Commit dbff1f18 authored by Matthieu Constant's avatar Matthieu Constant
Browse files

corrected equations

parent 92f33f44
Pipeline #3734 failed with stage
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......@@ -123,7 +123,7 @@ The weak form of these equations is obtained by integrating it multiplied by som
\begin{align*}
N_f\times\left\lbrace\begin{aligned}
\langlC\rho_k\dpartial{\ub}{t}\psi_i\ranglC+\langlC\rho_k\nb\cdot\left(\dfrac{\ub\ub}{q_k}\right)\psi_i\ranglC\\
\langlC\dpartial{q}{t}\psi_i\ranglC+\langlC\nb\cdot\ub\psi_i\ranglC+\langlC\varepsilon\nb\cdot\bm{r}_k\ranglC\end{aligned}\right.&\begin{aligned}
\langlC\dpartial{q}{t}\psi_i\ranglC+\langlC\nb\cdot\ub\psi_i\ranglC+\langlC\varepsilon\nb\cdot\bm{r}_k\psi_i\ranglC\end{aligned}\right.&\begin{aligned}
&=-\langlC q_k\nb p \psi_i\ranglC+\langlC\nb\cdot\taub q_k\psi_i\ranglC+\langlC q_k\rho_k \gb\psi_i\ranglC\phantom{\dpartial{\ub}{t}}\\
&=0\phantom{\dpartial{q}{t}}\end{aligned}\\
\langlC \left(1-c-\sum_{k=1}^{N_f}q_k\right)\psi_i\ranglC&\begin{aligned}&=0\end{aligned}
......@@ -131,19 +131,19 @@ N_f\times\left\lbrace\begin{aligned}
These equations can be simplified by integrating by part and applying the divergence theorem to the flux terms:
\begin{scriptsize}
\begin{tiny}
\begin{align*}
N_f\times\left\lbrace\begin{aligned}
\langlC\rho_k\dpartial{\ub}{t}\psi_i\ranglC+\langlC\rho_k\nb\cdot\left(\dfrac{\ub\ub}{q_k}\right)\psi_i\ranglC\\
\\
\langlC\dpartial{q_k}{t}\psi_i\ranglC\end{aligned}\right.&\begin{aligned}
\langlC\dpartial{q_k}{t}\psi_i\ranglC+\langlC\varepsilon q_k\nb p\cdot\nb\psi_i\ranglC-\langlC\langlC\varepsilon q_k\nb p\psi_i\cdot\bm{n}\ranglC\ranglC\end{aligned}\right.&\begin{aligned}
&=\langlC p\nb q_k\psi_i\ranglC-\langlC\langlC pq_k\psi_i\bm{n}\ranglC\ranglC-\langlC\taub\cdot\nb q_k\psi_i\ranglC+\langlC\langlC\taub q_k\psi_i\cdot\bm{n}\ranglC\ranglC+\langlC q_k\rho_k \gb\psi_i\ranglC\phantom{\dpartial{\ub}{t}}\\
\\
&=\langlC\varepsilon\bm{r}_k\cdot\nb\psi_i\ranglC+\langlC\langlC\varepsilon\bm{r}_k\psi_i\cdot\bm{n}\ranglC\ranglC+\langlC\ub\cdot\nb\psi_i\ranglC-\langlC\langlC \ub \psi_i\cdot\bm{n}\ranglC\ranglC\phantom{\dpartial{q}{t}}\end{aligned}\\
&=-\langlC\varepsilon\nb\cdot(q_k\rho_k\bm{g})\psi_i\ranglC+\langlC\ub\cdot\nb\psi_i\ranglC-\langlC\langlC \ub \psi_i\cdot\bm{n}\ranglC\ranglC\phantom{\dpartial{q}{t}}\end{aligned}\\
\\
\langlC \left(1-c-\sum_{k=1}^{N_f}q_k\right)\psi_i\ranglC&\begin{aligned}&=0\end{aligned}
\end{align*}
\end{scriptsize}
\end{tiny}
The fully developed 2D equations considering that the vertical velocity is $v$ while the horizontal velocity is $u$ are given by:
......@@ -164,9 +164,9 @@ $\langlC\bm{f}_{q_p}^0,\psi_i\ranglC$&&&$=0$\\[1em]
\bm{f}_{\bm{u}}^0&=\rho_k\left( \dpartial{\ub}{t}+ \ub\dfrac{\nb\cdot\ub}{q}+\dfrac{\bm{A}_k\cdot\ub}{q}\right)-p\nb q_k+\mu\dfrac{\bm{B}_k\cdot\nb q_k}{q_k}-q_k\rho_k\bm{g}\\[1em]
\bm{f}_{\bm{u}}^1&= -q_kp\ib+\mu \bm{B}_k\\[1em]
\bm{f}_{\bm{ub}}&=\bm{B}_k\cdot\bm{n}_k+q_k p\bm{n}\\[1em]
\bm{f}_{q_k}^0&=\dpartial{q_k}{t}\\[1em]
\bm{f}_{q_k}^1&=\varepsilon\left( q_k\nb p-q_k\rho_k \bm{g}\right)-\ub\\[1em]
\bm{f}_{q_k b}&=\varepsilon \left(q_k\nb p\cdot\bm{n}_k-q_k\rho_k\bm{g}\right)+\ub\\[1em]
\bm{f}_{q_k}^0&=\dpartial{q_k}{t}+\varepsilon\nb\cdot( q_k\rho_k\bm{g})\\[1em]
\bm{f}_{q_k}^1&=\varepsilon\left( q_k\nb p\right)-\ub\\[1em]
\bm{f}_{q_k b}&=-\varepsilon q_k\nb p\cdot\bm{n}_k+\ub\\[1em]
\bm{f}_{q_p}^0&=1-c-\sum_{k=1}^{N_f}q_k
\end{align*}
......
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