Stable, entropy-pressure compatible subsonic Riemann boundary condition for embedded DG compressible flow simulations

One approach to reducing the computational cost of simulating transitional compressible boundary layer flow is to adopt a near body reduced domain with boundary conditions enforced to be compatible with a computationally cheaper three-dimensional RANS simulation. In such an approach it is desirable to enforce a consistent pressure distribution which is not typically the case when using the standard Riemann inflow boundary conditions. We therefore revisit the Riemann problem adopted in many DG based high fidelity formulations. Through analysis of the one-dimensional linearised Euler equations it is demonstrated that maintaining entropy compatibility with the RANS simulation is important for a stable solution. It is also necessary to maintain the invariant for the Riemann outflow boundary condition in a subsonic flow leav-ing one condition that can be imposed at the inflow boundary. Therefore the entropy-pressure enforcement is the only stable boundary condition to enforce a known pressure distribution. We further demonstrate that all the entropy compatible inflow Riemann boundary conditions are stable providing the invariant compatible Riemann outflow boundary condition is also respected. Although the entropy-pressure compatible Riemann inflow boundary condition is stable from the one-dimensional analysis, two-dimensional tests highlight divergence in the inviscid problem and neutrally stable wiggles in the velocity fields in viscous simulations around the stagnation point. A two-dimensional analysis about a non-uniform baseflow assumption provides insight into this stability issue (ill-posedness) and motivate the use of a mix of inflow boundary conditions in this region of the flow. As a validation we apply the proposed boundary conditions to a reduced domain of a wing section normal to the leading-edge of the CRM-NLF model taken out of a full three-dimensional RANS simulation at Mach 0.86 and a Reynolds number of 8.5 million. The results show that the entropy-pressure compatible Riemann inflow boundary condition leads to a good agreement in pressure distribution.