The Use of Asymptotic Methods in Boundary-Layer and Interfacial Phenomena

Abstract

We present four studies that use asymptotic methods to understand physical phenomena in boundary-layer and interfacial flows. First, we present solutions for the shapes of static equilibria in rectangular channels that have been perturbed by isolated ridges and grooves and by scattered bump protrusions and intrusions. We solve the Young--Laplace equation to quantify the sensitivity of the meniscus shape to the perturbations using a combination of numerical computations and asymptotic techniques for a linearised model when the amplitude of the perturbations is small relative to the channel height. For small-amplitude ridge/groove and bump perturbations, we derive an equation for the induced pressure difference over the meniscus that depends solely on the boundary data. Thus the total pressure difference over the meniscus (and therefore the mean curvature) can be found without solving the Young--Laplace equation. Mirror symmetric ridge and bump perturbations which change the volume of the channel cause a change in the mean curvature of the meniscus which leads to long-range curvature of the contact line. For ridge perturbations, we show that this long-range curvature matches onto the contact line of a droplet with the same mean curvature as the meniscus. We use this information to choose specific combinations of perturbations to engineer contact line shapes. We present preliminary results for bump intrusions/protrusions which show that the direction of deformation of a meniscus changes as it passes over a bump. Next, we present an asymptotic description of nonlinear equilibrium and travelling-wave solutions of the Navier--Stokes equations in incompressible unsteady boundary-layer and compressible parallel boundary-layer flows. The solutions take the form of self-sustaining vortex-wave interaction-type states, known as free-stream coherent structures, with the nonlinear interaction between the vortex and the wave taking place in a layer close to the free-stream. The interaction produces streaky disturbances that can grow exponentially due to interaction with the base flow. We first extend the asymptotic theory of Deguchi & Hall (2014a) to show that free-stream coherent structures can be embedded in unsteady two-dimensional boundary layers. The time evolution of the structure is affected strongly by the unsteady base flow, and ultimately it can only persist for a finite time. Next, we describe free-stream coherent structures in compressible parallel boundary layer flows in the subsonic and moderate supersonic regimes. These flows are more industrially relevant to laminar flow control than the previously studied incompressible flows. The key result is that the equations for the nonlinear interaction of the vortex and wave in the layer near the free-stream are identical to those obtained in the incompressible problem, but the velocity field now also drives a passive thermal field. The resulting disturbances to both the velocity and temperature fields can then grow exponentially towards the wall; the maximum amplitude of the disturbances depends on the Mach number and the Prandtl number.

Date of Award	31 Dec 2022
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Andrew Hazel (Supervisor) & Oliver Jensen (Supervisor)