Concept in physics
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. They are named in recognition of the work carried out by Scottish engineer and physicist William John Macquorn Rankine and French engineer Pierre Henri Hugoniot.
The basic idea of the jump conditions is to consider what happens to a fluid when it undergoes a rapid change. Consider, for example, driving a piston into a tube filled with non-reacting gas. A disturbance is propagated through the fluid somewhat faster than the speed of sound. Because the disturbance propagates supersonically, it is a shock wave, and the fluid downstream of the shock has no advance information of it. In a frame of reference moving with the wave, atoms or molecules in front of the wave slam into the wave supersonically. 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. Because the mean free path length is assumed to be negligible in comparison to all other length scales in a hydrodynamic treatment, the shock front is essentially a hydrodynamic discontinuity. The jump conditions then establish the transition between the pre- and post-shock flow, based solely upon the conservation of mass, momentum, and energy. The conditions are correct even though the shock actually has a positive thickness. 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.
Governing Equations.
In a coordinate system that is moving with the discontinuity, the Rankine–Hugoniot conditions can be expressed as:
where "m" is the mass flow rate per unit area, "ρ"1 and "ρ"2 are the mass density of the fluid upstream and downstream of the wave, "u"1 and "u"2 are the fluid velocity upstream and downstream of the wave, "p"1 and "p"2 are the pressures in the two regions, and "h"1 and "h"2 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
formula_1
to vanish both upstream and downstream of the discontinuity. Here, formula_2 is the mass production rate of the "i"-th species of total "N" species involved in the reaction.
Combining conservation of mass and momentum gives us
formula_3
which defines a straight line known as the Michelson–Rayleigh line, named after the Russian physicist (usually anglicized as Michelson) and Lord Rayleigh, that has a negative slope (since formula_4 is always positive) in the formula_5 plane. Using the Rankine–Hugoniot equations for the conservation of mass and momentum to eliminate "u"1 and "u"2, the equation for the conservation of energy can be expressed as the Hugoniot equation:
formula_6
The inverse of the density can also be expressed as the specific volume, formula_7. Along with these, one has to specify the relation between the upstream and downstream equation of state
formula_8
where formula_9 is the mass fraction of the species. Finally, the calorific equation of state formula_10 is assumed to be known, i.e.,
formula_11
Simplified Rankine–Hugoniot relations.
The following assumptions are made in order to simplify the Rankine–Hugoniot equations. 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
formula_12
where formula_13 is the universal gas constant and the mean molecular weight formula_14 is assumed to be constant (otherwise, formula_14 would depend on the mass fraction of the all species). If one assumes that the specific heat at constant pressure formula_16 is also constant across the wave, the change in enthalpies (calorific equation of state) can be simply written as
formula_17
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. Eliminating temperature using the equation of state and substituting the above expression for the change in enthalpies into the Hugoniot equation, one obtains an Hugoniot equation expressed only in terms of pressure and densities,
formula_18
where formula_19 is the specific heat ratio, which for ordinary room temperature air (298 KELVIN) = 1.40. An Hugoniot curve without heat release (formula_20) is often called a "shock Hugoniot", or simply a(n) "Hugoniot". Along with the Rayleigh line equation, the above equation completely determines the state of the system. These two equations can be written compactly by introducing the following non-dimensional scales,
formula_21
The Rayleigh line equation and the Hugoniot equation then simplifies to
formula_22
Given the upstream conditions, the intersection of above two equations in the formula_23-formula_24 plane determine the downstream conditions; in the formula_23-formula_24 plane, the upstream condition correspond to the point formula_27. If no heat release occurs, for example, shock waves without chemical reaction, then formula_28. The Hugoniot curves asymptote to the lines formula_29 and formula_30, which are depicted as dashed lines in the figure. As mentioned in the figure, only the white region bounded by these two asymptotes are allowed so that formula_31 is positive. Shock waves and detonations correspond to the top-left white region wherein formula_32 and formula_33, 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). Deflagrations, on the other hand, correspond to the bottom-right white region wherein formula_34 and formula_35, that is to say, the pressure decreases and the specific volume increases across the wave; the pressure decrease a flame is typically very small which is seldom considered when studying deflagrations.
For shock waves and detonations, the pressure increase across the wave can take any values between formula_36; the steeper the slope of the Rayleigh line, the stronger is the wave. On the contrary, here the specific volume ratio is restricted to the finite interval formula_37 (the upper bound is derived for the case formula_38 because pressure cannot take negative values). If formula_39 (diatomic gas without the vibrational mode excitation), the interval is formula_40, in other words, the shock wave can increase the density at most by a factor of 6. For monatomic gas, formula_41, the allowed interval is formula_42. For diatomic gases with vibrational mode excited, we have formula_43 leading to the interval formula_44. In reality, the specific heat ratio is not constant in the shock wave due to molecular dissociation and ionization, but even in these cases, density ratio in general do not exceed a factor of about .
Derivation from Euler equations.
Consider gas in a one-dimensional container (e.g., a long thin tube). 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:
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where
Assume further that the gas is calorically ideal and that therefore a polytropic equation-of-state of the simple form
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is valid, where formula_19 is the constant ratio of specific heats formula_51. This quantity also appears as the "polytropic exponent" of the polytropic process described by
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For an extensive list of compressible flow equations, etc., refer to NACA Report 1135 (1953).
Note: For a calorically ideal gas formula_52 is a constant and for a thermally ideal gas formula_52 is a function of temperature. In the latter case, the dependence of pressure on mass density and internal energy might differ from that given by equation (4).
The jump condition.
Before proceeding further it is necessary to introduce the concept of a "jump condition" – a condition that holds at a discontinuity or abrupt change.
Consider a 1D situation where there is a jump in the scalar conserved physical quantity formula_54, which is governed by integral conservation law
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for any formula_55, formula_56, formula_57, and, therefore, by partial differential equation 
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for smooth solutions.
Let the solution exhibit a jump (or shock) at formula_58, where formula_59 and formula_60, then
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The subscripts 1 and 2 indicate conditions "just upstream" and "just downstream" of the jump respectively, i.e. formula_61 and w\left(x_s + \epsilon\right) ;/math ;.}} formula_62 is the therefore sign.
Note, to arrive at equation (8) we have used the fact that formula_63 and formula_64.
Now, let formula_65 and formula_66, when we have formula_67 and formula_68, and in the limit
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where we have defined formula_69 (the system "characteristic" or "shock speed"), which by simple division is given by
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Equation (9) represents the jump condition for conservation law (6). A shock situation arises in a system where its "characteristics" intersect, and under these conditions a requirement for a unique single-valued solution is that the solution should satisfy the "admissibility condition" or "entropy condition". For physically real applications this means that the solution should satisfy the "Lax entropy condition"
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where formula_70 and formula_71 represent "characteristic speeds" at upstream and downstream conditions respectively.
Shock condition.
In the case of the hyperbolic conservation law (6), we have seen that the shock speed can be obtained by simple division. However, for the 1D Euler equations (1), (2) and (3), we have the vector state variable formula_72 and the jump conditions become
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Equations (12), (13) and (14) are known as the "Rankine–Hugoniot conditions" for the Euler equations and are derived by enforcing the conservation laws in integral form over a control volume that includes the shock. For this situation formula_73 cannot be obtained by simple division. However, it can be shown by transforming the problem to a moving co-ordinate system
(setting formula_74, formula_75, formula_76 to remove formula_77) and some algebraic manipulation (involving the elimination of formula_78 from the transformed equation (13) using the transformed equation (12)), that the shock speed is given by
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where formula_79 is the speed of sound in the fluid at upstream conditions.
Shock Hugoniot and Rayleigh line in solids.
For shocks in solids, a closed form expression such as equation (15) cannot be derived from first principles. Instead, experimental observations indicate that a linear relation can be used instead (called the shock Hugoniot in the "u"s-"u"p plane) that has the form
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where "c"0 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 "u"p = "u"2 is the particle velocity inside the compressed region behind the shock front.
The above relation, when combined with the Hugoniot equations for the conservation of mass and momentum, can be used to determine the shock Hugoniot in the "p"-"v" plane, where "v" is the specific volume (per unit mass):
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Alternative equations of state, such as the Mie–Grüneisen equation of state may also be used instead of the above equation.
The shock Hugoniot describes the locus of all possible thermodynamic states a material can exist in behind a shock, projected onto a two dimensional state-state plane. It is therefore a set of equilibrium states and does not specifically represent the path through which a material undergoes transformation.
Weak shocks are isentropic and that the isentrope represents the path through which the material is loaded from the initial to final states by a compression wave with converging characteristics. In the case of weak shocks, the Hugoniot will therefore fall directly on the isentrope and can be used directly as the equivalent path. In the case of a strong shock we can no longer make that simplification directly. 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