6.1. MMS02 manufactured solutions

The manufactured solutions for the two-phase, 3D, curl-based functions with constant volume fraction are listed below.

Gas pressure:

(6.8) $$\begin{split}p_{g} = p_{g0}& + p_{\text{gx}}\cos\left( A_{p_{\text{gx}}}\text{πx} \right) + p_{\text{gy}}\cos\left( A_{p_{\text{gy}}}\text{πy} \right) + p_{\text{gxy}}\cos\left( A_{p_{\text{gxy}}}\text{πxy} \right) \\ &+ p_{\text{gz}}\sin\left( A_{p_{\text{gz}}}\text{πz} \right) + p_{\text{gyz}}\sin\left( A_{p_{\text{gyz}}}\text{πyz} \right) + p_{\text{gzx}}\cos\left( A_{p_{\text{gzx}}}\text{πzx} \right)\end{split}$$

Gas velocity components:

(6.9) $$\begin{split}u_{g} =& A_{w_{\text{gy}}}\pi w_{\text{gy}}\cos\left( A_{w_{\text{gy}}}\text{πy} \right) + A_{w_{\text{gxy}}}\pi w_{\text{gxy}}x\cos\left( A_{w_{\text{gxy}}}\text{πxy} \right) \\ &- A_{v_{\text{gyz}}}\pi v_{\text{gyz}}y\cos\left( A_{v_{\text{gyz}}}\text{πyz} \right) + A_{w_{\text{gyz}}}\pi w_{\text{gyz}}z\cos\left( A_{w_{\text{gyz}}}\text{πyz} \right) \\ &+ A_{v_{\text{gz}}}\pi v_{\text{gz}}\sin\left( A_{v_{\text{gz}}}\text{πz} \right) + A_{v_{\text{gzx}}}\pi v_{\text{gzx}}x\sin\left( A_{v_{\text{gzx}}}\text{πzx} \right)\end{split}$$

(6.10) $$\begin{split}v_{g} =& - A_{w_{\text{gxy}}}\pi w_{\text{gxy}}y\cos\left( A_{w_{\text{gxy}}}\text{πxy} \right) + A_{u_{\text{gyz}}}\pi u_{\text{gyz}}y\cos\left( A_{u_{\text{gyz}}}\text{πyz} \right) \\ &+ A_{w_{\text{gx}}}\pi w_{\text{gx}}\sin\left( A_{w_{\text{gx}}}\text{πx} \right) - A_{u_{\text{gz}}}\pi u_{\text{gz}}\sin\left( A_{u_{\text{gz}}}\text{πz} \right) \\ &- A_{u_{\text{gzx}}}\pi u_{\text{gzx}}x\sin\left( A_{u_{\text{gzx}}}\text{πzx} \right) + A_{w_{\text{gzx}}}\pi w_{\text{gzx}}z\sin\left( A_{w_{\text{gzx}}}\text{πzx} \right)\end{split}$$

(6.11) $$\begin{split}w_{g} =& A_{v_{\text{gx}}}\pi v_{\text{gx}}\cos\left( A_{v_{\text{gx}}}\text{πx} \right) - A_{u_{\text{gyz}}}\pi u_{\text{gyz}}z\cos\left( A_{u_{\text{gyz}}}\text{πyz} \right) \\ &+ A_{u_{\text{gy}}}\pi u_{\text{gy}}\sin\left( A_{u_{\text{gy}}}\text{πy} \right) + A_{u_{\text{gxy}}}\pi u_{\text{gxy}}x\sin\left( A_{u_{\text{gxy}}}\text{πxy} \right) \\ &- A_{v_{\text{gxy}}}\pi v_{\text{gxy}}y\sin\left( A_{v_{\text{gxy}}}\text{πxy} \right) - A_{v_{\text{gzx}}}\pi v_{\text{gzx}}z\sin\left( A_{v_{\text{gzx}}}\text{πzx} \right)\end{split}$$

Solids velocity components:

(6.12) $$u_{m} = u_{m0}\operatorname{}\left( \frac{\pi}{2}\left( x + y + z \right) \right)$$

(6.13) $$v_{m} = v_{m0}\operatorname{}\left( \frac{\pi}{2}\left( x + y + z \right) \right)$$

(6.14) $$w_{m} = w_{m0}$$

Gas and solids temperature:

(6.15) $$\begin{split}T_{g} = T_{g0}& + T_{\text{gx}}\cos\left( A_{T_{\text{gx}}}\text{πx} \right) + T_{\text{gy}}\cos\left( A_{T_{\text{gy}}}\text{πy} \right) + T_{\text{gxy}}\cos\left( A_{T_{\text{gxy}}}\text{πxy} \right) \\ &+ T_{\text{gz}}\sin\left( A_{T_{\text{gz}}}\text{πz} \right) + T_{\text{gyz}}\sin\left( A_{T_{\text{gyz}}}\text{πyz} \right) + T_{\text{gzx}}\cos\left( A_{T_{\text{gzx}}}\text{πzx} \right)\end{split}$$

(6.16) $$\begin{split}T_{m} = T_{m0}& + T_{\text{mx}}\cos\left( A_{T_{\text{mx}}}\text{πx} \right) + T_{\text{my}}\cos\left( A_{T_{\text{my}}}\text{πy} \right) + T_{\text{mxy}}\cos\left( A_{T_{\text{mxy}}}\text{πxy} \right) \\ &+ T_{\text{mz}}\sin\left( A_{T_{\text{mz}}}\text{πz} \right) + T_{\text{myz}}\sin\left( A_{T_{\text{myz}}}\text{πyz} \right) + T_{\text{mzx}}\cos\left( A_{T_{\text{mzx}}}\text{πzx} \right)\end{split}$$

Gas and solids volume fractions:

(6.17) $$\begin{split}\varepsilon_{g} = 1 - \big(& \varepsilon_{m0} + \varepsilon_{\text{mx}}\cos\left( A_{\varepsilon_{\text{mx}}}\text{πx} \right) + \varepsilon_{\text{my}}\cos\left( A_{\varepsilon_{\text{my}}}\text{πy} \right) + \varepsilon_{\text{mxy}}\cos\left( A_{\varepsilon_{\text{mxy}}}\text{πxy} \right) \\ & + \varepsilon_{\text{mz}}\sin\left( A_{\varepsilon_{\text{mz}}}\text{πz} \right) + \varepsilon_{\text{myz}}\sin\left( A_{\varepsilon_{\text{myz}}}\text{πyz} \right) + \varepsilon_{\text{mzx}}\cos\left( A_{\varepsilon_{\text{mzx}}}\text{πzx} \right) \big)\end{split}$$

(6.18) $$\begin{split}\varepsilon_{m} = \varepsilon_{m0}&+ \varepsilon_{\text{mx}}\cos\left( A_{\varepsilon_{\text{mx}}}\text{πx} \right) + \varepsilon_{\text{my}}\cos\left( A_{\varepsilon_{\text{my}}}\text{πy} \right) + \varepsilon_{\text{mxy}}\cos\left( A_{\varepsilon_{\text{mxy}}}\text{πxy} \right) \\ &+ \varepsilon_{\text{mz}}\sin\left( A_{\varepsilon_{\text{mz}}}\text{πz} \right) + \varepsilon_{\text{myz}}\sin\left( A_{\varepsilon_{\text{myz}}}\text{πyz} \right) + \varepsilon_{\text{mzx}}\cos\left( A_{\varepsilon_{\text{mzx}}}\text{πzx} \right)\end{split}$$

Solids granular temperature:

(6.19) $$\begin{split}\theta_{m} = \theta_{m0}& + \theta_{\text{mx}}\cos\left( A_{\theta_{\text{mx}}}\text{πx} \right) + \theta_{\text{my}}\cos\left( A_{\theta_{\text{my}}}\text{πy} \right) + \theta_{\text{mxy}}\cos\left( A_{\theta_{\text{mxy}}}\text{πxy} \right) \\ & + \theta_{\text{mz}}\sin\left( A_{\theta_{\text{mz}}}\text{πz} \right) + \theta_{\text{myz}}\sin\left( A_{\theta_{\text{myz}}}\text{πyz} \right) + \theta_{\text{mzx}}\cos\left( A_{\theta_{\text{mzx}}}\text{πzx} \right)\end{split}$$

The parameters appearing in the manufactured solutions are as follows:

Table 6.4 Parameters in MMS02 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	$\varepsilon_{\text{mx}}$	0.0
$p_{\text{gy}}$	-50.0	$v_{\text{gz}}$	5.0	$T_{\text{gx}}$	10	$\varepsilon_{\text{my}}$	0.0
$p_{\text{gz}}$	20.0	$v_{\text{gxy}}$	-3.0	$T_{\text{gy}}$	-30	$\varepsilon_{\text{mz}}$	0.0
$p_{\text{gxy}}$	-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	$A_{\varepsilon_{\text{mx}}}$	0.5
$A_{p_{\text{gy}}}$	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