^{1,a)}and Jing-Tang Yang

^{2,b)}

### Abstract

We developed a method to extract the energetically dominant flow features in a complicated fish wake according to an energetic point of view, and applied singular-value decomposition (SVD) to two-dimensional instantaneous fluid velocity, vorticity and (vortex-detector) data. We demonstrate the effectiveness and merits of the use of SVD through an example regarding the wake of a fish executing a fast-start turn. The energy imparted into the water by a swimming fish is captured and portrayed through SVD. The analysis and interpretation of complicated data for the fish wake are greatly improved, and thus help to characterize more accurately a complicated fish wake. The velocity vectors and Galilean invariants (i.e., vorticity and ) resulting from SVD extraction are significantly helpful in recognizing the energetically dominant large-scale flow features. To obtain successful SVD extractions, we propose useful criteria based on the Froude propulsion efficiency, which is biologically and physically related. We also introduce a novel and useful method to deduce the topology of dominant flowmotions in an instantaneous fish flow field, which is based on combined use of the topological critical-point theory and SVD. The concept and approach proposed in this work are useful and adaptable in biomimetic and biomechanical research concerning the fluid dynamics of a self-propelled body.

the National Science Council of the Republic of China partially supported this work under Contract Nos. NSC 96-2628-E-002-256-MY3 and NSC 96-2628-E-002-258-MY3. We thank Yu-Chun Lin for discussion on our work.

I. INTRODUCTION

II. EXPERIMENTS AND SVD TECHNIQUES

A. Fish

B. Kinematics and DPIV measurement setup

C. SVD

D. The SVD of velocity data

E. SVD of vorticity and data

III. RESULTS OF SVD ANALYSIS

A. SVD results of velocity

B. SVD results of vorticity

C. Kinetic energy and enstrophy preservation

D. SVD results of

IV. METHOD TO DEDUCE THE TOPOLOGY OF DOMINANT FLOWMOTIONS

A. Topological critical-point theory

B. Examples demonstrating utility of the method

V. DISCUSSION

A. SVD analysis of the complicated fish wake

B. Wake structure of fish fast-start turn

C. Circulation alteration

VI. CONCLUSIONS

### Key Topics

- Rotating flows
- 68.0
- Vortex dynamics
- 61.0
- Kinematics
- 28.0
- Critical point phenomena
- 26.0
- Vector fields
- 23.0

## Figures

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

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

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

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

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).

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).

(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.

(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.

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.

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.

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

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

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

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

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

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

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).

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).

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

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

(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.

(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.

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

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

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

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

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

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

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.

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.

(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.

(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.

(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.

(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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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

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

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

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