COMPARISON BETWEEN EXPERIMENTAL AND NUMERICALRESULTS OF FLOOD FLOW IN A COMPOUND MEANDERINGCHANNEL
Alex George Mutasingwa, Graduate student, Hiroshima University
Shoji Fukuoka, Professor, Department of Civil and Env. Eng., Hiroshima University
Akihide Watanabe, Ass. Professor, Department of Civil and Env. Eng., Hiroshima University(1-4-1, Kagamiyama, Higashi Hiroshima, 739, Japan; Tel./Fax; +81-824-24-7821)
While numerous steady compound channel flow studies have been conducted, unsteadycompound channel flow has received relatively little attention. This paper presents acomparison between 2-dimensional unsteady flow numerical model and experimental results
for flow variables in a compound meandering channel. Extensive flood flow experimental
data obtained from a compound meandering channel experiments in the Hydraulic Laboratory,
Hiroshima University is used for comparison with the numerical model. The numericalmethod is based on a finite volume discretization on a staggered grid with upwind scheme influx, it handles drying and wetting process for a flood plain, has the ability to handle complex
geometry and discontinuities, which are the main requirements for modeling compoundchannel flows. Comparison between measured data and model simulations show generally
good agreement. This shows how useful the model can be in prediction of flood propagationin channels.
Key word: unsteady flow, compound meandering channel and numerical computation
1.

INTRODUCTION

Various laboratory and numerical investigations were performed to study flow phenomenon
in open channels, in most of these studies, steady flow investigations have been done (ASCE
T.C, 1996; Fukuoka et. al, 1998; Sofialidis and Prinos, 1998; Zhao et. al 1994). Zhao et. al,
1994 observed that existing 2-dimensional numerical models have limited range ofapplicability in which they were developed. This results in uncertainty of computational
results when the 2-dimensional model is applied outside its basic simulation features. From
this viewpoint, water resources engineers, river and hydraulic engineers still demandreasonably reliable and accurate 2-dimensional dynamic models for practical engineering
planning and designs.
Unsteady flow in a multistage channel involves a stage in which the flow overtops the bankand inundates the flood plain so that part of the flow is carried by the flood plains. The
relative depth varies from zero at bankful to some value depending on the dischargehydrograph. Low relative depths are associated with large velocity difference between main
channel and flood plain that results to lateral momentum transfer across the interface between
main channel and flood plain (Wormleaton et. al, 1990). To quantify this lateral flow
exchange in one-dimensional models, weir equations are normally used (Cunge et. al, 1980;
Kun-Yeun et. al, 1998), or apparent shear stress computation approach (Fukuoka et. al, 1990;
Wormleaton et. al, 1990). However, these approaches have some uncertainties on the choiceof weir discharge coefficients (Shome et. al, 1990) and determination of mixing coefficientsin the apparent shear stress approach.
This paper presents a comparison between unsteady flow laboratory experimental results and2-dimensional numerical simulations. It describes an improved numerical model, which was1

developed earlier by Nagata et. al, 1999. The original Nagata’s approach could not simulate
well laboratory unsteady compound meandering channel flow results. Two major items of themethod were modified; the way of implementation of turbulent shear stress (instead of thesemi-empirical Reynolds stress approach of Nezu and Nakagawa, 1993), boundary conditions
and initial conditions. For upstream and downstream boundaries zero-order extrapolation
method was changed to method of characteristic that behaves well according to the directionof waves (Hirsch, 1998). This model is based on finite volume on a staggered grid withupwind scheme in flux. Steady and unsteady flows can be simulated; different flow regimes
such as sub-critical, supercritical, flow discontinuities can be handled. Comparison between
measured data (Fukuoka et. al, 2000) and model simulations show good agreement.
2. MODEL DESCRIPTION
2.1
Governing Equations
Two-dimensional depth integrated momentum equations were transformed into natural
coordinate system. In addition, by mathematical manipulation using metrics, equation [1a][1c] below with contravariant fluxes was obtained.
∂  h  ∂  Q ξ  ∂  Q η 
+
= 0.0
 +
∂t  J  ∂ξ  J  ∂η  J 

[1a]∂ξ y ∂ξ  vh  ∂ξ y∂  Q ξ  ∂  UQ ξ  ∂  VQ ξ  uh  ∂ξ x
U
=
U
+
−
+ V x  −+V+∂η  J  ∂ξ∂η ∂t  J  ∂ξ  J  ∂η  J  J  ∂ξ2
 ξ 2x + ξ 2y ∂z ξ x ηx + ξ y η y ∂z  τξ
 − b + ξ x ∂ (τ h )
− gh
+
xx

J
∂ξ
J
∂η  ρJ
J ∂ξ


2
ξ η
ξ η +ξ η
2ξ ξ
ξ η ∂
+ x x
(τ xx h ) + ξ x ∂ τ yyh + y y ∂ τ yyh + x y y x ∂ τ xy h + x y ∂ τ xy h
J ∂η
J ∂ξ
J ∂η
J
∂η
J ∂η

(

)

(

)

(

)

(

[1b]

)

∂η y ∂η  vh  ∂η y∂  Q η  ∂  UQ η  ∂  VQ η  uh  ∂ηx
U
=
 U
+
−
+ V x  −
+V
+







∂η  J  ∂ξ∂η ∂t  J  ∂ξ  J  ∂η  J  J  ∂ξ
 ξ x η x + ξ y η y ∂z η2x + η2y ∂z  τ η
 − b + ξ x ηx ∂ (τ h )− gh 
+
xx

J
∂ξ
J
∂η  ρJ
J ∂ξ


+

[1c]

ξ η
η2
ξ η +ξ η2η η
η2x ∂(τ xx h ) + y y ∂ τ yyh + y ∂ τ yy h + x y y x ∂ τ xy h + x y ∂ τ xy h
J ∂η
J ∂ξ
J ∂η
J
∂ξ
J
∂ξ

(

)

(

)

(

)

(

)

Where, t is time, h and z (=h+zb) are water depth and water level elevation respectively, zb is
the ground level elevation, u and v are depth averaged velocities in x and y directions
respectively, g is the acceleration due to gravity. The turbulent shear stress terms in x, x-y andy directions are evaluated as in equation [2a] below. The depth averaged turbulent eddy
viscosity coefficient, ε was evaluated by using the commonly used parabolic eddy viscosity
formula in [2a] below. This approach has been adopted due to computational ease of
application and its contribution to many practical applications. κ is the Von Karman constant(=0.41) and u ∗ is the shear velocity.
 ∂u ∂v  ∂v κhu ∗ ∂u 
 ; τ yy = 2ε ; ε =τxx = 2ε  ; τ xy = τ yx = ε+; u ∗ = τbx / ρ ,
∂
y
∂
x
∂
y
6
 ∂x 


 

[2a]

Expressions for boundary shear stress and Manning’s roughness coefficient are shown in [2b].
ks is the equivalent roughness. The metrics and contravariant velocities (U, V), discharge (Qξ,Qη) and contravariant boundary shear stress terms (τbξ, τbη) are defined as shown in [2c]below.

2τbx =

ρgn 2u u 2 + v 2
1; τby =

ρgn 2 v u 2 + v 2
1

h 3

1 
; n = h 6
 h 
g  6.0 + 5.75 log10      k s   
 

h 3

ξ x = Jy η ; ηx = − Jyξ ; ξ y = −Jx η ; ηy = Jx ξ ; J = 1
U = ξ x u + ξ y v;

V = ηx u + ηy v;

−1

[2b]

(x ξ yη − xη yξ )ξQη = ηx uh + ηy vh

Q = ξ x uh + ξ y vh;[2c]τξb = ξ x τbx + ξ y τby and τηb = ηx τbx + ηy τby

2.2
Method of solution
Equation [1a] to [1c] above were solved by an explicit numerical scheme that is second orderaccurate in time and first order accurate in space. The numerical scheme is an improved
version Nagata’s approach (1999) which uses Adam’s Bashforth method (Kreyzig, 1999) andupwind scheme in flux for time and space advancement respectively. Finite volume
discretization is employed on a staggered grid arrangement for velocity components in order
to avoid pressure discontinuities at the interface. The grid used was non-orthogonal. On the
open boundaries (upstream and downstream boundaries), measured discharge hydrograph andmeasured water depth hydrograph was given respectively. Additionally, on the upstream anddownstream boundaries, a unidirectional flow characteristic boundary condition was
implemented to compute velocities. For the solid wall boundaries, a slip condition was
implemented and no flux crossing through the wall. Wetting and drying was considered bypredefining a minimum water depth (hst = 0.001m). When the water depth in the cell underconsideration and its surrounding cells was less than hst then fluxes in that cell during thattime step were assigned a value of zero, then computation advanced to next time step.
Stability requirement was not explicitly accounted for in the numerical model, time step waschosen sufficiently small and constant throughout the computation time based on trials.
2.2
P2-07

1.1
FLOW

P3-07

P2-01

P3-01

P3-10

0
0

5

[m]

10

h
Dr =(h-hf )/h

P5-07

P4-07
P4-01

P5-13

P5-01

15

20

hf =0.045m

0.5m

Figure 1 Channel layout plan and measuring points and cross section
0.08
Water depth(m)Q(m3/s)

0.03
0.02
0.01

Upstream discharge
Exp.1
Exp.2

0.06
0.04

Downstream water depth
Exp.1

0.02

Exp.2

0

0
0

800

1600
2400
Time(secs)

3200

4000

0

800

1600
2400
Time(secs)

3200

4000

Figure 2 Upstream input discharge hydrograph and downstream water depthhydrograph for Exp.1 and Exp.2

2.3
Model Application
Numerical simulation was done on experimental data (Fukuoka et. al, 2000) for thecompound meandering channel shown in Figure 1. The experimental channel was 21.5m long,
0.045m-bankfull depth, and 0.5m wide sine generated main channel with each wavelength34.1m. Total width of the channel including flood plains was 2.2m and channel slope was
1/500. Measurement points were located at P2-01, P2-07, P3-01, P3-07, P4-01, P4-07, P5-01,
P5-07 and P5-13 as indicated in Figure 1. Two sets of experimental runs, Exp.1 and Exp.2 inthe same channel conditions were used to verify the simulation results. Measured upstream
discharge hydrograph and downstream water depth hydrograph for the respective Exp1 andExp2, shown in Figure 2 were used as open boundary input conditions in the numericalsimulations.
3.

RESULTS AND DISCUSSION

0.6

0.6

0.4

0.4
Velocity(m/s)

Velocity(m/s)

3.1
Effect of roughness
In these numerical simulations, water depth on the upstream boundary condition wascomputed by uniform flow Manning’s formula using input discharge hydrograph. The streamwise velocities for upstream boundary computed by 1-dimensional method of characteristic,required accurate Manning’s roughness coefficient distribution with time. Also, in order toobtain accurate upstream boundary water depth from discharge, proper Manning’s roughness
distribution with time was required. Composite temporal Manning’s roughness coefficient forthe compound section at the upstream boundary was calculated from measured water surface
profiles using one-dimensional non-uniform flow equation. These distributions with time areshown in Figure 4 for Exp.1 and Exp.2.

measured u,P3-07m25
measured v,P3-07m25
computed u,P3-07m25
computed v,P3-07m25

0.2

measured u,P4-07m25
measured v,P4-07m25
computed u,P4-07m25
computed v,P4-07m25

0.2

0

0

-0.2

-0.2
0

800

1600

2400

3200

0

4000

800

Time (secs)

1600

2400

3200

4000

Time (secs)

Figure 3 Comparison between measured and simulated velocity hydrographs for Exp1.

Manning's value

3.2
Velocity –hydrographs
Computed velocity hydrographs presented in Figure3 shows discrepancies when relative
depth is small (210s < t < 600s and 3300s < t < 3800s). However, the velocity hydrograph oflarge relative depth coincide well. Three reasons under investigation in this study account forthis discrepancy at small relative depths. First, during transition from in-bank flow to overbank flow, large variations of roughness coefficient account for this discrepancy. In theprevious steady flow studies (Myers and Brennan, 1990) roughness variability with relative
0.03
0.02
0.01

Exp.1

Exp.2

0
0

800

1600 2400
Time(secs)

3200

4000

Figure 4 Temporal distribution of upstream Input Manning’s roughness coefficient4depth was observed. Second, order 1 upwind schemes are not sufficiently accurate in mostpractical purposes (Hirsch, 1998). Third, explanation is the depth averaged turbulent eddy
viscosity coefficient is inaccurate at the transition from in-bank flow to over-bank flow near
the main channel and flood plain interface. This is because of three-dimensional nature offlow in compound channels.

0.1

0.1

0.08

0.08
Water depth(m)Water depth(m)3.3Water depth-hydrographs
The comparison of experimental and simulated water depth hydrograph results at centerline of
the main channel are indicated in Figure 5. Since agreement between the comparison is
generally good, the model might be used for prediction of flood wave height, as it propagates

0.06
measured,P2-01m25
computed,P2-01m25

0.04
0.02

0.06
measured,P5-01m25
computed,P5-01m25

0.04
0.02

0

0
0

800

1600

2400

3200

4000

0

800

Time (secs)

1600

2400

3200

4000

Time (secs)

0.1

0.08

0.08
Water depth(m)Water depth(m)(a) Comparison between measured and simulated water depth hydrograph for Exp1
0.1

0.06
measured,P3-01m25
computed,P3-01m25

0.04

0.06
0.04

0.02

0.02

0

0
0

800

1600

2400

3200

measured,P4-01m25
computed,P4-01m25

4000

0

800

1600

2400

3200

4000

Time (secs)

Time (secs)(b) Comparison between measured and simulated water depth hydrograph for Exp2

1

0.6

0.6
0.4
0.2
0
-0.2
0

0.4

0.8

1.2

1.6

2

0.4
0.2
2

0.8

x(N/m )

Turbulent shear stress in x-

2

Boundary shear stress(N/m )Figure 5 Comparison between measured and simulated water depth hydrograph

0

-0.2
-0.4
-0.6
0

0.4

0.8

1.2
Y(m)

1.6

2

Y(m)

TWB(t=210.2s)
TWB(t=1021s)
TWB(t=3303.3s)

TWB(t=600.6s)
TWB(t=3063.06s)

BU2A(t=210.2s)
BU2A(t=1021s)
BU2A(t=3303.3s)

BU2A(t=600.6s)
BU2A(t=3063.06s)

Figure 6 Temporal variation of boundary shear stress and turbulent shear stress in x-x
direction (BU2A) at cross section P3-07

5

in channels. However, for application of the model to natural rivers further investigation isneeded.
3.4
Shear stress-hydrographs
From Figure 6, it is clear that maximum values of boundary shear and turbulent shear stress in
x-x direction are located around the interface between main channel and flood plain. Thisoccurs when the flow is above bankful (210s < t < 3600s). This shows the interaction between
main channel and flood plain when the flow inundates the flood plain. The unsymmetricalpattern of the transverse distribution of the shear stresses arises due to meander channel planform.
4. CONCLUSION
Comparison between the measured and the simulated water depth hydrographs showed good
agreements, this demonstrate how useful the model can be for flood prediction in channels.
This unsteady flow study shows additional temporal change of those hydraulic variables andparameters need to be considered during flood flow. The accuracy of upstream anddownstream input data is very important for unsteady flow computations.
REFERENCES

1. ASCE Task Committee, 1998, River width adjustment. II: Modeling, Journal of HydraulicEngineering, ASCE, Vol124-9, pg903-917.

2. Cunge, J A, Holly, F M Jr and Verwey, A, 1980, Practical aspects of computational river
hydraulics, Pitman Publishing Limited, London, U.K.
3. Erwin Kreyzig, 1999, Advanced Eng. Mathematics, 8th Edition”, John Wiley and Sons
4. Hirsch C,(1994), Numerical Computation of Internal and External Flows, Volume2,
Computational methods for Inviscid and Viscous Flows, John Wiley & Sons
5. Fukuoka, S and Watanabe, A, 1998, 3- dimensional analysis on flows in meandering compoundchannels, Journal of Hydraulic, Coastal and Environmental Eng., JSCE, No586/II-42,pg. 39-50
6. Fukuoka, S, Watanabe, A, Okabe, H and Seki, K, 2000, Effects of Unsteadiness, Plan-form andCross Sectional form of Channels on Hydraulic Characteristics of Flood Flow, Annual Journal of
Hydraulic Engineering, JSCE, Vol.44, pg867-872.
7. Fukuoka S and Fujita K, 1990, Flow resistance due to lateral momentum transport acrossvegetation in the river course, International Conference on Physical Modeling of Transport andDispersion, ASCE.
8. Kun-Yeun Han, Jong-Tae Lee, Jae-Hong Park, 1998, Flood inundation analysis resulting from
Levee-break, Journal of Hydraulic Research, Vol36-5, pg. 747-759
9. Myers, W R C and Brennan, E K, 1990, Flow Resistance in Compound Channels, Journal of
Hydraulic Research, Vol. 28(2), pg. 141-155
10. Nagata, N, Hosoda, T and Muramoto Y, 1999, Characteristics of River Channel Processes withBank Erosion and Development of Their Numerical Models, Journal of Hydraulic, Coastal andEnvironmental Engineering, JSCE, Vol. No. 621/II-47, page 23-39
11. Nezu, I and Nakagawa, H, 1993, Turbulence in open channel flows, IAHR Monograph, Balkema,
Rotterdam, pg. 53-56
12. Shome M L and Steffler P M, 1998, Lateral flow exchange in transient compound channel flow,
Journal of Hydraulic Engineering, ASCE, Vol124-1, pg77-80
13. Sofialidis D and Prinos P, 1998, Compound Open Channel Flow Modeling with Non-Linear Low
κ-ε Models, Journal of Hydraulic Engineering, ASCE, Vol124-4, pg253-262
14. Wormleaton P R and Merrett D J, 1990, An improved method of calculation for steady uniform
flow in prismatic main channel/flood plain sections, J of Hydr. Research, Vol28-2, pg157-174
15. Zhao D H, Shen H W, Tabios III G Q, Lai J S and Tan W Y, 1994, Finite Volume Twodimensional Unsteady Flow Model for River Basins, J of Hydr. Eng., ASCE, Vol120-7, p863-883

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