NONLINEAR DYNAMICS AND INSTABILITIES OF VISCOELASTIC FLUID ... · NONLINEAR DYNAMICS AND...

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NONLINEAR DYNAMICS AND INSTABILITIES OF VISCOELASTIC FLUID FLOWS by Li Xi A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Chemical Engineering) at the UNIVERSITY OF WISCONSIN – MADISON 2009

Transcript of NONLINEAR DYNAMICS AND INSTABILITIES OF VISCOELASTIC FLUID ... · NONLINEAR DYNAMICS AND...

Page 1: NONLINEAR DYNAMICS AND INSTABILITIES OF VISCOELASTIC FLUID ... · NONLINEAR DYNAMICS AND INSTABILITIES OF VISCOELASTIC FLUID FLOWS by Li Xi A dissertation submitted in partial fulfillment

NONLINEAR DYNAMICS AND

INSTABILITIES OF VISCOELASTIC FLUID

FLOWS

by

Li Xi

A dissertation submitted in partial

fulfillment of the requirements for the degree of

Doctor of Philosophy

(Chemical Engineering)

at the

UNIVERSITY OF WISCONSIN – MADISON

2009

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Acknowledgments

Above all, I would like to thank my advisor Professor Michael D. Graham, for bringing

me into this area of research and exposing me to all these opportunities; and most

importantly, for his great passion of teaching, which has helped me grow in the past

five years in terms of both research capability and academic scholarship.

I am also indebted to Professor Fabian Waleffe, for many inspiring discussions that

benefited my research on viscoelastic turbulence. Same for Doctor John F. Gibson,

who generously shared his ChannelFlow code for Newtonian flows, based on which

my viscoelastic code was developed. He also offered some very helpful advice on my

numerical algorithm.

I have enjoyed working with many former and current members of the Gra-

ham group, including: Samartha G. Anekal, Yeng-Long Chen, Juan P. Hernandez-

Ortiz, Aslin Izmitli, Pieter J. A. Janssen, Rajesh Khare, Wei Li, Mauricio Lopez,

Hongbo Ma, Pratik Pranay, Christopher G. Stoltz, Patrick T. Underhill and Yu Zhang.

Wei Li, in particular, offered many discussions in my first two year that helped me

greatly in understanding some quite obscure topics.

Finally, I am grateful to my family for the endless support they provided me

during all these years.

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Research projects presented in this dissertation are financially supported by the

National Science Foundation, and the Petroleum Research Fund, administered by the

American Chemical Society.

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Abstract

This dissertation focuses on the fluid dynamics of dilute polymer solutions, with an

emphasis on nonlinear flow behaviors and instabilities in different parameter regimes.

Even at a very low concentration, flexible polymer solutes can introduce strong vis-

coelasticity into the fluid, causing flow instability at very low Reynolds number. At

high Reynolds number, the coupling between inertial and elastic effects bring forth fur-

ther complex dynamics. We study two representative problems in these two regimes,

respectively: elastic instabilities involving stagnation points at low Reynolds number,

and dynamics of viscoelastic turbulent flows at relatively high Reynolds number.

In the low Reynolds number case, interior stagnation point flows of viscoelastic

liquids arise in a wide variety of applications including extensional viscometry, poly-

mer processing and microfluidics. Experimentally, these flows have long been known

to exhibit instabilities, but the mechanisms underlying them have not previously

been elucidated. We computationally demonstrate the existence of a supercritical

oscillatory instability of low-Reynolds number viscoelastic flow in a two-dimensional

cross-slot geometry. The fluctuations are closely associated with the “birefringent

strand” of highly stretched polymer chains associated with the outflow from the stag-

nation point at high Weissenberg number. Additionally, we describe the mechanism

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of instability, which arises from the coupling of flow with extensional stresses and

their steep gradients in the stagnation point region.

In turbulent flows, the observation that a minute amount of flexible polymers re-

duces turbulent friction drag has been long established. However, many aspects of

the drag reduction phenomenon are not well-understood; in particular, the existence

of the maximum drag reduction (MDR) asymptote, a universal upper limit of drag

reduction, remains a mystery. Our study focuses on the drag reduction phenomenon

in the plane Poiseuille geometry in a parameter regime close to the laminar-turbulent

transition. By minimizing the size of the periodic simulation box to the lower limit

for which turbulence persists, the essential self-sustaining turbulent motions are iso-

lated. In these “minimal flow unit” (MFU) solutions, consistent with previous exper-

iments, a series of qualitatively different stages are observed, including one showing

the universality of MDR: i.e. the mean flow is universal with respect to changing

polymer-related parameters. Before this stage, an additional transition exists be-

tween a relatively low degree (LDR) and a high degree (HDR) of drag reduction.

This transition occurs at about 13-15% of drag reduction, and is characterized by a

sudden increase in the minimal box size of sustaining turbulence, as well as many

qualitative changes in flow statistics. The observation of LDR–HDR transition at

less than 15% drag reduction shows for the first time that it is a qualitative transi-

tion instead of a quantitative effect of the amount of drag reduction. Spatiotemporal

flow structures change substantially upon this transition, suggesting that two dis-

tinct types of self-sustaining turbulence dynamics are observed. In LDR, similar as

Newtonian turbulence, the self-sustaining process involves one low-speed streak and

its surrounding streamwise vortices; after the LDR–HDR transition, multiple streaks

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are present in the self-sustaining structure and complex intermittent behaviors of the

streaks are observed. This multistage scenario of LDR–HDR–MDR recovers all key

transitions commonly observed and studied at much higher Reynolds numbers.

The asymptotic upper-limit of drag reduction observed in MFU is much lower

than the experimentally-found MDR; however, an important progress has been made

toward the understanding of the latter. In all stages of transition, even in the Newto-

nian limit, we find intervals of “hibernating” turbulence that display many features

of the experimental MDR asymptote in polymer solutions: weak streamwise vortices,

nearly nonexistent streamwise variations and a mean velocity gradient that quan-

titatively matches experiments. As viscoelasticity increases, the frequency of these

intervals also increases, while the intervals themselves are unchanged, leading to flows

that increasingly resemble MDR. This observation would inspire future research that

might finally solve the puzzle of MDR.

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Contents

Acknowledgments i

Abstract iii

List of Figures ix

List of Tables xxiii

1 Overview: the scope of study 1

Part I Dynamics at low Re: oscillatory instability in vis-

coelastic cross-slot flow 7

2 Introduction: elastic instabilities and viscoelastic stagnation-point

flows 8

3 Cross-slot geometry, governing equations and numerical methods 15

4 Results: viscoelastic cross-slot flow and its oscillatory instability 20

4.1 Steady states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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4.2 Periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Instability mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Conclusions of Part I 48

6 Future work: nonlinear dynamics of viscoelastic fluid flows in com-

plex geometries 50

Part II Dynamics at high Re: viscoelastic turbulent flows

and drag reduction 56

7 Introduction: viscoelastic turbulent flows and polymer drag reduc-

tion 57

7.1 Fundamentals of polymer drag reduction . . . . . . . . . . . . . . . . 57

7.2 Previous direct numerical simulation (DNS) studies . . . . . . . . . . 61

7.3 Traveling waves and the nonlinear dynamics perspective of turbulence 64

7.4 Multistage transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.5 About this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8 DNS formulation and numerical method 77

8.1 Flow geometry and governing equations . . . . . . . . . . . . . . . . . 77

8.2 Numerical procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9 Methodology: minimal flow units (MFU) 83

10 Results: observations during multistage transitions 88

10.1 Overview of the multistage-transition scenario . . . . . . . . . . . . . 88

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10.2 Flow statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

10.3 Polymer conformation statistics . . . . . . . . . . . . . . . . . . . . . 109

10.4 Spatio-temporal structures . . . . . . . . . . . . . . . . . . . . . . . . 113

11 Toward an understanding of the dynamics: active and hibernating

turbulence 126

11.1 Intermittent dynamics in MFU . . . . . . . . . . . . . . . . . . . . . 126

11.2 Generalization to full-size turbulent flows: a preliminary investigation 143

12 Conclusions of Part II 150

13 Future work: dynamics of viscoelastic turbulence and drag reduction

in turbulent flows 154

13.1 Hibernation statistics: effect on the LDR–HDR transition . . . . . . . 155

13.2 A hypothetical dynamical-scenario . . . . . . . . . . . . . . . . . . . 156

13.3 Development of methodology . . . . . . . . . . . . . . . . . . . . . . 162

13.4 Further extensions: other drag-reduced turbulent flow systems . . . . 167

A Numerical algorithm for the direct numerical simulation of viscoelas-

tic channel flow 172

Bibliography 185

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List of Figures

2.1 Symmetry-breaking instability in viscoelastic cross-slot flow (Arratia

et al. 2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

(a) Dye convection pattern. . . . . . . . . . . . . . . . . . . . . . . 10

(b) Contours of velocity magnitude (colors) and streamline (dark

lines) measured by particle image velocimetry (PIV) . . . . . . 10

3.1 Schematic of the cross-slot flow geometry. . . . . . . . . . . . . . . . . 16

4.1 Contour plots of steady state solution: Wi = 0.2 (only the central part

of the flow domain is shown). . . . . . . . . . . . . . . . . . . . . . . 22

(a) ‖u‖ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

(b) ∂ux/∂x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

(c) tr(α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Contour plots of steady state solution: Wi = 50 (only the central part

of the flow domain is shown). . . . . . . . . . . . . . . . . . . . . . . 23

(a) ‖u‖ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

(b) ∂ux/∂x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

(c) tr(α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Profiles of tr(α) along y = 0. . . . . . . . . . . . . . . . . . . . . . . . 24

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4.4 Profiles of tr(α) along x = 0 in the region very near the stagnation

point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.5 Effect of Wi on the size of the birefringent strand (tr(α) > 300 is

considered as the observable birefringence region). . . . . . . . . . . . 25

(a) Birefringent strand width W . . . . . . . . . . . . . . . . . . . . 25

(b) Birefringent strand length L. . . . . . . . . . . . . . . . . . . . 25

4.6 Profiles of ux along y = 0. . . . . . . . . . . . . . . . . . . . . . . . . 26

4.7 Average extension rate (∂ux/∂x)avg (averages taken in the domain

−0.1 < x < 0.1,−0.1 < y < 0.1). . . . . . . . . . . . . . . . . . . . . . 27

4.8 Evolution of the birefringence strand width W after a small initial

perturbation on the steady state; inset: enlarged view of 2500 6 t 6 3200. 30

4.9 Two dimensional projection of the dynamic trajectory from the steady

state to the periodic orbit at Wi = 66: ux at (0.5, 0) v.s. W . . . . . . 30

4.10 Left-hand axis: root-mean-square deviations of the birefringent strand

width W at periodic orbits, normalized by steady state values; right-

hand axis: oscillation periods. . . . . . . . . . . . . . . . . . . . . . . 31

4.11 Perturbation of the x-component of velocity, u′x with respect to steady

state at the periodic orbit: Wi = 66. The region shown is 0 < x < 1,

−1 < y < 0; the stagnation point is at the top-left corner. (To be

continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

(a) t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

(b) t = 1.68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

(c) t = 3.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

(d) t = 5.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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4.11 (Continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

(e) t = 6.71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

(f) t = 8.38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

(g) t = 10.06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

(h) t = 11.74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.12 Perturbation of the y-component of velocity, u′y with respect to steady

state at the periodic orbit: Wi = 66. The region shown is 0 < x < 1,

−1 < y < 0; the stagnation point is at the top-left corner. (To be

continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

(a) t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

(b) t = 1.68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

(c) t = 3.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

(d) t = 5.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.12 (Continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

(e) t = 6.71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

(f) t = 8.38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

(g) t = 10.06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

(h) t = 11.74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.13 Perturbation of the xx-component of polymer conformation tensor, α′xx

with respect to steady state at the periodic orbit: Wi = 66. The region

shown is 0 < x < 1, −1 < y < 0; the stagnation point is at the top-left

corner. The edge of the steady state birefringent strand is the line

y ≈ −0.05. (To be continued). . . . . . . . . . . . . . . . . . . . . . . 38

(a) t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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(b) t = 1.68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

(c) t = 3.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

(d) t = 5.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.13 (Continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

(e) t = 6.71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

(f) t = 8.38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

(g) t = 10.06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

(h) t = 11.74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.14 Time-dependent oscillations at (0,−0.05). Top view: perturbations of

variables normalized by steady-state quantities; bottom view: magni-

tudes of terms on RHS of Equation (4.4). . . . . . . . . . . . . . . . . 43

4.15 Schematic of instability mechanism (view of the lower half geometry).

Thick arrows represent net forces exerted by polymer molecules (dumb-

bells) on the fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

(a) Thinning process of the birefringent strand. . . . . . . . . . . . 45

(b) Re-thickening process of the birefringent strand. . . . . . . . . 45

6.1 The microfluidic flip-flop device (Groisman et al. 2003). . . . . . . . . 52

(a) Overall Geometry. The auxiliary inlets (comp. 1 and comp. 2)

are for flow-rate measurement purpose. . . . . . . . . . . . . . . 52

(b) Blowup near the intersection during the instability. Only fluids

from one of the two inlets are dyed. . . . . . . . . . . . . . . . . 52

7.1 Schamatic of the Prandtl-von Karman plot. Thin vertical lines mark

the transition points on the typical experimental path shown as a thick

solid line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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7.2 Experimental data of maximum drag reduction (MDR) in pipe flow,

from different polymer solution systems and pipe sizes, plotted in the

Prandtl-von Karman coordinates (Virk 1971, 1975). It can be shown

that 1/√f = U+

avg/√

2, Re√f =

√2Reτ (f in this plot is the friction

factor, which is denoted as Cf in this dissertation; Re in this plot is the

Reynolds number based on average velocity: Reavg ≡ ρUavgD/η, D is

the pipe diameter). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.3 Newtonian ECS solution at Re = 977 in plane Poiseuille flow; sym-

metric copies at both walls are shown. Slices show coutours of stream-

wise velocity, dark color for low velocity; the isosurface has a constant

streamwise vortex strength Q2D = 0.008, definition of Q2D is given

in Section 10.4. This plot is published by Li & Graham (2007); the

solution is originally discovered by Waleffe (2003). . . . . . . . . . . . 65

7.4 Dynamics of turbulence in the solution state space in a plan Couette

flow (Gibson et al. 2008). . . . . . . . . . . . . . . . . . . . . . . . . . 67

(a) Dynamical trajectory of a turbulent transient in a MFU visu-

alized in the state space using coordinates proposed by Gibson

et al. (2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

(b) Same trajectory (dotted line) visualized in the context of TW

solutions (solid dots, except uLM, which is the laminar state) and

their unstable manifolds (solid lines). . . . . . . . . . . . . . . . 67

7.5 The Virk (1975) universal mean velocity profile for MDR in inner

scales: U+mean = 11.7 ln y+ − 17.0. . . . . . . . . . . . . . . . . . . . . . 70

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8.1 Schematic of the plane Poiseuille flow geometry: the box highlighted

in the center with dark-colored walls is the actual simulation box, sur-

rounded by its periodic images. . . . . . . . . . . . . . . . . . . . . . 78

8.2 Schematic of the finitely-extensible nonlinear elastic (FENE) dumbbell

model for polymer molecules. . . . . . . . . . . . . . . . . . . . . . . 78

9.1 Summary of simulation results: “Turbulent” indicates that at least

one simulation run gives sustained turbulence within the given time

interval (Newtonian and β = 0.97, b = 5000). . . . . . . . . . . . . . . 85

10.1 Variations of the average streamwise velocity with Wi at different β

and b values (average taken in time and all three spatial dimensions);

the corresponding DR% is shown on the right ordinate. Solid sym-

bols represent points in the asym-DR stage (defined in the text); the

horizontal dashed line is the average of all asym-DR points. . . . . . . 89

10.2 Mean velocity profiles (Newtonian and β = 0.97, b = 5000). . . . . . . 93

10.3 Spanwise box sizes used in this study for various parameters. Solid

symbols represent points in the asym-DR stage. . . . . . . . . . . . . 94

10.4 Variations of spanwise box size at different DR%. Solid symbols rep-

resent points in the asym-DR stage. . . . . . . . . . . . . . . . . . . . 96

10.5 Mean velocity profiles of 15 different asym-DR states (Wi: 27 ∼ 30

for β = 0.97, b = 5000, Wi: 32 ∼ 36 for β = 0.99, b = 10000 and

Wi: 40 ∼ 50 for β = 0.99, b = 5000). . . . . . . . . . . . . . . . . . . . 98

10.6 Deviations in mean velocity profile gradient from that of Newtonian

turbulence (β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23;

asym-DR: Wi = 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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10.7 Magnitude of mean velocity profile gradient at y+ = 40. Solid symbols

represent points in the asym-DR stage. . . . . . . . . . . . . . . . . . 100

10.8 Profiles of the Reynolds shear stress (Newtonian and β = 0.97, b =

5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR: Wi = 29. . . . . 101

10.9 Deviations in Reynolds shear stress profiles from that of Newtonian

turbulence (β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23;

asym-DR: Wi = 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

10.10Magnitude of Reynolds shear stress at y+ = 40. Solid symbols repre-

sent points in the asym-DR stage. . . . . . . . . . . . . . . . . . . . . 102

10.11Profiles of root-mean-square streamwise and wall-normal velocity fluc-

tuations (Newtonian and β = 0.97, b = 5000). LDR: Wi = 17, 19;

HDR: Wi = 23; asym-DR: Wi = 29. . . . . . . . . . . . . . . . . . . . 104

10.12Profiles of root-mean-square wall-normal velocity fluctuations (Newto-

nian and β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23;

asym-DR: Wi = 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

10.13Profiles of root-mean-square spanwise velocity fluctuations (Newtonian

and β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-

DR: Wi = 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

10.14Profiles of root-mean-square velocity fluctuations and Reynolds shear

stress at 15 different asym-DR states (Wi: 27 ∼ 30 for β = 0.97,

b = 5000, Wi: 32 ∼ 36 for β = 0.99, b = 10000 and Wi: 40 ∼ 50 for

β = 0.99, b = 5000). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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10.15Normalized profiles of the trace of the polymer conformation tensor

(β = 0.97, b = 5000). Pre-onset: Wi = 16; LDR: Wi = 17, 19; HDR:

Wi = 23; asym-DR: Wi = 27, 29. . . . . . . . . . . . . . . . . . . . . 109

10.16Averaged trace of the polymer conformation tensor (average taken in

time and all three spatial dimensions). Solid symbols represent points

in the asym-DR stage. . . . . . . . . . . . . . . . . . . . . . . . . . . 110

10.17Position of the maximum in the tr(α) profile. Solid symbols represent

points in the asym-DR stage. . . . . . . . . . . . . . . . . . . . . . . 112

(a) Dependence on DR% . . . . . . . . . . . . . . . . . . . . . . . . 112

(b) Dependence on Wi . . . . . . . . . . . . . . . . . . . . . . . . . 112

10.18Dynamics of the self-sustaining turbulent structures in a selected New-

tonian simulation (Re = 3600, L+x = 360, L+

z = 140). Top panel:

spatial-temporal patterns of the wall shear rate (∂vx/∂y taken at x = 0;

two periodic images are shown for each case. Note: the mean value is 2

owning to the fixed pressure gradient constraint.); bottom panel: (left

ordinate and thick line) spatially-averaged velocity and (right ordinate

and thin line) average wall shear rate (average taken in the z-direction

at x = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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10.19Dynamics of the self-sustaining turbulent structures in a selected LDR

simulation (Re = 3600, Wi = 19, β = 0.97, b = 5000, L+x = 360,

L+z = 150). Top panel: spatial-temporal patterns of the wall shear

rate (∂vx/∂y taken at x = 0; two periodic images are shown for each

case. Note: the mean value is 2 owning to the fixed pressure gradient

constraint.); bottom panel: (left ordinate and thick line) spatially-

averaged velocity and (right ordinate and thin line) average wall shear

rate (average taken in the z-direction at x = 0). . . . . . . . . . . . . 116

10.20Dynamics of the self-sustaining turbulent structures in a selected HDR

simulation (Re = 3600, Wi = 23, β = 0.97, b = 5000, L+x = 360,

L+z = 180). Top panel: spatial-temporal patterns of the wall shear

rate (∂vx/∂y taken at x = 0; two periodic images are shown for each

case. Note: the mean value is 2 owning to the fixed pressure gradient

constraint.); bottom panel: (left ordinate and thick line) spatially-

averaged velocity and (right ordinate and thin line) average wall shear

rate (average taken in the z-direction at x = 0). . . . . . . . . . . . . 117

10.21Dynamics of the self-sustaining turbulent structures in a selected asym-

DR simulation (Re = 3600, Wi = 29, β = 0.97, b = 5000, L+x = 360,

L+z = 250). Top panel: spatial-temporal patterns of the wall shear

rate (∂vx/∂y taken at x = 0; two periodic images are shown for each

case. Note: the mean value is 2 owning to the fixed pressure gradient

constraint.); bottom panel: (left ordinate and thick line) spatially-

averaged velocity and (right ordinate and thin line) average wall shear

rate (average taken in the z-direction at x = 0). . . . . . . . . . . . . 118

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10.22Typical snapshots of the flow field (Re = 3600, β = 0.97, b = 5000,

L+x = 360). (Reg) denotes snapshots chosen from “regular” turbulence,

and (LS) denotes snapshots of “low-shear” events. Translucent sheets

are the isosurfaces of vx = 0.6vx,max; opaque tubes are the isosurfaces of

Q2D = 0.3Q2D,max. The values of vx and Q2D for each plot is shown in

its caption. Note that (LS) states typically have much lower Q2D values

than (Reg) states. The bottom wall of each snapshot corresponds to

the wall shear rate patterns shown in Figures 10.18, 10.19, 10.20 and

10.21 at corresponding time. (To be continued). . . . . . . . . . . . . 119

(a) Newtonian (Reg), L+z = 140;

t = 8500, vx = 0.25, Q2D = 0.025. . . . . . . . . . . . . . . . . . 119

(b) Newtonian (LS), L+z = 140;

t = 4600, vx = 0.27, Q2D = 0.012. . . . . . . . . . . . . . . . . . 119

(c) LDR (Reg): Wi = 19, L+z = 150;

t = 5900, vx = 0.26, Q2D = 0.024. . . . . . . . . . . . . . . . . . 119

(d) LDR (LS): Wi = 19, L+z = 150;

t = 8200, vx = 0.29, Q2D = 0.0079. . . . . . . . . . . . . . . . . 119

10.22(Continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

(e) HDR (Reg): Wi = 23, L+z = 180;

t = 7700, vx = 0.31, Q2D = 0.026. . . . . . . . . . . . . . . . . . 120

(f) HDR (LS): Wi = 23, L+z = 180;

t = 7300, vx = 0.31, Q2D = 0.0089. . . . . . . . . . . . . . . . . 120

(g) asym-DR (Reg): Wi = 29, L+z = 250;

t = 8500, vx = 0.27, Q2D = 0.018. . . . . . . . . . . . . . . . . . 120

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(h) asym-DR (LS): Wi = 29, L+z = 250;

t = 8900, vx = 0.31, Q2D = 0.0050. . . . . . . . . . . . . . . . . 120

11.1 Mean shear rates at the walls (“b”–bottom, “t”–top) and bulk velocity

Ubulk as functions of time for typical segments of a Newtonian sim-

ulation run (Re = 3600, L+x = 360, L+

z = 140). Rectangular signals

in the middle panel indicate the hibernating periods at the wall of the

corresponding side, identified with the criterion explained in the text.

Dashed lines show the line 〈∂vx/∂y〉 = 1.80. Time average of 〈∂vx/∂y〉is 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

11.2 Mean shear rates at the walls (“b”–bottom, “t”–top) and bulk velocity

Ubulk as functions of time for typical segments of a high-Wi simulation

run (Re = 3600, Wi = 29, β = 0.97, b = 5000, L+x = 360, L+

z = 250).

Rectangular signals in the middle panel indicate the hibernating peri-

ods at the wall of the corresponding side, identified with the criterion

explained in the text. Dashed lines show the line 〈∂vx/∂y〉 = 1.80.

Time average of 〈∂vx/∂y〉 is 2. . . . . . . . . . . . . . . . . . . . . . . 129

11.3 Level of drag reduction and spanwise box size as functions of Wi (New-

tonian and β = 0.97, b = 5000). . . . . . . . . . . . . . . . . . . . . . 130

11.4 Time scales (left ordinate) and fraction of time spent in hibernation

(right ordinate) as functions of Wi (Newtonian and β = 0.97, b =

5000): TA is the mean duration of active periods; TH is the mean

duration of hibernating periods; FH is the fraction of time spent in

hibernation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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11.5 A hibernation event (200 6 t 6 600 in Figure 11.2). Thick black lines

are mean wall shear rates and bulk velocity Ubulk at Wi = 29. Thin

colored lines are from Newtonian simulations started at the correspond-

ing colored dots, using velocity fields from the Wi = 29 simulation as

initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

11.6 Instantaneous mean velocity profiles of selected instants before, dur-

ing and after a typical hibernating period (marked with grid-lines in

Figure 11.5). Profiles for the bottom half of the channel are shown; su-

perscript “*” represents variables nondimensionalized with inner scales

based on instantaneous mean shear-stress at the wall of the correspond-

ing side. Black lines show important asymptotes: “viscous sublayer”,

U∗mean = y∗; “Newtonian log-law”, U∗mean = 2.44 ln y∗+5.2 (Pope 2000);

“Virk MDR”, U∗mean = 11.7 ln y∗ − 17.0 (Virk 1975). . . . . . . . . . . 135

11.7 Comparison between hibernation in Newtonian and high-Wi viscoelas-

tic flows (the Newtonian simulation is the one starting from t = 260 in

Figure 11.5). Instantaneous mean velocity profiles for instants in hi-

bernation (c) and after turbulence is reactivated (e) are show (marked

with grid-lines in Figure 11.5). Profiles for the bottom half of the

channel are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

11.8 Flow structures at selected instants before, during and after a typi-

cal hibernating period (marked with grid-lines in Figure 11.5). Green

sheets are isosurfaces vx = 0.3, pleats correspond to low-speed streaks;

red tubes are isosurfaces of Q2D = 0.02, Q2D is defined in Section 10.4.

Only the bottom half of the channel is shown. . . . . . . . . . . . . . 137

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11.9 Instantaneous profiles of αxx (streamwise polymer deformation) for in-

stants marked in Figure 11.5. Profiles for the bottom half of the channel

are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

11.10Instantaneous profiles of αyy (wall-normal polymer deformation) for

instants marked in Figure 11.5. Profiles for the bottom half of the

channel are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

11.11Instantaneous profiles of αzz (spanwise polymer deformation) for in-

stants marked in Figure 11.5. Profiles for the bottom half of the chan-

nel are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

11.12Instantaneous profiles of Reynolds shear stresss for instants marked in

Figure 11.5. Profiles for the bottom half of the channel are shown. . . 140

11.13Flow structures of a typical snapshot in a full-size Newtonian simula-

tion (Re = 3600, L+x = 4000, L+

z = 800). Green sheet is the isosurface

of vx = 0.3; red tubes are isosurfaces of Q2D = 0.02. Only the bottom

half of the channel is shown. . . . . . . . . . . . . . . . . . . . . . . . 144

11.14Flow structures of a typical snapshot in a full-size viscoelastic simula-

tion near MDR (Re = 3600, Wi = 80, β = 0.97, b = 5000, L+x = 4000,

L+z = 800). Green sheet is the isosurface of vx = 0.3; red tubes are

isosurfaces of Q2D = 0.02. Only the bottom half of the channel is shown.145

11.15Contours of streamwise velocity in a plane 25 wall units above the

bottom wall for the Newtonian snapshot shown in Figure 11.13 . . . . 146

11.16Contours of streamwise velocity in a plane 25 wall units above the bot-

tom wall for the viscoelastic (near MDR) snapshot shown in Figure 11.14147

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13.1 Schematic of near-transition turbulent dynamics: intermittent excur-

sions toward certain saddle points and the laminar-turbulence edge

structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

13.2 Schematic of the edge-tracking method based on repeated bisection (Skufca

et al. 2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

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List of Tables

4.1 Terms on the right-hand side of Equation (4.4). . . . . . . . . . . . . 42

A.1 Numerical coefficients for the Adams-Bashforth/backward-differentiation

temporal discretization scheme with different orders-of-accuracy (Peyret

2002). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

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1

Chapter 1

Overview: the scope of study

Viscoelastic fluids, materials that exhibit both viscous and elastic characteristics upon

deformation, have been an interesting object of study to researchers in various areas.

The most common type of fluids with viscoelasticity is polymeric liquids: melts and

solutions of polymers. Owning to the elasticity of polymer molecules, as well as the

nontrivial polymer-solvent and polymer-polymer interactions, behaviors of these liq-

uids under certain flow conditions and deformations can be drastically different from

those of Newtonian fluids; some classical examples are discussed in Bird, Armstrong

& Hassager (1987). Another example of viscoelastic fluids is surfactant solutions with

worm-like micelles formed (Larson 1999, Walker 2001). Similar as polymer molecules,

these semi-flexible chain-like micelles can be deformed and reoriented by the flow; in

addition, the capability of dynamical break-up and reformation of the micellar struc-

ture, and the possibility of forming super-molecular aggregates under flow, make the

dynamics of surfactant solutions even more complicated (Cates & Candau 1990, Liu

& Pine 1996, Zakin et al. 1998, Butler 1999).

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As suggested by the title of this dissertation, our study resides within the scope

of fluid dynamics of viscoelastic fluids. In particular, we are interested in nonlinear

behaviors of flowing viscoelastic fluids in different parameter regimes. For Newtonian

fluids, nonlinear flow behaviors are driven purely by inertia, the impact of which

can be measured with a single dimensionless parameter, the Reynolds number Re

(Re ≡ ρUl/η, here ρ and η represent the density and viscosity of the fluid, while

U and l are the characteristic flow velocity and length scale of the geometry) (Bird

et al. 2002). Significant nonlinear behaviors are only expected for Re > O(1); with

Re high enough, the flow eventually becomes fully turbulent. For viscoelastic fluids,

inertia is no longer the sole source of nonlinearity. Microscopic structures of the fluid,

including (take polymeric liquids for instance) individual polymer molecules as well

as high-order structures (e.g. clusters and networks) of polymers (in concentrated

solutions and melts), interact in a nontrivial way with the macroscopic momentum

and mass balances. Therefore nonlinearity can be significant even at very low Re.

Among many types of viscoelastic fluids mentioned above, we limit our attention

to dilute solutions of flexible linear polymer chains in this dissertation. By “dilute

solutions” we refer to those with concentration much lower than the overlap con-

centration (Rubinstein & Colby 2003): in these systems, polymer molecules are so

far apart from one another that they do not “feel” the existence of others; therefore

interactions between polymer molecules are negligible, and polymer dynamics under

flow is relatively simple. Individual polymer molecules can be oriented and stretched

by the flow; once the flow-induced strain is released, they have a tendency of relax-

ing toward the coiled configuration, which they prefer at equilibrium. During these

processes, when polymer molecules change configuration, they apply a drag force on

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3

the solvent around; in terms of the macroscopic momentum balance, this polymer

feedback to the flow is described as an additional contribution to the stress, which

is well-captured by the FENE-P constitutive equation (Bird, Curtis, Armstrong &

Hassager 1987) for dilute solutions. These interactions between the macroscopic flow

and microscopic polymer dynamics introduce additional nonlinearity into the system.

Another dimensionless parameter, the Weissenberg number Wi ≡ λγ, is introduced,

which by definition is the time scale of the relaxation of polymer λ nondimensionlized

by the inverse of the characteristic strain rate γ of the flow. When Wi > O(1), poly-

mer relaxation lags significantly behind changes in fluid deformation, and the fluid

has a stronger “memory” effect, or elasticity, which could result in various nonlinear

behaviors and instabilities. Instabilities can occur even at extremely low Re, where

inertial effects are negligible; these types of instabilities driven completely by elastic-

ity are very often mentioned as “purely-elastic” or “inertia-less” instabilities (Larson

1992, Shaqfeh 1996, Groisman & Steinberg 2000, Larson 2000). At high Re where

inertia by itself can trigger flow instabilities, the coupling between elastic and inertial

effects can cause intriguing nonlinear behaviors inaccessible with mere contribution

from either of them (Rodd et al. 2005). In particular, turbulent flows of viscoelas-

tic fluids show qualitatively different dynamics at high Wi from that in Newtonian

turbulence (Xi & Graham 2009c).

There are of course numerous problems of interest in the whole parameter space

of Re and Wi. In this dissertation, we select two representative examples for case

study: one at the low Re limit and one at the high Re limit. In the low Re regime,

we choose a cross-slot geometry and study the instability mechanism involving stag-

nation points (Xi & Graham 2009b). Among different types of elastic instabilities

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4

reported in various experimental conditions, the best-understood are those so-called

“hoop-stress” instabilities, which occurs in viscoelastic fluid flows with curved stream-

lines (Larson 1992, Pakdel & McKinley 1996, Shaqfeh 1996, Graham 1998). Mean-

while, instabilities are reported in flow geometries involving stagnation points as well,

including those in rheometric flows (Chow et al. 1988, Muller et al. 1988) and mi-

crofluidics (Arratia et al. 2006). However, understanding of these instabilities was

very limited. In the cross-slot geometry we study, a stagnation point is created in

the center; experimental research reported symmetry-breaking and oscillatory insta-

bilities in flows of polymer solutions (Arratia et al. 2006). Our goal is to understand

these instabilities via numerical simulation, in the hope that the resulting mechanism

can be applicable to a wider range of instabilities involving stagnation points. On the

other hand, in terms of computational methodology, these problems are extremely

challenging: around a stagnation point there is typically a strong extensional flow

field, in which polymer molecules are highly stretched; fully resolving the stress field

without losing numerical stability is a difficult task. This makes the stagnation point

flow an excellent test problem for numerical methods. We are interested in developing

a generalizable method of computing viscoelastic fluid flows in complex geometries

(which is very common in microfluidic applications), using a finite element package,

Comsol Multiphysics ; and the stagnation point flow naturally becomes a first problem

to look at.

For high Re, we are interested in viscoelastic turbulent flows, especially in the

regime where Re is close to but above the value of laminar-turbulence transition (Xi

& Graham 2009c,a). As a well-established experimental observation, flexible polymer

solutes at a very low concentration (O(10 ∼ 100) ppm) can reduce the friction drag

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of turbulent flows by as much as 80% (Virk 1975, Graham 2004, White & Mungal

2008). This phenomenon is of obvious practical interest because of the potential

energy savings it can bring in fluid transport applications. On the theoretical side, this

problem lies between two challenging areas: turbulence and polymer dynamics, study

of which bears the prospect of advancing the knowledge in both of them. Despite the

long history of study (since its original discovery in the 1940s by Toms (1948, 1977)),

understanding of polymer drag reduction remains very limited, especially for systems

with high Wi and large extent of drag reduction. In particular, the existence of a

universal upper-limit of drag reduction (for a given Re), the “Virk maximum drag

reduction” (Virk 1975), remains a mystery.

Computer simulation has been proven an powerful tool of reproducing the full 3D

flow fields of turbulence in both Newtonian (Moin & Kim 1982, Kim et al. 1987) and

viscoelastic systems (Sureshkumar & Beris 1997, Dimitropoulos et al. 1998), which

makes a valuable supplement to experimental research where accurate measurement

of time-dependent 3D fields, especially the stress field, is very difficult. Most previous

computational studies in viscoelastic turbulence mainly focus on statistical descrip-

tions of the flow. In this study, we take a nonlinear-dynamics approach and try to

understand the effect of polymer on individual coherent structures (Robinson 1991)

in turbulent flows. From a dynamical-system perspective, the temporal evolution of

a turbulent coherent structure is depicted as a complex transient trajectory (Jimenez

et al. 2005, Kerswell & Tutty 2007, Gibson et al. 2008) in the state space, built

around solution objects, such as traveling waves (TWs) (Nagata 1990, Waleffe 1998,

2001, 2003, Faisst & Eckhardt 2003, Wedin & Kerswell 2004). This view has ben-

efited research in Newtonian turbulence greatly in the past 10 ∼ 15 years in terms

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of understanding the self-sustaining mechanism of turbulence (Hamilton et al. 1995,

Waleffe 1997, Jimenez & Pinelli 1999) and the laminar-turbulence transition process

(see e.g. Skufca et al. (2006), Wang et al. (2007), Duguet et al. (2008)). Less has

been done on the viscoelastic turbulence side. Past work (Stone et al. 2002, Stone

& Graham 2003, Stone, Roy, Larson, Waleffe & Graham 2004, Li, Stone & Graham

2005, Li, Xi & Graham 2006, Li & Graham 2007) has studied the effects of polymer

on one class of traveling waves, the “exact coherent states” (Waleffe 1998), which,

according to above, correspond to the static building blocks of the trajectory. Fo-

cus of the current study is shifted to the dynamical side: we look at the dynamical

trajectory corresponding to the predominant coherent structures in near-transition

viscoelastic turbulence; and study the influence of polymer on temporal behaviors of

these structures. The goal is to interpret the dynamics of viscoelastic turbulence in

the context of the recent progresses made in Newtonian turbulence and viscoelastic

traveling waves reviewed above, and thus obtain a physical picture of the mechanism

of drag reduction

The following contents of this dissertation are thus divided into two independent

parts, each of which contains a review of previous studies, a summary of formulation

and methods, discussion of results, conclusions and a proposal for future research.

Although the scope covered by these two projects is only a small subset of the area

of viscoelastic fluid dynamics, these examples are representative enough that the

methodology we apply in these studies and the understanding we acquire about dy-

namics of viscoelastic fluid flows in different parameter regimes, can prospectively

impact a broader range of research.

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Part I

Dynamics at low Re: oscillatory

instability in viscoelastic cross-slot

flow

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8

Chapter 2

Introduction: elastic instabilities

and viscoelastic stagnation-point

flows

While Newtonian flows become unstable only at high Reynolds number Re, when

the inertial terms in momentum balance dominate, flows of viscoelastic fluids such as

polymer solutions and melts are known to have interesting instabilities and nonlinear

dynamical behaviors even at extremely low Re. These “purely-elastic” instabilities

arise in rheometry of complex fluids as well as in many other applications (Lar-

son 1992, Shaqfeh 1996). Recent studies of viscoelastic flows in microfluidic devices

broaden the scope of these nonlinear dynamical problems in low-Re viscoelastic fluid

dynamics (Squires & Quake 2005). The small length scales in microfluidic devices

enable large shear rates, and thus high Wi (Weissenberg number, Wi ≡ λγ, where λ

is a characteristic time scale of the fluid and γ is a characteristic shear rate of the

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flow), at very low Re. Instabilities are not always undesirable, especially when the

accompanying flow modification is controllable and can thus be utilized in the design

and operation of microfluidic devices. Specifically, instabilities have been found and

flow-controlling logic elements have been designed in a series of microfluidic geome-

tries, e.g. flow rectifier with anisotropic resistance (Groisman & Quake 2004), flip-flop

memory (Groisman et al. 2003) and nonlinear flow resistance (Groisman et al. 2003).

Another prospective application of these instabilities is to enhancement of mixing at

lab-on-a-chip length scales (Groisman & Steinberg 2001), where turbulent mixing is

absent due to small length scales and an alternative is needed.

The best understood of these instabilities are those that occur in viscometric flows

with curved streamlines: e.g. flows in Taylor-Couette (Muller et al. 1989), Taylor-

Dean (Joo & Shaqfeh 1994), cone-and-plate (Magda & Larson 1988) and parallel-

plates (Magda & Larson 1988, Groisman & Steinberg 2000) flow geometries. In

these geometries, the primary source of instability is the coupling of normal stresses

with streamline curvature (i.e. the presence of “hoop stresses”), leading to radial

compressive forces that can drive instabilities (Magda & Larson 1988, Muller et al.

1989, Larson et al. 1990, Joo & Shaqfeh 1994, Pakdel & McKinley 1996, Shaqfeh

1996, Graham 1998). Similar mechanisms drive instabilities in viscoelastic free-surface

flows (Spiegelberg & McKinley 1996, Graham 2003).

Attention in this study focuses on a different class of flows, whose instabilities

are not well-understood – stagnation point flows, like those generated with opposed-

jet (Chow et al. 1988, Muller et al. 1988), cross-slot (Arratia et al. 2006), two-roll

mill (Ng & Leal 1993) and four-roll mill (Broadbent et al. 1978, Ng & Leal 1993)

devices. Figure 3.1 shows a schematic of a cross-slot geometry. A characteristic

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(a) Dye convection pattern. (b) Contours of velocity magnitude (colors) andstreamline (dark lines) measured by particle imagevelocimetry (PIV)

Figure 2.1: Symmetry-breaking instability in viscoelastic cross-slot flow (Arratia et al.2006).

phenomenon in these stagnation point flows is the formation of a narrow region of

fluid with high polymer stress extending downstream from the stagnation point. This

region can be observed in optical experiments as a bright birefringent “strand” with

the rest of the fluid dark (Harlen et al. 1990). Keller and coworkers (Chow et al.

1988, Muller et al. 1988) reported instabilities in stagnation point flows of semi-dilute

polymer solutions generated by an axisymmetric opposed-jet device. Specifically, for

a fixed polymer species and concentration, upon a critical extension rate (or critical

Wi) polymer chains become stretched by flow near the stagnation point and a sharp

uniform birefringent stand forms. The width of this birefringent strand increases with

increasing Wi until a stability limit is reached, beyond which the birefringent strand

becomes destabilized and changes in its morphology are observed. At higher Wi,

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the flow pattern and birefringent strand become time-dependent. Recent tracer and

particle-tracking experiments of stagnation point flow in a micro-fabricated cross-slot

geometry by Arratia et al. (2006) show instabilities of dilute polymer solution at low

Re (< 10−2). In their experiments fluid from one of the two incoming channels is

dyed and a sharp and flat interface between dyed and undyed fluids is observed at

low Wi. Upon an onset value of Wi, this flow pattern loses its stability: spatial

symmetry is broken but the flow remains steady (Figure 2.1). The interface becomes

distorted in such a way that more than half of the dyed fluid goes to one of the

outgoing channels while more undyed fluid travels through the other. At even higher

Wi the flow becomes time-dependent and the direction of asymmetry flips between

two outgoing channels with time. Particle-tracking images in the time-dependent flow

pattern indicate the existence of vortical structures around the stagnation point.

Another class of stagnation point flows is associated with liquid-solid or liquid-gas

interfaces, such as flows passing submerged solid obstacles, around moving bubbles

or toward a free surface. For example, McKinley et al. (1993) reported a three-

dimensional steady cellular disturbances in the wake of a cylinder submerged in a

viscoelastic fluid. Around a falling sphere in viscoelastic fluids, fore-and-aft symmetry

of velocity field is broken and the velocity perturbation in the wake can be away from

the sphere, toward the sphere or a combination of the two depending on the polymer

solution (Hassager 1979, Bisgaard & Hassager 1982, Bisgaard 1983).

Remmelgas et al. (1999) computationally studied the stagnation point flow in a

cross-slot geometry with two different FENE (finitely-extensible nonlinear elastic)

dumbbell models. Using the two models, they studied the effects of configuration-

dependent friction coefficient on polymer relaxation and the shape of the birefringent

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strand. Their simulation approach was restricted to relatively low Wi (O(1)) with

symmetry imposed on the centerlines of all channels. Harlen (2002) conducted simula-

tions of a sedimenting sphere in a viscoelastic fluid to explore the wake behaviors. He

explained the experimental observations of both negative (velocity perturbation away

from the sphere) and extended (velocity perturbation toward the sphere) wakes in

terms of combined effects of the stretched polymer in the birefringent strand following

the stagnation point behind the sphere and the recoil outside of the strand. Neither

of these analyses directly addressed instabilities of these flows. In the recent work

of Poole et al. (2007), a stationary symmetry-breaking instability in the cross-slot

geometry has been predicted by conducting simulations using the upper-convected

Maxwell model. This instability is similar to the first steady symmetry-breaking in-

stability in the experiments of Arratia et al. (2006). However, the question as to why

the flow field becomes time-dependent in different geometries involving stagnation

points still needs to be addressed.

Various approximate approaches have been taken in the past to obtain an under-

standing of the instabilities observed in experiments. Harris & Rallison (1993, 1994)

investigated the instabilities of the birefringent strand downstream of a free isolated

stagnation point through a simplified approach, in which polymer molecules are mod-

eled as linear-locked dumbbells, which are fully stretched within a thin strand lying

along the centerline. Polymer molecules contribute a normal stress proportional to

the extension rate only when they are fully stretched (i.e. in the strand); otherwise the

flow is treated as Newtonian. The lubrication approximation is applied for the Newto-

nian region and the effects of birefringent strand are coupled into the problem through

point forces along the strand. Two instabilities are reported. At low Wi (≈ 1.2−1.7),

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a varicose disturbance is linearly unstable, in which the width of birefringent strand

oscillates without breaking the symmetry of the flow pattern. At higher Wi another

instability is observed in which symmetry with respect to the extension axis breaks

and the birefringent strand becomes sinuous in shape and oscillatory with time, with

zero displacement at the stagnation point and increasing magnitude of displacement

downstream from it. Symmetry with respect to the inflow axis is always imposed.

The mechanism of these instabilities is explained: perturbations in the shape or po-

sition of the birefringent strand affect the stretching of incoming polymer molecules

such that they enhance the perturbation after they become fully stretched and merge

into the strand. This mechanism is close to the one we are about to present later

in this study with regard to the importance of flow kinematics and the extensional

stress. However, in their linear stability analysis with which the instability mecha-

nism is investigated, the spatial dependence of the birefringent strand in the outflow

direction is neglected. Therefore although this factor is included in their numerical

simulation, it is not taken into consideration in their explanation of the instability.

As will be shown later, according to our simulations this spatial dependence of the

birefringent strand plays an important role.

In this study, we present numerical simulation results of viscoelastic stagnation

point flow in a two-dimensional cross-slot geometry. With increasing Wi, we observe

the formation and elongation of the birefringent strand across the stagnation point.

At high Wi, we find the occurrence of an oscillatory instability. These results resemble

the experimental observations of oscillatory birefringent width by Muller et al. (1988)

and the varicose instability predicted by Harris & Rallison (1994). By analyzing the

perturbations in both velocity and stress fields, a novel instability mechanism based

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on normal stress effects and flow kinematics is identified.

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Chapter 3

Cross-slot geometry, governing

equations and numerical methods

We consider a fourfold symmetric planar cross-slot geometry, as shown in Figure 3.1.

Flow enters from top and bottom and leaves from left and right. For laminar New-

tonian flow, two incoming streams meet at the intersection of the cross, and each

of them splits evenly and goes into both outgoing channels, generating a stagnation

point at the origin near which an extensional flow exists. We use round corners at

the intersections of channel walls in order to avoid enormous stress gradients at the

corners, which cause numerical difficulties.

The momentum and mass balances are:

Re

(∂u

∂t+ u ·∇u

)= −∇p+ β∇2u+ (1− β)

2

Wi(∇ · τ p) , (3.1)

∇ · u = 0. (3.2)

Parameters in Equations (3.1) and (3.2) are defined as: Re ≡ ρUl/ (ηs + ηp), Wi ≡

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x

y

l

l

l

Figure 3.1: Schematic of the cross-slot flow geometry.

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2λU/l and β ≡ ηs/ (ηs + ηp), where ρ is the fluid density, for a dilute polymer solution

we assume it to be the same as the solvent density; ηs is the solvent viscosity and

ηp is the polymer contribution to the shear viscosity at zero shear rate; U and l are

characteristic velocity and length scales of the flow. Here l is chosen to be the half-

channel width and the definition of U is based on the pressure drop applied between

the entrances and exits of the channel. Specifically, U is defined to be the centerline

velocity of a Newtonian plane Poiseuille flow under the same pressure drop in a

straight channel with length 20l, which is comparable to the lengths of streamlines

in the present geometry. According to this definition, the nondimensional pressure

drop in our simulation is fixed at 40 and the centerline Newtonian velocity in cross-

slot geometry is typically slightly lower than 1 since the extensional flow near the

stagnation point has a higher resistance than that in a straight channel. The polymer

contribution to the stress tensor is denoted τ p and is calculated with the FENE-P

constitutive equation (Bird, Curtis, Armstrong & Hassager 1987):

α

1− tr(α)b

+Wi

2

(∂α

∂t+ u ·∇α−α ·∇u− (α ·∇u)T

)=

(b

b+ 2

)δ, (3.3)

τ p =b+ 5

b

1− tr(α)b

−(

1− 2

b+ 2

). (3.4)

In Equations (3.3) and (3.4), polymer chains are modeled as FENE dumbbells (two

beads connected by a finitely-extensible-nonlinear-elastic spring). Here α ≡ 〈QQ〉is the conformation tensor of the dumbbells where Q is the end-to-end vector of the

dumbbells and 〈·〉 represents an ensemble average. The parameter b determines the

maximum extension of the dumbbells: i.e. the upper limit of tr(α).

At the entrances and exits of the flow geometry, normal flow boundary conditions

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are applied: i.e. t · u = 0 where t is the unit vector tangential to the boundary.

Pressure is set to be 40 at entrances and 0 at exits. No-slip boundary conditions are

applied at all other boundaries. Boundary conditions for stress are only needed at the

entrances, where the profile of α is set to be the same as that for a fully developed

pressure-driven flow in a straight channel with the same Wi. Unless otherwise noted,

several parameters are fixed for most of the results we report here: Re = 0.1, β = 0.95

and b = 1000, which means we focus on dilute solutions of long-chain polymers at

low Reynolds number.

The discrete elastic stress splitting (DEVSS) formulation (Baaijens et al. 1997,

Baaijens 1998) is applied in our simulation: i.e. a new variable Λ is introduced as

the rate of strain and a new equation is added into the equation system:

Λ = ∇u+ ∇uT . (3.5)

A numerical stabilization term γ∇ · (∇u+ ∇uT −Λ)

is added to the right-hand-

side of the momentum balance (Equation (3.1)), and it is worthwhile to point out

that this term is only nontrivial in the discretized formulation and does not change

the physical problem. In this term, γ is an adjustable parameter and γ = 1.0 is

used in our simulations. The velocity field u is interpolated with quadratic elements,

while pressure p, polymer conformation tensor α and rate of strain Λ are interpolated

with linear elements. Consistent with Baaijens’s conclusion (Baaijens 1998), DEVSS

greatly increases the upper limit of Wi achievable in our simulations. Quadrilateral

elements are used for all variables. Our experience shows that quadrilateral elements

have great advantages over triangular ones, yielding much better spatial smoothness

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in the stress field at comparable degrees of freedom to be solved. Another merit

of quadrilateral elements is the capability of manual control over mesh grids. This

is extremely important when certain restrictions, such as symmetry, are required.

In our simulation, finer meshes are used within and around the intersection region

of the geometry, and the mesh is required to be symmetric with respect to both

axes. Within a horizontal band (−0.2 < y < 0.2) across the stagnation point, very

fine meshes are generated to capture the sharp stress gradient along the birefringent

strand. The streamline-upwind/Petrov-Galerkin (SUPG) method (Brooks & Hughes

1982) is applied in Equation (3.3) by replacing the usual Galerkin weighting function

w with w+ δhu ·∇w/‖u‖, where h is the geometric average of the local mesh length

scales and δ is an adjustable parameter, set to δ = 0.3 in our simulations. This

formulation is implemented using the commercially available Comsol Multiphysics

software.

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Chapter 4

Results: viscoelastic cross-slot flow

and its oscillatory instability

4.1 Steady states

Steady-state solutions are found for all Wi investigated (0.2 < Wi < 100) in our

study. For Wi 6 60 steady states are found by time integration and for those with

larger Wi Newton iteration (parameter continuation) is used because of possible loss

of stability, as we describe below. At low Wi the velocity field is virtually unaffected

by the polymer molecules. Velocity contours at Wi = 0.2 are plotted in Figure 4.1(a);

for clarity only part of the channel is shown. A stagnation point is found at the center

of the domain ((0, 0)). In both incoming and outgoing channels, the flow is almost the

same as pressure driven flow in a straight channel. No distinct difference can be ob-

served for the incoming and outgoing directions in velocity field. Figure 4.1(b) shows

contours of extension rate at Wi = 0.2, in which a region dominated by extensional

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flow is found near the stagnation point. High extension rate is also found near the

corners due to the no-slip walls. The magnitude of polymer stretching can be mea-

sured by the trace of its conformation tensor tr(α), and is plotted in Figure 4.1(c).

At low Wi, the extent to which polymers are deformed is barely noticeable, but it

can be clearly seen that polymers are primarily stretched in either the extensional

flow near the stagnation point and corners, or the shear flows near the walls. At

high Wi (Wi = 50, Figure 4.2), the situation is very different. Polymers are strongly

stretched by the extensional flow near the stagnation point and this stretching effect

by extensional flow overwhelms that of the shear flow. A distinct band of highly

stretched polymers (the birefringent strand) forms (Figure 4.2(c)). Since the polymer

relaxation time in this case is larger than the flow convection time from stagnation

point to the exits, this birefringent strand extends the whole length of the simu-

lation domain. The resulting high polymer stress significantly affects the velocity

field (Figure 4.2(a)). Regions with reduced velocity extend much farther away in the

downstream directions of the stagnation point than in the low Wi case, especially

along the x-axis, where high polymer stress dominates. Correspondingly, a reduction

in the extension rate near the stagnation point is observed, most noticeably along the

birefringent strand (Figure 4.2(b)).

Figures 4.3 and 4.4 show profiles at various values of Wi of tr(α) along the out-

flow (x-axis) and inflow (y-axis) directions of this stagnation point (note the difference

in scales in the two plots). For increasing Wi the length of the region with highly

stretched polymer keeps increasing due to the increased relative relaxation time (Fig-

ure 4.3). In high Wi cases (Wi = 30 and Wi = 100), polymers are not fully relaxed

even when they reach the exit of the simulation domain. The cross-sectional view of

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(a) ‖u‖

(b) ∂ux/∂x

(c) tr(α)

Figure 4.1: Contour plots of steady state solution: Wi = 0.2 (only the central part ofthe flow domain is shown).

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(a) ‖u‖

(b) ∂ux/∂x

(c) tr(α)

Figure 4.2: Contour plots of steady state solution: Wi = 50 (only the central part ofthe flow domain is shown).

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Figure 4.3: Profiles of tr(α) along y = 0.

Figure 4.4: Profiles of tr(α) along x = 0 in the region very near the stagnation point.

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(a) Birefringent strand width W .

(b) Birefringent strand length L.

Figure 4.5: Effect of Wi on the size of the birefringent strand (tr(α) > 300 is consid-ered as the observable birefringence region).

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Figure 4.6: Profiles of ux along y = 0.

tr(α) profiles along the y-axis (Figure 4.4) show interesting non-monotonic behaviors.

Although the height of the profile (tr(α)max) keeps increasing upon increasing Wi,

the width of the Wi = 100 case is smaller than that of Wi = 30, resulting in a steeper

transition section between low and high stretching regions. If we arbitrarily define

tr(α) > 300 as the observable birefringence region, the width W and the length L of

the birefringent strand (measured on the inflow and outflow axes, respectively) can

be plotted as functions of Wi, as in Figure 4.5 (values of L for Wi > 30 are not shown

since they exceed the length of the simulation domain). A clear non-monotonic trend

is observed in the plot of birefringence width, where W increases sharply at relatively

low Wi and peaks around Wi = 40. After that W decreases mildly but consistently

with further higher Wi. This non-monotonic trend is consistent with experimental

observations of birefringence in opposed-jet devices (Muller et al. 1988).

Similarly, a non-monotonicity is also found in the change of velocity field with Wi.

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Figure 4.7: Average extension rate (∂ux/∂x)avg (averages taken in the domain −0.1 <x < 0.1,−0.1 < y < 0.1).

Velocity profiles along the outflow axis are plotted in Figure 4.6. Magnitude of the

outflow velocity is obviously reduced for high-Wi flows, consistent with the breakup

of fore-aft (along the streamlines) symmetry in velocity distributions observed in

Figure 4.2(a). Comparing the profiles of Wi = 5, Wi = 30 and Wi = 100, one

can find that this suppression of outgoing flow is also non-monotonic with increasing

Wi. Changes in velocity field affect the polymer stress field via changes in the strain

rate. Shown in Figure 4.7 is the value of extension rate, averaged within a box

around the stagnation point (−0.1 < x < 0.1,−0.1 < y < 0.1), as a function of Wi.

As Wi increases, the extension rate decreases at low Wi but increases at high Wi,

with a minimum found around Wi = 40. Besides, most of experimental results are

presented in terms of Deborah number (De), defined as the product of the polymer

relaxation time and an estimate of the extension rate near the stagnation point.

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Noticing that the average (nondimensionlized) extension rate changes within a very

narrow range (around 0.55 ∼ 0.6), a conversion De = 0.3Wi can be adopted for

comparison of our results with experimental ones.

Some understanding of this non-monotonicity can be gained by looking at Fig-

ure 4.3. Here it can be seen that for Wi . 30, the birefringent strand is not yet “fully

developed” in the sense that the polymer stretching is not yet saturating near full

extension. Thus the evolution of the velocity field in this regime of Wi reflects the sig-

nificant changes that occur in the stress field in this regime. At higher Wi, however,

the polymer stress field in the strand is saturating, and thus not changing signif-

icantly. Furthermore, at these high Weissenberg numbers, the relaxation of stress

downstream of the stagnation point diminishes, decreasing the gradient ∂τxx/∂x and

thus decreasing the effect of viscoelasticity on the flow near the stagnation point.

4.2 Periodic orbits

We turn now to the stability of the steady states that have just been described. Rather

than attempting to compute the eigenspectra of the linearization of the problem,

an exceedingly demanding task, we examine stability by direct time integration of

perturbed steady states. The perturbations take the form of slightly asymmetric

pressure profiles at the two entrances (0.1% maximum deviation from the steady state

value) that are applied for one time unit, then released. As an example, temporal

evolution of the birefringent strand width W starting from the perturbed steady-

state at Wi = 66, measured on the inflow axis, is plotted in Figure 4.8. The system

stays near the steady-state solution for a long time (> 1000), before the tiny initial

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perturbation grows to a noticeable extent. In the range 1000 < t < 3100, this

deviation oscillates with increasing amplitude; then it reaches a limit. After that

the system fluctuates around the steady state with a fixed magnitude and frequency,

i.e. it approaches a limiting cycle. Figure 4.9 shows a two dimensional projection of

the trajectory of the same process. Here the velocity magnitude at a point near the

stagnation point (0.5, 0) is plotted against the birefringent strand width W measured

on the inflow axis. The system starts at the steady state with W = 0.1593 and

ux|(0.5,0) = 0.2687 and spirals outward with time after the perturbation. Eventually

the trajectory merges into a cycle (the outer dark cycle in the Figure 4.9). This clearly

identifies the existence of a stable periodic orbit. Note the anti-correlation between

ux|(0.5,0) and W , i.e. when the flow speeds up near the stagnation point, the strand

thins and vice versa. Although a finite asymmetric perturbation has been introduced

in the simulation results presented here, it is worth to mention that in order to trigger

the instability, the initial perturbation does not have to be in this particular form,

nor does it have a finite threshold. We have tested another form of perturbation

in which we add zero-mean random noises of different orders-of-magnitude onto the

initial steady state solutions and the instability can always be observed.

Figure 4.10 shows the root-mean-square deviations over one period of W from its

steady-state values, normalized by the corresponding steady-state values Ws.s., as a

function of Wi for all the cases where we found periodic orbits. Time integrations for

Wi > 74 did not converge due to the enormous stress gradient around the corners of

the no-slip walls and the consequent numerical oscillations downstream. Data points

for Wrms computed from our simulations are fitted with a function of the form a(Wi−Wic)

p, with p fixed at 1/2. Very good agreement is found for our simulation data with

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Figure 4.8: Evolution of the birefringence strand width W after a small initial per-turbation on the steady state; inset: enlarged view of 2500 6 t 6 3200.

Figure 4.9: Two dimensional projection of the dynamic trajectory from the steadystate to the periodic orbit at Wi = 66: ux at (0.5, 0) v.s. W .

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Figure 4.10: Left-hand axis: root-mean-square deviations of the birefringent strandwidth W at periodic orbits, normalized by steady state values; right-hand axis: os-cillation periods.

the 1/2 power law, characteristic of a supercritical Hopf bifurcation (Guckenheimer &

Holmes 1983). The critical Weissenberg number Wic is identified to be 64.99 by this

fitting. Also shown in Figure 4.10 are periods of oscillations, where a slight decrease

with increasing Wi is found. This is interesting since it indicates that some time scale

other than the polymer relaxation time sets the period of oscillations.

Simulations have also been conducted at other values of β and b. Within the dilute

regime, Wic has a strong dependence on the polymer concentration (∝ (1− β)) and

the bifurcation occurs at much higher Wi for more dilute solutions. (In the Newtonian

limit β → 1, Wic must diverge.) For example, for β = 0.96, Wic lies between 80 and

82. Simulations for lower β, i.e. higher concentration, are not feasible at this point

due to numerical instabilities. For b values not very far way from 1000, changing the b

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parameter barely affects Wic. By changing the b parameter downward to 900, Wic is

almost unchanged. However, for further smaller b values, the dependence is stronger

and Wic increases with decreasing b.

As mentioned earlier, time-dependent instabilities have been observed in viscoelas-

tic stagnation-point flows in both opposed-jet and cross-slot geometries. In particular,

the birefringent stability found by Muller et al. (1988) is very similar to the one re-

ported in this study. In their optical experiments with semi-dilute aPS solutions, the

width of the birefringent strand oscillates rapidly between two values in a certain

range of extension rate. Compared with their experiments, as well as the asymptotic

model of Harris & Rallison (1994), our simulation predicts a higher critical Wi. This

could be as least partially attributed to the low concentration we are looking at. In

the cross-slot geometry, time-dependent oscillations are found for De > 12.5 (Arra-

tia et al. 2006), which is of the same order-of-magnitude as what we have observed

(Dec ≈ 0.3Wic = 19.5). Although symmetry is not imposed in our simulations, we

do not observe any symmetry-breaking instability, which according the experiments

should occur at a much lower De. This might be related to the constant-pressure drop

constraint we applied between entrances and exits. In both the experiments (Arra-

tia et al. 2006) and simulations (Poole et al. 2007) where asymmetry is observed,

there are no restrictions on the pressure at the boundaries, and the constant-flow rate

constraint is applied instead (at the flow entrances).

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4.3 Instability mechanism

We turn now to the spatiotemporal structure of the instability and its underlying

physical mechanism. We will denote the deviations in velocity, pressure and stress

with primes, while steady-state values will be denoted with a superscript “s”:

u = us + u′, (4.1)

p = ps + p′, (4.2)

α = αs +α′. (4.3)

Figures 4.11, 4.12 and 4.13 illustrate u′x, u′y and α′xx, respectively, at intervals of 1/8

period, corresponding to the periodic orbit at a Weissenberg number close to the

bifurcation point (Wi = 66). Time starts from an arbitrarily chosen snapshot on the

periodic orbit and only a quarter of the region near the stagnation point is shown;

behavior in the rest of the domain can be inferred from the reflection symmetry across

the axes.

At the beginning of the cycle (Figure 4.11(a)), u′x is positive in the region very

close to the stagnation point while it is negative in most of the downstream region.

As time goes on, this positive deviation near the stagnation point grows into a “jet”,

a region of liquid moving downstream away from the stagnation point faster than

the steady-state velocity, as shown in Figures 4.11(b), 4.11(c) and 4.11(d). Corre-

spondingly, by continuity, the inflow toward the stagnation point is also faster as

shown in Figures 4.12(a)–4.12(d). Note that very near the stagnation point devia-

tions from steady state remain small. At the beginning of the second half of the cycle

(Figure 4.11(e)), the jet extends further downstream and grows to the full width of

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(a) t = 0 (b) t = 1.68

(c) t = 3.35 (d) t = 5.03

Figure 4.11: Perturbation of the x-component of velocity, u′x with respect to steadystate at the periodic orbit: Wi = 66. The region shown is 0 < x < 1, −1 < y < 0;the stagnation point is at the top-left corner. (To be continued).

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(e) t = 6.71 (f) t = 8.38

(g) t = 10.06 (h) t = 11.74

Figure 4.11: (Continued).

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(a) t = 0 (b) t = 1.68

(c) t = 3.35 (d) t = 5.03

Figure 4.12: Perturbation of the y-component of velocity, u′y with respect to steadystate at the periodic orbit: Wi = 66. The region shown is 0 < x < 1, −1 < y < 0;the stagnation point is at the top-left corner. (To be continued).

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(e) t = 6.71 (f) t = 8.38

(g) t = 10.06 (h) t = 11.74

Figure 4.12: (Continued).

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(a) t = 0 (b) t = 1.68

(c) t = 3.35 (d) t = 5.03

Figure 4.13: Perturbation of the xx-component of polymer conformation tensor, α′xxwith respect to steady state at the periodic orbit: Wi = 66. The region shown is0 < x < 1, −1 < y < 0; the stagnation point is at the top-left corner. The edge ofthe steady state birefringent strand is the line y ≈ −0.05. (To be continued).

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(e) t = 6.71 (f) t = 8.38

(g) t = 10.06 (h) t = 11.74

Figure 4.13: (Continued).

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the channel. Meanwhile, in the region closer to the stagnation point, velocity devia-

tions drop (Figures 4.11(e), 4.12(e)) and start to change sign (Figures 4.11(f), 4.12(f)).

Consequently, the growth of the jet ends and a “wake”, a region of fluid moving slower

than the steady-state velocity, emerges from the stagnation point (Figures 4.11(f)–

4.11(h) and 4.12(f)– 4.12(h)). Similarly, as the wake grow larger, velocity devia-

tions near the stagnation point change signs and a new cycle starts (Figures 4.11(a)

and 4.12(a)).

The velocity deviations are closely related with those of the stress field (Fig-

ure 4.13). Generally speaking, “jets” are accompanied by negative α′xx and thus

thinning of the birefringent strand and “wakes” are associated with the birefringent

thickening. The largest deviations are found at the edges of the birefringent strand

where ∂αsxx/∂y is largest. Note that deviations in the stress field are always small

along the centerline of the birefringent strand because there polymer molecules are

almost fully stretched and the huge spring force is sufficient to resist any perturba-

tions.

One may notice the small spatial oscillations in the stress field deviations, charac-

terized by alternating high and low stress stripes, along the outflow direction. These

oscillations, apparently unphysical and centered around zero, also exist along the bire-

fringence strand in steady-state solutions, though they are not easy to see from the

contours in Figure 4.2(c) as they are overwhelmed by the high tr(α) in the birefringent

strand. Unfortunately, as shown by recent studies (Renardy 2006, Thomases & Shel-

ley 2007), spatial non-smoothness is inevitable in numerical simulations of viscoelas-

tic extensional flow upon certain Wi, owning to the singularities in stress gradients.

These singularities could not be fully resolved by any finite mesh size and this prob-

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lem would always show up in numerical solutions of high Wi viscoelastic stagnation

point flows. However, we do not expect these oscillations to qualitatively affect our

observations for a couple of reasons. First, non-smoothness has been observed in our

simulation at Wi values much lower than the critical Wi of this instability. Second,

observable non-smoothness is always found some distance away from the stagnation

point in the downstream direction while the instability is dominated by the physics in

the close vicinity of the stagnation point and, since FENE-P is a convective equation,

we do not expect anything occurring downstream to affect upstream dynamics. Last,

and most importantly, simulations with different meshes display different mesh size

dependent stripes, while the nature of the instability remains virtually unchanged.

Insight into the mechanism of this instability can be gained by examining the

linearized equation for α′xx:

∂α′xx∂t

=− 2

Wi

α′xx

1− tr(αs)b

− 2

Wi

αsxxtr(α

′)

b(

1− tr(αs)b

)2

− usx

∂α′xx∂x− us

y

∂α′xx∂y− u′x

∂αsxx

∂x− u′y

∂αsxx

∂y

+ 2αsxx

∂u′x∂x

+ 2αsxy

∂u′x∂y

+ 2α′xx∂us

x

∂x+ 2α′xy

∂usx

∂y.

(4.4)

In the following analysis, terms on the right-hand-side (RHS) of Equation 4.4 are

named “RHS∗”, where “∗” is determined by the order of appearance on the RHS.

Terms and their physical meanings are summarized in Table 4.1. To understand

the mechanism of the instability, magnitudes of these terms at the point (0,−0.05)

are plotted as a function of time during roughly a period in the bottom view of

Figure 4.14. Terms RHS3, RHS5, RHS8 and RHS10 are zero by symmetry and not

plotted. This position is right at the edge of the birefringent strand and as shown

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Term Formula Physical Significance

RHS1 − 2Wi

α′xx

1− tr(αs)b

Relaxation.

RHS2 − 2Wi

αsxxtr(α′)

b(1− tr(αs)b )

2 Relaxation.

RHS3 −usx∂α′

xx

∂x

Convection of conformation deviations by thesteady-state x-velocity.

RHS4 −usy∂α′

xx

∂y

Convection of conformation deviations by thesteady-state y-velocity.

RHS5 −u′x ∂αsxx

∂x

Convection of the steady-state conformation by x-velocity deviations.

RHS6 −u′y ∂αsxx

∂y

Convection of the steady-state conformation by y-velocity deviations.

RHS7 2αsxx

∂u′x

∂x

Stretching caused by deviations in the extensionrate.

RHS8 2αsxy∂u′

x

∂yStretching caused by deviations in the shear rate.

RHS9 2α′xx∂us

x

∂x

Stretching caused by deviations in the extensionalstress.

RHS10 2α′xy∂us

x

∂y

Stretching caused by deviations in the shearstress.

Table 4.1: Terms on the right-hand side of Equation (4.4).

in Figure 4.13, it is also where significant deviations in the stress field are observed.

Time-dependent oscillations at other places, including off the symmetry axis x =

0, have also been checked and nothing that would qualitatively affect our analysis

was seen. Correspondingly, deviations in polymer conformation, inflow velocity and

extension rate, normalized by steady-state values, are plotted in the top view of

Figure 4.14.

Consistent with our earlier observations, deviations in the velocity field (u′y and

∂u′x/∂x) and deviations in stress field (α′xx) are opposite in sign for most of the time

within the period. Among the terms plotted, RHS4, RHS6, RHS7 and RHS9 are

much larger than the relaxation terms, RHS1 and RHS2, and dominate the dynamics.

(Relaxation terms are large at the very inner regions of the birefringent strand and

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Figure 4.14: Time-dependent oscillations at (0,−0.05). Top view: perturbations ofvariables normalized by steady-state quantities; bottom view: magnitudes of termson RHS of Equation (4.4).

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44

that is why oscillations in the stress field there are barely noticeable.) Moreover,

RHS4, RHS6 and RHS9 are mostly in phase with α′xx and thus tend to enhance the

deviations while RHS7 is out of phase with α′xx and hence damps the deviations. It is

the joint effect of these competing destabilizing and stabilizing forces that gives the

oscillatory behavior of the system. Finally, notice that among the three destabilizing

terms, RHS6 is the one that leads the phase and thus guides the instability.

Based on these observations from Figure 4.14, a mechanism for the instability can

be proposed, which is illustrated schematically in Figure 4.15. At the beginning of the

cycle (t = 0), u′y is slightly above zero, indicating that the inflow speed is faster than

that in the steady state. As a consequence, RHS6 becomes negative first, followed by

RHS4 and RHS9. In particular, a faster incoming convective flow brings unstretched

polymer molecules toward the stagnation point (corresponding to RHS6), as depicted

in Figure 4.15(a). These polymer chains have less time to get stretched and when they

reach the edges of the birefringent strand (e.g. dumbbell B), they are less stretched

compared with the steady state. As a result, fluid around dumbbell B has lower stress

than at the steady state, corresponding to a thinning of the birefringent strand.

Meanwhile, since dumbbell B contains smaller spring forces than its downstream

neighbors A and A’, the net forces (thick arrows) exerted by polymer on the fluid

point outward, generating jets downstream from the stagnation point. (In other

words, when the stress at the center is lower, the net stress divergence points outward,

which increases momentum in the downstream directions.) By continuity, more fluid

has to be drawn toward the stagnation point and the initial deviation in u′y is then

enhanced. However, as the flow speeds up in the vicinity of the stagnation point, the

extension rate also starts to increase. This effect (corresponding to RHS7) tends to

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(a) Thinning process of the birefringent strand.

(b) Re-thickening process of the birefringent strand.

Figure 4.15: Schematic of instability mechanism (view of the lower half geometry).Thick arrows represent net forces exerted by polymer molecules (dumbbells) on thefluid.

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stretch polymer molecules more and stabilize the deviations, as shown in Figure 4.14.

Eventually this effect will be able to overcome that of RHS6 as well as RHS4 and

RHS9, and the stress near the stagnation point starts to increase after it passes the

minimum at around t = 3.5, which causes a re-thickening of the birefringent strand

as illustrated in Figure 4.15(b). By a similar argument as that above, dumbbell C has

higher spring forces than B and B’, the dumbbells which were passing near the center

when stress was at minimum, and the net polymer forces point inward, which starts

to suppress the jets. Inflow velocity decreases as the birefringent strand thickens,

and this gives incoming polymer molecules more time to be stretched, which further

thickens the birefringent strand. Eventually αxx will come back to the steady state

value at around t = 7.2. However, since all the deviations are not synchronized, a

negative deviation is found in uy; and an identical analysis with opposite signs can

be made for the second half of the cycle.

Within this mechanism, a sharp edge of the birefringent strand, i.e. large magni-

tude of ∂αxx/∂y (∼ O(104) in our simulations), is required so that a small u′y can give

a sufficiently large RHS6 to drive the instability. This is made possible by the kine-

matics of the flow near the stagnation point, where the incoming polymer molecules

are strongly stretched within a short distance. Another similar effect is that stress

derivatives are stretched in the outgoing direction and thus greatly weakened as fluid

moves downstream; therefore the instability is dominated by physics in the vicinity of

the stagnation point. In the earlier mechanism for the so-called “varicose instability”,

given by Harris and Rallison (Harris & Rallison 1994), the importance of extensional

stress and flow kinematics, especially the role of the convection of incoming molecules,

was also recognized. However, the picture described in their work is not the same as

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47

ours due to the simplifications in their model. Their linear stability analysis ignored

the x-dependence of the birefringent width while in our simulations, x-dependence of

the stress field is closely related to the changes in velocity field. Besides, their analysis

did not identify a restoring force for the deviations and the oscillatory behavior could

not be explained.

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Chapter 5

Conclusions of Part I

Using a DEVSS/SUPG formulation of the finite element method, we are able to sim-

ulate viscoelastic stagnation point flow and obtain steady-state and time-dependent

solutions at high Wi. For Wi 1, a clear birefringent strand is observed. The width

of this birefringent strand increases with increasing Wi until Wi ≈ 40 after which it

declines gradually. This also results in a non-monotonic trend in the modification of

the velocity field.

At around Wi = 65 the steady state solution loses stability and a periodic orbit

becomes the attractor in phase space. Flow motion of the periodic orbit is char-

acterized by time-dependent fluctuations, specifically, alternating positive (jet) and

negative (wake) deviations from the steady-state velocity in the regions downstream

of the stagnation point. A mechanism is proposed which, taking account of the in-

teraction between velocity and stress fields, is able to explain the whole process of

the oscillatory instability. Extensional stresses and their gradients, as well as the

flow kinetics near the stagnation points, are identified as important factors in the

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49

mechanism. This mechanism is different from that of the “hoop-stress” instabilities,

which occur in viscometric flows with curved streamlines, and we expect that this

mechanism could be extended to explain various instabilities occurring in viscoelastic

flows with stagnation points.

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Chapter 6

Future work: nonlinear dynamics

of viscoelastic fluid flows in

complex geometries

With the method we develop in this study, we are able to obtain numerically-stable

and smooth solutions even for very high Wi. Based on the steady-state and time-

dependent solutions, we proposed a novel mechanism for an oscillatory instability

involving a stagnation point. Compared with the experimental results by Arratia

et al. (2006), we do not observe any symmetry-breaking in our simulations. Even

with asymmetric initial perturbations, time intergration would eventually lead to axis-

symmetric steady states or periodic orbits. A probable cause for this inconsistency

is the constant-pressure-drop constraint we applied between the entrances and exits.

The Arratia et al. (2006) experiments were performed under the constant-incoming-

flow-rate constraint, with no restriction on the pressure drop. Same constraint was

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51

used in the simulation of Poole et al. (2007) where symmetry-breaking was observed

in their steady-state solutions of the Stokes equation (Navier-Stokes equation at the

Re → 0 limit (Deen 1998)) coupled with the “upper-convected Maxwell” (UCM)

constitutive equation (Bird, Armstrong & Hassager 1987). Mechanism of symmetry-

breaking was not elucidated in that study, which we propose as the next goal of

our study on low-Re viscoelastic flows. Recall in Arratia et al. (2006) that steady

symmetry-breaking observed at moderate Wi would develop into a second instability

of fluctuating asymmetry at higher Wi. It is possible that the mechanism of the second

instability is a combination of the yet-unknown symmetry-breaking mechanism and

the oscillatory instability mechanism proposed in this study. Therefore understanding

the symmetry-breaking could be a key step toward the full understanding of both

instabilities.

Beyond the cross-slot flow, our method can be extended to many other flow ge-

ometries. The biggest advantage of the finite element method is that different flow

geometries can be implemented with minimal efforts. Since stagnation-point flows are

among the most difficult to simulate for viscoelastic fluids (stress field turns singular

at high Wi (Renardy 2006, Thomases & Shelley 2007, Becherer et al. 2009)), our

method should be numerically stable for a variety of geometries, an important merit

of a method designed for microfluidic applications.

As reviewed in Chapter 2, instabilities are observed in many different geometries

involving stagnation points. Compared with the relatively better-understood class of

“hoop-stress” instability, it is interesting to see if there is any commonness on the

mechanism level among instabilities in all these stagnation-point flows. One close

example is the so-called microfluidic “flip-flop” device (Groisman et al. 2003). Its

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52

(a) Overall Geometry. The auxiliary inlets (comp. 1 andcomp. 2) are for flow-rate measurement purpose.

(b) Blowup near the intersection during the instability. Onlyfluids from one of the two inlets are dyed.

Figure 6.1: The microfluidic flip-flop device (Groisman et al. 2003).

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53

geometry (Figure 6.1(a)) is also of the cross-channel form, and is symmetric with

respect to the incoming axis, but asymmetric with respect to the outgoing axis. Two

incoming channels are different in width, and both connect to the intersection via a

contraction; after the intersection the flow diverges through two expansions toward the

exits. Metastable asymmetric states (Figure 6.1(b)) are observed at high Wi where

stream from either of the incoming channels almost all exits from one outgoing chan-

nel. This instability appears similar to the symmetry-breaking instability observed

by Arratia et al. (2006), but the contractions and expansions near the intersection

further complicate the problem. Besides flows with isolated stagnation points studied

here, instabilities in stagnation-point flows near solid-liquid interfaces may also share

similar mechanisms. These problems include many interesting nonlinear phenomena

in flows around immersed solid objects (Bisgaard & Hassager 1982, McKinley et al.

1993, Harlen 2002), where one stagnation point exists at the separatrix of streamline

in front of the object and another at the merging axis behind the object.

Viscoelastic fluid flows without stagnation points are of interest as well. For

dilute polymer solutions, strong nonlinear effects are expected in extensional flows.

Kinematics of many flow types encountered in microfluidics include both shear and

extension components (Pipe & McKinley 2009), e.g. flow-focusing (Oliveira et al.

2009), contraction and expansion (Groisman et al. 2003, Groisman & Quake 2004,

Rodd et al. 2005). Instabilities have been observed in many of them. Beyond the

scope of viscoelastic fluid dynamics, microfluidics has been applied extensively in

the manipulation and separation of individual bio-macromolecules (see, e.g. Perkins

et al. (1997), Dimalanta et al. (2004) and Chan et al. (2004)). Although the FENE-P

(dumbbell) model we use is too coarse-grained to capture certain degrees-of-freedom

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in the dynamics of these molecules under flow, our method is still useful in terms of

obtaining a crude estimation of the stretching and orientation of these molecules, and

thus aid in the design of experiments and more refined computational studies.

On the methodology level, our current method is limited to 2D geometries. How-

ever, most microfluidic devices are built to be three-dimensional, i.e. the depth is

at the same order-of-magnitude as the width. Some applications even require a 3D

flow geometry to function: e.g. the microfluidic chaotic mixer (Stroock et al. 2002).

Dimensionality can affect nonlinear dynamics as well: although some instabilities are

be purely two-dimensional, such as the oscillatory instability studied here, many oth-

ers need all three dimensions to develop. The “hoop-stress” instability (Larson et al.

1990, Pakdel & McKinley 1996, Shaqfeh 1996, Graham 1998) is one example (which

has also been applied in microfluidics for enhancing mixing (Groisman & Steinberg

2001)). Same for the inertio-elastic instabilities observed in the contraction-expansion

microchannel fabricated by Rodd et al. (2005), where streamlines are clearly overlap-

ping and crossing in a 2D projection. Extending the current method to 3D geometries

would greatly expand its power of predicting nonlinear phenomena in microfluidics

(which unfortunately would also cost much more computational resource; the current

study is mostly performed on a desktop PC, and each simulation takes from a few

hours to one day).

Another extension to consider is to include liquid-liquid and liquid-gas interfaces

in the simulation. Multiphase flows are very commonly seen in microfluidic devices,

in the forms of drops, bubbles, free surfaces and immiscible streams (Anna et al. 2003,

Garstecki et al. 2004, Stone, Stroock & Ajdari 2004, Atencia & Beebe 2005, Squires &

Quake 2005). In the case of viscoelastic fluids, coupling between viscoelasticity and

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interfacial dynamics can cause further complexity in nonlinear dynamics (see, e.g.

Arratia et al. (2008a,b), Sullivan et al. (2008)). To simulate these flows, in addition

to the relatively simple task of enabling free-slip or pratial-slip boundary conditions,

the main challenge is to include mobile boundaries in the computational model.

In summary, there are mainly three directions of utilizing and expanding the

achievements of the current study: (1) to further study the mechanisms of instabili-

ties involving stagnation points, especially the symmetry-breaking instability and the

potential similarities among different types of stagnation-point flows; (2) to apply

the current numerical method directly to more general geometries and understand

different types of instabilities; (3) to improve the method and adapt it to the needs

of more general microfluidic applications.

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Part II

Dynamics at high Re: viscoelastic

turbulent flows and drag reduction

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Chapter 7

Introduction: viscoelastic

turbulent flows and polymer drag

reduction

7.1 Fundamentals of polymer drag reduction

It has been experimentally observed that by introducing a minute amount of flexible

polymers (at concentrations of O(10 − 100) ppm by weight or even lower) into a

turbulent flow, the turbulent friction drag can be substantially reduced (Virk 1975,

Graham 2004, White & Mungal 2008), resulting in a higher flow rate for a given

pressure drop. The percentage drop of the friction factor can be as high as 80%

in turbulent flows in straight pipe or channel geometries. Since its initial discovery

in the 1940s (Toms 1948, 1977), the phenomenon of polymer drag reduction has

been an active area of study due to its practical and theoretical significance. It

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58

log(Reτ)

U+ av

g

Wi↑

LaminarL­T

Trans. Pre­Onset Intermediate DR MDR

Lam

inar

Flo

w(P

oise

uille

's L

aw)

Maximum Drag Reduction

(Virk's Asymptote)

Newtonian Turbulence

(Prandtl­von Kármán Law)

Figure 7.1: Schamatic of the Prandtl-von Karman plot. Thin vertical lines mark thetransition points on the typical experimental path shown as a thick solid line.

can obviously be utilized to improve energy efficiency in various fluid transportation

applications. Moreover, unraveling the physical mechanism of the phenomenon in

terms of the complex interactions between turbulence and polymer molecules would

not only expand our knowledge of polymer dynamics in fluid flows, but also provide

additional insight into the nature of turbulence itself.

Bulk flow data obtained from drag reduction experiments are very often plotted

in Prandtl-von Karman coordinates, i.e. a plot of average velocity U+avg ≡ Uavg/uτ

versus friction Reynolds number Reτ ≡ ρuτ l/η. (Here, ρ is the fluid density, η is

the total viscosity, and l is a characteristic length scale of the flow geometry; the

friction velocity uτ ≡√τw/ρ is a characteristic velocity scale for near-wall turbu-

lence, where τw is the mean wall shear stress; the superscript “+” denotes quantities

nondimensionlized with inner scales, i.e. velocities scaled by uτ and lengths scaled

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Figure 7.2: Experimental data of maximum drag reduction (MDR) in pipe flow,from different polymer solution systems and pipe sizes, plotted in the Prandtl-vonKarman coordinates (Virk 1971, 1975). It can be shown that 1/

√f = U+

avg/√

2,

Re√f =

√2Reτ (f in this plot is the friction factor, which is denoted as Cf in

this dissertation; Re in this plot is the Reynolds number based on average velocity:Reavg ≡ ρUavgD/η, D is the pipe diameter).

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by η/ρuτ .) A schematic Prandtl-von Karman plot for Newtonian and polymeric flow

is shown in Figure 7.1. In a typical experiment where the polymer solution system

and the pipe/channel size are fixed, measurements made under different Reτ are con-

nected to form a line called an “experimental path”. Along an experimental path,

Re and Wi vary simultaneously, while their ratio, defined as the elasticity number

El ≡Wi/Re, remains constant. (Here, Re ≡ ρUl/η is the Reynolds number using the

characteristic bulk flow velocity U as the velocity scale; Wi ≡ λγ is the Weissenberg

number, which is the product of polymer relaxation time λ and a characteristic shear

rate γ. Note that γ ∝ U/l, thus El ∝ λη/ρl2 is constant when the polymer solu-

tion system and flow geometry is fixed.) A typical experimental path is sketched in

Figure 7.1 as a thick solid line. With increasing Reτ , the flow system undergoes a

series of transitions among several qualitatively different stages, including: laminar

flow, laminar-turbulence transition, turbulence before the onset of drag reduction

(pre-onset), intermediate drag reduction and the maximum drag reduction (MDR).

The boundaries of each stage (i.e. the transition points) are marked with thin verti-

cal lines for the experimental path denoted with the thick solid line. The last stage

(MDR) is so named because it is invariant with changing polymer species, molecular

weight, concentration and geometric-confinement length scale (pipe diameter or chan-

nel height) (Virk et al. 1967, Virk 1971, 1975, Graham 2004, White & Mungal 2008).

Experimental paths of different polymer solution systems and pipe (or channel) sizes

are sketched in dashed lines. Although changing the polymer solution system and the

confinement length scale, i.e. via changes in El, would affect the slope in the inter-

mediate DR stage as well as the points of transition, all experimental paths collapse

into a single straight line after they reach the MDR stage (Figure 7.2). This line,

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commonly referred to as the Virk’s MDR asymptote, sets the universal upper limit of

drag reduction when polymer is used as the drag-reducing agent. Note that once this

asymptote is reached, the friction drag is solely dependent on Reτ . This universality

of the MDR stage is perhaps the most intriguing problem in polymer drag reduction.

The study of polymer-induced drag reduction thus can be divided into several

important questions: (1) what is the mechanism by which polymers alter turbulence

and reduce drag; (2) what are the qualitative changes underlying these multistage

transitions; and in particular (3) why is there a universal upper limit on drag reduction

(MDR) and what is the nature of turbulence in that regime?

7.2 Previous direct numerical simulation (DNS)

studies

None of these questions has been completely answered to date; however, advances in

computer simulations of viscoelastic turbulent flows in the past decade have substan-

tially advanced the understanding of drag reduction. Beris and coworkers pioneered

the direct numerical simulation (DNS) of viscoelastic turbulent flows (Sureshkumar &

Beris 1997, Dimitropoulos et al. 1998) using the FENE-P (Bird, Curtis, Armstrong &

Hassager 1987) constitutive equation. Most major experimental observations in the

intermediate drag reduction regime (after onset and before MDR), including the onset

of drag reduction, thickened buffer layer, wider streak spacing, and changes in the ve-

locity fluctuations and Reynolds shear stress profiles, were qualitatively reproduced.

Since then DNS has been adopted as a powerful tool to access the details of velocity

and polymer stress fields, and thus to infer the mechanism by which polymers reduce

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drag. By inspecting the instantaneous snapshots of velocity fluctuations and polymer

force fields, as well as the correlation between the two, De Angelis et al. (2002) claimed

that polymer suppresses turbulence by counteracting the velocity fluctuations. (This

mechanism is also predicted by Stone et al. (2002), Stone, Roy, Larson, Waleffe & Gra-

ham (2004) and Li & Graham (2007) with a different means, as we discuss below.)

Similar results on the velocity-polymer force correlations were reported by Dubief

et al. (2004, 2005), which showed that polymer forces are anti-correlated with veloc-

ity fluctuations in the transverse directions, while in the streamwise direction these

two quantities are positively correlated in the viscous sublayer and anti-correlated for

the rest of the channel. Based on this they suggested that polymer molecules suppress

the vortical motions and meanwhile are stretched by these near-wall vortices; when

they are convected toward the wall to the high-speed streaks during the “sweeping”

events, they release the energy back to the flow and thereby aid in the sustenance

of turbulence. Another common practice to interpret DNS data is to examine the

transport equations of kinetic energy and Reynolds stresses, and describe the effects

of polymer in terms of the changes it causes to different contributions to the energy

budgets. Min, Yoo, Choi & Joseph (2003) proposed that the kinetic energy of the

turbulent flow is transferred to elastic energy by stretching the polymer molecules

very close to the wall; these stretched molecules are lifted upward to the buffer and

log-law layers to release energy back to the flow. Ptasinski et al. (2003) evaluated the

budget of each component of the turbulent kinetic energy, and found that polymer

suppresses pressure fluctuations and thus impedes energy transfer among different

components via the pressure-rate of strain term in the Reynolds stress budgets.

The studies mentioned above primarily rely on the statistical representations of

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the three-dimensional fields. Quantities being investigated are averaged in time as

well as in the two periodic dimensions, and the analyses are mostly based on mean

profiles with dependence on the wall-normal coordinate only. Although this approach

assures the statistical certainty of the results, it eliminates most of the structural

information about the turbulent flow motions. On the other hand, in the near wall

region turbulent flows are known to be dominated by coherent structures (Robinson

1991), where most drag reduction effects caused by polymer take place. Further

understanding of the interplay between turbulent structures and polymer dynamics

requires the capability of isolating the coherent structures from the complex turbulent

background.

Information about these coherent structures can be extracted from DNS solu-

tions a posteriori. For example, the Karhunen-Loeve analysis (or proper orthogonal

decomposition) (Holmes et al. 1996, Pope 2000) has been applied to viscoelastic tur-

bulent flows for this purpose (De Angelis et al. 2003, Housiadas et al. 2005). Given

a set of statistically independent snapshots from the time-dependent solution of the

turbulent flow, this method constructs a series of mutually orthogonal modes, or

eigen-states, which form an optimal decomposition of the original solution in the

sense that the leading modes always contain the largest amount of turbulent kinetic

energy. These studies showed that viscoelasticity modifies the turbulent flow by in-

creasing the amount of energy carried by the leading modes, or the energy-containing

modes. However further study is still needed to connect this finding with the complex

process of the polymer-turbulence interactions. More recently, conditional averaging

has been used to sample the predominant structures around certain local events that

contribute substantially to the turbulent friction drag (Kim et al. 2007). These re-

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sults confirmed that polymer inhibits vortical motions, both streamwise vortices in

the buffer layer and hairpin vortices further away from the wall, by applying forces

that counter them. This is consistent with many other studies (De Angelis et al. 2002,

Stone et al. 2002, Stone, Roy, Larson, Waleffe & Graham 2004, Dubief et al. 2005, Li

& Graham 2007). Using these sampled structures as the initial conditions for time

integration, evolution of the hairpin vortices was simulated (Kim et al. 2008), and

it was found that viscoelasticity not only suppresses the primary vortices but also

prevents secondary vortices from being created.

7.3 Traveling waves and the nonlinear dynamics

perspective of turbulence

In the past decade, the discovery of three-dimensional fully nonlinear relative steady-

state solutions, or traveling wave (TW) solutions, to the Navier-Stokes equation,

made the a priori study of the coherent structures a reality. These solutions are

steady states of the Navier-Stokes equation typically in a reference frame moving at a

constant speed, and they are found in all canonical wall-bounded geometries for New-

tonian turbulent flows (plane Couette, plane Poiseuille and pipe geometries) (Waleffe

1998, 2001, 2003, Faisst & Eckhardt 2003, Wedin & Kerswell 2004, Pringle & Ker-

swell 2007, Viswanath 2007). These TWs usually appear in the form of low-speed

streaks straddled by streamwise vortices, which closely (in both structure and length

scales) resemble the recurrent coherent structures in near wall turbulence. In par-

ticular, an optimal spanwise box size of 105.51 wall units was reported for the TW

solution found by Waleffe (2003) in the plane Poiseuille geometry (Figure 7.3), which

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Figure 7.3: Newtonian ECS solution at Re = 977 in plane Poiseuille flow; symmetriccopies at both walls are shown. Slices show coutours of streamwise velocity, dark colorfor low velocity; the isosurface has a constant streamwise vortex strengthQ2D = 0.008,definition of Q2D is given in Section 10.4. This plot is published by Li & Graham(2007); the solution is originally discovered by Waleffe (2003).

is remarkably close to experimentally observed near-wall streak spacing of about 100

wall units (Smith & Metzler 1983). Transient structures that look very similar to

these solutions have been experimentally observed (Hof et al. 2004). In the context of

drag reduction, past work has examined one family of these TW solutions, the “exact

coherent states” (ECS) (Waleffe 1998, 2001, 2003) (see Figure 7.3), of viscoelastic

turbulent flows in both plane Couette (Stone et al. 2002, Stone & Graham 2003,

Stone, Roy, Larson, Waleffe & Graham 2004) and plane Poiseuille geometries (Li,

Stone & Graham 2005, Li, Xi & Graham 2006, Li & Graham 2007). Not only do the

viscoelastic ECS solutions show drag reduction compared with their Newtonian coun-

terparts, they also capture many characteristics of drag-reduced turbulence, including

reduced vortical strength and changes in turbulence statistics. Consistent with DNS

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results (De Angelis et al. 2002, Dubief et al. 2005, Kim et al. 2007), polymer influ-

ences the flow structures and causes drag reduction in ECS by counteracting velocity

fluctuations and vortical motions (Stone et al. 2002, Stone, Roy, Larson, Waleffe &

Graham 2004, Li & Graham 2007). Viscoelasticity also changes the minimal Re at

which ECS exist; under fixed Re and with high enough Wi, these solutions are totally

suppressed by the polymer (Stone, Roy, Larson, Waleffe & Graham 2004, Li, Xi &

Graham 2006, Li & Graham 2007). Based on these studies, a simple framework con-

taining different stages of the ECS solutions in the parameter space, which includes

the laminar-turbulence transition, the onset of drag reduction and the annihilation of

ECS, was proposed (Li, Xi & Graham 2006, Li & Graham 2007). With the hypothesis

that the annihilation of ECS is linked with MDR, this framework covered most key

transitions in viscoelastic turbulent flows.

Although viscoelastic ECS solutions do provide new insight into the problem of

drag reduction, they are only fixed points (i.e. steady states) in the state space. On

the other hand, the dynamics of the coherent structures is a complex time-dependent

trajectory, so further investigation into the coherent turbulent motions requires the

study of transient solutions. DNS studies mentioned earlier belong to this category,

but in most of them periodic simulation boxes much larger than the characteristic

length scales of the coherent structures are used. Transient solutions obtained from

that approach typically involve a large number of coherent structures convoluted with

one another, and include the long range spatial correlations between them, which

makes the identification and analysis of individual coherent structures difficult. The

most straightforward way to isolate the transient solution corresponding to an indi-

vidual coherent structure is the “minimal flow unit (MFU)” approach: i.e. by limiting

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(a) Dynamical trajectory of a turbulent transient in a MFU visualized in the state spaceusing coordinates proposed by Gibson et al. (2008).

(b) Same trajectory (dotted line) visualized in the context of TW solutions (solid dots,except uLM, which is the laminar state) and their unstable manifolds (solid lines).

Figure 7.4: Dynamics of turbulence in the solution state space in a plan Couetteflow (Gibson et al. 2008).

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the simulation box to the smallest size that still sustains the turbulent motion, only

the very essential elements of the self-sustaining process of turbulence will be in-

cluded in the simulation. This approach was first adopted by Jimenez & Moin (1991)

in Newtonian turbulent flows. The minimal spanwise box size in inner scales they

found, L+z ≈ 100, is in very good agreement with the experimental measurement of

the streak spacing in the viscous sublayer (Smith & Metzler 1983), and this value

is insensitive to the change of Re. The minimal streamwise box size they reported

is dependent on Re and falls in the range of 250 . L+x . 350, which is also consis-

tent with experimental measurements of the streamwise structure spacings (Sankaran

et al. 1988). By comparing transient trajectories of MFU simulations with various

TW solutions in certain 2D projections of the state space, Jimenez et al. (2005) de-

scribed the dynamical process of MFU in plane Poiseuille and Couette geometries

as a combination of relatively long-time stays in the vicinity of the TWs (“equilib-

rium”) and intermittent excursions away from these states (“bursting”). A different

result is obtained in pipe flows: Kerswell & Tutty (2007) proposed several correlation

functions as quantitative measurements of the distance between transient solutions

and TWs, and observed that the transient turbulent trajectories only visit the TWs

about 10% of the time and more complex objects in the state space, such as periodic

orbits, are necessary for a good approximation of the time-dependent solutions. In

the plane Couette geometry and using coordinates constructed with upper-branch

ECS solutions and symmetry arguments, Gibson et al. (2008) visualized the trajec-

tories of MFU solutions together with the TW states and their unstable manifolds

in a geometrical view of the state space, with which the connection between tran-

sient solutions and the dynamical structure formed by TWs can been clearly seen

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(Figure 7.4).

All these studies on MFU are focused on Newtonian turbulent flows. Despite the

simplicity behind the MFU idea, this approach has not been applied in the study

of viscoelastic turbulence and drag reduction, partially due to the additional degrees

of freedom in the parameter space when polymer is introduced. While Newtonian

flows can be characterized by a single parameter Re, this is no longer true in polymer

solutions where polymer species, molecular weight and concentration can also affect

the flow dynamics and hence minimal box sizes. To search for minimal box size

therefore becomes a highly computationally demanding task when variations in all

parameters are taken account of. Since the term MFU is often generalized by other

authors (e.g. Min, Yoo, Choi & Joseph (2003), Ptasinski et al. (2003), Dubief et al.

(2005)) to describe DNS in relatively small, but not necessarily minimal, boxes, here

we clarify that in this study, the term “minimal flow unit ” or MFU refers exclusively

to a flow determined via a size minimization process. That is, for each parameter

setting, different box sizes should be tested in order to determine a minimal size

at which turbulence persists. In this study, as we will discuss in Chapter 9, the

minimization process is only taken in the spanwise direction, while the streamwise

box size is fixed at the value of the Newtonian MFU. The goal of the current work is

to find the MFU of viscoelastic turbulence under a variety of parameters and observe

the transitions among different stages in terms of drag reduction behaviors.

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Figure 7.5: The Virk (1975) universal mean velocity profile for MDR in inner scales:U+

mean = 11.7 ln y+ − 17.0.

7.4 Multistage transitions

A classical picture of multistage transitions in viscoelastic turbulence includes: pre-

onset turbulence, intermediate DR (after onset and before MDR) and MDR (Fig-

ure 7.1). Compared with the extensive studies of the intermediate DR regime sum-

marized earlier, the research on MDR is very limited. Though there is a certain

degree of understanding of the phenomenon of how polymer additives reduce tur-

bulent drag, the origin of the universal upper limit in the MDR stage remains very

poorly understood.

Early theory of Virk (1975) assumed that drag reduction only occurs in the buffer

layer; as viscoelasticity increases, thickness of this layer increases, and MDR is reached

when the buffer layer dominates the whole flow geometry. This view is similar to the

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conclusion drawn from the elastic theory by Sreenivasan & White (2000), that at

MDR the length scale of turbulence structures affected by polymer is comparable

with that of the flow geometry, and indeed this view is consistent with the results

presented below. Based on these views, phenomenological models have been devel-

oped to predict mean velocity profiles, in which quantitative agreement with the Virk

MDR profile was reported (Benzi et al. 2006, Procaccia et al. 2008). These theories

achieved various levels of success in predicting many experimental results; however,

discrepancies are still found with some other observations, as discussed by White &

Mungal (2008). In addition, all these theoretical studies are based on average (in both

space and time) quantities, the lack of information in these models about turbulent co-

herent structures and their spatiotemporal behavior limits their ability to contribute

to a physical picture of the dynamics underlying experimental observations.

Among the few DNS studies on MDR, most efforts are dedicated to reproducing

the Virk mean velocity profile of MDR (Ptasinski et al. 2003, Dubief et al. 2005, Li,

Sureshkumar & Khomami 2006): i.e. they look for parameter settings under which the

mean velocity profile of DNS is the same as or close to that of experimentally observed

MDR at Re far from transition, which according to Virk (1971, 1975) is universal in

inner scales for a wide range of Re (Figure 7.5). The only exception to our knowledge

is the work of Min, Choi & Yoo (2003), where the convergence of DR% with increasing

Wi is used to identify MDR. In that study, DR% of several Wi are calculated with

other parameters held fixed, and the last two points on the high Wi end show almost

the same DR%. (The percentage of DR, DR% ≡ (Cf,s − Cf)/Cf,s × 100%, where

Cf ≡ 2τw/(ρU2avg) is the friction factor of the viscoelastic fluid flow, and Cf,s is the

friction factor of the flow of pure solvent.) As mentioned earlier, MDR is a stage

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where the friction factor is only dependent on Re, and is unaffected by variations

in Wi and other polymer-related properties; therefore the problem of MDR is the

mechanism by which the same friction factor is preserved at fixed Re with changing

polymer parameters. This mechanism cannot be studied without simulation data at

MDR for a range of different parameter settings. Furthermore, whether one should

expect the same mean velocity profile in DNS studies as that of Virk is uncertain:

first, most experiments on MDR are conducted at relatively high Re, and the lack of

experimental measurements in the regime close to the laminar-turbulence transition

makes it hard to conclude whether the Virk profile is valid at Re comparable to those

in many DNS studies; second, the widely used FENE-P constitutive equation is a

highly simplified model for polymer molecules and how well it can quantitatively

predict the mean velocity at MDR is still unknown. In the simulations of Min, Choi

& Yoo (2003), the mean velocity profile after drag reduction reaches the limit at high

Wi is clearly lower than the Virk MDR profile; Dubief et al. (2005) also reported that

the Virk MDR profile is only obtained in a relatively small simulation box, and is

not found in large-box simulations; a small box is also used in the study of Ptasinski

et al. (2003). The only DNS study that predicts mean velocity profiles comparable

to Virk’s in large simulation boxes is Li, Sureshkumar & Khomami (2006). However,

they did not report the universal convergence of the mean velocity. In the present

work, we use the criterion that the friction factor converges with Wi to identify

regimes representing the asymptotic behavior of MDR.

In recent years, an additional distinction was noticed within the intermediate DR

regime between a low degree of drag reduction (LDR) and a high degree of drag

reduction (HDR). This difference was investigated by Warholic, Massah & Hanratty

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(1999) in their channel (plane Poiseuille) flow experiments, where differences between

LDR and HDR appear in several flow statistical quantities, including: (1) mean ve-

locity profile: LDR has the same log-law slope as Newtonian turbulent flows while

HDR shows larger slope of the log-law; (2) streamwise velocity fluctuation profile:

at LDR the magnitude of fluctuations (in inner scales) increases with DR% and the

location of the peak shifts away from the wall, while at HDR fluctuations are greatly

suppressed compared with Newtonian turbulent flows; (3) wall-normal velocity fluc-

tuation profile: fluctuations are suppressed in both cases, but at LDR there is still a

recognizable maximum in the profile while at HDR the maximum is not observable;

(4) Reynolds shear stress profile: at LDR the Reynolds shear stress decreases with

DR but the profile retains the same slope as that of Newtonian turbulent flows at

large distance away from the wall, while at HDR the Reynolds shear stress is almost

zero across the channel and the slope farther away from the wall also changes sig-

nificantly. Some of these differences have also been noted by several other groups

through both experiments (Ptasinski et al. 2003) and simulations (Min, Choi & Yoo

2003, Ptasinski et al. 2003, Li, Sureshkumar & Khomami 2006). Most authors tend

to treat these differences as quantitative effects of the percentage of drag reduction

DR%, and DR% ≈ 30%− 40% is commonly adopted as the separating point between

LDR and HDR.

7.5 About this study

As stated earlier, in this work we look for MFU solutions, i.e. transient solutions

containing the minimal self-sustaining structures, of viscoelastic turbulent flows. A

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wide range of the parameter space is sampled in order to provide a complete picture

of the different stages in terms of drag reduction behaviors. Note that although ex-

perimental measurements are typically made following paths with constant El, along

which Re and Wi are varying simultaneously (ref. Figure 7.1), in this study (as in

many others, for example, Sureshkumar & Beris (1997), Min, Choi & Yoo (2003),

and Li, Sureshkumar & Khomami (2006)), we focus on the behavior as a function of

Wi while holding Re fixed. As shown by the vertical arrow in Figure 7.1, one can

still visit all different stages of transition, on different experimental paths, by varying

Wi under fixed Re; the advantage of doing so is that the MDR stage can be easily

identified as a plateau on the bulk flow rate versus Wi curve. Our results show that all

the stages of transition previously reported from both experiments and full-size DNS

studies, including pre-onset turbulence, LDR, HDR and a high-Wi regime showing

the asymptotic behavior of MDR, are observed in these transient solutions in MFUs.

In particular, we do identify a regime in which DR% converges with increasing Wi,

which should recall the experimental observation of MDR. We have varied all the

parameters (except Re) in the system and there is no observable difference in the

bulk flow rate with changing parameters once that asymptotic stage is reached. This

is to our knowledge the first report of a universal upper-limit of DR% in numerical

simulations, which matches the qualitative experimental hallmark of MDR: the bulk

flow rate is only a function of Re. In addition, all simulation results reported in this

study are obtained at a Re lower than any previously published DNS study, close to

the laminar-turbulence transition. The fact that all these key stages of viscoelastic

turbulence can be studied in the parameter regime close to the laminar-turbulence

transition, as predicted in earlier work (Li, Xi & Graham 2006, Li & Graham 2007), is

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important not only from the computational point of view (computational cost grows

rapidly with increasing Re), but also in terms of the understanding of the turbulent

structures (at Re this low, the near-wall coherent structures dominate the whole flow

geometry and are easier to observe). We also need to mention that the highest DR%

reached in our simulations in only in the range of 20−30%, which is clearly below the

separating point between LDR and HDR identified in other studies. The fact that

the LDR–HDR transition exists under such low DR% indicates that it is a transition

between two qualitatively different stages during the drag reduction process instead

of a quantitative difference caused by the amount of drag reduction.

Following contents of this part are organized as follows. Chapter 8 summarizes the

mathematical formulation and numerical method. In Chapter 9 we discuss in detail

the process of finding minimal flow units. Discussion of our observations during these

transitions (Chapter 10) is divided into several sections: we start with an overview of

the multistage-transition scenario in MFU solutions for a variety of polymer-related

parameters (Section 10.1); then we present flow and polymer confirmation statis-

tics (Sections 10.2, 10.3) at different stages for the sake of comparison with existing

publications; finally in Section 10.4 we study the spatiotemporal structure of the self-

sustaining dynamics, which provides insight into the changes in turbulence dynamics

accompanying the multistage transitions.

Although the asymptotic MDR-like stage observed in Chapter 10 recover the uni-

versality of MDR, its degree of drag reduction is much lower than the experimentally-

observed Virk MDR. The Virk MDR profile can not be statistically reproduced in

our current MFU study. Nevertheless, a more careful inspection of the spatiotempo-

ral dynamics of MFU does point a new direction of understanding the long-lasting

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mystery of Virk MDR. Analysis of these results are included in Chapter 11, the mo-

tivation of which will only be clear after the discussions in Section 10.4. Conclusions

of our study on viscoelastic turbulence are summarized in Chapter 12, after which,

some discussions will be given about how these results, especially those reported in

Chapter 11, may lead to new levels of understanding of the Virk MDR, as well as

many other problems in drag-reduced turbulence (Chapter 12).

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77

Chapter 8

DNS formulation and numerical

method

8.1 Flow geometry and governing equations

We consider a channel flow (plane Poiseuille, see Figure 8.1) geometry in which the

flow is driven by a constant mean pressure gradient. The x, y and z coordinates are

aligned with the streamwise, wall-normal and spanwise directions, respectively. The

no-slip boundary condition is applied at the walls and periodic boundary conditions

are adopted in the x and z directions; the periods in these directions are denoted Lx

and Lz. All lengths in the geometry are nondimensionalized with the half channel

height l of the channel and the velocity scale is the Newtonian laminar centerline

velocity U at the given pressure drop. Time t is scaled with l/U and pressure p with

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x

y

z

Lx

L z

2l

Figure 8.1: Schematic of the plane Poiseuille flow geometry: the box highlighted inthe center with dark-colored walls is the actual simulation box, surrounded by itsperiodic images.

Q 0Q max

= (b/3)1

/2 Q 0

Retractio

n Forces

Stretching

Equilibrium

Figure 8.2: Schematic of the finitely-extensible nonlinear elastic (FENE) dumbbellmodel for polymer molecules.

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79

ρU2. The conservation equations of momentum and mass give:

∂v

∂t+ v ·∇v = −∇p+

β

Re∇2v +

2 (1− β)

ReWi(∇ · τ p) , (8.1)

∇ · v = 0. (8.2)

Here, Re ≡ ρUl/(ηs + ηp) (ρ is the total density of the fluid, and (ηs + ηp) is the

total zero-shear rate viscosity; hereinafter, subscript “s” represents “solvent”, i.e.

the Newtonian fluids, and subscript “p” represents the polymer contribution) and

Wi ≡ 2λU/l, which is the product of the polymer relaxation time λ and the mean

wall shear rate. Under this definition, the friction Reynolds number, defined as Reτ ≡ρuτ l/(ηs + ηp), can be directly related to Re: i.e. Reτ =

√2Re. The viscosity ratio

β ≡ ηs/(ηs + ηp) is the ratio of the solvent viscosity and the total viscosity. For dilute

polymer solutions, 1−β is approximately proportional to the polymer concentration.

The last term on the right-hand-side of Equation (8.1) captures the polymer effects

on the flow field, where the polymer stress tensor τ p is modeled by the FENE-P

constitutive equation (Bird, Curtis, Armstrong & Hassager 1987):

α

1− tr(α)b

+Wi

2

(∂α

∂t+ v ·∇α−α ·∇v − (α ·∇v)T

)=

(b

b+ 2

)δ, (8.3)

τ p =b+ 5

b

1− tr(α)b

−(

1− 2

b+ 2

). (8.4)

In Equations (8.3) and (8.4), polymer molecules are modeled as FENE dumbbells: two

beads connected by a finitely-extensible nonlinear elastic (FENE) spring (Figure 8.2).

The variable α is the nondimensional polymer conformation tensor α ≡ 〈QQ〉, where

Q is the end-to-end vector of the dumbbells. The parameter b defines the maximum

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80

extensibility of the dumbbells; max (tr (α)) 6 b.

In this study, we fix Re = 3600 (Reτ = 84.85) and span the parameter space

at three different (β, b) pairs, (0.97, 5000), (0.99, 10000) and (0.99, 5000), with a

large range of Wi for each β and b. The importance of β and b becomes apparent

in considering the exetensibility parameter Ex, defined as the polymer contribution

to the steady-state stress in uniaxial extensional flow, in the high Wi limit. For the

FENE-P model, Ex = 2b(1 − β)/3β. For a dilute solution (1 − β 1), significant

effects of polymer on turbulence are only expected when Ex 1. For the three sets

of β and b given above, the values of Ex are 103.09, 67.34 and 33.67, respectively.

8.2 Numerical procedures

The coupled problem of Equations (8.1), (8.2), (8.3) and (8.4) is integrated in time

with a 3rd-order semi-implicit time-stepping algorithm: linear terms are updated

with the implicit 3rd-order backward differentiation method and nonlinear terms are

integrated with the explicit 3rd-order Adams-Bashforth method (Peyret 2002). The

continuity equation (Equation (8.2)) is coupled with the momentum balance (Equa-

tion (8.1)) with the influence matrix method (Canuto et al. 1988). The alternating

form is used to evaluate the inertia term in Equation (8.2): we switch between the

convection form v ·∇v and the divergence form ∇ · (vv) upon each time step (Zang

1991).

The Fourier-Chebyshev-Fourier spatial discretization is applied in all variables

and nonlinear terms are calculated with the collocation method. The numerical grid

spacing for the streamwise direction is δ+x = 8.57, and in the spanwise direction we

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81

adjust the number of Fourier modes according to the varying box width (as discussed

in Chapter 9) to keep the grid spacing roughly constant, in the range of 5.0 6 δ+z 6

5.5; in the wall-normal direction 73 Chebyshev modes are used, which gives δ+y,min =

0.081 at the walls and δ+y,max = 3.7 at the channel center at Re = 3600. The time step

size is determined from the CFL stability condition: for the simulations reported in

this study, since the spatial grid spacing is fixed, a constant time step δt = 0.02 is

used.

An artificial diffusivity term 1/(ScRe)∇2α is added to the right-hand side of

Equation (8.3), a common practice to improve numerical stability in pseudo-spectral

simulations of viscoelastic fluids (Sureshkumar & Beris 1997, Dimitropoulos et al.

1998, Housiadas & Beris 2003, Ptasinski et al. 2003, Housiadas et al. 2005, Li,

Sureshkumar & Khomami 2006, Kim et al. 2007). In this study, we use a fixed

value of the Schmidt number, Sc = 0.5, which gives a constant artificial diffusiv-

ity of 1/(ScRe) = 5.56 × 10−4. The magnitude of this artificial diffusivity is at the

same order of those used by previous studies of other groups, typically O(10−4); an

additional diffusive term at this order of magnitude should not affect the numerical

solutions significantly while it helps to the numerical stability greatly. With the intro-

duction of this term, an additional boundary condition is needed for Equation (8.3),

for which we used the solution without the artificial diffusivity (same as many other

DNS studies, e.g. Sureshkumar & Beris (1997)): i.e. we update the α values at the

walls without the artificial diffusivity term first; using these results as the boundary

values, we solve Equation (8.3) with the artificial diffusivity term added to update

the α field for the rest of the channel.

Numerical parameters listed above apply to most of the results in this part, with

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82

exceptions of those discussed in Section 11.2. Detailed formulation for the numerical

scheme is provided in Appendix A.

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Chapter 9

Methodology: minimal flow units

(MFU)

The dimensions of the simulation box (L+x , L+

z ) determine the longest wavelengths

captured in the numerical solutions. As introduced in Section 7.3, the MFU approach

finds the transient solutions of Equations (8.1), (8.2), (8.3) and (8.4) that correspond

to the self-sustaining coherent structures by finding the smallest box in which turbu-

lent motions are sustained. Note that this minimal box size is in general a function of

all parameters in the system, i.e. Re, Wi, β and b, this minimization process has to be

performed for each different parameter combination. In Newtonian MFUs, a roughly

constant value L+z ≈ 100 is found for different magnitudes of Re whereas L+

x decreases

with increasing Re (Jimenez & Moin 1991). Experimentally measured steak spacings

in turbulent flows of polymer solutions are larger than the 100 wall units found for

Newtonian turbulent flows, and also increase with increasing DR% (Oldaker & Tie-

derman 1977, White et al. 2004). This observation is consistent with large-box DNS

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84

results, where the length scales of spanwise spatial correlation functions increase with

increasing DR% (Sureshkumar & Beris 1997, De Angelis et al. 2003, Li, Sureshkumar

& Khomami 2006). Therefore, L+z larger than that of the Newtonian MFUs is ex-

pected in our search for viscoelastic MFUs. Viscoelasticity increases the correlation

length scales in the streamwise direction as well. In particular, Li, Sureshkumar &

Khomami (2006) reported that the streamwise correlation length is increased by more

than an order of magnitude when DR% increases from 0 to 60% or more. As a result,

a significantly longer simulation box is required to capture all these long-range corre-

lations at high DR%. Consistently, the optimal length scales, in both streamwise and

spanwise directions, of viscoelastic ECS solutions increase with increasing Wi (Li &

Graham 2007).

A rigorous search of MFUs should consider the parameter dependence of both L+x

and L+z , a task involving impractically large number of simulation runs. In this study,

we fix L+x = 360 and focus on the variation of L+

z only. Although both length scales

depend on parameters, L+z is arguably the quantity of more interest: the dominant

structures at the Re we study are the streamwise streaks and the streamwise vortices

aligned alongside them, thus L+z directly restricts the streak spacing and the size of

the vortices whereas L+x only imposes a periodicity in the longitudinal direction. The

fact that we are able to find sustained turbulence in various stages of transitions at

fixed L+x = 360, which is in the range of Newtonian-MFU streamwise sizes (Jimenez

& Moin 1991), indicates that the minimal streamwise box size may not change as

much as the streamwise correlation length does.

Note that there is not a widely-accepted definition of “sustained turbulence”;

in fact, the question of whether turbulence sustains indefinitely after the laminar-

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85

0 5 10 15 20 25 30 35100

150

200

250

300

350

Wi

L+ z

↓ Single-wall Turbulence ↓

LaminarTurbulent

Figure 9.1: Summary of simulation results: “Turbulent” indicates that at least onesimulation run gives sustained turbulence within the given time interval (Newtonianand β = 0.97, b = 5000).

turbulence transition or eventually decays after some long but finite life time is still

subject to controversy (Hof et al. 2006, Willis & Kerswell 2007). Here we take a

pragmatic approach to this issue by checking the persistence of turbulent motion

within a fixed time interval. In all results reported in this study, we use a statistically

converged MFU solution at an adjacent parameter (typically with a slightly different

Wi and/or L+z ) as the initial condition, and declare that sustained turbulence is found

if the turbulent motions do not decay after 12000 time units, which is at the same order

as but larger than the longest time scale in the system (O(Re)). Figure 9.1 summarizes

our results with Newtonian runs and viscoelastic runs at β = 0.97, b = 5000. With the

exception of one Newtonian run where we use L+z = 105.51 (this is the size of the ECS

solution when it starts to appear in a “optimal” box (Waleffe 2003)), at each Wi we

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86

test different L+z with an increment of ∆L+

z = 10, and whether sustained turbulence

is found or not is recorded with filled and open symbols, respectively. Consistent

with all previous studies, L+z of MFU has an obvious dependence on Wi and increases

almost monotonically with Wi. There is some roughness on the boundary between

the regions where turbulence persists and where it does not, however, this seeming

inconsistency is a natural consequence of the sensitivity of near-transition turbulence

to initial conditions: with the same parameters and box size, some initial conditions

will laminarize and some will not. Nevertheless, the laminar-turbulent boundary in

Figure 9.1 should still serve as a reasonable estimate of the smallest spanwise size

of the self-sustained turbulent motions. Similarly, some simulation runs with box

sizes larger than the minimal values still laminarize, especially at the high Wi side.

Results reported in the rest of this study are primarily from simulation runs with

the minimal L+z , i.e. on the boundary of filled and open symbols in Figure 9.1.

The exceptions are those with L+z < 140, where L+

z = 140 is used instead of the

actual minimal values, because it is found that at Re close to the laminar-turbulence

transition, when L+z is relatively small, turbulence very often tends to sustain near

only one wall of the channel, while near the other wall the flow is almost laminar. One

explanation is when Re is very low, the size of the coherent structure is comparable to

and sometimes larger than the half channel height, so that the channel is geometrically

not high enough to accommodate structures at both walls. This kind of “single-wall

turbulence” was also reported by Jimenez & Moin (1991) at Re near the laminar–

turbulence transition, and is highly undesirable in our study since the flow statistics

are strongly biased by the laminar side. Empirically, we find that this problem does

not show up for L+z > 140. This truncation of L+

z at the low L+z limit would of course

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render our simulations there inconsistent with the definition of MFU; fortunately, as

shown later, this problem does not affect the viscoelastic turbulence after the onset

of drag reduction, the regime we are most interested in.

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Chapter 10

Results: observations during

multistage transitions

10.1 Overview of the multistage-transition scenario

In this section, we present MFU simulation results of viscoelastic flows at various

parameters. Most of the results are presented in the form of statistical averages

(averages in time as well as in either the x and z dimensions or all three spatial

coordinates depending on the figure). Each viscoelastic simulation run is 12000 time-

units long, and we discard the solution of the first 4000 time units to avoid any

possible initial-condition dependence. Temporal averages are taken in the last 8000

time units of each simulation run. The Newtonian simulation is 20000 time units long

and the last 16000 time units are included in the statistics.

The foremost quantity of interest with regard to drag reduction is the average

streamwise velocity, as plotted in Figure 10.1 against Wi at different β and b. As we

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89

0 5 10 15 20 25 30 35 40 45 50 550.31

0.32

0.33

0.34

0.35

0.36

0.37

0.38

Newtonian

DR

Ons

et

Uavg

Wi

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

DR

%

β = 0.97, b = 5000, Ex = 103.1

β = 0.99, b = 10000, Ex = 67.3

β = 0.99, b = 5000, Ex = 33.7

Asymptotic Upper Limit of DR

Figure 10.1: Variations of the average streamwise velocity with Wi at different β andb values (average taken in time and all three spatial dimensions); the correspondingDR% is shown on the right ordinate. Solid symbols represent points in the asym-DRstage (defined in the text); the horizontal dashed line is the average of all asym-DRpoints.

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90

report the simulation runs with the minimum L+z that sustains turbulence as long

as L+z > 140, the box size for different data points in Figure 10.1 are in general

different; the specific box size used for each data point is reported in Figure 10.3.

The corresponding amount of drag reduction, measured in terms of the percentage

drop of the friction factor DR%, is marked on the right ordinate. The error bars

on the plot show the error estimates of the time-averaged quantity with the block-

averaging method (Flyvbjerg & Petersen 1989). All three curves from different β

and b are qualitatively similar and here we start by taking the β = 0.97, b = 5000

curve as an example. At Wi 6 16, Uavg remains at the same level as the Newtonian

turbulent flow, which apparently belongs to the pre-onset turbulence stage. After the

onset, DR% increases monotonically with Wi until Wi > 27, where it starts to level

off and converges to a limit. Within the range 27 6 Wi 6 30, Uavg is approximately

independent of Wi. Recall in Section 7.4 that MDR is identified by the convergence

of the friction factor (subject to the statistical fluctuations in the data), and thus

Uavg in this plot, upon increasing Wi, this range of Wi hence corresponds to the

MDR stage for β = 0.97, b = 5000. As discussed below, one main difference of this

asymptotic upper-limit of drag reduction from the experimentally-observed MDR,

is that its mean velocity profile is much lower than the Virk (1975) MDR profile.

For the convenience of discussion, we will refer to this stage as “asym-DR” in the

following text, instead of MDR, despite that it recovers the most important feature

of the latter. Simulation runs with Wi > 30 all eventually become laminar within

the 12000 time-unit interval, regardless of the L+z chosen. On the remaining two

curves, β = 0.99, b = 10000 and β = 0.99, b = 5000, the onset of drag reduction also

occurs at about Wionset & 16, but the increasing slope with Wi is different. The trend

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91

of changing slope is consistent with changes in the extensibility number; higher Ex

corresponds to steeper rise of Uavg after onset. At the high Wi end, asym-DR stages

can be identified in both curves, at 32 6 Wi 6 36 and 40 6 Wi 6 50 respectively,

after which the flow laminarizes. There is no discernible difference in Uavg among the

asym-DR stages for all three curves. Despite the range of parameters, all of them give

DR% ≈ 26%, i.e. the friction drag at asym-DR is constant for a given Re in spite of

variations in Wi, β and b. This is to our knowledge the first time this universal aspect

of MDR is reproduced in numerical simulations. Figure 10.1 summarizes the whole

data set we will present and discuss in the rest of this chapter, from which we clearly

see that major components of the transitions in viscoelastic turbulent flows, including

the pre-onset stage, intermediate DR and a universal asymptotic upper limit of drag

reduction, are well-captured by the transient solutions in MFUs, even at Re very close

to the laminar-turbulence transition.

Housiadas & Beris (2003) reported full-size DNS results at Reτ = 125 (Re =

7812.5), β = 0.9, b = 900 (Ex = 66.67) and various Wi up to 125. With these

parameters, the onset occurs at Wionset ≈ 6, smaller than but within the same order

of magnitude as our estimation in MFUs. Studies on ECS solutions (Li, Xi & Graham

2006, Li & Graham 2007) predict that Wionset = O(10) and decreases slowly with

increasing Re. As to the dependence of drag reduction on Wi, Housiadas & Beris

(2003) found that Uavg increases monotonically with Wi for the whole range of Wi they

studied; however, the slope drops greatly at Wi ≈ 50. They did not see a complete

convergence of Uavg for the range of Wi they studied. It is unclear with which stage

in our MFU solutions their slowly-growing stage (50 . Wi 6 125) matches. Since

DR% keeps on increasing, it should be naturally categorized in the intermediate

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92

DR stage (before asym-DR); however, we do not observe any appreciable change in

the increment slope against increasing Wi in our study. One possibility is that this

slowly-growing stage does not exist at the Re we study; since our Re is lower and

our simulations only include the structures in the buffer layer and part of the log-

law layer, the decrease in slope may be related to those structures excluded in our

simulation but included in theirs. Meanwhile we can not disprove the other possibility

that this stage corresponds to our asym-DR stage. Since our asym-DR stages only

span limited ranges of Wi, even if there indeed is a weak dependence on Wi, the

actual difference in Uavg must be small. Although our time average is taken in the

interval of 8000 time units, longer than most previous studies, the interval is still at

the same order of the longest time scale (O(Re)) in the system. Therefore our error

bars are not small enough to let us discern subtle changes in Uavg, if any exist. To

further reduce the errors would require increasing the length of each simulation run

by multiple times, which is practically infeasible in terms of the computational cost.

Nevertheless, since we have multiple (4− 6) points for each β and b, where we do not

see any consistent dependence on Wi, for the present we will treat these solutions as

time-dependent coherent structures with the same DR%.

The mean velocity profiles of several typical points on the β = 0.97, b = 5000

curve in Figure 10.1 are plotted in Figure 10.2; for comparison, the asymptotic lines

of the viscous sublayer (U+ = y+), the log-law layer of Newtonian turbulent flows

(U+ = 2.44 ln y+ + 5.2) (Pope 2000) and the universal profile of MDR summarized

by Virk (U+ = 11.7 ln y+ − 17.0) (Virk 1975), are also shown on the same plot. All

profiles from our simulations collapse well on that of viscous sublayer at y+ 6 5.

Further away from the wall, the Newtonian profile deviates from the U+ = y+ line

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93

100

101

0

2

4

6

8

10

12

14

16

18

20

y+

U+

Newtonian, L+z = 140

Wi = 16.0, L+z = 140

Wi = 17.0, L+z = 150

Wi = 19.0, L+z = 150

Wi = 23.0, L+z = 180

Wi = 27.0, L+z = 210

Wi = 29.0, L+z = 250

Viscous Sublayer

Log-law for Newtonian Flows

Virk’s MDR Profile

Figure 10.2: Mean velocity profiles (Newtonian and β = 0.97, b = 5000).

in the buffer layer. Even though Re is too low in the present simulations for the

log-law layer to be fully developed, the Newtonian profile still lies very close to the

semi-empirical log-law at y+ & 50. Among the viscoelastic cases, except that of

Wi = 16 which belongs to the pre-onset stage, the mean velocity profiles are all

elevated compared to the Newtonian case outside the viscous sublayer. The last two

curves, Wi = 27 and Wi = 29, are selected from the asym-DR stage and they collapse

well onto each other, although they are still notably lower than the Virk MDR profile.

We will further discuss the mean velocity profiles in Section 10.2.

In Figure 9.1 we presented the dependence of MFU box sizes on Wi at β = 0.97,

b = 5000; in Figure 10.3 we show the L+z values for all data points in Figure 10.1.

For the reason explained in Chapter 9, we use a minimum of L+z = 140 if the actual

minimal box size is smaller than this value. This truncation affects at most up to

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94

0 5 10 15 20 25 30 35 40 45 50 55140

160

180

200

220

240

260

Wi

L+ z

β = 0.97, b = 5000

β = 0.99, b = 10000

β = 0.99, b = 5000

Figure 10.3: Spanwise box sizes used in this study for various parameters. Solidsymbols represent points in the asym-DR stage.

Wi 6 16 for β = 0.97, b = 5000 and β = 0.99, b = 10000, and Wi 6 24 for

β = 0.99, b = 5000, which mostly belongs to the pre-onset stage. At higher Wi, L+z

is larger than 140 and should faithfully reflect the size of the minimal self-sustaining

coherent structures, which increases with increasing Wi with some uncertainty owing

to the initial-condition dependence (Chapter 9). A somewhat surprising finding is

that this trend persists in the asym-DR stage: L+z changes with Wi despite the

converged mean velocity profile and flow rate. This result suggests that different

points within the asym-DR stage in Figure 10.1 are distinguishable from one other,

i.e. they are not identical solutions, but rather different dynamical structures with

the same average velocity. We will further examine the similarities and differences

among these solutions at the asym-DR stage in the later sections of this chapter.

Comparing results at different β and b, the L+z in the asym-DR stage are close in

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95

magnitude, and they all fall into the range of 200 − 260, about twice the size of a

Newtonian MFU.

A natural question one may raise is about the legitimacy of comparing Uavg from

different parameters (Wi, β, b) in Figure 10.1 while the box size is changing. We would

like to address this point by first pointing out that for DNS study in general, unlike Re,

Wi, β or b, L+z itself is not a physically meaningful input parameter for the simulation

that can be adjusted independently; it is indeed an artificial restriction, same as L+x ,

on the spatial periodicity of the solution. Therefore the most straightforward way of

treating the box size is to make it much larger than the longest possible correlation

in the physical system; changing box size in that regime will have no effect on the

statistics of the solution. This is what we have referred to as “full-size” DNS. For box

size smaller than that, the concern of solution statistics being affected is alway present.

There is no reason to believe that a fixed L+z would be better than a varying one in

this sense. On the contrary, intrinsic length scales exist in turbulent flows, over which

coherent structures are spatially recurrent; in Newtonian turbulence, this length scale

is about 100 wall units in the spanwise direction as reviewed in Section 7.4. Small-box

simulations should take advantage of this inherent spatial-periodicity: box-sizes that

are multiples of these length scales should in principle minimize the box-size effects.

In viscoelastic flows, these length scales vary with polymer properties, and box size

should be adjusted accordingly. Finally, remember that the purpose of using MFU is

to isolate individual coherent structures and analyze their dynamics, while inevitably

sacrificing statistical accuracy to some extent. It is a model-based approach subject

to future verifications in full-size DNS.

Previous discussions focus on the dependence of the bulk flow rate and the length

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96

0 5 10 15 20 25 30140

160

180

200

220

240

260

Asy

mpt

otic

Upp

erLim

itof

DR

:26.

0%

DR%

L+ z

β = 0.97, b = 5000

β = 0.99, b = 10000

β = 0.99, b = 5000

LDR

HDR

Figure 10.4: Variations of spanwise box size at different DR%. Solid symbols representpoints in the asym-DR stage.

scales of MFUs on various parameters (Wi, β, b); here we examine the existence of

possible structure-flow rate correlations by plotting L+z against DR% in Figure 10.4. It

is interesting to note that the dependence of L+z on DR% is insensitive to the changes

in β and b: data points from different β and b roughly fall onto a single relationship,

i.e. for any given DR% before the asym-DR stage, the corresponding values of L+z

for different β and b are very close to one another. Note that the step size in our

MFU search is ∆L+z = 10; the discrepancies among the L+

z ∼ DR% relationships of

different β or b are smaller than the methodological uncertainty. Figure 10.4 shows

that the structural length scale increases monotonically after the onset of DR until

asym-DR is reached at DR% ≈ 26%, where L+z seems to diverge: i.e. L+

z increases

with approximately constant DR% and eventually turbulence does not sustain even

at larger boxes.

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97

Within the intermediate DR stage, an additional transition can be identified at

DR% ≈ 13%− 15% by a sharp change in the slope of the increasing L+z with DR%.

Note that in Figure 9.1, L+z is about 140 wall units at the onset of DR, and after

that L+z only increases by about 10 wall units when DR% reaches ∼ 13− 14%. From

DR% ≈ 15% to just before asym-DR (DR% ≈ 25%), L+z increases from ∼ 160

to ∼ 200 wall units. This transition divides the intermediate DR stage into two

parts, to which we will refer as LDR and HDR in the following text. As mentioned

in Section 7.4, the terms LDR and HDR are commonly used by other authors for

DR% . 35% and DR% & 35% respectively, whereas in this study they are used

to describe a qualitative transition within the intermediate stage. This transition is

further discussed below.

In summary, we have found transient viscoelastic turbulence solutions in MFU at

various Wi, β and b at a Re close to the laminar-turbulent transition, each of which

lasts > 12000 units in time. By studying the parameter-dependence of the bulk flow

Uavg and the structural length scale L+z , the whole multistage transition sequence,

including pre-onset, LDR, HDR and an asym-DR stage corresponding to the MDR

regime, where DR% reaches a universal upper limit, is observed, even though the

highest DR% we observe is less than 30%.

10.2 Flow statistics

We start our discussion of turbulent flow statistics by revisiting the mean velocity

profiles in Figure 10.2. The six viscoelastic runs shown in that plot are selected from

the pre-onset (Wi = 16), LDR (Wi = 17, Wi = 19), HDR (Wi = 23) and asym-

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98

100

101

0

2

4

6

8

10

12

14

16

18

20

y+

U+

Viscous Sublayer

Log-law for Newtonian Flows

Virk’s MDR Profile

Figure 10.5: Mean velocity profiles of 15 different asym-DR states (Wi: 27 ∼ 30 forβ = 0.97, b = 5000, Wi: 32 ∼ 36 for β = 0.99, b = 10000 and Wi: 40 ∼ 50 forβ = 0.99, b = 5000).

DR (Wi = 27, Wi = 29) stages, respectively. The two curves at asym-DR overlap

each other. In Figure 10.5, mean velocity profiles of all runs in the asym-DR stage,

including those of other Wi not shown in Figure 10.2, and those at different β and

b, are plotted together. All these profiles from different Wi, β and b collapse well

onto a single curve. This profile is clearly lower than Virk’s MDR profile, but is

universal to different polymer properties in our simulations. Within the intermediate

DR stage (Figure 10.2), there is also a difference between LDR and HDR. The two

LDR profiles (Wi = 17 and Wi = 19), although shifted upward compared with the

Newtonian profile, still keep roughly the same slope in the log-law layer. Most of the

drag reduction occurs in the buffer layer, while the log-law layer seems unaffected and

stays parallel with the Newtonian log-law, which is thus described as the “Newtonian

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99

0 10 20 30 40 50 60 70 80−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

y+

dU

+

dy+

-(dU

+

dy+

) new

t

Newtonian, L+z = 140

Wi = 17.0, L+z = 150

Wi = 19.0, L+z = 150

Wi = 23.0, L+z = 180

Wi = 29.0, L+z = 250

Figure 10.6: Deviations in mean velocity profile gradient from that of Newtonianturbulence (β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR:Wi = 29.

plug” by Virk (1975). In the HDR stage (Wi = 23), consistent with the experimental

observations of Warholic, Massah & Hanratty (1999), a change in the log-law slope

can also be noticed, although it is not as large as those reported at higher Re, where

DR% is much higher. The log-law slope of the HDR profile is higher and lies between

that of the Newtonian turbulence and that of asym-DR.

To see this difference more clearly, in Figure 10.6 we plot deviations in the gra-

dients of the mean velocity profiles from that of the Newtonian profile for several

selected runs. Note that with the constant-pressure-drop constraint, the mean wall

shear stress should be the same for all runs; in Figure 10.6 the mean shear rate values

at y+ = 0 of viscoelastic solutions are slightly higher than that of the Newtonian

solution owing to the shear-thinning effect. Beyond the viscous sublayer, drag reduc-

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100

0 5 10 15 20 25 300.04

0.05

0.06

0.07

0.08

0.09

0.1

DR%

dU

+/d

y+

| y+=

40

β = 0.97, b = 5000

β = 0.99, b = 10000

β = 0.99, b = 5000

LDR

HDR

Figure 10.7: Magnitude of mean velocity profile gradient at y+ = 40. Solid symbolsrepresent points in the asym-DR stage.

tion is reflected in the increase of the gradient. For LDR (Wi = 17, 19), this increase

is mainly localized in the buffer layer and a reflection of the curves can be noticed

at y+ ≈ 40 after which the deviations are rather small. In HDR and asym-DR, the

change of gradient is large and clear across the channel, except in the viscous sublayer

(y+ 6 5) where no big change is expected. This difference is not specific to the con-

ditions shown in Figures 10.2 and 10.6; it also exists between LDR and HDR when

β = 0.99, b = 10000 and β = 0.99, b = 5000. In Figure 10.7 we plot the magnitude

of dU+/dy+, measured at y+ = 40, versus DR% for all MFU runs. The dependence

of mean velocity profile gradient on DR% is roughly the same (within statistical un-

certainty) at different values of β and b. A distinction in this trend can be noticed

between relatively low and high DR%: significant increase of the gradient above the

buffer layer is only observed at DR% & 14%, before which change in the gradient is

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101

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

y+

-v′ xv

′ y/u

2 τ

Newtonian, L+z = 140

Wi = 17.0, L+z = 150

Wi = 19.0, L+z = 150

Wi = 23.0, L+z = 180

Wi = 29.0, L+z = 250

Figure 10.8: Profiles of the Reynolds shear stress (Newtonian and β = 0.97, b = 5000).LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR: Wi = 29.

small. This change well coincide with the LDR–HDR transition as identified from

Figure 10.4.

Recall that in Section 10.1, we defined the stages of LDR and HDR according to

the sudden change in the L+z vs. DR% relationship; here we demonstrated that this

transition corresponds well to the change in the log-law slope observed by other groups

between low DR% and high DR% at higher Re (Warholic, Massah & Hanratty 1999,

Min, Choi & Yoo 2003, Ptasinski et al. 2003, Li, Sureshkumar & Khomami 2006).

This is why we choose to use the terms “LDR” and “HDR”, notwithstanding that

our highest DR% is less than 30%. The fact that this transition can be observed at

DR% ≈ 13 − 15% suggests that this corresponds to a qualitative transition in the

process of drag reduction instead of the quantitative effect of DR%. Consequently,

we also expect that the DR% of the LDR–HDR transition should be a function of Re.

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102

0 10 20 30 40 50 60 70 80−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

y+

-v′ xv

′ y/u

2 τ+

(v′ xv

′ y/u

2 τ) n

ew

t

Newtonian, L+z = 140

Wi = 17.0, L+z = 150

Wi = 19.0, L+z = 150

Wi = 23.0, L+z = 180

Wi = 29.0, L+z = 250

Figure 10.9: Deviations in Reynolds shear stress profiles from that of Newtonianturbulence (β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR:Wi = 29.

0 5 10 15 20 25 300.39

0.41

0.43

0.45

0.47

0.49

DR%

-v′ xv

′ y/u

2 τ| y+

=40

β = 0.97, b = 5000

β = 0.99, b = 10000

β = 0.99, b = 5000

HDR

LDR

Figure 10.10: Magnitude of Reynolds shear stress at y+ = 40. Solid symbols representpoints in the asym-DR stage.

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103

Similarly, a distinctive change also occurs in the Reynolds shear stress profiles

during the LDR–HDR transition. As shown in Figure 10.8, Reynolds shear stress is

suppressed with increasing drag reduction. Comparing the profiles of LDR (Wi = 17,

19) and HDR, asym-DR (Wi = 23, 29), one can notice that at LDR, −v′xv′y/u2τ is

suppressed mainly in the buffer layer (5 . y+ . 30), and in the region y+ & 40 the

deviation is barely noticeable; whereas at HDR and asym-DR, suppression is observed

even near the center. Deviations of −v′xv′y/u2τ with respect to the Newtonian profile

are plotted in Figure 10.9, where this difference is clearer: in HDR and asym-DR,

magnitude of deviation is substantial across the entire channel except the viscous

sublayer. This distinction between local and global suppression of the Reynolds shear

stress is also observed at the LDR–HDR transitions at the other values of β and b we

studied. As shown in Figure 10.10, at y+ = 40 (above the buffer layer), Reynolds sheer

stress is substantially suppressed only after the LDR–HDR transition, which occurs at

DR% ≈ 13−15%. Warholic, Massah & Hanratty (1999) reported that the magnitude

of Reynolds shear stress is significantly lower in HDR than in Newtonian flow, and it

eventually drops to almost zero at DR% > 60%. While in some other studies, non-

zero (though still significantly smaller than the Newtonian case) Reynolds shear stress

was reported even for cases with more than 70% drag reduction (Warholic, Massah

& Hanratty 1999, Min, Choi & Yoo 2003, Ptasinski et al. 2003, Li, Sureshkumar

& Khomami 2006). Based on our study, these seemingly contradicting results can

be well reconciled: Figures 10.8 and 10.10 show that −v′xv′y/u2τ remains at the same

order of magnitude as the Newtonian value even in our asym-DR stage. Therefore, the

quantitative magnitude of −v′xv′y/u2τ is not the key difference between LDR and HDR;

it instead might be affected by both DR% and Re. It is the location where −v′xv′y/u2τ

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104

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3

y+

v′2 x

1/2/u

τ

Newtonian, L+z = 140

Wi = 17.0, L+z = 150

Wi = 19.0, L+z = 150

Wi = 23.0, L+z = 180

Wi = 29.0, L+z = 250

Figure 10.11: Profiles of root-mean-square streamwise and wall-normal velocity fluc-tuations (Newtonian and β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23;asym-DR: Wi = 29.

is suppressed that qualitatively indicates the transition. Indeed, despite the difference

in the magnitude of Reynolds shear stress reported in those studies (Warholic, Massah

& Hanratty 1999, Min, Choi & Yoo 2003, Ptasinski et al. 2003, Li, Sureshkumar &

Khomami 2006), one common observation is Reynolds shear stress is substantially

suppressed near the channel center only after the LDR–HDR regime. This agreement

is yet another evidence that this transition, initially identified in the L+z vs. DR%

plot (Figure 10.4), corresponds to the LDR–HDR transition observed in other studies

at much higher Re.

The root-mean-square velocity fluctuation profiles are shown in Figures 10.11,

10.12 and 10.13. After the onset of drag reduction, the streamwise velocity fluctua-

tions (Figure 10.11) increase with Wi until asym-DR is reached; meanwhile the peak

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105

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y+

v′2 y

1/2/u

τ

Newtonian, L+z = 140

Wi = 17.0, L+z = 150

Wi = 19.0, L+z = 150

Wi = 23.0, L+z = 180

Wi = 29.0, L+z = 250

Figure 10.12: Profiles of root-mean-square wall-normal velocity fluctuations (New-tonian and β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR:Wi = 29.

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y+

v′2 z

1/2/u

τ

Newtonian, L+z = 140

Wi = 17.0, L+z = 150

Wi = 19.0, L+z = 150

Wi = 23.0, L+z = 180

Wi = 29.0, L+z = 250

Figure 10.13: Profiles of root-mean-square spanwise velocity fluctuations (Newtonianand β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR: Wi = 29.

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106

of the profile moves away from the wall, reflecting the thickening of the buffer layer.

Both the wall-normal (Figure 10.12) and spanwise (Figure 10.13) velocity fluctuations

are suppressed with increasing Wi.

As to the LDR–HDR transition, the spanwise velocity fluctuation profiles show

most notable differences between these two stages. In Figure 10.13, the LDR profiles

resemble that of the Newtonian turbulence in the shape, though they are lower in

the magnitude. In particular, one can notice two bulges at y+ ≈ 16 and y+ ≈ 46

between which the curves are concave. This subtle concavity is absent in the HDR

and asym-DR stages, and in those stages this part of the curve is roughly straight.

Therefore, unlike turbulence in LDR where the spanwise velocity fluctuations are al-

most uniformly suppressed across the channel, in HDR and asym-DR stages, more

suppression occurs in the buffer layer and the lower edge of the log-law layer. This is

also observed in data at other values of β and b, but has not previously been reported

in the literature. Meanwhile, Warholic, Massah & Hanratty (1999) reported exper-

imentally that there is a maximum in the wall-normal velocity fluctuation profiles

when DR% . 35% whereas when DR% is high, the maximum becomes unrecogniz-

able. It is unclear though whether this is a quantitative effect of the substantially

suppressed wall-normal velocity fluctuations, as at high DR% their v′2y1/2/uτ magni-

tude is one order of magnitude smaller than that of the Newtonian profile, and the

noise of measurements can be comparable with the actual velocity fluctuation. In our

results (Figure 10.12), there is a very subtle maximum at y+ ≈ 45 in the Newtonian

profile as well. As Wi increases, this bulge decreases in height and shrinks in size,

with the lower edge moving away from the wall. At the HDR and asym-DR stages,

the profile is almost flat after the initial uprising region near the wall, and the bulge

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107

becomes unrecognizable. This effect, however, is not as obvious as the changes in the

spanwise velocity fluctuations.

It has also been reported experimentally that notable differences can be observed

in the streamwise velocity fluctuations between low DR% and high DR% (Warholic,

Massah & Hanratty 1999): when DR% . 35%, v′2x1/2/uτ increases with DR% and

the peak of the profile moves away from the wall; at high DR%, v′2x1/2/uτ is greatly

suppressed compared with the Newtonian flows. However, as shown in Figure 10.11,

this non-monotonicity is not observed in our MFU simulations; instead, our v′2x1/2/uτ

profiles at different stages all follow the former (low DR%) case in experiments. DNS

studies from other groups reported contradictory results on whether or not this non-

monotonic trend exists in streamwise velocity fluctuations (Ptasinski et al. 2003, Min,

Choi & Yoo 2003, Li, Sureshkumar & Khomami 2006). The origin and significance of

this discrepancy are not understood, but the fact that in those studies, comparisons

between different DR% were made under different constraints (constant flow rate

vs. constant pressure drop) may have contributed to the complexity in this issue.

Our observation (that the trend is monotonic) is consistent with Li, Sureshkumar &

Khomami (2006), where the constant-pressure-drop constraint was also applied.

We have shown earlier that in the asym-DR stage, the mean velocity profiles con-

verge to a single curve (Figure 10.5); here we resume the discussion of the turbulence

statistics in this stage. In Figure 10.14 we plot the RMS velocity fluctuations (left or-

dinate) and Reynolds shear stress (right ordinate) profiles for all the simulation runs

in the asym-DR stage (corresponding to the solid data points in Figure 10.1) with a

variety of Wi, β and b. The profiles of wall-normal and spanwise velocity fluctuations

converge for different parameters. The situation of the streamwise component is a bit

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108

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3

3.5

y+

v′2

1/2/u

τ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-v′ xv

′ y/u

2 τ

v′z

v′y

v′x

-v′xv

′y

Figure 10.14: Profiles of root-mean-square velocity fluctuations and Reynolds shearstress at 15 different asym-DR states (Wi: 27 ∼ 30 for β = 0.97, b = 5000, Wi: 32 ∼ 36for β = 0.99, b = 10000 and Wi: 40 ∼ 50 for β = 0.99, b = 5000).

complicated: the profiles from different parameters are very close to one another near

the wall and reach maxima at very similar values in the buffer layer; while beyond

the buffer layer, they spread out. To detect any possible parameter dependence of

v′2x1/2/uτ , we have examined the distributions of its magnitudes with respect to Wi, β

and b. Even though the v′2x1/2/uτ profiles do not merge in the asym-DR stage, there

is no identifiable trend of dependence of v′2x1/2/uτ on any of the parameters: v′2x

1/2/uτ

neither increases nor decreases with increasing Wi consistently in the asym-DR stage,

and the same applies for the other two parameters (β and b). Therefore we believe

that this dispersion of v′2x1/2/uτ profiles in Figure 10.14 is a result of statistical uncer-

tainty: it might take much longer simulation runs to obtain reliable averages on the

streamwise velocity fluctuations than many other quantities we have discussed. As

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109

0 10 20 30 40 50 60 70 800

0.05

0.1

0.15

0.2

0.25

y+

tr(α

)/b

Wi = 16.0, L+z = 140

Wi = 17.0, L+z = 150

Wi = 19.0, L+z = 150

Wi = 23.0, L+z = 180

Wi = 27.0, L+z = 210

Wi = 29.0, L+z = 250

Figure 10.15: Normalized profiles of the trace of the polymer conformation tensor(β = 0.97, b = 5000). Pre-onset: Wi = 16; LDR: Wi = 17, 19; HDR: Wi = 23;asym-DR: Wi = 27, 29.

to the Reynolds shear stress, the convergence is very good over most of the channel

except in a small region near the maxima of the profiles at y+ ≈ 30; this discrepancy,

as we have also examined, is again due to statistical uncertainty.

10.3 Polymer conformation statistics

We turn now to the statistics of the polymer conformation tensor. Figure 10.15 shows

the mean profiles of the trace of the polymer conformation tensor α, which physically

corresponds to the square of the end-to-end distance of the polymer chains, normalized

by its upper limit b, for several selected Wi with β = 0.97 and b = 5000. Perhaps the

most interesting observation is that although it is expected that polymers are more

highly stretched as Wi increases, this trend goes on in the asym-DR stage. The two

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110

0 5 10 15 20 25 30 35 40 45 50 550

0.1

0.2

0.3

0.4

tr(α

) avg/b

Wi

β = 0.97, b = 5000

β = 0.99, b = 10000

β = 0.99, b = 5000

Figure 10.16: Averaged trace of the polymer conformation tensor (average taken intime and all three spatial dimensions). Solid symbols represent points in the asym-DRstage.

curves belonging to the asym-DR stage in Figure 10.15 do not overlap: i.e. tr(α) keeps

on increasing with Wi even though the mean velocity (as well as many other velocity

statistical quantities) converges. This trend is confirmed in Figure 10.16, where the

average tr(α) normalized by b is plotted against Wi for all simulation runs reported

here. Data points in the asym-DR stage are filled. For every β and b, tr(α)avg/b

increases monotonically with Wi: the slope is relatively low at Wi ∼ O(1); after

the onset of drag reduction (Wi & 16), the curves are steeper and tr(α)avg/b roughly

rises in straight lines; tr(α)avg/b continues to increase at approximately constant slope

even after asym-DR is reached. In addition, the ranges of tr(α)avg/b at the asym-DR

stages of different β or b are far apart from one another even though their Uavg is very

close: for example, at β = 0.99 and b = 10000, tr(α)avg/b is more than twice as large

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111

as that of β = 0.99 and b = 5000, and almost three times the magnitude at β = 0.97

and b = 5000. Similar to our findings, Housiadas & Beris (2003) reported in their

DNS studies that while the increase of mean velocity slows down at high Wi, tr(α)

continues to increase with Wi at about the same rate.

Another observation from Figure 10.15 is that the profile changes shape with

increasing Wi. At relatively low Wi, tr(α) decreases monotonically with distance

away from the wall y+. At higher Wi, the profile becomes non-monotonic with a

maximum some distance from the wall (in the buffer layer). This distance increases

with increasing Wi. This observation can be explained kinematically. The process of

near-wall polymer stretching is a combined effect of shear flow in the viscous sublayer

and extensional flow in the buffer layer. The former is relatively more effective in

stretching polymers at low Wi and the latter dominates at higher Wi; consequently

the maximum location reflects the shift of the dominant kinematic effect.

Although the separation between the maximum location of tr(α) and the wall

in Figure 10.15 might be thought to coincide with the LDR–HDR transition, this

agreement is totally fortuitous: unlike the changes in turbulent flow statistics we

studied earlier, this accordance between the Wi where the maximum shifts away

from the wall and the Wi at the LDR–HDR transition is specific to the choice of

β = 0.97 and b = 5000. In Figure 10.17 we plot the location of the maximum of

tr(α) against DR% and Wi, respectively, for all β and b we studied. One can see

from Figure 10.17(a) that although the detachment of the maxmum from the wall

occurs at DR% ≈ 15% for β = 0.97 and b = 5000, close to the LDR–HDR transition,

it takes place at much lower DR% under other β and b (far before the LDR–HDR

transition). Meanwhile, the dependence on Wi for different β and b is very close. This

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112

−5 0 5 10 15 20 25 300

2

4

6

8

10

12

y+

ofth

em

axim

umof

tr(α

)

DR%

β = 0.97, b = 5000

β = 0.99, b = 10000

β = 0.99, b = 5000

(a) Dependence on DR%

0 5 10 15 20 25 30 35 40 45 50 550

2

4

6

8

10

12

y+

ofth

em

axim

umof

tr(α

)

Wi

β = 0.97, b = 5000

β = 0.99, b = 10000

β = 0.99, b = 5000

(b) Dependence on Wi

Figure 10.17: Position of the maximum in the tr(α) profile. Solid symbols representpoints in the asym-DR stage.

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113

is consistent with the above explanation that this displacement of the maximum is a

Wi-effect: polymer react to different local kinematics differently with increasing Wi;

the small differences between data from different β and b values is accounted for by

differences in local strain rates at the same Wi. The lack of correlation between the

maximum location of tr(α) profiles and DR%, and the increasing tr(α) in the asym-

DR stage where the mean velocity converges, suggest that the mean deformation of

polymer chains is a process independent of the transitions among LDR, HDR and

asym-DR (or MDR in experiments).

Polymers exert their influence on the flow field through the polymer force term,

fp = 2(1 − β)/(ReWi)(∇ · τ p). Consequently, one might intuitively expect fp to

saturate in the asym-DR stage, instead of α or τ p, so that polymer would contribute

equally to the momentum balance (Equation (8.1)) despite the differences in the

magnitude of polymer stress. However fp profiles do not converge in the asym-

DR stage either, although the discrepancies of fp among different parameters are

significantly smaller than those of tr(α).

10.4 Spatio-temporal structures

Above we discussed statistical representations of the velocity and polymer conforma-

tion fields of MFU solutions during the multistage transitions. As MFU solutions

contain the structural information of the essential self-sustaining process of turbu-

lence, we study here the spatial and temporal images of these transient structures.

Figures 10.18, 10.19, 10.20 and 10.21 show the spatial-temporal patterns in z and t

of the shear rate ∂vx/∂y at the lower wall y = −1, at fixed streamwise location of

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114

x = 0, taken from one selected run for each of Newtonian turbulence, LDR, HDR and

asym-DR. The choice of x is arbitrary since the system is translation-invariant in x.

The distribution in the z direction of the wall shear rate is recorded every time unit

and plotted in color-scale in the axes of t and z+. A length of 8000 time units of each

simulation run, after turbulence reaches the statitistically-steady range, is included

in the plot. To aid interpretation, two periods in z are shown. Along with the wall

shear rate patterns, the spatially-averaged velocity Ubulk is also shown. Note that the

time-dependence of Ubulk is physically meaningful only in minimal flow units; in a

full-size DNS solution, the spatial average of any quantity should in principle be the

same as the ensemble average, and should be invariant with time. Also plotted is the

z average of the wall shear rate 〈∂vx/∂y〉z as a function of time; note that the time

average of this quantity is 2 owing to the fixed pressure gradient constraint.

Figure 10.22 shows representative snapshots of the velocity field during differ-

ent stages. Two snapshots are selected for each simulation run that is shown in

Figures 10.18, 10.19, 10.20 and 10.21, marked with (Reg) and (LS) in their cap-

tions according to the criterion to be discussed below. In each of them, isosur-

faces for two quantities are plotted in a 3D view of the simulation box. The flat

translucent sheets with pleats are isosurfaces of streamwise velocity vx, taken at

the magnitude of 0.6vx,max, where vx,max is the maximum value of vx in the do-

main for the given snapshot. The pleats correspond to low-speed streaks, where

slowly-moving fluid near the wall is lifted upward toward the center. The tube-like

objects with opaque dark colors are the isosurfaces of a measure of the streamwise-

vortex strength Q2D, whose definition we now describe. We apply a modified version

of the Q-criterion of vortex identification (Jeong & Hussain 1995, Dubief & Del-

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115

Fig

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116

Fig

ure

10.1

9:D

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ics

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(Re

=36

00,

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00,L

+ x=

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+ z=

150)

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esar

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ote:

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117

Fig

ure

10.2

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(Re

=36

00,

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=0.

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00,L

+ x=

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118

Fig

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10.2

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00,

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119

(a) Newtonian (Reg), L+z = 140;

t = 8500, vx = 0.25, Q2D = 0.025.(b) Newtonian (LS), L+

z = 140;t = 4600, vx = 0.27, Q2D = 0.012.

(c) LDR (Reg): Wi = 19, L+z = 150;

t = 5900, vx = 0.26, Q2D = 0.024.(d) LDR (LS): Wi = 19, L+

z = 150;t = 8200, vx = 0.29, Q2D = 0.0079.

Figure 10.22: Typical snapshots of the flow field (Re = 3600, β = 0.97, b = 5000,L+x = 360). (Reg) denotes snapshots chosen from “regular” turbulence, and (LS)

denotes snapshots of “low-shear” events. Translucent sheets are the isosurfaces ofvx = 0.6vx,max; opaque tubes are the isosurfaces of Q2D = 0.3Q2D,max. The valuesof vx and Q2D for each plot is shown in its caption. Note that (LS) states typicallyhave much lower Q2D values than (Reg) states. The bottom wall of each snapshotcorresponds to the wall shear rate patterns shown in Figures 10.18, 10.19, 10.20 and10.21 at corresponding time. (To be continued).

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120

(e) HDR (Reg): Wi = 23, L+z = 180;

t = 7700, vx = 0.31, Q2D = 0.026.(f) HDR (LS): Wi = 23, L+

z = 180;t = 7300, vx = 0.31, Q2D = 0.0089.

(g) asym-DR (Reg): Wi = 29, L+z = 250;

t = 8500, vx = 0.27, Q2D = 0.018.(h) asym-DR (LS): Wi = 29, L+

z = 250;t = 8900, vx = 0.31, Q2D = 0.0050.

Figure 10.22: (Continued).

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121

cayre 2000, Wu et al. 2005): i.e. by comparing the magnitudes of the vorticity

tensor and the rate-of-strain tensor, one can identify the local regions manifesting

strong vortical motions. For low Re, the buffer layer structure dominates the tur-

bulence, so we use the Q-criterion in the y-z 2D plane only to focus on vortices

aligned along the mean flow direction. Specifically, we compute the 2D versions

of the rate-of-strain tensor Γ2D ≡ (1/2)(∇v2D + ∇vT2D) and the vorticity tensor

Ω2D ≡ (1/2)(∇v2D −∇vT2D), where ∇v2D ≡ (∂vy/∂y, ∂vz/∂y; ∂vy/∂z, ∂vz/∂z); then

calculate the quantity Q2D ≡ (1/2)(‖Ω2D‖2 − ‖Γ2D‖2). Positive magnitudes of Q2D

would indicate regions having streamwise vortices; in Figure 10.22 the isosurfaces of

Q2D = 0.3Q2D,max are shown, where Q2D,max is the maximum value of Q2D in the

domain for the given snapshot. Note that this varies substantially among different

snapshots; the isosurface value for each image is reported in the caption.

A typical coherent structure of Newtonian turbulence contains a pair of stream-

wise vortices staggered alongside one sinuous low-speed streak, e.g. the structure at

the bottom wall of Figure 10.22(a). The dynamics around a single streak is sufficient

to make a self-sustaining process: the vortices on different sides of the streak rotate

in opposite directions so that the low-speed fluid near the wall between them is lifted

upward, forming the streak; instabilities of the streak will bring forth streamwise

dependence into its morphology, which through nonlinear interactions further main-

tains the vortices (Hamilton et al. 1995, Waleffe 1997, Jimenez & Pinelli 1999). In

Figures 10.18, 10.19, 10.20 and 10.21, low-speed streaks correspond to minima of the

wall shear rate distributions in the z direction, which in contour plots are observed

as dark stripes. The Newtonian MFU solution (Figure 10.18) contains one almost

continuous streak during the whole time range shown, which confirms that a self-

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122

sustaining process involving one streak (and the vortices around it), lasting for a very

long life-time, dominates the dynamics of the transient solution. With the transla-

tion invariance in z, the streak is not bound to any position and is free to drift in the

spanwise direction. However, there are still certain periods (e.g. 6200 . t . 6800 and

7900 . t . 8600) when the streak appears to be quiescent and stays with the same

z location for a fairly large amount of time; in some other time intervals the streak

can be very active and move rapidly in the transverse direction (e.g. 5000 . t . 6200

and 7300 . t . 7900). The LDR stage (Figure 10.19) is qualitatively similar to the

Newtonian case with one continuous streak dominating the dynamics for a long time

period. In the particular case we show, there is only one break point, at t ≈ 7400,

where the first streak decays and meanwhile a second streak is growing. The minimal

spanwise box size to sustain turbulence is however slightly larger, which indicates that

the self-sustaining coherent structure is wider in size, resulting in an increase of streak

spacing. In HDR, as shown in Figure 10.20, the number of streaks in the minimal

box varies between one and two, and complex dynamics are seen from time to time.

These dynamics are also evident in asym-DR (Figure 10.21) where more frequently it

involves two streaks although a single streak can sometimes also been found. These

complex activities and dynamics of the streaks are observed through various events

that change the topology of the streak patterns, including: emergence of new streaks

(e.g. t ≈ 11300, z+ ≈ 120 in Figure 10.20 and t ≈ 6500, z+ ≈ 30 in Figure 10.21);

decay of existing streaks (e.g. t ≈ 8200, z+ ≈ 25 in Figure 10.20); merger of multiple

(typically two) streaks into one (e.g. t ≈ 6600, z+ ≈ 160 in Figure 10.21) and division

of one streak into multiple streaks (e.g. t ≈ 9600, z+ ≈ 125 in Figure 10.21). This

transition from single-streak dynamics to multiple-streak dynamics at the LDR–HDR

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123

transition suggests that the underlying self-sustaining mechanism of turbulence may

have changed; complex dynamics involving interactions between streaks might be

essential in sustaining turbulent motions in HDR and asym-DR stages.

Recall in Figure 10.4 that when the LDR–HDR transition occurs, the dependence

of L+z on DR% undergoes an abrupt transition; this can be interpreted based on

the observations in Figures 10.18, 10.19, 10.20 and 10.21. In the LDR stage, the

underlying self-sustaining process is qualitatively the same as the Newtonian turbu-

lence, which involves the nonlinear interactions between a single low-speed streak

and the streamwise vortices on its both sides (Hamilton et al. 1995, Waleffe 1997,

Jimenez & Pinelli 1999). Viscoelasticity reduces the drag by weakening the vortical

motions (Li, Xi & Graham 2006, Li & Graham 2007) and the increase of L+z is caused

merely by the enlargement of the coherent structures (Li & Graham 2007). After the

LDR–HDR transition, viscoelasticity is strong enough to suppress the “Newtonian”

coherent structures (as predicted by earlier ECS study of Li, Xi & Graham (2006)

and Li & Graham (2007)), and the process involving a single isolated streak cannot

sustain turbulence for a very long time (see the relatively shorter streak segments

in Figures 10.20, 10.21). As a result a new self-sustaining process involving inter-

streak interactions arises, the details of which have yet to be elucidated. Therefore

the increase of L+z in the HDR stage involves both the contribution from the enlarged

structure by viscoelasticity, and the extra room needed to accommodate more streaks.

As to the turbulent dynamics reflected by the evolutions of Ubulk and the mean

wall shear rate (bottom panels of Figures 10.18, 10.19, 10.20 and 10.21), one interest-

ing observation is that there are certain moments in the self-sustaining process when

the change of Ubulk can be inferred by the shear rate at the wall. Specifically, during

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124

these moments, the wall shear rate is low in magnitude and its curve remains rela-

tively smooth for O(100) time units; meanwhile the mean velocity increases steadily.

Examples of these events include: t ≈ 4400 of Figure 10.18, t ≈ 4200, 4900 and 8000

of Figure 10.19, t ≈ 6200, 6900, 7300, 8800 and 10100 of Figure 10.20 and t ≈ 5500,

5800, 8100, 8800, 9700, 11000, 11300 and 11600 of Figure 10.21. By comparing these

temporal evolution plots with the spatial-temporal wall shear rate patterns (top pan-

els of Figures 10.18, 10.19, 10.20 and 10.21), one may find that these events usually

correspond to the moments when the patterns are blurry: i.e. the wall shear rate

has relatively small variance in both space and time. Besides, these events appear

to occur more often as DR% increases; to quantify their frequency of occurrence,

and its dependence on Wi, simulations much longer in time are required, which will

be presented in Chapter 11. To a first approximation, the correlation between bulk

velocity and the wall shear rate can be interpreted as such: since the driving force of

the flow, the mean pressure gradient, is fixed, the change of the total momentum in

the flow unit is mainly determined by the rate momentum is consumed at the wall;

when shear rate at the wall is low, there is less momentum being transferred to the

wall by viscous shear stress, which makes it easier to accumulate momentum in the

flow unit and increase the mean velocity.

In the 3D views of velocity fields shown in Figure 10.22, one of the two snapshots

presented for each run are taken from one of these “low-shear” events, marked as (LS)

in the caption; and the other is from a regular turbulence cycle, marked as (Reg).

The typical snapshot of “regular” Newtonian turbulence (Figure 10.22(a)) has been

discussed above. At LDR (Figure 10.22(c)), the structure is qualitatively similar

with one sinuous streak near each wall surrounded by streamwise vortices. At HDR

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125

(Figure 10.22(e)) and asym-DR (Figure 10.22(g)), this type of streak-vortex struc-

ture is still observed, though very often two streaks can be observed near each wall.

Compared with these snapshots of “regular” turbulence (Figures 10.22(a), 10.22(c),

10.22(e), 10.22(g)), those taken during the “low-shear” events (Figures 10.22(b),

10.22(d), 10.22(f), 10.22(h)) in general have much lower vortex strength, as reflected

by lower Q2D magnitudes. Meanwhile, the streaks are less wavy in shape: the x-

dependence of the streak morphology is weak. (This would explain the increased

smoothness of the 〈∂vx/∂y〉z v.s. t curve: since average is taken only in the z-direction

at a fixed x position, dependence of ∂vx/∂y on x will be reflected in temporal fluc-

tuations owning to the convection of flow structures). As discussed above, these

“low-shear” events occur more frequently as DR% increases, therefore we expect that

in a full-size system the probability of observing relatively straight streaks is higher in

HDR and asym-DR (or MDR in experiments) stages, while at lower DR% the streaks

should be mostly wavy. This is consistent with the observation by Li, Sureshkumar

& Khomami (2006) in full-size DNS that long straight streaks are more predominant

when the HDR regime is reached. The nature of these “low-shear” events is as yet

unclear. How these events are triggered and what roles they play in the self-sustaining

processes of turbulence will be important for further understanding of drag reduction

by polymers. Some further investigation into this dynamical feature of turbulence in

MFUs will be presented in Chapter 11.

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126

Chapter 11

Toward an understanding of the

dynamics: active and hibernating

turbulence

11.1 Intermittent dynamics in MFU

Observations in Chapter 10 demonstrate that MFU solutions almost (with the ex-

ception of the Virk MDR profile in time-averages) fully recover the qualitative tran-

sitions previously reported in experiments and full-size DNS studies, most of which

were conducted at much higher Re. Since MFU contains isolated individual coher-

ent structures only, instead of populations of them, temporal intermittency of the

self-sustaining process is more readily identifiable via this approach. Attempting to

interpret these transitions in terms of temporal dynamics in MFU is the natural next

step to take. In particular, spatiotemporal structures presented in Section 10.4 show

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127

intermittent time periods with substantially low magnitudes of wall shear rate in all

stages of transition. These periods occur much more frequently in HDR and asym-

DR stages, where complex dynamics are also observed in the self-sustaining process.

Motivated by these results, we further focus on these periods in this chapter. Results

presented below will show that although in the current MFU study we are not able

reach comparable level of drag reduction with experimental MDR, new directions

that might lead to a final understanding of the mechanism of MDR will be pointed

out based on the study of these dynamics.

Figures 11.1 and 11.2 show time series of instantaneous bulk average velocity Ubulk

and area-averaged shear rate 〈∂vx/∂y〉 at the top and bottom walls for Newtonian

flow and viscoelastic flow at Wi = 29 (where DR% = 26 and L+z has increased from

140 to 250; results in this chapter are all obtained at Re = 3600, β = 0.97, b = 5000

and L+z = 360), respectively. (Note that here average for ∂vx/∂y is taken in both

x and z directions). In the Newtonian case one occasionally observes long-lasting

periods during which the shear rate at one or both walls is substantially lower than

the average value of 2 – for example the time interval 5000 < t < 5500. By mo-

mentum conservation, the bulk velocity increases during these periods. For reasons

that will emerge as the discussion proceeds, these periods will be termed “hiberna-

tion”. Turbulence outside these periods will be termed “active”. As Wi increases,

it is observed that hibernation periods become increasingly frequent – since the bulk

velocity increases during these periods their high frequency contributes substantially

to drag reduction. Note that the flow does not closely approach the laminar state

(Ubulk = 2/3) during hibernation periods.

To systematically identify hibernation events, two criteria are used: (1) area-

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128

23

〈∂vx/∂y〉t 0.350.

4

Ubulk

010

0020

0030

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〈∂vx/∂y〉b

t

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ure

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(“b”–

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sign

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ion

expla

ined

inth

ete

xt.

Das

hed

lines

show

the

line〈∂v x/∂y〉=

1.80

.T

ime

aver

age

of〈∂v x/∂y〉i

s2.

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129

23

〈∂vx/∂y〉t 0.350.

4

Ubulk

010

0020

0030

0040

0050

0060

00123

〈∂vx/∂y〉b

t

Fig

ure

11.2

:M

ean

shea

rra

tes

atth

ew

alls

(“b”–

bot

tom

,“t

”–to

p)

and

bulk

velo

cityU

bu

lkas

funct

ions

ofti

me

for

typic

alse

gmen

tsof

ah

igh

-Wi

sim

ula

tion

run

(Re

=36

00,

Wi

=29

=0.

97,b

=50

00,L

+ x=

360,L

+ z=

250)

.R

ecta

ngu

lar

sign

als

inth

em

iddle

pan

elin

dic

ate

the

hib

ernat

ing

per

iods

atth

ew

all

ofth

eco

rres

pon

din

gsi

de,

iden

tified

wit

hth

ecr

iter

ion

expla

ined

inth

ete

xt.

Das

hed

lines

show

the

line〈∂v x/∂y〉=

1.80

.T

ime

aver

age

of〈∂v x/∂y〉i

s2.

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130

0 5 10 15 20 25 300

5

10

15

20

25

30

Wi

DR

%

100

120

140

160

180

200

220

240

260

L+ z

Figure 11.3: Level of drag reduction and spanwise box size as functions of Wi (New-tonian and β = 0.97, b = 5000).

averaged wall shear rate at one or both walls drops below a cutoff value 〈∂vx/∂y〉|cutoff =

1.8; and (2) it stays there for longer than a certain amount of time ∆tcutoff = 50. Hi-

bernating periods identified with these criteria are shown in the middle (bulk velocity)

panels of Figures 11.1 and 11.2 as rectangular signals, on the top or bottom of the

plot according to the wall(s) on which the criterion is satisfied. This criteria is so

chosen since it captures the main phenomenological characteristics of these periods.

Although the choices of cutoff values are to some extent arbitrary, we have found

that changing these values within a reasonable range does not qualitatively affect the

following discussion.

With these periods clearly identified, we can now quantify the dependence of their

frequency and duration on viscoelasticity. Since turbulence hibernation occurs inter-

mittently, and the time scales between two adjacent occurrences can be rather long,

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131

0 5 10 15 20 25 300

200

400

600

800

1000

1200

Wi

Tim

eSc

ales

0.1

0.15

0.2

0.25

0.3

FH

TA

TH

FH

Figure 11.4: Time scales (left ordinate) and fraction of time spent in hibernation(right ordinate) as functions of Wi (Newtonian and β = 0.97, b = 5000): TA is themean duration of active periods; TH is the mean duration of hibernating periods; FH

is the fraction of time spent in hibernation.

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132

especially for Newtonian and LDR turbulence, much longer simulations (compared

with results in Chapter 10) are needed for satisfactory statistics. Extended amount

of MFU simulations are thus performed for selected Wi and fixed values of β = 0.97,

b = 5000. These results are presented in Figure 11.4, as functions of Wi, in terms

of: the mean duration of the hibernation periods TH, mean duration of active periods

TA, and fraction of time spent in hibernation FH. The corresponding DR% and box

size (set to be the same as in Chapter 10) are shown in Figure 11.3 for reference.

For each Wi, multiple runs with independent initial conditions are included in the

average; each of them lasts for a minimum of 8000 time units after the statistically-

converging regime is reached; the total amount of time included in each data point

ranges from 32000 to 148600 time units (O(10Re) or longer). Error bars for TA and

TH are computed assuming that each transition between active and hibernating pe-

riods are independent from one another; error bars for DR% and FH are estimated

with the block-averaging method (Flyvbjerg & Petersen 1989) with a fixed block size

of 4000 time units (O(Re)).

Several important observations emerge from these results. First, the average du-

ration TH of a hibernating period is almost completely insensitive to Wi. In contrast,

the average time TA between two neighboring hibernating periods decreases sub-

stantially after onset of drag reduction. Accordingly, the fraction of time spent in

hibernation is determined only by TA, since TH does not depend on Wi. These re-

sults indicate that in the high Wi regime, viscoelasticity compresses the lifetime of an

active turbulence interval, facilitating the occurrence of hibernation, while having no

effect on hibernation itself. The net outcome is hibernation becomes an increasingly-

significant component of the overall turbulent dynamics in the high-Wi regime. Also

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133

200 250 300 350 400 450 500 550 6001

2

3

t

〈∂v x

/∂y〉 b

0.35

0.40

Ubulk

2

3

〈∂v x

/∂y〉 t

a b c d e

Figure 11.5: A hibernation event (200 6 t 6 600 in Figure 11.2). Thick black linesare mean wall shear rates and bulk velocity Ubulk at Wi = 29. Thin colored lines arefrom Newtonian simulations started at the corresponding colored dots, using velocityfields from the Wi = 29 simulation as initial conditions.

noted in Figure 11.4 is that substantial decrease in TA, and thus increase in FH, are

only observed at Wi > 19 (obviously higher than Wionset, which is below 16 as shown

in Figure 11.3), which coincides with the value where LDR–HDR transition occurs as

reported in Chapter 10. This suggests that those qualitative changes in flow statistics

between LDR and HDR might be linked with the increased frequency of hibernation.

Further investigation of the effect of hibernation on flow statistics is proposed for our

future work (Section 13.1).

The insensitivity of TH to Wi suggests that flow during hibernation does not

strongly stretch polymer molecules. Indeed, as shown below (Figure 11.10), at

Wi = 29 the peak value of 〈αyy〉, which is closely associated with streamwise vor-

tex suppression (Stone, Roy, Larson, Waleffe & Graham 2004, Li & Graham 2007,

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134

Procaccia et al. 2008), drops from about 210 in active turbulence to about 5 during

hibernation, a 40-fold reduction. These results suggest that hibernation should be

very similar in the Newtonian and viscoelastic cases. To test this possibility, velocity

fields from time instants before and during a hibernation event at Wi = 29 were used

as initial conditions for a Newtonian simulation, the trajectories of which were then

compared with those from the original viscoelastic simulation. Figure 11.5 illustrates

the original viscoelastic trajectory (thick black line) as well as Newtonian trajectories

(colors) started at various times. For the Newtonian run starting before any sign

of hibernation is observed (t = 205), active turbulence is sustained. However, the

runs started from later times show that once the system begins to enter hibernation,

removing the polymer stress does not cause turbulence to revert to an active state,

although the depth and duration of hibernation are weakly dependent on the time at

which their initial conditions are taken from the original viscoelastic run. In short,

while polymer increases the probability of entering hibernation, it has little effect on

flow within the hibernation region itself.

To better understand the above results, we examine more closely the hibernat-

ing period shown in Figure 11.5. Several time instants are selected as marked: (a)

is an instant right before turbulence enters hibernation; (b) is one on the path to-

ward hibernation; (c) and (d) are within hibernation; (e) is after turbulence becomes

reactivated. Figure 11.6 shows instantaneous area-averaged velocity profiles in the

bottom half of the channel for these instants, plotted in inner units based on the in-

stantaneous wall shear stress at the bottom wall (denoted by the superscript ∗ rather

than +). In active turbulence (a and e), the profiles fluctuate substantially. Profiles

for instants completely in hibernation (c and d) are fundamentally different. In par-

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135

100

101

0

5

10

15

20

25

y∗

U∗ m

ean

abcde

Viscous sublayer

Newtonian log-law

Virk MDR

Figure 11.6: Instantaneous mean velocity profiles of selected instants before, duringand after a typical hibernating period (marked with grid-lines in Figure 11.5). Profilesfor the bottom half of the channel are shown; superscript “*” represents variablesnondimensionalized with inner scales based on instantaneous mean shear-stress atthe wall of the corresponding side. Black lines show important asymptotes: “viscoussublayer”, U∗mean = y∗; “Newtonian log-law”, U∗mean = 2.44 ln y∗ + 5.2 (Pope 2000);“Virk MDR”, U∗mean = 11.7 ln y∗ − 17.0 (Virk 1975).

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136

100

101

0

5

10

15

20

25

y∗

U∗ m

ean

Wi = 29, instant (c)

Wi = 29, instant (e)

Newtonian, instant (c)

Newtonian, instant (e)

Viscous Sublayer

Log-law for Newtonian Flows

Virk’s MDR Asymptote

Figure 11.7: Comparison between hibernation in Newtonian and high-Wi viscoelasticflows (the Newtonian simulation is the one starting from t = 260 in Figure 11.5).Instantaneous mean velocity profiles for instants in hibernation (c) and after turbu-lence is reactivated (e) are show (marked with grid-lines in Figure 11.5). Profiles forthe bottom half of the channel are shown.

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137

Figure 11.8: Flow structures at selected instants before, during and after a typicalhibernating period (marked with grid-lines in Figure 11.5). Green sheets are isosur-faces vx = 0.3, pleats correspond to low-speed streaks; red tubes are isosurfaces ofQ2D = 0.02, Q2D is defined in Section 10.4. Only the bottom half of the channel isshown.

ticular, in the range 15 . y∗ . 40, both profiles show a clear log-law relationship

with a slope very close to the MDR asymptotic slope of 11.7 reported by Virk (1975)

(also shown on the plot). The Newtonian hibernation periods are very similar and

the Virk MDR slope is observed there as well. In Figure 11.7 we can see that at in-

stant (c) mean velocity profiles are almost indistinguishable between the Newtonian

and Wi = 29 simulations. Their difference becomes noticeable only after turbulence

returns to active periods.

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138

0 10 20 30 40 50 60 70 80 900

200

400

600

800

1000

1200

1400

y∗

αx

x

abcde

Figure 11.9: Instantaneous profiles of αxx (streamwise polymer deformation) for in-stants marked in Figure 11.5. Profiles for the bottom half of the channel are shown.

Figure 11.8 shows flow structures corresponding to these time instants. Within

active periods ((a) and (e)), turbulence shows highly 3D coherent structures consist-

ing of streamwise vortices and low-speed streaks (Jimenez & Moin 1991, Robinson

1991, Waleffe 1997). During hibernation ((c) and (d), also (b)), streamwise vortices

are significantly weaker; low-speed streaks are still observed, but are weak and only

weakly dependent on x. Weak streamwise vorticity and three-dimensionality are also

distinct characteristics of flow in the MDR regime (Virk 1975, White et al. 2004,

Housiadas et al. 2005, Li, Sureshkumar & Khomami 2006, White & Mungal 2008).

The weak effect of viscoelasticity on hibernating turbulence may lie in its nearly

streamwise-invariant kinematics. In the limiting case of a streamwise invariant steady

flow, material lines cannot stretch exponentially (Ottino 1989); accordingly, polymer

stretch in such a flow will not be substantial. As shown in Figures 11.10 and 11.11,

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139

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

y∗

αyy

abcde

Figure 11.10: Instantaneous profiles of αyy (wall-normal polymer deformation) forinstants marked in Figure 11.5. Profiles for the bottom half of the channel are shown.

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

y∗

αzz

abcde

Figure 11.11: Instantaneous profiles of αzz (spanwise polymer deformation) for in-stants marked in Figure 11.5. Profiles for the bottom half of the channel are shown.

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140

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

y∗

-<v′∗ x

v′∗ y

>

abcde

Figure 11.12: Instantaneous profiles of Reynolds shear stresss for instants marked inFigure 11.5. Profiles for the bottom half of the channel are shown.

compared with active periods, in hibernation intervals polymer is almost undeformed

in the transverse directions; while in active turbulence, transverse polymer deforma-

tion is known to suppress streamwise vortices (Stone, Roy, Larson, Waleffe & Graham

2004, Dubief et al. 2005, Li & Graham 2007, Procaccia et al. 2008). During hiber-

nation, deformation is noticeable only in the streamwise direction (Figure 11.9), the

direction of mean flow. Unlike in active periods, αxx profile in hibernation is mono-

tonic, and decreases with distance from the wall, which reflects the distribution of

the mean shear rate.

Finally, the Reynolds shear stress during hibernation drops to very low values

relative to active turbulence (Figure 11.12); the peak value during hibernation is

about 0.3 compared to values near unity in active turbulence. Again, this result

is consistent with observations near and in the MDR regime (Warholic, Massah &

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141

Hanratty 1999, Warholic et al. 2001, Ptasinski et al. 2001, 2003).

The qualitative picture that emerges from these simulations is thus the follow-

ing. Active turbulence generates substantial stretching of polymer molecules. The

resulting stresses act to suppress this turbulence and drive the flow toward a very

weakly turbulent hibernating regime. During hibernation the polymer molecules are

no longer strongly stretched and they relax toward their equilibrium conformations.

Eventually the hibernation ends, as new turbulent fluctuations begin to grow, and

the system transits back into active turbulence. The active turbulence again stretches

polymer chains and the (stochastic) cycle repeats.

In this picture, experimental observations in which the Virk MDR mean velocity

profile is found correspond to a limiting situation – not achieved at the low Reynolds

number and small boxes studied here – where the fraction of time and space occupied

by active turbulence becomes small enough that the hibernating regime dominates

the statistics. Active turbulence cannot vanish entirely, because it is known experi-

mentally (Warholic, Massah & Hanratty 1999, Ptasinski et al. 2003) that on average,

the polymer molecules carry a substantial fraction of the mean shear stress (they

must, if the time averaged Reynolds shear stress is to be small), and since hibernat-

ing turbulence does not stretch polymers, some active turbulence must remain. These

considerations lead to a picture of turbulence in the MDR regime as a state in which

hibernating turbulence is the norm, and active turbulence arises intermittently in

space and time only to be suppressed by the polymer stretching that it induces. MDR

is asymptotically independent of polymer properties because hibernating turbulence,

which dominates the statistics of MDR, is fundamentally a Newtonian phenomenon.

Only a study of turbulence in MFUs would allow for a picture this clean to emerge:

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142

in a larger flow domain there are likely to be some regions where the turbulence is

active and some where it is hibernating, but without knowing in advance about these

regions they would be difficult to identify.

The present study focused on MFU flows at low Reynolds number, where there

is not yet a large separation between inner and outer scales. This approach allowed

the collection and analysis of an extensive data set in a regime where flow structures

are relatively simple. Remarkably, even this regime displays clear signatures of the

features of MDR commonly associated with higher Reynolds numbers. Future work

should use simulations at high Re to carefully evaluate the hypothesized picture just

presented.

In addition, attention must be focused on the hibernating turbulence phenomenon.

Recently, Waleffe has identified a class of nonlinear traveling wave solutions to the

Navier-Stokes equations in the plane Couette and Poiseuille geometries that share

many characteristics with hibernating turbulence, specifically weak streamwise vor-

tices and weak streamwise dependence (Wang et al. 2007). Indeed, at least one family

of these solutions has vanishing streamwise dependence as Re→∞. These states are

saddle points in phase space and it may be that hibernating turbulence is a trajectory

moving transiently in the vicinity of one of these saddles.

These hypotheses, that MDR turbulence is fundamentally hibernating turbulence,

and that hibernating turbulence is closely related to nonlinear traveling wave struc-

tures in Newtonian flow, point toward a fundamentally new direction for research in

the field of turbulent drag reduction by additives, which will be the topic of our future

work (Chapter 13).

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143

11.2 Generalization to full-size turbulent flows: a

preliminary investigation

Although the above scenario of the transition toward MDR depends on temporal in-

termittency observed in individual coherent structures isolated by the MFU approach,

it is verifiable in experiments and full-size DNS studies. A full-size turbulent flow (in

either experiments or DNS) consists of a large population of coherent structures, each

of which evolves through the same (at least qualitatively) series of dynamical phases

observed in MFU, such as the active and hibernating intervals discussed above. In

general, evolution of these structures is not in phase with one another: for a given

instant, some of them may show characteristics of active turbulence, while others

might be in hibernation. Therefore, spatial intermittency would be expected in snap-

shots of these flow fields. The fraction of time spent in hibernation for one coherent

structure in a long-time simulation FH (plotted in Figure 11.4), would be reflected in

the fraction of coherent structures caught in hibernation within a randomly picked

snapshot of a sufficiently large simulation box at the statistically-steady regime (i.e.

fraction of space where hibernation is observed).

Some preliminary results of full-size DNS are presented here to illustrate this

interchangeability between temporal and spatial intermittency. These simulations

are performed in a periodic box with L+x = 4000 and L+

z = 800. According to

previous studies (Dubief et al. 2004, Li, Sureshkumar & Khomami 2006), with a box

size at this magnitude, effects of finite box size on flow statistics are negligible. This

simulation box is significantly larger, in both streamwise and spanwise directions,

than those used in the MFU study presented above, and should accommodate a

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144

Fig

ure

11.1

3:F

low

stru

cture

sof

aty

pic

alsn

apsh

otin

afu

ll-s

ize

New

tonia

nsi

mula

tion

(Re

=36

00,L

+ x=

4000

,L

+ z=

800)

.G

reen

shee

tis

the

isos

urf

ace

ofv x

=0.

3;re

dtu

bes

are

isos

urf

aces

ofQ

2D

=0.

02.

Only

the

bot

tom

hal

fof

the

chan

nel

issh

own.

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145

Fig

ure

11.1

4:F

low

stru

cture

sof

aty

pic

alsn

apsh

otin

afu

ll-s

ize

vis

coel

asti

csi

mula

tion

nea

rM

DR

(Re

=36

00,

Wi

=80

=0.

97,b

=50

00,L

+ x=

4000

,L

+ z=

800)

.G

reen

shee

tis

the

isos

urf

ace

ofv x

=0.

3;re

dtu

bes

are

isos

urf

aces

ofQ

2D

=0.

02.

Only

the

bot

tom

hal

fof

the

chan

nel

issh

own.

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146

Fig

ure

11.1

5:C

onto

urs

ofst

ream

wis

eve

loci

tyin

apla

ne

25w

all

unit

sab

ove

the

bot

tom

wal

lfo

rth

eN

ewto

nia

nsn

apsh

otsh

own

inF

igure

11.1

3

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147

Fig

ure

11.1

6:C

onto

urs

ofst

ream

wis

eve

loci

tyin

apla

ne

25w

all

unit

sab

ove

the

bot

tom

wal

lfo

rth

evis

coel

asti

c(n

ear

MD

R)

snap

shot

show

nin

Fig

ure

11.1

4

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148

number of coherent structures. A 240 × 73 × 90 numerical grid is used for spatial

discretization, this corresponds to δ+x = 16.67 and δ+

z = 8.89. Time step is in the

range of 0.03125 6 δt 6 0.04. Since the spatial and temporal resolutions are both

lower (still at the same level as those in previous studies, e.g. Jimenez & Moin (1991),

Housiadas & Beris (2003), Housiadas et al. (2005) and Li, Sureshkumar & Khomami

(2006)) than those used in the MFU study (Section 8.2), a slightly lower Schmidt

number (corresponding to a larger artificial diffusivity value) of Sc = 0.03 is used.

Figure 11.13 is a snapshot of a typical full-size Newtonian turbulent flow field (only

the bottom half of the channel is shown). Characteristic streak-vortex structures are

observed across the whole domain. Each pair of them are not identical, but they are

qualitatively similar. Almost all of them show features of active turbulence, including

strong vortical motions and wavy streaks. Streak structures are better observed in

the contour plot (Figure 11.15) of streamwise velocity in the x-z plane, taken at a

position within the buffer layer (y+ = 25). Alternating high- and low-speed streaks

are clearly identified, most of which contain wrinkles and other features at relatively

small wavelengths (O(100)).

For viscoelastic turbulent flows near MDR, flow structures are very different. Fig-

ures 11.14 and 11.16 are a snapshot taken from a high-Wi run; the DR% for this

snapshot is 57% (DR% for Virk (1975) MDR is slightly above 60% at this Re). In

most of the domain, streaks are elongated and regulated, only very weak streamwise-

dependence is observed; intensity of streamwise vortices is lower than the isosurface

level and these vortices in the regions with weak streak-waviness are thus not visual-

ized. These observations should recall the characteristics of hibernating turbulence.

As noted above, polymer cannot stabilize hibernating turbulence, hence intermittent

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149

occurrence of active turbulence is expected. Indeed, scattered patches of structures

resembling active turbulence, i.e. those showing strong vortices and wavy streaks, are

still observed.

Further analysis of this spatial intermittency is due in our future work. The

important message from results in this chapter is that the answer to the four-decade-

long puzzle of MDR might lie in the spatial and temporal intermittency in turbulent

flows, which has been screened out in most previous studies that typically focused on

spatially and temporally averaged profiles.

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150

Chapter 12

Conclusions of Part II

In this study, we study viscoelastic turbulent flows under a variety of conditions.

These solutions are obtained from the minimal flow unit approach and represent the

essential coherent structures for the self-sustaining process of turbulent motions. The

box size is minimized in the spanwise direction with fixed streamwise wavelength. The

minimal box size to sustain turbulence increases with increasing Wi for fixed β and b,

and the correlation between this length scale and the bulk flow rate is approximately

universal with respect to varying β and b at fixed Re (Figure 10.4). At a Re close to

the laminar–turbulence transition, all key stages of transition, reported previously in

experiments and simulations at much higher Re, are observed in the MFU solutions,

including pre-onset turbulence, LDR, HDR and an asym-DR stage that reproduces

the universal aspect of experimental MDR. The onset of drag reduction (transition

between the pre-onset and LDR stages) is observed at Wi & 16. The LDR–HDR

transition occurs at around DR% ≈ 13−15% under different β and b, which we expect

to be a function of Re. The discovery of the LDR–HDR transition at the current

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low Re and especially, at a relatively low DR%, indicates that this is a qualitative

transition between two stages of viscoelastic turbulent flows and not a quantitative

effect of the amount of drag reduction. Drag reduction reaches its upper limit at

DR% ≈ 26% in the asym-DR stage, where DR% converges upon increasing Wi. This

upper limit is universal with respect to different β and b, and it is to our knowledge the

first time the universality of MDR with respect to polymer parameters is examined

in numerical simulations. After the asym-DR stage, which persists for a finite range

of Wi at given β, b and Re, the flow returns to the laminar state.

The LDR–HDR transition is associated with a change in the underlying dynamics

of the self-sustaining process of turbulence. At the LDR stage, the essential coherent

structure to sustain turbulence is similar to that of Newtonian turbulence, which con-

sists of one undulating low-speed streak and its surrounding counter-rotating stream-

wise vortices. At the HDR stage, the essential structure is more complicated and

involves more than one streak; inter-streak interactions may be important. Never-

theless, the streamwise streaks and vortices are still the major components of the

self-sustaining process in all turbulent stages in our MFU solutions. This change of

the basic structure is reflected in the length scale of the MFU, resulting in a sudden

change in the slope of the L+z ∼ DR% curve: the minimal box size increases more

sharply with DR% at the HDR stage compared with the LDR stage. Several qual-

itative changes in flow statistics are observed during this transition, including: (1)

change of the log-law slope in the mean velocity profile, from the Newtonian log-law

to a larger slope; (2) disappearance of the concavity in the root-mean-square span-

wise velocity fluctuation profile; (3) change in the location of the suppression of the

Reynolds shear stress profile, which is suppressed locally (in the buffer layer) at LDR

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while globally (in most of the channel) at HDR and asym-DR. These changes cannot

be correlated with any observed qualitative transitions in the statistics of the polymer

conformation tensor.

At the asym-DR stage, the mean velocity profiles converge onto a single curve

at the given Re. The Reynold stresses either converge to a limit or at least lose

their dependence on Wi, β and b, and fluctuate within certain ranges. In contrast,

polymer is increasingly stretched by the flow with increasing Wi despite the converged

flow rate, and the polymer conformation tensor continues to dependent on Wi, β

and b. In the asym-DR stage, the spatiotemporal flow structure seems similar as

that of the HDR stage; the self-sustaining process also shows complex dynamics

involving multiple streaks. The minimal length scale in z to sustain turbulence keeps

on increasing with Wi in the asym-DR stage; however, the length scale of the MFU

solutions in the asym-DR stage under different β and b all approximately fall in the

range of 200 6 L+z 6 260.

This study shows that the drag reduction process with varying parameters is com-

posed of several key stages of transition, which are present in both fully developed

turbulence (according to other studies) and the laminar-turbulence-transition regime.

The mechanism of these transitions, especially the LDR–HDR transition and the ex-

istence of a universal asym-DR, is as yet unclear. Spatiotemporal images of turbulent

coherent structures suggest that a shift of the underlying self-sustaining mechanism

occurs at the LDR–HDR transition. Further study of this change will be important

in understanding drag reduction behaviors in HDR and MDR regimes. In addition,

the capability of isolating the minimal transient solutions, and the knowledge that

these transitions can all be studied in the near-transition regime, will greatly facilitate

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future insight into the polymer drag reduction phenomenon.

Dynamics of turbulence in MFU, both Newtonian and viscoelastic, show inter-

mittent occurrence of relatively-long periods when substantially-lower magnitudes of

wall shear rate are observed. These “hibernating” periods display many features of

experimentally-observed MDR in polymer solutions, including weak streamwise vor-

tices, nearly nonexistent streamwise variations, strongly suppressed Reynolds shear

stress, and most importantly, a mean velocity gradient that quantitatively matches

experiments. Frequency of hibernation increases significantly in the high-Wi regime,

where polymer compresses the lifetime of an active turbulence interval, and facili-

tates the occurrence of these periods. Once inside hibernation, polymer is weakly

stretched by the flow, and has little effect on hibernation itself. These results point

toward a fundamentally-new direction of understanding turbulence in the high-Wi

regime, especially the maximum drag reduction.

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Chapter 13

Future work: dynamics of

viscoelastic turbulence and drag

reduction in turbulent flows

Our study on viscoelastic turbulence raises more questions than it answers. Although

it does not provide a complete mechanism for MDR, nor does it offer a clear explana-

tion of the LDR–HDR transition, a new direction has be pointed to understand the

regime of high–Wi viscoelastic turbulence. In particular, the resemblance between

the intermittent turbulence hibernation and experimentally-observed MDR provides

an important clue that might lead to the eventual revelation of the nature of this

long-lasting mystery. Further knowledge is needed about this newly-recognized hi-

bernating period: understanding its nature and its connection with MDR is the most

important task in the future.

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13.1 Hibernation statistics: effect on the LDR–

HDR transition

With the current numerical method and data, the short-term plan is to quantify

the differences between active and hibernating turbulence. Results on turbulence

behaviors during hibernation discussed in Chanpter 11 is based on a typical exam-

ple, although other instances have been examined to ensure that those observations

are not specific to the selected set of data, more general analysis on the flow and

polymer conformation statistics in both active and hibernating turbulence should be

performed. As reviewed in Chapter 7, the LDR–HDR transition is marked by a series

of qualitative transitions in turbulence statistics and flow structures: e.g. increased

log-law slope in the mean velocity profile, reduced Reynolds shear stress across the

channel and dramatically weakened three-dimensionality in the streak-vortex struc-

tures (Warholic, Massah & Hanratty 1999, Ptasinski et al. 2003, White et al. 2004, Li,

Sureshkumar & Khomami 2006). Many of these observations consist with the char-

acteristics of hibernating turbulence. A natural conjecture is that the LDR–HDR

transition is caused by an event leading to the frequent occurrence of hibernation.

Before this transition, hibernation is rare and active turbulence dominates the statis-

tics. In HDR and MDR regimes, hibernation makes a substantial contribution to the

overall statistics, which causes all the changes reported in previous studies; meanwhile

active turbulence remains qualitatively the same, although quantitatively it should

have a smooth dependence on Wi as well (as seen from the dependence of LDR statis-

tics on Wi, where hibernation frequency is almost constant). To test this hypothesis,

one needs to effectively divide data from each time series into the two categories,

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such that statistics for either of them can be computed separately, and the difference

between these two regimes can be compared with statistical certainty.

13.2 A hypothetical dynamical-scenario

From a nonlinear dynamics point of view, the distinct separation between hibernating

and active turbulence in the solution state space usually suggests the existence of cer-

tain solution objects governing the hibernation dynamics, which are located far away

from the major TWs that construct the latter (Guckenheimer & Holmes 1983). In the

simplest case, hibernation is caused by intermittent visits of the proximity of some

saddle point: TW solution with both stable and unstable dimensions (Figure 13.1).

The system is pulled toward the saddle point via trajectories going along the stable

manifold; it turns near the saddle and is ejected away toward active turbulence along

the direction of the unstable manifold. In Newtonian turbulence, these excursions

are rare: the system is trapped in active turbulence for long time periods before it

hits the orbit toward the saddle. At high Wi, active turbulence can sustain for much

shorter time, and these orbits are visited more frequently. Differences in duration

and “depth” of hibernation among individual instances are accounted for by differ-

ent closeness between the incoming orbits (orbits entering hibernation from active

turbulence) and the stable manifold. Although an one-saddle scenario is shown in

Figure 13.1, it may also involve more than one saddles, or even more complex solution

objects such as periodic orbits.

Characteristics of hibernating turbulence observed in Chapter 11 recall us to the

concept of “lower-branch” TWs. In the regime near the critical Re for the laminar-

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Laminar Flow

Active Turbulence

TW (Saddle)

Stable

Man

ifold

Unstable Manifold

Edge Structure

Edg

e S

truct

ure

Hib

erna

tion?

Figure 13.1: Schematic of near-transition turbulent dynamics: intermittent excursionstoward certain saddle points and the laminar-turbulence edge structure.

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turbulence transition, TWs typically appear in pairs through saddle-node bifurca-

tions (Waleffe 1998, 2001, 2003). By lower-branch solutions we refer to the ones in

each pair that are relatively closer to the laminar state: i.e. those have lower turbu-

lence intensity. Jimenez et al. (2005) summarized TW solutions obtained by various

groups in Newtonian plane Couette flow and concluded that these solutions can all be

categorized as either lower-branch or upper-branch solutions. TWs of both categories

are in the form of a sinuous low-speed streak straddled by a pair of staggered counter-

rotating streamwise vortices, but the lower-branches generally have smaller transverse

velocity fluctuations, much weaker vortical motions and less streamwise waviness in

the streak. Upper-branch TWs are widely believed to be the building blocks of the

chaotic structure of active turbulence (Waleffe 2001, 2003, Jimenez et al. 2005, Gib-

son et al. 2008), while characteristics of lower-branch TWs are remarkably similar to

the hibernating turbulence discovered in this study. In the context of the scenario

illustrated in Figure 13.1, it is likely that the saddle(s) dominating hibernation dy-

namics belong to the category of lower-branch TWs. Previous studies on viscoelastic

ECS (Stone et al. 2002, Stone & Graham 2003, Stone, Roy, Larson, Waleffe & Gra-

ham 2004, Li, Stone & Graham 2005, Li, Xi & Graham 2006, Li & Graham 2007)

showed that upper-branch ECSs are strongly suppressed by polymer stress, and at Wi

sufficiently high, they are completely eliminated. This is consistent with the current

observation that the average life time of active turbulence is significantly shortened

at high Wi. Meanwhile, since lower-branch TWs have much weaker vortical struc-

tures and less three-dimensionality, they might not stretch polymer substantially to

generate enough polymer stress, and these solutions may be largely unaffected with

increasing Wi. This would explain the invariant duration time scale of hibernation

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for different Wi, and the similarity between Newtonian and viscoelastic hibernation

observed in this study.

Recent studies in Newtonian turbulence suggested that lower-branch TWs play an

important role in the laminar-turbulence transition. For ECS in plane Couette flow,

Wang et al. (2007) showed that the stable manifold of lower-branch ECS forms part

of the separating boundary between basins of attraction of laminar and turbulence

states. This solution has only one unstable dimension (Waleffe 2003). One side of

the unstable manifold points to the laminar state and the other leads to turbulence

(see Figure 13.1). Initial states “above” the stable manifold will become turbulent

and those below will laminarize. This separatrix is commonly known as the “edge

structure”, and has been widely studied recently (Skufca et al. 2006, Schneider et al.

2007, Duguet et al. 2008, Viswanath & Cvitanovic 2009). Although the lower-branch

ECS and its stable manifold form part of the edge structure, they probably are not the

only contribution. Skufca et al. (2006) observed that the stable manifold of a periodic

orbit coincides with the edge at low Re, and at higher Re a higher-dimensional chaotic

object is involved. Study of Duguet et al. (2008) in pipe flow suggested that a few TWs

and the heteroclinic connections between them are the key structures that organized

the edge. Although highly fractal in shape (Schneider et al. 2007), this laminar-

turbulence edge is presumed to be a surface insulating states in the turbulence side

from the laminar attractor. Based on the above discussion on lower-branch TWs,

we may assume that the edge is also hardly affected by polymer, and would prevent

turbulence from laminarization even at very high Wi. In this case, turbulence would

eventually be stuck near the edge structure as it is moved toward the laminar side

with increasing viscoelasticity; the dynamics of the edge would persist as Wi further

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increases. This should echo the puzzle of experimentally observed MDR upper-bound.

After connecting all these threads, a picture would emerge presenting the dynam-

ics underlying the transitions of viscoelastic turbulence at moderate Re. In Newtonian

flows, a number of upper-branch TWs form a chaotic saddle of active turbulence. Tur-

bulence stays active for long time, while occasionally embarks on excursions toward

the laminar state. These trajectories can extend no further than the edge surface,

and would be reflected back near certain saddle structures on the edge (lower-branch

TWs or others). These excursions are observed as hibernating turbulence. In di-

lute polymer solutions, at low Wi (pre-onset stage), polymer has little effect on any

components of the turbulence dynamics. When Wi exceeds Wionset, polymer is signif-

icantly stretched in the active turbulence regime. Upper-branch TWs are modified:

polymer stress weakens the streamwise vortices, and the friction factor of these so-

lutions is reduced (Stone et al. 2002, Stone & Graham 2003, Stone, Roy, Larson,

Waleffe & Graham 2004, Li, Stone & Graham 2005, Li, Xi & Graham 2006, Li &

Graham 2007). Changes in these TWs collectively cause drag reduction in active

turbulence. At LDR, hibernation still occurs on a occasional basis; its frequency

starts to increase at the LDR–HDR transition. The cause of this transition is un-

clear. One straightforward possibility is: as Wi increases, upper-branch TWs and

active turbulence is moved in the state space; when they are close enough to the edge

structure, formation of certain dynamical objects, such as heteroclinic orbits between

certain upper- and lower-branch solutions, greatly facilitates the visits of edge struc-

ture and thus hibernation. Another possibility is with sufficient viscoelasticity, some

of the TWs are eliminated (Stone et al. 2002, Li, Xi & Graham 2006, Li & Graham

2007), and more exits are created in the domain of active turbulence. This change

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in hibernation frequency is reflected in various experimentally-measurable quanti-

ties (Warholic, Massah & Hanratty 1999). At sufficiently high Wi, turbulence stay in

hibernation for the majority of time. Since polymer is only mildly deformed during

hibernating turbulence, it is not able to keep turbulence stay in hibernation. Active

turbulence occurs intermittently, which is quickly quenched by polymer stress. Hiber-

nating turbulence dominates experimental measurements due to the large fraction of

time it occupies. Since polymer is largely ineffective in changing flow structure dur-

ing turbulence hibernation, this would be the upper-limit of polymer-induced drag

reduction.

The notion of “edge state” is mentioned by Benzi et al. (2005, 2006) and Procaccia

et al. (2008) in their phenomenological model of MDR. However, we need to clarify

that their “edge state” is fundamentally different from the edge structure in our

scenario. In their work, “edge” refers to the limit of turbulent kinetic energy (TKE)

reducing to zero, where their model for mean velocity profile approaches the Virk

MDR profile. The edge structure in our discussion is an actual object in the solution

state space, and there is no indication so far about its quantitative features. Studies

on Newtonian turbulence show that flow structures on the edge closely resemble many

TWs (Schneider et al. 2007, Duguet et al. 2008), which clearly have finite TKE. Also,

according their model, Reynolds shear stress is proportional to TKE, which in our

simulation although reduces, does not approach zero during hibernation. What is in

common between their model and our simulation, however, is that the universality of

MDR is rooted in Newtonian turbulence. Our observation that hibernating turbulence

exists in Newtonian flows, and is unaffected by polymer, is the key element that could

potentially explain the universality of MDR.

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All current results are obtained at a relatively low Re, close to the critical Re

(≈ 1000) of laminar-turbulence transition. Turbulent structures in this regime are

relatively simple. In addition, at Re this low, the active turbulence regime is close

to the laminar-turbulence edge in the state space; if the above scenario is true, this

might be the reason the intermittent hibernation dynamics is easier to observe in our

study. At higher Re where most experiments are performed, hibernation in Newtonian

turbulence might be very rare, which would only become frequent at very high Wi.

Nevertheless, after the dynamics at near-transition-Re is further understood in the

future, effect of increasing Re should also be investigated; the whole physical picture

should be verified in the high-Re regime.

13.3 Development of methodology

The scenario above consists of two hypotheses: first, hibernating turbulence is built

around certain remote (w.r.t. active turbulence) TW solution(s), which might belong

to the category of lower-branch TWs; second, these TWs form at least part of the

laminar-turbulence edge structure. To verify this picture, the first step to take is

to find the corresponding TWs responsible for hibernating dynamics, analyze their

linear stability, and study their connection with the turbulence trajectory. During a

hibernation period, there has to be certain amount of time when the system passes

through the vicinity of these solution objects; if these are indeed TWs (steady-states

in traveling reference frames), using properly selected snapshots during hibernation

as initial guesses, Newton iteration should converge to these solutions. Therefore, de-

veloping an algorithm of finding steady-state solutions in viscoelastic turbulent flows

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is the first task we propose in this section. Although our past work has been success-

ful in numerically finding one class of viscoelastic TWs (ECS), the algorithm used in

that study is restricted to solutions with certain imposed symmetry-conditions (Stone,

Roy, Larson, Waleffe & Graham 2004, Li & Graham 2007). All other previous stud-

ies on viscoelastic turbulence were based on transient solutions (see Chapter 7). A

general algorithm of solving for viscoelastic TW solutions is thus due. Given the vis-

coelastic DNS (time-integration) code we have already developed, the Newton-Krylov

method is the more preferable algorithm (Sanchez et al. 2004, Viswanath 2007, 2009,

Gibson et al. 2008), which, instead of computing the Jacobian matrix directly, es-

timates its product with the state vector using a time-integration algorithm. This

method has been successfully applied in Newtonian turbulence problems in finding

steady states, traveling waves, periodic orbits and relative periodic-orbits (which al-

lows phase shifts) (Viswanath 2007, 2009, Gibson et al. 2008).

Initial conditions for the Newton iteration should be taken at several important

instances during typical hibernation periods, including turning points of significant

signals such as Ubulk, 〈∂vx/∂y〉, and the mean velocity profile slope. Solution objects

in control of the hibernating orbits could then be identified; although the case of

TW is discussed in Section 13.2, the Newton-Krylov method mentioned above is not

limited to TWs. In comparison, solutions dominating the active turbulence regime

should also be studied. Two aspects of these solutions are of interest. The first is how

do these solutions quantitatively recover the flow structures and statistics observed

in either hibernating or active turbulence. In particular, we are interested in if there

is one of them that can quantitatively match with the experimental observations

during MDR. The second problem to investigate is the effect of viscoelasticity on

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these solutions: much weaker dependence on the viscoelasticity is expected for those

governing the hibernating regime.

Beyond the study of these solutions themselves, their connection with the dynam-

ical trajectory should also be inspected. The relevant importance of each solution

to different stages of the trajectory can be determined by the frequency at which

these solutions are visited, and the distance between them and the trajectory during

each visit. The closeness of these solutions to a given instant on the trajectory, and

in general the distance between any two states in the state space, can be measured

by a form of inner product with the translational (and also rotational for the pipe

geometry) symmetry taken account of (Kerswell & Tutty 2007). A clearer view of

how these solutions determine the state-movement on the trajectory would require an

effective projection of the high-dimensional state space onto a 2D or 3D coordinate

system. Gibson et al. (2008) projected the transient trajectory of a Newtonian plane

Couette flow onto a set of orthonormal basis-states constructed with the upper-branch

ECS (Waleffe 2003) and its symmetric copies; and the geometry of the state space

was clearly visualized (Figure 7.4).

Computation of unstable eigenvalues and eigenvectors can be achieved through

time integration as well using Arnoldi iteration (Viswanath 2007, Gibson et al. 2008).

This would enable us to include unstable manifolds into the visualization discussed

above, along which the trajectory moves away from each solution. With these in-

formation, heteroclinic orbits, trajectories connecting the unstable manifold of one

solution to the stable manifold of another, could be numerically found (Duguet et al.

2008, Halcrow et al. 2009). These orbits determine the transitions from one TW to

another, thus would be important in the understanding of transitions between active

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and hibernating turbulence, as well as the movement of state within the latter.

As to the second hypothesis, dynamical trajectories embedded on the edge struc-

ture can be computed directly with a time integration algorithm using a bisection-

based edge-tracking method (Skufca et al. 2006, Schneider et al. 2007, Duguet et al.

2008). This method is illustrated in Figure 13.2. For a given form of perturbation on

the laminar state with one adjustable parameter measuring its amplitude, through

bisection, one can find two amplitudes that are close to one other to the required nu-

merical precision, while one of the corresponding states is below the edge, the other is

above. Using these two states as initial conditions, time integration will generate two

trajectories stay close to the edge for a fairly long amount of time; both trajectories

are good approximations to the actual edge they move along. As time proceeds, these

trajectories start to diverge, both moving away from the edge; once the distance be-

tween them grow larger than the required precision, a new pair of initial conditions

should be obtained by another bisection process, starting from which the tracking

would continue. This process is repeated for the desired length of the edge trajectory.

Using this method, we can readily find trajectories on the edge for viscoelastic

flows with our current DNS code. The proposed edge-tracking study should address

two questions: first, how close is the edge to hibernating turbulence; second, how

does viscoelasticity affect the edge. For both questions, beyond analyzing the edge

structure statistically, better understanding could be achieved when edge-tracking is

carried out in conjunction with the search and stability analysis of TWs discussed

above. On one hand, same as above, TWs and other solution objects forming the

edge can be found by carefully selecting the initial conditions (Duguet et al. 2008),

which can be compared with those found governing hibernating turbulence. On the

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Figure 13.2: Schematic of the edge-tracking method based on repeated bisec-tion (Skufca et al. 2006).

other, given a TW found relevant to hibernating turbulence, if this particular solution

is indeed on the edge, one can also find its stable manifold on the edge using the

method proposed by Viswanath & Cvitanovic (2009).

In summary, many methods have been developed by the Newtonian turbulence

community to analyze the nonlinear dynamics governing the laminar-turbulence tran-

sition; most of them are readily adaptable to viscoelastic problems. The overall goal

is to obtain a clear view of the dynamical structure in the state space that causes all

these transitions in viscoelastic turbulence, especially the unique turbulence structure

and statistics in HDR and MDR, and the university of the latter.

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13.4 Further extensions: other drag-reduced tur-

bulent flow systems

Polymer is not the only type of drag-reducing agent, drag reduction in turbulent flows

are observed in many other fluid systems, such as fiber suspensions (Metzner 1977),

worm-like-micelle-forming surfactant solutions (Shenoy 1984, Zakin et al. 1998), and

even liquids with injected micro-bubbles (Madavan et al. 1984). Surfactant-induced

turbulent drag reduction is a particularly interesting extension to the current study.

Practically speaking, unlike polymer molecules which gradually degrade upon strong

deformations, and lose their drag-reducing capability (Culter et al. 1975, Vanapalli

et al. 2005), scission of worm-like micelles is non-permanent. Micelles broken in

a strong flow can regain their formation after the deformation is released. This

is particularly desirable in closed circulation flow systems, such as district heating

and cooling systems. In addition, solutions of worm-like micelles are fundamentally

viscoelastic fluids with more complicated interactions between the microscopic micelle

structures and macroscopic flow behaviors than dilute polymer solutions (Cates &

Candau 1990, Butler 1999, Raghavan & Kaler 2001, Qi & Zakin 2002); one would

thus expect both similarities and disparities between the drag-reducing mechanisms of

these two systems. With all the understanding we are about to acquire of turbulence

in dilute polymer solutions, the mechanism of surfactant-induced drag reduction could

be more accessible.

Perhaps the most interesting difference between these two drag-reducing systems

is that the maximum drag reduction limit in surfactant solutions can be higher than

in polymeric fluids (Bewersdorff & Ohlendorf 1988, Chara et al. 1993, Zakin et al.

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1996, 1998). This clearly indicates a different drag-reducing mechanism, at least in

the high-extent of drag reduction limit, from the polymer system. If the hypothetical

picture in Section 13.2 would be verified, this difference would become even more

intriguing: the understanding of how surfactant might change the dynamics near the

edge would bring further insight into the problem of laminar-turbulence transition.

Mean velocity profiles of surfactant solutions appear similar to polymeric fluids at

low-extent of drag reduction (Bewersdorff & Ohlendorf 1988, Zakin et al. 1998), while

in high-extent of drag reduction, they can be qualitatively different. For cases close

to or even above the Virk MDR limit, the “S-shape” profile is often reported: the

profile is lower than the Virk MDR in the buffer layer, at y+ ≈ 30 it starts to raise

with a much larger slope, and later crosses the Virk MDR profile (Chara et al. 1993,

Myska & Zakin 1997). Other shapes have been observed in different experimental

conditions (Warholic, Schmidt & Hanratty 1999, Itoh et al. 2005, Tamano et al. 2009).

Li, Kawaguchi, Segawa & Hishida (2005) even reported that for a fixed experimental

setup, the mean velocity profile and other turbulence statistics quantities can be

qualitatively different at the same level of drag reduction, in different regimes of

transition (note that in surfactant solutions, along an experimental path, DR% can

change non-monotonically with increasing Re; see e.g. Qi & Zakin (2002) and Li,

Kawaguchi, Segawa & Hishida (2005)). Since in near-wall turbulence, different types

of coherent structures dominate different layers away from the wall (Robinson 1991),

these complexities observed in mean velocity profiles suggest a variety of complicated

interactions between worm-like micelles and different turbulent coherent structures.

As to the mechanism of the micro-structure-flow interaction, many studies (Bew-

ersdorff et al. 1989, Myska & Zakin 1997, Myska & Stern 1998, Warholic, Schmidt &

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Hanratty 1999) suggested that drag reduction is closely linked with the formation of

“shear-induced structures” (SIS) (super-molecular aggregates of worm-like micelles

formed under flow (Cates & Candau 1990, Liu & Pine 1996, Butler 1999, Forster

et al. 2005)); while some other studies indicated that the ability of forming SIS is not

a necessity for drag reduction (Myska & Zakin 1997, Qi & Zakin 2002). The role of

SIS in surfactant drag reduction is another major unsolved problem in this area.

Understanding of surfactant drag reduction is very limited even compared with

that of polymer drag reduction. Dynamics of worm-like micelles under flow are so

complicated that a satisfactory micro-mechanical model (like the bead-spring model

for flexible linear polymer (Bird, Curtis, Armstrong & Hassager 1987) for computer

simulation is still missing. Even if there is one, it is unlikely to be simple enough that

a constitutive equation can be derived analytically, which is necessary for DNS studies

with the state-of-the-art computation capacities. Among the very few computational

studies on turbulence of surfactant solutions, e.g. Yu et al. (2004) and Yu & Kawaguchi

(2005), constitutive equations for polymer were used; only the parameters were fitted

with rheological data of drag-reducing surfactant solutions. These simulations were

not able to capture the qualitatively-different dynamics in surfactant solutions.

Since surfactant solutions are also viscoelastic fluids, semi-empirical constitutive

equations can be built based on polymer models, with qualitative features of worm-

like-micelle dynamics included. Bautista et al. (1999), Manero et al. (2002) and Boek

et al. (2005) proposed a constitutive equation, which based on the Oldroyd-B equa-

tion for polymer solutions (Bird, Armstrong & Hassager 1987), included an additional

partial-differential equation taking account of the dynamical destruction and reforma-

tion of micellar structures. Change of structure is parameterized as a varying micellar

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contribution to the shear viscosity, the destruction rate is proportional to the rate of

work done by the flow on micellar structures, and the reformation rate is determined

by the distance from the current state to equilibrium. Simple as it is, this model

demonstrates how different conceptual elements of surfactant-solution dynamics can

be incorporated by modifying a viscoelastic constitutive equation for polymer solu-

tions. More features can be included in the model in a similar manner; in particular,

the effect of SIS can be modeled by a shear-rate dependent term contributing to the

structure change. A good starting point to study surfactant turbulent flow is to per-

form DNS with the Boek-improved Bautista-Manero model (Boek et al. 2005). By

comparing the results with experimental observations, the constitutive model can be

further improved by including more features of surfactant dynamics, or better pa-

rameterization of the micellar structure. In addition, DNS results with and without

the SIS contribution should be compared to determine the importance of SIS relative

to other surfactant-specific features such as the dynamical destruction-reformation

process.

Another extension to the study of polymer drag reduction is the active control of

turbulence. Besides these intrusive drag-reducing agents like polymer and surfactant,

drag reduction can also be achieved with more controllable engineering techniques,

such as mechanical actuators (Rathnasingham & Breuer 2003), temperature varia-

tion (Yoon et al. 2006), blowing and suction of fluids through the wall (Choi et al.

1994), and electrical forces (Du & Karniadakis 2000). Previous feedback control

strategies focus on empirically-determined objective functions (e.g. Lee et al. (1998)).

With the recent understandings of the nonlinear dynamics in the regime of laminar-

turbulence transition, more rational schemes can be developed: these schemes can

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either aim at restricting the flow state to the region close to the edge, or bringing

the state over the edge to laminar flow (Kawahara 2005, Wang et al. 2007). Gener-

ally speaking, further knowledge of the dynamical structures in the state space has

a two-fold impact on improving the design of active turbulence control strategies:

first, knowing the characteristics of important solution objects, measurements can be

better designed to provide a good estimation of the system state; second, given infor-

mation about the current state, a clearer objective of control can be generated with

the knowledge of the state space geometry: e.g. to move the system to the closest

relatively-stable low-drag state.

Our proposed study on polymer drag reduction would also benefit the design of

more rational control strategies. For example, since hibernating turbulence is believed

to be a Newtonian structure, with a better knowledge of how polymer increases its

frequency of appearance, we can design a control strategy to maximize the probabil-

ity of hibernating turbulence without adding polymer into the fluid. Furthermore,

polymer drag-reducing agents can be applied in conjunction with active control tech-

niques to achieve larger drag reduction: the former can bring the turbulence close to

MDR whereas the latter can potentially break the MDR limit.

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Appendix A

Numerical algorithm for the direct

numerical simulation of viscoelastic

channel flow

This appendix provides the detailed algorithm for the direct numerical simulation

(DNS) of viscoelastic flows in the plane Poiseuille geometry, used in Part II of this

dissertation. This algorithm is an extension of that of the Newtonian DNS code Chan-

nelFlow, developed and maintained by Gibson (2009) (see also Canuto et al. (1988)),

to the viscoelastic system. A summary of the numerical method and parameters used

in this study is provided in Section 8.2; listed here is the corresponding formulation

for the method. For convenience, the equation system to be solved (Equations (8.1),

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(8.2), (8.3) & (8.4)) is relisted below:

∂v

∂t+ v ·∇v = −∇p+

β

Re∇2v +

2 (1− β)

ReWi∇ · τ p, (A.1)

∇ · v = 0. (A.2)

α

1− tr(α)/b+

Wi

2

(∂α

∂t+ v ·∇α−α ·∇v − (α ·∇v)T

)=

(b

b+ 2

)δ,

(A.3)

τ p =b+ 5

b

1− tr(α)/b−(

1− 2

b+ 2

). (A.4)

We start by discussing the numerical algorithm of solving the Navier-Stokes equa-

tion: (A.1) & (A.2). The velocity and pressure fields are decomposed into the base

and perturbation components:

v = Uex + v†, (A.5)

p = Πx+ p†. (A.6)

Hereinafter, † indicates the perturbation component. Constant Π is the mean pressure

gradient; for plane Poiseuille flow with the fixed-pressure-drop constraint, its value is

−2/Re. The base flow velocity profile U = U(y) is chosen to be that of the laminar

plane Poiseuille flow, i.e. U(y) = 1− y2 (note: walls locate at y = ±1); ex is the unit

vector in the streamwise direction (similarly, ey and ez, which will appear below, are

unit vectors in wall-normal and spanwise directions, respectively). Plugging (A.5) &

(A.6) into (A.1) & (A.2), we obtain the partial differential equations for perturbation

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variables v† and p†:

∂v†

∂t= −v ·∇v −∇p† − Πex +

β

Re

∂2U

∂y2ex +

β

Re∇2v† +

2 (1− β)

ReWi∇ · τ p, (A.7)

∇ · v† = 0. (A.8)

We introduce simplified notations for the terms on the right-hand side of (A.7):

N ≡ v ·∇v, (A.9)

Lv† ≡ β

Re∇2v†, (A.10)

C ≡(β

Re

∂2U

∂y2− Π

)ex, (A.11)

S ≡ 2 (1− β)

ReWi∇ · τ p. (A.12)

Here, N is the inertia term (nonlinear); Lv† is the viscosity term (linear); C is the

constant term; S is the contribution of the divergence of polymer stress (nonlinear).

Equation (A.7) is then simplified as:

∂v†

∂t= −N −∇p† + Lv† +C + S. (A.13)

Taking Fourier transform in x and z directions on both sides of the equation, we

obtain:

∂v†

∂t= −N − ∇p† + Lv† + C + S, (A.14)

where ∼ denotes variables in Fourier space in x and z dimensions, and in physical

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space in the y dimension. Differential operators in (A.14) are defined as:

∇ = ∇kx,kz ≡ 2πikxLxex +

∂yey + 2πi

kzLzez, (A.15)

∇2 = ∇2kx,kz

≡ ∂2

∂y2− 4π2(

k2x

L2x

+k2z

L2z

), (A.16)

L = Lkx,kz ≡β

Re∇2kx,kz

. (A.17)

As introduced earlier in Section 8.2, the semi-implicit Adams-Bashforth/backward-

differentiation scheme is used for temporal discretization. Linear terms, Lv† and

−∇p†, are discretized with the implicit backward-differentiation method; nonlin-

ear terms, −N and S, are discretized with the explicit Adams-Bashforth method.

Detailed discussion of this scheme is given in Peyret (2002) (pages 131-132, Sec-

tion 4.5.1(b)); for (A.14), the discretized time-stepping equation is:

1

∆t

(ηv†,n+1 +

k−1∑j=0

ajv†,n−j

)=

k−1∑j=0

bj

(−Nn−j

+ Sn−j)

+ Lv†,n+1 + C − ∇p†,n+1,

(A.18)

which, after rearrangement, becomes

η

∆tv†,n+1 − Lv†,n+1 + ∇p†,n+1

=k−1∑j=0

(− aj

∆tv†,n−j − bj

(N

n−j − Sn−j))

+ C

≡ Rn.

(A.19)

Here, n is the index of the current step; n+1 is the index of the next step, i.e. that of

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Order η a0 a1 a2 a3 b0 b1 b2 b3

1 1 −1 1

2 3/2 −2 1/2 2 −1

3 11/6 −3 3/2 −1/3 3 −3 1

4 25/12 −4 3 −4/3 1/4 4 −6 4 −1

Table A.1: Numerical coefficients for the Adams-Bashforth/backward-differentiation temporal discretization scheme with different orders-of-accuracy (Peyret 2002).

quantities to be solved. For an algorithm with k-th order accuracy in time, solutions

at k previous steps, including that at the n-th step, are needed at each time step.

These known solutions are indexed with n− j (0 6 j < k). Numerical coefficients η,

aj and bj are listed in Table A.1. Hereinafter, Rn

denotes the summation of terms

known at the n-th time step: i.e. terms do not involve quantities at the to-be-solved

(n+ 1)-th step.

Expanding (A.19) with (A.17), we obtain:

β

Re

∂2

∂y2v†,n+1 −

(4π2 β

Re

(k2x

L2x

+k2z

L2z

)+

η

∆t

)v†,n+1 − ∇p†,n+1 = −Rn

. (A.20)

For each (kx, kz) pair, Equation (A.20) is a differential equation with derivatives in y

only. The following quantities are constant for a given wavenumber pair:

ν ≡ β

Re, (A.21)

λ = λkx,kz ≡ 4π2 β

Re

(k2x

L2x

+k2z

L2z

)+

η

∆t. (A.22)

With the simplified notation above, (A.20) is rewritten below together with the

continuity equation in Fourier space (take Fourier transform in x and z dimensions

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on both sides of (A.8)) and the no-slip boundary conditions at both walls:

ν∂2v†

∂y2− λv† − ∇p† = −R, (A.23)

∇ · v† = 0, (A.24)

v† |y=±1= 0. (A.25)

The above equation is referred to as the tau-equation. For each time step, the tau-

equation is solved for each wavenumber pair (kx, kz). Note that time step indices n

and n+ 1 are omited from these equations; at each time step, v† and p† are unknown

quantities to be solved, and R is known with information from solutions at previous

time steps.

Kleiser & Schumann (1980) proposed an elegant way, the influence matrix method,

of solving the tau-equation with both the divergence-free and boundary conditions sat-

isfied analytically, which is discussed in detail in Canuto et al. (1988) (pages 216-221,

Section 7.3.1). Below we summarize the basic ideas of this method. To separate equa-

tions for v† and p†, we take divergence of (A.23) and apply (A.24) to obtain (A.26);

then we take the y-component of (A.23) to get (A.28). Boundary conditions, (A.27)

and (A.29), are obtained by evaluating (A.24) at the no-slip walls, and taking the

y-component of (A.25), respectively. Here are the equations and boundary conditions

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for p† and v†y:

∂2p†

∂y2− κ2p† = −∇ · R, (A.26)

∂v†y∂y|y=±1= 0, (A.27)

ν∂2v†y∂y2− λv†y −

∂p†

∂y= −Ry, (A.28)

v†y |y=±1= 0, (A.29)

where κ is a constant for a given wavenumber pair:

κ2 = κ2kx,kz

≡ 4π2 β

Re

(k2x

L2x

+k2z

L2z

). (A.30)

Equations (A.26), (A.27), (A.28) & (A.29) are called the A-problem. This problem

is not ready to solve since there are no boundary conditions for p†, while there are

two boundary conditions for v†y at each wall. If we could replace boundary conditions

(A.27) with Dirichlet boundary conditions for p† (see (A.32)), these equations would

be much easier to solve. This hypothetical problem equivalent to the original A-

problem, called the B-problem, is listed below:

∂2p†

∂y2− κ2p† = −∇ · R, (A.31)

p† |y=±1= P±, (A.32)

ν∂2v†y∂y2− λv†y −

∂p†

∂y= −Ry, (A.33)

v†y |y=±1= 0. (A.34)

Of course boundary values for the pressure field P± are unknown. However, it can

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be shown that a general solution to the B-problem can be constructed with a par-

ticular solution from the inhomogeneous version of the B-problem with homogeneous

boundary conditions (the B’-problem):

∂2p†p∂y2

− κ2p†p = −∇ · R, (A.35)

p†p |y=±1= 0, (A.36)

ν∂2v†y,p∂y2

− λv†y,p −∂p†p∂y

= −Ry, (A.37)

v†y,p |y=±1= 0, (A.38)

and basis solutions from two corresponding homogeneous problems, the B+-problem:

∂2p†+∂y2

− κ2p†+ = 0, (A.39)

p†+ |y=−1= 0, (A.40)

p†+ |y=+1= 1, (A.41)

ν∂2v†y,+∂y2

− λv†y,+ −∂p†+∂y

= 0, (A.42)

v†y,+ |y=±1= 0; (A.43)

and the B−-problem:

∂2p†−∂y2

− κ2p†− = 0, (A.44)

p†− |y=−1= 1, (A.45)

p†− |y=+1= 0, (A.46)

ν∂2v†y,−∂y2

− λv†y,− −∂p†−∂y

= 0, (A.47)

v†y,− |y=±1= 0. (A.48)

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All these equations are readily solvable with a standard numerical scheme. Take

the B’-problem for example, (A.35) is a complex Helmholtz equation with Dirichlet

boundary conditions (A.36), which can be solved with the Chebyshev-tau method

(see Canuto et al. (1988), pages 129-133, Section 5.1.2; this method is included in

the ChannelFlow code by Gibson (2009)). Once p†p is known, (A.37) and (A.38) is

another comlex Helmholtz equation system solvable with the same method. Same

procedures are performed for B+- and B−-problems ; note that these two problems do

not vary from one time step to the next, and only need to be solved once in the whole

simulation run. The general solution to the B-problem is thus:

p†

v†y

=

p†p

v†y,p

+ δ+

p†+

v†y,+

+ δ−

p†−

v†y,−

. (A.49)

There are two undetermined coefficients in the solution, δ+ and δ−, because the

boundary values of pressure in the B-problem are unknown. These coefficients can be

determined by matching the solution to the yet-unused boundary condition (A.27) in

the A-problem (which is equivalent to the B-problem):

∂v†y,+/∂y |y=+1 ∂v†y,−/∂y |y=+1

∂v†y,+/∂y |y=−1 ∂v†y,−/∂y |y=−1

δ+

δ−

= −

∂v†y,p/∂y |y=+1

∂v†y,p/∂y |y=−1

. (A.50)

Equation (A.50) is known as the influence matrix equation. After we obtain the

numerical solutions of v†y and p† for the A-problem, an additional step called tau-

correction is performed. This step corrects the discretization error in the above pro-

cedure, and is found necessary to maintain numerical stability. Detailed discussion of

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tau-correction is found in Canuto et al. (1988) (Section 7.3.2), and not repeated here;

the tau-correction procedure is also included in the ChannelFlow code. Plugging p†

into the x and z components of (A.23), we get the complex Helmholtz equations for

v†x and v†z, both solvable with the Chebyshev-tau method. This completes the solution

for the Navier-Stokes equation ((A.1) & (A.2)).

The FENE-P equation for the polymer conformation tensor field (A.3) is easier

to solve. The equation, with the artificial diffusivity term 1/(ScRe)∇2α added, is

rearranged as:

∂α

∂t=− v ·∇α+α ·∇v + (α ·∇v)T

− 2

Wi

α

1− tr (α) /b+

2

Wi

b

b+ 2δ +

1

ScRe∇2α.

(A.51)

We again simplify the notation by defining:

Np ≡ −v ·∇α+α ·∇v + (α ·∇v)T − 2

Wi

α

1− tr (α) /b, (A.52)

Cp ≡ 2

Wi

b

b+ 2δ, (A.53)

Lpα ≡ 1

ScRe∇2α. (A.54)

Here Np denotes all nonlinear terms; Lpα is the linear term; and Cp is the constant

term. The simplified convection-diffusion equation is:

∂α

∂t= Np +Cp + Lpα. (A.55)

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Taking Fourier transform in x and z directions, we obtain:

∂α

∂t= Np + Cp + Lpα, (A.56)

where,

Lp = Lp |kx,kz≡1

ScRe∇2kx,kz

. (A.57)

Same as above, a semi-implicit scheme is used for temporal discretization: the

nonlinear term Np is discretized with the explicit Adams-Bashforth method; the

linear term Lpα is discretized with the implicit backward-differentiation method.

The resulting time-stepping equation is:

1

∆t

(ηαn+1 +

k−1∑j=0

ajαn−j

)=

k−1∑j=0

bjNn−jp + Lpα

n+1 + Cp. (A.58)

After rearrangement, it becomes:

η

∆tαn+1 − Lpα

n+1 =k−1∑j=0

(− aj

∆tαn−j + bjN

n−jp

)+ Cp

≡ Rn

p.

(A.59)

Here Rn

p denotes terms that can be calculated with information known at the n-th

step. Numerical coefficients are the same as those given in Table A.1. Expanding

(A.59) with:

Lp =1

ScRe

(∂2

∂y2− 4π2

(k2x

L2x

+k2z

L2z

)), (A.60)

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we obtain:

1

ScRe

∂2

∂y2αn+1 −

(4π2

ScRe

(k2x

L2x

+k2z

L2z

)+

η

∆t

)αn+1 = −Rn

p. (A.61)

Once boundary values of the αn+1 tensor are known, each component of (A.61) is

a complex Helmholtz equation that can be solved with the Chebyshev-tau method.

The boundary values are obtained by updating (A.58) without the artificial diffusivity

term Lpαn+1:

1

∆t

(ηαn+1 +

k−1∑j=0

ajαn−j

)=

k−1∑j=0

bjNn−jp + Cp, (A.62)

which, after rearrangement, gives:

αn+1 =∆t

η

(k−1∑j=0

(− aj

∆tαn−j + bjN

n−jp

)+ Cp

). (A.63)

This equation can be explicitly computed.

The overall procedure is as follows. At the beginning of each time step, inverse

Fourier transform is performed for all fields, and the nonlinear terms, N , S and Np,

are computed directly at each grid point. Note that for the computation of N , the

alternating form is used:

Nn =

∇ · (vv) divergence form, when n is odd;

v ·∇v convection form, when n is even.(A.64)

Among several forms for evaluating this term discussed in Zang (1991), the alternat-

ing form offers the best combination of efficiency, accuracy and numerical stability.

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Forward Fourier transform is then performed for all fields including results of these

nonlinear terms. A loop over every (kx, kz) is started. In each step of the loop, Rn,

Rn

p and boundary conditions for αn+1 are computed, after which the tau-equation

((A.23), (A.24) & (A.25)) for velocity and pressure fields, and Helmholtz equations

for the polymer conformation tensor field (A.61) are constructed and solved. At each

time-step and for each wavenumber pair (kx, kz), a total of 10 complex Helmholtz

equations are solved: 4 for velocity and pressure fields (2 in the B’-problem, 2 more

for v†x and v†z), and 6 for the FENE-P constitutive equation (only 3 out of the 6 off-

diagonal components need to be computed because of the symmetry of the tensor).

Other than the 1st-order algorithm, all higher-order algorithms require initial

conditions at more than one consecutive time steps. For a typical situation where

initial condition at only one instant is available, initialization of the algorithm is

required. We use lower-order algorithms to initialize higher-order ones. For example,

if the 3rd-order algorithm is used as the main algorithm (as in all simulations presented

in Part II), and if we denote the initial condition as step n = 0, we compute the n = 1

solution with the 1st-order algorithm; then we use the n = 0 and n = 1 solutions to

compute the n = 2 solution with the 2nd-order algorithm. After these steps, sufficient

solutions at previous steps are available for the 3rd-order algorithm. Increasing the

order-of-accuracy in time slightly increases the computation time, it however requires

substantially larger memory space to store additional previous steps. In general, 2nd-

or higher-order algorithms are recommended for reasons of numerical stability and

accuracy.

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