Uncertainty Quantification and Propagation in Numerical Simulations of Flow-Structure Interactions

1. Uncertainty Quantification and Propagation in Numerical Simulations of Flow-Structure Interactions • Didier Lucor • Laboratoire de Modélisation en Mécanique UPMC - UMR CNRS 760 • Boite 162, 4 place Jussieu Tel: 33 (0)1 44 27 87 12 • 75252 Paris Cedex 05 Fax: 33 (0)1 44 27 52 59 • France email: lucor@lmm.jussieu.fr

2. DNS of 3D turbulent flow past a rigid cylinder at Re=10000 • Re=10000 • DoF:200 Millions • Number of Processors:512 • Use of multi-level parallelism (MPI-MPI or OpenMP-MPI) Dong & Karniadakis, JFS, (2005).

3. Exponential shear case Uniform case Linear shear case

4. DNS-Experiments comparison of a turbulent flow past a rigid stationary cylinder Re=3900 DNS: Ma & Karniadakis, JFM, (2000). Experiments: Ong & Wallace, Experiments in Fluids (1996). Energy spectrum based on the transverse velocity component of the flow field in the wake (x/D=7).

5. Sources of uncertainty Random inflow condition (stochastic process) • Parameters, simulation constants, material properties • Transport coefficients, physical properties • geometry • Boundary conditions, initial conditions • Physical laws, numerical schemes Random structural parameters Uncertain boundary conditions

6. generalized Polynomial Chaos (gPC) Not limited to a Gaussian distribution! There exists a unique correspondence between the PDF of the stochastic input and the weightingfunction of the orthogonal polynomials. Inner product:

7. Polynomials choice Uniform distribution approximation using the Gaussian/Hermite Chaos.

8. : random space dimension : highest polynomial order gPC summary Example: : Gaussian distribution : Hermite polynomials N=2; P=2 with not limited to Gaussian distributions! Mean: Variance:

9. σU Noisy inflow past an oscillating cylinder 0% Uncertainty at the inflow velocity boundary condition Lucor & Karniadakis, Phys. Rev. Lett. (2005). • Dramatic change in the vortices arrangement in the wake. • The shedding-mode switches from a (P+S) pattern to a (2S) mode in the presence of uncertainty. • For a given level of uncertainty, the change is more pronounced for higher Reynolds numbers. 10% Deterministic forced motion 20% 30%

10. Instantaneous vorticity field RMS values Lucor & Karniadakis, PRL, (2005).

11. Uncertainty in flow-structure interaction • Objectives: Uncertainty propagation and quantification in flow-structure interactions coupled phenomena. Sensitivity of the solution to the different random inputs. Stochastic response surfaces. Reliability and robustness of the structures to random perturbations. • Technical approach: Intrusive and non-intrusive use of the generalized Polynomial Chaos; Karhunen-Loève stochastic process representation. Development of efficient and accurate stochastic numerical codes DNS-gPC & LES-gPC. Large-scale parallel numerical simulations. • Applications: Different sources of uncertainty: - advection velocity (écoulement aux bords) - Source term - Initial conditions - physical properties of the structure - geometry - Boundary conditions Incompressible 2D & 3D turbulent flows in complex stationary or moving geometry. Linear & nonlinear structural models, higher Re numbers. DNS: Dong & Karniadakis, JFS, (2005).

13. Turbulence et simulation aux grandes échelles (LES) • Objectifs: • Propager et quantifier les incertitudes dans les petites échelles (sous-maille) de l'écoulement. • Quel est l’espace engendré par un modèle sous-maille? Quelles sont les quantités statistiques les moins sensibles (les plus robustes) donc les plus fiables? • Construction de nouveaux modèles sous-maille. Etude de la sensibilité de la solution aux différents paramètres des modèles sous-maille. • Approche technique: • Utilisation intrusive ou non-intrusive des polynômes de chaos généralisés et représentation de Karhunen-Loève. • Ecriture d’un code de calcul stochastique (LES-PCg) et comparaison/validation avec un code (DNS-PCg) existant. • Calculateurs parallèles haute performance. • Applications: • Ecoulements turbulents ouverts (de type sillage) et écoulements pariétaux à haut nombre de Reynolds.