Model Development

Introduction

Many flow phenomena of great engineering importance are unsteady in characters, and cannot be reduced to steady flow by changing the viewpoint of the observer. A complete theory of unsteady flow is therefore required, and will be reviewed in this section. The equations of motion are not solvable in the most general case, but approximations and numerical methods can be developed which yield solutions of satisfactory accuracy.

Review of Governing Equations

The law of continuity for unsteady flow may be established by considering the conservation of mass in an infinitesimal space between two channel sections (figure 1). In unsteady flow, the discharge, Q, changes with distance, x, at a rate , and the depth, y, changes with time, t, at a rate . The change in discharge volume through space dx in the time dt is ( ) dx dt. The corresponding change in channel storage in space is T dx ( ) dt = dx ( ) dt in which A = Ty. Because water is incompressible, the net change in discharge plus the change in storage should be zero; that is

( ) dxdt + T dx ( ) dt = ( ) dxdt + dx ( ) dt = 0.

Simplifying, + T = 0 (1) or + = 0 (2)

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