Scalar transport in turbulent shear flows

Turbulent diffusion can be defined as the study of how a fluid in turbulent motion transports foreign substances that it contains. There are many examples, including smoke, acid rain, and other pollution in the atmosphere, salt in the sea and estuaries, and hot water in factory cooling systems. The foreign substance may have properties (especially its density and overall volume) that affect the motion of the ambient fluid, but it is often the case that the contaminant (as the foreign substance will be called in these lectures) is passive, i.e. the motion of the ambient fluid is the same as it would be in the absence of the contaminant. In practice nearly all cases of atmospheric dispersion fall into this category, as do many industrial applications. On the other hand the presence of dissolved salt has a profound effect on the behaviour of estuaries. These lectures will deal primarily with passive contaminants, although much of what is said applies qualitatively or, sometimes with minor modifications, even quantitatively to non-passive diffusion. Host fluid flows are, in practice, incompressible, i.e. the density of each fluid element is invariant during the motion (or can be regarded as invariant for all practical purposes). Variations of density from fluid element to fluid element are, of course, quite consistent with incompressible behaviour (e.g. in the atmosphere), but they will not be relevant in these lectures. The ambient fluid will therefore be taken as having constant uniform density p. The velocity field at position x and time t in the ambient fluid will be denoted by T(x,t), where (by incompressibility).........