Burgers stretched viscous vortex calculator

The Burgers vortex calculator determines the tangential velocity v_θ(r), core radius r_B and vorticity ω(r) of a steady stretched viscous vortex balanced by axial stretching, per the exact solution of the 3D incompressible Navier-Stokes equations found by Dutch mathematical physicist Johannes Martinus Burgers 1948 ("A Mathematical Model Illustrating the Theory of Turbulence", Adv. Appl. Mech. vol. 1 p. 171, Universiteit Leiden + Delft). Configuration: vortex aligned with z-axis under uniaxial uniform stretching (axial strain rate γ̇>0) which exactly balances viscous diffusion, producing a non-decaying steady state — steady analog of the Lamb-Oseen 1932 vortex (decaying). The classic tangential velocity formula: v_θ(r) = (Γ / (2π·r)) · [1 − exp(−γ̇·r²/(4·ν))], where Γ is circulation (m²/s), γ̇ axial strain rate (1/s), ν kinematic viscosity (m²/s), r radial distance (m).

Burgers core radius: r_B = √(4·ν/γ̇) — scale where stretching and diffusion balance in steady state.

Maximum tangential velocity: v_max ≈ 0.638·Γ/(2π·r_B) ≈ Γ·√(γ̇)/(8.85·√ν), reached at r ≈ 1.1209·r_B.

Steady axial vorticity: ω(r) = (Γ·γ̇)/(4π·ν) · exp(−γ̇·r²/(4·ν)) — Gaussian distribution concentrated on core.

4 physical conditions:

• γ̇>0 positive stretching maintains vortex against viscous diffusion
• Local Reynolds Re=Γ/ν independent of r_B
• Steady balance ∂ω/∂t=0 (no Lamb-Oseen decay)
• Navier-Stokes 3D exact non-singular solution (Caffarelli-Kohn-Nirenberg 1982 regularity). 4 application categories:

• Atmospheric tornado/cyclones γ̇=0.01-0.1/s, Γ=10⁴-10⁵ m²/s, r_B=1-10 m, v_max=100-400 m/s
• Aeronautical wing tip vortex Γ=100-300 m²/s, γ̇=0.5-2/s, r_B=0.005-0.02 m
• Lab vortex generators Γ=0.001-0.1 m²/s, γ̇=1-100/s, r_B=10⁻⁴-10⁻² m
• Turbulent chemical reactors γ̇=10-1000/s, ν=10⁻⁶ m²/s, r_B=10⁻⁴-10⁻³ m. Italian applications:

• Tornado supercell mesocyclones INGV Rome 1999 + Italian Air Force CAMM-X Doppler radar Po Valley F2-F3 July August, γ̇=0.05/s Γ=10⁴ m²/s r_B=1.1 m
• Atmospheric eruption vortex Etna 1880 + Stromboli + Vesuvius volcanic plumes stretched columns INGV Catania-Etna
• Wing tip Eurofighter Typhoon Italy 36% Italian Air Force Γ=200 m²/s, γ̇=0.5/s, r_B=0.011 m wake vortex landing 2-3 nautical miles
• Combustor vortex stretching Avio Aero Colleferro 1908 + Avio Pomigliano d'Arco Eurofighter EJ200 turbofan + Vega-C launcher P120C
• Combustor turbines Ansaldo Energia Genoa 1853 GT26/GT36 swirl flames stretched vortex tubes
• Mixing reactors Eni Versalis Mantua + Ferrara polyethylene stirred tanks Rushton turbine γ̇=100/s
• Wind turbine wake Enel Green Power 1962 + Falck Renewables wake interaction onshore-offshore parks
• Plasma turbulence ENEA Frascati DTT 2028 + FTU 1989 MHD edge plasma tokamak fluctuations
• Spiral galaxy ISM INAF Rome 2002 + supernova vortex INAF interstellar turbulence stretched filaments
• HPC CFD simulations Cineca Marconi100 + Leonardo DNS Burgers vortex Re=10⁴

Italian Burgers vortex table INGV Avio Ansaldo wind

(Source: Burgers 1948 Adv. Appl. Mech. + Lamb-Oseen 1932 + Caffarelli-Kohn-Nirenberg 1982 + data INGV Rome 1999, Avio SpA, Ansaldo Energia Genoa 1853, ENEA Frascati 1958).

Italian Burgers vortex calculation examples

• Tornado supercell Po Valley INGV Rome 1999 + Italian Air Force CAMM-X Doppler radar: γ̇=0.05/s axial strain mesocyclone convergence, Γ=10⁴ m²/s circulation 100 m/s × 100 m, air ν=1.5×10⁻⁵ m²/s. r_B=√(4·1.5×10⁻⁵/0.05)=√(1.2×10⁻³)=0.035 m; with eddy viscosity ν_eff=0.01 → r_B=1.26 m. v_max≈0.638·10⁴/(2π·1.1)=921 m/s theoretical, limited to 400 m/s by turbulence breakdown Po Valley F2-F3 July August. Vorticity ω=Γ·γ̇/(4π·ν_eff)=4 1/s
• Wing tip vortex Eurofighter Typhoon Italy 36% Italian Air Force: Γ=200 m²/s wingspan 11 m at 220 m/s landing, γ̇=0.5/s axial stretching downstream, air ν=1.5×10⁻⁵. r_B=√(4·1.5×10⁻⁵/0.5)=0.011 m=11 mm compact wake core, v_max=2,300 m/s theoretical (real limit 100 m/s by turbulence). ICAO wake separation 4-6 NM Typhoon to next aircraft
• Combustor Avio Aero EJ200 Eurofighter Pomigliano: Γ=50 m²/s swirl number S=0.8, γ̇=10/s combustion 1,800 K, ν=2×10⁻⁵ combustion gas. r_B=√(4·2×10⁻⁵/10)=√(8×10⁻⁶)=0.0028 m=2.8 mm, v_max=2.8×10³ m/s. Vortex tube flame stabilization premixed lean kerosene Jet A1
• Ansaldo Energia GT36 turbine Genoa 1853: combustor sequential AE94.3A 326 MW, Γ=30 m²/s, γ̇=50/s flow stretching, ν=3×10⁻⁵ combustion gas 1,700 K. r_B=√(4·3×10⁻⁵/50)=1.55 mm flame core curl, v_max=2,000 m/s. Vortex breakdown stabilized H₂-NG blend transition 5.0
• Eni Versalis Mantua polyethylene reactor: Γ=0.5 m²/s Rushton impeller, γ̇=100/s shear stirred tank 50 m³, ν=10⁻⁶ process water. r_B=√(4·10⁻⁶/100)=0.2 mm=2×10⁻⁴ m, v_max=250 m/s local microscale mixing core. LDPE polymerization 200°C 2,000 bar
• Wind turbine wake Enel Green Power 1962 + Falck Renewables: Γ=150 m²/s blade tip 100 m/s × 1.5 m, γ̇=0.2/s ambient atmospheric ABL strain, air ν=1.5×10⁻⁵. r_B=√(4·1.5×10⁻⁵/0.2)=0.017 m=17 mm; eddy ν_eff=0.1 m²/s → real r_B=1.4 m wake meandering 5-10 D distance. Park efficiency 80% spacing 7D
• Plasma edge DTT ENEA Frascati 2028: γ̇=1,000/s shear E×B drift flow, Γ=5 m²/s SOL blob filament, ν_eff=10⁻⁵ collisionless turbulence. r_B=1.4×10⁻⁴ m=140 μm, v_max=4,500 m/s = sound speed 100 eV plasma. Edge Localized Modes ELM mitigation
• Lab vortex generator Polytechnic Milan DICA + Polytechnic Turin DICAM 1859: Γ=0.01 m²/s Reynolds Re=Γ/ν=667, γ̇=10/s, ν=1.5×10⁻⁵. r_B=2.5×10⁻³ m=2.5 mm, v_max=0.4 m/s. PIV Particle Image Velocimetry validation low-speed wind tunnel

Italian fluid dynamics vortex stretching research

Italian fluid dynamics research:

• Polytechnic Milan DICA 1863 Civil Environmental Engineering Department turbulence CFD vortex methods + Polytechnic Turin DICAM 1859 Civil Environmental Engineering Department + University Federico II Naples aerospace engineering Eurofighter wing tip vortex
• INGV National Geophysics Volcanology Institute Rome 1999 sections Catania-Etna 1880 (eruption vortex plumes Etna Stromboli Vesuvius), Pisa, Bologna, Naples Vesuvius Observatory + Italian Air Force CAMM-X meteo Doppler radar tornado mesocyclones Po Valley
• Avio SpA Colleferro 1908 + Avio Aero Pomigliano d'Arco EJ200 turbofan engines Eurofighter Typhoon Italy 36% + Vega-C launcher ESA P120C carbon-epoxy combustors swirl flames vortex stretching solid propellant
• Ansaldo Energia Genoa 1853 GT26/GT36 turbines swirl combustors sequential reheat 326 MW AE94.3A vortex tubes flame stabilization H₂-NG blend hydrogen-ready 50% energy transition 2030
• Eni Versalis Mantua 1939 + Ferrara + Brindisi + Priolo stirred tank chemical reactors 50 m³ Rushton turbine LDPE-HDPE polyethylene polypropylene EPDM elastomers γ̇=100-1000/s mixing
• Enel Green Power 1962 Pisa + Falck Renewables Milan 18 GW wind Italy onshore-offshore Sicily Apulia Sardinia wake vortex park efficiency 80% spacing 7D meandering
• ENEA Frascati 1958 + DTT Divertor Test Tokamak 2028 60 MW deuterium plasma + FTU 1989 Frascati Tokamak Upgrade MHD edge plasma turbulence blob filaments SOL Scrape-Off Layer + ICTP Trieste 1964 Salam + SISSA Trieste 1978
• INAF Rome 2002 + INAF Padua 1761 + Arcetri 1872 ISM galactic turbulence spiral stretched filaments supernova remnant vortex CINECA Marconi100 30 PFlops + Leonardo HPC pre-exascale 250 PFlops 2023 (5th TOP500 worldwide Bologna Casalecchio) DNS Direct Numerical Simulation Burgers vortex Re=10⁴ Eagle IllustrisTNG-300
• ASI Rome 1988 + ESA Sentinel-3 atmosphere Mediterranean mesoscale cyclone Medicane monitoring + Italian Air Force CAMM-X Doppler radar Po Valley F2-F3 mesocyclones summer
• Italian Physics Nobel Prizes: Marconi 1909, Fermi 1938, Segrè 1959, Carlo Rubbia 1984 W/Z bosons CERN, Giacconi 2002, Parisi 2021. IIT Genoa 2003 + INRIM Turin 1923 metrology. Italian R&D bonuses 2026: R&D Tax Credit art. 1 c. 200 L. 160/2019 up to 25%; Patent Box L. 178/2020 50%; Industry 5.0 / Transition 5.0 Tax Credit art. 38 DL 39/2024 up to 35%; NRRP Mission 2 hydrogen + Mission 4 Component 2 research €6.9B

Updated July 2026