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Turbulence modeling II: Anisotropy Considerations

Jordanian-German Winter Academy 2006. Turbulence modeling II: Anisotropy Considerations. Bettina Frohnapfel LSTM - Chair of Fluid Dynamics Friedrich-Alexander University Erlangen-Nuremberg, Germany. OVERVIEW. How to solve turbulent flow problems?

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Turbulence modeling II: Anisotropy Considerations

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  1. Jordanian-German Winter Academy 2006 Turbulence modeling II: Anisotropy Considerations Bettina Frohnapfel LSTM - Chair of Fluid Dynamics Friedrich-Alexander University Erlangen-Nuremberg, Germany

  2. OVERVIEW • How to solve turbulent flow problems? • Review of Reynolds averaging and the need for turbulence models • Transport equations of the Reynolds stresses • Is turbulence isotropic? • Introduction of the anisotropy invariant map • Introduction of two-point correlation technique • Model formulation of anisotropy invariant turbulence model

  3. HOW TO SOLVE THE BASIC EQUATIONS IN TURBULENT FLOWS • DNS: Direct Numerical Simulation • Solves the unsteady Navier-Stokes equations directly. • RANS Model: (Reynolds Averaged Navier-Stokes Model) • Motivation: Engineers are normally interested in knowing just a few quantitative properties of a turbulent flow. • Method: Using Reynolds-Averaged form of the Navier-Stokes equations with appropriate turbulence models. • LES: Large Eddy Simulation Motivation:The large scale motions are generally much more energetic than the small scales, and they are the most effective transporters of the conserved properties. Method: Large Scales -> Solve, Small Scales ->Model

  4. REYNOLDS AVERAGING AND THE CLOSURE PROBLEM continuity equation: RANS:

  5. Time-Averaged Equations Modeling Reynolds-Averaged Closure Model 10 variables, 4 equations (Unclosed) Closed! Additional Equations for TURBULENCE MODELLING + = Eddy Viscosity Model Reynolds Stress Model PDF Model (probability density function) accuracy complexity

  6. EDDY VISCOSITY VS. REYNOLDS STRESS MODELS Eddy viscosity models: k-e model: solve transport equations for k and e Reynolds stress models: solve transport equation for Reynolds stresses

  7. REYNOLDS STRESS MODEL Transport equation for Reynolds stresses: Pij Tij eij Dij Pij transport equation contains three unknown correlations that need to be modeled !

  8. IS TURBULENCE ISOTROPIC? Flow structure in the wake behind a bullet Artificially produced patterns Flows visualisation of small-scale structure of turbulence at large Re (a) isotropic pattern, (b) and (c) are anisotropic patterns

  9. ANISOTROPY INVARIANT MAP IIa= aijaji IIa two component turbulence IIIa= ajkakjaij axisymmetric turbulence k- e Model IIIa Lumley and Newman 1977 Lumley 1978 isotropic turbulence

  10. TURBULENT CHANNEL FLOW IN AI MAP wall y=0 channel centerline y=d y=0 y=0 y=0 y=d y=d y=d Decaying Reynolds number

  11. y=0 y=d TURBULENT PIPE FLOW IN AI MAP y=0 y=d IIa IIa y=0 y=d IIIa IIIa

  12. TRAJECTORIES THROUGH AI MAP flow through sudden expansion turbulent scalar transport with mixing x H

  13. TWO POINT CORRELATION TECHNIQUE Chou (1945), Kolovandin and Vatutin (1969-1972) separate effects of local character from large scale fluid motions B A

  14. REYNOLDS STRESS MODEL transport equation for the Reynolds stresses where transport equation for the homogeneous part of the dissipation

  15. MODELLING OF DISSIPATION EQUATION interpolation functions:

  16. MODELLING OF TRANSPORT EQUATION FOR REYNOLDS STRESSES

  17. DYNAMIC EQUATIONS FOR INHOMOGENEOUS ANISOTROPIC TURBULENCE

  18. TEST CASES FOR HOMOGENEOUS FLOWS as given by Stanford conference (1980), compiled by Ferzinger

  19. TEST CASES FOR HOMOGENEOUS FLOWS axisymmetric strain plain strain

  20. TEST CASES FOR HOMOGENEOUS FLOWS imposed rotation simple shear accuracy of predictions for homogeneous flows: + 5% in kinetic energy and +10% in Reynolds stresses

  21. INHOMOGENEOUS FLOWS Flow Over a Backward-Facing Step reattachment length - experimentally: 6.1H Velocity profiles at stations x/H = 4, 6.5, 8, 14, 32. From left to right n = 0,1,2,3,4 and 5

  22. INHOMOGENEOUS FLOWS Flow Over a Backward-Facing Step – Reynolds stress components

  23. INHOMOGENEOUS FLOWS Flow over a periodic arrangement of hills The mesh of hill flow test case.

  24. SWIRL FLOW - EXPERIMENTAL DATA

  25. SWIRL FLOW - SIMULATION DATA k-e - Model AIRSM - Model mean velocity profiles at x/D = 3, 10, 17.3, 37.3, 44.8, 52.3, 81.7, 98.4

  26. inlet channel evaluation plane inlet valves cylinder outlet valves INDUSTRIAL COMPUTATIONS Illustration of an exemplary geometry used for the stationary blow experiment Cross section of the velocity distribution in the tumble channel (AI-computations)

  27. CONCLUSIONS AND OUTLOOK • Reynolds stress models provide more accurate predictions than eddy viscosity models but need more computational time • In Reynolds stress models anisotropy considerations can be taken into account • AIRSM - Anisotropy Invariant Reynolds Stress Model is based on nearly no empirical input • AIRSM shows good prediction quality and can be improved systematically • Computer power is constantly increasing so that direct numerical simulations (DNS) can be increasingly used for engineering predictions

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