Fourth-order velocity statistics

Fluid Dynamics Research 28 (2001) 1–22

Fourth-order velocity statisticsReginald J. Hill ∗, James M. Wilczak

National Oceanic and Atmospheric Administration, Environmental Research Laboratories,Environmental Technology Laboratory, 325 Broadway, Boulder, CO 80303-3328, USA

Received 13 October 1998; received in revised form 1 June 1999; accepted 1 July 1999

Abstract

An investigation of several fourth-order velocity statistics is described. Whereas local isotropy and local scaling areapplicable to structure functions expressible as averages of di�erences of velocity, local isotropy and local scaling areinapplicable to the structure functions that we study. Data from wind tunnel grid turbulence show the behavior of thefourth-order statistics for nearly isotropic turbulence. The scaling relations predicted by the joint Gaussian assumption(JGA) are considered, as are those from the statistical independence assumption (SIA). The basis of the JGA is thatvelocities at several points are joint Gaussian random variables, whereas the basis of the SIA is that locally averagedvelocity is statistically independent of velocity di�erence. The JGA and SIA relate fourth-order statistics to the second-ordervelocity structure function, as well as to the velocity covariance. For various fourth-order statistics, the predictions of theJGA and SIA are compared with data. These comparisons quantify how accurately our fourth-order statistics follow thescaling dependence on second-order velocity structure functions and on velocity covariance as predicted by the JGA andSIA. Our measured structure functions are in agreement with the scaling predicted by the SIA with no exceptions andwith that predicted by the JGA with two exceptions. As is known, one exception is the structure function that obeys localisotropy [e.g., 〈(ui − u′i)

4〉] (which the SIA does not predict). The other exceptions are called anomalous components[e.g., 〈(u2i −u′2i )(u2j −u′2j )〉]. These anomalous components are shown to be sensitive indicators of intermittency for locallyisotropic turbulence, whereas they are indicators of anisotropy for the case of anisotropic turbulence. The anomalouscomponents are in reasonable agreement with the scaling predicted by the SIA but not with the JGA. c© 2001 The JapanSociety of Fluid Mechanics and Elsevier Science B.V. All rights reserved.

PACS: 47.27.Jv; 47.27.Gs

Keywords: Turbulence; Higher-order correlations; Structure functions

1. Introduction

Fourth-order velocity statistics are important for studies of space–time correlations (M�unch andWheelon, 1958; Tennekes, 1975; Nelkin and Tabor, 1990), acceleration correlations (Lin, 1953;Hill and Thoroddsen, 1997), pressure–velocity statistics (Limber, 1951; Schumann and Patterson,

∗ Corresponding author.E-mail address: [email protected] (R.J. Hill).

0169-5983/00/$20.00 c© 2001 The Japan Society of Fluid Mechanics and Elsevier Science B.V.All rights reserved.PII: S0 1 6 9 - 5 9 8 3 ( 0 0 ) 0 0 0 1 7 - 4

2 R.J. Hill, J.M. Wilczak / Fluid Dynamics Research 28 (2001) 1–22

1978; Hill, 1996a), pressure statistics (Batchelor, 1951; Uberoi, 1953; George et al., 1984; Hill andWilczak, 1995), decay of turbulence (Proudman and Reid, 1954), and acoustic generation (Zhouet al., 1995; Zhou and Rubinstein, 1996). For example, in a previous study (Hill and Wilczak,1995), we derived the formulas for the pressure structure function based on several fourth-ordervelocity statistics, and Uberoi (1953) expressed the pressure correlation as integrals of componentsof the fourth-order velocity correlation.The spectrum of the streamwise component of the fourth-order (as well as higher-order) velocity

correlation has been studied by Van Atta and Wyngaard (1975). They gave a derivation of theirexperimental observations on the basis of the assumption that velocities at several points have a jointGaussian probability density. A consequence of the JGA is that the probability distribution of anycomponent of velocity di�erence is Gaussian. Van Atta and Park (1972), among others, measured theprobability density of di�erences of the streamwise velocity component for several spacings within theinertial and energy-containing ranges. Their measurements depart greatly from Gaussian probabilityfor spacings in the inertial range, with increasing departure from Gaussian for progressively smallerspacings, whereas for spacings in the energy-containing range, their measurements are close to aGaussian probability density. Van Atta and Wyngaard (1975) discussed a number of experiments thatdiscredit the JGA for statistics at small-length scales. Frenkiel and Klebano� (1967, 1973) comparedmeasured higher-order (including fourth-order) statistics of velocity uctuations with predictions fromjoint-Gaussian probability, thereby quantifying the departure from the JGA for their data. Batchelor(1951) and Van Atta and Wyngaard (1975) noted that the JGA cannot be precisely correct becauseit predicts that odd-order moments are zero. However, the JGA continues to be of interest forsimpli�cation of fourth-order velocity moments (e.g., Zhou and Rubinstein, 1996).An alternative assumption to the JGA is that volume-averaged velocity is statistically independent

of local velocity di�erence where the averaging volume has a size comparable with this spacingand contains the two points. This assumption seems plausible only if the turbulence is locallyhomogeneous. In contrast, for inhomogeneous turbulence, such as the case of the edge of a jet,it is clear that there is statistical dependence of volume-averaged velocity and velocity di�erence.Our subject is fourth-order velocity statistics; statistical independence is used for the insight it cangive about fourth-order statistics. We do not consider the consistent use of statistical independencehypotheses for other purposes.Although the above assumption involves volume-averaged velocities, to produce tractable results

we simplify by using two-point velocity averages. The two-point velocity di�erence is de�ned by

�j ≡ uj − u′j; (1)

where primed and unprimed velocities uj and u′j refer to velocities at spatial points x and x′,

respectively. Following Frenkiel and Klebano� (1967, 1973) and Praskovsky et al. (1993), uj andu′j are uctuations of velocity with zero mean. The two-point velocity average is de�ned by

ai ≡ 12(ui + u′i): (2)

Solving Eqs. (1) and (2) for ui and u′i , one can then express any two-point fourth-order velocitystatistic in terms of sums of products involving di�erences (�’s) and two-point averages (a’s). Anassumption of statistical independence of ai and �j then expresses the fourth-order statistic in termsof lower-order statistics. We abbreviate this statistical independence assumption as SIA.

R.J. Hill, J.M. Wilczak / Fluid Dynamics Research 28 (2001) 1–22 3

An assumption similar to the SIA is that velocities and derivatives are statistically independentfor very large Reynolds numbers. In fact, SIA implies this latter assumption because if x′ ap-proaches x, then ai approaches the single-point velocity, and �j=|x−x′| becomes a derivative. Lumley(1965) and Wyngaard and Cli�ord (1977) used the assumption of statistical independence of veloc-ity and velocity derivatives to obtain corrections to Taylor’s frozen- ow hypothesis, and Tennekes(1975) used statistical independence of large- and small-scale velocities to derive the Eulerian timemicroscale. Praskovsky et al. (1993) used an assumption of statistical independence that di�ers fromSIA in an essential manner, which we discuss in Section 6.The subjects of this paper are fourth-order velocity statistics, not the JGA and SIA in themselves.

The JGA and SIA are useful tools because they give scaling predictions for these statistics in termsof the velocity covariance and the second-order velocity structure function. These scaling predictionsare not speci�c to the inertial range, or any other range of spacings. As we will show, these scalingpredictions give insight into the nature of the fourth-order statistics. Evaluating the accuracy of theJGA and SIA for statistics of all orders is not the goal of this paper.Van Atta and Wyngaard (1975) and Dutton and Deaven (1972) show that higher-order spectra of

the streamwise velocity component have a k−5=3 power law in the inertial range. Praskovsky et al.(1993) show a r2=3 inertial-range power law for higher-order structure functions of the streamwisecomponent, which agrees with the observation by Van Atta and Wyngaard, and the same r2=3 powerlaw for higher-order structure functions composed entirely of one transverse velocity component.Mixed higher-order structure functions, i.e., those composed of both transverse velocity componentsor of streamwise and one transverse component have not been studied previously. Data presented hereshows that mixed higher-order functions, other than those that we call the anomalous components,behave like the others such that the expectation that they too have the r2=3 power law has support,but is not yet proved. We do not investigate power-law exponents, so we do not show compensatedpower-law plots. There is not a strong need for such further investigation given the previous studiesof higher-order statistics, but we do note in Section 6 the implications of those previously establishedexponents for inertial range power laws of the statistics that we study.In contrast to the previous studies, we formulate our fourth-order statistics as tensors. This has

several advantages. For example, the incompressibility conditions on the statistical tensors as well asthe algebraic relationships between the tensors as well as their relationships to other statistics (such asthe pressure structure function) can be used to critically examine the SIA and JGA. Further, we canidentify and investigate all tensor components of interest (an example being those components thatare nonzero under isotropy), and isotropic relationships between tensor components yield restrictionson the SIA formulas.In Section 2, we de�ne the fourth-order statistics, and we give the algebraic relationships between

them. In Section 3, we give the predictions of the JGA and SIA for the fourth-order statistics. We alsodiscuss in Section 3 which statistics do not obey local isotropy if the scaling relationships suggestedby the JGA and SIA are found to agree with experiment. (Appendix A states relationships obtainedby combining local isotropy of Dij(r) with the JGA and SIA. Appendix B gives the derivation ofthe SIA formulas.) In Section 4, we present experimental evidence for the scaling predicted by theJGA and SIA; nearly isotropic data from a wind tunnel is used. Section 5 emphasizes the specialproperties of the anomalous components. Comparison with previous studies is given in Section 6.Section 7 gives our conclusions.


2. De�nitions and relationships

From points x and x′, we have the vector spacing r = x − x′. For homogeneous turbulence,statistics depend only on r, whereas for inhomogeneous turbulence, statistics depend on r and x. Forsimplicity, we always suppress the argument x, even though our results apply in the inhomogeneouscase.The fourth-order velocity correlation is de�ned by

Rijkl(r) ≡ 〈ui uj u′k u′l〉: (3)

Angle brackets denote an average. We consider the fourth-order structure functions,

Sijkl(r) ≡ 〈(uiuj − u′iu′j)(ukul − u′ku′l)〉 (4)

and

Dijkl(r) ≡ 〈(ui − u′i)(uj − u′j)(uk − u′k)(ul − u′l)〉; (5)

as well as the statistics,

Bijkl(r)≡ 〈(ui − u′i)ujukul〉+ 〈(u′i − ui)u′ju′ku′l〉= 〈(ui − u′i)(ujukul − u′ju′ku′l)〉 (6)

and

Mijkl(r) ≡ Bijkl(r) + Bjikl(r) + Bkijl(r) + Blijk(r): (7)

The statistics, Eqs. (4)–(7), have re ection symmetry; that is, they are unchanged by the interchangeof primed and unprimed points. Clearly, Sijkl(r) and Rijkl(r) are related by

Sijkl(r) = 〈uiujukul〉+ 〈u′iu′ju′ku′l〉 − Rijkl(r)− Rklij(r); (8)

which is an identity because it is the average of an algebraic identity. An algebraic identity is

Dijkl(r) =−Sijkl(r)− Sikjl(r)− Siljk(r) +Mijkl(r): (9)

No assumptions are used to obtain Eqs. (8) and (9); they are identities. Any data used to evaluateboth sides of Eqs. (8) and (9) must produce equality to within computer round o�. The statistic

Wikjl(r) ≡ 〈ui(uj − u′j)uk(ul − u′l)〉+ 〈u′i(uj − u′j)u′k(ul − u′l)〉 (10)

is related to an algebraic combination of Eqs. (4) and (5).For future reference, we de�ne the velocity covariance as

�ij ≡ 12(〈uiuj〉+ 〈u′iu′j〉): (11)

The second-order velocity structure function is

Dij(r) ≡ 〈(ui − u′i)(uj − u′j)〉: (12)

We use the preferred coordinate system, which is Cartesian with its 1-axis aligned along the sep-aration vector r. When we refer to speci�c components of the tensors, such as S1111; S1122; D22; D33,we imply that these components are taken along axes of the preferred coordinate system. Greekindices are used to denote a general index for a component resolved in the preferred coordinatesystem [e.g., S��(r)]. No summation is implied by repeated Greek indices.


3. The r-dependence of fourth-order velocity statistics and JGA and SIA scaling relationships

We present relationships derived from the JGA and SIA; we emphasize that those relationshipsare not limited to the inertial range. We brie y describe the inertial-range r-dependence predictedby those relationships. We then discuss which statistics do not obey local isotropy if the scalingrelationships agree with experiment.Quantities that are derived using the JGA or SIA are denoted by the superscript JG or SI, re-

spectively. The JGA formulas for the fourth-order statistics of interest are easily derived from themoment-generating function of the joint Gaussian distribution, which is in Section 8:3 of Batchelor(1960). They are

SJGijkl(r) = �ilDjk(r) + �jkDil(r) + �ikDjl(r) + �jlDik(r)

− 12[Dil(r)Djk(r) + Dik(r)Djl(r)]; (13a)

BJGijkl(r) = �klDij(r) + �jlDik(r) + �jkDil(r); (13b)

M JGijkl(r) = 2[�ijDkl(r) + �klDij(r) + �jlDik(r)

+�ikDjl(r) + �jkDil(r) + �ilDjk(r)]; (13c)

DJGijkl(r) = Dik(r)Djl(r) + Dil(r)Djk(r) + Dij(r)Dkl(r): (14)

Also, RJGijkl (r) follows immediately from Eqs. (8) and (13a) using RJGijkl (0)=�ij�kl+�ik�jl+�il�jk . The

simpli�cations of Eqs. (13a)–(13c), (14) based on local isotropy of Dij(r) are given in Appendix A,Eqs. (A.2a)–(A.2i). We have expressed these statistics in terms of Dij(r) rather than the velocitycorrelation tensor because, for r in the viscous or inertial ranges, the order of magnitude of the termsis immediately apparent in Eqs. (13a)–(13c) and (14) from use of the viscous-range formula forDij(r) or its inertial-range formula (i.e., D��(r)˙ �2=3r2=3�� substituted into Eqs. (A.2a)–(A.2i)).Corresponding formulas from the SIA are derived in Appendix B. First, algebra is used to express

the de�nitions in Eqs. (4) and (6) in terms of �j and ai by substituting Eqs. (1) and (2) intoEqs. (4) and (6); next, the only aspect of the SIA used is 〈aiaj�k�l〉 = 〈aiaj〉〈�k�l〉. We examinein Appendix B the incompressibility conditions that Bijkl(r) and Mijkl(r) must satisfy [Eqs. (B.11),(B.12)]. Some terms must be neglected from the SIA predictions for Bijkl(r) and Mijkl(r) to satisfy theincompressibility conditions [compare Eqs. (B.9), (B.10) with (B.17), (B.18)]. These neglected termsare small at all r relative to the total BSIijkl(r) and M

SIijkl(r), and we have veri�ed that discarding them

makes no di�erence in our �gures presented here. When these small terms are neglected, Eq. (9) isno longer satis�ed, whereas the SIA did and must satisfy Eq. (9). Thus, as suggested by Eq. (9) andas discussed in Appendix B, one-third of the small terms neglected from M SI

ijkl(r) must be subtractedfrom SSIijkl(r) such that Eq. (9) is once again exact for the SIA. Subtracting these small terms makesno noticeable di�erence in our �gures that show SSIijkl(r), with the exception of components of Sijkl(r)that we will name the anomalous components. For the anomalous components, the modi�cation ofSSIijkl(r) leads to an interesting observation. As shown in Appendix B, the modi�cation to enforce theincompressibility condition on Bijkl(r) and Mijkl(r) causes the fourth-order divergence of SSIijkl(r) toresult in the exact pressure structure function. In contrast, the fourth-order divergence of Eq. (13a)


produces the pressure structure function in the JGA as given by Obukhov (1949), Obukhov andYaglom (1951), and Batchelor (1951). From Appendix B, the SIA formulas, including the bene�cialmodi�cations based on enforcing the incompressibility condition, are

SSIijkl(r) = �ilDjk(r) + �jkDil(r) + �ikDjl(r) + �jlDik(r)

− 16[Dil(r)Djk(r) + Dik(r)Djl(r)− 2Dij(r)Dkl(r) + 2Dijkl(r)]; (15)

BSIijkl(r) = BJGijkl(r) ≡ BJGSIijkl (r); (16)

M SIijkl(r) =M

JGijkl(r) ≡ M JGSI

ijkl (r): (17)

The superscript JGSI in Eqs. (16) and (17) emphasizes that Eqs. (13b), (13c) apply to both as-sumptions, and we use JGA and SIA to refer to both assumptions. Of course, the SIA does notsimplify Dijkl(r) because Dijkl(r) is the average of �i�j�k�l and cannot be expressed in terms of theai by means of algebra (other than by the trivial act of introducing terms that sum to zero). Unlikethe JGA, the SIA does not predict Eq. (14); thus SIA escapes demonstrable incompatibility withintermittency theory because the SIA does not predict a simpli�ed formula for Dijkl(r). On the otherhand, it is easily shown that components of DJGijkl(r) have the opposite sign of intermittency exponentrelative to their established exponent for the inertial range. Therefore, the JGA is incompatible withthe intermittency theory, as is well known.Consider SSIijkl(r), S

JGijkl(r), B

JGSIijkl (r), and M

JGSIijkl (r). The presence of �ij in Eqs. (13a)–(13c) and (15)

suggests that these tensors do not obey local isotropy unless, of course, isotropy holds at all scalessuch that �ij is an isotropic tensor; this is examined in more detail in Appendix A. Also, Eqs. (13a)–(13c) and (15) predict that those components of SSIijkl(r), S

JGijkl(r), B

JGSIijkl (r), and M

JGSIijkl (r) that are

nonzero under local isotropy are proportional to �2=3r2=3 in an inertial range [that is, proportional toa diagonal component of Dij(r)], as well as proportional to either one diagonal component of �ij orto a weighted sum of two diagonal components of �ij (see Appendix A). An exception is SJG��(r)for � 6= � (no sum on Greek indices); it is zero under the combination of isotropy and the JGA,but S��(r) for � 6= � is not required to be zero under either isotropy alone or the JGA alone. Theother exception is SSI��(r) for � 6= � [see Eq. (A.1b)].For the case of very large Reynolds numbers and very small spacings, the last terms (in square

brackets) in Eqs. (13a) and (15) are very much smaller than the terms that are proportional tovelocity covariance. Comparing Eq. (13a) with Eq. (15) we see that SSIijkl(r) equals S

JGijkl(r) in this

asymptotic case, with one exception. For the anomalous components and locally isotropic turbulencethe terms in Eqs. (13a) and (15) that are proportional to velocity covariance vanish [Eqs. (A.1b),(A.2b)]; the JGA and SIA predictions for the anomalous components therefore di�er even in theaforementioned asymptotic case.

4. Experimental results: grid turbulence

We evaluate the JGA and SIA expressions using nearly isotropic turbulence data obtained from gridturbulence produced in a wind tunnel. The data, tunnel, and grid are described by Thoroddsen (1995).An X-con�guration hot-wire anemometer produced the streamwise velocity component and one


Table 1Values of � in 10−4 m2 s−3, R�T (dimensionless), U in m s−1, and �ij in m2 s−2 for the gridturbulence data

� R�T U �11 �12 �22

3700 208 10.1 0.135 −0.0036 0.119

cross-stream velocity component. These are assigned the subscripts �=1 and �=2, respectively. En-ergy dissipation rate, Reynolds number, mean velocity, and velocity covariance are given in Table 1.The Kolmogorov microscale � was 0.31 mm, and the integral scale was 108 mm (Thoroddsen, 1995).The hot wires were 1.2 mm long and the signals were �ltered at 5 kHz, so the spatial scale over whichvelocity was averaged is about 1 mm. The data were sampled at 10 kHz, so, using Taylor’s hypothe-sis with the mean velocity of 10:1 m s−1, the samples are spaced 1 mm in the streamwise direction.The velocity covariances in Table 1 show that the grid turbulence was nearly isotropic. We

performed several tests of isotropy. For instance, the numerators of the ratios D12(r)=D11(r), D1112(r)=D1111(r), and D1222(r)=D1111(r) should be zero in isotropic turbulence, and for local isotropy theyshould approach zero as r decreases. These ratios were less than 0.02 for r ¡ 100 mm, with theexception of a rapid decrease for r ¡ 3 mm caused by spatial averaging. For isotropic turbulence(or for local isotropy as r decreases), the numerator of the ratio [(r=2)(dD11(r)=dr) + D22(r) −D11(r)]=D11(r) should vanish on the basis of incompressibility. For 7 mm¡r¡ 58 mm, this ratiowas less than 0.015. There is a systematic increase of the ratio as r increases further, and the ratiois 0.065 at the integral scale of r = 108 mm. For r ¡ 5 mm, there is a rapid decrease of this ratioto a minimum of −0:16 at r = 2 mm, presumably caused by spatial averaging.Statistics were corrected for inaccuracy of Taylors’s hypothesis using the equations in Hill (1996b).

The correction was found to be unnoticeable for all statistics, so uncorrected statistics are shown.Estimates of errors caused by wire orientation and miscalibration showed these errors to be negli-gible. The statistical reliability of all our statistics was determined by calculating joint probabilitydistributions and graphing the integrands that produce the statistics by integration of a function mul-tiplying the joint probability distribution. For instance, the integrand for D1122(r) at a given r is(u1 − u′1)2(u2 − u′2)2 times the joint probability distribution of (u1 − u′1) and (u2 − u′2). There was nosystematic underestimation of any of the statistics.The power spectra of both velocity components [shown in Thoroddsen (1995)] have power laws

extending over slightly more than a decade in wave number (Thoroddsen, 1995); the power-lawexponents are shallower than −5=3; they are about −5=3+0:19. This deviation from −5=3 is expectedfor R� = 208, in comparison with measurements by Mydlarski and Warhaft (1996) and with slopesof velocity spectra collected in the �fth �gure of Sreenivasan (1996). Our R� value is in the rangethat Mydlarski and Warhaft (1996) characterize as being strong turbulence for which the probabilitydensity of velocity-component di�erences are super Gaussian and for which they show that theirdata is consistent with Kolmogorov’s re�ned similarity hypothesis and for which they establishthat the power-law scaling is well developed and approaches 5=3 as R� increases further. In the gridturbulence used here, the apparent power law holds over a factor of 15 in spacing and wave number,so there is signi�cant separation of energy-containing and dissipation scales.In Fig. 1, we show D1111(r) and D11(r) and the 4=3 and 2=3 power laws to demonstrate the

shallowness of the power law in these structure functions and the factor of about 15 in the power-law


Fig. 1. The fourth-order functions D1111(r); S1111(r); and M1111(r), evaluated using grid turbulence data (units are m4 s−4)and the second-order functions D11(r) (units m2 s−2). As indicated, all functions except D1111(r) are multiplied by 10 toincrease their separation from D1111(r). The curves correspond to the functions listed from top to bottom, respectively, onthe ordinate. Lines of slope 2=3 and 4=3 osculate the curves for D11(r) and D1111(r), respectively.

scaling range. We also present in Fig. 1 the higher-order functions S1111(r) and M1111(r) to illustratetheir magnitude and shape. Of the statistics that we studied, only components of Dijkl(r) and Dij(r)obey Kolmogorov scaling; this will become evident. The ordinate in Fig. 1 is intentionally not scaled,partly because it would be misleading to use Kolmogorov scaling for quantities S1111(r) and M1111(r)that do not scale in that manner.We studied four fourth-order structure functions, each having 81 components. Presenting data

for all these components is excessive. Therefore, we present data for only those components ofSijkl(r); Bijkl(r), and Mijkl(r) that are nonzero under the assumption of local isotropy of Dij(r) asgiven in Appendix A. Of course, this does not mean that we use isotropy. We avoid repetition; forinstance, 4B��(r) =M��(r), so separate graphs for B��(r) are not given.We examine the accuracy of the JGA and SIA formulas by graphing the ratio of each fourth-order

component to its JGA formulas [Eqs. (13a)–(13c) and (14)] and to its SIA formulas [Eqs. (13b),(13c) and (15)–(17)]. The symbol G denotes that the graphed quantity shows the accuracy of theJGA, and for Bijkl(r) and Mijkl(r), it shows the accuracy of the SIA as well [see Eqs. (16) and (17)].Speci�cally, we take � 6= � and de�ne

GS��(r) ≡ S��(r)={4��D��(r)− [D��(r)]2}; (18a)

GS��(r) ≡ S��(r)/{

��D��(r) + ��D��(r) + 2��D��(r)− 12D��(r)D��(r) + [D��(r)]

2};

(18b)

GS��(r) ≡ S��(r)={4��D��(r)− [D��(r)]2}; (18c)


Fig. 2. Comparison of M1111(r); M1122(r); M2222(r), and B1122(r) evaluated using grid turbulence data (long-dashed,medium-dashed, short-dashed, and solid curves, respectively), with their JGA and SIA predictions by plotting theratios in Eqs. (18d)–(18f ).

GB��(r) ≡ B��(r)=[��D��(r) + 2��D��(r)]; (18d)

GM��(r) ≡ M��(r)= [12��D��(r)] ; (18e)

GM��(r) ≡ M��(r)= {2[��D��(r) + ��D��(r) + 4��D��(r)]} : (18f)

The symbol I denotes that the graphed quantity shows the accuracy of SSIijkl(r); that is,

IS��(r) = S��(r)=SSI��(r) = S��(r)/[4��D��(r)− 1

3D��(r)

]; (19a)

IS��(r) = S��(r)=[��D��(r) + ��D��(r) + 2��D��(r)

−16{D��(r)D��(r)− [D��(r)]2 + 2D��(r)}

]; (19b)

IS��(r) = S��(r)/[4��D��(r)− 1

3{[D��(r)]2 − D��(r)D��(r) + D��(r)}

]: (19c)

We substitute our measured statistics into Eqs. (18a)–(18f) and (19a)–(19c) and graph the results.The denominators in Eqs. (18a)–(18f) and (19a)–(19c) are the scaling predictions of the JGA

and SIA.Fig. 2 shows GM1111(r); GM1122(r); GM2222(r), and GB1122(r). These quantities are con�ned to nearly

10% of unity, thereby showing agreement with JGA and SIA scaling predictions. One notices,however, an increasing departure from a constant value for r ¡ 10−1 m in Fig. 2. Such increasingvariation with r indicates some violation of the JGA and SIA scaling with D��(r) that appears inthe denominators of Eqs. (18d)–(18f). Because of the good local isotropy of the grid turbulence,


Fig. 3. Comparison of S1111(r); S1212(r), and S2222(r) with their JGA predictions (solid curves) and SIA predictions (dashedcurves). Thin curves are GS1111(r) and IS1111(r) from Eqs. (18a) and (19a). Medium curves are GS1212(r) and IS1212(r)from Eqs. (18b) and (19b). Thick curves are GS2222(r) and IS2222(r) from Eqs. (18a) and (19a).

if the terms D��(r) [which would be zero for local isotropy, see Eqs. (A.2e)–(A.2g)] are deletedfrom Eqs. (18d), (18f), then the resultant curves would be indistinguishable from the correspondingcurves graphed in Fig. 2.Fig. 3 shows GS1111(r); GS1212(r); GS2222(r); IS1111(r); IS1212(r), and IS2222(r). Similar to Fig. 2,

Fig. 3 shows systematic deviation from a constant for r ¡ 10−1 m, which has the same implicationsregarding scaling with D��(r) as described for Fig. 2. However, the curves in Fig. 3 are everywherewithin 10% of unity, thereby showing the level of agreement with JGA and SIA scaling predictions.If the terms D��(r) were deleted from the expressions for GS1212(r) and IS1212(r) in Eqs. (18b) and(19b), then the resultant curves would be indistinguishable from the corresponding curves graphedin Fig. 3. As in the case of Fig. 2, this results from the good local isotropy of the grid turbulence.Fig. 4 shows S1122(r) and its JGA and SIA scaling predictions [which are the denominators of

Eqs. (18c) and (19c), respectively, as obtained from Eqs. (13a) and (15)]. Also shown is the SIA pre-diction for locally isotropic turbulence as given in Eq. (A.1b), i.e., (−1=3)[D��(r)−D��(r)D��(r)],as obtained by neglecting those terms in Eq. (19c) containing D��(r) for � 6= �. There is littledistinction between the full SIA prediction and its counterpart for locally isotropic turbulence [Eq.(A.1b)]; this attests to the local isotropy of the grid-generated turbulence. As mentioned, the JGAprediction vanishes in the isotropic case. In Fig. 4, the JGA curve is multiplied by 4 to raise it onthe graph; clearly, the JGA prediction is much smaller in magnitude than the data and the SIA pre-diction. At r=0:007 m, the JGA prediction has a local minimum (not a zero crossing) that is beyondthe range of the ordinate. The JGA prediction is positive everywhere in Fig. 4, whereas S1122(r),the SIA prediction, and the SIA’s locally isotropic approximation Eq. (A.1b) are all negative (theiradditive inverses are plotted).In addition, we have examined three-axis sonic anemometer data from a height of 7 m in the

atmospheric surface layer, which data are not shown here for brevity. That data set consists of


Fig. 4. The solid curve is −S1122(r). The long-dashed curve is the negative of the SIA prediction in the denominatorof Eq. (19c). The short-dashed curve is the negative of the local isotropy simpli�cation for the SIA in Eq. (A:1b) (i.e.[D1122(r) − D11(r)D22(r)]=3). The medium-dashed curve is 4 times the JGA prediction in the denominator of Eq. (18c)(i.e., 4{4�12D12(r)− [D12(r)2]}).

three cases of very anisotropic turbulence and distinctly di�erent anisotropy for each case, and islimited to 0:8 m¡r¡ 90, which range includes the large-scale end of the inertial range (r ¡ 3:5 m)through the energy containing range. Our comparison of the grid turbulence data with the atmosphericdata for the statistics S��(r); S��(r); M��(r); M��(r); B��(r), and B��(r) [the latter equalsM��(r)=4] supports the JGA and SIA scaling for the following reasons. The anisotropy checks thescaling even when contributions from �� and D��(r) (for � 6= �) in Eqs. (18b)–(18f), (19b)–(19c)are signi�cant. The grid turbulence values of �11 and �22 lie between those of the atmospheric cases,whereas the value of � for the grid turbulence is 168 to 15 times larger than for the atmosphericcases. Nevertheless, the scaling with the JGA and SIA predictions holds for both atmospheric andgrid turbulence. In order words, the scaling has been checked for cases having very di�erent Reynoldsnumbers but comparable �11; �22, and �33.

5. The anomalous components of Sijkl (r)

We have seen that the components S��(r)= 〈(u2�− u′2� )2〉 and S��(r)= 〈(u�u�− u′�u′�)2〉 are wellpredicted by the JGA and SIA. In contrast, the components S��(r)=〈(u2�−u′2� )(u2�−u′2� )〉 are in verypoor agreement with the JGA. In addition, the components S��(r) and S��(r) are much greater inmagnitude (by factors of 50–30 for the grid turbulence) than the S��(r). For these reasons, we namethe components S��(r) the anomalous components. These anomalous components are required forthe use of velocity data in the evaluation of expressions for the pressure–velocity–velocity statistic(Hill, 1996a).The anomalous components have the distinction of being zero under combined application

of isotropy and the JGA. Fig. 4 shows that the anomalous components are well represented by


(−1=3) [D1122(r)− DJG1122(r)], which is a measure of the deviation of velocity di�erences from jointGaussian random variables and is deduced from the SIA [cf. Eqs. (A.1b) and (A.2i)]. Deviation fromGaussian probability is a characteristic of the intermittency phenomenon. This makes the anomalouscomponents interesting and worthy of further study because their entire value in isotropic turbulenceis a characteristic of intermittency. Other statistics typically used to study intermittency have a portionof their power law attributed to the intermittency phenomenon. Perhaps there are other higher-orderstatistics that are useful for study of intermittency because they too vanish under the combined as-sumptions of isotropy and joint Gaussian probability, but do not vanish if either assumption is takenseparately. In contrast to the usefulness of anomalous components for quantifying intermittency ofisotropic turbulence, our examination (not shown here) of anisotropic atmospheric data con�rmsthe expectation that the anomalous components have values representative of anisotropy for veryanisotropic turbulence as opposed to values indicative of intermittency for isotropic turbulence. Weexpect that if the JGA denominator in Eq. (18c) is comparable or larger than S��(r), then theanomalous components will exhibit anisotropy e�ects.

6. Comparison with previous work

Van Atta and Wyngaard (1975) considered “higher-order” velocity correlation functions and theirspectra. They limited their study to the streamwise velocity component; this corresponds to settingi= j= k= l=1. They presented data from both a laboratory mixing layer and from the atmosphere;that data scaled as predicted by the JGA. The atmospheric data showed that higher-order spectra ofthe longitudinal velocity component vary as k−5=3 in the inertial range. This corresponds to S1111(r)being proportional to r2=3. Recently, Sanada and Shanmugasundaram (1992) used direct numericalsimulation of the Navier–Stokes equations to show results in agreement with the experimental resultsof Van Atta and Wyngaard (1975).Using dimensional analysis, Van Atta and Wyngaard (1975) deduced that higher-order velocity

spectra vary as k−5=3 in the inertial range. Applicability of their dimensional analysis supposes thatlocal scaling and local isotropy apply to S1111(r). For S1111(r), their result gives

S1111(r)˙ �2�−1=3r2=3; (20)

where �2 is a second-order viscous dissipation rate, which plays a central role in their dimensionalanalysis. Their dimensional analysis is not deterministic because the number of parameters is onemore than the number of applicable dimensions. Thus, as they noted, their analysis suggests butdoes not prove the k−5=3 law for higher-order spectra.Van Atta and Wyngaard (1975) also used the JGA to deduce that certain higher-order spectra

vary as k−5=3. For S1111(r), their result corresponds to

SJG1111(r) = 4�11D11(r)˙ �11�2=3r2=3; (21)

in agreement with a k−5=3 spectrum. The dissipation rate �2 in Eq. (20) is not necessarily proportionalto the velocity variance in Eq. (21). Consequently, in Eqs. (20) and (21), they proposed two mutuallyexclusive theories. If the JGA is presumed to be dubious when applied to the inertial range, it mayseem that Eq. (20) is the more plausible theory. On the other hand, applying the SIA, we obtainEq. (15), which, given local isotropy, supports Eq. (21). For the inertial-range case, the SIA is moreplausible and less restrictive than the JGA, yet the SIA supports the JGA prediction in Eq. (21).


Praskovsky et al. (1993) investigate higher-order structure functions using high-Reynolds-numberdata. They show that those components consisting entirely of either the streamwise component andthose of the cross-stream component obey the 2=3 power law in the inertial range. This agreeswith the �nding by Van Atta and Wyngaard (1975) and extends it to cross-stream components,although mixed components were not investigated. These studies show that S��(r) has the 2=3inertial-range power law for both streamwise and transverse components. Because D��(r) is muchless than either M��(r) or S��(r) in an inertial range, Eq. (9) requires that M��(r) have thesame inertial-range power law as S��(r). Thus, these inertial-range slopes are not in doubt despitethe moderate Reynolds number of our data. Now consider the mixed components; because bothD��(r) and the anomalous components S��(r) are much less than either M��(r) or S��(r) in aninertial range, Eq. (9) requires that M��(r) has the same inertial-range power law as S��(r). Be-cause our moderate-Reynolds-number data shows that these mixed components behave like M��(r)and S��(r), the data suggests, but does not prove, that M��(r) and S��(r) also have the r2=3

inertial-range power law for high Reynolds numbers.We interpret our results and the experiments by Van Atta and Wyngaard (1975) and Praskovsky

et al. (1993) to mean that measurement of a power law does not, in itself, imply that the statisticobeys local isotropy or local scaling. Nelkin (1994) reaches a similar conclusion.M�unch and Wheelon (1958) derived the Eulerian acceleration correlation on the basis of the

Navier–Stokes equation and the JGA. The JGA is adequate in their application because of theimportant role of energy-containing scales of motion. On this basis, they derived that the “largeeddies pick up and carry the small ones between points separated by a distance 1=k in a timeshort compared with the decay time of the small eddies”. More recently, this property has becomeknown as random sweeping. Tennekes (1975) assumed this random sweeping of small scales by theenergy-containing motions to derive the mean-squared Eulerian acceleration and Eulerian frequencyspectrum of velocity. By relating Tennekes’ result for the mean-squared Eulerian acceleration to thespectrum of stress, Nelkin and Tabor (1990) obtained the spectral equivalent of Eq. (21).In addition to demonstrating the power laws of higher-order structure functions as mentioned

above, Praskovsky et al. (1993) compare their measured structure functions with predictions fromthe assumption that velocity ui at one point and the di�erence (uj− u′j) are statistically independent,and they showed by means of their data that this statistical independence assumption is inaccurate.For this reason we abbreviate it by ISI; which is a mnemonic for inaccurate statistical independenceassumption. Appendix C compares ISI with SIA. Here, we show that the SIA is in better agreementwith the data of Praskovsky et al. (1993) than is the ISI. The aspect of the ISI that they used is that,for any exponents k and p, (ui)k and (�j)p are uncorrelated. Praskovsky et al. (1993) measured〈(umi −u′mi )2〉 for m=1–4 and 〈(ui−u′i)n〉 for n=2–4. For m=2, their former statistic is a componentof Sijkl(r), and for n=4, their latter statistic is a component of Dijkl(r). We need to de�ne the samequantities de�ned by Praskovsky et al. (1993). First, �(X ) is the root-mean-square of the quantity Xappearing in the subscripted parentheses. They de�ne a correlation in their Eq. (9) as follows:

�2;2(r) ≡ 〈(u2� − 〈u2�〉)(�2� − 〈�2�〉)〉�(u2�)�(�2�)

; (22)

wherein r is in the streamwise direction. In Praskovsky et al.’s Eq. (24) they de�ne the quantity

d(2)u� (r) ≡〈[u2� − u′2� ]2〉4〈u2�〉D��(r)

=S��(r)4��D��(r)

: (23)


In their Eqs. (25) and (26) they de�ne the quantity

g(2)u� (r) ≡〈[u2� − u′2� ]2=�2�〉

4〈u2�〉: (24)

On the basis of the SIA, the quantity g(2)ui (r) approaches unity at small scales, as shown in theirFig. 3, because our algebraic identity Eq. (B.5) shows that [u2�− u′2� ]=�2�=4a2�. Thus, assuming localhomogeneity and the SIA, we can apply Eq. (B.3) to obtain

g(2)u� (r) = 1−D��(r)4��

: (25)

That is, on the basis of the SIA, Eq. (25) shows that g(2)ui (r) must decrease with increasing r inan inertial range, which is what Praskovsky et al. observed in their Fig. 3. For the streamwisecomponent, use of the inertial-range formula for D11(r) and tabulated data in Praskovsky et al.(1993) shows that Eq. (25) agrees quantitatively with the data in their Fig. 3.For d(2)u� (r), the expression in Eq. (23) becomes

GS��(r) for 4��/D��(r)

or

IS��(r) for 12��/D��(r):

On that approximate basis we see that our Fig. 3 agrees with their Fig. 2, considering the di�erencein Reynolds numbers between the two data sets. That is, d(2)u� (r) is somewhat greater than unity atsmall r and decreases with increasing r.In contrast, Praskovsky et al. (1993) show that the ISI predicts that both g(2)u� (r) and d

(2)u� (r) are

unity and cannot decrease with increasing r, in contradiction to their data. They correctly concludethat ISI is incorrect. They refer to this aspect of their data as an alarming observation because ofthe implication for the random sweeping hypothesis. Since SIA qualitatively predicts the data, wesurmise that the failure is with the ISI and not with the random sweeping hypothesis.Now, return to their correlation coe�cient Eq. (22). From our SIA result Eq. (B.4) and algebraic

identity Eq. (B.2), we obtain, in their notation,

�2;2�(u2)�(�2) =14{D��(r)− [D��(r)]2}:

In the inertial range, this is a nonzero positive function that decreases with decreasing r, as Praskovskyet al. observed (their Fig. 5). In contrast, Praskovsky et al. (1993) showed that the ISI predicts that�2;2 =0. They interpret this as a violation of the random sweeping hypothesis. Again, the inaccuracyappears to be in the ISI prediction rather than with the random sweeping hypothesis.As emphasized by Praskovsky et al. (1993): “The sweeping hypothesis is based on the assump-

tion of statistical independence of large- and small-scale motion”. We believe that comparison ofhigher-order structure functions and spectra with the predictions of speci�c implementations of thestatistical independence assumption, such as SIA and ISI, show the weaknesses of these imple-mentations rather than a weakness of the random sweeping hypothesis. On this basis, the randomsweeping hypothesis provides no motivation for more detailed comparison of the SIA with the dataof Praskovsky et al. (1993). In e�ect, Lin (1953) demonstrated the approximate validity of Tennekes’(1975) basis for random sweeping, i.e., Tennekes’ Eq. (1).


Praskovsky et al. (1993) and Sreenivasan and Stolovitzky (1996) investigate powers of �i con-ditionally averaged on velocity at a point. The latter authors �nd that conditional averaging on avelocity near an endpoint of the interval x to x′ causes a statistical dependence of �i on the velocity.This is analogous to our �nding in Appendix C as to why ISI is less accurate than SIA.

7. Conclusions

We have investigated the fourth-order statistics Sijkl(r); Bijkl(r), and Mijkl(r). Although the JGAand SIA are not subjects of our study, they are useful for suggesting testable scaling relationships.Experimental evaluation of the JGA and SIA predictions show that Sijkl(r); Bijkl(r), and Mijkl(r)scale with components of the velocity variance; therefore, they do not obey local isotropy unlessthe turbulence is isotropic at all scales, and they do not obey local scaling in any case. With theexception of the anomalous components, the studied components of Sijkl(r); Bijkl(r), and Mijkl(r)follow the scaling predictions of the JGA and SIA reasonably well.The anomalous components, S��(r) for � 6=�, are discussed in Section 6. The anomalous compo-

nents are sensitive characteristics of intermittency for the case of locally isotropic turbulence. On theother hand, they are strongly a�ected by local anisotropy. By applying the SIA to nearly isotropicgrid turbulence data, we found a close correspondence between the anomalous components and(−1=3)[D��(r)−DJG��(r)]; this quantity is a measure of the deviation of velocity di�erences fromjoint Gaussian random variables. Such deviation is a characteristic of intermittent random variables.Components of Dijkl(r) depart greatly from the JGA prediction as spacing is decreased; this fact

has been observed in many previous studies (Frenkiel and Klebano�, 1967,1973; Van Atta andChen, 1970; Dutton and Deaven, 1972). However, the SIA does not predict a simpli�ed formula forDijkl(r), so the SIA is not contradicted by measurements of Dijkl(r).On the basis of this study, and in agreement with Nelkin (1994) and Praskovsky et al. (1993),

we believe that local isotropy and local scaling should be applied only to statistics composed en-tirely of di�erences of the basic hydrodynamic quantities; such quantities are velocity, pressure, andtemperature. For instance, even di�erences of products of these basic quantities, such as appear inSijkl(r), make local isotropy and local scaling inapplicable. Kolmogorov’s (1941,1962) hypotheses,as well as dimensional analysis based on local scaling, should be applied only to statistics composedentirely of di�erences of the basic quantities.We conclude that dimensional analysis based on local properties is not applicable to the tensors

Sijkl(r); Bijkl(r), and Mijkl(r), or their spectra. Thus, we agree with Dutton and Deaven (1972) thattheir extension of Kolmogorov’s (1941) dimensional analysis is not applicable. Similarly, we agreewith the statement by Nelkin and Tabor (1990) that “higher-order” correlations are not to be de-termined from dimensional analysis. In disagreement with Van Atta and Wyngaard (1975), we �ndthat dimensional analysis based on higher-order dissipation rates is inapplicable.We �nd that some statistics are well approximated by use of the JGA and SIA [speci�cally,

S��(r); S��(r); B��(r); B��(r); M��(r); M��(r), and probably their higher-order analoguesas well], whereas others are well known to be approximated by use of dimensional analysis withintermittency corrections [speci�cally, D��(r); D��(r), etc., and their higher-order analogues], andyet others are approximated by the SIA but not by the JGA [speci�cally, S��(r) and higher-orderanalogues]. The latter statistics are called anomalous.


We give the �rst study of the statistical tensors Bijkl(r) and Mijkl(r), and of the mixed compo-nents of Sijkl(r). These tensors appear in the relationship between pressure statistics and fourth-ordervelocity statistics. We give the �rst identi�cation of the existence of anomalous components and theirproperties. We suggest that other higher-order statistical tensors also have anomalous components.The SIA, as we de�ne it, has never been used before. Despite its simplicity, the SIA is found to besuperior to the JGA for predicting fourth-order velocity statistics. For the purpose of critically ex-amining the predictions of the SIA or JGA, our analysis contains the �rst use of the tensor characterof our statistics in combination with the relationships between the tensors and their relationship tothe pressure statistics and their incompressibility conditions.

Acknowledgements

The authors thank S. Thoroddsen for use of his data and W. Otto for help with the calculations.This work was partially supported by ONR Contract nos. N00014-93-F-0038 and N00014-93-F-0047.

Appendix A Isotropic relationships

Using the isotropic formula for Dij(r), we give the following simpli�ed results for the SIA andJGA formulas [Eqs. (13a)–(17)]:

SSI��(r) = 4��D��(r)− 13D��(r); (A.1a)

SSI��(r) =13 [D��(r)D��(r)− D��(r)] for � 6= �; (A.1b)

SSI��(r) = 4��D��(r) + ��D��(r)− 16 [D��(r)D��(r) + 2D��(r)]; (A.1c)

SJG��(r) = 4��D��(r)− [D��(r)]2; (A.2a)

SJG��(r) = 0 for � 6= �; (A.2b)

SJG��(r) = ��D��(r) + ��D��(r)− 12D��(r)D��(r) for � 6= �; (A.2c)

BJGSI��(r) = 3��D��(r); (A.2d)

BJGSI��(r) = ��D��(r) for � 6= �; (A.2e)

M JGSI�� (r) = 12��D��(r); (A.2f)

M JGSI��(r) = 2[��D��(r) + ��D��(r)] for � 6= �; (A.2g)

DJG��(r) = 3[D��(r)]2; (A.2h)

DJG��(r) = D��(r)D��(r) for � 6= �: (A.2i)

All other components of these tensors vanish by isotropy, but not by assuming local isotropy forDij(r) alone. Eqs. (A.2d)–(A.2g) apply to both the JGA and SIA, as indicated by the superscript


JGSI. We used only local isotropy of Dij(r) to obtain Eqs. (A.1a)–(A.1c) and (A.2a)–(A.2i), notisotropy of �ij. That is, Eqs. (A.1a)–(A.1c) and (A.2a)–(A.2i) can be obtained from the denomi-nators in Eqs. (18a)–(18f) and (19a)–(19c) with D��(r) set to zero for � 6= �. Isotropy gives therelationships

S��(r) = S�� (r) + 2S� � (r); (A.3a)

B��(r) = 3�� (r); (A.3b)

M��(r) = 3M�� (r); (A.3c)

3D�� (r) = D (r); (A.3d)

where and � are 2 or 3, but not 1, and 6= �. Substituting Eqs. (A.1a)–(A.1c) and (A.2a)–(A.2g) into Eqs. (A.3a)–(A.3c) requires that �22 = �33. Thus, the tensors SSIijkl(r); S

JGijkl(r); B

JGSIijkl (r),

and M JGSIijkl (r) obey local isotropy only if �ij obeys isotropy; that is, isotropy must hold at all scales.

On the other hand, Dijkl(r) and DJGijkl(r) do obey local isotropy.

Appendix B. Formulas from the SIA

Formulas for SSIijkl(r); BSIijkl(r), and M

SIijkl(r) are derived on the basis that the average and di�erence

of velocities are statistically independent. Eq. (1) de�nes the two-point velocity di�erence as �j ≡uj − u′j, and Eq. (2) de�nes the two-point average as ai ≡ (1=2)(ui + u′i). The assumption that aiand �j are statistically independent is used (i.e., SIA is used). The only necessary assumption forthe results reported here is

〈aiaj�k�l〉= 〈aiaj〉〈�k�l〉: (B.1)

To use the SIA, one must express a given statistic in terms of �j and ai instead of in terms of thevelocity components; only algebra is needed for this purpose. First, we derive SSIijkl(r). An algebraicidentity is

12 (uiuj + u

′iu

′j) = aiaj +

14�i�j: (B.2)

Averaging Eq. (B.2) gives

〈aiaj〉= �ij − 14Dij(r): (B.3)

Therefore, Eqs. (B.1) and (B.3) give

〈aiaj�k�l〉= �ijDkl(r)− 14Dij(r)Dkl(r): (B.4)

An algebraic identity is

uiuj − u′iu′j = ai�j + aj�i: (B.5)

Substituting Eq. (B.5) for the factors in the de�nition [Eq. (4)] of Sijkl(r) gives

Sijkl(r) = 〈aiak�j�l + aial�j�k + ajak�i�l + ajal�i�k〉: (B.6)


Substituting Eq. (B.4) for each term in Eq. (B.6) gives

SSIijkl(r) = �ikDjl(r) + �jlDik(r) + �ilDjk(r) + �jkDil(r)

− 12 [Dik(r)Djl(r) + Dil(r)Djk(r)]: (B.7)

Next, we derive BSIijkl(r). An algebraic identity is

ujukul − u′ju′ku′l = ajak�l + akal�j + alaj�k + 14�j�k�l: (B.8)

Multiplying Eq. (B.8) by �i, averaging, and substituting Eq. (B.4) gives

BSIijkl(r) = �klDij(r) + �jlDik(r) + �jkDil(r) +14Tijkl(r); (B.9)

where, for simplicity of notation, we de�ne

Tijkl(r) ≡ [Dijkl(r)− DJGijkl(r)]:Recall from Eq. (14) that DJGijkl(r) = Dik(r)Djl(r) + Dil(r)Djk(r) + Dij(r)Dkl(r).To obtain M SI

ijkl(r), we need only substitute its de�nition, Eq. (7), into Eq. (B.9):

M SIijkl(r) = 2[�ijDkl(r) + �klDij(r) + �jlDik(r)

+�ikDjl(r) + �jkDil(r) + �ilDjk(r)] + Tijkl(r): (B.10)

Assuming local homogeneity and incompressibility, Eq. (6) gives (Hill and Wilczak, 1995)

Bijkl(r)|i = 0; (B.11)

and from Eqs. (7) and (B.11) (Hill and Wilczak, 1995),

Mijkl(r)|ijkl = 0: (B.12)

Performing the fourth-order divergence of Eq. (8) and assuming homogeneity gives (Hill andWilczak, 1995)

Sijkl(r)|ijkl =−2Rijkl(r)|ijkl: (B.13)

Performing the fourth-order divergence of Eq. (9) and substituting Eqs. (B.12) and (B.13) gives(Hill and Wilczak, 1995)

Dijkl(r)|ijkl =−3Sijkl(r)|ijkl (B.14a)

= 6Rijkl(r)|ijkl: (B.14b)

In addition to incompressibility, Eqs. (B.11), (B.12), and (B.14a) require local homogeneity, whileEqs. (B.13) and (B.14b) require the more restrictive assumption of homogeneity.From Eqs. (B.9) and (B.10), we have

BSIijkl(r)|i =18[Tijkl(r)|i] =

18 [Dijkl(r)|i − DJGijkl(r)|i] (B.15a)

and

M SIijkl(r)|ijkl = Tijkl(r)|ijkl = Dijkl(r)|ijkl − DJGijkl(r)|ijkl; (B.15b)

wherein the JGA result (Hill, 1994) is

DJGijkl(r)|ijkl = 3Dik(r)|jlDjl(r)|ik : (B.16)


Neither Eq. (B.15a) nor Eq. (B.15b) vanishes; therefore, they contradict Eqs. (B.11) and (B.12).Thus, the asymptotically small terms proportional to Tijkl(r) in Eqs. (B.9) and (B.10) cannot becorrect. Therefore, when using the SIA, we �nd that only the asymptotically largest terms shouldbe retained in Eqs. (B.9) and (B.10), so we have

BSIijkl(r) = �klDij(r) + �jlDik(r) + �jkDil(r); (B.17)

M SIijkl(r) = 2[�ijDkl(r) + �klDij(r) + �jlDik(r)

+�ikDjl(r) + �jkDil(r) + �ilDjk(r)]: (B.18)

Now, Eqs. (B.17) and (B.18) obey the incompressibility conditions of Eqs. (B.11) and (B.12).We next examine the accuracy of the asymptotically small terms in Eq. (B.7), i.e., the terms in

the square brackets. Performing the fourth-order divergence of both Eq. (B.7) and the correspondingJGA result, we have

SSIijkl(r)|ijkl = SJGijkl(r)|ijkl =−Dik(r)|jlDjl(r)|ik =− 1

3DJGijkl(r)|ijkl: (B.19)

Only the asymptotically small terms in Eqs. (B.7) and (13a) give Eq. (B.19); the asymptoticallylarge terms vanish by incompressibility.To determine if Eq. (B.16) or if Eq. (B.19) is accurate, we compare each with the exact relation-

ships in Eqs. (B.14a), (B.14b). Any of the three fourth-order divergences in Eqs. (B.14a), (B.14b)can be used to derive the pressure structure function DP(r). Hence, Eqs. (B.16) and (B.19) caneach be used to derive DJGP (r) in the JGA. From Eq. (B.19), both the JGA and SIA give the sameprediction for DJGP (r). The mean-squared pressure gradient � and the pressure-gradient correlationAij(r) are derived from DP(r) (Hill and Wilczak, 1995). Hill and Boratav (1997) compared DP(r)with DJGP (r) as calculated from numerical simulation (R�T = 82) as well as from the grid-turbulencedata used in the present work (R�T = 208). They found that DP(r) is greater than DJGP (r) by abouta factor of 3 for their Reynolds numbers. Hill and Thoroddsen (1997) compared Aij(r) with AJGij (r)using the present grid-turbulence data and found that Aij(r) is about 3AJGij (r) (this indicates that �is about 3�JG at R�T = 208). Hill and Wilczak (1995) showed that DP(r); �, and Aij(r) increaseyet further relative to their JGA counterparts as Reynolds number increases. The error in DP(r)caused by the asymptotically small terms in Eq. (B.7), which terms result in Eq. (B.19), motivatesa modi�cation of those asymptotically small terms in Eq. (B.7).Further motivation for a modi�cation of the asymptotically small terms in Eq. (B.7) is that dis-

carding Tijkl(r) in Eq. (B.10) to obtain Eq. (B.18) contradicts Eq. (9). Substituting Eqs. (B.7) and(B.10) into Eq. (9) gives the identity Dijkl(r)=Dijkl(r), corroborating the fact that the SIA does notpredict Dijkl(r). However, after discarding Tijkl(r) from Eq. (B.10) to obtain Eq. (B.18), substitutionof Eqs. (B.7) and (B.18) into Eq. (9) gives Dijkl(r) =DJGijkl(r). Therefore, enforcing the incompress-ibility condition, Eq. (B.12), requires a corresponding modi�cation to SSIijkl(r) such that Eq. (9) issatis�ed.From the preceding, we determined that (1) the last term in Eq. (B.7) is asymptotically much

smaller than the other terms and must be retained in some form because it produces the pressurestructure function; (2) neglecting the last term in Eq. (B.7) would cause the pressure structurefunction to vanish, and hence the mean-squared pressure gradient would vanish as well; (3) thepressure structure function produced by the last term in Eq. (B.7) coincides with that given by theJGA; and (4) incompressibility combined with Eq. (9) requires a modi�cation of the asymptotically


small terms in Eq. (B.7). This modi�cation is to subtract Tijkl(r)=3 from SSIijkl(r) in Eq. (B.7) andremove Tijkl(r) from Eqs. (B.9) and (B.10) such that Eq. (9) is the identity Dijkl(r) =Dijkl(r). Thatis, we have

SSIijkl(r) = �ilDjk(r) + �jkDil(r) + �ikDjl(r) + �jlDik(r)

− 16 [Dil(r)Djk(r) + Dik(r)Djl(r)− 2Dij(r)Dkl(r) + 2Dijkl(r)]: (B.20)

Now, Eq. (B.20) gives

−3SSIijkl(r)|ijkl = Dijkl(r)|ijkl;which agrees with Eq. (B.14a); therefore, Eq. (B.20) leads to the exact DP(r).As shown above, the fact that the right-hand sides of Eqs. (B.15a), (B.15b) are not zero, as

required by Eqs. (B.11) and (B.12), is an error closely related to the inaccuracy of the JGA inproducing pressure statistics. For instance, using results of Hill (1994) and Hill and Wilczak (1995),we see that the right-hand side of Eq. (B.15b) is (−1=3) of the Laplacian operating twice on[DP(r) − DJGP (r)]. Only the di�erence of the exact and JGA pressure structure functions causes theright-hand side of Eq. (B.15b) to be nonzero. An interesting speculation is that the inaccuracy ofthe asymptotically small terms is caused by use of only a two-point average [Eq. (2)].

Appendix C. Inaccuracy of statistical independence of ui and �j

If one assumes that velocity ui at spatial point x is statistically independent of �j = uj − u′j,then u′i (velocity at spatial point x

′) is statistically dependent on �j. A brief proof is given in thenext paragraph. The assumption that ui is statistically independent of �j is abbreviated ISI. Thepredictions of the ISI for our fourth-order statistics are given and compared with those of the SIA.Recall that ui and u′i are uctuations having zero mean. Statistical independence of ui and �j

implies that 〈ui�j〉= 〈ui〉〈�j〉=0; substituting ui = u′i +�i into 〈ui�j〉=0 gives 〈u′i�j〉=−〈�i�j〉=−Dij(r). Unless there is no turbulence, Dij(r) 6= 0 for all i and j in the general anisotropic case. Forthe case of isotropy at all scales, we have statistical dependence of u′i and �i because 〈u′i�j〉=Dij(r) 6=0 for i=j. Thus, 〈u′i�j〉 6= 0, implies that u′i and �j are statistically dependent. This is the reason thatISI is less accurate than SIA. We have no reason to choose ui over u′i as statistically independentof �j, so the assumption that either ui or u′j is statistically independent of �j is inconsistent.Now consider the predictions of ISI for our fourth-order statistics. From statistical independence of

ui and �j (for all i and j) and de�nitions Eqs. (6) and (7) one immediately obtains that Bijkl(r)= 0and Mijkl(r) = 0, which are very incorrect. This implies that ISI badly violates Eq. (9) becausethe right-hand side of Eq. (9) will be of order ��D��(r) for some components. Now consider theprediction for Sijkl(r) obtained form the ISI. An algebraic identity is

(uiuj − u′iu′j)(ukul − u′ku′l) = 12 [(uiuk + u

′iu

′k)�j�l

+(uiul + u′iu′l)�j�k + (ujuk + u

′ju

′k)�i�l

+(ujul + u′ju′l)�i�k]− �i�j�k�l: (C.1)


Averaging and assuming re ection symmetry such that, for instance, 〈u′iu′k�j�l〉 = 〈uiuk�j�l〉 andusing the ISI for all i and j, we have

S ISIijkl(r) = �ikDjl(r) + �jlDik(r) + �ilDjk(r) + �jkDil(r)− Dijkl(r): (C.2)

Alternatively, we can substitute into Eq. (C.1) the algebraic identity

u′iu′j = uiuj − ui�j − uj�i + �i�j

and use 〈ui�j�k�l〉= 0, etc., which follows from the ISI. We then obtain

S ISIijkl(r) = �ikDjl(r) + �jlDik(r) + �ilDjk(r) + �jkDil(r) + Dijkl(r): (C.3)

The asymptotically large terms in Eqs. (C.2), (C.3), and (15) are the same, but the asymptoticallysmall terms (the right-most terms) di�er; in fact, the latter terms in Eqs. (C.2) and (C.3) are ofopposite sign. That is, Eqs. (C.2) and (C.3) are inconsistent results obtained by using alternativeproperties of the ISI.To show that the asymptotically small terms in both Eqs. (C.2) and (C.3) are incorrect, we perform

the fourth-order divergence of Eqs. (C.2) and (C.3). We use the incompressibility condition,

Dij(r)|i = Dij(r)|j = 0: (C.4)

Performing the fourth-order divergence of Eqs. (C.2) and (C.3), we �nd that incompressibilityEq. (C.4) requires the asymptotically large terms to vanish, yielding

S ISIijkl(r)|ijkl =±Dijkl(r)|ijkl; (C.5)

where the minus and plus signs are from Eqs. (C.2) and (C.3), respectively. Substituting Eq. (C.5)for Sijkl(r)|ijkl in Eq. (B.14a) gives the contradiction

Dijkl(r)|ijkl =∓3Dijkl(r)|ijkl:Thus, the asymptotically small terms in Eqs. (C.2) and (C.3) are incorrect. The connection betweenthese asymptotically small terms and the pressure structure function is discussed in Appendix B.Eq. (C.5) produces very incorrect values of pressure structure function if substituted in formulasgiven in Hill and Wilczak (1995).

References

Batchelor, G.K., 1951. Pressure uctuations in isotropic turbulence. Proc. Cambridge Philos. Soc. 47, 359–374.Batchelor, G.K., 1960. The Theory of Homogeneous Turbulence. Cambridge University Press, Cambridge, England,197 pp.

Dutton, J.A., Deaven, D.G., 1972. Some properties of atmospheric turbulence. In: Rosenblatt, M., Van Atta, C. (Eds.),Statistical Models and Turbulence, Lecture Notes in Physics, Vol. 12. Springer, Berlin, pp. 402–426.

Frenkiel, F.N., Klebano�, P.S., 1967. Higher-order correlations in a turbulent �eld. Phys. Fluids 10, 507–520.Frenkiel, F.N., Klebano�, P.S., 1973. Probability distributions and correlations in a turbulent boundary layer. Phys. Fluids16, 725–737.

George, W.K., Beuther, P.D., Arndt, R.E., 1984. Pressure spectra in turbulent free shear ows. J. Fluid Mech.148, 155–191.

Hill, R.J., 1994. The assumption of joint Gaussian velocities as applied to the pressure structure function. NOAA Tech.Rep. ERL 451-ETL 66, NOAA=ERL=Environmental Technology Laboratory, Boulder, CO, 22 pp. [Available from theauthor or the National Technical Information Service, 5285 Port Royal Rd., Spring�eld, VA 22161.]


Hill, R.J., 1996a. Pressure–velocity–velocity statistics in isotropic turbulence. Phys. Fluids 8, 3085–3093.Hill, R.J., 1996b. Corrections to Taylor’s frozen turbulence approximation. Atoms. Res. 40, 153–175.Hill, R.J., Boratav, O.N., 1997. Pressure statistics for locally isotropic turbulence. Phys. Rev. E 56, R2363–R2366.Hill, R.J., Thoroddsen, S.T., 1997. Experimental evaluation of acceleration correlations for locally isotropic turbulence.Phys. Rev. E 55, 1600–1606.

Hill, R.J., Wilczak, J.M., 1995. Pressure structure functions and spectra for locally isotropic turbulence. J. Fluid Mech.296, 247–269.

Kolmogorov, A.N., 1941. The local structure of turbulence in incompressible viscous uid for very large Reynolds numbers.Dokl. Akad. Nauk SSSR 30, 301–305.

Kolmogorov, A.N., 1962. A re�nement of previous hypotheses concerning the local structure of turbulence in a viscousincompressible uid at high Reynolds number. J. Fluid Mech. 13, 82–85.

Limber, D.N., 1951. Numerical results for pressure–velocity correlations in homogeneous isotropic turbulence. Proc. Natl.Acad. Sci. USA 37, 230–233.

Lin, C.C., 1953. On Taylor’s hypothesis and the acceleration terms in the Navier–Stokes equations. Quart. Appl. Math.10, 295–306.

Lumley, J.L., 1965. Interpretation of time spectra measured in high-intensity shear ows. Phys. Fluids 8, 1056–1062.M�unch, G., Wheelon, A.D., 1958. Space-time correlations in stationary isotropic turbulence. Phys. Fluids 1, 462–468.Mydlarski, L., Warhaft, Z., 1996. On the onset of high-Reynolds-number grid-generated wind tunnel turbulence. J. FluidMech. 320, 331–368.

Nelkin, M., 1994. Universality and scaling in fully developed turbulence. Adv. Phys. 43, 143–181.Nelkin, M., Tabor, M., 1990. Time correlations and random sweeping in isotropic turbulence. Phys. Fluids A 2, 81–83.Obukhov, A.M., 1949. Pressure uctuations in turbulent ow. C.R. Acad. Sci. URSS 66, 17–20.Obukhov, A.M., Yaglom, A.M., 1951. The microstructure of turbulent ow. Prikl. Mat. Mekh. 15, 3–26. (Translation inNACA TM 1350, National Advisory Committee for Aeronautics, Washington, DC, June 1953.)

Praskovsky, A.A., Gledzer, E.B., Karyakin, M.Yu., Ye Zhow., 1993. The sweeping decorrelation hypothesis andenergy-inertial scale interaction in high Reynolds number ows. J. Fluid Mech. 248, 493–511.

Proudman, I., Reid, W.H., 1954. On the decay of a normally distributed and homogeneous turbulent velocity �eld. Philos.Trans. Roy. Soc., Ser. A 247, 163–189.

Sanada, T., Shanmugasundaram, V., 1992. Random sweeping e�ect in isotropic numerical turbulence. Phys. Fluids A 4,1245–1250.

Schumann, U., Patterson, G.S., 1978. Numerical study of pressure and velocity uctuations in nearly isotropic turbulence.J. Fluid Mech. 88, 685–709.

Sreenivasan, K.R., Stolovitzky, G., 1996. Statistical dependence of inertial range properties on large scales in ahigh-Reynolds-number shear ow. Phys. Rev. E 77, 2218–2221.

Sreenivasan, K.R., 1996. The passive scalar spectrum and the Obukhov–Corrsin constant. Phys. Fluids 8, 189–196.Tennekes, H., 1975. Eulerian and Lagrangian time microscales in isotropic turbulence. J. Fluid Mech. 67, 561–567.Thoroddsen, S.T., 1995. Reevaluation of the experimental support for the Kolmogorov re�ned similarity hypothesis. Phys.Fluids 7, 691–693.

Uberoi, M.S., 1953. Quadruple velocity correlations and pressure uctuations in isotropic turbulence. J. Aero. Sci. 20,197–204.

Van Atta, C.W., Chen, W.Y., 1970. Structure functions of turbulence in the atmospheric boundary layer over the ocean.J. Fluid Mech. 44, 145–159.

Van Atta, C.W., Park, J., 1972. Statistical self-similarity and inertial subrange turbulence. In: Rosenblatt, M., Van Atta,C. (Eds.), Statistical Models and Turbulence, Lecture Notes in Physics, Vol. 12. Springer, Berlin, pp. 402–426.

Van Atta, C.W., Wyngaard, J.C., 1975. On higher-order spectra of turbulence. J. Fluid Mech. 72, 673–694.Wyngaard, J.C., Cli�ord, S.F., 1977. Taylor’s hypothesis and high-frequency turbulence spectra. J. Atmos. Sci. 34,922–929.

Zhou, Y., Praskovsky, A., Oncley, S., 1995. On the Lighthill relationship and sound generation from isotropic turbulence.Theoret. Comput. Fluid Dyn. 7, 355–361.

Zhou, Y., Rubinstein, R., 1996. Sweeping and straining e�ects in sound generation by high Reynolds number isotropicturbulence. Phys. Fluids 8, 647–649.

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Fourth-order velocity statistics