Drag reduction by self-similar bending and a transition to forward flight by a symmetry-breaking instability.

摘要

The behavior of fluids and macroscopic solids in contact is fundamental to many phenomena in biology. In particular, the biomechanical structures of plants can be understood in terms of the fluid forces they must withstand to survive. Flexibility plays a dominant role, as the reconfiguration of a flexible body results in substantial reduction of fluid drag. The first part of this thesis considers this phenomenon in terms of two-dimensional free-streamline flows past a one-dimensional elastic body. We solve the coupled fluid-elastic equations numerically. At large flow speeds, a shape self-similarity emerges, with a scaling set by the balance of forces in a small "tip" region located at the body's support. The result is a transition from the quadratic scaling of drag with flow speed for rigid bodies to a new 4/3-power scaling as the body reconfigures.; We derive these behaviors in terms of an asymptotic expansion based on the length-scale of similarity. This analysis predicts that the body and wake are quasiparabolic at large velocities, and obtains the new drag law in terms of the drag on the tip region. We also consider variations of the model suggested by experiments, and find that the 4/3-drag-law persists with a simple modification.; The second part of this thesis considers dynamical fluid-body coupling. Many organisms locomote by flapping a wing or fin transverse to the body's direction of motion. A recent experiment has shown that a horizontal motion can arise spontaneously from vertical flapping as a symmetry-breaking bifurcation. Here we solve the Navier-Stokes equations for the flow induced by a flapping ellipse. We find a critical flapping Reynolds number above which the ellipse is unstable to horizontal motions. Just above the critical Reynolds number, the instability yields a quasi-periodic, back-and-forth oscillation with zero mean horizontal velocity. For larger Reynolds number, the body can enter a steady horizontal motion, characterized by a reverse von-Karman vortex street and a Strouhal number of 0.3. These phenomena are consistent with the experiment as well as efficient animal locomotion.