Modelling Fluid--Structure Interaction in an Initially Elliptical Elastic-Walled tube: Improved Onset Criterion for Self-Excited Oscillations

Abstract

We present a theoretical description of the fluid--structure interaction observed within a Starling resistor. The typical setup consists of a pre-stretched finite length thin-walled elastic tube mounted between two rigid tubes. The collapsible section is enclosed within a pressure chamber and a viscous fluid is driven through the system by imposing an axial volume flux at the downstream end. Valid within a long-wavelength thin-walled regime, we use our own results to model the wall mechanics. These results arise from the solution of a generalised eigenvalue problem, and avoid the need to invoke the ad-hoc approximations made in previous studies. The wall mechanics are then coupled to the fluid mechanics using the Navier--Stokes equations, under the assumption that the oscillations in the tube wall are of small amplitude, long wavelength and high frequency. We derive problems governing the leading-order steady and oscillatory fluid-structure interaction. At leading order, the system permits normal-mode oscillations of constant frequency and amplitude, which are obtained in the form of series solutions. Higher-order corrections govern the slow growth or decay of the oscillations, however (as in previous work) these growth rates can be determined by analysing the system's global energy budget without needing to compute the higher-order terms explicitly. Our results permit the first formal analysis of the errors incurred by neglecting contributions from higher-order azimuthal modes, and enable the determination of improved criterion for the onset of self-excited oscillations in the tube wall.

Figures (14)

• Figure 1: (a) The setup of an idealised Starling resistor. An initially elliptical elastic-walled tube is pinned between two rigid extensions. Fluid is driven through the system by imposing a steady dimensional axial volume flux of size $A_0^* \mathscr{U}$ at the downstream end $z^*=L$. (b) The base-state ellipses corresponding to the representative ellipticity parameter values $\sigma_0 = \infty, s_1,s_2,s_3,s_4$.
• Figure 2: Values of $\omega_0^{(1)}$ plotted throughout $(M, \tilde{F})$ parameter space, for $\sigma_0 = s_1, s_2, s_3, s_4$. The values were calculated by substituting the numerical solutions for $g$ and $h$ into (\ref{['omega_0_formula']}), with $z_1 = 0.1, z_2 = 0.9$. The values for $Q_1=q_1t_1$ were obtained from the numerical data for $q_nt_n$, which can be found in Netherwood2023deformations.
• Figure 3: Solutions for $p_{10}^{(j)}(z)$ with $z_1=0.1, z_2=0.9, \sigma _0 = 0.6, \tilde{F} =1$, and different values of the inertia parameter $M$ and mode number $j$. The solutions were obtained by substituting the numerical data for $g$ and $h$ into the analytical solution (\ref{['general_solution_p_10']}) with (\ref{['A_p10']})--(\ref{['C_p10']}) for $p_{10}^{(j)}$.
• Figure 4: Solutions for $\epsilon p_{20}$, plotted for $M=0,0.01,0.1,1$ with $\tilde{F}=1$, and $\sigma_0= s_1,s_2,s_3,s_4$ with $z_1=0.1$ and $z_2=0.9$. The solutions were calculated using the expression (\ref{['p_20_solution']}). The coefficients present in the solutions were determined analytically in terms of the numerical constants $h,g$ and $\omega$ within Maple. The second panel corresponds to $\sigma_0=0.6$ and allows for a comparison with walters2018effect and whittaker2010predicting. The values for $\epsilon Q_2 = q_2t_2$ were obtained from the numerical data for $q_nt_n$, which can be found in Netherwood2023deformations.
• Figure 5: The values $\epsilon\omega_1$, plotted throughout $(M, \tilde{F})$ parameter space, for $\sigma_0 = s_1, s_2, s_3, s_4$. The values were calculated using the expression (\ref{['correction_oscillation_frequency']}) with $z_1 = 0.1, z_2 = 0.9$ and the values for $Q_1$ were obtained from the numerical data for $q_nt_n$, which can be found in Netherwood2023deformations