Instabilities in a plane channel flow between compliant walls

Abstract

The stability of plane channel flow between compliant walls is investigated for disturbances which have the same symmetry, with respect to the channel centreline, as the Tollmien–Schlichting mode of instability. The interconnected behaviour of flow-induced surface waves and Tollmien–Schlichting waves is examined both by direct numerical solution of the Orr–Sommerfeld equation and by means of an analytic shear layer theory. We show that when the compliant wall properties are selected so as to give a significant stability effect on Tollmien–Schlichting waves, the onset of divergence instability can be severely disrupted. In addition, travelling wave flutter can interact with the Tollmien–Schlichting mode to generate a powerful instability which replaces the flutter instability identified in studies based on a potential mean-flow model. The behaviour found when the mean-flow shear layer is fully accounted for may be traced to singularities in the wave dispersion relation. These singularities can be attributed to solutions which represent Tollmien–Schlichting waves in rigid-walled channels. Such singularities will also be found in the dispersion relation for the case of Blasius flow. Thus, similar behaviour can be anticipated for Blasius flow, including the disruption of the onset of divergence instability. As a consequence, it seems likely that previous investigations for Blasius flow will have yielded very conservative estimates for the optimal stabilization that can be achieved for Tollmien–Schlichting waves for the purposes of laminar-flow control.

Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes

“…Nevertheless, onedimensional models can still provide useful insights, and capture the qualitative behaviour observed in higher dimensional models, as demonstrated in recent work by Stewart et al (2009). This paper also discusses the link between the widely studied 'local' eigenmodes (such as static divergence and travelling wave flutter) of flows in long flexible tubes and channels (Kumaran 1995;Davies & Carpenter 1997;Mandre & Mahadevan 2010) and the 'global' eigenmodes (dependent on upstream and downstream boundary conditions) arising in experimental systems such as the Starling resistor.…”

Enhanced effects from tiny flexible in-wall blips and shear flow

“…The function f in (2.4c) denotes the unknown scaled shape of the blip surface which is addressed further just below while the positive O(1) factor λ in (2.4d) is the given scaled incident WSS, namely Re −1/2 (∂u/∂y) at y = 0, in the surrounding boundary layer locally: see figure 1. Concerning (second) the blip surface shape as it responds to the fluid flow over the blip, the shape and the flow interact via the local pressure as in the models used by Carpenter & Garrad (1985), Davies & Carpenter (1997), Gajjar & Sibanda (1996), Pruessner (2013) and others. The assumptions made are primarily those of the widely used membrane-model type as in the references immediately above with particularly interesting background discussions of linearly elastic materials and allied facets relevant here being in Takagi & Balmforth (2011) as well as Xu, Billingham & Jensen (2014), Stewart, Waters & Jensen (2009), Vella, Kim & Mahadevan (2004, Singh, Lister & Vella (2014).…”

The cascade structure of linear instability in collapsible channel flows

“…As the Fourier transform picks up more than one mode, the modes that match with the eigensolutions are in bold. Davies & Carpenter (1997b), to be discussed later.…”

“…The work of Davies & Carpenter (1997b) suggests that the important structural properties which determine the system stability are tension and bending stiffness (the spring element in their model is absent here) for a massless wall. Using our nondimensional variables, their theory (see (34) in Davies & Carpenter 1997b) predicted that oscillations will occur with neutral frequency…”