High-order accurate large-eddy simulations of compressible viscous flow in cylindrical coordinates

Dynamic interactions of shock waves and turbulent structures occur in a wide range of applications. The accurate simulation of turbulence in these flows requires a numerical scheme with minimal dissipation whereas the shock-capturing requires a dissipative scheme to stabilise the solution in the shock-wave discontinuous regions. These contradictory requirements make simulations of these flows extremely challenging. A compatible implementation of the solid boundary conditions at the point of incepting the acoustic waves and their internalisation into hydrodynamic instabilities is also challenging. Here we present, a high-fidelity, massively parallel and accurate solver for both direct numerical simulations and large-eddy simulations of supersonic turbulent flows in cylindrical coordinates. A hybrid WENO/ high order central difference scheme is used for the spatial discretisation with a fourth-order five-stage 2N-storage Runge–Kutta for the time integration. The least square contraction of Lilly [1] is utilised for large-eddy subgrid-scale modelling. The solid surface boundary condition is implemented using the offset wall and ghost cell technique. The numerical implementation is validated through a wide range of test cases from simple to more complex cases. The numerical results show good agreement with analytical solutions and available experimental results. In the large-eddy simulation of a supersonic under-expanded impinging jet, it is observed that the conventional Ducros sensor leads to a more dissipative hybrid scheme. In this paper, a new shock sensor based on a WENO smoothness indicator is proposed and it is demonstrated to perform better especially in the simulation of the supersonic under-expanded impinging jet by limiting the expensive WENO scheme to the discontinuous regions and reducing the computational cost by approximately 10%.