ME 456

Prof. J. M. Powers

Homework 10

Due: Monday 26 April 1999

Consider the reactive Euler equations for a calorically perfect ideal gas as developed in class with $\gamma = {7 \over 5}, q = 1.0 \times 10^6~{J \over kg}, R = 287~{J \over kg~K}, P_o = 1 \times 10^5~Pa, T_o = 300~K, E = 1.0 \times 10^6~{J \over kg}.$

• If $D = 3000~{m \over s}$, find the pressure at N, S, and W.
• Find D_CJ, P_CJ, $\rho_{CJ}$, and T_CJ for both Chapman-Jouguet detonation and deflagration.
• Give a computer-generated plot of the complete reaction Hugoniot, the inert Hugoniot, and Rayleigh lines for D = D_CJ and $D = 3000~{m \over s}$
• Give a computer generated plot of the wave speed D as a function of the laboratory frame complete reaction velocity $u(\lambda=1)$.
• Plot $P(\xi)$, $u(\xi)$, and $M(\xi)$ for two wavespeeds: D = D_CJ (detonation branch) and $D = 3000~{m \over s}$. For the Mach number plots, take the Mach number to be defined in the steady wave frame. Take as a kinetics model
  
  $$\hat u {d \lambda \over d\xi} = k (1-\lambda) \exp \left(- {E \over R T} \right)$$
  
  
  with $k = 1 \times 10^8~s^{-1}$.

Give all supporting analysis. You may use any point in the course notes as a starting point, or you may want to give a complete development for your own benefit. You will likely need to use either a Fortran or Maple/Mathematica program to complete the analysis.