A Stochastic Projection Method for Fluid Flow: I. Basic Formulation

We describe the construction and implementation of a stochastic Navier–Stokes solver. The solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation. The uncertainty quantification scheme is adapted from the spectral stochastic finite element method (SSFEM), which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension. In the SSFEM formalism, the stochastic dependence is represented in terms of the polynomial chaos system, and the coefficients in the corresponding spectral representation are obtained using a Galerkin approach. It is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection–diffusion equations for the various uncertainty fields, and in a decoupled system of pressure projection steps. This leads to a very efficient stochastic solver, whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel.