Flow-Induced Flapping Foil Instability.

We have conducted a computational investigation of two-dimensional self-excited flapping-foil instability in the channel formed by two parallel plates, up to Reynolds numbers (based on average velocity and plate separation) of 400, and the effects on heat transfer. Our approach, for a massless foil, spatially discretizes the incompressible Navier-Stokes equations by a spectral-element method, while the dynamical equation describing foil deformation is discretized by a p-type finite element method. The spectral-element mesh for fluid temporally moves, based on an arbitrary Lagrangian-Eulerian (ALE) formulation, so that zero-thickness foil is always attached to the mesh and the no-slip condition is satisfied. Nonlinear terms in the incompressible Navier-Stokes equations are advanced explicitly in time, while linear terms are advanced implicitly. The structural equations are advanced in a weakly coupled sense, consistent with semi-implicit temporal discretization of Navier-Stokes, and the ALE moving-mesh strategy. We update the flow from Navier-Stokes, with a prescribed zero relative velocity on the foil, by superposing a series of Stokes solutions, each representing unit acceleration for one discretized degree of freedom of the foil, and weighted to satisfy the structural equation. This approach efficiently avoids the numerical instability of traditional partitioned fluid-structure interaction algorithms in the zero-mass case.