SECT. 6] TURBULENCE 803 



where fx is the coefficient of viscosity. For a fuller treatment, reference may be 

 made to Lamb (1932) or Proudman (1953), 



Since the effect of turbulence is to produce shearing stresses analogous to, 

 although usually much greater than, those due to molecular viscosity, it seems 

 natural to attempt to allow for the Reynolds stresses by introducing a kinematic 

 "eddy viscosity" N, analogous to the kinematic molecular viscosity v^fx/p. In 

 most cases, when dealing with the mean motion, the stresses due to molecular 

 viscosity may be neglected compared with the Reynolds stresses. Thus, if the 

 mean current U is in the OX direction and the shearing stresses are also in the 

 OX direction, it follows that 



— AT ^^ 



Tyx = - pUV = pJSy — 



^ (4) 



Tzx = — PUW = pJSlz -T—' 



cz 



where Tyx is taken to be the shearing stress on a surface perpendicular to OY 

 in the OX direction, and the stresses due to molecular viscosity have been 

 neglected. In general Ny will differ from Nz in magnitude and the coefficients of 

 eddy viscosity are not physical constants like the molecular viscosity v, but 

 depend on the type and scale of the motion and on the degree of stability. In a 

 given pattern of flow they will vary, in general, from one point to another. 



The Reynolds stress component — puw represents the mean rate of turbulent 

 transport of w-momentum by the w component of flow. Similar considerations 

 may be applied to a property of the fluid other than a component of momentum. 

 If 8' is the concentration per unit mass of fluid of any property, such as the 

 heat content or salinity, at any instant, 8' may be separated into a mean value 

 8 and a fluctuation s, i.e. 



8' = 8 + s. (5) 



Then across a plane perpendicular to OZ, for example, there will be a turbulent 

 transport of the property 8', given by pws. By analogy with molecular diffusion, 

 the turbulent transport may be regarded as a process of eddy diffusion. In the 

 example just considered, a coefficient of eddy diffusion, Kz, may be defined by 



Tsz = pws = -pKz^-' (6) 



cz 



where Tsz represents the turbulent transport of 8 in the OZ direction. Co- 

 efficients Kx and Ky may be defined similarly for turbulent diffusion in the OX 

 and OY directions. The coefficients Kx, Ky and Kz, like the corresponding 

 coefficients of eddy viscosity, will, in general, differ from one another, be 

 functions of position in the fluid, and depend on the type and scale of motion 

 and the stability. 



The existence of turbulence in a field of motion gives rise, therefore, to two 

 types of effect : the production of shearing stresses and the process of eddy 



