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i1520-0469-8-5-316 - 316 JOURNAL OF METEOROLOGY VOLUME 8...

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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 air-pollution 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 pressure-tube or cup anemometer, the result is given as a function of the horizontal component of the velocity. Similarly, in a wind tunnel a hot-wire 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 non-isotropic 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 x-component of the turbulent velocity a value between 125 and 71(2 + (15) and for the y-component 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 hot-wire 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 frequency-distribution 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 non-iso- 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,.1-y=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 well-known 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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