Rayleigh-Taylor instability RTI growth rate calculator

The Rayleigh-Taylor RTI calculator determines the exponential growth rate γ of the instability at the interface between two superposed fluids of different densities, when the denser fluid ρ₂ lies above the lighter fluid ρ₁ in an antigradient gravitational field, per the classical formulation of Lord Rayleigh 1883 "Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density" (Proceedings of the London Mathematical Society, vol. 14, p. 170) + generalization to accelerated fluids by Sir Geoffrey Ingram Taylor 1950 Cambridge "The instability of liquid surfaces when accelerated in a direction perpendicular to their planes" (Proc. Royal Society A, vol. 201, p. 192). The classical formula for inviscid surface-tensionless fluids: γ = √(A · g · k) [s⁻¹], with A = Atwood number = (ρ₂ - ρ₁) / (ρ₂ + ρ₁) (range -1 to +1; A>0 = denser above = unstable; A<0 = stable), g gravitational acceleration m/s² (or system effective acceleration), k = 2π/λ perturbation wavenumber (λ wavelength). The underlying physics: the perturbed interface h(x,t) = h₀·cos(kx)·exp(γ·t) grows exponentially when ρ₂ > ρ₁ since gravitational potential energy is released converting metastable into stable configuration (denser below).

The doubling time: τ_double = ln(2)/γ ≈ 0.693/γ. The maximum growth wavelength (cutoff) for fluids with surface tension σ and gravity: k_max = √(g·A·ρ_avg/(3σ)) for inviscid fluids (above this k surface tension stabilizes). Viscosity ν reduces γ and introduces cutoff k_visc ~ (g·A/ν²)^(1/3). The 4 RTI regime categories:

• Linear h/λ << 1, exponential growth γ = √(Agk) (duration 0-3·τ_double)
• Weakly non-linear h/λ ~ 0.1, bubble and spike formation (Layzer 1955)
• Self-similar turbulent h = α·A·g·t² with α ≈ 0.06 bubbles, α ≈ 0.03 spikes (Dimonte 2004 LANL)
• Multimode mixing layer ICF mixing zones. The Italian industrial applications:

• Inertial Confinement Fusion ICF DTT Frascati 2028 ENEA + FTU 1989 plasma fusion ENEA Frascati laser-compressed target RTI ablation front A~1 g~10¹⁵ γ~10⁹ s⁻¹
• Gas-turbine combustors Ansaldo Energia Genoa 1853 GT26 GT36 power plants RTI air-fuel turbulent mixing flame
• Avio SpA Colleferro 1908 Vega VV23 P120C launchers solid HTPB+AP propellant combustion RTI grain regression
• ITER Cadarache 2025 + DTT Frascati 2028 fusion reactors ELMs edge-localized modes plasma RTI MHD ballooning
• Explosive volcanism INGV Rome 1999 Etna 1880 magma chamber RTI gas bubble fragmentation
• Cumulonimbus atmospheric mixing tops cold-warm air RTI
• Salt finger oceanography halocline thermohaline double-diffusion
• Supernova SN 1987A INAF Rome 2002 Ni-rich ejecta beyond He-shell RTI mixing 56Ni→56Co→56Fe
• ABC laser Asterix-class ENEA Casaccia 1962 plastic ICF targets
• XENONnT underground detectors INFN-LNGS Gran Sasso liquid xenon suspension RTI vibration isolation. The typical RTI scales: lab water/oil λ=10 cm γ~18 s⁻¹; mercury/water λ=1 cm γ~22.8 s⁻¹; ICF NIF/DTT plasma DT ablation A~1 g=10⁹ γ~10⁶ s⁻¹; supernova ejecta A~1 g=10⁵ k=10⁻⁹ γ~0.1 s⁻¹ (Mpc scale); volcanism magmatic γ~0.01 s⁻¹

Italian RTI research table ENEA INFN INGV

(Source: Lord Rayleigh 1883 PLMS + G.I. Taylor 1950 Proc. Roy. Soc. A + Chandrasekhar 1961 Hydrodynamic and Hydromagnetic Stability + Dimonte 2004 LANL + ENEA Frascati DTT/FTU + INGV 1999).

Italian RTI calculation examples lab and fusion

• Water above oil kitchen lab Politecnico Milan 1863: ρ₁ = 900 (oil below), ρ₂ = 1000 (water above), A = (1000-900)/(1000+900) = 0.053 (low but positive → unstable). g = 9.81 m/s², λ = 0.1 m → k = 62.83 m⁻¹. γ = √(0.053·9.81·62.83) = √32.67 = 5.72 s⁻¹. τ_double = 0.693/5.72 = 0.121 s. Perturbation doubles in 121 ms (fast!)
• Mercury above water INRIM Turin 1923: ρ₁ = 1000, ρ₂ = 13550, A = (13550-1000)/(13550+1000) = 0.863 (high). g = 9.81, λ = 0.01 m, k = 628 m⁻¹. γ = √(0.863·9.81·628) = √5314 = 72.9 s⁻¹. τ = 9.5 ms (extremely fast)
• ICF DT plasma target DTT Frascati ENEA 2028: A ~ 1 (DT compressed vs corona), g = 10¹⁵ m/s² laser ablation, λ = 10⁻⁶ m → k = 6.28×10⁶. γ = √(1·10¹⁵·6.28×10⁶) = √6.28×10²¹ = 2.5×10¹⁰ s⁻¹. τ = 28 ps. RTI is the main obstacle to inertial fusion ignition (NIF Asterix Frascati ENEA)
• FTU plasma fusion ENEA Frascati 1989: compact tokamak 8 T, RTI MHD ballooning edge plasma instability. A = 0.98, g_eff = 10⁵ m/s² centrifugal, λ = 10⁻³ m. γ = √(0.98·10⁵·6283) = 2.5×10⁴ s⁻¹. τ = 28 μs
• Volcanism Etna fragmentation INGV Rome 1999 Etna 1880: ρ_magma = 2700, ρ_gas = 200, A = 0.862, g = 9.81, λ = 0.5 m. γ = √(0.862·9.81·12.57) = √106.3 = 10.3 s⁻¹. τ = 67 ms (volcanic explosion)
• Combustor turbine Ansaldo Energia Genoa 1853 GT36: ρ_air = 0.5 (preheated compressed), ρ_flame = 50 (CH4-air stoichiometric), A = 0.98, g_eff = 50 m/s² swirl centrifugal, λ = 0.01 m. γ = 175 s⁻¹. τ = 4 ms. RTI flame turbulent mixing
• Vega P120C Avio Colleferro 1908: solid HTPB+AP propellant. ρ_solid = 1800, ρ_gas = 5, A = 0.994, g_eff = 100 m/s² (combustion chamber), λ = 0.01 m. γ = 249 s⁻¹. τ = 2.8 ms grain regression rate 1 cm/s
• Supernova SN 1987A INAF Rome 2002 mixing 56Ni: A ~ 0.98, g_eff = 10⁵ m/s² shock-driven, λ = 10⁹ m (parsec scale). γ = √(0.98·10⁵·6.28×10⁻⁹) = √615 = 0.025 s⁻¹. τ = 28 s but on year scale. Mixing 56Ni→56Co→56Fe light curve observed HST INAF Trieste 1898

Italian RTI research ENEA INFN ASI INGV INAF

The Italian RTI research:

• ENEA Italian National Agency for New Technologies Rome 1960 (8 centers, 2,700 researchers): Frascati FTU 1989 compact tokamak plasma fusion 8 T (ITER analogous, RTI MHD ballooning); Frascati DTT Divertor Tokamak Test 2028 6 T 5.5 m R₀ liquid lithium divertor test (TRL 8 ITER/DEMO, RTI ablation front studies); Casaccia ABC laser 1962 Asterix-class plastic ICF targets keV plasma (RTI ablation); Brasimone fast reactors
• ITER Cadarache France 2025 + Eurofusion + Ansaldo Nucleare Genoa €23B tokamak fusion reactor 500 MW Q=10 (RTI ELMs edge plasma, Italian contribution F4E ENEA INFN Frascati 25%)
• INGV National Institute Geophysics Volcanology Rome 1999 (1,000 researchers, sections Catania-Etna, Naples-Vesuvius, Pisa, Bologna): Etna 1880 INGV Catania magma chamber RTI fragmentation; Stromboli paroxysm; Vesuvius surveillance; Campi Flegrei deformation (RTI magmatic dome)
• INAF Rome 2002 RTI astrophysics: SN 1987A Magellanic Cloud INAF Trieste 1898 ejecta mixing 56Ni; supernova remnants Cas A INAF Bologna; ICM cluster Bullet RTI; cumulonimbus tops INAF Catania-Etna 1880
• INFN-LNGS Gran Sasso Assergi 1987 1,400 m underground (XENONnT, DarkSide, GERDA, Borexino) liquid xenon suspension RTI critical vibration isolation
• Avio SpA Colleferro 1908 Vega VV23 + Vega-C + Vega-E launcher 4-stage P120C+Z40+Z9+AVUM (RTI solid HTPB+AP propellant combustion fragmentation grain regression)
• Ansaldo Energia Genoa 1853 gas turbines GT26 GT36 power plants (RTI swirl combustor NOx<25 ppm flame holding); Ansaldo Nucleare ITER ENEA F4E
• Politecnico Milan 1863 + Politecnico Turin 1859 CFD RTI multiphase OpenFOAM ANSYS Fluent (DICA/DIMEAS fluid mechanics course); University of Bologna 1088 (oldest worldwide) DEI fluid dynamics; Pisa Scuola Normale Superiore 1810 + ICTP Trieste 1964 + SISSA Trieste 1978 plasma physics RTI MHD
• Cineca Casalecchio Bologna 1969 Marconi100 IBM 30 PFlops + Leonardo HPC Atos 250 PFlops (5th TOP500 worldwide 2024) DNS multiphase RTI simulations NEK5000 OpenFOAM (preparation Aurora exascale)
• INRIM Turin 1923 metrology surface tension viscosity ρ SI standards (Du Noüy ring tensiometer ISO/IEC 17025)
• AGCOM Rome 1997 TLC regulation (not directly RTI, but wave propagation simulations)
• Italian Nobel laureates: Marconi 1909, Fermi 1938, Segrè 1959, Carlo Rubbia 1984, Riccardo Giacconi 2002. Carlo Rubbia 1984 W/Z bosons CERN UA1 fluid streamer chamber RTI. Italian R&D tax bonuses 2026: R&D Tax Credit L. 160/2019 art. 1 c. 200 up to 25%; Patent Box L. 178/2020 50%; Industry/Transition 5.0 art. 38 DL 39/2024 up to 35%; PNRR Mission 2 green hydrogen €3.2B + Mission 4 Component 2 research €6.9B ENEA INFN INAF

Updated July 2026