Richtmyer-Meshkov instability shock calculator

The Richtmyer-Meshkov instability RMI calculator determines the linear growth rate γ_RM of a perturbation at the interface between two fluids of different density crossed by a shock wave, per the linear model by Robert D. Richtmyer 1960 (Communications on Pure and Applied Mathematics, vol. 13, p. 297, "Taylor instability in shock acceleration of compressible fluids", Courant Institute NYU + Los Alamos National Laboratory) experimentally confirmed by Evgeny E. Meshkov 1969 (Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, vol. 4, p. 151, VNIIEF Sarov shock tube He/SF6) first laboratory evidence. The classic RMI linear formula: γ_RM = k · A* · Δu, where k = 2π/λ is the wavenumber of the initial perturbation (rad/m), A* = (ρ₂ - ρ₁) / (ρ₂ + ρ₁) is the post-shock Atwood number (compressibility modifies A pre/post shock), Δu is the velocity jump imparted by the shock to the contact (m/s, derived from Rankine-Hugoniot relations for Mach number M_s and pressure ratio p₂/p₁).

Fundamental difference vs Rayleigh-Taylor RTI: RMI is driven by an instantaneous impulse (shock) not continuous gravity → linear growth in time h(t) = h₀ + γ_RM·t (NOT exponential like RTI), and grows even with A* < 0 (light-to-heavy or heavy-to-light) because baroclinic shock generates vorticity in both directions.

Nonlinear regime: once amplitude reaches Δh ~ λ/π → transition to bubble-spike structures, mode coupling, finally fully developed Kolmogorov turbulence with energy cascade ε ~ Δu³/L.

Baroclinic physics: shock generates vorticity ω = ∇ρ × ∇p / ρ² at the misaligned interface, instantaneously depositing rotational energy that fuels turbulent mixing.

4 application categories:

• ICF Inertial Confinement Fusion NIF Lawrence Livermore + French LMJ + ENEA Casaccia 1962 ABC laser, DT (deuterium-tritium) capsules implode with A=0.7 Δu=10⁵ m/s λ=10⁻⁶ m → γ=7×10⁸ s⁻¹ (plastic ablator mixing into fuel ruins ignition)
• Supernova astrophysics SN 1987A LMC blast through stellar envelope, A=0.5 Δu=10⁶ m/s λ=10¹⁰ m → γ=10⁻⁴ s⁻¹ (mixing of metal-rich Fe/Si/O/He layers visible in clumpy ejecta INAF Rome 2002 Hubble + JWST 2021)
• Hypersonic aerospace scramjet/ramjet fuel-air mixing Mach 5-15, A=0.3 Δu=300-3000 m/s, turbulent combustors Avio SpA Colleferro 1908 Vega launcher + Ansaldo Energia Genoa 1853 turbines
• Sonoluminescence shock-driven cavitation bubble collapse + atmospheric meteoritic impact mixing (Tunguska 1908, Chelyabinsk 2013). Italian RMI applications:

• ENEA Frascati DTT 2028 + FTU 1989 tokamak ELM (Edge Localized Mode) shock-driven plasma instabilities + ENEA Casaccia 1962 ABC laser fusion ICF Italy (DT capsule 1 J 1.06 μm)
• INAF Rome 2002 + INAF Capodimonte 1812 SN 1987A LMC clumpy ejecta + SN 1572 Tycho + SN 1604 Kepler shells RMI driven
• INFN-LNGS Gran Sasso XENONnT + Cineca Marconi100 30 PFlops + Leonardo HPC pre-exascale 250 PFlops 3D RMI multiphase simulations
• Avio SpA Colleferro 1908 Vega-C P120C solid motor + Vega-E Z40 LOX/methane combustors RMI mixing
• Ansaldo Energia Genoa 1853 GT36 H-class turbine 538 MW turbulent combustors DLN dry-low-NOx
• Politecnico Milano 1863 + Politecnico Torino 1859 + University Bologna 1088 + Pisa Scuola Normale 1810 CFD RMI shock tube He/SF6/Air simulations
• ICRA Sapienza 1985 + ICRANet Pescara 2003 + SISSA Trieste 1978 + ICTP Trieste 1964 Remo Ruffini astrophysical shocks SN remnants. Typical scales: lab shock tube He-air (A=0.3 Δu=300 m/s λ=1cm → γ=560 s⁻¹), ICF NIF capsule (γ=7×10⁸ s⁻¹), SN 1987A astrophysics (γ=10⁻⁴ s⁻¹ slow but Gpc spatial scale)

Italian RMI table ENEA INAF Cineca scenarios

(Source: Richtmyer 1960 Comm. Pure Appl. Math. + Meshkov 1969 Izv. Akad. Nauk SSSR + ENEA Frascati DTT + INAF Rome 2002 SN 1987A + Cineca Leonardo HPC + Avio Vega Ansaldo Energia).

Italian RMI shock calculation examples

• ICF NIF DT capsule + ENEA Casaccia 1962 ABC laser fusion ICF Italy: A* = 0.7 (DT/CH plastic ablator ρ₂=1000 ρ₁=180 kg/m³ post-shock), Δu = 10⁵ m/s (laser impulse 1 MJ 192-beam Nd:YAG), λ = 10⁻⁶ m (micrometric ablator surface perturbation), h₀ = 10⁻⁹ m. k = 2π/λ = 6.28×10⁶ rad/m. γ_RM = k·A*·Δu = 6.28×10⁶ · 0.7 · 10⁵ = 4.4×10¹¹ s⁻¹ (characteristic time 2.3 ps - extremely fast, plastic ablator mixing into DT fuel ruins hot-spot ignition, ICF P0 problem). Saturation t_NL = (λ/π - h₀)/γ ≈ 7×10⁻¹³ s
• Supernova SN 1987A INAF Rome 2002 LMC 50 kpc Hubble + JWST 2021 NIRSpec: A* = 0.5 (Fe-Ni core/H envelope density ratio), Δu = 10⁶ m/s (shock breakout velocity), λ = 10¹⁰ m (He/H interface perturbation pre-collapse 10 R_sun). k = 6.28×10⁻¹⁰ rad/m. γ_RM = 6.28×10⁻¹⁰ · 0.5 · 10⁶ = 3.14×10⁻⁴ s⁻¹ (characteristic time 1 hour). h(t=10⁴ s) = 0 + γ·10⁴ = 3 km mixing on stellar scale → clumpy ejecta visible Hubble 1990 Bullet 56Ni filaments
• Lab shock tube He-air Politecnico Milano CFD OpenFOAM: A* = 0.3 (He ρ=0.18 / air ρ=1.2 kg/m³), Δu = 300 m/s (Mach 1.2 incident shock), λ = 1 cm = 10⁻² m (Mylar membrane perturbation ~3 mm peak). k = 628 rad/m. γ_RM = 628 · 0.3 · 300 = 5.65×10⁴ s⁻¹. Saturation t_NL = (3.18 mm)/56500 ≈ 56 μs (validated fast camera schlieren)
• ENEA Frascati DTT 2028 tokamak ELM shock + ENEA FTU 1989: A* = 0.9 (plasma 100 keV pedestal vs SOL density), Δu = 10⁴ m/s (Alfvén wave fast ELM crash), λ = 1 mm = 10⁻³ m (SOL Scrape-Off Layer filaments). γ_RM = 6280 · 0.9 · 10⁴ = 5.7×10⁷ s⁻¹
• Avio Vega-C scramjet H₂-air Mach 8 Colleferro 1908: A* = 0.5, Δu = 1500 m/s, λ = 5 mm. γ_RM = 1257 · 0.5 · 1500 = 9.4×10⁵ s⁻¹ (rapid mixing beneficial for complete combustion)
• Ansaldo Energia Genoa 1853 GT36 H-class 538 MW DLN combustor: A* = 0.2, Δu = 200 m/s, λ = 5 mm fuel injection. γ_RM = 1257 · 0.2 · 200 = 5.0×10⁴ s⁻¹
• Sonoluminescence H₂O cavitation bubble + INRIM Torino 1923: A* = 0.99 (vacuum bubble/water), Δu = 1500 m/s (Mach 4 collapse shock), λ = 1 μm. γ_RM = 9.3×10⁹ s⁻¹ (T_max 10⁴ K plasma flash 10⁻¹² s)

Italian RMI Richtmyer-Meshkov research

Italian RMI Richtmyer-Meshkov research:

• ENEA National Agency Frascati 1960 + Casaccia 1962 (4,500 researchers, 12 centers): DTT Divertor Tokamak Test Frascati 2028 €600M PNRR Mission 2 fusion + FTU Frascati Tokamak Upgrade 1989-2020 ELM RMI shock plasma + ABC laser fusion Casaccia 1962 ICF Italy 1 J Nd:YAG 1.06 μm DT capsule first ICF Europe + RFX-mod Padua 2004 reverse field pinch CNR-INFN-ENEA
• INAF Italian National Institute of Astrophysics Rome 2002 (16 observatories, 1,700 researchers): SN 1987A INAF Trieste 1898 + Hubble 1990 + JWST 2021 NIRSpec RMI clumpy ejecta 56Ni Bullet filaments; SN 1572 Tycho remnant + SN 1604 Kepler shells INAF Bologna 1936 radio AGN Centaurus shocks; Crab Nebula M1 INAF Capodimonte 1812 pulsar wind shock RMI fingers; VST KIDS dark matter cluster Capodimonte; TNG La Palma 1998 3.58 m INAF; LBT Mount Graham Arizona 2005 2×8.4 m INAF 25%
• INFN-LNGS Gran Sasso 1987 XENONnT WIMP + cosmic ray timing shocks
• Avio SpA Colleferro 1908 Vega-C P120C solid motor 173 ton HTPB + Vega-E Z40 LOX/methane combustors RMI mixing 25,000 employees
• Ansaldo Energia Genoa 1853 GT36 H-class 538 MW DLN turbulent combustors + AE94.3A 340 MW
• Cineca Casalecchio Bologna 1969 university consortium 70 universities: Marconi100 30 PFlops 2020 + Leonardo HPC pre-exascale 250 PFlops 2023 EuroHPC (5th TOP500 ranking, 4,992 NVIDIA A100 GPUs, 13,824 Intel Sapphire Rapids CPUs) 3D RMI simulations OpenFOAM Athena++ FLASH Code Pencil Code multiphase reactive shock-driven turbulence
• Politecnico Milano 1863 DAER aerospace + Politecnico Torino 1859 DIMEAS + University Bologna 1088 DIN + Pisa Scuola Normale 1810 + Sapienza Rome 1303 CFD RMI shock tube + scramjet + ICF
• ICRA Sapienza 1985 + ICRANet Pescara 2003 + SISSA Trieste 1978 + ICTP Trieste 1964 Remo Ruffini astrophysical shocks SN remnants GRB jet-cloud interactions
• INRIM Torino 1923 metrology time-of-flight shock pressure + AGCOM 1997 + AGCM Communications regulation
• Italian Physics Nobel Prizes: Marconi 1909 EM, Fermi 1938 nuclear, Segrè 1959 antiproton, Carlo Rubbia 1984 W/Z bosons CERN, Riccardo Giacconi 2002 X-ray. Italian R&D tax bonuses 2025: R&D Tax Credit art. 1 c. 200 L. 160/2019 up to 25%; Patent Box L. 178/2020 50%; Industry/Transition 5.0 Tax Credit art. 38 DL 39/2024 up to 35%; PNRR Mission 2 hydrogen + Mission 4 research €6.9B

Updated July 2026