 

The Thomas problem with online computation
Victor M. Ponce and Marcela I. Diaz
09 December 2016


ABSTRACT
An online calculator is developed
using the
MuskingumCunge method to solve the classical
Thomas problem of flood routing.
To enable it to study the effect of runoff diffusion,
the calculator varies: (1) peak inflow, (2) time base, and (3) channel length.
For peak inflow, the choice of q_{p} (cfs/ft) is:
(a) 200, (b) 500, and (c) 1,000.
For time base, the choice of T_{b} (hr) is: (a) 48,
(b) 96, and (c) 192.
For channel length, the choice of 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 (1934)
is a classical problem of flood routing. Thomas routed a sinusoidal flood wave
through a unitwidth channel of length L = 200 mi and slope S_{o} = 1 ft/mi. The baseflow is q_{b} = 50 cfs/ft,
the peak inflow is q_{p} = 200 cfs/ft,
and the time base of the hydrograph is T_{b}
= 96 hr.
In this paper we describe an online calculator for the Thomas problem using the
MuskingumCunge method (Ponce, 2014).
To enable it to study
the effect of runoff diffusion,
the calculator varies peak inflow, time base, and channel length.
The choice for peak inflow q_{p} (cfs/ft) is:
(a) 200, (b) 500, and (c) 1,000. The choice for time base T_{b} (hr) is: (a) 48,
(b) 96, and (c) 192. The choice for channel length L (mi) is: (a) 200, and (b) 500.
The accuracy of the MuskingumCunge has been documented elsewhere (Ponce et al., 1996).

2. METHODOLOGY
The MuskingumCunge method is a variant of the Muskingum method (Chow, 1959) in which
the routing coefficients are calculated based on hydraulic variables using the formulas derived
by Cunge (1969). The details of the methodology are given by Ponce (2014).
With reference to Fig. 1, the routing equation of the MuskingumCunge method is:
Q_{ j+1 } ^{n+1} =
C_{o} Q_{j} ^{n+1} +
C_{1} Q_{ j} ^{n} +
C_{2} Q_{j+1}^{n}  (1) 
Fig. 1 Spatial and temporal discretization
of the MuskingumCunge method.
in which j = spatial index, n = temporal index, and the routing coefficients, C_{0},
C_{1}, and C_{2} are calculated as follows
(Ponce and Yevyevich, 1978):
^{1 + C + D}
C_{0} = ^{_______________}
_{ 1 + C + D}
 (2) 
^{ 1 + C  D}
C_{1} = ^{______________}
_{ 1 + C + D}
 (3) 
^{ 1  C + D}
C_{2} = ^{_______________}
_{ 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 = ^{____________}
S_{o} c Δx
 (6) 
with Δt = temporal interval and Δx = spatial interval.
The dischargedepth rating is:
in which q = unitwidth discharge, d = flow depth, α = rating coefficient, and β = rating exponent.
The flood wave celerity c may be calculated as follows
(Seddon, 1900;
Ponce, 2014):
In the constantparameter MuskingumCunge method (Ponce, 2014),
the routing parameters C and D
are calculated based on the average inflow discharge q_{a}:
q_{b} + q_{p}
q_{a} = ^{____________}
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 q_{b} = 50 cfs/ft
Peak inflow q_{p} = 200 cfs/ft
Hydrograph time base: T_{b} = 96 hr
Channel length L = 200 mi
Channel slope S_{o} = 1 ft/mi
Time interval Δt = 12 hr
Space 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:
A choice of channel length ⇒ L (mi): (a) 200, and (b) 500.
A choice of peak inflows ⇒ q_{p} (cfs/ft): (a) 200, (b) 500, and (c) 1,000.
Note that this choice provides peakinflow/baseflow ratios ⇒ q_{p} /q_{b} : (a) 4, (b) 10, and (c) 20.
A choice of time base of the hydrograph ⇒ T_{b} (hr): (a) 48, (b) 96, and (c) 192.
The timebase multiplier adopted to determine total simulation time (TST) (hr) is 2.5.
Therefore,
the corresponding values of TST for the three selected values of T_{b} are: (a) 120 hr (5 days), (b) 240 hr (10 days),
and (c) 480 hr (20 days).
Table 1 shows the input data and routing parameters for the eighteen (18) possible combinations:
(2 channel lengths L) × (3
peak inflows q_{p}) × (3 time bases T_{b}).
The parameter N_{t} is the total number
of time intervals; N_{x} 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 for eighteen (18) possible combinations.

Run 
L (mi) 
q_{p} (cfs/ft) 
q_{p} / q_{b} 
q_{a} (cfs/ft) 
T_{b} (hr) 
Δt (hr) 
Δx (mi) 
C 
D 
TST (hr) 
N_{t} 
N_{x} 
(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
q_{p} = 200 cfs/ft
T_{b} = 96 hr
The results of the Online Thomas Problem
are shown in Fig. 2. They are:
For comparison, the analytical results
obtained by Ponce et al. (1996) using diffusion wave theory
are:
The agreement between numerical and analytical results is indeed remarkable.
Fig. 2 Results of Run 11 of Online Thomas Problem.

6. CONCLUSIONS
An Online Thomas Problem calculator using the
MuskingumCunge method (Ponce, 2014)
has been developed.
The calculator
varies peak inflow, time base, and channel length, enabling it to
analyze the effect of runoff diffusion on the outflow hydrograph.
The input choices are the following:

Peak inflow q_{p} (cfs/ft):
(a) 200, (b) 500, and (c) 1,000.
Hydrograph time base T_{b} (hr): (a) 48,
(b) 96, and (c) 192.

Total channel length L (mi): (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. Openchannel hydraulics. McGrawHill, 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, 205230.
Ponce, V. M. and V. Yevjevich. 1978. MuskingumCunge method with variable parameters. J. Hydraul. Div. ASCE, 104(HY12), 16631667.
Ponce, V. M. and F. D. Theurer. 1982.
Accuracy criteria in diffusion routing. J. Hydraul. Div., ASCE,
108(HY6), 747757.
Ponce, V. M., A. K. Lohani, and C. Scheyhing. 1996.
Analytical verification of MuskingumCunge routing.
Journal of Hydrology, Vol. 174, 235241.
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, 179243, June.

