^cftt(*1'l^7C' '■'■■■rs^fmrS^TT^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1370 SOME MEASUREMENTS OF TIME AND SPACE CORRELATION IN WIND TUNNEL By A. Favre, J. Gavlglio, and R. Dumas Translation of *Quelques Mesures de Correlation Dans le Temps et L'Espace en Soufflerie.* La Recherche Ae'ronautique No. 32, Mar. -Apr. 1953. Washington February 1955 "Y OF FLORIDA ' ^Y 17011 701 USA }i^i'li/o\ 3%oir3^9 NATIONAL ADVISORY COMMITIEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1570 SOME MEASUREMENTS OF TIME AND SPACE CORRELATION IN WIND TUNNEL*"'" By A. Favre, J. Gaviglio, and R. Dumas SUMMARY These researches are made at the Laboratoire de Mecanique de I'Atmosphere de I'l.M.F.M., for the Office National d'Etudes et de Recherches A^ronautiques (O.N.E.R.A. ) , with the aid of the Ministere de I'Air, and of the Centre National de la Recherche Scientifique (C.N.R.S.). We shall sum up the results which we have obtained by means of the apparatus for measurements of time and space correlation and of the spectral analyser in the study of the longitudinal components u-i of the turbulent velocities in a wind tunnel, downstream of a grid of meshes M (refs. 1 to 15) j and we shall give the first results in the case of a flat-plate boundary layer (ref. 10). The correlation RfVT/M, ^-[l^> ^^f^j ^^ measured between two velocities u]_ considered with a relative time delay T, at two points of space at a distance X-, from each other parallel to the general velocity V of the flow, and at a distance X^ orthogonally. I. TIME CORRELATION, TURBULENCE SPECTRA DOWNSTREAM OF A GRID For X-, = X2 = 0, only one hot-wire anemometer is used. Numerous time correlation, or autocorrelation curves, R(VT/M) have been drawn, the spectral curves being obtained on one hand by transforming them, and on the other hand by direct measurements with the analyser, for velocity u-, , behind two grids of meshes M = 3 l/k and M = 1 inch , successively, at distances of kO M downstream of the grids, and at various speeds and Reynolds mesh numbers B^ (refs. 2 to 9^ U^ and 12). "Quelques Mesures de Correlation Dans le Temps et L'Espace en Soufflerie." La Recherche Aeronautique No. 32, Mar. -Apr. 1953 , PP- 21-28. A communication to the 8th International Congress on Theoretical and Applied Mechanics, Istanbul, Aug. 1952. Analogous to those of the NBS (ref. 15), with elements being respec- tively of 1.6 and 0.5 centimeters diameter. 2 NACA TM 1570 As an example, figure 1 (ref . 12) represents an autocorrelation curve of the velocities n-, at kO M downstream of that grid of mesh M = 1 inch, the mean velocity being V = 12.27 meters per second, % = 21500. As an exajnple also, figures 2 and 5 (ref. 12) show the values of the spectral function F(n) obtained by Fourier transform of the pre- ceding autocorrelation curve or by direct measurements by means of the analyser (n frequencies). The autocorrelation curves have a limited radius of curvature at the origin (refs. 2 and 5)« The equivalent frequency N of the only sine wave, the autocorrelation curve of which would have the same curva- ture at the origin, can thus be measured and can show the decay rate of energy (ref. l6)5. On figure 2 we have drawn the spectral curve given by H. L. Dryden as a first approximation by transformation of the correlation ciirve f represented in an approximate way by an exponential curve (ref. 15) • In the spectra, from kO to 2000 cps, the experimental points are neighboring this empirical curve. For the lower frequencies, from 1 to ho cps, the measured energy is smaller.^ The same seems to hold true for the spectra of NBS and of NPL (ref. 13). On figure 5 are also reproduced the spectra measured by R. W. Stewart and A. A. Townsend (ref. I8) with slightly different grids. The same peculiarity appears for the lower frequencies in the case of low Reynolds numbers E^; for higher Reynolds numbers it is not detected; the band- pass of the spectral analyser used is limited to about 20 cps. As G. K. Batchelor (ref. 17) has shown, the tendency to isotropy — very strong for most frequencies — is very slight for the large scale components of homogeneous turbulence. In particular the large scale components of the turbulence behind a grid in a uniform flow are aniso- tropic. They are also nonhomogeneous (ref. I8) . They can be dependent on the grid shape . 5 N^ = -1-m^fJ = r n%(n) dn ^ -4 2 1 A^SO r„2„._. .^ U2 N = 1.55 cps A =0.^4-5 cm H. W. Liepmann (ref. I9) performed numerous measurements of X, under different test conditions, varying the rates of energy dissipation, the spectral curves, and the number of passages through zero to u-^ per unit time. ^In the case of that grid the rods of which are of circular section. NACA TM 1370 II. CCMPARISON OF TIME CORRELATION WITH LONGITUDINAL SPACE CORRELATION: G. I. TAYLOR'S HYPOTHESIS According to Taylor's hypothesis (ref . Ik) the time correlation curves R(VT/M) may be assimilated to the longitudinal space correlation curves R^Xt/M) or f, if the relative intensity of turbulence is very low, with the condition VT/M = X^/M and the general movement being rectilinear and uniform. We have measured, in the same experimental conditions, the time correlation and the longitudinal space correlation. It must be noted that the wake of the upstream wire disturbs the measurements of B.(XjJlA) but not of R(VT/M); in order to lessen this effect, a small lag X:? is used. The results agree with an approximation similar to that of the measurements, as shown, for instance, in figure k (refs. 9 and 12). Thus Taylor's hypothesis is directly verified in the case of the above-mentioned experiments made behind a grid in a wind tunnel by meas- urements of time correlation and longitudinal space correlation. III. TIME AND SPACE CORRELATION, LONGITUDINALLY The time and space correlation, longitudinally R^VT/M, ^±M) behind a grid of mesh M = 1 inch, has been measured in the course of two series of experiments (refs. 9 and 12). The first series of curves in figure 5 (ref. 9) relates to longi- dlnal distances X-,/M of 0.000 0.21+1 o.i+83 0.720 1.20 1.93 3.1^ ^.56 6.64 8.72 the mean velocity being V = 12.25 mps, the Reynolds number R. = 21500. The second series of curves in figure 5 (ref. 12) deals with longi- tudinal distances X /H of 0.000 0.236 0.473 0.946 1.892 3.78 7.57 the mean velocity being V = 12.27 mps, Rj^ = 21500. MCA TM 1370 The latter measurements have been made after several Improvements of the experimental apparatus5 (ref. 15) • Figure 6 gives the Isocorrelation curves corresponding to the second series of the above-mentioned measurements, which are in first approxi- mation ellipses whose axes have an inclination of about 45° and whose diaoneter ratios comprised between 0.027 and 0.055 are, on the average, of the order of 0.0^+, namely, R 0.90 0.80 0.70 0.60 0.50 0.40 a/b 0.048 0.054 0.055 0.055 0.028 0.027 One finds that the time and space correlation longitudinally reaches a maximtmi for each distance X /M when the delay is close to the time necessary to cover this distance at the general velocity: VT /m~ xJm The time and space correlation longitudinally downstream of a grid in a wind tunnel, with a delay compensative of the general movement R(VT/M = ^±h^} ^ll^)) retains high values even for distances which are great in comparison with those that would practically sxiffice to make null the longitudinal space correlation R^X-j^/m). IV. TIME AND SPACE CORREIATION, TRANSVERSELY The time and space correlation transversely R(VT/M, X,/mJ behind a grid of mesh M = 1 inch has been measured in the course of two series of experiments (refs. 4 to "J, 11, and 12). Figure 7 (ref. 11) shows the first results obtained for transversal distances X,/M of 0.000 0.059^ 0.0788 0.157 0.515 0.650 1.26 the mean velocity being V = 12.20 mps, ^ = 21500. -^New amplifiers of hot-wire anemometers, improvement of the compensa- tion by means of the square waves method, use of wire of 5kL in diameter, extension from 2.5 to 1 cps of the band-pass of the apparatus for meas- urements of time correlation, reduction of the intensity of the wind-tunnel turbulence from 0.00045 to O.OOO58. NACA TM 1370 5 Figure 8 (ref . 12) relates to the new measurements, made after the atove -mentioned improvements, for the same values of Xz/M, the mean velocity being V = 12.27 mps, % = 21500. Figure 9 gives the isocorrelation curves corresponding to the new measurements (ref. 12), which are in first approximation ellipses whose diameter-ratios comprised between O.36 and 0.^9 are, on the average, of the order of O.kk, namely, R 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 , 0.10 0.00 a/b 0.36 0.39 0.^4-3 0.49 0.1+9 0.k9 0.47 O.i+3 0.41 0.1+8 V. TIME AM) SPACE CORRELATION LONGITUDINALLY AND TRANSVERSELY, WITH CCMPENSATORY DELAY OF THE GENERAL MOVEMENT Figure 10 represents time and space correlation longitudinally and transversely r(vt/M, X-j/M, X,/m) for zero delay VT/M = and also for a delay compensative of the general movement VT/M ~ Xt/m behind a grid of mesh M = 1 inch for distances Xj/m of 0.000 0.236 0.1+73 0.91+6 1.89 3.78 7.57 the mean velocity being V = 12.27 mps, Rj^ = 21500. Figure 11 gives the corresponding isocorreZation curves. They are in first approximation ellipses whose diameter ratios decrease with R: R 0.90 0.80 0.70 0.60 0.50 a/b 0.31 0.25 0.20 0.15 0.12 One finds that the influence on the correlation between the com- ponents u-[^ of the velocities at two points behind a grid of the dis- tance between these points in the direction of the general movement is partly compensated, even for relatively great distances, by delays equal to the time necessary to cover this distance at the speed of the general movement . NACA TM 1370 VI. TIME CORRELATION, TURBULENCE SPECTRA IN THE BOUNDARY LAYER OF A FLAT PIATE The measurements are made at 0.91m from the leading edge of a flat plate; the mean velocity is V = 12.20 mps, the Reynolds number Rjj = 766000 (ref. 10). The grid being taken off, the preturbulence was of 0.00045 with four screens, and the boundary layer was laminar; a grid of M = 1 inch being set in, the preturbulence is of 0.01^80, and the boundary layer is turbu- lent (thickness 5 = 24 mm) . The autocorrelation has been measured for the component u, of the turbulent velocity, at various distances X^^/b from the plate (fig. 12): 0.06 0.12 0.25 0.50 0.75 1-00 5.62 An important evolution of these curves as a function of the distance to the plate takes place in the boundary layer. The smallest delay T for which the correlation is zero assumes the re spe ct i ve value s : 7.5 10 12 2k 8.3 5.5 5 ms On the contrary, the equivalent frequency N and the rate of energy decay change only slightly. Figure 13 gives the spectra corresponding to the above-mentioned experiments, obtained either by transformation of the autocorrelation curves or directly. They differ but little from those of the tiirbulence behind a grid for frequencies from kO to 2000 cps; for frequencies lower than hO cps, they show a marked evolution as a fiinction of the distance from the wall in the boundary layer. Translated by A. Favre NACA TM 1570 REFERENCES 1. Favre, A.: Appareil de mesures statistiques de la correlation dans le temps. VI° Cong. Intern. Mecan. Appl. 19^6, Paris. 2. Favre^ A.: Mesures statistiques de la correlation dans le temps. VII° Cong. Intern. Mecan. Appl. 19^8^ Londres. 5- Favre^ A.: Mesures statistiques de la correlation dans le temps. Premieres applications a I'etude de mouvements turbulents en soufflerie. 28/2/^4-9. a paraitre dans la serie des Publications de I'O.N.E.R.A. k. Favre, A.: Mesures de correlation dans le temps et I'espace en aval d'une grille de t\irbulence, pour la composante longitudinale de la Vitesse. I5/7/49. 5. Favre, A.: Nouvelles mesures de correlation dans I'espace et le temps en aval de grille de turbulence avec appareillage modif ie . 31/12/49. 6. Favre, A., Gaviglio, J., and Dumas, R.: Mesures de la correlation dans le temps et I'espace, et spectres de la turbulence en souf- flerie. Coll. Intern. Mecan. 1950, Poitiers. Publ. Sc . et Techn. Minist. Air, No. 251. 7. Favre, A., and Gaviglio, J.: -Mesures de la correlation dans le temps et I'espace, et spectres de la turbulence en soufflerie. (developpements) . 30/6/5O. 8. Favre, A., Gaviglio, J., and Dumas, R.: Correlation dans le temps et spectres de turbxilence, en veine reduite. Controle des mesures. 31/12/5O. 9. Favre, A., Gaviglio, J., and Dumas, R. : Correlation dans le temps et dans I'espace longitudinalement en aval d'une grille de turbulence. 51/12/50. 10. Favre, A., Gaviglio, J., and Dumas, R. : Mesures dans la couche limite des intensites de turbulence, et des correlations dans le temps; spectres. 31/3/51- 11. Favre, A., Gaviglio, J., and Dumas, R.: Correlation dans le temps et dans I'espace: transversalement, transversalement et longitudi- nalement avec retard compensateur du mouvement d' ensemble, en aval d'une grille de turbulence. 30/6/5I. 8 NACA TM 1570 12. Favre, A., Gaviglio, J., and Dumas, R.: Nouvelles mesiires de correlation dans le temps et I'espace, longitudinalement , trans- versalement, longitudinalement et transversalement en aval d'une grille de turbulence. IO/7/52. 15. Favre, A., Gaviglio, J., and Dumas, R.: Appareil de mesures de la correlation dans le temps et I'espace. VHI° Cong. Intern. Mecan. Theor. et Appl., Istanbul, 1952. Recherche Aeronautique No. 51» 1955. J.A.S. Ik. Taylor, G. I.: The spectrum of turbulence. Proc. Roy. Soc. London, Ser A, 164, 476, 1938. 15. Dryden, H. L.: A review of the statistical theory of turbulence. Quarterly of Appl. Math., Vol. I, No. 5, April 19^3. 16. Martinot-Lagarde, A.: Introduction au spectre de la turbulence. N.T. G.R.A. 55, Paris 19^6. 17. Batchelor, G. K. , and Stewart, R. W.: Anisotropy of the spectrum of turbulence at small wave-numbers. Quart. Joiirn. Mech. and Appl. Math., Vol. Ill, Pt. 1, 1950. 18. Stewart, R. W., and Townsend, A. A.: Similarity and self preserva- tion in isotropic turbulence. Phil. Trans. Roy. Sc. Ser. A, No. 867, Vol. 243, 12/6/51, London. 19. Liepmann, H. W., Laufer, J., and Liepmann, Kate: On the spectrum of isotropic turbulence. NACA TN 2^73, 1951. NACA TM 1570 > tt -I Ifl iX s r- OJ CO 1—1 II > t5 • 1-4 • LO ha rt II ^ ^ XI Pil ^ ^^ (1) C) f) a ■=tl 0) II ^ X2 CD -*-> ro • r-l a TD •1-1 .jc; +-> n . — 1 .g (1) u T-H u s Hi < (U 10 NACA TM 1570 10 VF ' L Cn x _) -- A :? 4 4i '^1^ • %^«- -•••IT ■* •^» *4 ^ 1 _ ^ •t TTl < -i-t _., — I — -4 '1: = !--■ \5 % v \ 0.1 H 1- ( ■)ht^,i 1 jaed fro m c 1 1 :orrela tion cvu ~ve 1 s V ^ - \ 1 1 1 1 1 1 1 11111:1 - A .. - ^ « Spectral analyser \* . \ V M I 1 i \ 0JD1 1 \ — 1 \ — 1 ■ 1 •N . — \u \ \ \ 001 _ ■ ^. .^ nU y 0.001 0.01 0.1 10 Figure 2.- Spectrum of turbulence downstream of a grid obtained by- Fourier transform of autocorrelation, or directly measured. V = 12,27 mps; M = 1 inch; distance = 40 M; Compensation square waves method, Ry[ = 21500. NACA TM 1370 11 *. = (e/^)« 200 ISO •^ 100 so X M • 00 + 80 X io<» • 40 2625 ,i'5!-fi-(;<4-. •-Sffl— I, X. 0-2 kjk. 60 y 6250 80 30 40 60 30 21000 . 10500 FCn) (a) Stewarl .-Townsend spectra. * * y*T hC J m "r^ — - — - t^ P' 4— _j ! — '3d k f- — ::V --t.S- — — ■ — * V • ' \ u 1 J - — \ ' — ' — — \ \ ') *. \. V \ ^ V \ _ !▲ • Spectral analyser n — ' — n-^ — — - + Obtained frogi correlation cur^ b-. _ — ,_ -■'s;;^ ' — 1 ■ — , ^.^ ■ lI: L-- ■*~ji ^ 1/1 « O o 000000 Oo OqoOOOoOo O f^ o »no oooooQmQOO ■' (b) Spectra obtained by Favre, Gaviglio, and Dumas. V = 12.27 mps; M = 1 inch; distance = 40 M; R]y[ = 21500; N = 435 cps. Figure 3.- Spectra of turbulence downstream of a grid by transform of autocorrelation or measured directly. 12 NACA TM 1570 ^i^ X|; » 1 s k ": s — - •" -s ^ X « — - - a — - •4 ■H +> nJ I H 2 ^1 V ^^ t( -._ r\ S 8 « 0) — _ - a a — 1~ - p. -p _^ (h -H '■ Jh T) — - - - 3 +J — - A •H (U t)0 — - - S g • ^ 5 , . • -11 =1 < jj ♦ ? ? • ^■x .^ tro: .^- -'1 o -4- 10 -d II S 10 cvj II > -p a a A^ u CO a 0) S -p m •H >>5 tro: § ■H +> cfl H (U u U 0) cd ft ra fH 5 1 •H > X 2 J- II 0) 1 -p CO •H tJ • •\ A 'i •H H II 2 & CVJ II > CO I to cS 1 ad longitud /pothesis. 3 x:' 9." ^^ s ♦ ^0 2 ■ I K , — ( CD • u s- £_ ^ CD f^ 1 -P -*-> ^ P 'H 0) f^ -- fe S^ « 3 1 Eh .„ d^ n '^ • • ■0 ' CD rt -^ _ J Xf ' 1?==:^ ■"<^ 1 ! -i.'- NACA TM 1570 13 kR(VT/M X,/M) N yO 2*1 An!"!" 7V^ I 20 "7 /1.«3 • M » I / ^^ t k lU \ / V —j- \ — 7 ~-7 ,X,/W aut± 3 . ■'' 72 V / \ / \ ^ ■• • • — . t' v\ \ ^ V / V / / \ . / \ A / /V\ V s/ > V N K \ .A / N s. ■ ^ •N, S^^^ \ V / \ / K . N y ^e^ »£! /^1^ rs^ ^:^; X A X X L ■ 1 i 1 — 1 ril 1 \jj%\ •=■ '- — — -f- ■ — ^ — ■- I 1 Y 1 V 1 ^^i.nv ^rTi r 1- ' (a) First meas\irement6 (ref. 9). V = 12.25 mps; M = 1 Inchj distance = 40 M; % = 21500. I RCVT/M X,/M ) Y /0.2 w ^a9 ■«w« — •j\ s/ \ V y / V / \ V ^ \ ^^ X "^«- N N. .-^ / \ v. -^ ^ **J a- --. ^ x::^ L/ V ^ _ » ^^^1 2 ) ^" 6 7 ' 9 (b) New measurements (ref. 12). V = 12.27 mps; M = 1 inch; distance = UO Mj Rj^ = 21500. Figure 5.- Tims and space correlation longitudinally R(VT/M, X^/M) for turbulence downstream of a grid. Ik NACA TM 1570 Eh ^ > II ■-i W a s O . — I d o •iH -a I — t O) ^1 u o o o > a TO 4_J a, w O O i-H 0^1 0) O) 42 CD I CD O) Vi W ;d M-i T3 NACA TM 1370 15 1 0.9 O.B R C VT/M , x,/M ; 1 0.6 V: a; • wire 1 _ 3 ^' 1 1 1 1 ^... ?**# L. 4 5 6 7 -- 6 , 9 1 2 * ••- *« -"• — •- VT/M 0.7 I v 0.4 », X /Mr 039': 0,2 0.1 — ^. Si >«.. r- 3 4 s 6 - 7 6 5 07 0.6 1 — -. -■• VT/M ; \ r \ OJ \ > ,1/ M= 0788 ^. 0.1 \ ■ \ s ^,. .», 3 4 5 6 7 6 5 0.5 1 2 ■*•' -•■ "*' VT/M ^ 3 \ \. X^/M = 0157 \l \t >« 3 4 5 =•' 6 7 S 9 1 i • ■"• — "" VT/M 0,2 ■'«. v 1 3/M.0315 V. - ' .^ =•: 4 5 6 7 6 9 ^1 2 r* ■1,0 630 -»• -• VT/M 3 ST 4 5 6 7 S 9 - 1 1 1 VT/M "3/"'= ' '" 1 ' 1 1 ^ 1 P 4 M r 1 r 1 a 9 VT/M Figure 7.- First series of measurements. Time and space correlation transversely R(VT/M, X3/M) for turbulence downstream of a grid. V = 12.20 mps; M = 1 inch; distance = 40 M; Rj^ = 21400. 16 NACA TM 1570 R CVT/M , X,/M) a9 0.6 0.7 1 1 1 1 -1- -V \ V Xj / M s 1 h V, 'c^ "^S ^f r*' •; 5 6 7 e 9 0,6 1 2 — 1- ■"'' ^^ ■1* ■ VT/M \ 0.6 • -\ 04 \ 1 \ X3/M = 0.039-; 0,2 *M 1 "if n ■•<, >|. i. ^ 5 6 7 e 9 1 2 ■~" ^7^ ^^ ■"" ■^"' VT/M V- 0.3 0.2 0.1 > ,/m = 0.0769 ■ 's ">< -. 3 4 5 6 7 s J OS 4 • 1 2 ^ '^ VT/M ••. '•• '\; Xj / M r 0,157 \ L!i ••-. *•§ 3 4 5 6 7 8 H*a 9 1 2~ VT/M 03 "i. 1 1 X^ /M; 0.315 ^•••-. ...1 I3 4 5 6 7 8 9 n 1 — 1 [2"^ M n ft^r ■ VT/M "^-^1...: "" 3 _ i — 5 ... 6 7 e 9 1 12 — •- VT/M L r 3 1 |3 r 1 p ' 1 r e ' 1 VT/M Figiore 8.- New measurements. Time and space correlation transversely R(VT/M, X3/M) for turbulence downstream of a grid. V = 12,27 mps; M = 1 inch; distance = 40 M; Ry^ = 21500. NACA TM 1570 17 1 JJJ''^ CO -1-1 ^ 0) > >., (D M > o o 1-H P^l o •r-H 1^ . 1 CD ^1 U O O O m Ui 73 ^ O O W O CD C/3 M fl c] •'-' CD " Eh ^^ (D > ho •rH 18 NACA TM 1370 f RCVT/M-vX^I, X,/M , X,/M) ' >L ^ 1 1 ■ 1 S*-! ^ ll 1— *sL n • ^ i \— s • 1 1 X,/M \ \ A » \ i i 0- N S, X./M = 0.946 1 ^>s^ •<*-•• L^*"^ ' U. — 2:y= - n •* "•— — . :^ 3: 1 U VM V Nc ^ X,/M = 1.892 \ Ix, ■^ r~ 1 — 1 — ' — — +^ f°- -^ — ^ In > ^h£1 1 u v^ »«. X X;/M=3.78| S^ 1 h^**^ 1 **> 8^,_ 1) ^ 1 ^ = hh = ^ "~T°" — ?■ H >«wl ~* I^^ 1 -. Xj/^ rs ^ ) M = 7.57 ^ •«J. 1 I e ^^ !5!»-0 , u ■— 0- ~ 1 1 1 1 ■| r n" 1 ] 1 r 1 1 X,/M' Figure 10. - Time and space correlation longitudinally and transversely R(VT/M -X^/M, X^/M, X3/M) with compensatory delay of general movement for turbulence downstream of a grid. V = 12.27 mps; M = 1 inch; distance = 40 M; Ryi = 21500. NACA TM 1370 19 1 v^ Figure 11.- Time and space isoc or relation longitudinally and transversely, with compensatory delay of general movement R(VT/M ~ X^/M, Xi/M, X3/M) for turbulence downstream of a grid. V = 12.27 mps; M = 1 inch; distance = 40 M; R^ = 21500; T ~ Xi/V. 20 NACA TM 1570 R(T) 1 1 I. L V OS A \- ^ ^ _K x.^it yyy -+5i7 '^ •< - ^' B r 2C _2; k' -y.. *« _ik__. ^ ^ » 0. 5 ■■•-»-*. ^-•^ • . " ■ 1 T I ). nsir "•^ \ X»X K'^- !!'■ V. 7 0. . |r43-— V '« .,.s:.-j ?i_ «' — - JM -. - — -. 3C .i^..=='k===.k... 50 SJ ^ - - — 1 1 \ ^,,^ _ xiic 3S i» "(1"= -T^ ■ di .._,-. .k_ _,_ ^ l2( u 3C y Id .i^ 50 55 _ _ 5 - - -v- rlrf --, - _, » i "t Ob 1^ ]t- x/i= dJsd •^ -J. bo n ! •~~^- 25 1 |(X_ 33 Vq , 65 5= 5^ i > °1 5 - rtC M5 • --1- 4 T tl osI~ 5 y I^^IIIX,^ D.'75^ • ^ 1 |1l1=L. Jl.. m--i >c ^ 1 3C ^ js^- V 1 \ n'7'\ » ^N ^J I.St s^ ^ \ S - -tN5N \ 1.00 t 562 -H _ _xx \ \. - ^ * >* ^ ^.^ \ \^ A -:k- uX V On N ^ "N. A> v^ > < Vi . >e X A' iin '^x.-x \\k ^^ A . HvW X ^'>,J ' < \a* >'(♦ V ^♦\' r ^ i-V ^\^ \+ ->x A -t ^ v\^ .Ill S-s^^t- ZL /L>Si.\ -i=^ -Jz^n^ hs^ ^^1^ t '-- - ■ t 1 1— Tl35m n 1 1 1 "M" zz:^' -z- ? :: : c > r\ IxV w\» Ar-jv W ' 4^ - Av-^° - Js A ^- ^\^ vX :: \i^ V t 1 ^\ 4\\ Ntv — r^ ^♦^ - ^«r, ^'i^* -m^ Y''^^4* n ___□. (X=)» •o "o >o Nooo>o 10 lo o 10 o looioo oooooio o o o 000000 0000 O O O n ^ Oi '^ tS CMojO)>>lO IOCSC0O.OCM "O O ij O \f >o "O ixoo °>o C>J vo O ^ ^ T- CN Figure 13,- Turbulence spectra in flat -plate boundary layer as a function of the distance from the wall. Rx = 766000. NACA-Langley - 2-28-55 - 1000 CO a,' M = rl 5 o o "J 3 O CO c m a>QJrtrt3<:rt-a;^ o. < 00 w „ w Q d , < a j K 2 OT a; ■£? S Pom "*^ C :2 ^ S "^ W J H •: U W !* rt ^O 3 Q a, rt-= i>) S "" csi (^ a u ci 52 > HI? z z Sz -^ - : S S 3: t1 c I ■^•S - c 0) = CO ° K S o hO rt C o Q> m O '. . o (U rt XJ 05 m O M i -g Z, 5 rt 0) " J-. o _g S £ C m T3 o ! ■§ ^ -g .b S > _ £ -2 o ^ _ o — =^ • c = ■■'-' " •S M to a, "■ 5 .H ■" -c ^ " o. (D ^ a, ■" < i3 d) • S ^ _ o (U < < < z CO aT^^ ^ 3. a"^ Q s^l 3 o S5 CD 1-3 O O CO c m C3i 3 . 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