Stochastic prediction of shear stress distributions at the bed of open channel bends.

Abstract

A stochastic-type numerical model is developed to predict the stable bed shear stress distributions and related bed topography in the vicinity of open channel bends having large radius of curvature to flow depth ratios. The flow regions described by the model include the bend sections and those portions of the downstream straight channel reaches along which the strong secondary flow mechanisms, generated in the former, remain an influential flow feature. In addition to the theoretical and numerical analyses the overall investigation included comprehensive laboratory studies on a representative physical model of the system. The physical model consisted of a 10 m long plexiglass channel, having a rectangular cross section 30 cm wide by 6 cm deep, that contained an adjustable bend (test) section approximately 2 m downstream from the entrance to the channel. Central bend angles of 45°, 60° and 75° with a mean radius of curvature to channel width ratio = 3 were adopted for the studies. The bed material used throughout the tests was a coarse sand having a median grain size (d50) = 0.7 mm and standard deviation (sigma) = 2.05. Point velocity and turbulence measurements were obtained using a Lazer Doppler Anemometer system. The major advantage of this particular system, over other velocity-measuring systems, is the remote sensing (optical) technique it employs. Since local disturbance of the flow field, at the point of measurement, is eliminated overall accuracy is enhanced. Measured shear stress distributions within the test sections were expressed in dimensionless format, i.e., local boundary shear stress was related to the corresponding theoretical uniform shear stress for a straight infinitely-wide rectangular channel operating under similar flow conditions. In the numerical analyses different models were employed to generate steady state bed shear stress distributions in the various test sections along selected pathlines that were parallel to the channel walls and comparison between predicted and measured stress distributions were used to examine the relative merits of the models developed for this purpose. The results of these comparative studies indicate that shear stress distributions, along any pathline parallel to the channel walls, can be predicted to any require accuracy using a finite Fourier series approach. Furthermore, to ensure an acceptable degree of accuracy in the prediction, a minimum number of harmonics in the analysis is recommended. This number was found to depend on the position of the respective pathlines (relative to the channel's inner wall) and also on channel geometry; for example the number increases as the radius of curvature of the pathlines and the bend angle decrease. Moreover, a second order autoregressive process appears appropriate in the analysis and the auto correlation function for all pathlines is found to be of the damped sine wave or damped exponential wave character. This conclusion is reflected in the periodic behaviour of the change in the bed shear stress along any pathline in the channel test section.