Skip to topic | Skip to bottom

Start of topic | Skip to actions

Numerical Simulations of Gaseous Detonations

Multi-component Euler Equations

The governing equations of gaseous detonations are the multi-component Euler equations with chemical reactive source terms. These equations can be written in conservation form with K inhomogeneous continuity equations

 \partial_t \, \rho_i + \sum_{n=1}^d \partial_{x_n} (\rho_i u_n ) = W_i\, \dot \omega_i \qquad {\rm for} \qquad i = 1,\dots,K \;,

d momentum equations

 \partial_t (\rho u_m) + \sum_{n=1}^d \partial_{x_n} (\rho u_n u_m + \delta_{n,m} \; p ) = 0 \qquad {\rm for} \qquad m = 1,\dots,d\;,

and an energy equation

 \displaystyle \partial_t (\rho E) +\sum_{n=1}^d \partial_{x_n} \left[u_n (\rho E+p)\right] = 0 \;.

According to Dalton's law, the total pressure p is the sum of the partial pressures pi, i.e.

 p(\rho_1,\dots,\rho_K,T) = \sum_{i=1}^K p_i = \sum_{i=1}^K \rho_i \frac{{\cal R}}{W_i} T = \rho \frac{{\cal R}}{W} T


 \sum_{i=1}^K \rho_i = \rho \quad {\rm and} \qquad Y_i = \frac{\rho_i}{\rho}\;. \

For detailed chemical reaction, all species are usually assumed to be thermally perfect gases with a caloric equation

 h(Y_1,\dots,Y_K,T) = \displaystyle \sum_{i=1}^K Y_i h_i(T) \quad {\rm with} \qquad \displaystyle h_i(T) = h_i^0 + \int_0^T c_{pi}(s) ds \;.

This model requires the computation of the temperature T from the implicit equation

 \sum_{i=1}^K \rho_i \, h_i(T) - {\cal R} T \sum_{i=1}^K \frac{\rho_i}{W_i} - \rho e = 0\;,

whenever the pressure has to be evaluated.


-- RalfDeiterding - 14 Dec 2004

You are here: Amroc > ClawpackHome > NumSimGasDet > MultiEulerEquations

to top

Copyright © 1997-2019 California Institute of Technology.