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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 $ D$ is

$\displaystyle D = Q^3 + R^2$ (98)

and where the definitions of $ Q$ and $ R$ 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)



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