National Aeronautics and
Space Administration
Langley Research Center
Hampton, Virginia 23681-2199
NASA/CR-1998-206909
ICASE Report No. 98-5
A Kinematically Consistent Two-Point Correlation
Function
J. R. Ristorcelli
ICASE
Institute for Computer Applications in Science and Engineering
NASA Langley Research Center
Hampton, VA
Operated by Universities Space Research Association
January 1998
Prepared for Langley Research Center
under Contracts NAS1-97046 & NAS1-19480
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A KINEMATICALLY CONSISTENT TWO-POINT CORRELATION FUNCTION
J. R. RISTORCELLI
*
Abstract.
A simple kinematically consistent expression for the longitudinal two-point correlation func-
tion related to both the integral length scale and the Taylor microscale is obtained. On the inner scale, in a
region of width inversely proportional to the turbulent Reynolds number, the function has the appropriate
curvature at the origin. The expression for two-point correlation is related to the nonlinear cascade rate, or
dissipation
ε
, a quantity that is carried as part of a typical single-point turbulence closure simulation. Con-
structing an expression for the two-point correlation whose curvature at the origin is the Taylor microscale
incorporates one of the fundamental quantities characterizing turbulence,
ε
, into a model for the two-point
correlation function.
The integral of the function also gives, as is required, an outer integral length scale
of the turbulence independent of viscosity. The proposed expression is obtained by kinematic arguments;
the intention is to produce a practically applicable expression – in terms of simple elementary functions –
that allow an analytical evaluation, by asymptotic methods, of diverse functionals relevant to single-point
turbulence closures. Using the expression devised an example of the asymptotic method by which functionals
of the two-point correlation can be evaluated is given.
Key words.
two-point correlation functions, turbulence modeling, functionals
Subject classification.
Fluid Mechanics, Aeroacoustics
1. Introduction.
In single-point turbulence closures one is often in the position of needing to approx-
imate functionals – integrals of functions of the two-point correlation function – Batchelor [1], Kraichnan
[2], Ristorcelli [3]. In aeroacoustical developments in which acoustic radiation due to turbulence is related
to the statistics of the turbulent source field one is in a similar position: the acoustic radiation is a function
of diverse two-point integrals, Proudman [4], Ribner [5], Lilley [6].
Batchelor [1], for example, in order to finish his development for the pressure variance in an isotopic
turbulence needed to evaluate the two functionals
I
(
f
) =
R
x
(
f
0
)
2
dx
and
I
(
f
) =
R
x
-
1
(
f
0
)
2
dx
. The function
f
(
x
) is the two-point longitudinal correlation function. Lilley [6] in an investigation of noise radiated from
isotropic turbulence required an evaluation of
I
(
f
) =
R
x
4
(
f
0
)
2
dx
. Proudman [4] in his statistical application
of Lighthill’s acoustic analogy required approximations to integrals such as
I
(
f
) =
R
f
(
f
000
+4
x
-
1
f
00
-
4
r
-
2
f
0
].
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- Fluid Dynamics, Taylor microscale, two-point correlation, W.C. Reynolds
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