如何计算质量守恒和能量守恒

2015年 10月 14日

让我们从质量守恒开始

\frac{\partial\rho}{\partial t}+\nabla \cdot (\rho {\bf{u}}) = 0

\int_\Omega \frac{\partial\rho} {\partial t}+\nabla \cdot (\rho {\bf{u}}) \ dV = \int_\Omega \frac{\partial\rho}{\partial t}\ dV +\int_{\partial\Omega} \rho{\bf{u}}\cdot {\bf{n}} \ dS

\int_\Omega \frac{\partial\rho}{\partial t} \ dV +\int_{\partialOmega}
\rho{\bf{u}}\cdot {\bf{n}} \ dS = 0

计算能量守恒

\frac{dE_\mathrm{{system}}}{dt}=Q_\mathrm{{exchange}}-P_\mathrm{{stress}}

P_\mathrm{{stress}}=\int_\Omega (\sigma: {\bf{D}}) \ dV

P_\mathrm{{stress}}=-\int_\Omega p(\nabla \cdot {\bf{u}}) \ dV+\int_\Omega (\tau : \nabla {\bf{u}}) \ dV

\frac{d}{dt}
\int_\Omega \rho E_0 \ dV +\int_{\partial\Omega} {\bf{e}}\mathrm{tot} \cdot {\bf{n}} \ dS = \int\Omega Q \ dV

\int_{\partial \Omega}
{\bf{e}}_\mathrm{tot}
\cdot {\bf{n}} \ dS = \int_\Omega Q_ \ dV

总累积能率 总净能率 \frac{d} {dt} \int_\Omega \rho E_0 \ dV \mathrm {ht.dEi0Int} \int_{\partial\Omega} (\rho{\bf{u}}E_0-k{\bf{\nabla}} T+{\bf{q}}_\mathrm{rad}-\sigma{\bf{u}}) \cdot {\bf{n}} \ dS \mathrm{ht.ntefluxInt} \int_\Omega Q \ dV \mathrm{ht.QInt}

\mathrm{ht.dEi0Int}
+ \mathrm{ht.ntefluxInt} = \mathrm{ht.QInt}