Research and scholarship
Research and scholarship
My research is still concerned with instabilities in fluid flows, and has included thermal boundary layers, three-dimensional boundary layers and swirling jets. I use asymptotic techniques to develop theories in large-Reynolds number and long-wavelength limits, and also to obtain large-time descriptions of initial value problems for determining absolute and convective instability characteristics. I also use numerical methods for solving viscous and inviscid stability equations.
Most of my research papers can be divided between three main areas.
1.Time series analysis
In the qualitative theory of dynamical systems one can represent complicated behaviour in terms of a state space, or phase space. The instantaneous state of the system is represented by a point in this space, and the evolution of the system is represented by the trajectory taken by this point. Thus, an equilibrium state is represented by a fixed point, a periodic behaviour by a closed orbit, a quasi-periodic behaviour by a path on a torus and chaotic behaviour by a path on a strange attractor with fractal properties. Remarkably, this trajectory, called a phase portrait, can often be reconstructed from a series of measurements of a single characteristic of the system (a time series).
Summary of results and selected publications in Time Series Analysis.pdf
2. The Blasius boundary layer
When a fluid flows at high speed past a solid surface a boundary layer forms in the fluid close to the surface. It is a region where viscosity is important and arises from the no-slip boundary condition on the fluid at a solid surface. Boundary layers are present however small the viscosity. In the absence of viscosity objects moving through a fluid experience no drag and no lift. It is the presence of boundary layers that always produces drag on a body and can also produce lift. A boundary layer can separate from a surface, dramatically increasing the drag (as when an aeroplane wing stalls). When the boundary layer is attached (like around a streamlined body) the amount of drag, and also the heat transfer characteristics, depends sensitively on whether the boundary layer is laminar or turbulent. Being able to predict the state of a boundary layer is of importance in the design of aeroplane wings and turbine blades in jet engines. Fuel consumption of commercial aircraft could be halved, and their range doubled, if laminar boundary layers could be maintained over their wings. However, laminar-turbulent transition, and turbulence, remain major outstanding problems in fluid mechanics, and affect many other flows in many other situations.
The simplest boundary layer arises when a flat plate is placed parallel to a uniform stream. This is called the Blasius boundary layer, and has been much studied experimentally, numerically and theoretically. Although it doesn't often arise in practical applications, and despite having certain peculiarities (an inflexion point at the wall), a greater understanding of its laminar-turbulent transition is expected to be helpful in understanding other boundary layers.
Summary of results and selected publications in The Blasius boundary layer.pdf
3. Absolute and convective instabilities
Unstable disturbances in a shear layer, like a boundary layer, might all propagate downstream, in which case the flow is called convectively unstable. If unstable disturbances travel both upstream and downstream in the shear layer then the flow is called absolutely unstable. Although this classification depends on the velocity of the reference frame being used, there is usually a frame of particular interest, e.g. the laboratory frame, or the frame moving with an aeroplane wing, and then the distinction is of crucial importance in determining the dynamics of the flow. Convectively unstable flows act as spatial amplifiers of whatever external disturbances are imposed on the flow. Absolutely unstable flows can generate their own intrinsic modes which are insensitive to the disturbance environment, and thus behave as self-excited oscillators. These modes are called global modes and arise through an interplay of local instability characteristics, nonlinearity and weak inhomogeneity of the basic flow in the streamwise direction. From a practical point of view, an unstable flow that is only convectively unstable will remain in a laminar state if the freestream turbulence level is low enough, but an absolutely unstable flow giving rise to a global mode will have the original basic flow replaced by the global mode regardless of how small the external disturbance levels are.
The local absolute/convective characteristics are obtained by assuming that the local basic velocity profile is independent of the streamwise coordinate and examining the solution produced by an impulsive localized disturbance to an otherwise undisturbed basic flow (the Green's function) at large times. The flow is absolutely unstable if there is growth in time in the rest frame. This behaviour can be determined using residue theory and saddle-point methods on the integrals appearing in the inverse Fourier-Laplace-type transforms of the initial value problem. At large times these integrals are dominated by the contribution from the highest saddle point whose valleys contain the real wavenumber axis. (In the Briggs-Bers interpretation, this dominant saddle-point is called the 'pinch-point' and represents a coalescence of upstream and downstream travelling waves). Contour deformations in the complex wavenumber plane are required to locate the dominant saddle, and in the case of convective instability this also gives information about the direction of propagation of waves (the so-called 'signalling problem').
Summary of results and selected publications in this area.pdf
