By Bernard LeMehaute
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Extra info for An Introduction to Hydrodynamics and Water Waves
In this case, since V does not vary with time, the corresponding inertia term av jot is zero. ) However, there are many unsteady motions in hydraulic engineering in which the local acceleration and the corresponding inertia terms are neglected. This occurs when the velocities are slow and their variations with time are very slow. For instance, in the case of a periodic motion in which the period Tis very long: oVjot ~ V/T Hence p(oVjot) would be negligible m comparison with other forces. Some particular cases where this approximation is valid are (1) flow in a porous medium: variation of the ground water table with respect to time; (2) flood wave in a river; (3) variation oflevel in a reservoir because of the variation of the upstream river flux, the spillway and bottom outlet control, and turbined discharge; and (4) emptying of a basin by a small valve.
2 It has been seen that the linear deformation velocity components are those given in Equation 4-1. ~~dx} -av d y oy ow dz az Two-dimensional motion ax dt Two similar expressions result for w and v. If u = dxjdt, v = dyjdt, and w = dzjdt, are substituted in these expressions and the result is multiplied by the density, the inertia forces are obtained. 1 The Case of Linear Deformation = ou dx = 1 a(v 2 ) -p-2 ay ow 1 o(w 2 ) pw-=-p-az 2 oz It should be noticed that the last group of expressions may be written as (ajox)(pu 2 j2).
4-3). This example is particularly helpful in understanding how the mathematical simplifications may be based on physical considerations. Hypothesis: u is large in comparison with v; the derivatives with respect to y are large compared to the derivatives with respect to x. The continuity equation for two-dimensional motion shows also that oujox and ovjoy are of the same order. 2). Hence, the 0 Y components of the convective inertia terms are negligible because au ox au ay ov ax ov ay u-+v-~u-+v- Similar approximations are made to analyze the development of a jet, and in the nonlinear long wave theory (Chapter 18).