Wave-mean flow interactions: from nanometre to megametre scales
Waves, which arise when restoring forces act on small perturbations, are ubiquitous in fluids. Their counterpart, mean flows, capture the remainder of the motion and are often characterised by a slower evolution and larger scale patterns. Waves and mean flows, which are typically separated by time- or space-averaging, interact, and this interaction is central to many fluid-dynamical phenomena. Wave-mean flow interactions can be classified into dissipative interactions and non-dissipative interactions. The former is important for small-scale flows, the latter for large-scale flows. In this thesis these two kinds of interactions are studied in the context of microfluidics and geophysical applications. Viscous wave-mean flow interactions are studied in two microfluidic problems. Both are motivated by the rapidly increasing number of microfluidic devices that rely on the mean-flow generated by dissipating acoustic waves - acoustic streaming - to drive small-scale flows. The first problem concerns the effect of boundary slip on steady acoustic streaming, which we argue is important because of the high frequencies employed. By applying matched asympototics, we obtain the form of the mean flow as a function of a new non-dimensional parameter measuring the importance of the boundary slip. The second problem examined is the development of a theory applicable to experiments and devices in which rigid particles are manipulated or used as passive tracers in an acoustic wave field. Previous work obtained dynamical equations governing the mean motion of such particles in a largely heuristic way. To obtain a reliable mean dynamical equation for particles, we apply a systematic multiscale approach that captures a broad range of parameter space. Our results clarify the limits of validity of previous work and identify a new parameter regime where the motion of particles and of the surrounding fluid are coupled nonlinearly. Non-dissipative wave-mean flow interactions are studied in two geophysical fluid problems. (i) Motivated by the open question of mesoscale energy transfer in the ocean, we study the interaction between a mesoscale mean flow and near-inertial waves. By applying generalized Lagrangian mean theory, Whitham averaging and variational calculus, we obtain a Hamiltonian wave-mean flow model which combines the familiar quasi-geostrophic model with the Young & Ben Jelloul model of near-inertial waves. This research unveils a new mechanism of mesoscale energy dissipation: near-inertial waves extract energy from the mesoscale ow as their horizontal scale is reduced by differential advection and refraction so that their potential energy increases. (ii) We study the interaction between topographic waves and an unidirectional mean flow at an inertial level, that is, at the altitude where the Doppler-shifted frequency of the waves match the Coriolis parameter. This interaction can be described using linear theory, using a combination of WKB and saddle-point methods, leading to explicit expressions for the mean-flow response. These demonstrate, in particular, that this response is switched on asymptotically far downstream from the topography, in contrast to what is often assumed in parameterisation.
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