Criterion for vortex breakdown on shock wave and streamwise vortex interactions

The interactions between supersonic streamwise vortices and oblique shock waves are theoretically and numerically investigated by three-dimensional (3D) Navier-Stokes equations. Based on the two inequalities, a criterion for shock-induced breakdown of the streamwise vortex is proposed. The simple breakdown condition depends on the Mach number, the swirl number, the velocity deficit, and the shock angle. According to the proposed criterion, the breakdown region expands as the Mach number increases. In numerical simulations, vortex breakdown appeared under conditions of multiple pressure increases and the helicity disappeared behind the oblique shock wave along the line of the vortex center. The numerical results are consistent with the predicted breakdown condition at Mach numbers 2.0 and 3.0. This study also found that the axial velocity deficit is important for classifying the breakdown configuration.

Figure 1

Schematic of an oblique shock wave interacting with a vortex at shock angle  $\beta$.

Figure 2

Shadow graphs of the flow during the interaction of a strong vortex with an oblique shock at $M_{\infty }=2.49$, by Smart et al. [10], reproduced with permission. Copyright 1998 Springer.

Figure 3

Conceptual diagram of an axial velocity profile and the velocity deficit.

Figure 4

An upstream streamwise vortex and the velocity components during interaction with an oblique shock wave.

Figure 5

Theoretical critical curves of the interaction between streamwise vortices and oblique shock waves for various shock angles $\beta$;  the interiors of curves (including the origin) indicate stable or no effect regions; exterior regions indicate unstable break regions: (a) $M_{\infty }=2.0$, (b) $M_{\infty }=3.0$.

Figure 6

Comparison of theoretical critical curves and experimental results [20] of vortex breakdown phenomena (swirl ratio versus upstream Mach number) during vortex interaction with a normal shock wave.

Figure 7

Comparison of theoretical critical curves and numerical results of vortex breakdown phenomena. The normal shock wave intercepts at $\beta ={90}^{\circ }$ at $M_{\infty }=2.0$: stable or no effect ($◯$); vortex breakage ($\times$); Mahesh's theory [21] (dotted line); modified vortex model (dashed line); present theory (solid line).

Figure 8

Contours of (top) helicity density, and (bottom) cross-sectional density contour lines at $Z=20$. Vortices are intercepted by a normal shock wave at $\beta ={90}^{\circ }$, namely NSVI: (a) $q=0.0$, $\mu =0.2$, (b) $q=0.4$, $\mu =0.0$, (c) $q=0.6$, $\mu =0.05$, at $M_{\infty }=2.0$.

Figure 9

Cross-sectional density contour lines at $Z=20$; (top) $M_{\infty }=2.0$, (bottom) $M_{\infty }=3.0$.

Figure 10

Isosurface of the second invariant of the velocity gradient tensor. Side view of the interaction between a streamwise vortex and an oblique shock wave intercepting at shock angle $\beta ={65}^{\circ }$, at $M_{\infty }=2.0$; (a) $q=0.2$, $\mu =0.2$, (b) $q=0.6$, $\mu =0.2$.

Figure 11

Velocity vectors on the cross section at $Z=20$, $M_{\infty }=2.0$: (a) $q=0.4$, $\mu =0.4$, and (b) $q=0.6$, $\mu =0.2$.

Figure 12

Cross-sectional contours of the absolute values of each term of the vorticity equation at $Z=20$: (a) stretching term, (b) dilatation term, and (c) baroclinic torque term. Parameters: $q=0.6$, $\mu =0.2$, $M_{\infty }=2.0$.

Figure 13

Temporal evolution of Mach number contours for $q=0.4$, $\mu =0.4$ at $M_{\infty }=2.0$ (black areas show subsonic regions).

Figure 14

Temporal evolution of Mach number contours for $q=0.6$, $\mu =0.2$ at $M_{\infty }=2.0$ (black areas show subsonic regions).

Figure 15

(Top) normalized axial vorticity profiles and (bottom) axial velocity profiles (normalized by local sonic speed), in the vicinity of an interaction between a streamwise vortex $q=0.4$, $\mu =0.4$ and an oblique shock wave intercepting at $\beta ={65}^{\circ }$, $M_{\infty }=2.0$.

Figure 16

(Top) normalized axial vorticity profiles and (bottom) axial velocity profiles (normalized by the local sonic speed), in the vicinity of an interaction between a streamwise vortex $q=0.6$, $\mu =0.2$ and an oblique shock wave intercepting at $\beta ={65}^{\circ }$, $M_{\infty }=2.0$.

Figure 17

Temporal evolution of the helicity density contours. Parameters: $q=0.2$, $\mu =0.2$, $M_{\infty }=2.0$.

Figure 18

Temporal evolution of the helicity density contours. Parameters: $q=0.6$, $\mu =0.2$, $M_{\infty }=2.0$.

Figure 19

(Top) static pressures and (bottom) normalized helicity densities of streamwise variations along the lines of changing vortex center. Shock angle $\beta ={65}^{\circ }$ at $M_{\infty }=2.0$. The yellow dotted lines indicate where the vortex center crosses with the shock wave.

Figure 20

Comparison of theoretical critical curves and numerical results of vortex breakdown phenomena. Vortices are intercepted by a shock wave at $\beta ={65}^{\circ }$: (a) $M_{\infty }=2.0$ and (b) $M_{\infty }=3.0$. Stable or no effect ($◯$); vortex breakage ($\times$);  present theory (solid line).