Effects of a nonadiabatic wall on hypersonic shock/boundary-layer interactions

High-fidelity numerical simulations of an impinging shock interacting with a turbulent boundary layer at free-stream Mach number ${\text{Ma}}_{\infty }=5.0$ are performed for three shock angles and a weakly cooled wall condition (wall-to-recovery temperature ratio of $T_w/T_r=0.8)$, matching the experimental conditions of Schülein [AIAA J. 44, 1732 (2006)]. Additional simulations are carried out with a heated wall $\left(T_w/T_r=1.9\right)$ to investigate the impact of the wall thermal condition on shock/boundary-layer interactions in the hypersonic regime. The free interaction theory is found to remain valid in the hypersonic regime even at different wall thermal conditions. The interaction length scaling proposed by Jaunet et al. [AIAA J. 52, 2524 (2014)] previously validated only at supersonic Mach numbers, is found to remain viable at higher Mach numbers only if the model coefficient is adjusted, thus showing that this coefficient is not universal.

Figure 1

Comparison of boundary layers BL-0.8-3800 and BL-0.8-5400 at station $x_{\mathrm{imp}}=40{\delta }_{\mathrm{in}}$ (lines) with the incompressible DNS results of Schlatter and Örlü [35] at ${\text{Re}}_{\tau }=359$ (triangles) and ${\text{Re}}_{\tau }=492$ (circles). (a) Van Driest transformed mean velocity profile (solid line). Simulation BL-0.8-5400 is shifted vertically. The linear $u^{+}=y^{+}$ and log-law $u^{+}=5.2+2.44ln{y}^{+}$ are also represented (dotted lines). (b) Density-scaled Reynolds stress components: longitudinal (solid line), wall-normal (dashed line), transverse (dotted-dashed line), and shear stress (dotted line).

Figure 2

Assessment of the boundary layer at station $x_{\mathrm{imp}}=40{\delta }_{\mathrm{in}}$. Van Driest transformed mean velocity $u^{+}$ plotted vs $y^{+}=y\sqrt{{\rho }_w{\tau }_w}/{\mu }_w$ compared to the experiments of Schülein [23] (symbols). The vertically offset lines are the velocity transformed according to Trettel and Larsson [34] $\left(u_{\text{TL}}^{+}\right)$ plotted vs the semilocally scaled $y_{\text{TL}}^{+}$. Cases BL-0.8-3800 (solid line), BL-1.9-3800 (dotted line), BL-0.8-5400 (dashed-dotted line), and BL-0.8-7000 (dashed line).

Figure 3

Isosurface of the $Q$ criterion colored by the streamwise velocity and dilatation isosurface (gray) for the simulation SBLI-0.8-14: $T_w/T_r=0.8$ and $\phi ={14}^{\circ }$.

Figure 4

Instantaneous (left) and mean (right) streamwise velocity field $\overline{u}$ at various incidence angle of the shock generator and $T_w/T_r=0.8$: $\phi =6^{\circ }$ (a), (b), $\phi ={10}^{\circ }$ (c), (d) and $\phi ={14}^{\circ }$ (e), (f). Contour levels are shown in the range $0<\overline{u}/u_{\infty }<1$. The white line denotes the zero level and the black one the sonic line. Isocontour of the mean dilatation is also shown in dark purple.

Figure 5

Mean flow properties across the interaction zone at various deflection angles and $T_w/T_r=0.8$: case SBLI-0.8-14 (dashed line), SBLI-0.8-10 (solid line), and SBLI-0.8-06 (dashed-dotted line). Symbols indicate experimental data from Schülein [23]. (a) Distribution of the mean skin friction coefficient. (b) Distribution of the mean wall pressure. (c) Distribution of the mean Stanton number.

Figure 6

Mean flow properties across the interaction zone at $T_w/T_r=0.8$ and deflection angle $\phi ={14}^{\circ }$: base grid (dashed line), fine grid (squares), and fine grid using Rome's code (dashed-dotted line). Circles indicate experimental measurements from Schülein [23]. In (a), the open circles are from oil film interferometry while the closed circles are from the measured velocity profile. (a) Distribution of the mean skin friction coefficient. (b) Distribution of the mean Stanton number.

Figure 7

Mean flow properties across the interaction zone at $\phi ={14}^{\circ }$ flow deflection: cases SBLI-1.9-14 (dotted line), SBLI-0.8-14 (dashed line), and SBLI-0.8-14-H (solid line). The symbols indicate experimental measurements from Schülein [23]. In (a), the open circles are from oil film interferometry while the closed circles are from the measured velocity profile. The plateau pressure prediction according to Zukoski [41] is also shown in (b). (a) Distribution of the mean skin friction coefficient. (b) Distribution of the mean wall pressure. (c) Distribution of the root-mean-square wall pressure. (d) Distribution of the mean wall heat flux. (e) Distribution of the mean Stanton number.

Figure 8

Instantaneous dilatation field $\xi ={\partial }_j{u}_j$ for simulation with $T_w/T_r=1.9$ and $\phi ={14}^{\circ }$ (case SBLI-1.9-14). Contour levels are shown in the range $-1<\xi <1$.

Figure 9

Mean turbulence kinetic energy $k=\widetilde{u_i^{″}{u}_i^{″}}/2$. Contour levels are shown in the range $0<k/u_{\infty }^2<0.04$ with $T_w/T_r=1.9$ and $\phi ={14}^{\circ }$ (case SBLI-1.9-14). The black line denotes the sonic line. Isocontour of the mean dilatation is shown in dark purple.

Figure 10

Correlation of maximum heating with shock strength. Heated walls (red circles), cooled walls (blue triangles), and SBLI-0.8-14-H (green squares).

Figure 11

Contours of premultiplied pressure spectrum $fE\left(f\right)$ for the SBLI-1.9-14 case with a heated wall. The spectrum is normalized in such a way that their integral over frequency is unity at each streamwise station.

Figure 12

Similarity function $\mathcal{P}$ [scaled pressure; Eq. (5)] plotted versus $\mathcal{X}$ [distance from separation point; Eq. (6)] evaluated for all SBLI cases. Cases: SBLI-0.8-06 (blue dashed line), SBLI-0.8-10 (blue dashed-dotted line), SBLI-0.8-14 (blue solid line), SBLI-1.9-06 (red dashed line), SBLI-1.9-10 (red dashed-dotted line), SBLI-1.9-14 (red solid line).

Figure 13

Normalized interaction length $L_{\mathrm{int}}$ as a function of the separation criteria by Souverein et al. [27] (top) and Jaunet et al. [17] (bottom). Simulations with cooled (blue triangles) and heated (red circles) wall; open symbols scaled using the original constants $(k_1=3$ in the left figure, $k_2=7.14$ in the right), filled symbols scaled using the modified constant $k_2=15$. The compilation of results discussed in Souverein et al. [27] is also reported: adiabatic walls (open right triangle [3, 4, 5, 53]) and heated walls (filled right triangle [54]). Experimental data from Jaunet et al. [17] at Mach $\left({\text{Ma}}_{\infty }=2.3\right)$ and Reynolds $\left({\text{Re}}_{\theta }=5\times {10}^3\right)$ numbers are also reported: adiabatic walls (open green diamond) and heated walls (filled green diamond).