Fluid mechanics

Turbulent flow and the Kármán vortex street

Turbulence features velocity fluctuations and enhanced mixing; flow past a cylinder can shed alternating vortices downstream.

Turbulence is unsteady motion with velocity fluctuations across length and time scales. As a fluid passes a cylinder, its boundary layer can separate and shed alternating vortices into the wake.

Re=\frac{\rho UD}{\mu},\qquad St=\frac{fD}{U}

Definition: Kármán street and Strouhal number

A vortex street is an alternating sequence of opposite-sign vortices in a wake. For a circular cylinder in some Reynolds-number regimes, $St=fD/U$ relates shedding frequency $f$ , diameter $D$ , and free-stream speed $U$ . A value near 0.2 is an approximation under certain conditions, not universal.

Vary the flow speed and observe a qualitative vortex wake behind a cylinder; this model does not numerically solve Navier–Stokes.

Boundary-layer separation

No slip slows fluid near the wall. An adverse pressure gradient can separate the boundary layer from a cylinder; downstream shear layers roll into alternating vortices. The wake depends on Reynolds number, geometry, disturbances, and inflow conditions.

Fluctuating forces and resonance

Alternating shedding causes fluctuating pressure and transverse force. If forcing approaches a structure’s natural frequency, resonance can amplify vibration; engineers account for wakes when designing piers, cables, and chimneys.

Example: Vortex-shedding frequency

Air at $U=10\,\mathrm{m/s}$ passes a cylinder of diameter $D=0.10\,\mathrm m$ . Assuming $St=0.20$ in an appropriate regime, estimate $f$ .

Solution

$f=StU/D=0.20(10)/0.10=20\,\mathrm{Hz}$ . Actual values depend on Reynolds regime and conditions.

Quick check

Example: Vortex-shedding frequency behind a cylinder

Over a range of Reynolds numbers, the Strouhal number $St=fD/U$ is nearly constant; for a circular cylinder in a relevant flow regime, a typical value is about $0.2$ . If $U=10,mathrm{m/s}$ and cylinder diameter $D=0.50,mathrm m$ , the estimate is $fapprox St,U/D=4,mathrm{Hz}$ . This is only an approximation in the appropriate regime; the actual frequency depends on Reynolds number, shape, inflow turbulence, and boundary conditions.

The Strouhal number is defined as:

In a Kármán street, vortices are typically:

References

G. K. Batchelor (1967). An Introduction to Fluid Dynamics