The Thomas problem
with online computation


Marcela I. Diaz and Victor M. Ponce


09 December 2016


ABSTRACT

An online calculator has been developed and tested using the Muskingum-Cunge method to solve the classical Thomas problem of flood routing. To enable it to study the effect of runoff diffusion, the calculator can vary peak inflow, time base, and channel length. The choice for peak inflow qp (cfs/ft) is: (a) 200, (b) 500, and (c) 1,000. The choice for time base Tb (hr) is: (a) 48, (b) 96, and (c) 192. The choice for channel length L (mi) is: (a) 200, and (b) 500. The results are in close agreement with existing analytical results of the Thomas problem.


1.   INTRODUCTION

The Thomas problem is a classical problem of flood routing. The problem is described in Thomas' original paper (1934). Thomas routed a sinusoidal flood wave through a unit-width channel of length L = 200 mi and slope So = 1 ft/mi. The baseflow is qb = 50 cfs/ft, the peak inflow is qp = 200 cfs/ft, and the time base of the sinusoidal hydrograph is Tb = 96 hr.

In this paper we develop and document an online calculator for the Thomas problem using the Muskingum-Cunge method (Ponce, 2014). The accuracy of the Muskingum-Cunge has been documented elsewhere (Ponce et al., 1996). To enable it to study the effect of runoff diffusion, the calculator can vary peak inflow, time base, and channel length. The choice for peak inflow qp (cfs/ft) is: (a) 200, (b) 500, and (c) 1,000. The choice for time base Tb (hr) is: (a) 48, (b) 96, and (c) 192. The choice for channel length L (mi) is: (a) 200, and (b) 500. Examples of online calculators for a diversity of hydraulic and hydrologic problems are given in onlinecalc.sdsu.edu.


2.   METHODOLOGY

The Muskingum-Cunge method is a variant of the Muskingum method (Chow, 1959) in which the routing coefficients are calculated based on hydraulic parameters using the formulas derived by Cunge (1969). The details of the methodology are given by Ponce (2014).

The routing equation of the Muskingum-Cunge method is:

Q j+1 n+1 = Co Qj n+1 + C1 Q j n + C2 Qj+1n(1)

 
(a) (b)

Fig. 1  Spatial and temporal discretization of: (a) Muskingum method, and (b) Muskingum-Cunge method.

in which j = spatial index, n = temporal index, and the routing coefficients, C0, C1, and C2 are calculated as follows (Ponce and Yevyevich, 1978):
                    

            -1 + C + D
C0  =  _____________
             1 + C + D

(2)
                    

             1 + C - D
C1  =  _____________
             1 + C + D

(3)
                    

             1 - C + D
C2  =  _____________
             1 + C + D

(4)

in which C is the Courant number:

              Δt      
C =  c  ______ 
              Δx
(5)

and D is the cell Reynolds number:

                  q      
D =    ____________ 
             So c Δx
(6)

with Δt = temporal interval and Δx = spatial interval.

The discharge-depth rating is:

q = α d β
(7)

in which q = unit-width discharge, d = flow depth, α = rating coefficient, and β = rating exponent.

The flood wave celerity c can be calculated as follows (Seddon, 1900; Ponce, 2014):

              q      
c =  β  _____ 
              d
(8)

In the constant-parameter Muskingum-Cunge method (Ponce, 2014), the routing parameters C and D are calculated based on the average inflow discharge qa:

              qb  +  qp       
qa =    ____________ 
                    2
(9)

The average inflow discharge is kept constant throughout the computation.


3.   THE THOMAS PROBLEM

The original Thomas problem had the following specifications:

  • Baseflow qb = 50 cfs/ft

  • Peak inflow qp = 200 cfs/ft

  • Time base of the sinusoidal hydrograph: Tb = 96 hr

  • Channel length L = 200 mi

  • Channel slope So = 1 ft/mi

  • Temporal interval Δt = 12 hr

  • Spatial interval Δx = 25 mi

  • Rating coefficient:   α = 0.688

  • Rating exponent:   β = 5/3

The value of Manning's friction coefficient, calculated from the rating curve, is n = 0.0297.


4.   ONLINE CALCULATOR

The online calculator resembles the Thomas problem, with important extensions:

  1. A choice of channel length L (mi): (a) 200, and (b) 500.

  2. A choice of peak inflows qp (cfs/ft): (a) 200, (b) 500, and (c) 1,000.

    Note that this choice provides peak-inflow/baseflow ratios qp /qb : (a) 4, (b) 10, and (c) 20.

  3. A choice of time base of the hydrograph Tb (hr): (a) 48, (b) 96, and (c) 192.

    The time-base multiplier adopted to determine total simulation time (TST) (hr) is 2.5. Therefore, the corresponding values of TST for the three selected values of Tb are: (a) 120 (5 days), (b) 240 (10 days), and (c) 480 (20 days).

Table 1 shows the input data and routing parameters for the eighteen (18) possible combinations (runs): (2 channel lengths L) × (3 peak inflows qp) × (3 time bases Tb). The parameter Nt is the total number of time intervals; Nx is the total number of space intervals.

The temporal and spatial intervals shown in Table 1 have been judiciously chosen such that the Courant number is reasonably close to 1 (Col. 9). This is necessary to preserve the accuracy of the numerical computation (Ponce and Theurer, 1982; Ponce, 2014).

Table 1.  Input data and routing parameters in eighteen (18) possible combinations (runs).
Run L
(mi)
qp
(cfs/ft)
qp / qb qa
(cfs/ft)
Tb
(hr)
Δt
(hr)
Δx
(mi)
C D TST
(hr)
Nt Nx
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
1 200 200 4 125 48 1.5 12.5 0.75 1.09 120 80 16
2 96 3 25 0.75 0.54 240 80 8
3 192 6 50 0.75 0.27 480 80 4
4 500 10 275 48 1.5 12.5 1.03 1.75 120 80 16
5 96 3 25 1.03 0.87 240 80 8
6 192 6 50 1.03 0.44 480 80 4
7 1,000 20 525 48 1.5 12.5 1.33 2.58 120 80 16
8 96 3 25 1.33 1.29 240 80 8
9 192 6 50 1.33 0.64 480 80 4
10 500 200 4 125 48 1.5 12.5 0.75 1.09 120 80 40
11 96 3 25 0.75 0.54 240 80 20
12 192 6 50 0.75 0.27 480 80 10
13 500 10 275 48 1.5 12.5 1.03 1.75 120 80 40
14 96 3 25 1.03 0.87 240 80 20
15 192 6 50 1.03 0.44 480 80 10
16 1,000 20 525 48 1.5 12.5 1.33 2.58 120 80 40
17 96 3 25 1.33 1.29 240 80 20
18 192 6 50 1.33 0.64 480 80 10


5.   EXAMPLE

Run 11 of Table 1 is used here as an example of the use of the calculator. For this case:

  • L = 500 mi

  • qp = 200 cfs/ft

  • Tb = 96 hr

The results of the Online Thomas Problem are shown in Fig. 2. They are:

  • Peak outflow:  qpo = 176.84 cfs/ft,

  • Time to peak [estimated]:  tp = 128 hr.

For comparison, the analytical results obtained by Ponce et al. (1996) using diffusion wave theory are:

  • qpo = 176.74 cfs/ft, and

  • tp = 48 + 79.82 = 127.82 hr,

which constitutes a remarkable agreement by any standard.

Fig. 2  Results of Run 11 of Online Thomas Problem.


6.   CONCLUSIONS

An Online Thomas Problem calculator using the Muskingum-Cunge method (Ponce, 2014) has been developed and tested. The calculator can vary peak inflow, time base, and channel length, enabling it to study the effect of runoff diffusion on the outflow hydrograph. The choice for peak [sinuosoidal] inflow qp (cfs/ft) is: (a) 200, (b) 500, and (c) 1,000. The choice for [hydrograph] time base Tb (hr) is: (a) 48, (b) 96, and (c) 192. The choice for [total] channel length L (mi) is: (a) 200, and (b) 500. The results are shown to be in close agreement with existing analytical results of the Thomas problem.


REFERENCES

Chow, V. T. 1959. Open-channel hydraulics. McGraw-Hill, New York.

Cunge, J. A. 1969. On the subject of a flood propagation computation method (Muskingum method), Journal of Hydraulic Research, Vol. 7, No. 2, 205-230.

Ponce, V. M. and V. Yevjevich. 1978. Muskingum-Cunge method with variable parameters. J. Hydraul. Div. ASCE, 104(HY12), 1663-1667.

Ponce, V. M. and F. D. Theurer. 1982. Accuracy criteria in diffusion routing. J. Hydraul. Div., ASCE, 108(HY6), 747-757.

Ponce, V. M., A. K. Lohani, and C. Scheyhing. 1996. Analytical verification of Muskingum-Cunge routing. Journal of Hydrology, Vol. 174, 235-241.

Ponce, V. M. 2014. Engineering hydrology: Principles and practices. Online text.

Thomas H. A.. 1934. The Hydraulics of Flood Movement in Rivers. Engineering Bulletin, Carnegie Institute of Technology, Pittsburgh, PA.

Seddon, J. A. 1900. River hydraulics. Transactions, ASCE, Vol. XLIII, 179-243, June.


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