MMS05 manufactured solutions ---------------------------- The manufactured solutions for the Free-slip wall BC, single phase, 3D, curl-based functions are listed below. Gas pressure: .. math:: :label: a1eq35 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) Gas velocity components: .. math:: :label: a1eq36 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] .. math:: :label: a1eq37 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] .. math:: :label: a1eq38 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] Gas volume fraction: .. math:: \varepsilon_{g} = 1.0 :label: a1eq39 The parameters appearing in the manufactured solutions are as follows: .. _tableA7: .. csv-table:: Parameters in MMS05 manufactured solutions. :widths: auto ":math:p_{g0}", "100.0", ":math:v_{g0}", "9.0", ":math:u_{g0}", "7.0", ":math:w_{g0}", "8.0" ":math:p_{\text{gx}}", "20.0", ":math:v_{\text{gx}}", "-5.0", ":math:u_{\text{gx}}", "3.0", ":math:w_{\text{gx}}", "-4.0" ":math:p_{\text{gy}}", "-50.0", ":math:v_{\text{gy}}", "4.0", ":math:u_{\text{gy}}", "-4.0", ":math:w_{\text{gy}}", "3.5" ":math:p_{\text{gz}}", "20.0", ":math:v_{\text{gz}}", "5.0", ":math:u_{\text{gz}}", "-3.0", ":math:w_{\text{gz}}", "4.2" ":math:p_{\text{gxy}}", "-25.0", ":math:v_{\text{gxy}}", "-3.0", ":math:u_{\text{gxy}}", "2.0", ":math:w_{\text{gxy}}", "-2.2" ":math:p_{\text{gyz}}", "-10.0", ":math:v_{\text{gyz}}", "2.5", ":math:u_{\text{gyz}}", "1.5", ":math:w_{\text{gyz}}", "2.1" ":math:p_{\text{gzx}}", "10.0", ":math:v_{\text{gzx}}", "3.5", ":math:u_{\text{gzx}}", "-2.0", ":math:w_{\text{gzx}}", "2.5" ":math:A_{p_{\text{gx}}}", "0.4", ":math:A_{v_{\text{gx}}}", "0.8", ":math:A_{u_{\text{gx}}}", "0.5", ":math:A_{w_{\text{gx}}}", "0.85" ":math:A_{p_{\text{gy}}}", "0.45", ":math:A_{v_{\text{gy}}}", "0.8", ":math:A_{u_{\text{gy}}}", "0.85", ":math:A_{w_{\text{gy}}}", "0.9" ":math:A_{p_{\text{gz}}}", "0.85", ":math:A_{v_{\text{gz}}}", "0.5", ":math:A_{u_{\text{gz}}}", "0.4", ":math:A_{w_{\text{gz}}}", "0.5" ":math:A_{p_{\text{gxy}}}", "0.75", ":math:A_{v_{\text{gxy}}}", "0.9", ":math:A_{u_{\text{gxy}}}", "0.6", ":math:A_{w_{\text{gxy}}}", "0.4" ":math:A_{p_{\text{gyz}}}", "0.7", ":math:A_{v_{\text{gyz}}}", "0.4", ":math:A_{u_{\text{gyz}}}", "0.8", ":math:A_{w_{\text{gyz}}}", "0.8" ":math:A_{p_{\text{gzx}}}", "0.8", ":math:A_{v_{\text{gzx}}}", "0.6", ":math:A_{u_{\text{gzx}}}", "0.9", ":math:A_{w_{\text{gzx}}}", "0.75"