Rankine–Hugoniot equation: Definition from Answers.com

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A schematic of a standing shock wave with the density

ρ

, velocity

u

, and temperature

T

in each region described by the Rankine–Hugoniot equations.

The Rankine–Hugoniot equation relate the density ratio across a shock wave to the pressure ratio and the fluid velocity ratio. It is named after physicists William John Macquorn Rankine and Pierre Henri Hugoniot and find applications in Enginnering Acoustics,

The equations are a mathematical formulation of the principles of conservation of mass, momentum and energy across a normal shock for a one-dimensional, steady flow of a fluid subject to the Euler equations. For an ideal gas, the equation of state, provides the fourth equation to completely solve for the downstream conditions.

In algebraic form, the equations are

$\rho_1u_1=\rho_2u_2\,$

$p_1+\rho_1u_1^2=p_2+\rho_2u_2^2$

$e_1+\frac{p_1}{\rho_1}+\frac{1}{2}u_{1}^2=e_2+\frac{p_2}{\rho_2}+\frac{1}{2}u_{2}^2$

where,

ρ is the fluid mass density,
u is the fluid velocity
e is the internal energy per unit mass for the fluid, and
p is the pressure.

and the subscripts 1 and 2 denote the conditions upstream and downstream of the shock respectively.

Note the three components to the energy flux: mechanical work, internal energy, and kinetic energy. Sometimes, these three conditions are referred to as the Rankine–Hugoniot conditions. The last equation is equivalent to Bernoulli's principle.

Eliminating the speeds $u 1$ and $u 2$ from the last two equations and using the equation of state gives the following relationship:

$\frac{\rho_2}{\rho_1}= \frac{1 + (\gamma+1)/(\gamma-1)p_2/p_1}{(\gamma+1)/(\gamma-1) + p_2/p_1} = \frac{u_1}{u_2}$

Thus, because the pressures are both positive, the density ratio is never greater than $(γ + 1) / (γ - 1)$ , or about 6 for air (in which $γ$ is about 1.4). As the strength of the shock increases, the downstream gas becomes hotter and hotter, but the density ratio $ρ 2 / ρ 1$ approaches a finite limit of 4 for a monatomic gas ( $γ$ = 5/3) and 6 for a diatomic gas ( $γ$ = 1.4).

References

Rankine, W. J. M. , On the thermodynamic theory of waves of finite longitudinal disturbances, Phil. Trans. Roy. Soc. London, 160, (1870), p. 277.
Hugoniot, H., Propagation des Mouvements dans les Corps et spécialement dans les Gaz Parfaits, Journal de 1’Ecole Polytechnique, 57, (1887), p. 3; 58, (1889), p. 1.
Salas, M. D. The Curious Events Leading to the Theory of Shock Waves Invited lecture at the 17th Shock Interaction Symposium Rome, Italy 4-8 September 2006.

External links

Gas Dynamics Toolbox Calculate normal shock wave parameters for mixtures of imperfect gases
The Rankine-Hugoniot Jump Equations

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