LBGK, ELBE, MRT-LB, and TRT-LB schemes: which one is better?

luo@odu.edu

Under the the topic of "LBM领域内部分活跃的专家", there has been a heated debate about the "advantages", if any, of the lattice BGK (LBGK) model. This issue was also discussed superficially (at the best) in the recent review of LBE in Annu. Review of Fluid Mech.

For those who are curious to know, attached please find a recent work of ours, in which we compare the LBGK, entropic LBE (ELBE), MRT-LB, and TRT-LB schemes in terms of accuracy, stability, and computational efficiency. The benchmark we use is the 2D lid-driven cavity flow. The solutions from a special method are used to compare with the LB solutions.

Our observation is that by far the ELBE is the most inferior one, followed by the LBGK. In terms of accuracy and stability, the ELBE and LBGK scheme are the same, while the LBGK scheme runs faster. Our conclusion is that, with no-slip boundary conditions, it is imperative to use the MRT or TRT LB schemes.

feixiang9

最近Freitas等人在CAF上面的一篇文章（http://dx.doi.org/10.1016/j.compfluid.2011.02.019）通过平板泊肃叶流动（plane Poiseuille flow）三维方腔流动以及通道湍流（turbulent channel flow）对BGK，MRT以及Cascaded lattice Boltzmann(CLB)三种格式进行了对比。
比较有趣的结果是：在进行槽道湍流进行较粗网格的直接数值模拟时，BGK得到的将结果反而比MRT和CLB的结果要好。MRT在计算中可能会出现高频振荡，从而使得数值计算无法进行；CLB在计算中则会出现非物理性的跳跃。

LBM各种格式的讨论以及稳定性理论的建立还需要更多的努力呀！：）

luo@odu.edu

Given the FACT that the LBGK model is only a special case of the corresponding MRT counterpart, it is hard to believe LOGICALLY that the LBGK model can outperform the MRT model. If one takes a careful look at the benchmark we have done, one should realize that choosing the relaxation rates can affect the results. Also, the initial and boundary conditions in the channel flow are important.

The implementation of the initial condition is incorrect in the paper by Freidas et al., because they use the distribution functions obtained by the LBGK model and impose them as the initial distributions for the MRT model. Because the nonequilibrium parts are NOT equation in the MRT and LBGK models, it of course won't work for the MRT model, and vice versa, that is, if they use the equilibrated f's from the MRT model and impose them as the initial conditions for the LBGK model, it won't work either.

To my surprise and dismay, the discussion about the boundary conditions is simply misleading (if not outright erroneous).

Well, my observation is that while there are only a few ways to do things right, there are much much more
(in fact infinite) ways to screw up.

feixiang9

多谢罗老师的回复。
疑问：
        我对Freidas  et al.对初始值设定的理解是：并不是将LBGK的分布函数之值直接赋给MRT来运算；而是LBGK计算得到的最终流场（速度、压力）做MRT的初始场进行计算。如果LBGK得到的流场是正确的话，那么这样的设定是没有问题的——因为我们在做数值计算值的时候，也常常用低雷诺数的计算结果作为高雷诺数的初始场。
      另外，关于反弹格式，您在2009年广州的ICMMES会议的中，讲的已经很明确了，我这边一直在用，精度和稳定性都很不错，在此也一并谢您了。

luo@odu.edu

WRONG! The equilibrated (NOT equilibrium) distribution functions from LBGK scheme do NOT equal to that of the MRT scheme, because the non-equilibrium parts depend on the relaxation rates. Thus, in work by Freitas et al., the initial conditions for the MRT and LBGK schemes are NOT equivalent. Thus, the LBGK scheme starts with equilibrated distributions, while the MRT schemes does NOT. Therefore there is an "initial layer" in the MRT simulations which takes long time to equilibrated. This in part explain the phenomenon, but there may be other causes.

wdlxmzd

3月12日我已将罗教授此文的PDF版传至LBM专题QQ群的群共享里面，大家也可以在群共享里面下载。

luo@odu.edu

Thank you for propagating our work.

feixiang9

多谢罗老师。
是不是可以这样理解：
第一、如果要比较格式的优劣的话，那么应该用相同的边界条件和初始条件进行计算；
第二、如果要使用已有的计算结果作为初始条件的话，应该很小心的注意初始条件的一致性（相容性）。

luo@odu.edu

Dear Feixiang9,

To compare several schemes, one must setup some objective standards. As for the boundary and initial conditions, they must be implemented correctly and consistently with each scheme. Some carefully analysis must be carried out. Clearly, the work by Freitas et al. may have quite a few "inconsistencies" in their implementations, and they are the direct consequence of the authors' lacking of understanding of the LB method.

pi_scu

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