Structures of flow with particles saltating over the permeable fixed rough bed
Yuya Takakuwa
Civil, Human and Environmental Engineering, Chuo University
1-13-27 Kasuga, Bunkyo-ku, Tokyo, Japan 112-8551
Tel. +81 (03) 3817 1615, Email: a18.ntwb@g.chuo-u.ac.jp
Shoji Fukuoka
Research and Development Initiative, Chuo University
Email: sfuku@tamacc.chuo-u.ac.jp
In this paper, first, the applicability of the numerical method (APM [2,3]) using the one-fluid model of solid-liquid
multiphase based on the Cartesian coordinate system is examined to the experiment of turbulent flow over the permeable
fixed rough bed. It is shown that vertical profiles of streamwise velocity and turbulence intensity follow the results of Manes’
experiment only if the volume fraction of solid phase is estimated with high accuracy, even if cell sizes are not small. Second,a three-dimensional numerical experiment is conducted about turbulent flow including particles saltating over thepermeable fixed rough bed. Then, effects of saltating particles on the main flow are analyzed. With saltation of particle, theslow velocity of the low position is transported to the higher position, and the streamwise velocity of the high position isconsiderably becomes small. Therefore, the saltation of particle has an influence such as the turbulence ejection to themain flow. In this condition, turbulence intensities of the flow including moving particles become larger than those of theclear flow.
Key Words: turbulent structures, effects of moving particles, the one-fluid model of solid-liquid multiphase, permeable
fixed rough bed, Euler-Lagrange simulation

1. Introduction
Gravel bed rivers are composed of large particles as gravels and cobbles, and particles of various sizes move on thegravel bed. Therefore, it is important to clarify the mechanism of interaction between the flow and moving particles in thevicinity of the gravel bed. Structures of flow with the moving particles change compared with those without particles, inaddition, measurements of flow around moving particles are difficult even in laboratory flumes.
In the calculation of flow with solid particles, it is not easy to set up fluid calculation cells adapting to the moving
boundary in every step. In recent years, the Euler-Lagrange solvers (MICS [1] and APM [2, 3]) using the one-fluid modelof solid-liquid multiphase were proposed to fluid calculation cells including solid phase in the Cartesian coordinate system.
Then, interactions between flow and particles are considered dynamically.
In this paper, first, 3D numerical experiments are conducted by APM about turbulent flow over the permeable fixedrough bed. The results of the numerical experiments are compared with those of laboratory experiment (Manes et al. [4]),and applicability and accuracy of the APM are confirmed. Second, the 3D numerical experiment is performed about
turbulent flow including moving particles over the permeable fixed rough bed, and changes in the flow structures due tothe moving particles are examined.

2. Numerical method
The basic equations of fluid motion are described as follows:
𝜕𝑢𝑖
=0𝜕𝑥𝑖

(1)

𝐷𝑢𝑖
1 𝜕𝑃
𝜕
= 𝑔𝑖 −+{2(𝜈 + 𝜈𝑡 )𝑆𝑖𝑗 }𝐷𝑡𝜌 𝜕𝑥𝑖 𝜕𝑥𝑗

(2)

1 𝜕𝑢𝑖 𝜕𝑢𝑗
𝑆𝑖𝑗 = (+)
2 𝜕𝑥𝑗 𝜕𝑥𝑖(3)

𝜈𝑡 = (𝐶𝑠 Δ)2 √2𝑆𝑖𝑗 𝑆𝑖𝑗(4)where 𝑢𝑖 : ith component of mass averaged velocity, P: sum of the pressure and isotropic component of SGS stress, 𝑔𝑖 : ith
component of gravitational acceleration, 𝜌: volume averaged density, 𝜈: kinematic viscosity, 𝜈𝑡 :SGS turbulent viscosity,

(b)

(a)

Fig. 1. (a) Methodology of fluid calculation for cells containing solid phase using the one-fluid model of solid-liquid
multiphase, and (b) concept of the subcell method to evaluate the volume fraction of solid phase.
𝑆𝑖𝑗 : strain rate tensor, 𝐶𝑠 : Smagorinsky constant and 𝛥: computational cell size.
Fig. 1 (a). shows the methodology of the one-fluid model of solid-liquid multiphase for cells containing solid phase, andthe cells are dealt as solid-liquid mixed fluid. Mass and momentum of mixed fluid are evaluated using the volume fractionof solid phase “α” (see Eq. (5-6)), and fluid motions are calculated.
(5)𝜌 = 𝛼𝜌𝑠 + (1 − 𝛼)𝜌𝑓𝜌𝑢𝑖 = 𝛼𝜌𝑠 𝑢𝑠,𝑖 + (1 − 𝛼)𝜌𝑓 𝑢𝑓,𝑖(6)where suffixes s and f denote solid phase and liquid phase, respectively. The volume fraction of solid phase influences
evaluation of the velocity of mixed fluid directly, and its precision has large influence on the velocity distribution in thevicinity of boundaries. The volume fraction of solid phase is evaluated by the subcell method (see Fig. 1 (b).).
Particle motions are calculated by the momentum equations and the angular momentum equations of rigid bodies:𝑀𝒓̈ 𝑮 = 𝑀𝒈 + 𝑭𝒇 + 𝑭𝒄

(7)𝝎̇𝒓 = 𝑰𝒓−1 {𝑹−1 (𝑵𝒇 + 𝑵𝒄 ) − 𝝎𝒓 × 𝑰𝒓 𝝎𝒓 }(8)

where the bold face letters indicate vector tensor and matrix, M: mass of particle, 𝒓𝑮 : position of center of gravity, g:gravitational acceleration, F: forces on particle surfaces, N: torque on particles, 𝝎: angular velocity, R: transformation
matrix (from the local coordinate system to the global coordinate system), and I: tensor of momentum inertia. Suffixes fand c denote fluid force and contact force, respectively, and suffix r denotes components in local coordinate systems ofeach particle.

3. Applicability and accuracy of APM for the turbulent flow over permeable fixed rough bedFig. 2. shows the methodology and measurement lines of the 3D numerical experiments reproducing the laboratory
experiment of Manes et al. [4]. The channel was 24d long, 10d width and 5 layers spheres of d=12[mm] were placed
regularly at the bed. Bed slope was ib=1/400 as same as Manes et al., and periodic boundary conditions were applied in thestreamwise and spanwise directions. The flow depth from the roughness top (Z=0) to free surface was h=42[mm] whichgave a relative submergence ratio of h/d=3.5. Densities of liquid phase and solid phase were 𝜌𝑓 = 1000 and 𝜌𝑝 =2500 [𝑘𝑔/𝑚3 ], respectively. Tab. 1. shows the calculation conditions of numerical experiments. The experiments of 6
cases were conducted [5] with different cell sizes Δ (d/Δ=5, 10 and 20) and subcell sizes Δsubcell (Δ/Δsubcell=2 and 6). In thispaper, case 2, 4, 5 and 6 are shown. Time steps for fluid calculation (dtf) were set so that the the Courant number were 0.1,
and those for sampling of turbulent flow were dtt=1.00×10-3[s].
Fig. 3 (a). shows time and space averaged velocities of numerical and laboratory experiments. But, the velocities inz/d>2 were not measured in the laboratory experiment [4]. In case 6 which were set fluid calculation cells and subcells
smallest, respectively, the velocity distribution reproduced in neighborhood of top of permeable layer, although it wasslightly bigger than the result of laboratory experiment. In addition, in case 4 which were set subcells small (Δ/Δsubcell=6),
and calculation fluid cells not small (d/Δ=10), the velocity distribution almost followed the result of laboratory experiment.
Then, turbulence intensities were compared (see Fig. 3 (b)). In case 4, peak values of streamwise and vertical componentsof turbulence intensity and heights taking the peak value almost reproduced.
From the above results, it was confirmed that flow structures over permeable fixed rough bed were explained by APM
only if the volume fraction of solid phase “α” was estimated with high accuracy (Δ/Δsubcell=6), even if cell sizes were notsmall (d/Δ=10).

Tab. 1. Conditions of numerical experiments.

(a)

case2

case4

case5

case6cell size Δ[mm]

2.4

1.2

0.60.6d/Δ

5102020subcell sizeΔsubcell[mm]

0.4

0.2

0.3

0.1Δ/Δsubcell

6(b)Fig. 2. (a) Computational domain and boundary
conditions, and (b) measurement lines.

6-4

2-4

6-4

1.25×10-4

dtf[s]

5.00×10

dtt [s]1.00×10-3 1.00×10-3 1.00×10-3 1.00×10-3

2.50×10

1.25×10u’rms/u*w’rms/u*

Top of spheres

Top of spheres(b)(a)
Fig. 3. The comparison of numerical results (curve) and experimental results [4] (symbol) of (a) Streamwise velocity,(b) Root-mean-square of velocity fluctuations (u’rms/u*: solid curves, and w’rms/u*: dashed curves). The friction velocity
was 0.0308[m/s].

4. Changes in flow structures with moving large particles
The 3D numerical experiment was performed in a 240d long, 20d width and 1/50 bed slope with the same structures ofpermeable layer as the numerical experiments of 3. Steady flow Qf=12.8[L/s] was given. Velocity distributions of the upperboundary condition were given to the reference section (X=0.47[m]) in every step to make the boundary layer develop in
the short distance. But, for the stability of the calculation, spanwise component of velocity was zero. Periodic boundary
condition was adapted in the spanwise direction. The flow depth of the clear flow was h=42[mm] from the roughness top(Z=0) to free surface. Spheres of d=12[mm] were thrown in the region of 0.6≤X≤1.0 [m] and Z=0.05[m] at random (particle
concentration: c=0.5%). Initial velocities of particle were u=1.2[m/s] to follow flow. Time step for fluid calculation
dtf=5.00×10-5[s], and that for sampling of turbulent flow dtt=4.00×10-4[s]. In addition, densities of fluid and particles
𝜌𝑓 =1000 and 𝜌𝑝 =2500 [kg/m3], coefficient of restitution e=0.7, elastic modulus E=8.0×1010[Pa], Poisson’s ratio 0.23,
coefficient of friction 0.2, and time step for particle motion dtp=2.50×10-7[s].
In this paper, the particle-A was focused (Fig. 4.), and changes in flow structures with saltating large particles wereexamined. Fluid around the saltating particle-A moved with it. Therefore, the upward velocity occurred in the upper sideof the particle-A (Fig. 4 (c)), and the slow water mass at the low position was transported to higher position (Fig. 4 (b)).
Fig. 5. shows time series of three components of velocity at the point P1 (see Fig. 4 (b) and (c)). Upward velocities occurred
momentarily at the P1 in the times (about 13.4 and 13.7[s]) when particles jumped at the position lower than P1, and thenthe streamwise velocities considerably became small. So, this means that saltations of large particles behave as theturbulence ejection to the main flow.
Finally, time and space averaged flow structures were examined (see Fig. 6.). The velocity distribution of flow including
particles became small slightly in comparison with that of the clear flow. In this case, effects of moving large particles on
averaged streamwise velocity were small because the particle concentration was small. On the other hand, the turbulence
intensities became larger than those of the clear flow in Z>0.5d, and this result was different from the result of the numericalexperiment (Chan-Braun et al. [6]). Because, in our numerical experiment, specific gravity was 2.5, and therefore saltatinglarge particles more disturbed main flow than case of Chan-Braun et al. using the particles of specific gravity 1.7.

(a)

(c)

(b)

Fig. 4. Snapshots of the simulation result, (a) Motions of particles, (b) and (c) Contours of streamwise and verticalvelocities around particle-A.

(a)Fig. 5. Time series of velocity fluctuation at the point-P1(see Fig. 4). In the times enclosed by dashed lines (about13.4, 13.6 and 13.7[s]), some particles passed a bottom
from the point P1.

(b)Fig. 6. The comparison of results of the flow including
particles (solid curve) and those of clear flow (dashedcurve) of (a) Time and space averaged streamwise
velocity, (b) Time and space averaged turbulence
intensities. The friction velocity of the flow including
particles was 0.095[m/s], and that of clear flow was0.091[m/s]. Streamwise velocity and concentration ofparticles are plotted in (a).

5. Conclusion
It was confirmed that flow structures over the permeable fixed rough bed were explained by APM only if the volume
fraction of solid phase was estimated with high accuracy (Δ/Δsubcell=6), even if cell sizes were not small (d/Δ=10).
Second, the 3D numerical experiments was conducted about turbulent flow including particles moving over permeable
fixed rough bed. It was shown that saltations of large particle behave as the turbulence ejection to the main flow. Moreover,in this condition, turbulence intensities of the flow including moving particles became larger than those of the clear flow.
Numerical experiments of higher particle concentrations are in progress and to be presented in the Conference.

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[2] Fukuoka, S., Fukuda, T. and Uchida, T.: Effects of sizes and shapes of gravel particles on sediment transports and bedvariations in numerical movable-bed channel, Advances in Water Resources, Vol.72, pp.84-96, 2014.
[3] Fukuda, T. and Fukuoka, S.: Interface-resolved large eddy simulations of hyperconcentrated flows using spheres andgravel particles, Advances in Water Resources, https://doi.org/10.1016/j.advwatres.2017.10.037, 2017.
[4] Manes, C., Pokrajac, D., McEwan, I. and Nikora, V.: Turbulence structure of open channel flows over permeable andimpermeable beds: A comparative study, Physics of Fluids, Vol.21, 125109, 2009.
[5] Takakuwa, Y. and Fukuoka, S.: Application of the one-fluid model of solid-liquid multiphase to turbulent flow with apermeable rough boundary, Proceeding of 32th CFD Symposium, E11-2, 2018 (in Japanese).
[6] Chan-Braun, C., Garcia-Villalba, M. and Uhlman, M.: Direct numerical simulation of sediment transport in turbulent
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