Rankine–Hugoniot conditions
The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, describe the relationship between the states on both sides of a shock wave or a combustion wave (deflagration or detonation) in a one-dimensional flow in fluids or a one-dimensional deformation in solids.
[2][3] The basic idea of the jump conditions is to consider what happens to a fluid when it undergoes a rapid change.
On a microscopic level, they undergo collisions on the scale of the mean free path length until they come to rest in the post-shock flow (but moving in the frame of reference of the wave or of the tube).
The bulk transfer of kinetic energy heats the post-shock flow.
The jump conditions then establish the transition between the pre- and post-shock flow, based solely upon the conservation of mass, momentum, and energy.
This non-reacting example of a shock wave also generalizes to reacting flows, where a combustion front (either a detonation or a deflagration) can be modeled as a discontinuity in a first approximation.
In a coordinate system that is moving with the discontinuity, the Rankine–Hugoniot conditions can be expressed as:[4] where m is the mass flow rate per unit area, ρ1 and ρ2 are the mass density of the fluid upstream and downstream of the wave, u1 and u2 are the fluid velocity upstream and downstream of the wave, p1 and p2 are the pressures in the two regions, and h1 and h2 are the specific (with the sense of per unit mass) enthalpies in the two regions.
If in addition, the flow is reactive, then the species conservation equations demands that to vanish both upstream and downstream of the discontinuity.
Combining conservation of mass and momentum gives us which defines a straight line known as the Michelson–Rayleigh line, named after the Russian physicist Vladimir A. Mikhelson (usually anglicized as Michelson) and Lord Rayleigh, that has a negative slope (since
The mixture is assumed to obey the ideal gas law, so that relation between the downstream and upstream equation of state can be written as where
is also constant across the wave, the change in enthalpies (calorific equation of state) can be simply written as where the first term in the above expression represents the amount of heat released per unit mass of the upstream mixture by the wave and the second term represents the sensible heating.
is the specific heat ratio, which for ordinary room temperature air (298 KELVIN) = 1.40.
If no heat release occurs, for example, shock waves without chemical reaction, then
, that is to say, the pressure increases and the specific volume decreases across the wave (the Chapman–Jouguet condition for detonation is where Rayleigh line is tangent to the Hugoniot curve).
On the contrary, here the specific volume ratio is restricted to the finite interval
Assume that the fluid is inviscid (i.e., it shows no viscosity effects as for example friction with the tube walls).
Furthermore, assume that there is no heat transfer by conduction or radiation and that gravitational acceleration can be neglected.
Such a system can be described by the following system of conservation laws, known as the 1D Euler equations, that in conservation form is: where Assume further that the gas is calorically ideal and that therefore a polytropic equation-of-state of the simple form is valid, where
In the latter case, the dependence of pressure on mass density and internal energy might differ from that given by equation (4).
(the system characteristic or shock speed), which by simple division is given by Equation (9) represents the jump condition for conservation law (6).
For physically real applications this means that the solution should satisfy the Lax entropy condition where
In the case of the hyperbolic conservation law (6), we have seen that the shock speed can be obtained by simple division.
[9][10][11][12][13][14] For shocks in solids, a closed form expression such as equation (15) cannot be derived from first principles.
Instead, experimental observations[15] indicate that a linear relation[16] can be used instead (called the shock Hugoniot in the us-up plane) that has the form where c0 is the bulk speed of sound in the material (in uniaxial compression), s is a parameter (the slope of the shock Hugoniot) obtained from fits to experimental data, and up = u2 is the particle velocity inside the compressed region behind the shock front.
It is therefore a set of equilibrium states and does not specifically represent the path through which a material undergoes transformation.
However, for engineering calculations, it is deemed that the isentrope is close enough to the Hugoniot that the same assumption can be made.
If the Hugoniot is approximately the loading path between states for an "equivalent" compression wave, then the jump conditions for the shock loading path can be determined by drawing a straight line between the initial and final states.
The point on the shock Hugoniot at which a material transitions from a purely elastic state to an elastic-plastic state is called the Hugoniot elastic limit (HEL) and the pressure at which this transition takes place is denoted pHEL.
Above the HEL, the material loses much of its shear strength and starts behaving like a fluid.
Rankine–Hugoniot conditions in magnetohydrodynamics are interesting to consider since they are very relevant to astrophysical applications.
