^{1,a)}and Charles H. K. Williamson

^{1}

### Abstract

By using a recently developed numerical method, we explore in detail the possible inviscid equilibrium flows for a Kármán street comprising uniform, large-area vortices. In order to determine stability, we make use of an energy-based stability argument (originally proposed by Lord Kelvin), whose previous implementation had been unsuccessful in determining stability for the Kármán street [P. G. Saffman and J. C. Schatzman, “Stability of a vortex street of finite vortices,” J. Fluid Mech.117, 171–186 (1982)10.1017/S0022112082001578]. We discuss in detail the issues affecting this interpretation of Kelvin's ideas, and show that this energy-based argument cannot detect subharmonic instabilities. To find superharmonic instabilities, we employ a recently introduced approach, which constitutes a reliable implementation of Kelvin's stability ideas [P. Luzzatto-Fegiz and C. H. K. Williamson, “Stability of conservative flows and new steady fluid solutions from bifurcation diagrams exploiting a variational argument,” Phys. Rev. Lett.104, 044504 (2010)10.1103/PhysRevLett.104.044504]. For periodic flows, this leads us to organize solutions into families with fixed impulse *I*, and to construct diagrams involving the flow energy *E* and horizontal spacing (i.e., wavelength) *L*. Families of large-*I*vortex streets exhibit a turning point in *L*, and terminate with “cat's eyes” vortices (as also suggested by previous investigators). However, for low-*I* streets, the solution families display a multitude of turning points (leading to multiple possible streets, for given *L*), and terminate with teardrop-shaped vortices. This is radically different from previous suggestions in the literature. These two qualitatively different limiting states are connected by a special street, whereby vortices from opposite rows touch, such that each vortex boundary exhibits three corners. Furthermore, by following the family of *I* = 0 streets to small *L*, we gain access to a large, hitherto unexplored flow regime, involving streets with *L* significantly smaller than previously believed possible. To elucidate in detail the possible solution regimes, we introduce a map of spacing *L*, versus impulse *I*, which we construct by numerically computing a large number of steady vortex configurations. For each constant-impulse family of steady vortices, our stability approach also reveals a single superharmonic bifurcation, leading to new families of vortex streets, which exhibit lower symmetry.

I. INTRODUCTION

II. USING KELVIN's ARGUMENT TO ESTABLISH STABILITY OF SPATIALLY PERIODIC FLOWS

A. Inferring stability from energy comparisons

B. Stability of finite-area von Kármán streets from bifurcation diagrams

III. NUMERICAL METHOD

IV. STEADY FINITE-AREA STREETS AND THEIR STABILITY

A. High-impulse streets

B. Low-impulse streets

C. Near-zero and zero-impulse streets

D. The transition between high and low-impulse streets

V. THE SPACING-IMPULSE MAP

VI. DISCUSSION

VII. CONCLUSIONS

### Key Topics

- Rotating flows
- 154.0
- Vortex stability
- 55.0
- Bifurcations
- 38.0
- Vortex dynamics
- 21.0
- Flow instabilities
- 20.0

## Figures

Sketch of the finite-area von Kármán vortex street. The vortex spacing within each row is *a*, the distance between rows is *b*, and the stagger is given by *d*. All distances are measured between vortex centroids.

Sketch of the finite-area von Kármán vortex street. The vortex spacing within each row is *a*, the distance between rows is *b*, and the stagger is given by *d*. All distances are measured between vortex centroids.

Normalized translational velocity for solution families obtained by fixing the street aspect ratio *b*/*a* and varying area *A*/*L* ^{2}, as computed by Saffman and Schatzman.^{13,18,20} Large-area solutions with *b*/*a* ≲ 0.36 proved costly to resolve in their calculations, and were therefore not computed.

Normalized translational velocity for solution families obtained by fixing the street aspect ratio *b*/*a* and varying area *A*/*L* ^{2}, as computed by Saffman and Schatzman.^{13,18,20} Large-area solutions with *b*/*a* ≲ 0.36 proved costly to resolve in their calculations, and were therefore not computed.

Map of nondimensional area versus street aspect ratio, as computed by Saffman and Schatzman.^{13,18} For large *b*/*a*, the fold points in Fig. 2 correspond to the black curve, while the limiting states are approximated by the red curve (gray in print) on the right. For small *b*/*a*, these results suggested that no turning point was present, and that solutions directly approached a limiting state as *A*/*L* ^{2} increased. The dashed lines indicate expected trends, for regions where solutions were not computed.

Map of nondimensional area versus street aspect ratio, as computed by Saffman and Schatzman.^{13,18} For large *b*/*a*, the fold points in Fig. 2 correspond to the black curve, while the limiting states are approximated by the red curve (gray in print) on the right. For small *b*/*a*, these results suggested that no turning point was present, and that solutions directly approached a limiting state as *A*/*L* ^{2} increased. The dashed lines indicate expected trends, for regions where solutions were not computed.

Schematic diagrams, illustrating how turning points are related to changes of stability, following Ref. 24. (a) Velocity-impulse diagram (computed at constant spacing *L*), and (b) energy-spacing diagram (at constant impulse). Whether the stability boundary corresponds to a loss or gain of stability can be inferred directly from the shape of the diagram.

Schematic diagrams, illustrating how turning points are related to changes of stability, following Ref. 24. (a) Velocity-impulse diagram (computed at constant spacing *L*), and (b) energy-spacing diagram (at constant impulse). Whether the stability boundary corresponds to a loss or gain of stability can be inferred directly from the shape of the diagram.

Construction of a symmetry-breaking imperfection for the Kármán street. The flow in one periodic strip (of width *L**) is made nonsymmetric through a two-step process. The neighborhood of a stagnation point (highlighted by a gray box in (a)) is first altered by introducing a point vortex of strength Γ_{PV} < 0 (marked by the larger bull's eye in (b) and (c)), which changes the local flow topology. This creates two new stagnation points near the original one. One can then introduce a further point vortex at one of these locations (as shown in (c)), thereby breaking all geometric symmetries in the flow.

Construction of a symmetry-breaking imperfection for the Kármán street. The flow in one periodic strip (of width *L**) is made nonsymmetric through a two-step process. The neighborhood of a stagnation point (highlighted by a gray box in (a)) is first altered by introducing a point vortex of strength Γ_{PV} < 0 (marked by the larger bull's eye in (b) and (c)), which changes the local flow topology. This creates two new stagnation points near the original one. One can then introduce a further point vortex at one of these locations (as shown in (c)), thereby breaking all geometric symmetries in the flow.

Energy-spacing diagram for a Kármań street, for varying *L**, and fixed impulse *I** = 2^{−3/2} ≃ 0.3536. Re-computing the steady states after introducing an imperfection, we find that the solution family is broken into two distinct branches (shown by the dashed lines in (c)), revealing an additional loss of stability (marked in by the open circle in (c)).

Energy-spacing diagram for a Kármań street, for varying *L**, and fixed impulse *I** = 2^{−3/2} ≃ 0.3536. Re-computing the steady states after introducing an imperfection, we find that the solution family is broken into two distinct branches (shown by the dashed lines in (c)), revealing an additional loss of stability (marked in by the open circle in (c)).

Energy-spacing diagram for a Kármán street with impulse *I** = 0.1. For these small values of *I**, we find a bifurcated branch before the turning point in *L**.

Energy-spacing diagram for a Kármán street with impulse *I** = 0.1. For these small values of *I**, we find a bifurcated branch before the turning point in *L**.

(a) Plot of translational velocity versus spacing for a Kármań street with impulse *I** = 5 × 10^{−5}. (b) Bifurcation diagram for *I** = 0.

(a) Plot of translational velocity versus spacing for a Kármań street with impulse *I** = 5 × 10^{−5}. (b) Bifurcation diagram for *I** = 0.

Possible limiting states that we obtained for the Kármán street, for progressively decreasing impulse: (a) “cat's eyes” (typical of large-impulse vortices), (b) “pie slices”, (c) “teardrops”, and (d) “bowling pins” (which occur for small impulse). Other limiting states of lower symmetry also exist.

Possible limiting states that we obtained for the Kármán street, for progressively decreasing impulse: (a) “cat's eyes” (typical of large-impulse vortices), (b) “pie slices”, (c) “teardrops”, and (d) “bowling pins” (which occur for small impulse). Other limiting states of lower symmetry also exist.

Spacing-impulse solution map, constructed using the results of Saffman and Schatzman.^{18} (This resembles quite closely the (*b*/*a*, *A* ^{2}/*L*) map shown earlier in Fig. 3.)

Spacing-impulse solution map, constructed using the results of Saffman and Schatzman.^{18} (This resembles quite closely the (*b*/*a*, *A* ^{2}/*L*) map shown earlier in Fig. 3.)

Spacing-impulse solution map, summarizing the results for the (symmetric) finite-area Kármán street obtained in this paper. Dashed curves denote bifurcations to lower symmetry solutions. Darker regions correspond to values of (*I**, *L**) for which a greater number of solutions exist. Vertical slices through this map correspond to one-parameter families, as illustrated by the basic curves in the energy-spacing plots of Figs. 6 and 7 (which show cases for *I** = 0.1 and 2^{−3/2} ≃ 0.3536). Horizontal slices through the map correspond to one-parameter families that can be represented in a velocity-impulse diagram. Over 30,000 steady states were calculated to construct this map.

Spacing-impulse solution map, summarizing the results for the (symmetric) finite-area Kármán street obtained in this paper. Dashed curves denote bifurcations to lower symmetry solutions. Darker regions correspond to values of (*I**, *L**) for which a greater number of solutions exist. Vertical slices through this map correspond to one-parameter families, as illustrated by the basic curves in the energy-spacing plots of Figs. 6 and 7 (which show cases for *I** = 0.1 and 2^{−3/2} ≃ 0.3536). Horizontal slices through the map correspond to one-parameter families that can be represented in a velocity-impulse diagram. Over 30,000 steady states were calculated to construct this map.

Examples of possible solution paths (shown by the blue lines) that can be obtained by starting with solutions having *I** = 0 and *L** < 1.0418, and increasing *I** (subsequently, one may try to also vary *L**, as shown by the vertical path). In this map, some of these steady vortices overlap with other flows previously shown in this paper. Paths (1) and (2) are described in the text.

Examples of possible solution paths (shown by the blue lines) that can be obtained by starting with solutions having *I** = 0 and *L** < 1.0418, and increasing *I** (subsequently, one may try to also vary *L**, as shown by the vertical path). In this map, some of these steady vortices overlap with other flows previously shown in this paper. Paths (1) and (2) are described in the text.

Sketch of the solution structure in the neighborhood of the point (*I**, *L**) = (0, 1.0418) (marked by an open circle in this and in the previous two figures). The steady states lie on a fold in the (*I**, *L**, *U**) space. The yellow region (light-gray in print) forms part of the map in Fig. 11. With the exception of a few results presented in Fig. 12, the blue region (dark-gray in print) remains essentially unexplored.

Sketch of the solution structure in the neighborhood of the point (*I**, *L**) = (0, 1.0418) (marked by an open circle in this and in the previous two figures). The steady states lie on a fold in the (*I**, *L**, *U**) space. The yellow region (light-gray in print) forms part of the map in Fig. 11. With the exception of a few results presented in Fig. 12, the blue region (dark-gray in print) remains essentially unexplored.

Sketch of the vortex shapes and variable definition for the computation of the “pie slice” vortices.

Sketch of the vortex shapes and variable definition for the computation of the “pie slice” vortices.

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