A vertex-based reconstruction for cell-centered finite-volume discretization on unstructured grids

•The cell and face gradients are obtained by the vertex-based reconstruction.•A vertex-based vectorial nonlinear weighting strategy is used for shock-capturing.•The nonlinear weighting strategy is efficient and robust.•The nonlinear weighting strategy can maintain the accuracy of the smooth solutions.•This method does not encounter the decoupling problem on quadrilateral grids. Recently, a vertex-based spatial reconstruction for unstructured cell-centered finite volume method has been proposed, showing advantages in accuracy, convergence, and efficiency. However, this method was only applied for inviscid flows. Moreover, for shock-capturing computations, the conventional cell-based limiter was the only choice to suppress numerical oscillations. In this work, the vertex-based reconstruction is extended for the solution of viscous flows, while automatically avoiding the typical decoupling problem. Moreover, a WENO-type nonlinear weighting strategy is designed and combined with the vertex-based reconstruction method, resulting in a simple and cost-efficient alternative to conventional slope limiters. In addition, the present method is combined with the recently proposed iterative near-boundary treatment, ensuring linear exactness even in the vicinity of boundaries, without sacrificing the overall computational efficiency. A series of numerical test cases involving viscous flows and shock waves provide evidence of the superior performance of the present method. The advantages of the present method are especially prominent on high aspect-ratio irregular triangular grids, which is a beneficial property for solving realistic fluid dynamic problems at high Reynolds numbers.