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#### 5.5.1.2 Energy equation

$\frac{\partial \overline{\rho }\left(\stackrel{˜}{h}+\frac{{\stackrel{˜}{v}}^{2}}{2}\right)-\overline{p}}{\partial t}+\frac{\partial \overline{\rho }\stackrel{˜}{{v}_{J}}\left(\stackrel{˜}{h}+\frac{{\stackrel{˜}{v}}^{2}}{2}\right)}{\partial {x}_{j}}-\overline{\rho }{g}_{i}\stackrel{˜}{{v}_{\iota }}-\frac{\partial \left(\overline{{\tau }_{\iota J}}+{R}_{ij}\right)\stackrel{˜}{{v}_{\iota }}}{\partial {x}_{j}}+\frac{\partial \overline{\stackrel{\to }{{q}_{J}}}+{\stackrel{\to }{q}}_{j}^{t}}{\partial {x}_{j}}=\overline{{q}_{ext}}$

$〈\frac{\partial \overline{\rho }\left(\stackrel{˜}{h}+\frac{{\stackrel{˜}{v}}^{2}}{2}\right)-\overline{p}}{\partial t}〉=\frac{\partial {〈\overline{\rho }\stackrel{˜}{h}〉}^{S}}{\partial t}+\frac{\partial {〈\overline{\rho }\frac{{\stackrel{˜}{v}}^{2}}{2}〉}^{S}}{\partial t}-\frac{\partial {〈\overline{p}〉}^{S}}{\partial t}$

Let us define the mass-weighted space-time average enthalpy H and the space-time average pressure P

$H\triangleq \frac{{〈\overline{\rho }\stackrel{˜}{h}〉}^{S}}{{〈\overline{\rho }〉}^{S}}\text{;}$

$P\triangleq {〈\overline{p}〉}^{S}=\stackrel{ˆ}{p}$

An approximation is necessary for the second term

$〈\overline{\rho }\frac{{\stackrel{˜}{v}}^{2}}{2}〉\cong {〈\overline{\rho }〉}^{S}\frac{{V}^{2}}{2}$

$\frac{\partial \overline{\rho }\left(\stackrel{˜}{h}+\frac{{\stackrel{˜}{v}}^{2}}{2}\right)}{}$

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