Strouhal Number (St): Oscillatory Flow Frequency Scaling
Use this Strouhal number deep dive alongside the Reynolds number guide and the Reynolds Number Calculator to align both unsteadiness and inertial scaling in your experiments. Connect oscillatory aerodynamic loads to the Froude number overview for complementary insights into other characteristic numbers before refining prototypes.
Definition and Fundamental Formulation
The Strouhal number (St) quantifies the relationship between oscillation frequency, flow velocity, and characteristic length in unsteady fluid dynamics. In its canonical form,
St = f · L / U
with f the vortex shedding frequency in hertz, L a reference length (such as cylinder diameter), and U the free-stream velocity. ISO 80000-11 categorizes Strouhal number as a dimensionless characteristic number useful for dynamic similarity when oscillations dominate.
The Strouhal number often appears alongside Reynolds number and Mach number in aerodynamic design because non-dimensional frequency scaling dictates how wake oscillations translate from subscale tests to full-scale configurations. Typical values span 0.15 to 0.3 for circular cylinders in the subcritical Reynolds regime, but deviate markedly for bluff bodies, airfoils at varying angles of attack, and biological propulsors.
Historical Development
Vincenc Strouhal documented the "singing wire" phenomenon in 1878 by noting that wires resonate at a tone proportional to wind speed. Shortly afterward, Lord Rayleigh linked the pitch to vortex shedding frequency, inaugurating systematic research on oscillatory flows. Theodore von Kármán provided the theoretical basis for alternating vortex streets in 1911, explaining why only particular spacing ratios satisfy momentum and continuity constraints. By the 1930s, Prandtl and Lighthill had incorporated Strouhal scaling into aeroelasticity, acoustic prediction, and fluid-structure interaction frameworks. Today, standards such as ISO 4866 for structural vibration reference Strouhal-driven excitation when prescribing measurement protocols.
Conceptual Foundations
Dimensional analysis and similarity
Non-dimensionalization of the unsteady Navier–Stokes equations reveals St as the coefficient scaling the temporal derivative term. Holding Strouhal constant while matching Reynolds and Mach numbers enables accurate dynamic similarity of vortex shedding, tonal noise, and lift fluctuations between model and prototype. Deviations in St indicate incorrect frequency scaling, leading to inaccurate predictions of buffeting, galloping, or lock-in phenomena.
Characteristic length selection
Engineers select L according to geometry and flow regime. For circular cylinders and cables, diameter is conventional. For airfoils, chord length governs Strouhal scaling for leading-edge vortex shedding, but trailing-edge tone research sometimes adopts trailing-edge thickness. Biological propulsion uses span or stroke amplitude to capture fin kinematics, while turbomachinery relies on blade pitch or hydraulic diameter. Documenting the choice of L is essential for reproducibility and cross-study comparisons.
Frequency measurement techniques
Strouhal analysis draws on hot-wire anemometry, particle image velocimetry, wall-pressure transducers, laser Doppler velocimetry, and accelerometers embedded in structures. Time-domain signals are transformed via Fourier analysis or wavelets to extract dominant shedding frequencies. The LC Resonant Frequency Calculator helps convert raw sample data into frequency metrics for quick checks before deeper spectral work.
Key Applications
Aeroelastic design and vibration control
High-lift devices, landing-gear fairings, and slender launch-vehicle components are susceptible to vortex-induced vibrations when structural natural frequencies align with Strouhal shedding frequencies. Designers therefore match Strouhal predictions to modal analyses, often using the Reynolds Number Calculator to quantify steady loads before superimposing unsteady contributions. Tuned mass dampers, fairings, and perforated shrouds are sized to detune the St-based excitation.
Metrology and instrumentation
Strouhal scaling governs the calibration of vortex-shedding flow meters used in gas and liquid pipelines. Manufacturers specify Strouhal number ranges where meter output is linear with volumetric flow rate. ISO 17089 guides uncertainty budgets that incorporate Reynolds dependence, ensuring that laboratory calibrations remain valid when fluid properties change.
Bio-inspired propulsion and locomotion
Fish swimming, insect flight, and flapping micro air vehicles operate within narrow Strouhal windows (approximately 0.2 to 0.4) for maximal propulsive efficiency. Comparing biological kinematics to idealized oscillators informs the design of oscillating foils, energy harvesters, and autonomous underwater vehicles. Coupling Strouhal insights with Prandtl and Nusselt numbers enables multi-physics optimization of thermal and hydrodynamic performance.
Importance for Contemporary Engineering
Monitoring and controlling Strouhal number is essential wherever unsteady loads, tonal noise, or frequency-sensitive processes occur. Offshore risers and cable-stayed bridges use Strouhal calculations to prevent fatigue. Urban planners apply Strouhal-based models to predict wind comfort around skyscrapers. Aerospace programs incorporate Strouhal scaling when designing landing-gear doors, fairings, and cavity treatments to mitigate tonal noise. The quantity also underpins benchmarking for data-driven surrogate models, where dimensionless features improve generalization across geometries.
Because Strouhal number depends on Reynolds and Mach numbers, maintaining consistent SI units for velocity (metre per second), length (metre), and frequency (hertz) is crucial. Cross-checking calculations with the hertz unit explainer and the metre definition article keeps documentation traceable to the International System of Units.
Looking Ahead
Advances in large-eddy simulation, reduced-order modeling, and machine-learning-based flow control rely on accurate Strouhal identification. Adaptive control surfaces can now modulate shedding frequency to damp oscillations, while energy harvesters tune Strouhal behavior for maximal power extraction. Documenting Strouhal number alongside Reynolds and Mach values remains best practice in reports, ensuring that future researchers can replicate conditions and build upon existing knowledge.