6.2. MMS03 manufactured solutions
The manufactured solutions for the two-phase, 3D, curl-based functions with variable volume fraction are listed below.
Gas pressure:
(6.20)\[\begin{split}p_{g} = p_{g0}& + p_{\text{gx}}\cos\left( A_{p_{\text{gx}}}\text{πx} \right) + p_{\text{gy}}\cos\left( A_{p_{\text{gy}}}\text{πy} \right) + p_{\text{gxy}}\cos\left( A_{p_{\text{gxy}}}\text{πxy} \right) \\
& + p_{\text{gz}}\sin\left( A_{p_{\text{gz}}}\text{πz} \right) + p_{\text{gyz}}\sin\left( A_{p_{\text{gyz}}}\text{πyz} \right) + p_{\text{gzx}}\cos\left( A_{p_{\text{gzx}}}\text{πzx} \right)\end{split}\]
Gas velocity components:
(6.21)\[\begin{split}u_{g} = \frac{1}{\varepsilon_{g}}\big[& A_{w_{\text{gy}}}\pi w_{\text{gy}}\cos\left( A_{w_{\text{gy}}}\text{πy} \right) + A_{w_{\text{gxy}}}\pi w_{\text{gxy}}x\cos\left( A_{w_{\text{gxy}}}\text{πxy} \right) \\
&- A_{v_{\text{gyz}}}\pi v_{\text{gyz}}y\cos\left( A_{v_{\text{gyz}}}\text{πyz} \right) + A_{w_{\text{gyz}}}\pi w_{\text{gyz}}z\cos\left( A_{w_{\text{gyz}}}\text{πyz} \right) \\
&+ A_{v_{\text{gz}}}\pi v_{\text{gz}}\sin\left( A_{v_{\text{gz}}}\text{πz} \right) + A_{v_{\text{gzx}}}\pi v_{\text{gzx}}x\sin\left( A_{v_{\text{gzx}}}\text{πzx} \right) \big]\end{split}\]
(6.22)\[\begin{split}v_{g} = \frac{1}{\varepsilon_{g}}\big[& - A_{w_{\text{gxy}}}\pi w_{\text{gxy}}y\cos\left( A_{w_{\text{gxy}}}\text{πxy} \right) + A_{u_{\text{gyz}}}\pi u_{\text{gyz}}y\cos\left( A_{u_{\text{gyz}}}\text{πyz} \right) \\
&+ A_{w_{\text{gx}}}\pi w_{\text{gx}}\sin\left( A_{w_{\text{gx}}}\text{πx} \right) - A_{u_{\text{gz}}}\pi u_{\text{gz}}\sin\left( A_{u_{\text{gz}}}\text{πz} \right) \\
&- A_{u_{\text{gzx}}}\pi u_{\text{gzx}}x\sin\left( A_{u_{\text{gzx}}}\text{πzx} \right) + A_{w_{\text{gzx}}}\pi w_{\text{gzx}}z\sin\left( A_{w_{\text{gzx}}}\text{πzx} \right) \big]\end{split}\]
(6.23)\[\begin{split}w_{g} = \frac{1}{\varepsilon_{g}}\big[& A_{v_{\text{gx}}}\pi v_{\text{gx}}\cos\left( A_{v_{\text{gx}}}\text{πx} \right) - A_{u_{\text{gyz}}}\pi u_{\text{gyz}}z\cos\left( A_{u_{\text{gyz}}}\text{πyz} \right) \\
&+ A_{u_{\text{gxy}}}\pi u_{\text{gxy}}x\sin\left( A_{u_{\text{gxy}}}\text{πxy} \right) + A_{u_{\text{gy}}}\pi u_{\text{gy}}\sin\left( A_{u_{\text{gy}}}\text{πy} \right) \\
&- A_{v_{\text{gxy}}}\pi v_{\text{gxy}}y\sin\left( A_{v_{\text{gxy}}}\text{πxy} \right) - A_{v_{\text{gzx}}}\pi v_{\text{gzx}}z\sin\left( A_{v_{\text{gzx}}}\text{πzx} \right) \big]\end{split}\]
Solids velocity components:
(6.24)\[u_{m} = \frac{1}{\varepsilon_{m}}\left\lbrack u_{m0}\operatorname{}\left( \frac{\pi}{2}\left( x + y + z \right) \right) \right\rbrack\]
(6.25)\[v_{m} = \frac{1}{\varepsilon_{m}}\left\lbrack v_{m0}\operatorname{}\left( \frac{\pi}{2}\left( x + y + z \right) \right) \right\rbrack\]
(6.26)\[w_{m} = \frac{1}{\varepsilon_{m}}w_{m0}\]
Gas and solids temperature:
(6.27)\[\begin{split}T_{g} = T_{g0} &+ T_{\text{gx}}\cos\left( A_{T_{\text{gx}}}\text{πx} \right) + T_{\text{gy}}\cos\left( A_{T_{\text{gy}}}\text{πy} \right) + T_{\text{gxy}}\cos\left( A_{T_{\text{gxy}}}\text{πxy} \right) \\
&+ T_{\text{gz}}\sin\left( A_{T_{\text{gz}}}\text{πz} \right) + T_{\text{gyz}}\sin\left( A_{T_{\text{gyz}}}\text{πyz} \right) + T_{\text{gzx}}\cos\left( A_{T_{\text{gzx}}}\text{πzx} \right)\end{split}\]
(6.28)\[\begin{split}T_{m} = T_{m0} &+ T_{\text{mx}}\cos\left( A_{T_{\text{mx}}}\text{πx} \right) + T_{\text{my}}\cos\left( A_{T_{\text{my}}}\text{πy} \right) + T_{\text{mxy}}\cos\left( A_{T_{\text{mxy}}}\text{πxy} \right) \\
&+ T_{\text{mz}}\sin\left( A_{T_{\text{mz}}}\text{πz} \right) + T_{\text{myz}}\sin\left( A_{T_{\text{myz}}}\text{πyz} \right) + T_{\text{mzx}}\cos\left( A_{T_{\text{mzx}}}\text{πzx} \right)\end{split}\]
Solids granular temperature:
(6.29)\[\begin{split}\theta_{m} = \theta_{m0} &+ \theta_{\text{mx}}\cos\left( A_{\theta_{\text{mx}}}\text{πx} \right) + \theta_{\text{my}}\cos\left( A_{\theta_{\text{my}}}\text{πy} \right) + \theta_{\text{mxy}}\cos\left( A_{\theta_{\text{mxy}}}\text{πxy} \right) \\
&+ \theta_{\text{mz}}\sin\left( A_{\theta_{\text{mz}}}\text{πz} \right) + \theta_{\text{myz}}\sin\left( A_{\theta_{\text{myz}}}\text{πyz} \right) + \theta_{\text{mzx}}\cos\left( A_{\theta_{\text{mzx}}}\text{πzx} \right)\end{split}\]
The parameters appearing in the manufactured solutions are as follows:
Table 6.5 Parameters in MMS03 manufactured solutions.
\(p_{g0}\) |
100.0 |
\(v_{\text{gx}}\) |
-5.0 |
\(w_{m0}\) |
5.0 |
\(\varepsilon_{m0}\) |
0.3 |
\(p_{\text{gx}}\) |
20.0 |
\(v_{\text{gy}}\) |
4.0 |
\(T_{g0}\) |
350 |
\(\varepsilon_{\text{mx}}\) |
0.06 |
\(p_{\text{gy}}\) |
-50.0 |
\(v_{\text{gz}}\) |
5.0 |
\(T_{\text{gx}}\) |
10 |
\(\varepsilon_{\text{my}}\) |
-0.1 |
\(p_{\text{gz}}\) |
20.0 |
\(v_{\text{gxy}}\) |
-3.0 |
\(T_{\text{gy}}\) |
-30 |
\(\varepsilon_{\text{mz}}\) |
0.06 |
\(p_{\text{gxy}}\) |
-25.0 |
\(v_{\text{gyz}}\) |
2.5 |
\(T_{\text{gz}}\) |
20 |
\(\varepsilon_{\text{mxy}}\) |
0.0 |
\(p_{\text{gyz}}\) |
-10.0 |
\(v_{\text{gzx}}\) |
3.5 |
\(T_{\text{gxy}}\) |
-12 |
\(\varepsilon_{\text{myz}}\) |
0.0 |
\(p_{\text{gzx}}\) |
10.0 |
\(A_{v_{\text{gx}}}\) |
0.8 |
\(T_{\text{gyz}}\) |
10 |
\(\varepsilon_{\text{mzx}}\) |
0.0 |
\(A_{p_{\text{gx}}}\) |
0.4 |
\(A_{v_{\text{gy}}}\) |
0.8 |
\(T_{\text{gzx}}\) |
8 |
\(A_{\varepsilon_{\text{mx}}}\) |
0.4 |
\(A_{p_{\text{gy}}}\) |
0.45 |
\(A_{v_{\text{gz}}}\) |
0.5 |
\(A_{T_{\text{gx}}}\) |
0.75 |
\(A_{\varepsilon_{\text{my}}}\) |
0.5 |
\(A_{p_{\text{gz}}}\) |
0.85 |
\(A_{v_{\text{gxy}}}\) |
0.9 |
\(A_{T_{\text{gy}}}\) |
1.25 |
\(A_{\varepsilon_{\text{mz}}}\) |
0.5 |
\(A_{p_{\text{gxy}}}\) |
0.75 |
\(A_{v_{\text{gyz}}}\) |
0.4 |
\(A_{T_{\text{gz}}}\) |
0.8 |
\(A_{\varepsilon_{\text{mxy}}}\) |
0.4 |
\(A_{p_{\text{gyz}}}\) |
0.7 |
\(A_{v_{\text{gzx}}}\) |
0.6 |
\(A_{T_{\text{gxy}}}\) |
0.65 |
\(A_{\varepsilon_{\text{myz}}}\) |
0.4 |
\(A_{p_{\text{gzx}}}\) |
0.8 |
\(w_{g0}\) |
8.0 |
\(A_{T_{\text{gyz}}}\) |
0.5 |
\(A_{\varepsilon_{\text{mzx}}}\) |
0.4 |
\(u_{g0}\) |
7.0 |
\(w_{\text{gx}}\) |
-4.0 |
\(A_{T_{\text{gzx}}}\) |
0.6 |
\(\theta_{m0}\) |
100.0 |
\(u_{\text{gx}}\) |
3.0 |
\(w_{\text{gy}}\) |
3.5 |
\(T_{m0}\) |
300 |
\(\theta_{\text{mx}}\) |
5.0 |
\(u_{\text{gy}}\) |
-4.0 |
\(w_{\text{gz}}\) |
4.2 |
\(T_{\text{mx}}\) |
15 |
\(\theta_{\text{my}}\) |
-10.0 |
\(u_{\text{gz}}\) |
-3.0 |
\(w_{\text{gxy}}\) |
-2.2 |
\(T_{\text{my}}\) |
-20 |
\(\theta_{\text{mz}}\) |
12.0 |
\(u_{\text{gxy}}\) |
2.0 |
\(w_{\text{gyz}}\) |
2.1 |
\(T_{\text{mz}}\) |
15 |
\(\theta_{\text{mxy}}\) |
-8.0 |
\(u_{\text{gyz}}\) |
1.5 |
\(w_{\text{gzx}}\) |
2.5 |
\(T_{\text{mxy}}\) |
-10 |
\(\theta_{\text{myz}}\) |
10.0 |
\(u_{\text{gzx}}\) |
-2.0 |
\(A_{w_{\text{gx}}}\) |
0.85 |
\(T_{\text{myz}}\) |
12 |
\(\theta_{\text{mzx}}\) |
7.0 |
\(A_{u_{\text{gx}}}\) |
0.5 |
\(A_{w_{\text{gy}}}\) |
0.9 |
\(T_{\text{mzx}}\) |
10 |
\(A_{\theta_{\text{mx}}}\) |
0.8 |
\(A_{u_{\text{gy}}}\) |
0.85 |
\(A_{w_{\text{gz}}}\) |
0.5 |
\(A_{T_{\text{mx}}}\) |
0.5 |
\(A_{\theta_{\text{my}}}\) |
1.25 |
\(A_{u_{\text{gz}}}\) |
0.4 |
\(A_{w_{\text{gxy}}}\) |
0.4 |
\(A_{T_{\text{my}}}\) |
0.9 |
\(A_{\theta_{\text{mz}}}\) |
0.7 |
\(A_{u_{\text{gxy}}}\) |
0.6 |
\(A_{w_{\text{gyz}}}\) |
0.8 |
\(A_{T_{\text{mz}}}\) |
0.8 |
\(A_{\theta_{\text{mxy}}}\) |
0.5 |
\(A_{u_{\text{gyz}}}\) |
0.8 |
\(A_{w_{\text{gzx}}}\) |
0.75 |
\(A_{T_{\text{mxy}}}\) |
0.5 |
\(A_{\theta_{\text{myz}}}\) |
0.6 |
\(A_{u_{\text{gzx}}}\) |
0.9 |
\(u_{m0}\) |
5.0 |
\(A_{T_{\text{myz}}}\) |
0.65 |
\(A_{\theta_{\text{mzx}}}\) |
0.7 |
\(v_{g0}\) |
9.0 |
\(v_{m0}\) |
5.0 |
\(A_{T_{\text{mzx}}}\) |
0.4 |
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