Simulation and stability analysis of oblique shock-wave/boundary-layer interactions at Mach 5.92

We investigate flow instability created by an oblique shock wave impinging on a Mach 5.92 laminar boundary layer at a transitional Reynolds number. The adverse pressure gradient of the oblique shock causes the boundary layer to separate from the wall, resulting in the formation of a recirculation bubble. For sufficiently large oblique shock angles, the recirculation bubble is unstable to three-dimensional perturbations and the flow bifurcates from its original laminar state. We utilize direct numerical simulation (DNS) and global stability analysis to show that this first occurs at a critical shock angle of $\theta =12.9^{\circ }$. At bifurcation, the least-stable global mode is nonoscillatory and it takes place at a spanwise wave number $\beta =0.25$, in good agreement with DNS results. Examination of the critical global mode reveals that it originates from an interaction between small spanwise corrugations at the base of the incident shock, streamwise vortices inside the recirculation bubble, and spanwise modulation of the bubble strength. The global mode drives the formation of long streamwise streaks downstream of the bubble. While the streaks may be amplified by either the lift-up effect or by Görtler instability, we show that centrifugal instability plays no role in the upstream self-sustaining mechanism of the global mode. We employ an adjoint solver to corroborate our physical interpretation by showing that the critical global mode is most sensitive to base flow modifications that are entirely contained inside the recirculation bubble.

Figure 1

Schematic of an oblique shock wave (red) impinging on a Mach 5.92 boundary layer. The adverse pressure gradient associated with the impinging shock causes the boundary layer to separate from the wall, forming a recirculation bubble (blue).

Figure 2

Contours of nondimensional pressure and density for a 2D oblique shock-wave/laminar-boundary-layer interaction with an incident shock angle of $\theta ={13}^{\circ }$.

Figure 3

Contours of streamwise velocity for a 2D oblique shock-wave/laminar-boundary-layer interaction with incident shock angles ranging from $\theta ={12}^{\circ }$ (top) to $\theta =13.4^{\circ }$ (bottom). For $\theta ={13}^{\circ }$, white contours indicate streamlines inside the recirculation bubble. The separation and reattachment locations are indicated by S and R, respectively.

Figure 4

Color contours of spanwise velocity shown on a wall-parallel plane close to the wall and three wall-normal planes for a 3D oblique shock-wave/boundary-layer interaction with $\theta ={13}^{\circ }$. Grayscale contours of the density gradient magnitude are shown on a streamwise slice on the left.

Figure 5

Streamwise variation of the spanwise PSD from the DNS with an incident shock angle of $\theta ={13}^{\circ }$. Color contours on a logarithmic scale represent the PSD of the streamwise velocity perturbation as a function of streamwise position and spanwise wave number. The horizontal dashed line indicates the dominant spanwise wave number $\beta =0.25$. The recirculation bubble, bounded by the separation and reattachment locations (vertical dashed lines), contains most of the perturbation energy.

Figure 6

Color contours of spanwise velocity for a SWBLI at $\theta =13.6^{\circ }$. Contours are shown on the same planes as in Fig. 4, including grayscale contours of density gradient magnitude on the left.

Figure 7

Eigenvalue spectrum of a SWBLI at $M_{\infty }=5.92$ from GSA with $\theta =12.6^{\circ }$ and $\beta =0.25$.

Figure 8

Stable global modes labeled in Fig. 7 colored by the real part of the nondimensional streamwise velocity perturbation. Here the dashed lines indicate the start of three different sponge layers.

Figure 9

Eigenvalue spectrum of a SWBLI at $M_{\infty }=5.92$ from GSA with $\theta ={13}^{\circ }$ and $\beta =0.25$. These conditions produce one eigenvalue in the unstable upper half plane. The unstable eigenvalue has zero frequency, indicating stationary instability.

Figure 10

Isosurface contours of streamwise velocity perturbations induced by the least-stable global mode for $\theta ={13}^{\circ }$ and $\beta =0.25$. Red and blue contours indicate positive and negative velocities, respectively. This mode corresponds to the eigenvalue in the unstable half plane in Fig. 9.

Figure 11

Maximum growth rate versus spanwise wave number for an oblique shock-wave/laminar-boundary-layer interaction at four different incident shock angles.

Figure 12

Temporal variation of the $L_2$-norm of an unstable stationary mode predicted by global stability analysis compared to the growth rate from the DNS after adding this global mode. This corresponds to an incident shock angle of $\theta ={13}^{\circ }$ and a spanwise wave number of $\beta =0.25$.

Figure 13

Rear view of the SWBLI recirculation bubble perturbed by the critical global mode. In both images, the gray separation stream surface bounds the recirculating flow contained in the bubble. (a) Magenta and green isosurfaces of positive and negative streamwise vorticity show a spanwise array of counterrotating streamwise vortices that form just under the apex of the recirculation bubble. (b) In response to the streamwise vortices, the recirculation bubble expands and contracts periodically in the span. The yellow surface represents another stream surface, taken closer to the core of the recirculation bubble. See [38] for a movie showing these figures from different angles.

Figure 14

(a) Side view of the recirculation bubble (gray) perturbed by the same global mode shown in Fig. 13. The front edge cuts through the spanwise location of maximum bubble strength (the spanwise position corresponding to the maximum streamwise extent of the bubble core as indicated by the yellow isosurface). At this spanwise location, the bubble pushes the impinging shock foot forward, producing a patch of positive pressure perturbation bounded by a red isosurface. Above the bubble, a shift in the expansion fan produces a negative pressure perturbation bounded by a blue isosurface. (b) Pressure perturbations produced by the impinging shock foot have the correct phase to drive the streamwise vortices (magenta and green isosurfaces) shown in Fig. 13. See [38] for a movie showing these figures from different angles.

Figure 15

Toroidal streamline patterns inside the recirculation bubble created by three-dimensional SWBLI instability. Corrugation of the impinging shock creates a region of high pressure just upstream of the point $F\left(u\right)$. This pushes streamlines at the periphery of the recirculation bubble away from this point and towards the point $F\left(s\right)$. As they recirculate, these streamlines (red) trace the exterior of a horizontally aligned torus. At $F\left(s\right)$, they are attracted to the center of the recirculation bubble and then quickly pass through the center of the torus to reemerge close to $F\left(u\right)$.

Figure 16

Topology of three-dimensional SWBLI instability. No-slip critical points form along the separation and reattachment lines. Likewise, three-dimensional instability transforms the center associated with the core of two-dimensional recirculation bubble to a series of stable and unstable spiraling foci. While this pattern is reminiscent of three-dimensional instability in low-speed recirculation bubbles, the stable and unstable nodes along the separation and reattachment lines do not align for SWBLI instability. This creates the toroidal flow patterns shown in Fig. 15.

Figure 17

Color contours of the Rayleigh discriminant $\mathrm{\Delta }$. Blue contours ($\mathrm{\Delta }<0$) indicate regions that can support centrifugal instability, whereas yellow contours ($\mathrm{\Delta }>0$) correspond to zones of stability with respect to centrifugal perturbations. The gray line represents the separation streamline. The black lines correspond to contours of streamwise vorticity from the critical global mode, showing the location of the streamwise vortices that support SWBLI instability.

Figure 18

Adjoint mode corresponding to the critical direct global mode shown in Fig. 10. Red and blue isosurface contours represent, respectively, positive and negative adjoint streamwise velocity perturbations.

Figure 19

Color contours of $\parallel {\widehat{\mathit{q}}}_j{\parallel }_E{\parallel {\widehat{\mathit{q}}}_j^{+}\parallel }_E$ quantifying the overlap between the direct and adjoint critical global mode. Red corresponds to regions where SWBLI instability is most sensitive to base flow modification, or the wave maker. The white line indicates the separation streamline, whereas the black lines represent contours of streamwise vorticity created by the critical global mode.

Figure 20

Eigenspectra that correspond to high (blue dots), medium (red circles), low (green triangles), and very low (magenta diamonds) resolution grids of the SWBLI for $\theta ={13}^{\circ }$ and $\beta =0.25$. A closeup of the unstable eigenvalue is displayed on the right.