IT 14 072

Examensarbete 15 hpDecember 2014

Evaluation of RANS turbulence models for the simulation of channel flow

André Hedlund

Institutionen för teknikvetenskaperDepartment of Engineering Sciences

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Evaluation of RANS turbulence models for thesimulation of channel flow

André Hedlund

The objective of this report is to investigate how RANS models perform on fully developed channel flow, for Re = 13 350, and the simulations are made with the open source software OpenFOAM. The velocity and turbulent kinetic energy profiles are compared with previously published DNS results. A short introduction to turbulence modelling is presented with focus on channel flow and the boundary layer. In total eleven models are evaluated, and the results are of varying quality. A convergence study is presented for two models, and reveals that the expected second order convergence is fulfilled for one of them, whereas the study for the other model is more ambiguous without a clear conclusion. The OpenFOAM case setups for each model and the results gathered from the simulations are publicly available.

Tryckt av: Reprocentralen ITCIT 14 072Examinator: Jarmo RantakokkoÄmnesgranskare: Gunilla KreissHandledare: Mattias Liefvendahl

Contents

1 Introduction 2

2 Theory of turbulence modelling 32.1 The turbulent viscosity hypothesis . . . . . . . . . . . . . . . 42.2 Fully developed channel flow . . . . . . . . . . . . . . . . . . 42.3 The boundary layer . . . . . . . . . . . . . . . . . . . . . . . 62.4 Turbulent viscosity models . . . . . . . . . . . . . . . . . . . . 7

2.4.1 k � " model . . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 k � ! model . . . . . . . . . . . . . . . . . . . . . . . . 92.4.3 Wall functions . . . . . . . . . . . . . . . . . . . . . . 9

3 Simulation 113.1 Test case setup . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 The mesh . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.2 Boundary and initial conditions . . . . . . . . . . . . . 123.1.3 Low Re-model setup . . . . . . . . . . . . . . . . . . . 143.1.4 High Re-model setup . . . . . . . . . . . . . . . . . . . 14

3.2 Convergence of the residuals . . . . . . . . . . . . . . . . . . . 15

4 Results 164.1 RANS model investigation . . . . . . . . . . . . . . . . . . . . 16

4.1.1 Low Re-models . . . . . . . . . . . . . . . . . . . . . . 164.1.2 High Re-models . . . . . . . . . . . . . . . . . . . . . . 18

4.2 Grid convergence study . . . . . . . . . . . . . . . . . . . . . 20

5 Discussion and conclusions 23

Acknowledgment 25

1

1 Introduction

Turbulence occur in many natural flows and is of paramount importance inmany engineering applications. The Navier-Stokes equations describe thedynamics of fluids and can rarely be solved analytically. For this reason, weare to a large extent dependent on numerical simulation for making fluidpredictions. Flow at a su�ciently low Reynolds (Re) number is laminar andthe Navier-Stokes equations can then be numerically solved directly. Forhigh Re-numbers, which typically occur in applications, the flow is turbulentand hence consists of fluid motion with a wide range of spatial and temporalscales. To numerically solve a turbulent flow it is important resolve theflow such that the smallest scales can be represented, therefore a directnumerical simulation (DNS) of turbulent flow is very costly and in mostcases unfeasible, see [1]. In the Reynolds-averaged Navier-Stokes (RANS)formulation, the velocity is decomposed into the mean velocity distribution,and the fluctuations around this mean. The equations are then solved for themean velocity, and the e↵ect of the fluctuations is modelled, see [2]. Withthis approach it is not necessary to resolve the smallest turbulent scales –and the computational cost can therefore be reduced significantly.

Channel flow is the case when a fluid flows between two parallel plates,such as a rectangular duct. Turbulent channel flow is a well investigatedcase both experimentally and numerically. Early DNS results for channelflow were presented by Kim, Moin and Moser [3], and a continuation of thiswork, which includes higher Re-numbers, where presented by Moser, Kimand Mansour [4].

The objective of this report is to evaluate the performance of severalRANS models for incompressible turbulent fully developed channel flow.The focus is to evaluate the models with respect to the velocity profile,turbulent kinetic energy profile and wall shear stress, and if it is possiblerank the models based on how they predict the mentioned quantities. Theresults will be compared with theoretical approximations (law of the wall)and the DNS results of Moser, Kim and Mansour [4].

The simulations are carried out with the OpenFOAM toolbox, an opensource software for solving continuum mechanics problems [5]. There areseveral RANS models implemented in OpenFOAM and each model has beendeveloped with a di↵erent objective in mind. This report will not try toexplain in depth why a certain model fails or succeeds in approximating thequantities investigated, it will simply state that a model failed or succeeded.

The results from the simulation and the OpenFOAM case setups arepublicly available at https://github.com/AndreHed/channelFlow.git.

2

2 Theory of turbulence modelling

This section describes the theory of turbulent flows necessary for describingturbulence modelling for fully developed channel flow.

The continuity equation is given by

r ·U = 0 (1)

and the Navier-Stokes equation is given by

DU

Dt

⌘ @U

@t

+U ·rU = �1

⇢

rp+ ⌫r2U (2)

where ⌫ = µ/⇢ is the kinematic viscosity. Here we have four unknowns: threecomponents of the velocity and the pressure; and we have four equations:the continuity equation (eq. (1)) and the three components of Navier-Stokesequations (eq. (2)). Hence, the system is closed and we can solve for theunknowns. [1]

As the Re-number increases the flow enters the turbulent regime, andsmaller and smaller eddies will form. The eddies correspond to small randomfluctuations in the variables describing the flow. Therefore, in a turbulentflow the velocity field, U(x, t), can be expressed by a mean and a fluctuatingpart, viz.

U(x, t) = hU(x, t)i+ u(x, t), (3)

this will be referred to as the Reynolds decomposition. The mean is givenby time averaging

hU(x, t)i = limT!1

1

T

Z T

0U(x, t) dt, (4)

which is an applicable averaging for the stationary flows considered in thisreport.

We can take the mean of equation (1) to obtain

@ hUii@xi

= 0. (5)

Taking the mean of equation (2) yields

D hUjiDt

⌘ @ hUji@t

+ hUii@ hUji@xi

= ⌫r2 hUji �@ huiuji@xi

� 1

⇢

@ hpi@xj

, (6)

which will be referred to as the Reynolds equations. The di↵erence betweenthe Navier-Stokes equations (eq. (2)) and the Reynolds equations (eq. (6))is that the term ⌧ij = huiuji is introduced, which is called the Reynoldsstresses and describe the stress arising from the fluctuating velocity field.It is important to note that we have introduced six new unknowns (⌧ij is a

3

symmetric tensor) without introducing additional equations, thus we haveten unknowns and only four equations – the system is therefore not closed. [2]

Half of the trace of the Reynolds stress is called the turbulent kineticenergy

k ⌘ 1

2huiuii . (7)

The turbulent kinetic energy can be described as the kinetic energy perunit mass in the oscillating velocity field. It will be used frequently in theremaining of the report and is one of the quantities that is investigated.

2.1 The turbulent viscosity hypothesis

The turbulent viscosity hypothesis resolves the unclosed problem that iscreated by the introduction of the Reynolds stresses. It states

huiuji =2

3k�ij � ⌫T

✓@ hUii@xj

+@ hUji@xi

◆(8)

where ⌫T is called the turbulent viscosity. The hypothesis inserted in equa-tion (6) yields

D hUjiDt

=@

@xi

⌫e↵

✓@ hUii@xj

+@ hUji@xi

◆�� 1

⇢

@

@xj

✓hpi+ 2

3⇢k

◆(9)

where ⌫e↵ = ⌫+⌫T. Comparing this result with equation (2) we can concludethat they are of the same form, therefore, if ⌫T is specified the modifiedReynolds equation (eq. (9)) is now in closed form.

There are two concerns with the hypothesis that should be discussed,

1. the accuracy of the hypothesis, and

2. the specification of the turbulent viscosity.

It has been proved that, unfortunately, the accuracy of the turbulent vis-cosity hypothesis is poor for many flows, however it still serves a purposesince it is reasonable for some simple flows. The turbulent viscosity can bewritten as

⌫T = l

⇤u

⇤, (10)

where di↵erent approaches for specifying the length l

⇤ and the velocity u

⇤

are presented in the following sections. [1]

2.2 Fully developed channel flow

Channel flow (see figure 1) is a simple flow bounded by two parallel platesseparated by a distance of 2� and the flow direction is along the x-axis. Forfully developed flow the velocity is dependent of only y and it is statistically

4

x

y

z

h = 2�Flow

Figure 1: Sketch of channel flow.

stationary and one-dimensional. Using the notation (U, V, W ) for the ve-locity field, it is easy to conclude that hW i = 0. The Re-number for thisflow is given by

Re =U · 2�⌫

, (11)

where U is the bulk velocity defined as

U ⌘ 1

�

Z �

0hUi dy. (12)

Because of the simplicity of the flow the mean continuity equation (eq. (5))and the mean momentum equations (eq. (6)) can be greatly simplified. Usingthe fact that hW i = 0 and that hUi is independent of x the mean continuityequation reduces to

@Ui

@xi=

d hV idy

= 0. (13)

The boundary condition at the walls are hV iy=0 = hV iy=2� = 0, thereforewe obtain that hV i is zero for all y.

The lateral mean momentum equation is deduced to

1

⇢

d hpidy

+d⌦v

2↵

dy= 0 (14)

by noting that the nonlinear terms are neglectable due to the boundary-layerapproximation and that the axial derivatives of the Reynolds stresses aresmall compared with the lateral derivatives. With the boundary condition⌦v

2↵y=0

= 0 we obtain

1

⇢

hpi+⌦v

2↵=

1

⇢

pw(x) (15)

where pw(x) = hp(x, 0, 0)i. Di↵erentiating equation (15) with respect to x

yieldsd hpidx

=dpwdx

. (16)

5

The axial mean momentum equation for fully developed channel flow,obtained in the same way as the lateral analog, is

@

@y

⇢⌫

@ hUi@y

� ⇢ huvi�=

@ hpi@x

. (17)

The total shear stress consists of the viscous stress and the Reynolds stress,viz.

⌧(y) = ⇢⌫

d hUidy

� ⇢ huvi . (18)

Using equation (16)-(18) we finally obtain

@⌧

@y

=@pw

@x

. (19)

Introducing ⌧w = ⌧(0) as the wall shear stress, the boundary conditions forequation (18) are ⌧(0) = ⌧w and ⌧(2�) = �⌧w, which yields [1]

d⌧

dy= �⌧w

�

. (20)

2.3 The boundary layer

The total shear stress (equation (18)) consists of two terms: the viscousstress ⇢⌫d hUi /dy and the Reynolds stress �⇢ huvi. Due to the no slipcondition the Reynolds stress is zero at the wall and the shear stress isdominated by the viscous stress in the vicinity of the wall. The wall shearstress is therefore described by

⌧w = ⇢⌫

✓d hUidy

◆

y=0

. (21)

Away from the wall the viscous stress is negligible compared with the Reynoldsstress and the total shear stress is therefore described by the Reynolds stress.

In the analysis of the flow close to the wall it is useful to define severalviscous scales from the viscosity ⌫, the wall shear stress ⌧w and the density⇢, these are summarised in table 1.

Due to the reciprocal action between the viscous stress and the Reynoldsstress, di↵erent layers can be identified in the vicinity to the wall. For y+ < 5the viscous sublayer is apparent and the velocity adheres to a linear relation

u

+ = y

+. (22)

As the y

+-values increases, the viscous stress becomes negligible comparedwith the Reynolds stress. For y+ > 30 there is a log-law region which obeysthe logarithmic law

u

+ =1

ln y+ +B, (23)

6

Table 1: Viscous scales.

Friction velocity u⌧ =

r⌧w

⇢

Viscous length scale �⌫ = ⌫

r⇢

⌧w=

⌫

u⌧

Friction Reynolds number Re⌧ =u⌧�

⌫

=�

�⌫

Wall units y

+ =y

�⌫=

u⌧y

⌫

Viscous velocity u

+ =hUiu⌧

Turbulent kinetic energy k

+ =k

u

2⌧

where (von Karman’s constant) and B are given by

= 0.41, B = 5.2. (24)

The region between the two layers (i.e. 5 < y

+< 30) is called the bu↵er

layer; in this region neither viscous nor turbulent processes dominate. Theselayers are presented in figure 2, and the theoretical laws are compared withDNS data. [1]

2.4 Turbulent viscosity models

The Reynolds decomposition (eq. (3)) separated the mean and the fluctua-tions. RANS models solve only for the mean quantities, and are thereforecomputationally much cheaper since there is no need to resolve the smallfluctuations of a turbulent flow. The Reynolds decomposition did howeverintroduce the Reynolds stresses, six new unknowns without additional equa-tions, which resulted in an unclosed system. This section will describe howto close the system by using the turbulent viscosity hypothesis and usingmodel transport equations to specify the turbulent viscosity ⌫T. There arevarious other ways to close the problem that are not considered in this report(e.g. Reynolds stress models and mixing-length model).

2.4.1 k � " model

The k�"model is a two-equation model - one transport equation is solved forthe turbulent kinetic energy, k, and one for the dissipation rate of turbulentkinetic energy, ". When these model equations are solved, the turbulent

7

100 101 102

100

101

y

+ = 5 y

+ = 30

viscous sublayer bu↵er layer log-law region

y

+

u

+

u

+ = y

+

u

+ = 1 ln x+B

DNS

Figure 2: The linear relation of the viscous sublayer and the logarithmic relationof the log-law region compared with the DNS data of Moser, Kim and Mansour [4].

viscosity can be computed by

⌫T = Cµk

2

"

, (25)

where Cµ is a model constant.The exact equation for k is given by

Dk

Dt

= �r ·T0 + P � ", (26)

where T0 is the flux of Reynolds stress and P is the rate of production ofturbulent kinetic energy, given by

P = �huiuji@ hUii@xj

. (27)

The mean derivative and P are in closed form and " will be modelled by aseparate equation, therefore only T0 need to be modelled, which can be donewith a gradient-di↵usion hypothesis. This results in the model transportequation for k

Dk

Dt

= r ·✓⌫T

�krk

◆+ P � ", (28)

8

where �k is a model constant.The dissipation rate is given by

" = ⌫

⌧@ui

@xj

@ui

@xj

�(29)

and the model transport equation for " is given by

D"

Dt

= r ·✓⌫T

�"r"

◆+ C"1

P"

k

� C"2"

2

k

. (30)

All the model constants introduced are given by

Cµ = 0.09, C"1 = 1.44, C"2 = 1.92, �k = 1.0, �" = 1.3. (31)

The inaccuracies of the model originates from the turbulent viscosityhypothesis and the " equation. [1]

2.4.2 k � ! model

The k�! model use a similar approach as the k� " model, but instead of "it uses the rate of dissipation of energy in unit volume and time, or merelythe specific dissipation rate, defined as

! ="

k

. (32)

The model equation for k is slightly modified

Dk

Dt

= r · [(⌫ + �⌫T )rk] + P � ", (33)

and the model transport equation for ! is given by

D!

Dt

= r · [(⌫ + �⌫T )r!] + ↵

P!

k

� �!

2, (34)

where the constants are given by

� = 12 , ↵ = 0.52, � = 0.072. (35)

This is the classic Wilcox k � ! model described in detail in [2].

2.4.3 Wall functions

For the models described, boundary conditions needs to be specified for thevelocity and the turbulent parameters. The boundary condition for the wallcan be problematic due to the fact that for y

+< 30 the gradients of the

velocity and " are large, therefore the mesh needs to be very fine close tothe wall to accurately resolve the gradients – which can be very costly. If

9

the mesh is not fine enough the models described above deficiently predictthe flow behaviour close to the wall.

Wall functions are introduced to mitigate the computational cost, andin the same time enable the model to predict accurate results. The wallfunction is utilised between the wall and the mesh point closest to the wall(matching point), and it will try to match the value at the matching pointwith the law of the wall described in section 2.3. Each turbulent parameterhas its own wall function which is derived from the law of the wall.

It is important to emphasise that if the mesh is very refined (i.e. thematching point lies before y

+< 1) wall functions are not necessary. [2]

10

3 Simulation

As described in section 2.2 fully developed channel flow is a one-dimensionalproblem. In OpenFOAM a case setup consists of: specifying the mesh, set-ting the boundary and initial conditions, and specifying the physical prop-erties. The di↵erent RANS models that are investigated in this report arelisted in table 2. The OpenFOAM solver that is used is boundaryFoam,which is a steady-state solver for one-dimensional turbulent flows.

Table 2: RANS models implemented in OpenFOAM that are investigated.

OpenFOAM Type Description

Low Re LaunderSharmaKE 2-eq. Modified k � " modelLamBremhorstKE 2-eq. Modified k � " modelLienCubicKELowRe 3-eq. Modified k � " modelLienLeschzinerLowRe 2-eq. Modified k � " model

High Re kEpsilon 2-eq. See sec. 2.4.1kOmega 2-eq. See sec. 2.4.2kOmegaSST 2-eq. Combination of k�" and k�!

RNGkEpsilon 2-eq. Modified k � " modelrealizableKE 2-eq. Modified k � " modelSpalartAllmaras 1-eq. Solves a transport eq. for ⌫v2f 4-eq. Solves for k, ", hvi2 and f

3.1 Test case setup

Fully developed channel flow is obtained by specifying the bulk velocity ofthe flow. The solver will add a pressure gradient to the momentum equation,and after each iteration it will correct the pressure gradient such that thespecified U is maintained. The bulk velocity together with the kinematicviscosity and half-channel width are the physical properties that completelydescribe the flow (eq. (11)), and in OpenFOAM these are specified in thefile transportProperties. The values that are used in the simulations aregiven in table 3.

Table 3: Physical properties for the simulation campaign.

� 1m⌫ 2⇥ 10�5m2/sU 0.1335m/sRe 13 350

11

2�

wall

wall

cyclic cyclic

x

y

z

Figure 3: OpenFOAM setup for the one-dimensional fully developed channel flow.

3.1.1 The mesh

OpenFOAM solves the equations with the finite volume method, therefore,the mesh is always three-dimensional. The fully developed channel flow issimulated as a one-dimensional problem (see fig. 3), with only one cell inthe x- and z-direction. In y-direction the number of cells Ny is variable andthe cells are distributed geometrically such that there is a higher density ofcells near the wall than in the centre of the channel. In OpenFOAM a ratiobetween the largest cell and the smallest cell is specified.

The half-channel width (�), Ny and the grading are specified in the fileblockMeshProperties, and the mesh is created with the blockMesh utility.

3.1.2 Boundary and initial conditions

Since the case is one-dimensional, the sides facing the z-direction are set toempty, which symbolises that no solution is required in that dimension. Thesides facing the flow direction, which could be seen as inlet and outlet, aregiven a cyclic boundary condition. The remaining sides correspond to thewalls (see fig.3).

The boundary conditions at the wall is specified for each variable. For thevelocity the no slip condition, (U, V,W ) = (0, 0, 0), is applied. Section 2.4.3described that wall functions may be utilised to reduce the computationalcost and improve the results. Wall functions are selected in the 0/ directoryfor each variable that is needed; the various wall functions implemented inOpenFOAM are presented in table 4.

12

Table 4: Wall functions

⌫T nutUWallFunction Provides a boundary condition for ⌫T

based on velocity.nutUSpaldingWallFunction Uses Spalding’s law to give a continu-

ous ⌫T profile, based on velocity.nutLowReWallFunction For low Reynolds number models. It

sets ⌫T = 0, and provides a function tocalculate y

+.nutkWallFunction Based on turbulent kinetic energy.

k kqRWallFunction For k, q and R for high Reynolds num-bers. Based on the zero-gradient con-dition.

kLowReWallFunction Provides a wall function condition forboth low and high Reynolds numbers.Operates in two modes based on y

+.

✏ epsilonWallFunction For high Reynolds numbers.epsilonLowReWallFunction Works for both low and high Reynolds

numbers. The model operates in twomodes based on an approximated y

+

value.

! omegaWallFunction Works for both low and high Reynoldsnumbers.

hvi2 v2WallFunction Works for both low and high Reynoldsnumbers.

Initial conditions are required for the velocity field, k, " and !. The tur-bulent kinetic energy is given by equation (7) which for isentropic turbulenceresults in

k =3

2u

2 (36)

where the fluctuating velocity can be approximated by the product of thebulk velocity and the turbulent intensity, viz.

u = tiU . (37)

For this case the turbulent intensity is approximately 5%. The initial con-dition for " is given by

" =C

3/4µ k

3/2

�

, (38)

and for ! equations (25) and (32) yield

! ="

Cµk. (39)

13

In OpenFOAM the boundary and initial conditions are specified in the0/ directory for the variables that are required. The variables required aredetermined by which solver and model that is used. [6]

3.1.3 Low Re-model setup

The low Reynolds number models are developed for flows with moderateRe-numbers where the mesh is refined all the way to the wall, therefore nowall functions are used. In table 4 there are, however, wall functions thatare specifically made for low Re-models; these are usually just a placeholderfor a regular boundary condition, such as zero gradient or no slip conditions.In this case the wall function for ⌫T will produce the same results as setting⌫T to be calculated from k and ".

Table 5 lists the settings for the low Re-models.

Table 5: Boundary conditions and mesh settings for low Re-models.

⌫T nutLowReWallFunction

k fixedValue = 0" fixedValue = 0U fixedValue = 0Ny 400grading 100

3.1.4 High Re-model setup

The high Reynolds number models are developed for high Re-numbers wherethe utilisation of wall functions are necessary to limit the computational cost.The Re-number that is investigated is relatively low and the logarithmicboundary layer is limited to a couple of hundred y

+ units, therefore wallfunctions are not necessary per se. In OpenFOAM the wall functions areimplemented such that they work e↵ectively for flows with low Reynoldsnumbers as well, therefore they are used as boundary conditions for thesimulations. In table 6 the settings for the high Re-models are displayed.

14

Table 6: Boundary conditions and mesh settings for high Re-models.

⌫T nutUWallFunction

k kqRWallFunction

" epsilonWallFunction

! omegaWallFunction

U fixedValue = 0Ny 400grading 100

3.2 Convergence of the residuals

Figure 4 shows the convergence of hUi, k and " with respect to the residualsfor one of the models that are investigated. The variables converge almostmonotonically until the residuals are reduced to a size where the machineepsilon prevents further reduction.

0 10000 20000 30000 40000 5000010�16

10�13

10�10

10�7

10�4

10�1

Iteration

Residual

hUik

"

Figure 4: The convergence of the initial residuals for the Launder-Sharma k � "

model with Ny = 100.

15

4 Results

In this section the RANS models are evaluated with respect to how accuratethey predict fully developed channel flow compared with the DNS data ofMoser, Kim and Mansour [4]. A grid convergence study with respect tothe wall shear stress and velocity is also presented. The friction Reynoldsnumber of the investigated flow is Re⌧ = 395, which correspond to theReynolds number, Re ⇡ 13 350. The OpenFOAM test setups for each modeland the results gathered from the simulations are available at https://

github.com/AndreHed/channelFlow.git.

4.1 RANS model investigation

The RANS models that are investigated in this report are given in table 2.The results presented are divided into low and high Re-models, this is mainlybecause the problem setup di↵ers between the two groups.

In table 7 the friction velocity, friction Reynolds number and smallesty

+ value are presented for all models that have been investigated. Note thatwhile the desirable friction Reynolds number is Re⌧ = 395, the resulting Re⌧varies a lot. Note also that the models produce reasonably small y+ values,therefore the mesh is su�ciently refined at the walls.

Table 7: The obtained friction velocity, friction Reynolds number and first y

+

value for the low Reynolds number models.

Model u⌧/mms�1 Re⌧ y

+/10�2

Low Re Launder-Sharma k � " 7.19 359 4.15Lam-Bremhorst k � " 7.66 383 4.42Lien cubic k � " 6.44 322 3.72Lien-Leschziner 6.42 321 3.71

High Re k � ! 7.87 393 4.54k � ! SST 7.71 386 4.45Spalart-Allmaras 7.55 377 4.36k � " 10.7 535 6.18RNG k � " 10.5 525 6.06Realizable k � " 5.84 292 3.37hvi2 � f 4.45 223 2.57

4.1.1 Low Re-models

In figure 5 the velocity and turbulent kinetic energy profile for low-Re modelsare compared with the DNS results. The results are normalised with boththe friction velocity and the bulk velocity (see table 1). The velocity profile

16

100 101 1020

10

20

y

+

u

+

0 0.2 0.4 0.6 0.8 10

2

4

y/�

k

+

100 101 102

100

101

y

+

u

+

0 0.2 0.4 0.6 0.8 10

0.5

1

y/�

hUiU

DNS; Launder-Sh.; Lam-Br.; Lien cubic; Lien-Le.

Figure 5: Simulating channel flow with low Reynolds number RANS models,where Re = 13 350.

in plus units is accurate for all models for y

+< 6, but for y

+> 10 the

results vary significantly. Both the Lien cubic and Lien-Leschziner modelincrease too much and at the channel centreline u

+ is up to 20 percent o↵.The Launder-Sharma and Lam-Bremhorst model is much better and deviateapproximately five percent or less at the centreline.

When the velocity is normalised with the bulk velocity the profiles areinstead inaccurate in the bu↵er layer, and at the centreline the di↵erenceis significantly less than when the results are normalised with the frictionvelocity.

The results for the turbulent kinetic energy are also not accurate, espe-cially where the maximum of the profile is. Note that an accurate resultfor the velocity does not imply that the result for k is accurate as well. Itis also important to note that while the Lam-Bremhorst model produced

17

the best results for both quantities, it is the model that is the most di�cultto achieve convergence for. The other models converged with initial condi-tions as described in section 3.1.2, whereas the Lam-Bremhorst model onlyconverged if the solution of one of the other models where used as initialconditions for U , k and ", and even then the convergence of " is very slow.

4.1.2 High Re-models

Figure 6 plots the results for the k � !, k � ! SST and Spalart-Allmarasmodels. These high Re-models predicted the quantities much better thanthe other high Re-models which are presented later. The profiles for u

+

shows the same behaviour as for the low Reynolds models in figure 5: fory

+< 6 the profiles agree very well with DNS data, whereas for y

+> 10

the profiles are dispersed. Even though the profiles disperse, it is not to thesame extent as for some of the low Reynolds models, the velocity does notvary significantly in comparison with the DNS data.

The results for k+ are similar to the results of the low Reynolds numbermodels in the sense that they fail to reach the same peak as the DNS data.Note that k is not required for the Spalart-Allmaras model, and thereforeno data is available.

When the velocity is normalised with the bulk velocity the profiles matchvery well with the DNS data. The models in figure 6 are successful inpredicting the velocity in both the viscous sublayer as well as in the log-lawregion.

In figure 7 the results for the k�", RNG k�", realizable k�" and hvi2�f

models are presented. These models generally mispredicted the investigatedquantities. Realisable k � " and hvi2 � f were particularly bad since theyboth had convergence issues and are dependent on the initial conditions.

Note the correlation between how the profile look in figures 6 and 7, andthe size of Re⌧ and u⌧ in table 7. For example, for k � " and RNG k � ",both produce a large Re⌧ , the u+ profiles are very similar to each other; andfor RNG k� " and hvi2 � f , both produce a small Re⌧ , the u

+ profiles havethe same characteristics.

18

100 101 1020

10

20

y

+

u

+

0 0.2 0.4 0.6 0.8 10

2

4

y/�

k

+

100 101 102

100

101

y

+

u

+

0 0.2 0.4 0.6 0.8 10

0.5

1

y/�

hUiU

DNS; k!; k!SST; Spalart-Al.

Figure 6: Simulating channel flow with high Reynolds number RANS models,where Re = 13 350.

19

100 101 1020

10

20

y

+

u

+

0 0.2 0.4 0.6 0.8 10

2

4

y/�

k

+

100 101 102

100

101

y

+

u

+

0 0.2 0.4 0.6 0.8 10

0.5

1

y/�

hUiU

DNS; k"; RNG k"; Realizable k"; hvi2 f

Figure 7: Simulating channel flow with high Reynolds number RANS models,where Re = 13 350.

4.2 Grid convergence study

The grid convergence was investigated with respect to the wall shear stress⌧w and the velocity. When OpenFOAM computes the wall shear stress itcomputes ⌧w/⇢, the values presented for the wall shear stress are thereforedivided by the density. Generally, the convergence rate is calculated by

p = ln|f2 � f1||f3 � f2|

�ln r (40)

where r is the mesh refinement ratio, fi is the variable that is investigatedand i = 3 is the finest and i = 1 is the coarsest mesh. The number ofcells that are used across the channel are Ny = 200, 600 and 1800. Eachquantity is calculated at the cell centre, therefore in order to get the cellcentres to overlap for the di↵erent meshes the mesh refinement ratio is r =

20

3. No grading is imposed on the mesh since that would result in non-overlapping cell centres for the di↵erent meshes. The solution is consideredto be converged when the residuals are below 1⇥ 10�9.

The convergence study is performed for the Launder-Sharma and Spalart-Allmaras models. Both of these models did produce accurate results (seefig. 5 and 6), and are therefore reasonable models to investigate. In table 8the results from the convergence study are presented, and equation 40 yieldsfor the wall shear stress

pLS = 1.81 pSA = 2.01. (41)

Note that the values for the wall shear stress are divided by density Theconvergence rate for the velocity vary considerably across the channel it istherefore more reasonable to plot the convergence rate, see figure 8. Forboth models extrema occur, this is explained by that the velocity profiles ofthe di↵erent meshes intersect each other. At the wall the no slip boundarycondition is imposed and the expected convergence rate in the vicinity of thewall is expected to be p = 2. It is troubling that for Launder-Sharma theconvergence rate is predominantly negative, which indicates divergence. Theresult for Spalart-Allmaras is approximately two across the entire channel,which is what is to be expected for the numerical schemes that are utilised.

Table 8: Convergence study.

Ny y

+⌧w/mm2 s�2 iterations

Launder-Sharma 200 2.07 55.5 50 000600 0.608 56.1 300 0001800 0.204 56.1 1 000 000

Spalart-Allmaras 200 1.92 58.8 50 000600 0.630 57.2 500 0001800 0.210 57.0 2 000 000

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1�4

�2

0

2

4

y

p

Launder-Sharma k � "

Spalart-Allmaras

Figure 8: The convergence rate for the velocity across the channel.

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5 Discussion and conclusions

The previous section presented the results of a simulation campaign madewith OpenFOAM for several RANS models, and the results were comparedwith the DNS results of Moser, Kim and Mansour [4]. The results di↵ersignificantly between the models, and while some models produce reason-ably good results, others fail to accurately predict the investigated quanti-ties. Apart from the flow quantities, the results of a grid convergence studyfor two models were also presented. The study verified that the Spalart-Allmaras model had second order convergence, whereas no convergence forthe Launder-Sharma k�" model could be established. Now follows a discus-sion of the results presented and statements about each models suitabilityfor simulating fully developed channel flow.

The results from the low Re-models agree well with the DNS data, butthere are significant di↵erences between the models. The Launder-Sharmaand Lam-Bremhorst models predicted the wall shear stress better than theLien cubic and Lien-Leschziner models, while for the turbulent kinetic energyprofile the Lam-Bremhorst and Lien cubic performed better than the othertwo. This could be the basis of concluding that the Lam-Bremhorst modelis the superior model for this case, but as mentioned in section 4.1.1, themodel has severe convergence problems, to the point that it only convergesif the initial conditions for hUi, k and " are mapped from the solution ofanother model such as Launder-Sharma.

The results from some of the high Re-models predicted the investigatedquantities well, whereas others mispredicted the profiles to the extent thatthey are not acceptable. The first distinction that can be made betweenthe models, is that the models based on the transport equations for k and" did not predict hUi, k and ⌧w very well, whereas the models based on thetransport equations for k and ! performed much better. The implementa-tion of the k � ! model in OpenFOAM is based on Wilcox’s (1988) k � !

model (see OpenFOAM source code available at [5]). This implementationis known to accurately predict the flow properties for wall bounded flow [2,p. 128]. Since channel flow is a wall bounded flow, it is reasonable that theresults, for the models based on transport equations for k and !, agrees wellwith the DNS results. The k�" model is known to poorly predict boundarylayers with strong pressure gradients (see [1, p. 461]), which agrees with thefact that the models based on the transport equations for k and " did notpredict the flow properties very well.

A surprising result is that the Spalart-Allmaras model performed verywell. Of the models that are investigated the Spalart-Allmaras model isthe only one-equation model and therefore the model with the simplestturbulence description. The model predicted the velocity profile just asaccurately as the k � ! and k � ! SST.

The results of the “hvi2 � f” model are not good, which could be the

23

result of an incorrect setup for the model, therefore the results presented inthe report may not be representative for the model. A peculiar notion is thatthe model is sensitive to the initial conditions, where a small perturbationcan produce a di↵erent solution, this has been noted before, see [7, p. 173].

The results of the models are ranked in table 9. The ranking is basedon the results presented in figures 5-7 and table 7, and only the top fivemodels are ranked. Based on the ranking in table 9, the RANS models thatare suitable for the simulation of fully developed channel flow are (in noparticular order): Launder-Sharma k � ", k � !, k � ! SST and Spalart-Allmaras.

Table 9: Ranking of the models for four di↵erent properties, the ranking is onlymade for the five best models for each property. Note that ’⇤’ signifies that themodel did rank among the top five and ’-’ means that no data is available.

Model Re⌧ hUi k Convergence

Launder-Sharma k � " 5 5 3 2Lam-Bremhorst k � " 3 2 1 *Lien cubic k � " * * 2 5Lien-Leschziner * * * *k � ! 1 4 4 3k � ! SST 2 3 5 4Spalart-Allmaras 4 1 - 1k � " * * * *RNG k � " * * * *Realizable k � " * * * *hvi2 � f * * * *

This investigation is for a constant Re-number, Re = 13 350, and furtherinvestigations for higher Re-numbers would be worthwhile. The OpenFOAMtest setups for each model and the results gathered from the simulationsare available at https://github.com/AndreHed/channelFlow.git, read-ers are encouraged to try the cases by themselves and any contribution orcomment is welcome.

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Acknowledgment

I would like to thank my supervisor Mattias Liefvendahl for providing mewith the subject of this thesis, and for his guidance in during the process ofwriting this thesis. I would also like to thank Timofey Mukha for taking histime to answer my questions and providing his insights in the subject.

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References

[1] S. Pope, Turbulent Flows. Cambridge University Press, 2000.

[2] D. Wilcox, Turbulence Modeling for CFD. DCW Industries, 2006.

[3] J. Kim, P. Moin, and R. Moser, “Turbulence statistics in fully developedchannel flow at low reynolds number,” J. Fluid Mech, 1987.

[4] R. D. Moser, J. Kim, and N. N. Mansour, “Direct numerical simulationof turbulent channel flow up to re= 590,” Phys. Fluids, vol. 11, no. 4,pp. 943–945, 1999.

[5] http://www.openfoam.com, October 2014.

[6] OpenFOAM Foundation, OpenFOAM - User Guide, 2.3.0 ed., 2014.

[7] D. Laurence, J. Uribe, and S. Utyuzhnikov, “A robust formulation ofthe v2- f model,” Flow, Turbulence and Combustion, vol. 73, no. 3-4,pp. 169–185, 2005.

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