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Extracting energetically dominant flow features in a complicated fish wake using singular-value decomposition
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10.1063/1.3122802
/content/aip/journal/pof2/21/4/10.1063/1.3122802
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/4/10.1063/1.3122802

Figures

Image of FIG. 1.
FIG. 1.

Vortex pair and fluid jet shed by a fish into the wake.

Image of FIG. 2.
FIG. 2.

Apparatus for DPIV measurement; the high-speed DPIV camera records images from a ventral view.

Image of FIG. 3.
FIG. 3.

Schematic illustration of the sequential variations of body posture of a fish executing a fast-start turn. For detailed description of the kinematics of fish fast-start turn, we refer a reader to Wakeling (Ref. 22).

Image of FIG. 4.
FIG. 4.

(a) Instantaneous flow-velocity field and (b) the vorticity contour measured in the wake of a fish executing a fast-start turn. The schematic drawings (not to scale) at the top of the figure present a ventral view of the swimming fish. The gray rectangle represents the data scope; the black curved line depicts the moving trajectory of the trailing tip of the fish tail.

Image of FIG. 5.
FIG. 5.

SVD-extracted velocity vector fields from original data shown in Fig. 4(a): (a) rank 6 (85.7% energy preserved), (b) rank 4 (79.3% energy preserved), (c) rank 3 (75.2% energy preserved), and (d) rank 1 (49.1% energy preserved). The bold black arrows in (c) indicate the fluid jets generated by the fish.

Image of FIG. 6.
FIG. 6.

Embedded small-scale flow motions truncated from the original wake velocity data shown in Fig. 4(a) using rank 3.

Image of FIG. 7.
FIG. 7.

Instantaneous velocity vector field and its SDV-extracted equivalent (rank 3) at an instant 0.364 s after the fish began to turn.

Image of FIG. 8.
FIG. 8.

Instantaneous velocity field and its SDV-extracted equivalent (rank 3) at an instant 0.46 s after the fish began to turn.

Image of FIG. 9.
FIG. 9.

Vorticity contour maps resulting from SVD manipulation of the original data shown in Fig. 4: (a) Derived from SVD-extracted velocity data (rank 3), (b) vorticity-SVD (rank 6), (c) vorticity-SVD (rank 3), and (d) vorticity-SVD (rank 1).

Image of FIG. 10.
FIG. 10.

(a) Plots demonstrating the preservation of kinetic energy and (b) enstrophy vs rank number used for reconstruction.

Image of FIG. 11.
FIG. 11.

(a) A contour map and (b) its equivalent derived from SVD-extracted velocity data using rank 3 reconstruction. White circles in (b) mark the locations of the large-scale vortices.

Image of FIG. 12.
FIG. 12.

Maps of contour derived from -SVD using (a) rank 13 and (b) rank 3. White circles mark the locations of the large-scale vortices.

Image of FIG. 13.
FIG. 13.

Plot to demonstrate the preservation of -energy vs rank number used for reconstruction.

Image of FIG. 14.
FIG. 14.

Classification of critical points. and indicate the real parts of the eigenvalues of the Jacobian matrix; and indicate the imaginary parts.

Image of FIG. 15.
FIG. 15.

Sketch illustrating the trajectory of the trailing tip of the fish tail. The unit of the specified coordinates of the initial and terminal points of the trajectory is meter. Thick arrows indicate fluid jets generated by the fish tail.

Image of FIG. 16.
FIG. 16.

(a) A raw instantaneous fluid velocity vector field and (b) its corresponding SVD extraction, using rank-4 reconstruction with 80.5% energy preserved. The dashed curve in (a) depicts the trajectory of the trailing tip of the fish tail, with the arrow indicating the moving direction.

Image of FIG. 17.
FIG. 17.

(a) A raw instantaneous fluid velocity vector field and (b) its corresponding SVD extraction, using rank 3 reconstruction with 79.1% energy preserved. The dashed curve in (a) depicts the trajectory of the trailing tip of the fish tail, with the arrow indicating the moving direction.

Image of FIG. 18.
FIG. 18.

Streamline plots of the SVD-extracted velocity vector fields. (a) and (b) correspond to the vector fields shown in Figs. 16 and 17, respectively. The filled black circles indicate the centers of identified critical points. The trajectory of the trailing tip of the fish tail is also shown.

Image of FIG. 19.
FIG. 19.

Topology of the dominant flow motions. (a) and (b) correspond to the vector fields shown in Figs. 16 and 17, respectively. Filled black circles denote critical points. Black curves are principal streamlines, with the arrows indicating the flow direction. Symbols , , and with numeral subscripts denote foci, nodes, and saddle points, respectively.

Image of FIG. 20.
FIG. 20.

Plot to demonstrate the ratio of enstrophy for SVD-extracted velocity fields to that for the raw velocity field shown in Fig. 4 vs rank.

Image of FIG. 21.
FIG. 21.

Flow velocity vector fields smoothed with (a) Gaussian filter, standard deviation 0.65, (b) Gaussian filter, standard deviation 0.65, (c) moving-average filter, and (d) moving-average filter.

Image of FIG. 22.
FIG. 22.

2D and 3D sketches illustrating the wake structure of a fish executing a fast-start turn. (a) Moving trajectory of the trailing tip of the fish tail, partitioned into four portions designated , , , and . (b) 2D wake structure (from a ventral view) that comprises 2D vortices and jets. (c) 3D wake structure that comprises linked vortex rings and their central jets. The curved arrows adjacent to the vortex rings indicate the direction of vortex rotation.

Image of FIG. 23.
FIG. 23.

Plot to demonstrate how fluid circulation alters, with respect to flow-velocity fields reconstructed using ranks in a range 2–10. The ratio is defined as the reconstructed circulation divided by the original circulation.

Tables

Generic image for table
Table I.

Fraction of energy preservation for SVD reconstructions using ranks in a range 1–10.

Generic image for table
Table II.

Type and eigenvalue of each identified critical point (CP).

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/content/aip/journal/pof2/21/4/10.1063/1.3122802
2009-04-22
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Extracting energetically dominant flow features in a complicated fish wake using singular-value decomposition
http://aip.metastore.ingenta.com/content/aip/journal/pof2/21/4/10.1063/1.3122802
10.1063/1.3122802
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