A high-order nodal discontinuous Galerkin method for solution of compressible non-cavitating and cavitating flows

Abstract

In this work, a high-order nodal discontinuous Galerkin method is applied and assessed for the simulation of compressible non-cavitating and cavitating flows. The one-fluid approach with the thermal effects is used to properly model the cavitation phenomenon. Here, the spatial and temporal derivatives in the system of governing equations are discretized using the nodal discontinuous Galerkin method and the third-order TVD Runge–Kutta method, respectively. Various numerical fluxes such as the Roe, Rusanov, HLL, HLLC and AUSM+-up and two discontinuity capturing methods, namely, the generalized MUSCL limiter and a generalized exponential filter are implemented in the solution algorithm. At first, the sinusoidal density wave problem which has a smooth solution is simulated and the effects of the numerical fluxes on the accuracy and performance of the nodal discontinuous Galerkin method are studied. Two problems, namely, the shock–density interaction (non-cavitating flow) and the two symmetric expansion waves (cavitating flow) are then computed and the effects of the numerical fluxes and the discontinuity capturing methods on the accuracy and computational cost of the solution are investigated. For non-cavitating flows, the high-pressure water–water shock tube and the low-pressure water–water shock tube are also simulated. Then, three cavitating flow problems, namely, the two symmetric expansion waves, the shock-condensation tube and the collapsing cavitation bubble are simulated to assess the accuracy and robustness of the solution algorithm. Results show that the solution methodology based on the high-order NDGM is accurate and robust for simulating the compressible non-cavitating and cavitating flows.