Governing equations on differential form: Difference between revisions

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--><noinclude><!--
--><noinclude><!--
-->{{#vardefine:secno|2}}<!--
-->{{#vardefine:secno|2}}<!--
-->{{#vardefine:eqno|31}}<!--
-->{{#vardefine:eqno|28}}<!--
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==== Conservation of Mass ====
==== Conservation of Mass ====


Apply Gauss's divergence theorem on the surface integral in Eqn. \ref{eq:governing:cont:int} gives
The continuity equation on integral form reads
 
{{InfoBox|<math>
\frac{d}{dt}\iiint_{\Omega} \rho dV+\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=0
</math>}}
 
Apply Gauss's divergence theorem on the surface integral gives


{{NumEqn|<math>
{{NumEqn|<math>
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{{NumEqn|<math>
{{NumEqn|<math>
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0
</math>}}
</math>|label=eq-cont-pde}}


which is the continuity equation on partial differential form.
which is the continuity equation on partial differential form.


==== Conservation of Momentum ====
==== Conservation of Momentum ====
The momentum equation on integral form reads
{{InfoBox|<math>
\frac{d}{dt}\iiint_{\Omega} \rho \mathbf{v} dV+\iint_{\partial \Omega} \left[(\rho\mathbf{v}\cdot\mathbf{n})\mathbf{v}+p\mathbf{n}\right]dS=\iiint_{\Omega}\rho \mathbf{f}dV
</math>}}


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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{{NumEqn|<math>
{{NumEqn|<math>
\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f}  
\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f}  
</math>}}
</math>|label=eq-mom-pde}}


which is the momentum equation on partial differential form
which is the momentum equation on partial differential form
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==== 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
The energy equation on integral form reads
 
{{InfoBox|<math>
\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=</math><br><br><math>\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV
</math>}}
 
Gauss's divergence theorem applied to the surface integral term in the energy equation gives


{{NumEqn|<math>
{{NumEqn|<math>
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{{NumEqn|<math>
{{NumEqn|<math>
\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
</math>}}
</math>|label=eq-energy-pde}}


which is the energy equation on partial differential form
which is the energy equation on partial differential form
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The governing equations for compressible inviscid flow on partial differential form:
The governing equations for compressible inviscid flow on partial differential form:


{{NumEqn|<math>
<div style="border: solid 1px;">
{{OpenInfoBox|<math>
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0
</math>}}
</math>|description=Continuity:}}


{{NumEqn|<math>
{{OpenInfoBox|<math>
\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f}
\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f}
</math>}}
</math>|description=Momentum:}}


{{NumEqn|<math>
{{OpenInfoBox|<math>
\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
</math>}}
</math>|description=Energy:}}
</div>


=== The Differential Equations on Non-Conservation Form ===
=== The Differential Equations on Non-Conservation Form ===
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{{NumEqn|<math>
{{NumEqn|<math>
\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\nabla
\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\nabla
</math>}}
</math>|label=eq-cont-pde-non-cons}}


where the first term of the right hand side is the local derivative and the second term is the convective derivative.
where the first term of the right hand side is the local derivative and the second term is the convective derivative.
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{{NumEqn|<math>
{{NumEqn|<math>
\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0
\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0
</math>}}
</math>|label=eq-pde-noncons-cont}}


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.
{{EquationNote|label=eq-pde-noncons-cont|nopar=1}} 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 ====
==== Conservation of Momentum ====
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{{NumEqn|<math>
{{NumEqn|<math>
\frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f}
\frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f}
</math>}}
</math>|label=eq-mom-pde-non-cons}}


==== Conservation of Energy ====
==== Conservation of Energy ====


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
The last equation on non-conservation differential form is the energy equation. We start by rewriting the energy equation on differential form {{EquationNote|label=eq-energy-pde}}, repeated here for convenience


{{NumEqn|<math>
{{NumEqn|<math>
\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
</math>}}
</math>|nonumber=1}}


Total enthalpy, <math>h_o</math>, is replaced with total energy, <math>e_o</math>
Total enthalpy, <math>h_o</math>, is replaced with total energy, <math>e_o</math>
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{{NumEqn|<math>
{{NumEqn|<math>
\rho\frac{\partial e_o}{\partial t} + e_o\frac{\partial \rho}{\partial t} + \rho\mathbf{v}\cdot\nabla e_o + e_o\nabla\cdot(\rho \mathbf{v}) + \nabla\cdot(p\mathbf{v})= \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho
\rho\frac{\partial e_o}{\partial t} + e_o\frac{\partial \rho}{\partial t} + \rho\mathbf{v}\cdot\nabla e_o + e_o\nabla\cdot(\rho \mathbf{v}) + \nabla\cdot(p\mathbf{v})=</math><br><br><math>= \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho
</math>}}
</math>}}


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{{NumEqn|<math>
{{NumEqn|<math>
\rho\frac{De_o}{Dt} + \nabla\cdot(p\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho
\rho\frac{De_o}{Dt} + \nabla\cdot(p\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho
</math>}}
</math>|label=eq-energy-pde-non-cons}}


==== Summary ====
==== Summary ====


Continuity:
<div style="border: solid 1px;">
 
{{OpenInfoBox|<math>
{{NumEqn|<math>
\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0
\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0
</math>}}
</math>|description=Continuity:}}


Momentum:
{{OpenInfoBox|<math>
 
{{NumEqn|<math>
\frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f}
\frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f}
</math>}}
</math>|description=Momentum:}}
 
Energy:


{{NumEqn|<math>
{{OpenInfoBox|<math>
\rho\frac{De_o}{Dt} + \nabla\cdot(p\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho
\rho\frac{De_o}{Dt} + \nabla\cdot(p\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho
</math>}}
</math>|description=Energy:}}
</div>


=== Alternative Forms of the Energy Equation ===
=== Alternative Forms of the Energy Equation ===
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</math>}}
</math>}}


Inserted in Eqn. \ref{eq:governing:energy:non}, this gives
Inserted in {{EquationNote|label=eq-energy-pde-non-cons|nopar=1}}, this gives


{{NumEqn|<math>
{{NumEqn|<math>
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</math>}}
</math>}}


Now, let's replace the substantial derivative <math>D\mathbf{v}/Dt</math> using the momentum equation on non-conservation form (Eqn. \ref{eq:governing:mom:non}).
Now, let's replace the substantial derivative <math>D\mathbf{v}/Dt</math> using the momentum equation on non-conservation form {{EquationNote|label=eq-mom-pde-non-cons}}.


{{NumEqn|<math>
{{NumEqn|<math>
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{{NumEqn|<math>
{{NumEqn|<math>
\rho\frac{De}{Dt} \cancel{-\mathbf{v}\cdot\nabla p} + \cancel{\mathbf{v}\cdot\nabla p} +  p(\nabla\cdot\mathbf{v}) = \dot{q}\rho\Rightarrow \rho\frac{De}{Dt} + p(\nabla\cdot\mathbf{v}) = \dot{q}\rho
\rho\frac{De}{Dt} \cancel{-\mathbf{v}\cdot\nabla p} + \cancel{\mathbf{v}\cdot\nabla p} +  p(\nabla\cdot\mathbf{v}) = \dot{q}\rho\Rightarrow</math><br><br><math>\Rightarrow\rho\frac{De}{Dt} + p(\nabla\cdot\mathbf{v}) = \dot{q}\rho
</math>}}
</math>}}


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{{NumEqn|<math>
{{NumEqn|<math>
\frac{De}{Dt} + \frac{p}{\rho}(\nabla\cdot\mathbf{v}) = \dot{q}
\frac{De}{Dt} + \frac{p}{\rho}(\nabla\cdot\mathbf{v}) = \dot{q}
</math>}}
</math>|label=eq-energy-pde-non-cons-b}}


Conservation of mass gives
Conservation of mass gives
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</math>}}
</math>}}


Insert in Eqn. \ref{eq:governing:energy:non:b}
Insert in {{EquationNote|label=eq-energy-pde-non-cons-b|nopar=1}}


{{NumEqn|<math>
{{NumEqn|<math>
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</math>}}
</math>}}


with <math>De/Dt</math> from Eqn. \ref{eq:governing:energy:non:b}
with <math>De/Dt</math> from {{EquationNote|label=eq-energy-pde-non-cons-b|nopar=1}}


{{NumEqn|<math>
{{NumEqn|<math>
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{{NumEqn|<math>
{{NumEqn|<math>
\frac{Dh}{Dt}=\dot{q} + \frac{1}{\rho}\frac{Dp}{Dt}
\frac{Dh}{Dt}=\dot{q} + \frac{1}{\rho}\frac{Dp}{Dt}
</math>}}
</math>|label=eq-energy-pde-non-cons-c}}


==== Total Enthalpy Formulation ====
==== Total Enthalpy Formulation ====
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</math>}}
</math>}}


From the momentum equation (Eqn. \ref{eq:governing:mom:non})
From the momentum equation {{EquationNote|label=eq-mom-pde-non-cons}}


{{NumEqn|<math>
{{NumEqn|<math>
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</math>}}
</math>}}


Inserting <math>Dh/Dt</math> from Eqn. \ref{eq:governing:energy:non:c} gives
Inserting <math>Dh/Dt</math> from {{EquationNote|label=eq-energy-pde-non-cons-c|nopar=1}} gives


{{NumEqn|<math>
{{NumEqn|<math>
\frac{Dh_o}{Dt}=\dot{q} + \frac{1}{\rho}\frac{Dp}{Dt}+\mathbf{v}\cdot\mathbf{f} -\frac{1}{\rho}\mathbf{v}\cdot\nabla p = \frac{1}{\rho}\left[\frac{Dp}{Dt}-\mathbf{v}\cdot\nabla p\right] + \dot{q} + \mathbf{v}\cdot\mathbf{f}
\frac{Dh_o}{Dt}=\dot{q} + \frac{1}{\rho}\frac{Dp}{Dt}+\mathbf{v}\cdot\mathbf{f} -\frac{1}{\rho}\mathbf{v}\cdot\nabla p =</math><br><br><math>=\frac{1}{\rho}\left[\frac{Dp}{Dt}-\mathbf{v}\cdot\nabla p\right] + \dot{q} + \mathbf{v}\cdot\mathbf{f}
</math>}}
</math>}}


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