804 BOWDEN [sect. 6 



diffusion. Whereas the turbulent shearing stresses react on the mean motion 

 and have an essentially dynamical effect, the turbulent diffusing processes 

 affect the distribution of a particular property of the fluid without reacting 

 directly on the flow, 



B. Mixing Length Theory 



There is an obvious analogy between the random motion of molecules con- 

 sidered in the kinetic theory of gases and the irregular movements of elements 

 of fluid which occur in turbulent flow. The introduction by Prandtl (1925) of 

 the concept of a "mixing length" I, analogous to the mean free path in the 

 kinetic theory of gases, led to a valuable line of development in the theory of 

 turbulence. From considerations of similarity between the turbulent motion 

 in various parts of the mean flow, von Karman (1930) was able to relate I to 

 the properties of the mean motion. For the special case of flow near a solid 

 boundary at 2 = 0, and in the region in which tzx does not vary appreciably with 

 z, the theory of Prandtl and von Karman leads to the well-known logarithmic 

 law for the velocity distribution : 



U ^ }- iZ^Y \n'-±^, (7) 



where to is the stress at the boundary, zq is a parameter known as the "roughness 

 length" and ko is von Karman's constant, having a numerical value of approxi- 

 mately 0.40. {toIpY'- is frequently denoted by U^ and termed the "friction 

 velocity". The logarithmic law has been much used in all branches of fluid 

 mechanics, including meteorology, and its validity in conditions of neutral 

 stability appears to be well established. Equation (7) implies that, within the 

 limits of z for which it is valid, 



I = ko{z + zo) 



(8) 

 Nz = koU^{z + ZQ). 



If the velocity U is measured at several values of the distance z and U plotted 

 against log 2, the shearing stress to and the roughness length 20 may be deter- 

 mined from (7). 



C. Diffusion by Continuous Movements 



The analogy between turbulent motion and the kinetic theory of gases is 

 imperfect, however, in that the interaction of an element of fluid with its sur- 

 roundings takes place continuously and the element itself gradually loses its 

 identity. An alternative approach was initiated by Taylor (1921) in his theory 

 of diffusion by continuous movements. He defined the function E{r) as the 

 coefficient of correlation between the turbulent velocity ut of a particle of fluid 

 at time t and the velocity ut+r of the same particle at a time t later, i.e. 



E{t) - {utut+r)lu2. (9) 



