Global instabilities and transient growth in Blasius boundary-layer flow over a compliant panel

Abstract

Abstract. We develop a hybrid of computational and theoretical approaches suited to study the fluid-structure interaction (FSI) of a compliant panel, flush between rigid upstream and downstream wall sections, with a Blasius boundary-layer flow. The ensuing linear-stability analysis is focused upon global instability and transient growth of disturbances. The flow solution is developed using a combination of vortex and source boundary-element sheets on a computational grid while the dynamics of a plate-spring compliant wall are couched in finite-difference form. The fully-coupled FSI system is then written as an eigenvalue problem and the eigenvalues of the various flow-and wall-based instabilities are analysed. It is shown that coalescence or resonance of a structural eigenmode with either a flow-based Tollmien-Schlichting Wave (TSW) or wall-based travelling-wave flutter (TWF) modes can occur. This can render the nature of these well-known convective instabilities to become global for a finite compliant wall giving temporal growth of system disturbances. Finally, a non-modal analysis, based on the linear superposition of the extracted temporal modes is presented. This reveals a high level of transient growth when the flow interacts with a compliant panel that has structural properties which render the FSI system prone to global instability. Thus, to design stable finite compliant panels for applications such as boundary-layer transition postponement, both global instabilities and transient growth must be taken into account.

Growth mechanisms of perturbations in boundary layers over a compliant wall

“…Due to the presence of such hydroelastic modes, even at low Reynolds numbers, compliant panels-with streamwise-length optimized between the growing hydroelastic modes and inhibited Tollmien-Schlichting modes-are prescribed for the purpose of drag-reduction [10,17]. Recently, [18] has studied the global (temporal) traveling-wave-flutter modes in the presence of finite-size compliant panels through a hybrid numerical technique where the temporal eigensystem is provided with an input of spatial eigenvalues from a separate spatial stability calculation. They have also studied the transient growth of a superposition of normal modes.…”

“…In this paper, we perform for the first time a transient growth analysis of the problem of hydrodynamic stability of a zero-pressure gradient boundary layer flow over a flat plate with a normal-velocity/normal-vorticity formalism for the study of three-dimensional (3D) modes. This formalism yields a tractable linear-eigenvalue problem without the need to resort to the traditional companion-matrix formulation, hence avoiding an unnecessary increase in the number of unknowns (see e.g., [16,[18][19][20]). Our new alternative formalism results in the linear appearance of an otherwise quadratic form of the eigenvalue parameter.…”