Sensitivity analysis for incompressible Navier--Stokes equations with uncertain viscosity using polynomial chaos method

Résumé

We present a stability estimate for the sensitivity of the incompressible Navier–Stokes equations under uncertainty in model parameters such as viscosity and initial or boundary conditions. The approach employs the stochastic Galerkin method, wherein the solution is represented using a generalized polynomial chaos expansion. The governing equations are projected onto stochastic basis functions, resulting in an extended coupled equation system. These coupled equations are challenging to solve numerically. A decoupling method is proposed to simplify their numerical resolution, which, along with the stability estimates, represents one of this study’s most valuable and original aspects. Finally, we present the lid-driven cavity numerical test to evaluate the polynomial chaos method and compare the solutions with the numerical data published in the literature.