Closed equation for the structure function of an isotropic turbulent velocity field

Equation for the structure function of a turbulent stationary isotropic velocity field and its solution in the inertial scale interval

“…1. A closed equation for the structure function of a turbulent isotropic nonstationary velocity field is obtained in [1,2]. In this paper, we attempt to obtain the stationary form of this equation and solve it in the inertial scale interval.…”

“…In order to obtain the stationary form of Eq. ( 22) in [2] for the structure function D(r), it is necessary to pass in this equation to the limit t > ~ and calculate the integral over the time variable. However, in so doing, it is necessary to take into account the fact that the integrand on the right side of this equation depends on time explicitly and through the function being sought.…”

“…One of the terms on the right side of Eq. ( 22) in [2] can be written in the form i i d' ~ k~(p,n, q)(p(p,'~), (1) It co dp (t --T) s 0 0…”

“…where k~ is defined by Eq. ( 28) in [2]; here and in what follows, we willunderstand the symbol q~ to mean the part of the integrand which depends on T through the functions sought and does not depend on T explicitly. Changing the variable of integration v according to the equation (t -~.…”

“…)-1 = z and using Eq. ( 28) in [2] for k~, Here, only one of a series of terms, all of which can be analyzed in a completely analogous manner, is written out.…”