This paper presents a short overview of the flow around a slender circular cylinder, the purpose being to place it within the frame of the distributed Hopf bifurcation problems described by the Ginzburg-Landau equation (GLE). In particular, the chaotic behavior superposed to a well tuned harmonic oscillation observed in the range Re > 270, with Re being the Reynolds number, is related to the defect-chaos regime of the GLE. Apparently new results, related to a Kolmogorov like length scale and the

Let a cylinder with a circular cross section in the plane

Equations (

Flow around circular cylinder: (a)

The periodic flow in the limit cycle regime can be expanded in its harmonic components by Fourier series decomposition; namely, if

The time average

Harmonic decomposition of the 2D velocity field—Re = 100.

Amplitude

It must be observed also that these numerical results indicate a

But this is not enough for the present purpose: the final goal is to solve the tri-dimensional (3D) problem for a

The hope is that such

A clue is given by the following observation: the

At

In the asymptotic solution that leads to Landau's equation (

The eigenvalue-eigenvector

Observing that the harmonic components

By placing (

The 2D systems (

Two results can be derived directly from the latter equality (see also (

Finally, once the 2D numerical solution

Notice that (

Summarizing: through the 2D simulation one obtains

The 2D flow around a slender
cylinder is

Strouhal number St =

In the range Re

The

The behavior is similar to the well known “drag crisis” in the range 10^{5}^{6}, although even sharper, and it should be also related to the chaotic (turbulent) flow observed when Re

The 2D

In this context, the GLE was first proposed as a

Normalizing time, space and amplitude by using

Behavior of GLE in the unstable region (

It seems then that the GLE, with recognized predictive ability in the

However, as seen in the first section,

As in a turbulent flow regime, the chaotic solution in the “defect chaos” regime is characterized by a cascade of length scales

Equation (

Wavenumber spectrum for several points in the dispersion plane

The intensity of the response can be also estimated by the wavenumber spectrum integral,

(a) Comparison between (

Figure

Now, if

This relation was obtained under the weak stationary assumption (

The expression (ii) in (

One expects then that

Contour lines of

In the flow problem, the linear dispersion coefficient

In this paper the possibility to solve asymptotically the flow around a slender cylinder using a 2D computation and the Ginzburg-Landau equation to obtain the 3D correction was elaborated, stressing the regime above Re

A practical problem where the present study may be relevant is related to the

The authors acknowledge the financial support from FINEP-CTPetro, FAPESP, PETROBRAS and CNPq.