According to Secs. 10.4.4 and 10.4.5, in the frame of the principal axes ( s ₁ , s ₂ , s ₃) of the rate of strain tensor e _ij with the principal values ? ₁ , ? ₂ , ? ₃, the longitudinal velocity difference is given by

where

with ? as the polar angle and the azimuthal angle to denote a point on a unit sphere. The average is replaced with the integral

Therefore, we have

In particular,

Using the above expressions, we immediately find

In the present case of incompressible flow, we have ? ₁ + ? ₂ + ? ₃ = 0.

For the second-order structure function F ₂( s), we note that

Using these together with ( ? ₁ + ? ₂ + ? ₃) ² = 0, we obtain

For the third-order structure function F ₃( s), we note first that

From these and similar relations, we obtain

Note that

Using these relations, we obtain

By the calculation, we have ? ? ⁶ ? _sp = 1/7, ? ? ² ? ⁴ ? _sp = 1 /35 and ? ? ² ? ² ? ² ? _sp = 1 /105. Substituting these, we finally obtain