Flow behind concave shock waves

Abstract

Curved shock theory is introduced and applied to calculate the flow behind concave shock waves. For sonic conditions, three characterizing types of flow are identified, based on the orientation of the sonic line, and it is shown that, depending on the ratio of shock curvatures, a continuously curving shock can exist with Type III flow, where the sonic line intercepts the reflected characteristics from the shock, thus preventing the formation of a reflected shock. The necessary shock curvature ratio for a Type III sonic point does not exist for a hyperbolic shock so that it will revert to Mach reflection for all Mach numbers. A demonstration is provided, by CFD calculations, at Mach 1.2 and 3.

Appendix: Coefficients for the curved shock equations

The curved shock equations are, Mölder [3],

$$\begin{aligned} \begin{array}{l} A_1 P_1 +B_1 D_1 +E_1 \Gamma _1 =A_2 P_2 +B_2 D_2 +CS_a +GS_b \\ {A}'_1 P_1 +{B}'_1 D_1 +{E}'_1 \Gamma _1 ={A}'_2 P_2 +{B}'_2 D_2 +{C}'S_a +{G}'S_b \\ \end{array} \end{aligned}$$

where the coefficients A, B, E, C, G and their primed and subscripted variants (14 in all) are given by,

$$\begin{aligned} \begin{array}{lll} A_1 =2\cos \theta ((3M_1^2 -4)\sin ^{2}\theta -(\gamma -1)/2)/(\gamma +1) \\ B_1 =2\sin \theta ((\gamma -5)/2+(4-M_1^2 )\sin ^{2}\theta )/(\gamma +1) \\ E_1 = 2\sin ^{3}\theta ((\gamma -1)M_1^2 +2)/(\gamma +1) \\ A_2 = \sin \theta \cos \theta /\sin (\theta -\delta ) \\ B_2 = -\sin \theta \cos \theta /\cos (\theta -\delta ) \\ C = -4\sin \theta \cos \theta /(\gamma +1) \\ G = 4\sin ^{2}\theta \cos \theta \sin \delta _1 /(\gamma +1)/\cos \left( {\theta +\delta _1 } \right) \\ {A}'_1 = M_1^2 \cos \delta \cos ^{2}\theta -(M_1^2 -1)\cos (2\theta +\delta ) \\ {B}'_1 = -\sin (2\theta +\delta )-M_1^2 \sin \delta \sin ^{2}\theta \\ {E}'_1 = (2+(\gamma -1)M_1^2 )\sin \delta \sin ^{2}\theta \\ {A}'_2 = (1+(M_2^2 -2)\sin ^{2}(\theta -\delta ))\left( {\sin \theta \cos \theta } \right) /\\ \qquad \left( {\sin \left( {\theta -\delta } \right) \cos \left( {\theta -\delta } \right) } \right) \\ {B}'_2 = -\sin (2\theta ) \\ {C}' = -\sin (2\delta )/(2\cos (\theta -\delta )) \\ {G}' = (\sin ^{3}\theta \sin \delta _1 /\sin \left( {\theta -\delta } \right) -\sin \theta \cos \theta \sin \delta _2 )/\\ \qquad \cos (\theta +\delta _1 ) \\ \end{array} \end{aligned}$$

where

$$\begin{aligned} \begin{array}{l} M_2^2 =\frac{\left( {\gamma +1} \right) ^{2}M_1^4 \sin ^{2}\theta -4\left( {M_1^2 \sin ^{2}\theta -1} \right) \left( {\gamma M_1^2 \sin ^{2}\theta +1} \right) }{\left[ {2\gamma M_1^2 \sin ^{2}\theta -\left( {\gamma -1} \right) } \right] \left[ {\left( {\gamma -1} \right) M_1^2 \sin ^{2}\theta +2} \right] } \\ \hbox { and }\delta =\delta _2 -\delta _1 \\ \end{array} \end{aligned}$$

The matrices [XY] are,

$$\begin{aligned} \left[ {AB} \right] =A_2 {B}_2^{\prime } -{A}_2^{\prime } B_2 \\ {[BC]}=B_2 {C}_2^{\prime } -{B}_2^{\prime } C_2 \\ {[CA]}=C_2 {A}_2^{\prime } -{C}_2^{\prime } A_2 \end{aligned}$$

The vorticity equation is, Mölder [3],

$$\begin{aligned} A_1 ^{\prime \prime }P_1 +B_1 ^{\prime \prime }D_1 +E_1 ^{\prime \prime }\Gamma _1= & {} A_2 ^{\prime \prime }P_2 +B_2 ^{\prime \prime }D_2 +E_2 ^{\prime \prime }\Gamma _2\\&+{C}''S_a +{G}''S_b \end{aligned}$$

The vorticity equation coefficients are,

$$\begin{aligned} \begin{array}{l} {A}''_1 =\left( {M_1^2 -1} \right) \tan \theta \sin \theta +\cos \theta \\ {B}''_1 =-2\sin \theta \\ {E}''_1 =\sin \theta \\ {A}''_2 =\left( {M_2^2 -1} \right) \tan \left( {\theta -\delta } \right) \sin \left( {\theta -\delta } \right) +\cos \left( {\theta -\delta } \right) \\ {B}''_2 =-2\sin \left( {\theta -\delta } \right) \\ {E}''_2 =\sin \left( {\theta -\delta } \right) \\ {C}''_2 =\tan \left( {\theta -\delta } \right) -\tan \theta \\ {G}''=[\sin \theta \tan \theta \sin \delta _1 -\tan \left( {\theta -\delta } \right) \sin \left( {\theta -\delta } \right) \sin \delta _2 ]/\\ \qquad \qquad \cos \left( {\theta +\delta _1 } \right) \\ \end{array} \end{aligned}$$