A Zeldovich spontaneous wave, also known as the Zeldovich gradient mechanism, is a theoretical type of reaction wave that can occur in a reacting substance, such as a gas mixture, where the initial temperature varies across different locations. This variation in temperature creates gradients that cause different parts of the substance to react at slightly different times, driving the wave's propagation. Unlike typical combustion waves, such as subsonic deflagrations and supersonic detonations, it's characterized by the absence of interactions between different parts of the substance, such as those caused by pressure changes or heat transfer. Introduced by Yakov Zeldovich in 1980 building on his earlier research, this concept is often cited to explain the yet-unsolved problem of deflagration to detonation transition (DDT), where a slow-moving subsonic flame (deflagration) accelerates to a supersonic detonation. Essentially, the Zeldovich spontaneous wave helps explain how a reaction can spread solely due to initial temperature differences, independent of factors like heat conduction or sound speed (provided the initial temperature gradients are small). While it simplifies real-world conditions by neglecting gas dynamic effects, it offers valuable insights into the fundamental mechanisms of rapid reactions. The wave's behavior is dependent on the initial temperature distribution.
Description of the spontaneous reaction wave Let T ( x , y , z ) {\displaystyle T(x,y,z)} be the initial temperature distribution, which is non trivial, indicating that chemical reactions at different points in space proceed at different rates. To this distribution, we can associate a function t a d ( x , y , z ) {\displaystyle t_{ad}(x,y,z)} , where t a d {\displaystyle t_{ad}} is the adiabatic induction period. Now, define in space some surface t a d ( x , y , z ) = c o n s t . {\displaystyle t_{ad}(x,y,z)=\mathrm {const.} } ; suppose if T = T ( x ) {\displaystyle T=T(x)} , then this surface for some constant will be parallel to y z {\displaystyle yz} -plane. Examine the change of position of this surface with the passage of time according to
t a d ( x , y , z ) = t . {\displaystyle t_{ad}(x,y,z)=t.}
From this, we can easily extract the direction and the propagation speed of the spontaneous front. The direction of the wave is clearly normal to this surface which is given by ∇ t a d / | ∇ t a d | {\displaystyle \nabla t_{ad}/|\nabla t_{ad}|} and the rate of propagation is just the magnitude of inverse of the gradient of t a d {\displaystyle t_{ad}} :
u s p = ∇ t a d | ∇ t a d | 2 , u s p = | u u p | = 1 | ∇ t a d | . {\displaystyle \mathbf {u} _{sp}={\frac {\nabla t_{ad}}{|\nabla t_{ad}|^{2}}},\quad u_{sp}=|\mathbf {u} _{up}|={\frac {1}{|\nabla t_{ad}|}}.}
Note that adiabatic thermal runaways at different places are not casually connected events and therefore u s p {\displaystyle u_{sp}} can assume, in principle, any positive value. By comparing u s p {\displaystyle u_{sp}} with other relevant speeds such as, the deflagration speed, u f {\displaystyle u_{f}} , the sound speed, c {\displaystyle c} and the speed of the Chapman–Jouguet detonation wave, u C J {\displaystyle u_{CJ}} , we can identify different regimes:
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