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