Simulation of Open-Channel Turbulence with Gravel Bed by LES
I.FUJITA 1, T.OKANISHI 1, T.ODA 1
11

Graduate School of Engineering, Department of Civil Engineering, Kobe University, Japan

Introduction

Open-channel turbulent flows with rough wall beds have been
investigated intensively in the past decades mainly by
experimental procedures; however, thanks to the developmentsof numerical techniques with respect to turbulence modellingsand computational resources, simulations of rough wallboundaries came to be possible in recent years (Hurther et al.
2007). In the actual river flow conditions, bottom boundary is a
mixture of gravels with various sizes which generates a wide
range of separation vortices at the bottom. Among such vortices,
those with relatively intense vorticities rise nearer to the water
surface and affect water surface fluctuations. However,interrelationship between water surface fluctuations andbehaviour to such a near-surface vortex has not been
investigated in detail so far both numerically and experimentally,
except for an experimental research conducted by Cooper et al.
(2006) who examined a relationship between hydraulicresistances and water surface characteristics. In this study, wepaid attention to a rough wall flow with randomly distributed
gravel particles. At the same time, a new idea of visualizing
turbulence structure by tracing a cloud of pseudo-particles
repeatedly placed in a horizontal sheet is proposed. The method
was applied to simulated results by using a large eddy
simulation (LES) model. The water surface was expressed by adensity function, thereby air flow phase as well as water flow
phase was calculated in the simulation. The bathymetrycomposed of gravel particles was reproduced by an immersedboundary method (IBM). In the new method, a large number ofpseudo-particles are firstly generated in a horizontal sheet
moving with their local flow directions.
2

Numerical Simulation Model

2.1
Basis Equations and boundary treatment
The numerical simulation is based on the large eddy simulation
model in a Cartesian grid system. The fundamental equations
used after the spatial filtering is as follows:
u i
0xi
uu in 1  u in
 2 ui   u i u j 1 puj i  fivt tx j xi   x j xi  xi
x 2j

(1)

2.2
SGS Model
In the LES application to complicated boundary configurations,the conventional Smagorinsky’s model is difficult to apply,
because the damping function used in the model is a function of
the distance from the boundary and it is difficult to measuresuch a distance for a complicated topography. For solving sucha problem, we applied the MTS SGS model developed byInagaki et al. (2002). This model takes into account of thedamping effect automatically by using a time scale calculated by
the harmonic averaging of the SGS time-scale component andthe time scale given by the strain rate. Simulation Condition
The average diameter of gravel particles considered here was1.5cm. The DED used in the present simulation is shown inFig.1. Although the individual gravel shape is not completelyreconstructed, the random distribution of bottom roughness canbe generated fairly well. The number of grid in LES was 85, 100
and 100 in streamwise, transverse and vertical directions,
respectively for the gravel bed flow simulation. The number ofvertical grid was relatively larger than conventional simulations
because the air phase calculation was required in the present
LES model. The LES was conducted for the Froude numbers of0.15, 0.3, 0.4 and 0.6 with a constant water depth of fivecentimeters. The range of the Reynolds number was between
10,500 and 21,000.
3Discussion

As previously mentioned, water surface fluctuation in turbulent
flow are partly caused by impingement of vortices against thewater surface. For investigating such a phenomenon,visualization of large-scale turbulence in a horizontal field nearthe water surface is useful. In the light of this simple idea, wedevised a method to use pseudo-particle tracers repeatedlyplaced in a horizontal plane. In this flow visualization, a large
number of pseudo-tracers are firstly generated in a horizontal
plane. After the particles are advected with their local velocity
vectors, other pseudo-tracers are generated at the same
horizontal location. While repeating this procedure at an
appropriate frequency, three dimensional coordinates of every(2)Here, xi is the coordinate in the i-th direction (i=1,2,3), ui is thegrid scale velocity in the i-th direction,  is the kinematicviscosity, t is the eddy viscosity and fi is the grid-scale external
force in the i-th direction. It should be noted that the unsteady
term in the filtered Navier-Stokes equation is discretized by aforward differencing. The bottom gravel boundary wasexpressed by an immersed boundary (IB) method proposed by
Fadln et al. (2000), the direct forcing method, into the LES
algorithm. The digital elevation data (DED) of the boundary wasobtained by a photogrammetric method using a pair ofphotographs of randomly distributed gravel particles taken by ahigh resolution digital camera at two different angles. The LES
code was developed so that it can include such a digitalbathymetry data as a bottom boundary condition automatically.

26.5cm27.1cm

Fig.1 DED for gravel bed

T=1.00
T=0.00T=0.50
Fig.2 Evolutions of membrane composed of pseudo-particles at y=0.8H for Fr=0.6; flow direction is from left to right.

T=0.00

T=1.00T=0.50

Fig.3 Evolutions birds-eye-view of the second invariant distribution of rate-of-strain tensor for Fr=0.6; flow direction is fromleft to right.
particle generated at any instance are stored on memory. The
specific feature of the method is that the maximum height of thepseudo-particle location at each horizontal coordinates ispreserved during the process in order to reveal the effects ofupwelling vortices. Therefore, when we view the cloud ofparticles from above the horizontal plane, some large-scale flow
structures become visible as a sort of topographic surface. Thisvisualization method is similar to a hydrogen-bubble technique,in which bubbles are generated in a horizontal membrane.
However, it is virtually impossible to conduct such an
experiment without disturbing the flow in a physical situation.
In the present simulation, 200 by 200 pseudo-particles weretraced at every non-dimensional time step of 0.05. As a typicalexample, the evolution of particle membrane located at y=0.8H
for the case of the Froude number 0.6 is shown in Fig.2. Here, H
is the water depth equal to the height of the liquid phase. Theflow direction is from left to right. In this figure, tens of large
scale structures emerged from below with time yielding amembrane deformed by numerous bump-like structures. Itshould be noted that each bump generated by a lump of
upwelling fluid has a circular shape, showing that separated
fluids moving upwards has a round nose at its tip.
Another feature is that there exists a region where bumps aremissing. This indicates there is a structure different instreamwise direction. To clarify this feature, water surface shapeis shown in Fig.3 together with the distribution of the second
invariant of the rate-of-strain tensor which is a typical quantitythat expresses the location of a vortex core. It is clearly seenthat the water surface is lowered a little in the downstream
region, which corresponds to the region with less bumps.
Therefore, we can see that upwelling vortices are reduced wherewater surface slope has a negative value even if its value issmall. The distribution of the second invariant shows vortices
generated at the bottom are broken into numerous small partsand some of them are ascending towards the water surface andthe number of such vortices are small with negative water slope
region, which agree with the visualized results shown in Fig.2.

4Conclusion

By using an LES model capable to treat water surface variations
and arbitrary bottom topography with gravel particles, thecharacteristics of turbulent flow structures were examined byintroducing a novel flow visualization method and threedimensional expressions of water surface and the second
invariant values. The commonly observed feature for different
Froude numbers, not explained above, is that the size of thebump becomes relatively smaller for a larger Froude number as
the Reynolds number also increases for the same condition. Thechange of the scale should be related to the change of theintegral scale of the flow. Another interesting feature made clear
in the present simulation was that the scale of each bumpvisualized in the present method seems comparable to the size ofthe bottom roughness, which suggests that a lump of fluid
separated at each roughness element has reached near the water
surface by preserving its relative scale because the water depth
is relatively shallow in the present simulation.
References
Hurther, D., Lemmin, U. and Terray, E. A., (2007) Turbulent
transport in the outer region of rough-wall open channel flows:
the contribution of large coherent shear stress structures (LC3S),
Journal of Fluid Mechanics, 574, pp.465-493.
Cooper, J. R., Tait, S. J., Horoshenkov, K. V. (2006)
Determining hydraulic resistance in gravel-bed rivers from thedynamics of their water surfaces, Earth Surface Processes andLandforms, Vol.31, Issue 14, 1839-1848.
Fadln,E. A., Verzicco, R., Orlandi, P. and Mohd-Yusof, J.
(2000) Combined Immersed- Boundary Finite-Difference
Methods for Three-Dimensional Complex Flow Simulations,
Journal of Computational Physics,161-1, 35-60.
Inagaki, M., Kondo, T. and Nagano, Y. (2002) A mixed-timescale SGS model with fixed model-parameters for practical LES,
Engineering Turbulence Modeling and Experiments 5 (Editedby Rodi, W and Fueyo, N.), Elsevier, 257-266.