Empirical Equation for Determination of Alternate Bar Height

A new empirical equation for alternate bar height is introduced. It is assumed that the bars are formed under a steady and uniform flow; the stage of interest is that where bars have grown to their fully developed state. The equation is developed on the basis of dimensional considerations and all data available to the authors; the formulation also incorporates findings by stability analysis. The data result from a total of 191 flume experiments reported in 16 different works carried out from 1961 to 2014 using either sand or gravel as bed material. Both fully rough and transitionally rough flows were used in the experiments. Overall, width-to-depth ratios and relative depths varied from 3.5 to 54.4 and 3.8 to 191, respectively; the ratios of bed shear stress to critical bed shear stress ranged from 1.1 to 14.7. Data from the Naka River, Japan, are also used. The equation correlates bar height, normalized by flow depth, with the excess width-to-depth ratio ($B/h$) with regard to the smallest, or critical, value of $B/h$ at which alternate bars occur as well as with the relative depth and, in the case of transitionally rough flows, also the grain-size Reynolds number. It is shown that when compared with previous empirical equations, the present equation produces a considerably improved overall alignment of the data with the perfect agreement line and a significant larger (nearly double) percentage of data falling within the 20% error range. The results by the present equation are also substantially more congruent with those derived from existing theoretical and numerical analyses of the development of alternate bars, and more specifically, those resulting from a stability analysis of the phenomenon. As a by-product of this work, a first comparative evaluation of existing equations for alternate bar height is also presented.