Research On Unstructured Shock-fitting Method For Inviscid Compressible Flow

Shock wave is a very important physics phenomenon that widely exists in the high-speed inviscid compressible flows.When the flow crosses a shock wave,several major flow variables will change significantly,which often has a great impact on the entire flow field characteristics.In the process of understanding shock wave propagation and interaction phenomena,numerical simulation is a very effective means.At present,there are two distinct methods to deal with the shocks in the flow field:the shock-capturing(S-C)method and the shock-fitting(S-F)method.As for the S-C method,the flow governing equations are solved uniformly in the whole flow field,and shock waves can be automatically captured without special treatment.Due to its high degree of automation,the S-C method has become a mainstream choice in engineering applications.However,there is always a contradiction between the computational accuracy and stability of the S-C method,and numerous studies have indicated that the spatial accuracy of post-shock smooth regions will be reduced to first-order because of the intrinsic defect of the S-C method.In contrast,the S-F method introduce the strict Rankine-Hugoniot jump relations to establish the relationship between the shock wave upstream/downstream flow variables,which ensures that a numerical scheme will not reduce its accuracy order in the post-shock smooth regions and avoids a series of shock wave instability phenomena caused by capturing shock waves.Nevertheless,since the S-F method must always explicitly track shock waves,and its automation and versatility are relatively poor,so the S-F method still has great challenges in engineering applications at present.In order to develop a more flexible S-F method,relevant researches have been carried out from three aspects in this thesis.Firstly,the shock wave detection(SWD)method,which is considered to be the foundation of the S-F method and can effectively improve the automation of the S-F methods,is investigated.A high-accuracy SWD method is proposed to solve the problem that the current SWD methods are not accurate enough to be directly applied to the position initialization process of fitted shock waves.More specifically,this SWD method includes three steps:coarse detection of shock waves,cluster analysis,and identification/fitting.At first,a series of grid cells located in the shock transition regions,namely shock-cells,are identified by a traditional SWD method.Moreover,the K-means clustering algorithm is introduced to divide these shock-cells into many clusters,and further classify these clusters according to their neighborhood characteristics.Finally,by designing a set of identification criteria,the spatial points on each shock branch can be determined,and then the high-quality shock lines/surfaces can be obtained piece by piece.It is worth noting that,as for two-dimensional problems,this SWD method can also automatically identify the approximate location of shock wave interaction points,so as to judge the shock wave patterns in the flow field.Numerical results show that the accuracy of this SWD method is greatly improved with low grid dependency compared to traditional methods,and it has high reliability in the complex two-dimensional steady and unsteady flows,and it also reflects a certain development potential in the extension to three-dimensional problems.On the basis of this SWD method,a new unstructured S-F method,namely adaptive discontinuity fitting(ADF)method,is proposed based on the cell-centered finite volume method and unstructured dynamic grids technique.ADF method combines advantages of the unstructured boundary fitting method and the embedded shock-fitting method,and only the flow variables at grid nodes are calculated uniformly.It is noted that the fitted shock wave can be not only regarded as the boundary of computational domain but also embedded in the flow field.On the one hand,as for relatively simple flow structures,the fitting solutions can be quickly obtained by directly starting the fitting calculations,and the approximate position of shock wave can be initialized based on a prior knowledge.On the other hand,as for the flows with complex shock wave interactions,the presented high-accuracy SWD method can be utilized to automatically determine the initial position of shock waves and interaction points from S-C solutions,thereby the fitting solutions can be calculated indirectly.This thesis introduces the key calculation processes of ADF method in detail from several aspects:marking discontinuity locations,initializing and correcting the flow variables of discontinuity nodes,motion of grid nodes,and flow field update.Moreover,the strict grid convergence tests are carried out to verify that ADF method can achieve the spatial second-order accuracy in the whole domain.Steady flow numerical results indicate that ADF method can effectively avoid the non-physical numerical oscillations that often occur in S-C solutions,and shows good applicability in complex two-dimensional shock-shock interaction problems,and also has good performances in the three-dimensional engineering problems.Finally,in view of the predicament that the current unstructured S-F methods are difficult in applying to the simulations of complex unsteady flow,several improvements for ADF method have been performed to further raise the capacity for dealing with unsteady flows.Due to related algorithm improvements involve several processes,e.g.the movement of shock waves along straight/curved boundaries,automatic reconstruction of discontinuity-node distribution,automatic reconstruction of the grids near discontinuities,and the movement of shock interaction points,the large displacement and deformation of fitted shock waves can be automatically tracked with high-accuracy.The fitting solutions of various shock propagation cases show that ADF method can not only effectively improve the calculation accuracy of unsteady flow field,but also extract more flow parameters compared with S-C methods.Moreover,the evolution map of moving shock waves in the flow field can be obtained more clearly.In a word,this thesis focuses on the research of the S-F method.A high-accuracy SWD method for S-F process is presented,and a more flexible unstructured S-F method is proposed,together with extension of the application range of S-F method in unsteady problems through several improvements.