The sinh-Poisson equation describes a stream function configuration of a stationary two-dimensional (2D) Euler flow. We study two classes of its exact solutions for doubly periodic domains (or doubly periodic vortex arrays in the plane). Both types contain vortex dipoles of different configurations, an elongated “cat-eye” pattern, and a “diagonal” (symmetric) configuration. We derive two new solutions, one for each class. The first one is a generalization of the Mallier–Maslowe vortices, while the second one consists of two corotating vortices in a square cell. Next, we examine the dynamic stability of such vortex dipoles to initial perturbations, by numerical simulations of the 2D Euler flows on periodic domains. One typical member from each class is chosen for analysis. The diagonally symmetric equilibrium maintains stability for all (even strong) perturbations, whereas the cat-eye pattern relaxes to a more stable dipole of the diagonal type.
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September 2004
Research Article|
September 01 2004
Vortex arrays for sinh-Poisson equation of two-dimensional fluids: Equilibria and stability Available to Purchase
D. Gurarie;
D. Gurarie
Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106
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K. W. Chow
K. W. Chow
Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong
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D. Gurarie
Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106
K. W. Chow
Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong
Physics of Fluids 16, 3296–3305 (2004)
Article history
Received:
November 12 2003
Accepted:
May 21 2004
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Comment on “Vortex arrays for sinh-Poisson equation of two-dimensional fluids: Equilibria and stability” [Phys. Fluids 16, 3296 (2004)]
Citation
D. Gurarie, K. W. Chow; Vortex arrays for sinh-Poisson equation of two-dimensional fluids: Equilibria and stability. Physics of Fluids 1 September 2004; 16 (9): 3296–3305. https://doi.org/10.1063/1.1772331
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