Travelling wave mathematical analysis and efficient numerical resolution for a one-dimensional model of solid propellant combustion

Abstract

We investigate a model of solid propellant combustion involving surface pyrolysis coupled to finite activation energy gas-phase combustion. Existence and uniqueness of a travelling wave solution are established by extending dynamical system tools classically used for premixed flames, dealing with the additional difficulty arising from the surface regression and pyrolysis. An efficient shooting method allows to solve the problem in phase space without resorting to space discretisation nor fixed-point Newton iterations. The results are compared to solutions from a CFD code developed at ONERA, assessing the efficiency and potential of the method, and the impact of the modelling assumptions is evaluated through parametric studies.

Keywords:

• solid propellant combustion
• nonlinear surface pyrolysis
• travelling wave solution
• dynamical system
• shooting method

Acknowledgments

The present research was conducted thanks to a Ph.D grant supported by Direction Générale de l'Armement (DGA), Ministry of Defence (E. Faucher, Technical Advisor), and Office National d'Études et de Recherches Aérospatiales (ONERA). The authors would like to thank V. Giovangigli for several helpful discussions, and the reviewers for their very useful remarks, which have helped improving the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The name eigenvalue is adopted here for two reasons. First, it is historically used in the papers on solid propellant theory as well as in the laminar flame theory, mainly without quotes. Second, we will investigate a non-linear eigenvalue problem for an elliptic operator with a non-linear source term as well as a non-linear dependency of the regression rate on the surface temperature, and thus a specific case of a general non-linear eigenvalue problem on the whole real line. It bears some similarity with the eigenvalue problem of a second order linear elliptic operator such as the Laplace operator on a compact interval with proper boundary conditions [29]: we look for both an eigenfunction of space related to a real eigenvalue of the operator. Its extension to the non-linear case has also been studied in the literature [15].