  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.

Fluid Dynamics;Computational Physics