Planar isotropy of passive scalar turbulent mixing with a mean perpendicular gradient
L. Danaila,
1
J. Dusek,
2
P. Le Gal,
1
F. Anselmet,
1
C. Brun,
3
and A. Pumir
3
1
IRPHE, 12 Avenue du Ge ´ne ´ral Leclerc, 13003 Marseille, France
2
IMF, 2 rue Boussingault, 67000 Strasbourg, France
3
INLN, 1361 route des Lucioles, 06560 Valbonne, France
Received 16 October 1998; revised manuscript received 23 March 1999
A recently proposed evolution equation Vaienti et al., Physica D 85, 405 1994 for the probability density
functions PDF’s of turbulent passive scalar increments obtained under the assumptions of fully three-
dimensional homogeneity and isotropy is submitted to validation using direct numerical simulation DNS
results of the mixing of a passive scalar with a nonzero mean gradient by a homogeneous and isotropic
turbulent velocity field. It is shown that this approach leads to a quantitatively correct balance between the
different terms of the equation, in a plane perpendicular to the mean gradient, at small scales and at large Pe ´clet
number. A weaker assumption of homogeneity and isotropy restricted to the plane normal to the mean gradient
is then considered to derive an equation describing the evolution of the PDF’s as a function of the spatial scale
and the scalar increments. A very good agreement between the theory and the DNS data is obtained at all
scales. As a particular case of the theory, we derive a generalized form for the well-known Yaglom equation
the isotropic relation between the second-order moments for temperature increments and the third-order
velocity-temperature mixed moments. This approach allows us to determine quantitatively how the integral
scale properties influence the properties of mixing throughout the whole range of scales. In the simple con-
figuration considered here, the PDF’s of the scalar increments perpendicular to the mean gradient can be
theoretically described once the sources of inhomogeneity and anisotropy at large scales are correctly taken
into account. S1063-651X9902108-X
PACS numbers: 47.27.Gs, 47.27.Ak, 47.27.Eq
I. INTRODUCTION
Understanding the statistical and dynamical properties of
high Reynolds number turbulent flows is currently the sub-
ject of many investigations. Kolmogorov theory 1, hereaf-
ter referred to as K41, has provided some major insight in
this field. It assumes local isotropy in high Reynolds number
turbulent flows. The K41 theory successfully predicts the
structure of the two-point correlation function of the velocity
field in the inertial range, intermediate between the large
anisotropic scales, where energy injection takes place, and
the small scales, where viscosity is important. Local isotropy
for the inertial range signifies simply that all the statistical
properties of the field are invariant under any rotation direc-
tion. In particular, this property implies the independence of
these scales with respect to the large scale forcing, in general
anisotropic and very specific nonuniversal. The K41 theory
leads to a scaling law for the velocity increments, S
p
= „u ( r ) -u (0) …
p
¯
p /3
r
p /3
, when the separation r is in the
inertial range, and where
¯
is the mean dissipation rate. Ac-
cordingly, these moments should generally determine self-
similar probability density functions PDF’s.
The K41 theory, originally derived for the velocity field
was later naturally extended to the passive scalar field, by
Oboukhov 2 and Corrsin 3. Indeed, it was reasonable to
argue that the passive scalar properties would be completely
determined by the driving field behavior.
Significant deviations of the high order velocity moments
from the K41 predictions were found experimentally 4. Un-
derstanding these deviations, attributed to the phenomenon
called ‘‘intermittency,’’ and predicting the corresponding
anomalous scaling laws, has become a major challenge. Sev-
eral ‘‘phenomenological models,’’ from the Kolmogorov
model K41 to the refined similarity hypothesis RSH5,
or the multifractal model 6, tried, with more or less suc-
cess, to refine the predictions qualitatively and quantitatively
for the velocity scaling law exponents see Ref. 7 for a
recent review. Recently, a general analysis of the turbulent
cascade was proposed in Ref. 8. A canonical distribution of
velocity differences at any scale was introduced, with the
help of a conserved quantity throughout the whole scale
range. Several of the previously mentioned models can be
recovered as particular cases in this general formulation. A
phenomenological description of the statistical properties of
the cascade was also presented in Ref. 9, providing an evo-
lution equation for the probability density function of the
velocity increment. A method to extract the exponents cor-
responding to the various irreducible representations of the
rotation group was proposed in Ref. 10, therefore address-
ing the issue of isotropy in turbulent flows. Alternatively, in
11, anomalous scaling exponents for the dynamic field
were obtained, by taking into account the interaction be-
tween random and large-scale coherent components of a tur-
bulent field.
Generally speaking, the passive scalar field was found to
exhibit a more anomalous behavior than the velocity field
12, being in this sense ‘‘more intermittent than the velocity
field.’’ Specifically, the probability density functions
PDF’s associated with the inertial and dissipative scales
deviate from self-similar behavior, and present wider than
exponential tails more pronounced than those for the velocity
field.
PHYSICAL REVIEW E AUGUST 1999 VOLUME 60, NUMBER 2
PRE 60 1063-651X/99/602/169117/$15.00 1691 © 1999 The American Physical Society