Transient growth analysis of oblique shock-wave/boundary-layer interactions at Mach 5.92

We study physical mechanisms that trigger transient growth in a high-speed spatially developing laminar boundary layer that interacts with an oblique shock wave. We utilize an approach based on power iteration, with the global forward and adjoint linearized equations, to quantify the transient growth in compressible boundary layers with flow separation. For a Mach 5.92 boundary layer with no oblique shock wave, we show that the dominant transient response consists of oblique waves, which arise from the inviscid Orr mechanism, the lift-up effect, and the first-mode instability. We also demonstrate that the presence of the oblique shock wave significantly increases transient growth over short time intervals through a mechanism that is not related to a slowly growing global instability. The resulting response takes the form of spanwise periodic streamwise elongated streaks, and our analysis of the linearized inviscid transport equations shows that base-flow deceleration near the reattachment location contributes to their amplification. The large transient growth of streamwise streaks demonstrates the importance of nonmodal effects in the amplification of flow perturbations and identifies a route for the emergence of similar spatial structures in transitional hypersonic flows with shock-wave/boundary-layer interactions.

Figure 1

Schematic of an oblique shock wave (red) impinging on a Mach 5.92 boundary layer, adapted from Ref. [11]. The adverse pressure gradient associated with the incident shock causes the boundary layer to separate from the wall, forming a recirculation bubble (blue). Here ${\theta }_1$ is the initial angle of the incident shock, while ${\theta }_2$ denotes the shock angle after the incident shock interacts with the bow shock.

Figure 2

Base flow contours of nondimensional streamwise velocity and density for a Mach 5.92 spatially developing boundary layer with ${\text{Re}}_{{\delta }_{\text{in}}^{*}}=9660$, adapted from Ref. [37].

Figure 3

Contour plots of the transient growth $G(\beta ,t)$ of a Mach 5.92 spatially developing boundary layer for streamwise lengths of the domain equal to (a) 59, (b) 118, (c) 176, and (d) 235. The streamwise lengths correspond to the factors 1/4, 1/2, 3/4, and 1 of the original length. These plots illustrate convective instability of the spatially developing boundary layer.

Figure 4

Maximum transient energy growth versus nondimensional spanwise wave number of a Mach 5.92 spatially developing boundary layer with five different streamwise extents for $L_x=235$.

Figure 5

Boundary-layer thickness (solid blue) and the inverse spanwise wave number (solid/dots red) corresponding to the maximum transient growth versus the streamwise length of the domain from the leading edge of the flat plate $L_{le}$. This corresponds to a Mach 5.92 spatially developing boundary layer with data from Figs. 3 and 4.

Figure 6

The real part of the normalized streamwise velocity component of the optimal (a) initial and (b) final states of a Mach 5.92 spatially developing boundary layer with $\beta =0.6$. These states correspond to the peak in Fig. 4 for a streamwise extent of $L_x=235$. The white lines indicate streamwise locations where the wall-normal profiles in Fig. 7 are extracted.

Figure 7

Normalized wall-normal profiles of the optimal (a) initial and (b) final states of a Mach 5.92 spatially developing boundary layer with $\beta =0.6$. The absolute value of the streamwise (solid blue), wall-normal (dashed red), and spanwise (dash-dot green) velocity components are plotted for each state along with the entropy (dots magenta). The initial and final states are extracted at $x=35$ and $x=185$, respectively.

Figure 8

Optimal final state in 3D of a Mach 5.92 spatially developing boundary layer with $\beta =0.38$. The streamwise domain length is set to $2L_x=470$. Isosurface contours represent the normalized streamwise velocity perturbation, where the red and blue contours indicate positive and negative velocities, respectively.

Figure 9

Contour plots of nondimensional streamwise velocity and density for a Mach 5.92 SWBLI with an incident shock angle of ${\theta }_1={13}^{\circ }$ at ${\text{Re}}_{{\delta }_{\text{in}}^{*}}=9660$, adapted from Ref. [37]. Here S and R denote the separation and reattachment points, respectively. The white contour lines indicate streamlines inside the recirculation bubble.

Figure 10

Maximum transient energy growth versus nondimensional spanwise wave number of a Mach 5.92 SWBLI with an incident shock angle of ${\theta }_1={13}^{\circ }$ and a streamwise extent of $235{\delta }_{\mathrm{in}}^{*}$.

Figure 11

Transient growth envelope (solid black line) of a Mach 5.92 SWBLI with ${\theta }_1={13}^{\circ }$ and $\beta =2.6$. We show growth curves (dashed blue line, dashed red line, and dashed green line) that originate from three different initial conditions corresponding to the fixed time intervals $t=150$, $t=214$, and $t=260$. These transient growth curves touch the envelope at exactly one point (solid circle).

Figure 12

Evolution of the Mach 5.92 SWBLI with ${\theta }_1={13}^{\circ }$ and $\beta =2.6$ from the (a) optimal initial condition to its (d) final state at $t=214$. The two intermediary states occur at (b) $t=150$ and (c) $t=182$. Contour levels, representing the real part of the normalized spanwise velocity perturbation, are identical in each frame. The black dashes indicate the start of three different sponge layers.

Figure 13

Contributions of the streamwise, wall-normal, and spanwise velocity perturbations as well as the entropy and pressure perturbations to the optimal transient growth of a Mach 5.92 SWBLI at ${\theta }_1={13}^{\circ }$ and $\beta =2.6$. Each curve that is associated with a perturbation variable has been normalized by the maximum growth at $t=214$.

Figure 14

Optimal initial and final states of a Mach 5.92 SWBLI with ${\theta }_1={13}^{\circ }$ and $\beta =2.6$. Isosurface contours, which represent the normalized streamwise velocity perturbation, are identical in each frame. The red and blue contours indicate positive and negative velocities, respectively, while the green contours indicate the size and position of the recirculation bubble.

Figure 15

Wave-packet responses of the (a) spatially developing boundary layer at $\beta =0.6$ and (b) oblique SWBLI at $\beta =2.6$. The dashed lines indicate the component of the wave packet that resides within a sponge layer. For the oblique SWBLI at ${\theta }_1={13}^{\circ }$ in (b), the red contours represent times when the main peak of the wave packet has crossed the apex of the recirculation bubble.

Figure 16

Comparison of the temporal change in the perturbation energy ${\partial }_{t}E$ (blue solid line) versus the production term $\mathcal{P}$ (red dashed line) integrated in space and plotted versus time for a Mach 5.92 SWBLI with ${\theta }_1={13}^{\circ }$ and $\beta =2.6$. The inset shows the spatial region where $\mathcal{P}>0$ at $t=190$.

Figure 17

(a) Spatiotemporal evolution of production terms contributing to the growth of streamwise perturbation energy that arises from lift-up and streamwise deceleration. The inset shows the 3D isosurfaces of the streamwise velocity along with separation streamline (in green). (b) Streamwise evolution of the time-averaged contribution. The points S and R mark the location of separation and reattachment in the 2D steady flow.