Dwight Barkley
This
research concerns the fundamental issue of vortex shedding from
a bluff body 
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 BenardvonKaman 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 nondimensional 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 NavierStokes 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 NaiverStokes 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 nondimensionalized 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  timeaveraged solutions of the NavierStokes 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 nondimensionalized 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. 