Energy balance equation

Formulation with total energy

This section is a summary of [1] (section VI.3 p. 119) and reminds the derivation of energy equation. The total energy which is contained inside a volume \(V\) is:

(284)\[\int_{V}\rho(e+\frac{1}{2}u^{2})dV\]

where \(e\) is the internal energy and \(\rho u^{2}/2\) is the kinetic energy of fluid. From first principle of thermodynamics, the time-variation of that energy is equal to the work of forces and the heat quantity exchanged between the system and the external medium:

(285)\[\frac{d}{dt}\int_{V}\rho(e+\frac{1}{2}u^{2})dV=\dot{W}+\dot{Q}\]

where \(\dot{W}\) is the work per time unit of non-conservative forces such as the volumic force and stress force applied at the surface of volume \(V\):

(286)\[\dot{W}=\int_{V}\rho\boldsymbol{g}\cdot\boldsymbol{u}dV+\int_{A}\boldsymbol{t}(\boldsymbol{n})\cdot\boldsymbol{u}dA\]

and \(\dot{Q}\) is the heat which is exchanged from the external to the system. Without source term, it is given by the heat flux by conduction \(\boldsymbol{q}\):

(287)\[\dot{Q}=-\int_{A}\boldsymbol{q}\cdot\boldsymbol{n}dA\]
(288)\[\frac{d}{dt}\int_{V}\rho(e+\frac{1}{2}u^{2})dV=\int_{V}\rho\boldsymbol{g}\cdot\boldsymbol{u}dV+\int_{A}\boldsymbol{t}(\boldsymbol{n})\cdot\boldsymbol{u}dA-\int_{A}\boldsymbol{q}\cdot\boldsymbol{n}dA\]

Once again we use the Reynolds’ transport theorem for the left-hand side and we transform both surfacic integral of right-hand side into volumic ones with Green-Ostrogradsky’s theorem:

(289)\[\int_{V}\left\{ \frac{\partial}{\partial t}\left[\rho(e+\frac{1}{2}u^{2})\right]+\boldsymbol{\nabla}\cdot\rho\boldsymbol{u}(e+\frac{1}{2}u^{2})\right\} dV=\int_{V}\rho\boldsymbol{g}\cdot\boldsymbol{u}dV+\int_{V}\left[-\boldsymbol{\nabla}\cdot(p\boldsymbol{u})+\boldsymbol{\nabla}\cdot(\overline{\overline{\boldsymbol{\tau}}}\cdot\boldsymbol{u})\right]dV+\int_{V}\boldsymbol{\nabla}\cdot\boldsymbol{q}dV\]

The volume is arbitray, hence

(290)\[\frac{\partial}{\partial t}\left[\rho(e+\frac{1}{2}u^{2})\right]+\boldsymbol{\nabla}\cdot\left[\rho(e+\frac{1}{2}u^{2})\boldsymbol{u}\right]=-\boldsymbol{\nabla}\cdot(p^{eos}\boldsymbol{u})+\boldsymbol{\nabla}\cdot(\overline{\overline{\boldsymbol{\tau}}}\cdot\boldsymbol{u})-\boldsymbol{\nabla}\cdot\boldsymbol{q}+\rho\boldsymbol{g}\cdot\boldsymbol{u}\]

Alternative formulations

Formulation with internal energy

(291)\[\frac{\partial(\rho e)}{\partial t}+\boldsymbol{\nabla}\cdot\left[\rho e\boldsymbol{u}\right]=-\boldsymbol{\nabla}\cdot\boldsymbol{q}-p^{eos}\boldsymbol{\nabla}\cdot\boldsymbol{u}+\overline{\overline{\boldsymbol{\tau}}}:\boldsymbol{\nabla}\boldsymbol{u}\]

Formulation with total enthalpy

(292)\[\frac{\partial}{\partial t}\left[\rho(h+\frac{1}{2}u^{2})\right]+\boldsymbol{\nabla}\cdot\left[\rho(h+\frac{1}{2}u^{2})\boldsymbol{u}\right]=-\boldsymbol{\nabla}\cdot\boldsymbol{q}+\frac{\partial p^{eos}}{\partial t}+\boldsymbol{\nabla}\cdot(\overline{\overline{\boldsymbol{\tau}}}\cdot\boldsymbol{u})+\rho\boldsymbol{g}\cdot\boldsymbol{u}\]

Formulation with enthalpy

(293)\[\frac{\partial(\rho h)}{\partial t}+\boldsymbol{\nabla}\cdot(\rho h\boldsymbol{u})=-\boldsymbol{\nabla}\cdot\boldsymbol{q}+\overline{\overline{\boldsymbol{\tau}}}:\boldsymbol{\nabla}\boldsymbol{u}+\frac{dp^{eos}}{dt}\]

Formulation with entropy

(294)\[T\left[\frac{\partial(\rho s)}{\partial t}+\boldsymbol{\nabla}\cdot(\rho s\boldsymbol{u})\right]=-\boldsymbol{\nabla}\cdot\boldsymbol{q}+\overline{\overline{\boldsymbol{\tau}}}:\boldsymbol{\nabla}\boldsymbol{u}\]

Formulation with temperature and \(c_p\)

(295)\[\frac{\partial(\rho c_{p}T)}{\partial t}+\boldsymbol{\nabla}\cdot(\rho c_{p}T\boldsymbol{u})=-\boldsymbol{\nabla}\cdot\boldsymbol{q}-T\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{\rho}\frac{dp}{dt}+\overline{\overline{\boldsymbol{\tau}}}:\boldsymbol{\nabla}\boldsymbol{u}+\rho T\frac{dc_{p}}{dt}\]

Formulation with temperature and \(c_v\)

(296)\[\frac{\partial(\rho c_{v}T)}{\partial t}+\boldsymbol{\nabla}\cdot(\rho c_{v}T\boldsymbol{u})=-\boldsymbol{\nabla}\cdot\boldsymbol{q}-T\left(\frac{\partial p}{\partial T}\right)_{\rho}\boldsymbol{\nabla}\cdot\boldsymbol{u}+\overline{\overline{\boldsymbol{\tau}}}:\boldsymbol{\nabla}\boldsymbol{u}+\rho T\frac{dc_{v}}{dt}\]

Bibliography