Editing Strouhal number (section)

{{Short description|Dimensionless number describing oscillating flow mechanisms}}
In [[dimensional analysis]], the '''Strouhal number''' ('''St''', or sometimes '''Sr''' to avoid the conflict with the [[Stanton number]]) is a [[dimensionless number]] describing oscillating flow mechanisms. The parameter is named after [[Vincenc Strouhal]], a Czech physicist who experimented in 1878 with wires experiencing [[vortex shedding]] and singing in the wind.<ref>Strouhal, V. (1878) [http://babel.hathitrust.org/cgi/pt?id=uc1.b4433702;view=1up;seq=230 "Ueber eine besondere Art der Tonerregung"] (On an unusual sort of sound excitation), ''Annalen der Physik und Chemie'', 3rd series, '''5''' (10) : 216–251.</ref><ref>{{cite book |last=White |first=Frank M. |title=Fluid Mechanics |edition=4th |year=1999 |publisher=McGraw Hill |isbn=978-0-07-116848-9 }}</ref> The Strouhal number is an integral part of the fundamentals of [[fluid mechanics]].

The Strouhal number is often given as

:<math> \text{St} = \frac{f L}{U}, </math>

where ''f'' is the frequency of [[vortex shedding]] in [[Hertz]],<ref>{{cite journal |last1=Triantafyllou |first1=M. S. |last2=Triantafyllou |first2=G. S. |last3=Gopalkrishnan |first3=R. |title=Wake mechanics for thrust generation in oscillating foils |journal=Physics of Fluids A: Fluid Dynamics |date=8 August 1991 |volume=3 |issue=12 |page=2835 |doi=10.1063/1.858173 |bibcode=1991PhFlA...3.2835T |url=https://d1wqtxts1xzle7.cloudfront.net/41159101/Wake_mechanics_for_thrust_generation_in_20160114-3088-11ap4b0.pdf20160114-19908-1joxyfi-libre.pdf?1452833192=&response-content-disposition=inline%3B+filename%3DWake_mechanics_for_thrust_generation_in.pdf&Expires=1723579455&Signature=CTCbNhtjpeCJtET5xsnyDOYZiYCORbcFGK2zbkOVE~aP0hUACbjHzg8htvWsESk2UCm0eWu8T5epc36v3vNXctm0cMwWg6~SH-Lj~q83rxMt0o-s2wSUwhaYmLdMwIejRuz0zibG8UjQKpkMYjUUGHptanipzrkMEY5x7pY8lUpmMHAQFrSkrmr2Ewmp2F4RauDXTRmAuIOxbs9NJSLEWYMuVCByyRSv5RPNBNHd~QyphUAtkfHfziQGFC3mmSRfIY4LGCOM302panuikGVGkaxAHBFLSZzGS4y7S1cZFKCPBMtRMAbcRvwoMJgM76kGctic3pQ5qrxG0gk87WZhyA__&Key-Pair-Id=https://doi.org/10.1063/1.858173 |access-date=13 August 2024 }}{{Dead link|date=August 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> ''L'' is the [[characteristic length]] (for example, [[hydraulic diameter]] or the [[Airfoil#Airfoil terminology|airfoil thickness]]) and ''U'' is the [[flow velocity]]. In certain cases, like heaving (plunging) flight, this characteristic length is the amplitude of oscillation. This selection of characteristic length can be used to present a distinction between Strouhal number and reduced frequency:

:<math> \text{St} = \frac{k A}{\pi c}, </math>

where ''k'' is the [[reduced frequency]], and ''A'' is amplitude of the heaving oscillation.

[[File:Lienhard StrouhalVsReynoldsVortex.jpg|upright=3|thumb|alt=Plot showing the variation of Strouhal number with Reynolds number for a circular cylinder in crossflow for Reynolds numbers from 50 to 10 million based on aggregated experimental data|Stouhal number variation with Reynolds number for a cylinder in cross-flow for Reynolds numbers based on aggregated experimental data<ref>{{cite report |last1=Lienhard |first1=John H |title=Synopsis of Lift, Drag, and Vortex Frequency Data for Rigid Circular Cylinders |date=1966 |department=College of Engineering, Research Division |series=Bulletin 300|publisher=Washington State University |location=Pullman, WA |url=https://engines.egr.uh.edu/sites/engines/files/talks/vortexcylinders.pdf| access-date= February 7, 2025}}</ref>]]

For large Strouhal numbers (order of 1), viscosity dominates fluid flow, resulting in a collective oscillating movement of the fluid "plug". For low Strouhal numbers (order of 10<sup>−4</sup> and below), the high-speed, quasi-steady-state portion of the movement dominates the oscillation. Oscillation at intermediate Strouhal numbers is characterized by the buildup and rapidly subsequent shedding of vortices.<ref>{{cite journal |first=Ian J. |last=Sobey |year=1982 |title=Oscillatory flows at intermediate Strouhal number in asymmetry channels |journal=[[Journal of Fluid Mechanics]] |volume=125 |pages=359–373 |doi=10.1017/S0022112082003371 |doi-broken-date=29 November 2024 |bibcode = 1982JFM...125..359S |s2cid=122167909 }}</ref>

For spheres in uniform flow in the [[Reynolds number]] range of 8×10<sup>2</sup> < Re < 2×10<sup>5</sup> there co-exist two values of the Strouhal number.  The lower frequency is attributed to the large-scale instability of the wake, is independent of the [[Reynolds number]] Re and is approximately equal to 0.2.  The higher-frequency Strouhal number is caused by small-scale instabilities from the separation of the shear layer.<ref name="Kim and Durbin, 1988">{{cite journal |last1=Kim |first1=K. J. |last2=Durbin |first2=P. A. |year=1988 |title=Observations of the frequencies in a sphere wake and drag increase by acoustic excitation |journal=[[Physics of Fluids]] |volume=31 |issue=11 |pages=3260–3265 |doi=10.1063/1.866937 |bibcode = 1988PhFl...31.3260K |doi-access=free }}</ref><ref name="Sakamoto and Haniu, 1990">{{cite journal |last1=Sakamoto |first1=H. |last2=Haniu |first2=H. |year=1990 |title=A study on vortex shedding from spheres in uniform flow |journal=Journal of Fluids Engineering |volume=112 |issue=December |pages=386–392 |doi= 10.1115/1.2909415|bibcode=1990ATJFE.112..386S |s2cid=15578514 }}</ref>