Turbulent flux of a scalar quantity averaged diffusion equation

Turbulent flux of a scalar quantity averaged diffusion equation 

We now look at dispersion over time, by a turbulent flow, of a passive scalar quantity introduced into the fluid. The quantity is said to be passive if its presence does not alter the flow. This could be, for instance, a dye, a pollutant, or a chemical constituent that does not alter the mechanical properties of the fluid (density, viscosity, etc). Temperature is regarded as a passive scalar quantity when its variations in space are sufficiently small for viscosity variations to remain negligible and for spatial variations of density not to cause movement within the fluid (absence of natural convection). 

Considering here a chemical constituent measured locally by its concentration c(t, X, y, z), the dispersion of that quantity in space and time is described by an advection-diffusion equation accounting for its conservation  

Assuming provisionally that the flow has no tuibulence-averaged component, velocity therefore accordingly gets reduced to turbulent fluctuations such as U cXt,x,y,z), Uy t,x,y,z), and U2 t,x,y,z). Concentration, however, may be represented as the srrm of the turbrrlence-averaged concentration and the concerrtration fluctrration  

The double correlations u c, Uy c, and u c have a very simple physical meaning. They are the average tnrbulent fluxes of concentration per unit area, produced by the action of turbnlence. c is the turbulent flux per unit area 

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