Global stability analysis of a double Taylor-Couette system using a higher-order meshless method

Meshless methods have recently gained popularity for numerically solving partial differential equations, such as those governing fluid flows, heat transfer, and species transport. They have also been employed for the linear stability analysis of fluid flows (Physics of Fluids, 36 (6): 064103 (2024), and Journal of Computational Physics, 474, 111756 (2023)). A numerical solver is developed using a meshless framework using polyharmonic spline–radial basis functions (PHS-RBF) with appended polynomials to efficiently compute the hydrodynamic stability of fluid flows in complex geometries. The key advantage of this method is that linear stability analysis can be performed of flows within complex geometries without the need for coordinate transformations, which result in complex Navier–Stokes operators involving transformation matrices. Instead, this approach uses a Cartesian system, resulting in the same Navier–Stokes operator regardless of the complexity of the geometry. This leads to significant computational efficiency for solving the eigenvalue problem.

The method has been first validated on concentric and eccentric Taylor–Couette flows. We show that the PHS-RBF method, combined with the Arnoldi algorithm significantly reduces the computation time for the eigenmodes. This study examines the global stability characteristics of a double Taylor–Couette system in which two circular cylinders co-rotate or counter-rotate within a rectangular enclosure. The computed eigenspectrum is similar to the well-studied concentric Taylor–Couette system, but the first unstable mode differs from the Taylor instability mode. Further, the critical Reynolds number and the critical axial wavenumbers are lower than the traditional Taylor–Couette flow.