806 BOWDEN [sect. 



E. Statistical Theory and Local Isotropy 



In the statistical theory of turbulence, various correlation functions are 

 used to relate a velocity component at one point with a velocity component at 

 another point. Thus if f{r) represents the correlation between the velocity 

 component Ux at a point x and the component Ux+r at a point x + r, 



f{r) = {UxUx+r)luK (17) 



A space correlation /(r) in the OX direction may be expressed as a time correla- 

 tion between the velocity at point x at time t and the velocity at the same point 

 at time ^ + t by putting r = Ur. Then 



/(r) = {utut^.)iu'^. (18) 



The correlation function /(r) is related by a Fourier transformation to the 

 spectrum function F{n), where F{n) dn is the proportion of the total turbulent 

 energy due to fluctuations with frequencies between n and n + dn (Taylor, 

 1938), i.e. 



f[r) = F{n) cos 27mT dn 



•'» (19, 



/•CO 



F{n) = 4 /(t) cos 277nT dr. 

 Jo 



The correlation functions which are derived from observations are usually of 

 the type (17) and (18) and are Eulerian, whereas the function R{t) defined in 

 (9), in connection with the theory of diffusion by continuous movement, is 

 Lagrangian. 



The concept of isotropic turbulence, i.e. a state of motion in which the mean 

 properties of the turbulence are independent of the axes of reference, was 

 introduced by Taylor (1935). In this case the Reynolds shearing stresses are 

 zero and the turbulence can neither gain energy from the mean motion nor 

 react on it. For this reason, isotropic turbulence, although studied extensively 

 both theoretically and in wind-tunnel experiments, found little application to 

 meteorological and oceanographic problems. 



A further advance was the introduction of the theory of locally isotropic 

 turbulence by KolmogoroflF (1941) and its development by von Weizsacker 

 (1948) and Heisenberg (1948). This assumes the existence of a continuous 

 spectrum of fluctuations which may be interpreted as a spectrum of eddy sizes. 

 The largest eddies receive energy from the mean motion and are anisotropic. 

 The theory of local isotropy applies particularly to eddies in an intermediate 

 range, which receive their energy from larger eddies, the motion of which is 

 considered as "relative mean motion" in this connection, and pass it on to 

 smaller eddies by the action of turbulent stresses due to the latter. This transfer 

 of energy may be expressed in terms of an effective eddy viscosity N, which is 

 a function of the fundamental volume, of typical length L, used to separate 



