Mache effect and combustion instability in solid rocket motor

A theoretical analysis of nonacoustic low-frequency instability (L* instability) of combustion in solid rocket motors (SRM) is considered. Such an instability takes place when the timescale of the thermal inertia of a combustion wave tau(T) is comparable with the timescale tau(W) of pressure relaxation in the motor cavity, An original model is proposed for unsteady combustion process in SRM. This model allows the flame temperature and temperature distribution in a gas flow to fluctuate with the chamber pressure [Mache effect (Mache, H., Die Physik der Verbrennungserscheinungen, Veit and Co., Leipzig, Germany, 1918)]. A laminar one-dimensional gas flow (without dissipation and mixing) is assumed. The propellant unsteady combustion is described using the Zel'dovich phenomenological approach and taking into account the variation of the combustion surface temperature (Novozhilov model) as well as flame temperature [Gostintsev and Sukhanov model (Gostintsev, Y, A., Sukhanov, L, A., and Pokhil, P P.,"On the Theory of Unsteady Combustion of Solid Propellant. Stability of the Process in Semi-Closed Volume," Journal of Applied Mechanics and Technical Physics, Vol. 12, No. 6, 1971, pp. 65-73)], Emphasis is on the dependence of the critical conditions of stable combustion on the temperature distribution in gas now under variable gas pressure (Mache effect). This problem is considered in detail for SRM having an end-burning charge. The result of the analysis shows that the Mache effect can significantly extend the instability region on the plane (k, chi = tau(W)/tau(T)), if chi < 2. Critical values of the temperature sensitivity of the combustion rate k at chi > 2 are close to the asymptote (at tau(W) much greater than tau(T)) that corresponds to the criterion of intrinsic instability of combustion under constant pressure. Application of these results to rocket motors with a common configuration of the propellant charge is discussed.