Simultaneous energy and entropy stability of volume-penalisation immersed-boundary methods for the compressible Navier--Stokes equations

Abstract

Immersed-boundary methods avoid body-fitted meshes, but the energy or entropy stability of the underlying flow discretisation does not automatically extend to the immersed treatment. This manuscript analyses the volume-penalised compressible Navier--Stokes equations for both energy and entropy stability, on the continuous and semi-discrete levels, and integrates the isolated source exactly in time. The source is formulated as a one-parameter family indexed by $θ$, whose momentum component relaxes the fluid velocity towards the prescribed solid velocity; $θ$ sets how much of the resulting kinetic-energy loss leaves the domain rather than being converted to internal energy. The two stability requirements impose opposing bounds on $θ$, with dissipation of the relative total energy requiring $θ\geq0$ and stability of the mathematical entropy requiring $θ\leq1$. Where penalisation acts, both conditions are therefore satisfied if and only if $0\leqθ\leq1$, and the symmetric value $θ=1/2$ divides the kinetic-energy loss equally between the domain and the internal energy. We conclude that any entropy-stable split-form discontinuous Galerkin spectral element method remains semi-discretely entropy stable once the source is added at the collocation nodes, for stationary and prescribed moving bodies alike.