Analytical study of stable meander rivers geometry

Channel geometry has been considered by numerous researchers over the years. Regime theory and hydraulic geometry are among the most important models, regarding the river engineering and river morphology, developed over the past centuries. There is an important condition for river engineers in which river shape, geometry section size and longitudinal slope of the river is in equilibrium with the hydraulic and sediment characteristics applied to the river. Each of the existing regime equations that are developed by different researchers would be reliable and applicable in certain conditions which are similar to the derivation conditions. Therefore, determination of regime equations to be used in a wider range of conditions is necessary for future researches. In fact, hydraulic geometry relationships describe the alluvial channel shape and determine changes of width, channel average depth, velocity and longitudinal slope of the river with discharge. This condition is considered for a cross-section and is called at-a-station hydraulic geometry. However, if changes the width, channel average depth, velocity and longitudinal slope of the river bed are considered over a river reach for a particular flow discharge, it is called downstream hydraulic geometry. This discharge could be bankfull discharge or mean annual discharge. In this study, the term “hydraulic geometry relations” refers to downstream hydraulic geometry relations and particular flow discharge is bankfull discharge. The main focus of this study is to analytically derive the hydraulic geometry equations the concept of secondary flow in meandering channels; Therefor, first, the principles and concepts of the hydraulic geometry are presented. Because of the importance of independent and dependent variables, they should be selected carefully. There are several parameters that control the rivers. Three parameters of flow discharge, bed particle size, and bed sediments load are more important than others so that changes of each parameter will influence the equilibrium and stability of the river. A new hydraulic geometry is developed to achieve the equilibrium state with four degrees of freedom of depth, width, bed longitudinal slope and velocity. Therefore, four equations are required to find the four unknowns (depth, width, bed longitudinal slope and velocity). Then, the hydraulic geometry equations are theoretically developed by using four governing equations: continuity, resistance to flow, bed load equations and secondary flow. The third equation which estimates sediment transport is a function of Shields function and Shields function equation can be used instead of this equation. The fourth equation considers the effect of secondary flow and is as the ratio of the radial shear stress to the longitudinal shear stress. Because of the numerous bends in such rivers, cylindrical coordinates should be used. A particle starts to move when its shear stress is higher than this value in threshold condition. Two stresses influence sediment particles in a meander: radial shear stress and longitudinal shear stress. Therefore, the angle between the resultant shear stress and the longitudinal shear stress is very important. By increasing the angle, the radial shear stress will increase and a stronger secondary flow would form. Therefore, the sediment particle will move from the outer bend to the inner and makes a sharper bend. By decreasing the angle, the longitudinal shear stress will increase and a weaker secondary flow will form, so that there is a milder bend. Therefore, to consider the effect of the secondary flow in the equations, the radial shear stress to the longitudinal shear stress ratio can be used. Thus, the independent variables of equations are: flow discharge, mean sediment size and slope. While, the dependent variables of equation are: the mean depth; surface width; mean flow velocity; and Shields parameter. Eatone and Milarer (2004) considered the longitudinal slope of the bed as the independent variable for two reasons: Measurement of the bed longitudinal slope is usually easier and cheaper than that of the sediment load in most rivers; and bed load functions are not necessarily capable of calculating accurate values. Therefore, parameters of the bed slope and sediment load were selected as the independent and dependent variables, respectively, in this case. Hey and Thorne (1986) field data for the UK rivers were applied to calibration the developed the model. A reasonable agreement between observed and calculated values was obtained partially. However, some discrepancies were also observed in the results which may be due to the assumptions made in the model. Finally, sensitivity analysis was conducted to figure out the parameter to which the model is most sensitive.