"Analytical
verification of Muskingum-Cunge routing".
The Muskingum-Cunge method of flood routing has for many years been successfully applied in flood routing, modeling and prediction (Natural Environment Research Council 1975; Ponce and Yevjevich, 1978). The Muskingum routing parameters were theoretically derived by Cunge (1969) based on the approximation error obtained by a Taylor series expansion of the grid function. Although the parameters so derived have been shown to work reasonably well in a variety of practical applications (Hydrologic Engineering Center, US Army Corps of Engineers, 1990), their link to the analytical expressions for wave celerity (Seddon, 1900) and hydraulic diffusivity (Hayami, 1951) has yet to be demonstrated in an actual routing application. This paper endeavors to test the Muskingum-Cunge model by comparing the analytically derived flood wave celerity and peak attenuation with those obtained using an actual numerical application of the Muskingum-Cunge model. The remarkable agreement between analytical and numerical results is an indication of the accuracy of the model.
The Muskingum-Cunge model of flood routing (Cunge, 1969; Natural Environment Research Council, 1975) is an improvement of the classical Muskingum model (Chow, 1959), in which the routing parameters are based on readily measurable hydraulic data (stage-discharge rating, channel slope, channel width, and reach length), instead of being based on historic flood discharge data. This allows extensive routing in ungaged streams with a reasonable expectation of accuracy, which would otherwise be possible only if the streams were gaged. Since it is impractical to gage more than a few streams, a method that is both accurate and does not rely on stream gaging holds substantial promise.
Ponce and Yevjevich (1978) improved the Muskingum-Cunge model by expressing the routing
parameters as the dimensionless Courant and cell Reynolds numbers, where the former is
the ratio of physical and numerical celerities (Seddon, 1900; O'Brien
in which With the Muskingum routing parameters defined in this way, the routing coefficients are (Ponce and Yevejevich, 1978):
and the routing equation is:
where
Cunge's formulation (Cunge, 1969) is based on diffusion wave theory,
the applicability of which was confirmed by Ponce In this paper we use the findings of Ponce and Simons (1977) to calculate analytical peak outflow and travel time for a series of sinusoidal hydrographs, and to compare these results with those obtained using the Muskingum-Cunge model (Eqs. 1-6).
The analytical diffusion wave celerity may be approximated as the Seddon speed, or Seddon celerity (Ponce and Simons, 1977):
in which
or, for a unit-width channel:
in which The wave attenuation follows an exponential decay function of the following form:
in which q = peak inflow,
_{pi}q_{o} = equilibrium flow, β_{I}_{ *} =
dimensionless amplitude propagation factor,
and the dimensionless elapsed time t_{*} = t u / _{o}L
[See Eq. 12 for the definition of _{o}L].
_{o}The amplitude propagation factor is the following (Ponce and Simons, 1977):
in which
in which d_{o} /S_{o},
in which d_{o} = equilibrium flow depth, and S_{o} = channel bed slope.
Since t_{*} = t (u_{o} /L_{o}), it follows that:
in which
This paper uses an extension of Thomas' (1934) classical flood routing
problem to test the agreement between analytically and numerically
derived peak outflow and travel time. The Thomas problem routes
a sinusoidal flood hydrograph of time base (i.e., wave period)
q
= 200 cfs/ft (cubic feet per second, per foot), and
baseflow _{pi}q = 50 cfs/ft through
a channel of length _{b}L =_{r}S_{o} = 1 ft/mi, and rating curve:
from which the Manning coefficient
A series of test cases was developed based on Thomas' problem,
using: (1) two wave periods q: 100, 200 and 500 cfs/ft,
i.e. three peak inflow-to-base flow ratios
_{pi}q /_{pi}q: 2, 4, and 10;
(3) two channel lengths _{b}L: 200 and 500 mi.
Twelve outflow hydrographs were generated using the constant-parameter
Muskingum-Cunge model (Ponce and Yevjevich, 1978),
for which the reference flow is defined as follows:
_{r}
For x = 6.25 mi, and Δt =
1.5 hr, respectively; for T = 96h_{b}x = 12.5 mi, and Δt =
3 hr. The chosen wave periods (48 and 96 hr) satisfy diffusion
wave applicability criteria (Ponce et al., 1978):
in which q = 500 cfs/ft),_{pi}
which guarantees that the routed waves remain within the realm of diffusion waves.
Table 1 summarizes the comparison of analytical and numerical
peak outflow T
for the chosen series of sinusoidal hydrographs. The objective is to
compare the analytical
results of Columns labeled (1) with the numerical
results of Columns labeled (2).
_{t}
Figure 1 shows inflow
and oufflow hydrographs for a typical case L = 500 mi, and
_{r}q /_{pi}q = 4)_{b}q_{o} = 125 cfs/ft,
peak outflow q = 176.70 cfs/ft, and
travel time _{po}T = 79.8 hr.
_{t}
On the basis of the foregoing analysis, it is concluded that Muskingum-Cunge routing accurately simulates flood wave propagation. In twelve (12) tests encompassing a wide range of conditions likely to be encountered in practice, the ratio of numerical to analytical peak outflows varied within the range 0.991-1.003, and the ratio of numerical to analytical travel times varied within the range 0.987-1.021. The agreement in indeed remarkable, considering that a constant reference flow was used in the computation of the analytical values. The findings underscore the utility of the Muskingum-Cunge method of flood routing for practical applications.
Chow, V. T., 1950. Open-channel Hydraulics. McGraw-Hill, New York.
Cunge, J. A., 1969. On the subject of a flood propagation computation method (Muskingum method). Hayami, S., 1951.On the Propagation of flood waves. Disaster Prey. Res. Inst, Kyoto Univ. Bull., 1: 1-16 Hydrologic Engineering Center, US Army Corps of Engineers, HEC-1, Flood hydrograph package. user's manual. Version 4.0, September, Davis, CA. Natural Environment Research Council. 1975. Flood Studies Report, Vol. 5: Flood Routing Studies. NERC, London.
O'Brien, G. G., Hyman, M. A. and Kaplan, S., 1950. A study of the numerical solution of partial differential equations.
Ponce, V. M., 1989.
Ponce, V. M. and Simons, D. B, 1977. Shallow wave propagation in open channel flow.
Ponce, V. M. and Yevjevich, V., 1978. Muskingum -Cunge method with variable parameter.
Ponce, V. M., Li, R. M. and Simons, D., 1978. Applicability of kinematic and diffusion models.
Seddon, J. 1900. River hydraulics.
Thomas, H. A., 1934. |

200627 06:20 |

Documents in Portable Document Format (PDF) require Adobe Acrobat Reader 5.0 or higher to view; download Adobe Acrobat Reader. |