Estimation of critical conditions for the detonation-to-deflagration transition

An estimate is proposed for the critical Mach number of the shock wave that can ensure the deflagration-to-detonation transition (DDT): $\mathrm{M}_{\mathrm{min}}\approx0.56\mathrm{M}_0$ for expanding waves and $\mathrm{M}_{\mathrm{min}}\approx0.33\mathrm{M}_0$ for plane waves propagating in a constant-section straight tube ($\mathrm{M}_0$ is the Mach number of an ideal Chapman—Jouguet detonation wave). The condition $\mathrm{M}>\mathrm{M}_{\mathrm{min}}$ ensures the DDT mode, whereas only laminar or turbulent burning without the DDT is observed for lower Mach numbers. The estimate is based on the equiprobable transition from the compressed state of the initial mixture both to the detonation and to the deflagration branch of the adiabat of reaction products (with respect to the initial state of the combustible mixture).