Mathematical modeling and simulation of aquatic and aerial animal locomotion

In this thesis we investigate the locomotion of fish and birds by applying both new and well known mathematical techniques. The two-dimensional model is first studied using Krasny’s vortex blob method, and then a new numerical method based on Wu’s theory1 is developed. To begin with, we will implement Krasny’s ideas for a couple of examples and then switch to the numerical implementation of the nonlinear analytical mathematical model presented by Wu. We will demonstrate the superiority of this latter method both by applying it to some specific cases and by comparing with the experiments. The nonlinear effects are very well observed and this will be shown by analyzing Wagner’s result for a wing abruptly undergoing an increase in incidence angle, and also by analyzing the vorticity generated by a wing in heaving, pitching and bending motion. The ultimate goal of the thesis is to accurately represent the vortex structure behind a flying wing and its influence on the bound vortex sheet. In the second part of this work we will introduce a three-dimensional method for a flat plate advancing perpendicular to the flow. The accuracy of the method will be shown both by comparing its results with the two-dimensional ones and by validating them versus the experimental results obtained by Ringuette in the towing tank of the Aeronautics Department at Caltech. In this thesis we will use ’Krasny’s method’ to refer to the method based on the vortex blob technique and ’Wu’s method’ to refer to the one implemented by Wu. Since we are not aware of an actual method developed by Krasny for an airfoil moving forward at a given angle of attack, we should rather use ’classical vortex method’ and ’semi-analytical method’ respectively to describe these two implementations. However, we will talk about ’Krasny’s method’ and ’Wu’s method’ for easier understanding.