1. A New Wall Function for Complex Turbulent Flows T. J. Craft, S. E. Gant, H. Iacovides, B. E. Launder and C. M. E. Robinson UMIST, Manchester, UK

2. Contents Introduction Overview of new wall function Results in 2-D/Axisymmetric flows –channel –impinging jet –spinning disc Generalization to 3-D curvilinear grids –Ahmed body flow –Rotor-stator flows Conclusions

3. Near-Wall Treatments in RANS Simulations Low-Reynolds-Number Model Integrate transport equations across the viscous sublayer to the wall using a fine near-wall grid +Good accuracy and conceptually simple –High computational cost due to highly elongated cells, high storage requirement Two main options in handling the wall boundary condition in turbulent flows: Standard Wall Functions Large near-wall cell in which profiles of velocity, temperature and turbulence parameters are assumed +Fast solution (typically 10 x faster than low-Re) low storage –Poor performance in complex flows and sensitive to near- wall cell size (difficult to obtain grid-independent result)

4. Summary of “Standard” Wall Functions Assumed log-law profiles for velocity and temperature Wall shear stress,, and average source terms (e.g., ) applied within wall-adjacent cell “Local equilibrium” between turbulence production and dissipation –Shear stress,, assumed constant/linear –Kinetic energy, k, assumed linear/quadratic –Length scale,, assumed linear

5. Wall-Function Research at UMIST Two new approaches recently developed at UMIST: Analytical WF: Adopts a simple prescribed viscosity (not velocity) that enables analytical integration of momentum and energy equations over the near-wall cell (Gerasimov et al.). Numerical WF: The present approach A simple prescribed viscosity is inadequate for really complex flows (e.g. recirculating flows)

6. Overview of Numerical Wall Function Near-wall control volume divided into subgrid volumes Wall-normal V-velocity from continuity within subgrid (and scaled) Upwind convection scheme used with saved subgrid values Transport equations solved across the subgrid for: –Mean-flow parameters: U, W, T –Turbulence parameters (e.g. k, )

7. Overview of Numerical Wall Function (cont.) Wall-parallel pressure gradient (dP/dx) calc’d from main-grid and assumed constant across subgrid and average source terms applied to main-grid as in standard wall-function treatments and average source terms ( ) calculated from subgrid solution

8. Subgrid Transport Equations Simplified low-Re transport equations for U, k, and T in plane Cartesian coordinates (convection terms in non-conservative form).

9. Numerical Solution of Subgrid Equations Similar to 1-D convection-diffusion problem Finite-volume method Central differences for diffusion, upwind for convection Tri-diagonal matrix algorithm (TDMA) Under-relaxation factors (U, k, , T) = (1.0, 0.9, 0.9, 1.0) One iteration of subgrid solver performed for each main-grid iteration: subgrid solution converges as main- grid converges Number of subgrid nodes must be sufficient for grid- independent result.

10. TEAM Code Finite-volume 2D/axisymmetric Cartesian grid Staggered velocity/scalar nodes SIMPLE pressure-correction algorithm Quadratic upstream-weighted (QUICK) or upwind differences for convection Central differences for diffusion

11. Channel Flow Results (Re = 100,000) Log-law predictions using the subgrid wall function with two different near- wall cell sizes:

12. Impinging Jet: grid and velocity vectors High-Re grid and velocity vectors, Re = 70,000, H/D = 4

13. Impinging Jet: low-Re and wall-function grids Close-up of near-wall grids: high-Re (left) and low-Re (right) DX=1 Low-Re Grid (90 x 70: grid-independent) wall Wall-Function Grids (45 x 70: four different near-wall cell sizes) DX=250DX=500

14. Impinging Jet Nusselt Number: Linear Numerical wall functionChieng & Launder wall function (Launder & Sharma model, Re = 70,000, H/D = 4)

15. Impinging Jet Nusselt Number: NLEVM Numerical wall functionChieng & Launder wall function (Craft et al. two-equation NLEVM, Re = 70,000, H/D = 4)

16. Computational Costs NLEVM (Craft et al.), axisymmetric impinging jet (Re = 70,000, H/D = 4), running on Silicon Graphics O 2 with same levels of under-relaxation and compiler optimisation in each case. Wall FunctionsLow-Re Craft et al. Chieng & Launder Numerical Number of Nodes 45 x 7045(+40) x 7090 x 70 CPU Time per Iteration (s) 0.1580.2600.324 No. of Iterations142613809116 Total CPU Time (s)2263592955 Relative CPU Time11.613.1

17. Spinning Free-Disc Flow Turbulent kinetic energy contours “Natural” transition from laminar to turbulent b.l. at Near-wall peak in radial velocity Tangential velocity exhibits log-law profile Linear model used in simulations

18. Free –Disc Computational Grids Low-Re grid 70 x 120 (axial x radial) Wall-function grid 22 x 120 (axial x radial)

19. Free-Disc Integral Nusselt Number Numerical wall function Chieng & Launder wall function

20. Free-Disc Radial Wall Shear Stress Numerical wall function Chieng & Launder wall function

21. Free-Disc Radial Velocity (Re  = 10 6 ) Low-Re Launder & Spalding WF Chieng & Launder WF Numerical WF “universal” log-lawexperiment

22. Free-Disc Computational Costs Linear k-  model for the spinning free-disc (Re   = 3.3  x 10 6 ) running on a single processor of an Origin 2000 with the same levels of under-relaxation and compiler optimisation. Model TestedNo. of NodesTime per Iter (s) No. of Iterations Total CPU Time (s) Relative CPU Time Launder & Spalding WF 120 x 28 0.0720091361 Chieng & Launder WF 120 x 28 0.0722081541.1 Numerical WF 120 x 28(+30) 0.1423633392.2 Low-Re 120 x 70 0.2019928399629.4

23. 3-D Flows: STREAM Code Lien & Leschziner 1994 Finite-volume Curvilinear structured multi-block grids Collocated storage with Rhie-Chow interpolation SIMPLE pressure-correction TVD version of QUICK scheme for convection terms Central differences for diffusion Fully-implicit or Crank-Nicolson Time scheme

24. Wall Function Generalization to 3-D Flows where (U, V, W) are grid-aligned contravariant subgrid velocity components in the (  directions, J is the Jacobian (equivalent to cell volume), g ii and g jj are metric tensors and the source term C includes pressure gradient, turbulence-equation sources and geometric terms. Generic form of subgrid transport equations  U, V, k  : Multi-block implementation using subgrid halo cells Efficient interpolation scheme to minimize cost of calculating geometric terms Same under-relaxation as for 2-D version x y z   

25. Ahmed Body Simplified car body with interchangeable rear slant angles. Previously studied at UMIST as part of EU project and subject of two ERCOFTAC workshops.

26. Ahmed body: flow regimes Flow features drag “crisis” at approx 30 o slant: < 30 o : flow attached on slant (high drag) > 30 o : flow separated at leading edge of slant (low drag)

27. Ahmed Body: flow domain Symmetry plane used No stilts Block decomposition (22 blocks)

28. Ahmed Body: standard WF’s Simulations using log-law wall functions at UMIST showed: -Linear k-  model attached for 25 o -Non-linear k-  model fully-separated for 25 o -Both models fully- separated for 35 o Aim: to see whether curvature of streamlines near surface of body has influential effect on formation of vortices and hence separation.

29. Ahmed Body: results

30. Ahmed body: rear slant Arrows indicate direction of velocity vector at near-wall main-grid node and subgrid nodes Significant skewing of velocity vector across near-wall cell due to side-edge vortex This effect would be ignored by log-law wall functions

31. Ahmed Body: Conclusions Numerical wall function successfully applied in complex 3D grid. However, no significant improvement in linear k-  results. Similar results have been obtained using full low-Re model treatments (from other research groups at ERCOFTAC workshops). Unable to get fully-converged results using non-linear k-  model with the new wall function, possible causes: -Unsteady flow behaviour -Problem with highly skewed near-wall cells -Positive feedback inherent with non-linear model Challenging test-case: at workshops none of the RANS or LES models correctly predicted flow patterns over rear slant.

32. Rotor-Stator Cavity Aim: to reproduce coherent structures observed experimentally by Czarny et al. using URANS. H R  Bottom disc rotates Top disc and walls stationary No through-flow

33. Rotor-Stator: 2D Simulations Itoh et al. test case: Re=  R 2 /    H/R  Numerical wall function results agree well with full low-Re model No problem with internal corner cells.

34. Rotor-Stator: 3D Simulations STREAM code parallelized using domain decomposition and MPI: -2-block axisymmetric grids running on local linux cluster -8-block version running on SGI Origin/Altix at CSAR Number of main-grid nodes in (R,  = (60,50,37) plus 30 subgrid nodes on each wall. Reynolds Number Re=  RH/  Disc spacing H/R=0.126

35. Rotor-Stator: Quasi-LES Using laminar viscosity only (deactivating turbulence model) Radial velocity isocontours (scale:  R max = 1.0) Axial velocity contours near stator surface

36. Rotor-Stator: URANS Eddy-viscosity contours: (red, yellow, green)  (40 , 30 , 20  ) Axial velocity contours near stator surface (scale:  R max = 1.0) Realizable k-  model

37. Rotor-Stator: URANS/LES Only OES shows continued presence of eddy structures Future work: examine filter-based (LES/URANS) approach using finer grids. Models tested: Linear k-  with/without Yap Realizable k-  Cubic non-linear k-  Linear Production (LP) (Laurence & Guimet) Organized Eddy Simulation (OES) (Braza) Filter-based URANS (Johansen et al.) (Using different initial turbulence levels for each case.)

38. Overall Conclusions The 2D test-cases showed that the new numerical wall function results agree well with full low-Re model treatments, in contrast to standard log- law wall functions. In 2D/axisymmetric flow the computational cost is approx. twice that of standard wall functions but an order-of-magnitude less than low-Re models. Storage requirements are roughly equivalent to a low-Re model. Changing the size of the near-wall main-grid cell has little/no effect (provided that there are sufficient subgrid cells). It can be applied to flows with internal/external corners (e.g. pipe expansion) and to 3D curvilinear grids.

39. Future Work Simplify the 3D-curvilinear wall function to use a local Cartesian grid instead of contravariant velocity components: -Easier mathematics -Simpler to code and debug -Fewer problems with skewed cells -Faster to compute and lower storage Investigate wall functions for LES: -Extensions to Balaras & Benocci wall function Continue rotor-stator calculations