多相流MRT格式的松弛因子

hellolegend

多相流模型是He的动理学格式（JCP, 1999, 152: 642）
附图是矩空间中的平衡态分布函数以及我准备用的松弛因子。

请教的问题是关于松弛因子何时取常数。
对f而言，序参数phi守恒，因而s_{fi}取1，而phi u_x 和 phi u_y没什么物理意义，因而其松弛因子不能取常数1或者0。
对g而言，第0行的量所对应的松弛因子就不该为常数1，第3行为rho u_x /3，能视为守恒量吗？

另外，在文献上看到用该模型的MRT格式中，研究人员将 i＝0，i＝3，i＝5时的松弛因子均设定为1。感觉比较困惑，请大家指点一下。

luo@odu.edu

In your notation, "f_0", "f_3" and "f_5" are CONSERVED moments, thus their relaxation rate MUST be 1 -- collision does NOT affect the conserved moments and that is the manifestation of conservation laws in terms of microscopic dynamics.

The relaxation rate for "g_0" does not seem to be correct.

-- LSL

[ 本帖最后由 luo@odu.edu 于 2013-10-25 22:09 编辑 ]

hellolegend

谢谢罗老师！

行向量
M0 = [1, 1, 1, 1, 1, 1, 1, 1, 1]
M3 = [0, 1, 0, -1, 0, 1, -1, -1, 1]
M5 = [0, 0, 1, 0, -1, 1, 1, -1, -1]

所对应的松弛因子均取常数1，而无需考虑其所对应的平衡态分布函数？

[ 本帖最后由 hellolegend 于 2013-10-27 18:30 编辑 ]

luo@odu.edu

These are the conserved moments: \rho, ux, and uy, right. By definition, the equilibrium MUST be the function of conserved variables ONLY. Thus, the equilibria of conserved quantities are themselves.

Hope this helps.

-- LSL

hellolegend

一项是不是守恒项由平衡态分布函数决定，比矩空间中平衡态分布函数为rho的项表示的是质量，服从质量守恒定律，自然是守恒项，因而其松弛因子可以是任意常数（经常选择0或者1）；平衡态分布函数为rho u的项表示的动量，自然遵从动量守恒定律，其松弛因子可以是任意常数而不影响结果。

在上面的图中，phi是一个参数，类似于体积分率，我可以理解其为守恒量。现在的问题是：如何理解phi u和rho RT + psi(rho) 【等于 pressure p】也是守恒量？

谢谢罗老师的耐心指教。

hellolegend

含作用力项的MRT（TRT）要求守恒项为非0常数，如果设定
s_0 = s_1 = s_2 = s_3 = s_5 = s_7 = s_8 = s_{nu}
s_4 = s_6 = s_q
也能满足TRT的要求，而不要区分那个是守恒项。

luo@odu.edu

Usually, there is an interface force in the multi-phase/component models (e.g., SC model), thus, the relaxation rate for the momentum affects the force and that is the defect in the LBGK model.

For models with reactions, the relaxation rate for \rho destroys the mass conservation.

The whole point of MRT is to set the relaxation rate correctly for each moment (or mode). This is also based on the basic linearized Boltzmann equation in kinetic theory (Gross-Jackson model) and projection method.

[ 本帖最后由 luo@odu.edu 于 2013-10-30 11:10 编辑 ]