One-dimensional flow with friction: Difference between revisions

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==== Energy equation ====
==== Energy equation ====
\noindent For an adiabatic one-dimensional flow we have that \\


\begin{equation}
For an adiabatic one-dimensional flow we have that
c_p T+\frac{u^2}{2}=const
\label{eqn:ttot:a}
\end{equation}\\


\noindent If we differentiate (\ref{eqn:ttot:a}) we get\\
<math display="block">
C_p T+\frac{u^2}{2}=const
</math>


\begin{equation}
If we differentiate (\ref{eqn:ttot:a}) we get
c_p dT+\frac{1}{2}d(u^2)=0.
\label{eqn:ttot:b}
\end{equation}\\


\noindent We replace $c_p$ with $\gamma R/(\gamma-1)$ and multiply and divide the first term with $T$ which gives us\\
<math display="block">
C_p dT+\frac{1}{2}d(u^2)=0.
</math>


\begin{equation}
We replace <math>C_p</math> with <math>\gamma R/(\gamma-1)</math> and multiply and divide the first term with <math>T</math> which gives us
 
<math display="block">
\frac{\gamma RT}{(\gamma-1)}\frac{dT}{T}+\frac{1}{2}d(u^2)=0.
\frac{\gamma RT}{(\gamma-1)}\frac{dT}{T}+\frac{1}{2}d(u^2)=0.
\label{eqn:ttot:c}
</math>
\end{equation}\\


\noindent Now, divide by $\gamma RT/(\gamma-1)$ and multiply and divide the second term by $u^2$ gives\\
Now, divide by <math>\gamma RT/(\gamma-1)</math> and multiply and divide the second term by <math>u^2</math> gives


\begin{equation}
<math display="block">
\frac{dT}{T}+\frac{(\gamma-1)}{2}M^2\frac{d(u^2)}{u^2}=0.
\frac{dT}{T}+\frac{(\gamma-1)}{2}M^2\frac{d(u^2)}{u^2}=0.
\label{eqn:ttot}
</math>
\end{equation}\\


\noindent We want to remove the $dT/T$-term in (\ref{eqn:ttot}). From the definition of Mach number we have that\\
We want to remove the <math>dT/T</math>-term in (\ref{eqn:ttot}). From the definition of Mach number we have that


\begin{equation}
<math display="block">
a^2M^2=u^2
a^2M^2=u^2
\label{eqn:Mach:a}
</math>
\end{equation}\\


\noindent which we can rewrite using the expression for speed of sound $(a^2=\gamma RT)$ according to\\
which we can rewrite using the expression for speed of sound <math>(a^2=\gamma RT)</math> according to


\begin{equation}
<math display="block">
\gamma RTM^2=u^2
\gamma RTM^2=u^2
\label{eqn:Mach:b}
</math>
\end{equation}\\


\noindent Differentiating (\ref{eqn:Mach:b}) gives us\\
Differentiating (\ref{eqn:Mach:b}) gives us


\begin{equation}
<math display="block">
\gamma RM^2 dT+\gamma RT d(M^2)=d(u^2)
\gamma RM^2 dT+\gamma RT d(M^2)=d(u^2)
\label{eqn:Mach:c}
</math>
\end{equation}\\


\noindent Now, if we divide (\ref{eqn:Mach:c}) by $\gamma RT M^2$ and use $a^2=\gamma RT$ and $a^2M^2=u^2$ we get\\
Now, if we divide (\ref{eqn:Mach:c}) by <math>\gamma RT M^2</math> and use <math>a^2=\gamma RT</math> and <math>a^2M^2=u^2</math> we get


\begin{equation}
<math display="block">
\frac{dT}{T}+\frac{d(M^2)}{M^2}=\frac{d(u^2)}{u^2}
\frac{dT}{T}+\frac{d(M^2)}{M^2}=\frac{d(u^2)}{u^2}
\label{eqn:Mach}
</math>
\end{equation}\\


\noindent Equation (\ref{eqn:Mach}) may now be used to replace the $dT/T$-term in equation (\ref{eqn:ttot})\\
Equation (\ref{eqn:Mach}) may now be used to replace the <math>dT/T</math>-term in equation (\ref{eqn:ttot})


\begin{equation}
<math display="block">
-\frac{d(M^2)}{M^2}+\frac{d(u^2)}{u^2}+\frac{(\gamma-1)}{2}M^2\frac{d(u^2)}{u^2}=0.
-\frac{d(M^2)}{M^2}+\frac{d(u^2)}{u^2}+\frac{(\gamma-1)}{2}M^2\frac{d(u^2)}{u^2}=0.
\label{eqn:ttot:Mach:a}
</math>
\end{equation}\\


\noindent which can be rewritten according to\\
which can be rewritten according to


\begin{equation}
<math display="block">
\frac{d(u^2)}{u^2}=\left[1+\frac{(\gamma-1)}{2}M^2\right]^{-1}\frac{d(M^2)}{M^2}
\frac{d(u^2)}{u^2}=\left[1+\frac{(\gamma-1)}{2}M^2\right]^{-1}\frac{d(M^2)}{M^2}
\label{eqn:ttot:Mach:b}
</math>
\end{equation}\\


\noindent Using the chain rule for derivatives, the last term may be rewritten according to\\
Using the chain rule for derivatives, the last term may be rewritten according to


\[\frac{d(M^2)}{M^2}=2M\frac{dM}{M^2}=2\frac{dM}{M}\]\\
<math display="block">
\frac{d(M^2)}{M^2}=2M\frac{dM}{M^2}=2\frac{dM}{M}
</math>


\noindent which gives\\
which gives


\begin{equation}
<math display="block">
\frac{d(u^2)}{u^2}=2\left[1+\frac{(\gamma-1)}{2}M^2\right]^{-1}\frac{dM}{M}
\frac{d(u^2)}{u^2}=2\left[1+\frac{(\gamma-1)}{2}M^2\right]^{-1}\frac{dM}{M}
\label{eqn:ttot:Mach}
</math>
\end{equation}\\


\subsection{The ideal gas law}
===== The ideal gas law =====
\noindent For a perfect gas the ideal gas law reads\\
\noindent For a perfect gas the ideal gas law reads\\


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