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<br />: dO + g aH + : 1.. (~~)+ g5 _ ~_ q" 0 <br />A dt OS A dS A A2 <br /> <br />(2) <br /> <br />where 0 is the discharge, A is the cross-sectional area of flow, t is the <br />time, s is the curvilinear coordinate along discharge centerline measured <br />fran the upstream entrance, q is lateral inflow rate per unit length, H is <br />the stage or water-surface elevation, and 5 is the energy gradient. The <br />upstream boundary conch tion for water routing is the inflow hydrogrphs and <br />the downstream condi tion is the stage-discharge relation or the base-level <br />variation. Techniques for rn.merical solution of Egs. 1 and 2 are described <br />in Chen (1973) and Fread (1971, 1974), among others. <br /> <br />In a curved channel, the total energy gradient, 5, in Eq. 2 is <br />partitioned into the longitudinal energy gradient, S', and the transverse <br />energy gradient, 5", due to secondary currents, i.e. <br /> <br />5 " 5' + 5" (3) <br /> <br />The 10ngitudimll energy gradient can be. evaluated using any valid flow <br />resistance relationship. If Manning's formula is employed, the roughness <br />coefficient n roust be selected by the modeler. However, if a formula for <br />alluvial bed roughness, e.g., Brownlie's fonnu1a (1983), is used, the <br />roughness coefficient is predicted by the formula. <br /> <br />Method for evaluating the transverse energy gradient by Chang (1983) is <br />used in the model. Because of the streamwise changing curvature, the <br />transverse energy loss varies with the growth and decay of secondary <br />currents along the channel. Analytical relationships pertaining to curved <br />channels are often based upon the mean channel radius, rc' Under the <br />strearnwise changing curvature, the application of such relationships is <br />limi ted to fully-developed transverse flow for which the curvature is <br />defined. 5treamwise variation of transverse flow, over much of the channel <br />length, is characterized by its growth and decay. In order to describe this <br />spatial variation, the mean flow curvature defined as the flow curvature <br />along the discharge centerline is anp10yed. It is assumed that analytical <br />relationships for developed transverse flows are applicable for developing <br /> <br />11 <br />