An implicit high-order k-exact finite-volume approach on vertex-centered unstructured grids for incompressible flows

Highlights: • High-order finite-volume method for incompressible flows on unstructured grids. • Incorporation of a spatially higher order of accuracy into fractional step methods. • Efficient k-exact multiple-correction approach for vertex-centered unstructured grids. • Performance benchmark of high-order method against conventional schemes. • Enabling higher order accuracy in space into complex production flow solver. We present a k-exact reconstruction method, which can be incorporated into vertex-centered unstructured finite-volume flow solvers to maintain a high-order accurate solution in space. The scheme is combined with a fractional step strategy for the solution of the incompressible Navier-Stokes equations with a fully implicit discretization of a Poisson equation for the pressure correction. This also involves a novel approach for the flux discretization in the k-exact framework. It is shown, that a third order of accuracy can be maintained for the discretization of convective fluxes and a second order of accuracy for diffusive fluxes, even on highly distorted grids. The scheme is implemented into ThetaCOM, a turbulent heat release extension of DLR's TAU code in its combustion version, showing its applicability to be used in a full production flow solver. The improved properties of our approach in terms of performance and accuracy are demonstrated with several benchmarks cases in both two and three spatial dimensions.