Governing equations on integral form: Difference between revisions
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\begin{figure}[ht!] | \begin{figure}[ht!] | ||
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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> | ||
<math | {{NumEqn|<math> | ||
-\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS | -\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS | ||
</math> | </math>}} | ||
Now, let's consider a small infinitesimal volume | Now, let's consider a small infinitesimal volume <math>dV</math> inside <math>\Omega</math>. The mass of <math>dV</math> is <math>\rho dV</math>. Thus, the mass enclosed within <math>\Omega</math> can be calculated as | ||
<math | {{NumEqn|<math> | ||
\iiint_{\Omega} \rho dV | \iiint_{\Omega} \rho dV | ||
</math> | </math>}} | ||
The rate of change of mass within <math>\Omega</math> is obtained as | The rate of change of mass within <math>\Omega</math> is obtained as | ||
<math | {{NumEqn|<math> | ||
\frac{d}{dt}\iiint_{\Omega} \rho dV | \frac{d}{dt}\iiint_{\Omega} \rho dV | ||
</math> | </math>}} | ||
Mass is conserved, which means that the rate of change of mass within <math>\Omega</math> must equal the net flux over the control volume surface. | Mass is conserved, which means that the rate of change of mass within <math>\Omega</math> must equal the net flux over the control volume surface. | ||
<math | {{NumEqn|<math> | ||
\frac{d}{dt}\iiint_{\Omega} \rho dV=-\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS | \frac{d}{dt}\iiint_{\Omega} \rho dV=-\iint_{\partial \Omega}\rho \mathbf{v}\cdot \mathbf{n} dS | ||
</math> | </math>}} | ||
or | or | ||
<math | {{NumEqn|<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>}} | ||
which is the integral form of the continuity equation. | which is the integral form of the continuity equation. | ||
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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}} | ||
<math | {{NumEqn|<math> | ||
\frac{d}{dt}(m\mathbf{v})=\mathbf{F} | \frac{d}{dt}(m\mathbf{v})=\mathbf{F} | ||
</math> | </math>}} | ||
What type of forces do we have? | What type of forces do we have? | ||
* Body forces acting on the fluid inside | * Body forces acting on the fluid inside <math>\Omega</math> | ||
** gravitation | ** gravitation | ||
** electromagnetic forces | ** electromagnetic forces | ||
| Line 71: | Line 82: | ||
Body forces inside <math>\Omega</math>: | Body forces inside <math>\Omega</math>: | ||
<math | {{NumEqn|<math> | ||
\iiint_{\Omega}\rho \mathbf{f}dV | \iiint_{\Omega}\rho \mathbf{f}dV | ||
</math> | </math>}} | ||
Surface force on <math>\partial \Omega</math>: | Surface force on <math>\partial \Omega</math>: | ||
<math | {{NumEqn|<math> | ||
-\iint_{\partial \Omega} p\mathbf{n}dS | -\iint_{\partial \Omega} p\mathbf{n}dS | ||
</math> | </math>}} | ||
Since we are considering inviscid flow, there are no shear forces and thus we have the net force as | Since we are considering inviscid flow, there are no shear forces and thus we have the net force as | ||
<math | {{NumEqn|<math> | ||
\mathbf{F}=\iiint_{\Omega}\rho \mathbf{f}dV-\iint_{\partial \Omega} p\mathbf{n}dS | \mathbf{F}=\iiint_{\Omega}\rho \mathbf{f}dV-\iint_{\partial \Omega} p\mathbf{n}dS | ||
</math> | </math>}} | ||
The fluid flowing through | The fluid flowing through <math>\Omega</math> will carry momentum and the net flow of momentum out from <math>\Omega</math> is calculated as | ||
<math | {{NumEqn|<math> | ||
\iint_{\partial \Omega} (\rho\mathbf{v}\cdot\mathbf{n}dS)\mathbf{v}=\iint_{\partial \Omega} (\rho\mathbf{v}\cdot\mathbf{n})\mathbf{v}dS | \iint_{\partial \Omega} (\rho\mathbf{v}\cdot\mathbf{n}dS)\mathbf{v}=\iint_{\partial \Omega} (\rho\mathbf{v}\cdot\mathbf{n})\mathbf{v}dS | ||
</math> | </math>}} | ||
Integrated momentum inside <math>\Omega</math> | Integrated momentum inside <math>\Omega</math> | ||
<math | {{NumEqn|<math> | ||
\iiint_{\Omega} \rho \mathbf{v} dV | \iiint_{\Omega} \rho \mathbf{v} dV | ||
</math> | </math>}} | ||
Rate of change of momentum due to unsteady effects inside <math>\Omega</math> | Rate of change of momentum due to unsteady effects inside <math>\Omega</math> | ||
<math | {{NumEqn|<math> | ||
\frac{d}{dt}\iiint_{\Omega} \rho \mathbf{v} dV | \frac{d}{dt}\iiint_{\Omega} \rho \mathbf{v} dV | ||
</math> | </math>}} | ||
Combining the rate of change of momentum, the net momentum flux and the net forces we get | Combining the rate of change of momentum, the net momentum flux and the net forces we get | ||
<math | {{NumEqn|<math> | ||
\frac{d}{dt}\iiint_{\Omega} \rho \mathbf{v} dV+\iint_{\partial \Omega} (\rho\mathbf{v}\cdot\mathbf{n})\mathbf{v}dS=\iiint_{\Omega}\rho \mathbf{f}dV-\iint_{\partial \Omega} p\mathbf{n}dS | \frac{d}{dt}\iiint_{\Omega} \rho \mathbf{v} dV+\iint_{\partial \Omega} (\rho\mathbf{v}\cdot\mathbf{n})\mathbf{v}dS=\iiint_{\Omega}\rho \mathbf{f}dV-\iint_{\partial \Omega} p\mathbf{n}dS | ||
</math> | </math>}} | ||
combining the surface integrals, we get | combining the surface integrals, we get | ||
<math | {{NumEqn|<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>}} | ||
which is the momentum equation on integral form. | which is the momentum equation on integral form. | ||
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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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;<math>E_3</math> Rate of change of energy of the fluid as it flows through <math>\Omega</math> | ;<math>E_3</math> Rate of change of energy of the fluid as it flows through <math>\Omega</math> | ||
<math | {{NumEqn|<math> | ||
E_1=\iiint_{\Omega} \dot{q}\rho dV | E_1=\iiint_{\Omega} \dot{q}\rho dV | ||
</math> | </math>}} | ||
where <math>\dot{q}</math> is the rate of heat added per unit mass | where <math>\dot{q}</math> is the rate of heat added per unit mass | ||
The rate of work done on the fluid in | The rate of work done on the fluid in <math>\Omega</math> due to pressure forces is obtained from the pressure force term in the momentum equation. | ||
<math | {{NumEqn|<math> | ||
E_{2_{pressure}}=-\iint_{\partial \Omega}(p\mathbf{n}dS)\cdot\mathbf{v}=-\iint_{\partial \Omega} p\mathbf{v}\cdot\mathbf{n}dS | E_{2_{pressure}}=-\iint_{\partial \Omega}(p\mathbf{n}dS)\cdot\mathbf{v}=-\iint_{\partial \Omega} p\mathbf{v}\cdot\mathbf{n}dS | ||
</math> | </math>}} | ||
The rate of work done on the fluid in $\Omega$ due to body forces is | The rate of work done on the fluid in $\Omega$ due to body forces is | ||
<math | {{NumEqn|<math> | ||
E_{2_{body\ forces}}=\iiint_{\Omega}(\rho\mathbf{f}dV)\cdot\mathbf{v}=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV | E_{2_{body\ forces}}=\iiint_{\Omega}(\rho\mathbf{f}dV)\cdot\mathbf{v}=\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
E_2=E_{2_{pressure}}+E_{2_{body\ forces}}=-\iint_{\partial \Omega} p\mathbf{v}\cdot\mathbf{n}dS+\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV | E_2=E_{2_{pressure}}+E_{2_{body\ forces}}=-\iint_{\partial \Omega} p\mathbf{v}\cdot\mathbf{n}dS+\iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV | ||
</math> | </math>}} | ||
The energy of the fluid per unit mass is the sum of internal energy <math>e</math> (molecular energy) and the kinetic energy <math>V^2/2</math> and the net energy flux over the control volume surface is calculated by the following integral | The energy of the fluid per unit mass is the sum of internal energy <math>e</math> (molecular energy) and the kinetic energy <math>V^2/2</math> and the net energy flux over the control volume surface is calculated by the following integral | ||
<math | {{NumEqn|<math> | ||
\iint_{\partial \Omega}(\rho \mathbf{v}\cdot\mathbf{n}dS)\left(e+\frac{V^2}{2}\right) | \iint_{\partial \Omega}(\rho \mathbf{v}\cdot\mathbf{n}dS)\left(e+\frac{V^2}{2}\right) | ||
</math> | </math>}} | ||
Analogous to mass and momentum, the total amount of energy of the fluid in <math>\Omega</math> is calculated as | Analogous to mass and momentum, the total amount of energy of the fluid in <math>\Omega</math> is calculated as | ||
<math | {{NumEqn|<math> | ||
\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)dV | \iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)dV | ||
</math> | </math>}} | ||
The time rate of change of the energy of the fluid in <math>\Omega</math> is obtained as | The time rate of change of the energy of the fluid in <math>\Omega</math> is obtained as | ||
<math | {{NumEqn|<math> | ||
\frac{d}{dt}\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)dV | \frac{d}{dt}\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)dV | ||
</math> | </math>}} | ||
Now, <math>E_3</math> is obtained as the sum of the time rate of change of energy of the fluid in | Now, <math>E_3</math> is obtained as the sum of the time rate of change of energy of the fluid in <math>\Omega</math> and the net flux of energy carried by fluid passing the control volume surface. | ||
<math | {{NumEqn|<math> | ||
E_3=\frac{d}{dt}\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)dV+\iint_{\partial \Omega}(\rho \mathbf{v}\cdot\mathbf{n}dS)\left(e+\frac{V^2}{2}\right) | E_3=\frac{d}{dt}\iiint_{\Omega}\rho\left(e+\frac{V^2}{2}\right)dV+\iint_{\partial \Omega}(\rho \mathbf{v}\cdot\mathbf{n}dS)\left(e+\frac{V^2}{2}\right) | ||
</math> | </math>}} | ||
With all elements of the energy equation defined, we are now ready to finally compile the full equation | With all elements of the energy equation defined, we are now ready to finally compile the full equation | ||
<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> | ||
</math> | \iiint_{\Omega}\rho\mathbf{f}\cdot\mathbf{v}dV+\iiint_{\Omega} \dot{q}\rho dV | ||
</math>|align=center}} | |||
The surface integral in the energy equation may be rewritten as | The surface integral in the energy equation may be rewritten as | ||
<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 | ||
<math | {{NumEqn|<math> | ||
\iint_{\partial \Omega}\rho\left[h+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS | \iint_{\partial \Omega}\rho\left[h+\frac{V^2}{2}\right](\mathbf{v}\cdot\mathbf{n})dS | ||
</math> | </math>}} | ||
Furthermore, introducing total internal energy <math>e_o</math> and total enthalpy <math>h_o</math> defined as | Furthermore, introducing total internal energy <math>e_o</math> and total enthalpy <math>h_o</math> defined as | ||
<math | {{NumEqn|<math> | ||
e_o=e+\frac{1}{2}V^2 | e_o=e+\frac{1}{2}V^2 | ||
</math> | </math>}} | ||
and | and | ||
<math | {{NumEqn|<math> | ||
h_o=h+\frac{1}{2}V^2 | h_o=h+\frac{1}{2}V^2 | ||
</math> | </math>}} | ||
the energy equation is written as | the energy equation is written as | ||
<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>}} | ||
==== Summary ==== | ==== Summary ==== | ||
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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> | |||
<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:}} | ||
<math | {{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:}} | ||
<math | {{OpenInfoBox|<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>|description=Energy:}} | ||
</div> | |||
