Governing equations on differential form: Difference between revisions
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=== Conservation of Mass === | === The Differential Equations on Conservation Form === | ||
==== Conservation of Mass ==== | |||
Apply Gauss's divergence theorem on the surface integral in Eqn. \ref{eq:governing:cont:int} gives | Apply Gauss's divergence theorem on the surface integral in Eqn. \ref{eq:governing:cont:int} gives | ||
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which is the continuity equation on partial differential form. | which is the continuity equation on partial differential form. | ||
=== Conservation of Momentum === | ==== Conservation of Momentum ==== | ||
As for the continuity equation, the surface integral terms are rewritten as volume integrals using Gauss's divergence theorem. | As for the continuity equation, the surface integral terms are rewritten as volume integrals using Gauss's divergence theorem. | ||
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which is the momentum equation on partial differential form | which is the momentum equation on partial differential form | ||
=== Conservation of Energy === | ==== Conservation of Energy ==== | ||
Gauss's divergence theorem applied to the surface integral term in the energy equation (Eqn. \ref{eq:governing:energy:int}) gives | |||
\[\oiint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=\iiint_{\Omega}\nabla\cdot(\rho h_o\mathbf{v})d\mathscr{V}\]\\ | \[\oiint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=\iiint_{\Omega}\nabla\cdot(\rho h_o\mathbf{v})d\mathscr{V}\]\\ | ||
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\noindent which is the energy equation on partial differential form\\ | \noindent which is the energy equation on partial differential form\\ | ||
==== Summary ==== | |||
\noindent The governing equations for compressible inviscid flow on partial differential form:\\ | \noindent The governing equations for compressible inviscid flow on partial differential form:\\ | ||
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\[\frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho\] | \[\frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho\] | ||
=== The Differential Equations on Non-Conservation Form === | |||
==== The Substantial Derivative ==== | |||
\noindent The substantial derivative operator is defined as\\ | \noindent The substantial derivative operator is defined as\\ | ||
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\noindent where the first term of the right hand side is the local derivative and the second term is the convective derivative.\\ | \noindent where the first term of the right hand side is the local derivative and the second term is the convective derivative.\\ | ||
==== Conservation of Mass ==== | |||
\noindent If we apply the substantial derivative operator to density we get\\ | \noindent If we apply the substantial derivative operator to density we get\\ | ||
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\noindent Eqn. \ref{eq:governing:cont:non} says that the mass of a fluid element with a fixed set of fluid particles is constant as the element moves in space.\\ | \noindent Eqn. \ref{eq:governing:cont:non} says that the mass of a fluid element with a fixed set of fluid particles is constant as the element moves in space.\\ | ||
==== Conservation of Momentum ==== | |||
\noindent We start from the momentum equation on differential form derived above\\ | \noindent We start from the momentum equation on differential form derived above\\ | ||
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\end{equation}\\ | \end{equation}\\ | ||
==== Conservation of Energy ==== | |||
\noindent The last equation on non-conservation differential form is the energy equation. We start by rewriting the energy equation on differential form (Eqn. \ref{eq:governing:energy:pde}), repeated here for convenience\\ | \noindent The last equation on non-conservation differential form is the energy equation. We start by rewriting the energy equation on differential form (Eqn. \ref{eq:governing:energy:pde}), repeated here for convenience\\ | ||
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%\end{equation}\\ | %\end{equation}\\ | ||
=== Alternative Forms of the Energy Equation === | |||
==== Internal Energy Formulation ==== | |||
\noindent Total internal energy is defined as\\ | \noindent Total internal energy is defined as\\ | ||
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\noindent Compare with the first law of thermodynamics: $de=\delta q-\delta w$\\ | \noindent Compare with the first law of thermodynamics: $de=\delta q-\delta w$\\ | ||
==== Enthalpy Formulation ==== | |||
\[h=e+\frac{p}{\rho}\Rightarrow \frac{Dh}{Dt}=\frac{De}{Dt}+\frac{1}{\rho}\frac{Dp}{Dt}+p\frac{D}{Dt}\left(\frac{1}{\rho}\right)\]\\ | \[h=e+\frac{p}{\rho}\Rightarrow \frac{Dh}{Dt}=\frac{De}{Dt}+\frac{1}{\rho}\frac{Dp}{Dt}+p\frac{D}{Dt}\left(\frac{1}{\rho}\right)\]\\ | ||
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\end{equation}\\ | \end{equation}\\ | ||
==== Total Enthalpy Formulation ==== | |||
\[h_o=h+\frac{1}{2}\mathbf{v}\mathbf{v}\Rightarrow\frac{Dh_o}{Dt}=\frac{Dh}{Dt}+\mathbf{v}\cdot\frac{D\mathbf{v}}{Dt}\]\\ | \[h_o=h+\frac{1}{2}\mathbf{v}\mathbf{v}\Rightarrow\frac{Dh_o}{Dt}=\frac{Dh}{Dt}+\mathbf{v}\cdot\frac{D\mathbf{v}}{Dt}\]\\ | ||
