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

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2 B. Frohnapfel, Jordanian German Winter Academy 2006 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

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3 B. Frohnapfel, Jordanian German Winter Academy 2006 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 HOW TO SOLVE THE BASIC EQUATIONS IN TURBULENT FLOWS

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4 B. Frohnapfel, Jordanian German Winter Academy 2006 REYNOLDS AVERAGING AND THE CLOSURE PROBLEM RANS: continuity equation:

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5 B. Frohnapfel, Jordanian German Winter Academy 2006 10 variables, 4 equations (Unclosed) Time-Averaged Equations + Modeling Additional Equations for = Eddy Viscosity Model Reynolds Stress Model PDF Model (probability density function) Closed! Reynolds-Averaged Closure Model complexity accuracy TURBULENCE MODELLING

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6 B. Frohnapfel, Jordanian German Winter Academy 2006 EDDY VISCOSITY VS. REYNOLDS STRESS MODELS Eddy viscosity models: k- model: solve transport equations for k and Reynolds stress models: solve transport equation for Reynolds stresses

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7 B. Frohnapfel, Jordanian German Winter Academy 2006 REYNOLDS STRESS MODEL Transport equation for Reynolds stresses: P ij T ij ij ij D ij transport equation contains three unknown correlations that need to be modeled !

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8 B. Frohnapfel, Jordanian German Winter Academy 2006 Flows visualisation of small-scale structure of turbulence at large Re (a) isotropic pattern, (b) and (c) are anisotropic patterns Flow structure in the wake behind a bullet Artificially produced patterns IS TURBULENCE ISOTROPIC?

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9 B. Frohnapfel, Jordanian German Winter Academy 2006 ANISOTROPY INVARIANT MAP two component turbulence axisymmetric turbulence isotropic turbulence II a = a ij a ji III a = a jk a kj a ij II a III a Lumley and Newman 1977 Lumley 1978 k- Model

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10 B. Frohnapfel, Jordanian German Winter Academy 2006 TURBULENT CHANNEL FLOW IN AI MAP Decaying Reynolds number y= y=0 y= y=0 wall y=0 channel centerline y=

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11 B. Frohnapfel, Jordanian German Winter Academy 2006 y=0 y= II a III a TURBULENT PIPE FLOW IN AI MAP y=0 y= y=0 y= II a III a

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12 B. Frohnapfel, Jordanian German Winter Academy 2006 TRAJECTORIES THROUGH AI MAP flow through sudden expansion x H turbulent scalar transport with mixing

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13 B. Frohnapfel, Jordanian German Winter Academy 2006 TWO POINT CORRELATION TECHNIQUE separate effects of local character from large scale fluid motions Chou (1945), Kolovandin and Vatutin (1969-1972) A B

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14 B. Frohnapfel, Jordanian German Winter Academy 2006 where REYNOLDS STRESS MODEL transport equation for the Reynolds stresses transport equation for the homogeneous part of the dissipation

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15 B. Frohnapfel, Jordanian German Winter Academy 2006 MODELLING OF DISSIPATION EQUATION interpolation functions:

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16 B. Frohnapfel, Jordanian German Winter Academy 2006 MODELLING OF TRANSPORT EQUATION FOR REYNOLDS STRESSES

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17 B. Frohnapfel, Jordanian German Winter Academy 2006 DYNAMIC EQUATIONS FOR INHOMOGENEOUS ANISOTROPIC TURBULENCE

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18 B. Frohnapfel, Jordanian German Winter Academy 2006 TEST CASES FOR HOMOGENEOUS FLOWS as given by Stanford conference (1980), compiled by Ferzinger

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19 B. Frohnapfel, Jordanian German Winter Academy 2006 TEST CASES FOR HOMOGENEOUS FLOWS axisymmetric strain plain strain

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20 B. Frohnapfel, Jordanian German Winter Academy 2006 accuracy of predictions for homogeneous flows: + 5% in kinetic energy and +10% in Reynolds stresses TEST CASES FOR HOMOGENEOUS FLOWS simple shear imposed rotation

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21 B. Frohnapfel, Jordanian German Winter Academy 2006 Flow Over a Backward-Facing Step Velocity profiles at stations x/H = 4, 6.5, 8, 14, 32. From left to right n = 0,1,2,3,4 and 5 reattachment length - experimentally: 6.1H INHOMOGENEOUS FLOWS

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22 B. Frohnapfel, Jordanian German Winter Academy 2006 INHOMOGENEOUS FLOWS Flow Over a Backward-Facing Step – Reynolds stress components

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23 B. Frohnapfel, Jordanian German Winter Academy 2006 Flow over a periodic arrangement of hills The mesh of hill flow test case. INHOMOGENEOUS FLOWS

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24 B. Frohnapfel, Jordanian German Winter Academy 2006 SWIRL FLOW - EXPERIMENTAL DATA

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25 B. Frohnapfel, Jordanian German Winter Academy 2006 SWIRL FLOW - SIMULATION DATA k- - Model AIRSM - Model mean velocity profiles at x/D = 3, 10, 17.3, 37.3, 44.8, 52.3, 81.7, 98.4

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26 B. Frohnapfel, Jordanian German Winter Academy 2006 Illustration of an exemplary geometry used for the stationary blow experimentexemplary inlet valves outlet valves cylinder evaluation plane inlet channel Cross section of the velocity distribution in the tumble channel (AI-computations) INDUSTRIAL COMPUTATIONS

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27 B. Frohnapfel, Jordanian German Winter Academy 2006 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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