ls

T.

Acceleration

berS).osedsedte igate

of these unsteady turbulent ows. For the cases of low and intermediate acceleration rates, these threemodels yield predictions of wall shear stress that agree well with the corresponding DNS data. For the

terestccurrent ofhe coms of un

frequency, amplitude and mean ow rates in the case of pulsatingow. Efforts on correlating the data on such ows have led to non-dimensional parameters representing the extent to which shearwaves generated attenuate in terms of wall units.

The experimental studies of Maruyama et al. [12], Lefebvre [13],He and Jackson [14], Greenblatt and Moss [15,16] and He et al. [17]are examples of research on non-periodic ows, while the

turbulencedue to the

He et al. [19] identied three stages in the developmentshear stress in ramp-type ow rate excursions through numstudies. The rst stage corresponds to the period of delay in turbu-lence response (frozen turbulence), occurring when inertial forcesare dominant. This stage covers the period when wall shear stressrst overshoots and then undershoots the corresponding quasi-steady values. He et al. [19] showed that a non-dimensionalparameter involving inner turbulence time scales associated withthe turbulence production correlates very well with the unsteadywall shear stress. It was shown by He and Ariyaratne [25] that

Corresponding author. Tel./fax: +44 114 2227756.

Computers & Fluids 89 (2014) 111123

Contents lists availab

rs

lseE-mail address: s.he@shefeld.ac.uk (S. He).Jackson [8] and the numerical studies of Scotti and Piomelli[9,10] and Cotton et al. [11] are all examples of such research.The research includes study of the ow behaviour for a range of

experience a longer delay. Eventually response ofexcursion is propagated towards the pipe centreof turbulent diffusion.0045-7930/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compuid.2013.10.037to theaction

of wallericalare mainly conducted through two main categories; periodic andnon-periodic.

Numerical and experimental techniques have been employed toinvestigate the turbulent ow features associated with periodicchanges of ow rate with time. Brereton and Mankbadi [1] andGndogdu and arpinlioglu [2] present comprehensive reviews.The experimental studies of Gerrard [3], Mizushina et al. [4], She-mer et al. [5], Mao and Hanratty [6], Tardu et al. [7] and He and

identied three delays associated with the response of turbulence.Delays in turbulence production, turbulence energy redistributionand turbulence radial propagation were found to be the key fea-tures of such unsteady turbulent ows. It was found that the rstresponse of turbulence to the imposed ow rate initiates from a re-gion close to the wall where turbulence production is highest (buf-fer layer). The axial component of the Reynolds stress is the rst torespond to the excursion while the other two normal componentsTurbulent owTurbulence modelsChannel owWall shear stress

1. Introduction

Unsteady turbulent ows are of iners because of their wide range of oneering disciplines. A large amouleading to a better understanding of tanisms present in such ows. Studiecase of high acceleration, the cReh model of Langtry and Menter (2009) [38] and the kemodel of Laun-der and Sharma (1974) yield reasonable predictions of wall shear stress.

2013 Elsevier Ltd. All rights reserved.

to turbulence research-nce across many engi-on-going research isplex turbulence mech-steady turbulent ows

numerical investigations of Chung [18], He et al. [19], Ariyaratneet al. [20], Seddighi et al. [21], Di Liberto and Ciofalo [22], Jungand Chung [23] and He and Seddighi [24] examined the effects ofsudden changes in pressure gradient or of linear ramp up/downin ow rates.

He and Jackson [14] focused their research on linearly increas-ing and decreasing ow rate in fully developed pipe ows. TheyKeywords:Unsteady

[28] and the cReh transition model of Langtry and Menter (2009) [38] capture well the key ow featuresA comparative study of turbulence mode

S. Gorji a, M. Seddighi a, C. Ariyaratne b, A.E. Vardy c,aDepartment of Mechanical Engineering, University of Shefeld, Shefeld S1 3JD, UKb Thermo-Fluid Mechanics Research Centre, University of Sussex, Brighton BN1 9QT, UKcDivision of Civil Engineering, University of Dundee, Dundee DD1 4HN, UKd School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UK

a r t i c l e i n f o

Article history:Received 19 January 2013Received in revised form 21 October 2013Accepted 28 October 2013Available online 1 November 2013

a b s t r a c t

The performance of a numnumerical simulations (DNsion of ow rate inside a clnolds number of 9308 (ba29,650. The acceleration raamong the models investi

Compute

journal homepage: www.ein a transient channel ow

ODonoghue d, D. Pokrajac d, S. He a,

of low-Reynolds number turbulence models is evaluated against directAll models are applied to an unsteady ow comprising a ramp-type excur-channel. The ow rate is increased linearly with time from an initial Rey-on hydraulic diameter and bulk velocity) to a nal Reynolds number ofs varied to cover low, intermediate and high accelerations. It is shown thatd, the ke models of Launder and Sharma (1974) and Chang et al. (1995)

le at ScienceDirect

& Fluids

vier .com/ locate /compfluid

stress behaves in a laminar-like manner. The second stage beginswith the generation of new turbulence which causes the wall shear

& Fstress to escalate. It is shown by He et al. [17] that a correlation ex-ists between the outer turbulence time scales and the critical Rey-nolds number at which transition from stage one to two occurs.The third stage includes the period when the wall shear stressasymptotically approaches the corresponding quasi steady trend.He et al. [19] also investigated the effects of uid properties onthe unsteady wall shear stress behaviour. For this purpose, twoow cases with different working uids (water and air) but identi-cal Reynolds range and acceleration rate were examined. It wasshown that the unsteady wall shear stress deviation from the cor-responding quasi-steady values is much smaller for air than forwater because of waters higher density.

The ability of Reynolds Averaged NavierStokes (RANS) modelsto predict the ow behaviour of steady/unsteady channel/pipeows has been investigated by a number of researchers. The stud-ies of Patel et al. [26], Myong and Kasagi [27] and Chang et al. [28]are some good examples of application of RANS models to steadypipe/channel ows. Sarkar and So [29] investigated the perfor-mance of different turbulence models for steady channel ows(along with Couette, boundary layer and back-step ows). Theyexamined ten different low-Reynolds number turbulence models,comparing their results with available DNS and experimental data.They observed that models with asymptotically consistent nearwall behaviour generally return better predictions of ow features.Asymptotic behaviour of the turbulent kinetic energy, its dissipa-tion rate and the Reynolds shear stress near a wall is explainedby Launder [30].

Performance of RANS models in unsteady ows have been stud-ied by Cotton et al. [11], Scotti and Piomelli [10], Tardu and DaCosta [31], Al-Sharif et al. [32], Khaleghi et al. [33] and Revellet al. [34]. The performance of turbulence models in predicting fea-tures of unsteady ows differ according to the turbulence modelformulations. In most cases researchers compare the performanceof different models against the available experimental or DNS data.Cotton et al. [11] examined the performance of the second-moment closure model of Shima [35] and the kemodel of Launderand Sharma [36] for both oscillatory at-plate boundary layer andpulsatile pipe ow. It was found that the second-moment closureschemes generally performed better in comparison with the kemodel examined. Scotti and Piomelli [10] assessed the perfor-mance of ve turbulence models against their own DNS data onpulsating ows (Scotti and Piomelli [9]), while Khaleghi et al.[33] investigated the performance of four turbulence models fora ramp-up pipe ow, comparing their results with the experimen-tal data of He and Jackson [14]. In each of these two studies, theperformance of an algebraic one-equation model, a ke model, akx model and a kev2 model were examined. It was concludedfrom both studies that kev2 model outperforms the rest. How-ever, these conclusions were based on investigations of only a lim-ited number of models among the various formulations.Furthermore, new turbulence models have recently been devel-oped which were not considered by previous researchers.

The present paper reports on a systematic study of the perfor-mance of a wide range of low-Reynolds number turbulence modelsused to predict the detailed ow characteristics of ramp-up-typeunsteady ows in a channel. Recent DNS results are used as bench-mark data for the assessment.

2. Methodologyduring this rst stage the unsteady component of the wall shear

112 S. Gorji et al. / ComputersThe study reported here involves the assessment of ten differ-ent turbulence models applied to three accelerating ow test cases.FLUENT 13.0 is used as the RANS solver for the numericalinvestigations.

The ow domain consists of a rectangular channel section withsmooth wall boundaries and the working uid is water(q = 1000 kg/m3, m = 1 106 m2/s). The channel is 8 m long and0.05 m high, giving a length to height ratio of L/H = 160 as shownin Fig. 1. Because of symmetry, the computational domain covershalf the channel height. In this study, only spatial fully developedow is of interest; hence, the results presented are taken at7.5 m from the inlet (L/H = 150, AB line in Fig. 1). Systematic meshsensitivity tests were carried out for each group of turbulencemodels to obtain mesh-independent solutions. These tests wereconducted by distributing 70, 100 and 180 control volumes inthe wall normal direction (y direction, shown in Fig. 1). It was con-cluded that distributing 100 control volumes non-uniformly alongthe wall normal direction is adequate to achieve mesh indepen-dent solutions. The number of control volumes used in the axialdirection (x direction, shown in Fig. 1) is 30 but this is of no signif-icance since only axially developed ow is of interest. This alsomeans that the level of turbulence intensity at the inlet is of no rel-evance as long as it is set to a sufciently high level to initiate tur-bulence in the pipe. In this work, it is set to be 5% in all simulations.The non-dimensional distance of the rst node from the wall ismaintained within the range of y+ = 0.30.9 (y+ = yus/m, us repre-senting friction velocity) during the excursion to ensure the low-Reynolds criterion for the models is satised.

In all test cases the ow rate is increased linearly from an initialsteady state Reynolds number (Re0 = Ub0Dh/m, Ub0 representing thebulk velocity) of 9308 to a nal Reynolds number (Re1) of 29,650.The length scale of the Reynolds number is based on the hydraulicdiameter, i.e. Dh = 2H, where H is the full height of the channel.We consider three acceleration time periods (T): Case A, 8.16 s(low acceleration); Case B, 2.86 s (intermediate acceleration);Case C, 0.02 s (high acceleration). Table 1 summarises theinitial and nal ow conditions of examined ow cases alongwith non-dimensional time scale Dt = T/(H/2)/Ub1 and ramp rate(dU/dt = (Ub1Ub0)/T). Although these simulations are carried outfor water, as long as the boundary conditions such as the initialand nal Reynolds numbers and non-dimensional acceleration rateare consistent, the choice of uid is of no signicance to theoutcome of the simulations.

The continuity and momentum transport equations along withthe Reynolds stress closure equations are solved for the computa-tional domain. The ow is assumed to be two-dimensional andCartesian coordinates are employed for the governing equations.

Continuity:

@Ui@xi

0 1

Momentum:

DUiDt

1q

@P@xi

@@xi

m@Ui@xj

uiuj

2

where linear eddy viscosity models employ a stressstrain relationas follows:

uiuj 2=3kdij mt @Ui@xj

@Uj@xi

3

where mt, the eddy viscosity, is obtained by solving a set of turbu-lence transport equations, the details of which are presented inthe next sections.

luids 89 (2014) 111123Only low-Reynolds number turbulence models can potentiallypredict the features of unsteady ows. Here we consider ten low-Reynolds turbulence models, which can be categorised into four

The performance of a correlation based transition turbulence

he c

& Fgroups: ke models, ke based models, the v2f model of Durbin[37] and the cReh transition model of Langtry and Menter [38].

2.1. ke models

Low-Reynolds number ke turbulence models are based onsolving transport equations for turbulent kinetic energy and its dis-sipation rate, as follow:

DkDt

@@xj

m mtrk

@k@xj

Pk ~e D 4

D~eDt

@@xj

m mtre

@~e@xj

Ce1f1 1Tt Pk Ce2f2

~eTt E 5

where Pk is the production of turbulent kinetic energy given by

Pk uiuj @Ui@xj

6

Eddy viscosity in kemodels is dened to be mt = Clflk2/e, Cl being aconstant and fl being the damping function. Tt in Eq. (5) is a turbu-

Fig. 1. Sketch of t

Table 1Test cases and ow conditions.

Flow case Re0 Re1 Dt T (s) dU/dt (m/s2)

A 9308 29,650 96.8 8.16 0.025B 9308 29,650 33.9 2.86 0.071C 9308 29,650 0.2 0.02 10.17

Note: Dt TH=2=Ub1; dUdt Ub1Ub0

T .

S. Gorji et al. / Computerslent time scale, ~e is the modied isotropic dissipation rate, D and Eare near wall correction functions for k and e equations,respectively.

The six ke turbulence models examined in this study are des-ignated as AB for Abid [39], LB for Lam and Bremhorst [40], LS forLaunder and Sharma [36], YS for Yang and Shih [41], AKN for Abeet al. [42] and CHC for Chang et al. [28]. Note that the performanceof FLUENTs built-in LS model is found to be unexpectedly poor andtherefore a User Dened Function (UDF) for LS developed byMathur and He [43] was also implemented. The FLUENT built-inimplementation of LS is designated LS-FLUENT and the UDF imple-mentations is designated LS-UDF in what follows.

A summary of the model constants, damping functions and nearwall correction functions are presented in Tables 24.

2.2. kx and shear stress transport (SST) kx models

FLUENT 13.0 employs the low-Reynolds number kx model ofWilcox [44], which solves two transport equations, one for turbu-lent kinetic energy (same as for the kemodels) and one for its spe-cic dissipation rate (x / e/k). Further details of this model can befound in Wilcox [44].

The kx Shear Stress Transport (SST) model developed by Men-ter [45] employs a blending function, which retains the near-wallx equation while switching to e equivalent further from the wall.Further details of the model can be found in Menter [45].

2.3. v2f model

Anisotropy of the turbulence stresses is not addressed in lineareddy viscosity turbulence models. Durbin [46] replaced the ad hocdamping functions of the ke models by introducing the wall-nor-mal stress vv as the velocity scale in the eddy viscosity formula-tion. An elliptic relaxation function is also solved to model theredistribution process of wall normal stress transport equation.However, due to difculties of implementation of the original for-mulation, major commercial codes tend to use more numericallystable formulations. The v2f version coded in FLUENT, which isexamined in this study, is the due to Iaccarino [47] but with defaultconstants of those proposed by Lien and Kalitzin [48].

2.4. cReh transition model of Langtry and Menter [38]

hannel geometry.

Table 2Constants for the turbulence models.

Model Cl Ce1 Ce2 rk re

AB 0.09 1.45 1.83 1.0 1.4LB 0.09 1.44 1.92 1.0 1.3LS 0.09 1.44 1.92 1.0 1.3YS 0.09 1.44 1.92 1.0 1.3AKN 0.09 1.5 1.90 1.4 1.4CHC 0.09 1.44 1.92 1.0 1.3

luids 89 (2014) 111123 113model of cReh Langtry and Menter [38] available in FLUENT 13.0is also considered. The model incorporates two extra transportequations into the SST model, one for the intermittency c andthe other for the transition onset momentum-thickness Reynoldsnumber fReht . Turbulent kinetic energy and its specic dissipationrate transport equations of the SST model are customised to in-clude the additional transport equations.

The following are the transport equations for intermittency andmomentum-thickness Reynolds number:

@qc@t

@qUjc@xj

Pc Ec @@xj

l ltrf

@c@xj

7

@qfReht@t

@qUjfReht

@xj Pht @

@xjrht l lt

@fReht@xj

" #8

where Pc and Ec are production and dissipation terms of the inter-mittency transport equation, respectively. Pht is the production termof momentum-thickness in the Reynolds number transport equa-tion. rf and rht are the constants of intermittency and momen-tum-thickness Reynolds number transport equations, respectively.

Intermittency is a measure of the regime of the ow. For in-stance, in a growing boundary layer over a at plate, intermittencyis zero before the transition onset and reaches a value of one when

& Fthe ow is fully turbulent. In order to determine the condition of a

Table 4D and E terms along with the boundary conditions.

Model D E Wall BC

AB 0 0 ew m @2k@y2

LB 0 0 ew m @2k@y2

LS2m @

k

p@y

22mmt @

2U@y2

2 ~ew 0YS 0 mmt @

2U@y2

2 ew m @2k@y2 AKN 0 0 ew m @

k

p@y

2CHC 0 0 ew m @2k@y2

Note: Ret k2me Rey yk

1=2

m y yuem ue me 0:25.Table 3Functions in the turbulence models.

Model fl

ABtanh0:008Rek 1 4Re3=4t

LB 1 exp0:0165Rey

2 1 20:5Ret LS exp 3:4

1Ret=50 2

YS

1 1= Retp 1 exp 1:5 104Rey

5:0 107Re3y1:0 1010Re5y

0B@1CA

2643750:5

AKN 1 5Re0:75t

exp Ret200 2 1 exp y14 2

CHC 1 exp0:0215Rey 2 1 31:66

Re5=4t

114 S. Gorji et al. / Computersdeveloping boundary layer, correlations exist between the locationof the transition onset and the free stream turbulence intensity,pressure gradient and transition momentum-thickness Reynoldsnumber (Abu-Ghannam and Shaw [49] and Mayle [50]). Both alge-braic and transport equations have been developed by researchersto determine the intermittency factor. Langtry and Menter [38]couples the transport equations for intermittency and transitiononset momentum-thickness Reynolds number to Menter [45] SSTmodel. The production term in the turbulent kinetic energy trans-port equation is modied to account for the changes in the inter-mittency of the ow.

The transition onset momentum-thickness Reynolds number ismainly responsible for capturing the nonlocal effects of turbulenceintensity and pressure gradient outside the boundary layer. Furtherdetails regarding the formulation of the model and its performancein different test cases can be found in Langtry and Menter [38].

3. Comparisons for steady ow

DNS results for two steady ow scenarios, corresponding to theinitial and nal Reynolds numbers (Re0 = 9308 and Re1 = 29650) ofthe unsteady ow cases to be discussed in the next section, areused to assess the performance of the ten different low-Reynoldsnumber turbulence models applied to two-dimensional, fullydeveloped, steady channel ow.

Fig. 2 shows the predications of mean axial velocity, turbulentkinetic energy, turbulent shear stress and turbulent viscosity withthe DNS results for the initial and nal Reynolds number ows.Bulk velocity (Ub) which represents the ratio of ow rate to crosssectional area and kinematic viscosity (m) are used to normalisethe mean and turbulence quantities.

All turbulence models with the exception of LS-FLUENT give anacceptable prediction of the axial velocity prole across thechannel. However, the performance of the various models is ratherdifferent for the predictions of turbulent kinetic energy, turbulentshear stress and turbulent viscosity.

Most models are successful in predicting the location of thepeak of turbulent kinetic energy for both steady ow cases. Thepredictions of the turbulent kinetic energy of LB, v2f and kxare the closest to DNS in the wall region, whereas the kinetic en-ergy predictions of AB, AKN, CHC, LS-UDF and cReh match betterthe DNS data in the core region. The last three models signicantlyunder-predict the peak of turbulent kinetic energy for both owseven though they show superior performance in predicting the un-steady ows (as discussed in Section 4). It is of interest to note thatcReh transition SST and the basic kx SST give rather differentpredictions of the turbulent kinetic energy even for these steadyfully developed turbulent ows. This is due to the fact that theintermittency becomes active in the inner and buffer layers ofthe wall shear ow where the local Reynolds number is low.

The turbulent shear stress is well predicted by AKN, LB, YS andcReh. This contrasts with AB, CHC, LS-UDF and v2f, which slightlyunder-predict, and LS-FLUENT, which signicantly over-predictsthe shear stress. The turbulent viscosity predictions of most ofthe ke models are quite good in the wall region, with AKN, LB,YS and kx being the closest to DNS. In the core region, however,all models except AKN and kx predict a monotonic increase ofeddy viscosity that is contrary to the DNS data. The over-predictionof eddy viscosity in this region is a known shortfall of many kemodels that results from the under-prediction of dissipation (Bil-lard and Laurence [51]). Myong and Kasagi [27] argue that in mostke models the chosen value of rk is too low in comparison to reand they therefore suggest using higher r /r ratios. In the absence

f1 f2

1.0 1 2=9exp Re2t36

1 exp Rek12

1.0 1 expRe2t 1.0 1 0:3 expRe2t

Retp

1Ret

pRet

p1

Ret

p

1.01 0:3exp Ret6:5

2 n o 1 exp y3:1 21.0 1 0:01 expRe2t [1 exp (0.0631Rey)]

luids 89 (2014) 111123k eof turbulent energy production in the core region, this ratio bal-ances the diffusion and dissipation terms in the turbulent kineticenergy and its dissipation rate transport equations. Among theke models investigated, AKN utilises the highest rk/re ratioimproving its eddy viscosity predictions in the core region.

It should be noted that in the framework of eddy viscosity mod-els the only connection between the turbulence model and themean ow eld is through the turbulent shear stress uv , which ismostly dependent on the eddy viscosity. It can be seen that uv iswell predicted by all models (excluding LS-FLUENT) for both thehigh and the low Reynolds number ows. This is true despite theturbulent viscosity being not well predicted in the core region bysome models. In fact, the most important part of the ow is thewall region, where the turbulent viscosity is predicted fairly wellby most of the models. In the core region, the role of turbulent vis-cosity is not of major importance to the performance of the models.

Results obtained for the higher Reynolds number ow are gen-erally more reliable (i.e. agree better with the DNS results) thanthose for the lower Reynolds number ow. We note that

& F

U/

Ub

LS-FLUENT

LS-UDF

v2-f

k-

k- SST

-Re

k/U

b2LS-FLUENT

LS-UDF

v2-f

k-

k- SST

-Re

S. Gorji et al. / Computersturbulence models are often tuned for relatively high Reynoldsnumber ows, and applying such models to relatively low Rey-nolds number ows can result in poor performance.

Fig. 3 presents the wall shear stress predicted by the variousturbulence models, together with the DNS results. Among the tur-bulence models considered, AKN, LB and kx are seen to yield themost accurate predictions of wall shear stress for both the lowerand higher Reynolds number ows.

In contrast to the LS-FLUENT, LS-UDF performs well overall,leading to our conclusion that the poor performance of the FLUENTbuilt-in LS model is due to its implementation within FLUENT, notdue to any inherent fault with the model itself. The results ob-tained from LS-FLUENT are not further presented or discussed inthe remainder of this paper.

0 200 40000.75

y+

AB

AKN

LB

YS

CHC

0 200 40000.02

y+

AB

AKN

LB

YS

CHC

0 200 40000.005

y+

uv/U

b2

AB

AKN

LB

YS

CHC

LS-FLUENT

LS-UDF

v2-f

k-

k- SST

-Re

0 200 400040

y+

t/

AB

AKN

LB

YS

CHC

LS-FLUENT

LS-UDF

v2-f

k-

k- SST

-Re

Fig. 2. Steady ows at Re = 9308 (short curves) and 29,650 (longer curves):Comparisons between predictions of various turbulence models (dashed line) withDNS data (solid line).4. Comparisons for unsteady ow

4.1. Key features of unsteady ow from DNS

The main features of the unsteady ows as predicted by DNS arerst summarised in this section in order to facilitate the assess-ment of the performance of the turbulence models in the next sec-tion. The discussion largely follows that of He and Jackson [14], Heet al. [17] and He and Seddighi [24], where more detailed discus-sion can be found.

Fig. 4 shows the DNS-predicted time histories of wall shearstress, turbulent viscosity, turbulent shear stress and turbulent ki-netic energy for the three unsteady ow cases. Considering rst thewall shear stress evolution for Case A (Fig. 4(a)), we can identify athree-stage development in the wall shear stress and turbulence.Stage 1 is initially dominated by large inertial effects, causing thewall shear stress to overshoot the corresponding quasi-steady val-ues. However, due to the delayed turbulence response, the growthrate of wall shear stress decreases during the nal moments ofstage 1. Stage 2 corresponds to the time period when the genera-tion of new turbulence causes the unsteady wall shear stressesto increase rapidly towards the corresponding quasi-steady values.During stage 3, the bulk ow is no longer accelerating and the wallshear stress gets gradually closer to the quasi-steady ow shearstress.

The ow acceleration is higher in Case B (Fig. 4(b)), and Re1 isreached while ow response is still in stage 1 (a). As a result ofthe sudden removal of the acceleration, a strong but negative iner-tial effect is imposed on the ow, which results in a sharp decreasein the wall shear stress (stage 1 (b)). Afterwards, the trend is re-versed when turbulence production starts to increase the wallshear stress, which eventually reaches the quasi-steady values.Increasing the acceleration even further (Case C, Fig. 4(c)) causesovershooting of the unsteady wall shear stress over the quasi-stea-dy wall shear stress to occur in an instant (stage 1 (a), not shownon the gure), because of the very sudden change in ow rate. Thisis then followed by a sharp reduction (stage 1 (b)). During stage 2,the wall shear stress rapidly increases again as a result of turbu-lence production. The wall shear stress approaches the correspond-ing quasi steady values in stage 3.

The DNS-predicted time-histories of turbulent viscosity at se-lected y0 locations (where y

0 yus0=m, us0 representing friction

velocity at Re0) are also shown in Fig. 4. It can be seen from theFig. 4 that turbulent viscosity close to the wall y0 5 remainsmore or less unchanged during stage 1, but begins to increase rap-idly at approximately 5, 4 and 2 s for cases A, B and C respectively,corresponding to the onset of stage 2. It can also be seen that thedelays of turbulent viscosity are roughly constant across the chan-nel for all three ow cases. However, response of turbulent viscos-ity to the imposed excursion in the wall region is of greaterimportance for modelling purposes as discussed in Section 4.2.

The response of the turbulent shear stress is consistent withthat of the turbulent viscosity. Turbulent shear stress very closeto the wall at y0 5 stays mainly constant during stage 1. Its re-sponse to the acceleration is initially observed in the wall region,while the delay period becomes progressively longer with distancefrom the wall.

The overall picture of the development of turbulent kinetic en-ergy with time is similar to that of turbulent shear stress. Delaysassociated with the response of turbulence increase with distancefrom the wall. However, the delay in turbulent kinetic energy inthe wall region is much shorter than that for the shear stress. Also,turbulent kinetic energy increases slowly during stage 1, while tur-

luids 89 (2014) 111123 115bulent shear stress and viscosity stay reasonably constant. Duringthis period, the streamwise turbulent normal stress uu increases

due to the stretching of the existing eddies, while the turbulentshear stress uv, the wall-normal stress vv and the spanwisestress ww are unaffected (He and Jackson [14]). This strong aniso-tropic behaviour in the near-wall turbulence is the key feature ofthe unsteady ow and is likely to pose a challenge for linear eddyviscosity models.

4.2. Performance of the turbulence models

Fig. 5 shows the RANS models predictions of wall shear stressbased on the various turbulence models for all three unsteady owcases; the benchmark DNS results are also shown. It is seen thatAB, CHC, LS, v2f and cReh (referred to as Group I hereafter) cap-ture the basic features of the ow exhibited by the DNS resultsrather well, whereas the other models (referred to as Group II)are not able to do so. All Group I models reproduce the three stage

Fig. 3. Predictions of steady ow wall shear stress for two Reynolds numbers by thevarious turbulence models (symbols) and by DNS (lines).

(a) Case A

0 5 10 150

1

2

3

4x 10

-5

t (sec)

t (m

2 /s)

y+

0=5

y+

0=15

y+

0=37

y+

0=83

0 5 10 150

1

2

3x 10

-4

t (sec)

uv (m

2 /s2 )

y+

0=5

y+

0=15

y+

0=37

y+

0=83

y+

0=141

0 5 10 150

0.5

1

1.5x 10

-3

t (sec)

k (m

2 /s2 )

0 5 10 150

0.1

0.2

0.3

t (sec)

w (N

/m2 )

Stage 1 Stage 2 Stage 3

Unsteady

Quasi-steady

(b) Case B

0 2 4 6 8 100

0.1

0.2

0.3

Stage 1-a Stage1-b

Stage 2 Stage 3

t (sec)

w (N

/m2 )

3

4x 10

-5

2 /s)

0 2 4 6 8 100

1

2

3x 10

-4

t (sec)

uv (m

2 /s2 )

Fig. 4. Time-histories of wall shear stress and turbulence quan

116 S. Gorji et al. / Computers & Fluids 89 (2014) 1111230 2 4 6 8 100

1

2

t (sec)

t (m

0 2 4 6 8 100

0.5

1

1.5x 10

-3

k (m

2 /s2 )t (sec)

tities in the three unsteady ow cases predicted by DNS.

& F (c) Case C

0 2 4 6 8 100

0.1

0.2

0.3

Stage1-b

Stage 2 Stage 3

t (sec)

w (N

/m2 )

3 x 10-4

S. Gorji et al. / Computersdevelopment: a delay stage followed by a rapid response in stage 2and then a slow adjustment phase (stage 3). Focusing on more de-tail, cReh appears to predict time scales that are very close tothose of DNS for all three ow cases. The LS and the CHC modelspredict time scales close to those of DNS, but are slightly shorteras the acceleration is increased. Note that CHC shows instabilityin the simulation of Case C. Although able to predict the generalfeatures of the unsteady turbulence response, the AB and v2f re-sults always show time scales that are much shorter than thoseof DNS. The Group II models fail to predict the main features ofthe unsteady ows, mostly because of their inability to predictthe delayed response of turbulence which controls the responseof the ow. All of the models (both groups) are able to capturethe initial overshoot of the shear stress in early stage 1 since thisbehaviour is due to the effect of the inertial forces, which is notstrongly dependent on turbulence (and hence not dependent on

0 2 4 6 8 100

1

2

t (sec)

uv (m

2 /s2 )

Fig. 4 (cont

0 5 10 150

0

0

0

0

0

0

0

0

0

0.3

t (sec)

w (N

/m2 )

00

0

0

0

0

0

0

0

0

0

0.3

w (N

/m2 )

AB

AKN

LB

YS

CHC

LS

v2-f

k-

k- SST

-Re(a)

t (

(b)

Fig. 5. Wall shear stress time-histories in unsteady ow cases; (a) Case A, (b) Case B and (0 2 4 6 8 100

1

2

3

4 x 10-5

t (sec)

t (m

2 /s)

1.5 x 10-3

luids 89 (2014) 111123 117the choice of turbulence model). We note once more that the sim-ulations of the LS model are based on the UDF version; predictionsbased on FLUENTs built-in LS model (not presented) are very dif-ferent from the results described above, with very small delaysthat are much like the predictions of Group II models.

The model predictions of turbulent viscosity at y0 5 due tothe various models are shown in Fig. 6. The turbulent viscositytime histories reect the trends in wall shear stress predicted byeach model. It is apparent that the characteristic delay in turbulentviscosity in the wall region is well reproduced by LS, CHC andcReh, and to some extent by AB and v2f. Once more, cReh slightlyover predicts the delays in all ow cases. CHC predicts the delayfor ow Cases A and B rather accurately, whilst predicting unreal-istic oscillation for ow Case C (not visible in the current scale ofthe gure). Even though v2f does not predict the delay period cor-rectly for any of the cases, it is the only model to return accurate

0 2 4 6 8 100

0.5

1

t (sec)

k (m

2 /s2 )

inued)

5 10sec)

AB

AKN

LB

YS

CHC

LS

v2-f

k-

k- SST

-Re

5 100

0

0

0

0

0

0

0

0

0

0.3

t (sec)

w (N

/m2 )

AB

AKN

LB

YS

CHC

LS

v2-f

k-

k- SST

-Re(c)

c) Case C. DNS (solid lines) and RANS with various turbulence models (dashed lines).

& F0

0

0

0

0

0.4

t (

m2 /s

)

CHC

LS

v2-f

k-

k- SST

-Re(a)x 10-5

0

0

0

0

0

0.4

t (

m2 /s

)

(b)x 10-5

118 S. Gorji et al. / Computersvalues of turbulent viscosity during the pre- and post-ramp periodsfor all three ows. The delays predicted by AKN, LB and YS aremuch shorter than those of DNS, whereas kx and kx SST returneven shorter delays. These observations are consistent with thepredictions of wall shear stress from the respective models, asshown in Fig. 5.

Figs. 79 show the comparisons for turbulent shear stress. Forbrevity results obtained from selected turbulence models onlyare presented. Focusing on the wall region rst y0 15, it isapparent that once more LS, CHC and cReh (some not shown) pre-dict the initial and the subsequent response rather well, whereasmodels AB, AKN, YS, LB and v2f (some not shown) predict muchshorter delays; hardly any delays are observed from the predic-tions of the kx and kx SST models (not shown). Consideringthe turbulent shear stress development in the core region, the

0 5 10 150

0

0

0

0

t (sec)

AB

AKN

LB

YS

00

0

0

0

0

t (

Fig. 6. Time-histories of turbulent viscosity at y0 5 in unsteady ows; (a) Case A, (b) C

0 5 10 150

1

2

3 x 10-4

t (sec)

(a) LB model

Thin Lines: RANSThick Lines: DNS

0 5 10 150

1

2

3 x 10-4

t (sec)

uv (m

2 /s2 )

uv (m

2 /s2 )

(c) CHC model

Fig. 7. Time-histories of turbulent shear stress at selected y0 for unstCHC

LS

v2-f

k-

k- SST

-Re

0

0

0

0

0

0.3

t (

m2 /s

)

CHC

LS

v2-f

k-

k- SST

-Re(c)

x 10-5

luids 89 (2014) 111123DNS data show progressively longer delays as one moves awayfrom the wall. All models are capable of capturing this feature, withsome models performing slightly better than others. The delay inthe core region results from the fact that the turbulence responseoccurs initially in the wall region, propagating towards the centrethrough diffusion. The above comparison shows that all models arecapable of reproducing this feature because the diffusion term isexplicitly included in the transport equations of the turbulent ki-netic energy and its dissipation rate. Once more CHC and cRehare outperforming the rest in reproducing the trends associatedwith the development of turbulent shear stress across the channel;CHCs prediction for Case A is nearly indistinguishable from theDNS data although it is less so for Case B. In Case C, CHCs instabil-ity in predicting turbulence quantities in the wall region is oncemore evident in the trends of the turbulent shear stress. The LS

5 10

sec)

AB

AKN

LB

YS

0 5 100

0

0

0

0

t (sec)

AB

AKN

LB

YS

ase B and (c) Case C. DNS (solid lines) and RANS with various models (dashed lines).

0 5 10 150

1

2

3 x 10-4

t (sec)

(b) v2-f model

uv (m

2 /s2 )

uv (m

2 /s2 )

0 5 10 150

1

2

3 x 10-4

t (sec)

(d) -Re modely+0=15

y+0=37

y+0=83

y+0=141

eady ow case A: RANS model (thin lines) and DNS (thick lines).

& F0 2 4 6 8 100

1

2

3 x 10-4

(a) LB model

Thin Lines: RANSThick Lines: DNS

uv (m

2 /s2 )

S. Gorji et al. / Computersmodel predicts the delay period fairly well, especially for cases Aand B, but it returns magnitudes that are lower than those ofDNS in the nal plateau (not shown). The trends obtained by theAB and v2f are close to those of DNS only in the core region, failingto predict the delays close to the wall.

Figs. 1012 show the predictions of turbulent kinetic energy forseveral wall-normal locations across the channel. Unlike turbulentshear stress, the turbulent kinetic energy shows a rather small de-lay in the wall region for the reasons explained in Section 4.1. It isinteresting that all models except CHC, LS and cReh can predictthis near wall response fairly well. v2f prediction of turbulentkinetic energy in the wall region is almost indistinguishable from

t (sec)

0 2 4 6 8 100

1

2

3 x 10-4

t (sec)

uv (m

2 /s2 )

(c) CHC model

Fig. 8. Time-histories of turbulent shear stress at selected y0 for unst

0 2 4 6 8 100

1

2

3 x 10-4

t (sec)

uv (m

2 /s2 )

(a) LB model

Thin Lines: RANSThick Lines: DNS

0 2 4 6 8 100

1

2

3 x 10-4

t (sec)

uv (m

2 /s2 )

(c) CHC model

Fig. 9. Time-histories of turbulent shear stress at selected y0 for unst0 2 4 6 8 100

1

2

3 x 10-4

(b) v2-f model

uv (m

2 /s2 )

luids 89 (2014) 111123 119the DNS data during the early stages of the three ows. cReh onthe other hand predicts a much longer delay, which is similar to thatof the turbulent shear stress (it is noted that the important require-ment for any eddy viscosity model is to faithfully represent the tur-bulent viscosity itself or the turbulent shear stress). In thisparticular case, since turbulent kinetic energy and shear stressshow different characteristics, it is actually desirable that the earlyresponse of the kinetic energy is not reproduced so that the turbu-lent shear stress uv can be well represented. This is what CHC, LSand cReh have done in order to capture the delay of the wall shearstress correctly. Clearly it is desirable for a model to decouple theprediction of turbulent shear stress and the prediction of turbulent

t (sec)

uv (m

2 /s2 )

0 2 4 6 8 100

1

2

3 x 10-4

t (sec)

(d) -Re model y+0=15

y+0=37

y+0=83

y+0=141

eady ow case B: RANS model (thin lines) and DNS (thick lines).

0 2 4 6 8 100

1

2

3 x 10-4

t (sec)

uv (m

2 /s2

)

(b) v2-f model

0 2 4 6 8 100

1

2

3 x 10-4

t (sec)

uv (m

2 /s2 )

(d) -Re modely+0=15

y+0=37

y+0=83

y+0=141

eady ow case C: RANS model (thin lines) and DNS (thick lines).

& F0

0.2

0.4

0.6

0.8

1

x 10-3

k (m

2 /s2 )

(a) k- modelThin Lines: RANSThick Lines: DNS

120 S. Gorji et al. / Computerskinetic energy so that both can be predicted faithfully. In principle,this is readily achievable with second momentum closure models.Regarding the performance of the models in the core region, allmodels reproduce the kinetic energy delay fairly well.

One of the main features of the turbulent viscosity formulationin the kemodels considered in this study is the damping function(fl), the major role of which is to reduce turbulent viscosity in thewall region. Fig. 13 shows the across-channel prole of dampingfunction fl at selected times, as predicted by four ke models(AB, YS, LB and CHC) for ow Case B. It is seen that the near-wallvalues of fl predicted by CHC is the only one that is sensitive tothe imposed excursion. fl in CHC begins to respond to the imposed

0 5 10 15t (sec)

0 5 10 150

0.2

0.4

0.6

0.8

1

x 10-3

t (sec)

k (m

2 /s2 )

(c) CHC model

Fig. 10. Time-histories of turbulent kinetic energy at selected y0 for un

0 2 4 6 8 100

0.5

1

1.5 x 10-3

t (sec)

k (m

2 /s2 )

(a) k- model

Thin Lines: RANSThick Lines: DNS

0 2 4 6 8 100

0.5

1

1.5 x 10-3

t (sec)

k (m

2 /s2 )

(c) CHC model

Fig. 11. Time-histories of turbulent kinetic energy at selected y0 for un0

0.2

0.4

0.6

0.8

1

x 10-3

k (m

2 /s2 )

(b) v2-f model

luids 89 (2014) 111123ow rate immediately after the rst stage of wall shear stress evo-lution (inertial-dominated period). The increased fl at the earlystages of the excursion keeps turbulent kinetic energy and there-fore turbulent shear stress low, reproducing the delay effectneeded.

A damping function does not exist in the v2f or in the cRehmodel. This correction function is replaced by the wall-normalstress component, as discussed in Section 2.3 for the v2f model.However, in the cReh model the production of turbulent kineticenergy is controlled via intermittency factor derived from its trans-port equation. Fig. 14 shows the temporal evolution of the inter-mittency at selected y0 for the three ow cases. It can be seen

0 5 10 15t (sec)

0 5 10 150

0.2

0.4

0.6

0.8

1

x 10-3

t (sec)k

(m2 /s

2 )

(d) -Re modely+0=15

y+0=37

y+0=83

y+0=141

steady ow case A: RANS model (thin lines) and DNS (thick lines).

0 2 4 6 8 100

0.5

1

1.5 x 10-3

t (sec)

k (m

2 /s2

)

(b) v2-f model

0 2 4 6 8 100

0.5

1

1.5 x 10-3

t (sec)

k (m

2 /s2

)

(d) -Re modely+0=15

y+0=37

y+0=83

y+0=141

steady ow case B: RANS model (thin lines) and DNS (thick lines).

& F0 2 4 6 8 100

0.5

1

1.5 x 10-3

k (m

2 /s2 )

(a) k- model

Thin Lines: RANSThick Lines: DNS

S. Gorji et al. / Computersthat the intermittency is reduced signicantly at the early stagesfollowed by a period of delay before increasing again. The turbu-lent viscosity trend shows similar behaviour. Such a reduction inthe intermittency leads to further reduction in turbulent kineticenergy and shear stress in the wall region.

5. Conclusions

The performance of ten eddy viscosity turbulence models inpredicting unsteady, ramp-up-type turbulent channel ows hasbeen examined by comparing predictions with DNS results forthe same ows. Three ramp-up ows with different accelerationrates have been considered. The key features of the unsteady ow

t (sec)

0 2 4 6 8 100

0.5

1

1.5 x 10-3

t (sec)

k (m

2 /s2 )

(c) CHC model

Fig. 12. Time-histories of turbulent kinetic energy at selected y0 for un

100 1020

0.2

0.4

0.6

0.8

1

y+0

f

(a) AB model0 sec1 sec2 sec3 sec4 sec

100 1020

0.2

0.4

0.6

0.8

1

y+0

f

(c) LB model

Fig. 13. Damping function proles at selected times, as predicte0 2 4 6 8 100

0.5

1

1.5 x 10-3

k (m

2 /s2 )

(b) v2-f model

luids 89 (2014) 111123 121as seen in the DNS data are the distinct delays in the response ofthe turbulent shear stress and turbulent viscosity to the imposedchange in the ow rate. These delays are in turn responsible forthe response of the wall shear stress. It is shown that the wall shearstress goes through a three-stage development. The rst stage isinuenced by the frozen turbulence and then inertia forces. Inthe early part of the rst stage inertial forces dominate, causingthe wall shear stress to overshoot the corresponding quasi-steadyvalues. Then the effect of frozen (or delayed) turbulence takes over,causing the wall shear stress to undershoot the quasi-steady val-ues. The second stage corresponds to rapid response of turbulence,causing rapid increase in wall shear stress. In the nal stage thewall shear stress approaches the quasi-steady value.

t (sec)

0 2 4 6 8 100

0.5

1

1.5 x 10-3

t (sec)k

(m2 /s

2 )

(d) -Re modely+0=15

y+0=37

y+0=83

y+0=141

steady ow case C: RANS model (thin lines) and DNS (thick lines).

100 1020

0.2

0.4

0.6

0.8

1

y+0

f

(b) YS model

100 1020

0.2

0.4

0.6

0.8

1

y+0

f

(d) CHC model

d by AB, YS, LB and CHC turbulence models for ow case B.

4t (

5

as p

122 S. Gorji et al. / Computers & Fluids 89 (2014) 111123Diffusion of turbulence from the wall to the core region is an-other feature of unsteady ows, leading to relatively long delaysin the response of turbulent kinetic energy and turbulent shearstress in the core region. However, the duration of the delay periodof turbulence response in the wall region is different for kinetic en-ergy and for turbulent shear stress because of the stretching of tur-bulence structures.

The following are the main conclusions regarding the perfor-mance of the various turbulence models examined in this study:

0 5 10 150

0.2

0.4

0.6

0.8

1

t (sec)

(a) Case A

0 20

0.2

0.4

0.6

0.8

1

y+0 = 3 y+0 = 6 y

+0 = 12.

Fig. 14. Time-histories of intermittency at selected y0 , The most important feature that needs to be modelled in orderto capture the overall behaviour of the ow is the delayedresponse of turbulent shear stress (and turbulent viscosity).Among the models examined, only the LS, CHC and cReh mod-els can capture this accurately, making them the only suitablemodels for such unsteady ows. LS and CHC achieve thisthrough an appropriately designed damping function (fl), whilecReh employs an intermittency parameter that responds suit-ably to the variation in ow rate. However, it should be pointedout that the performance of the CHC model in high accelerationow case was not satisfactory because of its instability. The ABand v2f model can also reproduce the basic trends but withmuch shorter time scales than expected.

All models reproduce the overshoot in wall shear stress over thecorresponding quasi-steady shear stress that occurs in the earlystage of the ow rate excursion, with only minor differencesbetween the predictions of each model. In fact, the overshootis an inertia-dominated effect that is not related to turbulence.For this reason we expect the predictions of the various turbu-lence models to be similar at this early stage.

The delay in the response of turbulence in the core region isgoverned by diffusion, represented explicitly in the transportequations of the ke/x models. As a result, such delays are rea-sonably well predicted by all models.

For accelerating ows, it is desirable that the early response ofturbulent kinetic energy is not reected in the modelpredictions unless turbulent kinetic energy and shear stressformulations are decoupled. It is also noted that due to the sim-ilarities between channel and pipe ows, performance of turbu-lence models are expected to be similar for both geometries.

Acknowledgments

The authors would like to acknowledge the nancial supportprovided by the Engineering and Physical Sciences Research Coun-

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

t (sec)

(b) Case B

6 8 10sec)

(c) Case C

y+0 = 15.5 y+0 = 18 y

+0 = 37.5

redicted by cReh for unsteady ow cases A, B and C.cil (EPSRC) through the Grant No. EP/G068925/1.

References

[1] Brereton GJ, Mankbadi RR. Review of recent advances in the study of unsteadturbulent internal ows. Appl Mech Rev 1995;48:189212.

[2] Gndogdu MY, arpinlioglu M. Present state of art on pulsatile ow theor(Part 2: Turbulent ow regime). JSME Int J Ser B: Fluids Thermal En1999;42:398410.

[3] Gerrard JH. An experimental investigation of pulsating turbulent water ow ia tube. J Fluid Mech 1971;46:4364.

[4] Mizushina T, Maruyama T, Shiozaki Y. Pulsating turbulent ow in a tube.Chem Eng Jpn 1973:6.

[5] Shemer L, Wygnanski I, Kit E. Pulsating ow in a pipe. J Fluid Mec1985;153:31337.

[6] Mao Z-X, Hanratty TJ. Studies of the wall shear stress in a turbulent pulsatinpipe ow. J Fluid Mech 1986;170:54564.

[7] Tardu SF, Binder G, Blackwelder RF. Turbulent channel ow with largeamplitude velocity oscillations. J Fluid Mech 1994;267:10951.

[8] He S, Jackson JD. An experimental study of pulsating turbulent ow in a pipEur J Mech B Fluids 2009;28:30920.

[9] Scotti A, Piomelli U. Numerical simulation of pulsating turbulent channel owPhys Fluids 2001;13:136784.

[10] Scotti A, Piomelli U. Turbulence models in pulsating ows. AIAA2002;40:53744.

[11] Cotton MA, Craft TJ, Guy AW, Launder BE. On modelling periodic motion witturbulence closures. Flow Turbul Combust 2001;67:14358.

[12] Maruyama T, Kuribayashi T, Mizushina T. Structure of the turbulence itransient pipe ows. J Chem Eng Jpn 1976;9:4319.

[13] Lefebvre PJ. Characterization of accelerating pipe ow. University of RhodIsland; 1987.

[14] He S, Jackson JD. A study of turbulence under conditions of transient ow inpipe. J Fluid Mech 2000;408:138.

[15] Greenblatt D, Moss EA. Pipe-ow relaminarization by temporal acceleratioPhys Fluids 1999;11:347881.y

yg

n

J

h

g

-

e.

.

J

h

n

e

a

n.

[16] Greenblatt D, Moss EA. Rapid temporal acceleration of a turbulent pipeow. J Fluid Mech 2004;514:6575.

[17] He S, Ariyaratne C, Vardy AE. Wall shear stress in accelerating turbulent pipeow. J Fluid Mech 2011;685:44060.

[18] Chung YM. Unsteady turbulent ow with sudden pressure gradient changes.Int J Numer Meth Fluids 2005;47:92530.

[19] He S, Ariyaratne C, Vardy AE. A computational study of wall friction andturbulence dynamics in accelerating pipe ows. Comput Fluids2008;37:67489.

[20] Ariyaratne C, He S, Vardy AE. Wall friction and turbulence dynamics indecelerating pipe ows. J Hydraul Res 2010;48:81021.

[21] Seddighi M, He S, Orlandi P, Vardy AE. A comparative study of turbulence inramp-up and ramp-down unsteady ows. Flow Turbul Combust2011;86:43954.

[22] Di Liberto M, Ciofalo M. Unsteady turbulence in plane channel ow. ComputFluids 2011;49:25875.

[23] Jung SY, Chung YM. Large-eddy simulation of accelerated turbulent ow in acircular pipe. Int J Heat Fluid Flow 2012;33:18.

[24] He S, Seddighi M. Turbulence in transient channel ow. J Fluid Mech2013;715:60102.

[25] He S, Ariyaratne C. Wall shear stress in the early stage of unsteady turbulentpipe ow. J Hydraulic Eng 2011;137:60610.

[26] Patel VC, Rodi W, Scheuerer G. Turbulence models for near-wall and lowReynolds number ows. AIAA J 1985;23:130819.

[27] Myong HK, Kasagi N. New approach to the improvement of je turbulencemodel for wall-bounded shear ows. JSME Int J 1990;33:6372.

[28] Chang KC, Hsieh WD, Chen CS. A modied low-Reynolds-number turbulencemodel applicable to recirculating ow in pipe expansion. J Fluids Eng TransASME 1995;117:41723.

[29] Sarkar A, So RMC. A critical evaluation of near-wall two-equation modelsagainst direct numerical simulation data. Int J Heat Fluid Flow1997;18:197208.

[30] Launder BE. Low-Reynolds number turbulence near walls. Report TFD/86/4:UMIST; 1986.

[31] Tardu SF, Da Costa P. Experiments and modeling of an unsteady turbulentchannel ow. AIAA J 2005;43:1408.

[32] Al-Sharif SF, Cotton MA, Craft TJ. Reynolds stress transport models in unsteadyand non-equilibrium turbulent ows. Int J Heat Fluid Flow 2010;31:4018.

[33] Khaleghi A, Pasandideh-Fard M, Malek-Jafarian M, Chung YM. Assessment ofcommon turbulence models under conditions of temporal acceleration in apipe. J Appl Fluid Mech 2010;3:2533.

[34] Revell AJ, Craft TJ, Laurence DR. Turbulence modelling of unsteady turbulentows using the stress strain lag model. Flow Turbul Combust 2011;86:12951.

[35] Shima N. Calculation of a variety of boundary layers with a second-momentclosure applicable up to a wall. In: Proceedings of 7th symposium on turbulentshear ows. Stanford University; 1989. p. Paper 53.

[36] Launder BE, Sharma BI. Application of the energy-dissipation model ofturbulence to the calculation of ow near a spinning disc. Lett Heat MassTrans 1974;1:1317.

[37] Durbin PA. Separated ow computations with the kev2 model. AIAA J1995;33:65964.

[38] Langtry RB, Menter FR. Correlation-based transition modeling for unstructuredparallelized computational uid dynamics codes. AIAA J 2009;47:2894906.

[39] Abid R. Evaluation of two-equation turbulence models for predictingtransitional ows. Int J Eng Sci 1993;31:83140.

[40] Lam CKG, Bremhorst K. Modied form of the k-epsilon model for predictingwall turbulence. J Fluids Eng Trans ASME 1981;103:45660.

[41] Yang Z, Shih TH. New time scale based ke model for near-wall turbulence.AIAA J 1993;31:11918.

[42] Abe K, Kondoh T, Nagano Y. A new turbulence model for predicting uid owand heat transfer in separating and reattaching ows-I. Flow eld calculations.Int J Heat Mass Transf 1995;37:13951.

[43] Mathur A, He S. Performance and implementation of the LaunderSharmalow-Reynolds number turbulence model. Comput Fluids 2013;79:1349.

[44] Wilcox DC. Turbulence modeling for CFD. California: DCW Industries; 1998.[45] Menter FR. Two-equation eddy-viscosity turbulence models for engineering

applications. AIAA J 1994;32:1598605.[46] Durbin PA. Near-wall turbulence closure modeling without damping

functions. Theoret Comput Fluid Dyn 1991;3:113.[47] Iaccarino G. Predictions of a turbulent separated ow using commercial CFD

codes. Trans Am Soc Mech Eng J Fluids Eng 2001;123:81928.[48] Lien FS, Kalitzin G. Computations of transonic ow with the m2f turbulence

model. Int J Heat Fluid Flow 2001;22:5361.[49] Abu-Ghannam BJ, Shaw R. Natural transition of boundary layers the effects

of turbulence, pressure gradient, and ow history. J Mech Eng Sci1980;22:21328.

[50] Mayle RE. The role of laminar-turbulent transition in gas turbine engines. JTurbomach 1991;113:50937.

[51] Billard F, Laurence D. A robust kev2/k elliptic blending turbulence modelapplied to near-wall, separated and buoyant ows. Int J Heat Fluid Flow2012;33:4558.

S. Gorji et al. / Computers & Fluids 89 (2014) 111123 123

A comparative study of turbulence models in a transient channel flow1 Introduction2 Methodology2.1 k models2.2 k and shear stress transport (SST) k mod2.3 ?2f model2.4 Re transition model of Langtry and Menter

3 Comparisons for steady flow4 Comparisons for unsteady flow4.1 Key features of unsteady flow from DNS4.2 Performance of the turbulence models

5 ConclusionsAcknowledgmentsReferences