Nonlinear oscillations in a two-dimensional spatially periodic flow

The stability of a two-dimensional spatially periodic flow of a homogeneous fluid, named the Kolmogorov flow, is studied in the absence of viscosity, that is, for very large values of the Reynolds number. To solve the stability problem, the Galerkin method with three basis Fourier harmonics was used. A system of ordinary differential equations for the amplitudes of the Fourier harmonics is formulated. The solution of the nonlinear system for amplitudes is obtained in terms of Jacobi elliptic functions. It is shown that the exponential growth of linear perturbations of the inviscid Kolmogorov flow is replaced by the regime of stable nonlinear oscillations or vacillations. Such vacillations were first observed experimentally in R. Hide's well-known laboratory studies. An analytical expression is obtained for the period of arising nonlinear oscillations. It is shown that, as a result of the instability, a system of closed vortex cells is formed in a periodic flow.