A Discrete Linear Stability Analysis Framework for Two-Dimensional Laminar Flows

Abstract

A discrete linear stability analysis framework for two-dimensional laminar flows is presented. Using two case studies involving analysis of thermal and laminar flows, the stability of flows in the discrete numerical sense is addressed. The two-dimensional base flow for various values of the controlling parameter (Reynolds number for flow past a square cylinder and Rayleigh number for double-glazing problem) is computed numerically by using the lattice Boltzmann method. The governing equations, discretized using the finitedifference method in two-dimensions and are subsequently written in the form of perturbed equations with twodimensional disturbances. These equations are linearized around the base flow and form a set of partial differential equations that govern the evolution of the perturbations. The eigenvalues, stability of the base flow and the points of bifurcations are determined using normal mode analysis. The eigenvalue spectrum predicts that the critical Reynolds number is 52 and the critical Rayleigh number is 6 1.88×10 for the square cylinder and double-glazing problem, respectively, The results are consistent with the previous numerical and experimental observations.