Problem 13 IYPT 2027: 13. Vortex pendulum

Problem Description & Analysis

Official Problem Statement

“A pendulum with a vertically oriented cylindrical bob is suspended above a channel of flowing water. When the bob is partially submerged, the flow may cause the pendulum to oscillate in a direction perpendicular to the flow. Explain this phenomenon and determine how it depends on relevant parameters.”

Karman Vortex Street and Transverse Force

A flow at speed $U$ past a circular cylinder of diameter $D$ sheds vortices alternately from each side in a Karman vortex street. The shedding frequency is set by the Strouhal number $St$:

$$\boxed{f_s = \frac{St \cdot U}{D}},\qquad St \approx 0.20\quad \text{for}\quad Re = 10^3 \text{–} 10^5$$

where $Re = UD/\nu$ is the Reynolds number and $\nu$ is the kinematic viscosity of water. The alternating vortex shedding produces an oscillating transverse (lift) force on the cylinder:

$$F_L(t) = \frac{1}{2}\rho U^2 D L_{\rm sub}\, C_L\,\sin(2\pi f_s t)$$

where $\rho$ is the water density, $L_{\rm sub}$ is the submerged length of the cylinder, and $C_L \approx 0.3$–$0.5$ is the lift coefficient.

Vortex-Induced Vibration (VIV) and Lock-in

The natural frequency of the pendulum (neglecting hydrodynamic effects) for a pendulum of length $L$:

$$f_n = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$$

When $f_s \approx f_n$, the transverse force drives the pendulum into resonance, producing large lateral oscillations. This is vortex-induced vibration (VIV).

The key non-dimensional parameter is the reduced velocity:

$$U_r = \frac{U}{f_n \cdot D}$$

Lock-in: over the range $U_r \approx 5$–$8$, the vortex shedding frequency synchronizes with the pendulum’s oscillation frequency even as the flow speed changes. The oscillation amplitude is maximized within the lock-in band:

$$A_{\max} \approx \frac{C_L\,\rho\, U^2\, D\, L_{\rm sub}}{2\,(2\pi f_n)^2\, m_{\rm eff}\, \zeta}$$

where $m_{\rm eff}$ is the effective pendulum mass and $\zeta$ is the damping ratio. Outside lock-in, the pendulum amplitude drops sharply and the shedding frequency reverts to $St \cdot U/D$.

Parameters

$U$ — flow speed (m/s): primary control parameter; $f_s \propto U$
$D$ — cylinder diameter (cm): $f_s \propto 1/D$; larger $D$ shifts lock-in to lower $U$
$L$ — pendulum length (cm): $f_n \propto 1/\sqrt{L}$; determines at which $U$ lock-in occurs
$L_{\rm sub}$ — submersion depth of cylinder (cm): lift force $F_L \propto L_{\rm sub}$
$Re = UD/\nu$ — Reynolds number: affects $St$ and $C_L$
$U_r = U/(f_n D)$ — reduced velocity: the criterion for lock-in ($U_r \approx 5$–$8$)

Experimental Setup

To measure transverse oscillation amplitude as a function of $U$ and $D$:

Build a water channel with adjustable flow speed (pump with control valve and flow meter).
Suspend the cylindrical pendulum over the channel. Keep the submersion depth fixed.
Increase flow speed from zero upward; record the transverse oscillation amplitude with a camera or potentiometer.
Plot $A/D$ versus $U_r$ — a clear peak near $U_r \approx 5$–$8$ should appear.
Repeat for several pendulum lengths $L$ (different $f_n$) to verify the dependence on $U_r$.
To determine $St$, measure the shedding frequency with a hot-wire or pressure sensor at flow speeds outside the lock-in range.

References

Blevins, R. D., Flow-Induced Vibration, 2nd ed., Van Nostrand Reinhold, 1990 — Chapter on cylinder VIV.
Williamson, C. H. K., Govardhan, R., “Vortex-Induced Vibrations,” Annu. Rev. Fluid Mech. 36 (2004) 413–455.
Sarpkaya, T., “A critical review of the intrinsic nature of vortex-induced vibrations,” J. Fluids Struct. 19 (2004) 389–447.
Zdravkovich, M. M., Flow Around Circular Cylinders, Vol. 1, Oxford, 1997.

13. Vortex pendulum