Mesh-free hydrodynamic stability*

We develop a high-order mesh-free hydrodynamic stability analysis tool for complex geometries using radial basis

function-based finite differences (RBF-FD). Polyharmonic spline RBFs with polynomial augmentations (PHS+poly)

are used to construct the discrete linearized Navier-Stokes and resolvent operators on scattered nodes. This scheme

enables accurate, stable, and computationally efficient discretizations of the large matrix problems arising in two-

dimensional hydrodynamic stability analysis. The study addresses the trade-off between computational efficiency

and accuracy and provides best practices. Furthermore, the practical treatment of boundary conditions, including

the pole singularity in cylindrical coordinates, is examined and discussed. The numerical framework is validated

across various hydrodynamic stability theoretical methods and flows. This includes conducting linear stability (LST),

resolvent (RA), and wavemaker (WM) analyses for the canonical cylinder flow at Reynolds numbers ranging from

47 to 180. Additionally, RA and WM analyses are performed for a laminar zero-pressure-gradient (ZPG) flat-plate

Blasius boundary layer at a Reynolds number of 0 ≤ Re ≤ 6 × 10⁵, as well as the turbulent mean transonic jet at

Mach number 0.9 and a Reynolds number of approximately 10⁶. The comparisons of these benchmark problems with

the literature demonstrate the broad applicability, accuracy, and robustness of the mesh-free framework. Lastly, the

pioneering application of RA-based WM analysis on the Blasius boundary layer and turbulent jet offers new insights

into modal and non-modal growth in these flows.