The interaction of Blasius boundary-layer flow with a compliant panel: global, local and transient analyses

Abstract

We study the fluid–structure interaction (FSI) of a compliant panel with developing Blasius boundary-layer flow. The linearised Navier–Stokes equations in velocity–vorticity form are solved using a Helmholtz decomposition coupled with the dynamics of a plate-spring compliant panel couched in finite-difference form. The FSI system is written as an eigenvalue problem and the various flow- and wall-based instabilities are analysed. It is shown that global temporal instability can occur through the interaction of travelling wave flutter (TWF) with a structural mode or as a resonance between Tollmien–Schlichting wave (TSW) instability and discrete structural modes of the compliant panel. The former is independent of compliant panel length and upstream inflow disturbances while the specific behaviour arising from the latter phenomenon is dependent upon the frequency of a disturbance introduced upstream of the compliant panel. The inclusion of axial displacements in the wall model does not lead to any further global instabilities. The dependence of instability-onset Reynolds numbers with structural stiffness and damping for the global modes is quantified. It is also shown that the TWF-based global instability is stabilised as the boundary layer progresses downstream while the TSW-based global instability exhibits discrete resonance-type behaviour as Reynolds number increases. At sufficiently high Reynolds numbers, a globally unstable divergence instability is identified when the wavelength of its wall-based mode is longer than that of the least stable TSW mode. Finally, a non-modal analysis reveals a high level of transient growth when the flow interacts with a compliant panel which has structural properties capable of reducing TSW growth but which is prone to global instability through wall-based modes.

Global stability and resolvent analyses of laminar boundary-layer flow interacting with viscoelastic patches

“…Note that the frequency of the coupled modes is slightly shifted towards lower frequencies because of added mass effects, compared with in vacuo frequencies. This part of the spectrum is in qualitative agreement with what was observed in the global mode analysis by Tsigklifis & Lucey (2017) in the case without damping: one branch of unstable modes was found in approximately the same frequency range. In the present case, more than one branch of structural modes is observed, as reported in figure 6: a second branch of structural modes is found at frequencies higher than approximately , since our solid model enables to carry many more types of modes than the spring-backed solid considered by Tsigklifis & Lucey.…”

“…Previous theoretical studies have addressed the problem either by assuming an infinite-length coating (Duncan, Waxman & Tulin 1985; Duncan 1988), or a finite-length coating having a very simple behaviour (spring-backed solid) coupled with inviscid flows (Lucey & Carpenter 1992, 1993) or linearised Navier–Stokes flows (Davies & Carpenter 1997; Stewart, Waters & Jensen 2009; Tsigklifis & Lucey 2017). On the other hand, the ability of compliant coatings to effectively delay transition in a practically interesting way is still debated (see Luhar, Sharma & McKeon (2015) for a review of some of the most recent advances).…”

“…On the other hand, TSW are more naturally analysed in terms of energy amplification of external disturbances. This dual approach was for instance adopted recently by Tsigklifis & Lucey (2017), who performed modal and transient growth analyses for the case of a viscous boundary-layer flow interacting with a finite-length coating modelled by a spring/damper-backed plate equation. In the same spirit, we consider here both modal, eigenvalue fluid-elastic analyses, such as that presented in Pfister, Marquet & Carini (2019), and an energy-amplification-based approach relying on the resolvent analysis.…”

Fluid–structure stability analyses and nonlinear dynamics of flexible splitter plates interacting with a circular cylinder flow

“…A linear and nonlinear analysis of the dynamics of an inverted-flap flapping in a low Reynolds number flow was also performed by Goza, Colonius & Sader (2018). The effect of a compliant wall on the growth of perturbations developing in a Blasius boundary layer was considered investigated by Tsigklifis & Lucey (2017) with modal and non-modal linear stability analyses of the fluid-structure interaction. In all of these studies, the elastic thin structure was modelled with a one-dimensional elastic beam.…”

Stability of fluid flows coupled by a deformable solid layer

“…This assumption is necessary for using the normal mode analysis employed in the present study. However, if the widths of the fluids and length of the DS are of comparable magnitude, then a fully global stability analysis is required (Theofilis 2011; Tsigklifis & Lucey 2017).…”