The application of random walk models to channel and river flows 1 Previous studies.

The random walk has been used in relatively few instances to model the dispersion characteristics of the flow in a channel. Recently the model was applied to the flow in the River Severn (Heslop and Allen, 1993). This study was primarily concerned with the longitudinal dispersion characteristics of the reach, and the tracer tests which were carried out showed skewed concentration curves with long tails in the upstream direction. This suggests that there were deadzones or long term correlations present in the flow (e.g. due to deadzones with large storage times or secondary circulation cells) which were not accounted for by the random walk model. The random walks in use were unable to reproduce the observed long tails, a shortfalling which it was said could be improved slightly with the inclusion of deadzone ‘storage’ effects near the modelled river bed, although results from such a model were not presented. Secondary circulation and dead zones are also present in the river flow which is investigated in chapters 5-8, and efforts are made to account for these features by the inclusion of a variable effective dispersivity coefficient for the random walk.

2.4.2 The scale dependency of the dispersion process.

The effects which the suggested inhomogeneities above can have on solute dispersion are exemplified by considering the case where the diffusion equation is used to predict the transport of the cross-sectional mean concentration of a tracer over a

large distance compared to any meander arc length in the reach. Here the length scales contributing to the constant of dispersivity are large because the large scale circulatory motions o f the secondary currents can be considered as random and part of the

dispersive motion. If the local concentration is of interest, as in this study, then the constant of dispersivity depends on the scales which are small enough that their motion can be considered random. However, due to the sweeping effect that the larger scales of motion have on these smaller scales, the dispersion equation at this scale is not enough to describe the evolution o f the localised concentration. The sweeping effect can give rise to dispersion which is non-Fickian, a problem which is addressed here by the inclusion of an effective memory to the motion of the particles, which can be modelled in many different ways (for example see Kinzelbach, 1990).

2.4.3 Estimating the integral length scale from Lagrangian measurements.

Most o f the time and length scales used in the different random walk models which are described below have been based upon the flume photography experiments carried out by Sullivan (1972), and were also employed by Allen (1982; 1992) and Heslop & Allen (1993). Alternatively the scales are based upon the measured Eulerian fluctuating velocity field in the channel flows. The measurements by Sullivan are the largest scale Lagrangian measurements available at the time of writing so far as is known.

Sullivan’s Lagrangian measurements were carried out using a camera which was moved along at the mean down stream velocity in a channel flow, recording the positions in two dimensions (three including camera position) of particles having

neutral buoyancy (0.5mm diameters). The flume was 8.95cm deep, with a working width and length of 0.46m and 2.45m. The experiment is outlined below.

The transverse and vertical positions of the particles were projected onto a plane perpendicular to the downstream co-ordinate. The particle paths in this plane could be resolved into a series o f circular arcs. Three successive co-ordinate positions were used to define a plane upon which a circular arc could be drawn through the three positions. The radius of the arc was then used to define an instantaneous length scale, r', which is shown in the sketch below:

r’ was non-dimensionalised by the depth, d = 8.95cm. The swept angle made by the arc was divided by two time intervals to define an instantaneous angular velocity, © (which was non-dimensionalised using [h/u*], where h is the channel depth and u* is the friction velocity). Next an experimental probability density function was defined by fitting the scale and shape parameters o f gamma distributions to the observed

distributions o f instantaneous length scales at ten different depths.

The instantaneous angular velocity of a particle was found to have a definite dependence on the instantaneous length scale. All the values of © which had the same instantaneous length scales, to within experimental accuracy, were averaged to produce a mean value co, and an experimental relationship was determined of the form given by the relation 2.24:

Flow is into the page

Sketch showing

construction o f r ’ The traced particle trajectory,

vertical T

A

|B projected onto the x/y plane is ABC. These three points form a unique arc, o f radius r ’.

cross-stream -»

m - 2.2(r')~°'m (2.24) The average of the absolute value o f the difference between individual co values and values given by the equation when corresponding values of r' were used was found to be ~ 20-30%.

An ensemble average angular velocity was then determined by inserting the ensemble mean observed length scale which was (r')= 0 .1 into 2.24. This was then used to determine an ensemble average fluctuating velocity magnitude in the y/z plane by putting (u'^ = (r ^G) ^ = 1.0975 in non-dimensional units. This estimate is

fundamentally based upon the Lagrangian length and inverse time scales estimated in the experiment. These values were used by Allen (1982 ; 1992) and Heslop and Allen (1993).

2.4.5 The random walk model applied to regions of shear.

A continuous range of step sizes can be used in the random walk, rather than using steps of equal sizes and applied in time steps equal to an estimation o f the integral Lagrangian timescale Tl. The major difference that this makes to the dispersion is that the coarseness o f the resulting field is reduced. Further, for the random walk having constant step sizes it becomes important to ensure that there are particle trajectories which take odd and even total numbers of steps between release and the sampling cross section since if the particles all take an even number of steps, then they are unable to settle at odd integer numbers o f displacements away from the release site. This effect is especially likely to occur when there are small velocity gradients perpendicular to the mean flow direction (such as in the transverse dimension for the in-bank flow, which will be described in the next section).

2.5 Flum e geom etry and flow conditions.

The hydrodynamic and dye dispersion data used in this report came from two different sets o f experiments which were carried out on the R ood Channel Facility at Hydraulics Research, Wallingford. The flume geometry will be described here, since the different random walk models (described in section 2.6) were chosen for the specific flows o f interest.

The tracer dispersion tests were carried out by Guymer et al.(1989) for in-bank and over-bank flows for several release points and different depths o f flow in a two stage, straight channel with geometry given in fig. 2.2:

Fig. 2.2 D iagram showing flume geometry for Flood Channel Facility, W allingford (not to scale).

^ | 2 -7

<---75.0 ---M -3 0 . 224.0 ►

{allmeasurements in cm.}

The two flows which were examined in this sensitivity analysis corresponded to flow depths o f 177mm (over-bank flow) and 100mm (in-bank flow), both having a side wall slope o f 2. The dye injection points considered in the analysis were channel side- bank top (depth = 15mm, y= 1.05m) for the over-bank flow, and centreline-water surface for the in-bank flow.

The ratio of the over-bank flow depth to the main flow depth can be considered as similar to that observed on natural rivers, with the side wall slope representing the slope of the inner bank between the flood plain and the main channel.

2.6 Sensitivity of the large scale dispersion characterstics to the form of the