有限要素法・流体力学による数値計算

\[ \begin{aligned} &\left( \frac{[C]}{5\Delta\tau} +V_x[C_x]+V_y[C_y]+V_z[C_z] \right)\{P\}^{\Delta\tau+\tau}\\[6pt] &\quad+\frac{1}{Ma^2} \left( [C_x]\{V_x\}^{\Delta\tau+\tau} +[C_y]\{V_y\}^{\Delta\tau+\tau} +[C_z]\{V_z\}^{\Delta\tau+\tau} \right) = \frac{[C]}{5\Delta\tau}\{P\}^{\tau} \end{aligned} \]

\[ \begin{aligned} \frac{[C]}{\Delta\tau}\{V_x\}^{\tau+\Delta\tau} &+\left(V_x[C_x]+V_y[C_y]+V_z[C_z]\right)\{V_x\}^{\tau+\Delta\tau}\\[6pt] &+\frac{1}{Re}\left\{ \left(2[S_{xx}]+[S_{yy}]+[S_{zz}]\right)\{V_x\}^{\tau+\Delta\tau} +[S_{yx}]\{V_y\}^{\tau+\Delta\tau} +[S_{zx}]\{V_z\}^{\tau+\Delta\tau} +[C_x]\{P\}^{\tau+\Delta\tau} \right\}\\[6pt] &= \frac{[C]}{\Delta\tau}\{V_x\}^{\tau} +\frac{2K^*}{We}n_x \begin{bmatrix}1\\1\\1\\1\end{bmatrix}\frac{S}{3} +g_x^* \begin{bmatrix}1\\1\\1\\1\end{bmatrix}\frac{V}{4} \end{aligned} \]

\[ \begin{aligned} \frac{[C]}{\Delta\tau}\{V_y\}^{\tau+\Delta\tau} &+\left(V_x[C_x]+V_y[C_y]+V_z[C_z]\right)\{V_y\}^{\tau+\Delta\tau}\\[6pt] &+\frac{1}{Re}\left\{ [S_{xy}]\{V_x\}^{\tau+\Delta\tau} +\left([S_{xx}]+2[S_{yy}]+[S_{zz}]\right)\{V_y\}^{\tau+\Delta\tau} +[S_{zy}]\{V_z\}^{\tau+\Delta\tau} +[C_y]\{P\}^{\tau+\Delta\tau} \right\}\\[6pt] &= \frac{[C]}{\Delta\tau}\{V_y\}^{\tau} +\frac{2K^*}{We}n_y \begin{bmatrix}1\\1\\1\\1\end{bmatrix}\frac{S}{3} +g_y^* \begin{bmatrix}1\\1\\1\\1\end{bmatrix}\frac{V}{4} \end{aligned} \]

\[ \begin{aligned} \frac{[C]}{\Delta\tau}\{V_z\}^{\tau+\Delta\tau} &+\left(V_x[C_x]+V_y[C_y]+V_z[C_z]\right)\{V_z\}^{\tau+\Delta\tau}\\[6pt] &+\frac{1}{Re}\left\{ [S_{xz}]\{V_x\}^{\tau+\Delta\tau} +[S_{yz}]\{V_y\}^{\tau+\Delta\tau} +\left([S_{xx}]+[S_{yy}]+2[S_{zz}]\right)\{V_z\}^{\tau+\Delta\tau} +[C_z]\{P\}^{\tau+\Delta\tau} \right\}\\[6pt] &= \frac{[C]}{\Delta\tau}\{V_z\}^{\tau} +\frac{2K^*}{We}n_z \begin{bmatrix}1\\1\\1\\1\end{bmatrix}\frac{S}{3} +g_z^* \begin{bmatrix}1\\1\\1\\1\end{bmatrix}\frac{V}{4} \qquad(i=1,2,3) \end{aligned} \]

\[ \begin{aligned} p_1&(v_{1x},v_{1y},v_{1z},p_1)\\[6pt] p_2&(v_{2x},v_{2y},v_{2z},p_2)\\[6pt] p_3&(v_{3x},v_{3y},v_{3z},p_3)\\[6pt] p_4&(v_{4x},v_{4y},v_{4z},p_4)\\[6pt] p_5&(v_{5x},v_{5y},v_{5z},p_5) \end{aligned} \]

\[ \left( \begin{array}{ccccccc} a_{11}&a_{12}&a_{13}&a_{14}&&a_{119}&a_{120}\\ a_{21}&a_{22}&a_{23}&a_{24}&&a_{219}&a_{220}\\ a_{31}&a_{32}&a_{33}&a_{34}&&a_{319}&a_{320}\\ a_{41}&a_{42}&a_{43}&a_{44}&&a_{419}&a_{420}\\ a_{51}&a_{52}&a_{53}&a_{54}&&a_{519}&a_{520}\\ a_{61}&a_{62}&a_{63}&a_{64}&&a_{619}&a_{620}\\ a_{71}&a_{72}&a_{73}&a_{74}&&a_{719}&a_{720}\\ a_{81}&a_{82}&a_{83}&a_{84}&&a_{819}&a_{820}\\ a_{91}&a_{92}&a_{93}&a_{94}&&a_{919}&a_{920}\\ a_{101}&a_{102}&a_{103}&a_{104}&&a_{1019}&a_{1020}\\ a_{111}&a_{112}&a_{113}&a_{114}&&a_{1119}&a_{1120}\\ a_{121}&a_{122}&a_{123}&a_{124}&&a_{1219}&a_{1220}\\ a_{131}&a_{132}&a_{133}&a_{134}&&a_{1319}&a_{1320}\\ a_{141}&a_{142}&a_{143}&a_{144}&&a_{1419}&a_{1420}\\ a_{151}&a_{152}&a_{153}&a_{154}&&a_{1519}&a_{1520}\\ a_{161}&a_{162}&a_{163}&a_{164}&&a_{1619}&a_{1620}\\ a_{171}&a_{172}&a_{173}&a_{174}&&a_{1719}&a_{1720}\\ a_{181}&a_{182}&a_{183}&a_{184}&&a_{1819}&a_{1820}\\ a_{191}&a_{192}&a_{193}&a_{194}&&a_{1919}&a_{1920}\\ a_{201}&a_{202}&a_{203}&a_{204}&&a_{2019}&a_{2020} \end{array} \right) \left( \begin{array}{c} v_{1x}\\ v_{1y}\\ v_{1z}\\ p_1\\ v_{2x}\\ v_{2y}\\ v_{2z}\\ p_2\\ v_{3x}\\ v_{3y}\\ v_{3z}\\ p_3\\ v_{4x}\\ v_{4y}\\ v_{4z}\\ p_4\\ v_{5x}\\ v_{5y}\\ v_{5z}\\ p_5 \end{array} \right)^{\tau+\Delta\tau} = \left( \begin{array}{c} v_{1x}\\ v_{1y}\\ v_{1z}\\ p_1\\ v_{2x}\\ v_{2y}\\ v_{2z}\\ p_2\\ v_{3x}\\ v_{3y}\\ v_{3z}\\ p_3\\ v_{4x}\\ v_{4y}\\ v_{4z}\\ p_4\\ v_{5x}\\ v_{5y}\\ v_{5z}\\ p_5 \end{array} \right)^{\tau} + \left( \begin{array}{c} f_{11}\\ f_{12}\\ f_{13}\\ f_{14}\\ f_{21}\\ f_{22}\\ f_{23}\\ f_{24}\\ f_{31}\\ f_{32}\\ f_{33}\\ f_{34}\\ f_{41}\\ f_{42}\\ f_{43}\\ f_{44}\\ f_{51}\\ f_{52}\\ f_{53}\\ f_{54} \end{array} \right) \]