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Gas Dynamics Equations Summary Source
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\newcommand{\Dxy}[2]{{d {#1} \over { d #2}} }
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\htmetainfo{description}{free textbook pdf and HTML open
source }
\htmetainfo{keywords}{isentropic flow, Converging-Diverging Nozzle,
gas dynamics calculator, nozzle efficiency, real gas isentropic
flow , stagnation properties}

\begin{document}

\begin{center}
\huge{Gas Dynamics Equations Summary}
\large{ Version 0.2 \\
\today \\
Genick Bar-Meir, Ph.D.}
\end{center}

\abstract{This document is a summary of the equations that
appeared in the book Fundamentals of compressible Flow Mechanics.
This summary supposed to use by professionals and students
who would like to handy summary of the equations without
going through the pages of the whole books.}

\section{Introduction}

Many have asked me to make a summary for the working questions
for the gas dynamics.
Due to time constrains, this document wasn't constructed.
Therefore, I found myself searching for an equation in the book
and I realized the importance and the urgency of this document.
At this stage, this collection is a quick--fix'' which
will be improved, hopefully, in the coming days.
These equations were collected from the book
Fundamentals of compressible Flow'' by Genick Bar-Meir
and translated by using latex2html versions 1.7.

\begin{rawhtml}
Gas Dynamics
Go to Potto Potto Home.

\end{rawhtml}
\begin{rawhtml}

version the book

\end{rawhtml}
\vspace{1in}

\section{Speed of Sound}
\label{SpeedSound}

\subsection{General}
The general equation of speed sound is
{\begin{align}
\Dxy{P}{\rho} = \left. {\Pxy{P}{\rho} } \right|_{s}
\label{sound:eq:insontropic2}
\end{align}}

\subsection{Ideal Gas}

Gas that obey the equation of state $P = \rho R T$,
the speed of sound is

{\begin{align}
c = \sqrt{ k R T}
\label{sound:eq:sound}
\end{align}}

Gas that obey the equation of state $P = z \rho R T$,
the speed of sound is

{\begin{align}
c = \sqrt{nz R T}
\label{sound:eq:speedSoundNonIdealGas}
\end{align}}

Where $n$ is defined as
{\begin{align}
n = \overbrace{C_p \over C_v}^{k}
\left( z + T \left( \partial z \over \partial T
\right)_{\rho}
\over z + T \left( \partial z \over \partial T
\right)_{P} \right)
\label{sound:eq:nDef}
\end{align}}

\subsection{Speed of Sound in Liquid}

{\begin{align}
c = \sqrt{elastic\; property \over inertial\; property}
= \sqrt{B \over \rho}
\label{sound:eq:sondLiquid}
\end{align}}
where
{\begin{align}
B = \rho {\Dxy{P}{\rho}}
\label{sound:eq:bulkModulus}
\end{align}}

\subsection{Speed of Sound in Solids}

{\begin{align}
c = \sqrt{ E \over \rho}
\label{sound:eq:solidExampl}
\end{align}}

where
E is Young's Modulus

\subsection{Sound Speed in Two Phase Medium}
For flow of mostly gas with drops of the other phase
(liquid or solid)

Let
{\begin{align}
{\rho \over \rho_a} = 1 + m
\label{sound:eq:appoximaetionR}
\end{align}}
where $m = {\dot{m}_b \over \dot{m}_a}$ is mass flow rate per gas
flow rate.
and the subscript a is for the gas phase and b for the liquid or
solid phase.

The equation of state is
{\begin{align}
{P \over \rho} = { R \over 1 + m} T
\label{sound:eq:combinedState}
\end{align}}

{\begin{align}
c = \sqrt{\gamma R_{mix} T}
\label{sound:eq:mixSound}
\end{align}}

where
{\begin{align}
\gamma = {C_p + mC \over C_v + mC}
\label{sound:eq:gamma1}
\end{align}}

and $R_{mix} = { R \over 1 + m}$

\section{Isentropic Flow}
\label{chap:variableArea}

{\begin{align}
{P_0 \over P } =
\left( { T_0 \over T} \right) ^ {k \over k -1} =
\left( 1 + { k -1 \over 2 } M^{2} \right)^ {k \over k -1}
\label{variableArea:eq:pressureDless}
\end{align}}

{\begin{align}
{\rho_0 \over \rho } =
\left( { T_0 \over T} \right) ^ {1 \over k -1} =
\left( 1 + { k -1 \over 2 } M^{2} \right)^ {1 \over k -1}
\label{variableArea:eq:densityDless}
\end{align}}
The star values
{\begin{align}
{T^{*} \over T_0} = {{c^{*}}^2 \over {c_0}^2} =
{2 \over k+1}
\label{variableArea:eq:TstarTzero}
\end{align}}
{\begin{align}
{P^{*} \over P_0} = \left(2 \over k+1 \right)^{k \over k-1}
\label{variableArea:eq:PstarPzero1}
\end{align}}
{\begin{align}
{\rho^{*} \over \rho_0} = \left(2 \over k+1 \right)^{1 \over
k-1}
\label{variableArea:eq:PstarPzero}
\end{align}}

\subsection{Relationships for Small Mach Number}

{\begin{align}
{P_0\over P} = 1 + {(k -1) M^2 \over 4} + {k M^4\over 8} +
{2(2-k)M^6 \over 48} \cdots
\label{variableArea:eq:PzeroReduced1}
\end{align}}
{\begin{align}
{\rho_0\over \rho} = 1 + {(k -1) M^2 \over 4} + {k M^4\over 8} +
{2(2-k)M^6 \over 48} \cdots
\label{variableArea:eq:RzeroReduced1}
\end{align}}
{\begin{align}
{P_0 - P \over {1 \over 2 } \rho U^2} = 1 +
\overbrace{{ M^2 \over 4} + {(2-k) M^4\over 24} + \cdots}
^{compressibility\; correction}
\label{variableArea:eq:PDiffReduced}
\end{align}}

{\begin{align}
M^{*} = {U \over c^{*} } = \sqrt{k+1 \over 2} M
\left( 1 - {k -1 \over 4} M^2 + \cdots \right)
\label{variableArea:eq:MstarReduced}
\end{align}}

{\begin{align}
{P_0 -P \over P} = {kM^2 \over 2} \left( 1 + {M^2 \over 4}
+ \cdots \right)
\label{variableArea:eq:PzeroReduced}
\end{align}}
{\begin{align}
{\rho_0 -\rho \over \rho} = {M^2 \over 2}
\left( 1 - {kM^2 \over 4} + \cdots \right)
\label{variableArea:eq:RzeroReduced2}
\end{align}}

{\begin{align}
{\dot{m} \over A} = \sqrt{k {P_0}^2 M^2 \over RT_0}
\left( 1 + {k-1 \over 4}M^2 + \cdots \right)
\label{variableArea:eq:RzeroReduced}
\end{align}}
The ratio of the area to star area is
{\begin{align}
{A \over A^{*}} = \left(2 \over k +1 \right)^{k +1 \over 2 (k-1)}
\left( {1\over M} + {k+1 \over 4}M + {(3-k) (k+1)\over 32 }
M^3 + \cdots \right)
\label{variableArea:eq:AstarReduced}
\end{align}}

{\begin{align}
{A \over A^{*}} = { 1 \over M}
\left(
{ 1 + {k -1 \over 2} M^{2} \over {k +1\over 2}}
\right) ^ {k+ 1 \over 2 (k -1 )}
\label{variableArea:eq:massFlowRateRatio}
\end{align}}

\subsection{ Isentropic Isothermal Flow Nozzle}

{\begin{align}
T_1 = T_2
\end{align}}
{\begin{align}
{{T_0}_1 \over {T_0}_2} =
{\left(1+{k -1\over2} {M_1}^2\right)\over
\left(1+{k -1\over2}{M_2}^2\right) }
= {\left(1+{k -1\over2} {M_1}^2\right)\over
\left(1+{k -1\over2}{M_2}^2\right) }
\label{variableArea:eq:isoTzeroR1}
\end{align}}

{\begin{align}
{P_2 \over P_1} = {\Huge e}^{k({M_1}^2 - {M_2}^2) \over 2}
= \left( {\Huge e}^{{M_1}^2} \over {\Huge e}^{{M_2}^2}\right)
^{k\over 2}
\label{variableArea:eq:isoPratio}
\end{align}}

{\begin{align}
{A_2 \over A_1} = { M_1 \over M_2}
\left( \mbox{\large e}^{{M_2}^{2}} \over
\mbox{\large e}^{{M_1}^{2}}
\right)^{k \over 2}
\label{variableArea:eq:isoAratio}
\end{align}}

{\begin{align}
{{P_0}_2 \over {P_0}_1} = { P_2 \over P_1}
\left( 1 + {k -1 \over 2}{M_2}^2 \over 1 +
{k -1 \over 2}{M_1}^2
\right)^{ k \over k-1} =
\left[ \mbox{\large e}^{{M_1}^{2}} \over \mbox{\large e}
^{{M_1}^{2}}
\right]^{k \over 2}
\label{variableArea:eq:isoTzeroR}
\end{align}}

The star values
{\begin{align}
T = T^{*}
\end{align}}

{\begin{align}
{ P \over P^{*}} = { \rho \over \rho^{*}}
= \mbox{\large e}^{(1-M^2) k \over 2}
\label{variableArea:eq:isoPratioStar}
\end{align}}

{\begin{align}
{ A \over A^{*}} = {1 \over M} \mbox{\large e}^{(1-M^2) k
\over 2}
\label{variableArea:eq:isoAratioStar}
\end{align}}

{\begin{align}
{ T_0 \over {T_0}^{*}} =
{2 \left( 1 + {k -1 \over 2}{M_1}^2 \right)
\over k +1 } ^ {k \over k-1}
\label{variableArea:eq:isoTzeroRatioStar}
\end{align}}

{\begin{align}
{ P_0 \over {P_0}^{*}} = {\Huge e}^{(1-M)k \over 2}
{2 \left( 1 + {k -1 \over 2}{M_1}^2 \right)
\over k +1 } ^ {k \over k-1}
\label{variableArea:eq:isoPzeroRatioStar}
\end{align}}

The initial stagnation temperature is denoted as
${T_{0}}_{int}$.

{\begin{align}
{T \over{T_{0}}_{int}} = { 1 \over 1 + {k-1 \over 2} M^2}
\label{variableArea:eq:isentropicTratio}
\end{align}}
{\begin{align}
{P \over{P_{0}}_{int}} = { 1 \over \left( 1 + {k-1 \over 2} M^2
\right) ^{k-1 \over k} }
\label{variableArea:eq:isentropicPratio}
\end{align}}

{\begin{align}
{F_{net} \over P_0 A^{*}} =
\overbrace{P_2A_2 \over P_0 A^{*}}^{f(M_2)}
\overbrace{\left( 1 + k{M_2}^2 \right)}^{f(M_2)}
- \overbrace{P_1A_1 \over P_0 A^{*}}^{f(M_1)}
\overbrace{\left( 1 + k{M_1}^2 \right)}^{f(M_1)}
\label{variableArea:eq:beforeDefa}
\end{align}}

{\begin{align}
{F \over F^{*}}
= {P_1A_1 \over P^{*}A^{*}}
{\left( 1 + k{M_1}^2 \right) \over \left( 1 + k \right) }
= {1 \over \underbrace{P^{*}\over P_{0} }_
{\left(2 \over k+1 \right)^{k \over k-1}}}
\overbrace{{P_1A_1 \over P_0A^{*} }
{\left( 1 + k{M_1}^2 \right)
}}^{\hbox{see function
\eqref{variableArea:eq:beforeDefa}}}
{1 \over \left( 1 + k \right) }
\label{variableArea:eq:ImpulseRatio}
\end{align}}

{\begin{align}
F_{net} = P_0 A^{*} (1+k)
{\left( k+1 \over 2 \right)^{k \over k-1}}
\left( {F_2 \over F^{*} } - { F_1 \over F^{*}}\right)
\label{variableArea:eq:NetForce}
\end{align}}
for isothermal

{\begin{align}
{F_2 \over F_1} =
{P_2 A_2 \over P_1 A_1} { 1 + {{U_2}^2 \over RT}
\over 1 + {{U_1}^2 \over RT }}
\label{variableArea:eq:isoRatioIdealgas}
\end{align}}

{\begin{align}
{F_2 \over F_1} =
{M_1 \over M_2} { 1 + k {M_2}^2 \over 1 + k {M_1}^2}
\label{variableArea:eq:isoRatioIdealgasISO}
\end{align}}
{\begin{align}
{F_2 \over F^{*}} =
{1 \over M_2} { 1 + k {M_2}^2 \over 1 + k }
\label{variableArea:eq:isoRatioIdealgasStar}
\end{align}}

\section{Normal Shock }
\label{chap:shock}
{\begin{align}
{T_0}_y = {T_0}_x
\end{align}}
{\begin{align}
{T_y \over T_x} = \left( { P_{y} \over P_{x}} \right)^{2}
\left( {M_y \over M_x} \right)^{2}
\label{shock:eq:nonDimMass}
\end{align}}

{\begin{align}
{P_y \over P_x} = {1 + k{M_{x}}^2 \over 1 + k{M_{y}}^2}
\label{shock:eq:pressureRatio}
\end{align}}

{\begin{align}
{{P_0}_y \over
{P_0}_x} =
{
P_y
\left( 1 + {k-1 \over 2} {M_y}^{2} \right) ^ {k \over k-1}
\over
P_x
\left( 1 + {k-1 \over 2} {M_x}^{2} \right) ^ {k \over k-1}
}
\label{shock:eq:totalPressureRatio}
\end{align}}

{\begin{align}
{M_y}^2 = {
{M_x}^2 + {2 \over k -1}
\over
{2k \over k -1} {M_x}^2 - 1
}
\label{shock:eq:solution2}
\end{align}}

{\begin{align}
\nonumber
{P_y \over P_x} & = {2k \over k+1 } {M_x}^2 - {k -1 \over
k+1} \\
{P_y \over P_x} & = 1 + { 2k \over k+1} \left({M_x}^2 -1
\right )
\label{shock:eq:pressureMx}
\end{align}}

{\begin{align}
{\rho_y \over \rho_x} = {U_x \over U_y} =
{( k +1) {M_x}^{2} \over 2 + (k -1) {M_x}^{2} }
\label{shock:eq:densityMx}
\end{align}}
{\begin{align}
{T_y \over T_x} = \left( {P_y \over P_x} \right)
\left( {k + 1 \over k -1 } + {P_y \over P_x} \over
1+ {k + 1 \over k -1 } {P_y \over P_x} \right)
\label{shock:eq:temperaturePbar}
\end{align}}

{\begin{align}
{\rho_x \over \rho_y} = { 1 + \left( {k +1 \over k -1} \right)
\left( {P_y \over P_x} \right)
\over \left( k+1 \over k-1\right) +\left( {P_y \over P_x}
\right)}
\label{shock:eq:densityPbar}
\end{align}}

Moving shocks

\section{Isothermal Flow}

{\begin{align}
\int_{0}^{L} { 4 f dx \over D} =
\int_{M^{2}}^{1/k} { 1 - kM{2} \over kM{2}} dM^{2}
\label{isothermal:eq:integralMach}
\end{align}}

{\begin{align}
{\fldmax} = { 1- k M^{2} \over k M^{2} } + \ln kM^{2}
\label{isothermal:eq:workingEq}
\end{align}}

{\begin{align}
{P_{0} \over P_{0}^{*}} =
{P \over P^{*}} \left[
{1 + { k -1 \over 2 } M ^ {2} \over
{ 1 + {k -1 \over 2k} } }
\right]
^ { k \over k -1 }
%\label{eq:}
\end{align}}

{\begin{align}
{P_{0} \over P_{0}^{*}} =
{1 \over \sqrt{k}} \left( {2k \over 3k- 1} \right)^{k \over k -1}
\left( 1 + {k -1 \over 2} M ^{2}\right)^{k \over k-1}
{ 1 \over M}
%\label{eq:}
\end{align}}
{\begin{align}
{T_{0} \over T_{0}^{*}} =
{ T \over T^{*}}
{ 1 + {k -1 \over 2} M ^{2} \over
1 + {k -1 \over 2k} } =
{2k \over 3k -1 } \left( 1 + {k -1 \over 2} \right) M ^{2}
\label{isothermal:eq:T0bar}
\end{align}}

{\begin{align}
\fld = \left. \fldmax \right|_{1} - \left. \fldmax \right|_{2} =
{ 1 - k{M_{1}}^{2} \over k {M_{1}}^{2}} -
{ 1 - k{M_{2}}^{2} \over k {M_{2}}^{2}} +
\ln \left( {M_{1} \over M_{2}} \right)^{2}
\label{isothermal:eq:workingFLD}
\end{align}}
For the case that $M_1 > > M_2$ and $M_1 \rightarrow 1$
equation \eqref{isothermal:eq:workingFLD} is
reduced into the following approximation
{\begin{align}
\fld = 2 \ln M_{1} -1 -
\overbrace{ 1 - k{M_{2}}^{2} \over k {M_{2}}^{2}}^{\sim 0}
\label{isothermal:eq:workingFLDApprox}
\end{align}}
{\begin{align}
M_1 \sim \hbox{\huge e}^{{1\over 2}\left(\fld +1\right)}
\label{isothermal:eq:workingFLDAppSol}
\end{align}}

\section{ Fanno Flow}
\label{chap:fanno}

{\begin{align}
{ 4 f dx \over D} = {{\left( 1 - M^2 \right) dM^2}
\over
{kM^4 ( 1 + {k-1 \over 2}M^2} ) }
\label{fanno:eq:fld-M}
\end{align}}

{\begin{align}
{4 \over D} \int^{L_{max}}_{L} f dx =
{1 \over k} {1 - M^2 \over M^2} +
{k+1 \over 2k}\ln {{k+1 \over 2}M^2 \over 1+ {k-1 \over
2}M^2}
\label{fanno:eq:fld-dM1}
\end{align}}

A representative friction factor is defined as
{\begin{align}
\bar{f} = { 1 \over L_{max}} \int ^{L_{max}} _{0} {f dx}
\label{fanno;eq:fDef}
\end{align}}

{\begin{align}
{4 \bar{f}L_{max}\over D} =
{1 \over k} {1 - M^2 \over M^2} +
{k+1 \over 2k}\ln {{k+1 \over 2}M^2 \over 1+ {k-1 \over 2}M^2}
\label{fanno:eq:solution}
\end{align}}
{\begin{align}
{P \over P^{*}} = { 1 \over M}
\sqrt{{k+1 \over 2}
\over
{ 1 + {k - 1 \over 2} M^{2}} }
\label{fanno:eq:Pratio}
\end{align}}

{\begin{align}
{ T \over T^{*}} = {c^{2} \over {c^{*}}^{2} } =
{{ k + 1 \over 2} \over
{ 1 + {k - 1 \over 2} M^{2}} }
\label{fanno:eq:Tbar}
\end{align}}

{\begin{align}
{\rho \over \rho^{*}} =
{ 1 \over M}
\sqrt{ { 1 + {k - 1 \over 2} M^{2}}
\over
{k+1 \over 2} }
\label{fanno:eq:Rhoratio}
\end{align}}

{\begin{align}
{ U \over U ^{*}} = \left( {\rho \over \rho^{*}}
\right)^{-1} =
M \sqrt{{k+1 \over 2}
\over
{ 1 + {k - 1 \over 2} M^{2}} }
\label{fanno:eq:Uratio}
\end{align}}

{\begin{align}
{P_{0} \over {P_{0}}^{*}} =
{ 1 \over M}
\left({ { 1 + {k - 1 \over 2} M^{2}}
\over
{k+1 \over 2} } \right)^{k +1 \over 2(k -1)}
\label{fanno:eq:stagnationPressreRatio1}
\end{align}}
{\begin{align}
{s - s^{*} \over c_p} =
\ln M^{2} \sqrt{\left({{k+1}\over 2 M^{2} \left( 1 + {k -1
\over 2 }M^{2}
\right) }\right)^{ k +1 \over k} }
\label{fanno:eq:entropySolution}
\end{align}}
{\begin{align}
{T_2 \over T_1} ={ \left. T \over T^{*} \right|_{M_2}
\over \left. T \over T^{*} \right|_{M_1} }
\label{fanno:eq:TratioExample}
\end{align}}

{\begin{align}
\left( {4f L_{max} \over D} \right)_{2} =
\left( {4{f} L_{max} \over D} \right)_{1} -
{4{f} L \over D}
\label{fanno:eq:fld2}
\end{align}}

\section{RAYLEIGH FLOW}
\label{chap:rayleigh}

{\begin{align}
{P^{*} \over P_1} = {1 + k{M_1}^{2} \over 1 + k}
\label{ray:eq:Pratioa}
\end{align}}
{\begin{align}
{T^{*} \over T_1} = {1 \over M^2}
\left( {1 + k{M_1}^{2} \over 1 + k} \right)^{2}
\label{ray:eq:Tratioa}
\end{align}}

{\begin{align}
{\rho_1 \over \rho^{*}} = {U^{*} \over U_1} =
{ {U^{*} \over \sqrt{kRT^{*}} } \sqrt{kRT^{*}} \over
{U_1 \over \sqrt{kRT_1} } \sqrt{kRT_1} }
= {1 \over M_1} \sqrt{ T^{*} \over T_1}
\label{ray:eq:rhoRa}
\end{align}}

{\begin{align}
{{T_0}_1 \over {T_0}^{*}} =
{T_1 \left( 1 + {k -1 \over 2} {M_1}^{2} \right)
\over T^{*} \left( {1 + k } \over 2 \right)} =
{ 2 ( 1 + k ) {M_1}^{2} \over (1 + kM^{2})^2}
\left( 1 + {k -1 \over 2} {M_1} ^2 \right)
\label{ray:eq:T0ratio}
\end{align}}

{\begin{align}
{{P_0}_1 \over {P_0}^{*}} =
{P_1 \left( 1 + {k -1 \over 2} {M_1}^{2} \right)
\over P^{*} \left( {1 + k } \over 2 \right)} =
{\left({ 1 + k \over 1 + k{M_1}^2}\right)}
\left( { 1 + k{M_1}^2 \over {(1 + k) \over 2}} \right)^{k
\over k -1}
\label{ray:eq:P0ratio}
\end{align}}

\section{Oblique-Shock}
\label{chap:oblique}

{\begin{align}
\tan \theta = {{U_1}_n \over {U_1}_t}
\label{oblique:eq:theta}
\end{align}}
{\begin{align}
\tan ( \theta - \delta ) = {{U_2}_n \over {U_2}_t}
\label{oblique:eq:thetaAlpha}
\end{align}}

{\begin{align}
\sin \theta = {{M_1}_n \over {M_1}}
\label{oblique:eq:M1n}
\end{align}}
{\begin{align}
\sin (\theta - \delta ) = {{M_2}_n \over {M_2}}
\label{oblique:eq:M2n}
\end{align}}
{\begin{align}
\cos \theta = {{M_1}_t \over {M_1}}
\label{oblique:eq:M1t}
\end{align}}

{\begin{align}
\cos (\theta - \delta ) = {{M_2}_t \over {M_2}}
\label{oblique:eq:M2t}
\end{align}}

{\begin{align}
\tan \delta = 2 \cot \theta
\left[{M_1}^{2} \sin^2 \theta - 1 \over
{M_1}^{2} \left(k + \cos 2 \theta \right) +2 \right]
\label{oblique:eq:sol}
\end{align}}

{\begin{align}
{\rho_2 \over \rho_1} = {{U_1}_n \over {U_2}_n} =
{ (k+1) {M_1}^{2} \sin^2\theta \over
(k-1) {M_1}^2 \sin^2\theta + 2}
\label{oblique:eq:rhoBar}
\end{align}}
{\begin{align}
{T_2 \over T_1} = {2k {M_1}^2 \sin^2\theta - (k-1)
\left[(k-1) {M_1}^2 + 2 \right]
\over (k+1)^2 {M_1}}
\label{oblique:eq:Tbar}
\end{align}}

The Rankine--Hugoniot relations are the same as the
relationship for the normal shock
{\begin{align}
{P_2 - P_1 \over \rho_2 - \rho_1} = k { P_2 - P_1 \over
\rho_2 - \rho_1}
\label{oblique:eq:RankineHugoniot}
\end{align}}

{\begin{align}
x^3 + a_1 x^2 + a_2 x + a_3=0
\label{oblique:eq:cubic}
\end{align}}
where
{\begin{align}
x = \sin^2 \theta
\label{oblique:eq:x}
\end{align}}
and
{\begin{align}
a_1 & = - {{M_1}^2 + 2 \over {M_1}^2} - k \sin ^2 \delta
\label{oblique:eq:a1} \\
a_2 & = - { 2{M_1}^2 + 1 \over {M_1}^4 } +
\left[ {(k+1)^2 \over 4}+ {k -1 \over {M_1}^2} \right]
\sin ^2 \delta
\label{oblique:eq:a2} \\
a_3 & = - {\cos ^2 \delta \over {M_1}^4}
\label{oblique:eq:a3}
\end{align}}

{\begin{align}
x_1 = - {1 \over 3} a_1 + (S +T )
\label{oblique:eq:x1}
\end{align}}
{\begin{align}
x_2 = - {1 \over 3} a_1 - \half (S +T ) + \half i \sqrt{3} (
S-T)
\label{oblique:eq:x2}
\end{align}}
and
{\begin{align}
x_3 = - {1 \over 3} a_1 - \half (S +T ) - \half i \sqrt{3} (
S-T )
\label{oblique:eq:x3}
\end{align}}
Where
{\begin{align}
S = \sqrt[3]{R + \sqrt{D}},
\label{oblique:eq:S}
\end{align}}
{\begin{align}
T = \sqrt[3]{R - \sqrt{D}}
\label{oblique:eq:T}
\end{align}}

and where the definition of the $D$ is
{\begin{align}
D = Q^3 + R^2
\label{oblique:eq:D}
\end{align}}
and where the definitions of $Q$ and $R$ are
{\begin{align}
Q = { 3 a_2 - {a_1 } ^2 \over 9}
\label{oblique:eq:Q}
\end{align}}
and
{\begin{align}
R = { 9 a_1 a_2 - 27 a_3 - 2 {a_1}^3 \over 54}
\label{oblique:eq:R}
\end{align}}

{\begin{align}
\sin ^2 \theta_{max} =
{ -1 + { k + 1 \over 4}{M_1}^2+ \sqrt{(k+1)
\left[ 1 + {k-1 \over 2} {M_1}^2 +
\left( {k+1 \over 2} {M_1} \right)^4 \right]}
\over k {M_1}^2}
\label{oblique:eq:thetaMax}
\end{align}}

A simplified case of the Maximum Deflection Mach Number's
equation for
large Mach number becomes
{\begin{align}
{M_{1n}} = \sqrt{ k+1\over 2k } M_{1} \quad \hbox{for}
\quad M_{1} >> 1
\label{oblique:eq:menikoffLarge}
\end{align}}

{\begin{align}
M_{1n} = {\sqrt{ (k+1) {M_1}^2
+1 +
\sqrt{({M_1}^2\left[{M_1}^2 (k + 1)^2 +8(k^2 -
1)\right]+16(1+k)}
}
\over 2 \sqrt{k} }
\label{oblique:eq:minikoffSol}
\end{align}}

{\begin{align}
{P_ 2 \over P_1} = { 2 k {M_1}^2 \sin ^2 \theta - (k -1)
\over k+1}
\label{oblique:eq:PR}
\end{align}}
The density ratio can be expressed as
{\begin{align}
{\rho_2 \over \rho_1 } = { {U_1}_n \over {U_2}_n}
= { (k +1) {M_1}^2 \sin ^2 \theta \over (k -1) {M_1}^2 \sin
^2
\theta + 2}
\label{oblique:eq:RR}
\end{align}}
{\begin{align}
{ T_2 \over T_1} = { {c_2}^2 \over {c_1}^2} =
{ \left( 2k {M_1}^2 \sin ^2 \theta - ( k-1) \right)
\left( (k-1) {M_1}^2 \sin ^2 \theta + 2 \right) \over
(k+1)
{M_1}^2 \sin ^2 \theta }
\label{oblique:eq:TR}
\end{align}}

{\begin{align}
{M_2}^2 = {(k+1)^2 {M_1}^4 \sin ^2 \theta -
4({M_1}^2 \sin ^2 \theta -1) (k {M_1}^2 \sin ^2 \theta +1)
\over
\left( 2k {M_1}^2 \sin ^2 \theta - (k-1) \right)
\left( (k-1) {M_1}^2 \sin ^2 \theta +2 \right)
}
\label{oblique:eq:M2}
\end{align}}
The ratio of the total pressure can be expressed as
{\begin{align}
{P_{0_2} \over P_{0_1}} = \left[
(k+1) {M_1}^2 \sin ^2 \theta \over
(k-1) {M_1}^2 \sin ^2 \theta +2 \right]^{k \over k -1}
\left[ k+1 \over 2 k {M_1}^2 \sin ^2 \theta - (k-1)
\right]
^{1 \over k-1}
\label{oblique:eq:P0R}
\end{align}}

\subsection{Given Two Angles, $\delta$ and $\theta$ }

{\begin{align}
{M_1}^2 = { 2 ( \cot \theta + \tan \delta )
\over \sin 2 \theta - (\tan \delta) ( k + \cos 2 \theta)
}
\label{oblique:eq:M1}
\end{align}}
{\begin{align}
{2(P_2 - P_1) \over \rho U^2} =
{2 \sin\theta \sin \delta \over \cos(\theta - \delta)}
\label{oblique:eq:reducedPressure}
\end{align}}

{\begin{align}
{\rho_ 2 -\rho_1 \over \rho_2} =
{\sin \delta \over \sin \theta \cos (\theta -\delta)}
\label{oblique:eq:reducedDensity}
\end{align}}

\section{Prandtl-Meyer Function}
\label{chap:Prandtl-Meyer}

{\begin{align}
\nu (M) & = \theta(M) - \theta(M_{starting}) \\ %\nonumber
&= \sqrt{k+1\over k-1}
\tan^{-1} \left( \sqrt{k-1\over k+1} \sqrt{ M^2
-1}\right)
- \tan^{-1} \sqrt{ M^2 -1}
\label{pm:eq:nuTheta}
\end{align}}

{\begin{align}
\nu_{\infty} = {\pi \over 2} \left[
\sqrt{k+1 \over k -1} - 1 \right]
\label{pm:eq:MaxTurning}
\end{align}}

\end{document}

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