46 where s0 is the slope of the channel bottom

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Unformatted text preview: d 2g 2g y HG V L τw y + dy S0 d x Fig. 10.13 Energy balance between two sections in a gradually varied open-channel flow. Sdx V+ Bottom slope S0 dV dx x x + dx But dA b0 dy, where b0 is the channel width at the surface. Eliminating dV/dx between Eqs. (10.46) and (10.47), we obtain dy 1 dx V2b0 gA S0 S (10.48) Finally, recall from Eq. (10.37) that V2b0/(gA) is the square of the Froude number of the local channel flow. The final desired form of the gradually varied flow equation is dy dx S0 S 1 Fr2 (10.49) This equation changes sign according as the Froude number is subcritical or supercritical and is analogous to the one-dimensional gas-dynamic area-change formula (9.40). The numerator of Eq. (10.49) changes sign according as S0 is greater or less than S, which is the slope equivalent to uniform flow at the same discharge Q S S0n f V2 Dh 2g V2 RhC2 n2V2 2 4/3 Rh (10.50) where C is the Chézy coefficient. The behavior of Eq. (10.49) thus depends upon the relative magnitude of the local bottom slope S0(x), compared with (1) uniform flow, y yn, and (2) critical flow, y yc. As in Eq. (10.38), the dimensional parameter 2 equals 1.0 for SI units and...
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This note was uploaded on 10/27/2009 for the course MAE 101a taught by Professor Sakar during the Spring '08 term at UCSD.

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