Instability and transition in curved channel flow

Abstract

A theoretical and numerical investigation of streamwise-oriented Dean vortices in curved channel flow is presented. The principal results are obtained from three-dimensional pseudospectral simulations of the incompressible time-dependent Navier-Stokes equations. With increasing Reynolds number, a sequence of transitions similar to that observed in non-turbulent Taylor-Couette flow is found. The transition from laminar curved channel Poiseuille flow to axisymmetric Dean vortex flow is studied using linear and weakly nonlinear analyses; these results are compared to the full simulations. Using the code, two transitions that cause the axisymmetric vortices to develop waves travelling in the streamwise direction at higher Reynolds numbers are discovered. The linear stability of axisymmetric Dean vortex flow to non-axisymmetric perturbations is examined. Associated with the two transitions are two different non-axisymmetric flows: undulating and twisting Dean vortex flow. Undulating vortices are similar to wavy Taylor vortices. Twisting vortices, with a much shorter streamwise wavelength, are dissimilar; to our knowledge, they have no counterpart in the Taylor-Couette problem. At sufficiently high Reynolds numbers, linear growth rates associated with twisting vortices far exceed those associated with undulating vortices. For the channel curvatures studied, angular speeds of both kinds of travelling waves are only weakly dependent on Reynolds number and wavenumber. A bifurcation limits the vortex spacings that can be examined and suggests an Eckhaus stability boundary. The development of wavy vortex flows from small-amplitude disturbances shows that full development of undulating vortices may require a streamwise distance greater than one circumference, whereas for sufficiently large Reynolds numbers, twisting vortices reach equilibrium amplitude within half this distance and are therefore more likely to be observed experimentally. We suggest twisting vortices are due to a shear instability.

Simulation of Laminar-Turbulent Transition in the Vicinity of a Wall.

“…Work on the plane channel flow by Bart Singer has been completed and a report on this work has been submitted to AFOSR (1). Similarly, the effort on the curved channel carried out by Warren Finlay is finished and a report has been sent to AFOSR (2). Work on the plane boundary layer by Kyung Soo Yang is essentially finished and the report is being written at the present time (3); it will be submitted shortly.…”

On the Bifurcations of Dean Flow between Porous Horizontal Cylinders with a Radial Flow and a Radial Temperature Gradient

“…For Dean flow stability, the numerical and experimental investigations of transitions to turbulence were carried out, for instance, in [22] [23] [24] [25]. In [22], the linear stability of axisymmetric Dean vortex flow to non-axisymmetric perturbations is considered.…”

“…In [22], the linear stability of axisymmetric Dean vortex flow to non-axisymmetric perturbations is considered. Two different nonaxisymmetric flows, undulating and twisting Dean vortex flow, are found to be associated with the two transitions.…”

Dean instability in double-curved channels

“…Following Ref. [1], we will refer to this type of flow as "curved channel Poiseuille flow". For increasing Reynolds number, the flow pattern is determined by centrifugal forces until it becomes unstable and bifurcates to secondary flow.…”

“…Numerical simulations of curved channel flow have also been popular, see for example Finlay et al [1,5,6], who extended Dean's stability criterion to channels of different aspect ratios η = r i /r o , using high-accuracy linear stability analysis. All the studies mentioned above deal with a very idealized channel geometry, namely with channels having a rectangular cross section, which are uniformly curved along the streamwise direction only.…”