Energy Conservation Equation for Compressible Flow
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This blog introduces various representations of the energy conservation in the governing equation system of compressible flows, where, except for the kinetic energy equation, all other forms are different manifestations of thermodynamic energy.
Terminology
- $e:(\mathrm{J/kg})$: thermodynamic energy or (specific) internal energy
- $E:(\mathrm{J/kg})$: (specific) total internal energy
- $h:(\mathrm{J/kg})$: (specific) enthalpy
- $H:(\mathrm{J/kg})$: (specific) total enthalpy
- $T:(\mathrm{K})$: temperature
Kinetic energy
General continuity equation:
\[\begin{equation} \partial_t \rho + (\rho u_j)_{,\,j} = \dot{m}. \end{equation}\]Cauchy momentum equation:
\[\begin{equation} \partial_t (\rho u_i) + (\rho u_i u_j)_{,\,j} = -p_{,\,i} + \sigma_{ij,\,j} + \rho f_i + \dot{m}u_i. \end{equation}\]Notice that
\[\begin{equation} \partial_t (\rho u_i) + (\rho u_i u_j)_{,\,j} = \rho \mathrm{D}_t u_i + u_i \dot{m}. \end{equation}\]The kinetic energy conservation can be expressed as
\[\begin{equation} \partial_t (\tfrac12 \rho u_i u_i) + (\tfrac12 \rho u_i u_i u_j)_{,\,j} = (-pu_j + \sigma_{ij}u_i)_{,\,j} + \rho f_i u_i + pu_{j,\,j} - \sigma_{ij} u_{i,\,j} + \tfrac12 u_i u_i \dot{m}. \end{equation}\]Internal energy
According to 1st law of thermodynamics,
\[\begin{equation} \begin{split} \partial_t (\rho e + \tfrac12 \rho u_i u_i) + (\rho u_j e + \tfrac12 \rho u_i u_i u_j)_{,\,j} &= \\ -q_{j,\,j} + \dot{Q} + \rho f_i u_i &+ ( -pu_j + \sigma_{ij}u_i)_{,\,j} + \dot{m}(e + \tfrac12 u_i u_i). \end{split} \end{equation}\]Subtracting the kinetic energy corresponding to (4), the equation of internal energy $e$ is obtained as
\[\begin{equation} \partial_t (\rho e) + (\rho u_j e)_{,\,j} = -q_{j,\,j} + \dot{Q} - pu_{j,\,j} + \sigma_{ij} u_{i,\,j} + \dot{m}e. \end{equation}\]Total internal energy
Based on (5), the equation of total internal energy $E$ is
\[\begin{equation} \partial_t (\rho E) + (\rho u_j E)_{,\,j} = -q_{j,\,j} + \dot{Q} + \rho f_i u_i + ( -pu_j + \sigma_{ij}u_i)_{,\,j} + \dot{m}E. \end{equation}\]Enthalpy
Using $h = e + pv$ or $\rho h = \rho e + p$,
\[\begin{equation} \partial_t (\rho h) + (\rho u_j h)_{,\,j} - \partial_t p - (u_j p)_{,\,j} = -q_{j,\,j} + \dot{Q} + \sigma_{ij} u_{i,\,j} - pu_{j,\,j} + \dot{m}(h - p/\rho). \end{equation}\]After rearranging, \(\begin{equation} \partial_t (\rho h) + (\rho u_j h)_{,\,j} = -q_{j,\,j} + \dot{Q} + \sigma_{ij} u_{i,\,j} + \mathrm{D}_t p + \dot{m}h - \dot{m}p/\rho. \end{equation}\)
Total enthalpy
Using $\rho H = \rho E + p$,
\[\begin{equation} \partial_t (\rho H) + (\rho u_j H)_{,\,j} = -q_{j,\,j} + \dot{Q} + \rho f_i u_i + (\sigma_{ij}u_i)_{,\,j} + \partial_t p + \dot{m}H - \dot{m}p/\rho. \end{equation}\]Temperature
Provided that fluid follows ideal gas law then
\[\begin{equation} \partial_t (\rho c_v T) + (\rho u_j c_v T)_{,\,j} = -q_{j,\,j} + \dot{Q} - pu_{j,\,j} + \sigma_{ij} u_{i,\,j} + \dot{m} c_v T. \end{equation}\]If all transport properties are constant, (11) becomes
\[\begin{equation} \partial_t (\rho T) + (\rho u_j T)_{,\,j} = (-q_{j,\,j} + \dot{Q} - pu_{j,\,j} + \sigma_{ij} u_{i,\,j})/c_v + \dot{m}T. \end{equation}\]