Rankine-Hugoniot condition

The Rankine-Hugoniot condition or Rankine-Hugoniot equation (after William John Macquorn Rankine and Pierre-Henri Hugoniot ) describes the behavior of shock waves using a one-dimensional hyperbolic conservation equation :

${\displaystyle u_{t}+f(u)_{x}=0}$

With

• the speed . ${\displaystyle u}$

Given two states and left and right of a collision , the condition states that the collision velocity is the equation ${\displaystyle u_{L}}$ ${\displaystyle u_{R}}$ ${\displaystyle s}$

${\displaystyle f(u_{L})-f(u_{R})=s\cdot (u_{L}-u_{R})}$

Fulfills. In the case of a scalar equation , this gives the impact velocity directly ${\displaystyle \left(u\in \mathbb {R} ^{1}\right)}$

${\displaystyle \Leftrightarrow s={\frac {f(u_{L})-f(u_{R})}{u_{L}-u_{R}}}}$.

The situation is more difficult with systems with . ${\displaystyle u\in \mathbb {R} ^{n};\,n\geq 2}$

• In the case of a linear equation , the condition arises that the impact velocity must be an eigenvalue of the matrix and the difference between the states must be an eigenvector of . This is not always possible, which then means that these states are connected by a sequence of discontinuities . ${\displaystyle u_{t}+A\cdot u_{x}=0}$ ${\displaystyle A}$ ${\displaystyle u_{L}-u_{R}}$ ${\displaystyle A}$
• This can also be applied to non-linear equations , in which case it should be noted that here the impact velocities change over time.

Conversely, in systems the set of states that can be connected to a given solid state by a single collision is called the Hugoniot locus .

Examples

Advection equation in 1D

A very simple conservation equation is given by the scalar flow:

${\displaystyle f\colon \mathbb {R} \to \mathbb {R} ,x\mapsto ax\quad {\textrm {mit}}\;a\in \mathbb {R} .}$

The jump condition thus results immediately . ${\displaystyle s=a}$

Burgers equation in 1D

The Burgers equation is defined by the following flow:

${\displaystyle f\colon \mathbb {R} \to \mathbb {R} ,x\mapsto {\frac {1}{2}}x^{2}.}$

The jump condition thus provides: . ${\displaystyle s={\frac {u_{L}+u_{R}}{2}}}$

Euler equations

In the case of the Euler equations , special relationships result. Elimination of velocity leads to:

${\displaystyle 2\cdot \left(h_{R}-h_{L}\right)=\left(p_{R}-p_{L}\right)\cdot \left({\frac {1}{\rho _{L}}}+{\frac {1}{\rho _{R}}}\right)}$

With

• the pressure ${\displaystyle p}$
• the density ${\displaystyle \rho }$
• the specific enthalpy . This is known as Hugonian adiabats (see below) ${\displaystyle h={\frac {p}{\rho }}+e}$
  • the internal energy per mass ( specific quantity ) ${\displaystyle e}$

If the equation of state is used for the ideal gas :

${\displaystyle p=\rho \cdot (\kappa -1)\cdot e}$

With

• the adiabatic exponent , ${\displaystyle \kappa }$

so it turns out

${\displaystyle \Rightarrow {\frac {p_{L}}{p_{R}}}={\frac {(\kappa +1)-(\kappa -1)\cdot {\frac {\rho _{R}}{\rho _{L}}}}{(\kappa +1)\cdot {\frac {\rho _{R}}{\rho _{L}}}-(\kappa -1)}}\quad \Leftrightarrow \quad {\frac {\rho _{L}}{\rho _{R}}}={\frac {p_{L}\cdot (\kappa +1)+p_{R}\cdot (\kappa -1)}{p_{L}\cdot (\kappa -1)+p_{R}\cdot (\kappa +1)}}}$.

Since the pressures are always positive, it follows for the density ratio:

${\displaystyle \Rightarrow {\frac {\rho _{L}}{\rho _{R}}}\leq {\frac {\kappa +1}{\kappa -1}}}$

For air with , the maximum density ratio is approximately 6. This result is clearly understandable, since an increase in pressure also leads to an increase in temperature, which partially counteracts the increase in density. While the impact strength (the overpressure ) can be arbitrarily large, the density ratio thus reaches a finite limit. ${\displaystyle \kappa \approx 1{,}4}$

However, in the case of strong impacts, high temperature can lead to dissociation or even to ionization and thus to an increase in the thermodynamic degrees of freedom and thus in turn to a lower value of . Therefore, the upper limit for the density ratio in real gases can be significantly higher than in ideal gas. ${\displaystyle \kappa }$

The first two conservation laws follow from the Euler equations or lead to them. With them the jump conditions for the speed and the density (or the pressure) at the shock front can be represented. The central idea of ​​Rankine and Hugoniot was to use the third law of conservation (conservation of energy ) to formulate a jump condition for entropy . This is discontinuous on the shock front:

${\displaystyle S_{1}-S_{0}>0}$.

It follows that a shock wave does not adiabatic (or isentropic is) process more, but the enthalpy a entropical contains (hugoniotsche Adiabatic, also known as shock adiabatic known):

${\displaystyle {\rm {d}}H=\int _{1}^{2}{\frac {{\rm {d}}p}{\rho }}+T\cdot {\rm {d}}S}$

in contrast to

${\displaystyle {\rm {d}}H=\int _{1}^{2}{\frac {{\rm {d}}p}{\rho }}}$

for a purely adiabatic compression.

literature

• H. Hugoniot: On the Propagation of Motion in Bodies and in Perfect Bodies in Particular , 1887, I. Journal de l'Ecole Polytechnique, Volume 57, pages 3-97.
• MA Meyers: Dynamic Behavior of Materials , 1994, John Wiley & Sons, New York, ISBN 0-471-58262-X .
• Randall J. LeVeque: Finite Volume Methods for Hyperbolic Problems , 2002, Cambridge Texts in Applied Mathematics, ISBN 0-521-00924-3 .
• WJM Rankine: On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance , 1870, Philosophical Transactions, London / Edinburgh, Volume 160, pages 270-288.

Web links

• Stanley P. Marsh: LASL Shock Hugoniot Data . In: Los Alamos Series on Dynamic Material Properties. University of California Press, Berkeley and Los Angeles, California, 1980, ISBN 0-520-04008-2 (PDF file; 25 MB).