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...nite-difference approximation in the predictor part and a backward finite-difference approximation in the corrector part are used. To dampen the high-frequency oscillations near the steep gradients, [=-=Jameson, et al., 1981-=-] is introduced as follows. A parameter νi is first computed using the computed flow depths at k + 1 time level νi = |hi+1 − 2hi + hi−1| |hi+1|+ 2|hi|+ |hi−1| νi+ 12 = κ max (ξi+1, ξi) (8− 36) in whic...

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...ng these channels by the finite-difference methods. The grid points usually do not coincide with the boundaries, thus requiring interpolation procedures which have proven to deteriorate the solution [=-=Roache, 1972-=-]. To avoid this problem, the coordinates 8-2 Governing Equations 253 may be transformed such that the coordinate axes coincide with the boundaries. For example, the following simple coordinate transf...

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...st describe the details of the MacCormack scheme, procedures for including the initial and boundary conditions and then present results for its verification. Numerical Solution The MacCormack scheme [=-=MacCormack, 1969-=-] is used to integrate numerically the transformed form of the governing equations (Eq. 8-5). Referring to the 262 8 COMPUTATION OF RAPIDLY VARIED FLOW Fig. 8-7. Circular-arc contraction 8-4 Computati...

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... observed hydrographs, the attenuation of a flood wave due to storage and friction may be included in the analysis. 17-8 Muskingum-Cunge Routing 491 Applicability The following criterion may be used [=-=Ponce et al. 1978-=-] for the applicability of the diffusion model: Kw = FrTSo √ g yo ≥ 30 (17− 33) in which Fr = reference flow Froude number and the other variables are as defined for the kinematic model. 17-8 Muskingu...

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...eviousely may be unsuitable for representing some reaches [Yoon and Padmanabhan 486 17 SPECIAL TOPICS 1993]; moreover, the relationship between the weighted flow and the storage is not always linear [=-=Geem, 2006-=-]. Gill [1978] suggested two forms of nonlinear Muskingum models which account for the nonlinearity in storage and flow relationship. S = K[XI + (1−X)O]mS = K[XIm + (1−X)Om] (17− 14) where m is an exp...

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...hi−1 + q2i−1 2gh2i−1 ) − 1 ∆x ( zi + hi + q2i 2gh2i ) (17− 45) Then the value of a was determined from a = S1.71e . The exponents in Eq. 17-39 were estimated from the sediment transport measurements [=-=Brush and Wolman, 1960-=-]. The conditions were identical in both the experiments except for the channel slope, and the difference in sediment transport rates could be related to the slope of the energy grade line by a power ...

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... avoid instability, they had to iterate the solution whenever the spatial gradient of change in bed level became too large. In this section, a one-dimensional, unsteady, coupled deformable bed model [=-=Bhallamudi and Chaudhry, 1991-=-] is presented. The complete Saint-Venant equations for water flow and the sediment continuity equation are solved simultaneously by the MacCormack explicit scheme [MacCormack, 1969]. The computed res...

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...e we derived in Chapters 12 and 13 for the absolute velocity of a disturbance as V ± c. However, extensive field measurements of the propagation of the crest of flood waves confirm this relationship [=-=Seddon, 1900-=-]. Equation 17-9 is a first-order partial differential equation with Q as the dependent variable and x and t as the independent variables. It describes the movement of a flood wave in terms of the rat...