| With the rapid development of computer hardware technology,computational fluid dynamics(CFD)methods for solving complex flow problems are gaining more and more attention.Traditionally,CFD usually uses a low-order method(first-and second-order numerical methods),which can meet the needs to a certain extent.However,for complex flow problems,such as vortex shedding,shock-boundary layer interaction,turbulent transition,etc.,the resolution of the low-order method hardly meets the desired expectations,and a high-order numerical method is needed for solving those kinds of problems.The flux reconstruction method(FR/CPR)is widely used as one of the high-order numerical methods because of its compactness,efficiency,and ease of programming.In this paper,a limiter was added to the framework of the flux reconstruction method to improve its resolution of the shocks and suppress the numerical perturbations around the shocks,while keeping the computational accuracy of the smooth region.This paper firstly reviews the history of the development of the numerical methods of computational fluid dynamics and introduces the family of the high-order numerical methods.While focusing on the design idea,numerical discretization,and algorithm implementation of the flux reconstruction method,it shows its high-precision characteristics with numerical cases.Secondly,the LD limiter is designed by expanding the detector range of the TVD limiter,which has no user-defined parameters and can effectively capture the shock waves and suppress the numerical perturbations around the shock waves for the low Mach number cases.Since the LD limiter’s capability of suppressing numerical perturbations is not sufficient in the case of strong shocks,a gradient-based smooth indicator is developed based on the absolute value of the gradient in the solution unit,which is used to exclude the solution unit containing smooth extrema that wrongly marked by the TVD detector.The above three-step method,TVD detector,gradient-based smooth extrema detector,and squeeze limiter,is named as the gradient-based limiter because the absolute value of the gradient of the solution unit is referenced in the process of capturing the shock wave.The gradient-based limiter is compact,accuracy preserving,parameter-free,and easy to insert into the flux reconstruction method,while still having high resolution for high Mach number flow problems containing strong shocks.Because of the excellent performance of the gradient-based limiter,its application to large eddy simulation is considered.The process applied to the large eddy simulation method puts higher bars on the limiter,because the numerical perturbations will be coupled with the physical perturbations generated by the vortex structure and the shocks,thus intensifying the instability of the numerical method,and also making it extremely difficult to capture the shocks,distinguish the vortex structure,and stabilize the simulation.Therefore,the gradient-based limiter is modified,and user-defined parameters are introduced to tune its performance,so that the modified gradient-based limiter can meet the needs of large eddy simulation. |
