Session HE: Multiphase and Particle-Laden Flows IV

Band patterns in thin film suspensions

A range of fascinating patterns can be observed when a thin, particle-laden flow has drained down an inclined substrate. If particles are on the order of 10 to 100 microns in diameter and behave, roughly, as hard spheres, the emerging pattern consists of horizontal bands, oriented perpendicular to the mean flow direction. We discuss the mechanisms of pattern formation, and set up a theory based on excluded-volume and viscous couplings between particles and solvent. Large particles are pinned in the thinning film by means of static friction with the substrate, mediated by surface tension effects at the fluid-air interface. Subsequently, pinned particles alter the film surface and provide an efficient instability mechanism. Additional particles become arrested and link-up the pinning points into band structures. These bands block the mean flow, followed by a phase of coarsening, where smaller particles sediment onto the band. The final band thickness is thus proportional to the captured feeding area.   

Dynamic Behaviour of Ventilated Hydrofoils.

In certain types of pumping applications oscillations are induced by operation with liquids containing a free gas load. In order to understand the physics of this process, a series of tests with a ventilated A 2D NACA 0015 hydrofoil were performed in the water tunnel at the St. Anthony Falls Laboratory of the University of Minnesota. The special bubble removal feature of the water tunnel allowed continuous ventilation without experiencing visible bubbles upstream the hydrofoil. These studies build on previous work on cavitation-induced oscillations. Gas injection studies were made over a range of gas flow rates and test section pressure. The results clearly show that lift oscillations increase in intensity when the gas load is increased. The point of maximum unsteadiness is also associated the rapid decline of the foil performance as measured as average lift. Further increase of the gas injection load gives a steady behaviour with almost no lift. These experiments are compared with traditional cavitation experiments. The similarities between gas injection- and cavitation induced unsteadiness on the hydrofoil are many, but the amplitude of lift oscillations found on the foil with gas injection corresponds to about 50{\%} of that found for cavitating hydrofoils. The fact that the oscillations are periodic leads to the consideration of both passive and active control.   

The singularity in particle-laden boundary layers

The classical ``dusty gas'' equations have been used recently in a number of investigations by the authors\footnote{See, most recently, Foster, Duck \& Hewitt (2006) {\it Proc. Roy. Soc A} {\bf 462}, 1145} to model boundary-layer flows of dilute suspensions of heavy particles. Though none of the difficulties of well-posedness that so often occur in more complicated particle-laden flow models seems to arise for this equation set, what does nearly always appear, and is now well documented in a variety of boundary layers, is a wall singularity that occurs at a finite distance from the leading edge, where the volume fraction is unbounded. The dusty-gas approximation replaces the quantity ``$1-\alpha$'' everywhere in the particle-laden equations by ``$1$''. One is forced to seek a more complicated model in order to remove the unphysical singularity, and there are plenty of candidates--including particle pressure, added mass, particle-particle interactions. From the point of view of modifying the theory in the simplest possible way, we restore ``$1-\alpha$'' where it had been replaced by ``1,'' and do nothing more. Such a procedure removes the singularity in boundary-layer flows, and we present computational and analytical results under such a change   

Broad hump or sharp cusp: two-layer withdrawal with unequal viscosities

In viscous two-layer withdrawal, the interface between two layers of liquid is deformed into a hump shape by a spatially converging withdrawal flow in the upper layer. Previous work found that, when the two layers have the same viscosity, increasing the withdrawal flux changes the shape of the steady-state interface in an unusual way: the hump curvature, kappa, increases dramatically but the hump height increases only weakly, proportional to ln(kappa). Here we show that this unusual weak coupling persists even when the lower layer is made more viscous than the upper layer. However, when the lower layer is less viscous than the upper layer, the coupling between the hump height and the hump curvature is modified from the logarithmic form.   

Stability Studies of Three-layer Hele-Shaw Flows

The stability of three-layer Hele-Shaw flows with middle layer having either constant or variable viscosity will be considered in this talk. We solve this problem for the case of constant viscosity exactly and obtain several results all of which are independent of the length of the middle layer: (i) a necessary and sufficient condition for instability of any mode with the wave-number (ii) a critical viscosity of the middle layer that gives the shortest bandwidth of unstable modes; and (iii) an upper bound on the growth rate of instabilities. Numerical evidence that supports the effectiveness of the upper bound is also presented. We also consider the case of variable viscosity in the middle layer and provide results on the upper bound of the growth rate of disturbances. Time permitting, we will also provide results on the growth rates in the presence of diffusion when the variable viscosity of the middle layer is due to varying polymer concentration. In particular, we will show analytically stabilizing effects of large enough diffusion.   

Vortex shedding behind a sphere: transition between the stationary non-axysimmetric instability and the hairpin shedding

We perform numerical simulations and also experiments in water channels, on flows behind a solid sphere in order to study the transitional flow, in the range of Reynolds number between 200 and 300, which shows successively stationary axysimmetrical properties, stationary non-axysimmetrical instability and global temporal instability. The numerical study, with spectral methods, shows how the symmetry breaking in the recirculation bubble induces streamwise vortex. In addition we experimentally study the influence of the real conditions of holding of the sphere, on the stationary instability, Finally, we perform very controlled experiments forcing the streamwise vortex, in order to follow the dynamics of formation of the hairpin vortex shedding.