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Most natural open-channel and overland flows belong to the class of hydraulically rough-bed flows. Although hydrodynamics of such flows has been studied extensively for the last two decades, there are still many unsolved problems awaiting clarification. In general, turbulent flow is modeled and studied through Reynolds averaged Navier-Stokes (RANS) equations. Despite the ability of these equations in modeling of turbulent flows, they have some deficiencies in natural flows (such as atmospheric flow or water flows in rivers and estuaries) where flow characteristics vary in multiple time and length scales. In addition, in some situations, these equations are not practicable due to the highly three-dimensional small-scale structure of the mean flow and turbulence, especially in the near bed region. Moreover, RANS equations are locally resolved the flow characteristics, which is in contrast to many hydraulics fundamental concepts such as uniformity, Manning coefficient, water discharge. To resolve this problem, the time averaging of the Navier-Stokes equations should be supplemented by spatial averaging in a plane parallel to the mean bed surface. After such an averaging a new system of equations will be obtained which are known as double averaged, or spatially averaged, Navier-Stokes equations. The double-averaging procedure gives new momentum and continuity equations for fluid, which are averaged in both time and space domains and which explicitly contain important additional terms such as form-induced stresses and, for the flow region below roughness tops, form and viscous drag terms. These type of Reynolds averaged Navier-Stokes equations have various applications in hydraulic studies. One of the main application of these equations is in heterogeneous turbulent flow above rough surfaces such as vegetated or gravel bed flow. The present study demonstrates applications of double averaged equations in study of rough bed flows. To this end, laboratory measurement were conducted in an open-channel laboratory flume. Experimental measurements cover an appropriate range of the bed roughness characteristics so that the channel aspect ratios, the ratio between the channel width and water depth (B⁄H), are 6.2 and 7.7. Bottom of the channel is roughened using two series of crushed stone which are spread randomly at the bed and then glued to the bed. Acoustic Doppler Velocimeter (ADV) is used to measure three components of velocity field. Both velocity profile and Reynolds shear stress are estimated based on the measured velocity time series. Results of these measurements show that velocity field in the near bed region shows strong spatial variation due to the rough bed elements protrusions. To properly take in to account this spatial variation in shear velocity, spatially averaged Reynolds shear stress profile can be reliably used for determination of the bed shear stress. Furthermore, a new method for determination of the vertical logarithmic profile of streamwise velocity is introduced using the bed shear velocity obtained from spatial averaging. This method is an iterative process in which parameters of logarithmic profile, i.e. zero-plane displacement (the bed origin displacement due to the rough elements presence) and constant if integration will be estimated based on the measured velocity profile. Results of experimental data analysis, using the new method, show that the logarithmic profile parameters can be efficiently determined.

Keywords: Rough bed, Velocity profile, Turbulent flow, Acoustic Doppler velocimeter (ADV), Double averaging.

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