Vorticity-production mechanisms in shock/mixing-layer interaction problems

Abstract

In this study, we investigate analytically the importance of different vorticity-production mechanisms contributing to the shock-induced vorticity caused by the interaction of a steady oblique shock wave with a steady, planar, supersonic, laminar mixing layer. The inviscid analysis is performed under the condition of a supersonic post-shock flow, which guarantees that the shock refraction remains regular. Special attention is paid to the vorticity production induced by a change in shock strength along the shock. Our analysis subdivides the total vorticity production into its contributions due to bulk or volumetric compression, pre-shock density gradients and variable shock strength. The latter is the only contribution dependent on the shock-wave curvature. The magnitudes of these contributions are analysed for two limiting cases, i.e., the interaction of an oblique shock wave with a constant-density shear layer and the interaction with a constant-velocity mixing layer with density gradients only. Possible implications for shock/mixing-layer interactions occurring in scramjet combustors are briefly discussed.

Appendix: Definition of the variables \(\mathcal {N}\) and \(\mathcal {D}\) appearing in (1)

Using ${\varLambda }:=\gamma -1$, ${\varOmega }:=\gamma +1$ and $A :=\frac{\sqrt{M_2^2-1}}{M_2^2}$ with the post-shock Mach number $M_2$, the numerator in (1) can be written as

$$\begin{aligned} \mathcal {N}:= & {} 4 M_1^5\, A \, \sin ^4 \theta [ 4 \gamma \sin ^2 \theta - {\varOmega }^2 ] \nonumber \\&- 8 M_1^3 \, \sin ^3 \theta [ \gamma {\varOmega }\cos \theta + 2 A {\varLambda }\sin \theta ] \nonumber \\&- 4 M_1 \sin \theta [ 4 A \sin \theta - (\gamma ^2-1) \cos \theta ], \end{aligned}$$

(24)

while the denominator reads

$$\begin{aligned} \mathcal {D} :=4 M_1^6 \sin ^3 \theta \, \mathcal {D}_6 + 2 M_1^4 \sin ^2 \theta \, \mathcal {D}_4 + 2 M_1^2 \, \mathcal {D}_2 + \mathcal {D}_0,\nonumber \\ \end{aligned}$$

(25)

with $\mathcal {D}_6$, $\mathcal {D}_4$, $\mathcal {D}_2$ and $\mathcal {D}_0$ defined as

$$\begin{aligned} \mathcal {D}_6:= & {} \gamma \sin \theta ({\varOmega }-2\gamma \sin ^2 \theta ) - A \cos \theta (4 \gamma \sin ^2 \theta - {\varOmega }^2), \end{aligned}$$

(26)

$$\begin{aligned} \mathcal {D}_4:= & {} 2 \gamma ({\varOmega }-4\sin ^2 \theta ) \nonumber \\&- {\varLambda }({\varOmega }-2\gamma \sin ^2 \theta ) + 8 A {\varLambda }\sin \theta \cos \theta , \end{aligned}$$

(27)

$$\begin{aligned} \mathcal {D}_2:= & {} 8 A \sin {\theta } \cos {\theta } + 4 \gamma \sin ^2 \theta -{\varLambda }({\varOmega }-4\sin ^2 \theta ), \end{aligned}$$

(28)

$$\begin{aligned} \mathcal {D}_0:= & {} - 4 {\varLambda }. \end{aligned}$$

(29)

The combination of (1) and (24)–(29) corrects the typographical errors contained in equation (9) of [6] (erroneous sign of the right-hand side of equation (9) in [6] and missing parentheses).