Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows

• Published in: Journal of computational physics 2010-08, Vol.229 (17), p.6084-6103
• Main Authors: Fu, S.C., So, R.M.C., Leung, W.W.F.
• Format: Article
• Language: English
• Subjects: ALGORITHMS; ARTERIES; BLOOD FLOW; BLOOD VESSELS; BODY; BOUNDARY CONDITIONS; CALCULATION METHODS; CARDIOVASCULAR SYSTEM; CHAOS THEORY; COMPRESSIBILITY; Computational fluid dynamics; Computational techniques; Computer simulation; COMPUTERIZED SIMULATION; DIFFERENTIAL EQUATIONS; EQUATIONS; Exact sciences and technology; Finite difference lattice Boltzmann method; Finite difference method; FINITE ELEMENT METHOD; FLUID FLOW; FUNCTIONS; INCOMPRESSIBLE FLOW; Incompressible Navier–Stokes; Mathematical analysis; MATHEMATICAL LOGIC; MATHEMATICAL METHODS AND COMPUTING; Mathematical methods in physics; Mathematical models; MATHEMATICAL SOLUTIONS; MATHEMATICS; MECHANICAL PROPERTIES; MONTE CARLO METHOD; NAVIER-STOKES EQUATIONS; NUMERICAL SOLUTION; ORGANS; PARTIAL DIFFERENTIAL EQUATIONS; Physics; POLYNOMIALS; SIMULATION; Stochastic; STOCHASTIC PROCESSES; TESTING; Uncertainty; VALIDATION; VISCOUS FLOW
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2010.04.041

With the advent of state-of-the-art computers and their rapid availability, the time is ripe for the development of efficient uncertainty quantification (UQ) methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data. The spectral stochastic finite element method (SSFEM) which is one of the widely used UQ methods, regards uncertainty as generating a new dimension and the solution as dependent on this dimension. A convergent expansion along the new dimension is then sought in terms of the polynomial chaos system, and the coefficients in this representation are determined through a Galerkin approach. This approach provides an accurate representation even when only a small number of terms are used in the spectral expansion; consequently, saving in computational resource can be realized compared to the Monte Carlo (MC) scheme. Recent development of a finite difference lattice Boltzmann method (FDLBM) that provides a convenient algorithm for setting the boundary condition allows the flow of Newtonian and non-Newtonian fluids, with and without external body forces to be simulated with ease. Also, the inherent compressibility effect in the conventional lattice Boltzmann method, which might produce significant errors in some incompressible flow simulations, is eliminated. As such, the FDLBM together with an efficient UQ method can be used to treat incompressible flows with built in uncertainty, such as blood flow in stenosed arteries. The objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the FDLBM with a SSFEM. Validation against MC solutions of channel/Couette, driven cavity, and sudden expansion flows are carried out.