A robust compact least-squares reconstruction method for compressible turbulent flow simulations of complex configurations

For the second-order finite volume method, implicit schemes and reconstruction methods are two main algorithms which influence the robustness and efficiency of the numerical simulations of compressible turbulent flows. In this paper, a compact least-squares reconstruction method is proposed to calculate the gradients for the distribution of flow field variables approximation. The compactness of the new reconstruction method is reflected in the gradient calculation process. The geometries of the face-neighboring elements are no longer utilized, and the weighted average values at the centroid of the interfaces are used to calculate the gradients instead of the values at the centroid of the face-neighboring elements. Meanwhile, an exact Jacobian solving strategy is developed for implicit temporal discretization. The accurate processing of Jacobian matrix can extensively improve the invertibility of the Jacobian matrix and avoid introducing extra numerical errors. In addition, a modified Venkatakrishnan limiter is applied to deal with the shock which may appear in transonic flows and the applicability of the mentioned methods is enhanced further. The combination of the proposed methods makes the numerical simulations of turbulent flow converge rapidly and steadily with an adaptive increasing CFL number. The numerical results of several benchmarks indicate that the proposed methods perform well in terms of robustness, efficiency and accuracy, and have good application potential in turbulent flow simulations of complex configurations.