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