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\documentclass[12pt]{article} Go to Potto Potto Home. \end{rawhtml} \begin{rawhtml} To download the pdf 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} About Potto Project
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2003 Dr. Genick Bar-Meir.
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