Scale-Adaptive Simulation (SAS) Turbulence Modeling. F.R. Menter, ANSYS Germany GmbH

Transcription

1 Scale-Adaptive Simulation (SAS) Turbulence Modeling F.. Menter, ANSYS German GmbH 1

2 Unstead ANS Based Models UANS (Unstead enolds averaged Navier Stoes) Methods UANS gives unphsical single mode unstead behavior Some improvement relative to stead state (ANS) but often not sufficient to capture main effects eduction of time step and refinement of mesh do not benefit the simulation UANS SAS (Scale-Adaptive Simulation) Method Extends UANS to man technical flows Provides LES -content in unstead regions Produces information on turbulent spectrum Can be used as basis for acoustics simulations SAS-UANS

3 Assumptions Two-Equation Models Largest eddies are most effective in mixing Two scales are minimum for statistical description of large turbulence scales Two model equations of independent variables define the two scales Equation for turbulent inetic energ is representing the large scale turbulent energ Second equation (e, w, L) to close the sstem Each equation defines one independent scale Both e- and w-equations describe the smallest (dissipate) eddies, whereas two-equation models describe the largest scales otta developed an exact transport equation for the large turbulent length scales. This is a much better basis for a term-b-term modelling approach 3

4 4 Classical Derivation Equation Models t x x c P x U t w t x x P x U t w w w w w w w w t The -equation: Can be derived exactl from the Navier-Stoes equations Term-b-term modelling The e- (w-) equation: Exact equation for smallest (dissipation) scales Model for large scales not based on exact equation Modelled in analog to - equation and dimensional analsis Danger that not all effects are included

5 Source Terms Equilibrium -w Model Onl one Scale in Sources (S~1/T) Input S ( ) ( U ) t t S cw t x x x ( w) ( U w) t w cw1s cww t x x w x Turbulence Model One input scale two output scales? Source terms do not contain information on two independent scales Output Output w 5

6 Determination of L in -w Model -equation: ) t ( U) x ( ( S c w ) w w Diffusion term carries information on shear-laer thicness Turbulent length scale proportional to shear laer thicness Finite thicness laer required Computed length scale independent of details inside turbulent laer No scale-resolution, as L t alwas large and dissipative S cw c 1 0 ( ) w w 0 w ~ S cs c L ~ t from w-equation ~ S S ~ ~ ~ w S 6

7 otta s Length Scale Equation To avoid the problem that the e (w) equation is an equation for the smallest scales, an equation for the large (integral) scales is needed. This requires first a mathematical definition of an integral length scale, L. In otta s (1968) approach this definition is based on two-point correlations Based on that definition of L, an exact transport equation can be derived from the Navier-Stoes equations (the actual equation is based on L) This exact equation is then modelled term-b-term otta, J.C.: Über eine Methode zur Berechnung turbulenter Scherströmungen, Aerodnamische Versuchsanstalt Göttingen, ep. 69 A14, (1968). 7

8 Two-Point Velocit Correlations Measurement of velocit fluctuations with two probes at two different locations For small r, all eddies contribute For large r, onl large scales contribute For r > L, correlation goes to zero Integral vs. r proportional to size of large eddies L ~ i ui '( x, t) u '( x r, t) u '( x, t) u '( x, t) i r Shifted Probe Fixed Probe r Eddies L r ~ 1 i 8

9 otta s -L Model Integral Length Scale: The integral of the correlations provides a quantit, L, with dimension length. L is based onl on velocit fluctuations and can therefore be described b the Navier-Stoes equations. Exact equation for L (or L,..) can be derived. L is a true measure of the size of the largest eddies L( x, t) c ( x, t, r) dr ii 1 L r ii r 9

10 10 Exact Transport Equation Integral Length-Scale (otta) Exact transport equations for F=L (boundar laer form): F F F vp v p dr r r dr r dr r x U dr x U x U t i i ii i i i i ' ' ) ( 16 3 ) ( 16 3 ) ( ) ( ) ( 1 1 Important term: F L(x) with i dr r x U 1 ) ( 16 3

11 Important term: Expansion of Gradient Function 3 U( x r ) U( x ) U( x ) U( x ) r r... 3 U ( x r ) 1 dr U ( x) 1 dr U( x) r 1 dr 1 3 U( x) 3 r 1 dr otta: U( x) r 1 dr 0 i 1 Due to smmetr of i with respect to r for homogeneous turbulence r 11

12 1 If z 3 =0 - No natural length scale No fundamental difference to other scale-equations Transport Equation Integral Length-Scale (otta) Transport equations for L: F F F F q c c x U L x U L uv x U t t L i i z z 3/ ) ( ) ( ) ( ) ( Equation has a natural length scale: / / U U c c L l z Problem 3 rd derivative: Non-intuitive Numericall problematic z 3 0

13 Virtual Experiment 1D Flow U r 1 dr 0? Logarithmic laer L t = ~ 1 xv( ux u x r ) uxv( x) const. v( x) ~ ( r ~ ) ( r ) ~ ( r I II 1 1 ~ ( r III II 1 1 ~ ( r ) ) ~ ~ ) ( r ~ ) ( r III I 1 1 ( r III III 1 1 w ) ) 1 asmmetric 13 r 1dr 0

14 With: New -Equation Model (KSKL) U 3/ 3/ 4 t P c t x L x x F U F F 1 t F z1p z L '' t U z 3 t x F F L t 1/ 4 c F Ui Ui Ui Ui U ' U ' ; U '' ; LvK x x x x x x U '' v. Karman length-scale as natural length-scale: L U ~ U / / L vk 14

15 SAS Model Derivation Using the exact definition and transport equation of otta, we re-formulated the equation for the second turbulence scale. We use a term-b-term modelling approach based on the exact equation. This results in the inclusion of the second velocit derivative U in the scale equation Based on U the scale equation is able to adust to resolved scales in the flow. The KSKL model is one variant of the SAS modelling concept, as these terms can also be transformed into other equations (e- or w). 15

16 Tranformation: Transformation of SAS Terms to SST Model F 1/ c 1 4 w Dw Dt 1 1/ c 4 D Dt F 1 c 1/ 4 1 F D Dt F DF Dt w D Dt w DF F Dt w U w t w 1 w w w L S w z S t x x w x F w x x w x x LvK Wilcox Model BSL (SST) Model SAS L vk U / U / 16

17 -D Stationar Flows: KSKL - ANS NACA-441 airfoil at 14 : trailing edge separation KSKL 17

18 Limitation of Growth b U Homogenous Shear Inhomogeneous Shear du d w ~ L const. du d du d w ~ const. du d L L vk Eddies grow to infinit Edd growth limited b LvK. 18

19 One Model Two Modes ANS Model L~ SAS L~ SAS U( ) U0 sin ANS 19

20 SAS Modell - D Periodic Hill Scale-Adaptation based on t t = h/u B 4 higher t higher t 0

21 SAS Modell - D Periodic Hill Time averaged velocit profiles U 1

22 Fluent-SAS Model Volvo Bluff Bod : Cold Case SAS-SST DES-SST Q = 1 e 6

23 VOLVO Cold Case Time-averaged U-velocit 3

24 Test case: Mirror Geometr EU proect DESIDE Testcase Plate dimensions LW=.41.6 Clinder Diameter : D = 0. m ear Face location : 0.9 m Free stream Velocit: 140 m/h.4 m 1.6 m e D : m/h 0.9 m Mach:

25 Test case: Mesh Mesh: Box around the Plate & Clinder Height of domain: 10 diameters (D=0.m) Coarse and fine meshes wall-normal distance around 1-3 *10-4 m obstacle edges resolution: step sizes around 0.0*D (height) *D (circumf.) Flow: Air as ideal gas 10D 5

26 SPL [db] SPL [db] SPL [db] Validation: Near field SPL Grid ~ 3 million nodes Sensor 11 Freestream Velocit = 140 m/h Experimental data SAS model Frequenc [Hz] Sensor 1 Freestream Velocit = 140 m/h Experimental data SAS model Frequenc [Hz] Sensors downstream the mirror Sensor 13 Freestream Velocit = 140 m/h Experimental data SAS model Frequenc [Hz]

27 Blow-Down Simulation SAS (SST) Mesh 1x10 7 control volumes hbrid unstructured Scale resolving results: SAS and DES show similar flow pattern SAS model does not rel on grid spacing SAS can be applied to moving meshes with more confidence Courtes VW AG Wolfsburg: O. Imberdis, M. Hartmann, H. Bensler, L. Kapitza VOLKSWAGEN AG, esearch and Development, Wolfsburg, German D. Thevenin Universit of Magdeburg 7

28 Flow Topolog and Mass Flow Mass flow ates Intae Valve Exp. ANS DES SAS 3 mm mm Courtes VW AG Wolfsburg: O. Imberdis, M. Hartmann, H. Bensler, L. Kapitza VOLKSWAGEN AG, esearch and Development, Wolfsburg, German D. Thevenin Universit of Magdeburg 8

29 Geometr of the Cavit D = 4 in L = 5 D, W = D L x x L x L z = 18 D x 17 D x 9 D M = 0.85 P = 6100 Pa T = K e = E 6 Massflow Inlet Pressure far field Outlet Smmetr 9

30 Mesh: 5.8 e 6 Cv double O-grid Inlet Side 30 Bottom

31 Turbulent structure b q-criterion Edd viscosit q = (q = 1 / (S:S W:W)) 31

32 Wave propagation b Fluctuating Densit Edd viscosit q = (q = 1 / (S:S W:W)) 3

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35 Testcase Description Experimental Test Facilit and Data The experimental data is provided b the Institute of Aerodnamics and Fluid Mechanics from TUM (not et released) Experiments are performed in a wind tunnel including a moving belt Courtes b TU Munich, Inst. of Aerodnamics 35

36 Computational Mesh 108,034,893 Cells Four efinement Boxes MF-Zones Number of Inflations First laer height Car mm oad mm 36 Courtes b TU Munich, Inst. of Aerodnamics

37 DrivAir Generic Car Model Courtes Tu Munich Currentl studied with ANSYS CFD (Fluent and CFX) Data not et public 37

38 Overall Summar SAS is a second generation UANS model It is derived on UANS arguments It can resolve turbulence structures with LES qualit A strong flow instabilit is required to generate new resolved turbulence Examples Flows past bluff bodies Strongl swirling flows (combustion chamber) Strongl interacting flows (mixing of two ets etc.) SAS Model is first and relativel save step into Scale- esolving Simulations (SS) modeling Worthwhile to tr Alternative Detached Edd Simulation (DES) 38