# High-Reynolds-Number Fluid Flow

Welcome to the high-Reynolds-number fluid flow group page. We are a small group consisting of two members at present (although we have a number of alumni):

This page is effectively a marker and rather out-of-date; it is still very much under construction. However, a number of people (including prospective students) have complained about the fact that we have no page! If you just want to read more WWW pages written by Stephen Cowley, then most of the Faculty of Mathematics pages (with the exception of the undergraduate pages) were initially written by him.

High-Reynolds-number fluid mechanics includes:

• the study of the drag, lift and/or heat-transfer properties of cars, ships, submarines, turbine blades and aircraft wings/fuselages,
• the forces experienced by buildings and other fixed structures (e.g. the legs of oil rigs), and
• the flow in the larger blood vessels.

In many of these examples separation/stall, the growth of instabilities leading to turbulence, and vortex-sheet roll-up are important phenomena.

One approach to understanding high-Reynolds-number flows is to construct asymptotic descriptions exploiting the inverse Reynolds number as a small parameter. Although systematic asymptotic analysis can sometimes be rather `elaborate', the resulting simplified equations capture essential physics, focus attention on key areas of the flow, and often have wide applicability (especially when combined with numerical solutions of the reduced equations).

Separation. When fluid flows rapidly past a rigid body, thin layers of high vorticity develop adjacent to the body surface. These `boundary layers' need not remain attached to the surface of the body - they can separate. In aerodynamics this process is often referred to as stall. A description of the start of three-dimensional unsteady separation has been obtained. Interests include the relationship between separation and (i) phenomena that arise in the interior of turbulent boundary layers, and (ii) cavitation.

Transition to Turbulence. Transition to turbulence is important because it can dramatically affect key properties of a flow, e.g. the drag on a body. Studies have focussed:

• on the growth of nonlinear Tollmien-Schlichting waves in the context of `bypass' transition (this work is also related to unsteady separation);
• on the growth of oblique waves in temporally and spatially developing boundary layers and shear layers;
• on the `phase-locked' mechanism of instability (in conjunction with Xuesong Wu of Imperial College).

Inter alia, the relationship between a number of different asymptotic approaches has been identified, and key assumptions highlighted. In addition:

• receptivity problems have been studied in order to examine how disturbances can enter boundary layers and influence transition to turbulence;
• similar asymptotic techniques have been used to study vortex breakdown (a phenomenon which fixes inter-aircraft distances near airports).

Vortex Sheets. Infinitely thin vortex sheets have been used for many practical calculations, such as the behaviour of wakes behind aircraft wings. However, because of Kelvin-Helmholtz instabilities, such models are ill-posed, and are often only valid for a limited time due to the formation of singularities. Such limitations apply even when there is a stable density jump across the vortex sheet. Asymptotic solutions have given an important insight into the nature of these ubiquitous singularities. Moreover, by introducing various regularisations into the idealised vortex sheet model, e.g. surface tension and/or viscosity, it should be possible to understand analytically how vortex sheets roll-up in spirals. Since (i) the thin shear layers that repeatedly form in high-Reynolds-number turbulent flow will be subject to Kelvin-Helmholtz instability, and (ii) it has been argued that spiral structures are generic features of turbulent flow, one of the aims of this study is to provide an insight into a fundamental building block of turbulence.

© Stephen J. Cowley, DAMTP, University of Cambridge.