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:
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).
Linear shear case
Exponential shear case
Uniform case
DNS-Experiments comparison of a turbulent flow past a rigid stationary
cylinder
Re=3900
Energy spectrum based on the transverse velocity component of the flow field in the wake (x/D=7).
DNS: Ma & Karniadakis, JFM, (2000).Experiments: Ong & Wallace, Experiments in Fluids (1996).
Sources of uncertainty
Parameters, simulation constants, material properties
Transport coefficients, physical properties
geometry
Boundary conditions, initial conditions
Physical laws, numerical schemes
Random inflow condition (stochastic process)
Uncertain boundary conditions
Random structural
parameters
generalized Polynomial Chaos (gPC)
Not limited to a Gaussian distribution!
There exists a unique correspondence between the PDF of the stochastic
input and the weighting function of the orthogonal polynomials.
Inner product:
Polynomials choice
Uniform distribution approximation using the Gaussian/Hermite Chaos.
gPC summary
with
: random space dimension
: highest polynomial order
Example:
: Gaussian distribution
: Hermite polynomials
N=2; P=2
not limited to Gaussian distributions!
Mean:Varianc
e:
Uncertainty at the inflow velocity boundary condition
Deterministic forced motion
Noisy inflow past an oscillating cylinder
30%
20%
10%
0%
σU
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.
Lucor & Karniadakis, Phys. Rev. Lett. (2005).
Instantaneous vorticity field RMS values
Lucor & Karniadakis, PRL, (2005).
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.
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.