Proceedings of International Forum for Green Asia 2019
5th International Exchange and Innovation Conference on Engineering & Sciences (IEICES 2019)

GA08 / P20
Normalization of gas transfer velocity at air-water interface
Sung Jin Kim, Yuji SugiharaInterdisciplinary Graduate School of Engineering Sciences, Kyushu University
Email: user0galaxy@gmail.com
Abstract: The turbulent motion that control the gas transfer velocity at the air-water interface is divided into two typesof turbulent scales, which are integral scale and microscale such as the Taylor microscale and Kolmogorov scale. Thegas transfer velocity can be expressed based on these turbulence scales. The turbulent scale to model the gas transfer
velocity is expected to vary with Reynolds number. The purpose of this study is to carry out the normalization of the airwater gas transfer velocity using a new turbulent time scale combining the two time scales. Comparing this normalization
with experimental data of oscillating grid turbulence shows that this result helps us describe the gas transfer velocity
over a wide range of Reynolds number.
Keywords: turbulence, gas transfer velocity, air-water interface, combined turbulent time scalescaling relation based on SDM is formally consistent
with SEM in this case.

1. INTRODUCTION
The gas flux on the air-water interface is generally
expressed in the form of the bulk expression using the gastransfer velocity and the difference of dissolved gas
concentration. Based on the surface renewal theory, thegas transfer velocity is expressed by using a turbulent
time scale, replacing the renewal rate, as follows:

kL = C

DHow should we judge whether LEM and SEM scaling
rules are valid or not for turbulent fields we consider?
One idea is to investigate how the dimensionless gas
transfer velocity varies depending on the turbulent
Reynolds number. Figure 1 shows the experimental
results of oscillating grid turbulence provided byHere, the turbulent energy and energy
Mitani[3].
dissipation rate at the air-water interface used for thenormalization have been evaluated using the semiempirical relations for the oscillating grid turbulence
proposed by Matsunaga et al.[4]. The plots in the figure
also display the experimental values. The red and bluelines shows the LEM relation (2) and the SEM relation(3), respectively. From this figure, the experimental
values seem to agree with the LEM relation in the rangeof low Reynolds number, whereas the SEM relation
appears in the high Reynolds number. This fact suggeststhat the turbulent scale controling the gas transfer
changes with Reynolds number.

(1)twhere D, t and C are the diffusion coefficient, turbulent
time scale and a coefficient of proportionality. Therefore,
the formulation for the gas transfer velocity varies
depending on turbulent time scale in this equation (1).
As the turbulent scale controlling the variation of gas
transfer, there are two types of models well known,
which are Large eddy model with the eddies at theintegral length scale and Small eddy model with theeddies at micro scales[1].
The gas transfer velocity is described using eachturbulent time scale and its dimensionless form becomest =tL =k®

ene

kLSc1/2µ Ret -1/2
1/2
k(2)

In this study, based on knowledge about these turbulent
scales, we discuss the normalization of the gas transfervelocity at the air-water interface and the empiricalexpression for the gas transfer velocity applicable over awider range of the Reynolds number.

kLSc1/2(3)µ Ret -1/4
k1/2
where k, e, n, and A are the turbulent energy and theenergy dissipation rate at the air-water interface, thekinematic viscosity coefficient on liquid side, and thecoefficient of proportionality. Also, Sc denotes Schmidt
number, and the turbulent Reynolds number Ret isdefined byt = ts = A

Ret =k2en®

2. Empirical expression of turbulent time scale andgas transfer velocity
Following the method of Yoshizawa, the combined timescale for multiple turbulent time scales may be shown as1(4)tFrom these dimensionless relations, we found the gastransfer velocity of LEM and SEM to show the scalinglaws proportional to -1/2 or -1/4 to the turbulent
Reynolds number, respectively. In addition to LEM andSEM, there is the surface divergence model (SDM) using
the surface divergence calculated from horizontal flow
velocities on the air-water surface. Tsumori andSugihara[2] showed that SDM is equivalent to usingTaylor micro scale as the turbulent scale, and that the2=1t

2s+ CL1(5)t L2

where CL is an empirical constant. Therefore, theturbulent time scale t gives the following relations:

t=ts

=ts

(6)

1 + C1 R
1 + CL (t s t L )The relations of equations (2) and (3) are used for tL andts, and C1 is a new empirical constant. In equation (5),the combined time scale is defined in the form of the sumof inverse squares, but we consider an empirical relation,- 21 -2-1et1.E-01

1.E-01
LEM/2
1/k/2
1c
SLk/k/2
1cSLk

Experimental data

1.E-02
1.E+02

1.E+03LEMSEM
Experimental data
CTM (a=0.105)

/2
1SEM

1.E-02
1.E+02

Ret 1.E+04

1.E+03

Ret 1.E+04

Fig. 1. Relation of dimensionless gas transfer
velocity with turbulent Reynolds number.

Fig. 2. Comparison between empirical model andexperimental data.

so that it may not be necessary to express the turbulent
time scale by the sum of inverse square. In order to moregenerally express the time scale under consideration ofequation (6), we propose the relation as[2] Tsumori, H., Sugihara, Y., “Lengthscales of motions
that control air-water gas transfer in grid-stirred
turbulence”, J. Marine Systems, 66, 6-18, 2007.
[3] Mitani, S., “Universal description of gas transfer
mechanism at the air-water interface”, Master’s thesis,
Dept. Earth System Science and Technology,
Interdisciplinary Graduate School of Engineering
Scieces, Kyushu university, 2009. (In Japanese)[4] Matsunaga et al., “Quantitative properties ofoscillating-grid turbulence in a homogeneous fluid,Fluid Dynamics Res., 25, 147-165, 1999.
[5] Yoshizawa, A.，”Chapter 3 Turbulent Transports
and Reynolds-Averaged Modeling”, Nagare, 30,
241-261, 2011. (In Japanese)

t=ts

(7)

(1 + C Re )1- n /2t

1/ nwith a constant being n = 1, 2, 3,…, but it is a lack ofphysical meaning except for n = 1 and 2.
Substituting the combined time scale of equation (7) intoequation (1) and deriving the gas transfer velocity, we
obtain the following relation:
1/(2n )

- n /2
éæ Ret ö ùkL Sc1/2-1/4(8)= a Ret ê1 + ç÷ úRek1/2
êëè tc ø úû
where a is an empirical constant, and Retc is the critical

Reynolds number at which the turbulence scale changes.
The above relation has been expressed based on SEM,
and the selection of n = 2 corresponds to the solution ofequation (6). Here, figure 2 shows the comparison
between equation (8) in the case of n=6 and theexperimental results shown in figure 1. From this figure,the dimensionless representation of equation (8) appears
to fit over a wide range of the turbulent Reynolds number.
We should note that as n increases, the kink near thecritical Reynolds number becomes clearer and the degreeof agreement between LEM and SEM increases,
independent of the validity of the value of n.
3. Summary
The gas transfer velocity at the air-water interface has
been modeled and normalized using an empirical
combined time scale based on two turbulent time scales.
Though the theoretical background is somewhat sparse,
such an empirical expression can be expected to apply tothe behavior of the gas transfer velocity over a wide rangeof the turbulent Reynolds number.
4. REFERENCES[1] Sugihara, Y., Sanjou, M., “Turbulence and Gas
Transfer in Interfacial Hydraulics -Progresses ofExperiments and Medels-“, Nagare, 30, 181-193,
2011. (In Japanese)- 22 -