Stochastic perturbation‐based finite element approach to fluid flow problems

Abstract

Purpose

To generalize the traditional 2nd order stochastic perturbation technique for input random variables and fields and to demonstrate for flow problems.

Design/methodology/approach

The methodology is based on an n‐th order expansion (perturbation) for input random parameters and state functions around their expected value to recover probabilistic moments of the response. A finite element formulation permits stochastic simulations on irregular meshes for practical applications.

Findings

The methodology permits approximation of expected values and covariances of quantities such as the fluid pressure and flow velocity using both symbolic and discrete FEM computations. It is applied to inviscid irrotational flow, Poiseulle flow and viscous Couette flow with randomly perturbed boundary conditions, channel height and fluid viscosity to illustrate the scheme.

Research limitations/implications

The focus of the present work is on the basic concepts as a foundation for extension to engineering applications. The formulation for the viscous incompressible problem can be implemented by extending a 3D viscous primitive variable finite element code as outlined in the paper. For the case where the physical parameters are temperature dependent this will necessitate solution of highly non‐linear stochastic differential equations.

Practical implications

Techniques presented here provide an efficient approach for numerical analyses of heat transfer and fluid flow problems, where input design parameters and/or physical quantities may have small random fluctuations. Such an analysis provides a basis for stochastic computational reliability analysis.

Originality/value

The mathematical formulation and computational implementation of the generalized perturbation‐based stochastic finite element method (SFEM) is the main contribution of the paper.

Citation

Kamiński, M. and Carey, G.F. (2005), "Stochastic perturbation‐based finite element approach to fluid flow problems", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 15 No. 7, pp. 671-697. https://doi.org/10.1108/09615530510613870