RECIRCULATING FLOW IN A SHALLOW BASIN

Altai W., Shafie M. and Chu, V. H.

Department of Civil Engineering and Applied Mechanics,

McGill University

817 Sherbrooke W., Montreal, Canada, H3A 2K6

Tel: 514-398-6863, Fax: 514-398-7361, E-mail: chu@civil.lan.mcgill.ca

Abstract: Recirculating flow in a shallow basin was studied experimentally for friction effect on turbulent exchange process. Dye was introduced into the basin during the experiments. Measurements of the dye concentration were made using a video imaging method. The turbulent intensities and the retention times in different regions of the recirculating flow were determined for a range of Froude number Fr = U/(gh)^1/2 varying from 0.14 to 0.68 and a range of bed-friction number S = c_f L/2h from 0.05 to 0.13. A core of relatively non-turbulent flow was observed to occupy the central region of the basin where the retention time scale was significantly greater than the rest of the basin. In the region outside of the core, the exchange process was governed by the circulation of eddies around the perimeter of the basin. The turbulent structure of the flow in small water depth was more coherent than the flow of greater water depth. However, the dimensionless mass-exchange flow rate was not significantly dependent on the degree of coherency of the turbulent motion.

Keywords: recirculating flow, turbulent mixing, coherent structure, bed-friction effect, open-channel flow

1    INTRODUCTION

Recirculating flows are observed in bays and harbors, and behind flow obstacles such as sand bars and spur dikes, when currents are separated from the rugged coast (Schmidt, 1990). Pollutants and sediments drawn into the recirculating region are likely to be trapped there because the velocity and the turbulent intensity in the region are low compared with the current along the coast. The rate of deposition of the particulate and flocculent matters in the region is directly related to the time available for the deposition.

In the present experimental investigation, the recirculating flow was produced in a square basin opened on one side to an open-channel main flow (see Figure 1). Dye was introduced at a point in the boundary layer upstream of the basin for a period of time. The supply of the dye was terminated after the dye concentration in the basin reaches a quasi-steady state. The subsequent reduction of the dye concentration in the basin is due to the exchange by the mixing layer between the recirculating flow in the basin and the main flow outside the basin.  The transient variation of the dye concentration in the basin was monitored by a video camera during the experiments. Figures 2a and 2b show the dye concentration distribution in the basin of two tests under different conditions. The dye in the basin is exchanged with the main flow through a mixing layer located on the edge of the basin between the recirculating flow in the basin and the main flow as shown in Figure 2c. Depending on the mixing layer and its interaction with the flow in the basin, the turbulent flow may be more coherent leading to higher exchange, or less coherent leading to lower exchange, of the dye in the basin with the main flow. A number of factors are affecting the structure of the turbulent motion and the rate of the exchange. These include the depth of the water in the basin and the thickness of the boundary layer in the main flow. The dimensionless parameters of significance are the Froude number, Fr = U/(gh)^1/2, the bed-friction number, S = c_fL/2h, and boundary layer thickness ratio, q/L. Two series of experiments were conducted. One in a large square basin of 89 cm by 89 cm and the other in a smaller square basin of 32 cm by 32 cm. The momentum thickness of the boundary layers was 0.475 cm and 0.25 cm for the large and the small basins, that is L/q = 187 and 128, respectively. The velocity and the water depth of the open-channel were varied to produce a range of gravity and friction effects covering a range of Froude number, Fr = U/(gh)^1/2, varying from 0.14 to 0.68 and a range of bed-friction number, S = c_f L/2h, varying from 0.05 to 0.13. Table 1 summarizes the conditions of the experiments.

2    EXPERIMENTAL RESULTS

The flow in the square basin is observed to be consisted of at least two main regions:   (1) a core region of essentially irrotational motion, and (2) an outer region of eddies circulating around the perimeter as shown in Figure 2. A transition region between the core and the outer region was also observed. At the leading edge of the mixing layer, waters from the side of the basin and from the side of the main flow are drawn into the mixing layer through the process known as turbulent entrainment. The mixed fluid in the form of a series of eddies then impinges onto the downstream edge of the basin. Part of the impinging eddies enters the basin and part follows the main flow. Eddies that enter the basin, move around the core region of the basin and eventually are re-entrained back into the mixing layer. The dye was removed first from the outer region by the circulation of eddies around the core and then from the core through a much slower process. The retention time in the core is significantly greater than the retention time in the outer region.

Figure 3a and 3b shows the retention times, t_o and t_c, obtained in the outer and core regions, -respectively.

To study the two stages of mass exchange between the main flow and the outer region, and then between the outer region and the core, the basin is divided into ten non-overlapping concentric strips, as shown in Figure 2d. Average dye concentration, , was obtained over the area of each strip. The variations of the average dye concentration with time are shown in Figure 4 in semi-logarithmic scale. The data of are plotted with dimensionless time tU/L. Ten sets of data are plotted. They are separated in the figure by a shift in time equal to . From left to right, the profiles of are obtained in strip number 1 to strip number 10. Strip number 1 is located in the very core and strip number 10 in the perimeter of the basin. The time scale of the decay is the retention time , that is the inverse slope of the data on the semi-logarithmic plot. The reduction of the dye concentration in the outer region of the basin was initially rapid and then slow down. The reduction in the core was the opposite. In the core, the reduction rate was initially slow but accelerated toward a higher rate in the final stage. Two sets of straight lines corresponding to the initial and the final stages of the reduction are fitted with the data in Figure 4. The solid lines fit the data in the final stage. The dash lines fit only the data of the initial stage (strips 7, 8, 9 and 10). The dimensionless retention times for the initial and final stages are  74.1 and  47.6, respectively. The rapid removal of the dye can be related to the circulation of eddies in the outer region whereas the final removal of the dye from the basin is controlled by the relatively slower release of the dye from the core. The retention times,  and , characterizes the slow rate of exchange in the core region and the fast rate of exchange in the outer recirculating region, respectively.

Figure 3a shows the dimensionless retention time, , obtained from the outer region. The mean and the standard deviation of the fifteen tests are . The data are scattered with its standard deviation equal to 32% of its mean value. Figure 3b shows the retention times, , obtained from the core. The mean and standard deviation of the fifteen tests are . The standard deviation in this case is 27% of its mean. The variation of the data from its mean value are quite large compared with the curve fitting errors of the data shown in Figure 4. The 27% to 32 % deviations from the mean values can not be attributed to measurement errors. We have to conclude that the variations in the retention time were due to other effects related to parameters such as bed-friction number, S = c_f L/2h, and boundary-layer thickness ratio, q/L. It is not entirely clear at this stage of the investigation how these friction and boundary-layer effects are relate to the exchange process.  Further analysis of the data would be necessary to gain better understanding of the mass-exchange processes in the recirculating flow.

3    CONCLUSION

Despite the scatter of the data, the experimental results have shown clearly the existence of a core region in the recirculating flow where the retention time is significantly greater than the outer region surrounding the core. This finding is helpful in understanding the pattern of sediment erosion and deposition along the coast. The data would be generally useful in the development of mathematical models for transport processes in inland and coastal waterways.

References

[1]    Altai W. and Chu V.H. (1997) “Retention Time in Recirculating Flow”, Proc. 27^th IAHR Congress, Vol. B1, pp. 9-14.

[2]    Booij. R. (1989) “Exchange of mass in harbours”, Proc. 23th IAHR Congress, Vol. D, pp. 69-74.

[3]    Langendoen, E. J., Kranenburg, C. and Booij, R. (1993) “Flow patterns and exchange of matter in tidal harbours”, J. Hydraulic Res., Vol. 32, pp. 259-270.

[4]    Schmidt, J. C. (1990) “Recirculating flow and sedimentation in the Colorado River in Grand Canyon, Arizona”, J. of Geology, AGU, Vol. 98, pp. 709-724.

   Table 1    Summary of the test conditions and results

Fig. 1    The dye concentration distribution in the basin as determined by the video imaging method. The concentration reduces with time after the supply of the dye to the basin from an upstream point in the boundary layer is terminated. The flow direction is as indicated by the arrow in the top-left-hand corner of the figure.

Fig. 2    Schematic diagram of the recirculating flow showing the core and the circulation of eddies in the outer region.

Fig. 3    Retention time associated with a) the rapid exchange process in the outer circulation region, b) the slow process in the final stage of the concentration decay.

Fig.4     Semi-logarithmic profiles of the dye concentration, , obtained in the ten             non-overlapping regions; data are from test 2-1, 4-1, 7-2 and 7-3, respectively. The  profiles are separated in the figure by a shift in time equal to .