1. © Fluent Inc. 12/18/2015 D1 Fluent Software Training TRN-98-006 Modeling Turbulent Flows

2. © Fluent Inc. 12/18/2015 D2 Fluent Software Training TRN-98-006  Unsteady, aperiodic motion in which all three velocity components fluctuate  mixing matter, momentum, and energy.  Decompose velocity into mean and fluctuating parts: U i (t)  U i + u i (t)  Similar fluctuations for pressure, temperature, and species concentration values. What is Turbulence? Time U i (t) UiUi u i (t)

3. © Fluent Inc. 12/18/2015 D3 Fluent Software Training TRN-98-006 Why Model Turbulence?  Direct numerical simulation of governing equations is only possible for simple low-Re flows.  Instead, we solve Reynolds Averaged Navier-Stokes (RANS) equations: where (Reynolds stresses)  Time-averaged statistics of turbulent velocity fluctuations are modeled using functions containing empirical constants and information about the mean flow.  Large Eddy Simulation numerically resolves large eddies and models small eddies. (steady, incompressible flow w/o body forces)

4. © Fluent Inc. 12/18/2015 D4 Fluent Software Training TRN-98-006 Is the Flow Turbulent? External Flows Internal Flows Natural Convection along a surface around an obstacle where Other factors such as free-stream turbulence, surface conditions, and disturbances may cause earlier transition to turbulent flow. L = x, D, D h, etc.

5. © Fluent Inc. 12/18/2015 D5 Fluent Software Training TRN-98-006 How Complex is the Flow?  Extra strain rates Streamline curvature Lateral divergence Acceleration or deceleration Swirl Recirculation (or separation) Secondary flow  3D perturbations  Transpiration (blowing/suction)  Free-stream turbulence  Interacting shear layers

6. © Fluent Inc. 12/18/2015 D6 Fluent Software Training TRN-98-006 Choices to be Made Turbulence Model & Near-Wall Treatment Flow Physics Accuracy Required Computational Resources Turnaround Time Constraints Computational Grid

7. © Fluent Inc. 12/18/2015 D7 Fluent Software Training TRN-98-006 Zero-Equation Models One-Equation Models Spalart-Allmaras Two-Equation Models Standard k-  RNG k-   Realizable k-  Reynolds-Stress Model Large-Eddy Simulation Direct Numerical Simulation Turbulence Modeling Approaches Include More Physics Increase Computational Cost Per Iteration Available in FLUENT 5 RANS-based models

8. © Fluent Inc. 12/18/2015 D8 Fluent Software Training TRN-98-006  RANS equations require closure for Reynolds stresses.  Turbulent viscosity is indirectly solved for from single transport equation of modified viscosity for One-Equation model.  For Two-Equation models, turbulent viscosity correlated with turbulent kinetic energy (TKE) and the dissipation rate of TKE.  Transport equations for turbulent kinetic energy and dissipation rate are solved so that turbulent viscosity can be computed for RANS equations. Reynolds Stress Terms in RANS-based Models Turbulent Kinetic Energy: Dissipation Rate of Turbulent Kinetic Energy: Turbulent Viscosity: Boussinesq Hypothesis: (isotropic stresses)

9. © Fluent Inc. 12/18/2015 D9 Fluent Software Training TRN-98-006  Turbulent viscosity is determined from:  is determined from the modified viscosity transport equation:  The additional variables are functions of the modified turbulent viscosity and velocity gradients. One Equation Model: Spalart-Allmaras Generation Diffusion Destruction

10. © Fluent Inc. 12/18/2015 D10 Fluent Software Training TRN-98-006 One-Equation Model: Spalart-Allmaras  Designed specifically for aerospace applications involving wall- bounded flows. Boundary layers with adverse pressure gradients turbomachinery  Can use coarse or fine mesh at wall Designed to be used with fine mesh as a “low-Re” model, i.e., throughout the viscous-affected region. Sufficiently robust for relatively crude simulations on coarse meshes.

11. © Fluent Inc. 12/18/2015 D11 Fluent Software Training TRN-98-006 Two Equation Model: Standard k-  Model Turbulent Kinetic Energy Dissipation Rate are empirical constants (equations written for steady, incompressible flow w/o body forces) Convection Generation Diffusion Destruction Convection Generation Diffusion

12. © Fluent Inc. 12/18/2015 D12 Fluent Software Training TRN-98-006 Two Equation Model: Standard k-  Model  “Baseline model” (Two-equation) Most widely used model in industry Strength and weaknesses well documented  Semi-empirical k equation derived by subtracting the instantaneous mechanical energy equation from its time-averaged value  equation formed from physical reasoning  Valid only for fully turbulent flows  Reasonable accuracy for wide range of turbulent flows industrial flows heat transfer

13. © Fluent Inc. 12/18/2015 D13 Fluent Software Training TRN-98-006 Two Equation Model: Realizable k-   Distinctions from Standard k-  model: Alternative formulation for turbulent viscosity where is now variable (A 0, A s, and U* are functions of velocity gradients) Ensures positivity of normal stresses; Ensures Schwarz’s inequality; New transport equation for dissipation rate,  : Generation Diffusion Destruction Buoyancy

14. © Fluent Inc. 12/18/2015 D14 Fluent Software Training TRN-98-006  Shares the same turbulent kinetic energy equation as Standard k-   Superior performance for flows involving: planar and round jets boundary layers under strong adverse pressure gradients, separation rotation, recirculation strong streamline curvature Two Equation Model: Realizable k- 

15. © Fluent Inc. 12/18/2015 D15 Fluent Software Training TRN-98-006 Two Equation Model: RNG k-  Turbulent Kinetic Energy Dissipation Rate Convection Diffusion Dissipation Generation where are derived using RNG theory (equations written for steady, incompressible flow w/o body forces) Additional term related to mean strain & turbulence quantities Convection Generation Diffusion Destruction

16. © Fluent Inc. 12/18/2015 D16 Fluent Software Training TRN-98-006 Two Equation Model: RNG k-   k-  equations are derived from the application of a rigorous statistical technique (Renormalization Group Method) to the instantaneous Navier- Stokes equations.  Similar in form to the standard k-  equations but includes: additional term in  equation that improves analysis of rapidly strained flows the effect of swirl on turbulence analytical formula for turbulent Prandtl number differential formula for effective viscosity  Improved predictions for: high streamline curvature and strain rate transitional flows wall heat and mass transfer

17. © Fluent Inc. 12/18/2015 D17 Fluent Software Training TRN-98-006 Reynolds Stress Model Generation Pressure-Strain Redistribution Dissipation Turbulent Diffusion (modeled) (related to  ) (modeled) (computed) (equations written for steady, incompressible flow w/o body forces) Reynolds Stress Transport Eqns. Pressure/velocity fluctuations Turbulent transport

18. © Fluent Inc. 12/18/2015 D18 Fluent Software Training TRN-98-006 Reynolds Stress Model  RSM closes the Reynolds-Averaged Navier-Stokes equations by solving additional transport equations for the Reynolds stresses. Transport equations derived by Reynolds averaging the product of the momentum equations with a fluctuating property Closure also requires one equation for turbulent dissipation Isotropic eddy viscosity assumption is avoided  Resulting equations contain terms that need to be modeled.  RSM has high potential for accurately predicting complex flows. Accounts for streamline curvature, swirl, rotation and high strain rates Cyclone flows, swirling combustor flows Rotating flow passages, secondary flows

19. © Fluent Inc. 12/18/2015 D19 Fluent Software Training TRN-98-006 Large Eddy Simulation  Large eddies: Mainly responsible for transport of momentum, energy, and other scalars, directly affecting the mean fields. Anisotropic, subjected to history effects, and flow-dependent, i.e., strongly dependent on flow configuration, boundary conditions, and flow parameters.  Small eddies: Tend to be more isotropic and less flow-dependent More likely to be easier to model than large eddies.  LES directly computes (resolves) large eddies and models only small eddies (Subgrid-Scale Modeling).  Large computational effort Number of grid points, N LES  Unsteady calculation

20. © Fluent Inc. 12/18/2015 D20 Fluent Software Training TRN-98-006 Comparison of RANS Turbulence Models

21. © Fluent Inc. 12/18/2015 D21 Fluent Software Training TRN-98-006 Near-Wall Treatments  Most k-  and RSM turbulence models will not predict correct near-wall behavior if integrated down to the wall.  Special near-wall treatment is required. Standard wall functions Nonequilibrium wall functions Two-layer zonal model Boundary layer structure

22. © Fluent Inc. 12/18/2015 D22 Fluent Software Training TRN-98-006 Standard Wall Functions Mean Velocity Temperature where and P is a function of the fluid and turbulent Prandtl numbers. thermal sublayer thickness

23. © Fluent Inc. 12/18/2015 D23 Fluent Software Training TRN-98-006 Nonequilibrium Wall Functions  Log-law is sensitized to pressure gradient for better prediction of adverse pressure gradient flows and separation.  Relaxed local equilibrium assumptions for TKE in wall-neighboring cells.  Thermal law-of-wall unchanged where

24. © Fluent Inc. 12/18/2015 D24 Fluent Software Training TRN-98-006 Two-Layer Zonal Model  Used for low-Re flows or flows with complex near-wall phenomena.  Zones distinguished by a wall- distance-based turbulent Reynolds number  High-Re k-  models are used in the turbulent core region.  Only k equation is solved in the viscosity-affected region.   is computed from the correlation for length scale.  Zoning is dynamic and solution adaptive.

25. © Fluent Inc. 12/18/2015 D25 Fluent Software Training TRN-98-006 Comparison of Near Wall Treatments

26. © Fluent Inc. 12/18/2015 D26 Fluent Software Training TRN-98-006 Computational Grid Guidelines Wall Function Approach Two-Layer Zonal Model Approach l First grid point in log-law region l At least ten points in the BL. l Better to use stretched quad/hex cells for economy. l First grid point at y +  1. l At least ten grid points within buffer & sublayers. l Better to use stretched quad/hex cells for economy.

27. © Fluent Inc. 12/18/2015 D27 Fluent Software Training TRN-98-006 Estimating Placement of First Grid Point  Estimate the skin friction coefficient based on correlations either approximate or empirical: Flat Plate- Pipe Flow-  Compute the friction velocity:  Back out required distance from wall: Wall functions Two-layer model  Use post-processing to confirm near-wall mesh resolution y 1 = 250 /u  y 1 = / u 

28. © Fluent Inc. 12/18/2015 D28 Fluent Software Training TRN-98-006 Setting Boundary Conditions  Characterize turbulence at inlets & outlets (potential backflow) k-  models require k and  Reynolds stress model requires R ij and   Several options allow input using more familiar parameters Turbulence intensity and length scale length scale is related to size of large eddies that contain most of energy. For boundary layer flows: l  0.4  99 For flows downstream of grids /perforated plates:l  opening size Turbulence intensity and hydraulic diameter Ideally suited for duct and pipe flows Turbulence intensity and turbulent viscosity ratio For external flows:  Input of k and  explicitly allowed (non-uniform profiles possible).

29. © Fluent Inc. 12/18/2015 D29 Fluent Software Training TRN-98-006 GUI for Turbulence Models Define  Models  Viscous... Turbulence Model options Near Wall Treatments Inviscid, Laminar, or Turbulent Additional Turbulence options

30. © Fluent Inc. 12/18/2015 D30 Fluent Software Training TRN-98-006 Example: Channel Flow with Conjugate Heat Transfer adiabatic wall cold air V = 50 fpm T = 0 °F constant temperature wall T = 100 °F insulation 1 ft 10 ft P Predict the temperature at point P in the solid insulation

31. © Fluent Inc. 12/18/2015 D31 Fluent Software Training TRN-98-006 Turbulence Modeling Approach  Check if turbulent  Re D h = 5,980  Developing turbulent flow at relatively low Reynolds number and BLs on walls will give pressure gradient  use RNG k-  with nonequilibrium wall functions.  Develop strategy for the grid Simple geometry  quadrilateral cells Expect large gradients in normal direction to horizontal walls  fine mesh near walls with first cell in log-law region. Vary streamwise grid spacing so that BL growth is captured. Use solution-based grid adaption to further resolve temperature gradients.

32. © Fluent Inc. 12/18/2015 D32 Fluent Software Training TRN-98-006 Velocity contours Temperature contours BLs on upper & lower surfaces accelerate the core flow Prediction of Momentum & Thermal Boundary Layers Important that thermal BL was accurately resolved as well P

33. © Fluent Inc. 12/18/2015 D33 Fluent Software Training TRN-98-006 Example: Flow Around a Cylinder wall 1 ft 2 ft air V = 4 fps Compute drag coefficient of the cylinder 5 ft 14.5 ft

34. © Fluent Inc. 12/18/2015 D34 Fluent Software Training TRN-98-006  Check if turbulent  Re D = 24,600  Flow over an object, unsteady vortex shedding is expected, difficult to predict separation on downstream side, and close proximity of side walls may influence flow around cylinder  use RNG k-  with 2-layer zonal model.  Develop strategy for the grid Simple geometry & BLs  quadrilateral cells. Large gradients near surface of cylinder & 2-layer model  fine mesh near surface & first cell at y + = 1. Use solution-based grid adaption to further resolve pressure gradients. Turbulence Modeling Approach

35. © Fluent Inc. 12/18/2015 D35 Fluent Software Training TRN-98-006 Grid for Flow Over a Cylinder

36. © Fluent Inc. 12/18/2015 D36 Fluent Software Training TRN-98-006 Prediction of Turbulent Vortex Shedding Contours of effective viscosity  eff =  +  t C D = 0.53 Strouhal Number = 0.297 where

37. © Fluent Inc. 12/18/2015 D37 Fluent Software Training TRN-98-006 Summary: Turbulence Modeling Guidelines  Successful turbulence modeling requires engineering judgement of: Flow physics Computer resources available Project requirements Accuracy Turnaround time Turbulence models & near-wall treatments that are available  Begin with standard k-  and change to RNG or Realizable k-  if needed.  Use RSM for highly swirling flows.  Use wall functions unless low-Re flow and/or complex near-wall physics are present.