6. Manufactured Solution Mathematical Forms¶
(Note: All solution variables are in SI units.)
The baseline manufactured solution selected for the verification study is a combination of sine and cosine functions and takes the following general form [25], [1]
where, L is a characteristic length (herein, selected equal to the domain length or L=1), and ϕ=[Pg,ug,vg,wg,us,vs,ws,Tg,Ts]T represents the set of primitive variables being tested for order of accuracy. The sinusoidal functions (fϕx, fϕy, etc.) selected are shown in Table 6.1
Variable, ϕ |
f_{ϕx} |
f_{ϕy} |
f_{ϕz} |
f_{ϕxy} |
f_{ϕyz} |
f_{ϕzx} |
---|---|---|---|---|---|---|
u_{g} |
sin |
cos |
cos |
cos |
sin |
cos |
v_{g} |
sin |
cos |
cos |
cos |
sin |
cos |
w_{g} |
cos |
sin |
cos |
sin |
sin |
cos |
u_{s} |
sin |
cos |
cos |
cos |
sin |
cos |
v_{s} |
sin |
cos |
cos |
cos |
sin |
cos |
w_{s} |
cos |
sin |
cos |
sin |
sin |
cos |
P_{g} |
cos |
cos |
sin |
cos |
sin |
cos |
T_{g} |
cos |
cos |
sin |
cos |
sin |
cos |
T_{s} |
cos |
cos |
sin |
cos |
sin |
cos |
\varepsilon_{s} |
cos |
cos |
sin |
– |
– |
– |
The frequency constants (a_{\phi x}, a_{\phi y}, a_{\phi xy}, etc.) and the amplitude constants (\phi_{0}, \phi_{x}, \phi_{\text{xy}}, etc.) are selected to ensure functions that are smooth but show reasonable periodicity and magnitude within the domain. The frequency constants selected are shown in Table 6.2.
Variable, ϕ |
a_{ϕx} |
a_{ϕy} |
a_{ϕz} |
a_{ϕxy} |
a_{ϕyz} |
a_{ϕzx} |
---|---|---|---|---|---|---|
u_{g} |
0.5 |
0.85 |
0.4 |
0.6 |
0.8 |
0.9 |
v_{g} |
0.8 |
0.8 |
0.5 |
0.9 |
0.4 |
0.6 |
w_{g} |
0.85 |
0.9 |
0.5 |
0.4 |
0.8 |
0.75 |
u_{s} |
0.5 |
0.85 |
0.4 |
0.6 |
0.8 |
0.9 |
v_{s} |
0.8 |
0.8 |
0.5 |
0.9 |
0.4 |
0.6 |
w_{s} |
0.85 |
0.9 |
0.5 |
0.4 |
0.8 |
0.75 |
P_{g} |
0.4 |
0.45 |
0.85 |
0.75 |
0.7 |
0.8 |
T_{g} |
0.75 |
1.25 |
0.8 |
0.65 |
0.5 |
0.6 |
T_{s} |
0.5 |
0.9 |
0.8 |
0.5 |
0.65 |
0.4 |
\varepsilon_{s} |
0.4 |
0.5 |
0.5 |
– |
– |
– |
The amplitude constants selected are shown Table 6.3.
Variable, ϕ |
ϕ_{0} |
ϕ_{x} |
ϕ_{y} |
ϕ_{z} |
ϕ_{xy} |
ϕ_{yz} |
ϕ_{zx} |
---|---|---|---|---|---|---|---|
u_{g} |
7 |
3 |
-4 |
-3 |
2 |
1.5 |
2 |
v_{g} |
9 |
-5 |
4 |
5 |
-3 |
2.5 |
3.5 |
w_{g} |
8 |
-4 |
3.5 |
4.2 |
-2.2 |
2.1 |
2.5 |
u_{s} |
7 |
3 |
-4 |
-3 |
2 |
1.5 |
2 |
v_{s} |
9 |
-5 |
4 |
5 |
-3 |
2.5 |
3.5 |
w_{s} |
8 |
-4 |
3.5 |
4.2 |
-2.2 |
2.1 |
2.5 |
P_{g} |
100 |
20 |
-50 |
20 |
-25 |
-10 |
10 |
T_{g} |
350 |
10 |
-30 |
20 |
-12 |
10 |
8 |
T_{s} |
300 |
15 |
-20 |
15 |
-10 |
12 |
10 |
\varepsilon_{s} |
0.3 |
0.06 |
0.1 |
0.06 |
– |
– |
– |
The baseline manufactured solutions presented above are used to generate manufactured solutions for the two-phase flow test cases. As an example, the manufactured solution for the test case presented in Section 2.6 is provided next.
The manufactured solutions for the scalar variables (P_{g}, T_{g}, and T_{s}) are simply obtained from Eq.6.1 and by substituting the appropriate functions and constants described above. For example, for the pressure variable (P_{g}), this function is as follow:
The manufactured solutions for velocity components of the gas phase are obtained by taking the curl of the baseline velocity vector field, i.e.,
where, for example, \phi\left( u_{g} \right) is the baseline manufactured solution obtained from Eq.6.1, the functions, and the constants described above for the variable u_{g}. This results in a divergence free velocity field because \nabla \cdot (\nabla \times \overrightarrow{H}) is identically zero for any vector field, \overrightarrow{H}. Thus, the manufactured solution for u_{g} is given as:
Similarly, the manufactured solution for v_{g} and w_{g} can be derived.
Finally, the manufactured solution for velocity components of the solids phase is selected as simply the following divergence free field:
Manufactured solutions for other MMS test cases presented are derived using the baseline manufactured solutions and appropriate constraints (divergence free field, boundary conditions, etc.). For a complete look at the MMS function and MMS source terms, please see the MMS_MOD.f file under the respective test case of the MFIX distribution.