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<br />II. THE MATHEMATICAL M0DEl <br /> <br />1. The Governing Equations <br /> <br />Unsteady flow encountered in flood-wave movement in a prismatic <br />channel is described by the Saint-Venant equations (2, 6) <br /> <br />. <br /> <br />1\ aV + BV 2i + ii 2i = 0 <br />ax ax at <br /> <br />(la) <br /> <br />, <br /> <br />av + V av + 9 1i = 9 (So-Sf) <br />at ax ax <br /> <br />(lb) <br /> <br />A bar over a variable indicate~ a dimensional quantity, unbarred variables <br />~r~ ~imen~i9nless. In Eqs. 1: A(y) = cros~-~e~tional area of flgw" <br />V(x,t) = Q/A = !v~rage velocity of flow; q(x,t) = discharge; y(x,t) = <br />depth of flow; B(y) =_top width_of flow; x = distance from the dam, <br />positive downstream; t = time; g = ratio of weight to mass; So = channel <br />bottom slope and Sf = friction slope. In this work Sf is evaluated using <br />the Manning relation <br /> <br />V = ~ 'R.2/3 Sf 112 (2) <br /> <br />where Cu = 1.0 in the metric system and Cu = 1.482 in the British system <br />of units; n = Manning roughness coefficient; R = AlP = hydraulic radius <br />and P = wetted perimeter. <br /> <br />2. Non-dimensionalization <br /> <br />_ Taking charac~eristic quantities: [ for distance, I for time, <br />Yo for depth and Vo for velocity, the fo?lowing dimension?ess variables <br />are defi ned <br /> <br />x = x/Lo ; t = t/Io; y = y/Vo; V = VIVo <br /> <br />A = A/Y~ ; B = 8/Yo; R = RIYo <br /> <br />Introducing these variables into Eqs. 1 and defining <br />[0 = To Vo; Yo = [0 So <br /> <br />(3a,b) <br /> <br />. <br /> <br />v = fuR 2/3 S 1/2 . <br />o n 0 0 . <br /> <br />90 <br />Co = {_ _ <br />g Yo <br /> <br />(3c,d) <br /> <br />, <br /> <br />2 <br />