Rankine–Hugoniot conditions

In a coordinate system that is moving with the discontinuity, the Rankine–Hugoniot conditions can be expressed as:[4]

${\displaystyle \rho _{1}\,u_{1}=\rho _{2}\,u_{2}\equiv m}$	Conservation of mass
${\displaystyle \rho _{1}\,u_{1}^{2}+p_{1}=\rho _{2}\,u_{2}^{2}+p_{2}}$	Conservation of momentum
${\displaystyle h_{1}+{\frac {1}{2}}u_{1}^{2}=h_{2}+{\frac {1}{2}}u_{2}^{2}}$	Conservation of energy

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

${\displaystyle \omega _{i,1}=\omega _{i,2}=0,\quad i=1,2,3,\dots ,N,\qquad {\text{Conservation of species}}}$

to vanish both upstream and downstream of the discontinuity. Here, ${\displaystyle \omega }$ 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

${\displaystyle {\frac {p_{2}-p_{1}}{1/\rho _{2}-1/\rho _{1}}}=-m^{2}}$

which defines a straight line known as the Michelson–Rayleigh line, named after the Russian physicist V. A. Mikhelson (usually anglicized as Michelson) and Lord Rayleigh, that has a negative slope (since ${\displaystyle m^{2}}$ is always positive) in the ${\displaystyle p-\rho ^{-1}}$ plane.[5] Using the Rankine–Hugoniot equations for the conservation of mass and momentum to eliminate u₁ and u₂, the equation for the conservation of energy can be expressed as the Hugoniot equation:

${\displaystyle h_{2}-h_{1}={\frac {1}{2}}\,\left({\frac {1}{\rho _{2}}}+{\frac {1}{\rho _{1}}}\right)\,(p_{2}-p_{1}).}$

The inverse of the density can also be expressed as the specific volume, ${\displaystyle v=1/\rho }$. Along with these, one has to specify the relation between the upstream and downstream equation of state

${\displaystyle f(p_{1},\rho _{1},T_{1},Y_{i,1})=f(p_{2},\rho _{2},T_{2},Y_{i,2})}$

where ${\displaystyle Y_{i}}$ is the mass fraction of the species. Finally, the calorific equation of state ${\displaystyle h=h(p,\rho ,Y_{i})}$ is assumed to be known, i.e.,

${\displaystyle h(p_{1},\rho _{1},Y_{i,1})=h(p_{2},\rho _{2},Y_{i,2}).}$

Hugoniot curves for ${\displaystyle \gamma =1.4}$. The shaded region is inaccessible since the Rayleigh line has a positive slope (${\displaystyle \mu <0}$) there.

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

${\displaystyle {\frac {p_{2}}{\rho _{2}T_{2}}}={\frac {p_{1}}{\rho _{1}T_{1}}}={\frac {R}{\overline {W}}}}$

where ${\displaystyle R}$ is the universal gas constant and the mean molecular weight ${\displaystyle {\overline {W}}}$ is assumed to be constant (otherwise, ${\displaystyle {\overline {W}}}$ would depend on the mass fraction of the all species). If one assumes that the specific heat at constant pressure ${\displaystyle c_{p}}$ is also constant across the wave, the change in enthalpies (calorific equation of state) can be simply written as

${\displaystyle h_{2}-h_{1}=-q+c_{p}(T_{2}-T_{1})}$

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,

${\displaystyle \left({\frac {\gamma }{\gamma -1}}\right)\left({\frac {p_{2}}{\rho _{2}}}-{\frac {p_{1}}{\rho _{1}}}\right)-{\frac {1}{2}}\left({\frac {1}{\rho _{2}}}+{\frac {1}{\rho _{1}}}\right)(p_{2}-p_{1})=q,}$

where ${\displaystyle \gamma }$ is the specific heat ratio. An Hugoniot curve without heat release (${\displaystyle q=0}$) 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,

${\displaystyle {\tilde {p}}={\frac {p_{2}}{p_{1}}},\quad {\tilde {v}}={\frac {\rho _{1}}{\rho _{2}}},\quad \alpha ={\frac {q\rho _{1}}{p_{1}}},\quad \mu ={\frac {m^{2}}{p_{1}\rho _{1}}}.}$

The Rayleigh line equation and the Hugoniot equation then simplifies to

${\displaystyle {\begin{aligned}{\frac {{\tilde {p}}-1}{{\tilde {v}}-1}}&=-\mu \\{\tilde {p}}&={\frac {[2\alpha +(\gamma +1)/(\gamma -1)]-{\tilde {v}}}{[(\gamma +1)/(\gamma -1)]{\tilde {v}}-1}}.\end{aligned}}}$

Given the upstream conditions, the intersection of above two equations in the ${\displaystyle {\tilde {v}}}$-${\displaystyle {\tilde {p}}}$ plane determine the downstream conditions; in the ${\displaystyle {\tilde {v}}}$-${\displaystyle {\tilde {p}}}$ plane, the upstream condition correspond to the point ${\displaystyle ({\tilde {v}},{\tilde {p}})=(1,1)}$. If no heat release occurs, for example, shock waves without chemical reaction, then ${\displaystyle \alpha =0}$. The Hugoniot curves asymptote to the lines ${\displaystyle {\tilde {v}}=(\gamma -1)/(\gamma +1)}$ and ${\displaystyle {\tilde {p}}=-(\gamma -1)/(\gamma +1)}$, 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 ${\displaystyle \mu }$ is positive. Shock waves and detonations correspond to the top-left white region wherein ${\displaystyle {\tilde {p}}>1}$ and ${\displaystyle {\tilde {v}}<1}$, 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 ${\displaystyle {\tilde {p}}<1}$ and ${\displaystyle {\tilde {v}}>1}$, 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 ${\displaystyle 0\leq {\tilde {p}}<\infty }$; 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 ${\displaystyle (\gamma -1)/(\gamma +1)\leq {\tilde {v}}\leq 2\alpha +(\gamma +1)/(\gamma -1)}$ (the upper bound is derived for the case ${\displaystyle {\tilde {p}}\rightarrow 0}$ because pressure cannot take negative values). If ${\displaystyle \gamma =1.4}$ (diatomic gas without the vibrational mode excitation), the interval is ${\displaystyle 1/6\leq {\tilde {v}}\leq 2\alpha +6}$, in other words, the shock wave can increase the density at most by a factor of 6. For monatomic gas, ${\displaystyle \gamma =5/3}$, the allowed interval is ${\displaystyle 1/4\leq {\tilde {v}}\leq 2\alpha +4}$. For diatomic gases with vibrational mode excited, we have ${\displaystyle \gamma =9/7}$ leading to the interval ${\displaystyle 1/8\leq {\tilde {v}}\leq 2\alpha +8}$. 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 ${\displaystyle 11-13}$.[6]

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:

${\displaystyle {\frac {\partial \rho }{\partial t}}\;\;=-{\frac {\partial }{\partial x}}\left(\rho u\right)}$

(1)

${\displaystyle {\frac {\partial }{\partial t}}(\rho u)\,=-{\frac {\partial }{\partial x}}\left(\rho u^{2}+p\right)}$

(2)

${\displaystyle {\frac {\partial }{\partial t}}\left(E^{t}\right)=-{\frac {\partial }{\partial x}}\left[u\left(E^{t}+p\right)\right],}$

(3)

where

• ${\displaystyle \rho ,}$ fluid mass density,
• ${\displaystyle u,}$ fluid velocity,
• ${\displaystyle e,}$ specific internal energy of the fluid,
• ${\displaystyle p,}$ fluid pressure, and
• ${\displaystyle E^{t}=\rho e+\rho {\tfrac {1}{2}}u^{2},}$ is the total energy density of the fluid, [J/m³], while e is its specific internal energy

Assume further that the gas is calorically ideal and that therefore a polytropic equation-of-state of the simple form

${\displaystyle p=\left(\gamma -1\right)\rho e,}$

(4)

is valid, where ${\displaystyle \gamma }$ is the constant ratio of specific heats ${\displaystyle c_{p}/c_{v}}$. This quantity also appears as the polytropic exponent of the polytropic process described by

${\displaystyle {\frac {p}{\rho ^{\gamma }}}={\text{constant}}.}$

(5)

For an extensive list of compressible flow equations, etc., refer to NACA Report 1135 (1953).[7]

Note: For a calorically ideal gas ${\displaystyle \gamma }$ is a constant and for a thermally ideal gas ${\displaystyle \gamma }$ 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).

Shock Hugoniot and Rayleigh line in the p-v plane. The curve represents a plot of equation (17) with p₁, v₁, c₀, and s known. If p₁ = 0, the curve will intersect the specific volume axis at the point v₁.

Hugoniot elastic limit in the p-v plane for a shock in an elastic-plastic material.

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 u_s-u_p plane) that has the form

${\displaystyle u_{s}=c_{0}+s\,u_{p}=c_{0}+s\,u_{2}}$

(16)

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 u_p = u₂ 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):[17]

${\displaystyle p_{2}-p_{1}={\frac {c_{0}^{2}\,\rho _{1}\,\rho _{2}\,\left(\rho _{2}-\rho _{1}\right)}{\left[\rho _{2}-s\left(\rho _{2}-\rho _{1}\right)\right]^{2}}}={\frac {c_{0}^{2}\,\left(v_{1}-v_{2}\right)}{\left[v_{1}-s\left(v_{1}-v_{2}\right)\right]^{2}}}\,.}$

(17)

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 loading path can be determined by drawing a straight line between the initial and final states. This line is called the Rayleigh line and has the following equation:

${\displaystyle p_{2}-p_{1}=u_{s}^{2}\left(\rho _{1}-{\frac {\rho _{1}^{2}}{\rho _{2}}}\right)}$

(18)

• Euler equations (fluid dynamics)
• Shock polar
• Mie–Grüneisen equation of state
• Engineering Acoustics Wikibook
• Atmospheric focusing