Effect of base-state curvature on self-excited high-frequency oscillations in flow through an elastic-walled channel

We investigate the development of flow-induced small-amplitude high-frequency oscillations occurring in a rigid fluid-conveying channel with a section of the upper wall replaced by a taut elastic sheet. We consider an axially nonuniform base state, caused by a negative transmural pressure (internal minus external), which results in the sheet bending inward, and then examine the evolution of oscillatory perturbations about this state. Due to the curvature of the base state, any normal displacement of the sheet will lead to a change in the axial stretching of the sheet, which is an order of magnitude higher than would be the case for perturbations to a uniform sheet. This stretching provides an additional restoring force that can be dominant. We derive a modified tube law to describe the wall mechanics with this additional effect and combine it with an existing fluid model to obtain a complete description of the system. At leading order, we obtain a one-dimensional eigenvalue problem for the frequencies and mode shapes of the oscillatory perturbations. The nonlinear interaction between the perturbations and the base-state curvature manifests itself as an additional integral term in the eigenvalue problem, corresponding to the total axial stretching in the wall. The normal modes are neutrally stable at leading order, and their slow growth or decay is determined by considering the global energy budget of the system. The stability of the system can be expressed in terms of a critical Reynolds number of the mean flow through the channel. We explain the behavior of the system as two key dimensionless parameters $\mathcal{F}$ and $\mathcal{K}$ are varied. These quantify the dimensionless axial tension and base-state curvature effects, respectively. Numerical simulations are used to identify distinct flow regimes in parameter space, which we explore further in detail using asymptotic analysis. This reveals that at large $\mathcal{K}$, i.e., strong base-state curvature effects, the leading-order axial profiles of the modes are forced to adjust in order to eliminate the need for significant stretching in the wall. This has the effect of stabilizing both the fundamental and, to a lesser extent, the first harmonic modes, which results in the first harmonic becoming the most unstable mode.

Figure 1

(a) Starling resistor, comprising a section of elastic-walled tube mounted between two rigid pipes and placed inside a pressure chamber. (b) Related collapsible-channel model considered here, comprising a 2D channel in which a section of one wall has been replaced by an elastic membrane.

Figure 2

Schematic illustrating (a) the axial-mode-1 and (b) the axial-mode-2 oscillations of a finite length of tube wall about an axially nonuniform buckled base state. The solid and dashed curves show, respectively, the base state and extrema of the oscillations. The oscillations in (a) necessarily involve significant changes in the length of the wall. With the right mode shape and axial motion, the oscillations in (b) can occur with no axial stretching or compression.

Figure 3

(a) Schematic of the rigid channel with a section of wall replaced with an elastic surface showing the dimensional lengths and coordinate orientation. The restoring force from the springs is a surrogate for azimuthal bending in the full three-dimensional tube system. (b) Horizontal and vertical displacements of a section of wall (ii) from its undeformed configuration (i).

Figure 4

Solutions (50) illustrating the base-state deformation under a uniform steady transmural pressure, showing the transition from small to large $\mathcal{F}$. The curves show the solutions for $\mathcal{F}=0.001$, 0.01, 0.1, 1, and 1000, with the arrow on the graph indicating the direction of increasing $\mathcal{F}$. The normalization $min\left\{\overline{\eta }\right\}=-1$ results from the definition of $d=\delta a$ as the maximum deformation in the base state. In dimensional terms, the base-state deformation is $\delta a\overline{\eta }\ll a$, where $a$ is the channel height. The dependence of $\delta$ on $\mathcal{F}$ and the other problem parameters can be seen in Eq. (52), with small-$\mathcal{F}$ and large-$\mathcal{F}$ approximations in (54) and (56).

Figure 5

Numerical simulations of (57) and (60) for the primary (P) and secondary (S) modes, illustrating (a) the oscillation frequency $\omega$ and (b) the scaled oscillation frequency $\omega /{\mathcal{K}}^{1/2}$ (plotted against the $\mathcal{F}\gg 1$ parameter ${\mathcal{F}}^{\prime }=\mathcal{F}/\mathcal{K}$) corresponding to the primary and secondary modes. Solutions are evaluated with $\mathcal{K}=0.01$ (green), 1.5 (blue), 100 (red), and 1000 (black) for varying $\mathcal{F}$.

Figure 6

Numerical solutions of (57) and (60) for the primary (P) and secondary (S) modes. (a) ${\text{Re}}_c$ for $\mathcal{K}=0.01$ (green), 1.5 (blue), 100 (red), and 1000 (black) for varying $\mathcal{F}$. (b) Critical curve on which the primary and secondary modes lose stability at the same value of ${\text{Re}}_c$. For points above this curve, the secondary mode is more unstable than the primary mode. The dashed line is the asymptote ${\mathcal{K}}_c=\mathcal{F}/{\mathcal{F}}_c^{\prime }=2.70\mathcal{F}$.

Figure 7

Total nonlinear axial stretching $I$, obtained using numerical solutions to (57) and (60) for the primary mode (solid curves), plotted against (a) $\mathcal{F}$ and (b) ${\gamma }^2\mathcal{K}=k_0\mathcal{K}/k_2\mathcal{F}$ for $\mathcal{K}=0$, $0.00001$, 0.001, 0.1, 1, and 10. The dashed curves are the values of $I$ obtained using the asymptotic approximation (77).

Figure 8

Amplitudes $\widehat{p}$ and $\widehat{\eta }$ of the oscillatory perturbations to the pressure and wall displacement as a function of the axial coordinate $z$ for the primary mode when $\mathcal{F}$ is small. The solid curves were obtained by solving (57) and (60) numerically and have been normalized such that ${\widehat{p}}_z\left(z_1\right)=1$, for $\mathcal{K}=1$, and $\mathcal{F}=0.00005$ (magenta), 0.0001 (cyan), 0.0005 (green), 0.001 (red), and 0.005 (blue). The arrows indicate the direction of increasing $\mathcal{F}$. For comparison, the case $\mathcal{K}=0$, $\mathcal{F}=0.00005$ is shown by the black curve. The circular points illustrate the approximate solution (A34) for $\mathcal{F}\ll 1$. In all cases they are almost indistinguishable from the corresponding solid curves. (Figure 4 illustrates the base state at $\mathcal{F}=0.001$ and provides an example of the form of the base state in the $\mathcal{F}\ll 1$ parameter regime about which these oscillations occur.) (a) Pressure perturbation amplitude $\widehat{p}$, which shows little variation across the range of parameters shown. The inset shows a magnified view of the upstream end to show the direction of the (small) changes. (b) Corresponding amplitudes of the oscillatory wall perturbation $\widehat{\eta }$, evaluated using (45). Also shown is $\widehat{\eta }$ plotted against the boundary-layer coordinates (c) $\widehat{\eta }$ plotted against the boundary-layer coordinate $\tilde{z}=\gamma (z-z_1)$. (d) $\widehat{\eta }$ plotted against the boundary-layer coordinate $\overline{z}=-\gamma (z-z_2)$.

Figure 9

(a) Oscillation frequency of the primary (blue solid curve) and secondary modes (red solid curve) plotted against $\mathcal{K}$ obtained by numerically solving (57) and (60) for $\mathcal{F}=1$. Dashed curves labeled I and II illustrate, respectively, the $\mathcal{K}\ll 1$ and $\mathcal{K}\gg 1$ approximate solutions derived in Appendixes pp2-s1 and pp2-s2. Crosses indicate the parameter values used to obtain the profile plots in Fig. 10. Also shown are the corresponding. (b) The corresponding critical Reynolds number ${\mathrm{Re}}_c$, evaluated using (71). (c) The corresponding nonlinear stretching $I$.

Figure 10

Amplitude profiles of the oscillatory perturbations $\widehat{p}$ and $\widehat{\eta }$ to the pressure and wall displacement for the primary (blue) and secondary modes (red) obtained by numerically solving (57) and (60) for $\mathcal{F}=1$ and $\mathcal{K}={10}^{-0.5}$ (dashed curves) and $\mathcal{K}={10}^{2.5}$ (solid curves); these parameter values are marked by crosses in Fig. 9. Solutions are normalized using (73). The red and blue circular points illustrate the corresponding $\mathcal{K}\ll 1$ and $\mathcal{K}\gg 1$ approximate solutions derived in Appendix pp2.

Figure 11

Amplitude profiles of the oscillatory perturbations $\widehat{p}$ and $\widehat{\eta }$ to the pressure and wall displacement obtained by numerically solving (57) and (60) for $\mathcal{F}=100$ and ${\mathcal{F}}^{\prime }=\mathcal{F}/\mathcal{K}=0.01$ (solid blue curves), 0.37 (red), and 10 (black). The critical value ${\mathcal{F}}^{\prime }={\mathcal{F}}_c^{\prime }\approx 0.37$ signifies the value at which there is a switch in the order the primary and secondary modes lose stability. Solutions are shown for (a) and (b) the primary mode and (c) and (d) the secondary mode. Solutions are normalized using (73). The circular points illustrate solutions to the reduced problem (C4) derived in Appendix pp3.

Figure 12

Schematic of the $(\mathcal{F},\mathcal{K})$ parameter space. The dotted line signifies the boundary between the $\mathcal{F}\ll 1$ and $\mathcal{F}=O\left(1\right)$ regions. To the left of the line the wall displacement presents a three-region structure [illustrated, for example, in Figs. 8–8]. Numerical solutions and asymptotic approximations of the solutions when $\mathcal{F}\ll 1,\mathcal{K}\gg {\mathcal{F}}^{3/2}$ (dark gray region), $\mathcal{K}\gg \mathcal{F}\ge 1$ and $\mathcal{K}\ll \mathcal{F}$ (light gray regions), and $O\left(\mathcal{K}\right)=O\left(\mathcal{F}\right)\gg 1$ (top right white region) are discussed in Sec. 4 and derived in Appendixes pp1, pp2, pp3. The curve in the top right quadrant illustrates the stability curve above which the secondary mode loses stability before the primary mode (hatched region).