Mesh-Free Hydrodynamic Stability

TL;DR

This paper introduces a mesh-free RBF-FD method for hydrodynamic stability analysis, enabling accurate eigenvalue computations in complex flow domains, validated on classical and turbulent flows, providing new physical insights.

Contribution

It develops a stable, efficient mesh-free RBF-FD framework for hydrodynamic stability, including novel boundary derivative stencils and pole treatment, applicable to a range of flow regimes.

Findings

Accurate eigenvalue results consistent with literature

Validated on flows from laminar to turbulent regimes

Provided new insights into modal and non-modal flow growth

Abstract

A specialized mesh-free radial basis function-based finite difference (RBF-FD) discretization is used to solve the large eigenvalue problems arising in hydrodynamic stability analyses of flows in complex domains. Polyharmonic spline functions with polynomial augmentation (PHS+poly) are used to construct the discrete linearized incompressible and compressible Navier-Stokes operators on scattered nodes. Rigorous global and local eigenvalue stability studies of these global operators and their constituent RBF stencils provide a set of parameters that guarantee stability while balancing accuracy and computational efficiency. Specialized elliptical stencils to compute boundary-normal derivatives are introduced and the treatment of the pole singularity in cylindrical coordinates is discussed. The numerical framework is demonstrated and validated on a number of hydrodynamic stability methods ranging from classical linear theory of laminar flows to state-of-the-art non-modal approaches that are applicable to turbulent mean flows. The examples include linear stability, resolvent, and wavemaker analyses of cylinder flow at Reynolds numbers ranging from 47 to 180, and resolvent and wavemaker analyses of the self-similar flat-plate boundary layer at a Reynolds number as well as the turbulent mean of a high-Reynolds-number transonic jet at Mach number 0.9. All previously-known results are found in close agreement with the literature. Finally, the resolvent-based wavemaker analyses of the Blasius boundary layer and turbulent jet flows offer new physical insight into the modal and non-modal growth in these flows.