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<br />
<br />they are not particularly demanding of computer
<br />resources. A recent workshop conducted by the
<br />Federal Energy Regulatory Commission (Fan
<br />1988) provided a comprehensive review of most. of
<br />the codes in active use or in development.
<br />The various 1- D models represent quite different
<br />conceptions of the physical processes of sorting and
<br />armoring, selective transport of sediment mix-
<br />tures, channel width adjustments, and flow hydro-
<br />dynamics. In addition, they differ significantly in
<br />the numerical techniques employed to obtain ap-
<br />proximate solutions to the nonlinear governing
<br />equations. At present, there does not seem to be one
<br />model, or family of models, that is best in the sense
<br />of incorporating the most complete description of
<br />I-D mobile-bed physical processes. Rather, each
<br />model tends to focus on the particular process
<br />judged most important by the model developer. For
<br />example, one code may can-y the distinction be-
<br />tween bedload and suspended load to a high level
<br />of refinement, assuming nonerodible banks,
<br />whereas another code may treat bedload and sus-
<br />pended-load distinctions with a great deal less care
<br />but adopt a highly refined procedure for channel
<br />width adjustment through bank erosion.
<br />The remainder of this section and the example
<br />Missouri River application that follows are focused
<br />on a particular I-D mobile-bed simulation code,
<br />CHARIMA (Holly et al. 1990).
<br />The central feature of any 1- D mobile-bed model
<br />is the so-called Exner equation, expressing conser-
<br />vation of sediment in a control volume of the river
<br />bed, as follows:
<br />az aQs 0
<br />(l-p)B- +- + s-
<br />at ax
<br />where p = bed-sediment porosity, B = bed width
<br />subject to erosion or deposition, z = average bed
<br />elevation, Qs = bedload transport, S = suspended-
<br />load source exchange between the bed and the
<br />water column, t = time, and x = longitudinal
<br />(streamwise) length coordinate. Equation 1 simply
<br />formalizes the fact that any net bedload or sus-
<br />pended-load inflow into the control volume across
<br />its control surface must result in a net increase of
<br />sediment material in the control volume, and this
<br />must cause a change in the bed elevation (erosion
<br />or deposition).
<br />If Equation 1 were considered as the pillar of a
<br />mobile-bed model, the rest of the model can be
<br />considered as providing the wherewithal for Equa-
<br />tion 1 to do its job. The bedload transport, Qs,
<br />depends strongly and nonlinearly on the flow hy-
<br />drodynamics, through the depth and velocity (dis-
<br />charge). Thus a parallel hydraulic computation
<br />
<br />FORREST M. HOLLY, JR. AND RoBERI' ETrEMA 419
<br />
<br />solves the de St. Venant equations for unsteady
<br />flow, expressing conservation of mass and moxnen-
<br />tum:
<br />
<br />(1)
<br />
<br />aA+~_o (2)
<br />at ax
<br />
<br />~ + :X(a~)+gA~+gA~-O (3)
<br />
<br />where A = cross-sectional flow area, Q = water
<br />discharge, a. = kinetic"energy correction coeffi-
<br />cient, g = gravitational acceleration, y = water-sur-
<br />face elevation, and K = channel conveyance. The
<br />dependent variables of these flow equations are
<br />the water discharge (Q) and water-surface eleva-
<br />tion (y), both varying spatially and temporally.
<br />The bedload transport (Qs) and the erosion com-
<br />ponent of the suspended-load source term (8) of
<br />Equation 1 depend not only on the flow conditions,
<br />but also on the bed composition, that is, the rep-
<br />resentation of various grain sizes, discretized as
<br />size classes, in the exposed material on the bed
<br />and available. for entrainment. The tracking of
<br />changes in the bed-material composition requires
<br />solution of several so-called sorting equations,
<br />similar to Equation 1 but expressing mass conser-
<br />vation for each of several sediment size classes in
<br />a layer of active bed-sediment mixing near the
<br />bed.
<br />Sediment in suspended-load transport obeys a
<br />mass-conservation law written for a control vol-
<br />ume that includes the water column. This advec-
<br />tion-diffusion equation is written for each size
<br />class as
<br />a a a ~
<br />at (CjA) + ax (CjQ) = 0; (AK ax) + Sj (4)
<br />
<br />where Cj = section-averaged suspended-load
<br />concentration for a size class j, K = diffusion
<br />coefficient, and Sj = the source-term exchange of
<br />sediment between the bed and the water column
<br />that appears in Equation 1; it also appears in the
<br />sorting equations for each size class. Note that the
<br />source term in Equation 1 actually represents the
<br />sum of source terms over all size classes in the
<br />model. This source depends strongly and
<br />nonlinearly on the local hydraulic conditions, the
<br />bed-s~diment and suspended-load composition,
<br />and the actual suspended-load concentration
<br />itself.
<br />These basic relations, all feeding Equation 1,
<br />are supplemented by additional ones representing
<br />bed armoring, transport-dependent bed rough-
<br />ness, channel geometry, bank erosion, and, of
<br />course, a bedload transport predictor. Most codes
<br />offer a choice of transport predictors; man.y are
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