New issues in LES of turbulent flows: multiphysics and uncertainty modelling

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1 New issues in LES of turbulent flows: multiphysics and uncertainty modelling Pierre Sagaut Institut Jean Le Rond d Alembert Université Pierre et Marie Curie- Paris 6, France Thanks to: J.C. Jouhaud, D. Lucor, J. Meyers, M. Meldi Ecole Centrale de Lyon October 30th, 2009

2 An example of complex flow 2

3 Motivation Simplified/incomplete boundary conditions CAD definition & mesh generation Physical modelling errors Complex system simulation with uncertainties Unknown/varying flow parameters Discretization errors 3

4 Outline 1. Response surface via Polynomial Chaos 2. Uncertainty in subgrid model calibration 3. Putting error/uncertainty bars on LES results 4

5 Uncertain system description What is the space of solutions spanned by uncertain parameters? f = f (t,z;x 1,x 2,...,x n ) df dt = G(t,Z;x 1,...,x n ) full solution space (t,z;x) Single fully deterministic solution (everything perfectly known/prescribed) 5

6 Uncertain system description (cont d) Probability Density Function P(x 1 ) P( f ) Solution P(x n ) x 1 f (x 1,...,x n ) x n Uncertain parameters Response surface f (x 1,...,x n ) Local sensitivity (x 1,...,x n ) 6

7 Generalized Polynomial Chaos Wiener (1938) : Homogeneous Chaos Theory Solution with uncertain random parameter Uncertain parameter Orthogonal polynomial basis functions Pseudo-spectral Galerkin projection of spanned solutions 7

8 Generalized Polynomial Chaos (cont d) Distribution Gaussian Gamma Beta/uniform Binomial Optimal polynomial basis Hermite Laguerre Legendre Krawtchouk gpc post-processing 8

9 Uncertain subgrid model calibration (Lucor, Meyers & Sagaut, J. Fluid Mech, 2007) Classical Smagorinsky-Lilly model Exact Smagorinsky constant expression (Meyers & Sagaut, J. Fluid Mech, 2007) Case-dependent parameters 9

10 Decaying HIT with uncertain Cs TKE decay Final TKE spectrum 10

11 Decaying HIT with uncertain Cs (cont d) Narrow pdf weakly sensitive mode PDF of energy spectrum wide pdf highly sensitive mode 11

12 Cs as a stochastic variable Meyers-Meneveau spectrum shape E(k) =C K ε 2/3 k 5/3 (kl) β f L (kl)f η (kη) f η (kη) = exp( α 1 kη) ( 1+ α 2(kη/α 4 ) α 3 1+(kη/α 4 ) α 3 ) f L (kl) = ( kl [(kl) p + α 5 ] 1/p ) 5/3+β+2 Uncertain parameters! 12

13 Cs as a stochastic variable Constrained problem with + 0 x 5/3 β C K f L (kη)f η (kηre 3/4 )d(kη) = 1 x 1/3 β Re 3β/4 C K f L (kηre 3/4 )f η (kη)d(kη) = 1/2 x 7/3 β Re 3β/4 C K f L (kηre 3/4 )f η (kη)d(kη) = 7S

14 Pdf of Cs Re λ =

15 Stochastic Cs analysis 15

16 Break: uncertain grid turbulence decay Corrsin & Comte-Bellot analysis E(k, t) Ak s A 1 L(t) t 2/(3+s) K(t) t 2(1+s)/(3+s) Re(t) t (1 s)/(3+s) 16

17 Break: uncertain grid turbulence decay Saturation effect (bounded physical domain): A 1 L(t) 1 K(t) t 2 Re(t) t 1

18 EDQNM/gPC analysis (Saffman spectrum) 18

19 Break: uncertain grid turbulence decay Isotropic turbulence decay exponent: Re 1 Re 1 Corresponding parameter tuning (k-ε model): Re 1 Re 1

20 Putting error bars on LES data In complex configurations: optimal values of subgrid models are not known best tuning of artificial viscosity parameter not known these two parameters are considered as uncertain parameters comparison with experimental data should account for possible numerical result variability 20

21 Case study (Jouhaud & Sagaut, J. Fluid Engng, 2008) 21

22 Response surface via Kriging Optimal linear unbiaised statistical predictor Based on sampling points (1 sample = 1 usual simulation) Several variants have been developed (cokriging, ) Kriging methods also provide an estimation of the interpolation error Sampling points can be generated dynamically to minimize the interpolation error (adaptive refinement) 22

23 Basic Kriging Method Estimator at position x Estimated function at position x s Covariance vector Covariance matrix a priori covariogram function: 23

24 Mean flow predicted by LES 24

25 Mean temperature field 25

26 Defining the best LES solution Which solution is the best LES solution? If some experimental data are available: some error functions can be defined solutions with the lowest error norm can be identified «clean» definition of the best LES solution(s) «best» LES a priori depends on the error measure 26

27 Error map Kriging-based response surface of error at X/D=8 L 1 norm L 2 norm 27

28 Error map (cont d) Kriging-based response surface of global error at both locations X/D=8 and X/D=1 L 1 norm L 2 norm 28

29 Best LES solution 29

30 Best LES solution 30

31 Conclusions Validation/certification not trivial! Uncertainties are ubiquitious in almost all application fields Mathematical tools do exist! Computational ressources now available Next step in modelling! 31

32 32

Fast Numerical Methods for Stochastic Computations

Fast Numerical Methods for Stochastic Computations Fast AreviewbyDongbinXiu May 16 th,2013 Outline Motivation 1 Motivation 2 3 4 5 Example: Burgers Equation Let us consider the Burger s equation: u t + uu x = νu xx, x [ 1, 1] u( 1) =1 u(1) = 1 Example: