Surface fitting at the flow boundaries

All o f the models described in chapter 2 incorporated the mean velocity field interpolations described below. The velocity data sets did not cover the entire flow domain, since velocity measurements could not be made right up to the boundaries. However the random walk model requires the complete flow domain {outside o f the viscous boundary layer) if the flow is to be modelled. Consequently a few ‘dummy’ data points were initially introduced at the boundaries having zero velocities in order that the surface fitted the entire flow domain.

The bi-cubic spline coefficients required to define the surfaces were determined using a program incorporating NAG subroutine E02DDF, and values o f the surface were then determined at user defined locations using NAG subroutine E02DFF so that they could be compared with the data.

Since the flume is symmetrical, only half of the flow field was used in the fitting, the opposite half was assumed to be a simple reflection about the centre line, although this would not be true for a non-axis symmetric flow geometry such as that for a meandering channel. This was conveniently implemented through temporarily changing the sign of the transverse position of the particle during the subroutine in which the velocity o f the particle was returned.

Similarly in the case of the over-bank region for the over-bank flow case, the polynomial coefficients were determined using NAG routines, E02ADF and E02DDF. 3.4.4.1 In bank flow interpolations close to the boundaries.

The closest velocity measurements which were made to any boundary were between 9 mm and 15mm away from the bed for the in-bank flow (Subsequent to these measurements, the velocities were measured closer to the wall, at 2mm, using the Preston tube technique, although these measurements were not available at the time of writing) . Dummy data points were put in place at the boundaries with zero velocities, although they were given zero weightings. The problem with giving these points zero velocities with weightings, was that for the velocity field to drop to zero in the space of a few millimetres, a large number of knots had to be used with the S factor having a very small value to give good closeness of fit. It is evident from fig. 3.5 that a large number of knots gave rise-

to a large degree of instability elsewhere in the flow domain. However, the exclusion of a weighted, zero velocity point at the boundaries introduces two problems:

(1) A contradiction to the non slip condition.

(2) It can also have the effect of reducing the velocity gradient close to the wall from its true value.

However, the contradiction of the non-slip condition was not experienced by the particles since they were reflected about a point a small distance from the

boundaries, this representing the modelled viscous sub-layer (discussed above). This distance represented the viscous sub-layer thickness on the smooth bed of the channel, and was estimated in section 3.5.2 to be 7mm.

The second problem of a reduction in the near bed velocity gradient, caused by excluding the weightings was therefore only a problem in the region of flow between Z = 7mm and z = 9mm or 15mm depending on the local value of the minimum depth of velocity measurement. It was assumed that the uncertainties arising from this form of approximation were smaller than those which were found to arise from using

weightings.

In the case of the fluctuating velocity field, the inclusion of weighted zero velocities at the boundary produced a peak in the velocities which was unphysical since it displaced the region of maximum turbulent energy production away from the

boundaries, whereas in reality the maximum is very close to the boundary (see Tritton, 1990 or Raupach, 1991). The fluctuating velocity field was therefore not extrapolated to zero at the boundaries, but rather the value of the fluctuating field at an adjacent

measurement site was adopted at the boundary. The same procedure was carried out with the interpolations of the Reynolds stresses.

Finally dummy data points with zero weighting were also placed at the water surface, where there were also no velocity measurements. The behaviour of the velocity profile close to the surface was chosen through criteria described in the sensitivity analysis in section 3.5.5.

3.4.4.2 Over-bank flow interpolations close to the boundaries.

The closest velocity measurements to the bed were 5 mm for the over-bank flow, and since the same value for the viscous sub-layer thickness as for the in-bank flow was used (7mm), neither of the problems which were discussed above were encountered with the over-bank flow model. Extrapolation of the flow domain was not required, and the interpolation was carried out as described above. In the over-bank region (shown in fig. 3.4), 6th order Chebychev polynomials were fitted to the transverse velocity distributions. These required further interpolation in the vertical direction, and this was done during the particle tracking model, through the use of a logarithmic profile derived from the point values in the vertical, the latter having been determined from the values of the polynomials. Table 3.2 shows the values of the polynomials, splines and data at measurement points at the join of the polynomials and the surface. The small discrepancy in these values was considered to be smaller than the uncertainties arising from the use of surfaces or the polynomials in the first instance.

Table 3.2 Values of the surface and polynomial fits to the over-bank flow where they join together above the bank top.

depth (m) value of surface (m/s) value of polynomial (m/s) recorded value (m/s) 0.155 0.421 0.398 0.412 0.16 0.446 0.455 0.460 0.17 0.462 0.440 0.465

Fluctuating velocity data was not available for the over-bank flow at the time of writing, although some large scale empirical relations determined by Knight and

Shiono (1990) were used and are discussed later.