Numerical Simulations of Gaseous Detonations

next up previous
Next: Reaction Terms of Detailed Up: Numerical Simulations of Gaseous Previous: Numerical Simulations of Gaseous

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

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

d momentum equations

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

and an energy equation

$\displaystyle \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.

$\displaystyle 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 $


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

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

$\displaystyle h(Y_1,\dots,Y_K,T) = \displaystyle \sum_{i=1}^K Y_i h_i(T)$   with$\displaystyle \quad
\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

$\displaystyle \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 evaluated.


Table of Contents      Home      Contact

last update: 06/01/04