude of

the forcing though increases with Reynolds number and as the turbulence is more

strongly modified.

This implies that for A+ > 20 we have loss of persistence of drag reduction,

possibly similar to that reported in the experiments [82]. In several other numerical

experiments (not presented here) we verified that X}max « 2y+ by varying the relative

regions of negative to positive force contributions keeping the net force equal to zero.

We found that the results do not depend on the exact form of the positive component,

but maximum amounts and persistence of drag reduction are ensured only if the

negative component (from the wall to a distance of A/2) is contained within distance

180

of about ten wall units. Experimentally, the consensus is that microbubbles should

be injected within the buffer layer but no quantitative evidence exists to support

this. Here, we found that, at least within the regime of Re* that we could simulate

accurately, the maximum retarding force should be located much closer to the wall.

7.3 Turbulence modification

7.3.1 M ean velocity profile

We now investigate how the streamwise forcing modifies the structure of the tur

bulent flow. The first consideration is the mean flow and how this changes in the

near-wall region. Figure(7.8) shows the mean flow at several different Reynolds num

bers with the forcing applied. These profiles are compared to the mean velocity of

the base flow at Re* — 380, without forcing, and the mean velocity is scaled by

u*, the friction velocity of the corresponding flow with no forcing. For all the forced

flows the mean velocity is markedly different from the standard profile. The gradient

at the wall is reduced but the slope increases sharply away from the wall, consistent

with the formation of an inflection in the mean velocity profile. For case AA, based

on A = 0.14 and I = 0.02 at Re* = 135, the reduction in drag is 25% and the change

in the mean velocity is less steep than for the other cases while at Re* = 633 (case

D3) the reduction in drag is 70%. At Re* = 633 (case D3), where the reduction in

drag is 70%, there is an apparent inner logarithmic region for 5 < y+ < 10. There

is a similar feature for Re* = 380 (case C3).

More revealing are the profiles for the mean velocity gradient shown in figure(7.9).

In all the forced flows shown A+ = 12 — 13.5 and correspond to significant levels of

reduction skin friction. The formation of a strong shear layer centered at y+ =

6 — 7 is clearly evident, with reduced mean shear both at the wall and beyond the

181

Figure 7.8: Mean velocity profiles, normalized with the friction velocity of the base flow:

(A) A+ = 13.5, I+ = 0.073 at Re* = 135; (B) case B3; (C) case C3; (D) case D3. The solid

line (S) represents the no-control case at Re* = 380.

*>p+3■o

0.5

Figure 7.9: Profiles of normalized mean velocity gradient in wall variables for Re* — 135

and A+ = 13.5,1+ = 0.109; case B3; case C3; and case D3. Solid line is the no-control result

at Re* = 380.

182


shear layer y+ > 12. The peak value of the mean velocity gradient, given in terms

of wall variables, increases with Reynolds number and the forcing amplitude. If

the turbulence is indeed strongly suppressed and there is a purely laminar, viscous

response to the near-wall forcing in (7.1) then the result would be a local additional

component to the mean shear

for 0 < y < A, and zero otherwise. This has a maximum at y+ = A+/2, with a

back to the prior discussion of the forcing levels, we can see that the peak value of

(dU\/dx2 )+ at Re* = 633 is close to the value of 7+A+/7t = 1.96 for this case (D3).

Other factors contributing to this near-wall maximum of the mean velocity gradient

are the ambient mean shear in response to the pressure-driven flow, which tends to

increase the peak value, while any Reynolds stress would tend to decrease the peak

value.

7.3.2 Reynolds stresses

Profiles of the rms velocity fluctuations at Re* = 380 for case C4 are compared in

figure(7.10) with the results for the unforced flow. All three peak values of u\ , u'2

and u'z are reduced by at least 40% for the forced flow. For the cases at the lower

Reynolds number, where the level of drag reduction is less, the reduction of the rms

fluctuations is smaller and at Re* = 135 the peak streamwise fluctuation is in fact

slightly higher for the forced case AA. Figure(7.11) shows the corresponding Reynolds

stress profiles. We see that for case C4 the peak value is reduced by 70% while the

location of the peak is shifted away from the wall. Even at the lowest simulated

Reynolds number, case AA with Re* — 135, there is a substantial reduction in the

Reynolds shear stress.

Close to the wall there is a more significant variation in the Reynolds shear stress

(7.4)

peak additional mean shear of 7+ \ +/-k when scaled by wall variables. Referring

183

2.5

<AEcc

0.5 ■

100 200 300 400

Figure 7.10: Profiles of rms velocity fluctuations at Re* = 380. Solid lines denote the

no-control results; dashed lines denote results for case C4.

(a)

1

0.9

0.8

0.7

0.6

£3V

0.5

0.4

0.3

0.2

0.1

0,(b)

1

0.9

0.8

0.7

0.6

3 0.5

0.4

0.3

0.2

0.1

0

Figure 7.11: Profiles of Reynolds shear stress at (a) Re* = 135, case AA; (b) Re* = 380,

case C4. Solid lines denote the no-control results.

184

-0.25

-0.75

100 125 15075

Figure 7.12: Profiles of Reynolds stress at Re* = 380 for cases C3, A+ = 8.7; C4, A+ = 13.7;

all for 1=0.2.

that is sensitive to the value of the length scale A used for the streamwise forcing.

Figure(7.12) shows the near-wall profiles at Re* — 380 for several values of A+ for a

fixed value of I = 0.2. For case C3, where A ; = 12.9 and there is a 70% reduction in

drag, the Reynolds stress is substantially reduced throughout the flow with a small

local maximum at around y+ = 13. If A 1 = 8.7 (case C2), the drag reduction is

much less, only 30%, and the change in Reynolds stress is smaller. For the somewhat

larger value of A+ = 13.7, there is a strong near-wall, local maximum even though

the Reynolds stress is reduced elsewhere. The reduction in drag is also slightly less

here. This local increase in Reynolds stress is linked to the formation of the shear

layer and the local maximum is approximately at y ' = 13, at the outer edge of the

shear layer.

The same general features are repeated in figure(7.13) for Re* = 633 with a fixed

forcing amplitude I = 0.4. Again the largest drag reduction, case D3 with A+ = 12,

is associated with a greatly reduced overall Reynolds shear stress. There is a small

local maximum in the Reynolds stress near the wall. For A+ = 13.3, case D4, the

near-wall maximum is much stronger and occurs at approximately y+ = 7.5, within

185


- 0.2

-0.4

>3- 0.6

100 150

Figure 7.13: Profiles of Reynolds stress at Re* = 633 for cases D2, A+ = 10.8; D3, A+ =

12.0; D4, A+ = 13.3; all for 1=0.4.

the shear layer.

The drag reduction effect is very sensitive to the wavelength A. As shown in

figure(7.12) and (7.13),there is an optimized value for A, which can achieve maximum

drag reduction.

7.3.3 Vorticity fluctuations

The rms vorticity fluctuations also exhibit significant near-wall variations when the

streamwise forcing is applied. Figure(7.14) to 7.16 shows profiles of the vorticity

fluctuations of both the base flow and the forced flows, cases C4 and C5, at Re* =

380. At the wall, the spanwise component lj '3 has the largest fluctuation levels,

which are associated with the fluctuations in the streamwise component of the wall-

shear stress. This is substantially reduced for case C4 as is and the wall-normal

component (jJ2, where the latter goes to zero at the wall because of the no-slip flow

conditions. With the forcing there is a simple, local maximum of u '2 at around

y+ = 8, closer to the wall than in the unforced flow. There is also a more pronounced

186


N o co n tro l c a s e C 4 c a s e C 50.3

3

20 4 0y+

Figure 7.14: Profiles of rms vorticity fluctuations at Re* = 380: cj[. Results for A+ = 12.9,

case C3; A+ = 13.7, case C4; no-control.

local maximum in at around y+ — 8. The case C4, where A is slightly larger, has

significantly increased vorticity fluctuations near the wall, while the fluctuations are

all reduced for y+ > 20.

These general features are consistent with the formation of a local shear layer

centered just above the wall at y+ ~ 6 — 7, due to the influence of the streamwise

forcing. This is more clearly evident if we examine the near-wall flow structure with

the second invariant Q for the velocity gradient, Q = (dui/dxj)(duj/dxi). A contour

of plot of the regions where Q+ < —0.0142 highlight the vortex structures in the

flow. The structure of the near-wall vortices is shown in figure(7.21), comparing the

base flow with the forced flow C4 at Re* = 380. The formation of vortex loops lifting

away from the wall is completely suppressed and instead there are regularly spaced

spanwise structures within the region y+ < 20. The standard turbulent processes

near the wall are replaced by those characteristic of a shear layer in transition.

When A+ is increased to 13.7, case C5, then the shear layer begins to create its

own turbulence structures. The corresponding contour plot of Q+ for case C5, see

figure(7.22), shows increased turbulence but the vorticity-dominated features are

confined to the near-wall region, y+ < 50. It would appear that at this forcing

187

0.3

3 0.2

20 40y+

Figure 7.15: Profiles of rms vorticity fluctuations at Re* = 380: u)'2. Results for A+ = 12.9,


0.5

0.4

0.3

3

0.2

0.1

y+

Figure 7.16: Profiles of rms vorticity fluctuations at Re* = 380: Results for A+ = 12.9,


188


Turbulent C harge R e*= 380 ,128X145X128------------ No control----------Lambda=0.034. 1=0.2

400

200

o>

-200

-400

-0.98 -0.9B -0.94 -0.9-0.92y

Figure 7.17: Turbulent charge at Re* = 380. Solid lines: no-control; dash lines:

I — 0.2, A = 0.034.

amplitude 7, case C4 corresponds to a near-optimal condition of marginal stability

and increasing the total forcing by increasing the length scale A leads to a breakdown

of the shear layer.

If the shear layer is too strong it promotes its own turbulence production, but

otherwise serves to screen the viscous sub-layer from the outer flow dynamics.

A different way of examining the modification of the near-wall turbulence is

offered by the turbulent charge, a concept introduced by [86]. In particular, the

turbulent charge is defined as the divergence of the Lamb vector, i.e. q = V • 1,1 =

w x u and is a completely kinematic quantity. Its relation to the flow dynamics is

through the Bernoulli energy function 4> = (u2/2) +p/p as follows

q = - v 2$. (7.5)

Therefore, the turbulent charge is connected with the curvature of the Bernoulli

189


energy function. This, in turn, implies that this energy $ is maximal where the

charge is positive. In the context of the turbulence energetics, its importance lies in

its localization in a narrow region very close to the wall, and it is dipolar as shown in

figure(7.17) for the no-control case. Within the viscous sublayer the turbulent charge

is negative and achieves its maximum value exactly at the wall. Beyond the sublayer

(up to y+ ~ 20), where the turbulent motion becomes most energetic the turbulent

charge is positive. Examining now the turbulent charge of the controlled flow in

figure (7.17), we see that the negative maximum charge has moved away from the

wall to the edge of the viscous sublayer while the positive charge is now concentrated

in a much narrower region around y+ = 10. The turbulent charge we have plotted

shows the instantaneous modification of the “flow dipole” in the near-wall region; a

particle-like force leads to exactly the opposite charge distribution.

7.3.4 Influence of force am plitude

When the amplitude I of the forcing is strong and the shear layer is clearly evident, as

in the previous examples, it is not difficult to see the changes in turbulence structure.

At lower levels of I, there may still be substantial reduction of the skin friction but

the changes in flow structure are less evident. Here we compare the results for cases

C2 and C4 at Re* = 380, where the length scale A is the same but the amplitude

I = 0.15 as opposed to 0.2. The level of drag reduction is lower, 47% instead of 70%

for case C4. The change in the mean velocity profile is less pronounced but there

is still a strong mean shear layer with a maximum shear (d U i /d x ^ = 1.3. The

profiles for the rms velocity fluctuations and Reynolds shear stresses for case C2 are

given in figure(7.18) and may be compared with those in figures(7.10, 7.11). Both

the rms fluctuations and Reynolds stress are reduced but not to the same degree as

before: the peak Reynolds stress is only 32% lower than for the unforced flow, even

though the level of drag reduction is greater than this.

Reduction in drag is closely linked to reductions in turbulent Reynolds shear

190


2.5

0.5

1.5

u ~ , N o co n tro l v'*, N o contro l w” , N o con tro l u * , 1=0.15 v " , 1=0.15 w * . 1=0.15

N o con tro l c a s e C4 c a s e C5

100 200y+

300 400 100 200y+

300 400

Figure 7.18: Profiles of rms velocity fluctuations and Reynolds stress at Re* = 380 for case

stresses but as pointed out by [39] both changes in the peak value and the profile

contribute. This is evident from taking integral moments of the equation for the

mean flow (7.1). The zero-order moment, which is a simple integral across the

channel, gives the usual result that in a final stationary state the mean pressure

gradient balances the effect of the combined mean shear stress from the two walls,

since the applied forcing exerts no net force on the flow. The second moment yields

where t ( ± / j ) is the mean shear stress at the respective walls and the first term on the

right equals the corresponding drag for laminar flow. Reductions in the Reynolds

stress nearer the walls are thus more significant. In the present case C2, the peak

Reynolds stress occurs further from the wall than for either case C4 or for the base

flow and so even though the peak value is significant its contribution to the overall

drag is limited. There is a small direct contribution of the forcing F\ in (7.6), tending

to reduce the drag. This may be compared to the mean shear stress of the base flow

pul and this ratio is

For case AA at Re* = 135, this ratio is —8.5%, contributing about one third to

the overall drag reduction. At higher Reynolds numbers the contribution is less

CIA

i [r(h) + r ( - h )] = ^ J h x2{-puiu2) dx2 + J x%Fx dx2 (7.6)

(7.7)

191


0.4

0 .35

c a s e C 2 c a s e C 2 c a s e C 2

0.3

0 .25

3

0.15

0 .05

20 40 50y+

Figure 7.19: Profiles of rms vorticity fluctuations at Re* = 380 for case C2.

significant as X/h = A+/i?e* and the value of A+ tends to be fixed.

The three components of the rms vorticity fluctuations for case C2 are shown

in figure(7.19) and are compared with those no control case. The spanwise vorticity

fluctuations oj':i are reduced overall but are relatively stronger for y+ < 10. The other

components are lower too with no strong local maxima. Examination of a contour

plot for Q+ shows no special features and it does not appear to be markedly different

from that of the flow without forcing.

Changes to the gradient of the Reynolds shear stress though can be linked to

vorticity flux terms. In (7.6) the term involving the Reynolds stress may be rewritten

to give

[r(h) + T( - h) } = J h X* { Fl + } dx2’ (7‘8)illustrating that near-wall reductions of the gradient of the Reynolds stress are im

portant for drag reduction. In general, the Reynolds stress gradient is

d(u^uj) _ 1 d(uju])+ 2 “ & T ' (7-9)

As the channel flow is homogeneous in the streamwise and spanwise directions and

192

by selecting i = 1, we obtain

= -v'u)'z + w'uj'y, (7.10)

where the terms v'u}'z and w'u'y represent the vortex-transport and vortex-stretching

contributions, respectively. Profiles for the three terms in 7.10 are shown in figure(7.20),

comparing the results for cases C4 and C2 with that of the base flow. As the ampli

tude I is increased the peak of the Reynolds stress gradient is reduced and shifted

towards the wall. The vortex stretching term w'u'y is also reduced and the minimum

again shifted towards the wall. For both, the most significant variations are confined

to y 1 < 30. The vorticity transport term v'lj'z is the wall-normal flux of spanwise

vorticity flucutations, which in the base flow extends over a wider range. Within the

shear layer, y+ < 13, this not significantly altered. However the vortex-transport

is reduced in the buffer layer and beyond, consistent with shear sheltering and a

reduction of turbulent transport across the shear layer.

The vortex-transport term FTTi has been identified by [61] as an important

part of the autonomous cycle of near-wall turbulence in a channel flow at low to

moderate Reynolds numbers. In their study they separately filtered and reduced the

vortex-transport and vortex-stretching terms to determine how this would modify

the generation of near-wall streaks. Their observation was that reducing the vortex-

transport term did significantly reduce the streaks and the skin friction, while the

filtering the vortex-stretching term M3ZD2 had little effect. The present results point

to other factors also being involved.

7.4 Flow Visualization

This is more clearly evident if we examine the near-wall flow structure with the

second invariant Q for the velocity gradient, Q = (dui/dxj)(duj/dxi). A contour of

plot of the regions where Q+ < —0.0142 highlight the vortex structures in the flow.

The structure of the near-wall vortices is shown in figure(7.21) for a portion of the

193

0.07

0.06

0.05

0.04

£ 0.03 3i 0 02

0.01

0-0.01

-0.02

iVI-o>c

0)H■EoQ_(Ac2H

Figure 7.20: Profiles of Reynolds stress gradient, vortex stretch and vortex transport for

case C3, case CIA and the base flow.

194


vortices, Q detection

150

100

50

vortices, Q detection

i;116a

Figure 7.21: Contours of Q at Re* = 380, no control (upper) and case C4 (lower).

flow domain, comparing the base flow with the forced flow C4 at Re* = 380. The

formation of vortex loops lifting away from the wall is completely suppressed and

instead there are regularly spaced spanwise structures within the region y+ < 20.

The standard turbulent processes near the wall are replaced by those characteristic

of a shear layer in transition. When A+ is increased to 13.7, case C5, then the shear

layer begins to create its own turbulence structures. The corresponding contour plot

of Q+ for case C5, see figure(7.22), shows increased turbulence but the vorticity-

dominated features are confined to the near-wall region, y+ < 50. It would appear

that at this forcing amplitude 7, case C4 corresponds to a near-optimal condition of

marginal stability and increasing the total forcing by increasing the length scale A

leads to a breakdown of the shear layer.

It is described in [57] that various configurations in which the presence of a thin

195

v o rtic es , Q d e te c tio n

1S0100

so

3(

Figure 7.22: Contours of Q at Re* = 380 for case C5.

shear layer in a turbulent flow will act to block the velocity fluctuations normal

to the layer and to limit the velocity correlations across the layer. In a turbulent

boundary layer this would screen the larger scale, outer region, turbulent eddies from

the direct influence of the wall. This mechanism of shear sheltering is considered

further by [110] in the context of polymer drag reduction and the presence of strong

shear in an expanded buffer layer. An important factor in shear sheltering is the

advection velocity of the eddy disturbance relative to the velocity within the shear

layer. In the present context, the shear layer is adjacent to the wall and strongly

influenced by viscous processes, making it difficult to compare directly wall-normal

velocity fluctuations above and below the shear layer.

Evidence for the mechanism of shear-sheltering may be seen by examining the

formation of low speed streaks in the near-wall zone in figure(7.23), where the con

tours for the streamwise velocity for case C4 are compared to those of the base flow.

At y+ — 5, there is a predominance of low speed (v,]) fluid for the forced flow with

elongation of the low speed streaks. At y+ = 10, there is now an absence of low

speed fluid in the forced flow while the individual regions of moderate or faster speed

fluid are both longer and wider. Even at y+ = 30, outside of the shear layer and the

196


spanwise structures seen in figure(7.21), there is again an absence of low speed fluid

in the forced flow. This indicates that the usual processes of exchange of high and

low speed fluid between the near-wall region and the buffer layer are greatly reduced

by the presence of the shear layer.

Figure (7.25) similar to (7.22), streamwise vortices dissappeared. The spanwise

vortices only located very close to the wall, about 10 wall units. The difference from

Re* = 400 is that there still some stream-wise vortices left, but very few.

Next we compare streaks between C3 and no control at y+ = 5, y ~ = 10 and

y- = 30. There are clear differences in all of them, suggesting that the effect of

control extend beyond y = 30.

We also show the cross section contour of streamwise velocity in figure (7.29) and

figure (7.30). They have same scaling level, and it is clear that control has compress

the low speed fluids in the region very close to the wall, and increase the velocity of

stream-wsie velocity in the central region of channel.

7.5 Summary and Discussion

In this chapter, we have presented a new technique for turbulent drag reduction,

providing an overview of the main observed features. The streamwise forcing leads

to the formation of a strong shear layer near each wall, reducing the mean velocity

gradient at the wall and shifting the maximum mean shear to a location at about

y+ = A+/2. The shear layer acts to screen the near-wall, viscous sublayer dynamics

from that of the outer flow. For larger forcing amplitudes a maximum drag reduc

tion of 70% is observed with a very large reduction of the Reynolds shear stress.

Substantial drag reduction is achieved even for lower amplitudes. The dynamics are

less dramatically altered but they are still consistent with a mechanism involving

shear sheltering. The streamwise forcing must be localized near the wall, preferably

197


Figure 7.23: Contours of streamwise velocity in the x\ — x?) plane at f?e* = 380, no control

(left) and case C4 (right): (a) y+ = 5; (b) y+ = 10; (c) y+ — 30.

198


vortex detection: Q

1500

1000600

400 500200

Figure 7.24: No control Q detection at Re* = 633.

vortex detection: Q

TUSim

1500

1000600

400 500200

Figure 7.25: Control case: A = 0.019, 1=0.4 at Re* = 633

199


Figure 7.26: Contours of streamwise velocity in the X\ — xs plane at y+ = 5, Re*

no control (left) and case C3 (right)

200


Figure 7.27: Contours of streamwise velocity in the x\ — X3 plane at y 1 = 10, Re* = 380,


201


i

CO LO Csl LO CO LO CM LO

Figure 7.28: Contours of streamwise velocity in the X\ — X3 plane at y ' = 30, Re* = 380,


202


Y

Re‘-30O

0 0.2 0.4 0.6 0.6

Figure 7.29: Contour in y-z plane at Re* = 380.

Y

R s‘- 3 8 0 ,1- 0 .2 , Lam bda-0.034

0 0.2 0.4 0.6 0.6

Figure 7.30: Control in y-z plane at Re* = 380, A = 0.034, 1=0.2.

203


fo r c in g

30.4

" p a ra b o l ic p ro f ile

0.5y

no contro l

Cl2•o<u3S8

NCOEoc forcing

t

Figure 7.31: (a) Time history of pressure drop during transition in channel flow at Re* =

135. All values are normalized with the value of the uncontrolled flow in the turbulent state,

(b) Streamwise elocity profile in the laminar state at time indicated by A.

with A+ < 13 — 20.

It is interesting to note that the same controlling force is also effective in reducing

the stresses during the transition process from laminar to turbulent flow. To this end,

we have simulated the transition process in a channel flow at Re* = 135 starting from

a laminar flow field. In figure(7.31) we plot the history of the pressure gradient for

the controlled transition as well as the natural transition. In order to accommodate

this transition a small amount of noise was added initially to the laminar field.

We see that during the transition there is an overshoot in the pressure drop (and

correspondingly in the wall shear stress), which is substantially higher than the

asymptotic mean value in the stationary state. However, the maximum peak in the

controlled case is below the level of the turbulent wall shear stress for the uncontrolled

case.

With regards to a practical implementation of this method, the first key issue is

how to produce a retarding force within a distance from the wall corresponding to

(A+/2) < 10. At high speeds (e.g., 20 m /s), this physical distance is less than 10

204

microns for a turbulent flow in water. The force amplitudes used here are smaller by

an order of magnitude than the force levels used in turbulence control via traveling

waves in [35]. Ideally, the excitation force should have its maximum away from the

wall at about y+ sa 5. However, we found in other DNS experiments with a Lorentz

force (produced by electromagnetic tiles as in [26] but of opposite direction) that

drag reduction is still achievable but at about half of the aforementioned levels. The

Lorentz force has its maximum, at the wall and decays exponentially to zero with the

distance from the wall; therefore, it never reverses direction and thus the required

pressure drop is larger than otherwise. The second key issue is then to induce a

positive streamwise force away from the wall that counter-balances the retarding

force in order to realize the maximum possible drag reduction. We have performed

several DNS to investigate the effect of the distribution of this positive force and we

have found that its exact form is not critical.

An example of the effect of a Lorentz type of force distribution applied at each

of the walls is shown in figure(7.32). The streamwise forcing is

Fi(y) = -P 1 {exp(~y/S) - 5/h} (7.11)

where y = h ± x 2 varying over 0 < y < h. This exerts no net force on the flow and

there is a small, uniform positive force to compensate for the negative force near

each wall. The results at Re* = 135 (case AA) show that for I = 0.02 and both

S/h = 0.017,0.04 there is about a 10-15% reduction in the skin friction.

Finally we comment on the power required to sustain the streamwise forcing and

the flow through the channel. The negative streamwise forcing adjacent to the wall,

in principle, extracts kinetic energy from the flow and the positive forcing further

away from the wall supplies energy to the flow. Overall there is a net power input

from the forcing that increases with the forcing amplitude. This will offset the power

savings from the drag reduction and the reduction in the mean pressure gradient.

At Re* = 135, with A = 0.1 and I = 0.02, the reduction in power needed for the

pressure gradient is 20% while the net power required for the forcing is 10% of the

205

No control

D

Exponential forcing delta=0.017Exponential forcing

delta=0.040.7

o 25 50 75 100t

Figure 7.32: Time history of normalized drag force with Lorentz forcing in channel flow at

Re* = 135.

power input for the base flow, giving a total power savings of about 10%. This drops

to 4% however if the kinetic energy is not extracted by the negative streamwise

forcing. At Re* — 380, case C4, there is an overall trade-off between the 70% power

savings from drag reduction and the power input for the forcing. This becomes a

net increase in power of 17% if the kinetic energy is not recovered from the flow by

the negative portion of the streamwise force. At Re* = 633, case D3, there is a net

increase in the power required.

These observations are not peculiar to this technique for drag reduction. Com

parable results are reported by [111] for drag reduction through spanwise oscillations

of the walls. An intermittent forcing of the flow, or some other modification of the

technique, would be needed to generate a power savings at higher Reynolds num

bers. The streamwise forcing is successful though in reducing turbulence levels and

this may be of value of itself. It is interesting to note what happens to the power

supplied to the flow when the skin friction is reduced substantially. In both cases

C4 and D3 the drag was reduced by 70%, the Reynolds shear stresses were greatly

206


reduced as was the overall turbulence production. A simple estimate shows that

the direct viscous dissipation of kinetic energy by the mean flow increases with the

forcing amplitude. Using (7.4) to calculate the additional viscous dissipation within

the shear layer, the ratio of the added dissipation rate to the power supplied by the

mean pressure gradient of the base flow pu^Us is (3/87r2)(/+)2(A+)3(u*/f7s). For

case C4 this is approximately 60% and 90% for case D3.

While one may speculate about the practical issues of implementing this stream-

wise forcing technique for drag reduction, the results presented raise interesting,

fundamental issues about the interaction of an imposed near-wall shear layer with

the turbulence.

207


Chapter 8

Slip Flow DNS

8.1 Background

From the results in Chapter 5, where we have explored the “density effect” of

bubbles, with the force monople representation, 10% drag reduction can be achieved

by adding small bubbles of about 10% in void fraction. This is for bubbles in size

range of d+ =27-40, which in flows at 10 m/sec might correspond to bubble diameters

of 54-100 microns, typical of smaller bubbles produced by current bubble injection

devices.

The level of drag reduction is substantially lower than that observed in experi

ments, suggesting that other physical factors, not considered so far, are important.

One possible mechanism not considered so far is the formation of a localized gas film

on the wall surface or a transient zero shear stress condition where a bubble contacts

with the wall. An observation from the experiments on microbubble drag reduction

is that at lower flow speeds the bubbles injected into a turbulent boundary layer

208


coalesce downstream to form a gas film adjacent to the wall. Sanders et al (2003)

[114] reported that at flow speeds of 6 m/sec the skin friction was reduced by up

to 90% as the bubbles merged to form an almost continuous sheet of gas close to

the wall. In experiments with a rough-wall turbulent boundary layer, Deutsch et al

(2003) [32] found that the levels of drag reduction that could be achieved were com

parable to those for a smooth-wall turbulent boundary layer at the same Reynolds

numbers. One may speculate as to whether small bubbles may become attached to

the roughness elements and provide a virtual gas film locally that screens the wall.

Motivated by this, and by the possible effects of hydrophobic surfaces on tur

bulent drag reduction, we consider the effects of a partial slip boundary condition

at the wall on the flow dynamics. Slip flow condition are usually considered in the

context of micro-flow systems and hydrophobic surfaces. Surface characteristics of

such systems are discussed by for example de Gennes (2002) [29] and Vinogradova

(1999) [138]. There are several papers that give experimental meaurements of the

effects of hydrophobic surfaces and skin friction reduction such as Watanabe et al

(1999) [139] and TYethaway & Meinhardt (2002) [126], and more recently by Ou et

al (2002) [105] for roughened microchannels. We characterize the possible effects of

a partial slip by a slip length parameter and investigate the reponse of the turbulent

flow with and without bubbles present in the flow.

8.2 Slip boundary condition

We assume that there are two mechanisms working together to produce drag

reduction effect. One contribution to the total drag reduction may be due to the

presence of the bubbles, e.g. at the level of 10% to 20%, as the simulations [134, 38]

have revealed. However, the majority of the drag reduction may be due to an

209


apparent partial slip of the flow with the wall. This is consistent with recent theories

on slippage in liquids [138, 29], according to which nanobubbles or a thin gas film

formed at the wall can lead to significant slip. This assumption is also consistent

with recent experimental findings reported in [114].

We impose a slip boundary condition [101] on the channel walls, which is ex

pressed by

In the above equation, uw is the stream-wise velocity on the walls, b is the slip

length, and the positive and negative signs are for the lower and upper walls, respec

tively. We investigate the slip effect on the friction drag by varying the slip length

systematically.

8.2.1 Verification

We have implemented equation(8.1) into our code, and verified it by solving the

following one-dimension Stokes problem.

( 8 . 1 )

d2u _ dp dy2 dx (8 .2)

0 , 2 / = 1

0 , 2/ = - 1 (8.3)

(8.4)

The accurate solution of (8.2) is:

u0 = —1/1.3(y2 - 1.2)

<gn = 2/1.3,//= -1

210

0 .9

0 .7

0.6

3 0 .5

R e = 1 .0 ,163S o lid -u= 0 , y= -1 ,1 , (du /d y = 2 .0 a t wall)D a sh -------u-0.1 ’ du /d y = 0 , y=-1

u+0.1 *d u /d y s0 , y = 1 , (d u /d y = 1 .5 6 5 1 3 2 a t wall)

0 .4

0 .3

0.2

*0.5 0 .5

y

Figure 8.1: Solution of slip test

f a = -2 /1 .3 , y = l

Figure (8.1) shows the solution of equation(8.2). Since the flow rate is constant,

the velocity profile becomes flatter than the case with no slip boundary condition.

8.3 Drag Reduction Under Different Slip Length b (Con

stant Slip Length)

Now we can investigate the drag reduction effect with this new boundary condi

tion. In figure(8.2), we plot the histories of drag and pressure drop of the turbulent

channel flow at Re* = 135 with slip lengths b+ = 5,10,15 and 50 in wall units

and a case with no-slip condition. The drag reduction effect is obviously associated

with the slip length b l . The notable feature is that using this boundary condition,

the maximum drag reduction can reach 80% or more, which is similar to the levels

211


reported in experiments. These results were obtained on 2ir x 2 x 2it domain and

64 x 65 x 64 mesh.

7

6

5

4

o>S 3 Q

2

1

0-1 0 10 20

T im e30 40

t

Figure 8.2: Drag Reduction and Pressure drop of channel flow with different slip

length at Re* = 135

Figure(8.3) shows the drag reduction effect at Re* = 380 with different slip

length. It is similar to the situation of Re* = 135, where the effects of drag reduction

are also dependent on the slip length. The simulation at Re* = 380 was done on

27t x 2 x 7r domain and 256 x 241 x 256 mesh.

Both figures show the drag reduction and pressure drop effect with different slip

length b+. The results are given in terms of b : . and show that this scaling by wall

variables provides a consistent correlation of the data.

Figure(8.4) shows the drag reduction percent with different slip length at different

Reynolds number. Same slip length gives similar drag reduction level at different

Reynolds number.

Figure(8.5) shows the mean velocity under different slip length at Re* = 135 in

both linear and logarithmic coordinates. As slip length increases, the mean profile

becomes flatter as the velocity gradient at the wall decrease and more drag reduction

is achieved. Since the mean profile has changed a lot, other turbulent statistics will

also change. We will see them in the following plots.

212


Figure 8.3: Drag Reduction and Pressure drop of channel flow with different slip

length at Re* — 400

100

-O 5 0

u> 4 0

3 0

20

2015

Figure 8.4: Drag Reduction vs. b+

213


0 .7

0.6R e*= 13 5 , 64-S o lid -------------D a s h ------------D ot---------------D ash D o t-—

•No S lip •S lip BC, b*=5 •S lip BC, b*=10 •S lip BC, b f= 15.Clin D r h f-c n

0 .5

0 .3

0.2

0 .5-0 .5

y

0 .7

0.6

0 .5

0 .4R e * = 1 3 5 ,6 4 5S o lid -------------D a sh -— — -—D ot---------------D a sh D o t-— _ ..r _____L o n g D a sh — S lip BC, b += 5 0

-N o S lip •S lip BC, b*=5 •S lip BC, b*=10•Clin D r K‘. 1 S

0 .3

0.2

0.1

0y

Figure 8.5: Mean velocity at Re* = 135.

214


Figure(8.6) shows the turbulent fluctuation and Reynolds stress under different

slip length. All components of turbulent fluctuation decrease, except the u' increase

at the wall due to slip boundary condition. Reynolds stress also decrease obviously.

With partial-slip, u is no longer zero at the wall. The peak values of both fluctuations

and Reynolds stress decrease. With the reduction in the mean velocity gradient there

is reduced turbulence production.

0.12Re = 135,64Solid------------- No SlipDash-------------Slip BC, b‘=5Dot Slip BC, b '=10DashDot------- Slip BC, b*=15DashDotDot—Slip BC, b*=500.1

0 .07

£ 0 .0 6cc0.05

0 .04

0 .03

0.02

0 .0 1 -i

54

R e= 135, 64Solid--------------No SlipDash--------------Slip BC, b*=5Dot---------------- Slip BC, b‘=10D ashDot-------- Slip BC, b*=15DashDotDot— Slip BC, b*=50

0.75

0.5

0 .25

-0.25

-0.5

-0.75

2 7

Figure 8.6: Statistics at Re* = 135. Turbulence fluctuation (left); Reynolds stress

(right)

Figure(8.7) shows the turbulent mean velocity gradient and rms vorticity at

215


b+ = 5. (jjz decreases, and all vorticity fluctuations decrease obviously. Since the

boundary condition has been changed, this defintely will modify the near wall tur

bulent structures, especially the vorticity distribution in the near wall region. Most

noticeable is a lower spanwise vorticity fluctuation at the wall, consistent with lower

fluctuations in the wall shear stress.

6

5

4

3

2

§ 10«9S -1

2

3

4

S6

y

'3

Figure 8.7: Statistics at Re* = 135. Mean vorticity (left); Vorticity fluctuation

(right)

All these statistics are consistent with the drag reduction effects obtained above.

Up to now, we only investigated the effect of slip in the streamwise direction.

In a channel flow, streamwise velocity is the dominant quantity of the three direc

tions. However, in real condition, if the wall material permits slip in the streamwise

direction, there is no reason to prohibit slip in other directions. In the following we

give the results comparing slip in streamwise direction only and in both streamwise

and spanwise directions. Figure(8.8) shows that under same slip length, turbulent

statistics change less if slip is allowed in both directions and less drag reduction

effect is obtained. Physically, if slip allowed in both directions, then the fluids tends

to change velocities in both directions. According to the divergence free condition,

the change in each direction will be less than the change if we only allow slip in

stream-wise direction. This is consistent with what we find.

216


2.5

2

1.5

Ecc1

0.5

0

16

14

12

10

3 66

2

0y

Figure 8.8: Statistics at Re* — 135. Mean vorticity (left); turbulence fluctuation

(right)

8.4 Combined Slip with Bubbles

Next we investigate the effects of combining slip and adding bubbles. Figure(8.9)

shows the history of drag and pressure drop at Re* = 135. 800 (a+ = 13.5) bubbles

have been added, with the slip-wall boundary condition. The mesh is 64 x 65 x 64

and domain size is 2 n x 2 x 2 n.

a t&aSolid------------No Slip, No bubblDash------------NoSlip, 800 bubiDot--------------Slip, No bubblesriaahrw_____Rlin Ann huhhUa

0.002

0.0015

a□0.001 lo SHpj SCO bubk

lip, No bubbles

0.0005

0,t

Figure 8.9: Re* — 135, 800 a+ = 13.5 bubbles, b+ = 1.35. Drag history (left);

Pressure drop (right)

Figure(8.10) shows the history of drag and pressure drop at Re* = 380. We

consider two configurations: without bubble and with 7200 bubbles. The bubble

217


t

Figure 8.10: Re* = 380, 7200 a+ — 20 bubbles, b+ = 1.35. Drag history (left);

Pressure drop (right)

radius is a+ = 19 in wall units, and the bubbles are initially seeded in a layer near

the wall. The flow rate in the channel is kept constant in all the cases. These

preliminary simulations were conducted on a 128 x 145 x 128 grid. It is observed

that the wall slip reduces the pressure drop (and correspondingly the friction drag)

significantly in presence or absence of bubbles. As the slip length increases the

reduction in drag also increases. We also observed an additive effect of the wall

slip and the bubbles on the drag reduction. That is, when both effects are present

the total reduction in drag is approximately the sum of those when only one effect

is accounted for. With 7200 bubbles and a slip length b+ = 1.35 the drag on the

channel wall is observed to decrease by about 30%.

Figure(8.11) compares turbulent velocity fluctuations and Reynolds stress by

adding 800 a+ = 13.5 bubbles and using b+ — 1.35 for the slip boundary condi

tion. They achieved similar level of drag reduction, about 10%. Their turbulent

fluctuations and Reynolds stress both decrease, and similar level of decrease has

been obtained. So the drag reduction effects and turbulent statistics decrease are

associated and consistent to each other, our results confirmed such connection.

218


2 .5 R e*= 135 , 64*Solid ••••N o Slip , N o B u b b lesD a sh -----------8 0 0 a = 0 .1 b u b b le sD ot------------- S lip BC, b*=1.35D ash D o t----- Slip BC, b * = 1 .3 5 ,8 0 0 a s0 .1 b u b b le s

Ecc

0.5

27 108 135 162 189 2 16 243 270

R e **135 , 64*So lid ----------- N o Slip , N o B u b b le sD a sh -----------8 0 0 a= 0 .1 B u b b lesD ot--------------Slip B C , b*=1.35D ash D o t-— Slip B C , b*=1.3 5 ,80(

0.4

0.2A>3V

- 0.2

-0.4

Figure 8.11: Re* — 135, b+ = 1.35. Turbulence fluctuation (upper); Reynolds stress

(low)

219


0 1 2 3 4 5 6

Figure 8.12: Streaks at y+ = 5 of Re* = 135.

8.5 Visualization

The partial slip boundary conditions significantly modify the near-wall structure and

the usual pattern of low speed streaks near the wall. Figures (8.12) and (8.13) show

the difference in the streamwise velocity contours at y+ = 5 for Re* = 135. The

contours have the same scaling so we can see that the number of streaks becomes less,

and the distance between them becomes larger. This mechanism is different from

that of adding micro-bubbles. Adding micro-bubble decreases the overall level of the

streamwise velocity in the near wall region. However, the slip boundary condition

increases the streamwise velocity in the near wall region and there is a decreases in

the velocity in the channel center. So the effect we see here is different from figure

(6.20). As the partial slip and bubbles gives about 30% drag reduction, the decrease

is more obvious than in the bubble case, where only 10% drag reduction is achieved.

Figure(8.14) to (8.16) compare the Q contours of no slip and slip boundary con

ditions. Q = —(dui/dxj)(duj/dxi) is the second invariant for the velocity gradient.

The threshold used for Re* = 135 is 0.3.

220


Figure 8.13: Streaks at y+ = 5 of Re* = 135, b+ = 2.7.

Q d e te c t io n

0 0

Figure 8.14: Q contour without slip BC at Re* — 135.

221


Q detection

0 0

Figure 8.15: Q contour with slip BC at Re* = 135, b+ = 1.35.

Q d e te c t io n

Figure 8.16: Q contour with slip BC at Re* = 135, b+ = 2.7.

222


Q detection

0 o

Figure 8.17: Q contour with slip BC at Re* = 135, b+ = 1.35 and 800 a+ — 13.5

bubbles.

Compared to figure(8.14), with the slip boundary condition the vorticity decrease

is obvious. The larger values of b+ reduce the vorticity. So there is less vorticity for

b+ = 2.7 than for b+ = 1.35. Slip weakens the formation of near-wall vortices.

Figure(8.17) shows the Q contour with 800 a+ — 13.5 bubbles at Re+ = 135 and

b+ = 1.35. It clearly shows that even fewer vortices exist than for the no bubble case

with the same slip condition. This is consistent with drag reduction effect obtained.

Threhold for Re* = 135 is again 0.3. The region shown is the full flow domain and

dimensions are in wall units.

Figure(8.18) to (8.20) compare the Q contours with no slip and slip boundary

conditions at Re* = 380.

Compared to figure(8.18), with the slip boundary condition the vorticity decrease

is obvious. As at low Reynolds number, the larger the value of b+ gives lower vorticity

level. So there is less vorticity for b+ = 2.7 than for b 1 = 1.35.

Figure(8.21) shows the Q contour with 6348, a+ = 19, bubbles at Re+ = 380 and

b+ = 2.7. It also clearly shows that even less vortices exist than for the no bubble

223


vo rtice s , Q d ete c tio n

150

100 1000

800

600

30 0 400200

200100

Figure 8.18: Q contour without slip BC at Re* = 380.

v o rtic e s , Q d e te c tio n

1 5 0

1005 0

1000

8 0 0

6 0 0

3 0 0 4 0 0200

200100

Figure 8.19: Q contour with slip BC at Re* = 380, b+ — 1.35.

224


•Hr.'Q0W,

■ft

&C

v°n,' w<%eCfc'O/J

eA/>° ^ ,Cocy, . .

Ae,

° ^ t, •^ */• SJjp Bq

*****%

'ss/(’Of)0 f t ^

19

Go,fyn•Qht OlV/A?/-©r

reAroO/Uct'on

PrOf)tib,'teaU/.^'th,out

S/Or,

case. This is consistent with drag reduction effect obtained.

8.6 Summary

In this chapter we have explored the effects of partial-slip boundary conditions

on the level of drag reduction. Slip in the streamwise direction, characterized by the

slip length b, results in increasing levels of drag reduction as b increases. Comparing

the results for Re* = 135 and 380 shows that the level of drag reduction is correlated

with b+, the slip length scaled by wall variables of the no-slip flows.

We also note that with slip conditions in both the streamwise and spanwise direc

tions there is higher skin friction than with slip in just the streamwise direction. Min

& Kim [100] has also investigated effects of slip issue recently. Though independent,

these results are similar to each other.

The amount of drag reduction is determind by the slip length b+. At large slip

length, up to 80% drag reduction can be realized, and this may explain the drag

reduction effect observed in experiment. We also investigated the drag reduction

effect by combining micro-bubbles and slip boundary condition. The computation

results proved that their effect can be added linearly, which suggested that maybe

these two effects lead to large reduction together in experiments. Statistics have

been compared in detail, and visualization has been given. All these quantities and

visualization are consistent to drag reduction phenomena.

226


Chapter 9

Summary and Conclusion

In this thesis, I have done work mainly in two directions.

The first is high Reynolds number DNS and LES simulation, the second is to

investigate several drag reduction techniques using DNS simulation.

In the first direction:

1. Developed a channel DNS code using different parallel methods. These mod

els included splitting the domain in streamwise or both streamwise and spanwise

directions, using MPI or OpenMP and different implementations of the FFT steps.

We have compared the parallel efficiency of these models, and an optimized model

has been obtained and used to do DNS later.

2. Using the code developed, DNS results at different Reynolds numbers Re* =

180,400,600,1000 have been obtained and compared with KMM’s and MKM’s data.

They match KMM’s and MKM’s results quite well. Detailed statistics and flow

visualization have been obtained, and further systematic investigation can be done

using these databases.

227


3. A Spectral Vanishing Viscosity (SVV) scheme has been used to implement

subgrid stress (SGS) terms in LES equations. This is a relatively new approach

compared to traditional LES methods. The results at low Reynolds number are

encouraging, and at high Reynolds number the results also approach to DNS results.

4. SVV has been implemented implicitly in order to increase the stability of the

scheme.

5. Standard SVV has been modified from a constant to variable e, using standard

and dynamic Smagorinsky models. This adds more artificial viscosity in the near

wall region, while less in the center region. The simulation proved that in order to

have similar results as DNS, enough high modes should be kepted at high Reynolds

number flow.

6. The SVV method has been extended to a 2d spectral element discretization

and similar results have been obtained for channel flow LES as in the earlier code.

Besides channel flow, I have also simulated the cylinder flow with LES, using both

standard SVV and Smagorinsky SVV. The Smagorinsky SVV shows a more reason

able viscosity distribution than standard SVV methods. But due to slowness of the

code, both of them have not converged yet at Re= 10,000.

In the second direction, I have done:

1. Force Coupling method (FCM) has been added into channel code, and be par

allelized using MPI. Velocity collision model has been used to avoid overlap between

bubbles and bubble with wall. The code has been validated and verified with both

experiment data and accurate solution in Stokes flow.

2. Up to 10% drag reduction has been obtained and sustained at Re* = 135,

and detailed statistics of turbulence and bubbles have been analyzed. These results

are consistent to the experiment results at this bubble void fraction level and low

Reynolds number. Drag reduction is connected to a Reynolds stress decrease, and

228


a decrease in turbulence fluctuations. We proposed several mechanisms which may

explain this drag reduction effect. The first one is we have found in all drag reduction

cases, there existed a near wall region, where the effective bubble force is negative.

This negative force retards the fluid in the near wall region, and reduced the drag.

The second one comes from density effect, due to the existence of the bubbles, the

density of the mixture is less than the liquid. So the momentum transfer is less than

before and this makes less high momentum fluid to reach the near wall region, and

the velocity is smaller than a pure liquid. The last possible mechanism is because

bubbles can not follow small-scale velocity fluctuation of fluids, so they reduce the

fluid velocity gradient in the near wall region. This means turbulence fluctuations

decrease, and the drag reduced. The full mechanism of micro-bubble drag reduction

is still an open question.

3. At Re* — 400, we achieved similar level of drag reduction, about 10%. It is

not consistent to the experiment results, maybe due to the reason of large bubble

size and small void fraction.

4. Inspired by the bubble force profile, we investigated a new constant force

control method. It achieved 30% drag reduction at Re* = 135, and 70% at Ac*=380

and 633. It is an efficient control method, as it only consumes 1/10 energy as that

used in traveling wave control.

5. Finally, motivated by the apparantly lower levels of drag reduction found

in the numerical simulation as compared to many experiments, we also proposed

another slip boundary mechanism to explain the large drag reduction effect observed

in experiment. A slip boundary condition can arise from effects of hydrophobic

surfaces or the formation of a thin gas film on the wall. The simulation results

show that a large level of drag reduction can be achieved by applying slip boundary

condition. The effect of combining slip boundary conditions with micro-bubbles has

also been investigated, and a detailed analysis has been carried out.

229


In concluding this thesis, we will now suggest some areas of future research which

continue some of the work presented here. As what we have done, the works will be

in two directions. In the first direction of DNS and LES simulation:

1. The code needs to be optimized more in order to reach faster speed. Both

memory and speed can be optimized further and higher Reynolds number can be

tried later. The data obtained at different Reynolds numbers needs to be analyzed

in more detail to get useful information, and further advanced analysis can be done

based on these databases.

2. Many flow phenomena related to high Reynolds turbulent flow can be inves

tigated by adding corresponding models into DNS method.

3. The SVV-LES model works fine at low Reynolds number, but more tests

should be done at high Reynolds number. Right now only the a posteriori tests have

been done, in order to analysis the model in detail, some a priori tests can also be

done.

4. More LES works can be done by combining SVV-LES model with other LES

models.

5. SVV-LES should be extended to complex geometry and code with 3d spectral

elements. Other complex flow problems, such as flow past a sphere can also be

investigated.

In the second direction of drag reduction techniques,

1. Microbubble flow can be simulated at higher Reynolds number, and both

higher void fractions and smaller bubbles can be added to the turbulent flow. This

needs a larger numerical mesh, and more processors.

2. A turbulent boundary flow, simulated using 2d spectral element and Fourier

230


methods can be investigated. A turbulent boundary flow is more general than a

channel flow and closer to the context of many applications.

3. A more realistic bubble model can be used to describe the interaction between

bubbles and turbulence, including higher-order representations of the bubbles and

possibly bubble deformation. The mechanism of drag reduction should be analyzed

in more detail.

4. Further analysis can be done of the shear-sheltering mechanism seen as the

results of applying constant forcing in the streamwise direction. This phenomena

also alters the transition to turbulence. Simulations have shown that there exists a

critical force magnitude, beyond which this control mechanism will lead to drag and

turbulence increase.

5. A slip boundary condition is plausible but work on the effect of partial-slip

boundary conditions is still at a preliminary stage. Further detailed investigation of

the modification to coherent structures near the wall needs to be done as well as the

linkage between near-wall vortices and turbulence production.

231


Appendix A: The Spectral//^ Method

The main discretization concepts of the spectral///,p method are reviewed in the

context of the 1-D inviscid Burgers’ equation enhanced with the SVV second-order

operator. The objective is to introduce the hierarchical trial basis of the method

which is derived from Jacobi polynomials and to discuss some of the implementation

details.

d . . d .u2 (x ,t) (9.1)

d , . d ,u2 (x ,t). d r du(x,t)),(9 .2)

Equation (9.1) is considered, in a domain —1 < x < 1, with a Dirichlet boundary

condition and a Neumann boundary condition, i.e. = g,u'{l,t) = h. The

initial condition is a sine wave u(x, 0) = sin(nx). The residual of equation (9.2) is

dR{ u ) = f Jn

= I w n dx ox

d d (u 2 (x, t) dx, (9.3)

where u is the trial solution, the set of which is denoted by S , and w £ V is a

test function. Each test function should satisfy u;(0) = 0 and be homogeneous on a

Dirichlet boundary. Here the spaces are defined as:

S = ju [ u 6 H l ,u{0) = #} , V = j«; | w £ H 1 , w(0) = 0 j .

Integrating equation (9.3) once by parts and setting R(u) = 0, gives

ew(1 )Qeh — f ew'Qeufdx — f Jn Jn

— I w n

d d ( u2 (x ,t) dx = 0.

Introducing the notation

( \ f n dw du ja(w,u) = / eQe — — dx,Jn dx dx

232

(9.4)

<•/ \ r \d / . d (u 2 (x ,t) \m = - L w stu{r't ]+d i { - ^ ~ ) d x ,

the above equation may be rewritten as

a(w,u) = f{w) + eQ(W{\)h. (9-5)

Searching for solutions in finite subspaces, i.e. S h ( S h C S ) , V h ( V h C V) equation

(9.5) may be rewritten as

a{wh,uh) = (whJ ) + eQtwh{l)h. (9.6)

Thus, the function uh is decomposed into a known component, uflD, which satis

fies the Dirichlet boundary condition and lies in the trial space, and an unknown

term, uhH, which lies in the test space and is zero on the Dirichlet boundary

i.e. uh = uhD + uhH. By reducing an infinite-dimensional problem to an 71-

dimensional one, each member of S h and V h is represented by a set of n basis

functions (0i, fa , ..., 0n)> 0p(O) = 0, admitting all linear combinations, i.e. wh =

C i + C202 + -Cn0n- Also U h = U h D + U h H = £0n+l + Ep=l dpcj)p, 0„+l(O) = 1.

Substituting uh for u and wh for w, equation (9.6) takes the form

n

V CpGp — o, p=i

wheren

G p = ^ ( ® ( 0 p j (l) q ) d q ~ { ( f t p i . / ) — e Q e 0 p ( l ) h + f l ( 0 p , 0 n + l ) < 7 -

Since this is true for any Cp, Gp is necessarily equal to 0 the above equation may be

rewritten as

~ fipf ~ f-Qe^pi^h + j = 0. (9.7)p \Q = i J

Equivalently to the Fourier representation of Tadmor (1989), Qn may be approxi

mated by a kernel Qp, of the form

(p- p)(p- p)Qp = e , mp < p < P. (9.8)

233

In essence, the multiplication of Fourier coefficients, in the Fourier method, is trans

lated into a multiplication of modal coefficients, hence an introduction of dissipation

at the high modes. Equation (9.7), therefore, takes the form

^2 f ^2 dqtQpQqfip&q ~ M ~ tQPM l )h + eQpQn+i(f>p(j)'n+1g j = 0. (9.9)V \ g = l J

The computational domain is subsequently divided into a number of elements

k. On each element, a set of local functions is introduced that provide pth order

accuracy for the solution over the kth element. In spectral/ hp methods, these local

functions are called basis functions and are invariably polynomials.

(1 - x ) 1+Q(l + x)1 +P2rUp{x) = Ap(l -a ;)“ (l + x)0 up(x),

The modal expansions adopted in this work are Jacobi polynomials, Pp ’ {x)

[67]. Jacobi polynomials are the family of polynomial solutions to a singular Sturm-

Louiville problem and for —1 < x < 1, can be written as

ddx Lv v* dx

with up{x) = Pp'P(x), Ap — —p(a + (3 + p + 1). Jacobi polynomials have the orthog

onality property

^ ( 1 + X)&{1 - x)aP ^ { x )P ^ { x )d x = C 8 pq

with C depending on a, ft, p. Thus, P “',3(r/;) is orthogonal to all polynomials of order

less than p, when integrating with (1 + x)&{\ — x)a and the modal expansion basis

is then defined as

M O = i ( i - 0 ( i + 0 ^ 1 2 (0 ,0 < p < p (9-io)

r .

M O — — 0

M O = = p

in the standard interval Q = {£ | — 1 < £ < 1}.

Unlike the nodal basis where every basis function is an N th order polynomial, in

the modal basis there is a hierarchy of modes starting from the linear, proceeding

234

Figure 9.1: Shape of modal expansion modes for a polynomial order of polynomial

order P = 5.

with the quadratic, cubic, etc. (figure 9.1).

Returning to the Burgers’ equation, equation (9.7) may be rewritten in matrix

form as

[A]x = B (9.11)

S . = 4>pf + — (-QpQn+ltfrp&n+ig

[A] = eQpQq4>'p4>q

X = dg

where [A]pq = f Qk eQpQq4>'p(j)'qdx.

So far only one element has been considered and thus convergence depends solely

on the increase in the polynomial order. Extending the above to multiple element

domains with varying coordinate systems requires a procedure to transform the

elemental matrices [A], x, B to their equivalent sub-matrices in the global multi

element domain. The global element Qg can be mapped to any elemental (or local)

235


A =

“ Ui~i ■ • * ■ ■ mm • m m m • mm ■

i « i ■ i • ^ ______ U3

■; ^ __ Usi ■ • ■ m m m m m k

U7

Interior nodes

Interior nodes Boundarynodes

Interior nodes

I n i ,,I3 1Element

1 1Element Element

1 2 3

Figure 9.2: Schematic of direct stiffness summation of local matrices to form the

global matrix A .

domain Q; via the transformation Xe(£) which expresses the global co-ordinate x in

terms of the local co-ordinate £, i.e.

X — -Xg(£) — X e —x + ^ g •

Therefore, the global expansion basis takes the form

= 4>(Xe(Q), (9.12)

w =£=*«>§•where | | = J _1, with J the Jacobian. Once all the local matrices have been trans

formed to global sub-matrices they need to be assembled, by summing contributions

from the elemental matrices. The procedure is illustrated in figure 9.2.

Matrix [A] is banded as a result of using local basis functions, with its non-zero

entries located in the N diagonals above and below the main diagonal. Each element

is placed on the matrix, as shown in figure 9.2, with the edges of each element added

to the neighboring elements. Due to the Galerkin approximation, matrix [A] is also

symmetric and positive definite.

The main aspects of spectral/hp method have been presented through the exam

ple of the solution of a 1-D Burgers’ equation. This may be summarized as:

236


1. Determine the number of elements and the number of modes.

2. Determine the global coordinates x.

3. Determine the local to global transformation matrix.

4. Calculate the elemental matrices [A], x,B_ for each element and transform the

elemental matrices to global sub-matrices.

5. Assemble the global matrices.

6. Solve the system of equations [A]x — B_.

7. Form the solution u{x) = Y l k = Si^o""1 'tk^i(x)-

It should be mentioned that when the Dirichlet boundary conditions are used, the

rows and columns containing the corresponding Dirichlet boundary points are not

included when inverting the matrix [A], since they have been condensed out and are

included in B_.

Appendix B: Dynamic Smagorinsky Model

The conceptual basis of LES as proposed by Leonard is a convolution of the exact

turbulent velocity field u with a filter kernel K that gives the resolved scale velocity

field u,

An evolution equation for u is obtained by convolving the incompressible Navier-

Stokes equations with the spatial filter kernel on the grid scale,

(9.13)

where A is the filter length, which is usually be taken to be the grid size.

eft(9.14)

237


where P = %>/p. This first filtering operation is taken as implicit in the formulation

(i.e., not explicitly carried out). Here, by assumption, filtering and differentiation

commute, i.e., du/dx = du/dx. although in general, they do not, and an additional

commutation error arises, in addition to other modelling errors. As in cnventional

turbulence modelling, the nonlinear terms are have to be modelled, because the

filtered dyad uu cannot be expressed in terms of the known resolved components u.

and an extra stress r is introduced to close the equations, such that t = uu — uu.

The momentum equations then become

— u + V • mZ = —V P + vW2u — V • t (9.15)

The turbulence modelling task is to estimate the subgrid-scale stress r from the

resolved velocity field u.

The dynamics methods seeks to exploit the expectation that in the inertial range

of the turbulence energy spectrum, the turbulence physics are statistically self-similar

when viewed at different length scales, and specificlly at the grid length scale A,

representative of the computational mesh, and at a larger test filter length scale A.

If the same turbulence model can be applied to the portions of the energyspectrum

that reside at sizes larger than these two length scales, then the model coefficients

should be the same in each case. The dynamic procedure provides the framework for

working back from the the grid- and test-filtered velocity fields to obtain the model

coefficient(s), which are then applied in estimation of the SGS stress t .

Applying a second filter with an associated size of A to the filtered Navier-Stokes

equation (9.15) leads to a similar stress tensor on the test-filter level T

— u + V • uu = -V P + uV2u — V ■ T (9.16)

with

T = ml — uu (9-17)

Assuming similar physics underly both stresses r and T, they can be estimated

238


with exactly the same model. Filtering of r gives f = uu — uu in which the first

term on the right matches the one in(9.17) and can therefore be eliminated, leading

to the relation known as ’Germano’s identity’

£ = T — u = uu — uu (9.18)

which can be used with any stand-alone SGS model - all terms, except any model

parameters, can be evaluated.

For simplicity and robustness, we employ the Smagorinsky (mixing length) eddy

viscosity model for the deviatoric components of the SGS stress r, so that

r - i*r(r)7 = - 2 vtS = -2(cs A)2\S\S, (9.19)

where S is the traceless, symmetric, resolved-scale rate-of-strain tensor

5 = i [ V u + (Vu)r ], (9.20)

cs is the Smagorinsky constant and \S\ = (25 : 5)1/2. The isotropic component of

SGS stress is notionally combined with the filtered pressure, to obtain a modified

pressure II = P + gfr(r), which is then used in place of P in the evolution equation

(9.15).

Using the same model for T produces

T - ^ t r ( T ) I = -2(cs A)2\S\S, (9.21)

and introducing both modelled quantities, (9.19) and (9.21), into the deviatoric

components of (9.18) gives

£ - \ tr (£ ) I = -2(cs A)2M, (9.22)o

239


with

M = (A/A)2|5 |5 — \S\S, (9.23)

where typically A/A = 2 is assumed. In order to obtain a scalar dynamic estimate,

the tensor equation (9.22) is reducted by double contraction, in which process the

isotropic component of £ is eliminated, since M is deviatoric

£ : M = -2 (cs A)2M : M, (9.24)

from which the dynamic estimate can be extracted

f e ( l ' 1)A)2 = 4 s 7 ^ (9'25)

This procedure evaluates a local and time-dependent value of cs A and is updated

every timestep. note that the product (cgA) can be treated as a mixing length Is,

without explicitly specifying the length scale on the grid level - this is advantageous

in the current context as it enables us to bypass the need to define A.

240


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