When two species A and B are introduced through different parts of the bounding surface into a region of turbulent flow, molecules of A and B are brought together by the combined actions of the turbulent velocity field and molecular diffusion. A random flight model is developed to simulate the relative motion of pairs of fluid elements and random motions of the molecules, based on the models of Durbin (1980) and Sawford & Hunt (1986). The model is used to estimate the cross-correlation between fluctuating concentrations of A and B, $\overline{c_Ac_B}$, at a point, in non-premixed homogeneous turbulence with a moderately fast or slow second-order chemical reaction. The correlation indicates the effects of turbulent and molecular mixing on the mean chemical reaction rate, and it is commonly expressed as the ‘segregation’ or ‘unmixedness’ parameter $\alpha(=\overline{c_Ac_B}/\overline{C}_A\overline{C}_B) $ when normalized by the mean concentrations $\overline{C}_A$ and $\overline{C}_B$. It is found that α increases from near −1 to zero with the time (or distance) from the moment (or location) of release of two species in high-Reynolds-number flow. Also, the model (and experiments) agrees with the exact results of Danckwerts (1952) that $\overline{c_Ac_B}/(\overline{c^2_A}\overline{c^2_B})^{\frac{1}{2}} = -1$ for mixing without reaction. The model is then extended to account for the effects on the segregation parameter α of chemical reactions between A and B. This leads to α eventually decreasing, depending on the relative timescales for turbulent mixing and for chemical reaction (i.e. the Damköhler number). The model also indicates how a number of other parameters such as the turbulent scales, the Schmidt number, the ratio of initial concentrations of two reactants and the mean shear affect the segregation parameter α.
The model explains the measurements of α in previously published studies by ourselves and other authors, for mixing with and without reactions, provided that the reaction rate is not very fast. Also the model is only strictly applicable for a limited mixing time t, such that t [lsim ] TL where TL is the Lagrangian timescale, because the model requires that the interface between A and B is effectively continuous and thin, even if highly convoluted. Flow visualization results are presented, which are consistent with the physical idea underlying the two-particle model.