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5 ~ 5' + 5" . .. .., . ... ... ..... . co. ... " . .. . ..... . . ... '" . co co co. (3) The longitudinal energy gradient can be evaluated using any valid flow resistance relationship. This model employs the Manning formula for flow resistance. The transverse energy gradient is evaluated using the following equation for wide channels (9) 5" ~ (2.860 + ~7f) (J:;)' F' co co. co. co. . co co. co, . co co' co co co co (4) 0.565 + f r, in which f ~ Darcy-Weisbach friction factor; h, ~ flow depth at channel center line; r, ~ mean channel radius; and F ~ Froude number. Evaluation of Changing Flow Curvature.-Analytical relationships pertaining to curved channels are often based upon the mean channel radius, r,. Because of the streamwise changing curvature, application of such relationships is limited to the fully developed transverse flow for which the curvature is defined. Streamwise variation of transverse flow, over much of the channel length, is characterized by its growth and de- cay. In order to describe this variation, the mean flow curvature defined as the flow curvature along channel center line is employed. It is as- sumed that analytical relationships for developed transverse flows are applicable for developing transverse flows when the mean channel cur- vature, r" is replaced by the mean flow curvature, rf' Upon entering a bend, the mean flow curvature increases with the growth in transverse circulation. In a bend, the transverse flow becomes fully developed if the flow curvature approaches the channel curvature. In the case of ex- iting from a bend to the downstream tangent in which the channel cur- vature is zero, the flow curvature decreases with the decay of transverse circulation. The reason that the flow curvature lags behind the channel curvature during circulation growth and decay is attributed to the internal tur- bulent shear that the flow has to overcome in transforming from parallel flow into the spiral pattern and vice versa (22). From the dynamic equa- tion for the transverse velocity, the writer (to) has developed an equa- tion governing the spatial variation in transverse surface velocity, v, along channel center line. Details of this development are given in Appendix J. In finite difference form, the change in v over the distance as is given by v,+, ~ [Vi + ~ (~ -;~~) ~exp (F~ ~as) as] exp ( -F ~ ~ as) ,.......,............,....,................, (5) in which v ~ transverse surface velocity along channel center line; K ~ Karman's constant, which has the approximate value of 0.4; U ~ average velocity of a cross section; i, i + 1 = s-coordinate indices; F = a function of f and K; and the overbar denotes averaging over the distance between i and i + 1. Eq. 5 provides the spatial variation in v, from which the 646 24