Spectrum of turbulence in a contracting stream Page: 3 of 18
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SPECTRUM OF TiTRBUIENCI IN A CONTRACTING STREAM
represented as a set of three-dimensional Fourier integrals
q,(x1, 2, X)= f Q.(k, %, k3) eI(tlz+kr+ka) dkidk2dk3
-"
where a= 1, 2, or 3 and the significance of ki, k2, and kgs
will be brought out later. A continuous representation of
the turbulent field is obtained by allowing the Q. to vary
with time.
It will be convenient to abbreviat the Fourier integral toq f f)= fQa() e'dr (k)
and to introduce the companion equation
Q.(%= 8r qff.(8 e-@'td ()
f- f(x(la) -
(lb)
where k=1k,, k2, ki3; .=x,, zx, xz; dr(.)=-dxxdSCd3. The
second equation allows, in principle at least, the coefficients
Qa(k) to be calculated. Mathematically, q~,() and Q,(k) are
termed three-dimensional Fourier transforms of each other;
use will be made of this relation later.
The velocity components q, are connected by the condition
of continuity. In many cases of practical interest, these
turbulent velocities originate from boundary layers and the
wakes of obstacles in flows of low subsonic speed, so that
associated density fluctuations may be ignored; this is still
permissible when the turbulence so produced is transported
by a high-speed stream. Thus the incompressible form of
the continuity equation may be used and the result is
Q1ik1+ Q2k + Q3k=0 O
This relation may be written more compactly asZ Q, k,= O
a(2)
Physical interpretation.-The amplitude components Q.
are complex in general. According to equation (ib), then,
the requirement that the velocity components q, be real
implies that Qa(-k) is the complex conjugate of Q(9k). If
corresponding terms for _ and -ik in equation (la) are paired,
their sum is thus equal to the real quantity2(Re Q.) cos (k-.)-2(Im Q.) sin (k.x)
(3)
The imaginary parts cancel in the pairing, which implies that
they contribute nothing to the integral. Expression (3)
represents a pair of plane standing waves, a cosine wave and a
sine wave, with normals in the direction ik= (k, ik2, 3),
where x= (xi, x2, x3) is the radius vector to any point. The
vector k is termed the wave-number vector and its magnitude
k simply the wave number; the corresponding wave length is
27r divided by the wave number. Since k is perpendicular tothe wave front, it is sometimes referred to herein as the 'wave
normal.'
The continuity condition, equation (2), states that both
the real part (Re Qa) and the imaginary part (Ira Q,) of the
amplitude vector Q=Q,=Q. (Q, 2, Q) are perpendicular to
the wave nbrmal k; that is, both waves of expression (3) are
transverse. For each wave any one of the parallel planes
containing both the local velocity vector q and the wave-
number vector k (which is perpendicular to Q and hence toq)
is called the plane of polarization. The cosine wave (real
part) and sine wave (imaginary part) may be polarized in
different planes in general; the necessary and sufficient con-
dition that they be polarized in the same plane isRe Q e Q2 Re Q3
Im Q Im Qs Im Q3(4)
Equations (1) are now seen to represent a superposition of
plane sinusoidal waves (Fourier components) with all orienta-
tions of the wave fronts (all directions of the wave normal k)
and all wave lengths (all wave numbers k). Each wave is
transverse, and all planes of polarization, are permitted.
For each value of k there exist a cosine vave and a sine
wave; their respective amplitudes and planes of polarization
are different in general. The complex amplitude components
Q.(k) express, in their real and imaginary parts, how the
respective amplitudes and planes of polarization vary with
the wave-front orientation and the wave number.
Mean values: correlation tensor.--Consider the spatial 3
mean value of the product of the velocity component g,, at x
and the velocity component qp' at z'=S-+r as x varies but
the separation r of the two points remains fixed during the
averaging process; this mean value is called a velocity cor-
relation and is given the symbol R(r). There are nine such
correlations, corresponding to a=1l, 2, 3, and 3=1, 2, 3.
The form R4(r.) has been shown to transform like a second-
order tensof and has been designated (sometimes divided by
q2) as the 'correlation tensor' (ref. 6).
Evaluation of correlation tensor in terms of spectral
quantities.-The basis of the idea that the correlation tensor
might be expressed in terms of individual-wave parameters
is drawn from reference 4. In that paper mean-square values
of gi, q2, q3 in terms of such parameters are discussed; these
constitute the diagonal terms of R, (r) evaluated at r-=0.
The following derivation of Rop(r) amounts to a generaliza-
tion of the derivation of R,(0) given in that reference.
Assume that the turbulence is confined to a large, but not
infinite, parallelepiped of edges 2DI, 2D2, 2D3 and vanishes
everywhere outside.4 The space average Ro4) is to be
: If the statistical properties of the turbulence are independent of position (homogeneous
turbulence) and time-independent, an average at a given time over all space equals an average
at a given point (or pair of correlated points) over all time; a proof is given in appendix B.
If the statistical properties vary slowly with time, the space average will still approximate a
time average over an interval just long enough to smooth out the fluctuations.
4 The space integral of the square of any component velocity is thereby bounded, which is a
prerequisite for the existence of a Fourier integral, equations (1); in other words, equation
(Ib) shows that Q~ must depend on the volume 8DIDs of the region within which q differs
from zero: If this volume is infinite Q,. Is infnite, and the Fourier ntegral, equation (la). does
not exist.101
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Ribner, H. S. & Tucker, M. Spectrum of turbulence in a contracting stream, report, August 30, 1951; (https://digital.library.unt.edu/ark:/67531/metadc60473/m1/3/: accessed April 22, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.