Two-pHase modElling for Sediment dynamIcS in geophysical flows
THESIS-2016
September 12-14, 2016, Tokyo – Japan

Quasi-3D two-phase modelling of fluid and sediment dynamics
Organizing
committees
based ona non-hydrostatic depth integrated modelwith a dynamic rough wall law
Tatsuhiko Uchida
International Scientific Committee

Research and Development Initiative, Chuo University
1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
Balachandar, S. (University of Tel.
Florida,USA)+81-3-3817-1617

Blondeaux, P. (University of Genova, Italy)
utida@tamacc.chuo-u.ac.jp
Brocchini, M. (UniversitàE-mail:
Politecnicadelle Marche, Italy)
Dong, P. (University of Dundee, U.K.)
Elghobashi, S. (University of California Irvine, USA)
Shoji Fukuoka
Guillou, S. (University
Caen,France) Initiative, Chuo University
Researchofand
Development
Hsu, T.J. (Universityof Delaware,USA) Tokyo 112-8551, Japan
1-13-27
Kasuga, Bunkyo-ku,
Tel.
+81-3-3817-1625
Izumi, N. (Hokkaido University,
Japan)
Jenkins, J. T. (Cornell University, USA)
E-mail: sfuku@tamacc.chuo-u.ac.jp
Meiburg, E. (University of California, USA)
Nguyen, K.D. (Laboratory for Hydraulics Saint-Venant, France)
Pham Van
D. (Laboratory
fortheHydraulics
Saint-Venant,France)
The assumptionof Bang,
the equilibrium
flow invicinity ofthe bed is questionable
to calculate sediShimizu,
Y.
(Hokkaido
University,
Japan)
ment transport for complex flows, such as dam-break flows and flows around structures. This study
proposed aSimonin,
new two-phase
depth-integrated
model forthe applications
O. (Institut
National Polytechniquede Toulouse,
France) of large scale sedimenttransport phenomena
and bed morphology
geophysical flows. The model for fluid phase is based on
Soldati, A. (Universityof Udine, in
Italy)
the non-hydrostaticwithemploying a dynamic rough wall law.
Toorman,quasi-3D
E.A. (KUmethod
Leuven,Belgium)
It was confirmed
comparisonswith previous formulae of bed load and suspended load
Uhlmann,throughM. (KITthe
Karlsruhe,
Germany)that the present model provided similar results for equilibrium conditions. Then, the present model
Wang, G.Q. (Tsing-Hua University, China)
was applied to the experiment on dam-break flow on movable bed. The comparisons between the experimental results and results calculated by the present model and the previous models demonstrate
the validity
the present Committee
model and advantages of evaluating vertical velocity distributions and inLocalofOrganizing
troducing two-phase model.
Fukuoka, S. (Chair) (Chuo University)

Key Words: two-phase model, dynamic wall law, depth integrated model, bottom velocity computation
Gotoh,
H. (Kyoto University)
method, vertical velocity distribution, dam-break flow

Izumi, N. (Hokkaido University)
Kajishima, T. (Osaka University)
1. Introduction
Kashiyama, K. (Chuo University)
T. bed(National
Graduate
InstituteforthePolicyFor the Mizuyama,
conventionalvariation
analysis
method,bed Studies)
shear stress, which is used as the bed
Nakagawa,(Kyoto University)
tractive force
acting onH.sedimentparticles to calculate sediment transport rate, has been evaluated with
Sakai,wall
M. (TheUniversity
Tokyo)
an equilibriumlaw (EWL)
andoftheassumption of the equilibrium flow condition near the bed.
S. case(TheofUniversity
Tokyo)
However, Sato,for thecomplex offlows,
such as dam-break flows and flows around hydraulic structures, the assumptionof the equilibrium
flow is not suitable even for the flow in the vicinity of the bed
Sekine, M. (Waseda
University)
which includes
roughnesslayer underthe bed surface [1], [2]. As long as EWL is employed, nonShimizu,
Y. (Hokkaido
University)
equilibrium
motionsofsediment
particles
Ushijima, S. (Kyoto University) and dynamic interactions between fluid and sediment motions cannot
be evaluatedcorrectly.
Uchida,T. (Secretary)(Chuo University)
Because it is difficult or even impossible to fully understand fluid-sediment dynamics and interactions only by laboratory experiments, a high or fully resolution computational fluid dynamic model
coupled with sediment particles equations has been developed [3]. Although the numerical experiSecretariat
ments withthese models have a great role for investigating particle-scale dynamics between fluid flow
Researchand motions,
Development
Chuoand sedimentparticle
it canInitiative
be more (RDI),
attractive
forUniversity
practical applications in large domains todevelop a reliable depth-integrated model. The quasi-3D models with the ability of evaluating variations in vertical distributions of velocity have been developed for bed variation analysis in curved or
meandering channels [4]. Some two-dimensional two-phase models are proposed for dam-break flow
over mobile bed [5]. However, to our knowledge, no quasi-3D two-phase models, which can evaluate169

2. Quasi-3D two-phase model
2.1. Fluid phasezz

Main calculation domain for GBVC methodzszb

Main calculation
domainusx
Bottomuvx

u x (z )Velocity distribution
with forth-orderpolynomial equation

Roughness
surface

zt

ubx

Bed

zB

Suspended
sediment

ubx

zb

Vortex layer

uvxs

Transition layer

urx

urxs

Roughness layer

Bedload layer

Fluid phase

Sediment phase

Fig.1 Computational domain of the present quasi-3D two-phase model100

qB/(sgd2)1/2

both 3D flow structure and momentum transfer between fluid and sediment motions withthe framework of 2D model, have been
available so far.
In this study, a new two-phase depthintegrated model is derived for the calculation of large scale sediment transport phenomena and bed morphology in geophysicalflows. The validity of the present model andadvantages of evaluating vertical velocity
distributions and introducing two-phasemodel are discussed though the comparisonswith previous experimental results for dambreak flow on mobile bed [6] and existing
depth integrated model with conventional
single phase sediment transport models [4].

10

Meyer‐Peter & Muller (1948)
Ashida & Michiue (1972)
Eq. (1)Cb (zb=0.05h)

1
The calculation method for fluid phase isbased on the General Bottom Velocity Com0.1
putation method with employing Dynamic
0.01
rough Wall Law (GBVC4-DWL [2]), taking
into account interactions with sediment phase.
0.001
The calculation domain is divided into three
0.01
0.1
1
/(sgd)layers in vertical directions: main calculation(a)
Bedloadrate
domain, vortex layer and roughness layer, as
1
shown in Fig.1. For the vertical distribution
0.1
of horizontal velocity, the fourth-degree polynomial velocity distribution is adopted with
0.01
the constrain conditions of depth averaged
0.001
velocity, water surface and bottom velocity,
Itakura & Kishi (1980)
and the vertical gradient of velocity at water
Garcia & Parker (1991)
0.0001surface and bottom. These quantities to evalEq. (2)0.00001
uate vertical distributions of velocity and
0.1
1
10
100
u/w0
pressure in the main calculation domain are(b) Sediment concentration for Rp=10
calculated with several depth-integrated
Rp= (sgd3/v2)1/2, v=kinematic viscosity, h=water depth,
equations which are derived from Reynold’s
ks=equivalent roughness heightaveraged continuity, momentum and vorticiFig.2 Comparisons with previous formulae forty equations. The bottom boundary conditions of the main calculation domain for equilibrium conditions
depth integrated momentum and vorticity equations are given by the DWL with the continuity andmomentum equations for the vortex and roughness layers, instead of the uniform velocity distribution
under the bottom with the EWL. The avoidance of assuming the equilibrium flow condition near thebed, in which sediment particles are driven actively by fluid flow, is a significant advantage in thepresent method to take into account interactions between fluid and sediment motions: two-phase modelling. The detail of the GBVC4-DWL method may be found in the previous literature [2].

2.2. Sediment phase

The model for sediment phase newly introduced in this study has a different sub-model developedfor each layer. The bedload layer is assumed to be in the roughness layer. The particle skeleton stressis considered to be significant for the bed load layer with dense sediment concentration. A two-phasemodel, which consists of continuity and momentum equations for both fluid and sediment phases withthe variation in the thickness of the layer under the assumption of the constant sediment concentration

170

1.5s

Initial water depth: 0.25 m

Gate

0.3s
1.2s
0.6s

3

Exp.

0.9s

0.9s

4

2D

GBVC3‐EWL

1.2s

1.5s

1.5s

5 x (m) from the channel upstream end 6

GBVC4‐DWL

Present model (GBVC4‐DWL+2 phase model)

Single phase model (non-equilibrium bedload formulae [4] )Fig.3 Water front propagations in the experiment [6] and several computation methods[5], is applied to the bedload layer, considering interactions with the upper transition layer. The existing formulae are adopted for evaluating bed load entrainment rate E and deposition rate D, along theline of Greco at al. [5]. The hydrodynamic force acting on bedload sediment is calculated with roughness and vortex layers velocity without EWL which has been employed for previous sedimenttransport models [4],[5]. Over the bedload layer, we consider the transition layer in the vortex layer, inwhich a dilute two-phase model without particle-collision stress is introduced, calculating sedimentconcentration and velocity. In the main computational domain, a single-phase model is adopted with adepth integrated advection-diffusion equation of sediment concentration. The interactions between thelayers are dynamically evaluated without assuming equilibrium flow conditions. For equilibrium flow
condition, the governing equations of the present model for sediment phase yield following bedload
rate qB (1) and sediment concentration at the bottom Cb (zb=0.05) (2):2/3
 cp 
CB1  a ( w0 / u* )4  d qB(1), Cb (2)
 cL    *   *c   * 2 
1  w0 / u* 1  (1  a )( w0 / u* )3C D cL sgd
 cd 
where, s=submerged specific gravity of sediment, g=acceleration of gravity, d=sediment particle diameter, cL=14, cp=0.02, cd=0.025, *=u*2/sgd, u*=shear velocity, *c=dimensionless critical Shields
stress, d=dimensionless dynamic friction coefficient (d=0.6), CD=drag coefficient (CD=0.4),
CB=sediment concentration at the bed (CB=0.6), w0=settling velocity, =/6, a=0.1. It is confirmed
from Fig. 2 that the bed load rate and sediment concentrate calculated by present model for equilibrium flow condition are within the range of those estimated by previous formulae [7].

3. Model application to dam-break flow over mobile bed

The present model (GBVC4-DWL for two phase model) is applied to the experiment on dam-breakflow on movable bed with a left-side sudden enlargement [6]. The experiment was conducted in theflat bed channel with 6 m long, 0.25 m to 0.50 m width, a gate 1m upstream from the enlargement
section, as shown in Fig. 3. The uniform sand with d50=1.72 mm was used. Refer to the literature [6]for details. For the calculations, the above experimental conditions are given with equivalent roughness ks=d50 and the depth of the log law origin dz0=0.3d50 for evaluating bed roughness. Numericalcalculations with 2D, GBVC3-EWL and GBVC4-DWL coupled with the previous single phase nonequilibrium bed load model [4] are conducted for the same condition and parameters used in the present model.
Fig.3 shows comparisons of water front propagation between experimental results [6] and several
computation results. The water front for 0.3 sec. is considered to be influenced by the gate opening
operations and initial complex sediment motion near the gate. The water fronts after 0.6 sec. computedwith the single phase model are propagated considerably far from that of experiment. The effect ofsophistication of fluid phase model for the main computational domain on the water front propagationis relatively small (2D vs GBVC3-EWL). The BVC4-DWL with single phase model still underestimates movable bed resistance. On the other hand, the present model provides good agreement withexperimental results. The above indicates that the non-equilibrium flow and momentum exchange
between fluid and sediment phase near the bed are essential to evaluate the resistance of movable bed
for dam-break flows. Fig.4 shows bed topography after dam break flow by experiment [6] and several

171

computation methods. For the calculation withsingle phase model (b)-(c), sediment depositionareas occur downstream compared to experiment.
The calculation results in Fig.4 show the effect of
fluid phase models on the calculation of bed topography. However, it should be noted from the comparison with measured result that the sophisticationof the fluid phase model does not necessarily improve the bed variation calculation unless an adequate sediment phase model is employed. It is investigated in the preliminary calculations that the
bed variation calculation results by BVC4-DWL
with single phase model depend on the definitionof the bed tractive force for computing bedload rate.
The calculation result by the present model explains well about experimental bed topography
except for the deposition height and local scour
along the left bank investigated for the experiment.

4. Conclusions

A new two-phase depth-integrated model basedon non-hydrostatic quasi-3D flow model with adynamic wall law (GBVC4-DWL) is proposed forthe applications of large scale sediment transportphenomena and bed morphology in geophysicalflows. The comparisons for dam-break flow overmovable bed between the experimental results andresults calculated by the present model and theprevious models demonstrate the validity of thepresent model and advantages of evaluating vertical velocity distributions and introducing twophase model.

[2]
[3]
[4]
[5]
[6]
[7]

0.2

0

(a) Measured [6]

0.4

0.2

0

(b) 2D (Single phase model)

0.4

0.2

0

(c) GBVC3-EWL (Single phase model)

0.4

0.2

0

(d) GBVC4-DWL (Single phase model)

0.4

0.2

0

References
[1]

0.4

(d) Present model:
Uchida, T. and Fukuoka, S. (2015) A new calcuGBVC4-DWL(Two phase model)
lation method for local three dimensional flows
by using the non-Hydrostatic depth integrated
Be variation (m)model (BVC method) with dynamic wall-law for-0.03 -0.02 -0.01 0
0.01 0.02 0.03
rough bed ， Journal of Japan Society of Civil
Fig.4 Comparisons of bed topography
Engineers, Ser. B1 (Hydraulic Engineering), Vol.
caused by dam break flow
71, No. 2, pp.43-62, 2015, in Japanese.
Uchida, T., Fukuoka, S., Papanicolaou, T. and Takiris, A.G. (2016): Non-hydrostatic quasi-3D model
coupled with dynamic rough wall law for simulating flow over rough bed with submerged boulders, Journal of Hydraulic Engineering, accepted.
Kidanemariam, A. G. and Uhlmann, M. (2014) Interface-resolved direct numerical simulation of theerosion of a sediment bed sheared by laminar channel flow, International Journal of Multiphase Flow,
Vol.67, 174-188.
Uchida, T. and Fukuoka, S. (2014) Numerical calculation for bed variation in compound-meanderingchannel using depth integrated model without assumption of shallow water flow, Advances in Water Resources, Vol.72, pp.45-56, 2014.
Greco, M., Iervolino, M., Leopardi, A., and Vacca, A. (2012) A twophase model for fast geomorphic
shallow flows.” Int. J. Sediment. Res., 27(4), 409–425.
Goutiere, L., Soares-Frazão, S. and Zech, Y. (2011) Dam-break flow on mobile bed in abruptly widening
channel: experimental data, Journal of Hydraulic Research, 49:3, 367-371.
Garcia, M. H. (2008) Sedimentation engineering: Processes, management, modeling and practice, ASCE,
Reston, VA..

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