Three-Dimensional High-Order Reconstruction Scheme for Genuinely Three-Dimensional Riemann Solver

The genuinely three-dimensional Riemann solvers proposed by Balsara and other practitioners exhibit better multidimensionality than the one-dimensional Riemann solvers when simulating multidimensional flows. However, typical reconstruction methods designed for structured grids make it difficult to reconstruct the flow quantities at grid vertices, which are required by the genuinely three-dimensional methods. To improve the genuinely three-dimensional schemes’ resolution for small-scale structures, a three-dimensional third-order reconstruction method named the Taylor-series-based weighted essentially nonoscillatory (TWENO) scheme is proposed. This method assembles seven polynomials of small stencils into a weighted three-dimensional third-order polynomials by employing the weighted essentially nonoscillatory method. Several numerical test cases are simulated to validate the proposed method. The three-dimensional Euler problem demonstrates that the TWENO scheme achieves third-order accuracy. The three-dimensional Riemann problems, the double Mach reflection problem, and the forward-facing step case indicate that the TWENO scheme is capable of resolving more small-scale structures than the second-order reconstruction with Barth–Jespersen (BJ) limiter method while stably capturing strong discontinuities. In the hypersonic cases, the hypersonic heating loads of the TWENO scheme are in better agreement with the theoretical or experimental values than with those of the BJ method as the boundary-layer grids become coarse.