Modelling and analysis of the global stability of Blasius boundary-layer flow interacting with a compliant wall

Abstract

Theoretical and experimental studies have shown that compliant walls are able to reduce the growth rates of unstable Tollmien-Schlichting waves (TSWs) that are the conventional route to boundary- layer transition in low-disturbance environments. Accordingly, transition can be postponed by an appropriately designed compliant coating adhered to an otherwise rigid surface, thereby leading to a potentially significant reduction of skin-friction drag in marine applications. The more compliant the wall, the greater is the suppression of TSWs. However, the compliant wall can also support unstable wall-based waves that typically occur when the wall is too compliant and thereby undermine the overall flow- stabilization strategy. Accordingly, when designing useful complaint coatings, it is necessary to take into account all of the possible instabilities of the fluid-structure interaction (FSI) system. The majority of previous studies utilize local-stability analyses based upon the assumptions of an infinitely long compliant wall and parallel-flow to identify and characterise the system instabilities while numerical simulation has been used for walls of finite extent. In contrast, we carry out a bi-global linear stability analysis in the present study of the FSI system. We model the flow using a combination of vortex and source boundary-element sheets on a computational grid while the dynamics of a plate-spring, Kramer-type, complaint wall are represented in finite-difference form. The assembled FSI system is then couched as an eigenvalue problem and the eigenvalues of the various flow- and wall-based instabilities are analyzed for a range of system parameters. The key findings of the study are that coalescence - or resonance - of one of the structural eigenmodes with either the most unstable TSW or a travelling-wave flutter (TWF) mode can occur. This renders the convective nature of these instabilities to become global for a finite compliant wall. A local analysis of the temporally unstable modes shows that besides the TSW and TWF modes, a divergence-type mode associated with the structural behaviour can additionally yield global instability. Finally, a non-modal analysis reveals that the behaviour of flow-based instabilities over a structurally damped compliant wall in response to an initial disturbance shows slightly lower transient growth and energy advection than occurs over a rigid wall in the sub-resonance combination of wall and flow parameters. However, for system conditions that yield resonance-type behaviour, transient growth is significantly larger than that which occurs over a rigid wall.