We introduce an immersed high-order discontinuous Galerkin method for solving
the compressible Navier-Stokes equations on non-boundary-fitted meshes. The
flow equations are discretised with a mixed discontinuous Galerkin formulation
and are advanced in time with an explicit time marching scheme. The
discretisation meshes may contain simplicial (triangular or tetrahedral)
elements of different sizes and need not be structured. On the discretisation
mesh the fluid domain boundary is represented with an implicit signed distance
function. The cut-elements partially covered by the solid domain are integrated
after tessellation with the marching triangle or tetrahedra algorithms. Two
alternative techniques are introduced to overcome the excessive stable time
step restrictions imposed by cut-elements. In the first approach the cut-basis
functions are replaced with the extrapolated basis functions from the nearest
largest element. In the second approach the cut-basis functions are simply
scaled proportionally to the fraction of the cut-element covered by the solid.
To achieve high-order accuracy additional nodes are introduced on the element
faces abutting the solid boundary. Subsequently, the faces are curved by
projecting the introduced nodes to the boundary. The proposed approach is
verified and validated with several two- and three-dimensional subsonic and
hypersonic low Reynolds number flow applications, including the flow over a
cylinder, a space capsule and an aerospace vehicle