Prediction of homogeneous shear flow and a backward-facing step flow with some linear and non-linear...

Communications in Nonlinear Science

and Numerical Simulation 10 (2005) 315–328

www.elsevier.com/locate/cnsns

Prediction of homogeneous shear flow and abackward-facing step flow with some linear and

non-linear K–e turbulence models

X.-D. Yang a, H.-Y. Ma a,*, Y.-N. Huang b

a Department of Physics, Graduate School of the Chinese Academy of Sciences, P.O. Box 3908, Beijing 100039, Chinab State Key Laboratory for Turbulence and Complex Systems/Department of Mechanics and Engineering Science,

Peking University, Beijing 100871, China

Received 18 November 2002; received in revised form 30 April 2003; accepted 2 July 2003

Available online 28 October 2003

Abstract

In this paper we present a numerical study of homogeneous turbulent shear flow and turbulent flow pasta backward-facing step with a few linear and non-linear turbulence models. Two linear models are the

standard K–e model of Launder and Spalding and the non-equilibrium model of Yoshizawa and Nisizima.

The non-linear models include three quadratic models of Speziale, Shih, Zhu and Lumley, and Huang, and

the cubic model of Craft, Launder and Suga. In particular, the capability of capturing the non-equilibrium

effect of turbulence and turbulence anisotropy based on these models is assessed in detail. The numerical

results indicate that the model of Yoshizawa and Nisizima and the model of Huang predict the homoge-

neous turbulent shear flow correctly with the introduced terms that depict the non-equilibrium effect.

Moreover, the non-linear models account for the anisotropy of the normal Reynolds stresses reasonablywell and yield considerable improvement in predicting the backward-facing step flow over the linear

models.

� 2003 Elsevier B.V. All rights reserved.

PACS: 47.27.Eq; 47.27.Jv; 47.27.Nz

Keywords: Eddy-viscosity models; Homogeneous shear flow; Flow past a backward-facing step; Non-equilibrium effect;

Anisotropy of the Reynolds stress

* Corresponding author.

E-mail address: [email protected] (H.-Y. Ma).

1007-5704/$ - see front matter � 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.cnsns.2003.07.001

mail to: [email protected]

316 X.-D. Yang et al. / Communications in Nonlinear Science and Numerical Simulation 10 (2005) 315–328

1. Introduction

It is generally believed that the Navier–Stokes (NS) equations can describe the turbulent flowsof a Newtonian fluid, which in fact constitutes the fundamental framework for the study ofturbulence over the past century. Even though since its commencement in the early 1980s thedirect numerical simulation of turbulence has evinced its powerful and impressive capability indealing with the turbulent flows with a simple geometric boundary, this approach is computa-tionally very much expensive for high Reynolds number turbulent flows because the computa-tional cost increases as Re3 and, as a result, besides other potential approaches such as large eddysimulation, in the foreseeable future the numerical simulations of turbulent flows that are ofpractical importance and interest would still have to rely on the approach of the Reynolds–av-eraged NS equations, i.e., the so-called RANS approach in which the Reynolds stress must bemodelled and given in advance. In fact, among the various RANS models for turbulence, the two-equation (K–e) models have found their broad applications of feasibility in scientific research aswell as in engineering practice for the predictions of the complex turbulent flows. However, it isfound that a number of currently well-used linear and non-linear K–e models are often good atpredicting some benchmark turbulent flows but not the case at all when applied to other typicalturbulent flows. For instance, the standard K–e (SKE) model (cf. [10]), like all other linear eddy-viscosity models, is not able to capture the turbulent secondary flows in a fully developedturbulent flow through a straight tube of non-circular cross-section, which is an interestingphenomenon of practical importance in engineering and, being a typical turbulent secondary flow,it has been well documented in the literature (cf. [1]); in reality, for a model to capture suchphenomenon it would require non-zero normal Reynolds stress differences over the cross-section(cf. [6]). Remarkably, Speziale [17] developed a non-linear K–e model that includes the SKE modelas a special case, in which a quadratic term of the mean stretching (rate of strain) tensor plays akey role in capturing the turbulent secondary flow through a straight tube of non-circular cross-section.

Over the past two decades many linear and non-linear K–e models have been developed to meetthe challenges of practical application and requirement in turbulence modelling. Under thesecircumstances, it is necessary to carry out detailed numerical simulations to assess the capabilityof these models and compare their performance in predicting the benchmark turbulent flows so asto validate their accuracy and identify their advantages and shortcomings respectively. To thisend, recently, Yang and Ma [18] have done a series of numerical simulations of some typicalturbulent flows and made a careful comparison of the numerical results obtained by employing anumber of linear and non-linear K–e models with the experimental data. Clearly, withoutknowing about the deficiency as well as the advantage of these previously proposed models, itwould be impossible to develop a better closure model of broader capability in predicting thecomplex turbulent flows.

In this work, we shall present a numerical investigation of the homogeneous turbulent shearflow and the fully developed turbulent flow past a backward-facing step with a non-linear K–emodel proposed by Huang [9] and a few previously developed linear and non-linear K–e models,i.e., the SKE models, the non-linear model of Speziale [17], the non-equilibrium linear model ofYoshizawa and Nisizima (YN) [19], the model of Shih, Zhu and Lumley (SZL) [16], and the non-linear cubic model of Craft, Launder and Suga (CLS) [3], and the performance of these models

X.-D. Yang et al. / Communications in Nonlinear Science and Numerical Simulation 10 (2005) 315–328 317

will be assessed in comparison with the numerical results. In particular, the accuracy in predictingthe reattachment point in the fully developed turbulent flow over a backward-facing step and thecapability in capturing the non-equilibrium effect of the homogeneous turbulent shear flow will beevaluated in detail.

2. Governing equations and the models tested

We shall focus on the turbulent flows of an incompressible Newtonian fluid. In this case, themean continuity equation and the Reynolds–averaged NS equations read respectively

div�v ¼ 0; ð1Þ

qD�v

Dt¼ divð�Tþ qsÞ þ qB; ð2Þ

where an overbar represents the ensemble average, q is the constant mass density, B is the pre-scribed body force density, D=Dt denotes the material time derivative associated with the meanvelocity field �v, �T ¼ ��p1þ 2l�D, �D ¼ 1

2½grad�vþ ðgrad�vÞT� is the mean stretching tensor, �p is the

mean pressure, l is the viscosity, and s :¼ �v0 v0 is the Reynolds stress tensor.Ever since the pioneering work of Prandtl [14] in which he first developed the mixing-length

closure model for turbulence, developing the closure models for turbulence of practical feasibilityand relative generality in engineering applications and in scientific research has long been being anarduous task, which often encounters difficulties in selecting the appropriate kinematic quantitiesto describe the complex turbulent flows and in identifying model coefficients based on the ex-perimental data obtained from some well-devised experiments. Even so, the past eight decades haswitnessed the important advances as well as some presently insurmountable difficulties involved inturbulence modelling; readers are referred to a paper of Bradshaw [2], which reviews and ad-dresses in depth the difficulties, many aspects of turbulence modelling, and the prospects forfuture research in turbulence.

In a remarkable short note, Rivlin [15] addressed the similarity between the turbulent flows of aNewtonian fluid and the laminar flows of non-Newtonian fluids. Later, based on the reasoning ofrational mechanics, Lumley [11] proposed a general constitutive equation for the Reynolds stress,which depends on the whole history of the mean velocity field. In a follow-up research of the workof Huang and Kasagi [8] and Huang and Rajagopal [7], recently, within the context of continuummechanics and as an illustration of the approach, a non-linear K–e model has been developed inHuang [9] based on a proposed general constitutive equation that depends upon the history of themean deformation of turbulence but with a fading memory, which takes the form:

s ¼ � 2K31þ 2Cl

K2

��D� c1C

2l

K3

�2�D2

�� 1

3trð�D2Þ1

�þ c2C

2l

K3

�2�Dþ c3Cl

K2

�3ðK _�� 2 _K�Þ�D; ð3Þ

where Cl ¼ 0:09, c1 ¼ 2:896, c2 ¼ 2:784, c3 ¼ 0:843, �W ¼ 12½grad v� ðgrad vÞT� is the mean spin

tensor, and denotes the Jaumann derivative, i.e., �D¼ _�D� �W�Dþ �D �W. Hereinafter, for sim-

plicity in writing, an over dot also denotes the material time derivative associated with the meanvelocity field �v, i.e., D=Dt.


In the following study, we shall employ this non-linearK–emodel (3) to do numerical calculationsof homogeneous turbulent shear flow and fully developed turbulent flow over a backward-facingstep, and compare its performance with the following popular linear and non-linear K–e models:

(1) The SKE model (cf. [10]):

s ¼ � 2K31þ 2Cl

K2

e�D; ð4Þ

where Cl ¼ 0:09.(2) The non-linear K–e model of Speziale [17]:

s ¼ � 2K31þ 2Cl

K2

e�D� 4CDC2

l

K3

e2�D2

�� 1

3trð�D2Þ1

�� 4CEC2

l

K3

e2�D}

"� 1

3trð�D

}Þ1#; ð5Þ

}
where Cl ¼ 0:09, CD ¼ CE ¼ 1:68, and �D ¼ _�D� ðgrad�vÞ�D� �Dðgrad�vÞT is the Oldroyd derivativeof �D.(3) The K–e model of YN [19]:
s ¼ � 2K31þ 2mt

1þ CG1Ke

DDt ln mt

�D; ð6Þ

where mt ¼ ClK2=e, Cl ¼ 0:09, and CG1 ¼ 1:3.(4) The quadratic non-linear K–e model of SZL [16]:

s ¼ � 2K31þ 2Cl

K2

e�D� b2

K3

e2½ �W�D� �D �W�; ð7Þ

and

C�l ¼ 1

A0 þ A�sU �ðK=eÞ ; and b2 ¼

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 9C�2

l trð�D2ÞðK=eÞ2q

1þ 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�trð�D2Þtrð �W2ÞðK=eÞ2

q ; ð8Þ

where A0 ¼ 6:5, A�s ¼

ffiffiffi6

pcos/, / ¼ 1

3arccos

ffiffi6

ptrð�D3Þ

½trð�D2Þ�3=2, and the shear parameter is U � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

trð�D2Þ � trð �W2Þq

.

(5) The non-linear cubic model of CLS [3]:

s ¼ � 2K31þ 2eCl

K2

e�D� b1

K3

e2�D2

�� 1

3trð�D2Þ1

�� b2

K3

e2ð �W�D� �D �WÞ

� b3

K3

e2�W2

�� 1

3trð �W2Þ1

�þ c1

K4

e3trð�D2Þ�Dþ c2

K4

e3trð �W2Þ�D

þ c3K4

e3�W2 �D

�þ �D �W2 � trð �W2Þ�D� 2

3trð �W2 �DÞ1

�þ c4

K4

e3ð �W�D2 � �D2 �WÞ; ð9Þ

and

eCl ¼ 0:3ð1� exp½�0:36= expð�0:75gÞ�Þ1þ 0:35g3=2

; and g ¼ maxðeS ; eXÞ ð10Þ


where
eS ¼ ðK=eÞ½2trð�D2Þ�12; and eX ¼ ðK=eÞ½�2trð �W2Þ�
12; ð11Þ

and b1 ¼ �0:4eCl, b2 ¼ 0:4eCl, b3 ¼ �1:04eCl, c1 ¼ c2 ¼ 40:0eC3l, c3 ¼ 0, and c4 ¼ �80:0eC3

l.

3. Numerical results and discussion

The conventional modelled K equation and e equation are employed in the following study.

3.1. Homogeneous turbulent shear flow

Homogeneous turbulent shear flow is a typical turbulent flow that displays significantly the so-called non-equilibrium effect of turbulence. Developing a closure model to capture this interestingphenomenon has drawn for years the attention of the workers in the field of turbulence modelling(cf. [9,19,20]), since a closure model�s capability in predicting well the non-equilibrium effect ofturbulence would play an important role for its performance in predicting the complex turbulentflows.

Here, we shall evaluate the performance of the above listed models in predicting the evolutionof the turbulent kinetic energy in homogeneous turbulent shear flow in comparison with the directnumerical simulation (DNS) data of Matsumoto et al. [12], in which the shear rateS ¼ oU=oy ¼ 30, where y is the coordinate normal to the mean velocity U in the x-direction.

In homogeneous turbulent shear flow, the K equation and e equation simply become

oKot

¼ �KSa12 � e; ð12Þ

oeot

¼ �Ce1eSa12 � Ce2e2

K; ð13Þ

where S is the shear rate, Ce1 ¼ 1:44, and Ce2 ¼ 1:92. Note that the anisotropy tensor aij is definedas

aij ¼�vivj � 2

3Kdij

K: ð14Þ

The expressions of the normalized shear stress a12 for different models are given in Table 1.The fourth-order Runge–Kutta method is used for solving the above turbulent transport

equations. The computation is initiated from St ¼ 2 to avoid the dependence on the initial valuefor the DNS (cf. [12,13,19]). The initial conditions K0 ¼ 10:34745 and e0 ¼ 60:2102 at St ¼ 2 areadopted.

Fig. 1 shows the evolution of the turbulent kinetic energy K with the dimensionless time t� ¼ St.Model predictions are compared with the DNS data. The SKE model overpredicts the timeevolution of turbulence to a great extent––thus, it does a poor job in capturing the non-equi-librium effect. In addition, the model of Speziale in fact gives the same results as those by the SKEmodel. The SZL and the CLS models perform well in capturing the treads given by the DNS data,

Table 1

The expression of a12 for different models in homogeneous shear flow

Model a12

SKE/Speziale a12 ¼ �ClKSe

� , Cl ¼ 0:09

SZL a12 ¼ �C�l

KSe

� CLS a12 ¼ �eCl

KSe

� YN a12 ¼ �Cl

KSe

� =ð1þ CG1ð2 _Ke � K _eÞ=e2Þ, Cl ¼ 0:09

Huang a12 ¼ �ClKSe

� � c3ClK

2e3 ðK _e � 2 _KeÞS, Cl ¼ 0:09

Fig. 1. Time evolution of the turbulent kinetic energy.


especially the CLS model, although they do not include terms with _K and _e which reflect the non-equilibrium effect; actually, the quadratic term and the cubic term make no contributions to a12 inthese two models and the good agreement with the experimental result is due to the coefficient C�

l

and eCl�s dependence on the mean strain and the mean vorticity. It is shown that in order to matchup the DNS data, C�

l=eCl should be reduced to about 0.04. In both the model of YN and the

model of Huang, time and spatial non-equilibrium of turbulence is represented by the termscontaining _K and _e. The model of Huang predicts the results in good agreement with the DNSdata. The model of YN slightly underestimates the time evolution of turbulent kinetic energy inthis test case of strong shear but, nevertheless, it permits a narrow range of flow conditions toobtain good results (cf. [13,19]).

3.2. Turbulent flow past a backward-facing step

Separation and reattachment are such important and common phenomena in turbulent flowsthat their predictions are extremely crucial in many engineering applications, e.g., in the designingof aircraft, turbines and other engineering devises. The fully developed turbulent flow past abackward-facing step, especially with a high ratio of the step height to the tunnel exit height,


provides a benchmark example to test the accuracy of the closure models in predicting the re-attachment location, the mean velocity profiles, and the Reynolds stress distributions along theexit channel. Indeed, it would be hard to believe that a closure model which fails to captureproperly the reattachment point in turbulent flow past a backward-facing step can do a better jobat predicting the unsteady turbulent flow separation which appears ubiquitously in the complexturbulent flows.

In this case, we shall carry out a numerical investigation of the above-mentioned K–e models ontheir performance in predicting the fully developed turbulent flow past a backward-facing stepwith a ratio of 1:9 of the step height to the tunnel exit width, and compare the numerical resultsobtained herein with the experimental data in Driver and Seegmiller [4]. In their experimentsDriver and Seegmiller investigated the two-dimensional, steady, incompressible and turbulentflow past a backward-facing step. A diagram of this flow is shown in Fig. 2. The geometry has astep height to tunnel exit height ratio of 1:9 and the Reynolds number based on the step heightand the experimental reference free-stream velocity is 37, 423, which is high enough to ensure thatthe boundary would be fully turbulent before passing over the step.

All models are computed with the same code based on the finite volume method with non-orthogonal grids (cf. [5]). Variable storage is co-located and cell-centered, with Rhie–Chow in-terpolation for cell-face mass fluxes. The SIMPLE pressure-correction algorithm is used to obtainthe pressure field. The convection and diffusion terms of all the equations, including the mo-mentum equations and the model transport equations for turbulence quantities, are approximatedby the second-order central differencing scheme. The deferred correction technique has been usedfor the discretization of the convection term. The Stone�s SIP (strongly implicit procedure)method is employed with under-relaxation factors. To stabilize the iteration, a time marchingprocess has been adopted. Convergence is judged by monitoring the magnitude of the absoluteresidual sources of mass and momentum, normalized by the respective inlet fluxes. The iteration iscontinued until all above residuals fall below 0.05%.

The inlet plane of solution domain is located at the step (x=H ¼ 0). At the inlet, the meanvelocities U , V and the turbulent normal stresses uu and vv are extrapolated from the measureddata. Turbulent kinetic energy K is calculated with the assumption of

ww ¼ 1

2ðuuþ vvÞ; ð15Þ

Fig. 2. Diagram of backward-facing step flow.


and e is calculated by

Table

Comp

Gri

54·108

216

e ¼C3=4

l K3=2

L; L ¼ minð0:21Dy; 0:085dÞ; ð16Þ

where Dy is the distance from the wall and the boundary-layer thickness d ¼ 1:5H (cf. [16]). Theexit plane of the solution domain is at x=H ¼ 40, where the streamwise gradient is assumed to bezero for all variables.

In order to check the grid-independence in our computation, three different grid levels, i.e.,54· 41, 108 · 82, and 216 · 164, have been tested for the backward-facing step flow with the SKEmodel. The results of the computed reattachment points for different grid levels are given in Table2.

The comparisons of the computed skin friction coefficient, and the mean streamwise velocityprofile for different grid levels are shown in Figs. 3 and 4 respectively.

In view of the comparisons it is clear that the grid of 108· 82 is sufficient for the presentcomputation. So after the grid independence�s having been examined, a grid with 108· 82 cells isemployed, which is refined around the region of the measured reattachment point x=H ¼ 6:26.Near the wall, the first grid point is within the logarithmic region of yþ > 30 for the use ofstandard wall functions to account for the viscous sublayer.

Fig. 5 gives the skin friction coefficient distributions downstream of the step. It is clear that theSKE model overpredicts the negative peak of Cf and underpredicts the recirculation region. Themodel of Speziale overpredicts the results downstream of the reattachment point. The results ofthe CLS model as shown are in good agreement with the experimental data overall. The locationof the reattachment point is characterized by Cf ¼ 0. Table 3 lists the computed reattachment

2

arison of the predicted reattachment points for different mesh numbers

d number Reattachment point (x=H )

41 3.6

· 82 4.9

· 164 4.9

Fig. 3. Skin friction coefficient for different grid levels.

Fig. 4. The mean streamwise velocity profile at x=H ¼ 5:0 for different grid levels.

Fig. 5. Skin friction coefficient distribution.

Table 3

Comparison of the predicted reattachment points

Model Reattachment point (x=H )

SKE 4.9

Speziale 5.2

YN 5.3

Huang 5.7

SZL 5.7

CLS 5.9

Experiment 6.26


points in comparison with the measured ones. In fact, the reattachment point is a critical pa-rameter often taken to assess the overall performance of turbulence models. It is seen that all six

Fig. 6. Streamline patterns.



models tested underestimate the recirculation region; however, the non-linear models show theirbetter overall performance than the linear models� in this regard.

Fig. 6 shows the streamline patterns obtained by different models. It appears that no modelpredicts the corner recirculation region, which is due to the use of wall functions near the wall. Itis clear that the models of Huang, SZL and CLS obtain larger recirculation bubbles and thepredicted reattachment points are close to those given by the experiments (Table 3).

Fig. 7 shows the streamwise mean velocity profiles at x=H ¼ 2, x=H ¼ 5 and x=H ¼ 8 near thereattachment point. The differences among the results of different models are not substantial. Thenumerical results of all these models are in an overall agreement with the experimental data.However, the non-linear models show somewhat slower recovery near the reattachment point,while the SKE model predicts faster recovery and underpredicts the recirculation region.

Fig. 8 is the profile of the normal Reynolds stress uu. At x=H ¼ 2, the model of Spezialeoverpredicts the results to a great extent. The model of Huang agrees with the experiments prettywell, especially near the wall. The model of YN and the SZL model also predict a good trend ofmeasured data in the near wall region. At x=H ¼ 5, the uu profiles obtained by all the models havea vertical displacement, which may be caused by the underprediction of the reattachment point.The results of x=H ¼ 5 and x=H ¼ 8 indicate that the non-linear models give better results thanthe linear models.

Fig. 9 gives the profile of the normal Reynolds stress vv. At x=H ¼ 2, all linear models as shownoverpredict the vv near the wall, while the non-linear models predict the near-wall behaviors of vvbetter, especially the model of Huang and the model of Speziale. However, at x=H ¼ 5 andx=H ¼ 8, the model of Speziale and the SZL model underestimate vv by a large margin.

In addition, it is found that near the wall the linear models of SKE and YN give unrealisticresults vv > uu at all the three locations. By contrast, the non-linear models predict at least thequalitatively correct results as shown, due to the contributions of the non-linear terms whichrepresent the anisotropy of the normal Reynolds stresses.

Fig. 7. Streamwise velocity profiles.

Fig. 9. Normal Reynolds stress vv profiles.

Fig. 8. Normal Reynolds stress uu profiles.


Fig. 10 shows the profile of the shear Reynolds stress �uv. At x=H ¼ 2, the model of Huangagrees pretty well with the experimental data. All non-linear models tested herein, except themodel of Speziale, yield better results than the linear models, especially in the near wall region. Infact, in comparison with the experiments, it is seen that the model of Speziale overpredicts whilethe models of SZL and CLS underestimate the peak value of the shear Reynolds stress. Atx=H ¼ 5, all models give almost the similar results, namely, all of which have a vertical shift. Atx=H ¼ 8, all models give more or less the same trend with the experimental data, except for themodel of YN, which shows a bit off the trend at the place about one step of distance away fromthe wall, like in the case of x=H ¼ 5. It is seen that the model of Huang and the CLS model givethe best overall results.

Fig. 10. Shear Reynolds stress �uv profiles.


4. Conclusions

Six linear and non-linear K–e models have been assessed of their performance in predicting twotypical turbulent flows, namely, homogeneous turbulent shear flow and fully developed turbulentflow over a backward-facing step.

It is found that both the model of YN and the model of Huang predict the homogeneousturbulent shear flow pretty correctly with the contribution of the terms which depict the non-equilibrium effect. Owing to the functional dependence of the coefficient C�

l=eCl on the mean strain

and the mean vorticity, the numerical results given by the SZL model and the CLS model alsoagree quite well with the experimental data. So it is worthwhile to investigate further whichmechanism actually reflects better the non-equilibrium effect.

In the case of fully developed turbulent flow over a backward-facing step, the non-linear modelstested in this work yield considerably better predictions than those obtained by the linear models,which are mainly due to the contributions of those non-linear terms representing the anisotropy ofthe normal Reynolds stresses; moreover, in view of the numerical results, it appears that the modelof Huang [9] combines the advantages in predicting the turbulent non-equilibrium effect (cf. [19])and the anisotropy of the normal Reynolds stresses (cf. [17]).

Acknowledgements

The financial support provided by the National Natural Science Foundation of China(90205002) is gratefully acknowledged.

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