Low-dissipation BVD schemes for single and multi-phase compressible flows on unstructured grids

Solving compressible flows containing both smooth and discontinuous flow structures still remains a big challenge for finite volume methods, especially on unstructured grids where one faces more difficulties in building high-order polynomial reconstruction and limiting projection to suppress numerical oscillations in comparison with the case of structured grids. As a result, most of the current finite volume schemes on unstructured grids are of second order and too dissipative to resolve fine structures of complex flows. In this paper, we report two novel hybrid schemes to resolve vortical and discontinuous solutions on unstructured grids by reducing numerical dissipation. Different from conventional shock capturing schemes that use polynomials and limiting projections for reconstruction, the proposed schemes employ two second-order schemes, i.e. a polynomial and a sigmoid function as candidate reconstruction functions to approximate smooth and discontinuous solutions respectively. As the polynomial function, the MUSCL (Monotone Upstream-centered Schemes for Conservation law) scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted, while being a sigmoid function, the multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used to mimic the discontinuous solution structure. With these candidates for reconstruction, a single-step boundary variation diminishing (BVD) algorithm, which aims to minimize numerical dissipation, is designed on unstructured grids to select the final reconstruction function. The resulting two variant schemes, MUSCL-THINC/QQ-BVD schemes with two and three candidates respectively, are algorithmically simple and show great superiority to other existing schemes in capturing discontinuous and vortical flow structures for single and multiphase compressible flows on unstructured grids. The performance of the proposed schemes has been extensively verified through benchmark tests of single and multi-phase compressible flows, where discontinuous and vortical flow structures, like shock waves, contact discontinuities and material interfaces, as well as vortices and shear instabilities of different scales, coexist simultaneously. The numerical results show that the proposed schemes that hybrids two second-order schemes are capable of capturing sharp discontinuous profiles without numerical oscillations and resolving vortical structures along shear layers and material interfaces with significantly improved solution quality superior to other schemes of even higher order reconstructions.