1014 gradually varied flow for five classes of

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Unformatted text preview: pect to the normal depth yn and the critical depth yc. In type 1 solutions, the initial point is above both yn and yc, and in all cases the water-depth solution y(x) becomes even deeper and farther away from yn and yc. In type 2 solutions, the initial point lies between yn and yc, and if there is no change in S0 or roughness, the solution tends asymptotically toward the lower of yn or yc. In type 3 cases, the initial point lies below both yn and yc, and the solution curve tends asymptotically toward the lower of these. Figure 10.14 shows the basic character of the local solutions, but in practice, of course, S0 varies with x and the overall solution patches together the various cases to form a continuous depth profile y(x) compatible with a given initial condition and a given discharge Q. There is a fine discussion of various composite solutions in Ref. 3, chap. 9; see also Ref. 18, sec. 12.7. Numerical Solution The basic relation for gradually varied flow, Eq. (10.49), is a first-order ordinary differential equation which can be easily solved numerically. For a given constantvolume flow rate Q, it may be written in the form dy dx S0 4/3 n2Q2/( 2A2Rh ) 2 3 1 Q b0/(gA ) (10.51) subject to an initial condition y y0 at x x0. It is as...
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