Large-scale secondary structures in duct flow

Abstract

The spatial growth of small perturbations developing on a fully developed base flow in a duct with two inhomogeneous cross-flow directions is examined. For a laminar mean flow it is shown that optimally configured vortices at an upstream cross-section induce large transient amplification of a disturbance energy norm downstream. Such a linear growth is a likely initial stage of transition in ducts for which the cross-section is square or of moderate aspect ratio, asymptotically stable according to conventional eigen-analyis. With the increase of the cross-sectional aspect ratio the transient growth results of the plane channel flow case are approached. The optimization methodology is then employed to study the transient amplification of large-scale secondary structures developing in a square duct for which the unidirectional base motion has a turbulent-like profile. Under the hypothesis that fine-scale turbulence does not affect the growth of large-scale secondary flows, it is shown that cross-stream vortices are sustained by the mean flow and grow maximally over a streamwise distance of several thousand viscous units of length.

Numerical study of magnetohydrodynamic duct flow at high Reynolds and Hartmann numbers

“…In the hydrodynamic case, we see a structure similar to those observed at smaller Re, for example, by Gavrilakis (1992) and Galletti & Bottaro (2004). The secondary structures form a slightly distorted pattern of eight counter-rotating vortices in the corners of the duct.…”

“…The value of the smallest wall-normal grid step in the z A preliminary verification of the model was conducted by reproducing the results of Galletti & Bottaro (2004) for the transient growth in the hydrodynamic duct and the results of Boeck et al (2007) and Krasnov et al (2008) for turbulent MHD flows in channels. We have also conducted a grid sensitivity study.…”

Marginally turbulent flow in a square duct

“…Although some insight about vortex kinematics in duct flow was provided by Kawahara & Kamada (2000), a detailed investigation of the dynamics of coherent structures in such flow has to our knowledge not been reported in the literature. Galletti & Bottaro (2004) and Bottaro, Soueid & Galletti (2006) have employed a parabolized linear formulation (in the latter reference a simple mixing-length model accounts for the Reynolds stresses) to describe the transient growth of a perturbation field which under certain circumstances resembles the experimentally observed secondary flow pattern in the square duct. However, their linear analysis of the mean field seems to be more appropriate to model a transitional path toward turbulence rather than to describe a fully developed turbulent state.…”

Recirculation cells in a wide channel

“…They reveal secondary flow structures in the plane normal to the primary flow 6,7,9,10 , among which are the stationary helicoidal flows observed in straight channels (Prandtl refers to them as "secondary flow of the second kind " 11 ). They were originally associated with duct corners 10,12,13 , where they were attributed to the turbulence anisotropy 3,12,[14][15][16][17] . Recent experiments reported their existence in wide channels (aspect ratio of about 10), where they arrange themselves as pairs of counter-rotating vortices aligned with the stream direction 6-8 .…”