Rankine–Hugoniot conditions

The Rankine - Hugoniot condition or Rankine - Hugoniot equation (after William John Macquorn Rankine and Pierre -Henri Hugoniot ) describes the behavior of shock waves of a one-dimensional hyperbolic conservation equation. Given two states, and the left and right of an impact, says the condition that the shock velocity, the equation

Met. In the case of a scalar equation it directly provides the shock speed

For systems with the situation is more difficult. In the case of a linear equation gives the condition that the shock velocity is an eigenvalue of the matrix needs to be and the difference between the states is an eigenvector of. This is not always possible, which then means that the states are connected by a sequence of discontinuities. This can also be applied to non-linear equations, which is then to note that change here the collision velocities with time.

Conversely refers to systems, the amount of states that can be associated with a given solid state by a single impact, the Hugoniot locus.

Euler equations

In the case of the Euler equations, special relationships are. It is the density, pressure and speed. With the internal energy per unit mass is called; in the ideal gas equation of state so therefore ( is the adiabatic exponent ).

Elimination of the velocity leads to the following relationship:

Said. Which is referred to as adiabatic hugoniotsche. Now, if the equation of state is used for the ideal gas results in

Since the pressures are always positive, it follows that the density ratio can never be greater than. For air at about 1.4, the ratio is about 6 This result is intuitively understandable because an increase in pressure leads to an increase in temperature, which partially counteracts the increase in density. While the shock strength ( the pressure ) can be arbitrarily large, the density ratio reaches a finite limit. However, high temperature may result in the dissociation, or even strong shocks for the ionization and thus the increase in the thermodynamic degrees of freedom and thus to a smaller value. Therefore, this limit may be much higher than for the ideal gas in real gases.

The first two conservation laws follow from the Euler equations or lead to these. With them, the jump conditions for the velocity and the density (or pressure) can be displayed at the shock front. The central idea of Rankine and Hugoniot was now the use of the third conservation law ( conservation of energy ) in order to formulate a jump condition for the entropy. The former is discontinuous at the shock front:

It follows that a shock wave is not adiabatic ( isentropic or ) process more and includes an enthalpy entropical:

In contrast to

For an adiabatic compression. The hugoniotsche adiabat is also known as shock adiabatic.