Turbulence onset is considered in viscous incompressible flows. Development of fluctuations in an infinite flow with a constant gradient of mean velocity ∂U1/∂x2 = const., U2 = U3 = 0 is rigorously treated.It is shown that two-dimensional eddy fluctuations, with infinitesimal initial amplitude A and scale of initial eddies l, increase in this flow so that the maximum ratio max ε(t)/ε(0) of their energy at the moment t to the initial energy exceeds any prescribed value as the Reynolds number R = (∂U1/∂x2) l2/v increases. The analysis of the non-linear equations obtained in the paper which describe development of fluctuations with a finite amplitude leads to the conclusion that there exists a’ stability barrier’ Ã (R) for the initial amplitude of eddy fluctuations. If A < Ã (R), then fluctuations decay as t → ∞, and if A > Ã(R) the energy of fluctuations does not decay. As R → ∞, Ã(R) → 0 according to the inequalities \[ l(\partial U_1/\partial x_2)K_2/R^{\frac{2}{3}}\leqslant \tilde{A}(R)\leqslant l(\partial U_1/\partial x_2)\,K_1/R^{\frac{2}{3}}. \] It is shown that the non-linear mechanism of preventing turbulence from decay involves generation of large-scale turbulent oscillations which then transmit energy to small-scale motions.The described mechanism of turbulence onset from small eddies in shear flows appears to be of universal character. It is interesting that several qualitative characteristics of turbulence observed in various shear flows can be rigorously deduced even in a model where disturbances remain two-dimensional.

Hasen, E. M. (1967). A non-linear theory of turbulence onset in a shear flow. Journal of Fluid Mechanics, 29(4), 721–729.