Flow structure and unsteadiness in a highly confined shock-wave–boundary-layer interaction

Numerical simulations were carried out to examine a ${24}^{\circ }$ compression ramp interaction in which the confinement effect of sidewalls was very strong. The ratio of turbulent boundary-layer thickness to domain width at the start of the interaction was $\delta /w=0.12$. Inflow conditions of $M_{\infty }=2.25$ and ${\mathrm{Re}}_{\theta i}=1800$ were chosen to match previous simulations. Comparing to the limited data available from similar computations, the centerline separation length scale $L_s/\delta$ was seen to vary with the flow confinement parameter $\delta /w$, as previously recognized in reflected shock interactions. Significant differences in flow structure were observed between these computations with confining sidewalls and previous computations with periodic boundary conditions at the lateral boundaries of the domain. The most intense turbulent fluctuations occurred in the corners of the domain. Symmetric and antisymmetric motions of the separation region were identified in the instantaneous flow using conditional averages and spatial correlations. The presence of the sidewalls was found to influence both the mean and unsteady flow. Accurately capturing the sidewall flows is judged to be essential in realistic modeling of confined flows.

Figure 1

Computational mesh of $4711\times 1421\times 1401$ points; every 32nd point is shown for clarity.

Figure 2

Overview of the flow field, showing the instantaneous density field in several planes.

Figure 3

Behavior of the shock sensor at a particular instant. Black indicates locations where the sensor is triggered.

Figure 4

Density field in the $x/{\delta }_0=80$ plane: (a) instantaneous and (b) mean.

Figure 5

Density field in the plane normal to ramp, $x/{\delta }_0\approx 111$: (a) instantaneous and (b) mean. The coordinate $n$ is the distance from the ramp surface.

Figure 6

Density field in the $z/{\delta }_0=10$ plane: (a) instantaneous and (b) mean.

Figure 7

Turbulent boundary-layer profiles on the floor ($x/{\delta }_0=80$ and $z/{\delta }_0=10$) and sides of the domain ($x/{\delta }_0=80$ and $y/{\delta }_0=10$) compared to DNS [21, 46] and to experiment [44]: (a) outer coordinates and (b) van Driest transformed inner coordinates. The horizontal axis represents the wall-normal coordinate in each case.

Figure 8

Profiles of the Reynolds stress on the floor ($x/{\delta }_0=80$ and $z/{\delta }_0=10$) and sides of the domain ($x/{\delta }_0=80$ and $y/{\delta }_0=10$) compared to DNS [21] and to experiment [44, 49]: (a) streamwise normal stress $\overline{\rho }\overline{u^{\prime 2}}$, (b) transverse normal stress $\overline{\rho }\overline{v^{\prime 2}}$, (c) spanwise normal stress $\overline{\rho }\overline{w^{\prime 2}}$, and (d) shear stress $-\overline{\rho }\overline{u^{\prime }{v}^{\prime }}$.

Figure 9

Turbulent boundary-layer profiles on the floor ($x/{\delta }_0=80$ and $z/{\delta }_0=10$) and on the ramp ($x/{\delta }_0\approx 111$ and $z/{\delta }_0=10$): (a) outer coordinates and (b) van Driest transformed inner coordinates.

Figure 10

Skin friction on the left ($z/{\delta }_0=0$) and floor (minimum $y$ coordinate) boundaries of the computational domain: (a) instantaneous values and (b) mean values with trajectories.

Figure 11

Skin friction in the region of separation: (a) and (b) examples of instantaneous skin friction, (c) mean skin friction with trajectories, and (d) root-mean-square skin friction.

Figure 12

Probability of low instantaneous skin friction magnitude $P(C_f<1.0\times {10}^{-3})$.

Figure 13

Profiles along the centerline of the interaction compared to the analogous case with periodic sidewall boundary conditions [37]: (a) mean pressure coefficient and (b) mean streamwise skin friction coefficient.

Figure 14

Wall pressure in the region of separation: (a) mean pressure $\overline{p}/p_{\infty }$ and (b) root-mean-square pressure fluctuation intensity ${\left(\overline{p^{\prime 2}}\right)}^{1/2}$.

Figure 15

Mean pressure contours $\overline{p}/p_{\infty }$ on floor of computational domain, along with regions used for conditional averages: (a) condition 1, looking for high pressures at the edges, and (b) condition 2, looking for high pressure on the right side.

Figure 16

Deviation from overall mean for conditional averages of the pressure field: (a) high pressures at the edges (32.8% of samples) and (b) complementary average (67.2% of samples).

Figure 17

Deviation from overall mean for conditional averages of the pressure field: (a) high pressures on the right side (40.5% of samples) and (b) complementary average (59.5% of samples).

Figure 18

Spatial correlation of wall pressure: (a) reference point $x_r/{\delta }_0=91.0$, $z_r/{\delta }_0=10.0$, (b) reference point $x_r/{\delta }_0=91.0$, $z_r/{\delta }_0=0.1$, (c) reference point $x_r/{\delta }_0=106.0$, $z_r/{\delta }_0=10.0$, and (d) reference point $x_r/{\delta }_0=106.0$, $z_r/{\delta }_0=0.1$.

Figure 19

Spatial correlation of wall pressure for the case with periodic sidewall boundary conditions [37]: (a) reference point $x_r/{\delta }_0=86.9$, $z_r/{\delta }_0=5.0$ and (b) reference point $x_r/{\delta }_0=108.7$, $z_r/{\delta }_0=5.0$.