understanding and mathematical formulation of the relevant physical processes.

Notwithstanding this fundamental difficulty, design engineers have an immediate need for

reliable numerical simulations, and hydraulic research engineers have targeted alluvial river

hydraulics as a prime area for continuing fundamental and applied research. Out of this

fortunate confluence of interest have arisen a variety of simulation techniques and software

systems, as well as many apparently successful simulations of prototype situations.

The most basic one-dimensional description of water and sediment flow in an alluvial river

consists of four relations: conservation of water; conservation of water momentum;

conservation of sediment; and sediment-transport relationships. These equations form a

nonlinear partial-differential system that in general cannot be solved analytically.

When the water wave propagation effects are of secondary importance for sediment-transport

phenomena, the system of equations can be simplified by assuming that the water flow

remains quasi-steady during a certain interval of time.

Most published 1-dimensional software systems for the solution of the water- and

sediment-flow equations use one form or another of the finite-difference method, in which time

and space derivatives are approximated by differences of nodal values of grid functions that

replace the continuous functions, leading to a system of algebraic equations. Some authors

have used the finite-element method, but in one dimension there does not appear to be any

strong reason for doing so. In any case, the quality and reliability of numerical models for bed

evolution are determined primarily by the sediment-transport formulation and mechanisms

adopted for sorting, armoring, and so forth. The particular numerical method used, as long as

it is consistent with the partial-differential equations and is stable, has only a secondary effect

on simulation quality.

Whether the full unsteady set of equations or the quasi-steady set of equations is solved

numerically, two basic approaches are possible: coupled or uncoupled. In the coupled case, a

simultaneous solution of both water and sediment equations is sought. This is evidently the

physically proper way to proceed, because the water-flow and sediment-transport processes

occur simultaneously. However, the simultaneous solution may involve certain computational

complications, especially when the sediment-transport flow resistance equation involves not

just an analytic mathematical expression, but a whole series of procedures and computations

to simulate alluvial channel processes such as armoring, sorting, and bed forms.

The uncoupled procedure has arisen essentially to circumvent the computational difficulties of

the coupled approach. The uncoupling of the liquid and solid transport occurs during a short

computational time step. First the water-flow equations are solved to yield new values of depth

and velocity throughout the reach of interest, assuming that neither the bed elevation nor the

bed-sediment characteristics change during the time step. Then the depths and velocities are

taken as constant, known inputs to the sediment continuity and transport equations; these

equations then become relatively easy to solve numerically, yielding the new bed elevations.

When the overall model includes bed-sediment sorting or armoring, these processes are

simulated in a third uncoupled computational phase using new depths, velocities, and bed

elevations as known inputs. Although it is difficult to quantify the error associated with this

artificial uncoupling of simultaneous, mutually dependent processes, it is intuitively obvious that

the uncoupling is justified only if bed elevations and bed-material characteristics change very

little during one time step. Experience in the use of uncoupled models, with both the unsteady

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