NUMERICAL SIMULATION OF LOCAL SCOUR PROCESS DUE TOSUBMERGED JET DOWNSTREAM OF A GROUNDSILL
Tatsuhiko Uchida(1), Shoji Fukuoka(2)(1)

Department of Social and Environmental Engineering, Graduate School of Engineering, Hiroshima University,
1-4-1 Kagamiyama, Higashi-hiroshima, 739-8527, Japan,
phone: +81 82 424 7847, fax: +81-82-424-7847, e-mail: utida@hiroshima-u.ac.jp(2)
Research and Development Initiative, Chuo University, 1-3-1 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan,
phone: +81-3-3817-1625, fax: +81-3-3817-1625, e-mail: sfuku@tamacc. ac.jp

ABSTRACT
Local scours downstream of groundsills undermine the foundation of the structure. There
are two different processes of local scour downstream of groundsills due to the flow over the
structure. The duration of the developing process due to submerged jet is so short that ourinterest is only in the deformation of scour shape and the maximum scour depth. In this study,
scour shape deformation process is computed by the equilibrium sediment transport model
because of its simplicity, not emphasizing the rate of absolute sediment transport. The vertical
two-dimensional model for the flow in the local scour has been developed, which employs
sigma coordinate system to take into account of the variation of water surface, bed height. We
confirmed that the model can compute both the scour processes under the submerged jet flow
and wave jump flow. And it is found that the maximum local scour depth by the experiment isreproduced by the computation.
Keywords: scour process, maximum scour depth, vertical two-dimensional model, submerged
jet flow, wave jump flow, sediment transport
1 INTRODUCTION
Local scours just downstream of structures may undermine their foundations, revetment
works and levees. Since complete protection against local scour is almost impossible, areliable design method by the prediction of the maximum depth and the shape of local scour ishighly required to minimize the damage of the structure. To this end, it is necessary to clarify
the flow field and the local scouring just downstream of the structure based on experiments
and numerical simulations.
Local scouring is a complex phenomenon depending on the flow characteristics. It
develops owing to the turbulence of a hydraulic jump [Farhoudi and Smith (1985)], theturbulence downstream of bed protective works [Hoffmans (1993) and Kanda et al. (1996)]
and at the presence of plunging jet. Local scour downstream of a groundsill, which is the main
topic in this study, is caused by a submerged jet, a type of plunging jet. Most of the studies
dealing with the prediction of the local scour at the presence of plunging jets have been
carried out experimentally [Stein et al. (1993), Bormann and Julien (1991), Hoffmans (1998),
Mazurek and Rajatnam (2005)]. Their prediction methods are relatively simple and practical.
However, the applicability of their methods to scour depth downstream of a groundsill is notclear, because the flow pattern downstream of the structure varies considerably according to
the bed deformation. Hence, a numerical model which can simulate both flow and sedimenttransport is to be refined. Kawashima and Fukuoka (1995) developed a two-dimensional
numerical model for shallow water flow to compute the bed variation around a groundsill.
Their model, however, did not reproduce the local scour well, because the model did not take
into account of the vertical fluid motions and pressure distribution.
There are two different processes of local scouring downstream of a groundsill(see Fig.1).
These phenomena are the same as reported by Suzuki et. al (1982). One is a developing

Flow

Flow

(a) Decaying process under wave-jump flow(b) Developing process under submersed-jet flow

Fig.1 The process of local scouring just downstream of a groundsill
z

Flow

0.10m
x

0.035mInitial bed
Sand bed

Sand bed
Groundsill

Fig.2 The view of the groundsill
installed in the channel

Table 1 Experimental conditions
Channel length
Channel width
Initial bed slope
Experimental discharge
Water depth at downstream end
Diameter of sand bed

8.0 m
0.30 m
1/166.7
0.0072 m3/s
0.054 m
0.8 mmprocess where the local scour enlarges by the action of the submerged jet flow. This isimportant to design the structures, because the maximum scour depth is generated in thisprocess. Stein et al. (1993) pointed out that the sediment transport under plunging jet isgreatly different from that in equilibrium flow. Non-equilibrium sediment transport with a lotof suspended sediment can be seen in Fig.1(b). Traditional sediment transport models such as
Meyer and Peter-Muller formula do not correctly estimate the sediment transport from thelocal scour. Jia et. al (2001) considers the non-equilibrium sediment transport and fluctuating
lift force acting on the sand in their sediment transport model. However, the duration of thedeveloping process is so short that the main interest lies not in the speed of bed deformation
but in the shape and the maximum depth of the scour.
In the following, a numerical model for the flow in the local scour is developed tocompute longitudinal bed shear distribution reasonably with a sediment transport model todiscuss the deformation of local scour shape. The reliability of the present model is discussed
through the comparison with the experimental data.
2

EXPERIMENTTo understand the mechanism of a local scour downstream of a groundsill and to verify
the performance of the numerical model, an experiment is conducted. Table 1 shows theexperimental conditions. A groundsill is installed in the experimental channel as shown in
Fig.2. The top of the groundsill is set 0.035 m above the initial bed for the bed scour. The bed
shear is not beyond the critical tractive force upstream of the groundsill and no sediment is
supplied from upstream of the scour in the experiment.
A local scour develops after sufficient time and two different scour processes appear
alternately. One is a developing process under a submersed jet flow and the other is adecaying process with a wave jump flow. These processes were recorded by digital video
camera. Fig.1 shows two different processes in the experiment. Velocity components aremeasured using an electro-magnetic velocimetry.

z & covariant σ axis

contravariant σ axis

~
W

V

w
dz , Δσφ~⎛U ⎞ ⎛ 1
0 ⎞⎛ u ⎞⎜ ~ ⎟ = ⎜⎜⎜W ⎟ − tan φ 1 ⎟⎟⎜⎜ w ⎟⎟
⎠⎝ ⎠
⎝ ⎠ ⎝

x & contravariant ξ axis~
U ,uΔξ , Δσ (= dz)：grid spaces of covariant ξ , σ axisφΔξdσ = Δσ ⋅ cos φdξ = Δξ ⋅ cos φComputational gridcovariant ξ axisdξ , dσ：grid spaces of contravariant ξ , σ axisu, w：fluid velocity of x, z component~
U：fluid velocity of contravariant ξ component~
W：fluid velocity of covariant σ component

Fig.3 The definition of the sigma coordinate system
3 NUMERICAL MODEL
The governing equations in the present numerical model are expressed in the sigma
coordinate system to take into account of the variation of water surface and bed shape. A
vertical two-dimensional model is developed. Fig.3 explains the sigma coordinate system
where the covariant ξ-axis can be taken in the optional direction and the covariant σ-axis in
the vertical direction (z-axis). Many models that use the body fitted coordinate system, havedefined the vectors in the computational space. In the present model, however, vectors aredefined in the physical space to easily specify the proper boundary conditions. Fig.3 also
shows the vectors in the sigma coordinate system in the physical space. The momentum
equations and the continuity equation are written as in Eqs.(1) ,(2) and (3).
∂u ~ ∂u∂u∂p ⎞∂1 ⎛ ∂p1 ⎧∂~~(dξτ~ξσ )⎫⎬+ U ~ + W − wg
= − ⎜⎜ ~ − tan φ ⎟⎟ +(1)
⎨ (dzτ ξξ ) +∂t∂z∂z ⎠ dξdz ⎩ ∂ξ∂σρ ⎝ ∂ξ
∂ξ
⎭

(

)

(

)

1 ∂p
1 ⎧∂
∂
∂w ~ ∂w∂w~~(dξτ~zσ )⎫⎬+ U ~ + W − wg
= −g −+⎨ (dzτ zξ ) +ρ ∂z dξdz ⎩ ∂ξ
∂t
∂z
∂σ
∂ξ
⎭

( )

(

)

∂
∂
~
~
dzU +
dξW = 0
∂ξ
∂σ
Here,
∂
∂
∂
∂
, p: pressure, τ~ij : shear stress tensor given by Eq. (4), wg :,=~=dξ ⋅ ∂ξ∂z dz ⋅ ∂σ∂ξmoving grid velocity in the vertical direction.
⎧~
⎛ ∂u
∂u ⎞
⎪τ ξξ = 2ν t ⎜⎜ ~ − tan φ ⎟⎟
∂z ⎠
⎝ ∂ξ
⎪
⎪τ~ = − tan φτ~ + τ~
ξξ
zξ
⎪⎪ ξσ
⎧ ∂u ∂w⎨~∂w ⎫⎪τ zξ = ν t ⎨ ∂z + ∂ξ~ − tan φ ∂z ⎬
⎩
⎭
⎪
~
⎪
⎧ ∂W
1 ⎛ ∂w∂w∂φ ⎞⎫⎪τ~zσ = ν t ⎨− tan φ ~ +
+u
⎜
⎟⎬
2
∂z ⎠⎭
∂ξ cos φ ⎝ ∂z
⎪⎩
⎩ ∂z

(2)

(3)

(4)

Grid of water surface~
Section for W, w

Computational water fuface~
Section for UΔζ (u > 0 )groundsill

Fig.4 An example of the computational gridin the sigma coordinate systemΔζ (u < 0 )Point for water surface~
Point for velocity of U

Fig.5 Cross-sectional area of passing through fluid

vt is given by Eq. (5), which is based on Smagorinsky (1963).
ν t = (C s Δ) 2 ⋅ S

(5)

Here,
1

⎛ ~ ∂u~ ∂u~ ∂w~ ∂w ⎞ 2 ~~⎟⎟ , τ ij = 2ν t S ij , C s = 0.2
Δ = dξ ⋅ dz , S = ⎜⎜ 2Sξξ ~ + 2Sξσ
+ 2S zξ ~ + 2 S zσ
∂z
∂z ⎠
∂ξ
∂ξ
⎝
Governing equations are solved by the finite difference method, using the staggered grid
system. Fig.4 shows an example of the computational grid. An upwind scheme of the thirdorder accuracy is used for the convection terms in the ξ direction. The other terms are
discretized using the central differencing scheme of the second order accuracy. The pressure
field is calculated with a HSMAC scheme (Hirt and Cook, 1972). The level of water surface
and the vertical velocity on the water surface are calculated together with the pressure, using
the continuity equation integrated over the water depth and the kinematics boundary conditionat the water surface. Shear stresses acting on the bed and the groundsill are determined byapplying the logarithmic law for rough beds at the node nearest to the relevant boundaries.
The computational grid is reconstructed at each time step. The Heun method of the second
order accuracy is adopted for the time integration.
Water surface in the control volume is depicted by the straight line of the averagewater surface gradient. However, the grid line of the water surface, which is constructed byconnecting the averaged water surfaces in the control volume, is different from thecomputational water surface, as shown in Fig.5. In the present model, cross-sectional area for
the fluid to pass through is introduced. To compute the fluid transport terms in the ξ direction
at the top control volume, vertical sectional length dz’=dz+Δζ is used instead of dz. There are
two computational water surfaces at the cross-section accounting the ξ direction velocity. Δζis calculated by the upstream water surface, because the fluid is transported from the upstream.
The above method is also adopted in the calculation near the bed. As in the case of water
surface, two computational bed levels are defined at the cross-section. The higher bed levelof the two is used, because the fluid cannot pass through the bed.
As the boundary conditions, the discharge is given at the upstream end and the water
depth at the downstream end of the channel.
4 CALCULATION OF THE FLOW FIELDS IN LOCAL SCOUR
In the downstream of a groundsill, wave-jump and submersed-jet are alternatedcorresponding to the bed variation as shown in Fig.1. Numerical models have to reproduce
these phenomena. The present model is at first checked whether it can generate two different

0.1

0.5m/sHeight from the top of the groundsill (m)

0.1

0

実験値
Measured
計算値
Computed

0.05

-0.1

Measured

0

-0.2
-0.1

0.1

0.3

0.5

(m)

0.7

0.5m/s

0.1

-0.05
0

-0.1
-0.1

-0.15
-0.5

0

0.5

1

1.5

Computed

2
-0.2
-0.1

Longitudinal distance from theback of the groundsill (m)

0.1

0.3

0.5

Maximum velocity pass(a) Comparison of water surface

(m)

0.7

MeasuredComputed(b) Comparison of velocity distribution

z (m)Fig.6 Comparison of the flow field for submersed-jet between measured and computed result
0.1

0.1

0.1

0.1

0.05

0.05

0.05

0.05

MeasuredComputed
0
-0.2

0.2

0.6

1

0
-0.2

0.2

0.6

1

0
-0.2

0.2

0.6

1

0
-0.2

0.2

0.6

u (m/s)u (m/s)u (m/s)u (m/s)(a) x=0.19m(b) x=0.27m(c) x=0.35m(d) x=0.43m

1

Fig.7 Comparison of streamwise velocity distribution between measured and computed results
flows. In order to calculate two different flows generated under the same hydraulic conditions,
initial condition and computation method are changed. For the calculation of a wave jumpflow, the flow develops in the condition of high water-depth at downstream end, then thewater-depth at downstream end is decreased to that in the experiment. On the other hand, forthe calculation of submerged-jet flow, the submersed-jet flow is established in the conditionof low water-depth at downstream end, then the water-depth at downstream end is increased
to that in the experiment.
Fig.6 shows the comparison of (a) longitudinal water surface profiles and (b) velocity
vectors in the local scour between measured and computed results at the presence ofsubmersed jet flow. The flow over the groundsill runs into the water surface. At thedownstream of a groundsill, high velocity occurs near the bed and recirculating flow near thewater surface. These important flow characteristics in the local scour are well explained by
the present model. Fig.7 shows the comparison of the vertical distribution of x-direction
velocity between measured and computed results. Although the computed results show
slightly different values, the present model gives good agreement with measured velocity
distributions.
Fig.8 shows the comparison of (a) longitudinal water surface profiles and (b) velocity
vectors in the local scour for the case of wave-jump flow. The flow past the groundsill runs

0.1

0.5m/sHeight from the top of the groundsill (m)

0.1

0

実験値
Measured
計算値
Computed

0.05

-0.1

Measured
0

-0.2
-0.1
0.1

0.1

0.3

0.5

0.7
0.5m/s

(m)

-0.05
0

-0.1
-0.1

-0.15
-0.5

0

0.5

1

1.5

Computed

2

Longitudinal distance from theback of the groundsill (m)(a) Comparison of water surface

-0.2
-0.1

0.1

Reverse flow area

0.3

0.5

Measured

0.7(m)Computed(b) Comparison of velocity distribution

Fig.8 Comparison of the flow field for wave jump flow between measured and computed result
along the water surface, forming standing waves. For the wave-jump flow, a large area of thereverse flow is produced near the bed. The accuracy of the calculated result near water surface
is not good. Since a wave-jump flow can coexist with a submersed-jet flow, water surface isunstable. In the computation under such condition, the velocity under water surface isdecreased as shown in Fig.8, and then the flow becomes stable and a small wave-jump is
produced. Although a finite differences scheme of the third order accuracy is used forconvective terms, the numerical viscosities stemming from the sigma coordinate system andcalculation method adopted under the water surface are not negligible to calculate the flownear the abrupt change of the water surface. However, the velocity near the bed, which plays
an important role to induce a local scour, is almost reproduced by the present model.
The above results demonstrate that the coexistence of a wave jump and a submersed jet
flow is satisfied in this model. Because the model explains the two different characteristics ofthe velocity distribution near the bed well, the model can be applied to calculate the local
scour just downstream of a groundsill.
5 CALCULATION OF SCOUR PROCESS DUTE TO SUBMERGED JET
The performance of the present model is tested against the experimental results. Sedimenttransport rate from the local scour bed under submerged jet flow is different from thatcomputed by the equilibrium sediment transport model owing to the pressure fluctuations andthe suspended sediment. In fact, the absolute sediment transport rate and the deformation
speed of the scour obtained by the following computation is underestimated by one tenth of
those of the experiments. To improve this situation, the law of the wall on roughness bed andsediment transport model are modified. However, the duration of the submerged jet flow is so
short that the emphasis is not placed on the speed of bed deformation rather on the scour
shape deformation and the maximum scour depth.
Scour shape deformation is computed by an equilibrium sediment transport model of
Ashida and Michiue formula (1972), taking into account of the gravity effect by the criticaltractive force on inclined bed and additional tractive force due to gravity [Fukuoka andYamasaka (1983)]. The moving grid speed on the bed, which is a part of the convective
velocity in the sigma direction (see Eq.(3)), is calculated by the continuity equation for the
sediment. If the angle of the bed surface is calculated beyond its repose angle, the bed surface

Water surface
Maximum velocity pass

Flow

Bed

Groundsillx
Bed shear stress τ
Bed shear stress τComputational range forbed deformation

Fig.9 Computational range for bed deformation
0.1

0.1

0

0

-0.1

-0.1

t*=0.05
-0.2
-0.1

0.1

0.3

0.5

0.7

t*=1.00
-0.2
-0.1

0.1

0.1

0

0

-0.1

-0.1

t*=0.20
-0.2
-0.1

0.1

0.3

0.5

0.7

-0.2
-0.1

0.1

0.1

0

0

0.1

0.3

0.5

0.1

0.3

0.5

0.7

t*=1.05
0.7

-0.1

-0.1

t*=2.00*t =0.40
0.1

-0.2
-0.1
0.1

0

0

-0.1

-0.1

-0.2
-0.1

0.1

0.3

0.5

0.7

0.1

0.3

0.5

t*=50

t*=0.80
-0.2
-0.1

0.1

0.3

0.5

0.7

0.7

-0.2
-0.1

0.1

0.3

0.5

Fig.10 The calculated flow field in the local scour and the transition process of
the local scour

0.7

0.5m/sangle is made to have the repose angle in the calculation. The initial condition (t=0.0s) for thecomputation of the local scour is given as the calculated flow field for a submerged jet flow
over the measured bed shape (see Fig.6). The relative time t* is defined as the nondimensionalized duration time by the submerged jet tm (t*=t/tm). The deformation of scour
shape is limited to as shown in Fig.9.
Fig.10 shows a series of the calculated flow field in the transition process of the local
scour. The sand just downstream of the groundsill is transported towards upstream anddownstream soon after the flow has changed into a submersed jet (t*=0.05). Since the pressure
at downstream of a groundsill is decreased by bed scouring, the reattachment point moves in
the upstream direction (t*=0.20). Then a dune is produced in the downstream part of the local
scour with the flow separation on the top of the dune (t*=0.40). After that, the maximum scour
depth becomes deeper with time, moving the top of the dune towards downstream (t*=0.80,1.00). The reattachment point of flow, after the flow has changed into a wave-jump, moves to-0.05-0.1

Submersed jet*
t = 0.0*
t = 0.2*
t = 0.4*
t = 0.6*
t = 0.8*
t = 1.0
Wave jump*
t = 1.1*
t = 1.2

-0.15

-0.2
0

0.2

0.4

Height from the top of the groundsill (m)Height from the top of the groundsill z (m)

-0.05

-0.1

Submersed jett *= 0.0*
t = 0.2t *= 0.4*
t = 0.6*t *= 0.8
t = 1.0
Wave jump*
t = 1.3*
t = 1.7

-0.15

-0.2
0

0.6

Longitudinal distance from the back of the groundsill x (m)

0.2

0.4

0.6

Longitudinal distance from the back of the groundsill x (m)(b) Computed(a) Measured
Fig.11 The transition process of the local scour in just downstream of the groundsill
0.015

-0.1

-0.15
Maesured 1
Measured 2
Measured 3
Measured 4
Computed
-0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Height from the top of the groundsill z (m)Height from the top of the groundsill z (m)

-0.05

0.01

0.005

Submersed
jet*t = 0.0*t *= 0.2
t = 0.4*
t *= 0.6
t = 0.8*
t = 1.00-0.005
0

0.2

0.4

0.6

Longitudinal distance from the back of the groundsill x (m)Longitudinal distance from the back of the groundsill x (m)Fig.12 The comparison of maximum depth ofthe local scour

Fig.13 The transition process of the water
surface in just downstream of thegroundsill by computation

the top of the dune, and then the local scour decays due to reverse flow (t*=2. 00). The present
numerical model can explain both processes. It is also made clear in the computation that after
the long duration of the decaying process under wave jump flow, the bed shape enter thedeveloping process with a submerged jet, which is similar to the experiment (t*=50).
Fig.11 shows the transition processes of the local scour in (a) the measurement and (b)computation. Fig.12 shows the shapes of the local scour at the time when the flow change
from a submersed jet to a wave jump. Various shapes of the local scour are measured in thesimilar flow condition. This results from the instability of the flow transition phenomenon,
which involves the transition to a wave jump from a submersed jet and to a submersed-jet
from wave jump. However, the scour shape past the groundsill upto the top of the dune issimilar. Therefore, the phenomenon that the flow changes from a submerged jet to a wave
jump is subject to the effects of shape in this region. Since the shape of the back side of thedune is decided by the repose angle of the sand, the calculated dune is a little smaller than that

of the measurement, and no sand is transported toward the downstream of the local scour. It issafe to say that the present model reproduces the scour shape from the groundsill to the top of
the dune, which is subject to the developing process of the local scour as indicated above.
And the maximum bed scour depth for the experiment is reproduced by this computation.
Fig.13 shows the calculated transition process of water surface just downstream of thegroundsill. The water surface just downstream of the groundsill goes down due to bed scour atthat cross-section soon after that the flow has become a submersed jet. Then the profile of thewater surface is produced at the backward of the bed wave profile (compare Fig.11 (b)). The
water surface at downstream part of the local scour goes down due to the dune formation withthe flow separation. Then, the reverse flow area under the water surface is decreased withtime, because fluid momentum is transported toward water surface with the developing of thedune. Therefore, the wave profile of the water surface goes upstream, although the wave
profile of the bed goes downstream. Then the phase difference of wave profiles between thewater surface and the bed shape is reduced. The transition from a submersed-jet to a wave
jump occurs when the wave profile of the water surface is laid a little forward of the bed wave
profile.
As mentioned above, sediment transportation process on the local scour bed underplunging jet shows large difference from that calculated by equilibrium sediment transportmodels because of the existence of the pressure fluctuations and suspended sediment. The
present model assuming equilibrium sediment transport is, however, successful in predicting
scour process and the maximum scour depth. The reason is that the scour process in thepresent condition is produced directly by the longitudinal distribution of bed shear stress. In
other words, it is important to predict local scour process with a submerged jet that the bed
shear distribution is computed by a reliable model for the flow, which can represent thevertical velocity distribution in the local scour.
6 CONCLUSION
1.
A vertical two-dimensional model, which employs the sigma coordinate system totake into account of the variation of water surface and bed height, is developed. The
calculation near the boundary is designed to evaluate abrupt change of the boundary
shape.
2.
The transition to wave jump and submersed jet flow can be computed by the present
model. And it can explain the two different characteristics of the velocity distribution
near the bed.
The developing process of the local scour under the submerged jet flow and the3.
decaying process with the formation of wave jump flow are reproduced in thecomputation. The maximum depth of local scour that occurs at the time when the flow
pattern changes from submersed jet to wave jump is reproduced by the model. And it isclarified that to predict local scour process with a submerged jet it is quite important toevaluate the bed shear distribution and the vertical velocity distribution in the local scour.
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