In 1925 Prandtl proposed the mixing length theory of turbulent flow by analogy with the kinetic theory of gases so that the Reynold's shear stress term -p u'v' can be expressed as (details can be found in references 1 to 7 ) -Although it is now accepted that the concept of a mixing length does not adequately represent the correct physical picture of the structure of turbulence in detail, mixing length theories are nevertheless most useful to engineers as a means of correlating and extrapolating experimental data. The reason for this, as discussed by Hinze ( 3 ) , is simply that more correct theories that can be used successfully from a practical engineering point of view are not available.The purpose of this note is to show that the mixing length theory can be made more useful if it is reformulated and some elementary mathematical properties of the result are recognized.Let us first briefly discuss some of the limitations of Prandtl's mixing length theory. First, it assumes that a lump of fluid retains its identity over a certain distance after which it loses its momentum to the surroundings and adopts the properties of its environment. This oversimplified picture fails to give physical insight into the structure of turbulent flow because no attempt is made to explain how and why a fluid lump will retain its identity and the mechanism by which it will adopt the properties of the surrounding. It assumes that the mixing length and eddy diffusion depend only on local conditions in the flow. However, it can be seen from the experimental investigations of Clauser (8) and Corino and Brodkey ( 9 ) , among others, that eddies originate in a region near the wall ( y + < 7 0 ) .These eddies then move towards the turbulent core region and are at the same time convected downstream. Consequently, the turbulent core region (y+ > 7 0 ) contains eddies which originated within the generation region ( y + < 70) at various upstream positions. But this assumption will not be a serious limitation if the flow is axially homogeneous, for example, as in the case of fully developed turbulent flow in a pipe. Second, it assumes that the diffusion and convection of turbulent energy are negligibly small, so that the turbulent energy generated locally is equal to the dissipation. However, it has been found ( 3 ) that the diffusion and convection terms in the energy equation are, in general not negligible. Third, it assumes that the rate of transport is proportional to the gradient of the mean velocity. Therefore, a coefficient of eddy diffu-
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