Effect of surface temperature strips on the evolution of supersonic and hypersonic Mack modes: Asymptotic theory and numerical results

In this paper, we perform an asymptotic study of the effect of local surface temperature (heating or cooling) strips on oncoming inviscid Mack instability in supersonic or hypersonic boundary layers, which is presented for a canonic problem to shed light on the physical mechanisms by which surface imperfections impact on boundary-layer transition to turbulence. Assuming the Reynolds number to be sufficiently large, the change of the Mack amplitude due to interaction with the temperature strip is quantified by an explicit model developed by the multiscale analysis. It is revealed that the temperature strip plays an equivalent role as a roughness element by producing mean-flow distortions described by the triple-deck structure. However, the former renders a more complicated interaction with the Mack modes since the lower deck is compressible to leading-order accuracy. Based on this model, a systematic study for different control parameters is conducted. A heating strip enhances (or suppresses) the Mack modes with frequencies below (or above) a critical value, whereas a cooling strip plays the opposite role. The critical frequency is found to be the most unstable frequency of the second mode, instead of the synchronization frequency as was found in some previous works for both roughness and temperature-strip configurations. A few phenomena in contrast to the roughness configuration are observed and readily explained by the asymptotic model. The asymptotic predictions agree favorably with the Harmonic linearized Navier-Stokes calculations and the direct numerical simulations, especially when the wall temperature is low.

Figure 1

Physical model to be studied and the characteristic length scales, where the red region at the wall denotes the heating/cooling strip. Panel (b) shows the multiscale structure of the rectangular zone of panel (a).

Figure 2

Dependence on $\omega$ of $c_r$ (left column) and $-{\alpha }_i$ (right column) of the 2D Mack modes, where the velocities of the fast and slow acoustic waves, $1\pm 1/M$, are marked in the left column. (a), (b) case M1T1; (c), (d) case M1T4; (e), (f) case M2T1; (g), (h) case M2T4; (i), (j) case M2T9. The horizontal lines in the right column represent the zero-growth lines.

Figure 3

Streamwise evolution of $|\tilde{A}|$ (a), $-\tilde{A}/{\mathrm{\Theta }}_m$ (b), and $|I_{T1}|$ (c) for $(T_{\infty },T_w)=(65.15K,4.4)$ and $D=1$ with different ${\mathrm{\Theta }}_m$. For the solid curves in panel (a), $|\tilde{A}|=-\tilde{A}$.

Figure 4

Comparison of the lower-deck base flow between the triple-deck calculation (red solid lines) and DNS result (green dashed lines) for case M1T1R1-1. (a) Streamwise evolution of the wall velocity shear ${\tilde{U}}_{\tilde{Y}}{|}_w$, the wall temperature gradient ${\tilde{T}}_{\tilde{Y}}{|}_w$ and the wall pressure $\tilde{P}$, where $\tilde{P}$ is shifted for plotting convenience; (b) temperature profiles at five representative stations.

Figure 5

Comparison of the mean-velocity distortion $\overline{u}-U_B$ between the triple-deck solution (red solid lines) and DNS (green dashed lines) for cases M1T1R1-1 (a) and M1T4R1-1 (b).

Figure 6

Dependence on $\omega$ of the prefactor ${\mathcal{H}}_1$ for $(M,R,Te)=(4.5,50000,65.15K)$ (a), (b), where the vertical dashed lines represent the neutral frequencies, and the circles and rectangles denote the frequencies of the most unstable modes and the synchronization points, respectively.

Figure 7

Dependence of the efficiency functions $K_{0,r}$ (left column) and $K_{0i}$ (right column) on $\omega$ for $(M,R,T_{\infty })=(4.5,50000,65.15K)$ (a), (b) and $(M,R,T_{\infty })=(5.92,131820,48.69K)$ (c), (d). The horizontal dashed lines in panels (b) and (d) are for $K_{0,i}=0$. The vertical dashed lines, circles, and rectangles are the same as those in Fig. 6. The horizontal colored bars in panels (c) and (d) represent the unstable frequency bands of the second modes.

Figure 8

(a) The perturbation temperature (colored contours) and the mean-temperature distortion (continuous lines) for case M1T4R1-1 (upper-half plane) and the flat-plate case (lower-half plane), where $\omega =1.12$. (b) Normalized temperature amplitude obtained by the DNS (continuous curves) and HLNS (circles) approaches for case M1T4R1-1.

Figure 9

Amplitude evolution normalized by that of the flat-plate case for case M1T4R1-1: (a) for $\omega$ =1.16; (b) for 1.40.

Figure 10

Variation of ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ on $x_N\equiv {\epsilon }^{-1}{X}_N$ for case M1T4R1-1.

Figure 11

Comparison of the transmission coefficient $\left|\mathcal{T}\right|$ between the asymptotic prediction and HLNS calculation for $({\mathrm{\Theta }}_m,d/{\delta }_{99})=(0.5,8)$. (a) $(M,R,T_{\infty })=(4.5,50000,65.15K)$, (b) $(M,R,T_{\infty })=(4.5,50000,226.5K)$, (c) $(M,R,T_{\infty })=(5.92,131820,48.69K)$, (d) $(M,R,T_{\infty })=(5.92,26364,226.5K)$. Solid lines: asymptotic predictions; dashed lines: HLNS calculations; crosses in (a) DNS results. The vertical dot-dot-dashed lines mark the most unstable mode, and the horizontal colored bars mark the unstable frequency band of the second mode.

Figure 12

Comparison of $\left|\mathcal{T}\right|$ between the asymptotic prediction and HLNS calculation for different ${\mathrm{\Theta }}_m$ (a) and $D$ (b), where $(M,T_{\infty },T_w,R)=(4.5,226.5K,1.1,50000)$. Solid lines: asymptotic predictions; dashed lines: HLNS calculations.

Figure 13

Comparison of $\left|\mathcal{T}\right|$ between the asymptotic prediction and HLNS calculation for $R=50000$, 25 000, and 10 000, where $(M,T_{\infty },T_w,{\mathrm{\Theta }}_m,d/{\delta }_{99})=(4.5,226.5,1.1,0.5,8)$. Solid lines: asymptotic predictions; dashed lines: HLNS calculations.

Figure 14

Asymptotic predictions of the growth rate (a) and the transmission coefficient (b) on the spanwise wave number $\beta$ for case M1T8R1-1.

Figure 15

Comparisons of the present results with reference results: (a) for a supersonic case [42]; (b) for a subsonic case [44].