One-dimensional flow with friction: Difference between revisions

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


\noindent continuity:\\
continuity:


\[\frac{d}{dx}(\rho u)=0\]\\
<math display="block">
\frac{d}{dx}(\rho u)=0
</math>


\noindent momentum:\\
momentum:


\[\rho u\frac{du}{dx}+\frac{dp}{dx}=-\frac{2}{D}f\rho u^2\]\\
<math display="block">
\rho u\frac{du}{dx}+\frac{dp}{dx}=-\frac{2}{D}f\rho u^2
</math>


\noindent energy:\\
energy:


\[\frac{d}{dx}h_o=0\]
<math display="block">
\frac{d}{dx}h_o=0
</math>


\noindent From chapter 3.9 we have the following expression for the momentum equation for one-dimensional flow with friction (equation (3.95))\\
From chapter 3.9 we have the following expression for the momentum equation for one-dimensional flow with friction (equation (3.95))


\begin{equation*}
<math display="block">
dp+\rho u du=-\frac{1}{2}\rho u^2 \frac{4 f dx}{D}\ (3.95)
dp+\rho u du=-\frac{1}{2}\rho u^2 \frac{4 f dx}{D}
\end{equation*}\\
</math>


\noindent For cases dealing with calorically perfect gas, (3.95) can be recast completely in terms of Mach number using the following relations\\
For cases dealing with calorically perfect gas, (3.95) can be recast completely in terms of Mach number using the following relations


\begin{itemize}
* speed of sound: <math>a^2=\gamma p/\rho</math>
\item[] speed of sound: $a^2=\gamma p/\rho$
* the definition of Mach number: <math>M^2=u^2/a^2</math>
\item[] the definition of Mach number: $M^2=u^2/a^2$
* the ideal gas law for thermally perfect gas: <math>p=\rho R T</math>
\item[] the ideal gas law for thermally perfect gas: $p=\rho R T$
* the continuity equation: <math>\rho u=const</math>
\item[] the continuity equation: $\rho u=const$
* the energy equation: <math>C_p T+u^2/2=const</math>
\item[] energy equation: $c_p T+u^2/2=const$
\end{itemize}


==== Continuity equation ====
==== Continuity equation ====
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