\J/kVh7M 3 5# 
 
 4.1EAD0M 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1350 
 
 THE MICROSTRUCTURE OF TURBULENT FLOW 
 
 By A. M. Obukhoff and A. M. Yaglom 
 
 Translation 
 
 "Mikrostructura turbulentnogo potoka," Prikladnaya Matematika i 
 
 Mekhanika, Vol. XV, 1951. 
 
 Washington 
 
 June 1953 
 
 
 
°b3\}V 2^ 
 
 NACA TM 1350 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 
 
 THE MICROSTRUCTURE OF TURBULENT FLOW* 
 
 By A. M. Obukhoff and A. M. Yaglom 
 
 In 1941 a general theory of locally isotropic turbulence was pro- 
 posed by Kolmogoroff which permitted the prediction of a number of laws 
 of turbulent flow for large Reynolds numbers . The most .important of 
 these laws, the dependence of the mean square of the difference in vel- 
 ocities at two points on their distance and the dependence of the coef- 
 ficient of turbulence diffusion on the scale of the phenomenon, were 
 obtained by both Kolmogoroff (references 1 and 2) and Obukhoff (ref- 
 erence 3) in the same year. At the present time these laws have been 
 experimentally confirmed by direct measurements carried out in aero- 
 dynamic wind tunnels in the laboratory (references 4 and 5) , in the 
 atmosphere (references 6 and 7), and also on the ocean (reference 8). 
 In recent years in the Laboratory of Atmospheric Turbulence of the 
 Geophysics Institute of the Soviet Academy of Sciences, a number of 
 investigations have been conducted in which this theory was further 
 developed. The results of several of these investigations are pre- 
 sented in this paper . 
 
 The fundamental physical concepts which are the basis of 
 Kolmogoroff ' s theory may briefly by summarized as follows. A turbulent 
 flow at large Reynolds numbers is considered to be the result of the 
 imposing of disturbances (vortices or eddies) of all possible scales of 
 
 "Mikrostructura turbulentnogo potoka," Prikladnaya Matematika i 
 Mekhanika, Vol. XV, 1951, pp. 3-26. 
 
 The applications of these laws to certain problems of the physics 
 of the atmosphere may be found in references 9 and 10. 
 
 2 
 In addition to the results contained in the present article, ref- 
 erence may also be made to the theoretical investigation of the struc- 
 ture of the temperature field (or of the concentrations of any neutral 
 additive) in the turbulent flow, presented in references 11 and 12. 
 The applications of the latter results may be found in references 13 
 and 14. 
 
 For a more detailed presentation see reference 15. 
 
NACA TM 1350 
 
 magnitude. Only the very largest of these vortices arise directly from 
 the instability of the mean flow. The scale L of these large vortices 
 is comparable with the distance over which the velocity of the mean flow 
 changes (for example, in a turbulent boundary layer, with the distance 
 from the wall). 
 
 The motion of the largest vortices is unstable and gives rise to 
 smaller vortices of the second order; vortices of the second order give 
 rise to still smaller vortices of the third order, and so forth, down 
 to the smallest vortices which are stable (i.e. the characterizing 
 Reynolds number is less than the critical value) . Since for all vor- 
 tices, except the smallest ones, the characteristic Reynolds number is 
 large, the viscosity has no appreciable effect on their motion. The 
 motion of all vortices that are not too small is therefore not associ- 
 ated with any marked dissipation of energy; the vortices of the n^h 
 order use practically all the energy which is received from the vor- 
 tices of the (n - l)"th order to form the vortices of the (n + l)th order. 
 However, the motion of the smallest of the existing vortices is 
 "laminar" and depends essentially on the molecular viscosity. In these 
 very small vortices the entire energy that is transferred along the 
 vortex cascade goes over into heat energy. 
 
 The motion of all the vortices, except for the very largest, may 
 be assumed homogeneous and isotropic . Any directional effect of the 
 mean flow ceases to be appreciable for vortices of a relatively low 
 order. It is also of importance that this motion may be assumed quasi- 
 stationary, that is, a change in the statistical characteristics of the 
 motion of the vortices under consideration proceeds very slowly in com- 
 parison with the periods characteristic of these vortices. It follows 
 that the motion of all vortices whose scales are considerably less than 
 L (the microstructure or local structure of the flow) must be subject 
 to certain general statistical laws which do not depend on the geometry 
 of the flow and on the properties of the mean flow. The establishment 
 of these general laws, which have a wide range of applicability, con- 
 stitutes the theory of local isotropic turbulence. 
 
 In the investigation of the laws of the local structure, consider- 
 ations from the theories of similitude and dimensions are of great value. 
 It is only these considerations which permit obtaining a number of essen- 
 tial results. To apply these ideas it is necessary, first of all, to 
 separate out those fundamental magnitudes on which the local structure 
 of the flow may depend. On account of the homogeneous and isotropic 
 character of the motion of the vortex system under consideration, the 
 
 ^The length L coincides with the length of the mixing path intro- 
 duced in the semiempirical theory of turbulence. 
 
NACA TM 1350 
 
 characteristics of the mean motion (of the type of length characteris- 
 tics, velocity characteristics, etc.) do not enter among these funda- 
 mental magnitudes. Therefore, only two magnitudes remain, the mean 
 dissipation of energy per unit time per unit mass of the fluid e , 
 which determines the intensity of the energy flow transferred along a 
 cascade of vortices of different scales, and the kinematic viscosity v, 
 which plays an essential role in the process of dissipation. 5 These 
 two magnitudes thus play a fundamental part in the theory that is pre- 
 sented herein. 
 
 The dimensions of e and v are: 
 
 [e] = L 2 T -3 
 
 [v]= lV 1 
 
 From these two magnitudes, it is evidently possible to form a single 
 combination in the dimension of length 
 
 3 l/4 
 
 * - (f ) 
 
 The length t) determines an internal scale characteristic of the local 
 structure. By use of the previously described physical picture of tur- 
 bulent motion, it is possible to identify t] with the scale of the 
 smallest vortices in which a dissipation of energy occurs (since this 
 picture does not contain any other characteristic length) . The scale r 
 was first introduced in the work of Kolmogorof f (reference l) ; it is 
 termed the internal (or local) scale of turbulence (in contrast to the 
 external scale L) . 
 
 In the further analysis of the microstructure, two limiting cases 
 may be considered separately to advantage: the case of scales much 
 larger than t) and that of scales much smaller than r\. First, the 
 system of vortices with dimensions much smaller than L but much greater 
 than the scale t\ of the smallest vortices is considered. The motion 
 of these vortices, as has already been pointed out, should not depend 
 
 The fluid is assumed everywhere to be incompressible and to have 
 a constant density p. The magnitude p is not included herein among 
 the fundamental magnitudes because in the main part of the paper (sec- 
 tions 1 and 3), the purely kinematic characteristics of the flow, which 
 of course cannot depend on the density, will be considered. When, how- 
 ever, the structure of the pressure field (section 2) is investigated, 
 it is necessary to add p to e and v. Information on the fundamen- 
 tal magnitudes on which the local structure of the temperature field 
 may depend is found in references 11 and 12. 
 
NACA TM 1350 
 
 on the viscosity v, a circumstance which immediately facilitates the 
 obtaining of concrete results by computation of the dimensions. In the 
 second extreme case, for scales of motion much less than r\, the motion 
 may be assumed laminar. However, in the intermediate range of scales 
 of the order of t\, the theory of dimensions gives, as a rule, less 
 concrete results. Thus, for example, it follows from this theory that 
 any nondimensional function of the distance determined by the local 
 structure should be a universal function of t/t\. The form of this 
 function for values of the argument of the order of unity remains how- 
 ever undetermined . 
 
 In the present paper an attempt is made to describe quantitatively 
 the structure of the fundamental hydrodynamic fields (pressure, velocity, 
 and acceleration^) for all distances less than L (i.e., for the entire 
 range for which the theory of Kolmogoroff applies). For this purpose 
 some additional hypotheses are introduced which have a certain experi- 
 mental basis. The asymptotic formulas for r>>T) and for r<<-q 
 obtained are in agreement with known earlier results where all the 
 undetermined numerical coefficients that figure in these results are 
 expressed in terms of a single constant S (asymmetry or skewness 
 factor) , the value of which has been experimentally determined by 
 Townsend (reference 4) . The nondimensional magnitude S (as well as 
 the magnitudes e and v) enters only in the expression for the char- 
 acteristic scales so that with an accuracy up to the choice of units 
 the measurements of the structure of all the fields considered under 
 the assumed hypotheses are described by universal functions not depend- 
 ing on any experimental data (see figs. 1 to 3; the meaning of these 
 functions will be explained in a later discussion) . 
 
 The investigation of the structure of the velocity field (section l) 
 is the work of A. M. Obukhoff; the investigation of the pressure field 
 (section 2) was started by Obukhoff (reference 16) and continued by 
 A. M. Yaglom; the investigation of the acceleration field (section 3) 
 was carried out by Yaglom. Several results of the present work were 
 first published in the form of separate short communications (refer- 
 ences 7, 16, and 17). 
 
 1. Computation of structural functions of velocity field . In order 
 to be able to make use of the concepts of locally isotropic turbulence in 
 investigating the velocity field of a turbulent flow, it is first neces- 
 sary to separate out those characteristics of the field which depend 
 only on the local structure. The true velocity v will essentially be 
 determined by the mean flow. In the theory of turbulence the usual 
 decomposition of the true velocity v into the mean velocity v and 
 the fluctuating velocity v' = v - v gives a component v 1 not depend- 
 ing on this mean flow; but the theory does not solve the problem pro- 
 posed since the value of v 1 will be determined mainly by the very 
 
 6 The acceleration of the flow is considered herein to be the total 
 acceleration dv/dt of the fluid particles moving in space. 
 
NACA TM 1350 
 
 large vortices, the scale of which is comparable with L. However, as 
 was first noted by A. N. Kolmogoroff (reference l) , the above mentioned 
 required that a separation of the characteristics be effected by con- 
 sidering the difference of the velocities at two sufficiently near 
 points (i.e., the relative motion of two neighboring elements of the 
 fluid). It is clear that this difference will not be affected by the 
 large vortices which transport the pair of points under consideration 
 as a whole. Hence, in the theory of local isotropic turbulence, the 
 following functions are taken as the fundamental quantitative charac- 
 teristics of the structure of the velocity field: 
 
 D i(j (M,M') = [v-fo') - v^Mj^VjCM') - Vj (MJ] (i,j" = 1,2,3) (l.l) 
 
 where vj_(m) is the i component of the velocity vector v(M) at the 
 point M, and the bar above a symbol denotes the average value. The 
 function Dj*(M,M') is termed the structural function of the velocity 
 field. According to the preceding discussion, for a distance r 
 between the points M and M' much less than L, this function depends 
 only on the local structure of the flow. On account of the homogeneity 
 and isotropy of the motion of the vortices with scales much less than 
 L, the function D^fM^'), for r<<L, is an invariant tensor function 
 of the vector MM' and may therefore be represented in the form 
 
 D i(j (M,M') = ACr)^ + B(r)6 iJ (1.2) 
 
 where £]_> £g> an( ^ ?3 are ^ ne components of the vector MM' (so that 
 
 2 2 2 
 £-, + 4 2 + ? 3 = r) and 8. . = 1 for i = j and 8. . = for 
 
 i 4 j- 
 
 When first v, = v. = v where v is the projection of the 
 1 j n n r o 
 
 velocity vector on a certain direction perpendicular to the vector 
 MM' and then Vj_ = Vj = v^ where Vt is the projection of v on the 
 direction of the vector MM' are set into this formula, it is readily 
 shown that equation (l.2) may be represented in the form 
 
 D^MO . %(r) ' Dnn(r) <*€« + D m (r)6 i1 (1.3) 
 
 r 
 
 where the functions D,,(r) and D nn (r) (the longitudinal and trans- 
 verse structural functions) have the simple physical meaning: 
 
NACA TM 1350 
 
 d 7?^^ = r v i( M ') - v i( M )i 2 
 
 11 L J (1.4) 
 
 D nn( r ) = [ v n( M ') " v n M] 
 
 The determination of these functions, V^ir) and D nn( r )> will he 
 the main object of this section. 
 
 In the theory of local isotropic turbulence it is possible to con- 
 sider the functions D^^r) and D (r) as independent of the time. 
 As a matter of fact, a quasi-stationary statistical regime in a region 
 of sufficiently small turbulence scale is assumed. From the consider- 
 ations of the theory of similarity, it follows that in the range of 
 applicability of the theory of locally isotropic turbulence (i.e., for 
 r<<L), the functions D^fc) and D nn (r) are representable in the 
 form 
 
 D n (r) = 4^1 d„ff 
 
 "' (1.5) 
 
 D nn (r) = V^i d nn ^ 
 
 . 3 -I.1/4 , . 
 
 where t] = (v e ) is the internal scale of turbulence and d,,(x) 
 
 and d nn (x) are universal functions. Formulas (l.5) may also be 
 represented in the form 
 
 In the theory of isotropic turbulence, the correlation functions 
 (longitudinal and transverse) are usually employed. 
 
 B 2Z (r) = VM)v*(M') 
 
 B nn( r ) = v n ( M ) v n( M ') 
 
 The structural functions in the isotropic case are connected with the 
 correlation functions by the following relations: 
 
 D„(r) = 2(B(0) - B n (r)) 
 
 E nn (r) = 2(B(0) - B nn (r)) 
 where B(o) = B n (0) = B nn (0). 
 
NACA TM 1350 
 
 D 7^>= u l 2 P2zg-) 
 
 yiJ (1.6) 
 
 where 
 
 (1.7) 
 
 The numerical factors k-, and k ? can "be chosen by inspection and 
 will always be assumed to be of the order of unity, and ^^^(x) and 
 (3 (x) are new universal functions the graphs of which are obtained from 
 the graphs of the functions d,,(x) and ^ (x) *>y a simple change of 
 scales along the x and y axes . 
 
 Since for r>>T) the functions T)-,-,(r) and D (r), on account 
 of the stated physical considerations, should not depend on the viscos- 
 ity v, the asymptotic equations should hold 
 
 d (x) ~ x 2 / 3 
 
 i I 
 
 d nn (x) - x 2 / 3 
 
 for x>>l (1-8) 
 
 The same equations also hold, of course, in relation to the 
 functions p^^ ( x ) and Pnn( x )- Whence it follows that for r>>Ti 
 
 n / % „ 2/3 2/3 
 D,,(r) « Ce ' r ' 
 11 (1.9) 
 
 „ , , „, 2/3 2/3 
 D nn (r) - C'e ' r ' 
 
NACA TM 1350 
 
 (the so-called 2/3 law). In the other extreme case, for r<<T}, the 
 difference of the velocities v(M') - v(M) will be of the first order 
 of smallness with respect to r (for such distances the velocity at a 
 point of the flow is continuous and is a differentiable function of 
 the coordinates), so that in this case 
 
 D(r) - Ar 2 
 " (1.10) 
 
 D m (r) - AT 2 
 
 The more complete theory based on the equations of hydrodynamics 
 is now discussed. First use of the equation of continuity 
 
 3 A 
 
 Zov. 
 ^ = (1.11) 
 
 i=l **! 
 
 shows with little difficulty that 
 
 aD,,(r) 
 
 C nn(r)= E n^ + \ ~^~ (l ' 12) 
 
 dr 
 and that 
 
 [v(M') - v(M)][p(M') - p(M)] = (1-13) 
 
 where p(M) is the pressure at point M (see, for exmaple, refer- 
 ences 2 and 15 and compare also references 18 and 19). Now with the 
 aid of equations (1.12) and (1.13) and the equations of motion 
 
 dv. r — > dv. -. N 
 
 _i + ^ v , *- - i^ + vAv. (i = 1,2,3) (1.14) 
 
 ot fz^ J ox. p ox- x 
 
 J— X J J- 
 
NACA TM 1350 
 
 it may be shown that the function Dj^r) is connected with the struc- 
 tural function of the third order 
 
 hi^ r) - = h (w) - v * (M) l (1 - 15) 
 
 by the known relation of Kolmogoroff (reference 2) 
 
 8 
 
 ^H-."T-f" (1 " 16 ' 
 
 In the case of homogeneous and isotropic turbulence, the equation 
 relative to the correlation functions (references 18 and 19) is easily 
 derived from equation (1.14) : 
 
 aB n ( bB ui . 4 
 
 "3t" = V - ^ 7- + T lU 
 
 where 
 
 When the correlation functions are replaced by the structural 
 functions given by the formulas in the previous footnote (and by an 
 analogous formula for ^hn)> "the following is obtained: 
 
 from which equation (1.16) is obtained after a single integration with 
 respect to r. In a similar manner, equations (1.12) and (1.13) may 
 be obtained from known results relative to isotropic turbulence. It 
 may likewise be shown that equations (1.12), (1.13), and (1.16) are 
 also valid within the framework of the theory of a locally isotropic 
 flow. 
 
10 WACA TM 1350 
 
 For r«T], the term D, ? ,(r) may be neglected in this relation 
 (since for these values of r the function Dmtr) will be of 
 
 third-order smallness with respect to r) and therefore, equations (1.16) 
 and (1.12) give the solutions 
 
 D (t) = — £ r 2 
 
 11^' "15 v 
 
 D nn( r ) = 15 V rZ 
 
 for r«T] (1-17) 
 
 This is an improvement in the accuracy of relations (1.10). On 
 the other hand, for r>>T) the term with the viscosity may be rejected 
 since 
 
 D m (r) = - - er for r»t] (l-18) 
 
 The nondimensional magnitude, the asymmetry of distribution of the 
 probabilities for the longitudinal component of the velocity difference 
 is now introduced 
 
 ~Wz (1 - 19) 
 
 \?n^l 
 
 From the considerations of the theory of dimensions, it follows 
 that for r>>T] the magnitude S should have a constant value (it 
 can depend only on r and on e , but from these two magnitudes it is 
 not possible to obtain any nondimensional combination) . From equa- 
 tions (1.19), (1.18), and (1.12) it follows that for r>>r\ 
 
NACA TM 1350 11 
 
 , , / 4 \ 2 / 3 2 / 3 2 / 3 
 
 hz^ = (-5S £ r 
 
 D (P) , i f. ±W 3 e 2/3 r 2 / 3 
 
 (1.20) 
 
 The coefficients C and C of formulas (1.9) are thus connected 
 with the asymmetry S by the following simple relations: 
 
 C- l-L 
 
 2/3 
 
 (1-21) 
 
 4 
 C = - C 
 3 
 
 It follows that S is always negative : S = - | S | . Formulas 
 (1.17), (1.9), and (1.21) were obtained by A. K. Kolmogoroff in 1941 
 (references 1 and 2). Up to that time, the results obtained from the 
 equations of hydrodynamics only slightly improved the accuracy of the 
 results obtained previously from a dimensional analysis and they 
 referred only to the two extreme cases: r>>T] and r<<^. In the 
 matter of the computation of T>->-,(r) for the intermediate values of r, 
 
 the single relation (1.16) is of course not sufficient. In this rela- 
 tion are two unknown functions B-,-,(r) and Dttt^), and therefore 
 
 still another relation between them is required for their determination. 
 The theory does not give this needed relation, but an attempt may be 
 made to derive it from experimental data. 
 
 At the present time, results are known of the direct measurements 
 of the magnitude S for various distances, conducted by Townsend 
 (reference 4) in wind-tunnel tests at very high Reynolds numbers for 
 the purpose of checking the theory of Kolmogoroff. These measurements 
 have shown that the asymmetry S may, with a sufficient degree of 
 accuracy, be assumed as constant not only for r>>T) but in general 
 for all values of r lying within the range of applicability of the 
 theory of locally isotropic turbulence. The experimental value of S 
 
12 
 
 WACA TM 1350 
 
 for all values of r is approximately -0.4. This experimental fact 
 provides the additional relation between D^^Cr) and D-,-,-,(r), which 
 
 permits the determination of these functions uniquely for all values of 
 r . 
 
 Thus the asymmetry S is assumed constant. From equations (1.16) 
 and (1.19) 
 
 -^| S |[D n (r)]^ = f 
 
 er 
 
 (1.22) 
 
 where |S | is constant. This equation in the function 
 
 V r > 
 
 with 
 
 coefficients depending on v, t, and |S| is considerably simplified 
 if transfer is made to nondimensional magnitudes and the as yet unde- 
 termined numerical factors k-, and kp are in the expressions for the 
 
 scales (i.e., use is made of formulas (1.6) and (1.7)). Then for 
 
 e u (x), 
 
 k 2 <*p. 
 
 n 
 
 dx 
 
 l S l k 2 3 [PziW] 3/2 =|V 
 
 (1.23) 
 
 The magnitudes e and v no longer enter into this equation, 
 
 For a corresponding choice of the constants k, 
 
 possible to eliminate the experimental constant 
 p,,(x) an equation with numerical coefficients. 
 
 choose k n and k^ 
 
 such that 
 
 and 
 
 it is also 
 
 S I and obtain for 
 It is convenient to 
 
 ! S l k l k 2 
 
 tf 2 
 
 k ° 
 2 1 
 
 15 r~2 = 
 
 K 2 
 
 (1.24) 
 
 The experimentally determined values of S fluctuate between 
 the limits -0.36 and -0.42. This scatter lies within the limits of 
 accuracy of the measurements. As the most probable value of S 
 Townsend gives the value -0.38. However, this value may not be assumed 
 reliable for purposes of this report. 
 
NACA TM 1350 
 
 13 
 
 that is, to set 
 
 4 
 4V¥ 1 
 
 v _ 4^2 1 
 K 2 _ 
 
 The equation for 377 ( x ) is then 
 
 d0 (x) 
 
 5.035 
 
 1.838 
 
 (1.25) 
 
 dx 
 
 I »„W 
 
 3/2 
 
 = X 
 
 (1.26) 
 
 Equation (l.26) together with the initial condition P-,(0) = 
 
 uniquely determines the nondimensional longitudinal structural function 
 P2t(x) which describes the structure of the velocity field. ^ 
 
 The structure of a turbulent flow may likewise be described with 
 the aid of the spectral energy distribution. In this case, E(p) 
 denotes the energy of the system of disturbances the wave number of v 
 which is larger than p (the scale of disturbance is inversely propor- 
 tional to the wave number) . In the statistical theory of homogeneous 
 (stationary) processes and fields, it is shown that there exists a one 
 to one correspondence between the correlational (structural) functions 
 and the functions E(p); the formulas that permit expressing one of 
 these functions in terms of the other approximate in type the Fourier 
 transformation (cf. references 20 and 3l) . The 2/3 law for the struc- 
 tural functions, equations (1.9), is equivalent to the ratio of the 
 spectral function E(p) for P<<P-,> to the magnitude p"2/3 (i.e., 
 
 the ratio of the spectral density dE(p)/dp = E'(p) to the magnitude 
 
 p~ ' ) . The scale r\ corresponds in the spectral theory to the critical 
 wave number p-, = l/rj . The 2/3 law was first obtained in this form by 
 
 A. M. Obukhoff (reference 3) in 1941. The complete description given 
 in the text of the structural function D^fr) is equivalent to the 
 
 determination of the spectral function not only for p«Pn but also, 
 
 in general, for all values of p. There are a number of attempts 
 (references 3, 21, 22 and 5) at a direct theoretical computation of the 
 function E(p) for all p. The results thereby obtained are however 
 difficult to compare with experimental data. 
 
14 NACA TM 1350 
 
 The corresponding nondimensional transverse structural function B (x) 
 is determined from the relation (1.12) which, after substitution from 
 equation (1.6), may be represented in the form 
 
 Figure 1 shows the graphs of the graphs of the functions B 77 (x) 
 and 3 nn ( x )> where p,,(x) was determined with the aid of numerical 
 integration- 1 -! of equation (1.26) for the conditions 377(0) = ®> and 
 P nn (x) was computed with the aid of p^ Z ( x ) from relation (1.27). 
 The dotted curves denote the asymptotic values of these functions for 
 small and large values of x: 
 
 P»M- h z 
 
 B (x) = X 2 
 
 for x<<1 (1.28) 
 
 for x>>1 (1.29) 
 
 |3 (*) 
 
 2/3 
 
 nn 
 
 x 
 
 These formulas correspond to the asymptotic equations (1.17) and (1.20) 
 for the structural functions. The particularly simple form of the 
 asymptotic formulas for the function P nn ( x ) permits a very simple 
 determination of the magnitudes of tu and u-, of equation (1.6) from 
 
 the transverse structural function B (x) which was obtained from 
 
 nn 
 
 For large values of x (for x>8), it is convenient to make 
 use of the asymptotic expansion for B 77 (x): 
 
NACA TM 1350 15 
 
 12 
 experiment. It is for this reason that the previously mentioned values 
 
 for the coefficients k, and k ? were chosen. 
 
 A direct comparison of the computed curves with the experimental 
 curves obtained in wind-tunnel measurements is technically difficult 
 to make because of the smallness of the scale r\. In wind-tunnel 
 measurements it is thus usually possible only to check the agreement 
 with the 2/3 law (see for example references 4 and 5) . With relation 
 to the results which refer to the trend of the curve for r ~ t^j 
 
 it is necessary to be satisfied with an indirect check of the type used 
 in checking the accuracy of the constancy of the asymmetry factor. 
 From this point of view measurements in the free atmosphere are evi- 
 dently more convenient because here the scale r\-. is somewhat larger 
 
 (of the order of several mm). Nevertheless, such experiments are very 
 complicated and up to this time only one investigation containing data 
 referring to scales of the order of r\-, is known. This is the investi- 
 gation of Godecke (reference 23) in which the mean absolute differences 
 in velocity in a. direction perpendicular to the base (which corresponds 
 to the transverse structural function) is measured for distances of r 
 varying from 0.1 to 80 centimeters at an altitude of 1.15 meters [YJ. 
 The evaluation of these data, (reference 7) has shown that they are in 
 good agreement with the theoretical curve obtained herein for (3 n (x) 
 where r\j_ = 0.54 centimeter and u, = 2.02 centimeters per second. 
 
 2. Computation of structural function of pressure field . The 
 study of the local structure of the field of pressures in a. turbulent 
 
 12 , . 
 
 Technically, the measurement of D nn (r) can be affected much 
 
 more simply than the measurement of D^( r ) . For this reason D nn (r) 
 
 is generally measured in experimental work. Approximation of the curve 
 obtained for D (r) to a parabola, for small values of r to a parab- 
 ola and to the 2/3 law for large values of r gives precisely the mag- 
 nitudes of t] and u 2, the coordinates of the point of intersection 
 
 of these two asymptotic expressions. The above construction is con- 
 veniently carried out on logarithmic scale; the parabola, and the 2/3 law 
 are thereby represented by two straight lines (cf. reference 7). 
 
16 NACA TM 1350 
 
 13 
 flow is considered in this section. As a quantitative characteristic 
 
 of this structure, as in the case of the velocity field, the corres- 
 ponding structural function is chosen 
 
 TI(M,M') = [p(M') - p(M)J 2 (2.1) 
 
 In the case of a locally isotropic flow, the function H(M,M'), 
 for a distance r between the points M and M' much less than the 
 external scale of turbulence L, will depend only on r: 
 
 II(M,M') = H(r) (2.2) 
 
 and will be entirely determined by the local structure of the flow. 
 From considerations of the theory of dimensions it follows that 
 
 n(r) = ^(~~\ (2-3) 
 
 where 
 
 q^ = D u 2 = k g 2p Vv£ (2.4) 
 
 the numerical coefficients k-j_ and k 2 being assumed to coincide with 
 
 the coefficients in equation (1.25) and rt(x) being a universal func- 
 tion. Further, since for r>>T)-, the structural formula II (r) should 
 not depend on the viscosity V ; *he asymptotic equation is 
 
 n(x) ~ x 4 / 3 for x>>1 (2.5) 
 and therefore 
 
 Il(r) ~ c 2e 4 / 3 r 4 / 3 - o 2 [D u (r)] 2 for r»T, 1 (2.6) 
 
 -"-^From the fact that when deriving the fundamental equation connect- 
 ing the second and third moments of the velocity field of an isotropic 
 (locally isotropic) incompressible flow, the pressure is excluded (see 
 references 18 and 19 and also equation (1.13)), it does not follow that 
 in an isotropic (locally isotropic) turbulent flow fluctuations of the 
 pressure are absent. Such an erroneous conclusion has been drawn by 
 M. D. Millionshtchikov (reference 24). 
 
NACA TM 1350 17 
 
 It will now be shown how the numerical coefficient in this formula 
 and, in general, the entire trend of the function «(x) may be approx- 
 imately computed. 
 
 For this purpose use is made of equations (1.14). If the i^* 1 
 equation is differentiated with respect to x^ and summed over i, then 
 on account of relation (l.ll) the terms with dv^/cH and with Av-^ 
 drop out and 
 
 E 
 
 ijj=i 
 
 
 ±Ap (2.7) 
 
 or 
 
 ,3 ^ v . £) v _ 
 
 Ap = -p 2_, _i J (2.8) 
 
 i,j=l 5x7 5xT 
 
 (the equation of continuity is again applied) . 
 
 From equation (2.8) it is not difficult to derive the differential 
 equation for the function H(r) . It is simplest to proceed as follows. 
 At first the assumption is made that the velocity field and pressure 
 field are statistically homogeneous and isotropic (and not only locally 
 homogeneous and locally isotropic). In this case, the left and right 
 sides of equation (2.8), written out for the point M with coordinates 
 x , x , x , are multiplied correspondingly by the left and right sides 
 
 of the analogous equation for the point M' with coordinates xj, xA, 
 
 x', and the result is averaged and after taking into account the fact 
 
 that in the case of a homogeneous and isotropic pressure field 
 
 Ap (M)Ap (M ' ) = A 2 [p (M) p (M ' )J 
 
 where when differentiation is carried out on the right side with 
 respect to the components £± = x { ~ x i °^ ^he vector MM' 
 
 ^ Sv.(M) ov.(M) dv (M') bv (M') 
 
 A 2 P (M)p(M') = p 2 . 2_, -^ £— -gL ^L (2.9) 
 
 i,j,k,I oXj ° x i dx 2 ox k 
 
 It should now be noted that in t he case of a homogeneous and iso- 
 tropic flow the correlation function p(M)p(M' ) is connected with the 
 structural function (2.l) by the relation (see previous footnote): 
 
18 NACA TM 1350 
 
 TT(r) = 2[p 2 - p(M)p(M')J (2.10) 
 
 Equation (2.9) may therefore be rewritten in the form 
 
 A 2 n(r) 5 dlnirl + 4 d 5 H(r) 
 
 dr 4 r dr 3 
 
 ■2p 
 
 , v-^ dv, (M) Sv.(M) dv,(M') Sv 7 (M') 
 
 ±,f^,i ^7 dx i ^T dx k 
 
 This is the required equation. It also has a meaning in the case 
 of locally homogeneous and locally isotropic (but not homogeneous and 
 isotropic) flow, and with the aid of more complicated considerations 
 it may also be derived without the assumption of homogeneity and iso- 
 tropy. 
 
 The structural function XI (r) is thus seen to be a solution of 
 equation (2.1l), in the right side of which appears a combination of 
 four moments of the derivatives of the velocity field. Unfortunately 
 these moments are not known, and in order that any use may be derived 
 from equation (2.1l), it is necessary to make an additional assumption 
 which will permit computing these moments. The assumption adopted herein 
 is that proposed by M. D. Millionshtchikov (reference 24) which states 
 that the fourth moments of the velocity field are expressed in terms 
 of the second moments in the same manner as in the case of the normal 
 Gaussian distribution. As a first approximation this assumption 
 appears to be an entirely natural one. This assumption finds a certain 
 justification in the measurements of Townsend (reference 4) which show 
 that the experimental value of the fourth moment for the velocity deriv- 
 ative dv, /ctoc, differs by no more than 15 percent from the value com- 
 puted by the measured value of the second moment on the assumption of 
 normal distribution. 
 
 For any four chance magnitudes w-, , w ? , w, , and w, subject to a 
 four-dimensional normal-distribution law, the equation holds (see for 
 example, reference 25): 
 
 W 1 W 2 W 3 W 4 = V 1 W 2 W 3 W 4 + W 1 W 3 W 2 W 4 + W 1 W 4 W 2 W 3 
 
 ^- 4 It is noted that in the recent work of Heisenberg (reference 21) 
 a hypothesis with regard to the spectral functions of an isotropic tur- 
 bulent flow precisely equivalent to that proposed by M. D. Millionsctchikov 
 was used. 
 
NACA TM 1350 19 
 
 When this formula is applied to the product of the four derivatives 
 of the velocity field which enter into the right side of equation (2.1l), 
 the following equation is obtained: 
 
 Z 
 
 Sv 1 (M) dv.(M) dv k (M') dv^M') dv^M) dv.(M) dv k (M') dv-j(M') 
 
 Sx . Sx . Sx ' dx.' -,- 4 w i dx . Sx . <5x,' dx,' 
 
 ,— . Sv.(M) dv, (M') dv.(M) dv, (M') ,- — Sv.(M) dv, (M-) dv.(M) Sv, (M') 
 
 \ _J: 5 J " + y 1 • 1 £ (2.12) 
 
 ,■ r~^ l dx . Sx ' dx . Sx,' ,- fV 7 Sx . <W' Sx . Sx ' 
 
 The fi rst t e rm on the right-hand side of the equation is propor- 
 tional to Ap(M) Ap (M ' ) . In the case of a locally isotropic flow, it 
 is easily verified that this term becomes zero, as can be derived, for 
 example, from equations (1.3) and (1.12). The last two terms of equa- 
 tion (2.12) are equal to each other. It is further noted that in the 
 case of a locally isotropic flow 
 
 Sv j( M ) 5v k( M ') 1 5 Dj k (M,M') 
 
 d Xi chc< 2 bKM, 
 01 j 1 
 
 where D., is the structural function in equation (l.l) and 
 
 K- = xl - x.. From this it follows that for the assumption made about 
 J J J 
 
 the relation of the second and fourth moments, equation (2.1l) may be 
 
 represented in the form 
 
 £5fel + ±S=Sfel... p * Z Jj^^L. ( 2.i 4) 
 
 dr 4 r dr 3 l,J,k,l ^j ^2 S ?i ^k 
 
 The function on the right side of this equation depends, of course, 
 only on r 
 
 ^ 5 2 D (M,M ! ) h 2 D ,(M,M') 
 *(M,M«) = 2. /* r^- - «(p) (2.15) 
 
20 NACA TM 1350 
 
 With the aid of equations (1.3) and (1.12), equation (2.15) may be 
 reduced, after rather long transformations, to the form 
 
 »(r) = 4^lf ♦ SO f^i !5k + ifelY + 4 Si ^» (2.16) 
 
 r 2 V dr / r to dr 2 Var 2 ) dr dr 3 
 
 In accordance with the definition (2.1) , the function H(r) is 
 even and assumes the value zero for r = 0. Therefore, the two "bound- 
 ary conditions which result are: 
 
 11(0) = (2.17) 
 
 TI'(O) = (2.18) 
 
 As a third boundary condition use is made of 
 
 Hill +0 for r^» (2.19) 
 r 2 
 
 Equation (2.14), for the conditions of equations (2.17), (2.18), 
 and (2.19), has a unique solution which is the required structural 
 function. 16 
 
 Since all linearly independent solutions of the homogeneous equa- 
 tion corresponding to equation (2.14) are found without difficulty 
 (they are 1, r, r and r" ) , the required solution of the nonhomo- 
 geneous equation can be constructed with the aid of Green's functions. 
 It is easily verified that in the case of the boundary conditions 
 expressed in equations (2.17), (2.18), and (2.19), this function for 
 equation (2.14) has the form 
 
 1 cz 
 
 It may be shown that this condition is required so that the 
 
 correlation between the differences in the values of the pressures at 
 
 two pairs of points will approach zero as one pair of points recedes 
 
 infinitely from the other (the distance between the points for each 
 
 pair is assumed to be fixed). 
 
 It may appear strange that only three boundary conditions are 
 used, whereas equation (2.14) is of the fourth order. The fact is, 
 however, that equation (2.17) is a double condition: Zero is a singu- 
 lar point of equation (2.14) and therefore one boundary condition will 
 be the requirement that the function have regularity at zero. 
 
WACA TM 1350 21 
 
 2Li for r ^ C 
 
 G(r,C) = < (2.20) 
 
 ~3 ,,2 ,,4 
 
 L_ _ ££_ _ L_ for r ^5 
 
 -■2 2 6r 
 
 The required solution for H(r) may be represented in the form 
 
 n(r) = -p 2 / G(r,s)$te)d€ (2.21) 
 
 The function ^(r), given by equation (2.16), may, on account of 
 relations (l.6) and (1.7), be represented in the form 
 
 where kn , kg, and r\-, are determined from equations (1.25) and (l.7) 
 and (p(x) is the universal function: 
 
 , ( x) = i. (*_hi) + *o d 2n£hi + Jthif + , d .hi£hi 
 
 x 2 V<w x dx dx 2 V^ 2 y dx dx 3 
 
 (2.23) 
 
 When equations (2.20) and (2.22) are substituted in equation (2.2l) 
 and a change of variables is made (cf. equations (2.3) and (2.4)) 
 
 H(r) = k„ 4 p 2 vtJ— \ (2.24) 
 
 1] 
 
 where 
 
 'W-JTt^^^^t 1 '^ 
 
 00 
 
 (2.25) 
 
 1 P X .3 
 2 
 
22 NACA TM 1350 
 
 Thus the universal function Jt(x) of equation (2.3) is connected 
 with the function 3,7 ( x ) > which was computed in the preceding section, 
 by use of relations (2.25) and (2.23). Equation (l.26) expresses the 
 derivative dp,-,/dx in terms of the function p,-,(x). When this equa- 
 tion is applied several times, the second and third derivatives of 
 these functions can be expressed in terms of PttTx) and therefore 
 also the function <p(x) . Thus, with knowledge of the function (377 (x) 
 
 from section 1, cp(x) can be determined and all the integrals in equa- 
 tion (2.25) can be numerically computed, that is, the function «(x), 
 which determines (due to a relation with equation (2.3)) the structure 
 of the pressure field can be computed. The graph of the function it(x) 
 thus obtained is given in figure 2. 
 
 The dotted curves in figure 2 show the asymptotic behavior of 
 n(x) for small and large values of x. Since the motion of the fluid 
 for scales much smaller than t]-, is laminar, for x<<1 
 
 jt(x) = ax 2 (2.26) 
 
 (See the analogous derivation for the structural functions of the vel- 
 ocity field.) The coefficient a in this formula can easily be 
 obtained from equation (2.25) as follows. From equation (1.28), 
 (3,,(x) » x /*2 for x<<1. Hence the function (p(x), for such values 
 of x, may be considered as constant: <p(x) = 30. When this value is 
 substituted in equation (2.25), the term proportional to x gives 
 only the last of the integrals in equation (2.25) and 
 
 f\oo 
 
 a = I / £cp(0d£ - 0.8 (2.27) 
 
 (This value has been obtained with the aid of numerical integration.) 
 The curve 
 
 tt(x) = 0.83 x 2 
 
 is the first of the asymptotic curves drawn in figure 2. 
 
 In the second limiting case, for x>>1* equation (l.29) shows 
 Pj-jCx) " 3x 2 / 3 /4, and therefore <|>(x) - 7x- 8 / 3 /l8. Equation (2.25) is 
 
 now represented in the form 
 
NACA TM 1350 23 
 
 '. sl + *d + ilk t- e / 3 d t + r" ^ i r 8 / 3 
 
 «w -J I- V + ^ + kHs ?~°' J d? + j x v fa 6 ' «? + 
 
 4^(0 " 7^ r 8/3 )^ (2.28) 
 
 18 
 
 It is not difficult to see that the values of the • integrals on the 
 right side of equation (2.28) for x->-°° will not increase any faster 
 than the first degree of x, so that the .principal term of the asymptotic 
 formula for «(x) will be the term 9xV 3 /l6. Thus, the numerical 
 coefficient in equation (2.5) is equal to 9/l6 and the symbol of the 
 asymptotic equation (2.5) means only that 
 
 -*kL-- r *W -1 for x»l 
 ± 4/3 [Puto] 2 
 16 X 
 
 or 
 
 2 
 P 
 
 H(r) 
 
 -=^r = 1 for r>>T], 
 
 The difference it(x) - 9x / /l6, however, increases without limit 
 
 as x increases, 
 
 To obtain the succeeding terms of the asymptotic formula for 
 it(x), equation (2.28) is further transformed: 
 
24 NACA TM 1350 
 
 
 (2.29) 
 
 Here the integrals over the range from to °° converge very 
 rapidly and may be numerically computed while the last integral over 
 the range from x to m may be evaluated for x>>1 with the aid of 
 the asymptotic formula given in a previous note. It should be noted 
 that this integral adds only an insignificant increment to the constant 
 term of the asymptotic formula for «(x). Finally, vith an accuracy 
 up to terms approaching zero as x-+ °° , 
 
 it(x) « ^- x ' - 0.08x + 0.85 for x»l (2. 30) 
 lo 
 
 This is the equation for the asymptotic curve for large values of 
 x plotted in figure 2 . 
 
 No knowledge of any experimental data, on the structure of the pres- 
 sure field which could be compared with the results obtained herein is 
 known to the authors. It should be remarked that the computations pre- 
 sented previously show that the mean square values of the differences 
 in pressures are found to be so small, as a rule, that their measure- 
 ment would be associated with very great experimental difficulties. It 
 does not follow from this, however, that the computation of the struc- 
 tural function of the pressure field is practically useless. In the 
 following section it will be shown that the values of the local pressure 
 gradients thereby obtained are very large so that the accelerations pro- 
 duced by the fluctuations of pressure may play an essential role in 
 processes which arise in turbulent flow. 
 
NACA TM 1350 25 
 
 3 . Computation of correlation functions of acceleration field . A 
 study of the acceleration field of the fluid particles in a turbulent 
 flov is now undertaken. This field differs from the fields considered 
 in the previous sections in that the very smallest and not the largest 
 vortices are essentially responsible for values of the acceleration at 
 a point, as is the case for the velocity and pressure fluctuations. For 
 this reason, in the case of the field of accelerations of the local 
 flow structure, not only the statistical characteristics of the differ- 
 ence in values of the field at two points (e.g., the structure function) 
 are determined, but also the statistical characteristics of the values 
 of the field. The most important of these characteristics is the corre- 
 lation function, the mean value of the product of the values of the 
 field at two points (i.e., in the case under consideration, the mean 
 value of the product of the acceleration components). The computation 
 of this correlation function is the main concern in this section. 
 
 The value of the correlation function at zero is determined first, 
 that is, the mean square of the acceleration of a fluid particle at a 
 single point. This magnitude is the numerical characteristic of most 
 interest of the acceleration field. From the equations of motion (1.14), 
 the acceleration components of the fluid particle 
 
 dv. dv. ,3 dv. 
 
 l 
 
 ~ + 2 v i ^ (i = 1 > 2 ' 3 ) (S' 1 ) 
 
 v i = sr = 5T + Af v j sr 
 
 1 7 
 
 From considerations of the theory of dimensions it follows that 
 
 to vortices of the scale of Z, where Z>>t), there corresponds the 
 characteristic period T^ = (Z /e) ' such that the velocity character - 
 
 l/3 
 
 istic for these vortices is equal to v^ = Z/Tj = (el) and the char- 
 acteristic acceleration is w. = Z/t, 2 = (e 2 /^) 1 ' 3 . Thus it is observed 
 
 that when the scale of lengths is decreased, characteristic velocity 
 decreases while the characteristic acceleration increases . From this 
 it follows that the very small vortices of scales Z ~ t\ are mainly 
 responsible for the value of the acceleration at a point of the flow 
 (for such vortices, the dimensional considerations adduced herein do not 
 correspond, of course, to actual conditions, for the motion of these 
 vortices essentially depends on the viscosity) 
 
 ■^It is clear that the correlation function is a more significant 
 characteristic of the field than the structural function. Knowledge of 
 the correlation function always allows determination of the structural 
 function also. The converse does not hold true. 
 
26 NACA TM 1350 
 
 are equal to 
 
 w, = - ±f£- +vAv, (3.2) 
 
 1 n ox. x 
 
 from which is obtained 
 
 3 3 3 3 
 
 1=1 o 2 1=1 V3x i' ° 1=1 Sx l 1-1 
 
 The first and third term on the right side of this equation may 
 be expressed, without difficulty, in terms of the structural functions 
 of the velocity and pressure fields, equations (l.l) and (2.1): 
 
 ^ (dp \ 2 _ 1 y- 5 2 U(0) _ 5 d 2 n(0) (3 4) 
 
 h WJ ' 2 U ^ ± 2 " 2 dr 2 
 
 3 
 
 Z (Av f = -lA 2 ^D^CO)) (3.5) 
 
 i=l 1 2 \i=l / 
 
 The middle term on the right side may "be expressed through the 
 interrelated structural functions 
 
 D ip (M,M') = [v.(M') - v.(M)J[p(M') - p(M)] (i = 1,2,3) (3.6) 
 
 of the velocity and pressure fields. Since in the case of incompressible 
 local isotropic flow these functions should be equal to zero (see equa- 
 tion (1.13)), the middle term on the right side of equation (3.3) 
 becomes zero, and therefore 
 
 ±^-f,^4- s &^°) (3 - 7) 
 
 i=l 2p dr \i=l / 
 
 But on account of equations (2.24), (1.7), (1.25), (2.26), and 
 (2.27) 
 
NACA TM 1350 27 
 
 iim. . *A *„-■& ^ ,,.. (0) . i "il\" um ) p 2 v -iA ,3/* 
 
 ar 2 k/ ^ k! 2 g 
 
 
 
 l?4.96pV l/2 » 5/Z - 2fS P^" l/2 ^ (3 ' 8) 
 
 s| — |S| 
 
 Hence 19 
 
 3 d 2 n(0 1 ^ f dp\ 2 1.1 -1/2 3/2 
 
 -5- 9 = ~o > h^r- = T5T v e ( 3 - 9 ) 
 
 p 2 dr 2 P 2 ^ \ dx i/ S 
 
 Further use is made of the fact that for any choice of coordinate 
 systems 
 
 3 
 
 Y, Dii(r) = D n (r) + 2D m (r) (3.10) 
 
 i=l 
 
 and of equation (1.12), the following is obtained; 
 
 ,4 A J>\f . . dD,,(r)\ 
 
 A" 
 
 2 > . t, /_«v _ /d* L 4d J L / ^ , „ ZI 
 
 &*iiM- ZI + ?Z3 »»,«♦* 
 
 ^\i^ d4 V r ) 24 d3D n (r) , , 
 = r t± + 11 tk + ±i U (3.11) 
 
 dr 5 dr 4 r dr 3 
 
 With the aid of formulas (l.6) and (l.7), the change from D-^(r) 
 to the nondimensional function (3,,(x) gives 
 
 The computation of the magnitude | grad p|' for locally isotropic 
 turbulence is also contained in the work of Heisenberg (reference 21) . 
 The method of Heisenberg is based on the employment of the spectral 
 function E(p) and requires considerably more complicated computations. 
 Moreover, in the final formula of Heisenberg, magnitudes enter which 
 cannot be separately measured in tests . 
 
28 NACA TM 1350 
 
 a 2 £ Dli( r) . % v " 5 / 2 Mjl &(JL 
 
 1=1 kl « \l! "K 
 
 (3.13) 
 
 It is now noted from equation (1.25) that 
 
 k 2 = | S | V^2 
 
 k x 4 120^^5 
 
 and that J3- 7 (x) is an even function of x which may be expanded in 
 the neighborhood of zero in a power series in x : 
 
 3 (x) = b x 2 + b x 4 + . . . (3.14) 
 
 From equations (3.12), (3.13), and (3.14) the following is obtained: 
 
 M± „ J . ism v- 5 / 2 ^ 840 _ tV^I -5/ 2 e 3/2 
 
 Vi=l X1 7 120 Vs 45 z 
 
 (3.15) 
 
 By use of this method, only the determination of the coefficient 
 \>2 in equation (3.14) remains. From the first of equations (1.28) it 
 follows that b^ = l/2. When the expansion (3.14) is substituted in 
 equation (1.26) and the coefficients of r are equated (or, what is 
 equivalent, differentiating equation (l.26) with respect to r three 
 times and then setting r '= 0) , the following equation is readily 
 obtained: 
 
 bo = - — ~ (3.16) 
 
 The substitution of this value of bp in equation (3.15) gives 
 
 " T*l£ D ll<^ - v 2 S ^? = JM v -1/2 ,»A , o.3| S |v- l/2 e 3/2 
 \i=l j i=l 6^15 
 
 (3.17) 
 
NACA TM 1350 29 
 
 Since |S | * 0.4, it follows from a comparison of equation (3.9) 
 with equation (3.17) that the acceleration of the fluid particles in a 
 turbulent flow is essentially determined by the fluctuating pressure 
 gradients and not by the friction forces. The term with IT 1 ' (0) in 
 equation (3.7) is more than 20 times as large as the term depending on 
 the viscosity. It shall be seen that this greatly simplifies the compu- 
 tation of the correlation functions of the acceleration field. 
 
 When equations (3.9) and (3.17) are substituted in equation (3.7), 
 the following formula is obtained for the computation of the mean square 
 of the. acceleration w : 
 
 »o 2 
 
 1=1 
 
 / \2 /n i ,\ -l/2 3/2 
 
 (w ± ) « l±i± + 0.3|s| v ' e' (3.18) 
 
 Since |s| ■ 0.4, equation (3.18) may be replaced by the simple 
 relation 
 
 -1/2 3/2 
 w Q 2 » 3v e (3.19) 
 
 This general relation permits the estimation of the order of mag- 
 nitude of w n in specific cases of turbulent flow without difficulty. 
 
 As an example, formula (3.19) is applied to the computation of the 
 mean square acceleration in certain turbulent flows behind a screen (or 
 grid) in wind tunnels and in turbulent atmosphere. In the case where 
 isotropic turbulence was produced by screens in wind tunnels, the dissi- 
 pation e may be defined either as 
 
 e = . 3 v dv£ 
 e 2 dx 
 
 2 
 where v' is the mean square of the velocity fluctuation, V the 
 
 mean velocity, x the distance from the screen, or as 
 
 15v v* 2 
 
 £ = 
 
 x 2 
 
 where X is the length introduced by Taylor, experimentally determin- 
 able by inscribing a parabola in the graph of the correlation function 
 B,,(r). When the dissipation e is known, w can be computed from 
 the formula 
 
 w Q = 2.77 e 3 / 4 cm/sec 2 (3.20) 
 
30 
 
 NACA TM 1350 
 
 obtained by substituting the air viscosity 
 tion (3.19). 
 
 v = 0.15 sq cm/sec in equa- 
 
 In particular, when use is made of some of the data given by 
 Townsend (reference 4) (these data refer to the flow in a wind tunnel 
 behind a square screen with size of mesh M = 6 inches at a distance 
 x = 30.5 M from the screen for various values of the velocity V") , the 
 following values for e and Wq are obtained; 
 
 V m sec -1 
 
 2 i 
 e cm sec~ J 
 
 _2 
 Wq cm sec 
 
 12.2 
 
 60.5 
 
 60.4 
 
 24.4 
 
 312.4 
 
 206.8 
 
 30.5 
 
 559.8 
 
 320.3 
 
 From this table it is observed that the instantaneous values of 
 the acceleration in turbulent flow behind the screen will be of the 
 order of several meters per second per second. 
 
 The application of formula (3.19) or (3.20) to the computation of 
 the accelerations in a turbulent atmosphere is rendered difficult by 
 the fact that at the present time there are no available measurements 
 of energy dissipation for this case. However, for the degree of accur- 
 acy of the computations, much justification exists for employing an 
 estimate of the magnitude of e for a turbulent atmosphere by the 
 formulas of the theory of the logarithmic boundary layer. It is known 
 (reference 15) that for the logarithmic boundary layer 
 
 £ - 
 
 1 
 
 X 
 
 y 
 
 (3.21) 
 
 where y is the distance from the wall, x is a nondimension al con - 
 stant (Karman constant) equal approximately to 0.4, and v^ =\ % q ) /o 
 (to is "the friction stress, p the density) is the so-called dynamic 
 velocity determined by the difference of the mean velocities at two 
 points or by the mean velocity at one point and the magnitude of the 
 roughness. Substitution in formula (3.19) of expression (3.2l) for 
 the dissipation and v = 0.15 sq cm/sec gives a computational formula 
 which determines the mean square acceleration in a logarithmic boundary 
 air layer : 
 
 9/4 
 
 w^ 
 
 5.5 
 
 v* 
 
 cm 
 
 "374 2 
 y ' sec 
 
 (3.22) 
 
NACA TM 1350 
 
 31 
 
 Since v^. is proportional to the mean velocity V, 
 
 w ~ V^/ 4 
 
 (3.23) 
 
 that is, Wq increases rapidly with V. For the example, the ma.gnitude 
 of the roughness is assumed to be h n = 3 cm (it is noted incidentally 
 that the computations following depend relatively little on the magni- 
 tude of the roughness) and the mean velocity of the wind at the height 
 150 cm is denoted by V. Then 
 
 v = 
 
 xV 
 
 ln(y/h ) 
 
 « 0.1 V 
 
 (3.24) 
 
 and for the mean square acceleration 
 following values are obtained : 
 
 w r 
 
 at various velocities V the 
 
 V, m sec~l 
 
 1 
 
 3 
 
 5 
 
 6 
 
 8 
 
 _2 
 
 v_j cm sec 
 
 22 
 
 260 
 
 830 
 
 1200 
 
 2400 
 
 The mean square acceleration under the conditions considered for 
 a mean velocity of the wind V = 5.5 m/sec thus attains the magnitude 
 of the acceleration of gravity g, and for a greater wind velocity may 
 considerably exceed this acceleration. It is natural to assume that 
 such large accelerations may play a significant part in many physical 
 processes in the atmosphere (e.g., in the phenomenon of the condensation 
 of fogs) . 
 
 The computation of the correlation function of the acceleration 
 field is now considered: 
 
 A i .(M,M') = w i (M)w,(M') 
 
 (3.25) 
 
 Again, substitution of equation (3.2) gives 
 
 A.j(M,M') 
 
 _1_ dp dp' 
 
 2 5x7 SxT 
 p x J 
 
 v Av.A'v! 
 i J 
 
 (3.26) 
 
 The magnitudes without the primes refer to point M and those with 
 primes to the point M'. The middle term on the right side may be 
 neglected for the same reasons for which the middle term on the right 
 side of equation (3.3) was previously rejected, and the first and third 
 
32 
 
 NACA TM 1350 
 
 terms may easily be expressed in terms of the structural functions (l.l) 
 and (2.1). Therefore, 
 
 b-p dp' 1 d II(M,M') 
 5xT SxT 2 dg ££ 
 
 J 1 J 
 
 (3.27) 
 
 where £. and £ . are the components of the vector MM' and 
 
 Av.A'v'. = - ± A^D..(M,M') ( A = -A_ + JL_ + _L_] (3.28) 
 
 1 A 2 T 
 
 1 3 
 
 ij 
 
 ^l 2 ^2 2 ^3 2 
 
 The transformation of equations (3.27) and (3.28) follows. Since 
 
 / \ /T 2 " 2 2 
 
 H(M,M') depends only on the distance r = M?]_ + ?;? + ^3> 
 
 S 2 TI(M,M') _ a J dn(r) £jl = j d 2 H(r) 1 dU(r)\ g i g j 1 dll(r) 
 
 "5|ToT~ = 5^7 \ dr ' r 
 
 dr £ 
 
 r dr 
 
 .2 ' r dr ij 
 
 (3.29) 
 
 Replacement of D. .(M,M') by means of equations (1.3) and (1.12) 
 
 yields 
 
 which gives the following; 
 
 .2 
 
 A^jCM^O = D 1 (r) -£-* + D 2 (r)& i(j 
 
 (3.30) 
 
 where 
 
 d M = 12 ^n + 12 ^n 
 
 1 r 3 dr r 2 dr 2 
 
 d4D 22 r d %Z 
 dr 4 2 dr 5 
 
 (3.31) 
 
 D P (r) 
 
 dD 
 
 d 3 D-, 
 
 i_ n _ i_ H + 8 u ^U 
 
 3dr 2 -, 2 r-,3 
 r r dr dr 
 
 + 5 
 
 d^D 
 
 dr 
 
 4 2 ,5 
 
 dr 1 " 
 
 Thus 
 
 A ltJ (M,M') = A x (r) -lli + A 2 (r) Sij 
 
 (3.32) 
 (3.33) 
 
NACA TM 1350 33 
 
 where 
 
 A lM . i {^fM - i ^4 - # B l(r , (3.34) 
 
 A 2 (r) - 1 S^l-^DaCp) . (3.35) 
 
 2p r 
 
 and D-i(r) and Dg^-O are determined by formulas (3.3l) and (3.32). 
 
 The functions A-^(r) and Ag(r) are expressed in terms of the 
 
 longitudinal and transverse correlation functions of the acceleration 
 field determined by the equations 
 
 A u (r) = w 1 (M)w I (M') 
 
 (3.36) 
 
 A nn (r) = w n (M)v n (M') 
 
 where wi (M) and wt(M') are the projections of the accelerations at 
 the points M and M' on the direction of the vector MM'j and w n (M) 
 and w n (M') are the projections of the accelerations at these points 
 in a direction perpendicular to the vector MM' . In fact, the acceler- 
 ation field of a locally isotropic turbulent flow is isotropic in the 
 usual sense, and therefore 
 
 A . j(M , M .) . Au(r j A "" (r C ± «j + A nn (r)6 iJ ' (3.37) 
 
 (see reference 19 and equation (l.3) herein). Comparing equations 
 (3.33) and (3.37) and taking into account equations (3.34) and (3.35) 
 yields 
 
 A n (r) = A l( r) + A g (r) - -L ^M . *f (^ + ^fr)) 
 
 £p dr 
 
 (3.38) 
 
 ^M-A.M.^^Mri-^D^r) (3.39) 
 
 2p r 
 
 In formulas (3.38) and (3.39) it is possible, in the usual manner, 
 to pass to nondimensional functions. These may be further computed 
 with the aid cf the results of sections 1 and 2. 
 
34 NACA TM 1350 
 
 It may be noted that in these computations the terms with D-^(r) 
 and Dp(r) may be neglected without introducing any appreciable error. 
 In fact, it was shown previously that for r = the terms depending 
 on the viscosity, that is, the terms containing D]_(r) and D2(r), are 
 negligibly small compared with the terms determining the pressure gra- 
 dients. With increasing r both terms decrease asymptotically, the 
 terms depending on the viscosity decreasing much more rapidly than 
 those determined by the pressure gradient. From formulas (2.6) and 
 (1.9) it follows that for r>>Ti 1 
 
 (3.40) 
 
 
 l 2 H(r) 
 dr 2 
 
 r 
 
 -2/3 
 
 1 
 
 r 
 
 dTT(r) 
 dr 
 
 r' 
 
 -2/3 
 
 T, t \ " 10 / 3 
 
 IMr) ~ r ' 
 
 r. ( \ " 10 / 3 
 
 D 2 (r) ~ r ' 
 
 (3.41) 
 
 Thus, for both small and large r, the terms of equations (3.38) 
 and (3.39) containing v are considerably smaller than the terms 
 depending on H(r). In this connection, the investigation of the struc- 
 ture of the acceleration field in a turbulent flow permits the rejection 
 of terms with viscosity in the equations of motion, and the assumption 
 that 
 
 w. = - — 
 
 1 P 
 
 i|E_ (i- 1,2,3) (3.42) 
 
 33F 
 
 A..(M,M0 ^-i ^f 1 (3.43) 
 
 1J 2p 2 ^i^j 
 
 For the longitudinal and transverse correlation functions (3.36), 
 there is then obtained 
 
 2p dr 
 
 A (r) - -L- dII ( r ) 
 W ] 2p 2 r <* 
 
 (3.44) 
 
MCA TM 1350 35 
 
 With the aid of formulas (2.24), (1.7), and (1.25), the change to 
 nondimensional magnitudes is made, and using equation (2.25) 
 
 *»M " ^ "- 1/2 » 3/ S:(f)- W ^ < 3/2 "»© (3 - 45) 
 
 4 
 . , k 2 -l/2 3/2 / r \ 0.45 -1/2 3/2 / r \ ,, ._, 
 
 nn k x 2 nn W I s ! nn Vb. 
 
 where a_ 7 (x) and a (x) are universal functions which are given by 
 the formulas 
 
 a u (x) --3 / £ 4 <p(0^ + § P £cp(0<H (3.47) 
 6x Jo Jx 
 
 - nn (x) - i / Vq>(0^ " "H r^vM**. + I / 5«pU)^ 
 
 Jd 12x Jo Jx 
 
 (3.48) 
 
 As in the case of the velocity and pressure fields, for x<<1 and^for 
 x>>1, it is possible to obtain for the functions introduced in the 
 theory described herein simple asymptotic formulas. It is clear first 
 of all that 
 
 ct n (0) = a m (0) = I / k(5)d£ » 0.83 , (3.49) 
 
 If in formulas (3.47) and (3.48) x is assumed much less than 1 
 (x<<l), use may be made of the fact that for these values of x, as 
 shown in the first formula of equation (1.28), P 77 (x) =» x 2 /2, and there- 
 fore cp(x) = 30; whence 
 
 ^W = ^°) -f * 2 
 
 for x<<1 (3.50) 
 
 1 2 
 a (x) « a (0) - tt x 
 
 nn nn 2 
 
36 NACA TM 1350 
 
 In the second extreme case, for x>>1, the asymptotic behavior 
 of aj2(x) and a (x) is determined with the aid of formulas (2.30) 
 and (3.44). ™ 
 
 / V 1 -2/3 
 
 ( \ 5 -2/3 
 a (x) = — x ' 
 
 nn v 8 
 
 for x>>l (3.51) 
 
 The computation of the functions aj-^x) and a nn (x) for x~l 
 may be carried out numerically by using the data contained in sections 
 1 and 2. It is convenient in place of aj-^x) and a nn (x) to intro- 
 duce the normalized functions 
 
 R nn( x ) - 
 
 c^x) 
 11 (3.52) 
 
 These functions are equal respectively to the correlation coeffi- 
 cient of the longitudinal and transverse components of the acceleration 
 at two points a distance r = xr\-, from each other. The graphs of the 
 
 functions Rjj(x) and R nn (x), which were determined by numerical inte- 
 gration of the integrals appearing in the right sides of equations (3.47) 
 and (3.48), are shown in figure 3. It is seen that the longitudinal 
 correlation function R2i(x) rapidly decreases, and for x^l.l it 
 may practically be considered equal to zero. The function Rnn( x )> on 
 the contrary, decreases at a relatively slow rate, and for x = 3 is 
 approximately equal to 0.17. When the magnitudes of these functions are 
 estimated for relatively large values of x (of the order of 10 and 
 above), formulas (3.5l) may be used. From these formulas, when x = 10, 
 for example, Rn^lO) = 0.03. (in fig. 3 the range of applicability of 
 formulas (3.51J is not represented, since to do so it would be necessary 
 to choose a much smaller scale.) 
 
 It may be noted further that the form of the correlation functions 
 of the acceleration field shown in figure 3 differs sharply from the 
 form of the correlation functions of the velocity field for isotropic 
 turbulence. In the case of the velocity field, the graph of the longi- 
 tudinal correlation function is generally located above the graph of 
 the transverse function and the axis of the abscissas intersects the 
 second and not the first of these curves. This difference in behavior 
 of the correlation functions for the velocities and accelerations is 
 
NACA TM 1350 
 
 37 
 
 explained by the fact that the velocity field in an incompressible fluid 
 is a solenoidal vector field, whereas the acceleration field is con- 
 sidered as a potential vector field (see equation (3.42)). From this 
 it follows that the functions R,, (x) and R nn ( x ) are interconnected 
 by the relation 
 
 dR„„(x) 
 
 R,,(x)=R (x) + x " n 3.53 
 
 1 1 v ' nn * ' dx 
 
 [This relation, which is a necessary and sufficient condition for the 
 isotropic potential vector field having the correlation functions 
 R 7 ,(x) and R (x), was obtained by A. M. Obukhoff , while the correla- 
 
 tion functions B^-^r) and B nn (r) of the velocity field satisfy the 
 Karman condition (cf. reference 19 and equation (l.l3)J: 
 
 B nnM- B nW + f4^— (3 - 54) 
 
 Conditions (3.53) and (3.54), in addition to the factor l/2 in 
 the second term on the right, differ in the interchange of the roles 
 of the longitudinal and transverse functions. It is not surprising, 
 therefore, that the functions R, -, (x) and R (x) behave in a manner 
 opposite to the behavior of the functions B-ij^r) and B nn (r). 
 
 In conclusion, the authors wish to express thanks to 
 A. V. Perepelkina and Y. V. Prokhorova., who carried out the numerical 
 computations for sections 2 and 3. 
 
 Translated by S. Reiss 
 National Advisory Committee 
 for Aeronautics 
 
 REFERENCES 
 
 1. Kolmogoroff, A. N. : The Local Structure of Turbulence in Incom- 
 
 pressible Viscous Fluid for Very Large Reynolds Numbers. DAN 
 (SSSR), vol. XXX, no. 4, 1941. 
 
 2. Kolmogoroff, A. N. : Dissipation of Energy in the Locally Isotropic 
 
 Turbulence. DAN ( SSSR ) , vol. XXXII, no. 1, 1941. 
 
 3. Obukhoff, A.: On the Energy Distribution in the Spectrum of a Tur- 
 
 bulent Flow. Izv. AN SSSR, ser. geogr. i geofiz, vol. XXXII, 
 no. 1, 1941. 
 
38 NACA TM 1350 
 
 4. Townsend, A. A.: Experimental Evidence for the Theory of Local Iso- 
 
 tropy. Proc . Cambridge Phil. Soc . , vol. 44, pt. 4, Oct. 1948, 
 pp. 560-565. 
 
 5. Von Karman, Theodore: Progress in the Statistical Theory of Turbu- 
 
 lence. Proc. Wat. Acad, Sci., vol. 34, no. 11, Nov. 15, 1948, 
 pp. 530-539. 
 
 6. Richardson, L. F.: Atmospheric Diffusion Shown on a Distance- 
 
 Neighbour Graph. Proc. Roy. Soc . (London), ser. A, vol. 110, 
 no. 756, 1926. 
 
 7. Obukhoff, A.: Local Structure of Atmospheric Turbulence. DAN SSSR., 
 
 T. 67, No. 4, 1949. 
 
 8. Stommel, H.: Horizontal Diffusion Due to Oceanic Turbulence. Jour. 
 
 Marine Res., vol. 8, no. 3, 1949. 
 
 9. Krasilnokov, V. A.: On the Propagation of Sound in Turbulent Atmos- 
 
 phere. DAN(SSSR), T. 47, No. 7, 1945. 
 
 10. Yudin, M. I. : Problems of the Theory of Turbulence and Wind Struc- 
 
 ture with Application to the Problem of the Vibrations of an 
 Airplane. Gidrometizdat, 1946. 
 
 11. Obukhoff, A.: Structure of the Temperature Field in a Turbulent 
 
 Flow. Izv. AN SSSR, ser. geogr. i geofiz, vol. XIII, no. 1, 
 1949. 
 
 12. Yaglom, A. M.: On the Local Structure of the Temperature Field in 
 
 Turbulent Flow. DAN ( SSSR ) , T. 69, No. 6, 1949. 
 
 13. Krasilnokov, V. A.: On the Effect of Fluctuations in the Coeffi- 
 
 cient of Refraction in the Atmosphere on the Propagation of Ultra- 
 short Radio Waves. Izv. AN SSSR, ser. geogr. i geofiz, T. 13, 
 No. 1, 1949. 
 
 14. Krasilnokov, V. A.: On the Fluctuations of the Angle of Incidence 
 
 in the Phenomenon of Star Twinkling. DAN(SSSR), T. 65, No. 3, 
 1949. 
 
 15. Landau, L. D., and Livshitz, E. M. : Mechanics of Dense Media. 
 
 Gostekhizdat, 1944. 
 
 16. Obukhoff, A.: Pressure Fluctuations in Turbulent Flow. DAN(SSSR), 
 
 T. 66, No. 1, 1949. 
 
NACA TM 1350 39 
 
 17. Yaglom, A. M. : On the Field of Accelerations in Turbulent Flow. 
 
 DANISH), T. 67, Wo. 5, 1949. 
 
 18. de Karman Theodore, and Howarth Leslie: On the Statistical Theory 
 
 of Isotropic Turbulence. Proc . Roy. Soc . (London), ser. A, 
 vol. 164, no. 917, Jan. 21, 1938, pp. 192-215. 
 
 19. Loitsianskii, L. G. : Some Basic Laws of Isotropic Turbulent Flow. 
 
 Trudy TSAGI (CAHl) Rep. No. 440, 1939. (Central Aero-Hydrodynamical 
 Inst. (Moscow), 1939.) (Available as NACA TM 1079.) 
 
 20. Yaglom, A. M. : Homogeneous and Isotropic Turbulence in a Viscous 
 
 Compressible Fluid. Izv. AN SSSR, ser. geogr. i geofiz. T. 12, 
 No. 6, 1948. 
 
 21. Heisenberg, von W. : Zur statistischen Theorie der Turbulenz . 
 
 Zschr. f. Phys., Bd. 124, H. 7-12, 1948. 
 
 22. Kovasznay, Leslie S. G.: Spectrum of Locally Isotropic Turbulence. 
 
 Jour. Aero. Sci., vol. 15, no. 12, Dec. 1948. 
 
 23. Godecke, K.: Messungen der Atmospharische Turbulenz. Ann. d. 
 
 Hydrographie . Heft 10, 1936. 
 
 24. Millionshtshikov, M. D.: On the Theory of Homogeneous Isotropic 
 
 Turbulence. Izv. AN SSSR, seria geogr. i geofiz, vol. XXXII, 
 no. 9, 1941. 
 
 25. Obukhoff, A.: Theory of the Correlation of Vectors. Uchenye 
 
 zapiski MGU., no. 45, 1940. 
 
40 
 
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