LOS CONTEXTOS POLÍTICOS DE LAS REFORMAS A LA EDUCACIÓN SECUNDARIA DE 1993 Y

The impact process is the basis of both the continuation of particle motion and the cessation of particle motion. In continuing motion, the fraction of particle

momentum conserved at an impact influences the initial conditions of the next

saltation. While in the cessation of particle motion, it determines under what condition the particle ceases to move.

Cessation of motion of particles has been observed to be caused by different mechanisms, and these depend on the mode in which the particle is moving. After saltation and sliding, particles can be stopped by collisions with other particles which were close to head on (Drake et al., 1988). After rolling, deposition usually follows deceleration of the particle, particle speed dropping until it falls into a crevice or cannot clear the next obstacle (as observed by Francis, 1973, Drake et al., 1988).

Observations of the impact process have been made by Gordon et al. (1972) in a 2-dimensional flume using spheres as the mobile particles, travelling over a mobile bed of spheres. These results showed a loss of the normal component of momentum on impact, with tangential momentum being conserved. Observations by Abbott & Francis (1977) showed no coupling between pre- and post-impact particle trajectory. However their observations usually included a period of rolling between trajectories, which could explain this decoupling (Naden, 1987b).

Observations of impact in the aeolian environment have been made by Willetts & Rice (1985), observing the transport of sand in a wind tunnel; Werner (1987), observed the impact of single grains of quartz; Mitha et al. (1986), simulated quartz grains using ball bearings, observing the impact of a ball bearing into a bed formed of ball bearings. In addition to these observations numerical simulations of the impact process in aeolian saltation have been made by Werner & Haff (1988) and Anderson & Haff (1988). In these numerical experiments the effects of the impact of particles into a bed of discs, free to move horizontally and vertically, were examined. The aim of all this work was to determine appropriate splash functions to use in models of aeolian sediment transport. The splash function relates the effect of an impacting grain to the resulting outgoing grains. The splash functions determined from observations and those from numerical simulation give qualitative agreement (McEwan et al., 1992). Outgoing grains due to an aeolian impact consist of the grain rebounding from the impact and a number of other grains ejected from the bed as a result of the impact.

Observations of the fluvial environment indicate that ejection of grains from the bed due to impact was rare (Drake et al., 1988). If only rebounding grains were considered, the observations show particles impacting at low angles and rebounding at much higher angles with a reduced velocity (McEwan et al., 1992). Simple geometric models such as that of Rumpel (1985) modelled the impact between a moving particle and a single bed particle in a bed which was not mobile but was also not rigid, that is, the momentum of impact could be dissipated without modifying the bed. In such a case a perfect collision would conserve all the tangential momentum while the normal component of momentum would be transferred to the bed, Figure 2.8. A model of this type predicts the observed increase in angle of the rebounding particle along with the reduction of particle velocity. Such a collision shows a decreasing ratio of outbound to incident particle velocities as the angle between particle centres increased. Since the leaving angle increases with angle between particle centres, analysis of the observed ratio of particle velocities

Tangential

Normal Particle velocity

a) Velocity components immediately before impact

Tangential

w.

b) Velocity components immediately after impact (Rumpel, 1985)

should show a negative correlation. McEwan et al. (1992) using the data of Willetts & Rice (1985) found this to be the case.

The numerical simulations described in Werner & Haff (1988) and Anderson & Haff (1988) were performed for discs constrained to move in two dimensions in an aeolian environment. Calculations were performed to determine the effects of the impact of individual discs. Similar calculations have been performed for a fluvial environment and are described in Jiang & Haff (1993); these allow for the effects of a high density fluid by increasing the damping amongst the particles. The calculations were performed for the effects of a shearing flow on the bed, rather than being driven by the impact of individual particles. The effects of the shearing flow were calculated on any particles exposed to the flow; the effects of interactions of particles were calculated between all particles. Though the interactions of particles were calculated and the model of these interactions is described, the results of individual impacts are not described. Since the model is of mass movement of sediment particles the details of the shearing flow and interactions of particles are described in Section 6.2.5.

23.4 Two-phase flow

By examining the movement of only single particles in water the interaction between flow and particle is reduced to the influence of the flow on the particle. Any momentum extracted from the flow is small and has little effect on the flow profile.

By contrast, turbulent eddies in the flow can affect the motion of particles. However particles in a flow do not respond immediately to a change in the surrounding velocity field but respond over a period of time, the relaxation or characteristic

response time. The response time was defined by Hinze (1972) as the time required for the relative particle velocity to fall to half its initial value. For a high particle Reynolds number this gives an expression for the response time

t. K u

where ur initial is the initial value of the relative particle velocity, K is of 0(1) and is related to the expression for drag and CA is the added mass coefficient

where CD is the coefficient of drag for the particle. For a sediment particle in water p ,/p is 0(1), and equating urinitial to g m, the standard deviations of the velocity fluctuations gives an expression

the motion of eddies will influence particles.

The slow response of heavy particles to turbulent fluctuations has an effect other than causing relative motion and hence additional fluid forces acting on the particle. Since a heavy particle is always in motion relative to the fluid the flow sampled by the particle varies continuously. This has been called the ' crossing

trajectories' effect by Csanady (1963). A heavy particle will not remain in an eddy as a fluid particle would, but leave it; the autocorrelation of the velocity of a heavy particle will therefore fall more rapidly than that of a fluid particle which will show the

correlation of the flow itself.

2 A . J 2 l+cM

A ^ p * JA

where A is the length scale of the eddies. Thus if

2.4 Bed

In rivers the bed affects and is affected by the flow. Simplifying the system under consideration to one in which only the movement of single particles is considered removes the possibility of the bed being modified. In the fluvial environment, Sekine & Kikkawa (1992) found that the position of particle centres formed a Gaussian distribution about a mean bed height. In Furbish (1987) and Robert (1988) detailed measurements of the variation of bed surface heights, in the cross stream and streamwise directions respectively, and statistical descriptions of these are given. These studies describe the roughness heights present directly without

attempting to characterise the bed in terms of the distribution of particle sizes present. Robert (1988) found two scales present within the measurements of bed roughness, corresponding to the grain roughness scale and the scale of small structures on the bed. These measurements were made in either rivers or flumes with a mobile bed, after it had been worked by a flow.

The observations that have been made of the movement of individual particles moving in isolation have been either for fixed beds, with the particle forming the roughness glued in place, or for a bed below its threshold for initial motion and hence with no bed particles moving. The range of the roughness elements used in these experiments has also been simplified, to either a single size (spheres: Meland & Norrman, 1966, Murphy & Hooshiari, 1982) or a single size fraction (gravel: Francis, 1973, Abbott & Francis, 1977). Under these circumstances the bed roughness can be characterised by a single length scale, derived from the particle size.

Observations have been made with mobile beds, in flume by Fernandez Luque & van Beek (1976), and in the field by Drake et al. (1989) and Williams (1990). The observations of Fernandez Luque & van Beek (1976) were of a mobile bed formed of a single type and size of sediment particle, the bed can therefore still be characterised using a single length scale. The observations of Drake et al. (1989) and Williams

(1990) were for a bed formed of a range of particle sizes, in Drake et al. (1989) only qualitative descriptions of particle behaviour were given, while in Williams (1990) distributions showing ranges of conditions present are shown but the behaviour of individual particles is not