6.4. MMS05 manufactured solutions
The manufactured solutions for the Free-slip wall BC, single phase, 3D, curl-based functions are listed below.
Gas pressure:
(6.35)\[\begin{split}p_{g} = p_{g0} &+ p_{\text{gx}}\cos\left( A_{p_{\text{gx}}}\text{πx} \right) + p_{\text{gy}}\cos\left( A_{p_{\text{gy}}}\text{πy} \right) + p_{\text{gxy}}\cos\left( A_{p_{\text{gxy}}}\text{πxy} \right) \\
&+ p_{\text{gz}}\sin\left( A_{p_{\text{gz}}}\text{πz} \right) + p_{\text{gyz}}\sin\left( A_{p_{\text{gyz}}}\text{πyz} \right) + p_{\text{gzx}}\cos\left( A_{p_{\text{gzx}}}\text{πzx} \right)\end{split}\]
Gas velocity components:
(6.36)\[\begin{split}u_{g} = x^{3}\big[& A_{w_{\text{gy}}}\pi w_{\text{gy}}\cos\left( A_{w_{\text{gy}}}\text{πy} \right) + A_{w_{\text{gxy}}}\pi w_{\text{gxy}}x\cos\left( A_{w_{\text{gxy}}}\text{πxy} \right) \\
&- A_{v_{\text{gyz}}}\pi v_{\text{gyz}}y\cos\left( A_{v_{\text{gyz}}}\text{πyz} \right) + A_{w_{\text{gyz}}}\pi w_{\text{gyz}}z\cos\left( A_{w_{\text{gyz}}}\text{πyz} \right) \\
&+ A_{v_{\text{gz}}}\pi v_{\text{gz}}\sin\left( A_{v_{\text{gz}}}\text{πz} \right) + A_{v_{\text{gzx}}}\pi v_{\text{gzx}}x\sin\left( A_{v_{\text{gzx}}}\text{πzx} \right) \big]\end{split}\]
(6.37)\[\begin{split}v_{g} =& v_{g0} + x^{3}\big[ - A_{w_{\text{gxy}}}\pi w_{\text{gxy}}y\cos\left( A_{w_{\text{gxy}}}\text{πxy} \right) + A_{u_{\text{gyz}}}\pi u_{\text{gyz}}y\cos\left( A_{u_{\text{gyz}}}\text{πyz} \right) \\
&+ A_{w_{\text{gx}}}\pi w_{\text{gx}}\sin\left( A_{w_{\text{gx}}}\text{πx} \right) - A_{u_{\text{gz}}}\pi u_{\text{gz}}\sin\left( A_{u_{\text{gz}}}\text{πz} \right) \\
&- A_{u_{\text{gzx}}}\pi u_{\text{gzx}}x\sin\left( A_{u_{\text{gzx}}}\text{πzx} \right) + A_{w_{\text{gzx}}}\pi w_{\text{gzx}}z\sin\left( A_{w_{\text{gzx}}}\text{πzx} \right) \big] \\
&+ 3x^{2} \big[ - w_{g0} - w_{\text{gx}}\cos\left( A_{w_{\text{gx}}}\text{πx} \right) - w_{\text{gz}}\cos\left( A_{w_{\text{gz}}}\text{πz} \right) \\
&- w_{\text{gzx}}\cos\left( A_{w_{\text{gzx}}}\text{πzx} \right) - w_{\text{gy}}\sin\left( A_{w_{\text{gy}}}\text{πy} \right) \\
&- w_{\text{gxy}}\sin\left( A_{w_{\text{gxy}}}\text{πxy} \right) - w_{\text{gyz}}\sin\left( A_{w_{\text{gyz}}}\text{πyz} \right) \big]\end{split}\]
(6.38)\[\begin{split}w_{g} =& w_{g0} + x^{3}\big[ A_{v_{\text{gx}}}\pi v_{\text{gx}}\cos\left( A_{v_{\text{gx}}}\text{πx} \right) - A_{u_{\text{gyz}}}\pi u_{\text{gyz}}z\cos\left( A_{u_{\text{gyz}}}\text{πyz} \right) \\
&+ A_{u_{\text{gxy}}}\pi u_{\text{gxy}}x\sin\left( A_{u_{\text{gxy}}}\text{πxy} \right) + A_{u_{\text{gy}}}\pi u_{\text{gy}}\sin\left( A_{u_{\text{gy}}}\text{πy} \right) \\
&- A_{v_{\text{gxy}}}\pi v_{\text{gxy}}y\sin\left( A_{v_{\text{gxy}}}\text{πxy} \right) - A_{v_{\text{gzx}}}\pi v_{\text{gzx}}z\sin\left( A_{v_{\text{gzx}}}\text{πzx} \right) \big] \\
&+ 3x^{2}\big[ v_{g0} + v_{\text{gy}}\cos\left( A_{v_{\text{gy}}}\text{πy} \right) + v_{\text{gxy}}\cos\left( A_{v_{\text{gxy}}}\text{πxy} \right) \\
&+ v_{\text{gz}}\cos\left( A_{v_{\text{gz}}}\text{πz} \right) + v_{\text{gzx}}\cos\left( A_{v_{\text{gzx}}}\text{πzx} \right) \\
&+ v_{\text{gx}}\sin\left( A_{v_{\text{gx}}}\text{πx} \right) + v_{\text{gyz}}\sin\left( A_{v_{\text{gyz}}}\text{πyz} \right) \big]\end{split}\]
Gas volume fraction:
(6.39)\[\varepsilon_{g} = 1.0\]
The parameters appearing in the manufactured solutions are as follows:
Table 6.7 Parameters in MMS05 manufactured solutions.
\(p_{g0}\) |
100.0 |
\(v_{g0}\) |
9.0 |
\(u_{g0}\) |
7.0 |
\(w_{g0}\) |
8.0 |
\(p_{\text{gx}}\) |
20.0 |
\(v_{\text{gx}}\) |
-5.0 |
\(u_{\text{gx}}\) |
3.0 |
\(w_{\text{gx}}\) |
-4.0 |
\(p_{\text{gy}}\) |
-50.0 |
\(v_{\text{gy}}\) |
4.0 |
\(u_{\text{gy}}\) |
-4.0 |
\(w_{\text{gy}}\) |
3.5 |
\(p_{\text{gz}}\) |
20.0 |
\(v_{\text{gz}}\) |
5.0 |
\(u_{\text{gz}}\) |
-3.0 |
\(w_{\text{gz}}\) |
4.2 |
\(p_{\text{gxy}}\) |
-25.0 |
\(v_{\text{gxy}}\) |
-3.0 |
\(u_{\text{gxy}}\) |
2.0 |
\(w_{\text{gxy}}\) |
-2.2 |
\(p_{\text{gyz}}\) |
-10.0 |
\(v_{\text{gyz}}\) |
2.5 |
\(u_{\text{gyz}}\) |
1.5 |
\(w_{\text{gyz}}\) |
2.1 |
\(p_{\text{gzx}}\) |
10.0 |
\(v_{\text{gzx}}\) |
3.5 |
\(u_{\text{gzx}}\) |
-2.0 |
\(w_{\text{gzx}}\) |
2.5 |
\(A_{p_{\text{gx}}}\) |
0.4 |
\(A_{v_{\text{gx}}}\) |
0.8 |
\(A_{u_{\text{gx}}}\) |
0.5 |
\(A_{w_{\text{gx}}}\) |
0.85 |
\(A_{p_{\text{gy}}}\) |
0.45 |
\(A_{v_{\text{gy}}}\) |
0.8 |
\(A_{u_{\text{gy}}}\) |
0.85 |
\(A_{w_{\text{gy}}}\) |
0.9 |
\(A_{p_{\text{gz}}}\) |
0.85 |
\(A_{v_{\text{gz}}}\) |
0.5 |
\(A_{u_{\text{gz}}}\) |
0.4 |
\(A_{w_{\text{gz}}}\) |
0.5 |
\(A_{p_{\text{gxy}}}\) |
0.75 |
\(A_{v_{\text{gxy}}}\) |
0.9 |
\(A_{u_{\text{gxy}}}\) |
0.6 |
\(A_{w_{\text{gxy}}}\) |
0.4 |
\(A_{p_{\text{gyz}}}\) |
0.7 |
\(A_{v_{\text{gyz}}}\) |
0.4 |
\(A_{u_{\text{gyz}}}\) |
0.8 |
\(A_{w_{\text{gyz}}}\) |
0.8 |
\(A_{p_{\text{gzx}}}\) |
0.8 |
\(A_{v_{\text{gzx}}}\) |
0.6 |
\(A_{u_{\text{gzx}}}\) |
0.9 |
\(A_{w_{\text{gzx}}}\) |
0.75 |