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8 Oblique-Shock

$\displaystyle \tan \theta = {{U_1}_n \over {U_1}_t}$

(78)

$\displaystyle \tan ( \theta - \delta ) = {{U_2}_n \over {U_2}_t}$

(79)

$\displaystyle \sin \theta = {{M_1}_n \over {M_1}}$

(80)

$\displaystyle \sin (\theta - \delta ) = {{M_2}_n \over {M_2}}$

(81)

$\displaystyle \cos \theta = {{M_1}_t \over {M_1}}$

(82)

$\displaystyle \cos (\theta - \delta ) = {{M_2}_t \over {M_2}}$

(83)

$\displaystyle \tan \delta = 2 \cot \theta \left[{M_1}^{2} \sin^2 \theta - 1 \over {M_1}^{2} \left(k + \cos 2 \theta \right) +2 \right]$

(84)

$\displaystyle {\rho_2 \over \rho_1} = {{U_1}_n \over {U_2}_n} = { (k+1) {M_1}^{2} \sin^2\theta \over (k-1) {M_1}^2 \sin^2\theta + 2}$

(85)

$\displaystyle {T_2 \over T_1} = {2k {M_1}^2 \sin^2\theta - (k-1) \left[(k-1) {M_1}^2 + 2 \right] \over (k+1)^2 {M_1}}$

(86)

The Rankine-Hugoniot relations are the same as the relationship for the normal shock

$\displaystyle {P_2 - P_1 \over \rho_2 - \rho_1} = k { P_2 - P_1 \over \rho_2 - \rho_1}$

(87)

$\displaystyle x^3 + a_1 x^2 + a_2 x + a_3=0$

(88)

where

$\displaystyle x = \sin^2 \theta$

(89)

and

$\displaystyle a_1$	$\displaystyle = - {{M_1}^2 + 2 \over {M_1}^2} - k \sin ^2 \delta$	(90)
$\displaystyle a_2$	$\displaystyle = - { 2{M_1}^2 + 1 \over {M_1}^4 } + \left[ {(k+1)^2 \over 4}+ {k -1 \over {M_1}^2} \right] \sin ^2 \delta$	(91)
$\displaystyle a_3$	$\displaystyle = - {\cos ^2 \delta \over {M_1}^4}$	(92)

$\displaystyle x_1 = - {1 \over 3} a_1 + (S +T )$

(93)

$\displaystyle x_2 = - {1 \over 3} a_1 -$ $\displaystyle \mbox{$\frac{1}{2}$}$ $\displaystyle (S +T ) +$ $\displaystyle \mbox{$\frac{1}{2}$}$ $\displaystyle i \sqrt{3} ( S-T)$

(94)

and

$\displaystyle x_3 = - {1 \over 3} a_1 -$ $\displaystyle \mbox{$\frac{1}{2}$}$ $\displaystyle (S +T ) -$ $\displaystyle \mbox{$\frac{1}{2}$}$ $\displaystyle i \sqrt{3} ( S-T)$

(95)

Where

$\displaystyle S = \sqrt[3]{R + \sqrt{D}},$

(96)

$\displaystyle T = \sqrt[3]{R - \sqrt{D}}$

(97)

and where the definition of the is

$\displaystyle D = Q^3 + R^2$

(98)

and where the definitions of

and

are

$\displaystyle Q = { 3 a_2 - {a_1 } ^2 \over 9}$

(99)

and

$\displaystyle R = { 9 a_1 a_2 - 27 a_3 - 2 {a_1}^3 \over 54}$

(100)

$\displaystyle \sin ^2 \theta_{max} = { -1 + { k + 1 \over 4}{M_1}^2+ \sqrt{(k+1... ...ver 2} {M_1}^2 + \left( {k+1 \over 2} {M_1} \right)^4 \right]} \over k {M_1}^2}$

(101)

A simplified case of the Maximum Deflection Mach Number's equation for large Mach number becomes

$\displaystyle {M_{1n}} = \sqrt{ k+1\over 2k } M_{1} \quad \hbox{for} \quad M_{1} » 1$

(102)

$\displaystyle M_{1n} = {\sqrt{ (k+1) {M_1}^2 +1 + \sqrt{({M_1}^2\left[{M_1}^2 (k + 1)^2 +8(k^2 - 1)\right]+16(1+k)} } \over 2 \sqrt{k} }$

(103)

$\displaystyle {P_ 2 \over P_1} = { 2 k {M_1}^2 \sin ^2 \theta - (k -1) \over k+1}$

(104)

The density ratio can be expressed as

$\displaystyle {\rho_2 \over \rho_1 } = { {U_1}_n \over {U_2}_n} = { (k +1) {M_1}^2 \sin ^2 \theta \over (k -1) {M_1}^2 \sin ^2 \theta + 2}$

(105)

$\displaystyle { T_2 \over T_1} = { {c_2}^2 \over {c_1}^2} = { \left( 2k {M_1}^2... ...( (k-1) {M_1}^2 \sin ^2 \theta + 2 \right) \over (k+1) {M_1}^2 \sin ^2 \theta }$

(106)

$\displaystyle {M_2}^2 = {(k+1)^2 {M_1}^4 \sin ^2 \theta - 4({M_1}^2 \sin ^2 \th... ...\sin ^2 \theta - (k-1) \right) \left( (k-1) {M_1}^2 \sin ^2 \theta +2 \right) }$

(107)

The ratio of the total pressure can be expressed as

$\displaystyle {P_{0_2} \over P_{0_1}} = \left[ (k+1) {M_1}^2 \sin ^2 \theta \ov... ... -1} \left[ k+1 \over 2 k {M_1}^2 \sin ^2 \theta - (k-1) \right] ^{1 \over k-1}$

(108)

Subsections

8.1 Given Two Angles, $\delta$ and $\theta$

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