ALBERT

Description: Surface wave motions in a container with a square base, which is subject to a vertical oscillation, are considered when the amplitude of the oscillation is small and the frequency of the oscillation is close to twice the natural frequency of the system. Subcritical wave motions are found for single modes as well as mixed modes. Here, single modes are described by either one of the two horizontal coordinates whereas mixed modes depend on both coordinates. It is found that in some subcritical region a stable single mode and a stable mixed mode coexist, producing complex basins of attraction. © 1989, Cambridge University Press. All rights reserved.

Description: A comparison of several commonly used turbulence models (including the K-e model and three second-order closures) is made for the test problem of homogeneous turbulent shear flow in a rotating frame. The time evolution of the turbulent kinetic energy and dissipation rate is calculated for these models and comparisons are made with previously published experiments and numerical simulations. Particular emphasis is placed on examining the ability of each model to predict equilibrium states accurately for a range of the parameter Q/S (the ratio of the rotation rate to the shear rate). It is found that none of the commonly used second-order closure models yield substantially improved predictions for the time evolution of the turbulent kinetic energy and dissipation rate over the somewhat defective results obtained from the simpler K-e model for the unstable flow regime. There is also a problem with the equilibrium states predicted by the various models. For example, the K-e model erroneously yields equilibrium states that are independent of Q/S while the Launder, Reece & Rodi model and the Shih-Lumley model predict a flow relaminarization when Q/S 0.39-a result that is contrary to numerical simulations and linear spectral analyses, which indicate flow instability for at least the range 0 Q/S 0.5. The physical implications of the results obtained from the various turbulence models considered herein are discussed in detail along with proposals to remedy the deficiencies based on a dynamical systems approach. © 1989, Cambridge University Press. All rights reserved.

Description: The flow in a channel with an oscillating constriction has been studied by the numerical solution of the Navier-Stokes and Euler equations. A vorticity wave is found downstream of the constriction in both viscous and inviscid flow, whether the downstream flow rate is held constant and the upstream flow is pulsatile, or vice versa. Closed eddies are predicted to form between the crests/troughs of the wave and the walls, in the Euler solutions as well as the Navier-Stokes flows, although their structures are different in the two cases. The positions of wave crests and troughs, as determined numerically, are compared with the predictions of a small-amplitude inviscid theory (Pedley & Stephanoff 1985). The theory agrees reasonably with the Euler equation predictions at small amplitude (ε≲ 0.2) as long as the downstream flow rate is held fixed otherwise a sinusoidal displacement is superimposed on the computed crest positions At larger amplitude (ε= 0.38) the wave crests move downstream more rapidly than predicted by the theory, because of the rapid growth of the first eddy (‘eddy A’)attached to the downstream end of the constriction. At such larger amplitudes the Navier-Stokes predictions also agree well with the Euler predictions, when the downstream flow rate is held fixed, because the wave generation process is essentially inviscid and the undisturbed vorticity distribution is the same in each case. It is quite different, however, when the upstream flow rate is fixed, as in the experiments oi Pedley & Stephanoff, because of differences in the undisturbed vorticity distribution in the growth rate of the vorticity waves and in the dynamics of eddy A. A furthei finite-amplitude effect of importance, especially in an inviscid fluid, is the interactior of an eddy with its images in the channel walls. © 1989, Cambridge University Press. All rights reserved.

Description: Experimental data are presented that demonstrate the existence of a family of gravitational water waves that propagate practically without change of form on the surface of shallow water of uniform depth. The surface patterns of these waves are genuinely two-dimensional and fully periodic, i.e. they are periodic in two spatial directions and in time. The amplitudes of these waves need not be small; their form persists even up to breaking. The waves are easy to generate experimentally, and they are observed to propagate in a stable manner, even when perturbed significantly. The measured waves are described with reasonable accuracy by a family of exact solutions of the Kadomtsev-Petviashvili equation (KP solutions of genus 2) over the entire parameter range of the experiments, including waves well outside the putative range of validity of the KP equation. These genus-2 solutions of the KP equation may be viewed as two-dimensional generalizations of cnoidal waves. © 1989, Cambridge University Press. All rights reserved.

Description: It is shown that hydrodynamic interactions between non-Brownian, non-spherical, sedimenting particles give rise to an increase in the number of neighbouring particles in the vicinity of any given particle. This result suggests that the suspension is unstable to particle density fluctuations even in the absence of inertia; a linear stability analysis confirms this inference. It is argued that the instability will lead to convection on a lengthscale (nl)-1/2, where l is a characteristic particle length and n is the particle number density. Sedimenting suspensions of spherical particles are shown to be stable in the absence of inertial effects. © 1989, Cambridge University Press. All rights reserved.

Description: A theoretical description of the low-Reynolds-number collision and rebound of two rigid or elastic spheres separated by a thin layer of viscous fluid with pressure-dependent physical properties is presented. It has previously been shown by Davis et al. (1986) that the hydrodynamic pressure which builds up in the thin fluid layer must become large enough to elastically deform the spheres near the axis of symmetry, if they are to rebound subsequent to colliding. Under these extreme pressures, however, it is expected that the fluid may also compress and that its viscosity may increase by several orders of magnitude. It is shown that these pressure-dependent effects may significantly alter the minimum separation reached during approach of the spheres, as well as the maximum separation and relative velocity attained during rebound of the spheres. In particular, the pressure buildup during the collision process is predicted to become sufficiently large under some conditions so that the corresponding viscosity increase causes the fluid in the gap between the colliding spheres to behave nearly as a solid and to limit the close approach of the opposing surfaces. Also, the storage of energy via the compression of the fluid in the gap allows rigid spheres to bounce as this energy is released subsequent to their collision. However, it is found that this rebound is very weak relative to that which is predicted for elastic spheres. © 1989, Cambridge University Press. All rights reserved.

Description: A solid long slender body with curved centreline is held at rest in a fluid undergoing a uniform flow. Assuming that the Reynolds number Re based on body length is fixed, the force per unit length on the body is obtained as an asymptotic expansion in terms of the ratio k of the cross-sectional radius to body length. In the limit of large Re, this result is no longer valid and an asymptotic expansion in KRe is necessary. A uniformly valid solution is obtained from these two expansions. The total force and torque acting on a body with a straight centreline are explicitly determined. The limiting cases of small and large Re are studied in detail. © 1989, Cambridge University Press. All rights reserved.