Turbulence, Heat and Mass Transfer 6 K. Hanjali´ c, Y. Nagano and S. Jakirli´ c (Editors) c 2009 Begell House, Inc. Lagrangian properties of diffusion in the theory of turbulent combustion V. L. Zimont and G. Pagnini CRS4, Polaris Bldg. 1, 09010 Pula (CA), Italy, zimont@crs4.it , pagnini@crs4.it Abstract — In this paper we analyze two effects of the Lagrangian nature of turbulent transfer that are usually ignored in the theory of turbulent combustion. They are (i) nonequilibrium behavior of the turbulent diffusion coefficient, which is important for modeling of initial stage of combustion in the SI engine, and (ii) the existence of a traveling front of turbulent diffusion with finite speed, which controls the velocity of the steady state flame in strong turbulence. First we derive the parabolic diffusion equation with both the diffusion and chemical source terms expressed by Lagrangian characteristics of turbulence. We show that chemical transformation does not influence the turbulent diffusion coefficient and this result is important for combustion theories that relate the formation of the initial flame with the development of the diffusion coefficient. Second, an hyperbolic diffusion equation based on hydrodynamics is derived and we analyze relationship between velocities of the turbulent diffusion front and the speed of the steady state premixed flame. In particular, in the case of the flamelet combustion mechanism and strong turbulence, we state that the flame speed is very closed to the theoretical value of the diffusion front velocity, which is equal to the root mean square of turbulent velocity fluctuations. 1. Introduction The classical paradigm of diffusion transfer modeling is based on parabolic equations with the turbulent diffusion coefficient D t that is expressed in terms of Eulerian turbulent char- acteristics D t = const 〈u ′2 〉 1/2 L, where 〈u ′2 〉 1/2 and L are the root mean square of turbulent velocity fluctuation and the Eulerian integral legnthscale of turbulence, respectively. In the case of homogeneous and stationary turbulence D t is here constant. However, such Eulerian approach misses qualitative effects that are caused by the Lagrangian nature of turbulent transfer. In this paper we analyze two of these effects: i) the time-dependence of the diffusion coefficient due to finite value of the Lagrangian timescale of turbulence, and ii) the existence of the traveling turbulent diffusion front due to limited values of the instantaneous fluctuations of the velocity, which requires to use an hyperbolic equation instead of a parabolic one. Though both these effects are discussed in the scientific literature (see, for example, [1]), they are usually ignored in the engineering handbooks [2] and this neglect is justified in many practical applications because the relaxation time of the diffusion coefficient is small and, at the same time, the front speed is large with respect to the reference time and velocity of the process, respectively. But these effects play an important role in the theory of turbulent premixed combustion. In fact, an accurate description of nonequilibrium behavior of the turbulent diffusion coefficient, i.e. its growing in time, is important during the initial transient stage of the combustion process in the SI engine and laboratory bombs with spark ignition (as well as in initial instants of the Bunsen flame),