A turbulence model in unbounded smooth shear flows: The weak turbulence approach

Abstract

We discuss a new concept of the subcritical transition to turbulence in unbounded smooth (noninflectional) spectrally stable shear flows. This concept (the so-called bypass transition) follows from considering the nonnormality of the linear dynamics of vortex disturbances in shear flows and is most easily interpreted by tracing the evolution of spatial Fourier harmonics (SFHs) of the disturbances. The key features of the concept are as follows: the transition of the flow by only finite-amplitude vortex disturbances despite the fact that the phenomenon is energetically supported by a linear process (the transient growth of SFHs); the anisotropy of processes in the k space; the onset of chaos due to the dynamical (not stochastic) process—nonlinear processes that close the transition feedback loop by the angular redistribution of SFHs in the k space. The evolution of two-dimensional small-scale vortex disturbances in a parallel flow with a uniform shear is analyzed within the weak turbulence approach. This numerical test analysis is carried out to prove the most problematic statement of the concept, the existence of a positive feedback caused by the nonlinear process. Numerical calculations also show the existence of a threshold: if the amplitude of the initial disturbance exceeds the threshold value, the self-maintenance of disturbances becomes realistic. The latter is a characteristic feature of the flow transition to the turbulent state and its maintenance.