The transition of the flow in a duct of square cross-section is studied. Like in the similar case of the pipe flow, the motion is linearly stable for all Reynolds numbers; this flow is thus a good candidate to investigate the 'bypass' path to turbulence. Initially the so-called 'linear optimal perturbation problem' is formulated and solved, yielding optimal disturbances in the form of longitudinal vortices. Such optimals, however, fail to elicit a significant response from the system in the nonlinear regime. Thus, streamwise-inhomogeneous, sub-optimal disturbances are focussed upon; nonlinear quadratic interactions are immediately evoked by such initial perturbations and an unstable streamwise-homogeneous large amplitude mode rapidly emerges. The subsequent evolution of the flow, at a value of the Reynolds number at the edge between fully developed turbulence and relaminarization, shows the alternance of patterns with two pairs of large scale vortices near opposing parallel walls. Such edge states bear a resemblance to optimal disturbances.
A linear approach, inspired by hydrodynamic stability theory, is used to describe the formation of large-scale coherent vortices for the turbulent flow that develops in a duct of square cross section. A set of equations for smallamplitude coherent motion is derived and closed with a simple mixing-length strategy. The initial condition that maximizes a chosen functional (related to either the kinetic energy of the coherent motion or the rate of turbulence production) is found through a direct/adjoint numerical approach borrowed from optimal control theory. It is found that different kinds of secondary flows can appear in the duct cross section, sustained by the mean shear. Some of these optimal states display a symmetry about the bisectors and the diagonals of the duct, in agreement with experimental observations and direct numerical simulations.
The generation of a thick fully turbulent boundary layer is investigated in a lowspeed wind tunnel at a nominal zero pressure gradient over the Reynolds number range 0.145 × 10 6 ≤ Rex ≤ 0.58 × 10 6. The wind tunnel floor natural boundary layer is laminar with thickness δ between 5.76 mm and 8.13 mm. Different tripping devices are tested to trigger transition so to double the boundary layer thickness and provide a fully established turbulent velocity profile. Using a trip wire significantly increases δ but leads to an unsatisfactory velocity profile. Using a sandpaper strip slightly increases δ but keeps the boundary layer laminar. Using a strip of sharp-edged silicon granules doubles the boundary layer thickness that increases up to 20 mm and the mean velocity profiles are a good fit to the logarithmic law of the wall over the outer region of the boundary layer. The spectral decay of turbulent kinetic energy in this outer layer is exponential and close to −5/3, indicating turbulence equilibrium. This work is of practical interest to wind tunnel practitioners for generating equilibrium thick turbulent boundary layers at low Reynolds numbers.
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