Growth mechanisms of perturbations in boundary layers over a compliant wall

The temporal modal and nonmodal growth of three-dimensional perturbations in the boundary layer flow over an infinite compliant flat wall is considered. Using a wall-normal velocity and wall-normal vorticity formalism, the dynamic boundary condition at the compliant wall admits a linear dependence on the eigenvalue parameter, as compared to a quadratic one in the canonical formulation of the problem. As a consequence, the continuous spectrum is accurately obtained. This enables us to effectively filter the pseudospectra, which is a prerequisite to the transient growth analysis. An energy-budget analysis for the least-decaying hydroelastic (static divergence, traveling wave flutter, and near-stationary transitional) and Tollmien-Schlichting modes in the parameter space reveals the primary routes of energy flow. Moreover, the maximum transient growth rate increases more slowly with the Reynolds number than for the solid wall case. The slowdown is due to a complex dependence of the wall-boundary condition with the Reynolds number, which translates into a transition of the fluid-solid interaction from a two-way to a one-way coupling. Unlike the solid-wall case, viscosity plays a pivotal role in the transient growth. The initial and optimal perturbations are compared with the boundary layer flow over a solid wall; differences and similarities are discussed.

Figure 1

Schematic diagram: the thick solid line represents the mean-velocity profile with respect to $y$; dashed line shows a modal-perturbed compliant wall; dotted line is the displacement thickness with respect to $x$.

Figure 2

Spectra and eigenfunctions for $\text{Re}=1000,\alpha =0.1,\beta =0.1,m=2,K=0.3,B=3.2,T=0.75$: (a) $d=0$; (b) $d=1$; (c) $d=300$; (d) normal velocity $v^{\prime }$ and vorticity ${\eta }^{\prime }$ of the marked discrete modes in panel (a); (e) normal velocity of the least-decaying continuous modes in panel (a); and (f) streamwise and normal components, $u^{\prime }$ and $v^{\prime }$ respectively, of the marked discrete modes in panel (c).

Figure 3

Contours of $c_i=\text{Im}\left(c\right)$, with $c\equiv \omega /\alpha$, in the $\text{Re}-\alpha$ plane with $\beta =0,T=0.1,B=0.2$, and $m=2$: (a) $d=1,K=0.05$; (b)–(d) for $d=10,20,100$ respectively with other parameters identical to those in panel (a); (e) $d=10,K=0.005,B=0.2,T=0.1$; and (f) $d=10,K=0.5,B=0.2,T=0.1$.

Figure 4

Contours of $c_i=\text{Im}\left(c\right)$ for a rigid wall: (a) in $(\text{Re},\alpha )$ plane with $\beta =0$; (b) in $(\alpha ,\beta )$ plane with $\text{Re}=1000$.

Figure 5

Contours of $c_r\times 100$ for the case of Fig. 3.

Figure 6

$c_r$ and $c_i$ for $\text{Re}=200,\alpha =0.1,\beta =0,m=2,B=0.2,T=0.1$: (a) versus $d$ with $K=0.05$; (b) versus $K$ with $d=1$.

Figure 7

Isocontours of $c_i$ in the $\alpha \text{−}\beta$ plane with $K=0.05,B=0.2,T=0.1$: (a) $\text{Re}=1000$ and $d=1$; (b) $\text{Re}=1000$ and $d=10$; (c) $\text{Re}=200$ and $d=1$; (d) $\text{Re}=200$ and $d=10$.

Figure 8

Energy transfer rates in terms of constituent channels: red, transfer rate by mean shear; blue, transfer rate by viscous dissipation; green, the transfer rate by wall damping; cyan, the transfer rate via forcing by normal stress on the membrane; black, the total transfer rate: (a) “transitional” mode with $\text{Re}=200,\beta =0,m=2,d=10,B=0.2,T=0.1,K=0.05$; (b) TS mode with parameters same as panel (a), but for $\text{Re}=1000$; (c) transitional mode with $\text{Re}=200,\alpha =0.25,m=2,d=1,B=0.2,T=0.1,K=0.05$; (d) same as panel (c) but for $\text{Re}=1000$; (e) TWF mode with $\text{Re}=1000,\beta =0,m=2,d=0,B=0.2,T=0.1,K=0.05$; and (f) TWF with $\text{Re}=1000,\alpha =0.1,m=2,d=0,B=0.2,T=0.1,K=0.05$.

Figure 9

Phase difference, $\varphi =({\varphi }_1-{\varphi }_2)$ (top) and growth rate $\text{Im}\left(\omega \right)$ (bottom) vs $\alpha$ for $\text{Re}=1000,\beta =0,m=2,d=10,B=0.2,T=0.1,K=0.05$ [same parameters as in Fig. 8]: (a) least-decaying mode; (b) second least-decaying mode. The dashed line marks the value of $\pi /2$ and $3\pi /2$.

Figure 10

Transient growth: (a) Black line is $G\left(t\right)$ for $\text{Re}=1000,m=2,d=1,K=0.3,B=3.2,T=0.75\alpha =0.001$ and $\beta =0.09$; the red line in the inset corresponds to the solid-wall case. (b) $G_{\text{max}}$ versus number of modes $K_1$ in the superposition. (c) The red part of the spectrum contributes to 99% of $G_{max}$. (d) $G_{\text{max}}$ and $t_{\text{max}}$ vs Re. (e) $G_{max}$ in $\alpha \text{−}\beta$ plane with other parameters as in panel (a) and $G_{\text{max}}$ vs $K$.

Figure 11

Initial and optimal perturbations for $\text{Re}=1000,m=2,d=1,K=0.3,B=3.2,T=0.75$: panels (a) and (b) are initial and optimal patterns of perturbation velocities, respectively, with $\alpha =0.002,\beta =0$ in the $(x,y)$ plane (red lines show streamlines); panel (c) shows contours of streamwise optimal perturbation velocity in $(y,z)$ plane with $\alpha =0.00025,\beta =0.09$; panels (d) and (e) show initial and optimal patterns of perturbation velocities, respectively, corresponding to panel (c); and (f) indicates the initial and optimal wall fluctuations.

Figure 12

Initial (a) and optimal (b) perturbations in the case of flat rigid wall with $\text{Re}=1000,\alpha =0.1$, and $\beta =0$ in $(x,y)$ plane (red lines show streamlines).

Figure 13

Transient energy budget for $\text{Re}=1000,m=2,d=1,K=0.3,B=3.2,T=0.75$: (a) $\alpha =2.5\times {10}^{-4},\beta =0.09$; (b) $\alpha =0.002,\beta =0.01$.