ALBERT

Description: During the motion of a fluid interface undergoing Rayleigh-Taylor instability, vorticity is generated on the interface baronclinically. This vorticity is then subject to Kelvin-Helmholtz instability. For the related problem of evolution of a nearly flat vortex sheet without density stratification (and with viscosity and surface tension neglected), Kelvin-Helmholtz instability has been shown to lead to development of curvature singularities in the sheet. In this paper, a simple approximate theory is developed for Rayleigh-Taylor instability as a generalization of Moore's approximation for vortex sheets. For the approximate theory, a family of exact solutions is found for which singularities develop on the fluid interface. The resulting predictions for the time and type of the singularity are directly verified by numerical computation of the full equations. These computations are performed using a point vortex method, and singularities for the numerical solution are detected using a form fit for the Fourier components at high wavenumber. Excellent agreement between the theoretical predictions and the numerical results is demonstrated for small to medium values of the Atwood number A, i.e. for A between 0 and approximately 0.9. For A near 1, however, the singularities actually slow down when close to the real axis. In particular, for A = 1, the numerical evidence suggests that the singularities do not reach the real axis in finite time. © 1993, Cambridge University Press. All rights reserved.

Description: The motion of vortex sheets is susceptible to the onset of the Kelvin-Helmholz instability. There is now a large body of evidence that the instability leads to the formation of a curvature singularity in finite time. Vortex blob methods provide a regularization for the motion of vortex sheets. Instead of forming a curvature singularity in finite time, the curves generated by vortex blob methods form spirals. Theory states that these spirals will converge to a classical weak solution of the Euler equations as the blob size vanishes. This theory assumes that the blob method is the result of a convolution of the sheet velocity with an appropriate choice of a smoothing function. We consider four different blob methods, two resulting from appropriate choices of smoothing functions and two not. Numerical results indicate that the curves generated by these methods form different spirals, but all approach the same weak limit as the blob size vanishes. By scaling distances and time appropriately with blob size, the family of spirals generated by different blob sizes collapse almost perfectly to a single spiral. This observation is the next step in developing an asymptotic theory to describe the nature of the weak solution in detail. © 2006 Cambridge University Press.

Description: In the absence of significant body forces the passive manipulation of fluid interfacial flows is naturally achieved by control of the specific geometry and wetting properties of the system. Numerous 'microfluidic' systems on Earth and 'macrofluidic' systems aboard spacecraft routinely exploit such methods and the term 'capillary fluidics' is used to describe both length-scale limits. In this work a collection of analytic solutions is offered for passive and weakly forced flows where a bulk capillary liquid is slowly drained or supplied by a faster capillary flow along at least one interior edge of the container. The solutions are enabled by an assumed known pressure (or known height) dynamical boundary condition. Following a series of assumptions this boundary condition can be in part determined a priori from the container dimensions and further quantitative experimental evidence, but not proof, is provided in support of its expanded use herein. In general, a small parameter arises in the scaling of the problems permitting a decoupling of the edge flow from the global bulk meniscus flow. The quasi-steady asymptotic system of equations that results may then be easily solved in closed form for a useful variety of geometries including uniform and tapered sections possessing at least one critically wetted interior edge. Draining, filling, bubble displacement and other imbibing flows are studied. Cursory terrestrial and drop tower experiments agree well with the solutions. The solutions are valued for the facility they provide in computing designs for selected capillary fluidics problems by way of passive transport rates and meniscus displacement. Because geometric permutations of any given design are myriad, such analytic tools are capable of efficiently identifying and comparing critical design criteria (i.e. shape and size) and the impact of various wetting conditions resulting from the fluid properties and surface conditions. Sample optimizations are performed to demonstrate the utility of the method. © 2011 Cambridge University Press.

Description: Deep water waves in two-dimensional flow can have curvature singularities on the surface profile; for example, the limiting Stokes wave has a corner of 2π/3 radians and the limiting standing wave momentarily forms a corner of π/2 radians. Much less is known about the possible formation of curvature singularities in general. A novel way of exploring this possibility is to consider the curvature as a complex function of the complex arclength variable and to seek the existence and nature of any singularities in the complex arclength plane. Highly accurate boundary integral methods produce a Fourier spectrum of the curvature that allows the identification of the nearest singularity to the real axis of the complex arclength plane. This singularity is in general a pole singularity that moves about the complex arclength plane. It approaches the real axis very closely when waves break and is associated with the high curvature at the tip of the breaking wave. The behaviour of these singularities is more complex for standing waves, where two singularities can be identified that may collide and separate. One of them approaches the real axis very closely when a standing wave forms a very narrow collapsing column of water almost under free fall. In studies so far, no singularity reaches the real axis in finite time. On the other hand, the surface elevation y(x) has square-root singularities in the complex x plane that do reach the real axis in finite time, the moment when a wave first starts to break. These singularities have a profound effect on the wave spectra. © 2011 Cambridge University Press.

Description: The equations for the two-dimensional motion of a layer of uniform vorticity in an incompressible, inviscid fluid are examined in the limit of small thickness. Under the right circumstances, the limit is a vortex sheet whose strength is the vorticity multiplied by the local thickness of the layer. However, vortex sheets can develop singularities in finite time, and their subsequent nature is an open question. Vortex layers, on the other hand, have motions for all time, though they may develop singularities on their boundaries. Fortunately, a material curve within the layer does exist for all time. Under certain assumptions, its limiting motion is again the vortex sheet, and thus its behaviour may indicate the nature and possible existence of the vortex sheet after the singularity time. Similar asymptotic results are obtained also for the limiting behaviour of the centre curve as defined by Moore (1978). By examining the behaviour of a sequence of layers, some physical understanding of the formation of the curvature singularity for a vortex sheet is gained. A strain flow, induced partly by the periodic extension of the sheet, causes vorticity to be advected to a certain point on the sheet rapidly enough to form the singularity. A vortex layer, however, simply bulges outwards as a consequence of incompressibility and subsequently forms a core with trailing arms that wrap around it. The evidence indicates that no singularities form on the boundary curves of the layer. Beyond the singularity time of the vortex sheet, the limiting behaviour of the vortex layers is non-uniform. Away from the vortex core, the layers converge to a smooth curve which has the appearance of a doubly branched spiral. While the circulation around the core vanishes, approximations to the vortex sheet strength become unbounded, indicating a complex, local structure whose precise nature remains undetermined. © 1990, Cambridge University Press. All rights reserved.