On Numerical Instabilities Of Shock-capturing Methods

Despite significant advances in both CFD and computational techniques,numerical simulation of hypersonic flow still presents significant challenges.Current numerical methods applied to hypersonic flow simulation often encounter shock instability problems,such as the carbuncle phenomenon.For simple flow problems,the accurate flow field is generally predictable,and the identification of shock anomalous structures is relatively easy.However,practical simulations often involve complex geometries and flow phenomena.If anomalous phenomena such as shock instabilities occur,it is almost impossible to distinguish between an accurate flow field and a spurious flow field.This difficulty greatly reduces the reliability of existing CFD methods and affects their application in hypersonic flow.Therefore,studying the numerical stability of the shock-capture method is of great significance for improving the reliability of the existing CFD method and achieving accurate prediction of hypersonic flow.Based on the combination of linear perturbation analysis and numerical experiments,the numerical dissipative analysis of the typical Godunov-type shock-capturing methods is carried out,and the numerical dissipative characteristics of different types of shockcapture methods for linear degenerate waves are clarified.It is found that the numerical viscosity of the entropy wave and the shear wave have different effects on the shock stability.The numerical dissipation related to the entropy wave can not effectively suppress the numerical shock instability,but the one corresponding to the shear wave is able to stabilize the shock effectively.Numerical experiments are carried out to study the shock stability of a large number of shock-capture schemes.It is found that the disturbance causing the shock instability originates from the numerical shock structure.If the mass flux before the shock is consistent with that after the shock,then the shock-capture scheme is stable.The mechanism of numerical shock instability is clarified with a linear perturbation analysis.It is found that the disturbance causing the shock instability is generated inside the shock structure and propagates along with the entropy wave and the acoustic wave to the downstream of the shock wave,which leads to the non-physical disturbance error after the shock wave.This leads to the inconsistency of the mass flux before and after the shock wave which eventually leads to the occurrence of shock instability.Starting from the second law of thermodynamics,a comprehensive approach based on entropy generation analysis and numerical experiments is used to clarify the relationship between numerical entropy generation and shock stability.The comprehensive study also clarify the intrinsic mechanism of numerical shock instability.It is found that the numerical shock instability problem is caused by improper entropy production inside the numerical shock structure.If the shock-capture method ensures sufficient entropy inside the shock structure,the shock captured by the numerical scheme will be stable.Based on the research conclusions,this paper proposes an entropy control method for the numerical shock instability problem of the low-dissipation shock-capturing methods.This method is general and can be applied to a variety of shock-capturing schemes.Different from the existing modifications that improve shock stability,the entropy control method does not rely on additional numerical viscosity to suppress the shock instability,thus it does not affect the accuracy of the original flux functions,which is very suitable for the calculation of complex hypersonic flow problems.In order to meet the requirements of the numerical method for all-speed flows,this paper analyzes the numerical characteristics of the shock-capturing methods under the low Mach incompressible limit.The all Mach correction method is used to extend the shockcapturing method for compressible flow to the low Mach incompressible flow regime.Combined with the entropy control method for suppressing numerical shock instability,this paper develops a simple framework for constructing an all-speed scheme.The allspeed framework is general and suitable for a variety of shock-capturing methods.The accuracy and robustness of the typical all-speed schemes are verified by a series of complex test cases.The experimental results show that the all-speed shock-capturing methods can simulate various complex flow problems accurately and stably in the all-speed flows.