Governing equations on integral form: Difference between revisions
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[[Category:Compressible flow]] | <!-- | ||
[[Category:Governing equations]] | -->[[Category:Compressible flow]]<!-- | ||
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==== The Continuity Equation ==== | ==== The Continuity Equation ==== | ||
{{ | {{QuoteBox|Mass can be neither created nor destroyed, which implies that mass is conserved}} | ||
The net massflow into the control volume <math>\Omega</math> in Fig. \ref{fig:generic:cv} is obtained by integrating mass flux over the control volume surface <math>\partial \Omega</math> | The net massflow into the control volume <math>\Omega</math> in Fig. \ref{fig:generic:cv} is obtained by integrating mass flux over the control volume surface <math>\partial \Omega</math> | ||
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==== The Momentum Equation ==== | ==== The Momentum Equation ==== | ||
{{ | {{QuoteBox|The time rate of change of momentum of a body equals the net force exerted on it}} | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
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==== The Energy Equation ==== | ==== The Energy Equation ==== | ||
{{ | {{QuoteBox|Energy can be neither created nor destroyed; it can only change in form}} | ||
<math display="block"> | <math display="block"> | ||
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{{NumEqn|<math> | {{NumEqn|<math> | ||
\dfrac{d}{dt}\iiint_{\Omega}\rho\left(e+\dfrac{V^2}{2}\right)dV+\iint_{\partial \Omega}\left[\rho\left(e+\dfrac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math> | |||
\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV | |||
</math>|align=center}} | |||
</math>}} | |||
The surface integral in the energy equation may be rewritten as | The surface integral in the energy equation may be rewritten as | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\iint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=\iint_{\partial \Omega}\rho\left[e+\frac{p}{\rho}+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS | \iint_{\partial \Omega}\left[\rho\left(e+\frac{V^2}{2}\right)(\mathbf{v}\cdot\mathbf{n}) + p\mathbf{v}\cdot\mathbf{n}\right]dS=</math><br><br><math>\iint_{\partial \Omega}\rho\left[e+\frac{p}{\rho}+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS | ||
</math>}} | </math>|align=center}} | ||
and with the definition of enthalpy <math>h=e+p/\rho</math>, we get | and with the definition of enthalpy <math>h=e+p/\rho</math>, we get | ||
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{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV | \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>}} | </math>}} | ||
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The integral form of the governing equations for inviscid compressible flow has been derived | The integral form of the governing equations for inviscid compressible flow has been derived | ||
<div style="border: solid 1px;"> | |||
{{OpenInfoBox|<math> | |||
{{ | |||
\frac{d}{dt}\iiint_{\Omega} \rho dV+\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=0 | \frac{d}{dt}\iiint_{\Omega} \rho dV+\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS=0 | ||
</math>}} | </math>|description=Continuity:}} | ||
{{ | {{OpenInfoBox|<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 | \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>}} | </math>|description=Momentum:}} | ||
{{OpenInfoBox|<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>|description=Energy:}} | ||
\frac{d}{dt}\iiint_{\Omega}\rho e_o dV+\iint_{\partial \Omega}\rho h_o(\mathbf{v}\cdot\mathbf{n})dS=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV | </div> | ||
</math>}} | |||
