Three-dimensional stability of periodic arrays of counter-rotating vortices

We study the temporally developing three-dimensional stability of a row of counter-rotating vortices defined by the exact solution of Euler's equations proposed by Mallier and Maslowe [Phys. Fluids 5, 1074 (1993)]. On the basis of the symmetries of the base state, the instability modes are classified into two types, symmetric and anti-symmetric. We show that the row is unstable to two-dimensional symmetric perturbations leading to the formation of a staggered array of counter-rotating vortices. For long wavelengths, the anti-symmetric mode is shown to exhibit a maximum amplification rate at small wave numbers whose wavelengths scale mainly with the period of the row. This mode could be interpreted as due to the Crow-type of instability extended to the case of a periodic array of vortices. For short wavelengths, symmetric and anti-symmetric instability modes are shown to have comparable growth rates, and the shorter the wavelength, the more complex the structure of the eigenmode. We show that this short wavelength dynamic is due to the elliptic instability of the base flow vortices, and is well modeled by the asymptotic theory of Tsai and Widnall. The effect of varying the Reynolds number was also found to agree with theoretical predictions based on the elliptic instability.