Stability of jets and wakes confined by compliant walls

A spatiotemporal stability analysis is conducted on a flow representing both jets and wakes, subject to confinement by identical compliant walls. The walls are placed at equal distances from the fluid center line for a range of wall and flow parameters. By following the position of special saddle points (pinch points) of the dispersion relation in the complex wave-number plane, the absolute and convective instability stability properties of the flow are determined for various system parameters. The compliant walls are shown to modify the shear-induced instabilities, which exist in the rigid wall case, as well as introduce new additional instabilities originating from the presence of the wall itself. It is observed that under certain system parameters, these wall-induced modes become the dominant instability present in the system and can induce an absolute instability into flows which are only convectively unstable when confined by rigid walls, as well as extending the region of absolute instability to large confinement parameters. Results are presented for both a piecewise linear velocity profile and a smooth velocity profile, with the results of the two studies in qualitative agreement.

Figure 1

Schematic of the plug flow velocity profile with shear layers, which is considered in this paper. Both fluids have density $\rho$. The inner fluid has velocity $U_1\underline{\widehat{\mathbf{x}}}$ and width $2h_1$, while the outer fluid has velocity $U_2\underline{\widehat{\mathbf{x}}}$ and width $h_2+\delta$, where $\delta$ is the shear layer width. Fluid interfaces are represented by thick black lines at $z=\pm h_1$.

Figure 2

A diagram of the spring-backed elastic plate compliant wall model considered here. The springs all share the same spring constant, and $\xi$ is the displacement of the elastic sheet from the undisturbed position represented by the dashed line.

Figure 3

Plots of $c_i\left(\alpha \right)$ for four different wall configurations: (a) rigid wall, (b) $(B,T,K,m,d)=(2,0,10,0.1,0)$, (c) $(B,T,K,m,d)=(0.015,0.025,0.01,0.1,0)$, and (d) $(B,T,K,m,d)=(1,1.5,1.5,0.1,0.55)$. The red line signifies the dominant shear-induced mode, while the green line gives the subdominant wall-induced mode.

Figure 4

Plots of the absolute values of sinuous pressure eigenmodes $|\widehat{p}|$ as a function of $z$ for the wall configurations given in Fig. 3. Here we fix $\mathrm{\Lambda }=3, h=1.5$, and $\alpha =1$ and normalized such that $|\widehat{p}|=1$ at $z=1$ for the mode represented with the solid lines. Solid lines represent the dominant shear-induced mode while dashed lines represent the subdominant wall-induced mode. The vertical black dashed-dot line represents the fluid interface at $z=1$.

Figure 5

The plot on the left shows rigid wall ($Q\to \infty$) temporal instability growth rate ${\omega }_i$ as a function of $\alpha$ for varicose modes for $\delta =1$ (blue line), $\delta =0.5$ (green line), and $\delta =0.1$ (red line). The right plot is the equivalent phase velocity $c_i={\omega }_i/\alpha$. Here $\mathrm{\Lambda }=1$ and $h=1.5$. This result is for the rigid wall case, and hence the two values of $\omega$ are either real or complex conjugate pairs.

Figure 6

Regions of AI (gray) in $(\frac{1}{\mathrm{\Lambda }},h)$ space for (a) varicose modes and (b) sinuous modes in the rigid wall limit where $\delta \to 0$. White regions represent regions of CI.

Figure 7

Plot of ${\omega }_i$ contours in the $\alpha$ plane for the varicose mode in the (a) $Q\to 0$ limit and (b) $Q\to \infty$ limit for $(\mathrm{\Lambda },h)=(1,0.5)$, which corresponds to case 2. In (a) we define saddle 1 (red cross), saddle 2 (cyan cross), and saddle 3 (black cross), where only saddles 1 and 2 are pinch points, while in (b) the dominant saddle is given by the white cross.

Figure 8

Case 2: Plot of the varicose AI growth rates ${\omega }_i$ of saddle 1 (red), as we vary (a) flexural rigidity $B$ and (b) the damping coefficient $d$. Dashed lines represent growth rates in the rigid wall limit. Note the existence of two new modes in (a) represented by green and purple curves. Panels (c) and (d) represents the evolution of the wave number ${\alpha }_r$ for each mode as we vary $B$ and $d$, respectively.

Figure 9

Case 3: Plot of the sinuous AI growth rates ${\omega }_i$ of saddle 1 (red) and saddle 2 (green) as we vary (a) flexural rigidity $B$ and (b) the damping coefficient $d$. Dashed lines represent growth rates in the rigid wall limit. Note that in (a) a new saddle appears, represented by the purple line.

Figure 10

Contours of constant ${\omega }_i$ in the $\alpha$ plane as we vary the flexural rigidity $B$ for a varicose mode with parameters $(\mathrm{\Lambda },h)=(1,1.5)$ and $(m,K,d,T)=(0.1,10,0,0)$. Here (a) $B=0$, (b) $B=0.1$, (c) $B=0.5$, and (d) $B=0.75$, and the black and magenta marked saddles are wall-induced modes and are a consequence of having the compliant walls, while the blue cross represents the location of the flow or wall-enhanced mode $W{I}_1$. The black and white dashed lines represent branch cuts.

Figure 11

(a) Plot of the growth rate ${\omega }_i$ of the three saddle points in Fig. 10 as a function of the confinement parameter $h$. The other fixed parameters are $(\mathrm{\Lambda },S)=(1,1)$ with wall parameters $K=10, m=0.1, T=d=0$, and $B=0.1$ (red line), $B=1$ (green lines), and $B=2$ (blue lines). The rigid wall limit result is given by the black line. The dashed lines represent the growth rates of the new $W{I}_1$ saddle found by adding compliant walls, while the dotted line (only for $B=2$) represents the $W{I}_2$ wall mode. Plot (b) shows the dominant growth rate curves only, maintaining the line style to highlight which mode takes dominance, while (c) gives the values of ${\alpha }_r$ for these dominate modes.

Figure 12

Sinuous mode: Plots of AI growth rate ${\omega }_i$ as a function of confinement $h$ as we vary shear layer thickness (a) $\delta =0.5$, (b) $\delta =0.35$, and (c) $\delta =0.15$, where the dots represent ${\omega }_i^{\mathrm{num}}$ values. The other flow parameters are as in Fig. 14 with $B=2$. In panel (d) we plot ${\omega }_r$ for the case $\delta =0.15$, corresponding to panel (c).

Figure 13

Varicose mode: Plots of AI growth rate ${\omega }_i$ as a function of confinement $h$ as we vary shear layer thickness (a) $\delta =0.5$, (b) $\delta =0.35$, and (c) $\delta =0.15$, where the dots represent ${\omega }_i^{\mathrm{num}}$ values. The other flow parameters are as in Fig. 11 with $B=2$. In panel (d) we plot ${\omega }_r$ for the case $\delta =0.15$, corresponding to panel (c).

Figure 14

Plot of (a) the dominant growth rates ${\omega }_i$ and (b) the corresponding values of ${\alpha }_r$ for three relevant saddle points for the sinuous mode, as a function of the confinement parameter $h$. The fixed parameters are $\mathrm{\Lambda }=-1$ with wall parameters $(K,m,d,T)=(10,0.1,0,0)$ and $B=0.1$ (red line), $B=1$ (green lines), and $B=2$ (blue lines). The rigid wall limit result is given by the black line. The solid line give the $S{I}_1$ saddle growth rates, the dashed lines give the $W{I}_1$ saddle growth rate and the dotted line represents the $W{I}_2$ wall mode.

Figure 15

Plot of $({\omega }_i,{\alpha }_r)$ in the $h\to \infty$ limit for [(a) and (b)] the varicose mode in Fig. 11 and [(c) and (d)] the sinuous mode in Fig. 14 as a function of $B$.

Figure 16

Regions of AI for (a) varicose and (b) sinuous modes, when the flow is confined by compliant walls with $(B,K,m)=(2,10,0.1)$ and $d=T=0$. Gray regions indicate AI while white regions depict CI. Dotted lines indicate AI-CI boundaries found from the rigid wall case, while solid lines represent the AI-CI boundaries of the compliant wall case.

Figure 17

A plot of the smooth base flow (35) for $(\mathrm{\Lambda },\delta ,h)=(1,0.4,1)$ with $\mathrm{\Delta }=0.15$ (red line), $\mathrm{\Delta }=0.1$ (green line), $\mathrm{\Delta }=0.05$ (blue line), and $\mathrm{\Delta }=0.01$ (magenta line).

Figure 18

Plot of ${\omega }_i$ for the dominate saddles for varicose modes using the smooth profile with $(\mathrm{\Lambda },h)=(1,1)$ and wall parameters $(B,K,T,m,d)=(2,10,0,0.1,0)$ with (a) $(\delta ,\mathrm{\Delta })=(0.5,0.025)$, (b) $(\delta ,\mathrm{\Delta })=(0.5,0.1)$, (c) $(\delta ,\mathrm{\Delta })=(0.15,0.025)$, and (d) $(\delta ,\mathrm{\Delta })=(0.15,0.1)$. The dots represent ${\omega }_i^{\mathrm{num}}$ values, and the piecewise linear equivalent results are in Figs. 13 and 13, respectively.