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", " ", " "