Optimal computational parameters for maximum accuracy and minimum cost of Arnoldi-based time-stepping methods for flow global stability analysis

Author(s)

Mathias, Marlon Sproesser ; Faraco de Medeiros, Marcello Augusto

• Document type: Journal article
• Source: THEORETICAL AND COMPUTATIONAL FLUID DYNAMICS; v. 36, n. 6, p. 24-pg., 2022-11-06.

Abstract

Global instability analysis of flows is often performed via time-stepping methods, based on the Arnoldi algorithm. When setting up these methods, several computational parameters must be chosen, which affect intrinsic errors of the procedure, such as the truncation errors, the discretization error of the flow solver, the error associated with the nonlinear terms of the Navier-Stokes equations and the error associated with the limited size of the approximation of the Jacobian matrix. This paper develops theoretical equations for the estimation of optimal balance between accuracy and cost for each case. The 2D open cavity flow is used both for explaining the effect of the parameters on the accuracy and the cost of the solution, and for verifying the quality of the predictions. The equations demonstrate the impact of each parameter on the quality of the solution. For example, if higher-order methods are used for approaching a Frechet derivative in the procedure, it is shown that the solution deteriorates more rapidly for larger grids or less accurate flow solvers. On the other hand, lower-order approximations are more sensitive to the initial disturbance magnitude. Nevertheless, for accurate flow solvers and moderate grid dimensions, first-order Frechet derivative approximation with optimal computational parameters can provide 5 decimal place accurate eigenvalues. It is further shown that optimal parameters based on accuracy tend to also lead to the most cost-effective solution. The predictive equations, guidelines and conclusions are general, and, in principle, applicable to any flow, including 3D ones. (AU)

• FAPESP's process: 17/23622-8 - Effect of a gap on natural transition to turbulence in a boundary layer
• Grantee: Marcello Augusto Faraco de Medeiros
• Support Opportunities: Scholarships abroad - Research
• FAPESP's process: 18/04584-0 - Computational study of the effect of a small cavity on the transition of a compressible boundary layer
• Grantee: Marlon Sproesser Mathias
• Support Opportunities: Scholarships in Brazil - Doctorate
• FAPESP's process: 19/15366-7 - Experimental study of the effect of a cavity on boundary layer transition
• Grantee: Marcello Augusto Faraco de Medeiros
• Support Opportunities: Regular Research Grants
• FAPESP's process: 13/07375-0 - CeMEAI - Center for Mathematical Sciences Applied to Industry
• Grantee: Francisco Louzada Neto
• Support Opportunities: Research Grants - Research, Innovation and Dissemination Centers - RIDC