Supervisor: Dr G. P. King (School of Engineering, Warwick)
The explosive or subcritical onset of turbulence is an important unsolved problem in fluid dynamics. By subcritical, we mean that a flow passes abruptly from a laminar or spatially ordered state to turbulence when the Reynolds number is increased past a critical value. Turbulent bursts take the form of isolated spots or patches of turbulence that are randomly created in time and space. In open flows, such as channel and pipe flows and boundary layer flows over flat plates, the turbulent bursts decay or advect out of the experimental apparatus. In closed systems, such as Rayleigh-Benard convection or the flow between rotating cylinders (Taylor-Couette flow), the turbulent bursts are characterised by a decay time that becomes longer as a critical Reynolds number is approached, reminiscent of phase transition behaviour. Furthermore, for certain parameter values the bursting takes the form of a repetitive laminar-turbulent burst cycle.
An understanding of the burst cycle in Taylor-Couette flow is beginning to emerge from the Navier-Stokes simulations of Coughlin & Marcus. They propose the following mechanism based on their calculations and observations in H L Swinney's laboratory at the University of Texas. Counter-rotating cylinders produce a nodal surface r = r* where the azimuthal velocity is zero. At sufficiently large inner cylinder speeds, the fluid in the inner layer r < r* is centrifugally unstable to a spiral vortex flow, while the shear flow in the outer layer is centrifugally stable. At a sufficiently large fixed speed of the inner cylinder, a secondary linear instability sets in giving rise to an azimuthally travelling modulation of the primary spiral vortex flow near the nodal surface. The modulation grows in size and extends into the region r > r*. This growing modulation takes on the role of a finite amplitude perturbation to the shear layer in the outer region and triggers a burst through a shear instability. When the burst depletes the energy of the outer region, the three-dimensional flow structure collapses to the laminar state, and the process begins again. In terms of a phase transition analogy, the burst cycle may correspond to a "nonlinear feedback of the order parameter onto the control parameter" scenario for self-organised criticality (SOC).
Turbulent bursts (of energy and mass) also occur in plasma confinement systems, such as the earth's magnetosphere, the solar corona, accretion disks and, in the laboratory, tokamaks.
Mathematical projects will involve a combination of analytic and numerical work.
Experimental projects will investigate investigate the turbulent bursts by making velocity field measurements using particle imaging velocimetry and/or ultrasound velocimetry - two different methods for measuring the space-time evolution of the fluid.
Here is an example of an ultrasound velocity measurement taken from work with Dr Y. Takeda (PSI, Switzerland) using an ultrasound velocity profiler (UVP). This instrument measures velocities in the ultrasound beam direction at 128 spatial positions "instantaneously" as a function of time. Doppler methods are employed that allows the sign of the velocity to be determined (i.e., whether the motion is directed away or towards the ultrasound transducer). The following figure shows a plot of the velocity data obtained from the UVP for wavy Taylor-Couette flow. The vertical axis is space (axial position) and the horizontal axis is time. Red and yellow indicate positive velocities (motion away from the detector) and green and blue the opposite.
PhD projects suitable for both mathematical and experimentally inclined students are offered.
The long-range transport and mixing of material tracers in stirred vessels,
as well as the atmosphere and oceans, can occur either through vigorous
local instabilities and turbulence, or because long-term trajectories of
advected parcels of fluid are chaotic, even when the basic flow pattern
may be smooth and regular. Such chaotic advection processes can also be
observed and studied on a laboratory scale using experimental systems and
numerical models. Recent theoretical work at Warwick University has shown
that certain symmetries in the flow field can lead to a quantification
of the mixing due to chaotic advection ( http://link.aip.org/link/?phf/13/2522
). In this project, the aim is to investigate the application of diagnostics
derived from flow symmetries to predict and quantify transport and mixing
by chaotic advection in Taylor-Couette
flows. The project will investigate material transport and particle
trajectories in numerical models and study the feasibility of generalising
the diagnostics to flow in other geometries as well as the atmosphere and
oceans. Some laboratory experimentation may also be included in the programme,
though the project would particularly suit a mathematical or computational
physicist.
