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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1451 ON THE STATISTICAL THEORY OF TURBULENCE* By W. Heisenberg The interpretation of turbulence presented in the preceding paper by v. Weizsaecker is treated mathematically with the aid of the custom ary method of Fourier analysis. The spectrum of the turbulent motion is derived to the smallest wave lengths, that is, into the laminar region; the mean pressure fluctuations and the correlation functions are calculated. Finally, an attempt is made to derive the constant which is characteristic for the energy dissipation in the statistical turbulent motion from the hydrodynamic equations. In the statistical theory of turbulence developed by G. I. Taylorl and v. K&arman,2 the irregular turbulent motion of a fluid is described by several characteristic functions between which simple mathematical relations exist: the "spectral" distribution of energy to waves of dif ferent wave length, the correlations between the velocities at points along a prescribed displacement in space or time, and the like. The reports of G. I. Taylor contain detailed empirical and theoretical data on these functions. In his preceding report, v. Weizsaecker3 derived the most important of these functions, the spectral distribution of energy for the limiting *"Zur statistischen Theorie der Turbulenz." Zeitschrift fuir Physik, vol. 124, 1948, pp. 628657. 1Taylor, G. I.: Proc. Roy. Soc. A 151, 421 (1955); 156, 507 (1956); 164, 15 (1958); 164, 476 (1958). 2Karman, Th. v.: Journ. Aero. Sci. 4, 151 (1957). SC. F. v. Weizsaecker: ZS f. Phys., being published. V. Weizsaecker's paper and the present treatise have been written in close collaboration during the time of our stay in England in 1945. Only after the articles had been finished, Mr. G. I. Taylor kindly told us (spring 1946) that essential ideas in these articles had been found and published already by Kolmogoroff: Compt. Rend. Acad. Sc. USSR 50, 501 (1941); 52, 16 (1941); and Onsager (Phys. Revue 68, 286, 1945). Compare a report by G. K. Batchelor at the VI. Internat. Congress F. angew. Mechanik, Paris 1946. Approximately at the same time we learned, furthermore, about a paper by Prandtl and Wieghardt which contains similar concepts and has mean while been published in the Gottingen Academy reports (Nachrichten der Akademie der Wissenschaften in G'dttingen, Math.Physikal KLasse of the year 1945, p. 6). The present paper may therefore be regarded only as a supplement and completion of these earlier investigations. 2 NACA TM 1431 case of large Reynolds numbers on the basis of similitude considerations. The following sections of this paper first will translate v. Weizsaecker's considerations into the accustomed terms of Fourier analysis and, with the aid of this translation, study the discontinuity of the spectrum for high frequencies due to molecular viscosity. Then conclusions will be drawn for the correlation functions and the pressure fluctuations, and finally, a derivation of the fundamental constant of the energy dissipa tion will be attempted. 1. Representation of v. Weizsaecker's Considerations in the Terms of Fourier Components At sufficiently high Reynolds numbers the energy dissipation for the turbulent motion takes place in such a manner that the large turbu lence elements lose energy due to the fact that, for them, the energy and momentum transfer by small turbulence elements has the effect of an additional viscosity (cf. for instance Prandtl4). Under steadystate conditions, energy thus is continuously trans ferred from larger to smaller turbulence elements, with the spectral region of a certain wave length always receiving from larger waves as much energy as it gives off to smaller waves. For maintenance of this equilibrium a certain energy distribution is necessary which, when the molecular friction is neglected, is represented according to v. Weizsaecker by the law P = P = F(k)dk (1) 2 2 J F(k) k5/3 (k wave length signifies the wave number, vo = V or measure of the mean velocity.5) This spectrum F(k) k5/5 is bounded on two sides. For small wave numbers, that is, large wave lengths, for one reason or another, it will no longer be possible to regard the flow as isotropically turbulent, for the largest turbulence elements are governed by the geom etry of the devices which generate the turbulence. This end of the spec trum for small k therefore cannot in any case be the subject of a purely statistical theory. For large k, in contrast, the spectrum is bounded 4L. Prandtl, Str6mungslehre (Flow theory). Vieweg, Braunschweig, 5rd edition 1942, p. 105 ff. 50ur vO differs from the vO in v. Weizsaecker's report by a numerical factor of the order of magnitude of 1. NACA TM 1451 by the molecular viscosity. For large k, finally, the molecular vis cosity will become larger than the apparent turbulent viscosity, and the spectrum then will drop off very rapidly. For the calculations, we shall use the following notation: the velocity v in a normalized volume V is to be expanded into a Fourier series v = e c = nx, nx, ny, nz integers) (2) k Therein vk = v*k and the number of the "natural vibrations" between 6 __2 Z'S k and k + Ak is given by 4itk Then one obtains IvT,.21 =. h,2d V Iv,..21 = F(k)dk (3) 22 k (2(r)3 thus F(k) = (2,)2k2VIk2l (4) From div v = 0 there follows (.kt) = 0() Let us call the coefficient of the molecular viscosity, p. The mean energy loss then is, because of div v = 0 with the assumption that the bounding surfaces are at rest: 6This method is somewhat less apparent, but mathematically more convenient than the customary expansion with respect to sin and cos. It amounts formally to the limiting condition that vjv/6x, at a bounding surface of the volume V are to have the same values as at the opposite bounding surface. NACA TM 1451 S = P(rotv)2 (6) = ' i 2 = ^ F(k)2k2dk (7) If the spectrum obeys in a large region the law F(k) k5/5, the total energy is determined by the largest turbulence elements. We may assume for instance that the law k5/3 is valid down to a minimum wave number ko; for smaller k we shall assume F(k) = 0. Then V02 = 2 0 F(k)dk = 2 CO C dk = 3Cko2/3 (8) Thus CVO 2/3 3 kO and 2 k 32/5 F(k) = k02/ for k > kO v02kO2/3(2it)3 vk2 6lVkl/ (10) V. Weizsaecker considers the energy loss Sk of that portion of the total spectrum the wave numbers of which lie below k. For these turbulence elements, the turbulence elements of smaller wave length (<2n/k) have the effect of an additional viscosity. One may therefore generally write k Sk + 7k)f F(k') 2k'2dk' (11) where rk is to designate the additional turbulent viscosity; it iq produced by the cumulative action of all turbulence elements with wave lengths <21r/k. With respect to dimensions, iTk is according to Prandtl the product of density, mixing length, and velocity where the NACA TM 1431 mixing length will be comparable to the diameter of the turbulence ele ments in question whereas the velocity of the turbulence elements is given for instance by vO(kO/k)1/5. With reference to v. Weizsaecker's report one will therefore put Ik = oP / dk' (12) (K is a numerical factor) The expression below the integration sign is essentially determined by dimensional considerations; but one could of course imagine that, for instance, the waves k' in the proximity of k enter into the integral with somewhat different weights than the waves with large k values; that is, the integrand could depend, in addition, on the dimensionless number k'/k. However, because of the homogeneous form of the spectrum F(k) k5/3, one may include all these uncertainties in the numerical factor K and give to the integral arbitrarily the exact form (12). This method is unobjectionable in the region of th6 k5/3law but becomes inaccurate at the ends of the region where the geometry or the molecular friction modifies the spectrum. But even at the latter limit (12) will still be a good approximation which at least qualitatively correctly represents the effect of friction. The constant K in equation (12) must be exactly determined by the hydrodynamic equations; it has the same numerical value in all cases where one may speak of statistical isotropic turbulence, and does not depend in any way on the geometry of the flow. The theoretical deter mination of this important number will be attempted in section 5. There, it will also be shown that the turbulent energy dissipation actually may be written as a double integral of the type k po Sdk' dk with the integrand signifying the energy 0 k transferred from k' into k'' per unit time (equation (89)). This integral is more complicated than the simplifying expression (13) which results from (11) and (12); however, for the following considerations (11) and (12) may be regarded as sufficient approximations. For Sk one thus obtains Sk = + pK dk" ) F(k') 2k'2dk' (15) The decisive step of v. Weizsaecker's consideration is the state ment that this expression for k > ko must be independent of k: 6 NACA TM 14351 Sk = S = const (for k > ko) (14) because the total energy lies, almost entirely, in the longwave region of the spectrum, and the energy "transport" thus must become independent of k. Equation (15) may therefore be interpreted as the determining equa tion of the turbulent spectrum F(k) which must yield for the region of large Reynolds numbers the k5/5law, and for still larger k values the fading of the spectrum due to the molecular viscosity. 2. Shape of the Spectrum in the Region of the Smallest Turbulence Elements We put first [i/P = v and differentiate (15) with respect to k. Then there results / r m k^ dk. F(k'))F(k) k2 = S00 X (k F(k')k'2dk k J5' 0 (15) Then we define new variables x and w by the equations x = g ko F(k) = F(ko)eW; w = w(x) (16) Therewith (15) is transformed into 72 2 e WF0 w x) e 2 2 ex wdx (17) The constant v = (cf. equation (9)) is essentially the reciprocal Reynolds number of the total flow and therefore always very small; if the Reynolds number itself were small, the flow could not be turbulent at all. By repeated differentiation there originates from (17)  1 dw + w x e 2 x w x = 2e 2 2 (18) NACA TM 14351 7 In this equation, one may approximately evaluate the integral m .w+x x w(x) + (xx) + and breaking off after the second term. Since the exponential function rapidly decays, one thus obtains a good approx imation w+x po w+x 2 dxe 2 g 2 e (19) x 1 + (9 dx By substitution into (18) there finally results (i fi w+x One recognizes from (20) immediately the variation of the spectrum. For not too large x and w the first term in the sum may be neglected and one obtains 7( x 2 (1 + dw, i.e. F(k) ^= FO (21) dx) dx 3 kO as must be the case according to v. Weizsaecker's theory. For large x and w, in contrast, the first term predominates; therefore one must then have 7 F const k7 (22) dx In the region of the smallest turbulence elements the spectrum therefore decays very rapidly, namely with the seventh power of the wave number. Only in the transitional region from (21) to (22) are numerical calculations necessary in order to determine the solution of (20). Since one may put for smaller x, that is, for the region 1 << x << 12 A (25) w = x 3 NACA TM 1431 (therewith not only (18), but also (17) then is satisfied with sufficient accuracy), one may calculate, progressing from point to point, dw/dx according to (20) from w, and therewith derive w for higher x. It is sufficient to perform the numerical calculation for a particular large value of the constant, for instance 2K FO = a. For another value b one may then obtain w by a simple similitude transformation: wb(x) =wa + lg+ ) Ig (24) as one recognizes by substitution into (20) and (25). Figure 1 shows the result of the numerical calculation for L F = about 1000. The numerical integration shows that for this value in the region x > 5 (25) w(x) 7x 21.85 More generally, one obtains therefore in this region of the k7' law: w(x) = 7x + 5.0 4lg( F/0) (26) that is, F(k) = 0.0496 F 0 (27) A serviceable interpolation formula which is correct in the two limiting cases and does not result in any large errors in the transitional region, either, reads: F(k) = FO 1 (28) If one defines IL = it/kg as the "diameter of the largest turbulence elements" and introduces as the Reynolds number of the total flow RO pv (29) NACA TM 1431 one obtains according to (9), (27), and (29) kg = 0.16ko(RO)5/4 (30) One may denote Ls = it/ks as the "diameter of the smallest turbulence elements" and obtains Ls = 6.25IO(ROK)/4 (51) By (9), (28), and (30) the form of the spectrum is determined for the entire kregion. For the actual flows, of course, the shape of the spectrum will be different for small kvalues (k~ko) since there the geometry of the tests plays a role, for instance, the shape of the grids by means of which the turbulence is produced. In order to be able to sensibly carry out the comparison with experiments, one will then introduce a quantity ko in such a manner that, for instance in the domain of the k5/law (thus for ks >> k >> ko), the formula F(k) = kO2( ) (32) becomes correct. The quantity ko thus defined then does not give any direct statement regarding the variation of the spectrum for the smallest kvalues. Generally, though, the spectrum will greatly deviate from the k 5/3law in the region k ~kO For k > ks, too, the spectrum will not unlimitedly retain the form k7. The wellknown investigations of BurgersY make it very probable that at sufficiently small Reynolds numbers finally no turbulent motions what ever exist. On the other hand, the k7law shows such rapid decay that the region k > k. is practically insignificant. A somewhat larger error will arise, particularly in the transitional region, due to the inaccuracy of equation (15) itself; but it is probably not worth while to apply already at this point the much more complicated equation of section 5 to the problem stated here. The correct equations would at any rate lead to somewhat different numerical factors in (27), (30), and (51). 7Burgers, J. M.: Verh. d. Kgl. Nied. Akad. d Wiss. 17, Nr. 2, 1 (1959); 18, Nr. 1, 1 (1940). NACA TM 14531 For the comparison with experiment one needs the energy distribution with wave number for a certain direction, for instance with kx, since the spectra have been measured experimentally by Simmons8 and Dryden9 by means of the fluctuations of the velocity with time in an airstream which is guided past the measuring point at a constant velocity U which is large relative to vO. Also, this spectrum has a different form according to whether the Fourier expansion of vx or Vy is concerned. Experimentally, first the spectrum for yx is required; however, we shall also derive the spectrum for vy since it will be necessary later on, in the calculation of the correlation functions. Since according to equation (5) (vkI) = 0, one obtains B v 2( kx2) vx 2 1 2 33) The spectrum of x in kx which we shall designate by Fx(kx) becomes therefore xx moo y z 1 F(k) _0 41tk2 2 k2 / = dk (k2 _ k2)F(k) (34) In a similar manner, the spectrum for Vy becomes F+ + +k dkydkz 1 (k) = kx (k2 + kx2)F(k) (35) 8o kx 0 / From (34) and (35) there follows for (a) ko << k << ks: (ko 5/3 9Fotko 5/3 6Fnfko 5/3 F(k) = FO )/ thus Fx(kx) = and Fy(kx) = 6 (56) Simmons, U. Salter: Proc. Roy. Soc. A 165, 73, (1958). 9Dryden, Schubauer, Mock u. Skramstad: Nation. Adv. Comm. Aero., Nr. 581, 1938; Dryden, H. L.: Proc. V. Intern. Congr. f. Applied. Mech. Cambridge, Mass., p. 362, 1958. NACA TM 1431 (b) k >> ks: k05/5k 16/5 F(k) = F k s 126 Fx) EE kO5/ksl6/5k7 PM kx) = 1 k/ l6/3k (^ As a serviceable interpolation formula (which, however, is somewhat less accurate than (28) in the transitional region), one may again put Fxx) 9FO/ko5/ x \8/5 (38) x = xg j + ()) (ks8) with kgxx = 0.645 ks (59) and Fy(kx) 6F (O/[1 +( (40) with ksYX = 0.793 ks (41) Before the comparison with experiment is carried out in detail, we want to raise the question at what critical Reynolds numbers the transition from the k5/law to the k7law takes place, that is, if one wants to express it in this manner the transition from the really turbulent motion proper to the laminar motion. One may regard as the critical Reynolds number for this, perhaps, the expression RS = (42) wherein, according to v. Weizsaecker, vs = v( ) From (29), (50), and (51) one then obtains R 10.2 Rs K (43) 12 NACA TM 1431 The numerical value of K will be discussed later on. At any rate, the transition therefore takes place at a certain numerical value of the Reynolds number, as was to be expected from general similitude considera tions. In figure 2, the measurements of the spectrum Fx(kx) by SimmonslO are compared with the theory. The measurements in question are intensity measurements on an airstream which flows past the measuring point at the velocities U = 456 cm/sec (0), 608 cm/sec (X), 1060 cm/sec (0), and which has been made turbulent by a grid of 7.6 cm mesh width; the meas urement was made 2.1 m behind the grid. The measured points of Simmons are plotted individually only in the right part of the figure, in the left half the approximate scatter of the measured points is indicated by a vertical line. The abscissa is k in cm1 (in logarithmical scale), the ordinate Fx(kx), likewise logarithmically, in arbitrary units. If one assumes U/vO to have the same value in all three meas uring series, which is confirmed by other measurements by Taylor, one obtains, in the case of a suitable selection of this ratio, the three curves plotted in the figure. Qualitatively, the Simmons data are well represented by the curves, particularly also the divergence of the three test series in the shortwave part of the spectrum. In details, however, there exist considerable discrepancies; one recognizes from the figure that the range of validity of the k5/3law is here so small that a reliable check is not possible. The reason for this is the smallness of the Reynolds number RO. For k = 1 cm1 the diameter of the turbu lence elements is 5 cm, thus about half as large as the mesh width of the grid; in this region, the turbulence is not yet fully isotropic, therefore the k5/3 law cannot yet be valid. However, already for k = 4 cm1 the influence of the molecular viscosity becomes noticeable, and the intensity drops off markedly. The related measurements by Dryden quoted before which extend over a large spectral region have been made at Reynolds numbers so small that the validity of the k~5/5law can hardly be checked. Therefore it would be desirable that similar meas urements be carried out at very much larger Reynolds numbers. For the ratio U/v0, one obtains from the adjustment of the theoretical curves to the measuring points of U/vO = 553 if one identifies TO with the mesh width of the grid. This value agrees well with measurementsll of this ratio in similar tests if one assumes K to be about 0.5. 10Simmons, U. Salter: Proc. Roy. Soc. A 165, 753 (1938). 11Cf. G. I. Taylor, Proc. Royal Soc. A, 164, 486, (1958). NACA TM 1451 Another and probably more accurate determination of K is obtained from the damping with time of the turbulence, already theoretically treated completely by Taylor (cited before). For the total energy loss per cm5 and second: S, there results from (9), (15), and (14) S = pK v3k (44) For the damping with time of VO there must thus apply d.(4KF)= ,O3k (45) with the solutionl2 vo(0) vo(t) = V (46) 1 + K kOVO(0)t Taylor who essentially derived this equation reports on measurements by Simmons in which U/vO(t) was ascertained as a function of t = x/U (x equals distance of the measuring point from the grid). From (46) follows U U x U 68 x (47)  +  + 0.68r (47) vo(t) vo(O) 8 JL ()M LO If one puts vo = u' 5 (u' = 2v according to Taylor) and identifies LO with the mesh width, there follows from Taylor's measurements K = 0.85, from the corresponding measurements by Dryden a somewhat smaller value. However, because of the uncertainty regarding the value to be inserted for L(), this determination is probably still uncertain by about 50 per cent. 35. The Correlation Functions Taylor and von Karman (cited before) studied the correlations which exist between the velocities at two points at a given distance. The two 12Footnote at the time of proof correction: For this solution, ko = const. is presupposed which certainly is not the case for larger times. The problem of damping is investigated more closely in a paper of the author about to be published (Proc. Roy. Soc. A.). NACA TM 1431 correlation functions R1(x) and role, are defined as R1(x) = VL VL2 vx2 RL~xJ= *X(P)~ R2(x) which therein play the main R2(x) vy(Pi)vy(P2) R2(x) v 2 with the point P2 displaced with respect to the point Pi distance x in the xdirection. These functions are, according to Taylor, in with the spectra: a simple relationship S 00 kxFx(kx)cos kxx So J0 dkxFx(kx) (49) dkxFy(kx)cos kxx 0 O ,00 0 dkxFy(kx) With the aid of equations (34) and (55), (49) is transformed into Rl(x) = R2(x) = 3 00 dkF(k)(sin kx kx cos kx)k~5x3 foo 00 dkF(k) S 00 Co dkF(k)(k2x2sin k x + kx cos kx sin kx)k~3x3 S00 cO (50) dkF(k) From these expressions one recognizes immediately the correctness of von Karma'n's relationship R2 = R1+ + (51) (48) by the Rl(x) R2 (x) NACA TM 14351 The formulas (50) may be approximately evaluated in the two limiting cases x < 1/ks and 1/ks < x < 1/kO. When x < , it is advisable ks to expand the integrands with respect to powers of x. The first terms of the expansion then lead to the quantity j l dkF(k)k2 k2 0= (52) SdkF(k) 0 which may easily be calculated from (13), (44), and (29): T2=_ ,Roko2 (55) Thus one obtains: x22 R1(x) = 1 + . 10 For x < 1/ks (54) R2(x) = 1 x 2k2 + Taylor defined a length A by the equation 2 5 (55) j2 and designated it as a measure for the magnitude of the smallest turbu lence elements. According to (55), A becomes 1 A = 2.71 LO= 0 .4.4Ls(ROK) (56) It must be emphasized that A is not identical with the quantity L, (equation (51)) which we have denoted as "diameter of the smallest turbulence elements" and that A also depends on 1L and VO in a NACA TM 1431 manner different from that of Ls. A comparison of (51) and (56) shows that for sufficiently large Reynolds numbers Ls becomes A. In the opposite limiting case 1/k5s < x < l/kO one obtains from (50) Rl(x) = 1 0.645(kox)2/5 + (57) R2(x) = 1 0.858(kOx)2/5 + Here these first terms of the expansion do not depend on the special form of the spectrum in the proximity of kO; only for x ~ 1/kO does the form of the spectrum in the proximity of kO becomes important; there, however, the problem may no longer be treated with purely statistical methods. The formulas (54) and (57) thus give essentially a complete description of the correlations in so far as they may be regarded as a consequence of statistical isotropic turbulence. The formulas (54) and (57) also show clearly that the correlation function does not have the same form in all flows but that rather, in the case of variations of the parameters, the inner and outer parts of the function undergo different similitude transformations. This point has been stressed particularly by Taylorl5 in contrast to a different con jecture of von Karman (cited before). For comparison with experiment, the measurements of R1(x) and R2(x) made by Simmons have been plotted (circles and dots, respectively) in figure 5; furthermore, the theoretical curves calculated according to the exact formula (50) are shown. Here again LO has been identified with the mesh width of the grid and A has been calculated from the spectrum for U = 1060 cm/sec. The experimental values agree, in fact, with the theoretical ones very exactly at the smaller values of x, actually more exactly than could have been expected in view of the uncer tainty of IO. Beginning from xk ,~ 1 the deviation of the experimental points from the theoretical curves becomes noticeable which was to be expected from the derivation. The variation for larger xvalues depends on the behavior of the spectrum in the proximity of kO which cannot, in principle, be represented by our formulas. But even for larger xvalues the deviations from the theoretical curves remain small. 15G. I. Taylor, Jour. Aero. Sci. 4, No. 8, 311, 1957. NACA TM 1451 4. The Pressure Fluctuations While studying the diffusion in a turbulent airstream, Taylor (cited before) has derived a relationship between the correlation function for the diffusion and the root mean square value of the pressure gradient. We shall investigate therefore also the root mean square values of the pressure fluctuations from the viewpoint of the theory here described. With reference to equation (2) one may expand the pressure into a Fourier series p = pkeir,p = pk k (58) and the fundamental hydrodynamic equation i = (Vv) VP + [I Av p p  is transformed into k t ixt~v k =P k2vk k?  (59) Because of k'k = ky 0 it follows i [ k  i(yk'k) YI k k' (60) Pk = (Yk')(Ivkk:,1)  k  For the root mean square values of pressure and pressure gradient there results grad2p =2 k2j p2 k (61) We are interested, first, in this latter mean value:  I k2v p k p2 lk21 k NACA TM 1451 grad2 p = X k' 1 (Ykpk)(2kkk)( ktsk)(Skrt") (62) k k' k 1kk' ~ The superscript bar indicating approximation to the mean signifies here simply approximation to the mean in time. If one wants to calculate mean values of the type (62), one must take as a basis some kind of "assumption of disorder" regarding the turbulent motion. One may start from the fact that the amplitudes yk in the course of time fluctuate by a value given by (10) or (28), respectively, so that the time average v v is given simply by the spectrum (28). The phases of the vk, however, will in the course of time pass through all possible values; all values of the phase will occur, on the average, with the same frequency. If one could regard the phases pertaining to different wave numbers as completely independent statistically, there would, in taking the mean of such products of four factors v 1 v2 v, be left only the terms in which every two wave numbers are equal and opposite; thus, terms of the type v k k k and these mean values could be replaced by the pro ducts of the mean values of the squares of the amplitudes: vvkl k vk = k ik v2 v2 (65) vk'tlr :17 1171 z!!2742 Actually, however, statistical correlations will exist between the phases pertaining to different wave numbers since the waves mutually influence one another. In section 5 we shall attempt to estimate such correlations in a simple case. In spite of the presence of the correlations, however, we are inclined to assume that in a sum of the kind (62) the terms of the type (63) make the largest contribution since their mean value is differ ent from zero even in the first approximation, without any assumption regarding the behavior of the waves with respect to time whereas the other mean values attain a value different from zero only because of the finer fluctuations of various waves. We believe therefore that one obtains a serviceable approximation if one takes only the terms of the type (65) in (62) into consideration. Then there results grad2p = 2p2 v )(" )  (64) k'k" (k)' k")2 NACA TM 1451 In taking the mean with respect to the directions of the vk one assumes again all directions perpendicular to k for vk to be equally probable  it is expedient to use the relation (a2)(2kb) 1 (65) Thus one obtains g 2T p2 [k2k2 (k 2]' 2 (6) graAdp jv 2 vk ,2 (66) 2 k ,k'1 I (kv _)2 If one sets (k'k?") = k'k"'t, one can perform the integration with respect to t and finds gradp >1,2 Y\ k,,2 I k'k" k ) (67) where (s) + 1 i[.s2 + . s4 s6 + (g2)lg s 16s3 3 3 2s 1 __sj (68) For 0 s < 1 there applies approximately Vy(s) P:s s2 + (69) S15( 7 21 If one transforms the sums into integrals and substitutes the spec trum (28) into (67), there follows finally grad2p f2= 21 21 81 F k 2. k r k ', k F r r,, (70) NACA TM 1451 One recognizes from (70) that the integrals converge at small kvalues, and that one may therefore perform the integration from k = 0 without a considerable error. This shows that grad2p is actually determined by the behavior of the spectrum at large k, that is, by the smallest turbulence elements. Had we calculated p2, we would have found, on the contrary, that the integral diverges at small values of k, thus that its value is deter mined entirely by the largest turbulence elements. Therefore, the value of p2 cannot at all be calculated according to the method used here; for, first, in the case of small k, the spectrum has a form dependent on the geometry; and second, it would surely by quite unjustified to consider, for the largest turbulence elements, only the mean values of the type (65) since the geometry certainly impresses definite phase relations upon the system of the largest eddies. Equation (70) now becomes grad2 = 2 2 4, 4/5,3 2/3 f 0 d / dq T (71 grap Ok k 0 OT3 0 12T3 + 8/12l + 8/2 (71) The double integral at the right was estimated, according to a gra phical method, to be 0.765; thus there follows finally (cf. (50)) grad2p = 017p2v4kO S4ks2/5 (72) 0= 05.Op2V k02 (73) Taylor (cited before) had expressed the conjecture that grad2p should have the same order of magnitude as p2 v0 2(v/ox)2 thus the order of magnitude p O 4kO2/5ks/5. One recognizes now from (72) that grad2p must be considerably smaller, the more so, the larger the ratio ks/kO. The length AT defined by Taylor: ap 2 ( )2 )= 2p2 (74) NACA TM 1431 that is, 7 4 grad p = 2 p2 (75) must, therefore, for large Reynolds numbers, become considerably larger than the length ?h of equation (55). From (75), (75), and (56) follows AT = X0.42ROS (76) It is true that this result does not agree with the experimental findings. Taylor indicates, for a test which Simmons had performed adjoining similar experiments by Schubauerl4, /I\ = 0.5 where one must assume approximately 5ROK 5.9 according to the test conditions. Thus one must raise the question whether the result (72) has perhaps been falsified by the fact that only the terms of the type (65) were taken into consideration in taking the mean. However, one can easily see that this may perhaps affect the numerical factor in (72) but that the dependence of kO and ks, that is, the dependence of A\ on (ROK) is in no way con nected with this approximation. For already equation (62) shows that on the right side the normalization factor v0 4k04' must appear, because of equation (10). After the mean has been taken, this factor is supple mented by a factor of the dimension k2/5 which obviously can be at most of the order ks2/5 ; it must be, because the pertaining integral with respect to k would diverge like k2/3 if the decay of the spectrum would not set in for k ~ kg with k7. There would remain the possi bility that only the numerical factor in (72) has been estimated as too low due to the consideration solely of the terms of the type (65). But it is hard to imagine that the correct expression would increase by more than a tenfold which would be necessary for interpretation of the experiments. Perhaps the contradiction may be cleared up in the following manner: The main contribution to grad2p stems from wave numbers of the order ks, thus, from turbulence elements whose diameter measures a few millimeters. In Simmons' test, the airstream is heated by a hot wire of 20 cm length stretched across the wind tunnel, and then the distribution of the heated air is measured at a certain distance behind the hot wire. Precisely the smaller distances (5 to 15 cm) are decisive for the determination of ? .* 14Schubauer: Rep. Nat. Adv. Comm. Aero. Nachr. Nr. 524, 1955. 22 NACA TM 1431 It suggests itself to assume that the hot wire itself produces in the airstream a small vortex street and additional turbulence, with the turbulence elements probably having a length of a few millimeters that is, the wire increases precisely the intensity of the turbulence in the spectral region which exerts the strongest influence on grad2p. The additional turbulence in the immediate proximity of the wire is probably much more intense than the original turbulence of the same wave range. However, this additional turbulence is rapidly damped, of course, and it is surely difficult to estimate whether this turbulence alone can explain the discrepancy between (76) and the empirical ,.value. 5. Energy Dissipation for Normal Isotropic Turbulence The investigations of the preceding section are already closely con nected with a basic problem of the statistical turbulence theory: namely, with the determination of the energy dissipation in the case of normal energy distribution, that is, the determination of the constant K in equation (12). In this problem the molecular friction may be neglected entirely. The fundamental hydrodynamical equations could therefore be presupposed in the form Y = (x)Z 1vp v = 0 Furthermore, a suitably singledout partial volume of the fluid is to be selected as normalization volume which under certain circumstances is moved simultaneously with the fluid, corresponding, for instance, to the mean value of the velocity with respect to the volume. We assume therefore that the volume moves with the velocity u. Then (60) is transformed into = iZ (') kk is(ki ) + i (1) (77) For the calculation of the energy dissipation, one has to ascertain how the intensity x21 of a certain natural vibration (or perhaps better: the sum of such squares of amplitudes with respect to a small spectral region Ak:/ .2 varies in the course of time. One NACA TM 1451 recognizes from (77) that one needs for this purpose time averages with respect to products of the type Yk v1, v5 (78) wherein k = k. Because of the statistically uniform distribution of the phases, these mean values would disappear if there would not exist statistical correlations between the phases pertaining to different k which stem from the mutual influence of the various waves, as has been explained already in section 4. In order to ascertain these correlations, one must somehow express in the equations the influence exerted upon a wave with given k by waves with a different k; one may do this for instance by representing one of the three amplitudes in (78) as a time integral over vk and expressing vk in turn by a sum over two other vk according to (77). Then one obtains products of four amplitudes yk each of which, however, must partly be taken at different times. For such pro ducts the considerations of section 4 are valid according to which one obtains a first approximation by taking only products of the type (65) into consideration. Of course, one could continue the procedure in prin ciple and attempt to calculate the other mean values of quadruple products by tracing them back to sixfold ones, etc. However, such calculations would probably become much too complicated; the higher terms probably also would make a lesser contribution, and we shall thus be content with the first step. For these calculations, one will obviously need mean values of the type Yk(t)vk(t + T) and we define therefore k+Ak v t + T.)vk(t  Rk(t,T = k kA(79) k  NACA TM 1451 The summation over a small spectral region Ak has been included into the definition of Rk(t,T) so that the magnitude of the normaliza tion volume does not directly enter into Rk(t,T) and that the mean is taken equally over all directions of k. Evidently the spectral region Ak must be selected wide enough that many natural vibrations of the normaliza tion volume still may be accommodated in it (that is, k2AkV > 1), yet very small comparedi to k itself. These requirements are for the turbu lence elements of the order of magnitude V themselves no longer compatible but for those, the statistical methods cannot be applied anyway. The whole procedure thus can be carried out only if it is found that the large turbu lence elements practically do not any more contribute to the mean values to be investigated. In order to obtain from the hydrodynamical equations information about the quantities Rk(t,T), it suggests itself to examine the following expression: t+ k 2 1 i v2 vk(k ) + i(uk) k 7 (80) 22) k' t+T In this expression one can replace v, by a time integral with respect to k' r)+ T T t4 j t +2 t+T' k' = d' + k(T) = d'k'2 + kr(T) (81) T 0 If T is selected sufficiently large, the correlation between vkt(t) and vk'(T) will disappear; it is therefore expedient to per form, after substitution into (80), the limiting process T)m. If, in addition, one takes the mean with respect to the directions thus elimi nating the term with u one obtains from (80) and (81): NACA TM 1451 k+Ak k+Ak t+Tr k k k' YO k"  I g t+ k(t ) Z 4 dT'(kt k) L1" k  / t+1T ' (k k') v2k k' t+ I ti\ 2 1S ) k k (82) k'2 K V k If one furthermore, as in section 4, takes into consideration the terms of the type (65) only, also replaces k k' by k', and integrates with respect to the cosine of the angle between k and this vector, there follows: k+6k k+Ak V Z I Z (t + 1) 1 = d'2 ( dT 11 Rk1 t ,T x Rk,(t + ,T)k'k5(k2 k'2)2kk(kI + k k2k'2  (k2 + k'2)(k2 k,2)1g k + k' (85) This equation presents the possibility of expressing the differential dRk(t,T) dRTk(t,) quotients with respect to time k(tT) and k(,) by the Rk them dt dT selves; when the Rk are known, one can, moreover, calculate the energy dissipation from (853), putting T = 0. For this purpose we shall assume that the entire turbulence phenomenon is either stationary or is damped very slowly so that the times during which the intensity vk2 noticeably varies are very long compared with the fluctuation period of vk. The notation vk22 represents, therefore, the mean value over a time which is certainly much longer than the fluctuation period but is very much shorter than the damping time. d~k ((t, T) The equation for gives a measure for the fluctuations of dt the quantity Rk(t,T) as a function of t about its time mean: Rk(T) = Rk(t,) (84) NACA TM 1431 One may surmise that these fluctuations are small in the region of small T which is determined by the small turbulence elements, and that they increase with growing T; this question will be further investigated later on. Before carrying out the further calculations one has to determine how the partial volume V and its velocity u are to be chosen. One could first try to put u = 0 and to identify V with the total volume. How ever, one would obtain an erroneous picture of the actual conditions: The decrease with time of the correlation function Rk(t,T) = Rk(T) as a function of T is determined in this coordinate system by the largest turbulence elements, and is therefore very rapid. One can show that the correlation function in this coordinate system is given with sufficient approximation by k2v2T2 kV kVOT i0 kvT 1 2 Rk(T) = 1 e dxex The calculations which have led to this expression need not be discussed in more detail since the expression is not further used later. The physical interpretation of the expression is given by the following con sideration: The function Rk(T) in it decreases after a time of the order 21(; that is the time during which, for instance, precisely an kvo eddy of the wave length 23, due to the high velocity in the largest tur k' bulence elements, passes by the point of observation. The fact that the correlation function decreases after that time signifies therefore simply that the velocity in the largest turbulence elements is of the order of magnitude vO, but, statistically, fluctuates about values of this order. This phenomenon is not connected with the disintegration of the eddy of the wave length 2. If the F(k) ~ k5/3law is valid, it is rather to be expected, according to the similitude considerations of v. Weizsaecker, that the disintegration of the eddy takes place only after time intervals of the order 21cvOk2/3kO1/. On the other hand, however, the energy dissipation is connected with the disintegration of the eddies, not with the motion on a large scale. If one wants to describe the disintegration of the eddies in equations, one must move the coordinate system at the same time. One then must make the linear dimensions of the partial vol ume V somewhat, but not very much larger than 2J and move it simul k taneously in proportion to the mean velocity within it. We shall assume NACA TM 1451 experimentally that one can give for every k a volume V corresponding to it in such a manner that Vk5 becomes independent of k and that for the thus selected, simultaneously moved volume elements, the correlation function R(t,T) is a universal function of the variables vOk2/ k1/3t and vOk2/5k1/5T as is to be expected according to v. Weizsaecker's similitude consideration. We shall show that the relations for Rk(t,T) arising from (85) actually can be satisfied by this assumption if the Fk are distributed according to the k~5/3law. If one puts Rk(t,T) = g(,Tj) wherein v= 2/ 1/3t v= 2/5k 1/ ekk0T k ' ii = y k (85) there follows from (85) dg( ,n) = 3 yf(y) d P g( Lrr ^k+ n f )y2/3,  S 2, + S ( ) ,y2/3 5, dyf(y) df,' [ T' i)g + (86) g L + T + ')2/3I ,2/3 where f(y) = (l y2 + y) ( y2)(l y4)lg + Y y 14/3 \ (l 3 y ] (87) dg(,TI) d 8 Jo I Iy2/5,,,y2/3) NACA TM 1451 These equations actually do no longer contain the constants ko, vO. The reason is that the integral with respect to y converges for small as well as for large y; f(y) disappears sufficiently for y = 0 as well as for yy. Therefore one may take the integral over k', instead of from kO, simply from 0, without considerable error; moreover, the convergence of the integral for large values of k shows that the molecular friction actually is of no importance in this problem; the behavior of the spectrum in the region of the smallest turbulence elements is unimportant for the correlation functions R(t,T) and the energy dissipation for medium kvalues. Before attempting the numerical solution of (86), we shall use the equation (85) for calculating the energy dissipation in the approximation here aspired to. For this purpose we put T = 0 and integrate the equa tion (85) with respect to k between two arbitrary limits K1 and K2: d f2 4,k2dk = 2 j I J dT'R1(t ,T)Rk,(t ,T) Tt K 2 'k2n)5jK: k k' JQ L( 2 2 (k2 k'2) kk( 4 + 2k2 (k2 + k2 )(k2 k Ik k1] (88) The integrand on the right side is an antisymmetrical function in k and k'. If one calculates the variation with time of the total energy, that is, if one puts K1 = 0, K2 = m, there results therefore zero, as far as the integral on the right converges at all. That is, the total energy is constant in time; this is a necessary requirement since the molecular friction has not been taken into consideration. However, if one considers the variation with time of the energy which is contained in the part of the spectrum lying between K1 and K2, the integral on the right side may be transformed in the following manner (we shall call the antisymmetrical integrand simply J): fK2 2 f K2 rK2 k K1 K2 ( k2d k = dk dk'J = dk dk'J dk dk'(J) (89) Kt K 2 0 j o j J 1 K 1 1 S2 NACA TM 1451 In the first of the two integrals on the right J is always posi tive; in the second (J) is always positive. From this notation there follows that the first integral may be interpreted as the energy which, per unit time, flows from smaller wave numbers (k' < Kl) into the region between K1 and K2; the second integral as the energy which flows toward larger wave numbers (k' > K2). If one puts, in particular, K1 = 0 and K2 > kO, the second integral represents the entire energy dissipation; for the normal spectrum [F(k) ~ kb'5 it must prove to be independent of K2. Thus one obtains from (85), (86), and (89) for the energy dissipa tion the expression K2 SLv3k oJ' ikf dyf(y) fr ( d ) g I2/5,2/3) = vo b dy(lgy)f(y) J ad s( g y2/5,IY2/3) (90) T4 o o 2 2) This expression is actually independent of K2 as it must be. Since the entire energy dissipation according to equation (44) is also given by pK vgkOx, there follows K = 1J dy(lgy)f(y) J drg( g y2/5y2/5) (91) From this equation K can be calculated numerically if the function g(,f) is known. We now turn to the treatment of the equation system (86). This sys tem represents a considerable simplification compared to the initial equation (77) in so far as it does not any more contain any dimensional quantities, and is already derived from the equilibrium spectrum k 5/. On the other hand, (86) also still contains statements regarding the fluctuations of the g(t,r) as a function of the and is, for that reason, doubtlessly too complicated to permit rigorous solutions. One could attempt to completely neglect the fluctuations in a first approxi mation, and to calculate with the mean values. Unfortunately, however, NACA TM 1431 it turns out that the contribution of the fluctuations in certain regions is large. One recognizes this from the second equation (86). For, if one puts g(,r) = g() + ng(Q,T) (92) where g(r) signifies the mean value over of g(r) = g(,T]) (95) there follows from the time mean of the second equation (86): Sdyf(y) d.T1j[g(r' rTO + gk' + Tr)gri y2/ )1 dyf (y) fdr f rAg(^ g(S n' +  2  )A ( y2I3, TI/5 I y2/5)] (94) This relationship shows that the always be small. It is true that the  = 0; this follows from the relation fluctuations Ag((,Tj) cannot left side of (94) vanishes for f(l) = y 4/f(y) \" (95) 3 f T8 J + 1 TV.y21,,y13 NACA TM 1431 which will have to be discussed later, and signifies that the spectrum k is actually in equilibrium; however, for larger qld the left side assumes appreciable values. Therefore, it is doubtful whether one will obtain a sufficient approximation if, in the transition from (82) to (83), one takes only the mean values of the type (65) into consideration. How ever, I did not succeed in improving here the approximation or in obtaining more'than a very crude estimation of (86). One may perhaps assume for such an estimation that, for large values of 7T, the first term of the summation on the right side of (94) is much larger than the second. For in the first one, the integral is taken over the region 7T' T] which probably contributes a great deal whereas in the second, for large T1, the factors Ag have already strongly decayed in the entire integration range. One may therefore attempt the assumption, at least for large q, that the second term of the summation on the right side of (94) may be neglected. In this approximation the time average of the first equation (86) then becomes, with use of (94): dg I)= _ (fy dy) dTI'g( + T1)g iy2/5) (for > > 0) (96) One may utilize this equation, for instance, in such a manner that one assumes a plausible form for g(Q), leaving the scale in the rdirection undetermined at first, and determining it subsequently so that the equation (96) is valid as exactly as possible for large T1. In this way one will describe the steepness of the decrease of g(Tr) for large T with some correctness, and precisely this steepness is decisive for the value of K. In the practical execution of the calculation it is expedient to introduce, in the place of y and f(y), new variables s = y2/; cp(s)ds = f(y)dy (97) Then there applies as one recognizes from (87) (compare also (95)): cp) = sp(s) (98) NACA TM 1431 This equation is based on the fact that the energy dissipation from the wave number lk to the wave number k coincides with the one from k a to ak, except for a factor qualified by the similitude transformation. Furthermore, there then applies in good approximation (s) = (1 (for 0 s l) (99) and for larger s one can reduce cp(s) by means of (98) to the range o < s < 1. Figure 4 gives a plausible curve for g(Tr), and in addition the right side of (96) as J d and g'(rj). The scale is selected in such a manner that the two last curves coincide for large il. Considerable differences then exist for small r1 but there the equation (96) also can no longer be correct. If one substitutes the function g(r) thus obtained into (91) and neglects the fluctuations, there results for K: K J ds(lg s)cp(s) 1drg(T)g(Ts) = 0.98 (100) This crude estimation therefore gives the correct order of magnitude for K, but the exact value may well differ from 0.98 by as much as a fac tor 2. The calculations of this section thus have not led to an exact calculation of the constant K but they did provide a qualitative mathe matical representation of the processes on which the energy dissipation is based. Perhaps it will be possible to arrive at a rather exact experi mental determination of K by means of a comprehensive discussion of the various experiments of Simmons, Dryden (cited before), Prandtll5, and others regarding the spectrum and the damping of turbulence. Translated by Mary L. Mahler National Advisory Committee for Aeronautics 15Prandtl, L.: Proc. VI. Intern. Congr. f. Appl. Mech. Cambridge, Mass., 1938, p. 340. NACA TM 1451 55 X 0 2 4 6 5 1 Figure 1. Representation of the function w(x). NACA TM 1451 S" 0 1 0 0 x 0 0 1 2 3 4 5 10 20 kx cm1 Figure 2. The turbulent energy distribution as a function of the wave number. NACA TM 1431 35 1.0 0.8 06 ^ 0.4 S\R( x) 0.2 R2(x) 0. 0 2 3 4 5 6 x  Figure 3. The correlation functions. NACA TM 1431 0.8 0.4 0. I 0.2 0.3 0 5 10 15 20 Figure 4. The correlation function g(r). NACA Langley Field, Va. ""N h S . & R* C44 bD L; cd a* s <;a a Su B Cc a C~M' ^a N 02) .cw o4 C4 n n*.ra4 .0 C: I. ho m W :3I s I1 d l U N E. 3 2 E  Now a ZL P C4 P 1 a Li I0 z .M 0 ra 03 Ow'O' uuif EdH M m. h6 (D Lfl cc~c a a 0=g0 v3 5r E uS bS m Er r S<"a a 433~ L r, c B r Sz 4 2202.5. P m d02 a 0 "s ( cd '0s C y '4o~u o  5,.a S S o w, c Z c 1E "6 .10 C) o i M C. M C md 0 CU a0 ..a' B03 l.ri d A U g 0.. cQ:n Cu !'D r .0 to ; = e E w w 03 cd a m L. oi 03 :3 C:5 c ci W r, w cd ct " anj iu Th g^ 8 i ^a C L. 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S ig bO1 II5 ! cd C40 a n o> =r L. 4mZ SS g z" Id Co 2 pB 0 '= * >>c z o4 z z oJ " a .C t . to B _ o Ci 0 w w I c da  ra a.co o I C O S V;, Ca S a d a aM c Ci CJ cc W !! 0 rz a 0 E e" CL 0 g0 rt S; a U. ..! Su p~~a S a a i^o SiU.SCiZ^o 25 B a d Q S a.! ^"is~r5 3 tiiE 2 .5 a. i3 Z sgas T*.Sg~ 04 0 N *s ~ U *. s s0?  a K^ . s " 3 ; i n  0 ^~ '.4 C, >ZNC co N i i 0. U0 :3 .i 10H .1rP DO Cd m c Ci0 WL4'' 0 N'a 504 a50 g4 a b ULJ 5" gW . "3 4 ' A4 C O ^ar> u u .E  P.MW a0 L r U. 3 c .!I E, >l^S agS S a &. = u 0 IO .2 a  0 C W i Ci u. m 0 s ai ".EQ i". 0 0. r Sai. i5 Ci.5 O as s" "1 ^ " do 0i a B :5'Sa too C: 0^" rt>E t., o c 0 a ?. g ar 2 Q A, Cd M iZ >1 M n UNIVERSITY OF FLORIDA 3 126208106652 3 
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