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Research Papers: Fundamental Issues and Canonical Flows

# Coherent Structures in Shallow Water Jets

[+] Author and Article Information
A.-M. Shinneeb

Department of Mechanical and Materials Engineering, Queen’s University, 130 Stuart Street, Kingston, ON, K7L 3N6, Canadashinneeb@me.queensu.ca

J. D. Bugg

R. Balachandar

Department of Civil and Environmental Engineering, University of Windsor, 401 Sunset Avenue, Windsor, ON, N9B 3P4, Canadarambala@uwindsor.ca

J. Fluids Eng 133(1), 011203 (Jan 28, 2011) (14 pages) doi:10.1115/1.4003194 History: Received February 04, 2010; Revised December 06, 2010; Published January 28, 2011; Online January 28, 2011

## Abstract

This paper reports an experimental investigation of a round jet discharging horizontally from a vertical wall into an isothermal body of water confined in the vertical direction by a flat wall on the bottom and a free surface on top. Specifically, this paper focuses on the effects of vertical confinement on the characteristics of large vortical structures. The jet exit velocity was 2.5 m/s, and the exit Reynolds number was 22,500. Experiments were performed at water layer depths corresponding to 15, 10, and 5 times the jet exit diameter (9 mm). The large-scale structures were exposed by performing a proper orthogonal decomposition (POD) analysis of the velocity field obtained using a particle image velocimetry system. Measurements were made on vertical and horizontal planes—both containing the axis of the jet. All fields-of-view were positioned at an axial location in the range $10. The number of modes used for the POD reconstruction of the velocity fields was selected to recover $∼40%$ of the turbulent kinetic energy. A vortex identification algorithm was then employed to quantify the size, circulation, and direction of rotation of the exposed vortices. A statistical analysis of the distribution of number, size, and strength of the identified vortices was carried out to explore the characteristics of the coherent structures. The results clearly reveal the existence of numerous vortical structures of both rotational senses in the jet flow, and their number generally decreases in the axial direction while their size increases. The size of vortices identified in the vertical plane is restricted by the water depth, while they are allowed to increase in size in the horizontal plane. Moreover, the results show a significant decrease in the number of small vortices for the shallowest case in the horizontal plane, with a corresponding increase in the number of large vortices and a significant increase in their size. This behavior was accompanied with an increase in the vortex circulation in the horizontal plane and a reduction in the circulation in the vertical plane. This is indicative of the dominance of the pairing process due to shallowness. Moreover, the balance between the positive and negative vortices in the vertical plane changed because of the formation of negative (clockwise) vortices near the solid wall at downstream locations.

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## Figures

Figure 1

Schematic description of the apparatus

Figure 2

Variation in the correlation coefficient C′(ti,1000) of the velocity field at 1000 s with the velocity fields at all other times ti. For clarity, only the correlation values of 100 velocity fields before and after the velocity field at 1000 s are shown.

Figure 3

The effect of sample size M on the eigenvalue spectra for the first ten modes. The eigenvalue problem was calculated for ensembles of different sizes taken from a single data set.

Figure 4

Illustration of the effect of the threshold T on (a) the mean size Rmean and (b) the mean circulation Γmean of vortices. These mean values are calculated from the vortices that exist in 16-grid-unit intervals of the axial distance x indicated.

Figure 5

Examples of POD-reconstructed velocity fluctuation fields for the H/D=15 case: (a) vertical plane and (b) horizontal plane. The circles represent the size of identified vortices. Arrowheads indicate the direction of rotation.

Figure 6

Examples of POD-reconstructed velocity fluctuation fields for the H/D=5 case: (a) vertical plane and (b) horizontal plane. The circles represent the size of identified vortices. Arrowheads indicate the direction of rotation.

Figure 7

Grayscale contour that shows the distribution of the number of vortex centers that exist in eight-grid-unit intervals in both x and y (or z) directions calculated from 2000 velocity fields on (a) a vertical plane and (b) a horizontal plane of the shallowest case H/D=5. Note that y/D=z/D=0 represents the centerline of the jet.

Figure 8

Variation in the normalized number of vortices Ns/Nf in the axial direction x/D for the shallow jet cases on (a) the vertical plane and (b) the horizontal plane. The results are compared with the free jet case of Shinneeb (6).

Figure 9

The distribution of vortex size R in the axial direction x. Each figure represents data extracted from 2000 velocity fields on (a) a vertical plane and (b) a horizontal plane of the shallowest case (H/D=5). Note that positive R/D represents positive rotational sense.

Figure 10

Percentage of vortices for (a) the free jet and shallow jets on the vertical plane of depths; (b) H/D=15, (c) H/D=10, and (d) H/D=5. Each figure shows three profiles (i), (ii), and (iii) taken at different ranges of axial distances.

Figure 11

Percentage of vortices for the shallow jets on the horizontal plane of depths: (a) H/D=10 and (b) H/D=5. Each figure shows three profiles (i), (ii), and (iii) taken at different ranges of axial distances.

Figure 12

Variation in the normalized mean radius Rmean/D of vortices in the normalized axial direction x/D on (a) the vertical plane and (b) the horizontal plane

Figure 13

Distribution of vortex circulation Γ of the shallowest case (H/D=5) extracted from 2000 velocity fields on a vertical plane of two adjacent fields-of-view: (a) 28<x/D<44 and (b) 44<x/D<61. Note that positive Γ/DUe represents positive rotational sense.

Figure 14

Distribution of vortex circulation Γ of the shallowest case (H/D=5) extracted from 2000 velocity fields on a horizontal plane of two adjacent fields-of-view: (a) 29<x/D<45 and (b) 46<x/D<61. Note that positive Γ/DUe represents positive rotational sense.

Figure 15

Variation in the normalized mean circulation Γmean/DUe of vortices in the normalized axial direction x/D on (a) the vertical plane and (b) the horizontal plane

Figure 16

Distribution of normalized circulation Γ/DUe associated with the identified vortices of different sizes at two adjacent fields-of-view for the free jets (a) and (b), the shallowest jet case (H/D=5) on the vertical planes (c) and (d), and the shallowest jet case on the horizontal planes (e) and (f)

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