Growth mechanisms of perturbations in boundary layers over a compliant wall

Abstract

The temporal modal and nonmodal growth of three-dimensional perturbations in the boundary layer flow over an infinite compliant flat wall is considered. Using a wall-normal velocity/wall-normal vorticity formalism, the dynamic boundary condition at the compliant wall admits a linear dependence on the eigenvalue parameter, as compared to a quadratic one in the canonical formulation of the problem. As a consequence, the continuous spectrum is accurately obtained. This enables us to effectively filter the pseudospectra, which is a prerequisite to the transient growth analysis. An energy-budget analysis for the least-decaying hydroelastic (static-divergence, traveling-wave-flutter and near-stationary transitional) and Tollmien-Schlichting modes in the parameter space reveals the primary routes of energy flow. Moreover, the maximum transient growth rate increases more slowly with the Reynolds number than for the solid wall case. The slowdown is due to a complex dependence of the wall-boundary condition with the Reynolds number, which translates into a transition of the fluid-solid interaction from a two-way to a one-way coupling. Unlike the solid-wall case, viscosity plays a pivotal role in the transient growth. The initial and optimal perturbations are compared with the boundary layer flow over a solid wall; differences and similarities are discussed.

Effect of surfactant on the linear stability of a shear-imposed fluid flowing down a compliant substrate

“…As the present study deals with two-dimensional disturbance, the investigation of three-dimensional modal and non-modal stability analyses (Malik et al. 2018) as well as the investigation of nonlinear surface wave dynamics based on the depth averaged method (Ruyer-Quil & Manneville 2000) will be the possible extensions of the present work and are kept for future consideration.…”

“…Basically, their theoretical study was developed in exploring the travelling wave flutter instability pointed out by Carpenter & Garrad (1985, 1986). Recently, the temporal modal and non-modal growth of three-dimensional disturbances in the boundary-layer flow over an infinite compliant flat wall was deciphered by Malik, Skote & Bouffanais (2018) based on the normal velocity and normal vorticity formulations. As discussed by them, the maximum transient growth rate increases slowly with the Reynolds number in comparison with the rigid-wall case.…”

Modal analysis of a viscous fluid falling over a compliantwall

“…where ζ , E Y and ν are, respectively, the damping ratio, Young's modulus and the Poisson ratio of the compliant wall. Finally, at the compliant wall, the kinematic and no-slip boundary conditions are employed [12,21] v = ∂ t η, u = 0, w = 0, at y = η(x, z, t).…”

“…Carpenter & Gajjar [11] took the same problem further to develop the asymptotic analysis in boundary layers on anisotropic and isotropic flexible substrates for two-dimensional and three-dimensional perturbations. The non-modal and modal stability of three-dimensional perturbations in the boundary layer on an infinite flexible substrate was recently investigated by Malik et al [12] by using the formalism of normal velocity and normal vorticity, respectively. It was reported that the maximum energy growth rate enhances gradually as long as the Reynolds number amplifies.…”

A linear system for pipe flow stability analysis allowing for boundary condition modifications

“…However, this method comes with a task of changing the trial and test bases if the boundary conditions are other than no-slip. In some occasions, the boundary conditions are even time dependent (see for example, [23]). Finding appropriate solenoidal trial and test bases that satisfy arbitrary boundary conditions and the properties prescribed by the ansatz of Priymak & Miyazaki is not a simple task, in general.…”