The energetics of self-excited oscillations in collapsible channel flows

In this thesis, we study the energetics of two-dimensional flow through a flexible-walled channel, where we mainly consider two models.The first model we consider is a fluid-membrane model in a long domain where the upper wall is replaced by elastic membrane under external pressure. The normal viscous stress, wall damping, wall inertia and membrane tension are all included in the membrane equation. We establish the corresponding eigenvalue problem of this model and trace the neutrally stable curves of this system across the parameter space. In agreement with previous work, we identify three different modes of instability (i:e: Tollmien-Schlichting waves (TS), traveling-wave flutter (TWF) and static divergence (SD) waves). We classify these instabilities into two classes (i:e: class A and class B form Benjamin [3]) using wall damping. Class A waves are destabilised by wall damping while class B waves are stabilised by wall damping. Furthermore, we consider the energy budget of the fully nonlinear system as well as that of the linearised system in order to determine whether the energy budget can be used to distinguish these different classes of instability. We found that the concept of ‘activation energy’ that connects with instability mode classification (Landahl [45], Cairns [17]) is not easily identified with terms in our energy budget. In particular, this wave energy is not equal to the work done by the fluid on the wall in our energy budget, as has previously been attributed to TWF. The second model we consider is a finite length fluid-beam model formed from a two dimensional channel, where one segment of the upper wall is replaced by a plane strained elastic beam subject to an external pressure. A parabolic inlet flow with constant flux is driven through the channel. We apply the finite element method with adaptive meshing to solve the fully nonlinear system numerically. We demonstrate the stability of the system after small stimulation, where the system exhibit large amplitude self-excited oscillations. In addition, large amplitude vorticity waves are found in the downstream segment of the flexible wall. The energy budget of this fully nonlinear system is calculated; the energy budget of the system balances the kinetic energy, the rate of working of external pressure and the dissipation energy over one oscillation. Moreover, we form the corresponding eigenvalue problem of the fluid-beam model by linearising the system about the corresponding static state to second order. A finite element method (similar to that of the fully nonlinear system) is employed to solve for the linear eigenfunctions. The observation of stability calculated from the eigenvalue problem are consistent with that calculatedfrom the fully nonlinear problem. We identify the stability of the system and establishthe neutral stability curve in the parameter space spanned by the beam extensional stiffness and Reynolds number. Two modes of instabilities are identified (i:e: mode-2 and mode-3, here the system is mode-i when the oscillation to the elastic beam contains i-half wavelengths). Finally, we derive the energy budget of the linearised system at second order. The energy budget of the linearised system exhibits a balance between the averaged second order dissipation energy, the work done by non-linear Reynolds stresses and the rate of working of perturbation fluid stress on the elastic wall. We anticipate that the precise balance of energy might serve as a robust method to distinguish the different modes of oscillation, although this has yet to be confirmed.