On the non-linear stability of the 1:1:1 ABC flow

Abstract ABC flows which can be considered as prototypes for the study of the onset of three-dimensional spatio-temporal turbulence are known both analytically and numerically to be linearly unstable. We analyze the nonlinear evolution of the ABC flow A 1 with A = B = C = 1 and with characteristic wavenumber k0 = 1 in the interval of Reynolds number 13 ≤ R ≤ 50. We solve numerically the forced Navier-Stokes equations with periodic boundary conditions for up to 9.9 × 104 eddy turnover times. Bifurcations towards progressively more complex flows obtain, with a relaminarization window, loss of symmetries, and chaotic oscillations probably revealing an underlying heteroclinic structure. In the chaotic regime, only three steady solutions emerge besides A1; they consist of a perturbed ABC flow A 2 with A = B ≠ C plus cyclic permutations. At 23 ≤ R ≤ 50 an unstructured temporal chaos is observed with the flow still dominated by the largest scales.

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