1. RATIONALE

1.01
The Muskingum method can calculate diffusion by varying the parameter X.

1.02
A numerical solution of the linear kinematic wave equation using a third-order accurate scheme leads to pure flood hydrograph translation, with no diffusion.

1.03
Other numerical solutions of the linear kinematic wave equation invariably produce a certain amount of numerical diffusion and/or dispersion.

1.04
The routing equations for the Muskingum method and the linear kinematic wave are strikingly similar.

1.05
Unlike the kinematic wave, the diffusion wave does have the capability to describe physical diffusion.

1.06
From these propositions, Cunge showed in 1969 that the Muskingum method is a linear kinematic wave solution.

1.07
Thus, the flood wave attenuation shown by the calculation is due to the numerical diffusion of the scheme itself.

1.08
To prove this assertion, the kinematic wave equation is discretized on the x-t plane in a way that resembles the Muskingum method, that is, in a linear mode, and centering the spatial derivatives and off-centering the temporal derivatives by means of the weighting factor X:

1.09

1.10
in which c = βV

1.11

1.12
is the kinematic wave celerity.

1.13
Solving this equation for the unknown discharge leads to the routing equation:

1.14

1.15
The routing coefficients are:

1.16

1.17

1.18

1.19
By defining the travel time K:

1.20

1.21
it is confirmed that the linear kinematic wave solution and the Muskingum method are the same.

1.22
This kinematic wave solution is referred to as the Muskingum-Cunge method.

1.23
The Muskingum parameter K is the flood wave travel time, that is, the time it takes a given discharge to travel the reach length Δx with the celerity c.

1.24
The Courant number is the ratio of physical celerity to grid celerity:

1.25

1.26
For X = 0.5 and C = 1, the third-order accurate kinematic wave solution is obtained, that is, the exact solution, with no numerical solution or dispersion.

1.27
For X = 0.5 and C ≠ 1, the second-order accurate kinematic wave solution is obtained, with numerical dispersion only.

1.28
For X < 0.5 and C ≠ 1, the second-order accurate kinematic wave solution is obtained, with numerical diffusion and dispersion.

1.29
For X < 0.5 and C = 1, the first-order accurate kinematic wave solution is obtained, with numerical diffusion only.

1.30
The numerical properties of the Muskingum-Cunge method are summarized in this table...

2. METHODOLOGY

2.01
In the Muskingum-Cunge method, the numerical diffusion is used to simulate the physical diffusion of the actual flood wave.

2.02
By expanding the discrete function

2.03

2.04
in Taylor series, the numerical diffusion coefficient of the Muskingum scheme is derived:

2.05

2.06
in which ν_n = numerical diffusion coefficient of the Muskingum-Cunge method.

2.07
This equation reveals the following:

2.08
1. For X = 0.5, there is no numerical diffusion, although some numerical dispersion remains for Courant numbers other than 1.

2.09
2. For X > 0.5, the numerical diffusion is negative, that is, numerical amplification, which explains the odd behavior of the Muskingum method for this range of X values.

2.10
In fact, for X > 0.5, the presence of wave oscillations can lead to unrealistic negative flows.

2.11
2. For X < 0.5, the numerical diffusion is positive.

2.12
3. For Δx = 0, clearly the trivial case, the numerical diffusion is zero.

2.13
A predictive equation for X can be obtained by matching the hydraulic diffusivity of the diffusion wave with the numerical diffusion coefficient of the Muskingum scheme.

2.14
This leads to the following expression for X:

2.15

2.16
Therefore, the routing parameter X can be calculated as a function of the following physical and numerical properties: (1) reference discharge per unit width, (2) bottom slope, (3) kinematic wave celerity, and (4) reach length.

2.17
This equation was obtained by matching diffusivities, a second-order process.

2.18
Thus, it does not account for dispersion, a third-order process.

2.19
Therefore, in order to simulate wave diffusion properly, it is necessary to minimize numerical dispersion by keeping the Courant number as close to 1 as practicable.

2.20
A salient feature of the Muskingum-Cunge method is its grid independence.

2.21
This sets it apart from other kinematic wave solutions, which feature uncontrolled numerical diffusion and dispersion.

2.22
If the numerical dispersion is minimized, the calculated outflow hydrograph will be essentially the same, regardless of how many subreaches are used in the computation.

2.23
The Muskingum-Cunge method is expressed in a reduced form by defining the cell Reynolds number as a ratio of physical and numerical diffusivities:

2.24

2.25
Therefore, the Muskingum parameter X is:

2.26

2.27
The substitution of Courant and cell Reynolds numbers into the Muskingum-Cunge routing coefficients leads to:

2.28

2.29

2.30

2.31
The calculations can proceed in two ways: (1) linear mode, and (2) nonlinear mode.

2.32
In the linear mode, the method resembles the Muskingum method, with the difference that the routing parameters are based on measurable flow and channel characteristics, instead of historical flow data.

2.33
The computation of the routing parameters is based on suitable reference flow values, such as peak flow or average flow.

2.34
In the nonlinear mode, the routing parameters are allowed to vary with the flow, enabling the modeling of the nonlinear features of flood waves.

3. EXAMPLE

3.01
Assume an inflow hydrograph with peak flow 1000 m³/s, baseflow 0 m³/s, bottom slope = 0.000868, peak flow area = 400 m², peak flow top width = 100 m, rating exponent β = 1.6, reach length = 14,400 m, and time interval &Delta t = 1 hour.

3.02

3.03
The mean velocity is:

3.04

3.05
The wave celerity is:

3.06

3.07
The unit-width discharge is:

3.08

3.09
The Courant number is:

3.10

3.11
The cell Reynolds number is:

3.12

3.13
The routing coefficients are:

3.14

3.15
The routing computations are shown in this table.

3.16
Column 1 shows the time in hours.

3.17
Column 2 shows the inflow hydrograph.

3.18
Columns 3 to 5 show the weighted flows.

3.19
Column 6 shows the outflow hydrograph.

3.20
Note that the peak inflow is 1000 m³/s at time = 5 hr, while the peak outflow is 963.6 m³/s at time = 6 hr.

3.21
The flood peak has attenuated about 3%, and the time base has increased to 13 hr.

4. ASSESSMENT

4.01
The Muskingum-Cunge method is a physically based alternative to the Muskingum method.

4.02
Unlike the Muskingum method where the parameters are calibrated based on streamflow data, that is, hydrologic data, in the Muskingum-Cunge method the parameters are based on flow and channel characteristics, that is, hydraulic data.

4.03
This makes possible channel routing without the need for parameter calibration.

4.04
More importantly, it makes possible extensive channel routing in ungaged stream networks with a reasonable expectation of accuracy.

4.05
Like the Muskingum method, the Muskingum-Cunge method is limited to diffusion waves in natural streams without significant backwater effects.

4.06
In the Muskingum method, the routing parameters are reach averages, reflecting its lumped or storage basis.

4.07
Conversely, in the Muskingum-Cunge method, the routing parameters are evaluated at channel cross sections, reflecting its distributed kinematic wave basis.

4.08
Thus, for the Muskingum-Cunge method to improve on the Muskingum method, the cross sections used in the evaluation must be representative of the reach under consideration.

4.09
In the nonlinear mode of the Muskingum-Cunge method, the parameters are allowed to vary with the flow.

4.10
Thus, the Muskingum-Cunge method can account for the flood wave nonlinearities, which are markedly absent in the Muskingum method.

4.11
The nonlinear method is particularly applicable to the modeling of complex flood flows in large basins.

4.12

4.13

4.14

Narrator: Victor M. Ponce

Music: Fernando Oñate

Editor: Flor Pérez

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Visualab Productions

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