Case 13.2: Flow in a 3-D diﬀuser - description of …...1 13th ERCOFTAC SIG15 Workshop on Reﬁned Turbulence Modelling , Graz University of Technology, Graz, Austria, September

1

13th ERCOFTAC SIG15 Workshop on Refined Turbulence Modelling,Graz University of Technology, Graz, Austria, September 25-26, 2008

Case 13.2: Flow in a 3-D diffuser- description of the computational method -

G. Kadavelil1, M. Kornhaas2, S. Saric1, D. Sternel2, S. Jakirlic1,∗, M. Schafer2

1Chair of Fluid Mechanics and Aerodynamics, Petersenstr. 30, 64287 Darmstadt2Chair of Numerical Methods in Mechanical Engineering, Dolivostr. 15, 64293 Darmstadt

Technische Universitat Darmstadt, Germany

AbstractAn incompressible flow in a 3-D diffuser with deflected upper wall (with an angle of expansionα = 11.3o) and one side wall (with an angle of expansion α = 2.56o) at the inlet-duct-height-based Reynolds number of Reh = 10000 (Fig. 1), for which the experimentally obtained referencedatabase was provided by Cherry et al. (2008), was studied computationally by using the LES(Large Eddy Simulation), DES (Detached Eddy Simulation) and HLR (Hybrid LES/RANS) meth-ods. The focus of the investigation was on the adverse pressure gradient effects evoked by theduct expansion on the size and shape of the three-dimensional flow separation pattern.

Figure 1: Geometry of diffuser 1 (Cherry et al., 2008)

Computational methods / turbulence modelsThe computations were performed with:

• LES (Large Eddy Simulation): the sub-grid scales were modelled by the Smagorinsky (1963)formulation utilizing the dynamic determination of the model coefficient proposed by Ger-mano et al. (1991)

∗[email protected]

Case 13.2-4

2

• DES (Detached Eddy Simulation): a seamless hybrid LES/RANS approach employing theone-equation turbulence model by Spalart and Allmaras (S-A, 1994), based on the transportequation for turbulent viscosity νt, to model the influence of the smallest, unresolved scaleson the resolved ones (e.g. Travin et al., 2002) in the LES sub-region of the solution domain.The same (RANS) model was used to model the near-wall layer. The smooth transitionfrom the near-wall RANS layer to the off-wall LES region was achieved by switching thewall distance d in the destruction term in the νt-equation to the representative grid spacing∆DES in accordance with the formulation:

min (d, CDES∆DES) with ∆DES = max (∆x, ∆y, ∆z)

• HLR (Hybrid LES/RANS): a zonal (with a variable interface), two-layer hybrid approachcombining LES method in the outer layer and RANS method in the near-wall layer

In the latter method, the low-Reynolds number k − ε model due to Launder and Sharma (1974)was applied in the near-wall region. The subgrid-scale model due to Smagorinsky is used in thecore flow. The model coupling is realised via the turbulent viscosity, representing an approachwhich enables the solution obtained by using one system of equations. Depending on the flowzone, the turbulent viscosity is either computed from the RANS formulation:

µt = Cµ fµk2/ǫ (1)

or from the Smagorinsky model:µt = C2

s∆2∣

∣S∣

∣ (2)

where the Smagorinsky constant Cs takes the value of 0.1, ∆ represents the filter width and Sthe strain rate. The interface values for k and ǫ representing actually the boundary conditionsfor the corresponding equations in the RANS region are obtained by estimating the subgrid-scalekinetic energy and dissipation:

kSGS =νt|S|

0.3=

C2s ∆2|S|2

0.3, εSGS = νt|S|

2 = C2

s ∆2|S|3 (3)

Such a procedure provides the continuity in k and ǫ profiles across the interface. Consequently,a fairly smooth transition of the turbulent viscosity µt is ensured in accordance to the equations(1), (2) and (3).The decision whether the viscosity is to be computed by RANS or LES formulations depends onthe position of the interface between the two domains. The interface can either be positioned ata certain y+ or at a certain wall distance (Fig. 2). In both cases this value can be fixed or varyaccording to the control parameter

k∗ =

⟨

kmod

kmod + kres

⟩

(4)

which represents the ratio (fraction) of modelled to the total turbulent kinetic energy in the LESregion, averaged over all grid cells at the interface belonging to the LES domain. As soon asthis value exceeds about 20 % the interface is moved farther from the wall. In contrast, theinterface will be moved towards the wall in the case of values below 20. More details about theHLR method are given in Jakirlic et al. (2006) and Kniesner et al. (2007).

Numerical Method. The computational results presented were obtained by using the in-housecode FASTEST-3D (Flow Analysis by Solving Transport Equations Simulating Turbulence) whichuses a finite volume method for block-structured, body-fitted, non-orthogonal, hexahedral meshes,

Case 13.2-5

3

[3]. Block interfaces are treated in a conservative manner, consistent with the treatment of in-ner cell-faces. Cell centered (collocated) variable arrangement and cartesian vector and tensorcomponents are used. The equations are linearised and solved sequentially using an iterativeILU method. The velocity-pressure coupling is ensured by the pressure-correction method basedon the SIMPLE algorithm which is embedded in a geometric multi-grid scheme with standardrestriction and prolongation, Briggs et al. (2000). The code is parallelized applying the MessagePassing Interface (MPI) technique for communication between the processors. The convectivetransport of all variables was discretized by a second-order, central differencing scheme for LESand DES. In the case of the HLR method for k− and ε− equation some upwinding is used byapplying the so called ”flux blending” technique. Time discretization was accomplished applyingthe (implicit) Crank-Nicolson scheme.

LES. The solution domain comprising a part of the development duct (5h), the diffuser section(15h), the outlet straight duct (12.5h) as well as the convergent part (≈ 9h) is meshed withalmost 4 Mio. grid cells. Two simulations with and without SGS model have been performed.The simulation with a finer grid with twice as many control volumes (CV) in each grid direction,and therefore almost 32 million CV (including the periodic inlet duct for the inflow generation),is in progress. The wall boundary layers are resolved with y+ values of approximately O(1).According to the experimental setup of Cherry et al. [?] the fully developed turbulent channelflow has been computed with respect to the inflow generation. These inlet data are generatedby a simultaneously running periodic channel flow simulation of a channel with the same crosssection as the diffusor inlet. To allow the flow through the diffusor to influence the flowfield inthe development channel, a part (5 channel hights) of this channel has been modelled in front ofthe diffusor. The turbulent flow fields in a cross section in the periodic channel are copied to thisinlet location. Fig. 2 displays a slice through the computational domain and the periodic channelwhere the contours show the unsteady streamwise velocity component u. The convergent partbehind the diffusor (Fig. 1) was taken into account in order to avoid the use of convective exitboundary conditions. The time step size chosen corresponds to the Courant number being smallerthan one. This leads to a time step on the coarser grid of 1.1×10−4 s. Averaging was perfomedfor approximately 80000 time steps. The simulations were carried out on 16 IBM Power 5 CPUsand a load ballancing efficiency of 100%. This leads to a computational time per time step ofapproximately 14s.

Figure 2: On generation of turbulent inlet data utilizing a simulataneously running periodic chan-nel simulation. Contours show the streamwise velocity component U .

DES and HLR. The inflow data were generated by a precursor simulation of the fully developedduct flow using the respective models. The solution domain for both simulations DES and HLRcomprised a part of the development duct (3h), the diffuser section (15h) and the straight outletduct (12.5h). At its outlet cross-section the convective outflow conditions were applied. No-slipboundary conditions were applied at the walls. The grid applied in both simulations contained224x62x134 cells (approximatelly 1.86 Mio. grid cells in total). The dimensionless time step∆t = 0.028 (normalized by the inlet channel parameters Ubulk = 1m/snd h = 1cm) was used

Case 13.2-6

4

in the computations providing the CFL number less than unity throughout the solution domain(CFLmax ≈ 0.76). The details about the near-wall resolution can be gathered from Fig. 3displaying the wall-adjacent cell size in wall units along the lower and upper walls within theentire flow domain considered.

0

20

40

60

80

100

120

-0.5 0 0.5 1 1.5 2

∆x

+,∆

z+

x/L

∆x+ lower wall

∆z+ lower wall

∆x+ upper-wall

∆z+ upper wall

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1 1.5 2

∆y

1+

x/L

∆y1+ lower wall

∆y1+ upper wall

0

20

40

60

80

100

-0.5 0 0.5 1 1.5 2

∆x

+,∆

z+

x/L

∆x+ lower wall

∆z+ lower wall

∆x+ upper-wall

∆z+ upper wall

0

0.2

0.4

0.6

0.8

1

-0.5 0 0.5 1 1.5 2

∆y

1+

x/L

∆y1+ lower wall

∆y1+ upper wall

Figure 3: Wall-adjacent cell size in wall units along both walls corresponding to the HLR (upper)and DES (lower) simulations (L = 15h - diffuser length)

The final position of the interface separating the near-wall (RANS) and off-wall (LES) regionsdetermined in accordance with the criterion explained earlier (Eq. 4) corresponded to y+ ≈ 50 inthe HLR simulation. The development of the interface position in the DES framework expressedin dimensionless wall distance along both walls is depicted in Fig. 4

0

20

40

60

80

100

-0.5 0 0.5 1 1.5 2

yin

t+

x/L

y+

int lower wally

+int upper wall

Figure 4: RANS-LES interface in DES simulation (L = 15h - diffuser length)

Case 13.2-7

5

ResultsSelected variable profiles at 14 streamwise locations in all characteristic flow regions in two ver-tical planes correspodning to z/B = 1/2 (B = 3.33cm) and z/B = 7/8 as well as the pressurecoefficient distribution and the contour plots of the axial velocity and streamwise stress compo-nents at five streamwise cross-sectional areas are shown in the following figures.

References

1. Briggs, W.L., Van Emden Henson and McCormick, S.F. (2000): A Multigrid Tutorial.SIAM 2000

2. Cherry, E.M., Elkins, C.J. and Eaton, J.K. (2008): Geometric sensitivity of three-dimensionalseparated flows. Int. J. of Heat and Fluid Flow, Vol. 29, pp. 803-811

3. FASTEST-Manual (2005): Chair of Numerical Methods in Mechanical Engineering, De-partment of Mechanical Engineering, Technische Universitat Darmstadt, Germany 2005

4. Germano, M., Piomelli, U., Moin, P. and Cabot,W.H. (1991): A dynamic subgrid-scaleeddy viscosity model Phys. of Fluids A, Vol. 3(7), pp. 1760-1765

5. Launder, B.E., and Sharma, B.I. (1974): Application of the Energy-Dissipation Model ofTurbulence to the Calculation of Flow Near a Spinning Disc. Letters in Heat and MassTransfer, Vol. 1, pp. 131-138

6. Jakirlic, S., Kniesner, B., Saric, S. and Hanjalic, K. (2006): Merging near-wall RANS modelswith LES for separating and reattaching flows. ASME Joint U.S.-European Fluids Engi-neering Summer Meeting: Symposium on DNS, LES and Hybrid RANS/LES Techniques,Miami, FL, USA, July, 17-20, Paper No. FEDSM2006-98039

7. Kniesner, B., Saric, S., Mehdizadeh, A., Jakirlic, S., Hanjalic K., Tropea, C., Sternel, D.,Gauß, F. and Schafer, M. (2007): Wall treatment in LES by RANS: method developmentand application to aerodynamic flows and swirl combustors. ERCOFTAC Bulletin, No.72, pp. 33-40

8. Spalart, P.R., and Allmaras, S.R. (1994): A one-equation turbulence model for aerodynamicflows. La Recherche Aerospatiale, No. 1, pp 5-21

9. Travin, A., Shur, M., Strelets, M. Spalart, P.R. (2002): Physical and numerical upgradesin the detached-eddy simulation of complex turbulent flow. In: Fluid Mechanics and itsApplication, Friedrich, R. and Rodi, W. (Eds.), Vol. 65, pp. 239254.

Case 13.2-8

pressure coefficient Cp

lower wall, central plane (z/B=1/2)

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

cp

x/L

EXP

LES

HLR

DES

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5

-2 0 2 4 6 8 10 12 14 15.5 17 18.5 20 21.5

y/h

U/Ubulk

x/h

Cherry, Eaton and Elkins (2008)

Rb=10000, Ubulk=1 m/s

z/B=1/2

0

0.5

1

1.5

2

2.5

3

3.5

4

0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10

-2 0 2 4 6 8 10 12 14 15.5 17 18.5 20 21.5

y/h

urms/Ubulk

x/h



z/B=1/2

1Case 13.2-9

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5

-2 0 2 4 6 8 10 12 14 15.5 17 18.5 20 21.5

y/h

U/Ubulk

x/h



z/B=7/8

0

0.5

1

1.5

2

2.5

3

3.5

4

0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10

-2 0 2 4 6 8 10 12 14 15.5 17 18.5 20 21.5

y/h

urms/Ubulk

x/h



z/B=7/8

2Case 13.2-10

Contours of streamwise velocity at cross-sections x/H = 2, 5, 8, 12 and 15

Experiment

DES (1.8 Mio CV’s)

HLR (1.8 Mio CV’s)

LES (4 Mio CV’s)

3Case 13.2-11

Contours of urms/Ub ∗ 100 at cross-sections x/H = 2, 5, 8, 12 and 15

Experiment

DES (1.8 Mio CV’s)

HLR (1.8 Mio CV’s)

LES (4 Mio CV’s)

4Case 13.2-12

A Hybrid LES–URANS Approach based on an Explicit

Algebraic Reynolds Stress Model

Michael Breuer

Lehrstuhl fur Stromungsmechanik, Universitat Erlangen–Nurnberg,

Cauerstr. 4, D–91058 Erlangen, Germany

[email protected]

In the present study a hybrid LES–URANS approach based on an explicit algebraic Reynolds stressmodel (EARSM) designed by Wallin and Johansson [10] for pure RANS applications is suggestedand evaluated. The model is applied in the RANS mode with the aim of accounting for the Reynoldsstress anisotropy emerging especially in the near–wall region.

The hybrid approach relies on a unique modeling concept and a dynamically adapted interface. Inboth modes a transport equation for the turbulent kinetic energy is solved, either for the trace ofthe modeled stresses (kmod) in the viscosity–affected near–wall RANS region or for the subgrid–scale(SGS) contributions (ksgs) in the LES region. That represents the basis for determining the velocityscale. The closure is completed by algebraic relations for the length scale in the near–wall RANSregion and the assumption that the length scale in LES is naturally given by the filter width. Inprevious studies [1, 5] the one–equation SGS model of Schumann [9] in the outer LES region wascombined with the linear eddy–viscosity one–equation model suggested by Rodi et al. [8] for the

viscosity–affected near–wall RANS layer. Since the wall–normal velocity fluctuations (v′2)1/2 arebetter suited to characterize the near–wall turbulent motion than kmod [8], they are used as velocityscale in the RANS model. Introducing an algebraic equation relating the wall–normal velocityfluctuations to kmod [8] assures that the transport equation does not have to be modified.

The linear eddy–viscosity model (LEVM) has shown several deficits [1, 5]. Thus, in order to moreappropriately account for the near–wall anisotropy, an explicit algebraic Reynolds stress model iscoupled to the one–equation model in the RANS mode. For the implementation into the CFD codethe anisotropy–resolving closure can be formally expressed in terms of a non–linear eddy–viscositymodel (NLEVM). The EARSM of Wallin and Johansson [10] was chosen because of its near–walltreatment ensuring realizability of the individual stresses. Furthermore, its extra computationaleffort is small still requiring solely the solution of one transport equation for the turbulent kineticenergy. The EARSM provides an algebraic relation for the Reynolds stress tensor [10], which canbe introduced in the momentum equation as

u′

iu′

jmod= kmod

(

2

3δij − 2Ceff

µ Sij + a(ex)ij

)

.

Here a(ex)ij represents an extra anisotropy tensor which is computed explicitly at low computational

costs based on the normalized mean strain and rotation tensors. In the EARSM [10] the value of Cµ

within the relation for νt is not a constant but dynamically calculated, thus denoted Ceffµ . However,

the EARSM itself is not complete as the different scales are not defined. These are supplied by anadditional scale–determining part, i.e., the model by Rodi et al. [8]. Hence in this formulation thek–equation is still needed but the additional term takes the anisotropy of the stresses appropriatelyinto account. Additionally, the enhanced representation of the Reynolds stresses can be introducedinto the turbulent diffusion term in the k–equation. Thus, in addition to the classical gradient–diffusion model the diffusion model of Daly and Harlow [4] is implemented. A further consequence

Case 13.2-13

of applying an EARSM is on the production term in the k–equation, which can now be calculatedon the basis of the more consistent Reynolds stress formulation including the anisotropy term.Hence, the production and diffusion terms and subsequently kmod are improved. In the presentformulation the predefinition of RANS and LES regions is avoided and a gradual transition betweenboth methods is assured. A dynamic interface criterion is suggested which relies on the modeled

turbulent kinetic energy and the wall distance (y∗ = k1/2mod ·y/ν ≤ Cswitch,y∗) and thus automatically

accounts for the characteristic properties of the flow. An enhanced version guaranteeing a sharpinterface without RANS islands was also taken into account. The interface behavior was thoroughlyinvestigated and it was shown that the method automatically reacts on dynamic variations of theflow field.Both model variants, i.e. LEVM and EARSM, have been tested on the basis of the standard planechannel flow at Reτ = 595 and 2003 and even more detailed on the flow over a periodic arrangementof hills at Reb = 10, 595 [2, 6]. The superiority of the hybrid approach based on EARSM over thelinear eddy–viscosity version could be demonstrated.For the 13th ERCOFTAC–SIG15 Workshop on Refined Turbulence Modeling in Graz (Sept. 25–26, 2008) the 3D diffuser test case was predicted based on this hybrid LES–URANS approach.Additionally, a pure LES prediction was carried out.

References

[1] Breuer, M., Jaffrezic, B., Arora, K. (2008) Hybrid LES–RANS Technique Based on a One–

Equation Near–Wall Model, J. Theoretical and Computational Fluid Dynamics, 22(3–4), pp.157–187, (2008).

[2] Breuer, M., Aybay, O., Jaffrezic, B. (2008) Application of an Anisotropy Resolving Algebraic

Reynolds Stress Model within a Hybrid LES–RANS Method, Seventh Int. ERCOFTAC Work-shop on DNS and LES: DLES-7, Trieste, Italy, Sept. 8–10, 2008.

[3] Chen, H.C., Patel, V.C. (1988). Near–Wall Turbulence Models for Complex Flows Including

Separation, AIAA Journal, 26(6), 641–648.

[4] Daly, B.J., Harlow, F.H. (1970) Transport Equations in Turbulence, Phys. Fluids, 13, 2634–2649.

[5] Jaffrezic, B., Breuer, M., Chikhaoui, O., Deng, G., Visonneau, M. (2007) Towards LES–RANS–

Coupling for Complex Flows with Separation, ESAIM Proc., CEMRACS 2005, CAA and CFDin Turbulent Flows, Marseille, France, July 18–Aug. 26, 2005, 16, 89–113.

[6] Jaffrezic, B., Breuer, M. (2008) Application of an Explicit Algebraic Reynolds Stress Model

within an Hybrid LES–RANS Method, J. of Flow, Turbulence and Combustion, in press.

[7] Lumley, J.L., Newman, G. (1977) The Return to Isotropy of Homogeneous Turbulence, J. FluidMech., 82, 161–178.

[8] Rodi, W., Mansour, N.N., Michelassi, V. (1993) One–Equation Near–Wall Turbulence Modeling

with the Aid of Direct Simulation Data, J. Fluids Eng., 115, 196–205.

[9] Schumann, U. (1975) Subgrid–Scale Model for Finite–Difference Simulations of Turbulent Flows

in Plane Channels and Annuli, J. Computat. Physics, 18, pp. 376–404.

[10] Wallin, S., Johansson, A.V. (2000) An Explicit Algebraic Reynolds Stress Model for Incom-

pressible and Compressible Turbulent Flows, J. Fluid Mech., 403, 89–132.

Case 13.2-14

13th ERCOFTAC-SIG15 Workshop: 3D Diffuser

H. Schneidera∗, D.A. von Terzia and W. Rodib

aInstitut fur Thermische Stromungsmaschinen, Universitat Karlsruhe (TH), 76128 Karlsruhe, GermanybInstitut fur Hydromechanik, Universitat Karlsruhe (TH), 76128 Karlsruhe, Germany

Abstract

LES with the standard Smagorinsky model employing wall-functions and wall-resolvingRANS calculations with the standard k-ω model of Wilcox and the Spalart–Allmaras modelwere used to compute the flow in asymmetric three-dimensional diffusers (test case 13.2).Using the same number of computational cells, the LES delivered very reasonable resultsin good agreement with experiments for both diffusers. In contrast, the RANS calculationsseverely over-predicted the extent of the recirculation zone and yielded substantially lessaccurate results.

Introduction

Reliable predictions of three-dimensional separation and reattachment can play an important rolein the design process of many engineering devices, in particular for asymmetric diffusers as theyoccur in turbomachinery. For such flows, slight changes in geometry or operating parameters canlead to a drastic alteration of the flow field and, therefore, strongly impact on aerodynamic lossesand overall performance of the device. For practical purposes, available simulation techniques in-clude Reynolds-Averaged Navier–Stokes (RANS) calculations and Large-Eddy Simulations (LES).The first is a well established tool in industry whereas the latter is in the process of being trans-ferred from academia. How well these distinct methods perform is scrutinized in the followingby applying them to test case 13.2 ”Flow in a 3-D diffuser” (see http://130.83.243.201/ercoftac-sig15/workshop2008.html for details).

Numerical Method

All simulations were performed with the Finite Volume flow solver LESOCC2 (Large Eddy Sim-ulation On Curvilinear Coordinates) developed at the University of Karlsruhe. This FORTRAN95 program solves the incompressible, three-dimensional, time-dependent, filtered or Reynolds-Averaged Navier–Stokes equations on body-fitted, collocated, curvilinear, block-structured grids.The viscous fluxes are always discretized with second-order accurate central differences whereasthe convective fluxes are approximated either with the same method for LES or a monotonicsecond-order upwind scheme for RANS calculations. Time advancement is accomplished by ei-ther an explicit, low-storage Runge–Kutta or a second-order accurate implicit method for LESand RANS, respectively. Conservation of mass is achieved by the SIMPLE algorithm with thepressure–correction equation being solved using the strongly implicit procedure (SIP) of Stone.The momentum interpolation method of Rhie and Chow is employed to prevent pressure–velocitydecoupling and associated oscillations. Parallelization is achieved via a domain decompositiontechnique with the use of ghost cells and MPI for the data transfer.

∗Email: [email protected], phone: +49 721 608-4703

1

Case 13.2-15

Computational Setup

The Reynolds number based on the bulk velocity and the height of the inlet channel was 10,000for both diffusers. The origin of the Cartesian coordinate system was placed at the intersectionof the two non-expanding walls and the beginning of the expansion. All values reported are madedimensionless using the inlet channel height H = 1 cm and bulk velocity U = 1 m/s as referencevalues. Simulations were carried out on two grids (G1 and G2) each consisting of roughly 1.6million cells, covering the solution domain x ∈ [−5; 23] with 448 × 60 × 60 cells in the stream-wise, vertical and lateral directions, respectively. Both grids feature equidistant grid spacing inthe streamwise direction. While one grid (G1) is equidistant for use with wall-functions, the sec-ond grid (G2) is stretched in the two wall-normal directions in order to allow for wall-resolvingsimulations. For both grids, the rounded corner at the inlet to the diffusers was replaced by asharp corner. The data presented was obtained using 112 processors on a HP Linux cluster ofthe University of Karlsruhe. Different boundary conditions, turbulence models and computationalgrids were used for the LES and RANS calculations. These are compiled in tables 1 and 2, andbriefly discussed in the following.

Boundary conditions DiscretizationInflow Outflow Temporal Spatial

LESunsteady, convective with explicit three-step second-orderturbulent buffer zone Runge–Kutta central differences

RANSuniform velocity homogeneous implicit second-order second-order

profile Neumann (reached steady-state) central/upwind

Table 1: Boundary conditions and discretization schemes.

Grid typeDiffuser Grid spacing

CPUhoursD1 D2 ∆x ∆y ∆z

G1equidistant with

LES LES 0.0625 0.0167 0.0555 15,000wall function

G2stretched with SA

- 0.06250.00167– 0.00167– 2,800

no-slip walls k − ω 0.06 0.26 1,500

Table 2: Turbulence modeling, computational grids with spacing in the inlet plane (x = 0), andcomputational cost.

LES was performed for both test diffusers (D1 and D2) using grid G1. The standard Smagorin-sky model with Cs = 0.065 and van Driest damping served as subgrid-scale model. At walls, anadaptive wall-function was used and at the outlet a convective boundary condition is enforced inconjunction with a buffer zone (x ∈ [22; 23]) in which the total viscosity is increased by a factorof 100. Unsteady turbulent inflow data was generated by enforcing periodicity in the inlet sectionwithin x ∈ [−5;−2] and using a controller to enforce the experimental mass flux. This is essen-tially a fully-developed channel flow simulation providing time-dependent realistic flow structuresfor the diffuser at x = −2. Adaptive time-stepping ensured a convective CFL limit of less than0.65 (with ∆t ≈ 0.01). In total 600,000 time steps were computed. Averaging started after roughly150 H/U resulting in an averaging time of more than 5000 H/U, i.e. more than 200 flow-throughtimes. Differences in the mean velocity profiles of the final results compared to those obtainedafter 100 flow-through times were minute.

For RANS simulations, the one-equation Spalart–Allmaras (SA) model and the standard two-equation k-ω model of Wilcox were used as turbulence closures. The simulations were wall-resolving on the stretched grid (G2) and only applied to diffuser D1. Boundary conditions were auniform inlet velocity profile, no-slip walls, and a homogeneous Neumann condition at the outlet.Using the implicit time–scheme, the time step was set to 0.01 and the simulations were converged

2

Case 13.2-16

to steady-state, i.e. 20,000 and 30,000 time steps were computed for the k-ω and SA models,respectively.

Results

z

y

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.5

1

1.5

2

2.5

3

3.5

4

z

y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0

0.5

1

1.5

2

2.5

3

3.5

z

y

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.5

1

1.5

2

2.5

3

3.5

4

z

y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0

0.5

1

1.5

2

2.5

3

3.5

Figure 1: Mean streamwise velocity contours of experiments (top) and LES (bottom) in a cross-section at x = 12 for diffusers D1 (left) and D2 (right); same velocity contours shown for allplots.

In Fig. 1 mean streamwise velocity contours in a cross-section at x = 12 are plotted fromexperimental data (top plots) and the LES results (bottom plots) for both diffusers, D1 and D2on the left and right hand side, respectively. The experimental data show that due to the changein the outlet geometry, the separation bubble moves from a horizontal alignment at the top wallto a predominantly vertical orientation on the right side. The LES data exhibit the same trendand good agreement with the region of maximum forward flow (note that the same contour levelsare shown). The zero contour line is highlighted and considering the large uncertainty in theexperiments for this quantity and the coarse grid and use of wall-functions in the simulations,the agreement between both approaches is rather quite remarkable. This is in contrast to theRANS results obtained for diffuser D1 as shown in Fig. 2 (again using the same contour levels).The k-ω model shifts the separation bubble to the right, hence the change in separation lines dueto geometric changes in such diffusers is unlikely to be captured. Although the SA model yields

3

Case 13.2-17

an almost horizontal separation bubble at the upper wall, the amount of reverse flow is vastlyoverestimated in size of the separation region and also in magnitude. Due to mass conservationthis leads also to higher forward velocities and the location of maximum velocity is shifted towardsthe lower wall. As a consequence, in mean streamwise velocity profiles (Fig. 3), the k-ω modeldiffers most from the experimental data inside the separated region whereas the SA model deviatesmore in the attached flow near the lower wall. However, both models differ already within therectangular inlet channel from the experiments, such that the inflow into the diffuser may at leastpartly be blamed for the failure of the RANS calculations.

z

y

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.5

1

1.5

2

2.5

3

3.5

4

z

y

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 2: Mean streamwise velocity contours of RANS calculations with k-ω (left) and Spalart–Allmaras (right) model in a cross-section at x = 12 for diffuser D1; same velocity contours areplotted as in the experimental data shown above.

U-velocity-mean

y

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

EXP_X0_Z2

SA_CS_X0_Z2

KO_CS_X0_Z2

U-velocity-mean

y

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

EXP_X6_Z2

SA_CS_X6_Z2

KO_CS_X6_Z2

U-velocity-mean

y

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

2.5

3

EXP_X12_Z2

SA_CS_X12_Z2

KO_CS_X12_Z2

Figure 3: Comparison of mean streamwise velocity profiles in diffuser D1 for RANS calculationswith the SA and k-ω models and experiments at z = 2; from left to right: x = 0, 6 and 12.

The failure of the SA model becomes more evident when looking at the mean wall-pressurecoefficient in the center plane of diffuser D1 at the lower wall (see Fig. 4). Again, the LES isin very reasonable agreement with the experimental data until the effects of the buffer domainbecome noticeable inside the outlet duct (x/L > 1.3). For the RANS predictions, the k-ω modeldelivers a qualitatively correct picture, but the SA model yields a too low pressure recovery whichis due to the larger separated flow region and the corresponding higher forward velocities in theattached flow.

4

Case 13.2-18

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4x/L

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Cp

Experiments (shifted)

LESRANS (k-ω)

RANS (Spalart-Allmaras)

Figure 4: Comparison of mean wall–pressure coefficient Cp = 2(p − pref )/(ρU2) in the center ofthe inlet plane along the lower wall of diffuser D1; x is normalized with the diffuser length L; pref

is taken at x = y = 0 and experimental data is shifted accordingly.

−5 0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

x/H

Area Fraction Separated

Experiment

LES

−5 0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

x/H

Area Fraction Separated

Experiment

LES

Figure 5: Fraction of cross-sectional area separated in experiments and LES for diffusers D1 (left)and D2 (right); RANS data of diffuser D1 deviates substantially (not shown).

As a figure of merit for assessing the quality of the LES serves Fig. 5. In this plot the fractionof cross-sectional area separated in experiments and LES is shown for both diffusers. The charac-teristics of diffuser D1 (left) are matched by the LES and deviations reside within experimentaluncertainties beyond the location of the maximum in this plot. Farther downstream a stronger de-viation is visible which is likely due to the presence of the outflow boundary and the buffer domain(x > 22) in the simulation. A longer computational domain may alleviate this deficiency. On theother hand, exchanging the round corner at the diffuser inlet by a sharp one had no detrimentaleffects (a tiny separation bubble is formed which is not visible in Fig. 5). The same holds by andlarge also for diffuser D2, except for a more significant deviation from the experiments that canbe seen in the center region of the plot. These differences are rather surprising since the meanstreamwise velocity profiles of the LES and experiments are in good agreement as is indicatedin Fig. 6 and 7. Fig. 7 shows profiles at the streamwise location where the largest deviation offraction of cross-sectional area separated occurs.

5

Case 13.2-19

U-velocity-mean

y

0 0.5 10

0.2

0.4

0.6

0.8

1

EXP_X-2_Z2

LES_CE_X-2_Z2

U-velocity-mean

y

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

EXP_X2_Z2

LES_CE_X2_Z2

U-velocity-mean

y

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

EXP_X6_Z2

LES_CE_X6_Z2

U-velocity-mean

y

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

EXP_X10_Z2

LES_CE_X10_Z2

U-velocity-mean

y

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

EXP_X14_Z2

LES_CE_X14_Z2

U-velocity-mean

y

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

EXP_X18_Z2

LES_CE_X18_Z2

Figure 6: Comparison of mean streamwise velocity profiles in diffuser D2 for LES and experimentsat z = 2; from left to right: x = −2, 2 and 6 (top row), 10, 14 and 18 (bottom row).

U-velocity-mean

y

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

EXP_X10_Z1

LES_CE_X10_Z1

U-velocity-mean

y

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

EXP_X10_Z2

LES_CE_X10_Z2

U-velocity-mean

y

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

EXP_X10_Z3

LES_CE_X10_Z3

Figure 7: Comparison of mean streamwise velocity profiles in diffuser D2 for LES and experimentsat x = 10; from left to right: z = 1, 2 and 3.

Conclusion

LES on a coarse equidistant grid using the standard Smagorinsky model and wall-functions isable to predict the three-dimensionally separated flow in asymmetric diffusers with fair accuracyat reasonable costs. On the other hand, both the k-ω and Spalart–Allmaras RANS models fallshort. It seems that the additional computer time required for the LES is well spent consideringthe higher reliability and accuracy of the results, in particular, since only the LES was able topredict the impact of the geometric changes on the flow field correctly.

6

Case 13.2-20

13th Ercoftac Workshop on Refined

Turbulence Modelling. Case 13.2: 3D

Diffuser

F. Billard1, J.C. Uribe1, D. Laurence 1,2

1School of Mechanical, Aerospace and Civil Engineering, the University ofManchester, Manchester, M16 1QD, UK - [email protected]

2EDF-Electricité de France, M.F.E.E. dpt., 6 quai Watier, 78400 Chatou,FRANCE

1 Introduction

The case 13.2 was computed using the Code_Saturne, developed at EDF (Ar-chambeau et al. (2004)). The code uses finite volume discretization and canhandle both structured and unstructured grids. Spatial discretization is basedon collocated cell centered storage and the time advancement uses a Rhie andChow filter on the projection step of the pressure.

2 Turbulence models

Four RANS models have been used in this case. The eddy viscosity modelsused are the k−ω SST of Menter (1994) (named SST hereafter) and two code-friendly versions of the v2

− f model: the ϕ − f model of Laurence et al.

(2004) (PHIFB) and the ϕ − α model of Billard et al. (2008) (PHIAL). Thesetwo versions are adaptations of Durbin’s formulation (1991), using the reducedvariable ϕ = v2/k. The ϕ− α uses the elliptic blending method introduced byManceau & Hanjalic (2002). Finally, the Reynolds Stress Model of Speziale et

al. (1991) (SSG) is also tested on this case.

3 Mesh

The results presented come from computations carried out on a block structuredmesh with (1089000) 242×50×90 control volumes over a domain which extentsfrom 4 units lenght before to 40 units length beyond the start of the diffuserexpension (up to 55 unit length for the SSG). The choice of the grid resolution

1

Case 13.2-21

4 Boundary conditions 2

Fig. 1: Close up of the mesh, refinement involved in the top-wall (top) and side-wall (bottom) expension

results from a grid refinement dependence study in the wall-normal directions.A finer mesh of 2230272 cells (242 × 96 × 96) was used for preliminary trialsfor that purpose. All the eddy viscosity models used are devised to resolve theviscous affected near-wall region, in all the computations and everywhere in thedomain, the distance of the first cell centre from the wall is below 1. The meshis however too fine near the wal for the SSG model to be used in its standardversion, thus requiring special near-wall treatment for this latter model.

4 Boundary conditions

For all the variables, except the pressure, dirichlet condition were given at inlet.The boundary values were obtained from a precusor computation of a periodicsquare duct, where a streamwise pressure gradient was imposed in order to reachthe same target mass flow rate as in the experiment. The outlet boundary wastreated with zero pressure gradient conditions. For the solid boundaries, all theeddy viscosity models have no-slip condition, and the SSG model uses scalablewall functions (Grotjans & Menter (1998)).

5 Numerical method

The code is collocated. The velocity and pressure fields are coupled by a predic-tion/correction method with a SIMPLEC algorithm. The conjugated gradientmethod is used to solve the Poisson equation for the pressure and the ellip-tic turbulent variable f or α whenever needed. An upwind first order schemeis used for the discretization of all the turbulent variables and a second ordercentered scheme is used for the velocity components.

Case 13.2-22

6 Results 3

Fig. 2: Streamwise velocity contours at location X=2cm, 8cm and 15cm, exper-imental (top) and ϕ− f (bottom)

6 Results

With no exception, all the RANS models tested predict the reciculation tooccur along the inclined side wall, whereas the experimental results show itappears along the inclined top wall. The figure 2 shows the streamwise velocitycontours at three different X locations in the diffuser, with increments of 0.05units velocity (the line corresponding to zero velocity is thicker). Only thePHIFB model is presented here along with the experimental results, but thecontours of all models are fairly similar. In both the experimental and thesimulation, the recirculation start in the upper right corner, but expends onthe top wall to become nearly 2 dimensionnal in the experiments, whereas itpropagates towards the top wall and the right bottom in all the simulations.However the extent of the near top wall region occupied by the recirculation dodepend on the model. Figure 3 shows streamwise velocity profiles at 14 differentX locations (scales by a factor of 2), all included in the mid-span plane. TheSST predicts the recirculation to reach far to earlier the mid-span top wall,whereas the recirculation predicted by the PHIAL never reaches the mid-spanof the top wall. However, in all the simulations, the separations location isnever before X=5cm (the earliest being for the SST model), whereas separationoccuring almost immediately at the diffuser inlet were reported for both RANSand LES calculations of Cherry et al. (2006), again, at mid-span locations. As

Case 13.2-23

6 Results 4

0 5 10 15x/H, 2 U/U

bulk

0

1

2

3

4

y/H

Exp.

v2-f (! - f )

v2-f (!!"!#"

k-$! SST

Rij SSG

15 20x/H, 2 U/U

bulk

0

1

2

3

4

y/H

Fig. 3: Velocity profiles for different midspan locations near the start (top) andthe end (bottom) of the diffuser.

for the PHIFB and the SSG models, predictions seem to be of a slightly betteragreement with the experiments. As for the bottom wall, all model predict aboundary layer thinner that the one observed in the experiments.

It is worth noting that the SSG model is the only one tested able to reproducethe secondary motion in the inlet square duct.

References

[1] Archambeau, F., Mechitoua, N. and Sakiz, M. (2004), A Finite VolumeCode for the Computation of Turbulent Incompressible Flows, IndustrialApplications, International Journal on Finite Volumes, Vol. 1.

[2] Durbin, P.A. (1991), Near-wall turbulence closure modelling without damp-

Case 13.2-24

6 Results 5

ing functions, Theoretical and Computational Fluid Dynamics, Vol.3, pp.1-13

[3] Billard, F., Uribe, J.C. and Laurence, D.R., (2008) A new formulation ofthe v2

−f model using elliptic blending and its applications to heat transferprediction, In. Proc. 7th Int. Symp. Engineering Turbulence Modelling andMeasurements, Limassol, Cyprus.

[4] Cherry, E.M., Iaccarino, G., Elkins, C.J. and Eaton, J.K., (2006) Sepa-rated flow in a three-dimensional diffuser: preliminary validation. Annual

Research Briefs, Center for Turbulence Research, Stanford Univ.

[5] Grotjans, H. and Menter, F. (1998), Wall functions for general applicationCFD codes. in Papailou et al., editor, ECCOMAS 98, pages 1112-1117.

[6] Laurence, D.R., Uribe, J.C. and Utyuzhnikov, S.V. (2004), A Robust For-mulation of the v2

− f model, Flow, Turbulence and Combustion, Vol.73,pp. 169-185

[7] Lien, F.S. and Durbin, P.A. (1996), Non-linear k − ε − v2 modelling withapplication to high-lift, In Proc. of the summer school program, Center for

Turbulence Research, Stanford Univ., pp.5-25

[8] Manceau, R. and Hanjalic, K. (2002), Elliptic blending model: A new near-wall Reynolds Stress Turbulence Closure, Physics of Fluids, Vol.14(2), pp.744-754

[9] Menter, F.R. (1994), Two-Equation Eddy-Viscosity Turbulence Models forEngineering Applications, AIAA Journal, pp. 1598-1605

[10] Speziale, C.G., Sarkar, S. and Gatski, T.B. (1991), Modeling the pressure-strain correlation of turbulence: an invariant dynamical system approach,Journal of Fluid Mechanics, Vol. 227, pp. 245-272

Case 13.2-25

13th ERCOFTAC/IAHR Workshop on Refined Turbulence Modelling

Institute of Fluid Mechanics and Heat Transfer, Graz University of Technology, Austria, September 25-26, 2008

APPLICATION OF AN ANALYTICAL WALL-FUNCTION TO A 3D DIFFUSER FLOW

(Case 13.2: Description of the Computations)

K. Suga, S. Nishiguchi

Department of Mechanical Engineering, Osaka Prefecture University,

1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan

e-mail: [email protected]

OverviewThe AWF (analytical wall function) originally proposed by Craft et al. (2002) is slightly modified and applied

to computations of a 3D diffuser flow (Case 13.2, Diffuser 1: Cherry et al., 2008) by the nonlinear eddy viscosity

model (Craft, Launder and Suga, 1996) and the TCL second moment closure (Craft and Launder, 2001). Although

the original form of the AWF does not contaminate flow field results, it sometimes leads to unphysical heat

transfer distribution near a corner of a 3D duct flow, particularly when it is coupled with a second moment closure.

The present study thus modifies the AWF form and evaluates its performance in a 3D square sectioned U-duct

flow as well as the test case flow. The results of Case 13.2 Diffuser 1 clearly indicate that the TCL model with the

present AWF performs reasonably well though the nonlinear eddy viscosity model only slightly improves poor

results of the standard model. k !

Analytical Wall-FunctionIn the AWF, the wall shear stress and scalar flux are obtained through the analytical solution of simplified

near-wall versions of the transport equations for the wall-parallel momentum and scalar. The main assumption

required for the analytical integration of the transport equations is a modelled variation of the turbulent viscosity

over a wall-adjacent computational-cell. For smooth wall flows, this is done using as the thickness of the

viscous sub-layer, and assuming that

"t vy

"t is zero for # vy y and then increases linearly:

, where , * *max{0, ( )}" $ %" t vy y * 1/ 2 /Py yk& ' % =c cµ, c =2.55 and cµ=0.09, and µ, ' , y, and kP are

respectively the molecular viscosity, the kinematic viscosity, the wall normal distance and the turbulence energy at

the node P. Then, with the assumption that the right hand side terms can be constant over the cell, the simplified

momentum and scalar equations in the wall adjacent cell:

( ) ( )2

* *, t

P

U PUU

y y k x

* +, , ' , ,x

* +" -" $ . -/ 0 / 0, , , ,1 21 2 (1)

( )2

* *,

Prt

P

U Sy y k x

3

* +, " ,4 ' ,5 6 *- 7 $ . 4 8 9/ 0+

/ 0, , ,: ; 11 2 2 (2)

can be easily integrated analytically to form the boundary conditions of the momentum and the scalar at the wall,

namely the wall shear stress and scalar flux. Note that the coordinate directions ,x y correspond to the

streamwise (wall parallel) and wall-normal directions, respectively. (See the original paper or Suga et al., 2006 for

the detailed treatments.)

Since the original forms of the AWF are obtained in 2D wall parallel flows, it is reasonable that 3D applications

of such a model require further discussions. When the 3D square sectioned U-bend duct flow (Fig.1) is considered,

the secondary flows near the duct corners are typically important and produce very different conditions from those

in the 2D wall parallel flows. In such a case, the streamwise direction differs from the wall parallel direction as

shown in Fig.2 and the gradient in the streamwise direction is:

x xx

< =

,> ,> ,>$ -

, ,< ,=, (3)

where x is the streamwise direction which is treated as the wall parallel direction in the original AWF. Due to

the large velocity gradient in the = direction, the convection terms in the equations (1) and (2) have peaky

distribution near the corners as shown in Fig.3. Thus, the predicted Nusselt number distribution has unphysical

peaks as in Fig.4. In order to eliminate those kinky profiles, the present study introduces a damping function S

f

1

Case 13.2-26

to the right hand side terms of equations (1) and (2) as:

( ) ( )

2

* *, t S

P

U PUU f

y y k x x

* +, , ' , ,*" -" $ . -/ 0 /, , , ,1 21 2

+0 (4)

( )

2

* *Prt S

P

U Sy y k x

3

* +, " ,4 ' ,5 6 * +- 7 $ . 4 8 9/ 0 f/ 0, , ,: ; 1 21 2. (5)

The form of S

f is

21exp( / ), ( )

2ij i jS S

kf S A S S h h$ $

!, (6)

where and the subscripts follow the wall coordinate. The vector is

defined as . The presently used coefficient is

/ij i j j iS U x U$ , , - , ,/ x ,i jih

(1,0,1)ih $

0.5S

A $ . This slight modification removes the

unphysical profiles in the Nusselt number distribution as shown in Fig.4.

Computational Methods and Results

The presently used turbulence models for the core flow regions are the standard model, the cubic

nonlinear model of Craft, Launder and Suga (1996) (CLS model) and the TCL second moment closure of

Craft and Launder (2001). The CLS model consists of the following model equation:

k !k !

23

1 13 31 2 3

2 2 234 5

2 2

6 7

( ) ( ) (

ij t ij iji j

t ik kj kl kl ij t ik kj jk ki t ik jk kl klij

t ki lj kj li kl t il lm mj il lm mj lm mn nl ij

t kl kl ij t kl kl

u u k S Hot

Hot c S S S S c S S c

c ( S S ) S c ( S S S )

c S S S c

? ?$ @ ' -

$ ' A @ - ' A B -B - ' A B B B B

- ' A - - ' A B - B B @

- ' A - ' A B B ,ijS

) (7)

which is the cubic stress-strain relation and / /ij i j j iU x U xB $ , , , , , /kA $ ! . The TCL model consists of

the cubic pressure-strain correlation of

( ) ( )C D

1 1 1 2

2

2 2

22

1'

3

10.6 0.3 0.2

3

3

7'

15 4

ij ij ik jk ij ij

j k i l jk l iij ij kk ij ij kk kl i k j k

l l

ij ij mi nj mn mn

c a c a a A A a

u u u u Uu u UP P a P S u u u u

k k x x

c A P D a a P D

Ac

E F5 6> $ ! - @ !G H8 9: ;I J

E F,5 6,K K5 6> $ @ - -G H8 98 9 , ,: ; : ;K KI J

-

5- :

2

2 2

1 1 10.1

3 2 3

20.05 0.1

3

10.1 6

3

ij ij kk ij ik kj ij kk

j mi m l mij kl kl jm im ij lm

j k i l l m k mij

P P a a a A P

u uu u u ua a P P P P

k k k

u u u u u u u u

k k

! "#$ # $ #% & ' ' %( )* + * +* +,- . - .. / 01

! "$ #2 2' & & ' %* +( )* +2 2- ./ 0

$ #& ' %* +* +

- .

3 4 3 4 213 0.2 ,

j k i l

kl kl kl kl

u u u uD kS D P

k

5& & ' 6

67

(8)

where 23 2/ , , ,

j i kij i j ij ij ij ij i k j k ij i k j k

k k j

U U Ua u u k A a a P u u u u D u u u u k

i

U

x x x x

8 8 8 89 ' % 9 9 ' ' 9 ' '

8 8 8

8and

A is Lumley’s flatness parameter. (One should refer to the original papers for further detailed model forms.)

The presently used computation code is the STREAM (Lien and Leschziner, 1994) and the third order MUSCL

type upwind scheme is used for convection terms. Fig.5 shows the computational grid used for the computations

of Case 13.2 Diffuser 1. Since the present computations use the AWF, a relatively coarse grid consisting of

non-uniform node points is applied. The inlet flow condition is obtained by solving a fully

developed rectangular duct flow whose cross section is the same as the inlet of Diffuser 1.

251 21 41: :

2

Case 13.2-27

Fig.6 compares the streamwise mean velocity profiles in the several sections of the three spanwise plane

sections. Although the CLS model tends to improve the results of the standard model, the agreement

with the experimental data is still very poor. The predicted profiles by the TCL model generally well accord

with the experimental data while those at some sections still have large margins to be improved (e.g. at

x/H=12,15 of z/B=3/4) and the predicted separation zone is smaller.

k ' ;

Concluding Remarks 1) The present modification for the analytical wall-function can improve unphysically predicted heat transfer

profiles near corners of a 3D duct flow by the original form. 2) The predictive performance of the TCL model with the present AWF is generally satisfactory in the turbulent

3D diffuser flow which is difficult to predict reasonably by the eddy viscosity models presently tested.

References Cherry, E.M., Elkins, C.J. and Eaton, J.K., 2008, Geometric Sensitivity of three dimensional Separated Flows. Int.

J. Heat Fluid Flow 29, 803-811.

Craft, T.J., Gerasimov, A.V., Iacovides H., Launder, B.E., 2002, Progress in the generation of wall-function

treatments. Int. J. Heat Fluid Flow 23, 148-160.

Craft,T.J.,Launder.B.E., Suga,K., 1996, Development and application of a cubic eddy-viscosity model of turbulence.

Int. J. Heat Fluid Flow, 17, 108-115.

Craft,T.J.,Launder.B.E., 2001, Principles and performance of TCL-based second-moment closures. Flow, Turb.

Combust., 66, 355-372.

Lien, F-S., Leschziner, M.A., 1994, A general non-orthogonal finite-volume algorithm for turbulent flow at all

speeds incorporating second-moment turbulence-transport closure, Part1: Numerical Implementation. Comp.

Meth. Appl. Mech. Engng., 114, 123-148.

Suga, K., Craft, T.J., Iacovides, 2006, H., An analytical wall-function for turbulent flows and heat transfer over

rough walls. Int. J. Heat Fluid Flow 27, 852-866.

θ

r

yx

Rc

Rc/D=3.357

θ=90

Ub D

o

Fig. 1 Square sectioned U-bend duct. Fig. 2 Velocity vector near a corner of a 3D duct flow.

!!!!

( )UUx

88

<( )U

x

8=

8<

Fig. 3 Distribution of the convection terms of the near wall cells.

!

!

3

Case 13.2-28

Original

Present

Fig. 4 Nusselt number distribution of the U-bend duct.

H

15H

4H

12.5H

3H

10H

R=6HR=6H

B4H

3H

R=6H

R=6H

11.3°

4°

X

Z

X

Y

B/H=3.33

251 21 41: :

Fig. 5 Computational grid for Case 13.2 Diffuser 1.

4

Case 13.2-29

Fig.6 Streamwise mean velocity distribution of Diffuser 1.

5

Case 13.2-30

13th ERCOFTAC!IAHR Workshop on REFINED TURBULENT MODELLING,

Institute of Fluid Mechanics and Heat Transfer (ISW), Graz University, of Technology (TUG),

Austria, September 25!26, 2008

D. Borello, K. Hanjali", F. Rispoli, G. Delibra, A. Alfieri

Abstract

We report on the simulation of the flow through a three-dimensional diffuser (experimentally

analyzed by Cherry et al., 2008) carried out with two different approaches: an elliptic

relaxation U-RANS model ( -f, Hanjali et al., 2004) and a hybrid LES/RANS model, which

couple the same -f (here adopted as near-wall model) with dynamic LES. Simulations were

performed on a 3.5 million cells structured grid using a fully validated FV code. The present

investigation is related to the analysis of the separation zone occurring along the upper wall

and the evolution of the mean field along the diffuser.

Physical Model

!f U!RANS model

The !-f is a 4-equations eddy-viscosity model based on Durbin's elliptic relaxation concept (Durbin et al., 1991), which

solves a transport equation for the velocity scale ratio !=v2/k, in combination with an elliptic relaxation function f.

t

j j

Df P

Dt k x x!

"! ! !"

#

$ %& '( () * + +, -. /. /( (, -0 12 3

(1)

2 21 2

1 2

3

PL f f c C !

4 5& ' & '6 * ) + *. /. /0 10 1

(2)

The eddy viscosity is defined as t c k7" ! 4) , where c7 =0.22, and 4 is the time scale, equal to k/5 away from a wall.

This model is more robust and less sensitive to non-uniformities and clustering of the computational grid respect to the

standard models because of a more convenient formulation of the equation for ! and especially of the wall boundary

condition for the elliptic function 22 /wf y"!) * . We recall here that the Reynolds stress components assume a

different trend approaching the wall. In particular, 2 4v y8 while

2 2 2,u w y8 and consequently 2k y8 . The !-f

model is able to reproduce this trend and has been fully validated computing several reference test cases (Borello et al.,

2008, Delibra et al. 2008). Because of these peculiar aspects, !-f can be considered well suited to act as near-wall

counterpart in a seamless hybrid LES/RANS approach.

Hybrid LES/RANS !f model

The hybrid LES/RANS formulation of !-f differs from the original U-RANS formulation as it includes in the k transport

equation an grid-detection function in term of the RANS and LES length scales ratio, i.e.:

9 :t

j j

Dk kv P

Dt x x" ;5

( () + + *

( (

& '. /0 1

(3)

where:

max 1,RANS

LES

L

L; )

& '. /0 1

3/ 2

1/ 3; 1.5( )tot

RANS LES

kL L x y z

5) ) < < < (4)

!"#$%&'()$*+,-$./$!$(01$2.334"#5$

Case 13.2-31

and ktot=kres+kmod. In other words, the dissipation rate 5 is evaluated from its transport equation in the conventional

RANS form when LLES>LRANS, and from k and < when LLES<LRANS so that the sink in the k-equation is ;5=max(5, k3/2/<).

Hence, for ; =1, the standard RANS model is in play, and for ;>1 we should have a one-equation RANS model with a

damped viscosity. On the other hand, the dynamic LES imposes a further switching based on the criterion:

max( , )RANS LES

t t t" " ") (5)

Because at the location where ;=1 the eddy-viscosity switching constraint (5) is still not satisfied, a “buffer” zone is

automatically established up to the position where constraint (5) is activated. Here the RANS is still in play but with an

automatic adjustment (through ; > 1) in the RANS eddy viscosity towards the LES SGS viscosity. This ensures a

continuous damping of "t until the dynamic SGS viscosity is reached when the computations switch to the true dynamic

LES. The buffer zone extends up to ; ? 1.5, covering only a few cells for the typical RANS and coarse-LES grids used

here. Fig 1 shows how the hybrid works: where > 1 (a) away from the walls, kMOD is destroyed (b) and becomes zero

(in the circle); in this zone !T is damped (d) and overcome by !SGS (e); the effective viscosity !EFF is given by eqn. (5)

and showed in (f).

Computational details

The computational results presented were obtained by using the in-house T-FlowS parallel Finite Volume code

originally developed at Department of Multi-Scale Physics, TU-Delft and currently advanced by our research group at

Dipartimento di Meccanica e Aeronautica, ‘Sapienza’ University of Rome (Niceno, 2001, Delibra et al., 2008). The

diffuser was discretized by a 3.5 million cells structured grid. The computational domain extends for 32.5 cm (starting 5

cm before the diffuser and ending 12.5 cm downstream the diffuser). Both simulations used the following settings: the

incompressible Navier- Stokes equation system is solved using the SIMPLE algorithm for velocity-pressure coupling;

we adopted SMART as convective scheme for all the variables and parabolic full-implicit time integration scheme

(Niceno, 2001). The non-dimensional time steps have been set to 10-2 for both cases. This allowed to have a CFL

number always less than 1. In all the simulations the convergence threshold parameters was set equal to 10-8 for the

field variables in the iterative cycle for resolving each time step. The tolerance value set for the SIMPLE algorithm was

set to 10-6. The U-RANS !-f solution after 8000 time steps has been used as initial condition field for the hybrid

simulation. In Table 1 we report on the time history for both computations. Note that the time steps of the hybrid model

account for the initial 8000 U-RANS time steps also.

Model Time steps Time steps used for

collecting statistics

U-RANS 13000 5000

Hybrid LES/RANS 12400 3000

Discussion of Computational Results

The analysis of the diffuser flow field was focused on the description of recirculation zone. Experiments returns the

start of recirculation in the middle of the diffuser, about 7 cm downstream the diffuser inlet. In our computations we

experienced similar results. In Fig. 2 we show the mean velocity and k profiles in five location placed along the diffuser

in the mid plane. It is worth to note that the solution of the main field is not fully resolved (at present both simulations

are running). However, the mean velocity profiles (figures on the left) are able to reproduce the presence of the

recirculation region since the section placed at X=12. The k distribution, Fig.2 left, shows not fully resolved profiles,

but it possible to argue that the hybrid simulation will better reproduce the variable profiles especially after the birth of

recirculation. The instantaneous plots of streamwise velocity (Figs.3-4) show the presence of the recirculation region

near the upper endwall. Fig.3 refers to a cut plane in the middle of the diffuser. Hybrid simulation returns a wider

extension of the recirculation region due to the influence of the LES forcing. Further it should be noted the presence of

an instantaneous backflow region predicted by hybrid model just downstream the diffuser inlet (Fig.3 down).

Finally, in Fig.4 we show instantaneous streamwise velocity profiles in a cut plane parallel to the upper endwall and

placed 2 mm far from the solid boundary. It is worth to note that in this region U-RANS model acts in both simulations

Case 13.2-32

and then similar profiles are obtained. However, U-RANS returns smooth velocity contours indicating a major influence

of eddy diffusivity respect to hybrid field.

Fig.1 Hybrid LES/RANS model: a) a field, b) kmod, c) kres, d) !T, e) !SGS, f) !EFF in a trasversal plane at x=8

Fig. 2 Mean flow feature: left - Streamwise velocity profiles, right - k profiles;

the examined sections are from left: x=0, x=4, x=8, x=12, x=18

References

Cherry E. M., Elkins C. J. and Eaton J. K., 2008, ‘Geometric Sensitivity of Three-Dimensional Separated

Flows’, Int. J. of Heat and Fluid Flows,, 29, pp. 803-811.

Durbin P. A., 1991, ‘Near wall turbulence closure modelling without damping functions’, Theor. Comp. Fluid

Dyn. 3, pp. 1-13.

Hanjali K., Popovac M. and Hadziabdic M., 2004, ‘A Robust Near-Wall Elliptic Relaxation Eddy Viscosity

Turbulence Model for CFD’, Int. J. of Heat and Fluid Flows, 25, pp. 1047-1051

Niceno B., 2001, ‘An unstructured parallel algorithm for Large Eddy and Conjugate Heat Transfer Simulations’,

Ph.D. Dissertation, TU-Delft

Delibra G., Borello D., Hanjali K. and Rispoli F., 2008, ‘U-RANS of flow in pinned passages relevant to gas

turbine blade cooling’, Engineering Turbulence Modelling and Measurements, Limassol, June 4-6 2008, Cyprus,

selected for publication on Int. J. of Heat and Fluid Flows.

a b c

d e f

Case 13.2-33

Delibra G., Borello D., Hanjali K. and Rispoli F., 2009, ‘U-RANS and Hybrid LES/RANS Computations of Tip

Leakage and Secondary Flows in Linear Axial Compressor Cascade’, to be presented at European Turbomachinery

Conference, Graz, Au.

Fig.3 Instantaneous streamwise velocity in the mid plane with streamwise velocity contours

up: U-RANS - down: hybrid

Fig.4 Instantaneous streamwise velocity in the mid plane with streamwise velocity contours

up: U-RANS – down: hybrid

Case 13.2-34

6th 13th ERCOFTAC/IAHR Workshop on Refined Turbulence Modelling, September 25-26, 2008, TU Graz, Austria

Test Case 2: 3D Diffuser

Muhamed Hadžiabdic

International University of Sarajevo, Faculty of Natural Sciences and Engineering

Paromlinska 66, 71 000 Sarajevo, Bosnia and Herzegovina

[email protected]

Introduction

The 3D diffuser that was experimentally investigated by Cherry, Iaccarino, Elkins and Eaton (2006) has been calculated by the

ζ − f RANS model. The focus of the computational investigation is the model performance for three-dimensional flows that

exhibited a high degree of geometric sensitivity.

Turbulence model

The ζ − f RANS model of Hanjalic, Popovac and Hadžiabdic (2004) is used for all computations. The ζ − f model is an

eddy-viscosity model based on Durbin’s elliptic relaxation concept. It solves a transport equation for the velocity scale ratio

ζ = υ2/k instead of the equation for υ2. The motivation behind the model development originated from the desire to improve

the numerical stability of the model, especially when using segregated solvers. Because of a more convenient formulation of

the equation for ζ and especially of the wall boundary condition for the elliptic function f , it is more robust and less sensitiveto non-uniformities and clustering of the computational grid.

Computational details

The computations were performed by using the in-house unstructured finite-volume computational code T-FlowS, with the cell-

centred collocated grid structure (Niceno 2001; Niceno and Hanjalic 2004). The second-order accurate MINMOD scheme

is used to discretize the convective terms in the governing equations. The SIMPLE algorithm is used for the pressure-velocity

coupling.

The used grid consisted of 1250000 cells. The mesh was hyperbolically clustered towards the walls. The maximum y+ and

z+ in the first wall cells were less than 1 throughout the computational domain. The mesh details are given in the table below.

The inflow was generated by separate, simultaneous calculation of the channel flow with the periodic boundary condition in

the stream-wise direction (see Fig.1). The development channel was 2H long, where H is the channel height, while the outlet

transition channel was 12H long. The convective outflow was imposed at the outlet boundary.

Nx in the development channel Nx in the diffuser Nx in the outlet transition Ny Nz Total

36 200 60 65 65 1.25× 106

Figure 1: Side view of the used mesh.

1 Case 13.2-35

6th 13th ERCOFTAC/IAHR Workshop on Refined Turbulence Modelling, September 25-26, 2008, TU Graz, Austria

References

Cherry, E.M., Iaccarino, G., Elkins, C.J. and Eaton, J. K. Separated flow in a three-dimensional diffuser: preliminary validation,

Center for Turbulence Research, Stanford University, Annual Research Brief 2006, pp. 31-40.

Hanjalic, K., Popovac, M. and Hadžiabdic, M. A robust near-wall elliptic relaxation eddy-viscosity turbulence model for CFD,

International Journal of Heat and Fluid Flow, vol. 25, p. 1047-1051, 2004

Niceno, B. An Unstructured Parallel Algorithm for Large Eddy and Conjugate Heat Transfer Simulations, Delft University of

Technology, Delft, The Netherlands, 2001

Niceno, B. and Hanjalic, K. Unstructured large-eddy- and conjugate heat transfer simulations of wall-bounded flows, in Model-

ing and Simulation of Turbulent Heat Transfer (Developments in Heat Transfer Series), editors M. Faghri and B. Sunden, WIT

Press, 2004

2 Case 13.2-36

© 2008 ANSYS, Inc. All rights reserved.

1ANSYS, Inc. Proprietary

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Florian M

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ANSYS G

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any, Otterfing

florian.m

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Andrey G

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NTS, St. Petersburg

agarbaruk@

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Pavel Smirnov

NTS, St. Petersburg

pavel.smirnov@

nts-int.spb.ru

Florian M

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ANSYS G

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any, Otterfing

florian.m

enter@

ansys.com

Andrey G

arbaruk

NTS, St. Petersburg

agarbaruk@

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Pavel Smirnov

NTS, St. Petersburg

pavel.smirnov@

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Case 13.2: Flow in a 3-D diﬀuser - description of …...1 13th ERCOFTAC SIG15 Workshop on Reﬁned Turbulence Modelling , Graz University of Technology, Graz, Austria, September

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