1. Computational Combustion Lab Aerospace Engineering Multi-scale Simulation of Wall-bounded Flows Ayse G. Gungor and Suresh Menon Georgia Institute of Technology Atlanta, GA, USA Supported by Office of Naval Research WALL BOUNDED SHEAR FLOWS: TRANSITION AND TURBULENCE Isaac Newton Institute for Mathematical Sciences Cambridge, UK September 11 th, 2008

2. Computational Combustion Lab Aerospace Engineering Motivation Flows of engineering relevance is at high Re – Wall bounded flows, wake and shear flows The cost of simulations that resolve all the scales of motion is of the order of Re 3 Almost 90% of this cost is a result of attempting to explicitly resolve near-wall boundary layers – Near-wall turbulence contains many small, energy containing, anisotropic scales that should be resolved DNS Computations of channel flows – 18 B grid points, Re  = 2003 (Hoyas et al., 2006) DNS Computations of turbulent separated flows – 151 M grid points, Re  = 395 (Marquillie et al., 2008) – DNS at lower Reynolds number (Experiment at Re  = 6500)

3. Computational Combustion Lab Aerospace Engineering Motivation Conventional LES requires very high near-wall resolution – Near-wall Models Use algebraic relationships to compute wall stresses Resolution requirement reduced significantly Additional source of errors due to the modeling the dynamics in the near-wall region – Zonal Approaches Two Layer Approach – Solves boundary layer equations and/or employ local grid refinement RANS-LES Approach – Uses RANS near the wall and LES in the core region Most of the cost-effective approaches do not properly resolve the turbulent velocity fluctuations near the wall Here, a two-scale approach for high-Re flows is discussed – attempts to resolve near-wall fluctuations

4. Computational Combustion Lab Aerospace Engineering Multi-Scale Simulation Approaches Multi-scale approaches: Dynamic multilevel method (Dubois, Temam et al.) Rapid Distortion Theory SS model (Laval, Dubrulle et al.) Variational multiscale method (Hughes et al.) Two-level simulation (TLS*) (Kemenov & Menon), extended for compressible flows (Gungor & Menon) Simulate both LS and SS fields explicitly – Computed SS field provides closure for LS motion All use simplified forms of SS equations – Some invoke eddy viscosity concept for SS motions TLS simulates the SS explicitly inside the LS domain *Kemenov and Menon, J. Comp. Phys., Vol. 220 (2006), Vol. 222 (2007) Gungor and Menon, AIAA-2006-3538

5. Computational Combustion Lab Aerospace Engineering Two-Level Simulation: Key Features Simulate both large- and small-scale fields simultaneously – large-scales (LS) evolve on the 3D grid – small-scales (SS) evolve on 1D lines embedded in 3D domain – 3D SS equations collapsed to 3x1D equations with closure Scale Separation approach employed – No grid or test filtering invoked – No eddy viscosity assumption invoked High-Re flows simulated using a “relatively” coarse grid – Efficient parallel implementation needed – Cost becomes acceptable for very high-Re flow – Potential application to complex flows

6. Computational Combustion Lab Aerospace Engineering Two Level Grid in the TLS-LES Approach Small scale equations are solved on three 1D lines embedded in the 3D domain  x LES  z LES  y LES  y SS  x SS  z SS x y z Large-Scale Grid Small-Scale Grid Resolution requirements Number of LES control volumes:N LES 3 N LES <N SS Grid points for TLS-LES:N LS 3 + 3N LS 2 N SS N LS <N LES, N LS <<N SS Grid points for DNS: N SS 3

7. Computational Combustion Lab Aerospace Engineering TLS v/s LES Two degrees of freedom in Conventional LES – Filter Width and Filter Type Two degrees of freedom in TLS: – Sampling/Averaging Operator (SS LS) – Interpolation Operator (LS SS) TLS does not require commutativity to derive LS Eqns. Full TLS approach described earlier – isotropic turbulence, free shear and wall-bounded flows* Here, a new hybrid TLS-LES approach demonstrated** – Application to wall bounded flows with separation * Kemenov and Menon, J. Comp. Phys. (2006, 2007) ** Gungor et al., Advances in Turbulence XI (2007)

8. Computational Combustion Lab Aerospace Engineering TLS – Scale Separation Operator L Exact Field is split into LS and SS fields: Continuous large scale field is defined by adopted LS grid: SS field is defined based on LS field from decomposition: Sampling at LS grid nodes Interpolation to the SS nodes

9. Computational Combustion Lab Aerospace Engineering Fully resolved signal (black) from a 128 3 DNS of isotropic turbulence study. The resolved field is represented with a 16 grid point. The top hat filtered LES field (red) obtained by taking a moving average of the fully resolved field over 8 points. The TLS LS field (green) truncated from the fully resolved signal. The TLS SS field (blue) obtained by subtracting the LS field from the fully resolved field. The longitudinal energy spectra of a fully resolved signal (black) and (a) LES energy spectra (red), (b) The TLS LS (green) and SS (blue) energy spectra. TLS has higher spectral support A priori analysis of scale separation operators (LES and TLS)

10. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES Wall Model The TLS equations are used in the near-wall region The LES equations are used in the outer flow All zonal approaches (Hybrid RANS-LES) use some form of domain decomposition Hybrid TLS-LES uses functional decomposition – No need for interface boundary conditions – Need to determine the transition region dynamically RANS LES prescribed y interface Hybrid RANS-LES Strategy TLS LES prescribed y interface for wall-normal lines TLS-LES Strategy

11. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES Formulation – Scale Separation Hybrid TLS-LES scale separating operator R defined as an additive operator that blends the LES operator F with the TLS operator L R = k F + (1 - k) L – LES operator F is the standard filtering operator – k is a transition function relating TLS and LES domains Step FunctionTanh Function

12. Computational Combustion Lab Aerospace Engineering Application of additive scale separation operator – Velocity components – Turbulent stress Hybrid TLS-LES Equations Hybrid Terms Hybrid terms also in RANS-LES formulation (Germano, 2004) Combination of time and space operation Here, the hybrid terms appear due to LES and TLS combination both are space operators !!

13. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES Equations Resolved / Large Scale Equations Continuity: Momentum: The unresolved term in the momentum equation TLS: LES: Specific closures for each model The scale interaction terms are closed if the small scale field is known any SGS model

14. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES Equations Small Scale Equations Represents the smallest scales of motion “Hybrid TLS-LES SS domain” – discrete set of points along 3- 1D lines – 3D evolution of small-scales in each line – Full 3D SS equations “collapsed” on to these 1D lines – Cross-derivatives modeled based on a priori DNS analysis – Channel and forced isotropic turbulence (Kemenov & Menon, 2006, 2007) Explicit forcing by the large scales on these 1D equations Continuity: Momentum:

15. Computational Combustion Lab Aerospace Engineering Numerical Implementation of SS Equations 1) Approximate LS field on each 1D SS line by linear interpolation 2) Evolve SS field from zero initial condition until the SS energy matches with the LS energy near the cut off 3) Calculate the unclosed terms in the LS equation Time evolution of the SS velocity and SS spectral energy

16. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES of Channel Flow Mean velocity profiles demonstrates the capability of the model Wall skin friction coefficient provides – good agreement with DNS – well comparison with Dean’s correlation

17. Computational Combustion Lab Aerospace Engineering Re  = 1200 Re  = 590 3-D energy spectra Red line : Instantaneous energy spectra Blue line : Volume average spectra Black line: k -5/3 slope Re  = 2400 Hybrid TLS-LES approach recovers both LS and SS spectra near the wall Hybrid TLS-LES of Channel Flow

18. Computational Combustion Lab Aerospace Engineering Numerical Solver Incompressible Multi-domain Parallel Solver – 4 th order accurate kinetic energy conservative form (used here) – 5 th order accurate upwind-biasing for convective terms – 4 th order accurate central differencing for the viscous terms – Pseudo-compressibility with five-stage Runge – Kutta time stepping – Implicit time stepping in physical time with dual time stepping – DNS, LES (LDKM), TLS-LES

19. Computational Combustion Lab Aerospace Engineering Turbulent Channel Flow

20. Computational Combustion Lab Aerospace Engineering Turbulent Channel Flow Coarse DNS – 192 x 151 x 128 – Well prediction of the mean velocity, turbulent velocity fluctuations and turbulent kinetic energy budget.

21. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES of Separated Channel Flow Hybrid TLS-LES(~0.18M) and LES(~1.6M) at Re  = 395 – DNS(~151 M) at Re  = 395 by Marquillie et al., J. of Turb., Vol. 9, 2008 – Experiment at Re  = 6500 by Bernard et al., AIAA J., Vol. 41, 2003 Spatial resolution (%75 coarser than DNS) – TLS-LES-LS (64 x 46 x 64) :  x +LS = 77.4,  z +LS = 19.2,  y +LS = 5.4 – TLS-LES-SS (8 SS points/LS):  x +SS = 9.6,  z +SS = 2.4,  y +SS = 0.68 – DNS (1536 x 257 x 384) :  x + = 3,  z + = 3,  y + | max = 4.8 Inflow turbulence from a separate LES channel study at Re  = 395 Total vorticity on a spanwise plane (LES) Streamwise vorticity on a horizontal plane (LES)

22. Computational Combustion Lab Aerospace Engineering Spanwise line in the separation region Time evolution of the SS velocity

23. Computational Combustion Lab Aerospace Engineering SS iterations: 20 SS vorticity magnitude isosurfaces colored with SS streamwise velocity SS iterations: 100 SS iterations: 300 SS evolution effect on the instantaneous flow simulations on each line optimal parallel approach

24. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES of Separated Channel Flow Hybrid TLS-LES grid is chosen very coarse deliberately Hybrid TLS-LES C p shows good agreement with DNS – ~%30 off from experiments (higher Re) for all studies Hybrid TLS-LES C f in reasonable agreement with DNS and LES – Separation is not properly predicted due to coarse LS resolution

25. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES of Separated Channel Flow Streamwise velocity fluctuation Wall-normal velocity fluctuation The authors would like to thank Dr. J.-P. Laval for providing the DNS data DNS-151M (circles and shaded contours), TLSLES (red), LES (green)

26. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES of Asymmetric Diffuser Flow Hybrid TLS-LES(~0.25M) and LES(~1.8M) at Re  = 500 – LES(~6.5 M) by Kaltenbach et al., J. of Fluid Mech., Vol. 390, 1999 – Experiment by Buice and Eaton, J. of Fluids Eng., Vol. 122, 2000 The main features of this flow – A large unsteady separation due to the APG – A sharp variation in streamwise pressure gradient – A slow developing internal layer Inflow turbulence from a separate LES channel study at Re  = 500 Inclination angle: 10 0

27. Computational Combustion Lab Aerospace Engineering Spatial resolution – TLS-LES-LS (110 x 56 x 40) :  x +LS = 54,  z +LS = 50,  y +LS = 5.72 – TLS-LES-SS (8 SS points/LS) :  x +SS = 6.7,  z +SS = 6.2,  y +SS = 0.72 – LES (278 x 80 x 80) :  x + = 25,  z + = 25,  y + = 0.98 – LES by Kaltenbach et al., 1999 (590 x 100 x 110) Step function (, F: LES, L: TLS operator) – pre-defined interface, Y+ TLS = 152 Hybrid TLS-LES of Asymmetric Diffuser Flow Isosurfaces of the second invariant of the velocity gradient tensor colored with local streamwise velocity predicted with LES model TLS LES

28. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES of Asymmetric Diffuser Flow C p along the lower and upper wall predicted reasonably well Hybrid TLS-LES shows reasonable agreement with the experiment Skin friction coefficient over the upper flat wall displays a strong drop and a long plateau starting near the separation region in the bottom wall, and a more gradual decrease downstream Overall, TLS-LES shows ability to predict separation regions without any model changes

29. Computational Combustion Lab Aerospace Engineering Hybrid TLS-LES of Asymmetric Diffuser Flow The total pressure decreases 30% in the streamwise direction due to frictional losses. Mean velocity predicted reasonably with the hybrid TLS-LES model – Separation location agrees well – But reattachment is observed further downstream Exp. (symbols), TLS-LES (dashed lines) LES (solid lines)

30. Computational Combustion Lab Aerospace Engineering Conclusion and Future Plans A generalized hybrid formulation developed to couple TLS-LES – New hybrid terms identified but they still need closure TLS as a “near-wall” model for high-Re flows used in a TLS-LES approach without the hybrid terms – Reasonable accuracy using “relatively” coarse LS grid – Potential application to complex flows with separation Efficient parallel implementation can reduce overall cost Next Step – Analyze the hybrid terms in the TLS-LES equations and develop models for hybrid terms – A priori analysis of SS derivatives for arbitrarily positioned SS lines