Onset of three-dimensionality in supersonic flow over a slender double wedge

We investigate the onset of three-dimensionality in a Mach 5 slender-double-wedge flow as the turn angle at the compression corner increases. Beyond a critical angle, the two-dimensional flow destabilizes to three-dimensional perturbations and results in growth of spanwise periodic flow structures. We carry out global linear stability analysis to identify the critical turn angle and the nature of the associated three-dimensional instability. At the critical angle, an unstable mode is present in the separation bubble and the reattached boundary layer. The mode is associated with streamwise streaks in wall temperature downstream of the corner. The existence of the unstable mode and its growth rate are confirmed with direct numerical simulations. It is found from budget analysis that streamwise deceleration of the recirculating flow plays a dominant role in the three-dimensional instability. Wave-maker analysis suggests that the instability does not have a centrifugal origin. Linear stability analysis of the steady-state flow at an angle beyond bifurcation is also carried out. The spanwise wavelength of the most unstable mode obtained at this turn angle compares well with experimental observations.

Figure 1

Illustration of the 2D stencil for linearized viscous flux computation.

Figure 2

Slender-double-wedge geometry and freestream conditions.

Figure 3

(a) Mach number field on the ${12}^{\circ }\text{−}{20}^{\circ }$ slender double wedge at the freestream conditions mentioned in Fig. 2 and (b) 2D streamlines visualizing the recirculation region marked with the separation (S), corner (C), and reattachment (R) points.

Figure 4

(a) Schematic showing the variation of the turn angle via rotation of the second ramp about the compression corner and (b) associated bifurcation of the 2D steady flow on the slender double wedge.

Figure 5

Growth rate vs angular frequency: shift in the eigenvalue spectrum at the least stable ${\lambda }_z$ with an increase in turn angle at bifurcation.

Figure 6

Unstable eigenmode at ${\lambda }_z/L=6.28$: 2D slices and 3D isosurfaces of different perturbation variables.

Figure 7

(a) Topology of the 3D bubble reconstructed with $u_i={\overline{u}}_i+\epsilon u_i^{\prime }$, $\epsilon =2$. The streamlines on the top are red ($u_i>0$) and those at the bottom are blue ($u_i<0$). (b) Simplified model of the 3D bubble structure with foci marked as squares (red denotes unstable and blue denotes stable).

Figure 8

The 3D bubble reconstructed with $u_i={\overline{u}}_i+\epsilon u_i^{\prime }$ with increasing mode amplitude $\epsilon =5$ (left column), $\epsilon =10$ (middle column), and $\epsilon =20$ (right column): (a)–(c) critical points on the wall streamlines (circles denote nodes, stars denote saddle points, and squares denote foci; red denotes unstable and blue denotes stable), (d)–(f) volume streamlines colored with $T^{\prime }$ (red is positive and blue is negative), and (g)–(i) the vortex tube passing through the stable foci.

Figure 9

Spanwise perturbation velocity $w^{\prime }$ in DNS run 3 at approximately $t{U}_{\infty }/L=6000$ (a) on the $z$ plane (GLSA mode overlaid for comparison) and (b) from the top view of near-wall planes.

Figure 10

Profiles of $w$ from run 1 at late times compared with $w^{\prime }$ of the most unstable eigenmode of the 2D steady-state flow for (a) $\xi /L=8$, (b) $\xi /L=13$, and (c) $\xi /L=18$.

Figure 11

(a) Comparison of the growth rate (for ${\lambda }_z/L=6.28$) of the unstable eigenmode between DNS and global linear stability analysis and (b) appearance of foci on near-wall streamlines in the DNS flow field (for run 5 with $t{U}_{\infty }/L\sim 2.3\times {10}^4$). Near-wall streamlines are in white and volume streamlines and the wall are colored by temperature.

Figure 12

(a) Wall-normal integrated rate of perturbation energy $E_{\mathrm{CE}}$ plotted along surface co-ordinate $\xi$ for the stable and the unstable turn angle and (b) budget of contribution from individual terms $F\left({\partial }_t{E}_{\text{CE}}\right)$ for $\theta =8^{\circ }$; Bezier fits are plotted. Asterisks indicate the corner ($\xi /L=12.2$) and base flow reattachment point ($\xi /L=23.6$); the separation point $\xi /L=0$ is outside the plot range.

Figure 13

Lines colored on a white-red scale mark the region where the contribution of the mean flow deceleration term to the perturbation energy is positive; mean flow streamlines are plotted in gray for comparison.

Figure 14

Dominant mechanism of instability in the recirculation region: (a) mechanism in the upstream half of the bubble ($\xi /L=8$) and (b) mechanism in the downstream half of the bubble ($\xi /L=17$); $\widehat{\xi }$ goes into the plane. The compression corner is at $\xi /L=12$. The $z$ axis is stretched by a factor of 1.5 for clarity of illustration.

Figure 15

(a) Direct mode corresponding to spanwise momentum perturbation and (b) the corresponding adjoint mode.

Figure 16

(a) Contours of receptivity to spatially localized feedback $\lambda$ in blue superimposed on the separation bubble streamlines (gray) and (b) regions with $\mathrm{\Delta }<0$ (red) plotted alongside the region with $\lambda >0.1{\lambda }_{\max }$ in blue.

Figure 17

(a) Growth rate of unstable perturbation modes vs spanwise wavelength ${\lambda }_z$ for $\theta ={10}^{\circ }$ and (b) eigenvalue spectrum corresponding to the most unstable spanwise wavelength ${\lambda }_z/L=2.3\mathrm{mm}$.

Figure 18

Spanwise component of the perturbation velocity $w^{\prime }$ for different unstable mode branches identified in Fig. 17 with blue denoting negative and red denoting positive values. Here (Re) and (Im) refer to real and imaginary components, respectively. The $w^{\prime }$ fields correspond to normalized eigenvectors of the nondimensional linear system.

Figure 19

(a) Streaks in wall temperature on the second ramp and (b) isosurface of temperature perturbation corresponding to the unstable eigenmode at ${\lambda }_z/L=2.2$ for $\theta ={10}^{\circ }$. The reattachment point corresponds to the 2D steady-state solution with adiabatic wall boundary conditions. The scales along the horizontal and vertical directions in (a) are the same [7].

Figure 20

Loci of the eigenvalues in the growth-rate–frequency–spanwise wavelength space. Projections on the ${\omega }_i\text{−}{\omega }_r$ and ${\lambda }_z\text{−}{\omega }_r$ planes are shown in gray.

Figure 21

Linearly unstable global mode for ${\text{Re}}_D=60$ and $M=0.1$ flow past a circular cylinder.

Figure 22

Least stable global mode with wave number $\beta =0.25$ obtained for oblique shock–boundary-layer interaction at $M=5.92$ [30].

Figure 23

Least stable global mode with wave number $\beta L_{\mathrm{sep}}=0.8$ obtained for oblique shock–boundary-layer interaction at $M=2.15$ [29].

Figure 24

(a) Schematic of the acoustic perturbation–normal shock interaction and (b) comparison of analytical transmission coefficient with that computed using third-order upwind discretization of linearized Euler equations in conserved variables.

Figure 25

Nondimensional velocity (red) and temperature profiles (blue) on three grids of increasing resolution (dotted line, 105 K; dashed line, 155 K; and solid line, 185 K); $\xi$ is the coordinate along the surface with $\xi =0$ at separation.

Figure 26

Grid convergence of the growth rates for the (a) $\theta =8^{\circ }$ and (b) $\theta ={10}^{\circ }$ cases.