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.