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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