A well-posed and stable stochastic Galerkin formulation of the incompressible Navier–Stokes equations with random data
- Uni Research, N-5007 Bergen (Norway)
- Department of Mathematics, Computational Mathematics, Linköping University, SE-58183 Linköping (Sweden)
- Aerospace Engineering Sciences, University of Colorado Boulder, CO 80309 (United States)
We present a well-posed stochastic Galerkin formulation of the incompressible Navier–Stokes equations with uncertainty in model parameters or the initial and boundary conditions. The stochastic Galerkin method involves representation of the solution through generalized polynomial chaos expansion and projection of the governing equations onto stochastic basis functions, resulting in an extended system of equations. A relatively low-order generalized polynomial chaos expansion is sufficient to capture the stochastic solution for the problem considered. We derive boundary conditions for the continuous form of the stochastic Galerkin formulation of the velocity and pressure equations. The resulting problem formulation leads to an energy estimate for the divergence. With suitable boundary data on the pressure and velocity, the energy estimate implies zero divergence of the velocity field. Based on the analysis of the continuous equations, we present a semi-discretized system where the spatial derivatives are approximated using finite difference operators with a summation-by-parts property. With a suitable choice of dissipative boundary conditions imposed weakly through penalty terms, the semi-discrete scheme is shown to be stable. Numerical experiments in the laminar flow regime corroborate the theoretical results and we obtain high-order accurate results for the solution variables and the velocity divergence converges to zero as the mesh is refined.
- OSTI ID:
- 22570220
- Journal Information:
- Journal of Computational Physics, Vol. 306; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); ISSN 0021-9991
- Country of Publication:
- United States
- Language:
- English
Similar Records
Stochastic Galerkin methods for the steady-state Navier–Stokes equations
CCM Continuity Constraint Method: A finite-element computational fluid dynamics algorithm for incompressible Navier-Stokes fluid flows