MMS03 manufactured solutions ---------------------------- The manufactured solutions for the two-phase, 3D, curl-based functions with variable volume fraction are listed below. Gas pressure: .. math:: :label: a1eq20 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: a1eq21 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] .. math:: :label: a1eq22 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] .. math:: 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] :label: a1eq23 Solids velocity components: .. math:: u_{m} = \frac{1}{\varepsilon_{m}}\left\lbrack u_{m0}\operatorname{}\left( \frac{\pi}{2}\left( x + y + z \right) \right) \right\rbrack :label: a1eq24 .. math:: v_{m} = \frac{1}{\varepsilon_{m}}\left\lbrack v_{m0}\operatorname{}\left( \frac{\pi}{2}\left( x + y + z \right) \right) \right\rbrack :label: a1eq25 .. math:: w_{m} = \frac{1}{\varepsilon_{m}}w_{m0} :label: a1eq26 Gas and solids temperature: .. math:: :label: a1eq27 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) .. math:: :label: a1eq28 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) Solids granular temperature: .. math:: :label: a1eq29 \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) The parameters appearing in the manufactured solutions are as follows: .. _tableA5: .. csv-table:: Parameters in MMS03 manufactured solutions. :widths: auto ":math:p_{g0}", "100.0", ":math:v_{\text{gx}}", "-5.0", ":math:w_{m0}", "5.0", ":math:\varepsilon_{m0}", "0.3" ":math:p_{\text{gx}}", "20.0", ":math:v_{\text{gy}}", "4.0", ":math:T_{g0}", "350", ":math:\varepsilon_{\text{mx}}","0.06" ":math:p_{\text{gy}}", "-50.0", ":math:v_{\text{gz}}", "5.0", ":math:T_{\text{gx}}", "10", ":math:\varepsilon_{\text{my}}", "-0.1" ":math:p_{\text{gz}}", "20.0", ":math:v_{\text{gxy}}", "-3.0", ":math:T_{\text{gy}}", "-30", ":math:\varepsilon_{\text{mz}}", "0.06" ":math:p_{\text{gxy}}", "-25.0", ":math:v_{\text{gyz}}", "2.5", ":math:T_{\text{gz}}", "20", ":math:\varepsilon_{\text{mxy}}", "0.0" ":math:p_{\text{gyz}}", "-10.0", ":math:v_{\text{gzx}}", "3.5", ":math:T_{\text{gxy}}", "-12", ":math:\varepsilon_{\text{myz}}", "0.0" ":math:p_{\text{gzx}}", "10.0", ":math:A_{v_{\text{gx}}}", "0.8", ":math:T_{\text{gyz}}", "10", ":math:\varepsilon_{\text{mzx}}", "0.0" ":math:A_{p_{\text{gx}}}", "0.4", ":math:A_{v_{\text{gy}}}", "0.8", ":math:T_{\text{gzx}}", "8", ":math:A_{\varepsilon_{\text{mx}}}", "0.4" ":math:A_{p_{\text{gy}}}", "0.45", ":math:A_{v_{\text{gz}}}", "0.5", ":math:A_{T_{\text{gx}}}", "0.75", ":math:A_{\varepsilon_{\text{my}}}", "0.5" ":math:A_{p_{\text{gz}}}", "0.85", ":math:A_{v_{\text{gxy}}}", "0.9", ":math:A_{T_{\text{gy}}}", "1.25", ":math:A_{\varepsilon_{\text{mz}}}", "0.5" ":math:A_{p_{\text{gxy}}}", "0.75", ":math:A_{v_{\text{gyz}}}", "0.4", ":math:A_{T_{\text{gz}}}", "0.8", ":math:A_{\varepsilon_{\text{mxy}}}", "0.4" ":math:A_{p_{\text{gyz}}}", "0.7", ":math:A_{v_{\text{gzx}}}", "0.6", ":math:A_{T_{\text{gxy}}}", "0.65", ":math:A_{\varepsilon_{\text{myz}}}", "0.4" ":math:A_{p_{\text{gzx}}}", "0.8", ":math:w_{g0}", "8.0", ":math:A_{T_{\text{gyz}}}", "0.5", ":math:A_{\varepsilon_{\text{mzx}}}", "0.4" ":math:u_{g0}", "7.0", ":math:w_{\text{gx}}", "-4.0", ":math:A_{T_{\text{gzx}}}", "0.6", ":math:\theta_{m0}", "100.0" ":math:u_{\text{gx}}", "3.0", ":math:w_{\text{gy}}", "3.5", ":math:T_{m0}", "300", ":math:\theta_{\text{mx}}", "5.0" ":math:u_{\text{gy}}", "-4.0", ":math:w_{\text{gz}}", "4.2", ":math:T_{\text{mx}}", "15", ":math:\theta_{\text{my}}", "-10.0" ":math:u_{\text{gz}}", "-3.0", ":math:w_{\text{gxy}}", "-2.2", ":math:T_{\text{my}}", "-20", ":math:\theta_{\text{mz}}", "12.0" ":math:u_{\text{gxy}}", "2.0", ":math:w_{\text{gyz}}", "2.1", ":math:T_{\text{mz}}", "15", ":math:\theta_{\text{mxy}}", "-8.0" ":math:u_{\text{gyz}}", "1.5", ":math:w_{\text{gzx}}", "2.5", ":math:T_{\text{mxy}}", "-10", ":math:\theta_{\text{myz}}", "10.0" ":math:u_{\text{gzx}}", "-2.0", ":math:A_{w_{\text{gx}}}", "0.85", ":math:T_{\text{myz}}", "12", ":math:\theta_{\text{mzx}}", "7.0" ":math:A_{u_{\text{gx}}}", "0.5", ":math:A_{w_{\text{gy}}}", "0.9", ":math:T_{\text{mzx}}", "10", ":math:A_{\theta_{\text{mx}}}", "0.8" ":math:A_{u_{\text{gy}}}", "0.85", ":math:A_{w_{\text{gz}}}", "0.5", ":math:A_{T_{\text{mx}}}", "0.5", ":math:A_{\theta_{\text{my}}}", "1.25" ":math:A_{u_{\text{gz}}}", "0.4", ":math:A_{w_{\text{gxy}}}", "0.4", ":math:A_{T_{\text{my}}}", "0.9", ":math:A_{\theta_{\text{mz}}}", "0.7" ":math:A_{u_{\text{gxy}}}", "0.6", ":math:A_{w_{\text{gyz}}}", "0.8", ":math:A_{T_{\text{mz}}}", "0.8", ":math:A_{\theta_{\text{mxy}}}", "0.5" ":math:A_{u_{\text{gyz}}}", "0.8", ":math:A_{w_{\text{gzx}}}", "0.75", ":math:A_{T_{\text{mxy}}}", "0.5", ":math:A_{\theta_{\text{myz}}}", "0.6" ":math:A_{u_{\text{gzx}}}", "0.9", ":math:u_{m0}", "5.0", ":math:A_{T_{\text{myz}}}", "0.65", ":math:A_{\theta_{\text{mzx}}}", "0.7" ":math:v_{g0}", "9.0", ":math:v_{m0}", "5.0", ":math:A_{T_{\text{mzx}}}", "0.4", " ", " "