EVALUATION OF A DIFFERENTIAL REYNOLDS STRESS MODEL...

11th World Congress on Computational Mechanics (WCCM XI)5th European Conference on Computational Mechanics (ECCM V)

6th European Conference on Computational Fluid Dynamics (ECFD VI)E. Oñate, J. Oliver and A. Huerta (Eds)

EVALUATION OF A DIFFERENTIAL REYNOLDS STRESSMODEL INCORPORATING NEAR-WALL EFFECTS IN A

COMPRESSOR CASCADE TIP-LEAKAGE FLOW

CHRISTIAN MORSBACH∗, MARTIN FRANKE† AND FRANCESCA DIMARE∗

Institute of Propulsion Technology, German Aerospace Center (DLR)∗Linder Höhe, 51147 Cologne, Germany, †Müller-Breslau-Str. 8, 10623 Berlin, Germany

e-mail: [email protected], [email protected], [email protected] page: http://www.dlr.de/at

Key words: Differential Reynolds Stress Model, Near-Wall Effects, Compressor Cascade,Turbulence Modelling

Abstract. The tip-leakage flow of a low speed compressor cascade at Ma = 0.07and Re = 400, 000 was simulated employing the Jakirlić/Hanjalić-ωh (JH-ωh) differentialReynolds Stress model (DRSM) and results are presented. The predictions are comparedwith those obtained using the SSG/LRR-ω DRSM and the Menter SST k-ω linear eddyviscosity model (LEVM). In addition to the mean flow quantities, the focus is on theReynolds stresses and their anisotropy. Both DRSMs show significant improvementscompared to the LEVM with respect to the mean flow quantities; however, details of theturbulence structure are more accurately predicted by the JH-ωh model.

1 INTRODUCTION

The flow in axial compressor rotors is highly complex due to, amongst other phenom-ena, the vortical motions which develop in the gap between the blades and the machine’scasing (tip-gap). Numerical simulations of such flows often rely on highly tuned lineareddy viscosity models (LEVM) despite the high anisotropy which characterises the tur-bulence field and plays a major role in many phenomena of practical interest. It wouldappear sensible to adopt, in these cases, an anisotropy-resolving modelling approach. Dif-ferential Reynolds stress models (DRSM) belong to this class of closures; however, reportson their application to realistic configurations are scarce. Gerolymos and co-workers wereamong the first to employ DRSMs to investigate complex configurations, ranging fromcascades [1] to multi-stage compressors [2]. Rautaheimo [3] conducted simulations of acentrifugal compressor, whilst Borello et al. [4] investigated the flow through a linear com-pressor cascade with tip-clearance. In all the above-mentioned studies, results obtained

1

Christian Morsbach, Martin Franke and Francesca di Mare

using DRSMs were found to be superior to those of a LEVM taken as reference, especiallyfor complex 3D flow features. Yet, despite the obvious advantages of DRSMs, they arestill not popular in industrial design applications.

In a previous paper, the present authors applied the SSG/LRR-ω DRSM to a compres-sor cascade flow and compared the results to those obtained with a LEVM and an explicitalgebraic Reynolds stress model [5]. It could be shown that the prediction of secondaryvelocities in the tip-gap flow and the shape of the tip-gap vortex could be improved bythe DRSM. However, there was still potential for improvement in the representation ofthe mean velocities and especially of the Reynolds stresses near the wall. This motivatedthe present investigation.

2 TURBULENCE MODELLING

In a Reynolds averaged Navier-Stokes (RANS) framework, the objective of turbulencemodelling is to determine the Reynolds stress tensor ρu′′i u

′′j . This can be accomplished

using closures entailing different levels of complexity. For standard industrial CFD ap-plications, the Boussinesq approximation is generally invoked, which defines a turbulentviscosity µT to relate the Reynolds stresses directly to the trace-free rate of strain S

∗ij. A

prominent example of such an approach is the Menter SST k-ω model [6], which will beused as reference in this paper. However, although the linear stress-strain coupling can bejustified for certain flow topologies, it cannot be expected to hold in general. In fact, thisis the reason for the inability of LEVMs to predict higher order effects such as stream-line curvature, rotation or three dimensional boundary layers. In these cases, individualcomponents of the rate of strain tensor influence differently and distinctively the variousterms appearing in the Reynolds stress budget, particularly the turbulence production.This mechanism cannot be captured by LEVMs since the production of turbulent kineticenergy in a Boussinesq context relies on the norm of the rate of strain tensor only.

One of the biggest advantages of DRSMs with respect to lower order closures is the ex-act turbulence production term, which appears in the transport equations for the Reynoldsstress tensor derived directly from the Navier-Stokes equations. However, the closure prob-lem is not automatically solved by differential models; it is only shifted, as still highercorrelations of fluctuating quantities are introduced. Models have to be found for thedissipation, redistribution due to pressure-strain interaction, and turbulent diffusion ofReynolds stresses. Two differential Reynolds stress models of different complexity will beevaluated in this paper, i. e. the SSG/LRR-ω and the JH-ωh model.

The SSG/LRR-ω model has been developed specifically in view of application to com-plex aerodynamic flows. It is a hybrid model in which the quadratic SSG [7] and the linearLRR [8] pressure-strain models are combined. Menter’s BSL ω-equation is employed forthe scale determining variable along with the blending function F1. The latter is also usedto blend between the LRR model (close to solid walls) and the SSG model (away fromwalls). For more details on the SSG/LRR-ω model, the interested reader is referred tothe original publications by Eisfeld and co-workers [9, 10]. The version used in this study

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is documented in [5].On the other hand, the Jakirlić/Hanjalić model has been developed with particular

attention to the exact reproduction of mean flow quantities and turbulent statistics inbuilding block flows. For this purpose, the terms in the Reynolds stress transport equa-tions as well as the dissipation rate equation were calibrated to reproduce the behaviourof their exact counterparts. In particular, using DNS data to compute the model terms,a system of equations was obtained and solved for the unknown model’s coefficients.Analysis showed that these can be expressed as functions of turbulence anisotropy in-variants and the turbulence Reynolds number [11, 12]. Originally, the model was basedon the dissipation rate �. Jakirlić showed that if the homogeneous dissipation rate �h

is used instead, wall limits for the normalised dissipation components are satisfied au-tomatically [13]. While adapting the model to be used in the context of scale-adaptivesimulations, Maduta suggested that the specific homogeneous dissipation rate ωh shouldbe employed [14]. The current model is based on this latest formulation and termedJH-ωh.

The transport equation for Reynolds stresses in the JH-ωh model reads:

Dρu′′i u′′j

Dt= ρPij − ρ�hij + ρ

(Πij,1 + Πij,2 + Π

wij

)+

∂

∂xk

[(1

2µ+

2CS3Cµ

µT

)∂ũ′′i u

′′j

∂xk

](1)

where Pij denotes the production, �hij the homogeneous dissipation and Πij the components

of the pressure-strain redistribution term. A simple gradient diffusion (SGD) approach isused to close the turbulent diffusion correlations. Indices occuring twice within a productimply summation over the three spatial directions. As mentioned above, the productionterm

Pij = −(ũ′′i u

′′k

∂ũj∂xk

+ ũ′′ju′′k

∂ũi∂xk

)(2)

is exact. The pressure-strain correlation is traditionally split into a slow part Πij,1, a rapidpart Πij,2, and a contribution due to the presence of solid walls Π

wij. Its components are

given by

Πij,1 = −C1�haij (3)

Πij,2 = k

[C3S

∗ij + C4

(aipSpj + ajpSpi −

2

3apqSpqδij

)+ C5 (aipWpj + ajpWpi)

](4)

Πwij = Cw1 fw

�h

k

(ũ′′ku

′′mnknmδij −

3

2ũ′′i u

′′knknj −

3

2ũ′′ku

′′jnkni

)(5)

+Cw2 fw

(Πkm,2nknmδij −

3

2Πik,2nknj −

3

2Πkj,2nkni

).

All closures are formulated in terms of the strain rate and vorticity tensors

Sij =1

2

(∂ũi∂xj

+∂ũj∂xi

), S∗ij = Sij −

1

3Sqqδij, Wij =

1

2

(∂ũi∂xj− ∂ũj∂xi

), (6)

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the Reynolds stress anisotropy tensor

aij =ũ′′i u

′′j

k− 2

3δij (7)

and the wall normal vector n. The influence of the wall is blended by a function dependenton the ratio of the turbulence length scale to the distance to the wall

fw = min

[k

32

2.5�hyn, 1.4

]. (8)

The anisotropy of the dissipation tensor is directly coupled to the anisotropy of theReynolds stress tensor

�hij = �h

[2

3δij + fsaij

]with fs = 1−

√AE2. (9)

All model coefficients are functions of invariants of aij and its dissipation counterparteij = fsaij. The second and third invariants as well as the two-component parameter aregiven by

A2 = aijaji, A3 = aijajkaki and A = 1−9

8(A2 − A3) (10)

and likewise for the dissipation anisotropy eij. The coefficients of the JH-ωh model are

summarised in Table 1. In contrast to the SSG/LRR-ω model, all pressure-strain termsare tensorially linear in the anisotropy tensor. The complexity of this model lies in thevariable coefficients. In the rapid part of the pressure-strain correlation Πij,2 these arechosen so that a classic isotropisation-of-production term with a variable coefficient C2 isobtained. Furthermore, the model introduces explicit modelling of near-wall effects whichEisfeld’s model does not consider.

The homogeneous dissipation rate is related to the specific homogeneous dissipationrate by

�h = Cµkωh (11)

with Cµ = 0.09. For this quantity, a transport equation can be derived from the transportequation for �h. The employed version of the equation

D(ρωh

)Dt

=∂

∂xi

[(1

2µ+ σωµT

)∂ωh

∂xi

]+ α

ρωh

2kPqq − βρ

(ωh)2

+2

k

(1

2µ+ σdµT

)max

[∂ωh

∂xi

∂k

∂xi, 0

]+C�3µ

Cµ

ũ′′pu′′q

�h∂2ũi∂xp∂xl

∂2ũi∂xq∂xl

(12)

differs from the exactly transformed version in several aspects. The most important oneis the limitation of the cross diffusion term to positive values in analogy to Menter’s BSL

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Table 1: Coefficients of Reynolds stress and dissipation rate equations.

Model Coefficient Model Coefficient

Πij,1 C1 = C +√AE2 Πwij C

w1 = max [1.0− 0.7C, 0.3]

C = 2.5AF14f Cw2 = min [A, 0.3]

F = min [0.6, A2] Diffusion CS = 0.082

f = min[(

ReT150

) 32 , 1]

ωh α = 0.44

ReT =ρk2

µ�hβ = 0.072

Πij,2 C3 =43C2 σω = 0.9091

C4 = C2 σd = 0.25C5 = −C2 C�3 = 0.3C2 = 0.8

√A

equation. This improves stability in complex test cases at the expense of the accuracy ofprediction of normal stresses in the viscous sublayer. Evaluation of the turbulent viscosityµT in a turbulent plane channel flow resulted in the formulation

µT = 0.144Aρk12 max [10ηK , L] with ηK =

(ν3

�h

) 14

and L =k

32

�h. (13)

It is used to model the diffusion of the Reynolds stresses and the dissipation rate. In theNavier-Stokes equations it serves a mere numerical stabilisation purpose by increasing thediagonal dominance of the implicit solution matrix [5].

At solid walls the Reynolds stresses vanish as prescribed by the no-slip condition. Forthe dissipation rate the Taylor microscale is employed to derive the following formulationfor ωh for the first cell away from the wall:

ωh∣∣first cell

=ν

Cµy2,

∂ωh

∂n

∣∣∣∣wall

= 0. (14)

The gradient is set to zero which is physically incorrect; however, this choice has noinfluence as long as the diffusion of ωh is computed using only directly neighbouringcells. This treatment in the JH-ωh model differs from the SSG/LRR-ω model, where ω isprescribed at the wall according to the suggestion by Menter [15].

3 NUMERICAL METHOD

All computations were performed using the DLR flow solver for turbomachinery ap-plications TRACE. TRACE is a hybrid grid, multi block, compressible, implicit Navier-Stokes code based on the finite volume method. It has been developed for over 20 years

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at the DLR Institute of Propulsion Technology and is designed to meet the specific re-quirements of simulating turbomachinery flows [16]. Within the RANS framework, theturbulence transport equations are solved with a second-order accurate, conservative, seg-regated solution method [17]. The source terms for the Reynolds stresses and dissipationrate are linearised and treated implicitly [5]. Due to the low Mach number of the com-pressor test case, a local low Mach preconditioning of the type proposed by Turkel wasemployed [18].

A key to a more robust solution method was the introduction of explicit realisabilityconstraints for all six Reynolds stress tensor components, as theoretically investigatedby Schumann [19]. Since the Reynolds stress equations are solved in a segregated man-

ner, it is possible that one of the normal components ũ′′αu′′α (no summation over Greek

indices) violates the realisability condition of positive normal stresses. This limit is ex-plicitly enforced; however, it is also very important to satisfy the constraints on the shearstresses resulting from the Cauchy-Schwarz inequality as otherwise the production term ofReynolds stresses would yield unphysical values which could lead to a diverging solution.Therefore, the shear stresses are also limited based on the limited normal stresses.

4 RESULTS

The implemented DRSMs were validated using a series of building block flows. In thiswork, the results of the simulations of the turbulent flow in a plane channel are reportedas an example. Figure 1 shows the results at Reynolds numbers based on friction velocityof Reτ = 180 [20] and Reτ = 2003 [21]. All quantities are made non-dimensional by thefriction velocity uτ . The velocity profile is predicted to a similar degree of accuracy by theMenter SST k-ω and the SSG/LRR-ω models. Especially the prediction of the logarith-mic region at y+ towards the channel centre line is improved by the JH-ωh model. Greatimprovements can be seen in the reproduction of the wall-normal velocity fluctuation v+

and the streamwise velocity fluctuation u+. The JH-ωh model is able to qualitatively cap-ture the peak in u+. On the other hand, quantitative agreement including the asymptoticbehaviour towards the wall is not as well achieved as it is by the original JH-� model [22].

The low speed compressor cascade investigated experimentally by Muthanna [23] andTang [24] (operated at Ma = 0.07 and Re = 400, 000) is representative of a turboma-chinery flow, characterised by complex 3D flow features. A detailed description of thenumerical setup was given in [5]. For all turbulence models tested in this work the sameinflow boundary conditions, i.e. isotropic turbulence, were used. The pressure distribu-tion on the blade at midspan, predicted by the different turbulence models, lies withinthe experimental scatter, confirming that the boundary conditions were chosen correctly.An overview of the flow topology is shown in Figure 2. The flow through the tip-gapof 1.65% blade height leads to the development of the tip-gap vortex visualised by thestreamlines. Qualitatively the velocity deficit in the vortex core predicted with the JH-ωh

model is compared to the measured data at x/ca = 0.98, ca being the axial chord length.Muthanna determined the centre of the vortex as the location of the maximal stream-

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Re =2003

Re =180

y+

U+

10-1 100 101 102 1030

5

10

15

20

25

30Menter SST k-SSG/LRR-JH- h

DNS Kim et al.DNS Hoyas & Jimenez

Re =2003

Re =180

y+

uv+

10-1 100 101 102 103

-1.5

-1

-0.5

0

w+

u+

v+

y+

u+ ,

v+ ,

w+

10-1 100 101 1020

1

2

3

4

5Re =180

w+

u+

v+

y+

u+ ,

v+ ,

w+

10-1 100 101 102 1030

1

2

3

4

5Re =2003

Figure 1: Turbulent plane channel flow at Reτ = 180 [20] and Reτ = 2003 [21]. Normalised velocityprofile, shear stress and normal stresses are compared for Menter SST k-ω, SSG/LRR-ω and JH-ωh

turbulence models.

wise vorticity [23]. From the simulation data, the vortex core was determined using theλ2 criterion. The position of the vortex core (spheres) at four measurement planes isshown in Figure 3. To facilitate comparison, its path is also projected onto the side wall(corresponding lines without symbols). From these results it can be argued that none ofthe employed turbulence models shows a clear advantage over the others and that all ofthem predict a path that is comparable to the experiment.

Tang measured the mean velocities and Reynolds stresses at various stations in thetip-gap [24]. Due to space constraints, only the station 5c was selected to be shown inthis paper as the most representative, with its location illustrated in Figure 3. To obtaina representation which is independent of the selected coordinate system, invariants ofthe Reynolds stress anisotropy tensor aij, given by equation (10), are plotted in Figure 4instead of Reynolds stress tensor components. Close to a solid wall, turbulence is expectedto tend towards the two-component limit with A = 0. Towards the end wall, experiments

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Figure 2: Illustration of tip-leakage flow in the Virginia Tech compressor cascade. Measured meanvelocity in blade passage is compared to prediction by JH-ωh DRSM.

Figure 3: Prediction of tip-gap vortex centre. Spheres represent vortex centre at different measurementplanes. Trajectory is projected to x-y plane for better comparability. Position for examined tip-gapmeasurements is shown in red.

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A2

z [m

]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-0.001

0

0.001

0.002

0.003

0.004

0.005

Menter SST k-SSG/LRR-JH- h

Experiment

End wall

Blade

A3

z [m

]

0 0.2 0.4 0.6 0.8 1-0.001

0

0.001

0.002

0.003

0.004

0.005

End wall

Blade

A

z [m

]

0 0.2 0.4 0.6 0.8 1-0.001

0

0.001

0.002

0.003

0.004

0.005

End wall

Blade

Figure 4: Reynolds stress anisotropy invariants in tip-gap at measurement position 5c.

UC / Uref

z [m

]

0 0.2 0.4 0.6 0.8 1-0.001

0

0.001

0.002

0.003

0.004

0.005

Menter SST k-SSG/LRR-JH- h

Experiment

End wall

Blade

VC / Uref

z [m

]

0 0.05 0.1 0.15 0.2-0.001

0

0.001

0.002

0.003

0.004

0.005

End wall

Blade

WC / Uref

z [m

]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2-0.001

0

0.001

0.002

0.003

0.004

0.005

End wall

Blade

Figure 5: Velocity components in tip-gap at measurement position 5c.

show such a trend except for the points closest to the wall, whereas towards the bladetip wall, no such trend can be observed, suggesting that the velocities were possiblynot measured down to the wall. It cannot be expected from the LEVM to correctlypredict turbulence anisotropy but also the high Reynolds SSG/LRR-ω DRSM is not ableto capture the peak of anisotropy at the end wall. The low Reynolds JH-ωh DRSMformulation, on the contrary, predicts this peak in line with the channel flow results anddisplays the correct two-component turbulence behavior at both solid walls in the tip-gap. Below the blade-tip boundary layer, turbulence reaches a nearly isotropic state withA→ 1, which is predicted by both DRSMs.

The mean velocity is shown in a coordinate system aligned with the chord of the blade.UC is in the direction of the chord, VC points in the spanwise direction and WC is theblade-to-blade direction. Figure 5 shows the mean velocity components in the specifiedcoordinate system normalised by the inflow velocity Uref. The measured velocities do not

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vanish at the blade tip wall which is in line with the findings concerning the turbulenceanisotropy. Whilst an almost symmetric chordwise velocity profile is predicted by theLEVM, experiments show a higher velocity near the blade tip than near the end wall.This asymmetry is predicted qualitatively by the JH-ωh and partly by the SSG/LRR-ωDRSM. Furthermore, improvements of the results obtained with DRSMs as compared toLEVM results can be seen especially in the secondary flow directions. However, althoughthe structure of the turbulence is predicted much better when DRSMs are employed, theimprovement in quantitative agreement of mean velocity data is still far from optimal.

5 CONCLUSIONS

The flow in a low speed compressor cascade with tip-gap was simulated using DRSMsof different complexity. Whereas the SSG/LRR-ω model employs coefficients that areblended by a single empirical function, the JH-ωh model introduces variable coefficientswhich are deduced from analysis of DNS data. Furthermore, the latter explicitly includesnear-wall effects in the pressure-strain term and through the use of the specific homo-geneous dissipation rate as scale determining variable. Improvements in the predictionof turbulence anisotropy could, therefore, be shown in the results obtained with the JH-ωh compared to the SSG/LRR-ω DRSM. This is in agreement with the improved normalstress prediction in the turbulent plane channel flow. Improvements in mean velocity com-ponents, however, could be shown compared to the LEVM but were not as pronounced,both DRSMs showed similar behaviour.

This leads to the real dilemma of turbulence modelling. In order to gain insight intoturbulence mechanisms and derive appropriate models, highly idealised flows focussing onvery few isolated effects have to be studied: basically all turbulence models are calibratedusing such flows. What distinguishes the various models is how accurately the flow featuresof building block flows can be reproduced. However, almost all flows to which the modelsare applied are highly complex and feature combinations of various effects. Since thegoverning equations are non-linear it can per se not be expected that a model calibratedfor a number of idealised flows yields satisfying results in a complex flow. Nevertheless,this procedure seems to be the only viable way to derive and calibrate turbulence models.

In the present study, the JH-ωh model appears to be clearly superior to the SSG/LRR-ω model in the prediction of the building block flow. Yet, this advantage cannot berecognised as pronounced in a complex 3D flow. This raises the question of the appro-priate level of complexity of DRSMs for practical simulations of such flows. It can onlybe answered by means of further investigations, such as the analysis of turbomachinerycomponents including rotating frames of reference, planned by the authors.

REFERENCES

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[23] Muthanna, C., The Effects of Free Stream Turbulence on the Flow Field through aCompressor Cascade, Dissertation, Virginia Polytechnic Institute and State Univer-sity, 2002.

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