C,5 • CONVENTIONAL APPROACH 



position in the field, and isotropy means that they are independent of 

 direction.'* For example, if we consider two points P and P' in such a 

 field, the velocity components Ur at P and u'^ at P', both in the direction 

 of PP', has a statistical correlation Uru'^ dependent only on the distance r 

 between P and P' , but independent of the coordinates of the point P and 

 the direction PP'. In a homogeneous anisotropic field, this correlation 

 would be unaltered by a translation of the vector PP' but would be 

 altered by a rotation. 



In particular, in a homogeneous isotropic field, the mean square value 

 of the three components of the velocity are equal to each other and are 

 the same throughout the field. Thus we have 



ul =^ = ^ = u^ (4-1) 



where Ui, u^, Us are the velocity components along the coordinate axes 

 Oa^i, 0x2, and Oxz, and u is the root mean square value. 



The statistical properties under consideration may be the spectrum, 

 the joint probability distributions of velocity and pressure, etc. In many 

 cases, we shall, however, be concerned with velocity correlations which 

 are the easiest to measure with hot wire instruments. 



The turbulent motion behind a grid in a wind tunnel has been found 

 to be approximately homogeneous and isotropic in the above sense. An- 

 isotropy is, however, generally found in the large scale eddies, and becomes 

 prominent when the Reynolds number is relatively low. 



C,5. Conventional Approach to the Statistical Theory of Turbu- 

 lence. Since a basic theoretical treatment of the frequency distribution 

 function has not yet been developed to an applicable stage (cf. [5]), cur- 

 rent statistical theories of turbulence are usually concerned with readily 

 measurable quantities. This has the advantage that experimental infor- 

 mation can be easily resorted to when purely theoretical considerations 

 become uncertain. Instead of dealing with the distribution functions, we 

 consider correlation functions, which can be more readily measured by 

 the hot wire technique. These are indeed the moments of the distribution 

 functions, as one can readily see from the formulas (Eq. 3-1 and 3-2) and 

 similar ones for correlations of higher orders. As higher and higher corre- 

 lations are known the over-all properties of the distribution functions are 

 known with increasing detail. Mathematically, the correlation represen- 

 tation can be shown to be equivalent to a spectral representation, con- 

 sidering energy distribution among various wave numbers or scales. The 

 two types of descriptions, however, exhibit different aspects of the same 

 physical phenomena. Both of them will therefore be used in the following 

 developments of the theory. 



* Indeed, general isotropy implies homogeneity, but the phrase "homogeneous 

 isotropic turbulence" is usually preferred as more descriptive. 



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