Rankine-Hugoniot shock wave compressible flow calculator

The Rankine-Hugoniot shock relations calculator determines the jump ratios across a stationary normal shock wave in a compressible fluid (ideal gas with adiabatic exponent γ = c_p/c_v), starting from the pre-shock Mach number M₁. The laws were formulated by Scottish engineer William John Macquorn Rankine 1870 "On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance" Phil. Trans. Roy. Soc. London (vol. 160, p. 277) and independently generalized by French physicist Pierre Henri Hugoniot 1887-1889 "Mémoire sur la propagation des mouvements dans les corps et spécialement dans les gaz parfaits" Journal de l'École Polytechnique (Cahiers 57-58), Paris Polytechnic, the first rigorous treatment of entropy jump across irreversible shock. The 4 fundamental formulas for ideal gas, derived from conservation of mass, momentum and energy across the discontinuity:

• Density: ρ₂/ρ₁ = (γ+1)·M₁² / ((γ-1)·M₁² + 2)
• Pressure: p₂/p₁ = 1 + (2γ/(γ+1))·(M₁² - 1)
• Temperature: T₂/T₁ = (p₂/p₁)·(ρ₁/ρ₂) = [1+(2γ/(γ+1))(M₁²-1)] · [(γ-1)M₁²+2]/((γ+1)M₁²)
• Post-shock Mach: M₂² = ((γ-1)M₁² + 2) / (2γM₁² - (γ-1)). Jump physics: shock is an irreversible discontinuity ~10⁻⁷ m thick (mean free path), in which entropy increases (ΔS > 0, second law of thermodynamics), velocity decreases (M₂ < 1, always subsonic post-normal shock), pressure/density/temperature increase. For diatomic air γ = 1.4 (N₂, O₂); for monoatomic gases γ = 5/3 (He, Ar, supernova plasma); for polyatomic vapors γ ≈ 1.3 (CO₂, H₂O, CH₄). Asymptotic limits: M₁ → 1⁺ ratios approach 1 (no shock, reversible acoustic wave); M₁ → ∞ strong shock limit: ρ₂/ρ₁ → (γ+1)/(γ-1) = 6 for γ=1.4 (max 6× compression, M₁-independent), p₂/p₁ → ∞ unlimited, T₂/T₁ → ∞ unlimited (kinetic energy → heat). Italian aerospace applications:

• CIRA Italian Aerospace Research Centre Capua 1984 SCIROCCO Mach 12 plasma wind tunnel hypersonic 70 MW (largest in Europe, re-entry simulation)
• Avio SpA Colleferro Rome 1908 Vega/Vega-C launcher 4 stages P80/Z40/Z9/AVUM Mach 6+ booster shock CFD
• Leonardo SpA Rome 1948 + Alenia Aermacchi Eurofighter Typhoon Italy 36% Mach 2.0 + Tornado IDS Cameri Mach 2.2
• MBDA Italy La Spezia 1996 tactical missiles Aspide Mach 4, Aster 30 Eurosam Mach 4.5, Marte Mach 0.8 anti-ship
• ASI 1988 missions Cassini-Huygens 1997 (Saturn ring shocks), BepiColombo 2018 (solar wind shock heliosphere), Galileo Galilei (cosmic rays)
• ESA Solar Orbiter 2020 ASI 25% in-situ measurements solar wind termination shock 90 AU Voyager 1 (2004) and 2 (2007)
• INAF Rome 2002 supernova remnants SN 1572 Tycho, SN 1604 Kepler, Cassiopeia A blast waves Sedov-Taylor self-similar M₁=10-100
• INFN Frascati 1957 + Cineca Bologna Marconi100/Leonardo HPC pre-exascale 250 PFlops MHD detonation plasma simulations and ITER instabilities

Italian aerospace Rankine-Hugoniot tables

(Source: Rankine 1870 Phil. Trans. Roy. Soc. + Hugoniot 1887-1889 J. École Polytechnique + NACA Report 1135 1953 compressible flow tables + manuals CIRA Capua 1984 and Polytechnic Milan 1863 hypersonic aerodynamics).

Italian Rankine-Hugoniot examples

• Eurofighter Typhoon Italy 36% Leonardo Cameri Mach 2.0 air γ=1.4: M₁=2.0, M₁²=4. ρ₂/ρ₁ = 2.4·4/(0.4·4+2) = 9.6/3.6 = 2.667; p₂/p₁ = 1+(2.8/2.4)·3 = 4.500; T₂/T₁ = 4.5/2.667 = 1.688; M₂² = (0.4·4+2)/(2.8·4-0.4) = 0.333 → M₂ = 0.577. Pressure behind shock 4.5× ambient (Cameri, AMX, Decimomannu tests)
• Tornado IDS Mach 2.2 Alenia Aermacchi: M₁²=4.84. ρ₂/ρ₁ = 2.952; p₂/p₁ = 5.480; T₂/T₁ = 1.857; M₂ = 0.547 subsonic
• Avio Vega launcher P80 Colleferro Rome 1908 Mach 6.0 high atmosphere: M₁²=36. ρ₂/ρ₁ = 5.268; p₂/p₁ = 41.833; T₂/T₁ = 7.941 (gas heated 7.94×); M₂ = 0.404. Vega-C front bow shock 800 K aerodynamic heating
• CIRA SCIROCCO Capua 1984 plasma wind tunnel Mach 12 hypersonic (70 MW): M₁²=144. ρ₂/ρ₁ = 5.799 (close to limit 6); p₂/p₁ = 167.833; T₂/T₁ = 28.944 (T₂ ≈ 8,700 K dissociated plasma); M₂ = 0.392. Tests Cosmo, IXV, PRIDE, Space Rider ESA heat shield
• Re-entry Space Shuttle/Cosmo Mach 25 high atmosphere ASI: M₁²=625. ρ₂/ρ₁ = 5.907 (very close to 6); p₂/p₁ = 728.5; T₂/T₁ = 123.32 (T₂ ≈ 30,000 K fully ionized plasma); M₂ = 0.388. Re-entry radio black-out
• MBDA Aster 30 Eurosam La Spezia missile Mach 4.5: M₁²=20.25. ρ₂/ρ₁ = 4.812; p₂/p₁ = 23.458; T₂/T₁ = 4.875; M₂ = 0.422. Missile nose heating 1,450 K
• Supernova remnant SN 1572 Tycho INAF Rome 2002 blast wave plasma γ=5/3 (monoatomic ionized): M₁=1000 strong shock. ρ₂/ρ₁ → (5/3+1)/(5/3-1) = 4.000 (plasma vs 6 air limit); p₂/p₁ ≈ 9.4×10⁵; T₂/T₁ ≈ 2.3×10⁵ (T₂ ≈ 10⁸ K X-ray emission Cassiopeia A). Sedov-Taylor self-similar R(t) ∝ t^(2/5)
• ITER plasma instability Cineca Bologna 1969 Marconi100 30 PFlops MHD simulations: detonation wave fusion D-T M₁≈30, plasma γ=5/3 → ρ₂/ρ₁ ≈ 3.99, T₂/T₁ ≈ 281

Italian aerospace shock waves research

Italian aerospace:

• CIRA Italian Aerospace Research Centre Capua 1984 350 researchers, hypersonic tunnels: SCIROCCO Mach 12 plasma 70 MW (largest in Europe, re-entry and heat shield simulation), GHIBLI Mach 5, PWT plasma wind tunnel; PRORA national program
• Polytechnic Milan 1863 DAER aerospace department CFD shock waves sub/supersonic tunnel; Polytechnic Turin 1859 DIMEAS aerodynamics + Master in Space Engineering; University Federico II Naples 1224 aerospace engineering DICII; Sapienza Rome 1303 + University of Pisa 1343 + University of Padua 1222
• Avio SpA Colleferro Rome 1908 1,000 employees, Vega launcher 4 stages solid + AVUM liquid, Vega-C 2022 (3.3 t LEO), Ariane 6 booster P120C, Mach 6+ stage CFD shock CIRA
• Leonardo SpA Rome 1948 ex-Finmeccanica 50,000 employees, Eurofighter Typhoon Italy 36% (Cameri AMX assembly + EJ200 engine), Tornado IDS Cameri 1980 Mach 2.2, M-346 trainer Aermacchi Mach 1, AW101 Helicopters Cascina Costa, F-35A/B Cameri FACO
• MBDA Italy La Spezia 1996 tactical missiles (Airbus 37.5% + BAE 37.5% + Leonardo 25% JV): Aspide Mach 4 SAM, Aster 30 SAMP/T Eurosam Mach 4.5 (Italy 50% + France 50%), Marte ER Mach 0.8 anti-ship, Meteor BVRAAM Mach 4, Storm Shadow/SCALP cruise
• Italian Space Agency ASI Rome 1988 250 employees, missions: Cassini-Huygens 1997 (Saturn arrival 2004 ring shocks), BepiColombo 2018 (Mercury arrival 2025 magnetopause), Galileo Galilei (L-band navigation constellation), COSMO-SkyMed gen.2 (X-band SAR), PRISMA 2019 hyperspectral, IXV Vega 2015 re-entry prototype
• ESA Solar Orbiter 2020 ASI 25% ESA-NASA mission perihelion 0.3 AU 10 instruments (SWA Italy + EPD), in-situ termination shock heliosphere; Voyager 1 (2004) and Voyager 2 (2007) crossed termination shock 90 AU
• INAF Rome 2002 supernova remnants: SN 1572 Tycho (radio + X-ray Cassini AGN), SN 1604 Kepler, Cassiopeia A SNR (350 years, Sedov-Taylor blast wave R∝t^2/5), SN 1987A LMC (40 light years shock)
• Italian Air Force CAMM-X meteo + INGV Etna 1999 volcanic explosive shocks Stromboli
• ICTP Trieste 1964 + SISSA Trieste 1978 + IIT Genoa 2003 CFD MHD plasma shock research
• Cineca Casalecchio Bologna 1969 Marconi100 (IBM Power9 30 PFlops 8th TOP500 2020) + Leonardo HPC pre-exascale (Atos BullSequana XH2000 250 PFlops 5th TOP500 2023) cosmological N-body simulations and CFD shock detonation. Italian aerospace tax bonuses 2025: R&D Tax Credit L. 160/2019 up to 25% research expenses; Patent Box L. 178/2020 50% IP income deduction; Industry 5.0/Transition 5.0 Tax Credit DL 39/2024 up to 35% green-digital twin tunnel investments; PNRR Mission 2 hydrogen (€3.2B) + Mission 4 research (€6.9B) + Mission 1 space (€1.7B)

Updated July 2026