Two Tales of Nonlinear Wave Motion: Wave Synchronization across a Fluid-Structure Interface and Wave Dispersion under Finite Strain
Mahmoud I. Hussein
Department of Aerospace Engineering Sciences, University of Colorado Boulder
Date and time:
Tuesday, October 13, 2015 - 4:00pm
Location:
ECOT 226
Abstract:
Wave motion lies at the heart of many disciplines in the physical sciences and engineering. For example, problems and applications involving light, sound, heat or fluid flow are all likely to involve wave dynamics at some level. In this seminar, I will present our recent work on two problems involving intriguing nonlinear wave phenomena.
The first topic is concerned with passive wave motion across a fluid-structure interface, for example, flow inside a pipeline or over an aircraft wing. There has been extensive research investigating nonlinear wave propagation in a laminar flow, such as Tollmien–Schlichting instability waves in a closed channel or a boundary layer. Similarly, there are numerous studies in the literature on elastodynamic waves in a solid, including near a surface or at an interface. However, I will argue, it is the coherent merging of wave fields across the fluid and solid media that presents the hardest challenge and offers the largest reward, if successful. In this context, we have demonstrated, by theory-guided design and simulations, that placing an enlarged crystal (a phononic crystal) underneath a flexible surface attached to a flow enables wave synchronization across the interface. The driving theory provides an accurate prediction of this behavior without the need for a flow simulation. This unprecedented capability allows us to achieve favorable, and predictable, alterations to the flow properties with consequences on drag reduction for air, sea and land vehicles, as well as numerous other applications.
The second topic examines the propagation of a large-amplitude wave in an elastic one-dimensional medium with a focus on the effects of inherent nonlinearities on the dispersion relation. Considering a thin rod, where the thickness is small compared to the wavelength, I will present an exact formulation for the treatment of two types of nonlinearity in the strain-displacement gradient relation: Green Lagrange and Hencky. The derivation starts with an implementation of Hamilton’s principle and terminates with an expression for the finite-strain dispersion relation in closed form. The derived relation is then verified by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) and short-term, pre-breaking dispersion (by Fourier transformations), as well as by perturbation theory. The results establish a perfect match between theory and simulation and reveal that an otherwise linearly nondispersive elastic solid may exhibit dispersion solely due to the presence of a nonlinearity. The same approach is also applied to flexural waves in an Euler Bernoulli beam, demonstrating qualitatively different stronger nonlinear dispersive effects compared to longitudinal waves. If time permits, I will briefly also present a method for extending this analysis to a continuous periodic thin rod. The method, which is based on a standard transfer matrix augmented with a nonlinear enrichment at the constitutive material level, yields an approximate band structure that accounts for the wave amplitude.