No / This paper presents an integration procedure for the Reynolds-averaged Navier-Stokes equations for the determination of the distribution of the streamwise velocity using the vertical component. This procedure is dedicated to the outer region and central part of channels. The proposed model is applicable to both rough and smooth flow regimes, provided the velocity at the inner-outer boundary has been properly defined. To generate a simplified expansion, a number of hypotheses are proposed, focusing in particular on the analytical modeling of the vertical component by adopting a negligible viscosity. The proposed hypotheses are validated by the experimental data existing in the literature. The proposed simplified expansion is studied through a sensitivity analysis and proved consistent in regards to model experimental data. The proposed model seems capable of demonstrating different kinds of flows, including dip phenomenon flow patterns.

Yes / The Prandtl second kind of secondary current occurs in any narrow channel flow causing velocity dip in the flow velocity distribution by introducing the anisotropic turbulence into the flow. Here, a study was conducted to explain the occurrence of the secondary current in the outer region of flow velocity distribution using a universal expression. Started from the basic Navier-Stokes equation, the velocity profile derivation was accomplished in a universal way for both smooth and rough open channel flows. However, the outcome of the derived theoretical equation shows that the smooth and rough bed flows give different boundary conditions due to the different formation of log law for smooth and rough bed cases in the inner region of velocity distribution. Detailed comparison with a wide range of different measurement results from literatures (from smooth, rough and field measured data) evidences the capability of the proposed law to represent flow under all bed roughness conditions.

Yes / Based on the previous studies, the shallow water equations (SWEs) model was proven to be insufficient to consider the flow turbulence due to its simplified Reynolds-averaged form. In this study, the k-ε model was used to improve the ability of the SWEs model to capture the flow turbulence. In terms of the numerical source terms modelling, the combined k-ε SWEs model was improved by a recently proposed surface gradient upwind method (SGUM) to facilitate the extra turbulent kinetic energy (TKE) source terms in the simulation. The laboratory experiments on both the smooth and rough bed flows were also conducted under the uniform and non-uniform flow conditions for the validation of the proposed numerical model. The numerical simulations were compared to the measured data in the flow velocity, TKE and power spectrum. In the power spectrum comparisons, a well-studied Kolmogorov’s rule was also employed to complement both the numerical and experimental results and to demonstrate that the energy cascade trend was well-held by the investigated flows. / The Major State Basic Research Development Program (973 program) of China (Grant Number 2013CB036402). Open Fund from the State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, China (Grant Number SKLH-OF-1103).