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.

对于有限体积方法而言，求解同时包含平滑和不连续流动结构的可压缩流仍然是一个巨大的挑战，特别是在非结构化网格上，与结构化情况相比，非结构化网格在构建高阶多项式重建和限制投影以抑制数值振荡方面面临更多的困难网格。结果，当前在非结构化网格上的大多数有限体积方案都是二阶的，并且对于解决复杂流的精细结构而言过于耗散。在本文中，我们报告了两种新颖的混合方案，通过减少数值耗散来解决非结构化网格上的涡旋和不连续解。与使用多项式和限制投影进行重构的常规冲击捕获方案不同，所提出的方案采用两个二阶方案，即 多项式和S形函数分别作为候选重构函数，分别近似平滑和不连续解。作为多项式函数，采用具有MLP（多维限制过程）斜率限制器的MUSCL（单调上游集中式守恒法）方案，同时作为S形函数，多维THINC（双曲线的正切值）具有二次曲面表示和高斯正交的捕获函数（所谓的THINC / QQ）用于模拟不连续解的结构。利用这些候选重建对象，在非结构化网格上设计了旨在最小化数值耗散的单步边界变化减小（BVD）算法，以选择最终的重建函数。结果产生了两种变体方案，分别具有两个和三个候选者的MUSCL-THINC / QQ-BVD方案在算法上很简单，并且在捕获非结构化网格上单相和多相可压缩流的不连续和涡流结构时，显示出优于其他现有方案的优越性。通过对单相和多相可压缩流进行基准测试，对所提出方案的性能进行了广泛验证，在该测试中，不连续和涡流结构，例如冲击波，接触不连续和材料界面，以及不同尺度的涡和剪切不稳定性，同时共存。