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Unformatted text preview: 316 JOURNAL OF METEOROLOGY VOLUME 8 FREQUENCY DISTRIBUTIONS OF VELOCITIES IN TURBULENT FLOW By F. N. Frenkiel Applied Physics Laboratory, Johns Hopkins University 1”
(Manuscript received 17 March 1951) ABSTRACT ‘ A conventional anemometer used in the atmosphere or in a wind tunnel measures velocities projected
on a plane independently of their directions in the plane. The relation between the frequency distribution
of those velocities and the frequency distribution of the turbulent components parallel to the direction of
the mean velocity is discussed. At small intensities of the longitudinal turbulence the two distributions are
approximately the same. However, at intensities of turbulence of the order of those measured in the atmos
phere, the difference between the two frequency distributions becomes appreciable. Some experimental frequency distributions of wind velocities are presented, and the intensities of atmos
pheric turbulence are determined from the experiments. 1. Introduction The importance of atmospheric turbulence in mete
orology has been realized for many years. The interest
in this ﬁeld increases now in connection with weather
control, various micrometeorological problems, and
particularly in relation to microwave propagation and
to air pollution [l]. The development of the uranium
industry and the construction of atomic piles empha
size still more the airpollution problems. Methods of
protection from radioactive contamination due to the
burst of an atomic bomb must also be investigated [2]. The meteorologist will have to evaluate on which
site a source of pollution will be the least dangerous.
He will have to determine when the local meteoro
logical conditions may become such as to produce a
danger of contamination from the polluting source. Let us consider, for instance, an area to be pro—
tected from pollutants emitted by a neighboring
source. In this area we can place a net of instruments
measuring the local mean velocities of the wind and
the character of its turbulence. A number of possible
trajectories of polluting clouds and their dispersion
along these trajectories can then be determined. To
each of these trajectories will correspond a distribution
of pollutant concentration. If any of the computed
concentrations reaches the dangerous proportion, the
micrometeorological conditions will be considered as
unfavorable. It will then be necessary to stop or to
reduce the emission of polluting aerosols during the
unfavorable time or to evacuate the dangerous area. To be effective, the determination of the local
meteorological conditions should be very rapid. The
computation will have to be done, therefore, on high
speed computing machines. Two elements of infor 1 Paper presented at the joint meeting of the American Meteor
ological Society and the Institute of the Aeronautical Sciences,
New York, 31 January 1951. 2 This work was supported by'the U. 5. Navy Bureau of Ord
nance under Contract NOrd 7386. mation will have to be fed into the computing ma
chines: the equations governing the dispersion in the
atmosphere, and the numerical values describing the
nature of the source, of the pollutants and of the
atmosphere. Although a fully satisfactory theory is
not yet completed, it is nevertheless possible to use
many of the available results (see papers by Sutton,
Calder and also [3]). In any case, it is necessary to
determine as correctly as possible the mean velocities,
in magnitude and in direction, and to measure the
characteristics of the turbulent fluctuations. In the
present paper, we shall discuss one question only: the
measurement of turbulent velocities with a conven
tional meteorological anemometer. 2. Frequency distributions of velocities When wind velocities are measured with a conven
tional pressuretube or cup anemometer, the result is
given as a function of the horizontal component of the
velocity. Similarly, in a wind tunnel a hotwire ane
mometer measures the component of the velocity
perpendicular to the direction of the hot wire. At
small intensities of turbulence, the difference between,
the horizontal component of the velocity, V” = [(71 + “92 + 7/2}, and the component parallel to the direction of the
mean velocity, u = 12 + u’, is small and, in general,
can be neglected. When, however, the intensity of
turbulence is large, as is most often the case in the
natural wind and in some artificial ﬂows (as for in
stance in jets and boundary layers), this difference can
no longer be ignored. We are going to determine the frequency distribu
tion of velocities projected on a plane (VH) under the
assumption that the components of the turbulent
velocity, u’, 'v’ and w’, are distributed according to a
normal law. This problem has already been treated OCTOBER 1951 by Hesselberg and Bjorkdal [4], Ertel and Jaw [5],
Kampé de F ériet [6] and Koo [7]. However, the
equations which they have derived were only applied
to the most simple case of isotropic turbulence and
lead to great difﬁculties in numerical applications to
the case of non—isotropic turbulence. In another paper
[8], the writer has shown how such equations can be
used in atmospheric turbulence, assuming isotropy of
the turbulence. In the present paper, the relations
for a particular case of nonisotropic turbulence are
derived and represented in a form which can be easily
used for numerical applications. Some numerical re
sults are given, and several experimental results are
presented for comparison. Assuming that the components of the turbulent
velocity are distributed according to a normal law,
one ﬁnds for the probability distributions of u’ and 11’ Prob[£<%<£+d£] =T Vﬂexp[ 23:2]«177, (2) where T, = Vii/12 and T1] = 4272/12 are, respectively,
the x— and the y'components of the intensity of
turbulence. In the present paper, it is assumed that the cross
correlation coefﬁcient W/ViTWVi—ﬁ (3) is small compared to unity. The case when the cross
correlation is not neglected will be considered at
another time. For the two—dimensional probability distribution,
we ﬁnd Prob[£<u—_<£+d£,n<v—_<n+dn]
u u
1 1 l + "W“
’— 21r Ksz K2 n, K = Ty/T, = Vii/V??? Expression (4) represents the joint probability of
having, at the same time, for the xcomponent of the
turbulent velocity a value between 125 and 71(2 + (15)
and for the ycomponent of this velocity a value
between an and 11(11 + (in). If we trace the vector
(u’2 + 7)”)é (see ﬁg. 1), (4) represents the probability
distribution for (u’2 + v”)* to have its tip inside the
square element dig, (in. (4) Where F. N. FRENKIEL 317 l
 E I
i n?“ V = V. $7
V”s (ﬁ+u')2+ \1'2 FIG. 1. Coordinate system. Since our anemometer measures the velocity VH, we
shall determine the probability distribution Prob [t < VH/a < t + dr] = D(§) dr, (5) which measures the probability of having the tip of
the vector VH within the annular area limited by the
concentric circles of radii f and f + (if. To ﬁnd the frequency distribution function D(§),
we shall determine the probability distribution of
V3 by replacing S and 11 in (4) by g“. Noting that
u’ = VH cos 0 — d and 71’ = VH sin 6, we have
5 = fees 0 ~ 1 and '0 = {sin 0, and after a simple
transformation we ﬁnd 1 g 1 K2+§2 _,21rKT2 [—2T2< x2 H ngr explg‘2 [<1K _ K2} c0526 (6)
2T“? 2 2
+ — cos 0]} d0.
f To simplify this equation, we must integrate the
expression 27r
J = ; exp (0 cos 6 + b cos2 0) d0. (7)
0
Developing the exponents in series, we ﬁnd
an—Zmbm 21r
J = Z Z——— cos" 0d0. n—O m=0 PU” +1)P(" — 27” +1) Since 1n the above expression the integrals with odd
powers of cogines are zero, we ﬁnd after integration aZn—Zmbm
1‘(m + 1) I‘(2n —— 21% +1)
I‘(1’L + 2) PM + 1) J=2 ri 1::
"°= "‘°= <8) 318 JOURNAL OF METEOROLOGY VOLUME 8 If we consider now the Bessel function of an imaginary gations (particularly for hotwire measurements in the argument of the zero order,
°° 1
I a = ——————— 02",
0( ) 7E) 1‘2(n + 1) 2211
we ﬁnd for its (23)th derivative w r(2n+ 1) I (28) = ____________.________.___ 211—28. 9
0 (a) 7311(2n—25’i—1) 112(n+1)22”a' ()
It can be easily shown that
J = 21r'1‘0(a, b), (10)
where
no b3
Ta,b = __.___I(2a)a. 10
0( ) EOF(5+1)0 () (a) We can now solve the integral which appears in (6),
and we ﬁnd 1
D(s‘) = exr> [ K (11) Fig. 2 represents the frequencydistribution func
tion D(§) when the longitudinal intensity of turbulence
T1 = 0.5 and for ratios K = Tu/Tx of 0.8, 1, 1.5, and 2.
In addition, a normal distribution curve is given for
comparison. The marked difference between the nor
mal distribution, the case of isotropic turbulence
(x = 1) and the frequency distribution for noniso
tropic turbulence should be noted. These differences
will still be very appreciable at smaller intensities of
turbulence. While in atmospheric measurements [9] we most
often meet the case of K 2 1, in wind—tunnel investi— “ 0 
0 0.2 0.4 0.6 0.8 L0 1.2 1.4 [.6 1.8 2.0 2.2 2.4; FIG. 2. Frequency distribution of horizontal velocities for
longitudinal intensity of turbulence T, = 0.5 and various ratios
x = T,,/Tz (Ty: transverse intensity of turbulence). Normal dis
tribution is given for comparison. ~ boundary layer) the case of K S 1 will be of interest. If we consider now the case of isotropic turbulence,
for which x =1 and T2 = T, = T, b = 0, To(a,b) = [0(a) and
(1 + m] Io(%) (12) This equation, which was already given by other
authors [5 ; 6], can be replaced by sz%J%W%—ﬁﬁhﬁﬂ,(m when 3‘/ T 2 is large enough (say of the order of 12).
If the intensity of turbulence is very small, Téﬁwm[~§%(1—§nl no The last equation is equivalent to (1), representing
the normal distribution function. The frequency distributions for isotropic turbulence
are given in ﬁg. 3. Curves corresponding to (12), (13),
and (14) are! represented for intensities of turbulence
of 0.5, 0.1, 0.05, and 0.01. It will be noticed that,
when T = 0.01, the three curves are superposed and
the frequency distribution function can be considered
as a normal distribution. f 1
DG) = 1:;exp[~ 2T2 DR“) 1* 3. Mean velocity and turbulent intensity To ﬁnd the correct mean velocity 12 and the correct
intensity of turbulence T from the measurements of
V3, it is necessary to take into account the difference
between the assumed normal distributions of the tur
bulent components u' , v’ , w' and the distribution
function D(§’). In the present paper, we shall only give approxi
mate relations for 12 and T in terms of VH. Developing
VH = [(12 + M)? + 2/2} in a series and neglecting the
terms of higher order, we ﬁnd E~1+5+”” (m
a 22 2(a)2
Upon averaging, this gives the relation [6]
mm x 1 + %T,,2. (16) Taking the square of (16) and again neglecting the
terms of higher order, we ﬁnd (VH)2/(71)2 z 1 + T112 (17) while, taking the square of (15) and averaging, we
obtain VIZ/(22)? z 1 + T; + T}. (18)
From the last two» equations we ﬁnd
(a)2 e (K2 + 1)(I7H)2 — KW. (19) OCTOBER 1951 F. N. FRENKIEL 319 I .O '7’
}Prob[§<\T/J'i < §+d§]= 0(5)“ _L] FIG. 3. Frequency distribution of horizontal velocities in isotropic turbulence for several values of intensity of turbulence. When the turbulence is isotropic, Figs. 4 and 5 represent experimental frequency dis
(mz z 2(VII)2 _ VHz (20) tributions measured, respectively, at the University
and of Texas in Austin and at the U. 5. Weather Bureau
2 _ 2 in Washin ton.
T2 ~ V” (VH) (21) g The mean velocity and the intensity of turbulence
indicated on the ﬁgures were obtained by applying the
approximate equations (20) and (21). It will be noticed
that the shape of the experimental distributions iS N 2(VH)2 — V112. These approximate equations give the mean velocity
to and the intensity of turbulence T in terms of VH,
measured with the anemometer. 15324—9]
If ..=I ,‘ry=o.5 and a sue—>1]:
fx=.5Ty =0.25 and ﬁnals—H } 
.f,=2,.1y=o.32 and 6530.9 ' 20 30 40 so mph ‘0 '5 20 25 30 mph FIG. 4. Experimental frequency distribution measured with FIG. 5. Experimental frequency distribution measured with
cup anemometer 24 ft above ground. Recorded at University of cup anemometer 100 ft above ground. Recorded at U. S. Weather 385?, Austin, on 16 November 1950 between 2134 and 2140 léureau, Washington, on 17 January 1951 between 1615 and 1649
. CT. 320 similar to the theoretical curves given in ﬁg. 2. It
seemed premature to use experiments for a more com
plete comparison with the theoretical results. It is
believed that measurements made under wellknown experimental conditions should be used for such a purpose. Acknowledgments.—The writer wishes to express his
thanks to Mr. J. R. Gerhardt of the University of
Texas and to Mr. G. W. Brier of the Weather Bureau
for making available the recordings of wind velocities,
and to Mrs. H. D. Sherwood for her assistance in the
numerical calculations. , REFERENCES 1. Hewson, E. W., 1950: Recommendations of the meteorology
panel, United States technical conference on air pollution.
Bull. Amer. meteor. Soc., 31, 269. 2. Los Alamos Scientiﬁc Laboratory, 1950: The eﬂects of atomic
weapons. Washington, U. S. Government Printing Ofﬁce. JOURNAL OF METEOROLOGY VOLUME 8 3. F renkiel, F. N., 1950: Introduction to some topics on turbulence.
College Park, University of Maryland, Lecture Series
No. 3 (prepared by S. I. Pai). 4. Hesselberg, T., and E. Bjorkdal, 1929: Uber das Verteilungs
gesetz der Windruhe. Beitr. Physik fr. Atmos., 15, 121. 5. Ertel, H., and J. Jaw, 1938: Uber die Bestimmung der Parame
ter im Verteilungsgesetz turbulenter Windschwankungen.
Meteor. Z., 56, 205. 6. Kampé de Fériet, J., 1945: Sur la moyenne des mesures dans
un écoulement turbulent, des anémométres dont les indi—
cations sont indépendantes de la direction de la vitesse.
Météorologie, 4, 133. 7. K00, C. C., 1945: The general laws of frequency distribution
of wind in a gust. Mem. not. res. Inst. Meteor., Acad.
Sinica, No. 14. 8. Frenkiel, F. N., 1945: Répartition des vitesses dans un écoule
ment de turbulence homogéne et isotrope. Toulouse, Groupe
ment des RecherchesAéronautiques (unpublished). 9. Frenkiel, F. N., 1947: Mesure de la diﬂ‘usion turbulente du
vent naturel dans le voisinage du.sol. C. R. Acad. Sci.
(Paris), 224, 98. ...
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