where Δt is the routing period (time interval). In the Muskingum-Cunge version, the parameters K and X are calculated by (2,3,4);

Δx K  =  _____            c

(6)

1                   q    X  =  ___ ( 1  -  __________ )           2              So c Δx

(7)

in which Δx = reach length (space interval); c = flood wave celerity; q = unit width discharge; and S_o = channel bed slope. Substituting Eqs. 6 and 7 into Eqs. 3-5:

1 + C - D C1  =  ______________              1 + C + D

(8)

-1 + C + D C2  =  _______________              1 + C + D

(9)

1 - C + D C3  =  ______________              1 + C + D

(10)

in which C = c Δt /Δx is the Courant number; and D = (q /S_o)/cΔx is a type of cell Reynolds number. Both C and D have physical and numerical significance, C being a ratio of celerities and D a ratio of diffusivities.

3.  VARIABLE PARAMETERS

Usually, Δt is fixed, and Δx and S_o are specified for each computational cell consisting of four grid points (Fig. 1). Therefore, it is necessary to determine the flood wave celerity, c, and the unit width discharge, q, for each computational cell. The values of c and q at grid point (j, n) are defined by:

dQ c  =  _____           dA

(11)

Q q  =  _____            B

(12)

in which Q = discharge; A = flow area; and B = top width. The following ways of determining c and q for use in calculating C and D are investigated:

1. Directly, by using a two-point average of the values at grid points (j, n) and (j + 1, n);

2. Directly, by using a three-point average of the values at grid points (j, n), (j + 1, n), and (j, n + 1); and

3. By iteration, using a four-point average calculation. To improve convergence, the values at (j + 1, n + 1) obtained by the three-point average method are used as the first guess of the iteration.

4.  EXAMPLE

The Muskingum-Cunge method with variable parameters is applied herein to the problem posed by Thomas (8) in his classical paper on flood routing. The problem consists of tracking the travel and subsidence of a flood wave of sinusoidal shape in a unit width channel with a steady-state rating curve given by:

q  =  0.688 d  5/3

(13)

The inflow hydrograph is defined by

π t I(t)  =  125 - 75 cos ( ____ ),          0 ≤ t ≤ 96;                                       48

(14a)

I(t) = 50,          t > 96

(14b)

in which t is in hours.

Thomas applied an approximate method to route the flood wave through a channel 200 miles (322 km) long, using a time interval of Δt = 12 hr. Thomas' approximate method neglects the inertia terms; therefore, his results are directly comparable to those of the Muskingum method (both the Thomas and Muskingum methods can be considered as numerical analogs of the diffusion wave equation).

Figure 2 and Table 1 summarize the results of computations using the Muskingum-Cunge method with variable parameters. For comparison purposes, the calculations using constant parameters for three reference values of discharge are also shown.

Fig. 2  Calculated hydrographs as described in Table 1.

The examination of Fig. 2 enables the following conclusions to be drawn:

1. The Muskingum-Cunge method with constant parameters (hydrographs B, C, and D) shows results that are dependent on the value of reference discharge chosen to evaluate the constant parameters. The higher the reference discharge, the faster the rate of travel and the lesser the subsidence of the flood wave (3). The calculated outflow hydrographs show negligible distortion from the initially sinusoidal shape, implying that the constant parameter assumption is tantamount to an assumption of linearity.

2. The Muskingum-Cunge method with variable parameters (hydrographs E, F, and G) shows results that fall within the range encompassed by the constant parameter calculations. The noticeable steepening of the rising limb of the calculated outflow hydrographs indicates that the nonlinearity of the phenomenon is being taken into account. The three-point and four-point methods give similar results; however, the two-point method shows a smaller peak and a somewhat slower rate of travel. Furthermore, the two-point method results in a significant loss of mass, as indicated in Col. 8 of Table 1.

3. The results of the Thomas and four-point variable parameter methods are comparable (hydrographs I and H). The three-point variable parameter hydrograph (not shown) is also very close to Thomas' results.

5.  SUMMARY AND CONCLUSIONS

The Muskingum-Cunge method in which the parameters K and X are allowed to vary in time and space is investigated. A three-point approach and an iterative four-point approach to the calculation of the variable parameters are shown to be sufficiently accurate in the simulation of flood flows. A two-point approach is shown to be inaccurate in the calculation of peak discharge and travel time. Furthermore, the two-point method results in a significant loss of mass.

APPENDIX. REFERENCES