An equality about the velocity derivative skewness in turbulence

We study velocity derivative skewness S of incompressible homogeneous isotropic turbulence. By using exact relations of isotropic turbulence and various typical models of second-order structure function DLL(r) and energy spectrum E(k), it is found that −S=C(kc/kd)2 when Taylor-microscale Reynolds number Rλ is high. Here, C is a coefficient, kc is the center wavenumber of energy dissipation spectrum, and kd is the Kolmogorov wavenumber. Therefore, the problem of Reynolds number dependence of S becomes the problem of Reynolds number dependence of kc/kd. In the inertial range, we have scaling DLL(r)∼rζ2 and E(k)∼k−(ζ2+1), ζ2 is the second-order inertial-range scaling exponent. Equality −S=C(kc/kd)2 is valid in the case of ζ2>2/3 (intermittency models of Kolmogorov’s 1962 theory) as well as in the case of ζ2=2/3 (Kolmogorov’s 1941 theory).