International Forum for Green Asia 2020

Numerical Investigation of the Effects of Surface Shear Stress and Thermal
Stratification on Surface Divergence in Open-Channel Flow
SungJin Kim1, Yuji Sugihara2 and Osama Eljamal2
1

Interdisciplinary Graduate School of Engineering Sciences, Kyushu University
6-1 Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan
2
Faculty of Engineering Sciences, Kyushu University
6-1 Kasuga-koen, Kasuga, Fukuoka 816-8580, Japan

Abstract
In order to accurately estimate the scalar transport at the water surface, it is significant to reveal
characteristics of turbulent structure on the water surface. The divergence of horizontal
velocities on the water surface, i.e., the surface divergence is an important parameter for
analyzing the air-water scalar transports such as heat and gas. We numerically simulate theopen-channel flow including the effects of the surface shear stress and the thermal stratificationby using the direct numerical simulation, and compare the numerical results with an existing
experimental model for the surface divergence. In addition, it is found that the surface
divergence can be expressed by the Taylor microscale in spite of a wide range of variations of
the surface shear stress and the thermal stratification.
1. Introduction
The accurate estimation of the climatechange is one of the most important issues inglobal environmental studies. The scalartransports across the water surface such as heatand gas become to be significant factors
controlling the climate change. It is believedthat the scalar transports depend on near-surface
turbulence, which is affected by various
environmental factors. For example, in naturalenvironments there are the shear stress acting on
the water surface and the thermal stratification.
The surface shear stress makes the velocity
gradient near the water surface steep, whereas
the thermal stratification forms the stablestratification and unstable stratification. The
surface divergence on the water surface has beenreported to be an important turbulent
characteristic quantity in describing the scalartransport[1][2]. However, the relation between
the scalar transport and the surface divergencehas not been clarified when the effects of thesurface shear stress and the thermal stratification
change the water surface condition.
The
purpose of this study is to analyze how botheffects of the surface shear stress and the thermal
stratification change the turbulent structure on
the water surface by means of the direct

Figure 1. Relation between the surface divergenceand Reynolds number

numerical simulation, i.e., the DNS.
2. Outline of Numerical Simulation
The basic equations for DNS are thecontinuity equation of incompressible fluid, theheat transport equation, and the Navier-Stokes
equation under the Boussinesq approximation.
Dimensionless parameters controlling the flow
state are the Reynolds number Ret, the Froudenumber Frt, the Richardson number Rit, thenondimensional surface shear stress ts* and the17International Forum for Green Asia 2020

Figure 2. Relation between the surface divergenceand macro length scale.

Figure 3. Relation between the surface divergence
and Taylor micro scale

Prandtl number Pr. In this study, the value of
Ret is fixed at 150. In the case of the negativevalue of Rit is caused by the unstablestratification, whereas in the case of the positive
value, the fluid field becomes the stablestratification. The definition of the surface
divergence in this study is expressed by thefollowing equation:b* =¶u1* ¶u3*+¶x1* ¶x3* surfacelS* = kS*3/2 e S*

where ks* and es* denote the turbulent energy andthe dissipation rate at the water surface,respectively. Figure 2 displays the scaling
relation of the rms of the surface divergencebased on the turbulent macro length scale. It isfound to be difficult to express universally thesurface divergence using the macro length scale.
Figure 3 also shows the relation of thesurface divergence with the Taylor microscale l*.
From this figure, it can be seen that there exists
a proportional relation between both despite theaction of the water surface shear stress and thethermal stratification. This fact suggest that theTaylor microscale may become a universal
parameter in defining the surface divergence.

(1)

3. Results and Discussion
Sanjou and Nezu[3] quantified the surface
divergence in an open channel flow as follows:

nb rms(2)= 6.0 ´ 102 Re-4/3
U S2
where brms is the rms of the surface divergence,n the kinematic viscosity and Us the surface
velocity, and Re is a Reynolds number based on
the surface velocity and the water depth. Figure
1 shows the comparison of the present numericalresults with the Sanjou’s experimental data andhis experimental model (the solid line). It canbe seen from this figure that the relation from thenumerical results depends on the action of thesurface shear stress on the water surface, though
the numerical results agree approximately withthe experimental ones.
A macro length scale of turbulence isexpressed by the following relation:(3)

References[1] McCready, M. J., E. Vassiliadou, and T. J.
Hanratty, “Computer simulation of turbulent
mass transfer at a mobile interface”, J. AIChE,1986, 32, 1108-1115.
[2] Banerjee, S., D. Lakehal, M. Fulgosi, “Surfacedivergence models for scalar exchange between
turbulent streams”, Int. J. Multiphase Flow,
2004, 30, 963-977.
[3] Sanjou, M., and I. Nezu, “Experimental study
on surface velocity divergence and interfacial
gas transfer in open-channel flows, J. JSCE,
2013, 1, 82-89.

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