Lacey Divergence Equations for Alluvial‐Canal Design

Besides discharge $Q$ and sediment characteristics (size $D_{50}$ and concentration $C$), the third physical factor governing hydraulic geometry of an alluvial canal is resistance to erosion of bank material. Lacey formulas provide one set of the parameters of hydraulic geometry through the wetted perimeter $P$, hydraulic mean radius $R$, and slope $S$, with implied values of sediment concentration $C$ and maximum safe boundary shear stress t\ds. But with varying values of $C$ and t\ds, many other sets of $P$, $R$, and $S$ are possible. Defining Lacey divergence as the ratio of actual value to value obtained with the Lacey formula and designating this ratio by subscript $r$, Lacey Divergence relations were obtained for velocity $V$ and $C$. Velocity ratio V\dr was found related only to slope ratio S\dr, whereas sediment concentration ratio C\dr was related to both S\dr and boundary shear stress ratio t\dr. Equations were then derived for P\dr, R\dr and S\dr in terms of C\dr and t\dr, which permit determination of $P$, $R$, and $S$ when design data in respect of $Q$, $D_{50}$, $C$, and t\ds are given.