Small-Scale Turbulence: How Universal Is It?

Transcription

1 5 th Astralasian Flid Mechanics Conference The University of Sydney, Sydney, Astralia -7 December Small-Scale Trblence: How Universal Is It? RA Antonia and P Brattini Discipline of Mechanical Engineering University of Newcastle, NSW, 8 AUSTRALIA Abstract Kolmogorov s similarity hypotheses and his /5 law are valid at very large Reynolds nmbers For flows encontered in the laboratory, the effect of a finite Reynolds nmber and of the inhomogeneity associated with the large scales can affect the behavior of inertial range scales significantly This paper focses on the sorce of inhomogeneity in two types of flows, those which are dominated mainly by a decay of energy in the streamwise direction and those which are forced, sally throgh a continos injection of energy at large scales Reslts based on a parameterization of the second-order velocity strctre fnction indicate that the normalized third-order strctre fnction approaches /5 more rapidly for forced than for decaying trblence These trends are spported by measrements and nmerical data in these two flow types Introdction There is little dobt that the similarity hypotheses proposed by Kolmogorov [6] (or K) and their sbseent revision ([8] or K6) have had a major impact on trblence research (eg, [,,6,7,9,5,58,6]) A fndamental element of these hypotheses is the assmption that the small scale motion, which incldes the dissipative and inertial ranges, is tropic Also, K and K6 reire that the Reynolds nmber is very large so that the small scale motion is independent of the invariably antropic large scale motion The major otcome of the first similarity hypothesis is the prediction ( δ ) n = fn ( r), () where the increment δ x ( + r) x ( ) ( is the velocity flctation along x ; the separation r is aligned with x, and the anglar brackets denote averaging) For each vale of n, f n is a niversal fnction, in the sense that it is expected to depend only on r r / η ( ( ) η v / ε is the Kolmogorov length scale, v is the kinematic viscosity and ε is the mean energy dissipation rate) The asterisk denotes normalization by the Kolmogorov velocity scale ( ) / K ν ε and/or η The second similarity hypothesis yields the famos inertial range ( η r L ; L is an integral length scale) reslt n ( δ ) = Cn r, (a) when K is sed ( C n is a niversal constant) With K6, ( δ ) n = Cn r ζ n, (b) where the exponents ζ n may depart from n / de to the effect of intermittency in the dissipation rate ( C n may differ slightly from C n ) An important exact relation between ( δ ) and ( δ ) was obtained by Kolmogorov [7], starting with the n Karman Howarth [5] eation for homogeneos tropic trblence, ( δ) = /5 ε r 6 ν d ( δ) dr () A detailed derivation of E() was provided by Batchelor [] More recently, the assmptions nderlying the Kolmogorov eation have been revised [6,] and formalized more rigorosly [] A matched asymptotic expansion approach has been sed by Lndgren [] to obtain the same reslt In the inertial range, the viscos term can be neglected and () redces to the /5 law, viz ( δ) = ( /5) ε r or ( δ ) = ( /5 ) r () It is important to nderline that () and (), as well as the hypotheses in K and K6, apply only at very large Reynolds nmbers It is not srprising, therefore, that for flows normally encontered in the laboratory, E() appears to be satisfied only in the dissipative range (typically r ) (see 7 of [9] and [5]); althogh the evidence is not altogether convincing especially when the tropic form of ε, viz ( ) ε = 5ν x (5) is sed in forming δ and r With a few exceptions, the laboratory data also indicate an asymptotic approach to E(a) (or E(b)) and E() as the Reynolds nmber (sally represented by and defined by Taylor microscale / ( / x) / λ ν, where λ is the ) increases When is finite, deviations from Es(), () and indeed (5) can be ite significant When is fixed, the deviations may also depend on the natre of the flow, ths casting dobt on any claim of niversality For the same flow and, departres from Es() and (5) can still depend on the initial conditions that are sed It seems reasonable to ascribe these deviations to a lack of homogeneity in laboratory flows, the sorce of inhomogeneity depending on a nmber of parameters, sch as the Reynolds nmber, type of flow and initial conditions In deriving E(), Kolmogorov ignored the nonstationary or nonhomogoneos term (space or time derivative term, respectively) This term has since been considered by a nmber of athors (eg, Danaila et al [], Lindborg [], Qian [5,55], Lndgren []) in the context of decaying homogeneos tropic trblence There have also been attempts at identifying this nonstationarity in more complicated flows, eg the centreline region of a flly developed channel flow (Danaila et al []), a homogeneos niform shear flow (Casciola et al [8], Danaila et al [], Qian [55,56]) and the region near the axis of a circlar jet (Danaila et al []) Forcing has been sed in physical experiments (eg, Moisy et al [6]) to achieve stationarity Stationary tropic trblence is often stdied nmerically by adding a forcing term to the Navier Stokes eation (eg, [6,])

2 It shold also be recalled that, at the relatively large Reynolds nmbers which occr in the atmospheric srface layer, the evidence in spport of the /5 law is rather inconclsive This is partly de to the ncertainty in estimating ε The data of Sreenivasan and Dhrva [6] for indicated, however, that there is no discernible range over which d ( δ ) dr is constant over a convincing range This raises some concern since the existence of the inertial range has been traditionally linked to the linear increase of ( δ ) with r These athors frther noted that an inertial range cold not be identified nambigosly from the local slopes of the even-order moments of δ They also stressed the difficlty of discssing the scaling of ( δ ) n effectively withot first nderstanding the effects of finite shear and finite The previos observations can only fel speclation abot the meaningflness of the scaling exponents that have been inferred from laboratory data and also the corresponding inferences regarding the departres of these exponents from the K predictions For example, Lindborg [] noted that in order to test intermittency theories experimentally, it is necessary to recover Kolmogorov s law with good accracy in a sfficiently broad range of scales Otherwise there is no reason to pt faith in possible observed deviations from the K theory In this paper, the focs is almost exclsively on the effect the inhomogeneity has on what is loosely referred to as the scaling range or restricted scaling range We adopt the approach previosly sed by Danaila et al [ ], which consists in extending E() to a statement which, in essence, expresses a scale-by-scale energy bdget for a particlar flow Previosly pblished data as well as new measrements for grid trblence and along the axis of a circlar jet are examined with the aim of antifying possible differences in the inhomogeneity among varios flows Specifically, we distingish between flows which are dominated by the decay of trblent energy along the main flow direction sch as grid trblence, jets and wakes and flows which have been forced In experiments, forcing can be achieved by stirring the flow with moving bondaries [6], introdcing body forces or with zero-net mass-flow-rate jets [,] In nmerical simlations, it is sally achieved by continosly injecting energy at low wavenmbers [] We also examine how the asymptotic K reslt may be reached in these two flow categories Inhomogeneity in Decaying and Forced Flows In this section, we give examples of how the inhomogeneity can affect relation () in decaying flows The modified form of this relation can be written as ( δ) = /5 ε r 6 v d ( δ) + I dr (6) where I represents the inhomogeneity associated with the large scale motion For decaying grid trblence, [] r d I / ( ) U r ω δ dω (7a) where U is the mean velocity in the x direction and ω a dmmy integration variable For forced trblence, Moisy et al [6] obtained (sing [5]) I /7 ε r / Lf, (7b) where L f is an integral scale which characterises the forcing A similar expression is given by Gotoh et al [] and Fkayama et al [6] Since I is negative, its presence in E(6) is to keep the magnitde of ( δ ) below its asymptotic vale of ε 5 As a conseence, measred vales of ( ) δ cannot exceed r /5, nless ε is evalated incorrectly Note that E(6) shold provide a more reliable indirect means of estimating ε A more general form of E(6) is the corresponding transport eation for the energy strctre fnction ( δ) ( δ) + ( δv) + ( δw) (v and w are velocity flctations in the y and z directions, respectively), eg [], d δ( δ) = / ε r ν ( δ) + I dr (8) with r d I ( / ) ( ) U r ω δ dω (9) for decaying homogeneos tropic trblence Note that now /5 has been replaced by / One advantage of E(8) over E(6) is that it allows a more meaningfl comparn with the, where θ is the scalar flctation (eg []) Another advantage, [], is that, for small r, it is consistent with a more general form of ε than ε, ie transport eation for ( δθ ) ε ( x) ( v x) ( w x) = ν / + / + / () whereas, at sfficiently large r, it reprodces the trblent energy bdget, viz U d ε = () (withot the need to assme tropy at large scales, ie = ; sch an assmption is invalidated in most grid trblence experiments becase v or is typically larger than w by nearly %) As noted by Antonia et al [9], the presence of I in E(6), or I in E(8), allows compliance with two important classical reslts The energy bdget is retrieved at large r, whilst the decay of ε or, eivalently, the enstrophy in homogeneos trblence, is correctly reprodced in the limit of r For reasons mentioned above we have preferred to validate E(8), instead of E(6), sing (new) measrements obtained in grid trblence and along the axis of a circlar jet Both these flows offer advantages in the present context In grid trblence, ε can be inferred reliably from E(); from an experimental viewpoint, this can be exploited seflly, eg for assessing any possible degradation to the high wavenmber part of the spectrm, eg [57] Along the axis of a circlar jet it can be shown that the strctre fnctions shold satisfy similarity, when normalized by the energy and Taylor microscale [6] Becase is constant along the axis, other combinations of normalising scales, sch as ( η, ) and ( L, ), shold also reslt in the K collapse of the strctre fnctions Experimental Details The grid trblence tnnel has a working section of 5mm 5 mm and is m long A biplane sare mesh

3 grid ( M = 8 mm with 76mm sare rods and a solidity of 5) is placed at the entrance section and measrements are made on the centreline at x = 5M, where the trblence is approximately locally homogeneos and tropic The mean streamwise velocity is abot 6ms The jet exits from a contraction (85: area ratio) The exit diameter, D, is 55mm Data are taken on the axis, at a distance of D downstream of the exit, where the mean velocity in the streamwise direction is abot 5ms The velocity was measred with in-hose hot wires, operated with in-hose constant temperatre anemometer circits The hot wire ( Pt % Rh ) was etched to a diameter of d w of 5µ m and an active length l w so that the ratio lw / d w was approximately The CTA circits were operated at an overheat ratio of 5, with a ct-off freency set at approximately 5kHz The sampling freency, f s, was 8kHz for the grid trblence and 6kHz for the jet measrements while the analog filters were set at f s / The anemometer signals were digitized into a PC sing a 6-bit AD board The geometrical angle between the wires was nearly 9 D The calibration of the X-wire was performed sing a look-p-table (LUT) method [5] A comparn between LUT and the effective angle method showed that the former gives more reliable reslts when the velocity is below abot 6ms - With LUT, the velocity and velocity derivative statistics, in both grid trblence and the far field of the circlar jet, were in closer (bt far from perfect) agreement with tropy than when the effective angle calibration was sed Corrections for the spatial attenation of the hot-wires (sing the method otlined in Zh and Antonia [6]) and Taylor s hypothesis (following []) have been applied Comparn of Experimental and Nmerical Data with Es(6) and (8) All the measred terms in E(8) are shown in Fig (grid trblence) and Fig (jet axis) All terms have been normalized by dividing by ε r The determination of ε, as given by E (), has been sed in each case For grid trblence, ε was within 5% of the vale estimated from measrements of via E () Use of ε reslted in E () being satisfied within % In the jet experiment, ε was in satisfactory (5%) agreement with the measred energy bdget along the axis This bdget inclded trblent advection and prodction via the normal stresses (the trblent diffsion was fond to be negligible) bt ignored the pressre diffsion term; althogh jstification for this latter assmption is lacking, the se of ε, instead of ε, reslted in a 7% imbalance in the bdget for [7], viz ) In estimating I se was made of similarity δ ( λ) = f r/ ( ) ( / ) δ δ = g r λ () () For grid trblence, where f and g determine the shape of the strctre fnctions and are dependent only on r / λ (not on x), r / λ dλ I U r f d ( / ) λ ( ω/ λ) ( ω/ λ) I On the jet axis, I is given by d + r / λ ( U / r ) λ ( ω/ λ) f d( ω/ λ) I r / λ du I I + I + r e h d () ( / ) λ ( ω/ λ) ( ) ( ω/ λ) I (5) where I and I are indicated in E () The prime denotes a derivative with respect to the similarity variable ( ω/ λ ) The extra term ( I ) represents the contribtion from the difference between the longitdinal and radial strctre fnctions, viz In forming ( ) ( λ) ( δ) = e r/ ( δv) = h( r/ λ) (6) (7) δw δ, we have assmed that ( δv) = ( ) so that ( δ) ( δ) ( δv) v = w, so that = + and conseently = + v Jstification for the similarity scales and λ was provided by George [9] who considered the spectral eations for decaying homogeneos tropic trblence and also Antonia et al [7] and Danaila et al [] who considered the transport eation for ( δ ) In the jet, the first term of E(), I, involving the decay of, was estimated after applying a least sares fit to the axial profiles of over the range 8 x/d The decay of the mean velocity was estimated in the same way, for the last term I Finally, a fit to the streamwise variation of λ was sed to estimate I A more general definition of λ (= 5 ν / ε, / and conseently / Rλ = λ ν []), which takes into accont the three velocity components, has been sed in analysing these data The linear vertical scale in Figs and provides a more severe test of how closely E(8) is satisfied by the measrements than the logarithmic scale sed in previos papers [9,5] The imbalance, or difference between / and the remaining normalized terms in E(8), is satisfactory, especially for Fig The imbalance (dashed lines in Figs and ) is larger for the jet than for grid trblence There may be several reasons for this More assmptions were made [] in the derivation of E(5) than for E(7a) The trblence intensity in the jet is abot 5%, mch larger than that for grid trblence (5%) This may introdce an additional ncertainty de to the se of Taylor s hypothesis (althogh modified, following []) and the neglect of the binormal component (the ot-of-plane velocity component not resolved by the X-wire) It is noteworthy, however, that at r λ the balance is almost exact, implying that ε can be estimated from E(8), once the second- and third-order strctre fnctions are known The major difference between Figs and reflects, to a large extent, the higher vale of in the jet Correspondingly, the maximm

4 vale of δ( δ) ε r is twice as large in Fig than in Fig Accordingly, the likelihood of a scaling range is bigger for Fig than Fig This range can be very roghly inferred from the intersections of the distribtion corresponding to this term with those for the viscos term and for I The locations of these crossings are denoted by r A and r h in the figres The difference ( rh r A ) is significant in Fig bt negligible in Fig, so that a scaling range wold be most tenos for Fig The separate contribtions to I are also inclded in Figs and The major contribtion comes from I, the term associated with the streamwise decay of ; in this sense, the central region of the jet resembles grid trblence for which this is the dominant term (see E(8)) One may expect a similar behavior to apply along the centreline of wakes, bt not in a pipe or a channel, becase of the streamwise inhomogeneity; here, the effect of trblent diffsion along the wall-normal direction cannot be ignored The contribtions from I are not negligible arond r λ (Fig) and r λ (Fig) I becomes largest at the largest r / λ The DNS data for box trblence of Fkayama et al [6] (replotted here on a linear scale) are shown in Fig Reslts for both decaying and forced trblence, obtained at the same ( 7), are inclded There is almost no difference between the viscos terms, reflecting the normalization by η (eg, []) The maximm vale of ( δ) / ε r is larger for forced trblence, reflecting the smaller vale of I in this case The difference between the forced and decaying vales of I is only likely to increase, as is increased Unfortnately, DNS data for decaying box trblence at large are not yet available That the inflence of I (or I ) on scales corresponding to the peak in ( δ) / ε r shold diminish as increases can be readily inferred from available data in decaying grid trblence The data in [6] are reprodced in Fig ; althogh the range is limited, the trend is nmistakable The magnitde of I, estimated here from the measred vales of ( δ ) and ( δ ) and assming the validity of E(5), decreases with at least as rapidly as that of the viscos term Terms in E(8) 5 5 r l r/λ hom Figre : Terms in E(8), divided by ε r, for grid trblence ( 5 ) YLVFRXVWHUPz, third-order strctre fnction; I ;, I ; +, I ; solid line: calclated third-order strctre fnction sing E(8); solid thick line: /; dashed line: sm of the viscos term, thirdorder strctre fnction term and I r h Terms in E(8) 5 5 r l r h r/λ hom Figre : Terms in E(8), divided by ε r, for the jet ( 6 ) V, viscos term; {, third-order strctre fnction; I ;, I ; +, I ;, I ; solid line: calclated third-order strctre fnction sing E(8); solid thick line: / limit; dashed line: sm of the viscos term, third-order strctre fnction term and I Terms in E(6) 8 6 r Figre : Terms in E(6), divided by ε r, sing the DNS data of Fkayama et al [6] at roghly the same ( 7) Filled-in symbols: forced trblence; open symbols: decaying trblence {, z: third-order strctre fnction; I ; T, V: viscos term; dashed line: sm of the viscos term, third-order strctre fnction term and I For a fixed, I (or I ) may not be niversal, even within the same flow type For example, in the similarity region of decaying grid trblence, the shape of the normalized form of I has been fond to depend on the geometry of the grid [] A less sensitive dependence on the initial condition at the nozzle exit of a rond jet has been reported in [7] In this case the third-order strctre fnctions, in the far field of the jet, remained nchanged nder modified initial conditions For the wake data of [] (where 5 wake generators were sed), Fig5, I was inferred indirectly from measrements of ( δ ) and ( δ ) by assming the validity of E(6) Althogh is nominally constant in Fig5, the peak of ( δ) / ε r and the magnitde of I exhibit significant variations as the initial conditions are changed It can be arged that different levels of large scale organisation in each flow can lead to differences in I, and they can prodce differences in the departre from the /5 law It is not difficlt to

5 imagine that sch departres can be reflected in differences in scaling exponents (eg, []) The reasonable collapse of the viscos term in Fig5 reflects mainly the constant vale of ; an improved collapse at small r wold be expected if r is normalized by η rather than λ The lack of collapse of the third-order strctre term in Fig5 seems all the more significant given that, in each wake, is constant along x, so that a similarity based on λ seems appropriate Terms in E(6) 8 6 r/λ Figre : Variation with of the terms in E(5), divided by ε r, for the grid trblence data of [6] ( increases between 7 and in the δ r ; dashed line: I direction of the arrows) Solid line: ( ) (calclated by difference from E(6)); dash-dotted line: viscos term; dashed horizontal line: /5 Terms in E(6) 8 6 r/λ Figre 5: Terms in E(6), divided by ε r, for the trblent wake data of [] at nominally the same ( ) SODWH{: circlar cylinder V: sare cylinder; : screen cylinder; +: screen strip; solid line: third-order strctre fnction; dash-dotted line: viscos term; dashed line: inhomogeneos term, calclated by difference from E(6); dash-dotted horizontal line: /5 Distribtions of δ( δ) were calclated from E(8) starting with the measred distribtions of ( δ ) and assming similarity based on λ and The agreement between these calclations (solid lines in Figs and ) and the measrements is ite satisfactory, providing some indirect spport for the validity of E(8) and the assmptions made in estimating its terms Extrapolation to Large R O Clearly, it is important to ascertain how ickly the maximm δ δ divided by ε r or ( δ ) divided vale of ( ) by ε r approaches its asymptotic vale of / or /5 This has been the objective of many investigations Or main interest here is to examine the dependence of this term in conjnction with that for the viscos and inhomogeneos terms in E(8) To this end, we follow the approach by Antonia et al [7] who sed a description of ( δ ) [5,9] which extends from the smallest (Kolmogorov) scale to the integral scale L, c c ( δ) = ( r 5 )( + βr) /( + ( r / r c ) ) (8) where r c is identified with the crossover between the dissipative and inertial ranges, c ( ζ /) and β L E(8) is essentially a modification, for finite Reynolds nmbers, of the model for ( δ ) first proposed by Batchelor [] with β= To obtain ( δ ) For tropic trblence, and L 5 / / ε λ, se is made of the tropic relation ( δ ) = + r ( δ ) CR d dr (9) / / λ = 5, / = ( /5 ), where L has been identified with / ε / ε The dimensionless energy dissipation rate C parameter C ε is expected to become constant at sfficiently large bt its magnitde shold depend on the initial conditions This is tre in both experiments, eg [], or simlations, eg [58] and references therein In general, one expects the shape of ( ) and δ to depend on the ( ) δ ( ) type and level of organisation in a particlar flow (eg []), which may in any case reflect the inflence of the initial conditions Here, we have assmed a vale of for C ε, as in [7] We have also assmed that ( ) / the Kolmogorov constant C assigned to r c 5C, with a vale of for The K vale of / was ζ, ie intermittency was ignored (calclations taking into accont intermittency on ( ) δ did not show significant differences) Althogh C, ζ and r c may vary slowly with (eg [5]) at small we believe that the present estimates of I (or δ δ δ ) shold be sfficiently reliable to provide a first-order indication of how the Kolmogorov s asymptotic reslt may be approached Calclations of ( δ) / ε r are shown in Fig6 together with those for the viscos term and I in E(6) As increases, the latter two distribtions shift to the left and right respectively, ths increasing the relevance of an inertial range Nonetheless, the attainment of sch a range reires vales of well in excess of = Even at I ) and ( ) (or ( )( ) 6 =, it wold appear that the extent of this range is limited; the difference between ( δ) / ε r and /5 is smaller than % over 8 decades in r / λ bt the shape

6 Ä of ( δ) / ε r indicates that this antity exhibits a maximm Perhaps not srprisingly, ( ) / δ / r appears to reach its assmed asymptotic vale of (the Kolmogorov constant) at (Fig7) The grid trblence data ( = ) of Zho & Antonia [6] have been inclded to show that relation (8) approximates the measred data adeately The variation with of the calclated maximm vale of ( ) δ r is shown in Fig8 The calclation (solid line) is in good agreement with measrements obtained in grid trblence and along the centreline of a circlar jet (note that here the / vale has been rescaled to /5 for or data) The plane jet data of Pearson and Antonia [5] lie below this line, partly becase ε has been sed instead of ε For the jet and grid trblence data of [8], ε was sed The large scatter associated with the active grid trblence data of [8] may in part reflect the ncertainty in the estimation of ε in this experiment The above comments sggest that, evidently, cation is reired when interpreting the good agreement of most of the experimental decaying trblence data with the model The model is nlikely to take flly into accont the differences in the initial conditions that have been fond to exist between different experiments The variation in the peak vale of ( ) δ r for varios wake data (fig5), at nominally the same, spports the idea that a niversal distribtion is nlikely This idea is consistent with the view (eg, []) that simple shear flows do not reach niversal asymptotic states The reslts from the model are in close agreement with the prediction (dash-dotted line) of Lndgren [] for > For lower vales of, the model based on E(8) is in satisfactory agreement with the grid trblence data of [6] (the difference between ε or ε and the vale of ε estimated with E() is ite small for these data) The measred and nmerical data obtained in forced trblence (denoted by filled-in symbols) seem to follow a separate distribtion (it is nlikely that this distribtion is nie in view of the docmented effects if initial conditions for this type of trblence); clearly the approach to /5 is mch more rapid for these data and a vale of for may be sfficient for this type of trblence to reach the Kolmogorov asymptote This vale is somewhat larger than the maximm vale ( 6 ) for DNS data of Gotoh et al []; this may need to be kept in mind when assessing DNS data with forcing For decaying trblence, a vale of in excess of 5 (perhaps even 6 ) may be needed before the K asymptotic state is reached This estimate is somewhat higher than that given in [6] where the Batchelor parameterization was sed for modelling ( δ ) The difference between r h and r A is of interest when eniring into the possible existence of a scaling range Fig9 indicates that log ( rh / r A ) varies linearly with log For, this range is eivalent to considering 5% of the maximm of δ r Qian [5] adopted a slightly different criterion ( ) for identifying the extent of the inertial range, bt obtained a similar linear dependence on (his figre 8) We recognize that there is inevitable arbitrariness in the definition of a scaling range; nonetheless, the dash-dotted line in Fig9 indicates that shold exceed 5 to achieve a decades platea in ( ) δ r The inset in Fig9 shows that the location of the maximm of ( ) δ r approaches a constant vale for r λ This trend appears to have been established for decaying trblence [8,,6] However, Lndgren [] showed that, in the case of linearly forced trblence, the maximm is at a higher vale, r λ, implying a different behavior for forced trblence Terms in E(6) 8 6 r l r/λ 6 r h = Figre 6: Variation with of the terms in E(6), divided by ε r, derived from a model of decaying tropic trblence, E(8) Solid line: third-order strctre fnction; dotted line: I ; dashed line: viscos term (δ ) r / = 6 r/λ 6 Figre 7: Kolmogorov-normalized second-order strctre fnction of, divided by r / Solid lines: model E(8);, measred grid trblence data of [6] at = Conclding Comments The limiting vales of /5 and /, which appear in the stationary form of the transport eations for ( δ ) and ( δ ), represent asymptotic states corresponding to large Reynolds nmbers For laboratory flows, deviations from these vales can be significant; even in the atmospheric srface layer, the deviations may not be negligible The inclsion of the nonstationarity or inhomogeneity (de mainly to the large scale motion) in the transport eation for ( δ ) and ( δ ) allows some assessment to be made on whether a scaling range, albeit of restricted extent, is possible Experimental and nmerical reslts obtained in a nmber of different flows indicate that the magnitde of the inhomogeneos term, I or I, can depend on varios parameters, inclding initial conditions In 6

7 particlar, across the scaling range, the magnitde of I (or I ) is smaller for forced than decaying-type trblence Consistently, δ r approaches /5 more the maximm vale of ( ) rapidly, as is increased, in forced than in decaying-type trblence This trend seems to be convincingly spported by both measred and nmerical data Antonia and Orlandi [] noted that significant differences also existed for statistics of smallscale passive scalar ( ) flctations between forced and decaying homogeneos tropic trblence One wold expect that the variation of the maximm vale of ( δ )( δθ ) r will exhibit a similar behavior, in terms of the difference between forced and decaying-type trblence, to that in Fig 8 max( (δ ) r ) Figre 8: Variation with of the maximm of the normalized thirdorder strctre fnction of Solid line: based on model E(8); DNS data for forced box trblence []; '6 GDWD IRU IRUFHG ER[ trblence [6]; Y, DNS data for decaying box trblence [6]; measred forced trblence data [6];, plane jet [5]; {, grid trblence [6];, vales measred by [8];, rond jet (present data); P, D wakes data []; dash-dotted line: model of []; grid trblence [8]; dashed horizontal line: /5 log(r h /r l ) (r/λ) max Figre 9: Dependence on of log(r h/r l), solid line The dash-dotted line corresponds to a range where ( ) δ r is 99% of /5 Inset: location of the maximm vale of the third-order strctre fnction, divided by ε r, as a fnction of Acknowledgements The spport of the Astralian Research Concil is acknowledged We are especially gratefl to L Danaila for her major contribtions to varios aspects of this work, in particlar the derivations of I in E(7a) and I in E(8) References [] Antonia, RA and Orlandi, P, Effect of Schmidt Nmber on Small-Scale Passive Scalar Trblence, Appl Mech Reviews, 56,, -8 [] Antonia, RA and Orlandi, P, Similarity of Decaying Isotropic Trblence with a Passive Scalar, J Flid Mech, 55,, -5 [] Antonia, RA, Old-Rois, M, Anselmet, F, Zh, Y, Analogy between predictions of Kolmogorov and Yaglom J Flid Mech,, 997, 95-9 [] Antonia, RA and Pearson, BR, Effect of Initial Conditions on the Mean Energy Dissipation Rate and the Scaling Exponent Phys Rev E 6,, [5] Antonia, RA, Pearson, BR and Zho, T, Reynolds Nmber Dependence of Second-Order Velocity Strctre Fnctions, Phys Flids,,, -6 [6] Antonia, RA and Smalley, RJ, Scaling Range Exponents from X-wire Measrements in the Atmospheric Srface Layer, Bondary-Layer Meteorology,,, 9-57 [7] Antonia, RA, Smalley, RJ, Zho, T, Anselmet, F, and Danaila, L, Similarity of energy strctre fnctions in decaying homogeneos tropic trblence, J Flid Mech, 87,, 5-69 [8] Antonia, RA, Zho, T, Danaila, L and Anselmet, F, Scaling of the mean energy dissipation rate eation in grid trblence, J of Trblence,,, [9] Antonia, RA, Zho, T, Danaila, L and Anselmet, F, Streamwise Inhomogeneity of Decaying Grid Trblence, Phys Flids,,, [] Antonia, RA, Zho, T and Romano, GP, Small-Scale Trblence Characteristics of Two-Dimensional Blff Body Wakes, J Flid Mech,, 67-9 [] Batchelor, GK, Kolmogoroff s Theory of Locally Isotropic Trblence, Proc Camb Phil Soc,, 97, [] Batchelor, GK, Pressre Flctations in Isotropic Trblence, Proc Camb Phil Soc, 7, 95, 59-7 [] Batchelor, GK, The Theory of Homogeneos Trblence, Cambridge University Press, Cambridge, Ma, 95 [] Birok, M, Sarh, B and Gokalp, I, An attempt to realize experimental tropic trblence at low Reynolds nmber, Flow Trbl Comb, 7,, 5-8 [5] Brattini, P and Antonia, RA, The Effect of Different X- wire Calibration Schemes on some Trblence Statistics, Accepted for pblication in Expts in Flids, 5 [6] Brattini, P, Antonia, RA and Danaila, L, Similarity in the Far Field of a Trblent Rond Jet, Accepted for pblication in Phys Flids, 5 [7] Brattini, P, Antonia, RA and Rajagopalan, S, Effect of initial conditions on the far field of a rond jet, To appear in proceedings of 5 AFMC, [8] Casciola, CM, Galtieri, P, Benzi, R, Piva, R, Scale-byscale bdget and similarity laws for shear trblence, J Flid Mech, 76,, 5- [9] Chassaing, P, Antonia, RA, Anselmet, F, Joly, L and Sarkar, S, Variable Density Flid Trblence, Klwer, [] Danaila, L, Anselmet F, Zho, T, Trblent Energy Scale- Bdget Eations for nearly Homogeneos Sheared Trblence, Flow Trbl Comb, 7,, 87- [] Danaila, L, Anselmet, F, Zho, T and Antonia, RA, A Generalization of Yaglom s Eation which acconts for the Large-scale Forcing in Heated Decaying Trblence, J Flid Mech, 9, 999, 59-7

8 [] Danaila, L, Anselmet, F, Zho, T and Antonia, RA, Trblent Energy Scale Bdget Eations in a Flly Developed Channel Flow, J Flid Mech,,, 87-9 [] Danaila L, Antonia, RA and Brattini, P, Progress in Stdying Small Scale Trblence sing exact two-point eations New J Phys, 6,, 8 [] Danaila, L, Brattini, P, Antonia, RA, Large-scale effects in a self-preserving, axisymmetric trblent jet,, In preparation [5] Dhrva, B, An experimental stdy of high Reynolds nmber trblence in the atmosphere PhD Thesis,, Yale University [6] Frisch, U, Trblence: the Legacy of AN Kolmogorov, Cambridge University Press, Cambridge, Ma, 995 [7] Fkayama, D, Oyamada, T, Nakano, T, Gotoh, T and Yamamoto, K, Longitdinal strctre fnctions in decaying and forced trblence, J Phys Soc Jpn, 69,, 7-76 [8] Gagne, Y, Castaing, B, Badet, C and Malécot, Y, Reynolds Nmber Dependence of Third-order Velocity Strctre Fnctions, Phys Flids, 6,, 8-85 [9] George, WK, The decay of homogeneos tropic trblence Phys Flids A,, 99, 9-59 [] George, WK and Davidson L, Role of initial conditions in establishing asymptotic flow behavior, AIAA J,,, 8-6 [] Gotoh, T, Fkayama, D and Nakano, T, Velocity field statistics in homogeneos steady trblence obtained sing a high-resoltion direct nmerical simlation, Phys Flids,,, 65-8 [] Hill, RJ, Applicability of Kolmogorov s and Monin s eations of trblence, J Flid Mech, 5, 997, 67-8 [] Hwang, W and Eaton, JK, Creating homogeneos and tropic trblence withot a mean flow, Expts in Flids, 6, -5, [] Kahalerras, H, Malécot, Y, Gagne, Y and Castaing, B, Intermittency and Reynolds nmber, Phys Flids,, 998, 9-9 [5] von Karman, T and Howarth, L, On the Statistical Theory of tropic trblence, Proc R Soc London A, 6, 98, 9-5 [6] Kolmogorov, AN, The Local Strctre of Trblence in Incompressible Viscos Flid for very Large Reynolds Nmbers, Dokl Akad Nak SSSR, 9, 99- [7] Kolmogorov, AN, Dissipation of Energy in Locally Isotropic Trblence, Dokl Akad Nak SSSR,, 9b, 9- [8] Kolmogorov, AN, A Refinement of Previos Hypotheses concerning the Local Strctre of Trblence in a Viscos Incompressible Flid at High Reynolds Nmber, J Flid Mech,, 96, 8-85 [9] Krien, KS and Sreenivasan, KR Antropic Scaling Contribtions to High-Order Strctre Fnctions in High Reynolds Nmber Trblence Phys Rev E, 6,, 6- [] Lavoie, P, Antonia, RA and Djenidi, L, The Effect of Grid Geometry on the Scale-by-Scale Bdget of Decaying Trblence, To appear in proceedings of 5 AFMC, [] Lindborg, E, A note on Kolmogorov s third-order strctre fnction law, the local tropy and the pressre-velocity correlation, J Flid Mech, 6, 996, -56 [] Lindborg, E, Correction to the For-fifths Law de to Variations of the Dissipation, Phys Flids,, 999, 5-5 [] Lndgren, TS, Linearly Forced Trblence, CTR Annal Research Briefs,, 6-7 [] Lndgren, TS, Kolmogorov Trblence by Matched Asymptotic Expansion, Phys Flids, 5,, 7-8 [5] Lndgren, TS, Kolmogorov Two-thirds Law by Matched Asymptotic Expansion, Phys Flids,,, 68-6 [6] Moisy, F, Tabeling, P and Willaime, H, Kolmogorov Eation in a Flly Developed Trblence Experiment, Phys Rev Lett, 8, 999, [7] Monin, AS, and Yaglom AM, Statistical Flid Mechanics, vol, 975, MIT press, Cambridge, MA [8] Mydlarski, L, and Warhaft, Z, On the onset of high- Reynolds-nmber grid-generated wind tnnel trblence, J Flid Mech,, 996, -68 [9] Nelkin, M, Universality and Scaling in Flly Developed Trblence, Advances in Physics,, 99, -8 [5] Nelkin, M, Trblence in flids, Am J Phys, 68,, -8 [5] Novikov, EA, Statistical balance of vorticity and a new scale for vortical strctres in trblence, Phys Rev Lett, 7, 99, 78 7 [5] Orlandi, P and Antonia, RA, Dependence of the Nonstationary Form of Yaglom s Eation on the Schmidt Nmber, J Flid Mech, 5,, 99-8 [5] Pearson, BR and Antonia, RA, Reynolds-nmber Dependence of Trblent Velocity and Pressre Increments, J Flid Mech,,, -8 [5] Qian, J, Inertial range and the finite Reynolds nmber effect of trblence, Phys Rev E, 55, 997, 7- [55] Qian, J, Slow Decay of the Finite Reynolds Nmber Effect of Trblence, PhysRev E, 6, 999, 9- [56] Qian, J, Scaling of strctre fnctions in homogeneos shear-flow trblence, Phys Rev E, 65,, 6 [57] Schedvin, JC, Stegen, GR and Gibson, CH, Universal Similarity at High Grid Reynolds Nmbers, J Flid Mech, 65, 97, [58] Sreenivasan, K, An pdate on the energy dissipation rate in tropic trblence, Phys Flids,, 998, [59] Sreenivasan, KR and Antonia, RA, The Phenomenology of Small-scale Trblence, Ann Rev Flid Mech, 9, 997, 5-7 [6] Sreenivasan, KR and Dhrva, B, Is there scaling in High- Reynolds-Nmber Trblence, Prog Theor Phys Sppl,, 998, - [6] Sreenivasan, KR, Dhrva, B and Gil, IS, The Effects of Large Scales on the Inertial Range in High-Reynolds- Nmber Trblence, 999, eprint arxiv:chao-dyn/996 [6] Zho, T and Antonia, RA, Reynolds nmber dependence of the small-scale strctre of grid trblence, J Flid Mech, 6,, 8-7 [6] Zho, T, Antonia, RA, Danaila, L and Anselmet, F, Approach to the For-fifths Law for Grid Trblence, J of Trblence,,, 5 [6] Zh, Y and Antonia, RA, Effect of Wire Separation on X- probe measrements in a Trblent Flow, J Flid Mech, 87, 995, 99-