Linear stability analysis of compressible boundary layer over an insulated wall: Existence of multiple new unstable modes for Mach number beyond 3

Abstract

Here, we investigate the linear spatial stability of a parallel two-dimensional compressible boundary layer on an adiabatic plate by considering 2D and 3D disturbances. We employ the Compound Matrix Method for the first time for compressible flows, which, unlike other conventional techniques, can efficiently eliminate the stiffness of the original equation. Our study explores flow Mach numbers ranging from low subsonic to supersonic cases, to investigate the effects of flow compressibility and spanwise variation of disturbances. We get some interesting results depending on the flow Mach number. Mack (AGARD Report No. 709, 1984) reported the existence of two unstable modes for Mach number greater than 3 from viscous calculations (the so-called second mode) that subsequently fuse to create only one unstable zone when Mach number increases. Our calculations show a series of unstable modes for a Mach number greater than 3. The number of such modes is much more than two (unlike what Mack reports). The number and the frequency extent of the corresponding unstable zones increase with an increase in M, which is significantly higher than subsonic or low-supersonic cases. While the shape of the neutral curves for the second unstable mode for a Mach number greater than 4 is similar to the fused neutral curve shown by Mack for a Mach number of 4.8, the characteristics of higher-order spatially unstable modes considering the viscous stability of supersonic boundary layers remain unreported to the best of our knowledge. The last one is the most novel element in the reported results.

Figures (9)

• Figure 1: The schematic of the $2D$ parallel flow-approximation showing a tentative variation of the velocity and temperature profiles of a wall-bounded shear layer. Here, $\delta_*$ represents the displacement thickness of the shear-layer, which is treated as constant here under the parallel flow approximation.
• Figure 2: $\Lambda_{1_r}$, $\Lambda_{3_r}$, and $\Lambda_{5_r}$ plotted in the $(\alpha_r,\alpha_i)$-plane, where $\alpha=\alpha_r+i\alpha_i$, for indicated values of $Re$ and $M$ when $\omega_r=0.1$ and $\beta=0$.
• Figure 3: (a) $\bar{U}(y)$, and (b) $\bar{T}(y)$, plotted as a function of $\hat{\eta}=\tilde{y}\sqrt{Re_x}/\tilde{x}$, where $Re_x=\tilde{\rho}_{\infty}\tilde{U}_{\infty}\tilde{x}/\tilde{\mu}_{\infty}$ is the Reynolds number based on the streamwise coordinate $\tilde{x}$. (c) The displacement thickness parameter $c_{\delta}$ plotted as a function of the free-stream Mach number $M$ where, $c_{\delta}=\delta_*\sqrt{Re_x}/\tilde{x}$.
• Figure 4: Comparison between the spatial stability analysis of 2D incompressible zero-pressure gradient Blasius and compressible boundary layer on the insulated wall at $M=0.1$. (a,b) Spatial eigenvalues for $Re=1500$ and $\omega_r=0.1$, (c) comparison of the corresponding Neutral curves, and (d,e) group velocity corresponding to mode-$1$ in the $(Re,\omega_r)$-plane.
• Figure 5: The Neutral curve plotted for indicated Mach number cases in (a) $(\sqrt{Re_x},\omega_r)$- and (b) $(Re,\omega_r)$-plane for $2D$ disturbance. Group velocity contours shown for (c) $M=0.6$ and (d) $M=2$ in the $(Re,\omega_r)$-plane. The corresponding neutral curve in frames (c,d) is represented as a dashed line.