International Journal of Heat and Fluid Flow 34 (2012) 62–69

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International Journal of Heat and Fluid Flow

journal homepage: www.elsevier .com/ locate / i jhf f

A modular RANS approach for modelling laminar–turbulent transitionin turbomachinery flows

Liang Wang a,b,⇑, Song Fu b, Angelo Carnarius a, Charles Mockett a,c, Frank Thiele a,c

a Institute of Fluid Mechanics and Engineering Acoustics, Berlin University of Technology, Müller-Breslau-Str. 8, 10623 Berlin, Germanyb School of Aerospace Engineering, Tsinghua University, Beijing 100084, Chinac CFD Software Entwicklungs- und Forschungsgesellschaft mbH, Wolzogenstr. 4, 14163 Berlin, Germany

a r t i c l e i n f o

Article history:Received 10 October 2010Received in revised form 1 September 2011Accepted 16 January 2012Available online 22 February 2012

Keywords:Transition modellingElliptic approachIntermittency factorTurbomachinery

0142-727X/$ - see front matter � 2012 Elsevier Inc. Adoi:10.1016/j.ijheatfluidflow.2012.01.008

⇑ Corresponding author at: Institute of Fluid Mechatics, Berlin University of Technology, Müller-Breslau-S

E-mail address: Liang.Wang@cfd.tu-berlin.de (L. W

a b s t r a c t

In this study we propose a laminar–turbulent transition model, which considers the effects of the variousinstability modes that exist in turbomachinery flows. This model is based on a K–x–c three-equationeddy-viscosity concept with K representing the fluctuating kinetic energy, x the specific dissipation rateand c the intermittency factor. As usual, the local mechanics by which the freestream disturbances pen-etrate into the laminar boundary layer, namely convection and viscous diffusion, are described by thetransport equations. However, as a novel feature, the non-local effects due to pressure diffusion are addi-tionally represented by an elliptic formulation. Such an approach allows the present model to respondaccurately to freestream turbulence intensity properly and to predict both long and short bubble lengthswell. The success in its application to a 3-D cascade indicates that the mixed-mode transition scenarioindeed benefits from such a modular prediction approach, which embodies current conceptual under-standing of the transition process.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

It is well known that the aerodynamic performance of turboma-chinery blades operating at chord-length-based Reynolds numbersof less than one million, as is typical of aircraft cruise conditions, isstrongly dependent on the transition modes occurring on the bladesurface where a mixture of laminar, transitional and turbulent flowoccurs. The determination of the transitional region is thus a veryimportant task in the design process.

Currently, the Reynolds-Averaged Navier–Stokes (RANS) ap-proach is still the main tool used for transition/turbulence model-ling in engineering applications. It is well known that low Reynoldsnumber turbulence models, with the aid of damping functions tocharacterise near-wall viscous effects on turbulence, have a certaintendency to simulate typical transition profiles, e.g. the sharp risein streamwise skin friction. However, it has recently been shownby Rumsey et al. (2006) and Rumsey (2007) that using such mod-els, the converged numerical solutions exhibit arbitrary depen-dence on initial conditions or other solution parameters and canbe susceptible to ‘‘pseudo-laminar’’ states. These appear betweenthe leading edge and the ‘‘transition onset’’, leading to the incorrectconclusion that the low Reynolds number model somehow pre-

ll rights reserved.

nics and Engineering Acous-tr. 8, 10623 Berlin, Germany.ang).

dicts the transition process when in fact, from a dynamical stand-point, it does not. This confirms the viewpoint of the previousreview by Savill (1996), as summarised from calculation experi-ences, that turbulence models which do not employ intermittencyprove to be very delicate and often extremely unreliable in the pre-diction of transition.

As a result, many correlation-based transition models adoptingthe intermittency concept have been proposed. The intermittencyfactor, c, defined as the probability of the flow being turbulent ina given spatial point, provides a general framework to model anytransition mechanism, from natural to by-pass and separation-induced processes, as the streamwise distribution of c in the tran-sitional region appears to be quite universal within a large range offreestream Reynolds number and Mach number (Dhawan andNarasimha, 1958; Mayle, 1996). Such distributions can bemodelled using either algebraic correlations (e.g. Dhawan andNarasimha, 1958) or transport equations (e.g. Vicedo et al.,2004). However, all such models must be coupled with a separatetransition-onset criterion involving non-local variables, such as themomentum thickness, h, giving rise to serious implementation dif-ficulties in modern CFD solvers. Models based on local variables arethus highly preferable for the purposes of industrial applications.

A successful example is the work of Menter et al. (2006), whichhas now been incorporated into commercial software packages. Inthis model, the value of the transitional h-based Reynolds number,which is used to determine the transition onset with an algebraiccriterion, is now obtained with a new transport equation instead

Nomenclature

a local sound speedCf skin frictioncr phase velocity of disturbancesd wall distance, mFSTI freestream turbulence intensity, FSTI = (2k/3)0.5/U1PK production term of K equationPr Prandtl number

Sij mean strain tensorxt transition onset location, mU1 freestream velocityc intermittency factorf length scale in transitional flowsleff effective viscositysnt characteristic timescale in the flow transition

L. Wang et al. / International Journal of Heat and Fluid Flow 34 (2012) 62–69 63

of being calculated by integration over the boundary layer. Anothertype of local formulation related to triggering transition is based onthe concept of laminar fluctuation energy, kL, introduced by Mayleand Schulz (1997). A one-equation model is used to describe such‘turbulent-like’ fluctuations, which are very different from true tur-bulent fluctuations and have been demonstrated both in experi-ment (Leib et al., 1999) and analysis (Jacobs and Durbin, 2001) inthe pre-transitional region. Recent examples of such models havebeen proposed e.g. by Walters and Leylek (2002) and Lardeauet al. (2004). It is noted that these methods still make use of thewall distance, which is strictly a non-local variable. However, thewall distance is routinely employed in industry-standard turbu-lence models.

Such local-variable-based models are however not validatedfor transition in supersonic flows or for crossflow-inducedtransition. One reason is that they rely strongly on empirical cal-ibration rather than deduction from the fundamental physicalphenomena responsible for the actual transition process. Forexample, the flow instability mode can vary considerably betweensupersonic boundary layers and incompressible flows. A newlocal-variable-based model formulated in terms of instabilitymodes has therefore been proposed recently, which can success-fully simulate 3-D high-speed aerodynamic flow transition(Fu et al., 2009; Wang and Fu, 2011). The goal of this work is toextend this model to capture the bypass and separation-inducedtransition effects common in separated flows, especially in turbo-machinery applications.

Studies on separated flow transition have revealed multipletransition modes that arise:

(1) Separation-induced mode: downstream of the separationlocation (xs), due to the Kelvin–Helmholtz (KH) instabilityof the separation velocity profile, transition occurs in thefree shear layer, whose turbulent part then entrains highermomentum fluid from the freestream more effectively thanthe laminar shear layer, leading to reattachment of the tur-bulent shear layer.

(2) Bypass mode: the freestream turbulence intensity (FSTI), e.g.the periodic passage of wakes from upstream blade rows,also affects transition and could even cause the boundarylayer to undergo transition ahead of any laminar separation.

(3) Natural mode: it is found by Hughes and Walker (2001) thatthe flow transition on compressor blades is likely to be pre-dominantly a natural growth process, rather than of thebypass type.

Due to the complexity of such mixed-mode transition scenarios,classical attempts to correlate the general transition-onset loca-tion, xtr, with one or two parameters, such as FSTI and the dimen-sionless pressure gradient, always encounter problems in casesthat deviate from those previously tested. Despite their lack of gen-erality however, such empirical correlations for xtr are stillwidespread.

Even if the xtr is accurately predicted, the calculation of transi-tional separation using conventional turbulence models coupledwith c consistently leads to over-predicted bubble lengths (Mayleand Schulz, 1997), because the turbulent spot formation rate hereis much larger than in pure bypass transition, which cannot beidentified by such models. To address this issue, many approacheshave recently been proposed. Some introduce additional sourceterms to the transport equation for c (e.g. Vicedo et al., 2004),while others force the distributions of c to exceed unity insidethe separated flow region (e.g. Koezulovic et al., 2007). Thesemethods pragmatically achieve the prediction of realistic reattach-ment locations, however, they are not formulated on the basis ofphysical reasoning.

Disturbances in the freestream flow in fact penetrate into thelaminar boundary layer not only by convection and viscous diffu-sion, referred to as local effects in spectrum, but also by pressurediffusion as the non-local mechanism (Mayle and Schulz, 1997).This offers an explanation for the delayed response of intermit-tency models to the freestream turbulence: they only considerthe transition triggered by the diffusion of the freestream turbu-lence into the shear layer. Mayle and Schulz (1997) thus proposedthe transport equation for kL, as further refined by Walters andLeylek (2002) and Lardeau et al. (2004), which considers the effectof pressure diffusion and which appears to be a promising meansof simulating both the onset and length of transition.

However, it is doubtful whether the non-local effect, which doesnot appear explicitly in any single-point model, can be modelledwith such a parabolic equation whose solution is only in the weakform. Therefore, with reference to the work by Durbin (1993) inwhich the non-local wall effect in the near-wall region is modelled,an elliptic equation for c is proposed here. This is expected to pro-vide a more natural way to introduce the effects of pressure fluctu-ation in the boundary layer: they enter into the boundary layer viathe solution of the governing equations, rather than via the trans-verse variations derived from the empirical correlations. Conse-quently, c is calculated by such an elliptic model coupled withthe original transport equation to take into account both the localand non-local effects outlined above.

2. Model formulation

2.1. Modelling of the effective viscosity

In the existing study, the total fluctuating kinetic energy, K, iscomposed of two parts: one relating to the non-turbulent fluctua-tions, kL, and one to the turbulence kT, i.e. K = kL + kT. Thus a sepa-rate transport equation for kL is proposed, albeit in the samemanner as the conventional form for kT (Mayle and Schulz,1997). However, the work of Rumsey et al. (2006) demonstratesthat the employment of two separate equations for kT and kL isnot necessary. It is found that a single equation for the totalfluctuating kinetic energy K is more cost-effective, which isadopted in the present work as

64 L. Wang et al. / International Journal of Heat and Fluid Flow 34 (2012) 62–69

@ðqKÞ@t

þ @ðqujKÞ@xj

¼ DK �gu00i u00j@ ~Ui

@xj|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}PK

�e ð1Þ

where DK, PK and e represent the energy diffusion, production anddissipation terms, respectively. The Boussinesq hypothesis on thesecond-order correlations is adopted in the same manner as in con-ventional eddy-viscosity models. Thus, for the Reynolds stress, thefollowing constitutive relation holds:

gu00i u00j ¼ �2leff

�qeSij �

13eSkkdij

� �þ 2

3dijK ð2Þ

Here, leff is the effective viscosity that can be conjectured from thefollowing exact shear stress relationships

gu00i u00j ¼ cdu0iu0j þ ð1� cÞu000i u000j þ cð1� cÞðcUi � UiÞðcUi � UjÞ ði–jÞð3Þ

where ^ and = denote the turbulent and the non-turbulent zoneaverages, respectively. The first and second terms on the right-handside stand for the momentum transport due to the turbulent andnon-turbulent fluctuations, respectively, while the last term repre-sents the mean velocity difference between the two fluids that alsocontributes to the momentum transport by the bulk convection mo-tion. Referring to the work by Cho and Chung for free shear flows(1992), leff is modelled as

leff ¼ ð1� cÞlnt þ clt þ Clg �qK2

x3 c�3ð1� cÞ @c@xk

@c@xk

ð4Þ

where the subscript nt denotes the non-turbulent part in the effec-tive viscosity and c bridges the non-turbulent and turbulent contri-butions. Clg = 0.1 is the model coefficient given by Cho and Chung(1992).

Since the modelling of the turbulent eddy viscosity lt can nowbe considered well-established, the present work adopts the SSTmodel (Menter, 1994). However, other eddy-viscosity modelscould readily be applied within this framework. Attention will befocussed on the modelling of the non-turbulent fluctuation, forwhich the technique of McDaniel et al. (2000) is adapted that reads

lnt ¼ Cl �qKsnt ð5Þ

where Cl is the model coefficient and snt represents the character-istic timescale associated with different instability-mode transition.

2.2. Modelling the characteristic timescales of the instability waves

Both experimental correlations and theoretical analyses (Arnaland Casalis, 2000) demonstrate that the formulation of the time-scale mentioned above would involve the boundary layer thick-ness, which, regarded as non-local, is calculated via integrationover the boundary layer. In this study, a local-variable-basedlength scale f is proposed as

f ¼ d2X=ð2EuÞ0:5 ð6Þ

Here, d is the distance to wall, X the absolute value of the mean vor-ticity, and Eu(=0.5 � |U|2) stands for the kinetic energy of the meanflow relative to the wall. Because |U| is not a Gallilean invariant, thisformulation is not generally applicable to moving geometries.

The turbulence length scale lT adopts the conventional defini-tion as K0.5/x and the bound of the length scale, lB = K0.5/(Cl|S|),is used to avoid the stagnation point anomaly (Medic and Durbin,2002). The effective transition length scale feff is then set as theminimum value among f, lT and C1lB, where C1 is a model constant.

Next, according to the stability analyses of the frequency of theKelvin–Helmholtz instability with the maximum amplification rate

in separated shear layers (Monkewitz and Huerre, 1982), its char-acteristic timescale, ssep, can be set as

ssep ¼ C1 � 1:21ðfeff =ð2EuÞ0:5Þ ð7Þ

Consequently, snt is expressed as

snt ¼ snt1 þ snt2 �12½1þ sgnðMrel � 1Þ� þ scrossðWsÞ þ ssep �

12

� ½1þ sgnðk1 þ 0:046Þ� ð8Þ

Here, the local relative Mach number Mrel = (U � cr)/a, a is the soundspeed and cr represents the phase velocity of disturbances with thesame value for all Mack modes. Ws is the magnitude of the cross-flow velocity perpendicular to the local inviscid streamline that isgenerated by the combination of curvilinear inviscid streamlinesand the viscous no-slip condition at the wall (Reed and Saric,1989). kf = (feff)2/m�(d|U|/ds) is the dimension-less pressure gradi-ent parameter, sgn (x) = |x|/x the sign function. snt1, snt2 and scross

stand for the timescales of first-mode (Mack), second-mode (Mack)and crossflow instabilities, respectively, as:

snt1 ¼ C8 � qf1:5eff =½ð2EuÞ0:5l�0:5 ð9Þ

snt2 ¼ C9 � 2feff =jUj ð10Þ

taucross ¼ C10 � ð4feff =jUjÞ � ðWs=jUjÞ0:5

� � exp �C11ðqfeff jUj=l� 44Þ2h in o

ð11Þ

However, since the test cases in the present paper are all sub-sonic and non-swept, the second-Mack mode and the crossflowmode do not exert any influence.

2.3. Calculation of the intermittency factor c

A transport equation for c has been developed by the authors as

@ðqcÞ@tþ @ðqujcÞ

@xj¼ @

@xjðlþ leff Þ

@c@xj

� �þ Pc � ec ð12Þ

Here Pc and ec represent the intermittency production and dissipa-tion term, respectively, which are modelled as follows:

Pc ¼ C3qFonset

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� lnð1� cÞ

q1þ C4

ffiffiffiffiffiffiffiffik

2Eu

s0@ 1AdjrEujm� 10�C6k

C7f

ec ¼ cPc ð13Þ

where the function Fonset is set to trigger the onset of transition andis given by

Fonset ¼ 1:0� exp �C5feff K

0:5jrKjmjrEuj

!ð14Þ

It is seen that Fonset is determined by the development of K andthe mean flow in the pre-transitional region.

Next, we rename c as cl, where the subscript l stands for the lo-cal. As mentioned in the introduction, the effect of disturbances pe-netrating into the laminar boundary layer by pressure diffusion isnon-local, which is equivalent to say that the information speed isinfinity. This means that a given point M0 in space will feel theinfluence of all points inside the sphere S0 of centre M0 and witha radius Lc. Lc is the correlation length scale of pressure effectson the transition process.

With reference to Durbin’s model (1993) in which the non-localwall effect in the near-wall region is modelled, we propose an

L. Wang et al. / International Journal of Heat and Fluid Flow 34 (2012) 62–69 65

elliptic equation for cnl to represent the non-local pressure diffu-sion effect as

cnl � L2cr2cnl ¼ Ccnl

ðK=EuÞ1=2Fd ð15Þ

Here, the length scale Lc = CL feff, CL is a model constant, the sensorfunction Fd is used to turn off the source term in the boundary layerand allows cnl to diffuse in from the freestream. Fd is equal to zero inthe boundary layer and one in the freestream, as defined as follows:

Fd ¼ 1� tanh ðleff =�qd2XÞ3h i

ð16Þ

The motivation for Eq. (15) is simply the notion that the non-lo-cal effect might indirectly be represented by an elliptic modelequation, because the exact representation of non-local processeswould require knowledge of two-point correlations. Furthermore,such a methodology can also reproduce the strong non-homogene-ity of the near-wall region.

Consequently, c is obtained as the union of these two probabil-ities as:

c ¼ cl [ cnl ¼ cl þ cnl � clcnl ð17Þ

Table 2

2.4. Summary of the present model

The present model consists of a transport equation for cl, i.e. thelocal component of the intermittency factor, an elliptic equationfor the non-local component cnl, and the equations for fluctuatingkinetic energy K and specific dissipation rate x that are modifiedfrom the SST K–x eddy-viscosity model, all of which are listed asfollows:

DðqclÞDt

¼ @

@xjlþ

leff

rc

� �@cl

@xj

� þ Pcl

ðFonsetÞ � ecl

cnl � L2cr2cnl ¼ Ccnl

ðK=EuÞ1=2Fd

DðqKÞDt

¼ @

@xjlþ

leff

rk

� �@K@xj

� þ PKðleff Þ � e

DðqxÞDt

¼ @

@xjlþ

leff

rx

� �@x@xj

� þ Px � ex þ Cdx

ð18Þ

Here, c = cl + cnl � clcnl is set as a weighting function between thenon-turbulent and the turbulent components of the effective vis-cosity leff, i.e. Eq. (4). In the pre-transitional region, leff � lnt anddifferent instability modes dominate the development of K. Thedevelopments of K and the mean flow then determine the valueof the transition-onset trigger Fonset, i.e. Eq. (14), in the productionterm of the cl equation. Fonset rapidly goes from zero to unity afterthe onset point. The increase of c is also affected by the FSTI in anon-local manner via the solution of the elliptic equation for cnl.In the transitional region, since lnt� lt, the flow develops accord-ing to the distribution of c. In the fully turbulent region, leff = lt, theSST model is recovered. All the model constant values are shown inTable 1.

Table 1Model constants.

Cl C1 C2 C3 C4 C5 C6

0.09 0.7 0.8 8e�5 0.07 1.2 0.03

C7 C8 C9 C10 C11 Ccnl CL

0.6 0.46 0.005 1e�3 5.0 0.2 0.1

3. Results and discussion

The model proposed above has been calibrated and validatedfor a reasonably wide range of transitional cases involving incom-pressible flows past sharp and semicircular leading-edge flat platesand blades exposed to different levels of freestream turbulentintensities (FSTI) either under zero or non-zero pressure gradients(PG) and a 3D stator compressor cascade.

The transition behaviour of the model is first tested for simplezero-PG flat-plate (sharp leading-edge) cases, in order to assessthe response to FSTI, and to compare the prediction of transitiononset and length with experimental data and available empiricalcorrelations. Following this, the zero-PG semicircular leading-edgeflat-plate cases (where the natural mode is fairly weak comparedto the separation-induced and bypass modes) are considered tocalibrate the model constants quoted in Sections 2.1 and 2.2. Next,to assess the applicability of the model to more realistic geome-tries, flat plates with adverse pressure gradients and exposed tovarying FSTI are computed. Finally, the steady RANS simulationof a 3D stator compressor cascade is performed.

All of the simulations presented here are performed using anin-house code where the RANS equations are solved on the non-staggered H-type and O-type grid systems. The convection anddiffusion terms in RANS are discretised with the UMIST-TVDscheme (Liou, 1996) and the central difference scheme respec-tively. To solve the pressure field, the SIMPLE algorithm is usedand in order to eliminate the pressure fluctuation associated withthe use of a non-staggered grid system, a momentum-interpolationtechnique is utilised to calculate the velocity on the finite volumefaces.

For all test case geometries, an initial mesh and a second meshrefined by a factor of 1.5 in each direction are constructed with afirst cell y+ value of one or less and a structured body-fitted meshin the boundary layer region. In all cases, steady-state solutionshave been obtained on the both meshes, which show negligible dif-ference and are therefore judged to be mesh independent. All casesare run to full convergence, determined based on a drop in residu-als of typically five orders of magnitude, as well as a flattening ofall residuals indicating that machine accuracy has been reached.

The various results reported herein correspond to the four dif-ferent models employed during the study. Results labelled SST cor-respond to those predictions in which no transition-prediction toolis used. Those of the present transition model are labelled SST-cl-cnl

and those when the elliptic approach is excluded SST-cl. Finally, theresults labelled by SST-I have been obtained using the standard SSTmodel with a manually-specified transition onset, xtr, obtainedfrom the SST-cl-cnl results, i.e. with leff = 0 upstream while leff = lt

downstream of xtr.

3.1. Sharp leading-edge flat plate with zero pressure gradient

The test cases chosen include the T3 series experiments (Coup-land, 1990a,b) from the European Research Consortium on Flow,Turbulence and Combustion (ERCOFTAC) database, and the well

Flow inlet conditions and computed transition-onset locations for the sharp leading-edge flat plates with zero pressure gradient.

Case S&K T3A T3B T3A-

U inlet (m/s) 50.1 5.4 9.4 19.8lt/l 5.0 12.5 100.0 8.72FSTI (%) inlet 0.18 3.5 6.5 0.874Computed xtr (mm) (SST-cl-cnl) 0.873 0.466 0.105 1.243Computed xtr (mm) (SST-cl) 0.873 6.255 3.328 2.221Measured xtr (mm) 0.86 0.46 0.10 1.31

x (m)

FSTI(%)

0.2 0.4 0.6 0.8 1

Experimental dataPresent calculation

T3B flat plate

Re∞ = 6.26E5 m-1

FSTI = 6.5%

(a)

Rex

Cf

200000 400000 600000 800000

SST - γl - γnlExperimental dataSST - γl

FSTI = 6.5%T3B flat plate

Re∞ = 6.26E5 m-1

(b)

2.5

3

3.5

4

4.5

5

5.5

0.001

0.002

0.003

0.004

0.005

0.006

Fig. 1. Streamwise decay of the freestream turbulence intensity (a) and skin frictiondistribution (b) for the test case T3B.

Table 3Flow inlet conditions and computed transition-onset and reattachment locations forthe semicircular leading-edge flat plates with zero PG. Here, (V) stands for thecalculations by Vicedo et al. (2004).

Case T3L1 T3L2 T3L3 T3L4

Red 3293 3293 3293 3293FSTI (%) inlet 0.17 0.63 2.39 5.34Computed xtr (mm) 23.7 20.2 18.1 16.0Computed xtr (mm) (V) – 12.4 12.2 11.8Measured xr (mm) 40 27 23 21Computed xr (mm) 39.0 29.2 24.2 20.5Computed xr (mm) (V) – 28.3 24.3 21.2

X (m)

Y(m)

0.01 0.02 0.03

Levelk / U∞

2:1

0.0053

0.0155

0.0257

0.0359

0.045110.055

Experimental data (T3L1):

Re∞ = 3.3E5 m-1

FSTI = 0.17%xs = 0.0049 mxr = 0.040 m

(a) SST - γl - γnl

X (m)

Y(m)

0 0.01 0.02 0.03

Levelk / U∞

2:1

0.0033

0.0095

0.0157

0.0219

0.027110.033

130.039

Experimental data (T3L1):

Re∞ = 3.3E5 m-1

FSTI = 0.17%xs= 0.0049 mxr = 0.040 m

(b) SST

-0.005

0

0.005

0.01

-0.005

0

0.005

0.01

Fig. 2. Streamline (above) and turbulent kinetic energy contours (below) for (a) thepresent model and (b) the baseline SST model.

66 L. Wang et al. / International Journal of Heat and Fluid Flow 34 (2012) 62–69

known Schubauer and Klebanof (S&K) experiment (1955), all ofwhich were conceived specifically for the validation of transitionmodels and have become a recognised standard in the researchcommunity. The T3 series (T3A-, T3A, and T3B) cases have zeropressure gradient with freestream turbulence levels of 0.874%,3.5% and 6.5% that correspond to transition in the bypass regime.The S&K test case has low freestream turbulence intensity and cor-responds to purely natural transition. The inlet quantities for allthe cases computed, besides q = 1.2 kg/m3, l = 1.8e�5 kg/ms, aresummarised in Table 2. Here the inlet values of lt/l are chosenin order to correctly reproduce the streamwise decay of freestreamturbulence reported in the experiments. An example of typicalagreement in the freestream is shown in Fig. 1a and this step hasbeen carried out for all the cases individually.

Fig. 1b gives the streamwise skin friction coefficient for the testcase T3B. It is seen that the flow transition profile is well capturedwith the present model, while this is missed entirely when theelliptic approach is excluded.

The transition-onset locations, xtr, as defined by the local min-ima of shear stress in the skin friction distribution, are listed forall cases in Table 2. It is seen that the intermittency model coupledwith the elliptic approach shows a significantly improved perfor-mance in the sensitivity of xtr to FSTI, with excellent agreementwith the experimental data. In contrary, exclusion of the ellipticcnl equation leads to a weak sensitivity to FSTI and the transitionoccurs approximately when Rex exceeds a critical value. This dem-onstrates that the additional modelling of the pressure diffusion offreestream turbulence into the boundary layer via the elliptic for-mulation is responsible for the improvements achieved.

3.2. Semicircular leading-edge flat plate with zero pressure gradient

The test cases chosen match the T3L1 to T3L4 validation casesfrom the ERCOFTAC database (Coupland, 1995). The flow condi-

tions specified are listed in Table 3. Here, the fixed Reynolds num-ber, Red, is based on the leading-edge diameter (0.01 m) and thefreestream velocity U1 = 5 m/s. The different FSTI values are mea-sured at 0.006 m downstream of the leading edge.

Figs. 2 and 3 show the separated flow predicted by means ofstreamlines and turbulence kinetic energy contours for the T3L1and T3L4 cases, respectively. It is firstly seen that the onset of sep-arated flow is detected at the point where the curved surfacemerges with the horizontal plate, at roughly xs = 0.0048 m, whichremains approximately constant for all cases.

For the T3L1 case, the SST model alone predicts the transitiononset at xtr = 0.0110 m, which is in clear disagreement with the va-lue of xtr = 0.0237 m obtained by the proposed transition model. Inthis model calculation, as a consequence of the xtr further down-stream, the turbulence generated due to the shearing effect ofthe separated flow takes longer to develop. Reattachment thus oc-curs further downstream (xr = 0.0390 m) than with the baselineSST model, resulting in an 85% increase in bubble length and im-proved agreement with the experimental data. Furthermore, sincethe bypass mode in this case is fairly weak compared to the sepa-ration-induced mode, the good agreement with experiment indi-cates that the modelling of ssep, i.e. Eq. (7), is effective.

X (m)

Y(m)

0 0.01 0.02 0.03

Levelk / U∞

2:1

0.0053

0.0155

0.0257

0.0359

0.045110.055

Experimental data (T3L4):

Re∞ = 3.3E5 m-1

FSTI = 5.34%xs = 0.0049 mxr = 0.021 m

(a) SST - γl - γnl

X (m)

Y(m)

0.01 0.02 0.03

Levelk / U∞

2:1

0.0053

0.0155

0.0257

0.0359

0.045110.055

Experimental data (T3L4):

Re∞ = 3.3E5 m-1

FSTI = 5.34%xs = 0.0049 mxr = 0.021 m

(b) SST - I

-0.005

0

0.005

0.01

-0.005

0

0.005

0.01

Fig. 3. Streamline (above) and turbulent kinetic energy contours (below) for (a) thepresent transition model and (b) the SST-I model.

Table 4Flow inlet conditions and computed transition-onset locations for the sharp leading-edge flat plates with non-zero pressure gradients.

Case T3C2 T3C3 T3C4 T3C5

FSTI (%) inlet 3.0 3.0 3.0 3.0lt/l 11.0 6.0 8.0 15.0Computed xtr (m) 0.834 1.173 1.173 0.352Measured xtr (m) 0.795 1.195 1.395 0.345Relative error of xtr (%) 4.90 1.88 2.50 2.03

Rex

Cf

50000 100000 150000-0.002

0

0.002

0.004

0.006

0.008

0.01 Experimental dataPresent calculation

FSTI = 3%T3C4 flat plate

Fig. 4. Comparison of computed and measured skin friction (Cf) for T3C4 case.

L. Wang et al. / International Journal of Heat and Fluid Flow 34 (2012) 62–69 67

Turning to the T3L4 case, despite the manual specification of thetransition onset to the same location, the extent of the separatedflow given by the present model is significantly smaller than thatof the SST turbulence model. This indicates that the strongly accel-erated turbulent spot formation in the transitional region is prop-erly simulated by the consideration of the entrainment mechanismin the leff modelling, i.e. Eq. (4).

Finally, Table 3 summarises the computed xtr and xr and mea-sured xr for all cases. The proposed model shows fairly good agree-ment with experiment, and with an increase in FSTI, the transitiononset obtained by this model is shifted significantly upstreamwhereas that predicted by a traditional K–x–c model (Vicedoet al., 2004), marked as V in Table 3, varies only slightly. In the lat-ter model, additional source terms are introduced to the transportequation for c, which results in excessive production of turbulentkinetic energy, hence forcing the boundary layer to reattach atthe location measured.

3.3. Flat plate with non-zero pressure gradient

In Section 3.2, the proposed model was shown to predict wellthe short bubble that has a local effect on the flow and only slightlyaffects the inviscid flow outside the bubble. Here, we assess its pre-dictive capability for the long bubble, which by contrast exhibits asignificant interaction with the external flow, such that the pres-sure distribution deviates markedly from the inviscid case.

The test cases chosen include the T3C2–T3C5 (flat plates withadverse PG) experiments from the ERCOFTAC database (Coupland,1990a,b). The computational domain for T3 cases is constructedwith a contoured upper surface, where the contour is chosen tomatch the experimental pressure distribution on the flat plate.The inlet quantities are summarised in Table 4.

Comparisons of computed and measured skin friction for theT3C4 case is given in Fig. 4. The model results for the T3C4 case

indicate a laminar boundary layer separation due to the adversepressure gradient, followed by the transition and reattachment ofthe boundary layer. All results show generally good agreementwith the experimental data.

3.4. 3D stator compressor cascade

The stator cascade chosen corresponds to the low speed cascadetest section at the Department of Aeronautics and Astronautics ofthe Berlin Institute of Technology (Zander et al., 2009). The highlyloaded Controlled Diffusion Airfoil (CDA) has a chord length ofL = 0.375 m. The stagger angle is h = 20�. In the experiment thetransition occurs in the free shear layer on its suction side, whichdetermines whether or not the separation bubble will reattach asa turbulent boundary and, ultimately, whether or not the blade willstall. Moreover, due to the relatively small pitch to chord ratio ofT/L = 0.4, the high turning angle of up to Db = 60�, and the lowaspect ratio of h/L = 0.8, strong secondary flow structures areobserved in the experiment. An overview of the stator cascadeparameters is shown in Fig. 5.

The oil-flow visualisation in Fig. 6a shows the general flow pat-tern over the projected suction side: the laminar boundary layerseparates at the blade surface S = 0.17, forming a quasi-2D laminarseparation bubble; the separated flow then undergoes transition,reattaching on the blade suction surface as a turbulent boundarylayer (approximately 0.25S); downstream of the turbulent reat-tachment, the flow is constricted by the occurrence of the cornervortices at the end walls and a 3D separation line is formed be-tween the secondary and the main flow, ending up at midspan(0.7S).

Fig. 6b compares the calculated wall streamlines to the under-lying oil-flow visualisation. It is seen that the laminar flow separa-tion and the reattachment, even in the near-end-wall region, arepredicted well. The turbulent separation position at midspan is incontrast predicted much later than the experiment (0.99S com-pared to 0.7S). The cause for this is believed to be the underlyinglinear eddy viscosity turbulence models, which are known toover-predict the strength of such corner vortices. This in turn leads

Fig. 5. Geometry of the stator cascade.

Fig. 6. General flow pattern over the projected suction side of the cascade. (a) Oil-flow visualisation and (b) superimposed simulated wall streamlines to (a).

Fig. 7. Contours of dimensionless turbulent kinetic energy (top) and intermittencyfactor (bottom) at midspan.

68 L. Wang et al. / International Journal of Heat and Fluid Flow 34 (2012) 62–69

to excessive induced downwards flow at midspan and delayed sep-aration. Future research will thus focus on predicting such turbu-lent separated-flow region using Detached-Eddy Simulation(Spalart et al., 2006) where the present transition model functionsin the RANS zones. Promising results for Detached-Eddy Simula-tion combined with manually-prescribed transition have alreadybeen obtained for this flow (Steger et al., 2011).

Fig. 7 shows the contours of dimensionless turbulent kinetic en-ergy and intermittency factor at midspan. It is seen that K ramps upwhen c exceeds 5–10%. The transition starts at 0.21S on the suctionside and at 0.08S on the pressure side, both of which agree wellwith the experimental data.

4. Conclusions

A local-variable-based K–x–c three-equation transition/turbu-lence model considering different instability modes is proposedand validated for a relatively wide range of transitional flow condi-tions corresponding to cases of practical relevance to turbomachin-ery flows. It takes into account not only the local effects by whichthe disturbances penetrate into the laminar boundary layer,namely convection and viscous diffusion as described by a trans-port equation for c, but also the non-local effect by pressure diffu-sion, as represented by an elliptic approach. This model, whichnow additionally considers bypass and separation-induced transi-tion effects, is an extension to the original transition-sensitivemodel (Fu et al., 2009; Wang and Fu, 2011), which itself takes intoaccount 3-D high-speed aerodynamic flow transition by the mod-elling of natural and crossflow modes.

The results show generally good agreement with experimentaldata, which applies both to global quantities such as separationlength and transition-onset location, as well as for local velocityprofiles and turbulent intensity contours. For the zero-PG sharpleading-edge flat-plate cases, the elliptic approach results in aproper response to FSTI. For the semicircular leading-edge cases,the modelling of the separation-induced mode and the entrain-ment in the transitional region are both effective. For thenonzero-PG cases, all new modelling components perform reason-ably well. For the 3D stator cascade case, the present model pre-dicts fairly accurate onset and reattachment positions of thelaminar separated flow on the suction side, and transition onsetlocations on both sides. The conclusion can therefore be made thatthe mixed-mode transition scenario benefits from such a modularprediction approach that is based on the current conceptual under-standing of the transition process.

With regard to the use of the intermittency model coupled withthe elliptic approach, as opposed to production term modificationsand prescribed intermittency models (Vicedo et al., 2004), thepresent approach has shown an improved performance when com-pared to earlier work on the same test case. The differences be-tween both approaches demonstrate that the inclusion of theintermittency model for the pressure diffusion of freestream tur-bulence into the boundary layer is the reason behind the improve-ments achieved.

In short, the model is based on an extremely simplified view oftransition physics, but can nonetheless make useful predictions by

L. Wang et al. / International Journal of Heat and Fluid Flow 34 (2012) 62–69 69

relying on a limited amount of simple and reliable experimentaldata. It suggests that the physics-based approach adopted here al-lows engineers to significantly extend RANS-based computationalanalysis capability by providing realistic transitional modelling ina relatively simple eddy-viscosity framework. Moreover, such aframework has an inherent potential to extend directly to De-tached-Eddy Simulation that can well resolve the both the bound-ary-layer and the free shear flows.

Acknowledgements

The investigations presented in this paper have been obtainedwithin the European research Project TATMo (Turbulence andTransition Modelling for Special Turbomachinery Applications).

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