Come calcolare il numero di Strouhal (St) rapporto adimensionale fluidodinamica che misura la frequenza di oscillazione (vortex shedding o flutter) rispetto al tempo convettivo del flusso: St = f * L / u. f frequenza oscillazione Hz (1/s); L lunghezza caratteristica m (diametro cilindro D, corda profilo c, larghezza struttura b); u velocità asintotica del flusso m/s.

Karman vortex street dietro a un cilindro circolare in flusso laminare-turbolento Re=u*D/nu: regime stabile Re10^7 post-supercritical St~0.21 turbulent.

Classificazione regime: St<0.01 very low frequency quasi-steady flow lento; 0.01<St<0.1 low frequency slow oscillation; 0.1<St<0.3 Karman vortex street typical cilindro Re 250-250k; 0.3<St<1 high frequency strong unsteadiness; St>1 very high extreme unsteady boundary layer. Storia: Vincenc Strouhal (Boemia Praga 1850-1922) Universita Karlova Praga + Universita Wurzburg + Universita Praga German fisico ceco austro-ungarico, 1878 paper Annalen der Physik 5:216 'Uber eine besondere Art der Tonerregung' Aeolian tones (toni eolici) cilindri sottili in vento + frequenza f proporzionale a u/D + nascita aerodinamica + acustica musicale tonifurum.

Theodore von Karman (Ungheria USA 1881-1963) Caltech GALCIT 1911 paper Gottinger Nachrichten + 1912 paper analisi stabilità scia vortice cilindro alternating staggered double-row vortex shedding pattern + 'Karman vortex street' eponimo + relazione Strouhal generalizzata + flutter aerodinamico. Roshko 1954 NACA Tech Note 3169 + Williamson 1996 Annual Review Fluid Mechanics 28:477 'Vortex Dynamics in the Cylinder Wake' classic review + Norberg 1994 Journal Fluid Mechanics 258:287 cylinder Strouhal.

Italia ricerca aerodinamica + ingegneria del vento: Politecnico Milano DAER Dipartimento Aerospaziale Ingegneria 1863 4000 students aeronautica spaziale + galleria del vento Politecnico GVPM Galleria Vento Politecnico Milano 14m x 4m boundary layer wind tunnel + Cidor Centro Ingegneria Vento + Mecc; Politecnico Torino DIMEAS Mater Energetica Aerospaziale 1859 5000 students + DIATI civile + GVT galleria vento Torino; Sapienza Roma DIMA Ingegneria Meccanica e Aerospaziale 1303 4000 + DICEA Civile Edile Ambientale; Federico II Napoli DIETI Ingegneria Elettrica Tecnologie Informazione 1224 5000; Bologna DIN Ingegneria Industriale 1088 4000; Padova DII 1222 3000; Pisa DICI 1343 2500; Politecnico Bari DEI 1990 2500; Trento DICAM 1962; Universita Genova DICCA 1481 2000 fluido dinamica navale + maritime; Salerno DIIN 1968; Firenze DICEA 1321; Cagliari DIMCM 1606. Italia istituzioni: CIRA Centro Italiano Ricerche Aerospaziali Capua Caserta 1985 280 employees research wind tunnel ipersonico Mach 12 + plasma + arc-jet scirocco facility 70 MW + atmospheric subsonic wind tunnel; CNR-INM Istituto Marina Roma + Genova 200 idrodinamica navale + galleria del vento + bacino prove modelli 470 m long INSEAN Roma;

CNR-IIA Inquinamento Atmosferico Monterotondo Roma + Sesto Fiorentino 250 atmospheric dispersion + wind engineering pollutants; INFN Roma 1951 + Frascati LNF 800; INRIM Torino 2006 250 metrologia. Italia industria aeronautica + spaziale + offshore: Leonardo Roma 1948 51000 defense + aeronautica Leonardo Aircraft Eurofighter Typhoon + Tornado + C-27J Spartan + ATR 42/72 ATR Aviazione Toulouse Italy + AgustaWestland helicopters + Selex ES Genova; Avio Colleferro Roma 1908 5000 propulsion Vega launcher + Ariane 6 stages composite chambers; Thales Alenia Space Italia Roma Torino L'Aquila 7800 satellite + thermal vacuum chamber TVAC space testing 8 m diameter Cannes/L'Aquila; ATR Aviazione Toulouse + Alenia Aeronautica Pomigliano d'Arco Napoli + ATR 42/72 turboprop regional airliner manufacturer 50/50 Airbus + Leonardo joint venture; Piaggio Aerospace Genova Villanova d'Albenga 1884 P.180 Avanti II twin turboprop business; D-Orbit Fino Mornasco Como orbital transfer ION Satellite Carrier; SITAEL Mola di Bari Puglia 600 small satellite manufacturing CubeSat + smallsat + Hall thruster electric propulsion; Argotec Torino 100 launcher integration.

Italia industria offshore + maritime + bridges: Saipem Milano 1957 32000 oil&gas EPC engineering procurement construction + offshore platforms + subsea pipelines + drilling rigs Castorone Castor 6 + Maxwell + dynamic positioning DP3; Fincantieri Trieste 1959 22000 cantieri navali Monfalcone Marghera Castellammare di Stabia Sestri Ponente Genova Genova Riva Trigoso + cruise liners MSC Costa Crociere + Marine Subsystems + military vessels U212 submarines; Astaldi Roma 1922 6000 EPC + offshore platforms; Salini Impregilo Webuild Milano 1906 80000 EPC global infrastructure + bridges + tunnels Stretto di Messina ponte 3.3 km design proposal; Fagioli Reggio Emilia heavy lift + transport + offshore. Italia bridges + civil engineering wind: Ponte Stretto Messina 2026-2032 progetto 3.3 km Webuild Salini Impregilo + Eurolink consortium + design longest single-span suspension bridge world; Ponte Morandi Genova ricostruzione 1100 m 2018-2020 Pier Luigi Cerlani Renzo Piano + Italferr Genoa Bridge + Webuild + Fincantieri Infrastructure; Viadotto Polcevera Genova 1100 m 18 piloni; Ponte Verrazzano-Narrows New York 1298 m influenza Strouhal Tacoma Narrows 1940 disastro flutter aeroelastico. Italia industria turbine eoliche: Enercon Italia + Vestas Italia + Siemens Gamesa Italia + Nordex Italia 600 wind farms 11 GW installed Italia 2024;

ANEV Associazione Nazionale Energia Vento Roma 100 wind energy Italia.

Italia industria HVAC + chimney engineering: chimneys industriale Eni 5 raffinerie + Versalis + Saipem + Tecnimont + Snam gas grid 30000 km gasdotti grandi diametri D=600-1200 mm vortex shedding wind engineering. Italia ricerca galleria del vento: Politecnico Milano GVPM 14x4 m boundary layer; Politecnico Torino GVT; CIRA Capua 280 wind tunnel + arc-jet; CNR-INM Roma INSEAN 470 m bacino + galleria del vento navale; Galileo Avionica Firenze + Leonardo wind tunnel test bench military aircraft.

Formula numero Strouhal

Definizione: St = f * L / u rapporto adimensionale frequenza oscillazione f (Hz) per lunghezza caratteristica L (m) su velocità u (m/s). Misura il rapporto tra tempo convettivo (L/u) e periodo di oscillazione (1/f).

Karman vortex street cilindro: regime laminare Re=u*D/nu con f frequenza shedding alternated. Re<47 wake stable no shedding; 47<Re<200 laminar shedding St=0.212*(1-21.2/Re) Roshko 1954; 200<Re<300 transition; 300<Re<3*10^5 subcritical St~0.21 (Karman classic); 3*10^5<Re<3*10^6 critical drag crisis St~0.45-0.50 jump; 3*10^6<Re10^7 post-supercritical St~0.27.

Significato fisico: St<>1 high-frequency oscillations small disturbances.

Numeri correlati: Reynolds Re=u*L/nu inerzia/viscosità; Mach M=u/c velocità/sound speed; Womersley alpha=L*sqrt(omega/nu) pulsatile flow viscosity; Keulegan-Carpenter KC=u_max*T/L oscillating; Reduced velocity U_r=u/(f_n*L) flutter aerodinamico ratio velocity to natural frequency.

Eolian tones (Aeolian tones): Strouhal 1878 telegraph wires + power lines emit Aeolian tone hum f=St*u/D when wind blows perpendicular to wire. D=4 mm wire u=10 m/s f=0.21*10/0.004=525 Hz audible; concert violin string analogy.

Storia e contesto

Vincenc Strouhal (Boemia Seec n.Praga 1850-1922) Universita Karlova Praga 1869 + Universita Wurzburg 1872 PhD physics + Universita Praga German 1882-1885 + Universita Karlova Praga professor 1885-1922; 1878 paper Annalen der Physik 5:216 'Uber eine besondere Art der Tonerregung' sullo studio degli Aeolian tones (toni eolici) prodotti da cilindri sottili in vento perpendicolare + relazione f = c * u/D dove c costante adimensionale dipendente da Reynolds (oggi numero di Strouhal St); pioneer aeroacoustics + experimental aerodynamics. Theodore von Karman (Budapest Ungheria USA 1881-1963) ETH Zurich + Universita Aachen + Caltech GALCIT 1930-1949 fondatore Aerojet 1942 + JPL Jet Propulsion Laboratory 1944 + Padre della Aerodinamica Moderna; 1911 paper Gottinger Nachrichten + 1912 paper Physikalische Zeitschrift analisi stabilità scia vortice cilindro alternating staggered double-row vortex shedding pattern stable formula geometry h/a=0.281 (h spaziature transverse, a longitudinal); 'Karman vortex street' eponimo + flutter aerodinamico.

Roshko 1954 NACA Tech Note 3169 'On the Drag and Shedding Frequency of Two-Dimensional Bluff Bodies' Caltech esperimenti dettagliati Re 50-10000 St-Re curve; Williamson 1996 Annual Review Fluid Mechanics 28:477 'Vortex Dynamics in the Cylinder Wake' classic review + Williamson 1989 NASA studies; Norberg 1994 J Fluid Mechanics 258:287 + Norberg 2003 J Fluids Structures 17:57 cylinder Strouhal; Bearman 1984 Annual Review FM 16:195 vortex shedding.

Italia ricerca aerodinamica + ingegneria vento: Politecnico Milano DAER Dipartimento Aerospaziale Ingegneria 1863 4000 students aeronautica spaziale + galleria del vento GVPM Galleria Vento Politecnico Milano 14x4 m boundary layer wind tunnel + Cidor Centro Ingegneria Vento + Mecc Mater Energetica + scuola Carlo Cercignani Boltzmann legacy; Politecnico Torino DIMEAS 1859 5000 + DIATI civile + GVT galleria vento Torino; Sapienza Roma DIMA + DICEA 1303 4000 students; Federico II Napoli DIETI 1224 5000; Bologna DIN 1088 4000; Padova DII 1222 3000; Pisa DICI 1343 2500; Politecnico Bari DEI 1990 2500; Trento DICAM 1962; Universita Genova DICCA 1481 2000 fluido dinamica navale + maritime + Cantieri Navali; Salerno DIIN 1968; Firenze DICEA 1321; Cagliari DIMCM 1606; Universita Brescia + Trieste + Napoli II Caserta + Catania + Palermo.

Italia istituzioni: CIRA Centro Italiano Ricerche Aerospaziali Capua Caserta 1985 fondato 280 employees research wind tunnel ipersonico Mach 12 SCIROCCO 70 MW high enthalpy plasma re-entry testing + arc-jet plasma + atmospheric subsonic wind tunnel + GHIBLI plasma 2 MW arc-jet + PLasma WInd Tunnel PWT atmospheric; CNR-INM Istituto Marina Roma + Genova 200 idrodinamica navale + galleria del vento + bacino prove modelli 470 m long INSEAN Roma test ship hull resistance + maneuvering + propulsion + structural dynamics; CNR-IIA Inquinamento Atmosferico Monterotondo Roma 250 atmospheric dispersion + wind engineering pollutants + atmospheric chemistry; INFN Roma 1951 5000 + Frascati LNF 800 DAFNE collider; INRIM Torino 2006 250 metrologia.

Italia industria aeronautica + spaziale: Leonardo Roma 1948 51000 defense + aeronautica Leonardo Aircraft Eurofighter Typhoon Tornado C-27J Spartan ATR 42/72 + AgustaWestland helicopters AW139 AW169 AW189 Augusta Bell + Selex ES Genova radar EW + Galileo Avionica Florence + Leonardo Wind Tunnel facility Florence Pisa subsonic transonic; Avio Colleferro Roma 1908 5000 propulsion Vega launcher + Ariane 6 stages composite chambers + Zefiro 9/23/40 Vega upper stages; Thales Alenia Space Italia Roma Torino L'Aquila 7800 employees joint venture Thales 67% Leonardo 33% satellite payloads + thermal vacuum chamber TVAC space testing 8 m diameter Cannes Italy + L'Aquila satellite payload integration; ATR Aviazione Toulouse + Alenia Aeronautica Pomigliano d'Arco Napoli ATR 42/72 turboprop regional airliner manufacturer 50/50 Airbus + Leonardo joint venture 1700 ATR delivered worldwide; Piaggio Aerospace Genova Villanova d'Albenga 1884 fondata + P.180 Avanti II twin turboprop business + military trainer; Galileo Avionica Florence + Leonardo Spazio Pisa + Leonardo space subsidiary; D-Orbit Fino Mornasco Como 200 employees orbital transfer ION Satellite Carrier OTV; SITAEL Mola di Bari Puglia 600 small satellite manufacturing CubeSat + smallsat + Hall thruster electric propulsion; Argotec Torino 100 employees launcher integration + LICIACube CubeSat NASA DART asteroid mission 2022.

Italia industria offshore + maritime + bridges: Saipem Milano 1957 32000 oil&gas EPC engineering procurement construction + offshore platforms Castor Castorone CMS Saipem 7000 + subsea pipelines + drilling rigs + dynamic positioning DP3; Fincantieri Trieste 1959 22000 cantieri navali Monfalcone Marghera Castellammare di Stabia Sestri Ponente Genova Genova Riva Trigoso + cruise liners MSC Costa Crociere Norwegian Cruise + military vessels U212 submarines + frigates FREMM aircraft carrier Cavour + Trieste; Astaldi Roma 1922 6000 EPC + offshore platforms + tunnels + bridges; Salini Impregilo Webuild Milano 1906 80000 EPC global infrastructure + bridges + tunnels + Stretto di Messina ponte 3.3 km design proposal 2026-2032 longest single-span suspension bridge world + Eurolink consortium; Fagioli Reggio Emilia heavy lift transport + offshore equipment installation.

Italia turbine eoliche: ANEV Associazione Nazionale Energia Vento Roma 100 employees wind energy Italia; Enercon Italia + Vestas Italia + Siemens Gamesa Italia + Nordex Italia 600 employees wind farms 11.6 GW installed Italia 2024 + 80% on-shore + 20% off-shore + new auctions FER 1 GSE Gestore Servizi Energetici Roma 1300 employees electricity grid management + renewable incentives.

Italia bridges + civil engineering wind: Ponte Stretto Messina progetto 2026-2032 3.3 km longest single-span suspension bridge world Webuild Salini Impregilo + Eurolink consortium + IHI Mitsubishi + Sercoss Acciaierie + Pizzi-Modesto + Studio De Miranda + Studio Pietrangeli + Saipem; Ponte Morandi Genova ricostruzione Genoa Bridge 1100 m 2018-2020 Renzo Piano + Italferr + Webuild + Fincantieri Infrastructure subsequent VLM Viadotto Polcevera; Italferr engineering Italian Railways subsidiary Roma 1700 employees civil engineering railway bridges + tunnels Italy + Europe; Sacaim Roma maritime + bridges; ANAS Roma 1928 6000 employees national highway + bridges + viaducts strade statali; ITALFERR FS Group; FS Italiane Roma 1905 90000 employees Trenitalia + RFI Rete Ferroviaria Italiana 17000 km railway bridges + tunnels.

Tabella valori Strouhal

• St = f * L / u: definizione
• St<0.01: very low quasi-steady
• 0.01<St<0.1: low slow oscillation
• 0.1<St<0.3: Karman vortex street
• 0.3<St<1: high frequency unsteady
• St>1: very high extreme
• Karman cilindro Re 250-250k: St~0.21
• Re<47: no shedding stable wake
• Re 47-200: St=0.212*(1-21.2/Re) Roshko
• Re 250k-3M: drag crisis St~0.30-0.50
• Re 3M-10M: supercritical St~0.30
• Re>10M: post-supercritical St~0.21
• Strouhal 1878 Praga: Annalen Physik 5:216
• Karman 1911 Gottingen: vortex street pattern
• Roshko 1954 NACA TN 3169: Caltech
• Williamson 1996 ARFM 28:477
• Italia Politecnico Milano DAER 1863: 4000 + GVPM 14x4 m
• Italia Politecnico Torino DIMEAS 1859: 5000 + GVT
• Italia Sapienza DIMA 1303: 4000
• Italia CIRA Capua 1985: 280 SCIROCCO 70 MW + GHIBLI 2 MW
• Italia CNR-INM Roma INSEAN: 470 m bacino
• Italia Leonardo 1948: 51000 Eurofighter
• Italia Avio Colleferro 1908: 5000 Vega Ariane
• Italia Saipem 1957: 32000 EPC offshore
• Italia Fincantieri 1959: 22000 cantieri navali
• Italia Webuild Salini 80000: bridges
• Italia ATR Pomigliano: 50/50 Airbus Leonardo
• Italia D-Orbit Como 200: orbital transfer
• Italia Ponte Messina: 3.3 km 2026-2032
• Italia ANEV wind 11.6 GW: 2024

Esempi numerici dettagliati

Esempio 1 - User-supplied frequency. f=10 Hz, L=0.05 m cilindro, u=1.0 m/s. St = 10*0.05/1 = 0.5. High frequency strong unsteadiness. Italia: laboratorio fluidodinamica Politecnico Milano DAER GVPM Galleria Vento Politecnico Milano 14x4 m boundary layer wind tunnel.

Esempio 2 - Re basso laminar shedding. Re=100, L=0.01 m, u=0.05 m/s. f = 0.212*(1-21.2/100)*0.05/0.01 = 0.836 Hz. St = 0.836*0.01/0.05 = 0.167. Karman vortex street regime. Italia: idrodinamica navale CNR-INM INSEAN bacino 470 m + Politecnico Genova DICCA fluidodinamica navale.

Esempio 3 - Karman vortex street classic. Re=5000, L=0.1 m cilindro, u=1.0 m/s. f = 0.21*1.0/0.1 = 2.1 Hz. St = 0.21. Karman vortex street regime classic. Italia: chimneys Eni 5 raffinerie + Versalis Mantova Ferrara + ponti viadotti ANAS 6000 + Italferr Roma 1700.

Esempio 4 - Supercritical regime. Re=500000, L=1.0 m struttura, u=10 m/s. f = 0.30*10/1 = 3.0 Hz. St = 0.30. Supercritical drag crisis regime. Italia: marine cables offshore Saipem 32000 + Fincantieri 22000 cantieri navali pipelines + Snam 30000 km gasdotti + bridges Webuild Stretto Messina 3.3 km design.

Esempio 5 - Chimney wind engineering. f=5 Hz, L=3.0 m diametro chimney, u=15 m/s wind. St = 5*3/15 = 1.0. Very high extreme unsteady. Italia: chimneys industriali Eni raffinerie + Versalis polimeri + Snam compressor stations + Tecnimont 6000 EPC chemical reactor + chimney engineering.

Limiti e considerazioni

1. Geometria assumption: St depends body shape - cilindro circolare St~0.21 Karman; cilindro quadrato St~0.13; airfoil dipende profilo + angle of attack; sphere St~0.18-0.22.
2. Reynolds dependence: St(Re) curve tabulata Roshko 1954 + Williamson 1996 + Norberg 2003 cilindro classic curve.
3. Surface roughness: rough cylinders shift drag crisis Re_crit lower + St curve modifications. Smooth vs rough D/k ratio matters.
4. Free stream turbulence: Tu intensity affects St + drag crisis transition. High Tu reduces transition Re.
5. Aspect ratio: finite length cylinders L/D < 100 end effects modify shedding + St.
6. Lock-in resonance: when f_shedding = f_natural structure -> lock-in resonance amplification. Tacoma Narrows 1940 disastro famoso flutter aeroelastico.
7. Multi-cylinder arrays: heat exchanger tube banks + bridge cables interference effects shift St.
8. Compressibility effects: high Mach M>0.3 compressible transonic shock-vortex interaction modifies St.
9. Three-dimensional: 3D wake structures Mode A (Re~190) Mode B (Re~260) span-wise instabilities.
10. Numerical CFD: DNS Direct Numerical Simulation + LES Large Eddy Simulation + URANS Unsteady RANS resolve vortex shedding f. RANS time-averaged misses unsteady.

Errori comuni

L definizione characteristic length: cilindro D diametro; corpo bluff body chord c; airfoil corda c. Convention application-specific. Errore confondere -> St sbagliato.

Frequency f in Hz: f = 1/T period; angular omega = 2*pi*f rad/s. Strouhal usa f Hz. Errore omega vs f.

Reynolds dependence ignorata: St(Re) varia. Subcritical St~0.21; supercritical St~0.30; transition jump. Use right value per Re.

Lock-in resonance: f_shedding ~ f_natural amplification flutter. Tacoma Narrows 1940 disastro aeroelastic flutter. Avoid lock-in via design tuning St ne natural frequency.

3D effects 2D assumption: 2D infinite cylinder St idealizzata; finite L/D < 100 end effects modify shedding pattern significantly.

Applicazioni pratiche

1. Wind engineering chimneys industriali: Italia Eni 5 raffinerie + Versalis Mantova Ferrara Brindisi Priolo Gargallo + Snam 30000 km gasdotti compressor stations + Tecnimont 6000 EPC + Saipem 32000 EPC.
2. Marine cables + offshore pipelines: Saipem 32000 oil&gas + Castorone Castor 6 + Maxwell + Snam Mediterraneo gas pipeline + ENI offshore Mediterraneo platforms + Eni Plenitude.
3. Bridges + civil structures: Webuild Salini Impregilo 80000 + Stretto Messina 3.3 km 2026-2032 + Ponte Morandi Genova ricostruzione + Italferr Roma 1700 + ANAS 6000 + RFI 17000 km.
4. Heat exchanger tube banks: Eni R&D San Donato 1500 + Tecnimont Maire 6000 EPC + Ansaldo Energia Genova 3000 + Versalis Eni petrochemical heat exchanger design vortex shedding tube bank vibration.
5. Airfoil flutter + aerospace: Leonardo Aircraft Eurofighter Typhoon + ATR 42/72 Pomigliano + Piaggio Aerospace P.180 Avanti II + Avio Vega Ariane 6 stages.
6. Turbine eoliche: ANEV 100 + Enercon Italia + Vestas Italia + Siemens Gamesa + Nordex 600 wind farms 11.6 GW Italia 2024 vibration tower + blade flutter analysis.
7. Cantieri navali: Fincantieri Trieste 22000 cantieri navali Monfalcone Marghera Castellammare di Stabia + cruise liners MSC Costa + frigate FREMM + U212 submarines vortex shedding hull design.
8. Hypersonic re-entry vehicle: CIRA Capua 280 SCIROCCO 70 MW + GHIBLI 2 MW arc-jet plasma re-entry testing + Sapienza DIMA + Politecnico Torino DIMEAS aerodinamica.
9. Aerodinamica automobilistica: Ferrari Maranello + Lamborghini Sant'Agata Bolognese + Pagani Modena + Pininfarina Cambiano + IDEA Institute Torino aerodinamica car body vortex shedding.
10. Universita ricerca aerodinamica: Politecnico Milano DAER GVPM 14x4 m + Politecnico Torino DIMEAS GVT + Sapienza DIMA + Federico II + Bologna + Padova + Pisa + Politecnico Bari + Genova DICCA naval + Salerno + Firenze + Cagliari.

Aggiornato a luglio 2026