When flow is turbulent, particles exhibit additional transverse motion which enhances the rate of energy and momentum exchange between them thus increasing the heat transfer and the friction coefficient.
Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual flow velocity v = (vx,vy) of every particle that passed through that point at any given time. Then one would find the actual flow velocity fluctuating about a mean value:
and similarly for temperature (T = T + T′) and pressure (P = P + P′), where the primed quantities denote fluctuations superposed to the mean. This decomposition of a flow variable into a mean value and a turbulent fluctuation was originally proposed by Osborne Reynolds in 1895, and is considered to be the beginning of the systematic mathematical analysis of turbulent flow, as a sub-field of fluid dynamics. While the mean values are taken as predictable variables determined by dynamics laws, the turbulent fluctuations are regarded as stochastic variables.
The heat flux and momentum transfer (represented by the shear stress τ) in the direction normal to the flow for a given time are