Abstract
The stability of flows in layers of finite thickness H is examined against small-scale three-dimensional (3D) perturbations and large-scale two-dimensional (2D) perturbations. The former provide an indication of a forward transfer of energy while the latter indicate an inverse transfer and the possibility of an inverse cascade. The analysis is performed using a Floquet-Bloch code that allows examination of the stability of modes with arbitrary large-scale separation. For thin layers, the 3D perturbations become unstable when the layer thickness H becomes larger than H>c1(νℓU/U)1/2=c1ℓURe−1/2, where U is the rms velocity of the flown, ℓU is the correlation length scale of the flow, ν is the viscosity, and Re=ℓUU/ν is the Reynolds number. At the same time, large-scale 2D perturbations also become unstable by an eddy viscosity mechanism when Re>c2, where c1,c2 are order 1 nondimensional numbers. These relations define different regions in parameter space where 2D and 3D instabilities can (co)exist and this allows us to construct a stability diagram. Implications of these results for fully turbulent flows that display a change of direction of cascade as H is varied are discussed.3 MoreReceived 1 June 2018DOI:https://doi.org/10.1103/PhysRevFluids.3.114601©2018 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasFlow instabilityTurbulenceFluid Dynamics
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