Finding the Bed Shear Stress on a Rough Bed Using the Log Law

Abstract

The procedure commonly used to determine the friction velocity from a measured velocity profile in turbulent flow over a hydraulically rough bed is to fit the log law to the velocity profile and adjust the displacement height to obtain a best-fit line to as many data points as possible in the inner layer. In practice, the process can be subjective and produce large uncertainty in the bed shear stress estimates and/or inconsistent results for the equivalent roughness height. In oscillatory flows, a temporal variation in the equivalent grain roughness is unrealistic because the roughness height should remain constant if the boundary Reynolds number is sufficiently large. An alternative method is presented in this study, in which the equivalent grain roughness is held constant, and the displacement height is varied until the value of the von Kármán constant obtained from the best-fit line is equal to the universally accepted value of about 0.4. The iterative process converges rapidly and is easier to apply than the traditional method, which requires the displacement height to be found by trial and error. The method was tested in steady, shallow uniform flows over a fixed bed of fine gravel. The channel slope was varied, and the velocity profile was measured using the particle image velocimetry (PIV) technique. Good agreement was found between the bed shear stress estimates obtained using the new method and the values calculated from the measured flow depth and channel slope when the ks/d90 ratio was taken from the literature for small values of the h/d90 ratio, where h is the flow depth, ks is the equivalent roughness height, and d90 is the grain diameter with 90% of finer particles, therefore verifying that the new method produced results consistent with published data. The method was then applied to velocity measurements under a solitary wave to obtain the temporal variation of bed shear stress on a gravel bed near the point of incipient wave breaking.