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发表于 2012-9-13 20:19:01
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回复 83# lwd1981 的帖子
Dear LWD1981,
Your question is the simplest one, and yet the hardest one for me to answer succinctly. But let me give a try anyway. The Boltzmann equation (BE) solves the distribution function (DF) in phase space, not the hydrodynamic variables (flow density, momentum, and temperature) in real space, which are only the first few velocity moments of the DF. The boundary conditions of the BE are of course imposed on DF instead of just the hydrodynamic variables. In the Navier-Stokes (NS) equations, the boundary conditions are directly imposed on the hydrodynamic variables, while in the BE, they are indirectly imposed on them, so to speak. The LBE inherits this feature from the BE, because it is derived from the BE. Thus, for any boundary conditions of the LBE, you have to check carefully what are the effects on the hydrodynamic variables. This is a nuisance really, especially for those who are familiar with CFD but not kinetic theory.
However, we do have a theory for the LBE, although it is often ignored. Let's use the boundary conditions for simple flows as an example. You take the LBE with whatever collision model you prefer: BGK, MRT, TRT, LKS, etc. For, say, the Poiseuille flow, the LBE becomes a set of algebraic equations, which can be EASILY solved ANALYTICALLY. When you compare the LBE solution with the analytic solution of the NS equations and you can immediately see the problem you have -- the boundary values of the hydrodynamic variables depend on the collision model you chose: While ALL LB models give a PERFECT parabola, but where u=0 (the no-slip boundary condition) depends on collision model, or more specifically, the relaxation times in the model. This is of course reasonable: the higher-order moments in the LBE (which do not appear in the NS equations) are fluxes, and they affect the hydrodynamic variables -- the relaxation times determine the dissipation of these fluxes, hence the transport coefficients (and beyond).
Once this principle is understood, you will choose a collision model which allows you to match the LBE solution with the NS one EXACTLY, and this is precisely what the MRT model does -- it provides the freedom to adjust/correct the boundary conditions on the hydrodynamic variables, while the LBGK model does not have this luxury -- it has ONLY one parameter which is determined by the Reynolds number (Re). While the LBGK model can yield a PERFECT parabola, its boundary location can be miles off the exact solution, depending on the relaxation parameter "tau". This problem of the LBGK model alone has confused many as whether the LBGK model is 1st or 2nd order accurate, and upon this confusion, an industry has been built -- so many papers have been published, and it's still going. Pure theoretical interest aside, this fatal defect of the LBGK model prevents it to be used for flows through porous media, for which the LBE is one of the most competitive means, if not the best one.
Lately, this erroneous artifact of the LBGK model has found a new life -- it has been cast (or re-cast) as the "kinetic" slip velocity in rarefied gas flow. The problem of this approach is: the flow fields are wrong, and it leads to the preposterous results which require the Knudsen number Kn < 0! Yet, you will see paper after paper on this.
In literature, you will find the pioneering theoretical work by Ginzburg, d'Humieres, and others from early '90's. Often people complain that these papers are difficult to read. This is true to some extend but it's not a very sound excuse. However, if you are interested in the algebraic system resulted from the LBE and its solution, you could read the paper by He et. al. (which can be download from my website, paper #9), which solves the LBGK model with a number of BCs. The same technique can be used for the MRT model and it has been done and published.
The bottom line is: a theory exists, whether it is accepted or not that's another matter. There are only a few gems buried in a desert of pebbles, while the almighty internet provides such a convenience, to find these gems still requires human intelligence -- I wonder if and when computer will do anything/everything so we will no longer need human intelligence. I much sympathize many students in dire situations -- only not too long ago I was in the same situation myself. My own experience and the current situation motivate and compel me to take a forceful stance against fables of falsehood and the charlatans who promote them.
-- LSL
[ 本帖最后由 luo@odu.edu 于 2012-9-13 22:30 编辑 ] |
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