Reynolds, Strouhal, and the Physics of Vortex Shedding for Sample Probes
First-principles guide to the dimensionless numbers that govern vortex-induced vibration of sample probe assemblies. Reynolds number flow regimes, Strouhal correlations, lock-in resonance, and Scruton number damping criteria.
TL;DR
Vortex shedding is the alternating release of vortices from the sides of any cylinder in a flow. The shedding frequency is set by the Strouhal number (St ≈ 0.22 in the subcritical regime). The flow regime is set by the Reynolds number. When the shedding frequency approaches the cylinder's natural mechanical frequency, lock-in resonance occurs and the cylinder can fail by fatigue. Whether it actually fails depends on the Scruton number — the damping capacity of the structure.
Reynolds Number — The Master Switch
Re = ρ × U × D / μ
Where ρ = fluid density, U = approach velocity, D = cylinder diameter, μ = dynamic viscosity.
Reynolds tells you which flow regime the cylinder lives in:
| Re range | Regime | Behavior |
| < 1 | Stokes | No vortices |
| 1 - 40 | Steady wake | Stable twin vortices |
| 40 - 150 | Laminar shedding | Alternating vortices begin |
| 150 - 3 × 10⁵ | Subcritical | Periodic shedding, St ≈ 0.21 |
| 3 × 10⁵ - 3 × 10⁶ | Critical / supercritical | Boundary layer transitions; St rises |
| > 3 × 10⁶ | Transcritical | Periodic shedding returns |
Most industrial sample probes operate squarely in the subcritical regime, where the Strouhal number is essentially constant. This is the regime ASME PTC 19.3 TW-2016 was written for.
Strouhal Number — The Frequency Setter
St = fs × D / U → fs = St × U / D
For a cylindrical probe in subcritical flow, St ≈ 0.22. So a 0.500" probe in a 30 ft/s gas stream sheds vortices at:
fs = 0.22 × 30 / (0.500/12) = 158 Hz
If the probe's natural mechanical frequency is anywhere near 158 Hz, the design is in lock-in territory.
Scruton Number — The Fatigue Decider
The Scruton number is the dimensionless damping capacity:
Sc = (4 × π × m × ζ) / (ρ × D²)
Where m = mass per unit length, ζ = damping ratio (typically 0.005 for steel in a process fluid), ρ = fluid density, D = cylinder diameter.
When Sc > 64, lock-in vibration cannot sustain itself even at exact frequency match — the structure damps out the oscillation faster than the wake can pump energy in. When Sc < 10, lock-in is virtually inevitable. The window between is the engineer's gray zone, and ASME PTC 19.3 gives explicit criteria for navigating it.
Lock-In Resonance — The Failure Mode
When the natural frequency fn and the shedding frequency fs come within roughly 30% of each other, the wake synchronizes to the structure rather than vice versa. The cylinder begins oscillating at fn, and the wake adjusts to match. This is lock-in and it is what kills probes.
Inside lock-in, the oscillation amplitude can grow to D/2 or more in the transverse direction. Bending stress at the probe root rises by orders of magnitude over the static drag stress. Fatigue crack initiation is fast.
Why the Configurator Cares
The SPA Configurator computes Re, St, fs, fn, and the fs/fn ratio for every geometry the user specifies. When fs/fn enters the PTC 19.3 caution zone, the wizard highlights the offending dimension and suggests:
1. Shortening the probe — raises fn fast (cantilever fn ∝ 1/L²)
2. Increasing the diameter — also raises fn, lowers fs
3. Adding a stop collar — converts the cantilever into a fixed-fixed beam, which doubles fn
4. Switching to a stiffer alloy — modest fn increase (E enters as √E)
Beyond the Subcritical Regime
When the Reynolds number crosses 3 × 10⁵, the boundary layer on the cylinder transitions from laminar to turbulent and the entire wake reorganizes. St jumps to ~0.45 briefly before settling. Critical-regime probes are rare in process service — they require gas velocities above ~150 ft/s on a 1" cylinder — but they do exist on choke valves and high-velocity steam laterals.