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<br />7 <br /> <br />Chapter II <br /> <br />BACKGROUND <br /> <br />. <br /> <br />Methods of Computing Water Surface Profiles <br /> <br />The profile of a water surface in a natural channel may be <br /> <br />computed by considering the conservation of energy between two adjacent <br /> <br />2 <br />sections. For a channel of slope e the total head above an arbi- <br /> <br />trary datum at the starting station is <br /> <br />v2 <br />H = "2g + <br /> <br />d cos e + z <br /> <br />(2-1) <br /> <br />where H is total head in terms of energy per unit weight, d is depth <br /> <br />of flow measured perpendicular to the channel bottom, z is the eleva- <br /> <br />tion of the channel bed above the arbitrary datum, " is the velocity <br /> <br />distribution coefficient and V is the mean velocity of flow in dimen- <br /> <br />sions of linear distance traversed per second of time. Refer to Fig. 2-1. <br /> <br />For the assumption that " and e stay constant from the starting <br /> <br />section to the next adjacent section, and taking the channel bed as the <br /> <br />x-axiS, differentiating Eq. (2-1) with respect to length along the <br /> <br />channel bed yields <br /> <br />dH d <br />dx = "dx <br /> <br />v2 <br />(2g) <br /> <br />+ cos e dd + dz <br />dx dx <br /> <br />(2-2) <br /> <br />If slopes are defined as the sine of the slope angle, dH/dx equals the <br /> <br />energy slope, <br /> <br />S ,and dz/dx is equal to the slope of the channel <br />o <br /> <br />bottom, Sf' Substituting these slopes in Eq. (2-2) and solving for <br /> <br /> <br />dd/dx yields the dynamic equation of gradually varied flow <br /> <br />dd _ <br />dx - cos <br /> <br />So - Sf <br />e + "d(Vz/dg)/dd <br /> <br />(2-3) <br /> <br />2 <br />Development of the flow profile analysis theory is covered in most <br />texts on Fluid Mechanics and is only reviewed here. <br />