From 343a439f3be4c13917b8d819e54f12bb5a51ebe5 Mon Sep 17 00:00:00 2001 From: James McClure Date: Fri, 3 Sep 2021 06:04:19 -0400 Subject: [PATCH] added color model formulation --- docs/source/userGuide/models/color/index.rst | 123 ++++++++++++------- 1 file changed, 81 insertions(+), 42 deletions(-) diff --git a/docs/source/userGuide/models/color/index.rst b/docs/source/userGuide/models/color/index.rst index db45f8c3..382f1ff7 100644 --- a/docs/source/userGuide/models/color/index.rst +++ b/docs/source/userGuide/models/color/index.rst @@ -67,94 +67,107 @@ Model Formulation Two LBEs are constructed to model the mass transport, incorporating the anti-diffusion + .. math:: :nowrap: $$ - \\begin{eqnarray} - A_q(\bm{x} + \bm{\xi}_q \delta t, t+\delta t) &=& w_q N_a \Big[1 + \frac{\bm{u} \cdot \bm{\xi}_q}{c_s^2} + A_q(\bm{x} + \bm{\xi}_q \delta t, t+\delta t) = w_q N_a \Big[1 + \frac{\bm{u} \cdot \bm{\xi}_q}{c_s^2} + \beta \frac{N_b}{N_a+N_b} \bm{n} \cdot \bm{\xi}_q\Big] \; - \\ - B_q(\bm{x} + \bm{\xi}_q \delta t, t+\delta t) &=& + $$ + +.. math:: + :nowrap: + + $$ + B_q(\bm{x} + \bm{\xi}_q \delta t, t+\delta t) = w_q N_b \Big[1 + \frac{\bm{u} \cdot \bm{\xi}_q}{c_s^2} - \beta \frac{N_a}{N_a+N_b} \bm{n} \cdot \bm{\xi}_q\Big]\;, - \\end{eqnarray} $$ -The number density is obtained directly for each fluid as the sum of the mass transport distributions +The number density for each fluid is obtained from the sum of the mass transport distributions + .. math:: :nowrap: $$ - \\begin{equation} N_a = \sum_q A_q\;, \quad N_b = \sum_q B_q\; - \\end{equation} $$ The phase indicator field is then defined as + .. math:: :nowrap: $$ - \\begin{equation} \phi = \frac{N_a-N_b}{N_a+N_b} - \\end{equation} $$ -The local fluid viscosity and density are determined based on linear interpolation +The fluid density and kinematic viscosity are determined based on linear interpolation + .. math:: :nowrap: $$ - \\begin{equation} - \nu = \frac{(1+\phi) \nu_n}{2}+\frac{(1-\phi) \nu_w}{2} \;, - \\end{equation} - $$ - -.. math:: - :nowrap: - - $$ - \\begin{equation} \rho_0 = \frac{(1+\phi) \rho_n}{2}+ \frac{(1-\phi) \rho_w}{2} \;, - \\end{equation} $$ +.. math:: + :nowrap: + + $$ + \nu = \frac{(1+\phi) \nu_n}{2}+\frac{(1-\phi) \nu_w}{2} \;, + $$ + +where + +.. math:: + :nowrap: + + $$ + \nu_w = \frac{1}{3}\Big(\tau_w - \frac{1}{2} \Big) \;, \quad + \nu_n = \frac{1}{3}\Big(\tau_n - \frac{1}{2} \Big) \;. + $$ + + These values are then used to model the momentum transport. The LBE governing momentum transport is defined based on a MRT relaxation process with additional terms to account for the interfacial stresses + .. math:: :nowrap: $$ - \\begin{equation} f_q(\bm{x}_i + \bm{\xi}_q \delta t,t + \delta t) - f_q(\bm{x}_i,t) = \sum^{Q-1}_{k=0} M^{-1}_{qk} \lambda_{k} (m_k^{eq}-m_k) + t_q \bm{\xi}_q \cdot \frac{\bm{F}}{c_s^2} \;, - \\end{equation} $$ -The moments are linearly indepdendent +The moments are linearly indepdendent: + .. math:: :nowrap: $$ - \\begin{equation} m_k = \sum_{q=0}^{18} M_{qk} f_q\;. - \\end{equation} $$ The relaxation parameters are determined from the relaxation time: + .. math:: :nowrap: $$ - \\begin{eqnarray} - \lambda_1 = \lambda_2= \lambda_9 = \lambda_{10}= \lambda_{11}= \lambda_{12}= \lambda_{13}= \lambda_{14}= \lambda_{15} = s_\nu\;, \\ + \lambda_1 = \lambda_2= \lambda_9 = \lambda_{10}= \lambda_{11}= \lambda_{12}= \lambda_{13}= \lambda_{14}= \lambda_{15} = s_\nu\;, + $$ + +.. math:: + :nowrap: + + $$ \lambda_{4}= \lambda_{6}= \lambda_{8} = \lambda_{16} = \lambda_{17} = \lambda_{18}= \frac{8(2-s_\nu)}{8-s_\nu} \;, - \\end{eqnarray} $$ The non-zero equilibrium moments are defined as @@ -163,34 +176,60 @@ The non-zero equilibrium moments are defined as :nowrap: $$ - \\begin{eqnarray} - m_1^{eq} &=& (j_x^2+j_y^2+j_z^2) - \alpha |\textbf{C}|, \\ - m_9^{eq} &=& (2j_x^2-j_y^2-j_z^2)+ \alpha \frac{|\textbf{C}|}{2}(2n_x^2-n_y^2-n_z^2), \\ - m_{11}^{eq} &=& (j_y^2-j_z^2) + \alpha \frac{|\textbf{C}|}{2}(n_y^2-n_z^2), \\ - m_{13}^{eq} &=& j_x j_y + \alpha \frac{|\textbf{C}|}{2} n_x n_y\;, \\ - m_{14}^{eq} &=& j_y j_z + \alpha \frac{|\textbf{C}|}{2} n_y n_z\;, \\ - m_{15}^{eq} &=& j_x j_z + \alpha \frac{|\textbf{C}|}{2} n_x n_z\;, - \\end{eqnarray} + m_1^{eq} = (j_x^2+j_y^2+j_z^2) - \alpha |\textbf{C}|, \\ + $$ + +.. math:: + :nowrap: + + $$ + m_9^{eq} = (2j_x^2-j_y^2-j_z^2)+ \alpha \frac{|\textbf{C}|}{2}(2n_x^2-n_y^2-n_z^2), \\ + $$ + +.. math:: + :nowrap: + + $$ + m_{11}^{eq} = (j_y^2-j_z^2) + \alpha \frac{|\textbf{C}|}{2}(n_y^2-n_z^2), \\ + $$ + +.. math:: + :nowrap: + + $$ + m_{13}^{eq} = j_x j_y + \alpha \frac{|\textbf{C}|}{2} n_x n_y\;, \\ + $$ + +.. math:: + :nowrap: + + $$ + m_{14}^{eq} = j_y j_z + \alpha \frac{|\textbf{C}|}{2} n_y n_z\;, \\ + $$ + +.. math:: + :nowrap: + + $$ + m_{15}^{eq} = j_x j_z + \alpha \frac{|\textbf{C}|}{2} n_x n_z\;, $$ where the color gradient is determined from the phase indicator field + .. math:: :nowrap: $$ - \\begin{equation} \textbf{C}=\nabla \phi\;. - \\end{equation} $$ and the unit normal vector is + .. math:: :nowrap: $$ - \\begin{equation} \bm{n} = \frac{\textbf{C}}{|\textbf{C}|}\;. - \\end{equation} $$ ****************************