Combinations Of Shock-fitting And Shock-capturing Methods Based On Unstructured Dynamic Grids

Shock wave is an important discontinuity phenomenon occurring in compressible flows.Through a shock wave,the flow variables suddenly change and the total mechanical energy decrease.The way to deal with shock waves is most important for a flow including shock waves.The shock-fitting technique is an old method to solve shock waves.Its theory is very clear and exact.At the dawn of CFD(Computational Fluid Dynamics),the shock-fitting technique was used to solve compressible flows with shock waves.It is proven to be an efficient and exact way to solve some simple shock waves.When used in complex flows,it is difficult to apply for many manual intervention.With the development of shock-capturing schemes,its large automation makes it more popular than the shock-fitting.Though the shock-capturing is a most popular tool,the researchers working with the shock-capturing have been devoting themself to improve the quality of shock solutions since the birth of shock-capturing technique.Since Harten proposed the TVD(Total Variation Diminishing)in 1983,a lot of progresses have been made and many capturing methods of high accuracy are proposed creatively.These schemes have been used in science and engineering widely.However,there are still many uncertainties behind the successes of the capturing methods and a lot of problems have yet been solved.These schemes which are proven to be of high accuracy have only first order in the vicinity of shock waves.And the contradiction between computational accuracy and stability still exists.It is difficult to use the schemes of high order accuracy in engineering widely for the influences of efficiency and stability.For now,CFD,mainly refers to shock-capturing,is not up to the standard of high accuracy,high efficiency and high stability.In a way,these problems strictly limit the development of shock-capturing methods.After comprehensive analysis of the difficulty of shock-fitting and the "aeipathia" of shock-capturing,the idea of shock-fitting is applied into a unstructured dynamic grids solver which is based on shock-capturing algorithms.And a simple computational program combining shock-fitting and shock-capturing methods,which is called "Unstructured Boundary Fitting solver,UBFs",is proposed.In this new technique,there are three main parts:(1)determining the shock boundary conditions by solving the Rankine-Hugoniot(R-H)relations;(2)using unstructured dynamic grids technique to track the shock waves;(3)solving the Euler/N-S equations in Arbitrary Lagrangian-Eulerian(ALE)form to obtain the internal flowfield.Though this shock-fitting technique also belongs to the boundary shock-fitting,it removes the limit of mesh topology for using unstructured grids and is able to deal with more complex flows with shock waves.In the smooth region,the design order of the spatial discretization scheme never reduces to first order.Thus,this simple combination of shock-fitting and shock-capturing resolves the problems of both the traditional shock-fitting and the shock-capturing.Both the principle and operation of this combination way are easy,and there are just a few of minimal changes needed for previous code.Although the UBFs is flexible enough,it can be well described for all kinds of shape regions,and also suitable for flow with complex shock structures.However,we find that this method is restricted to the boundary shock-fitting technique,it is difficult to describe the unsteady flow of topological changes such as shock generation/annihilation.In order to solve this problem,a new type of shock-fitting technique named "Embedded shock-fitting technique" is developed by defining grids' nature.A aerodynamics solver based on the embedded shock-fitting technique is developed and named MCFs(Mixed Cpaturing and Fitting solver).When the shock wave is fitted by MCFs,the computational grid is a complete grid,which does not need to be divided into several subregions according to the UBFs.The shock wave is judged by the natures of the grid nodes,and it is flexible to exchange the shock-fitting and shock-capturing methods.The MCFs sets up the advantages of the traditional shock-fitting techniques(the floating shock-fitting and boundary shock-fitting),makes the shock-fitting method more modular,and has less change to the previous solver,which accords with the condition of establishing the "general-purpose" shock-fitting solver.By using the UBFs and MCFs in the simple three-dimensional problems,the way of combining these two methods to deal with complex three-dimensional problems is determined.In the process of simulation,the UBFs is used to deal with the bow shock wave in order to reduce the quantity of computational grid,and the MCFs is used to deal with the internal shock wave in order to effectively utilize the flexibility of the method.The numerical experiments proves that this way is able to deal with the complicated embedded shock wave such as three-dimensional shock wave intersection/reflection reasonably and conveniently.