High-order simulations of non-ideal flows with efficient control of temporal and spatial accuracy

This work presents the simulation of complex transonic/supersonic viscous flows with non-ideal compressible conditions with a high-order accuracy both in time and space. The computational efficiency of a solver based on a discontinuous Galerkin spatial discretization is increased with algorithms to adapt the polynomial degree of the solution over the mesh and the time step-size during the simulation, i.e., the p- and Δt-adaptation algorithms. The proposed implementation of these algorithms and non-ideal equations of state allows a further decrease in the computational cost of the simulations. The look-up-tables are used instead of the direct resolution of the non-ideal equations of state and are generated after each adaptation of the polynomial degree of the solution to increase the accuracy of the interpolation of the thermodynamic variables/derivatives. If the flow phenomenon is steady, the Δt-adaptation algorithm is able to reach the convergence of pseudo-unsteady simulations with a computational cost comparable to that of steady simulations and an efficient time integration. Furthermore, a spectral filter is used on the dual graph of the mesh after each adaptation of the polynomial degree of the solution to decrease the differences in the polynomial degree between neighboring elements. The accuracy and computational efficiency of different variables used in the error estimator for the p-adaptation algorithm, i.e., pressure, temperature, velocity magnitude, speed of sound, entropy, and the fundamental derivative of gas dynamics, are compared on simulations with different free-stream and wall conditions, equations of state, and governing equations. The results also demonstrate the influence of negative values of the fundamental derivative of gas dynamics and cold-wall conditions on flow phenomena such as shock/boundary-layer interaction and buffet/vortex shedding.