A Depth Integrated Model for 3D Turbulence Flows Using Shallow Water
Equations and Horizontal Vorticity Equations
T. Uchida1 and S. Fukuoka2
1

Research and Development Initiative, Chuo University, 1-13-27 Kasuga, Bunkyo-ku,
Tokyo, 112-8551, Japan; Phone: +81-3-3817-1617; Email: utida@tamacc.chuou.ac.jp
2
Research and Development Initiative, Chuo University, 1-13-27 Kasuga, Bunkyo-ku,
Tokyo, 112-8551, Japan, Phone: +81-3-3817-1625; Email: sfuku@tamacc.chuou.ac.jp
ABSTRACT
There is a pressing need to check the safety degree of rivers, especially inchannel confluence sections, in which several problems will be occurred during
floods such as water level rising and bed scouring. A practical and reliable model forflows and bed variations in channel junctions are highly required. This study proposesa new methodology of computing velocity distributions on bed without theassumption of the hydrostatic pressure distribution in order to develop a bed variation
model for complex flow fields such as flows around a channel confluence andhydraulic structures. First, we introduce depth-integrated momentum equations and aturbulence energy transport equation with terms of deviation component velocities
from the depth averaged value by using a parabolic velocity distribution. Then, depthintegrated horizontal vorticity equations are derived to evaluate the deviation
component velocities. The adequacy of the model is discussed through thecomparisons with the experimental results for flows in a channel confluence.
INTRODUCTION
A safety degree of a river in Japan will be significantly reduced by the global
warming. There is a pressing need to check the safety degree of rivers, especially inchannel confluence sections, in which several problems will be occurred during
floods such as water level rising and bed scouring.
The previous studies for channel junctions are divided into two categories,
flows on fixed bed and bed variation (including channel deformation). The former
category includes many experimental studies for 3D flow structures with separation(e.g. Fujita & Komura, 1990; Biron et al., 1996; Weber et al., 2001). One of priorstudies on numerical analysis for flows with a channel confluence was presented by
Mcguirk & Rodi (1978). The model consists of depth averaged continuity andmomentum equations with k-ε model. A 3D turbulence model is generally required to
calculate flow configurations due to channel junctions. Recently, 3D numerical

33rd IAHR Congress: Water Engineering for a Sustainable Environment
c 2009 by International Association of Hydraulic Engineering & Research (IAHR)
Copyright °ISBN: 978-94-90365-01-1

33rd IAHR Congress: Water Engineering for a Sustainable Environment

analysis have been conducted (e.g. Weerakoon & Tamai, 1989; Huang et al., 2002).
However, applications of the 3D model are still limited for floods in natural rivers. So
a practical and reliable model for flows and bed variations in channel junctions arehighly required.
2D numerical model have been used for flows and bed variations in rivers
during flood events. Improved 2D model have been developed to evaluate the helicalflow due to the centrifugal force in curved and meandering channels. Especially, bed
surface velocity is important for bed variation analysis. The conventional evaluation
method of bed surface velocity due to stream curvatures (Engelund, 1974), fulldeveloped secondary flow model without considering effects of secondary flow onvertical distribution of stream-wise velocity, has been applied to bed variation
analyses in channels (e.g. Nishimoto et al., 1992). Refined secondary flow models
can consider non-equilibrium (e.g. Ikeda & Nishimura, 1986; Finnie et al., 1999) andmean flow redistribution effects (e.g. Blanckaert & de Vriend, 2003; Onda et al.,2006). As more versatile method for evaluating vertical velocity distributions, a quasi3D model, in which equations for vertical velocity distribution are computed, hasbeen developed (e.g. Ishikawa et al., 1986; Fukuoka et al., 1992; Jin & Steffler, 1993;
Yeh & Kennedy, 1993). The above mentioned models are categorized in theefficiently-simplified 3D model with the assumption of hydrostatic pressure
distribution for each objective of the computation. However, it is known that thesemodels are still-inadequate for local scouring due to complex flows such as flows
around a channel confluence and hydraulic structures, because they cannot consider
effects of flows due to non-hydrostatic pressure distribution on bed scouring (e.g. thedownward flow in front of structures). So, a reliable depth integrated models withconsidering vertical velocity distributions due to non-hydrostatic pressure distribution
is highly required.
This study proposes a new methodology of computing velocity distributions
on bed without the assumption of the hydrostatic pressure distribution in order todevelop a bed variation model for complex flow fields such as flows around a channel
confluence and hydraulic structures. First, we introduce depth-integrated momentum
equations and a turbulence energy transport equation with terms of deviation
component velocities from the depth averaged value by using a parabolic velocity
distribution. Then, depth-integrated horizontal vorticity equations are derived toevaluate the deviation component velocities. The adequacy of the model is discussed
through the comparisons with the experimental results (Weber et al., 2001) for flows
in a channel confluence.
GOVERNING EQUATIONS AND COMPUTATIONAL SCHEMES
Vertical velocity distribution. Velocity distributions on very thin layer on bed
surface, the idea is after the slip-velocity model (Engelund, 1974), are discussed inthis paper. The quadric curve (1) of vertical velocity distributions are obtained byusing depth averaged velocity Ui and slip-velocity ubi (velocity on the thin layer, z=zb),
if vertical velocity gradients on water surface (z=zs) are zero:1429

33rd IAHR Congress: Water Engineering for a Sustainable Environmentu 'i = ui − U i =

δui

U −u
(
(1 − 3η )1 − 3η ) =
2
2
2

i

bi

2

(1)

where ui =i direction velocity, η=(zs−z)/h, h=zs−zb, water depth.
For the equilibrium condition (uniform flow), the following equations are
obtained. With the eddy-viscosity model, shear stress τzi is written as:
3vδu idu '(2)
τ zi = τ 0i 2η = v i η =ηdzhwhere τ0i =i direction bed shear stress defined as:

τ 0i = C0 2U i U j 2 = cb 2ubi ubj 2(3)

δui for the equilibrium condition is given by:
h
2
2 h
(4)
= cb ubi ubj
3v
3v
With the depth averaged eddy-viscosity v=αu*h, α=κ/6, the relationship between cb
and C0 is described as:
cb = C0 / (1 − 2C0 / κ )(5)
In this paper, C0 is described by manning roughness coefficient:2
C0 = gn 2 / h1/ 3(6)

δui = C0 2U i U j 2

Depth averaged velocity vectors and water depth. Governing equations for water
depth h and depth averaged velocity vectors Ui (shallow water equations) withoccupancy ratio of water λ described as:∂λh ∂U j ⋅ λh
+
=0
(7)
∂t
∂x j
∂λhτ ij∂λU i h ∂λU iU j h∂z ττ+= − g s − 0i − swi +∂xi h Rsw λh∂x jλh∂tλh∂x j(8)

where τswi =shear stress on side wall, Rsw=hydraulic radius of side wall. τij is horizontal
shear stress:
τ ij = 2νSij − δuiδu j / 5 − 2 / 3 ⋅ δ ij ⋅ k(9)where k=depth averaged turbulence energy, Sij=strain ratio tensor of depth averaged
velocity vectors Ui. k is computed by Eq.(10), which is known as one-equation model(Rodi, 1984):
1 ∂  vhλ ∂k  Dsw
∂k
∂k
+

(10)
+uj
+ Pk − ε=∂t∂x j hλ ∂xi  σ k ∂xi  Rsw
where v=vm+Cµ k2/ε, ε=Cεk3/2/∆, Cµ =0.09, Cε/∆ =1.7/h (Nadaoka & Yagi, 1984). With
using vertical velocity distribution (1), the production term Pk is derived as:2Pk δu 
 2 1
2
(11)
= 2 S ij + δS ij  + Ch  i 
v
5


 h 
where δSij=strain ratio tensor of δui (=Ui −ubi, see Eq.(1)). Ch=9(αCε)4/Cµ3≈2.25. The
model is equivalent of zero-equation model for the equilibrium condition of k.

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33rd IAHR Congress: Water Engineering for a Sustainable Environment

Depth averaged horizontal vorticity vectors. With Stokes' theorem, δui is describedas:2
∂Wh  2 ≈ ε ij 3Ω j h
δui =  ε ij 3Ω j h +(12)3∂xi  3

where εij3=permutation symbol (ε123= −ε213=1, ε113= ε223=0), W=depth averaged verticalvelocity, Ωi =depth averaged vorticity vectors. Ωi is described by:∂λhDωij∂λΩ i h ∂λU j Ω i h(13)= ERi + Pωi +
+
λ∂t
λ∂x j
λ∂x j
where,∂U y ∂U x∂δu y ∂δu xv ∂Ω i δu j Ω i
Dωij =+−, Ωz =, δω z =−∂y∂x∂yσ ω ∂x j
4∂x
ERi = Ω j

 ∂z
δω 
∂z  Ω j h ∂δui 3 ∂U i h+ δui  Ω z − z − Ω j  u si s − ubi b  −∂x j 
∂x j
4 ∂x j 2 4  ∂x jPωi =production term of vorticity vectors. With wall law of eddy viscosity (Rodi,
1984), Pωi is approximated by:Ω − Ωi(14)Pωi = 2C pω vb 0 eihwhere,
3c ε u22222Ω ei = − b ij 3 bj , vb 0 = (αh ) 2 Ω bi , Ω bi = max(Ω i , Ω ei ) , Cpω=κ/α.
κhNumerical schemes. The explicit conservative CIP scheme for shallow water flows(Uchida & Kawahara, 2006) is adopted for computations of water depth and depth
averaged velocity vectors. The important points of the scheme are as follows. In themodel, to directly capture the effect of distributed parameters in the Cartesian
coordinate system, each control volume ij has three kinds of variables, i.e., the value
at the intersection of the grid (Point Value), the averaged value along the side of thegrid (Line-averaged Value) and the averaged value over the grid (Area-averaged
Value). Those values are computed all together based on CIP-CSL scheme (Nakamura
et al., 2001). All the variables in the governing equations are set on the same location.
The interpolating operation of the variables is not necessary in the present scheme.
Any boundary conditions can be taken into account without difficulty, because all thevariables and parameters are well-defined in a computational cell based on theControl Volume Approach. The utilization of multi-valuables on a computational cell
enables us not only to capture complex geometry even on the Cartesian coordinate
system, but also to compute flow transitions with high resolution and accuracy.
Details of numerical method and its performances are discussed in our previous paper(e.g. Uchida & Kawahara, 2006; Uchida et al., 2007). The above scheme is not
adopted for computations of the depth averaged turbulence energy and vorticity
vectors, only those averaged value of the computational cell are computed by

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33rd IAHR Congress: Water Engineering for a Sustainable Environment

traditional scheme. Because, shallow water equations control flow fields rather thandepth averaged turbulence energy equation and vorticity vectors equations, the mainrole of those equations is to decide horizontal shear stress terms of Eq.(8). And theaccuracy of diffusion terms and production terms of Eq.(10) and (13) is not high,compared with those of shallow water equations.
APPLICATION TO FLOWS IN A CHANNEL CONFLUENCE
Computational conditions. In this chapter, the present model is applied to flows
with a right angle channel confluence experiment (Weber et al., 2001), as shown in
Figure 1. For the constant total discharge Q=0.170 m3/s and channel width B=0.914m,
water level and velocity vectors were measured for various discharge ratio Q*. In thecomputation, discharges are given at upstream ends of the main channel and thetributary channel. Water levels are given at downstream end of the main channel. The
computational grid is shown in Figure 1, the grid size is 1/15 of channel width B.

Q= Qm+ Qt

y
B
B
Qt

x

Main channel
Qm=Q*Q

Tributary channel

Figure 1 Computational grid around a channel junction
Computational results and discussions. Figure 2 and 3 show comparisonsrespectively of water level profiles and velocity vectors between measured (Weber etal., 2001) and computed results for Q*=0.750, 0.417. The following is thecharacteristics of flows with a channel confluence. A separation zone develops andthe water level here decreases just downstream from the tributary channel withincreasing the tributary discharge (i.e. decreasing Q*). Two helical flows are created inwater paths from the main channel and the tributary channel. With increasing thetributary discharge, bed surface velocity vectors are bended to the left bank of themain channel due to the helical flow in the tributary channel water. And bed surface
velocity is accelerated due to flow contracting and vortex stretching effects
downstream of the channel junction. These are important factors of bed scouring anddecreasing just downstream from tributary entry.
The application of the conventional flow model with the simplified helicalflow to complex flows with a confluence is unrecommended because the relationshipbetween depth averaged and bed surface velocity vectors differ considerablyaccording to location, as shown in Figure 3. The present model can capture the above
complex characteristics of flows with a channel confluence. The separation zones of
the computation are larger than that of the experiment. 3D turbulence model by Hang
et al. (2002) gave better results in the separation zone. This result indicates that thefluid mixing effect in the present model is still-inadequate in the separation zone. The
further considerations to evaluate the fluid mixing effect by turbulence and vertical1432

33rd IAHR Congress: Water Engineering for a Sustainable Environment

0.3

0.305 0.31 0.315 0.32 0.325 0.33 0.335 0.34 0.345 0.35 0.355 0.36 0.365 0.37 0.375 0.38

1.5

0.34

1.5

1

0.5

Y

Y
-0.5

-0.5

32

0.36

0.3

0

6

0.

0

4
0.3 0.32

0.34

0.5

1

-1

-1
-5

-4

-3

-2

-1

0

-5

1

-4

-3

-2

-1

0

1

0

1

Computed result

Measured by Weber et al.(2001)(a) Q*=0.750

1.5

1.5
0.34

0.3

0.32

34
0.

0.34
2
0.3

0.32

0.5

0.34

Y

0.36

4

32

0.36

0.
32

0

0.3

0.

2
0.3

0.5

1

0 .3

0

6

32
0.

Y

1

-0.5
0.36

-0.5

-1

-1
-5

-4

-3

-2

-1

0

1

-5

-4

Measured by Weber et al.(2001)(b) Q*=0.417

-3

-2

-1

Computed result

Figure 2 Water level profiles.
1

1.5
1

1

0.5

0.5

0

0

-0.5

-0.5

-1
-5

-4

-3

-2

-1

0

1

1.5

1

-1
-5

-4

1

-2

-1

0

1

1.5
1

0.5

0.5

Y

1

1

Computed result

Measured by Weber et al.(2001)(a) Q*=0.750
1.5

-3

0

0

-0.5

-0.5

-1
-5

-4

-3

-2

-1

0

1

-1
-5

Measured by Weber et al.(2001)(b) Q*=0.417

-4

-3

-2

-1

0

1

Computed result

Figure 3 Depth averaged (solid black) and bed surface (dashed red) vectors.
velocity distribution in Eq.(9) and the diffusion effect of bed surface velocity byvertical velocity in Eq.(12) are required for the computation in the separation zone.
However, the flow characteristics inducing bed scouring out of the separation zone are
well-explained by the present model as well as full 3D model results (Hang et al.,1433

33rd IAHR Congress: Water Engineering for a Sustainable Environment

2002). The above indicates the present model is expected to be a practical flow model
for bed variation computations.
CONCLUSION

This study proposes a new methodology of computing a 3D turbulence flow inchannel confluences by depth-integrated equations of shallow water equations, aturbulence energy transport equation and horizontal vorticity equations. The
important characteristic of the present model is that vertical velocity distributions areevaluated without the assumption of the hydrostatic pressure distribution. The
complex flow characteristics inducing bed scouring around tributary entry are wellexplained by the present model. The present model is expected to be a practical flow
model for bed variation computations.
REFERENCES
Biron, P., Best, J.L. and Roy, A.G.(1996) “Effects of bed discordance on flow
dynamics at open channel confluence”, Journal of Hydraulic Engineering,
Vol.122, No.12, 1996.
Blanckaert, K. and de Vriend, H. J.(2003) “Nonlinear modeling of mean flow
redistribution in curved open channels.” Water Resources Research, Vol.39,
No.12, 1375, doi:10.1029/2003WR002068, 2003.
Engelund, F.(1974) “Flow and bed topography in channel bends.” Journal of
hydraulics division, Proc. of ASCE, Vol.100, HY11, pp.1631-1648.
Finnie, J., Donnell, B., Letter, J., and Bernard, R.S.(1999) “Secondary flow correction
for depth-averaged flow calculations.” J. Eng. Mech., ASCE, Vol.125, No.7,
pp.848-863, 1999.
Fujita, I. and Komura, S.(1990) “Three-dimensional structure of the flow at an openchannel confluence, Annual Journal of Hydraulic engineering, JSCE, Vol.34,
pp.301-306, in Japanese.
Fukuoka, S., Watanabe, A. and Nishimura, T.(1992) “On the groin arrangement inmeandering rivers, Proceedings of the Japan Society of Civil Engineers,
No.443/II-18㧘pp.27-36, in Japanese.
Huang, J., Weber, L.J. and G. L. Yong(2002) “Three-dimensional numerical study offlows in open-channel junctions.” Journal of Hydraulic Engineering, Vol.128,
No.3, pp.268-280.
Ikeda, S. and Nishimura, T.(1986) “ Three-dimensional flow and bed topography insand-silt meandering rivers”, Proceedings of the Japan Society of Civil
Engineers, Vol.369/II-5㧘pp.99-108, in Japanese.
Ishikawa, T., Suzuki, K. and Tanaka, M.(1986) “ Efficient numerical analysis of anopen channel flow with secondary circulations, Proceedings of the Japan
Society of Civil Engineers, Vol.375/II-6㧘pp.181-189, in Japanese.
Jin Y.-C. and Steffler, P.M.(1993) “Predicting flow in curved open channels by depthaveraged method.” J. Hydraul. Eng., ASCE, Vol.119, No.1, pp.109-124.

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33rd IAHR Congress: Water Engineering for a Sustainable Environment

Mcguirk, J.J. and Rodi, W.(1978) “A depth-averaged mathematical model for the nearfield of side discharges into open-channel flow.” J. Fluid Mech., Vol.86, part 4,
pp.761-781.
Nadaoka, K. and Yagi, H.(1998) “Shallow-water turbulence modeling and horizontal
large-eddy computation of river flow.” Journal of Hydraulic Engineering,
Vol.124, No.5, pp.493-500.
Nakamura, T., Tanaka, R., Yabe, T., and Takizawa, K. (2001) Exactly conservative
semi-Lagrangian scheme for multi-dimensional hyperbolic equations withdirectional splitting technique, Journal of Computational Physics 174, pp.
171–207.
Nishimoto, N., Shimizu, Y. and Aoki, K.(1992) “Numerical simulation of bed
variation considering the curvature of stream line in a meandering channel,
Proceedings of the Japan Society of Civil Engineers, Vol.456/II-21㧘pp.11-20,
in Japanese.
Onda, S., Hosoda, T. and Kimura, I. (2006) ”Refinement of a depth averaged modelin curved channel in generalized curvilinear coordinate system and its
verification”, Annual Journal of Hydraulic engineering, JSCE, Vol.50,
pp.769-774, in Japanese.
Rodi, W.(1984) Turbulence models and their application in hydraulics, 2nd ed.,
IAHR, The Netherlands.
Uchida, T., Kawahara, Y. (2006) Development of an explicit conservative CIP scheme
for shallow water flows, Journal of Applied Mechanics, JSCE, Vol. 9, pp. 917924, in Japanese.
Uchida, T., Kawahara, Y. and Ito, Y.(2007): Modeling of inundation flow in urbanizedarea using a high-resolution method with Cartesian mesh, Proceedings of 32nd
Congress of IAHR, CD-ROM, Venice, Italy㧚
Weber, L.J., Schumate, E.D. and Mawer, N.(2001) “ Experiments on flow at a
90 °open-channel junction.” Journal of Hydraulic Engineering, Vol.127, No.5,
pp.340-350.
Weerakoon, S.B., Tamai, N.(1989) “Three-dimensional calculation of flow in riverconfluences using boundary-fitted coordinate.” Journal of Hydroscience andHydraulic Engineering, Vol.7, No.1, pp.51-62.
Yeh, K.-C. and Kennedy, J..F.(1993) “Moment model of nonuniform channel-bend
flow. I:fixed beds.” J. Hydraul. Eng., ASCE, Vol.119, No.7, pp.776-795.

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