Untitled

Abstract

Numerical methods for tackling the inviscid instability problem are discussed. Convergence is demonstrated to be a necessary, but not a sufficient condition for accuracy. Inviscid flow physics set requirements regarding grid-point distribution in order for physically accurate results to be obtained. These requirements are relevant to the viscous problem also and are shown to be related to the resolution of the critical layers.In this respect, high-resolution nonlinear calculations based on the inviscid initial-boundary-value problem are presented for a model shear-layer flow, aiming at identification of the regions that require attention in the course of high-Reynolds-number viscous calculations. The results bear a remarkable resemblance with those pertinent to viscous flow, with a cascade of high-shear regions being shed towards the vortex-core centre as time progresses. In parallel, numerical instability related to the finite-time singularity of the nonlinear equations solved globally contaminates and eventually destroys the simulations, irrespective of resolution.

Massively Parallel Solution of the BiGlobal Eigenvalue Problem Using Dense Linear Algebra

“…On the other hand, experience with both spectral and finite difference methods for one-dimensional (ordinary-differential-equation-based) stability problems [16,17] has delivered a rule of thumb for the number of nodes required by a spectral and a finite difference numerical method to obtain results of the same accuracy. This rule of thumb depends on the order of the finite difference discretization; use of a sixth-order compact finite difference scheme requires a factor four higher number of nodes compared with a spectral method of equivalent accuracy [18]. Extrapolation of such results to the (two-dimensional, partial-differential-equation-based) BiGlobal EVP suggests that a high-order finite difference approach would require discretized arrays the leading dimension of which would be at least 1 order of magnitude higher than that quoted previously; such arrays would only be able to be treated by sparse-matrix techniques.…”

“…The conclusion drawn is that what was assumed to be second-order effects are more significant than implied by the scaling Eq. (18). The alternative time scaling is then constructed:…”