High-order simulations of isothermal flows using the local anisotropic basis function method (LABFM)

TL;DR

This paper advances mesh-free simulation techniques by developing high-order accurate operators within the LABFM framework, enabling precise direct numerical simulations of complex isothermal flows in diverse geometries.

Contribution

The paper introduces new developments in LABFM for basis functions, stencil optimization, stabilization, variable resolution, and boundary conditions, achieving up to 10th order accuracy for Navier-Stokes simulations.

Findings

High-order LABFM accurately simulates flows in complex geometries.

Excellent agreement with analytical, numerical, and experimental results.

Successful DNS of subsonic and transonic flows in porous media at Re=968.

Abstract

Mesh-free methods have significant potential for simulations of flows in complex geometries, with the difficulties of domain discretisation greatly reduced. However, many mesh-free methods are limited to low order accuracy. In order to compete with conventional mesh-based methods, high order accuracy is essential. The Local Anisotropic Basis Function Method (LABFM) is a mesh-free method introduced in King et al., J. Comput. Phys. 415:109549 (2020), which enables the construction of highly accurate difference operators on disordered node discretisations. Here, we introduce a number of developments to LABFM, in the areas of basis function construction, stencil optimisation, stabilisation, variable resolution, and high order boundary conditions. With these developments, direct numerical simulations of the Navier Stokes equations are possible at extremely high order (up to 10th order in characteristic node spacing internally). We numerically solve the isothermal compressible Navier Stokes equations for a range of geometries: periodic and channel flows, flows past a cylinder, and porous media. Excellent agreement is seen with analytical solutions, published numerical results (using a spectral element method), and experiments. The potential of the method for direct numerical simulations in complex geometries is demonstrated with simulations of subsonic and transonic flows through an inhomogeneous porous media at pore Reynolds numbers up to Re=968.