A note on the velocity derivative flatness factor in decaying HIT

We develop an analytical expression for the velocity derivative flatness factor, F, in decaying homogenous and isotropic turbulence (HIT) starting with the transport equation of the third-order moment of the velocity increment and assuming self-preservation. This expression, fully consistent with the Navier-Stokes equations, relates F to the product between the second-order pressure derivative (δ²p=δx²) and second-order moment of the longitudinal velocity derivative ((δu=δx)²), highlighting the role the pressure plays in the scaling of the fourth-order moment of the longitudinal velocity derivative. It is also shown that F has an upper bound which follows the integral of k*⁴E*_p(k*) where E_p and k are the pressure spectrum and the wavenumber, respectively (the symbol * represents the Kolmogorov normalization). Direct numerical simulations of forced HIT suggest that this integral converges toward a constant as the Reynolds number increases.