Characteristics of Loose Rough Boundary Streams at Near-Threshold

Laboratory experimental results on the turbulent flow characteristics, measured by an acoustic Doppler velocimeter, over loose rough boundaries at the near-threshold of motion are reported. The primary endeavor was to investigate the response of the turbulent flow field, having zero-pressure gradient, to the uniform gravel beds at the near-threshold of motion with reasonably wide range of relative roughness $\epsilon ∕h$ ( $0.02<\epsilon ∕h<0.15$ ; where $\epsilon =\text{Nikuradse}$ ’s equivalent roughness and $h=\text{flow}$ depth). Five uniform gravel sizes, ranging from $4.1\text{to}14.25\mathrm{mm}$ , were used in the experiments. The vertical distributions of time-averaged velocity, turbulence intensities and the Reynolds stress were measured and analyzed. The variation of the mixing-length is considerably linear with the depth within the inner layer, whose thickness is 0.23 times the boundary layer thickness $\delta$ , and the von Kármán constant is obtained as 0.35. The boundary shear stresses were determined by three independent methods, such as using the logarithmic law (the Clauser method), Reynolds stress profiles and bed slope. It is observed that in the inner layer, the logarithmic law of wall for the time-averaged streamwise velocity holds with the von Kármán constant 0.35 and a constant of integration 7.8; whereas in the outer layer, the law of the wake defines the velocity profiles with average value of the Coles’ wake parameter 0.11. Alternatively, velocity profiles based on $1∕n\text{th}$ power law are also put forward, where $n$ is expressed as a function of $\epsilon ∕\delta$ . The equation of the friction factor $f$ is obtained from the depth averaging of the velocity profile; and the collapse of the estimated $f$ and the measured data (experimental and field) is reasonable. The turbulence intensities being non-isotropic can be defined by an exponential law; whereas the Reynolds stress varies almost linearly with the depth. The values of the Shields parameter for the threshold of gravel motion obtained experimentally correspond closely with the curve obtained from the modified model of Dey considering the present logarithmic law of velocity. For all the aforementioned parameters, thorough statistical and error analyses of the experimental data are done to ascertain the adequacy of the proposed equations.