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