# 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:

 $$p_{g0}$$ 100 $$v_{\text{gx}}$$ -5 $$w_{m0}$$ 5 $$\varepsilon_{m0}$$ 0.3 $$p_{\text{gx}}$$ 20 $$v_{\text{gy}}$$ 4 $$T_{g0}$$ 350 $$\varepsilon_{\text{mx}}$$ 0.0 $$p_{\text{gy}}$$ -50 $$v_{\text{gz}}$$ 5 $$T_{\text{gx}}$$ 10 $$\varepsilon_{\text{my}}$$ 0.0 $$p_{\text{gz}}$$ 20 $$v_{\text{gxy}}$$ -3 $$T_{\text{gy}}$$ -30 $$\varepsilon_{\text{mz}}$$ 0.0 $$p_{\text{gxy}}$$ -25 $$v_{\text{gyz}}$$ 2.5 $$T_{\text{gz}}$$ 20 $$\varepsilon_{\text{mxy}}$$ 0.0 $$p_{\text{gyz}}$$ -10 $$v_{\text{gzx}}$$ 3.5 $$T_{\text{gxy}}$$ -12 $$\varepsilon_{\text{myz}}$$ 0.0 $$p_{\text{gzx}}$$ 10 $$A_{v_{\text{gx}}}$$ 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 $$A_{T_{\text{gyz}}}$$ 0.5 $$A_{\varepsilon_{\text{mzx}}}$$ 0.4 $$u_{g0}$$ 7 $$w_{\text{gx}}$$ -4 $$A_{T_{\text{gzx}}}$$ 0.6 $$\theta_{m0}$$ 100.0 $$u_{\text{gx}}$$ 3 $$w_{\text{gy}}$$ 3.5 $$T_{m0}$$ 300 $$\theta_{\text{mx}}$$ 5.0 $$u_{\text{gy}}$$ -4 $$w_{\text{gz}}$$ 4.2 $$T_{\text{mx}}$$ 15 $$\theta_{\text{my}}$$ -10.0 $$u_{\text{gz}}$$ -3 $$w_{\text{gxy}}$$ -2.2 $$T_{\text{my}}$$ -20 $$\theta_{\text{mz}}$$ 12.0 $$u_{\text{gxy}}$$ 2 $$w_{\text{gyz}}$$ 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 $$A_{w_{\text{gx}}}$$ 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 $$A_{T_{\text{myz}}}$$ 0.65 $$A_{\theta_{\text{mzx}}}$$ 0.7 $$v_{g0}$$ 9 $$v_{m0}$$ 5 $$A_{T_{\text{mzx}}}$$ 0.4