Usually, flow separation in a diffuser is sought to be avoided due to the invoked additional pressure loss. Other than in many strongly separated flows, such as ...
43rd AIAA Aerospace Sciences Meeting and Exhibit, January 10–13, 2005/Reno, NV
Large-Eddy Simulations of a Separated Plane Diffuser J. U. Schl¨ uter∗, X. Wu†, and H. Pitsch‡ Center for Turbulence Research Stanford University, Stanford, CA We present Large-Eddy Simulations (LES) of a weakly separated diffuser. The influence of mesh resolution and the choice of subgrid model is investigated. Three resolutions ranging from 1.5 to 6.5 million cells are computed. The influence of the subgrid model is studied on the coarsest mesh using no subgrid model, the standard Smagorinsky model, the dynamic Smagorinsky model and the dynamic localization model. Furthermore, some flow features are investigated using the results of the fine mesh simulation. The application of LES to a gas turbine prediffuser flow is briefly described.
I.
Introduction
D
iffusers are one of the standard challenges in fluid mechanics. The task of a diffuser is to decelerate the flow and to regain total pressure. Usually, flow separation in a diffuser is sought to be avoided due to the invoked additional pressure loss. Other than in many strongly separated flows, such as the flow over a backward facing step, the point of flow separation is not defined by the geometry but entirely by the pressure gradient. Hence, diffuser flows are very sensitive and are difficult to predict with numerical means. Diffusers have been studied extensively in the past, since this is a very common flow configuration. Apart from the characterization of diffusers, these flows are used to study fundamental physics of pressure-driven flow separations. Two laboratory incompressible diffuser flows have emerged as standard test-cases in a number of fundamental and modeling studies on spatially developing complex internal turbulent flows, namely, the study by Azad1 (henceforth referred to as the Azad diffuser) and the study from Obi et al.2 (henceforth referred to as the Obi diffuser). The Azad diffuser is an axisymmetric conical geometry with a total divergence angle of 8 ◦ and with fully developed pipe flow at the inlet. The inlet Reynolds number based on friction velocity and pipe diameter is 12,400. Extensive measurements have been performed on this flow.3, 4, 5 The results showed that sudden application of adverse pressure gradient at the diffuser throat affects the flow so drastically that the downstream mean and turbulent fields become unrecognizable in relation to the inlet condition. The Obi diffuser has an asymmetric planar configuration with a total expansion ratio of 4.7 and a single sided deflection wall of 10◦ (Fig. 1). The inlet was designed to be a fully developed turbulent channel flow, although in some of the experiments this condition was not achieved. The inlet Reynolds number based on friction velocity and channel half height is 500. Obi et al. studied the flow experimentally using a single component laser-Doppler anemometer.2 Buice & Eaton repeated the experiment and made hot-wire and pulsed-wire measurements in an identical flow configuration.6, 7 Further investigations on different aspects of flow separation were performed by Lim & Choi.8 The Obi diffuser has also been used as a test flow in a number of computational studies. These include the Reynolds-averaged Navier-Stokes (RANS) simulations9, 10 as well as LES studies.11 In the present study we want to concentrate on the numerical aspects of LES of diffuser flows and the prediction of flow separation. The point of separation is determined mainly by two factors: one is ∗ Research
Associate Associate ‡ Assistant Professor c 2005 by Center for Turbulence Research, Stanford University. Copyright Aeronautics and Astronautics, Inc. with permission. † Research
Published by the American Institute of
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Figure 1. Geometry of the diffuser.
certainly the pressure gradient, the other the level and nature of turbulence. The turbulent transport of mean momentum towards the near-wall regions delays the flow separation. An attempt to model diffuser flows has to take these two aspects into account. We chose to use LES, since in this approach the large-scale turbulent structures are resolved in time and space and can predict the turbulent transport more accurately than RANS approaches. In this study we want to examine the choice of mesh resolution and the choice of the subgrid model on the flow prediction. We use three different mesh sizes ranging from 1.5 to 6.5 million cells to assess the influence of mesh resolution. Furthermore, we use four different modeling approaches to determine the influence of the subgrid model. We will briefly describe how the results of the fine mesh computation can be used to assess fundamental aspects of turbulence, and we will present the application of LES to a diffuser in a gas turbine engine.
II.
LES Approach
The basic idea of LES is to resolve the larger scales of motion of the turbulence while approximating the smaller ones. To describe this in a mathematical form, a filter has to be applied to the equations of continuity and momentum transport: Z +∞ e (x, t) = Q Q (x, t) G (x − x0 ) dx0 (1) −∞
In most LES approaches a mesh filter is used and the filter is implicitly given by the mesh distribution. Applying this filter to the Navier-Stokes equation leads to the following equations for momentum u i : ∂ ug ∂p ∂ ∂ uei ∂ uei i uj + + = ρ¯ν (2) ∂t ∂xi ∂xi ∂xj ∂xj The convection term on the left hand side is not fully determined by the resolved spectrum of the LES computation. This term is split up in a resolved and an unresolved part: ug ei uej − τij i uj = u
and the unresolved part is usually put on the right hand side. This delivers the following equation: ∂τij ∂ uei ∂ ρ¯uei ∂ ρ¯uei uej ∂p ∂ ρ¯ν + + + = ∂t ∂xi ∂xi ∂xj ∂xj ∂xj 2 of 13 American Institute of Aeronautics and Astronautics Paper 2005-0672
(3)
(4)
Figure 2. x − y view of the coarse mesh. Every 3rd grid line shown.
The terms on the left hand side are resolved by the LES computation. On the right hand side an unresolved term τij remains which has to be modeled. This term can be seen as analogon to the Reynolds-stress tensor of RANS, but since in the LES formulation the larger length scales of turbulence are already resolved, it denotes the turbulent subgrid stresses and hence, is smaller than its counterpart in RANS.
III.
Subgrid Models
Most LES subgrid scale models use the eddy viscosity approach to model the subgrid turbulent stresses: 1 τij = −2νt S ij + τll δij 3 with S ij
1 = 2
∂ui ∂uj + ∂xj ∂xi
(5)
(6)
The definition of the eddy-viscosity νt depends on the employed model. In the current study we consider four different model approaches. A.
No Model
In this approach the subgrid stresses are neglected (νt = 0). Here, it is assumed that the numerical dissipation outweighs the influence of the subgrid model and hence, the subgrid stresses are set to zero. This approach requires an implicit time-advancement to ensure numerical stability. B.
Standard Smagorinsky Model
The standard Smagorinsky model12 was one of the first subgrid models used for LES: q νt = (C1 ∆x)2 2S˜ij S˜ij
(7)
with a constant C1 = 0.2. This constant has to be manually adjusted depending on the flow. This model has the advantage of simplicity and low computational costs and is still used for a number of applications. However, it is insufficient for a variety of flows, such as transitional flows, because applying this model on a laminar boundary layer delivers a νt > 0 and suggests turbulence where there is none. Additionally, in turbulent boundary layers, this model predicts turbulence in the viscous sub-layer very close to the wall. C.
Dynamic Smagorinsky Model
The dynamic Smagorinsky model13 computes the parameter C1 as a function of position from the information already contained in the resolved field. This has the advantage that nothing about the flow needs to be known beforehand and no adjustments of the model have to be made. Furthermore, the C1 may be also a function of space, which means that it can take different values in different flow situations, most notably in transitional flows, where C1 = 0 in laminar sections of the flow and C1 > 0 in turbulent sections. This model requires a test-filter (b.), which is larger than the LES filter. The subgrid stresses at the test-filter level are then given by: b Tij = ud g ˜i ub ˜j (8) i uj − u 3 of 13 American Institute of Aeronautics and Astronautics Paper 2005-0672
So far, Tij and τij are unknown. They are related by the Germano identity:13 Lij = Tij − τc ij
(9)
The Leonard term Lij can be computed from the resolved LES solution. It is considered that the scaling law can be used and the subgrid stress at the test level can be written as: b˜ S b˜ + 1 T δ c2 |S| Tij = −2C2 ∆ ij ll ij 3
(10)
This leads to an equation to determine C2 :
1 Lij = αij C2 − βd ij C2 + Lll δij 3
with
αij βij D.
b˜ S b˜ c2 |S| = −2C2 ∆ ij ˜ S˜ij = −2C2 ∆2 |S|
(11)
(12) (13)
Dynamic Localization Model
The derivation of the dynamic localization model is beyond the scope of the present study and we want to refer to the work of Ghosal et al.14 for details. In general, this model uses the same approach as the dynamic Smagorinsky model, but uses a variational formulation to determine C. This allows to generalize the dynamic procedure to flows that do not possess a homogeneous direction. Here, C = 0 is chosen as a constraint in the variational problem and an integral equation is obtained for the determination of C, which can be solved at each time-step. This procedure requires additional computational overhead, but showed promise in a number of turbulent flow applications.14
IV.
LES Flow Solver
We chose the unstructured LES flow solver CDP for the current investigation, which has been developed at the Center for Turbulence Research (CTR) at Stanford.15 The filtered momentum equations are solved on a cell-centered unstructured mesh with a second-order accurate central differences spatial discretization. 16 This algorithm is characterized by low numerical dissipation and a high level of stability. An implicit timeadvancement procedure is applied. A low-Mach number approximation is used and the Poisson equation is solved in order to determine the pressure field.
V.
Computational Meshes
We report the results of simulation using three different meshes. Although the LES code is an unstructured solver, we use structured (hexahedral) body-fitted meshes, since this geometry is rather simple and structured meshes allow for more control of the mesh distribution. The geometry of the diffuser is shown in Fig. 1. The computational domain extends upstream 10 channel half-widths h and downstream 100h. The span-wise extend is 8h. The first mesh uses 370 mesh points in axial direction, 70 in cross-wise direction, and 50 in span-wise direction (Fig. 2). This adds up to 1.3 million control volumes. The second mesh uses 400×80×80 cells adding up to 2.5 million control volumes, and the third mesh uses 590×100×110 cells adding up to 6.5 million cells. The height of the first cell on the wall is constant in all three meshes. Based on the experimental inflow data it was set to y + = 10 at the inflow.
VI.
Inflow Boundary Conditions: Channel Flow
In order to specify the inflow boundary condition and the resolved turbulence at the inflow, we use the procedure by Pierce & Moin.18 Here a separate LES computation of a periodic channel flow is performed 4 of 13 American Institute of Aeronautics and Astronautics Paper 2005-0672
25
3 20
15
2
10
1 5
0
1
10
100
1000
0
0
0.2
0.4
0.6
0.8
1
Figure 3. Inflow channel flow. Left hand side y + vs. u+ : • Moser et al 17 at Re = 395, Moser et al 17 at Re = 590, + Buice & Eaton6, 7 at Re = 490; Right hand side : y vs. rms velocities normalized by friction velocity; Moser 0 0 0 0 + + + + et al 17 at Re = 395: ◦ urms , vrms , • wrms ; Buice & Eaton6, 7 at Re = 490: + urms .
Figure 4. Flow visualization of the LES using the fine mesh. component. Blue isosurfaces denote flow reversal.
Isosurfaces of instantaneous axial velocity
using a massflux and Reynolds-number identical to that of the channel upstream of the diffuser. The flow data is then recorded in a y − z plane and stored in a database. This database is then fed as inflow data into the actual LES computation of the diffuser. This separate LES of a channel uses a 1283 mesh. The channel length is 12h, while the channel height and width is identical to that of the diffuser. A structured LES code specialized for this task is used, 18 which keeps the computational overhead of this simulation small. The computational costs to create this database is approximately 1% of the actual LES diffuser simulation using the Dynamic Smagorinsky model on a coarse mesh. Figure 3 shows the results for the channel simulation. The mean flow velocity profile shows a very good agreement with the experimental data. The turbulent fluctuations however, show some disagreements. Buice & Eaton6, 7 discuss the quality of their experimental data and conclude that the channel flow in the experiments is not fully developed. In order to match the experiments from Buice & Eaton as exactly as possible, the turbulent fluctuations of the created database were rescaled using a rescaling method for LES inflow data.19 This rescaled database delivers the inflow data for the actual diffuser simulations.
VII.
Results: Mesh Resolution
All LES simulations were perfomed on an IBM SP3. The simulations were run for five flow through times before collecting flow statistics for another five flow through times. Fig. 4 shows a flow visualization of the LES using a fine mesh. It can be seen that the flow is highly turbulent. At the inclined (lower) wall flow separation takes place.
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Figure 5. Axial mean velocity profiles. Symbols: experimental data from Buice & Eaton;6, 7 Lines: LES with different mesh resolution.
Figure 6. Axial mean velocity profiles. Close-up to the point of separation. Symbols: experimental data from Buice & Eaton;6, 7 Lines: LES with different mesh resolution.
Figure 7. Axial mean velocity fluctuations LES with different mesh resolution.
√ u02 . Symbols: experimental data from Buice & Eaton;6, 7 Lines:
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Figure 5 shows the axial velocity profiles for the mesh refinement study. All simulations use the dynamic Smagorinsky model. The velocity profile at x/H = −5 is the profile that is imposed at the inlet as boundary condition. Since the data-base delivering the inflow data was scaled to the experimental data, we can see here a perfect match. The velocity profiles downstream are results from the different LES simulations. As a trend, with increasing mesh refinement an increasing agreement of the LES with the experimental data can be stated. Of interest is especially the point of separation. Figure 6 shows a close-up to the velocity profiles behind the separation point. Here, differences between the different mesh resolutions are more apparent. However, even the coarse mesh simulation demonstrates a good overall agreement with the experimental data. The velocity fluctuations (Fig. 7) show a similar trend. At x/H = 6 all simulations show an overestimation of the turbulent fluctuations. Since this overestimation does not decrease much with increasing mesh resolution, we suspect the proximity of the inflow boundary conditions to contribute to this disagreement. Future simulations should have the inflow plane further upstream, so that the flow inside the LES domain has more time to settle. Table 1 relates the computational costs to the different mesh resolutions: Table 1. Mesh Refinement Study: Computational Costs per Flow Through Time
Case Coarse (1.5M cells) Intermediate (2.5M cells) Fine (6.5M cells)
CPU hours
Processors used
Wall clock time (h)
CPU time factor
250 520 3200
48 96 25
5.2 5.4 128
1 2.08 12.8
As can be seen, the computational costs for the simulations increase dramatically for an increased mesh resolution. This is an problem inherent to the LES approach, where mesh refinement in space leads to a refinement in time, since the smaller filter size requires a smaller time-step. Since the increase of mesh resolution creates a large computational overhead, we now want to have a look at different subgrid models to improve the simulation results for coarse meshes.
VIII.
Results: Subgrid Modeling
We now want to put different LES subgrid models to the test. We use the four different modeling approaches described in section III. For this study, we use the coarsest mesh from the previous study, since the influence of the subgrid model on the results is largest on this mesh. Figure 8 shows the results for the simulations. Again, all simulations were run for five flow through times before collecting flow statistics for another five flow through times. In Fig. 8, we did not include the results from the LES using the standard Smagorinsky model, since the results of this simulation showed dramatically different results. Using the standard Smagorinsky model, the flow separation was predicted on the straight (upper) wall instead on the inclined wall. Hence, the mean flow field does not show any agreement with the experiments. This example shows that the subgrid model can influence the outcome of the simulations dramatically, for the present case even so that the basic flow features are lost. The other three subgrid modeling approaches, using no model, the dynamic Smagorinsky model and the dynamic localization model, capture the presence of the separation at the inclined (lower) wall. All simulations predict the onset of separation reasonably well. The no-model approach overpredicts the extend of the flow separation. Without the subgrid stresses to dampen the effect of turbulence, the flow separation is too far upstream and the flow reattachment is missed. The dynamic Smagorinsky model does a better work, however, the extend of flow separation is underpredicted. The dynamic localization model demonstrates the best agreement. This can be seen especially at the point of separation (Fig. 9). Also, the flow reattachment is predicted remarkably well. The turbulent fluctuations (Fig. 10) show a good agreement for all models used, except for the location at x/H=6, which was discussed above. Differences between the models can be only determined in the far field, where the no-model approach fails to predict the reattachment.
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Figure 8. Axial mean velocity profiles. Symbols: experimental data from Buice & Eaton;6, 7 Lines: LES with different subgrid models.
Figure 9. Axial mean velocity profiles, close-up to the point of separation. Symbols: experimental data from Buice & Eaton;6, 7 Lines: LES with different subgrid models.
Figure 10. Axial mean velocity fluctuations LES with different subgrid models.
√ u02 . Symbols: experimental data from Buice & Eaton;6, 7 Lines:
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Figure 11. Friction coefficient for the straight (upper) wall. Eaton;6, 7 Lines: LES.
Symbols: experimental data from Buice &
Figure 12. Friction coefficient for the inclined (lower) wall. Symbols: experimental data from Buice & Eaton; 6, 7 Lines: LES.
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One of the best ways to locate the exact location and extend of flow reversal is by the determination of the friction coefficient at the wall. The friction coefficient is defined as cf =
τ
(14)
1 2 2 ρ ∞ U∞
∂u 2 the stress at the wall, ρ∞ the reference density and U∞ the reference velocity. With τ = ∂n Figure 11 shows the distribution of cf at the straigt (upper) wall. The flow is always attached and hence, the friction coefficient is always positive. All models show a reasonably good agreement with the experimental data at locations x/H > 12. Upstream the models demonstrate a remarkably different behavior. We also included the friction coefficient distribution from the fine mesh computation. Since the fine mesh computation has a very good agreement with the experimental data over the entire extend of the diffuser, we conclude that the abnormal behavior of the coarse mesh simulations is due to an under resolved pressure gradient at that location. Figure 12 shows the friction coefficient distribution at the inclined (lower) wall. Here, flow reversal takes place and the friction coefficient is negative in the extend of the recirculation zone. Again, we included the results of the fine mesh simulation, which show a very good agreement with experiments. The simulations using the dynamic Smagorinsky model and the dynamic localization model also demonstrate a reasonably well agreement. The no-model approach shows clearly an overprediction of the recirculation zone. Table 2 compares the computational costs for the different models. The CPU time factor is set to one for the case using the dynamic Smagorinsky model in order to facilitate the comparison with table 1. We can see that the models simpler than the dynamic Smagorinsky model do not have a distinctive advantage in computational costs. Considering the deterioration of the results shown in the previous figures, the use of these models seems not advisable for this kind of flow. The use of the dynamic localization model still comes with a strong penalty in computational costs. However, considering the improvement of the flow prediction with this model, its use can be considered in cases, where flow separation is important. The computational costs for this model are considerably lower than the use of a finer mesh.
Table 2. Model Study: Computational Costs per Flow Through Time
Case No model Standard Smagorinsky Dynamic Smagorinsky Dynamic localization model
CPU hours
Processors used
Wall clock time (h)
CPU time factor
195 227 250 616
48 48 48 12.8
4.1 4.7 5.2 48
0.78 0.9 1 2.5
IX.
Results: Flow Physics
We have stored the results from the fine mesh simulation in a data base. This high quality data allows to examine basic flow features of separated flows. The advantage of the simulation data is that time correlated data is available for the entire flow field. Work has been performed to identify structures20 following the vortex identification for incompressible flow by Jeong & Hussain.21 For this, we need the symmetric component and antisymmetric component of the three-dimensional velocity gradient tensor given by: 1 ∂ui ∂uj ∂uj 1 ∂ui , and ωij = , where i, j = 1, 2, 3 (15) sij = + − 2 ∂xj ∂xi 2 ∂xj ∂xi The coherent structures are then identified by using isosurfaces of negative λ2 , where λ2 is the second largest eigenvalue of ss + ωω. This allows to identify structures close to a solid wall (Fig. 13). Furthermore, investigations using the diffuser flow data base found the presence of an internal layer on the decelerated boundary layer at the straight upper wall.20
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Figure 13.
Three-dimensional view of the isosurfaces of one negative value of λ2 near the upper wall.
X.
Application: LES of a Gas Turbine Prediffuser
Now we want to apply LES of diffuser flows to an engineering application. The case that we want to report here is that of a prediffuser of a gas turbine engine. The role of the prediffuser is to decelerate the flow leaving the compressor and direct the flow towards the injectors of the combustion chamber. While these diffusers are usually not separated in order to improve the pressure recovery, the design of these diffusers require a tool that can predict flow separation. We performed an integrated RANS-LES computation on a real Pratt & Whitney gas turbine engine geometry (Fig. 14). The geometry consists of the last stage of the compressor, the prediffuser, the fuel injector and the combustion chamber. We use the Reynolds-Averaged Navier-Stokes (RANS) approach for the compressor section. The RANS simulation provides the inflow boundary conditions for the LES domain consisting of the prediffuser, the fuel injector and the combustion chamber. A more detailed description of this test-case can be found in the publication of Schl¨ uter et al.22 Here, we want to emphasize how the conclusions of the present study influenced the parameters chosen for this application. Previous simulations of the combustor flow used the standard Smagorinsky model. Due to the poor performance of this model in the current study, we used the dynamic Smagorinsky model instead. We chose not to use the dynamic localization model, since the computation of the prediffuser and the combustor requires a mesh of more than 3 million cells and the computational costs of a meaningful physical time-span would have been prohibitive. Instead, we refined the mesh in the prediffuser. In the current case, the time step was determined by the mesh size in the combustor, and hence, the mesh refinement in the diffuser did not require a smaller time step for the overall simulation. The simulation of the prediffuser in a gas turbine using LES allows to assess the flow features in this flow passage in more detail. The present study of a separated diffuser gives more confidence into the results and allows an assessment of the accuracy of the LES approach for diffusers.
XI.
Conclusions
We have performed LES computations of a separated diffuser flow. A mesh refinement study demonstrated the improved predictive capabilities of fine meshes, but also exposed a large penalty in computational costs. We then tested different LES subgrid modeling approaches and assessed the accuracy and computational costs of these models for a separated flows. We applied no model, the standard Smagorinsky model, the dynamic Smagorinsky model, and the dynamic localization model. The standard Smagorinsky model predicted the flow separation at the straight wall instead of the inclined wall, which led to a strong disagreement with experimental data. All other models showed a reasonably well agreement with experiments, with the dynamic localization model performing best.
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Figure 14. Integrated RANS-LES of a compressor-prediffuser-combustor assembly. 22
In the course of this study, we generated a database of a fine mesh diffuser flow, which allows to study basic turbulent flow phenomena. We were also able to apply the conclusions drawn out of this study to a prediffuser simulation of a gas turbine engine.
XII.
Acknowledgments
We thank the US Department of Energy for the support under the ASC program. We also thank Pratt & Whitney for providing the gas turbine engine geometry, helpful comments and discussions.
References 1 Azad, R. S., “Turbulent flow in a conical diffuser: a review,” Experimental Thermal and Fluid Science, Vol. 13, 1996, pp. 318–337. 2 Obi, S., Aoki, K., and Masuda, S., “Experimental and computational study of turbulent separating flow in an asymmetric plane diffuser,” Ninth Symp. Turbulent Shear Flows, 1993, Kyoto, Japan, August 16-19. 3 Okwuobi, P. A. C. and Azad, R. S., “Turbulence in a conical diffuser with fully developed flow at entry,” Journal of Fluid Mechanics, Vol. 57, 1973, pp. 603–622. 4 Trupp, A. C., Azad, R. S., and Kassab, S. Z., “Near wall velocity distributions within a straight conical diffuser,” Experiments in Fluids, Vol. 4, 1986, pp. 319–331. 5 Azad, R. S. and Kassab, S. Z., “Turbulent flow in a conical diffuser: overview and implications,” Phys. Fluids, Vol. A1, 1989, pp. 564–573. 6 Buice, C. U. and Eaton, J. K., “Experimental investigation of flow through an asymmetric plane diffuser,” CTR Annual Research Briefs, 1996, pp. 243–248, Center for Turbulence Research, Stanford University. 7 Buice, C. U., Experimental investigation of flow through an asymmetric plane diffuser , Ph.D. thesis, Stanford University, August 1997.
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8 Lim, S. and Choi, H., “Optimal shape design of a two-dimensional asymmetric diffuser in turbulent flow,” AIAA J., Vol. 42, 2004, pp. 1154–1169. 9 Durbin, P. A., “Separated flow computations with the k − − v 2 model,” AIAA Journal , Vol. 33, 1995, pp. 659–664. 10 Iaccarino, G., “Prediction of a turbulent separated flow using commercial CFD codes,” J. Fluids Eng., Vol. 123, 2001, pp. 819–828. 11 Kaltenbach, H. J., Fatica, M., Mittal, R., Lund, T. S., and Moin, P., “Study of flow in a planar asymmetric diffuser using large eddy simulation,” Journal of Fluid Mechanics, Vol. 390, 1999, pp. 151–185. 12 Smagorinsky, J., “General circulation experiments with the primitive equations, I, the basic experiment,” Mon. Weather Rev., Vol. 91, No. 3, 1963, pp. 99–152. 13 Germano, M., Piomelli, U., Moin, P., and Cabot, W., “A dynamic subgrid-scale eddy viscosity model,” Phys. Fluids, Vol. A, No. 3 (7), July 1991, pp. 1760–1765. 14 Ghosal, S., Lund, T. S., Moin, P., and Akselvoll, K., “A dynamic localization model for large-eddy simulations of turbulent flow,” Journal of Fluid Mechanics, Vol. 286, 1995, pp. 229–255. 15 Moin, P. and Apte, S., “Large-eddy simulation of realistic gas turbine combustors,” AIAA Paper , , No. AIAA 2004-0330, January 2004. 16 Mahesh, K., Constantinescu, G., and Moin, P., “A Numerical method for large-eddy simulation in complex geometries,” Journal of Computational Physics, Vol. 197, No. 1, June 2004, pp. 215–240. 17 Moser, R. D., Kim, J., and Mansour, N. N., “DNS of turbulent channel flow up to Re = 590,” Phys. Fluids, Vol. 11, τ 1999. 18 Pierce, C. D. and Moin, P., “Method for generating equilibrium swirling inflow conditions,” AIAA Journal , Vol. 36, No. 7, 1998, pp. 1325–1327. 19 Schl¨ uter, J. U., Pitsch, H., and Moin, P., “Large Eddy Simulation inflow conditions for coupling with Reynolds-averaged Flow Solvers,” AIAA Journal , Vol. 42, No. 3, March 2004, pp. 478–484. 20 Wu, X., Schl¨ uter, J., Moin, P., Pitsch, H., Iaccarino, G., and Ham, F., “Identification of an internal layer in a diffuser,” CTR Annual Research Briefs, 2004. 21 Jeong, J. and Hussain, F., “On the identification of a vortex,” Journal of Fluid Mechanics, Vol. 285, 1995, pp. 69–94. 22 Schl¨ uter, J. U., Wu, X., Kim, S., Alonso, J. J., and Pitsch, H., “Coupled RANS-LES Computation of a Compressor and Combustor in a Gas Turbine Engine,” AIAA paper , , No. AIAA 2004-3417, July 2004, 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit.
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