TRANSVERSE MIXING COEFFICIENTS OF DENSITY CURRENTIN COMPOUND CHANNEL

Hirokazu Fujiwara¹ and Masaki Sawamoto²

¹Department of Civil and Environmental Engineering,

Hachinohe National College of Technology,

Uwanotai 16-1, Tamonoki, Hachinohe 039-1192, Japan

Phone and Fax  +81-178-27-7311, E-mail fujiwara-z@hachinohe-ct.ac.jp

²Department of Civil Engineering, Tohoku University,

Aoba-yama 06, Sendai 980-8579, Japan

Phone and Fax  +81-22-217-7457, E-mail sawamoto@kaigan.civil.tohoku.ac.jp

Abstract: The Takase River estuary in Aomori Prefecture in Japan is a tidal river with the compound channel. The object of study is to understand mixing characteristics of salt water in compound channel reaches of the river estuary. This paper describes the comparison of the transverse mixing coefficients in compound channel with those in single section channel, and the consideration about the coefficient calculated from the field measurement data. Laboratory experiments and field measurements were carried out for this river. In the compound channel, the transverse velocity shear is produced because of the water depth difference between the main channel and the flood channel. The transverse mixing coefficients were obtained from these results using diffusion equation. In the experiments, the values of k_y/(u_*h) in compound channel were about 10 times of those in single section channel (k_y: diffusion coefficient, u_*: shear velocity, h: water depth). It was found that values of k_y/(u_*h) required from field measurement data were larger than those got from experimental data. This result agreed with the conventional researches. By this study, it was found that the mixing coefficient widely varied its values depended on channel cross-sectional shape, furthermore, k_y/(u_*h) of density current in compound channel was related to the product  and d/H ( : relative water level difference of fresh water and salt water, d: water depth of flood channel, H: water depth of main channel).

Keywords: tidal portion, compound channel, density current, diffusion coefficient

1    INTRODUCTION

The tidal river, where marine and freshwater coexist, has rich ecosystem. The precious natural environment has to be maintained. In the regional development, a balance of development and preserving of natural resource should be considered. The object of the present study is to understand mixing characteristics of salt water in compound channel reaches of the Takase River estuary.  The transverse mixing coefficients were calculated from both the laboratory experiments^{1), 2)} and the field measurement³⁾. Fischer⁴⁾ reported the conventional studies which transverse mixing coefficients of open channel calculated from experiments and field measurements. In the compound channel, the transverse velocity shear is produced because of the water depth difference between the main channel and the flood channel. Then, the salinity difference occurs by diffusive transportation. Therefore, it is anticipated that the transverse mixing coefficients of the compound channel differs from those of the single section channel.

Fig. 1    Map of Lake Ogawara and Takase River

Fig. 2    Cross section of compound channel reaches of Takase River(T.P. is Tokyo Bay mean sea level)

Figure 1 shows the map of the Lake Ogawara and the Takase River estuary in Aomori Prefecture, Japan. The marks, , , , A-0~A-5, B-0~B-5, C-0~C-4, D-0~D-5, and E-0~E-4, are the observation points. The river estuary connects the Pacific Ocean with the lake. The drainage area of the river is 866.9km², and the length of the channel is 63.7km. The surface area of the lake is 63.2km², and the average water depth is 11m. There is the flood-way at about 5.7km upstream from the river mouth. The salt water intrudes through the Takase River in ordinary time. Figure 2 shows the cross-section of the compound channel reaches in Fig.1.

2    CALCULATION METHOD OF TRANSVERSE MIXING COEFFICIENTS

2.1    Fundamental equation

When the water depth of flood channel in compound channel is shallow, we can calculate the transverse mixing coefficient as two-dimensional plane. Compared with u, v is very small, therefore, v is able to eliminate (where u is flow velocity of mainstream direction x, and v is flow velocity of transverse direction y.). In x direction, we considered that the salt water was only transported by the mainstream. When k_y (y direction diffusion coefficient) is constant, the diffusion equation is

   (1)

where t is time, and c is salinity.

2.2    Outline of experimental method

Figure 3 shows the sketch of experimental channel. This was made on the model of cross section in compound channel reaches of the Tkase River estuary. The freshwater tank, which corresponds to the lake, and the saltwater tank which corresponds to the sea, were installed in the both ends. When the gate in the saltwater tank side is opened, the salt water is intruded.

Fig. 3    Sketch of experimental channel

The distance of the downstream direction from the gate position is x, and the distance of the lateral width from the central axis of the channel is y. The relative density difference between fresh water and salt water is 0.002. The floor is horizontal. After the water levels of the tanks of the both ends are adjusted to the experimental conditions in each CASE (Table 1), the gate is opened to full width in the fixed time. The measurement points of the salinity, No.1~No.8 at both x=141cm and x=365cm, are shown in Fig.3(c). The surface velocities of No.2~No.8 are measured. The experiments in the single section channel were carried out similarly by closing the main channel in the compound channel.

                   Table 1    Experimental conditions

2.3    Calculation method

Equation (1) was replaced with finite difference equation, and k_y was calculated using an altering direction implicit (ADI) method from experimental data. The calculation was carried out for the zone of x=141~365cm and y=2~10cm on flood channel. The boundary conditions were the experimental data at x=141cm, y=2cm, and y=10cm. Figure 4 shows the conceptual sketch of the calculating area. k_y is arbitrary, and the salinity at x=365cm is calculated. The correlation coefficient r between these calculated and the experimental values are determined. By increasing k_y, r becomes the maximum point, which is the k_y of the CASE. The k_y of the field is calculated by the similar method.

Fig. 4    Calculating area

3    RESULTS AND DISCUSSIONS

3.1    Mixingcoefficients in laboratory experiments

The experimental conditions are shown in Table 1. Both CASE1 and CASE2 are compound channel, while both CASE3 and CASE4 are single section channel. CASE1, the water-level difference between salt water and fresh water is small, and CASE2, that is large. The example of these experimental results is shown in Fig.5. In the figure, (a) is the time series of surface velocity and relative concentration (C/C₀) at x=141cm, and (b) is at x=365cm. The sampling time interval was 0.2sec. C/C₀ is ratio of Cl^- concentration at the point to the initial value of the salt-water tank. Comparison of the time difference of C/C₀ between channel center and the sidewall, x=365cm was larger than x=141cm, because velocity u depended on y.

Fig. 5    Experimental results of CASE1

The mixing coefficients were calculated using the previous-mentioned time series. The experimental data used for the calculation were No.4~No.8 velocities at x=141cm, the C/C₀ of No.4 and No.8 at x=365cm, and average velocities of the two cross section. The C/C₀ of No.4~No.8 at the x=365cm was calculated by the increasing k_y one after another. The r between cross sectional mean value of the calculated C/C₀ and that of the experimental data was calculated. When r is a maximum, k_y is a solution. The time step t=0.8sec, and the grid sizes x=1.0cm and y=0.2cm. Figure 6 shows the time series of the experimental mean value of the C/C₀ at x=365cm and those of the calculation. In CASE1, k_y=3.4cm²/sec and r=0.981. Figure 7 shows those in CASE3, and the k_y was 0.3cm²/sec. For CASE2 and CASE4, the values of k_y were 2.1cm²/sec and 0.3cm²/sec respectively. Transverse mixing coefficients in the experiments are shown in Table 2. I is surface slope, and R is hydraulic radius, u_* is shear velocity, h is hydraulic depth, and k_y is transverse diffusion coefficient. In the experiments on the uniform channel in the oscillatory flow, Sumer and Fischer⁵⁾ reported that the k_y/(u_*h)=0.06~0.27. The values in CASE3 and CASE4 (single section channel) agreed with the result. However, the values in CASE1 and CASE2 (compound channel) were 2.06 and 0.96, and those were larger than their result. Comparison with the values of single section channel and compound channel, CASE1 was 12.8 times of CASE3, and CASE2 was 10.6 times of CASE4.

Fig. 6    Comparison between experimental value and calculated value (CASE1)

Fig. 7    Comparison between experimental value and calculated value (CASE3)

    Table 2    Transverse mixing coefficients in

3.2    Transverse mixing coefficient in field

The salinity was measured at , , and  points and at A, B, C, D, and E traverse lines in Fig.1. The sampling time interval was 1 minute. The calculation used the data of A and B traverse lines on August 8, 1999³⁾. Figure 8 shows the horizontal salinity distribution on this day. In the calculation, the mean velocity of the shoal in A and B traverse lines was used. A traverse line is correspondent to the experimental x=141cm, and the B line is correspondent to the x=365cm. Mean velocity of the shoal was calculated by using the water levels of the river mouth and the lake. In the calculation, t=60sec, x=5m, and y=3m. The k_y was 0.08m²/s, the u_* was 0.0155m/s, the mean of h was 0.831m, and the k_y/(u_*h) was 6.21. It became 3~6 times larger than the values calculated from the experimental data of the compound channel. Yotsukura et al.⁶⁾ reported that the k_y/(u_*h) was 0.60 in field of the Missouri River, which was 2~10 times larger than the experimental results in the laboratory by Sumer and Fischer⁵⁾. In the field, the mixing coefficient tends to larger than that of the laboratory level. Those results agreed with this study. In addition, the dimensionless mixing coefficients were 0.4~1.0 in the observation of oscillatory flow by Ward⁷⁾. The value of the shoal of the Takase River was 6.2~15.5 times lager than value by Ward. Table 3 shows the dimensionless mixing coefficients by conventional studies and by this study.

    Table 3    Transverse mixing coefficients by conventional studies and these results

Sumer and Fischer[5]	Laboratory, uniform channel		0.06~0.27
Ward[7]	Field, Cordova Bay, Gironde Estuary		0.4~1.0
Yotsukura et al.[6]	Field, Missouri River		0.6
This results	Lab. (mean value)	Single section	0.125
		Compound	1.51
	Field	Takase River	6.21

3.3    Relationship between field and experimental model

Okoye⁸⁾ reported that k_y/(u_*h) was a function of channel width W and water depth h. Okoye’s study agreed with the single section channel but not with the compound channel. Generally, characteristics of the cross sectional shape of the compound channel flow are expressed in the parameter of b/W, b/H, and d/H. Where b is the main channel width, W is the channel width, H is the water depth of main channel, and d is the water depth of flood-channel. The small change of these parameters largely affects the flow.  as the parameter of the density current and d/H as the parameter of the compound channel were introduced, where  is the relative water-level difference of fresh water and salt water. Figure 9 shows the relationship between ∙d/H and k_y/(u_*h). The regression formula in compound channel is

               (2)

The value of the field agreed with this equation in Fig.9.

4    CONCLUSIONS

The main findings got in this study are as follows.

(1) In the experiment of single section channel, the values of k_y/(u_*h) were 0.06~0.27 in the range of the conventional researches. (2) In the laboratory level, the values of k_y/(u_*h) in the flood-channel reaches of the compound channel were about 10 times of those in the single section channel. (3) The value of k_y/(u_*h) calculated from the field measurement data of the Takase River was larger than the values of the laboratory level, and it was 6.2~15.5 times of result by Ward. It was considered that this was the effect by the compound channel. (4) The k_y/(u_*h) of density current in compound channel was related to the product of  and d/H.

Acknowledgements

The authors wish to acknowledge the advices and helps by Profs. Mano and Tanaka, Department of Civil Engineering, Tohoku Univ., Prof. Ishikawa, Department of Environmental Science and Technology, Tokyo Inst. of Technology, and Dr. Nishida, Department of Civil Engineering, Osaka Univ.. Cooperation of Takase River Development Construction work office, Tohoku Regional Construction Bureau is thanked.

References

[1]    Fujiwara, H., Sawamoto, M. and Kamiyama, N.: Mixing Characteristics of Estuarine Density Current in Compound Channel, Proceedings of Coastal Engineering, JSCE, 42, pp.416-420, 1995. (in Japanese)

[2]    Fujiwara, H., Sawamoto, M. and Tanaka, H.: Comparison of Mixing Characteristics of Density Current between in Compound Channel and Single Section Channel, Annual Journal of Hydraulic Engineering, JSCE, 41, pp.515-520, 1997. (in Japanese)

[3]    Fujiwara, H., Ishikawa, T., Nishida, S., Tsuruta, Y. and Sawamoto, M.: Characteristics of salt water intrusion in compound channel of the Takase River, Annual Journal of Hydraulic Engineering, JSCE, 44, pp.1005-1010, 2000. (in Japanese)

[4]    Fischer, H. B.: Longitudinal dispersion and turbulent mixing in open-channel flow, Ann. Rev. Fluid Mech., Vol.5, pp.59-78, 1973.

[5]    Sumer, S. M. and Fischer, H. B.: Transverse mixing in partially stratified flow, J.Hyd.Div., Proc.ASCE, Vol.103, HY6, pp.587-600, 1977.

[6]    Yotsukura, N., Fischer, H. B. and Sayre, W. W.: Measurement of Mixing Characteristics of the Missouri River Between Sioux City, Iowa, and Plattsmouth, Nebraska, U.S.Geol.Survey Water-Supply Paper, 1899-G, 1970.

[7]    Ward, P. R. B.: Transverse dispersion in oscillatory channel flow, J.Hyd.Div., Proc.ASCE, Vol.100, HY6, pp.755-772, 1974.

[8]    Okoye, J. K.: Characteristics of transverse mixing in open-channel flows, Rep.KH-R-23, W. M. Keck Lab., Cal. Inst. Tech., 1970.