6.4. MMS05 manufactured solutions¶

The manufactured solutions for the Free-slip wall BC, single phase, 3D, curl-based functions are listed below.

Gas pressure:

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

Gas velocity components:

(6.36)$\begin{split}u_{g} = x^{3}\big[& A_{w_{\text{gy}}}\pi w_{\text{gy}}\cos\left( A_{w_{\text{gy}}}\text{πy} \right) + A_{w_{\text{gxy}}}\pi w_{\text{gxy}}x\cos\left( A_{w_{\text{gxy}}}\text{πxy} \right) \\ &- A_{v_{\text{gyz}}}\pi v_{\text{gyz}}y\cos\left( A_{v_{\text{gyz}}}\text{πyz} \right) + A_{w_{\text{gyz}}}\pi w_{\text{gyz}}z\cos\left( A_{w_{\text{gyz}}}\text{πyz} \right) \\ &+ A_{v_{\text{gz}}}\pi v_{\text{gz}}\sin\left( A_{v_{\text{gz}}}\text{πz} \right) + A_{v_{\text{gzx}}}\pi v_{\text{gzx}}x\sin\left( A_{v_{\text{gzx}}}\text{πzx} \right) \big]\end{split}$
(6.37)$\begin{split}v_{g} =& v_{g0} + x^{3}\big[ - A_{w_{\text{gxy}}}\pi w_{\text{gxy}}y\cos\left( A_{w_{\text{gxy}}}\text{πxy} \right) + A_{u_{\text{gyz}}}\pi u_{\text{gyz}}y\cos\left( A_{u_{\text{gyz}}}\text{πyz} \right) \\ &+ A_{w_{\text{gx}}}\pi w_{\text{gx}}\sin\left( A_{w_{\text{gx}}}\text{πx} \right) - A_{u_{\text{gz}}}\pi u_{\text{gz}}\sin\left( A_{u_{\text{gz}}}\text{πz} \right) \\ &- A_{u_{\text{gzx}}}\pi u_{\text{gzx}}x\sin\left( A_{u_{\text{gzx}}}\text{πzx} \right) + A_{w_{\text{gzx}}}\pi w_{\text{gzx}}z\sin\left( A_{w_{\text{gzx}}}\text{πzx} \right) \big] \\ &+ 3x^{2} \big[ - w_{g0} - w_{\text{gx}}\cos\left( A_{w_{\text{gx}}}\text{πx} \right) - w_{\text{gz}}\cos\left( A_{w_{\text{gz}}}\text{πz} \right) \\ &- w_{\text{gzx}}\cos\left( A_{w_{\text{gzx}}}\text{πzx} \right) - w_{\text{gy}}\sin\left( A_{w_{\text{gy}}}\text{πy} \right) \\ &- w_{\text{gxy}}\sin\left( A_{w_{\text{gxy}}}\text{πxy} \right) - w_{\text{gyz}}\sin\left( A_{w_{\text{gyz}}}\text{πyz} \right) \big]\end{split}$
(6.38)$\begin{split}w_{g} =& w_{g0} + x^{3}\big[ A_{v_{\text{gx}}}\pi v_{\text{gx}}\cos\left( A_{v_{\text{gx}}}\text{πx} \right) - A_{u_{\text{gyz}}}\pi u_{\text{gyz}}z\cos\left( A_{u_{\text{gyz}}}\text{πyz} \right) \\ &+ A_{u_{\text{gxy}}}\pi u_{\text{gxy}}x\sin\left( A_{u_{\text{gxy}}}\text{πxy} \right) + A_{u_{\text{gy}}}\pi u_{\text{gy}}\sin\left( A_{u_{\text{gy}}}\text{πy} \right) \\ &- A_{v_{\text{gxy}}}\pi v_{\text{gxy}}y\sin\left( A_{v_{\text{gxy}}}\text{πxy} \right) - A_{v_{\text{gzx}}}\pi v_{\text{gzx}}z\sin\left( A_{v_{\text{gzx}}}\text{πzx} \right) \big] \\ &+ 3x^{2}\big[ v_{g0} + v_{\text{gy}}\cos\left( A_{v_{\text{gy}}}\text{πy} \right) + v_{\text{gxy}}\cos\left( A_{v_{\text{gxy}}}\text{πxy} \right) \\ &+ v_{\text{gz}}\cos\left( A_{v_{\text{gz}}}\text{πz} \right) + v_{\text{gzx}}\cos\left( A_{v_{\text{gzx}}}\text{πzx} \right) \\ &+ v_{\text{gx}}\sin\left( A_{v_{\text{gx}}}\text{πx} \right) + v_{\text{gyz}}\sin\left( A_{v_{\text{gyz}}}\text{πyz} \right) \big]\end{split}$

Gas volume fraction:

(6.39)$\varepsilon_{g} = 1.0$

The parameters appearing in the manufactured solutions are as follows:

 $$p_{g0}$$ 100 $$v_{g0}$$ 9 $$u_{g0}$$ 7 $$w_{g0}$$ 8 $$p_{\text{gx}}$$ 20 $$v_{\text{gx}}$$ -5 $$u_{\text{gx}}$$ 3 $$w_{\text{gx}}$$ -4 $$p_{\text{gy}}$$ -50 $$v_{\text{gy}}$$ 4 $$u_{\text{gy}}$$ -4 $$w_{\text{gy}}$$ 3.5 $$p_{\text{gz}}$$ 20 $$v_{\text{gz}}$$ 5 $$u_{\text{gz}}$$ -3 $$w_{\text{gz}}$$ 4.2 $$p_{\text{gxy}}$$ -25 $$v_{\text{gxy}}$$ -3 $$u_{\text{gxy}}$$ 2 $$w_{\text{gxy}}$$ -2.2 $$p_{\text{gyz}}$$ -10 $$v_{\text{gyz}}$$ 2.5 $$u_{\text{gyz}}$$ 1.5 $$w_{\text{gyz}}$$ 2.1 $$p_{\text{gzx}}$$ 10 $$v_{\text{gzx}}$$ 3.5 $$u_{\text{gzx}}$$ -2 $$w_{\text{gzx}}$$ 2.5 $$A_{p_{\text{gx}}}$$ 0.4 $$A_{v_{\text{gx}}}$$ 0.8 $$A_{u_{\text{gx}}}$$ 0.5 $$A_{w_{\text{gx}}}$$ 0.85 $$A_{p_{\text{gy}}}$$ 0.45 $$A_{v_{\text{gy}}}$$ 0.8 $$A_{u_{\text{gy}}}$$ 0.85 $$A_{w_{\text{gy}}}$$ 0.9 $$A_{p_{\text{gz}}}$$ 0.85 $$A_{v_{\text{gz}}}$$ 0.5 $$A_{u_{\text{gz}}}$$ 0.4 $$A_{w_{\text{gz}}}$$ 0.5 $$A_{p_{\text{gxy}}}$$ 0.75 $$A_{v_{\text{gxy}}}$$ 0.9 $$A_{u_{\text{gxy}}}$$ 0.6 $$A_{w_{\text{gxy}}}$$ 0.4 $$A_{p_{\text{gyz}}}$$ 0.7 $$A_{v_{\text{gyz}}}$$ 0.4 $$A_{u_{\text{gyz}}}$$ 0.8 $$A_{w_{\text{gyz}}}$$ 0.8 $$A_{p_{\text{gzx}}}$$ 0.8 $$A_{v_{\text{gzx}}}$$ 0.6 $$A_{u_{\text{gzx}}}$$ 0.9 $$A_{w_{\text{gzx}}}$$ 0.75