Extension of Lighthill's slender-body theory to moderate aspect ratios

Calculating the fluid forces acting on a moving body at high Reynolds number is crucial in many fluid-structure interaction problems, such as fish swimming or flutter instabilities. To estimate these forces, Lighthill developed the slender-body theory, which assumes a potential flow and an asymptotically small aspect ratio. Yet, it is still unclear whether Lighthill's theory is still valid for aspect ratios of order one. To address this question, we solve numerically with a panel method the full three-dimensional problem of a rectangular plate deforming periodically in a potential flow. These numerical simulations are used to calculate the pressure jump distribution across the plate for different aspect ratios. We find that numerical simulations and slender-body theory give similar results far from trailing edge. Close to the trailing edge however, there is a discrepancy, which is due to the use of a Kutta condition in the simulations (i.e. no pressure jump at the trailing edge), while, in the slender-body theory, the pressure jump is non zero. We propose a simple extension of Lighthill's slender-body theory that accounts for this discrepancy. The usefulness of this extension is then discussed and illustrated with a generic fluid-structure interaction problem and with the flag instability problem.