12.2.2.3: Upstream Mach Number, \(M_1\), and Shock Angle, \(\theta\)
- Page ID
- 847
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The solution for upstream Mach number, \(M_1\), and shock angle, θ, are far much simpler and a unique solution exists. The deflection angle can be expressed as a function of these variables as
\(\delta\) For \(\theta\) and \(M_1\)
\[
\label {2Dgd:eq:Odelta-theta}
\cot \delta = \tan \left(\theta\right) \left[
\dfrac{(k + 1)\, {M_1}^2 }{ 2\, ( {M_1}^2\, \sin^2 \theta - 1)} - 1
\right]
\]
or
\[
\tan \delta = {2\cot\theta ({M_1}^2 \sin^2 \theta -1 ) \over
2 + {M_1}^2 (k + 1 - 2 \sin^2 \theta )}
\label{2Dgd:eq:Odelta-thetaA}
\]
Pressure Ratio
\[
\label {2Dgd:eq:OPR}
\dfrac{P_ 2 }{ P_1} = \dfrac{ 2 \,k\, {M_1}^2 \sin ^2 \theta - (k -1) }{ k+1}
\]
The density ratio can be expressed as
Density Ratio
\[
\label {2Dgd:eq:ORR}
\dfrac{\rho_2 }{ \rho_1 } = \dfrac{ {U_1}_n }{ {U_2}_n}
= \dfrac{ (k +1)\, {M_1}^2\, \sin ^2 \theta }
{ (k -1) \, {M_1}^2\, \sin ^2 \theta + 2}
\]
The temperature ratio expressed as
Temperature Ratio
\[
\label {2Dgd:eq:OTR}
\dfrac{ T_2 }{ T_1} = \dfrac{ {c_2}^2 }{ {c_1}^2} =
\dfrac{ \left( 2\,k\, {M_1}^2 \sin ^2 \theta - ( k-1) \right)
\left( (k-1) {M_1}^2 \sin ^2 \theta + 2 \right) }
{ (k+1)\, {M_1}^2\, \sin ^2 \theta }
\]
The Mach number after the shock is
Exit Mach Number
\[
\label{2Dgd:eq:OM2_0}
{M_2}^2 \sin (\theta -\delta) =
{ (k -1) {M_1}^2 \sin ^2 \theta +2 \over
2 \,k\, {M_1}^2 \sin ^2 \theta - (k-1) }
\]
or explicitly
\[
{M_2}^2 = {(k+1)^2 {M_1}^4 \sin ^2 \theta -
4\,({M_1}^2 \sin ^2 \theta -1) (k {M_1}^2 \sin ^2 \theta +1)
\over
\left( 2\,k\, {M_1}^2 \sin ^2 \theta - (k-1) \right)
\left( (k-1)\, {M_1}^2 \sin ^2 \theta +2 \right)
}
\label{2Dgd:eq:OM2}
\]
Stagnation Pressure Ratio
\[
\label {2Dgd:eq:OP0R}
{P_{0_2} \over P_{0_1}} = \left[
(k+1) {M_1}^2 \sin ^2 \theta \over
(k-1) {M_1}^2 \sin ^2 \theta +2 \right]^{k \over k -1}
\left[ k+1 \over 2 k {M_1}^2 \sin ^2 \theta - (k-1) \right]
^{1 \over k-1}
\]
Even though the solution for these variables, \(M_1\) and \(\theta\), is unique, the possible range deflection angle, \(\delta\), is limited. Examining equation (51) shows that the shock angle, \(\theta\), has to be in the range of \(\sin^{-1} (1/M_1) \geq \theta \geq (\pi/2)\) (see Figure Fig. 12.8). The range of given \(\theta\), upstream Mach number \(M_1\), is limited between \(\infty\) and \(\sqrt{1 / \sin^{2}\theta}\).
Fig. 12.8 The possible range of solutions for different parameters for given upstream Mach numbers.
Contributors and Attributions
Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.