Abstract: Spatiotemporal cascades of exposed and hidden
perturbations of the Couette flow are considered in
the range of Reynolds numbers A cascade model of exposed
perturbations, which are produced by the asymmetrically moving channel walls, is
based on the non-orthogonal expansion into the spatial
Taylor
submodes and temporal harmonic submodes. Three alternative approaches to the
cascade description of the exposed perturbations through a spatial cascade, a
temporal cascade, and a spatiotemporal cascade are discussed and compared. At
high Reynolds numbers, the exposed perturbations form two forced viscous sublayers on the moving walls, a free
viscous sublayer at a symmetry center of the Couette flow, and two multiscale
cascades of coherent structures, which spread as transversal waves in the core
of the Couette flow towards the symmetry center and compensate there each other.
A cascade model of hidden perturbations, which initially vanish with any
prescribed tolerance, is constructed by hyperbolic temporal submodes and
asymmetric spatial harmonic submodes. The temporal cascade of the hidden
perturbations results in a dynamic model of the spatial cascade, which is
reduced to an open-ended algebraic model for tensor coefficients of matrix
amplitudes of the hidden perturbations and a quantization condition, connecting
the excitation-relaxation parameters of the temporal cascade of coherent
structures with the wave numbers of their spatial cascade. At high Reynolds
numbers, the hidden perturbations model a uniform stochastic velocity profile. A
spontaneous emergence of the hidden perturbations is also justified by an
asymptotic Hamiltonian approach.
Keywords and phrases: multiscale cascades, the Couette flow, transition, coherent structures, exposed and hidden perturbations.
