The initial stage of transition in pipe flow: role of optimal base-flow distortions

Abstract

We explore the spatial growth of disturbances developing on top of a base flow given by the Hagen-Poiseuille profile, which has been modified by a small axisymmetric and axially invariant distortion. Such deviations from the ideal parabolic profile may, for instance, occur in experiments as a result of experimental uncertainties. The optimal distortion (i.e. the distortion with a prescribed norm that induces the maximum growth rate) is computed by a variational technique. Unstable modes are found to exist for very small values of the norm of the deviation at low Reynolds numbers, and the instability is governed by an inviscid mechanism. The growth of these modes and the ensuing transition to turbulence is then studied by means of direct numerical simulations. Two possible paths of transition are found, one based on the exponential amplification of axisymmetric disturbances and the subsequent formation of -vortices and the other based on the growth and breakdown of streamwise streaks.

Transition to turbulence in duct flow

“…There is an interesting connection here with the recent theory of 'minimal defects ; Biau & Bottaro (2004); Gavarini et al (2004);Ben-Dov & Cohen (2007). The secondary flow at t = 0.8 shown in the same figure displays symmetries about bisectors and diagonals, but this is not a generic occurrence and different initial conditions generate mean flow defects with other symmetries.…”

Optimal and robust control of streaks in pipe flow

“…This code computes the flow in a cylindrical pipe geometry by a finite-volume technique with secondorder accuracy for the spatial discretization. The code has already been applied by Gavarini, Bottaro & Nieuwstadt (2004) in order to simulate the spatial evolution of disturbances to the base flow and the ensuing transition to turbulence. At the inflow boundary, an optimal perturbation is imposed with a prescribed amplitude, whereas at the outflow, a fringe region is implemented to gradually damp the disturbances flowing out of the physical domain with minimal reflection.…”

Folded Edge of Turbulence in a Pipe

“…It has been recognized since Reynolds' original investigations [4] that finite amplitude disturbances are responsible for transition in practice. More recently, progress has been made in providing experimental estimates for the stability boundary between laminar and turbulent flow [5,6] and various scalings of the amplitude of disturbance required to cause transition have been found [7,8] some of which are in accord with theoretical predictions [9,10]. A reasonable question to ask is whether the boundary is sharp or, as in examples from lowdimensional dynamical systems it is folded or interleaved?…”