Full Equations (FEQ) Model for the Solution of the Full, Dynamic Equations
of Motion for One-Dimensional Unsteady Flow in Open Channels and Through
In unsteady-flow analysis, two governing algebraic equations must be explicitly solved because the flow and the elevation of the water surface are both unknown. One of the governing equations is the conservation of water volume, and the other is the conservation of water momentum. In steady-flow analysis, the equation for conservation of water volume was trivial because the flows were constant and were used to solve for the flows everywhere in the channel (known elevations were unnecessary). In unsteady-flow analysis, however, a governing equation of conservation of water volume must be explicitly solved for flows and elevations.
In unsteady-flow analysis, computational elements and algebraic approximations to the differential or integral terms in the governing equations must be used to develop two algebraic equations for each computational element written in terms of elevations and flows at the ends of the element. These governing equations are more complex than those for steady-flow analysis. For unsteady flow, a computational element with respect to time also must be considered, but it is simple: the time axis is divided into finite increments that, ideally, will be short enough so that the algebraic approximations of the differential and integral terms will be sufficiently accurate. Because of this dependence on time, the algebraic governing equations involve not only the unknown flow and elevation at two points along the channel but also at two points in time.
Control points with known relations between elevation and flow must be identified, as well as points of rapidly varied flow or of interaction between channels not described by the algebraic governing equations. As in steady-flow analysis, these points establish the limits of applicability of the governing equations with respect to distance along the channel and provide known values for the analysis. In unsteady-flow analysis, however, a starting time for the computations when all the flow values are known at the computational nodes (ends of the computational elements) must be established. Flow is assumed to be steady everywhere in the system at the starting time. This is the first major difference between steady flow and unsteady flow: a steady-flow analysis must be completed to establish the initial condition for the unsteady-flow analysis.
A second major difference between unsteady-flow analysis and steady-flow analysis is the information needed at the boundaries of the stream system. In steady-flow analysis, knowledge of one elevation at the downstream boundary is needed to start the computations for subcritical flow or at the upstream boundary for supercritical flow. A cursory analysis of the number of equations available in unsteady flow shows that more information is needed for unsteady-flow analysis. For example, a single channel with no special features is divided into 9 computational elements yielding 10 nodes. With 2 unknowns at each node, there are 20 unknowns but only 18 equations (2 per computational element). Thus, the unknowns cannot be determined without some additional information at the boundaries of the system. When the flow is subcritical, information at both the upstream and the downstream boundary of the system is needed. This information can be in one of three forms: flow known as a function of time, water-surface elevation known as a function of time, or a relation between flow and water-surface elevation. The upstream boundary is commonly flow known as a function of time (a hydrograph), and the downstream boundary is commonly a known relation between flow and water-surface elevation (a rating curve). The information supplied at a boundary is called a boundary condition.
The information supplied at a special feature internal to the stream system is often called an internal boundary condition. In unsteady-flow analysis, internal boundary conditions are approximated as steady-flow relations because the special features generally are short enough that the changes in momentum and volume of water within the special features are small. The isolation and description of the special features is a major component of unsteady-flow analysis.
The same computational problems can arise as for unsteady-flow analysis as steady-flow analysis because both analyses use algebraic approximations to the differential and integral terms. These approximations are developed for a computational element of finite length. If the computational element is too long, an incorrect solution results. The difference between the analyses is that in unsteady-flow analysis the computational problems are more complex and more frequent than in steady-flow analysis. The increased frequency is primarily because unsteady-flow analysis involves computations over a wide range of water-surface elevations, whereas most steady-flow analysis involves computations over a narrow range of water-surface elevations. Furthermore, the time dimension results in additional complications.
Similarities and differences between steady- and unsteady-flow analysis are summarized in table 1. The motion equation in this table is expressed by use of the principle of conservation of momentum.
[Q, flow rate; A, cross-sectional area; y, height of water surface above the minimum point in the cross section; x, distance along the channel; t, time; g, gravitational acceleration; q, inflow into channel over or through the sides (lateral flow); S0, bottom slope of the channel, positive with decline downstream; Sf, friction slope]