C • STATISTICAL THEORIES OF TURBULENCE 



for their deduction. Robertson [14] gave such a method based on a con- 

 sideration of invariants. It proved very useful in later developments, par- 

 ticularly in the study of homogeneous anisotropic turbulence. We shall 

 give below the development for the double velocity correlation tensor 

 as an illustration of the method. 



Consider two arbitrary unit vectors ai and hi. Then the correlation 

 between the velocity components Uitti at P and Ujhj at P' is a scalar 

 quantity Q, independent of rotation of the coordinate system: 



Q = UittiUjbj — u-Rijaibj (6-14) 



It must therefore be a scalar function of all the scalar quantities involved 

 in the problems, namely, all the scalar quantities formed with the vectors 

 Gi, hj, and rk. These invariants are the following: 



(i) ttiUi = 1, hjhj = 1, ThTk = r^ 



(ii) aibi, ttiVi, hiTi 



In addition, the determinant formed of these vectors is an invariant under 

 rotation. This may be written in the form 



(iii) eijkaibjVk 



where eijk is the alternating symbol : €ijk = 1 if (i, j, /c) is (1, 2, 3) or its cyclic 

 permutation, eijk = —1, if {i,jj k) is (1, 2, 3) or its cyclic permutation, and 

 €ijk = otherwise. 



We note that Q is a bilinear expression in the vectors a^ and bi. Hence, 



Q = Qi{r)aibi -t- Q2{r)airihjrj + Qz{r)eijkaibjrk 



since this is the most general bilinear form in ai and hi that can be formed 

 from the invariants cited above. If one now imposes the further condition 

 that Q must also be invariant under a reflection, which changes Vk into 

 — Tk, it is clear that Qs = 0, and hence 



Q = (Qi^ij + Q2rirj)aihj 



Since ai and hj are arbitrary unit vectors, it is at once clear that 



u'^Rij = Qi8ij + Q-iriTj (6-15) 



This may be identified with Eq. 5-3 if Qi and Q2 are related to f(r, t) and 

 g{r, t) as follows: 



Qi = u'g, Q2 = y.'^-^ (6-16) 



C,7. Dynamics of Isotropic Turbulence. The dynamics of isotropic 

 turbulence is governed by the Navier-Stokes equations of motion 



dUi . dUi 1 dp , . 



Ot OXj p OXi 



{ 208 ) 



