In order to understand the fundamental difference between laminar and turbulent mixing, consider the following example. In a room of length L  = 10 m, temperature can be diffused due to molecular diffusion and random motion of air. In both cases, estimate the time scale of the process. Which one is the more efficient? The thermal diffusivity of air is γ = 1.9.10^− 5 m² ⁄ s and you can assume that a weak motion of characteristic velocity u’ = 0.1 m/s is present in the room.

Laminar mixing is due to molecular diffusion. It follows the heat equation. where θ is the temperature and γ is the thermal diffusivity. In this example, one is not interested in the exact solution of Eq.(1↑). Simple dimensional analysis is sufficient to obtain a rough estimate of the diffusion process. so that the time scale due to molecular diffusion is roughly Plugging the room length and thermal diffusivity into Eq.(3↑), one obtains τ_lam ~ 5.10⁶ s. Molecular diffusion is an ineffective way of distributing the heat into the room. On the other hand, turbulent mixing is mostly due to velocity fluctuations. The properties are transported with the eddies, leading to the characteristic time Hence, turbulent diffusion is far more efficient than molecular diffusion. It is interesting to notice that the ratio of these two time scales is The thermal diffusitivity γ is related to the kinematic viscosity ν via the Prandtl number, with is typically P_r≃ν ⁄ γ≃0.7 for air in standard conditions. Hence, it directly follows that where R_e is the well known Reynolds number, defined here in terms of turbulent properties.