diff --git a/docs/source/userGuide/models/mrt/mrt.rst b/docs/source/userGuide/models/mrt/mrt.rst index 65fe57c7..b78d0987 100644 --- a/docs/source/userGuide/models/mrt/mrt.rst +++ b/docs/source/userGuide/models/mrt/mrt.rst @@ -30,6 +30,131 @@ The essential model parameters for the color model are Model Formulation **************************** +The LBE governing momentum transport is defined based on a MRT relaxation based on the D3Q19 discrete +velocity set, which determines the values :math:`\bm{\xi}_q` + +.. math:: + :nowrap: + + $$ + f_q(\bm{x}_i + \bm{\xi}_q \delta t,t + \delta t) - f_q(\bm{x}_i,t) = \sum^{Q-1}_{k=0} M^{-1}_{qk} \lambda_{k} (m_k^{eq}-m_k) + t_q \bm{\xi}_q \cdot \frac{\bm{F}}{c_s^2} \;, + $$ + + +The moments are linearly indepdendent: + +.. math:: + :nowrap: + + $$ + m_k = \sum_{q=0}^{18} M_{qk} f_q\;. + $$ + + +The non-zero equilibrium moments are + +.. math:: + :nowrap: + + $$ + m_1^{eq} = 19\frac{j_x^2+j_y^2+j_z^2}{\rho} - 11\rho \;, + $$ + +.. math:: + :nowrap: + + $$ + m_2^{eq} = 3\rho - frac{11}{2} \frac{j_x^2+j_y^2+j_z^2}{\rho} \;, + $$ + +.. math:: + :nowrap: + + + $$ + m_4^{eq} = -\frac 2 3 j_x \;, + $$ + +.. math:: + :nowrap: + + $$ + m_6^{eq} = -\frac 2 3 j_y \;, + $$ + +.. math:: + :nowrap: + + $$ + m_8^{eq} = -\frac 2 3 j_z \;, + $$ + +.. math:: + :nowrap: + + $$ + m_9^{eq} = \frac{2j_x^2-j_y^2-j_z^2}{\rho}\;, + $$ + +.. math:: + :nowrap: + + $$ + m_{10}^{eq} = -\frac{2j_x^2-j_y^2-j_z^2)}{2\rho_0} \;, + $$ + +.. math:: + :nowrap: + + $$ + m_{11}^{eq} = \frac{j_y^2-j_z^2}{\rho_0} \;, + $$ + +.. math:: + :nowrap: + + $$ + m_{12}^{eq} = -\frac{j_y^2-j_z^2}{2\rho_0} \;, + $$ + + +.. math:: + :nowrap: + + $$ + m_{13}^{eq} = \frac{j_x j_y}{\rho_0} \;, + $$ + +.. math:: + :nowrap: + + $$ + m_{14}^{eq} = \frac{j_y j_z}{\rho_0} \;, + $$ + +.. math:: + :nowrap: + + $$ + m_{15}^{eq} = \frac{j_x j_z}{\rho_0} \;, + $$ + +The relaxation parameters are determined based on the relaxation time :math:`\tau` +.. math:: + :nowrap: + + $$ + \lambda_1 = \lambda_2= \lambda_9 = \lambda_{10}= \lambda_{11}= \lambda_{12}= \lambda_{13}= \lambda_{14}= \lambda_{15} = s_\nu = \frac{1}{\tau} \;, + $$ + +.. math:: + :nowrap: + + $$ + \lambda_{4}= \lambda_{6}= \lambda_{8} = \lambda_{16} = \lambda_{17} = \lambda_{18}= \frac{8(2-s_\nu)}{8-s_\nu} \;, + $$ + + **************************** Example Input File