COMPARISON OF FINITE DIFFERENCE AND FINITE
ELEMENT SOLUTIONS OF UNSTEADY FLOW IN COMPOUND
CHANNELS
SAYYED ALI AYYOUBZADEH
Faculty of Irrigation Hydraulic Structures, Tarbiat Modares University, Tehran, Iran
ABDOLREZA ZAHIRI
PhD Student of Irrigation Hydraulic structures, Tarbiat Modares University, Tehran, Iran
Unsteady flow in compound open channels is complicated by large differences in
hydraulic properties (flow depth, resistance) and cross sectional geometry of the main
channel and the floodplains. In this paper, the flood routing in compound open channels
has been conducted using a depth-averaged flow model in collaboration with diffusion
wave model by finite difference numerical scheme. Implicit scheme has been used in this
study. Comparison of these results with the finite element ones of Abril [1] for a
hypothetical river with a symmetric compound channel show good agreement between
the two approaches. This model has extended to heterogeneous compound channels.
INTRODUCTION
Flood routing in rivers with wide floodplains has much concerns by the researchers
during the last few decades. In most of these studies, the floodplains have been
considered as ineffective sections with very low flow velocity. In addition, the onedimensional Saint-Venant equations have been used for flood routing. Due to large
differences in hydraulic properties (flow depth, resistance) and cross sectional geometry
of the main channel and the floodplains, flow hydraulics in these sections is so complex.
Momentum transfer between main channel and flood plains are so important and must be
considered in flood routing computations. One of the most practical models for analysis
of flow hydraulics in compound channels is the two-dimensional depth-averaged Shiono
and Knight model [14]. This model is based on the Navier-Stocks equations and solves
the lateral distribution of velocity and boundary shear stress. Recently, Abril [1] used the
finite element solution of this model jointed with the convection-diffusion equation for
flood routing computations in a symmetric hypothetical compound channel. In this paper,
the finite difference solution of Shiono-Knight model has been used for unsteady flow
simulation in compound river channels. Although the finite element solution has many
advantages related to finite difference solution, but the later is simple and has wide
applications in hydraulic engineering problems. The present model has extended to
symmetric compound channels with heterogeneous roughness.
1
2
GOVERNING EQUATIONS
Two-dimensional Shiono and Knight model
Flow hydraulics in compound channels can be described by two-dimensional depthaveraged mathematical model (Shiono and Knight [14]). This model is based on the
Navier-Stokes momentum equation. For streamwise motion:
 g H S0  
f
1

ud 2 1  2 
8
y
s
1/ 2

 ud 


2 f 
  H   u d

8
y 





 H (  U V )d
y
(1)
where ρ=fluid density; g = acceleration of gravity; H=local water depth; S0= channel
streamwise slope; f=Darcy-Wiscbach friction factor, ud=depth-averaged velocity;
s=channel side slope of the banks (1:s, vertical:horizental); =dimensionless eddy
viscosity; {UV}=velocity components in the {xy} directions, x=streamwise parallel to the
bed; and y=lateral direction.
The right hand side of the above equation is the secondary flow term, which is
significant in some cases, e.g. meandering compound channels (Shiono and knight [15],
Ervine et. al. [7]). Calibration processes carried out by Abril showed that the secondary
flow term in the main channel and floodplains may be estimated from equation (2):
 H ( UV )d
  s gS0 H
y
(2)
where  s is calibration coefficient. The value of this coefficient has been suggested
0.05 in main channel for inbank flow. In the overbank flow, these values are 0.15 and –
0.25 for main channel and floodplains respectively (Shiono and Knight [15], Knight and
Abril [10], Abril [1]). Shiono and Knight [15] have solved Eq. 1 analytically for
trapezoidal compound cross sections. Although the analytical solution of this equation
has also been used in irregular compound river channels (Ayyoubzadeh and Zahiri [5]),
but in this paper, the numerical solution is used (Ayyoubzadeh and Zahiri [4]).
Eq. 1 is a nonlinear partial differential equation. Using the transform of
u
1  u 2 and assumption of X  u 2 , a linear equation will be appeared:
u

d
y
2 y
 g H S0  
1/ 2
X
f
1
1  


2 f 
X 1 2 
  H  
   S  gHS 0
8
2y
8
y
s
 


(3)
The above equation can be solved numerically by implicit finite difference approach.
For calculation of Darcy-Wiscbach and dimensionless turbulent eddy viscosity
coefficients in main channel and floodplains, the Abril’s calibration relationships [1] are
used in this paper. These relationships for homogeneous compound channels for each
water depth are:
3
f f  fc R f
&
 f  c R
R f  0.669  0.331 Dr
R  0.20  1.2 Dr
0.719
1.44
(4)
(5)
(6)
where R refers to the ratio of floodplain to main channel’s value (friction factor and
dimensionless eddy viscosity) and Dr is relative depth (ratio of flow depth in floodplain
to the main channel). The c and f subscripts refer to the main channel and floodplain
respectively.
Friction factor in main channel for inbank flow is calculated based on the Manning’s
roughness coefficient. Abril [1] assumes a constant value of friction factor in main
channel for overbank flow and for all flow depths, equal to bankfull flow value. The most
appropriate calibration value for main channel’s eddy viscosity was 0.07 for all flow
depths (Abril [1]). With these information, Eqs. 4-6 are utilized for calculation of friction
factor and eddy viscosity of floodplains.
THE CONVECTION-DIFFUSION EQUATION
The Saint-Venant equations are the basic governing equations for flood routing in rivers.
Different forms of these equations have been used by various authors for flood routing
computations in compound channels (Abril [1], Ayyoubzadeh and Zahiri [6], Fread [8],
Mizanur and Chaudhry [13], Moussa and Bocquillon [14], Tuitoek and Hicks [17],). The
full dynamic wave, diffusion analogy and characteristic wave are some kinds of hydraulic
flood routing. In this paper, the diffusion analogy has been utilized. The convectiondiffusion equation is:
1 Q Q
  D Q 



0
c t
x
x  c x 
(7)
where Q is flow discharge, t is time, c is flood wave celerity and D is the flood
propagation coefficient. The c and D hydrodynamic coefficients are functions of flood
flow discharge:
1 dQ
T dh
Q
D
2TS 0
c
(8)
where T is top water surface width and h is flow depth. These coefficients are
calculated from stage-discharge rating curve of the compound channel.
Eq. 7 is a non-linear second-order differential equation and can be solved
numerically by implicit finite difference scheme to determine the unknown Q as function
of x and t. Of course, the solution is subject to boundary condition at the upstream end of
4
the subcritical unsteady flow reach and initial conditions of Q along the reach at start of
the time. These boundary and initial conditions are specified in the next sections.
APPLICATION OF MODEL IN A HYPOTHETICAL RIEVR
Ackers [3] defined a hypothetical symmetric compound cross section similar to natural
rivers with trapezoidal sections in main channel and floodplains. This hypothetical river
is used in this paper for flood routing computations. This river has a main channel of 15
m length and 1.5 m depth. The two floodplains are 20 m wide. Both side slopes of main
channel and floodplains are 1:1. The selected reach is 20 km long with longitudinal bed
slope of 0.003. The Manning roughness coefficients of 0.03 are assumed for both main
channel and floodplains.
Using the friction factors, turbulent eddy viscosity and secondary flow coefficients
specified in the previous section, the lateral distribution of depth-averaged velocity across
the compound channel can be solved by numerical finite difference solution of Eq. 1 for
any arbitrary flow depth. An example of this lateral distribution is shown in figure 1 for
flow depth of 2.0 m. Because of symmetry of the section, the results are illustrated only
for a half-cross section.
Velocity, Ud ( m /s )
.
.
.
.
.
.
Lateral Distance (m )
Figure 1. Lateral distribution of depth-averaged velocity for flow depth of 2.0 m
Depth of Flow ( m )
.
.
.
.
.
Finite Difference
.
Finite Element
.
Flow Discharge (m /s)
Figure 2. Stage-discharge curves for hypothetical river simulated by two numerical
methods
5
With lateral integration of this velocity distribution, the total flow discharge of
compound channel for flow depth of 2.0 m can be determined. Repeating this procedure
gives the stage-discharge curve of the section. This rating curve is shown in figure 2. In
this figure the rating curve simulated by finite element approach (Abril [1]) is seen, too.
As can be seen, the results of two numerical methods have close agreements. The
bankfull discharge of the section has estimated nearly to 53.4 cms by both two methods.
For flood routing computations in hypothetical river, firstly the variations of
convection and diffusion coefficients as function of flow discharge are calculated from
stage-discharge rating curve. Then the solution of Eq. 7 is obtained by implicit finite
difference scheme according to the initial and upstream boundary condition. The
convection and diffusion curves versus flow discharge are shown in figure. 3 for
hypothetical river. These curves are exactly similar to the Abril’s results [1]. As can be
seen, an abrupt reduction has occurred in these curves at bankfull flow discharge of 53.4
cms. This happens due to sudden increase of top water surface width once the flow
initially fills the main channel and enters to the wide floodplains. These curves are
essential for flood routing analysis since they express the speed and attenuation behaviour
of a flood wave routed in the river (Abril [1]).
C (m /s) & DS (m /s)
Diffusion
Convection
Discharge (m /s)
Figure 3. Variation of convection and diffusion coefficients vs. discharge in the
hypothetical river
INITIAL AND UPSTREAM BOUNDARY CONDITIONS
Abril has used a mathematical inflow hydrograph for flood routing in hypothetical river
by the form of:
 t

t  

Q (t )  Qbase  Q peak  Qbase 
exp 1 

t p  
 t p





(9)
where Qbase is the initial or base flow of the river before the flood occurring, Qpeak is
the peak flow discharge, tp is the time to peak discharge and  is the curvature parameter
which defines the width of the inflow hydrograph. The following values have been
6
utilized by Abril [1]: Qbase=10m3/s ; tp=15hr and   6 . The Qbase equal to 10m3/s was
assumed along the river at the start time as initial uniform flow condition. Two values
have been considered for peak flow discharges in this paper, one for inbank flow (40
m3/s) and another one for overbank flow (80 m3/s).
The implicit finite difference solution of flood routing for the hypothetical river is
shown in figure 4. In this figure both inbank and overbank outflows are illustrated. Fixed
space and time steps of 500m and 100s are adapted in this numerical solution. For
comparison, the finite element solutions of Abril [1] are observed, too. Close agreements
between two numerical solutions can be seen from this Figure for both inbank and
overbank flows. The peak discharge of outflow hydrographs and time to peaks are the
same in two methods. As Abril [1] has stated, no attenuation of the peak discharge is
observed in the outflow hydrographs 20 km downstream.
Discharge (m /s)
INFLOW
Outflow - Finite Difference
Outflow - Finite Elem ent
Time (hr)
Figure 4. Outflow hydrographs for hypothetical river simulated by two numerical
methods
.
Depth (m )
.
INFLOW
Outflow - Finite Difference
.
.
.
.
.
Time (hr)
Figure 5. Depth hydrographs at downstream end of the reach simulated by finite
difference scheme for inbank and overbank flow
7
The numerical solution of convection-diffusion equation (7), gives only the outflow
hydrographs for any arbitrary location along the river. However, using this outflow
hydrographs and the stage-discharge rating curve of the river, both simulated by the finite
difference scheme, the routed depth hydrographs can be calculated. Such calculations for
downstream end of the reach are illustrated in figure 5 for example. These results haven’t
presented by Abril [1] therefore only the finite difference solutions are shown here. The
peak flow depth of 1.81 m has obtained for inflow and outflow depth hydrographs.
EXTENDING THE MODEL FOR HETEROGENEOUS ROUGHNESS
The governing equations of (1) and (7) are solved here numerically for homogeneous
symmetric hypothetical river with compound cross section. Also, Abril [1] has solved
these equations for compound channel with homogeneous roughness. In the natural river,
however, the floodplains have roughness coefficients much greater than the main
channels due to high growth of vegetations on floodplains during the low flow in the
main channel. This heterogeneity makes the flow hydraulics in compound channels more
complex and should be considered in design of flood alleviation schemes (Lai et. al.
[12]). In this paper, the modified relationships for friction factors and secondary flow
contributions obtained by Abril and Knight [2] are used for flood routing computations in
heterogeneous compound channels.
Abril and Knight [2] developed these relationships for heterogeneous compound
channels:
f c  Rn2 f
 scHet   scHom 1  I c 
(10)
(11)
where fc is the corrected friction factor of floodplains for heterogeneity, f is the
floodplain’s friction factor obtained from Eq. 4 for homogeneous roughness, Rn is the
ratio of Manning’s roughness coefficients between floodplain and main channel,  s c Hom
and  s c Het refers to the main channel’s secondary flow coefficients in homogeneous
and heterogeneous compound channel and Ic is the percentage increase in main channel’s
secondary flow coefficient due to heterogeneity of compound channel with respect to the
corresponding value found in homogeneous conditions. It has been observed from
experimental data in compound channels with rough floodplains that the secondary
currents in the main channel are much increased by the increase in roughness of the
floodplains (Tominaga and Nezu [16]).
The value of Ic is only a function (third order polynomial) of the roughness ratio Rn
(Abril and Knight [2]):
I c  0.0347 Rn3  0.485Rn2  3.03Rn  2.57
(12)
8
In the above equation, the variations of Ic with respect to the relative depth are
ignored.
For heterogeneous compound channels, the same values of turbulent eddy viscosity
coefficients found in homogeneous conditions are used for main channel and floodplains,
because the results of the two-dimensional Shiono-Knight model have very low
sensitivity of these coefficients (Knight et. al. [11], Knight and Abril [10]).
For flood routing simulations, the same hypothetical river is considered except that
the Manning’s coefficient of 0.06 is assumed for floodplains, e.g. twice as rough as the
main channel (Rn=2). The coefficients fc=0.065 and c  0.07 are considered for all
overbank flow depths like the homogeneous conditions. The corrected values of main
channel’s secondary flow and floodplains’ friction factor are obtained from Eqs. 10-12.
Using this information, the flood routing computation carried out for the hypothetical
river with the pre-specified initial and upstream boundary conditions. In figure 6, the
finite difference solutions of the outflow hydrographs are illustrated for the reach of 20
km length. As can be seen, there is no attenuation of the peak flow discharge for
heterogeneous conditions.
INFLOW
Discharge (m /s)
OUTFLOW
Time (hr)
Figure 4. Outflow hydrographs for hypothetical river simulated by finite difference
method for heterogeneous roughness
CONCLUSIONS
In this paper, using the implicit finite difference solution of two-dimensional depthaveraged Shiono-Knight model combined with the convection-diffusion equation, the
flood routing for hypothetical compound channel with homogeneous roughness has been
simulated. For both inbank and overbank flow conditions, the outflow hydrographs are
obtained and compared against the finite element solutions of Abril [1]. The close
agreement between two numerical methods, showed the suitable applicability of the finite
difference scheme for flow hydraulic analysis of compound channels for both inbank and
9
overbank flows. Also, using the modified relationships of Abril and Knight [2], the
present model has extended to the heterogeneous compound channels.
ACKNOWLEDGMENTS
The authors greatly thank the Prof. Abril, for his very useful comments. Also thanks are
due to the committee of water engineering standards of the Khozestan Water and Power
Authority for financial supports.
REFFERENCES
[1] Abril, J. B., “Overbank flood routing analysis applying jointly variable parameter
diffusion and depth-averaged flow finite element models”, Proceedings of the
International Conference on Fluvial Hydraulics, Belgium, (2002), pp 161-167.
[2] Abril, J. B. and Knight, D. W., “Stage-discharge prediction for rivers in flood
applying a depth-averaged model”, Journal of Hydraulic Research, IAHR, (2004),
(To be published).
[3] Ackers, P., “Stage-Discharge functions for two-stage channels”, Water and
Environmental Management, Vol. 7, (1993), pp 52-61.
[4] Ayyoubzadeh, S. A. and Zahiri, A., “Numerical solution of depth-averaged lateral
distribution of velocity and bed shear stress in simple and compound channels”, 6th
River Engineering International conference, Ahwaz, Iran, Vol. 1, (2003), pp 285-293
(in Persian).
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Hydraulics of Compound Varying River Channels Using a Depth-Averaged 2D
Model”, Journal of Engineering Sciences, Vol. 14, No. 2, (2003), pp 103-116 (in
Persian).
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channels”, Int. Conference on Hydraulics of Dams and River Structures, Tehran, Iran,
(2004), pp 353-358.
[7] Ervine, D. A., Babaeyan-Koopaei, K. and Sellin, R. H. J., “Two-dimensional
solution for straight and meandering overbank flows”, Journal of Hydraulic
Engineering, ASCE, Vol. 126, (2000), pp 653-669.
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Rivers ’76, Third Annual Symposium of Waterways, Harbors and Coastal
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discharge in natural rivers with overbank flow”, International Conference on
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