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Ie I I ~ ,e COMPUTATION OF WATER-SURFACE PROFILES IN OPEN CHANNELS yo.... S. Yn --- Water surface passes through critical depth at 'lydraulicjumps MILO SLOPE Figure 8.-Backwater-curve trans.itions for steep slope to mild slope. characteristics such as slopes, roughnesses, widths, depths, or any combination of these between adjacent subreaches in a long reach. Obviously, in a long stretch of river that has been divided into shorter subreaches of "uni- form" characteristics, the water-surface pro- file is a series of transition curves from the normal-depth line in one subreach to the nor- mal-depth line in the adjacentsubreach. Unless there are radical changes in characteristics, as for example at control points and hydraulic jumps, one could speak of the uniform-flow profile in a long reach as an average of the numerous transition curves that are reflecting local nonuniformities in the channel. The normal depth line is determined using the backwater curves for channels of mild or steep slopes. Use of M1 and M2 curves A characteristic of the tranquil-flow MI and M2 backwater curves is that one can start at any point on either of them and, by solving the energy equation, determine the elevation of the water surface at another point farther up- stream. The intervening geometry and rough- ness, as well as the discharge, have to be known. As may be seen in figure lA, if the procedure were carried far enough upstream in incre- ments, this step-by-step computed profile would asymptotically approach the normal-depth line. This characteristic of the backwater curves is used in determining normal depth in a channel of mild slope. 5 Computed profiles that start with an eleva- tion higher than normal depth would be MI curves; those starting with elevations lower than normal depth would be M2 curves. When computed values of water-surface elevations along two profiles converge mathematically, it is generally assumed that the normal-depth profile has been reached. Computations along either profile continued farther upstream would be identical. regardless of whether the profiles had been two MI's, two M2's, or one of each, at the start of computations' at the down- stream end of the reach, Upstream from the point of convergence, the computed profile would define a locus of normal depth at each cross section, It would be the expected profile because the nonuniformities in the channel geometry and roughness would be translated to a series of minor, transitional backwater curves. Determination of normal depth for subcriti- cal flow in a channel by the procedure described above thus has two distinct phases. First, two starting backwater curves are assumed to exist in the channel. caused by different, assumed, control conditions downstream. Normal depth at a point(gaging station. bridge, or other place of interest) is determined when the two profiles have converged. The two computed profiles up to this point are imaginary and become useless after serving as a ploy for determining the normal-depth profile. The second phase begins with the point of convergence, All subsequent computations of the profile farther upstream represent the expected, or normal, water- surface elevation in this channel, to be used for inundation studies, flood-insurance studies, bridge-backwater studies, and so on. Use of 52 and 53 curves To determine normal depth in a steep chan- nel. the supercritical-f1ow S2 and S3 backwater curves are used. The procedures correspond to those described for MI and M2 curves, but in the downstream direction. Starting at any point on the S2 or S3 curve, the elevation of the water surface at any point farther downstream can be determined by solving the energy equa- tion. The intervening geometry and roughness, as well as the discharge, have to be known. As