Modelling uncertainty in incompressible flow simulation using Galerkin based generalized ANOVA

Abstract

This paper presents a new algorithm, referred to here as Galerkin based generalized analysis of variance decomposition (GG-ANOVA) for modelling input uncertainties and its propagation in incompressible fluid flow. The proposed approach utilizes ANOVA to represent the unknown stochastic response. Further, the unknown component functions of ANOVA are represented using the generalized polynomial chaos expansion (PCE). The resulting functional form obtained by coupling the ANOVA and PCE is substituted into the stochastic Navier–Stokes equation (NSE) and Galerkin projection is employed to decompose it into a set of coupled deterministic ‘Navier–Stokes alike’ equations. Temporal discretization of the set of coupled deterministic equations is performed by employing Adams–Bashforth scheme for convective term and Crank–Nicolson scheme for diffusion term. Spatial discretization is performed by employing finite difference scheme. Implementation of the proposed approach has been illustrated by two examples. In the first example, a stochastic ordinary differential equation has been considered. This example illustrates the performance of proposed approach with change in nature of random variable. Furthermore, convergence characteristics of GG-ANOVA has also been demonstrated. The second example investigates flow through a micro channel. Two case studies, namely the stochastic Kelvin–Helmholtz instability and stochastic vortex dipole, have been investigated. For all the problems results obtained using GG-ANOVA are in excellent agreement with benchmark solutions.