Low-frequency unsteadiness in hypersonic swept shock wave-boundary layer interactions

We carry out a numerical study of swept shock wave/turbulent boundary layer interaction in the hypersonic regime. Starting from a numerical/experimental benchmark case of a nearly adiabatic two-dimensional hypersonic interaction, a crossflow velocity component is added to the incoming flow to mimic three-dimensional interactions with cylindrical symmetry. We observe, for a fixed streamwise Mach number, monotonic increase of the extent of the interaction region for the swept cases. An attempt at extending the free-interaction theory to hypersonic swept interactions is made, which is found to apply only to the initial part of the interaction region. The spatiotemporal dynamics of wall pressure on mean separation line features large-scale pressure corrugations, which are advected at a phase speed which is a fraction of the mean crossflow velocity, if present. The characteristic wavelength of the corrugation is found to be a multiple of the separation bubble size. The numerically estimated peak frequencies well conform with the previously introduced formula for swept supersonic interactions [Ceci et al., J. Fluid Mech. 956, R1 (2023)]. Proper orthogonal decomposition is applied to investigate the spatial structure of the corrugation at the separation point and educe phase relations between the flow structure and pressure oscillations at the reattachment point. The present analysis leads us to conclude that the same phenomenology found in swept supersonic interactions also holds in the hypersonic case.

Figure 1

Schematic of SSBLI: ${\gamma }_0$ is the incoming flow sweep angle; ${\delta }_0$ is the incoming boundary layer thickness; $\beta$ is the shock angle, $\theta$ the deflection angle, and $x_{\mathrm{imp}}$ the nominal location of the shock impingement.

Figure 2

Numerical validation: streamwise distributions of time-averaged wall pressure (a) and friction coefficient (b). Solid lines denote the results of the present numerical simulations (red line: case G00_T14F; purple line: case G00_T14; black line: case G00_T14N), orange symbols denote numerical data [11], and dark green symbols denote experimental data [50].

Figure 3

Streamwise distribution of the inner-scaled mesh sizes for the baseline simulation in narrow domain G00_T14N (black lines) and for the finer one G00_T14F (red lines): $\mathrm{\Delta }{x}^{+}=\mathrm{\Delta }{z}^{+}$ (a), $\mathrm{\Delta }{y}_w^{+}$ (b).

Figure 4

Numerical validation: streamwise profiles of the nondimensional wall pressure r.m.s. $[\sqrt{\overline{p_w^{\prime }{p}_w^{\prime }}}/p_0$, (a)] and Stanton number $[c_h=q_w/\left[{\rho }_0{u}_0{c}_p(T_w-T_r)\right]$, (b)]. Solid lines denote the results of the present numerical simulations (red line: case G00_T14F; purple line: case G00_T14; black line: case G00_T14N), the orange symbols denote numerical data [11], and the dark green symbols denote experimental data [50].

Figure 5

Wall-normal profiles of Van Driest transformed, Favre-averaged streamwise velocity profiles ${\tilde{u}}_{VD}^{+}$ (left) and density-scaled stresses ${\widetilde{u_i{u}_j}}^{*}=\overline{\rho u_i^{\prime \prime }{u}_j^{\prime \prime }}/{\tau }_w$ (right) for different data sets at $x_r=45{\delta }_0$: baseline G00_T14 (purple solid line), finer mesh G00_T14F (red solid line), Volpiani et al. [11] (orange squares), Schlatter and Örlü [52] (cyan diamonds), and experiments of Schülein [50] (dark green circles).

Figure 6

Wall-normal profiles of Van Driest transformed, Favre-averaged velocity profiles ${\tilde{u}}_{s,VD}^{+}$ (a) and density-scaled stresses ${\widetilde{u_{i,s}{u}_{i,s}}}^{*}=\overline{\rho u_{i,s}^{\prime \prime }{u}_{i,s}^{\prime \prime }}/{\tau }_w$ (b) for different sweep angle, at the reference position $x_r=45{\delta }_0$. Unswept case (purple line), swept case with ${\gamma }_0={15}^{\circ }$ (green line); swept case with ${\gamma }_0={30}^{\circ }$ (cyan line). The subscript $s$ denotes quantities along the freestream direction.

Figure 7

Temperature-velocity relation for the unswept interaction (solid purple line), swept case with ${\gamma }_0={15}^{\circ }$ (solid green line), and swept case with ${\gamma }_0={30}^{\circ }$ (solid blue line) at $x_r=45{\delta }_0$. The hollow circles denote the relation proposed by Duan et al. [53].

Figure 8

Streamwise distribution of mean wall pressure (a) and $x$-projected friction coefficient ($c_{f,x}=2{\mu }_w{(\mathrm{d}\mathrm{u}/\mathrm{d}\mathrm{y})|}_{\mathrm{w}}/\left({\rho }_0{\mathrm{u}}_0^2\right)$, (b)). Unswept case (purple line), swept case with ${\gamma }_0={15}^{\circ }$ (green line), and swept case with ${\gamma }_0={30}^{\circ }$ (blue line).

Figure 9

Streamwise distribution of time-averaged of wall pressure root-mean square $\sqrt{\overline{p_w^{\prime }{p}_w^{\prime }}}/p_0$ and Stanton number $c_h$. Unswept case (purple line), swept case with ${\gamma }_0={15}^{\circ }$ (green line), and swept case with ${\gamma }_0={30}^{\circ }$ (cyan line).

Figure 10

Mean streamwise velocity field ($u$) at various crossflow angles, for $T_w/T_r=0.8$. Contour levels are shown in the range $0<u/u_{0,x}<1$; (a) ${\gamma }_0=0^{\circ }$, (b) ${\gamma }_0={15}^{\circ }$, and (c) ${\gamma }_0={30}^{\circ }$. A mean dilatation isoline (in black) is used to highlight the shock system.

Figure 11

Mean friction lines at the wall (left) and mean streamlines along the dividing streamline (right), for (a) ${\gamma }_0=0^{\circ }$, (b) ${\gamma }_0={15}^{\circ }$, and (c) ${\gamma }_0={30}^{\circ }$. The mean separation lines are shown in red, the shock impingement lines in green, and the reattachment lines in blue.

Figure 12

Numerical schlieren visualization based on the streamwise density gradient (top), and streamwise evolution of flow angles (bottom), for (a) ${\gamma }_0=0^{\circ }$, (b) ${\gamma }_0={15}^{\circ }$, and (c) ${\gamma }_0={30}^{\circ }$. The magenta dashed lines refer to the shape of the dividing streamline, and the magenta solid lines refer to the corresponding flow angle in the wall plane, the black dashed lines refer to the wall surface, and the black solid line to the wall-parallel friction lines. Dashed red, green, and blue lines denote the mean separation, impingement and reattachment positions, respectively.

Figure 13

Similarity function $F\left(s\right)$, as defined in Eq. (12), plotted by setting the origin ($x_i$) at the mean separation point ($x_{\mathrm{sep}}$, panels a, b), and at the location of maximum pressure gradient ($x_p$, panels c, d). The reference length is taken to be either $L^{*}={\delta }_i$ (a, c), or as the distance of the point where $F=4.22$ ($x_m$) from the origin (b), (d).

Figure 14

Premultiplied, normalized PSD of wall pressure for the baseline flow case G00_T14, determined from use of the Welch method with one, three, five, and seven Hamming windows with 50% overlap.

Figure 15

Premultiplied, normalized PSD of wall pressure fluctuations for flow cases G00_T14 (a), G15_T14 (b), and G30_T14 (c). The red line denotes the mean separation location, the green line the nominal shock impingement location, and the cyan line the mean reattachment location. Red crosses mark the position of the low-frequency peaks near the separation line.

Figure 16

Premultiplied, spanwise PSD of wall pressure fluctuations at the mean separation line, for various sweep angles. The spanwise wavelength ${\lambda }_z$ is scaled by the separation length. ${\kappa }_z=2\pi /{\lambda }_z$ is the spanwise wave number. The dashed line marks ${\lambda }_z=2L_{\mathrm{sep}}$.

Figure 17

Contour plots of frequency/wave number spectra of the wall pressure at the mean separation location. Dashed lines denote the linear relationship for the circular frequency $\omega ={\kappa }_z{w}_c$, with convection velocity $w_c=0.7u_{0,x}tan{\gamma }_0$. (a) ${\gamma }_0=0^{\circ }$; (b) ${\gamma }_0={15}^{\circ }$; (c) ${\gamma }_0={30}^{\circ }$. The cyan crosses mark the position of the low-frequency peaks.

Figure 18

(a) Premultiplied, normalized frequency spectra of wall pressure fluctuations at the mean separation line, for various sweep angles. Peaks are marked with crosses. (b) Peak frequency as a function of sweep angle: the solid and dashed lines denote the prediction of Eq. (16). Numerical values are marked with triangles.

Figure 19

Leading POD mode ${\varphi }_1(x,z)$ (left panel) and premultiplied PSD of associated temporal coefficient ${\psi }_1\left(t\right)$ (right panel) of wall pressure in the interaction region. (a) Case G00_T14, (b) case G30_T14.