Research Of Gas-Kinetic BGK Scheme And Its Application In The Near Continuum-Slip Flow Regimes

With the increasing development of aerospace techniques, the hybridalgorithms for entire Knudsen number flows have become one of the mostimportant directions in CFD. Both modern CFD techniques based on theNavier-Stokes(N-S) equations for continuum flows, and kinetic methods, such asdirect simulation Monte Carlo method (DSMC), will encounter computationaldifficulties when they are applied to entire Knudsen number flows. Due to thedifferences between their substances, the hybrid algorithms based on these twomethods also encounter some problems, for example, in the buffer zones, theinterface between different algorithms should be defined artificially orindentified automatically by a breakdown parameter. But these treatmentscontain strongly artificial factors, too experienced to be applied generally.Gas kinetic schemes provide us alternative choices. Differencing from thesolvers based on macroscopic quantities of fluid dynamics, the kinetic schemes’governing equation is Boltzmann equation. As the most fundamental equationdescribing the fluid motion, Boltzmann equation is valid for all Knudsen numberflows. Therefore, theoretically a unified kinetic scheme which is valid in thewhole range of Knudsen number, other than a hybrid scheme based on differentmethods, can be developed once the numerical discretization is properly designed.The unified scheme can treat the flow field in the same manner, needless toresolve different flow zones.Owing to the low efficiency caused by numerous float point operations,kinetic schemes are not commonly accepted by people in a long history. Based onthe integral solution of Boltzmann-BGK model equation and considering theparticle collision effect in the evolution stage, Kun Xu constructed a gas kineticBGK scheme for N-S equations successfully. The computational efficiency issimilar to the common N-S solvers. Recently, Kun Xu further developed a unifiedBGK scheme based on BGK-NS scheme with the discretization of velocity space.It is valid in all regions. In the low speed limit, UBGK is much more efficientthan DSMC and the discrete points in the velocity space can be reduced. Theresult is smooth without statistical noises. But in the high speed rarefied flowcomputations, DSMC and UBGK take similar computational time, because theremust be enough points in the velocity space.Considering the hardware conditions of computer in the present,applications of UBGK scheme directly to engineering is prohibitive. BGK-NS scheme and UBGK scheme have the same kinetic foundation. The former one isthe continuum limit of the latter. The computational time of BGK-NS scheme issimilar to common numerical schemes. So, we hope to first develop a hybridUBGK/BGK-NS algorithm covering different flow regimes for engineeringapplications. Although the BGK-NS scheme has been developed for almost twodecades, but from a view of engineering application, the job should still be delved.As a basis, it is necessary to carefully research and evaluate the scheme and itsability to application in hypersonic complex flows and in the near continuum-slipflow regions.The thesis is organized as following:Chapter one is the introduction. The background of present research isdescribed in detail. From two aspects: gas kinetic scheme and gas kinetic BGKscheme, the research progress at home and abroad is recommended. Finally thework done in this thesis is briefly introduced.Introduction of the numerical method is in chapter two. Firstly, somefundamental knowledge about gas kinetic theory and the BGK model equationare briefly introduced. Secondly, the famous KFVS scheme is introduced. In thefollow, Kun Xu’s gas kinetic BGK-NS scheme is well presented, including allformulas and its implementation. In the end, there are some importantstatements, including the extensions to multi-dimension, calculation of particlecollision time, Prandtl number fix, different limiting cases, some simplificationsand improvements by other researchers, and boundary conditions etc.In chapter three, some standard cases in CFD are tested to validate thecomputational codes, from inviscid to viscous; from low speed to high speed,from two dimensions to three dimensions and the performance of the scheme isresearched thoroughly. And the results show that, BGK-NS scheme hasexceptional performance, suitable to calculate various flows. Also, the codes arecorrect on the whole.The aim of chapter four is to exam the adaptability of BGK-NS scheme forcomplex flow simulations. Two benchmark cases are chosen including the lateraljet on a reentry vehicle in hypersonic freestream at high attack angle and thesupersonic flow around a grid-finned missile. There are two kinds of complexities,one is related to flow physics and the other geometries. The lateral jet on areentry vehicle in hypersonic freestream at high attack angle is a complex flowwith strong engineering background. And in the second case, a grid-finnedmissile contains very complex geometries. Good agreement is obtained betweenour results and the experimental and other simulating results in the literatures,indicating that BGK-NS scheme has great accuracy and ability to simulate various complex flows. This provides a good basis for our future work onconstructing a hybrid algorithm.In chapter five, two benchmark cases are chosen including hypersonic flowaround the hollow-cylinder-flare model and micro flow around a micro airfoilNACA0012, to quality for the applications of BGK-NS scheme in the nearcontinuum slip flow regimes. Good agreement is obtained between our resultsand the experimental and other simulating results in the literatures, indicatingthat BGK-NS scheme has latent capacity to be applied in the near continuum slipflow regimes. This also provides a good basis for our future work on constructinga hybrid algorithm.Conclusions are arranged in the sixth chapter. All the work done in theperiod of being a graduate student is summarized. The work in the future is alsopointed out.In the end, it’s a pleasure to give my gratitude and thanks to whomeveroffered me sincerely help and encouragement in the past three years. Thereferences cited in this thesis are listed in the last.