The cause of negative initial outflow with the Muskingum method

Abstract

The cause of negative or reduced outflow formation at the beginning of a Muskingum solution is examined in two steps. The first step involves a physical interpretation of the Muskingum weighted discharge and the storage equation, using theory based on an extension of the Kalinin-Milyukov method. The second step involves the derivation of an analytical solution for the weighted discharge based on linear systems analysis theory, and then subsequently the Muskingum solution from that analytical solution using the assumption of linear variation of discharge within the reach. It is proved that this assumption causes the formation of negative or reduced outflow at the beginning of a Muskingum solution.

A modified Muskingum routing approach for floodplain flows: Theory and practice

“…(15) and (16) are considered to provide reasonable estimates for K and x in the presented method. The satisfactory performance of the modified method further implies that the storage equation in Muskingum routing methods is a substitute for the momentum equation in hydraulic routing approaches in typical Irish rivers and therefore it is reasonable to relate the routing parameters to channel and flow characteristics (Perumal, 1992b). It should be noted however that the limitations of the approach are the same as in most hydrological or Muskingum flood routing methods and therefore, the results from this method should be confirmed if applied to river reaches where backwater and inertia effects are significant, where floodplain sinuosity is high or where significant lateral momentum exchanges between main channel and floodplain zones are influential.…”

Discussion of “Assessment and review of the hydraulics of storage flood routing 70 years after the presentation of the Muskingum method”

“…The only way to overcome this paradoxical illusion is to develop the linear form of the Muskingum storage equation using hydrodynamic principles based on the extension of the Kalinin-Miljukov (K-M) theory (Apollov et al, 1964), as illustrated by the author in his paper, and by this writer (Perumal, 1992b). Alternatively, the justification for the linear storage equation in the form of weighted discharge can also be arrived at directly from the St Venant equations based on the theory advocated by this writer (Perumal, 1994a), or by the method advocated by Todini (2007) and Price (2009).…”

“…The cause of the formation of initial dip in the Muskingum solution, especially when using a long reach as a single reach of the Muskingum method, has been brought out by Perumal (1992b) using the interpretation of the Muskingum method based on the extension of the K-M theory (Apollov et al, 1964). Based on the direct derivation of the Muskingum method from the St Venant equations, it was also shown by Perumal (1994a) that the initial dip in the solution develops when a long reach is used as a single reach of the Muskingum method.…”

“…Further, it was shown that when these schemes are considered as finite difference equivalents of the analytical solution of the Muskingum method, the Cunge (1969) scheme is better than that of Koussis (1978). The interpretation of the Muskingum parameters using the extension of the Kalinin-Miljukov theory, as presented in a Russian text book by Apollov et al (1964) 1 and reviewed in Koussis (2009), had already been published by this writer (Perumal, 1992b) in this very journal. Also, two of the very relevant papers by this writer (Perumal 1994a,b), by directly deriving a variable parameter Muskingum method from the St Venant equations, have also been presented in this journal.…”

Flood routing in a wide channel with a quadratic lag-and-route method

“…The Muskingum method is most frequently used in practice and is still the subject of research (Hjelmfelt, 1985;Perumal, 1992a;Gill, 1992). Its main appeal comes from work by Cunge (1969) who demonstrated that this method could be considered as being numerically related to the Saint-Venant equations via the diffusion wave equation.…”