Efficient implementation of high order unstructured WENO schemes for cavitating flows

In this article we present a new efficient implementation of high order accurate Godunov-type WENO finite volume schemes on unstructured triangular and tetrahedral meshes for the simulation of real fluids with complex equations of state. The main focus of this paper is the efficient computation of flows in compressible mixtures of liquid and vapor in multiple space-dimensions, including the onset of cavitation. In the present article we use the full equation of state (EOS) for water, vapor and wet steam in thermal equilibrium based on the industrial and scientific standard IAPWS-IF97, as well as the real equation of state for n-heptane provided by Span and Wagner. Since the direct evaluation of these very complex EOS is computationally very expensive, we propose a new robust, accurate and particularly efficient approach for their evaluation in high order Godunov type finite volume schemes. It is based on the L2 projection of the EOS (ρ, e) → (p, T) onto the space of piecewise polynomials of degree q using an adaptive mesh refinement (AMR) approach together with Cartesian cut-cells, which adjusts the grid of the phase space (ρ, e) so that the L∞ error of the L2-projection of the EOS is less than a given error threshold ∊. For a thorough validation of our numerical approach we construct several quasi-exact solutions to the Riemann problem for water and n-heptane using the corresponding detailed equations of state. We furthermore present numerical convergence results of third and fourth order schemes for an unsteady two-dimensional test problem for water with real EOS. We furthermore show the onset of cavitation on a simple model problem with an initially purely liquid fluid at ambient conditions at rest using a strong heat source term in the energy equation as well as the development of a cavitation bubble due to strong rarefaction. Finally, some numerical test problems are solved with the new approach on unstructured triangular and tetrahedral meshes in multiple space dimensions.