Theoretical Considerations on Turbulent Diffusion and Sedimentation

In many fluid flows of practical importance in engineering, agriculture, and meteorology, the presence in the flow of foreign matter which is transported by the flow is of great importance. Erosion by flood waters is one important example, the pollution of masses of water or atmosphere is another. The importance of the transported matter arises in various ways; in some cases principal interest centers on the quantity of an impurity removed by a flow, while in others a determination of the characteristics of the main flow itself depends to a large extent on the quantity and nature of solid matter entrained by the flow and carried along in it. The possibility of controlling or predicting such phenomena depends on a knowledge of the physical mechanisms of the processes which are involved. At the present time such knowledge is insufficient to cope with important practical problems. When the foreign matter consists of solid particles, as in problems of erosion and sedimentation, it is natural to seek an analogy with molecular processes by supposing that the history of each particle or group of particles is characterized by a randomness of the same type present in molecular kinetics. A diffusion theory is thereby obtained, the practical value of which is, of course, determined solely by its ability to explain observed phenomena. Under some circumstances both qualitative and quantitative agreement is found to be good; in other important cases the agreement is less than satisfactory. Similar analogies are used when one considers processes near boundaries at which foreign matter is entrained into and leaves the flow. One then draws analogy with the recognized theories of turbulent transfer of momentum, heat, and vorticity. These attempts likewise meet with varying degrees of success. Experiments in sedimentation (Vanoni, 1946) have shown not only limits of the theories which have so far been applied, but they have also indicated roughly the dependence of observed discrepancies on parameters in some cases where the analogy theories fail. These circumstances may facilitate attempts to obtain more refined theories. As a first step toward obtaining such refinements, the investigation described by the present report was conducted with a dual purpose : (1) to make an examination of the fundamental physical and mathematical features of classical molecular diffusion theory, with particular regard to the complications which are believed to be at the base of observed discrepancies in analogy theories and (2) to survey the work which has been done in allied fields, especially meteorology, in which similar problems have been faced for many years by competent physicists and mathematicians, in order to see which refinements may be carried over to the theory of sedimentation and turbulent diffusion and to suggest new lines of attack which may be fruitful. As the difficulties with the theory are largely due to great mathematical complications in any but the simplest problems, the discussion which follows places emphasis on mathematical techniques. In order to make the presentation more readable to workers in hydraulics who do not make frequent use of modern advanced techniques of mathematical physics, the discussion is confined to a moderately elementary level