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

Direct numerical simulations are employed to investigate a shock wave impinging on a turbulent boundary layer at free-stream Mach number $M_{\infty }=2.28$ with different wall thermal conditions, including adiabatic, cooled, and heated, for a wide range of deflection angles. It is found that the thermal boundary condition at the wall has a large effect on the size of the interaction region and on the level of pressure fluctuations. The distribution of the Stanton number shows a good agreement with prior experimental studies and confirms the strong heat transfer and complex pattern within the interaction region. An effort was also made to describe the unsteady features of the flow by means of wall pressure and heat flux spectra. Numerical results indicate that the changes in the interaction length due to the wall thermal condition are mainly linked to the incoming boundary layer, which is in agreement with previous experimental studies.

Figure 1

Comparison of the adiabatic $\left(T_w/T_r=1\right)$ boundary layer at station $x_0=35{\delta }_{\mathrm{in}}$ (lines) with the incompressible DNS results of Schlatter and Örlü [36] at ${\text{Re}}_{\theta }=670$ (triangles). (a) Van Driest–transformed mean velocity profile (solid line). 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), wall-normal (dashed), transverse (dot-dashed), and shear stress (dotted line).

Figure 2

Assessment of the boundary layer at station $x_0=35{\delta }_{\mathrm{in}}$. (a) Van Driest–transformed mean velocity $u^{+}$ plotted vs $y^{+}=y\sqrt{{\rho }_w{\tau }_w}/{\mu }_w$ (solid lines), compared to the velocity transformed according to Ref. [35], $u_{TL}^{+}$ plotted vs the semilocally scaled $y_{TL}^{+}$ (dashed lines, shifted vertically). (b) Density-scaled Reynolds stress components: longitudinal (solid), wall-normal (dashed), transverse (dot-dashed), and shear stress (dotted line). Black lines refer to the adiabatic case BL-1.0, red lines refer to the heated case BL-1.9, and blue lines refer to the cooled case BL-0.5.

Figure 3

The temperature-velocity relationship at station $x_0=35{\delta }_{\mathrm{in}}$ for BL-1.0 (solid), BL-1.9 (dotted), and BL-0.5 (dashed line). The circles $(\circ )$ indicate the Walz solution and the stars $\left(\star \right)$ denote the solution given by Zhang et al. [39].

Figure 4

Grid convergence test. Mean skin friction coefficient for cases SBLI-1.0-9.$5(1536\times 384\times 128$ grid points, solid) and SBLI-1.0-9.5-fine $(2048\times 490\times 170$ grid points, dashed line).

Figure 5

Contours of the instantaneous density $\rho$ (top) and wall-normal density gradient ${\rho }_y$ (bottom) fields for the simulation with adiabatic wall condition and $\phi =9.5^{\circ }$ SBLI-1.0-9.5.

Figure 6

Isosurface of the $Q$ criterion colored by the streamwise velocity and dilatation isosurface (gray) for the simulation with adiabatic wall condition and $\phi =9.5^{\circ }$ SBLI-1.0-9.5.

Figure 7

Distribution of the (a) mean skin friction coefficient, (b) mean wall pressure, and (c) Stanton number across the interaction zone at various incidence angle of the shock generator: $\phi =9.5^{\circ }$ (case SBLI-1.9-9.5, dotted), $\phi =8.0^{\circ }$ (case SBLI-1.9-8.0, solid), $\phi =6.5^{\circ }$ (case SBLI-1.9-6.5, dashed), $\phi =5.0^{\circ }$ (case SBLI-1.9-5.0, dash-dotted), and $\phi =3.5^{\circ }$ (case SBLI-1.9-6.5, dash-double-dotted line). Triangles indicate separation and reattachment position.

Figure 8

Mean streamwise velocity field $\overline{u}$ (left) and mean turbulence kinetic energy $k=\widetilde{u_i^{\prime \prime }{u}_i^{\prime \prime }}/2$ (right) at various wall-to-recovery-temperature ratios: cases SBLI-1.9-9.5, SBLI-1.0-9.5, and SBLI-0.5-9.5 from top to bottom. Contour levels are shown in the range $0<\overline{u}/u_{\infty }<1$ and $0<k/u_{\infty }^2<0.04$. The black line denotes the sonic line and the white line denotes the zero level.

Figure 9

Distribution of the (a) mean skin friction coefficient, (b) mean wall pressure, (c) root-mean-square wall pressure, and (d) mean wall heat flux across the interaction zone at various wall-to-recovery-temperature ratios and deflection angle $9.5^{\circ }$. Cases SBLI-1.9-9.5 (dotted), SBLI-1.0-9.5 (solid), and SBLI-0.5-9.5 (dashed lines).

Figure 10

Instantaneous contours of skin friction (top) and normalized heat flux (bottom) for heated SBLI-1.9-9.5 (left) and cooled SBLI-0.5-9.5 (right) cases. Dashed lines indicate the mean separation and reattachment positions.

Figure 11

Contours of premultiplied pressure spectra (top) and heat flux spectra (bottom) $fE\left(f\right)$ for flow case assuming heated SBLI-1.9-9.5 (left) and cooled wall SBLI-0.5-9.5 (right). The spectra are normalized in such a way that their integral over frequency is unity, at each streamwise station.

Figure 12

(a) Separation and (b) interaction length as a function of the deflection angle. $\left(▵\right)$ heated, $\left(◯\right)$ adiabatic, and $\left(\square \right)$ cooled wall. $\left(★\right)$ refers to the the adiabatic wall simulation at moderate Reynolds number.

Figure 13

Normalized interaction length as a function of separation criterions. $\left(▵\right)$ heated, $\left(◯\right)$ adiabatic, and $\left(\square \right)$ cooled wall. $\left(★\right)$ refers to the the adiabatic wall simulation at moderate Reynolds number. Compilation of results discussed in Ref. [42] is also reported: $\left(▸\right)$ heated walls [43] and $\left(▹\right)$ adiabatic walls [9, 11, 12, 44]. Experimental data from IUSTI in Marseille at Mach $\left(M_{\infty }=2.3\right)$ and Reynolds $\left({\text{Re}}_{\theta }=5\times {10}^3\right)$ numbers are also reported. $\left(\diamond \right)$ adiabatic walls [29] and $\left(♦\right)$ heated walls [29].

Figure 14

Mean turbulence kinetic energy $k=\widetilde{u_i^{\prime \prime }{u}_i^{\prime \prime }}/2$. Cases with $\phi =9.5^{\circ }$: heated (SBLI-1.9-9.5), adiabatic (SBLI-1.0-9.5), and cooled (SBLI-0.5-9.5) wall from top to bottom. Contour levels are shown in the range $0<k/u_{\infty }^2<0.04$. Coordinates are now scaled as $Y^{*}=y{g}_3/L$ and $X^{*}=(x-x_{\mathrm{imp}})/L$.

Figure 15

Mean streamwise velocity field $\overline{u}$. Cases with $\phi =9.5^{\circ }$: heated (SBLI-1.9-9.5), adiabatic (SBLI-1.0-9.5), and variable (adiabatic to heated) wall from top to bottom.

Figure 16

Distribution of the mean skin friction coefficient across the interaction zone for simulations with adiabatic (SBLI-1.0-9.5, solid), heated (SBLI-1.9-9.5, dashed), variable (SBLI-heat2adia-9.5, dash-dotted line), and variable (SBLI-adia2heat-9.5, dash-double dotted line) wall temperatures.

Figure 17

Distribution of the mean skin friction coefficient across the interaction zone for the adiabatic case at low (solid) and moderate (dash-dotted line) Reynolds number flows.

Figure 18

Normalized separation and reattachment positions normalized by ${\delta }_0$ vs deflection angle (a) and normalized by ${\delta }_0/g_3$ vs separation criterion (b). $\left(▵\right)$ heated, $\left(◯\right)$ adiabatic, and $\left(\square \right)$ cooled wall. Full symbols refer to the separation point and empty symbols to the reattachment point.

Figure 19

Distribution of the mean wall pressure across the interaction zone at various wall-to-recovery-temperature ratios and deflection angle $8.0^{\circ }$: cases SBLI-1.9-8.0 (dotted), SBLI-1.0-8.0 (solid), and SBLI-0.5-8.0 (dashed lines). Separation and reattachment locations are also shown with symbols.