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

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

d momentum equations

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

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.

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

with

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

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

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

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

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

whenever the pressure has to evaluated.

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last update: 06/01/04