ALBERT

Description: 〈div data-abstract-type="normal"〉〈p〉We experimentally investigate the two-phase interplay in an open-channel turbulent boundary layer laden with finite-size particles at global volume fractions between 4 and 25 %. The working fluid (water) and the dispersed phase (hydrogel spheres) have closely matched refractive indices, allowing us to measure the properties of both phases using particle image velocimetry and particle tracking velocimetry, respectively. The particles have a diameter of approximately 9 % of the channel depth and are slightly denser than the fluid. The negative buoyancy causes a strong vertical concentration gradient, characterized by discrete and closely spaced particle layers parallel to the wall. Even at the lowest considered volume fractions, the near-wall fluid velocity and velocity gradients are strongly reduced, with large mean shear throughout most of the channel height. This indicates that the local effective viscosity of the suspension is greatly increased due to the friction between particle layers sliding over one another. The particles consistently lag the fluid and leave their footprint on its mean and fluctuating velocity profiles. The turbulent activity is damped near the wall, where the nearly packed particles disrupt and suppress large-scale turbulent fluctuations and redistribute some of the kinetic energy to smaller scales. On the other hand, in the outer region of the flow where the local particle concentration is low, the mean shear produces strong Reynolds stresses, with enhanced sweeps and ejections and frequent swirling events.〈/p〉〈/div〉

Description: A homogeneous fluid of viscosityvis confined between two co-axial disks (vertical separationH) which rotate relative to a rotating system (angular velocity Ω). The resulting velocity field is studied for values of the parameterv/2ΩH2in the range 1·6 × 10−2to 1·8 × 10−3. The Rossby number, defined as the ratio of the relative angular velocity of the disks to the angular velocity of the system, ranged from 0·038 to 0·0041. The dependence of the resulting velocity field (interior and boundary-layer flow) on geometrical parameters, imposed surface and bottom velocities, and Ω, is in good agreement with the calculations of Stewartson and Carrier. In particular, when the two disks rotate with the same angular velocity, the width of the vertical shear layer at the edge of the disks is found to be proportional to Ω−0·25±0·02. When the disks rotate in opposite senses, a shear layer in the vertical velocity is observed which transports fluid from one disk to the other and whose width is proportional to Ω−0·40±0·10. The magnitude and shape of the observed vertical velocity is in fair agreement with a numerical integration of the theoretical results.

Description: When a fluid of low conductivity flows parallel to a plane wall from behind which projects a magnetic field, an electric potential field is established throughout the fluid. In this paper the potential field is obtained explicitly in terms of the velocity field when the latter is unidirectional and depends only on the coordinate normal to the wall. Experiments with a variety of velocity profiles are described, and the agreement with the theory is satisfactory. The effect of slow variation of the profiles in the direction of the flow is considered, and is shown to be unimportant under the conditions of the experiments.

Description: The horseshoe vortex formed around the base of a cylinder by a separating laminar boundary layer has been investigated experimentally. Smoke flow visualization shows that both steady and unsteady vortex systems exist. Pressure distributions beneath both types of vortex system have been measured and the variation of the horseshoe vortex position on the plane of symmetry upstream of the cylinder has been determined. Unsteady horseshoe vortex systems are shown to have a complex oscillatory behaviour and the nature of this oscillatory behaviour is described. Using smoke flow visualization techniques some measurements have been made of the velocity distributions within horseshoe vortex systems. © 1979, Cambridge University Press

Description: The motion of free surfaces in incompressible, irrotational, inviscid layered flows is studied by evolution equations for the position of the free surfaces and appropriate dipole (vortex) and source strengths. The resulting Fredholm integral equations of the second kind may be solved efficiently in both storage and work by iteration in both two and three dimensions. Applications to breaking water waves over finite-bottom topography and interacting triads of surface and interfacial waves are given. © 1982, Cambridge University Press. All rights reserved.

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: Moore (1979) demonstrated that the cumulative influence of small nonlinear effects on the evolution of a slightly perturbed vortex sheet is such that a curvature singularity can develop at a large, but finite, time. By means of an analytical continuation of the problem into the complex spatial plane, we find a consistent asymptotic solution to the problem posed by Moore. Our solution includes the shape of the vortex sheet as the curvature singularity forms. Analytic results are confirmed by comparison with numerical solutions. Further, for a wide class of initial conditions (including perturbations of finite amplitude), we demonstrate that 3/2-power singularities can spontaneously form at t = 0+ in the complex plane. We show that these singularities propagate around the complex plane. If two singularities collide on the real axis, then a point of infinite curvature develops on the vortex sheet. For such an occurrence we give an asymptotic description of the vortex-sheet shape at times close to singularity formation.

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: Steady uniform granular chute flows are common in industry and provide an important test case for new theoretical models. This paper introduces depth-integrated viscous terms into the momentum-balance equations by extending the recent depth-averaged μ(I)-rheology for dense granular flows to two spatial dimensions, using the principle of material frame indifference or objectivity. Scaling the cross-slope coordinate on the width of the channel and the velocity on the one-dimensional steady uniform solution, we show that the steady two-dimensional downslope velocity profile is independent of scale. The only controlling parameters are the channel aspect ratio, the slope inclination angle and the frictional properties of the chute and the sidewalls. Solutions are constructed for both no-slip conditions and for a constant Coulomb friction at the walls. For narrow chutes, a pronounced parabolic-like depth-averaged downstream velocity profile develops. However, for very wide channels, the flow is almost uniform with narrow boundary layers close to the sidewalls. Both of these cases are in direct contrast to conventional inviscid avalanche models, which do not develop a cross-slope profile. Steady-state numerical solutions to the full three-dimensional μ(I)-rheology are computed using the finite element method. It is shown that these solutions are also independent of scale. For sufficiently shallow channels, the depth-averaged velocity profile computed from the full solution is in excellent agreement with the results of the depth-averaged theory. The full downstream velocity can be reconstructed from the depth-averaged theory by assuming a Bagnold-like velocity profile with depth. For wide chutes, this is very close to the results of the full three-dimensional calculation. For experimental validation, a laser profilometer and balance are used to determine the relationship between the total mass flux in the chute and the flow thickness for a range of slope angles and channel widths, and particle image velocimetry (PIV) is used to record the corresponding surface velocity profiles. The measured values are in good quantitative agreement with reconstructed solutions to the new depth-averaged theory. © 2015 Cambridge University Press.