Modeling Rapidly Varied Flow in Tailwaters

Abstract

An understanding of the downstream propagation of sharp-fronted, large-amplitude waves of relatively short period is important for describing rapidly varying flows in tailwaters of hydroelectric plants and following the breach of a dam. We developed a numerical model of these waves by first identifying the primary physical processes and then performing an analysis of the solution. A linear analysis of the dynamic open channel flow equations provides relationships describing flow wave advection, diffusion, and dispersion in rivers. A one-dimensional diffusion wave model modified for application to tailwaters simulates the important physical processes and is straightforward to apply. The "modified equation" and von Neumann analyses provide insight into the effects of numerical parameters 0, Ax, and At upon stability and dissipative and dispersive behavior of the solution, but the Hirt analysis is found to yield incorrect phase relationships. The capability and accuracy of the model are enhanced when physical diffusion of a river wave is balanced by numerical diffusion in the model. Field studies were conducted in two greatly different tailwaters to assess our understanding of large-scale, rapidly varying flow waves. The accurate simulation of waves having wide-ranging amplitudes, shapes, periods, and base flows attests to the soundness of both the physical basis of the model and the numerical solution technique. These studies reveal that diffusion of short-period waves in natural, free-flowing rivers is significant and that inertia is negligible. FERRICK ET AL.' TAILWATER FLOW ditions required for stability of many numerical schemes are known, and numerical instability is generally apparent. The von Neumann and Hirt analyses are frequently used to develop stability criteria I-Roache, 1976]. Numerical solutions of the unsteady open channel flow equations typically exhibit errors in both amplitude and phase that may not be apparent without further analysis. Numerical dissipation or diffusion causes the Fourier components of the solution and the errors to be damped. Numerical dispersion results when the modeled wave celerity of some wavelength components differ from those of the governing equation, improperly modifying the wave form as the computation proceeds. The effects of numerical dissipation and dispersion upon the solution are subtle but gradually destroy the correspondence between model and prototype. An improved understanding of the dissipative and dispersive behavior of the numerical model enables the analyst to minimize or exploit these effects to enhance model accuracy and to better interpret computed results. Though only strictly applicable to linear equations, the "modified equation" [Warming and Hyett, 1974], Hirt and von Neumann analyses are used to relate the dissipative and dispersive behavior of the model to parameters of the numerical solution. A set of linear routings is used to demonstrate the model behavior predicted with the analysis and to verify the adequacy of an expression for numerical dif...

Comment on "A praxis-oriented perspective of streamflow inference from stage observations – the method of Dottori et al. (2009) and the alternative of the Jones Formula, with the kinematic wave celerity computed on the looped rating curve" by Koussis (2009)

“…An indication of the correctness of computing the KW celerity on the loop-shaped rating curve is that c(Q max ) = 0: the maximum discharge does not propagate downstream! However, care must be exercised in the iterative calculation of c, to ensure convergence (Koussis, 1975;Weinmann, 1977;Weinmann and Laurenson, 1979;Ferrick et al, 1984;Perkins and Koussis, 1996). We demonstrate this point below with an example of Weinmann (1977), also reported by Weinmann and Laurenson (1979).…”

Reply to Comment on "A dynamic rating curve approach to indirect discharge measurement by Dottori et al. (2009)" by Koussis (2009)

“…This fact is well known to Koussis who, in his original paper (1976) writes: "It must be noted here that in order to avoid numerical complications during the computation of the upper part of the rating curve (for example instabilities for h/ Q→0), one should substitute there for ( h/ Q) x the gradient of the steady flow rating curve." Moreover, in his recent comment Koussis states "However, care must be exercised in the iterative calculation of c, to ensure convergence (Koussis, 1975;Weinmann, 1977;Weinmann and Laurenson, 1979;Ferrick et al, 1984;Perkins and Koussis, 1996). "…”

Reply to the Discussion of “Assessment and review of the hydraulics of storage flood routing 70 years after the presentation of the Muskingum method” by M. Perumal

“…But even if a unique value were estimated following the traditional storage-weighted flow loop-minimization method, the routing results would differ, due to the different makeup of the schemes. Ferrick et al (1984) and Bowen et al (1989) give details on the exponential scheme.…”

“…An indication of the correctness of computing the kinematic wave celerity on the loop-shaped rating curve is that c k (Q max ) ¼ 0: the maximum discharge does not propagate downstream! However, care must be exercised in the iterative calculation of c k , to ensure convergence (Koussis, 1975;Weinmann, 1977;Weinmann & Laurenson, 1979;Ferrick et al, 1984;Perkins & Koussis, 1996).…”