Pk. Yeung et al., DYNAMICS OF DIRECT LARGE-SMALL SCALE COUPLINGS IN COHERENTLY FORCED TURBULENCE - CONCURRENT PHYSICAL-SPACE AND FOURIER-SPACE VIEWS, Journal of Fluid Mechanics, 283, 1995, pp. 43-95
As discussed in a recent paper by Brasseur & Wei (1994), scale interac
tions in fully developed turbulence are of two basic types in the Four
ier-spectral view, The cascade of energy from large to small scales is
embedded within 'local-to-non-local' triadic interactions separated i
n scale by a decade or less. 'Distant' triadic interactions between wi
dely disparate scales transfer negligible energy between the largest a
nd smallest scales, but directly modify the structure of the smallest
scales in relationship to the structure of the energy-dominated large
scales. Whereas cascading interactions tend to isotropize the small sc
ales as energy moves through spectral shells from low to high wavenumb
ers, distant interactions redistribute energy within spectral shells i
n a manner that leads to anisotropic redistributions of small-scale en
ergy and phase in response to anisotropic structure in the large scale
s. To study the role of long-range interactions in small-scale dynamic
s, Yeung & Brasseur (1991) carried out a numerical experiment in which
the marginally distant triads were purposely stimulated through a coh
erent narrow-band anisotropic forcing at the large scales readily inte
rpretable in both the Fourier- and physical-space views. It was found
that, after one eddy turnover time, the smallest scales rapidly became
anisotropic as a direct consequence of the marginally distant triadic
group in a manner consistent with the distant triadic equations. Beca
use these asymptotic equations apply in the infinite Reynolds number l
imit, Yeung and Brasseur argued that the observed long-range effects s
hould be applicable also at high Reynolds numbers. We continue the ana
lysis of forced simulations in this study, focusing (i) on the detaile
d three-dimensional restructuring of the small scales as predicted by
the asymptotic triadic equations, and (ii) on the relationship between
Fourier- and physical-space evolution during forcing. We show that th
e three-dimensional restructuring of small-scale energy and vorticity
in Fourier space from large-scale forcing is predicted in some detail
by the distant triadic equations. We find that during forcing the dist
ant interactions alter small-scale structure in two ways: energy is re
distributed anisotropically within high-wavenumber spectral. shells, a
nd phase correlations are established at the small scales by the dista
nt interactions. In the numerical experiments, the long-range interact
ions create two pairs of localized volumes of concentrated energy in t
hree-dimensional Fourier space at high wavenumbers in which the Fourie
r modes are phase coupled. Each pair of locally phase-correlated volum
es of Fourier modes separately corresponds to aligned vortex tubes in
physical space in two orthogonal directions. We show that the dynamics
of distant interactions in creating mall-scale anisotropy may be desc
ribed in physical space by differential advection and distortion of sm
all-scale vorticity by the coherent large-scale energy-containing eddi
es, producing anisotropic alignment of small-scale vortex tubes. Scali
ng arguments indicate a disparity in timescale between distant triadic
interactions and energy-cascading local-to-non-local interactions whi
ch increases with scale separation. Consequently, the small scales res
pond to forcing initially through the distant interactions. However, a
s energy cascades from the large-scale to the small-scale Fourier mode
s, the stimulated distant interactions become embedded within a sea of
local-to-non-local energy cascading interactions which reduce (but do
not eliminate) small-scale anisotropy at later times. We find that wh
ereas the small-scale structure is still anisotropic at these later ti
mes, the second-order velocity moment tenser is insensitive to this an
isotropy. Third-order moments, on the ether hand, do detect the anisot
ropy. We conclude that whereas a single statistical measure of anisotr
opy can be used to indicate the presence of anisotropy, a null result
in that measure does not necessarily imply that the signal is isotropi
c. The results indicate that non-equilibrium non-stationary turbulence
is particularly sensitive to ion-range interactions and deviations fr
om local isotropy.