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NASA Technical Me
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andum 84407
NASA-TM-84407 1
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Higher-Order
Derivative
Correlations
and
the
Alignment of
Sma
ll
-Scale
Structures
in
Isotropic
Nu
merical
Turbulence
Robert
M.
Kerr
De
ce
mber 1983
NI\S/\
National Aeronautics and
Space Admin
is
tration
B
RY
oPY
LANGLEY
RESEA
RCI
t.N rEH
LIBRARY NASA
HAMPTON. VIRGINIA
---
-
----
I
NASA
Technical Memorandum 84407
Higher-Order
Derivative
Correlations
and
the
Alignment
of
Small-
Scale
Structures
in
Isotropic
Numerical Turbulence
Robert
M.
Kerr, Ames
Research
Cen
ter,
Moffett
Field,
Cal
ifornia
NI\S/\
National Aeronautics and
Space Administration
Ames Research
Center
Moffett Field , California 94035
..
Higher-Order Derivative Correlations
and
the Alignment of
Small-Scale Structures
in
Isotropic Numerical Turbulence
ROBERT
M.
KERR
NASA
Ames
Research Center
M.S.
202A-l,
Moffett
Field,
CA
94035
August
1983
In a three-dimensional simulation higher-order derivative correlations,
including skewness
and
flatness (or kurtosis) factors, are calculated for
velocity
and
passive scalar fields
and
are compared with
structures
in
the
flow. Up to 128
3
grid points are used
with
periodic
boundary
conditions
in all
three
directions to achieve R). to 82.9.
The
equations are forced
to
maintain
steady-state
turbulence
and
collect statistics.
The
scalar-
derivative flatness
is
found to increase much faster
with
Reynolds number
than
the
velocity derivative flatness,
and
the
velocity-
and
mixed-derivative
skewnesses do
not
increase
with
Reynolds
number.
Separate
exponents
are
found for
the
various fourth-order velocity-derivative correlations,
with
the
vorticity-flatness exponent the largest.
This
does
not
support
a
major
assumption of the lognormal
and
fJ
models,
but
is
consistent with some
aspects of
structural
models of
the
small scales. Three-dimensional graphics
show
strong
alignment between
the
vorticity, rate-of-strain,
and
scalar-
gradient fields.
The
vorticity is concentrated in tubes
with
the
scalar
gradient
and
the
largest principal
rate
of
strain
aligned perpendicular to
the
tubes. Velocity spectra, in Kolmogorov variables, collapse to a single
curve
and
a
short
-5/3
spectral regime is observed.
§A. Introduction
The
classical approach
to
investigating small-scale intermit-
tency
in turbulence
is
through
the
higher-order derivative correlations such
as skewness
and
flatness factors. Experimentally this has been done for
both
the
velocity
and
temperature, which
is
a passive scalar when buoyancy
is
negligible.
The
long-range goal
is
to
improve our understanding of
the
struc-
ture
of
the
small scales, possibly leading
to
improved methods for subgrid
modelling.
An
intermediate objective has been to relate the derivative cor-
relations
to
dissipation correlations
and
corrections
to
Kolmogorov scaling.
It
is believed
that
this
is
possible because
the
small scales are universal;
that
is,
the
small scales have a
structure
that
is
independent of
the
large scales
and
can be modeled.
So
far only single-probe velocity and scalar statistics, such as
the
velocity-derivative flatness
and
skewness, have been measured
with
hot-
wire instruments. More complicated statistics have not been measured
and
1
alternative approaches, such as
flow
visualization, are limited because
they
do not have
the
flexibility necessary
to
distinguish small-scale structures
that
are
intermittent
in space and time.
Another approach
to
investigating
the
small scales
is
numerical simulation.
In
a simulation more detail can be obtained
than
in experiments
and
the
con-
ditions can be more closely controlled. For example, because all components
of
the
velocity are known, one
is
able
to
study correlations beyond
the
deriva-
tive skewness and flatness
and
computer graphics
can
display structures
not
accessible to experiments. The main disadvantage of a simulation
is
that
only
a limited range of length scales
is
allowed, which restricts
the
Reynolds num-
ber to very
low
values. But in low-Reynolds-number experiments
the
values
of most of
the
derivative correlations are significantly different from
their
uncorrelated or Gaussian values. Therefore, these statistics are accessib
le
t o
current numerical methods and computers. Siggia
(1
981a) used a numerical
simulation
to
calculate higher-order correlations of
the
velocity, and discussed
the
relation between small-scale vortex structures and intermittency. Our
approach will be to use a similiar simulation
to
look in more detail
at
the
velocity-derivative statistics and
the
statistics of a
pa
ssive scalar. Graphical
display of small-scale vorticity, rate-of-strain, and scalar-gradient
struct
ures
is
used to interpret these statistics and comparisons with phenomenological
theories
and
experiments are made.
There are two phenomenological approaches to predicting
the
small-scale
statistics: either by assuming a form for vortical structures or a form for the
energy cascade from large to small scal
es.
Two models
that
are based on
the
cascade of energy are
the
lognormal theory of Kolmogorov (1962) and
the
p-model of Frisch, Sulem, and Nelkin (1979). Both theories predict a cor-
rection to the
k-
5
/
3
inertial-range kinetic-energy spectrum of Kolmogorov
(1941). They also predict
that
as
the
Reynolds number grows
th
at
the
velocity fluctuations become increasingly localized, or intermittent, distribu-
tions become highly non-Gaussian,
and
the
higher-order correlations, such
as
the
derivative skewness and flatness factors, increase with Reynolds num-
ber with a power-l
aw
dependence. The power-law exponents depend on
the
details of each model
and
on
j.l
,
the
characteristic exponent of
the
dissipation-
dissipation correlation function in
the
inertial subrange,
rJ
< < r < < L,
<
E{X)E(X
+
r)
> = Ac
2
(L/
r)'"
(1)
(Monin and Yaglom 1975, p. 618), where A
is
a constant and L
is
defined
by (12). All correlations of a given order are predicted to have
the
same
2
power-law dependence. The lognormal model predicts
that
and
that
where
3
an = -J1n(n -
1)
4
(2)
(3)
(4)
(Frenkiel and Klebanoff 1975). The
,B-model
is similiar,
but
predicts
that
a
n
=
3j.t
(n
-1)
4-fJ
2
(5)
(Nelkin and Bell, 1978). The success of these models in predicting
the
ex-
perimentally observed dependence of derivative skewness and flatness on
Reynolds number
is
discussed in detail by Antonia, Satyaprakash,
and
Hussain (1982), who conclude
that
the
lognormal model
is
superior in this
respect. Neither of the corrections
to
the
k-
5
/
3
law proposed has been ob-
served, although
the
lognormal correction
is
much smaller and probably
not
much above statistical noise.
The value of
J1
can be found either by direct measurements or by using these
models
to
calculate back from an. Nelkin (1981) and Antonia, Satyaprakash,
and Hussain (1982) summarize
the
current evidence and conclude
that
most
previous estimates were too high. They suggest
that
J1
= 0.2.
Structural models assume
that
the
small scales are composed of tubes
or sheets of vorticity. Corrsin (1962) assumed
that
sheets dominated and
concluded
that
a4
= 1.5. Saffman (1968) also assumed
that
the
vorticity
would be found in sheets whose thickness would be
the
order of
the
Taylor
microscale (9),
but
that
within
the
sheets, dissipation would be localized in
regions characterized by
the
Kolmogorov length scale (lOa).
He
found
that
a4
= 1 and
a3
=
0,
that
is,
the
skewness
is
constant. A variation, proposed
by Tennekes (1968), assumes
that
the
dominant structures are tubes whose
thickness
is
the
Kolmogorov length scale,
but
which are subject only
to
the
large-scale strain. His results agree
with
those of Saffman. Experimentally,
a3
is
observed
to
be
very small, possibly zero,
but
a4
is
much less
than
the
predicted value of one.
3