Linear Analysis of the Cylinder Mean Flow

Dwight Barkley

This research concerns the fundamental issue of vortex shedding from a bluff body and in particular the selection of the vortex shedding frequency.

Onset of vortex shedding
Steady Flow Below the critical Reynolds number (Re_c = 46) the flow past a circular cylinder is steady and reflection symmetric. (Shown to the left is vorticity and separating streamlines at Re=40)
Vortex shedding Above Re_c the flow is time periodic. One observes the famous Benard-von-Kaman vortex street. (Shown to the left is a snapshot of the flow at Re=100)
If one measures, either in experiment or in numerical simulations, the oscillation frequency as a function of Reynolds number, one obtains the plot shown to the right. St is the Strouhal number, or non-dimensional frequency. A large body of research has been devoted to understanding this curve.

Onset of vortex shedding

Steady Flow

Below the critical Reynolds number (Re_c = 46) the flow past a circular cylinder is steady and reflection symmetric.

(Shown to the left is vorticity and separating streamlines at Re=40)

Vortex shedding

Above Re_c the flow is time periodic. One observes the famous Benard-von-Kaman vortex street.

(Shown to the left is a snapshot of the flow at Re=100)

If one measures, either in experiment or in numerical simulations, the oscillation frequency as a function of Reynolds number, one obtains the plot shown to the right. St is the Strouhal number, or non-dimensional frequency.

A large body of research has been devoted to understanding this curve.

Linear Stability Analysis of Base Flows
One can approach this problem by computing eigenvalues of steady solutions of the Navier-Stokes equations. These steady solutions are commonly referred to as base flows. (Shown to the right are two base flows: Re=40 top and Re=100 bottom. Both are steady solutions to the Naiver-Stokes equation. The top one is linearly stable while the bottom one is linearly unstable.)
The eigenvalues from linear stability analysis correctly give the wake frequency at onset, but rapidly diverge from St above onset. (Shown in red are non-dimensionalized eigenvalues. They are complex with the imaginary part giving a frequency (left) and the real part giving a growth rate (right).)

Linear Stability Analysis of Base Flows

One can approach this problem by computing eigenvalues of steady solutions of the Navier-Stokes equations. These steady solutions are commonly referred to as base flows.

(Shown to the right are two base flows: Re=40 top and Re=100 bottom. Both are steady solutions to the Naiver-Stokes equation. The top one is linearly stable while the bottom one is linearly unstable.)

The eigenvalues from linear stability analysis correctly give the wake frequency at onset, but rapidly diverge from St above onset.

(Shown in red are non-dimensionalized eigenvalues. They are complex with the imaginary part giving a frequency (left) and the real part giving a growth rate (right).)

Mean flows
Rather than computing eigenvalue for the base flows, one can compute eigenvalue of mean flows - time-averaged solutions of the Navier-Stokes equations. (Shown to the right is the mean flow at Re=100.)
The eigenvalues from linear analysis of mean flows correctly give the wake frequency in the nonlinear regime. Moreover, the mean flows are marginally stable. (Shown in blue are non-dimensionalized eigenvalues for the mean flow. The imaginary part agrees with St (left) and the real part shows marginal stability (right).)

Mean flows

Rather than computing eigenvalue for the base flows, one can compute eigenvalue of mean flows - time-averaged solutions of the Navier-Stokes equations.

(Shown to the right is the mean flow at Re=100.)

The eigenvalues from linear analysis of mean flows correctly give the wake frequency in the nonlinear regime. Moreover, the mean flows are marginally stable.

(Shown in blue are non-dimensionalized eigenvalues for the mean flow. The imaginary part agrees with St (left) and the real part shows marginal stability (right).)

Nonlinear Selection
Starting from the unstable base flow at Re=100, oscillations develop. As they do, the mean changes and frequency increases.
As the flow evolves from the base flow, Reynolds stresses grow. As they do the flow evolves to a point of marginal stability of the mean flow, at which point the frequencies is the nonlinear Strouhal number.

Bibliography