Beyond Part III

Young Researchers in Mathematics 2009

16-18 April 2009

Centre for Mathematical Sciences, Cambridge

Fluid Dynamics Session

Talks will be in MR15 on Thurday, in MR14 (combined with Astrophysics) on Friday morning, and in MR11 on Friday afternoon.



The Flow Generated by Oblique Impingement of Granular Material on an Inclined Plane - Christopher Johnson

We consider a steady stream of granular material impinging on, and subsequently flowing down, an inclined plane. The flow exhibits a wide range of behaviours, depending on the slope angle and fall velocity, ranging from kinetic gas-like flow to intermittent avalanche formation.

The flow exhibits a previously unreported steady state flow regime, with a teardrop-shaped hydrodynamic shock surrounding the area of impingement. We construct a depth-integrated hyperbolic flow model, similar to the shallow-water equations, which predicts with good agreement the experimental location and shape of the shock. A number of the experimentally observed features cannot be predicted by current shallow-layer models of granular materials, and we discuss these with reference to the constitutive rheology of thin-layer granular flows.

Saving lives with fluid mechanics - Andrew Hogg (keynote speaker)

Mathematical models of fluid flows are employed to describe a wide range of phenomena and make predictions with practical consequences in many applications. In this presentation I will report on two distinct research projects, with life-saving consequences. This first concerns snow avalanches and the design of barriers to deflect, retard and arrest the oncoming flow. The second concerns the direct injection of therapeutic drugs into the brain to treat tumours and degenerative diseases.

Thunderstorm Dynamics in the UK - Bethan White

Fluid phenomena are seen on all scales in the Earth's atmosphere, from large hurricanes to the individual convective cells associated with thunderstorms. The exact processes responsible for the initiation of such convection in the atmosphere are still widely unknown and the Convective Storm Initiation Project (CSIP) of 2005 was a field campaign which collected extensive data during convective storm events. One CSIP observation period revealed a series of Mesoscale Convective Systems (MCSs) which passed over the UK. Detailed data were collected for one of the MCSs and analysis of these data showed the flow within the storm to be highly structured. Convection was initiated from well above the surface, and took place above an undercurrent of air that was too cool to take part in convection. Two separate layers of air fed the convective updraughts, and the convection in the MCS had a slantwise nature. The UK Met Office's Unified Model (a numerical model designed for both weather and climate forecasting) was run at up to 1 km horizontal resolution for this case study. The model results have been compared with the observational data. Several MCSs were produced by the model, some of which bore close resemblance to that observed during CSIP. Preliminary results show that some of the observational features, are indeed seen in the model data. Numerical modelling work is extended to investigate the nature of the storm dynamics and coupling between systems.

Acoustic fluid-solid interaction across a boundary layer - Ed Brambley

This talk considers a fluid-solid boundary, with steady flow in the fluid parallel to the boundary. The fluid is compressible, and we introduce acoustic waves as linearized disturbances about the steady state. The solid is not necessarily fixed, and so may react to the acoustic disturbances.

If there were no flow, the boundary conditions between the fluid and the solid would be the matching of pressure and normal velocity at the surface. With flow, assuming the fluid to be inviscid, it was shown in the early 1970s that matching of normal velocity is incorrect. What is in fact required is the matching of normal particle displacement, as will be explained in the talk. This condition was later generalized to include surface curvature by Myers (1980), as is now known as the Myers boundary condition.

This talk will consider the effect of a thin laminar viscous boundary layer in the fluid at the boundary. This is in order to model sound in aircraft engines, and the damping of sound by aeroengine acoustic linings. It will be shown that the correct boundary condition at low frequency is continuity of normal mass flux, while at high frequency the correct boundary condition is the Myers boundary condition, with an important exception.

The turbulent equilibration of baroclinic jets: Asymmetric jets and barotropic governor - Ben Willcocks

A two-layer, quasi-geostrophic numerical model is used to investigate the lifecycles of unstable baroclinic jets. The waves grow and subsequently break, resulting in turbulence in both layers and finally the equilibration of the flow to a quasi-steady state. In previous work a theory was introduced to predict this equilibrated flow from the initial parameters. The idea is that potential energy is minimized subject to the dynamical constraints of energy and momentum conservation and a kinematic constraint that potential vorticity (PV) is completely homogenized within well-delineated 'mixing' regions in each layer. It was shown that this theory accurately predicts the final flow structures and characteristics of a range of symmetric jets.

The investigation is extended to jets in the presence of linear barotropic shear, the `barotropic governor', and to those with latitudinally asymmetric structures. The minimization principle itself is also extended to investigate other possible governing principles. Comparison of theoretical predictions and numerical results reveals that an alternative formulation of the theory, in which the area of the mixing zones over both layers is maximized, result in equally good agreement for symmetric jets. For asymmetric jets, the new theory is shown to result in significantly better predictions than minimization of potential energy.

Flow and Instability of Viscous Currents on a Cylinder or a Sphere - Daisuke Takagi

Layers of viscous fluid are considered both theoretically and experimentally to spread by gravity on the outer surface of a stationary sphere and separately a cylinder, whose axis is horizontal. After the instantaneous release of a constant volume of fluid from either a point source at the top of the sphere or a line source at the top of the cylinder, the resultant flow is initially axisymmetric or two-dimensional respectively. The structure of the flow is described using lubrication theory, when viscous forces dominate over both inertia and surface tension. The theory predicts the fluid thickness to remain uniform along the flow both near the top of the sphere and the cylinder. Laboratory experiments support the theoretical predictions until the thickness decreases to a volume-dependent length scale, when the advancing front of the flow splits into a series of rivulets. I will present photographs and video clips of the experiments, which were conducted by releasing golden syrup on a beach ball and a wheel.