Transport equations for the normalized moments of the longitudinal velocity derivative
${F_{n + 1}}$
(here,
$n$
is
$1, 2, 3\ldots$
) are derived from the Navier–Stokes (N–S) equations for shearless grid turbulence. The effect of the (large-scale) streamwise advection of
${F_{n + 1}}$
by the mean velocity on the normalized moments of the velocity derivatives can be expressed as
$C_1 {F_{n + 1}}/Re_\lambda$
, where
$C_1$
is a constant and
$Re_\lambda$
is the Taylor microscale Reynolds number. Transport equations for the normalized odd moments of the transverse velocity derivatives
${F_{y,n + 1}}$
(here,
$n$
is 2, 4, 6), which should be zero if local isotropy is satisfied, are also derived and discussed in sheared and shearless grid turbulence. The effect of the (large-scale) streamwise advection term on the normalized moments of the velocity derivatives can also be expressed in the form
$C_2 {F_{y,n + 1}}/Re_\lambda$
, where
$C_2$
is a constant. Finally, the contribution of the mean shear in the transport equation for
${F_{n + 1}}$
can be modelled as
$15 B/Re_\lambda$
, where
$B$
(
$=S^*{S_{s,n + 1}}$
) is the product of the non-dimensional shear parameter
$S^*$
and the normalized mixed longitudinal-transverse velocity derivatives
${{S_{s,n + 1}}}$
; if local isotropy is satisfied,
$S_{s,n + 1}$
should be zero. These results indicate that if
${F_{n + 1}}$
,
${F_{y,n + 1}}$
and
$B$
do not increase as rapidly as
$Re_\lambda$
, then the effect of the large-scale structures on small-scale turbulence will disappear when
$Re_\lambda$
becomes sufficiently large.
Low-Frequency Modulation of Turbulent Diapycnal Mixing by Anticyclonic Eddies Inferred from the HOT Time Series