A note on third–order structure functions in turbulence

Abstract

Starting from the Navier-Stokes equation, we rigorously prove that a modied third order structure functionS 3 (r) asymptotically equals 4 r = 3 (is the dissipation rate) in the large Reynolds number limit of Navier-Stokes equation in an inertial regime. From this result, we rigorously conrm the Kolmogorov four-fth law, without the Kolmogorov assumption on isotropy. Our denition of the structure function involves a solid angle averaging over all possible orientation of the displacement v ector y, besides space-time averaging. Direct numerical simulation for a highly symmetric ow for a Taylor Reynolds number of 155 shows that the ow remains signicantly anisotropic and that without solid angle averaging, the resulting structure functions approximately satisfy these scaling relations over some range of r = jyj for some orientation of y, but not for another.

Recovering isotropic statistics in turbulence simulations: The Kolmogorov 4/5th law

“…However, while the linear scaling in r of the third-order structure function is fairly robust, the coefficient exhibits only a slow trend toward 4/5 as indicated by the numerical work of [7]. It is clear that for anisotropic forcing, some choices of directions for the vector increment r are more "isotropic" than others [8].…”

“…However, the interpolation of square grid data over spherical shells has been deemed too expensive [8], or, when some such interpolation scheme is implemented, has not been used at sufficiently high Reynolds numbers as will allow for observations of the K41 type [17]. The new angle-averaged and "local" laws of [8,9,10] provide us with the theoretical impetus to investigate and extract the isotropic component of the flow in high-Reynolds number anisotropic turbulence. We use a novel means of taking the average over angles which avoids the expense and effort of interpolating the square-grid data over spherical shells.…”

“…Nie and Tanveer [8] proved that the 4/3rds and consequently, the 4/5ths laws can be recovered in homogeneous, but not necessarily isotropic flows,…”

“…5 correspond to projecting onto the j = 0 (isotropic) sector by integration over the sphere. The authors of [8] do not perform numerically the average over the sphere. However, they do point out that the direction of r matters strongly.…”

Cascades and wall-normal fluxes in turbulent channel flows

“…The basic result of (1.1) is that, at sufficiently large Reynolds numbers, an intermediate range of scales exists, away from energy injection and energy dissipation, where the energy flux across scales is identified as δu 3 /r. This expression provides a direct evaluation of the energy cascade through the inertial range (see Nie & Tanveer 1999;Aoyama et al 2005;Gotoh & Watanabe 2005;Ishihara, Gotoh & Kaneda 2009). …”