Discrete modelling of a liquid jet suspending a ball

Abstract

The steady Suspension of a rigid ball by a thin liquid upright jet in the presence of the gravity force is obtained numerically by a discrete model of inviscid incompressible fluids. Some kinematic characteristics of the moüon are studied. We find, in particular, that the period of ball oscillations has a minimnm at a certain jet-ball radius ratio. The dependence of the period on the ball density is weak. The results are compared with the experimental ones, and a new explanation of the retracing force nature is discussed.Since 1870, when Osborne Reynolds studied the Suspension of a ball by a vertical water jet, Uns effect has been known äs a classical example of the interesting hydrodynamical phenomenon. Yet only a few works on this problem have appeared ever since. It was considered by M. A. Lavrent'ev [7] (see also [8]), where the explanation of the effect was suggested on the basis of the analysis of the two-dimensional steady state jet flow around a cylinder. Since the flow was supposed to be inviscid and potential, some additional hypotheses concerning the detachment point position had to be introduced in order to select the unique solution. Namely it was assumed that the detachment point is always diametrically opposite to the attachment point. This phenomenon was observed in experiments [9] and considered to be caused by viscous forces. In [6] the similar principle was suggested: the total circulation over the cylinder surface must be equal to zero. With the above assumptions the retracing force nature becomes quite clear. Due to the reversibility and hence the symmetry of the problem, the direction of the outgoing jet must deviate from the incoming jet axis thus displacing a cylinder. In this case the retracing force arises because of the momentum conservation.In the above works the question of stability of a real time-dependent flow was not considered. It is an important problem because the System considered is not conservative and just the presence of a retracing force is not sufficient for the oscillations to be stable. Both analytical investigations of the problem and the direct numerical solution of the Euler equations present difficulties due to interaction of a body with a rather complicated free surface flow. That is why in the present paper (see also [5]) the socalled discrete models of incompressible fluids have been used to simulate numerically nonstationary 2D and 3D problems.The essence of the discrete approach is that the incompressible flow is simulated by a large number of particles with one or the other of constraints, which corresponds to the incompressibility and boundary conditions, and the governing equations are derived directly from a variational principle [3]. All these models are governed by the basic mechanical conservation laws even for coarse spatial discretization and this often allows one to get good results with a comparatively small number of degrees of freedom. The models used for the problem under consideration are free-Lagrange and allow one to simulate free surface flow...