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Chris Lacor Dept Mechanical Engineering

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33 pages
Chris Lacor Dept. Mechanical Engineering Vrije Universiteit Brussel With thanks to: Dr. S. Smirnov, Ir. Ch. Vandenhaute, Ir. M. Raes, Ir. G. Onorato Intrusive Polynomial Chaos for the Compressible Navier-Stokes Equations

  • input uncertainties

  • polynomials

  • compressible navier-stokes

  • uncertain input parameters

  • pc decomposition

  • basis functions

  • gramm-schmidt procedure

  • ??? ?

  • intrusive approach


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Intrusive Polynomial Chaos for the Compressible Navier+Stokes Equations
Chris Lacor Dept. Mechanical Engineering Vrije Universiteit Brussel
With thanks to: Dr. S. Smirnov, Ir. Ch. Vandenhaute, Ir. M. Raes, Ir. G. Onorato
Polynomial Chaos
Framework
– NODESIM European STREP project 1/11/06+1/03/10
– WP3.3: Polynomial Chaos Methods
– TUD: non+intrusive approach
– VUB: intrusive approach
Symposium on Accuracy and Uncertainty in Flow Simul ations, ONERA, Chatillon, December 3, 2010
Polynomial Chaos
• PC method relies on a probabilistic framework. • Uncertain input parameters are modelled as stochastic variables (or fields) with known distributions (PDF). • For each stochastic variable a new dimension ξ is introduced • Solution is sought in a form of PC decomposition. • A set of orthogonal multidimensional polynomials ψ is used as basis functions. The polynomial type depends on the PDF of the uncertainty, see further • The deterministic coefficients are derived by Galerkin projection of the governing equations. • The resulting set of equations is coupled due to non+linear terms in the Navier+ Stokes equations.
1(x,t,1( !,...,n( !! u1u(x,t,Π1(Κ!,...,Πn(Κ!! p1p(x,t,Π1(Κ!,...,Πn(Κ!!
N Λ(x,t,!Κ1Λk(x,t!Yk1,...,Πn! k10 N u(x,t,Κ!1uk(x,t!Yk1,...,Πn! k10 N p(x,t,Κ!1pk(x,t!Yk1,...,Πn! k10
Symposium on Accuracy and Uncertainty in Flow Simul ations, ONERA, Chatillon, December 3, 2010
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