Optimal control of diffuser shapes for non-uniform flow.

A simplified model is used to identify the diffuser shape that maximises pressure recovery for several classes of non-uniform inflow. We find that optimal diffuser shapes strike a balance between not widening too soon, as this accentuates the non-uniform flow, and not staying narrow for too long, which is detrimental for wall drag. Three classes of non-uniform inflow are considered, with the axial velocity varying across the width of the diffuser entrance. The first case has inner and outer streams of different speeds, with a velocity jump between them that evolves into a shear layer downstream. The second case is a limiting case when these streams are of similar speed. The third case is a pure shear profile with linear velocity variation between the centre and outer edge of the diffuser. We describe the evolution of the time-averaged flow profile using a reduced mathematical model that has been previously tested against experiments and computational fluid dynamics models. The model consists of integrated mass and momentum equations, where wall drag is treated with a friction factor parameterisation. The governing equations of this model form the dynamics of an optimal control problem where the control is the diffuser channel shape. A numerical optimisation approach is used to solve the optimal control problem and Pontryagin's maximum principle is used to find analytical solutions in the second and third cases. We show that some of the optimal diffuser shapes can be well approximated by piecewise linear sections. This suggests a low-dimensional parameterisation of the shapes, providing a structure in which more detailed and computationally expensive turbulence models can be used to find optimal shapes for more realistic flow behaviour.

Correction to 'Homogenized boundary conditions and resonance effects in Faraday cages'.

[This corrects the article DOI: 10.1098/rspa.2016.0062.].

Actively evolving subglacial conduits and eskers initiate ice shelf channels at an Antarctic grounding line.

Ice-shelf channels are long curvilinear tracts of thin ice found on Antarctic ice shelves. Many of them originate near the grounding line, but their formation mechanisms remain poorly understood. Here we use ice-penetrating radar data from Roi Baudouin Ice Shelf, East Antarctica, to infer that the morphology of several ice-shelf channels is seeded upstream of the grounding line by large basal obstacles indenting the ice from below. We interpret each obstacle as an esker ridge formed from sediments deposited by subglacial water conduits, and calculate that the eskers' size grows towards the grounding line where deposition rates are maximum. Relict features on the shelf indicate that these linked systems of subglacial conduits and ice-shelf channels have been changing over the past few centuries. Because ice-shelf channels are loci where intense melting occurs to thin an ice shelf, these findings expose a novel link between subglacial drainage, sedimentation and ice-shelf stability.

Homogenized boundary conditions and resonance effects in Faraday cages.

We present a mathematical study of two-dimensional electrostatic and electromagnetic shielding by a cage of conducting wires (the so-called 'Faraday cage effect'). Taking the limit as the number of wires in the cage tends to infinity, we use the asymptotic method of multiple scales to derive continuum models for the shielding, involving homogenized boundary conditions on an effective cage boundary. We show how the resulting models depend on key cage parameters such as the size and shape of the wires, and, in the electromagnetic case, on the frequency and polarization of the incident field. In the electromagnetic case, there are resonance effects, whereby at frequencies close to the natural frequencies of the equivalent solid shell, the presence of the cage actually amplifies the incident field, rather than shielding it. By appropriately modifying the continuum model, we calculate the modified resonant frequencies, and their associated peak amplitudes. We discuss applications to radiation containment in microwave ovens and acoustic scattering by perforated shells.

The speed of an inclined ruck.

Steady rucks in an elastic beam can roll at constant speed down an inclined plane. We examine the dynamics of these travelling-wave structures and argue that their speed can be dictated by a combination of the physical conditions arising in the vicinity of the 'contact points' where the beam is peeled off the underlying plane and stuck back down. We provide three detailed models for the contact dynamics: viscoelastic fracture, a thermodynamic model for bond formation and detachment and adhesion mediated by a thin liquid film. The results are compared with experiments.