Part I: Viscous evolution of point vortex equilibria; Part II: Effects of body elasticity on stability of fish motion

2011-08-08Vortex dynamics and solid-fluid interactions are two of the most important and most studied topics in fluid dynamics for their relevance to a wide range of applications from geophysical flows to locomotion in moving fluids. In this work, we investigate two problems in two parts: Part I studies the viscous evolution of point vortex equilibria; Part II studies the effects of body elasticity on the passive stability of submerged bodies. ❧ In Part I, we describe the viscous evolution of point vortex configurations that, in the absence of viscosity, are in a state of fixed or relative equilibrium. In particular, we examine four cases, three of them correspond to relative equilibria in the inviscid point vortex model and one corresponds to a fixed equilibrium. Our goal is to elucidate the dominant transient dynamical features of the flow. A multi- Gaussian “core growing” type of model is typically used in high fidelity numerical simulations, but we propose to implement it as a low-order model for the flow field. We show that all four configurations immediately begin to rotate unsteadily. We then examine in detail the qualitative and quantitative evolution of the structures as they evolve, and for each case show the sequence of topological bifurcations that occur both in a fixed reference frame, and in an appropriately chosen rotating reference frame. Comparisons between the cases help to reveal different features of the viscous evolution for short and intermediate time scales of vortex structures. We examine the dynamical evolution of passive particles in the viscously evolving flows and interpret it in relation to the evolving streamline patterns. Although the low-order multi-Gaussian model does not exactly coincide with the Navier-Stokes solution, the two results show remarkable resemblances in many aspects. ❧ In Part II, we examine the effects of body geometry and elasticity on the passive stability of motion in a perfect fluid. Our main motivation is to understand the role of body elasticity on the stability of fish swimming. The fish is modeled as an articulated body made of N links (assumed to be identical ellipses in 2D or identical ellipsoids in 3D) interconnected by hinge joints. It can undergo shape changes by varying the relative angles between the links. Body elasticity is accounted for via the torsional springs at the joints. The unsteadiness of the flow is modeled using the added mass effect. Equations of motion for the body-fluid system are derived using Newtonian and Lagrangian approaches for both hydrodynamically decoupled and coupled models in 2D and 3D. We specifically examine the stability associated with a relative equilibrium of the equations, traditionally referred to as the “coast motion” (proved to be unstable for a rigid elongated body model), and found that body elasticity does stabilize the system. Stable regions are identified based on linear stability analysis in the parameter space spanned by aspect ratio (body geometry) and spring constants (muscle stiffness), and the findings based on the linear analysis are verified by direct numerical simulations of the nonlinear system. Our result shows that even in the absence of fin movement and vortex interaction, shape change and muscle elasticity allow stable conditions of the coast motion