RankinehugoniotEdit
Rankine–Hugoniot conditions are the jump relations that govern how physical quantities change across a shock or other discontinuity in a compressible flow. They come from applying the conservation laws—mass, momentum, and energy—across a moving surface that separates two states of the fluid. These relations provide a compact, closed-form link between the upstream (pre-shock) and downstream (post-shock) conditions and are a cornerstone of gas dynamics and related fields.
Named after William Rankine and Pierre-Henri Hugoniot, the conditions are most clearly understood in the framework of the inviscid Euler equations. In that setting, the shock is treated as an infinitesimally thin surface where fluxes of conserved quantities are balanced on both sides. In practice, the Rankine–Hugoniot relations are used as a baseline to analyze high-speed flows, explosive phenomena, and astrophysical shocks, and they underpin numerical schemes that simulate discontinuities in fluids.
These jump conditions remain a powerful tool because they distill complex microphysics into a few fundamental, testable relationships. While modern engineering and physics routinely build on more elaborate models—incorporating viscosity, heat conduction, radiation, and chemical reactions—the Rankine–Hugoniot relations provide clear predictions and sanity checks for a wide range of problems, from supersonic aircraft design to modeling rapid astrophysical events.
Foundations and equations
Across a discontinuity moving with speed S normal to a surface, the one-dimensional Rankine–Hugoniot conditions for a compressible, inviscid fluid can be written in terms of the upstream state (1) and the downstream state (2) as:
- Mass: rho1 u1 = rho2 u2 = j
- Momentum: p1 + rho1 u1^2 = p2 + rho2 u2^2
- Energy: (rho1 e1 + 0.5 rho1 u1^2) u1 + p1 u1 = (rho2 e2 + 0.5 rho2 u2^2) u2 + p2 u2
Here rho is density, u is velocity in the frame where the shock is stationary, p is pressure, and e is the specific internal energy. In vector form, for a general three-dimensional flow with a normal n to the discontinuity, the relations reflect conservation of mass, momentum, and energy fluxes across the surface.
For a simple case often used in practice, a normal shock in a perfect gas with ratio of specific heats gamma, one can express the upstream Mach number M1 = u1/a1 (where a1 is the upstream sound speed) and obtain the classic relations:
- Pressure ratio: p2/p1 = 1 + 2 gamma/(gamma + 1) (M1^2 − 1)
- Density ratio: rho2/rho1 = ((gamma + 1) M1^2) / ((gamma − 1) M1^2 + 2)
- Temperature ratio: T2/T1 = [p2/p1] / [rho2/rho1]
- Downstream Mach number: M2^2 = [1 + (gamma − 1)/2 M1^2] / [gamma M1^2 − (gamma − 1)/2]
In the limit of very strong shocks (M1 >> 1), these reduce to simpler asymptotic forms such as p2/p1 ≈ (2 gamma/(gamma + 1)) M1^2 and rho2/rho1 → (gamma + 1)/(gamma − 1). The precise expressions become more intricate for non-normal shocks or non-ideal equations of state, but the essential structure—the balance of fluxes across a surface—persists.
The Rankine–Hugoniot framework generalizes to many situations, including oblique shocks (where the shock surface is inclined relative to the flow) and non-ideal gases where the equation of state deviates from the perfect-gas form. In those cases, the same conservation principles apply, but the algebra becomes more involved and requires the appropriate equation of state and possibly additional physics (e.g., radiation, ionization).
Key related concepts include the Euler equations (the governing equations for inviscid flow), the concept of Mach number (a measure of flow speed relative to sound), and the notion of a normal shock versus an oblique shock. In modern practice, the Rankine–Hugoniot relations are embedded in numerical methods such as finite volume method approaches and are central to solving the [Riemann problem] that arises at interfaces.
Applications
Rankine–Hugoniot relations are used across a wide spectrum of disciplines:
Aerospace and aerodynamics: they help predict the behavior of shocks around high-speed aircraft and in jet engines, supporting design decisions for performance and stability. They also provide benchmarks for computational models of compressible flow. See aerodynamics and compressible flow.
Detonation and explosives: in detonation theory, the jump conditions describe how pressure, density, and velocity change across the detonation front, informing safety margins and performance. See detonation.
Astrophysics: shocks are common in supernova remnants, star formation regions, and accretion flows. Rankine–Hugoniot relations underpin the basic accounting of energy and momentum transfer across these fronts. See supernova remnant and astrophysical shock.
Inertial confinement fusion and high-energy-density physics: radiative and shock fronts in dense plasmas are analyzed with these jump conditions as a starting point, before adding more complex physics. See inertial confinement fusion.
Computational fluid dynamics: numerical schemes for shock-capturing frequently rely on the Rankine–Hugoniot framework to define fluxes and to construct robust solvers, including various Riemann solver approaches and Godunov's method.
Limitations and real-world considerations
While the Rankine–Hugoniot relations are powerful, they are derived under idealized assumptions. Real flows often involve viscous and heat-conducting effects (Cartier, boundary layers, and finite dissipation), radiation, chemical reactions, and non-equilibrium thermodynamics. In such cases:
Viscosity and conduction: the pure jump conditions of the inviscid theory are complemented by the Navier–Stokes equations, which smear the discontinuity over a finite thickness and alter post-shock states slightly. See Navier–Stokes equations.
Real-gas and phase effects: at high pressures and temperatures, the equation of state deviates from the perfect-gas model, and phase changes or ionization can occur, modifying the jump relations. See equation of state.
Radiation and chemistry: radiative losses or gains, and fast chemical reactions, can carry energy across the front, changing the downstream state from the purely adiabatic prediction. See radiation hydrodynamics and chemical kinetics.
Non-ideal and relativistic regimes: in extreme environments (e.g., relativistic jets, some astrophysical contexts), the appropriate governing equations are the relativistic Euler equations, and the corresponding jump conditions differ from the classical ones. See relativistic hydrodynamics.