# Euler's equations 1d for pipe, Inlet boundary conditions

$\def\rmin{{\mathrm{in}}}$ $\def\l{\left}\def\r{\right}$ $\def\tagl#1{\tag{#1}\label{#1}}$ I am using the one-dimensional finite volume method to calculate the air flow in some tube. For subsonic flow one can predefine two quantities at the inlet.

The inlet is a wide opening to the ambient. The overall system deeply inside the tube is a bit more complicated. But, this does not really matter here. The system is described in some PhD and I am coding along the lines of this PhD. The author assumes isentropic flow at the inlet.

For the inlet pressure $p_\rmin$ he gave the formula \begin{align} p_\rmin + \frac{\gamma-1}\gamma\rho_0 \l(\frac{p_\rmin}{p_0}\r)^{\frac1\gamma}\frac{u_\rmin^2}2&= p_0 \tagl{A20amb} \end{align} with the ambient pressure $p_0=10^5 \rm Pa$, the ambient air density $\rho_0=1.23\rm \frac{kg}{m^3}$ and the given inlet velocity $u_\rmin$ . After solving this equation for $p_\rmin$ one can also determine the inlet density $\rho_\rmin$.

Pittyingly, I do not get this equation.

He motivated \eqref{A20amb} through the ideal gas law \begin{align} p=\rho(c_p-c_v)T, \tagl{A1amb} \end{align} the assumption of isentropic state transition \begin{align} \frac{p_\rmin}{p_0} = \l(\frac{\rho_\rmin}{\rho_0}\r)^\gamma \tagl{A3amb} \end{align} where \begin{align} \gamma := \frac{c_p}{c_v}, \tagl{A2} \end{align} and the energy balance \begin{align} c_p T_0 = c_pT_\rmin + \frac12 u_\rmin^2 \tagl{A5amb} \end{align} with the ambient temperature $T_0$.

Substitution of $T=\frac{p}{(c_p-c_v)\rho}$ from \eqref{A1amb} into \eqref{A5amb} delivers with $\rho_\rmin = \rho_0\l(\frac{p_\rmin}{p_0}\r)^{1/\gamma}$ from \eqref{A3amb} the equation \begin{align} c_p\frac{p_0}{(c_p-c_v)\rho_0} = c_p\frac{p_\rmin}{(c_p-c_v)\rho_0\l(\frac{p_\rmin}{p_0}\r)^{1/\gamma}} +\frac12 u_\rmin^2 \end{align} which can be transformed into \begin{align} 1 &= \l(\frac{p_\rmin}{p_0}\r)^{\frac{\gamma-1}\gamma} + (\gamma-1) \frac{\rho_0}{\gamma p_0}\cdot\frac{u_\rmin^2}2\\ 1 &= \l(\frac{p_\rmin}{p_0}\r)^{\frac{\gamma-1}\gamma} + \frac{\gamma-1}2\l(\frac{u_\rmin}{c_0}\r)^2 \tagl{p0} \end{align} where \begin{align} c_0=\sqrt{\frac{\gamma p_0}{\rho_0}} \tagl{speedOfSound} \end{align} is the speed of sound in the ambient.

This formula can be resolved for the inlet pressure: \begin{align} p_\rmin &= p_0 \l(1-\frac{\gamma-1}2\l(\frac{u_\rmin}{c_0}\r)^2\r)^{\frac{\gamma}{\gamma-1}} \tagl{solution} \end{align} I did not check whether \eqref{solution} is equivalent to \eqref{A20amb}. But, I do not think so because then the author would have used the explicite form \eqref{solution} to calculate $p_\rmin$. EDIT: I did check for differences. See below.

What am I missing?? Did I miss-interprete the assumptions?

Note that also \eqref{A20amb} can be re-formulated with the help of $c_0$: \begin{align} \l(\frac{p_\rmin}{p_0}\r) + \frac{\gamma-1}2 \l(\frac{p_\rmin}{p_0}\r)^{\frac1\gamma}\l(\frac{u_\rmin}{c_0}\r)^2 = 1 \tagl{A20mod} \end{align} If one divides by $\l(\frac{p_\rmin}{p_0}\r)^{\frac1\gamma}$ one nicely recognizes the difference between \eqref{A20amb} and \eqref{p0}: \begin{align} \l(\frac{p_\rmin}{p_0}\r)^{\frac{\gamma-1}\gamma} + \frac{\gamma-1}2\l(\frac{u_\rmin}{c_0}\r)^2 = \l(\frac{p_\rmin}{p_0}\r)^{-\frac1\gamma} \tagl{A20mod2} \end{align} Instead of the 1 on the left-hand side of \eqref{p0} the formula from the PhD-thesis has a $\l(\frac{p_\rmin}{p_0}\r)^{-\frac1\gamma}$.

• Sorry, in the equation-references worked in the preview but they do not work in the post. I try to improve them... – Tobias Mar 19 '14 at 18:02
• Possibly just a typo in your work, but $R=c_p-c_v$, not $c_v-c_p$ as you have above. Looks like you (or the author) carried that mistake through. – Aurelius Mar 31 '14 at 17:04
• @Aurelius Thank you very much. I corrected this. It was my mistake. – Tobias Mar 31 '14 at 18:02