One-dimensional flow with friction: Difference between revisions

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==== Differential Form ====
==== Differential Form ====


In order to remove the integral term in the momentum equation, the governing equations are written in differential form\\
In order to remove the integral term in the momentum equation, the governing equations are written in differential form


===== Continuity =====
===== Continuity =====


\[\rho_1 u_1=\rho_2 u_2=const\Rightarrow\]\\
<math display="block">
\rho_1 u_1=\rho_2 u_2=const\Rightarrow
</math>


\begin{equation}
<math display="block">
\frac{d}{dx}(\rho u)=0
\frac{d}{dx}(\rho u)=0
\label{eq:governing:cont}
</math>
\end{equation}\\


===== Momentum =====
===== Momentum =====


\[(\rho_2 u_2^2+p_2-\rho_1 u_1^2+p_1)=-\frac{4}{D}\int_0^L\tau_w dx\Rightarrow\]\\
<math display="block">
(\rho_2 u_2^2+p_2-\rho_1 u_1^2+p_1)=-\frac{4}{D}\int_0^L\tau_w dx\Rightarrow
</math>


\begin{equation}
<math display="block">
\frac{d}{dx}(\rho u^2+p)=-\frac{4}{D}\tau_w
\frac{d}{dx}(\rho u^2+p)=-\frac{4}{D}\tau_w
\label{eq:governing:mom:c}
</math>
\end{equation}\\


\[\frac{d}{dx}(\rho u^2+p)=\rho u\frac{du}{dx}+u\frac{d}{dx}(\rho u)+\frac{dp}{dx}=\left\{\frac{d}{dx}(\rho u)=0\right\}=\rho u\frac{du}{dx}+\frac{dp}{dx}\]\\
<math display="block">
\frac{d}{dx}(\rho u^2+p)=\rho u\frac{du}{dx}+u\frac{d}{dx}(\rho u)+\frac{dp}{dx}=\left\{\frac{d}{dx}(\rho u)=0\right\}=\rho u\frac{du}{dx}+\frac{dp}{dx}
</math>


\begin{equation}
<math display="block">
\rho u\frac{du}{dx}+\frac{dp}{dx}=-\frac{4}{D}\tau_w
\rho u\frac{du}{dx}+\frac{dp}{dx}=-\frac{4}{D}\tau_w
\label{eq:governing:mom:d}
</math>
\end{equation}\\


\noindent The wall shear stress is often approximated using a shear-stress factor, $f$, according to\\
The wall shear stress is often approximated using a shear-stress factor, <math>f</math>, according to


\begin{equation}
<math display="block">
\tau_w=f\frac{1}{2}\rho u^2
\tau_w=f\frac{1}{2}\rho u^2
\label{eq:tauw:b}
</math>
\end{equation}\\


\noindent and thus\\
and thus


\begin{equation}
<math display="block">
\rho u\frac{du}{dx}+\frac{dp}{dx}=-\frac{2}{D}f\rho u^2
\rho u\frac{du}{dx}+\frac{dp}{dx}=-\frac{2}{D}f\rho u^2
\label{eq:governing:mom:d}
</math>
\end{equation}\\


===== Energy =====
===== Energy =====


\[h_1 + \frac{1}{2}u_1^2=h_2 + \frac{1}{2}u_2^2=const\]\\
<math display="block">
h_1 + \frac{1}{2}u_1^2=h_2 + \frac{1}{2}u_2^2=const
</math>


\[h_{o_1}=h_{o_2}=const\]\\
<math display="block">
h_{o_1}=h_{o_2}=const
</math>


\begin{equation}
<math display="block">
\frac{d}{dx}h_o=0
\frac{d}{dx}h_o=0
\label{eq:governing:energy}
</math>
\end{equation}\\


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