6.2. MMS03 manufactured solutions

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

Gas pressure:

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

Gas velocity components:

(6.21)\[\begin{split}u_{g} = \frac{1}{\varepsilon_{g}}\big[& A_{w_{\text{gy}}}\pi w_{\text{gy}}\cos\left( A_{w_{\text{gy}}}\text{πy} \right) + A_{w_{\text{gxy}}}\pi w_{\text{gxy}}x\cos\left( A_{w_{\text{gxy}}}\text{πxy} \right) \\ &- A_{v_{\text{gyz}}}\pi v_{\text{gyz}}y\cos\left( A_{v_{\text{gyz}}}\text{πyz} \right) + A_{w_{\text{gyz}}}\pi w_{\text{gyz}}z\cos\left( A_{w_{\text{gyz}}}\text{πyz} \right) \\ &+ A_{v_{\text{gz}}}\pi v_{\text{gz}}\sin\left( A_{v_{\text{gz}}}\text{πz} \right) + A_{v_{\text{gzx}}}\pi v_{\text{gzx}}x\sin\left( A_{v_{\text{gzx}}}\text{πzx} \right) \big]\end{split}\]
(6.22)\[\begin{split}v_{g} = \frac{1}{\varepsilon_{g}}\big[& - A_{w_{\text{gxy}}}\pi w_{\text{gxy}}y\cos\left( A_{w_{\text{gxy}}}\text{πxy} \right) + A_{u_{\text{gyz}}}\pi u_{\text{gyz}}y\cos\left( A_{u_{\text{gyz}}}\text{πyz} \right) \\ &+ A_{w_{\text{gx}}}\pi w_{\text{gx}}\sin\left( A_{w_{\text{gx}}}\text{πx} \right) - A_{u_{\text{gz}}}\pi u_{\text{gz}}\sin\left( A_{u_{\text{gz}}}\text{πz} \right) \\ &- A_{u_{\text{gzx}}}\pi u_{\text{gzx}}x\sin\left( A_{u_{\text{gzx}}}\text{πzx} \right) + A_{w_{\text{gzx}}}\pi w_{\text{gzx}}z\sin\left( A_{w_{\text{gzx}}}\text{πzx} \right) \big]\end{split}\]
(6.23)\[\begin{split}w_{g} = \frac{1}{\varepsilon_{g}}\big[& A_{v_{\text{gx}}}\pi v_{\text{gx}}\cos\left( A_{v_{\text{gx}}}\text{πx} \right) - A_{u_{\text{gyz}}}\pi u_{\text{gyz}}z\cos\left( A_{u_{\text{gyz}}}\text{πyz} \right) \\ &+ A_{u_{\text{gxy}}}\pi u_{\text{gxy}}x\sin\left( A_{u_{\text{gxy}}}\text{πxy} \right) + A_{u_{\text{gy}}}\pi u_{\text{gy}}\sin\left( A_{u_{\text{gy}}}\text{πy} \right) \\ &- A_{v_{\text{gxy}}}\pi v_{\text{gxy}}y\sin\left( A_{v_{\text{gxy}}}\text{πxy} \right) - A_{v_{\text{gzx}}}\pi v_{\text{gzx}}z\sin\left( A_{v_{\text{gzx}}}\text{πzx} \right) \big]\end{split}\]

Solids velocity components:

(6.24)\[u_{m} = \frac{1}{\varepsilon_{m}}\left\lbrack u_{m0}\operatorname{}\left( \frac{\pi}{2}\left( x + y + z \right) \right) \right\rbrack\]
(6.25)\[v_{m} = \frac{1}{\varepsilon_{m}}\left\lbrack v_{m0}\operatorname{}\left( \frac{\pi}{2}\left( x + y + z \right) \right) \right\rbrack\]
(6.26)\[w_{m} = \frac{1}{\varepsilon_{m}}w_{m0}\]

Gas and solids temperature:

(6.27)\[\begin{split}T_{g} = T_{g0} &+ T_{\text{gx}}\cos\left( A_{T_{\text{gx}}}\text{πx} \right) + T_{\text{gy}}\cos\left( A_{T_{\text{gy}}}\text{πy} \right) + T_{\text{gxy}}\cos\left( A_{T_{\text{gxy}}}\text{πxy} \right) \\ &+ T_{\text{gz}}\sin\left( A_{T_{\text{gz}}}\text{πz} \right) + T_{\text{gyz}}\sin\left( A_{T_{\text{gyz}}}\text{πyz} \right) + T_{\text{gzx}}\cos\left( A_{T_{\text{gzx}}}\text{πzx} \right)\end{split}\]
(6.28)\[\begin{split}T_{m} = T_{m0} &+ T_{\text{mx}}\cos\left( A_{T_{\text{mx}}}\text{πx} \right) + T_{\text{my}}\cos\left( A_{T_{\text{my}}}\text{πy} \right) + T_{\text{mxy}}\cos\left( A_{T_{\text{mxy}}}\text{πxy} \right) \\ &+ T_{\text{mz}}\sin\left( A_{T_{\text{mz}}}\text{πz} \right) + T_{\text{myz}}\sin\left( A_{T_{\text{myz}}}\text{πyz} \right) + T_{\text{mzx}}\cos\left( A_{T_{\text{mzx}}}\text{πzx} \right)\end{split}\]

Solids granular temperature:

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

The parameters appearing in the manufactured solutions are as follows:

Table 6.5 Parameters in MMS03 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

\(\varepsilon_{\text{mx}}\)

0.06

\(p_{\text{gy}}\)

-50.0

\(v_{\text{gz}}\)

5.0

\(T_{\text{gx}}\)

10

\(\varepsilon_{\text{my}}\)

-0.1

\(p_{\text{gz}}\)

20.0

\(v_{\text{gxy}}\)

-3.0

\(T_{\text{gy}}\)

-30

\(\varepsilon_{\text{mz}}\)

0.06

\(p_{\text{gxy}}\)

-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

\(A_{\varepsilon_{\text{mx}}}\)

0.4

\(A_{p_{\text{gy}}}\)

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