Effects of a large-scale mean circulating flow on passive scalar statistics
in a model of random advection
Emily S. C. Ching,
1
C. S. Pang,
1
and Gustavo Stolovitzky
2
1
Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong
2
Center for Studies in Physics and Biology, Rockefeller University, New York, New York 10021
Received 24 December 1997
We study the effects of a large-scale mean circulating flow on passive scalar statistics in turbulent advection
using a two-dimensional lattice model. The incompressible advecting velocity field consists of a large-scale
circulation plus fluctuations. The latter are modeled by a random Gaussian field that has a finite correlation
time but is statistically independent at different lattice points. In the presence of the large-scale flow, we find
that the profiles of both the mean and the rms fluctuation of the passive scalar are modified significantly, and
have close resemblance to those observed experimentally in turbulent convection. The one-point probability
density functions PDFs of the passive scalar at the two sides are affected by the large-scale flow and become
more skewed. There is, however, not much change for the PDFs at other locations. Furthermore, the shape of
the PDFs of the scalar increments remains the same with or without the presence of the large-scale flow,
supporting the idea that large-scale effects can be filtered out by studying increments.
S1063-651X9808008-8
PACS numbers: 47.27.-i
I. INTRODUCTION
High-Rayleigh-number convection in fluid enclosed in a
cell is often taken as a system for investigating fluid turbu-
lence as the boundary conditions are well defined and the
flow can be changed from laminar to turbulent in a con-
trolled manner. Many interesting features have been uncov-
ered 1. A scaling state, which covers a wide range of Ray-
leigh number Ra from 10
8
to 10
15
, was revealed 2. This
turbulent regime is characterized by power-law dependence
of the heat flux and the size of local temperature fluctuations
with Ra, and has various features that have attracted much
attention. First, the scaling exponent of the heat flux is 2/7,
which is different from 1/3, the value one might obtain from
ideas of marginal stability 3, which suppose that the top
and the bottom thermal boundary layers do not interact. Sec-
ond, the probability distribution of temperature fluctuations
at the center of the cell was found to be exponential as
opposed to the Gaussian observed at lower values of Ra.
Finally, there is a persistent large-scale mean circulating flow
that spans the whole experimental cell 4, and the velocity
of the flow also scales with Ra. The presence of a large-scale
flow naturally induces an interaction between the two ther-
mal boundary layers. Two different theoretical models 5,6
have been proposed to derive the scaling laws. In particular,
the effect of the shear produced by the mean circulating flow
on the heat flux was studied explicitly in Ref. 6. However,
a complete understanding is still lacking 7–10. More re-
cently, it was further found that the distribution of tempera-
ture fluctuations become a superposition of two Gaussians
when the large-scale flow is perturbed 11.
In thermal convection, the temperature field takes part in
driving the flow and is so-called active. In this situation, the
velocity and the temperature fields are coupled in a compli-
cated manner. On the other hand, the problem of a passive
temperature field, i.e., a temperature field that is advected by
a given velocity field, is simpler. Insights or understanding of
some features of thermal convection may be gained by
studying a passive scalar subjected to a suitably prescribed
velocity field. As a result, various studies 12–18 were mo-
tivated to understand when the statistics of a randomly ad-
vected passive scalar will be non-Gaussian.
Along the same line of thoughts, it would be interesting to
study how the presence of a large-scale mean circulating
flow might affect the passive scalar statistics in turbulent
advection. In this paper, we report the results from a numeri-
cal study. We use a two-dimensional lattice model 17,19 in
which the fluctuation of the incompressible velocity field is
mimicked by a random Gaussian field that has a finite cor-
relation time but is statistically independent at different lat-
tice points. In earlier studies, it was found 17 that the pas-
sive scalar fluctuation becomes non-Gaussian for a certain
range of parameters of the model. Moreover, this change in
statistics is independent of the one-point statistics prescribed
for the velocity field 19. In the present study, we introduce
a large-scale mean circulating flow into the model and find
that its presence modifies the profiles of both the mean and
the rms fluctuation of the passive scalar significantly. Inter-
estingly, these profiles now show close resemblance to the
corresponding profiles for the temperature field observed ex-
perimentally in turbulent convection 7. The one-point prob-
ability density functions PDFs of the passive scalar at the
two sides are affected by the large-scale flow and become
more skewed. There is, however, not much change for the
PDFs at other locations. Furthermore, the shape of the PDFs
of the scalar increments remains the same with or without
the presence of the large-scale flow, supporting the idea that
large-scale effects can be filtered out by studying increments.
II. MODEL
We use a two-dimensional lattice model to study the ad-
vection of a passive scalar by a random incompressible ve-
PHYSICAL REVIEW E AUGUST 1998 VOLUME 58, NUMBER 2
PRE 58 1063-651X/98/582/19487/$15.00 1948 © 1998 The American Physical Society