ONE DIMENSIONAL MATHEMATICAL MODELING OF
POLLUTANT TRANSPORT IN COMPOUND OPEN CHANNELS
AYYOUBZADEH S, A
Water Structure Department, Tarbiat Modarres University, Jalale Ale Ahmad Ave
Tehran, 14115-11, Iran
FARAMARZ, M
Water Structure Department, Tarbiat Modarres University, Jalale Ale Ahmad Ave
Tehran, 14115-11, Iran
MOHAMMADI, K
Irrigation Department, Tarbiat Modarres University, Jalale Ale Ahmad Ave
Tehran, 14115-11, Iran
In this research, the transport of pollutant substances is studied by combining floodplain
flow hydraulics in compound open channels and a numerical solution for advectiondispersion equation. The hydraulic parameters such as transverse varying of depth mean
velocity, discharge and mean shear velocity in a typical and hypothetical river are
determined. The temporal and spatial variation of pollutant concentration is then
calculated by applying the obtained hydraulic parameters. In order to solve 1D
differential advection-dispersion equation, the finite difference techniques and the Gauss
Seidel iteration method are applied. The present mathematical model as POLLUTE1 is
then developed in Fortran code. The validity of the numerical model is obtained by
comparing the results with the corresponding results of the analytical solution under
proper initial and boundary conditions. The temporal and spatial variation of pollutant
concentrations are also compared with the results of familiar MIKE11 model. The
application of the model revealed that there is a significant difference between the
pollutant concentration results when interaction effect is taking into account in
comparison with when this effect is ignored.
INTRODUCTION
The performance of rivers in times of flood, and the need to understand sediment and
pollutant process in natural systems are the main reasons for studying rivers with
compound sections in Civil Engineering. On the otherhand, the hydrodynamic process at
work in such rivers is well known to be complex, particularly due to geometry and
roughness effect. A considerable amount of research work has been undertaken in recent
years to cope this complex hydrodynamic behavior. However, a few works could be
found in terms of the effect of this hydrodynamic process on mass transport. This paper
describes a numerical model solving the pollutant advection-diffusion equation for a
typical compound channel using a recent two dimensional hydrodynamic model for flow.
1
2
PREVIOUS WORKS
The literature of the problem considered here includes two main subjects namely
hydraulics of compound channels and transport process. Early investigation on
hydrodynamics of two stage open channels include Sellin (1964) and Zhelenzyakov
(1971) who demonstrated the presence of the vortices and their effect on velocity and
discharge at overbank flows [14], [12]. The UK Flood Channel Facility (FCF) performed
enormous studies on compound channels in various phases. See for example Myers and
Brennan (1990), Knight and Shiono (1990), Wormleaton and Merrit (1990) and Elliot
and Sellin (1990, Sellin, Ervin and Willets (1993), Greenhill and Sellin (1993), and
Ervin, Willets, Sellin and Lorena (1993) [14]. Phase A of the FCF program centered on
straight and skewed fixed boundary compound channels, and the results of this have been
presented by Ackers (1992, 1993) who introduced his coherence concept in account of
the interaction effect between the main channel and floodplains [5], [6]. Martin and
Myers (1991) presented an experimental study of a compound channel including velocity
distributions and stage discharge relationships [14]. They illustrated the effects of
momentum transfer between deep and shallow flows, which include reduction in main
channel velocity and discharge capacity, leading to a reduction in compound section
capacity at depth above bankfull. Bousmar and Zech (1999) presented a new theoretical
1D model of compound channel flows termed the Exchange Discharge Model (EDM)
[9]. Myers et al (2001) an experimental compound channel program carried out at the UK
Flood Channel Facility including fixed and mobile main channel boundaries together
with two-floodplain roughness [14]. Haidera and Valentine (2002) developed a new
method for predicting the total flow in compound channel [11]. The second subject of
the problem is dealt with pollutant transport. Mathematical modelling of the transport of
salinity, pollutants and suspended matter in shallow waters involves the numerical
solution of a convection–diffusion equation. Many popular finite difference methods,
such as the upwind scheme of Spalding (1972) and the flux-corrected scheme (Boris and
Book, 1973) are available for the solution of the depthintegrated form of the convection–
diffusion equation. Another widely used approach is the split-operator approach (Sobey,
1983; Li and Chen, 1989), in which the convection and diffusion terms are solved by two
different numerical methods. Noye and Tan (1988) used a weighted discretisation with
the modified equivalent partial differential equation approach for solving onedimensional convection–diffusion equations. Later, Noye and Tan (1989) extended this
technique to solve two-dimensional convection–diffusion equations. But the above
mentioned techniques have difficulty in solving three-dimensional problems, because of
extensive matrix inversions at each time step. Numerical studies show that the use of
central differencing for the convective terms of the convection–diffusion equation results
in negative species concentration. Lam (1975) points out that the central difference
approximation will overestimate the advective flux so much that it often causes a
negative concentration to appear in the adjacent cell. To avoid such a shortcoming of
3
central differencing, the upwind or donor cell method introduced by Gentry et al. (1966)
is generally used. To overcome the shortcomings of numerical dispersion, Leonard
(1979) introduced an upstream interpolation method, namely QUICK (Quadratic
Upstream Interpolation Convective Kinematics) f or one-dimensional unsteady flow.
Later, Leonard (1988) gave an improved version of the QUICK scheme, eliminating the
wiggles completely by introducing exponential integration into regions with sharp fronts.
Chen and Falconer (1994) showed that the QUICK scheme is only second-order accurate
in space and presented different forms of the third-order convection, second-order
diffusion for the solution of the convection–diffusion equation. A relatively recent
phenomenon is the development of three-dimensional transport models. Lardner and
Song (1991) used an algorithm splitting the horizontal convection and diffusion by an
implicit finite element method, treating the horizontal convection–diffusion explicitly and
vertical convection–diffusion by an implicit finite element method. It is to be noted that
Lardner and Song (1991) used a first-order upwind scheme for the convection terms of
the convection–diffusion equations. Sommeijer and Kok (1995) made a detailed study on
the use of various time-integration techniques for the numerical solution of
threedimensional convection–diffusion equations using finite differences. The numerical
model was validated by comparing the results obtained with analytical solutions for the
case of transport of a Gaussian pulse in unsteady and non-uniform flow. In the present
study, a third-order upwind difference scheme as given in Kowalik and Murty (1993) has
been used for the convection terms of the convection–diffusion equation. Earlier, the
authors Shankar et al. (1996) used a third-order upwind scheme for the convective terms
of the shallow water momentum equations.
OBJECTIVES AND PROCEDURE
The study case corresponds to the river with a symmetric compound channel and nonhomogeneous roughness proposed by Ackers (1993). The geometric characteristics of the
cross section are: main channel bottom width b=15m, main channel depth below lateral
berms h=1.5m, lateral wall slopes s=1:1 (horizontal: vertical), and floodplains widths
B=20m. The longitudinal bed slope S0=0.003 in a reach length of 20 km (corresponding
to Abril (2002) studies). Main channel Manning’s coefficient nmc=0.03 and floodplains
Manning’s coefficient nfp=0.06 are adopted. The 2D Shiono and Knight model was used
to determine the hydraulic parameters. Shiono and Knight (1988) developed an equation
to determine depth average velocity in compound channels using the Navier- Stocks
Equation [7]. Knight and Abril (1996) suggested equations for the computation of
floodplain (fp)/main channel (mc) ratios Rfand Rwhich are assumed to be function of
the relative depth, and hence obtain the dimensionless eddy viscosity  and local bed
friction f [3]. The main channel dimensionless eddy viscosity considered to the constant
with stage, that is mc=0.07[3].
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Numerical solution of advection-diffusion equation
The general equation for solute transport considering advection- dispersion without
source/sink term is as follow:
C
 2C
C
 D 2 u
t
x
t
(1)
Where C: concentration, u: velocity, D: dispersion coefficient.
Both explicit and implicit schemes employ the concentrations weighted at the old and
new time level to approximate the dispersion and advection terms of equation (1), leading
to the following finite difference at node I:


 D
n 1
n 1
n 1 
 2 C i 1  2C i  C i 1 
C i n 1  C i n

x


 

t
u
(1   )C i n 1  C i 1 n 1  (1   )C i 1 n 1  C i n 1
 
  x



 D
n
n
n 
 2 C i 1  2C i  C i 1 

x

 (1   ) 
 u
n
n
(1   )C i  C i 1  (1   )C i 1 n  C i n

 x














(2)

where andare time weighting factor and spatial weighting factor, respectively.
Equation (2), can be written as:
ai Ci 1n1  bi Ci n1  ci Ci 1n1   f i
(3)
where:
Dt
ut
  (1   )
2
x
x
Dt
ut
bi  2
  (1  2 )
1
x
x 2
Dt
ut
ci  
 
x
x 2
Dt
n
n
n
n
f i  Ci  (1   ) 2 Ci 1  2Ci  Ci 1
x
ut
n
n
n
n
 (1   )
(1   )Ci  Ci 1  (1   )Ci 1  Ci
x
ai  


(4)
(5)
(6)


(7)
The general form of equations (2) and (3) reduce to:   0 as Explicit scheme;
Implicit scheme; and   0.5 as Crank- Nicolson scheme. An analytical
solution of equation (1) with the initial condition: C(x,0)=0 and boundary conditions of
C(0,t)=C0 and C(,t)/x=0 is available as :
  1 as
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
 x  ut
C
 1 erfc
2
C0

 4 Dt



  exp  xu erfc x  ut 



D

 4 Dt 
(8)
where erfc is error function complementary.
RESULTS AND DISCUSSION
Numerical solution model of 2D Shiono and Knight equations are presented by
Ayyoubzadeh and Zahiri (2003) [8]. The model outputs include depth mean velocities,
shear velocity and discharge for each depth. The lateral distribution of depth mean
velocities for water depth of 3 m and stage-discharge diagram are illustrated in Figure 1.
The two curves in the stage-discharge diagram include the results of classic approach
ignoring any interaction effect denoted as 'Basic Method' and the other for the results
obtained from 2D Shiono and Knight model which takes interaction into account.
Figure 1. Lateral distribution of depth-mean velocity and stage discharge curve in the
Ackers hypothetical river
The temporal and spatial variation of pollutant concentrations in present model,
POLLUTE1, for one-fourth of simulation time and reach length for a depth ratio of 0.5
are plotted in Figures 2 and 3 as application examples. The analytical results also plotted
for the same cases to indicate the validity of the present numerical models. Furthermore
the results obtained from MIKE11 [10] which neglects the hydraulic behavior of the
compound channel flow are also plotted in the Figures. If the hydraulic results obtained
from the 2D compound channel flow model are exposed in the MIKE11, so the temporal
and spatial variation of pollutant concentration may be given as indicated in Figures 4
and 5 in comparison with those obtained from the present model. The final diagram in
Figure 6 shows the difference between spatial variation of the pollutant concentration
results in the cases when the interaction flow effect between main channel and
floodplains is taking into account and when it is ignored. The significant difference
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shown in the Figure indicates the importance role of compound channel flow and its
effects on pollutant transport.
Figure 2. Spatial variation of pollutant concentration in Ackers hypothetical river for
depth ratio 0.5 and one-fourth of total simulation time
Figure 3. Temporal variation of pollutant concentration in Ackers hypothetical river for
depth ratio 0.5 and one-fourth of total reach length
CONCLUSION
A numerical model is developed for determining the temporal and spatial variation in
compound channels. The effect of interaction between different parts of the channels is
Figure 4. Spatial variation of pollutant concentration in Ackers hypothetical river for the
same hydrodynamic conditions
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Figure 5. Temporal variation of pollutant concentration in Ackers hypothetical river for
the same hydrodynamic conditions
Figure 6. Spatial variation of pollutant concentration in Ackers hypothetical river with
and without interaction effect
taking into account using a 2D model of Shiono and Knight. Numerical solution of 1D
advection-diffusion equation is then established. The developed model is applied to a
hypothetical river conditions. The validity of the present model is shown by comparing
with the results of the analytical solution under certain boundary and initial assumed
conditions and also with the results of MIKE11 model under same conditions. The
graphical comparison presented in the paper highlights the effect of compound channel
flow hydraulics on pollutant transport.
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