Small viscosity asymptotics for the inertial range of local

Proc. Natl. Acad. Sci. USA Vol. 93, pp. 6749-6752, June 1996 Applied Mathematics

Small viscosity asymptotics for the inertial range of local structure and for the wall region of wall-bounded turbulent shear flow G. I. BARENBLATrtt AND ALEXANDRE J. CHORIN§ tMiller Institute for Basic Research, University of California, Berkeley, CA 94708; and §Department of Mathematics, University of California, Berkeley, CA 94720-3840 Contributed by Alexandre J. Chorin, March 12, 1996

ABSTRACT The small viscosity asymptotics of the inertial range of local structure and of the wall region in wallbounded turbulent shear flow are compared. The comparison leads to a sharpening of the dichotomy between Reynolds number dependent scaling (power-type) laws and the universal Reynolds number independent logarithmic law in wall turbulence. It further leads to a quantitative prediction of an essential difference between them, which is confirmed by the results of a recent experimental investigation. These results lend support to recent work on the zero viscosity limit of the inertial range in turbulence.

all the other components in incompressible flow, UL is the velocity component along the vector r joining two observation points x and x + r, s is the total rate of energy dissipation, r = Inr is the length of the vector r, A is an external length scale, e.g., the Taylor scale, and Re is a properly defined Reynolds number, for example, one based on the Taylor scale. The brackets (... .) denote an ensemble average. By the logic of the derivation of Eq. 2.1 the function .1 should be a universal function of its arguments, identical for all flows. Equation 2.1 is assumed to hold only at very high Reynolds numbers Re and very small r/A. It should be equally valid for different definitions of Re. As a result, Re can enter Eq. 2.1 only through lnRe (3). Indeed, a change in the definition of Re, for example, through a change in the choice of reference velocity, multiplies Re by a constant, say C. The property ln(CRe) = lnC + lnRe lnRe, valid for flows in which not only Re, but also lnRe is very large, shows that the scaling law (Eq. 2.1) is invariant in the definition of Re (asymptotic covariance, ref. 3). It is shown in ref. 3 that the logarithm is the only function that has the required invariance property. An expansion of t1 for flows in which lnRe is large gives (3)

1. Introduction

The analogy between the inertial range in the local structure of developed turbulence and the intermediate range in turbulent shear flow near a wall has long been noted (see e.g., refs. 1 and 2). We will use this analogy to shed light on both. In particular, our results lead to specific conclusions about the small viscosity asymptotics of scaling laws in wall-bounded turbulence, sharpening and quantifying the difference between power laws and their asymptotically logarithmic envelope. These conclusions are confirmed in recent experiments. The experimental results support recent work concerning the small viscosity (vanishing dissipation scale) limit of the Kolmogorov "2/3" law. We start with a brief discussion of the scaling of the inertial range and of the conditions under which it has a zero viscosity limit. We then discuss the conditions under which wallbounded turbulence has a zero viscosity limiting structure. The two sets of conditions are related. The existence of a zero viscosity limit leads to predictions that experimental results can verify. Note that as long as reference velocities and length scales have not been specified, it is possible only to speak of a zero viscosity limit and not of an infinite Reynolds number limit. Once a Reynolds number has been defined in each case, we assume that for a fixed large but finite Reynolds number there exists a "universal" state, Reynolds number dependent, but independent of the specific flow.

DLL = ((s)r)3[Ao + lnRe + kln2Re)] al

r

x

Ir

[2.2]

The classical "K-41" Kolmogorov theory (4) corresponds to Ao # 0; then, for large Re, the famous Kolmogorov 2/3 law is obtained 2

DLL = Ao((e)r)3.

[2.3]

In real measurements for finite but accessibly large Re, ai/lnRe is small in comparison with 2/3, and the deviation in the power of r in Eq. 2.2 should be unnoticeable. On the other hand, the variations in the "Kolmogorov constant" have been repeatedly noticed (see refs. 5-7). The asymptotic Eq. 2.3 can be obtained, as is usually done, by simply assuming the absence of the external length scale A and the viscosity v in the list of relevant governing parameters. Thus, Eq. 2.3 constitutes a case of complete sirhilarity, whereas Eq. 2.2 represents incomplete similarity (about complete and incomplete similarity, see refs. 8 and 9). Complete similarity is possible only ifAo / 0. IfAo * 0, one has a well-defined turbulent state with a 2/3 law in the limit of vanishing viscosity, and finite Re effects can presumably be obtained by expansion about that limiting state. The existence of such a limiting state was assumed by Onsager (10) in his

In the inertial range one has the following general scaling law 2

(ln2lRe)

A

2. The Scaling Laws in the Inertial Range

DLL = ((E )rY) (,Re),

IriRe +0

[2.1]

where DLL = ([UL(X + r) - UL(X)]2) is the basic component of the second order structure function tensor which determines The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

tPermanent address: DAMTP, University of Cambridge, Silver Street, Cambridge CB3 9EW, United Kingdom. 6749

6750

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Proc. Natl. Acad. Sci. USA 93

independent derivation of the 2/3 law. Like Kolmogorov (4), Onsager assumed that the structure function was independent of the viscosity v, but while Kolmogorov's more elaborate analysis left open the possibility that there would be no inertial range without viscosity serving as an energy sink on small scales, Onsager simply assumed that viscosity could be set equal to zero in the equations of motion and yet the 2/3 law would hold. In recent years the belief that a finite viscosity operating on small scales was needed to yield the 2/3 law has been supported by the idea that, in the limit of vanishing v, the equations of motion admit the Hopf-Lee equipartition ensemble (11, 12) as an ergodic invariant measure (see e.g., ref. 13). In that ensemble, energy is equipartitioned among all scales of motion and the 2/3 law is invalid. A finite viscosity v would thus be needed to destroy equipartition and produce Eq. 2.3; there would be no turbulence in the limit of zero viscosity and Ao would be zero. However, recent work (14, 15) has shown that the Hopf-Lee ensemble is irrelevant to turbulence, and that one can indeed obtain Eq. 2.3 in the zero viscosity limit; thus, the question whether Ao does or does not vanish is the touchstone for the validity of contrasting approaches to turbulence theory, as typified, for example, by refs. 13 and 14. We shall use the analogy between the inertial range and the wall region to clarify the issue. 3. The Scaling Laws for Wall-Bounded Shear Flows

For the wall region of turbulent shear flow the fundamental relationship analogous to Eq. 2.1 is

ayu

(-v,)Re

=

[3.1]

Here u is the ensemble averaged velocity at the distancey from the wall, u * is the so-called friction or dynamic velocity, Re is again a properly defined Reynolds number, for example, for pipe flow R = ud/ v, where u is the mean velocity (discharge rate divided by the cross-section area), d is the diameter of the pipe, and v is the kinematic viscosity of the fluid. The asymptotic form of Eq. 3.1 in the case of incomplete similarity, corresponding to Eq. 2.2, is 1

Ayu=y +

o

+

lnRe + O(ln2Re))(U;

O(n2Re

)n? )

[3.2]

Again, the simple assumption of disappearing molecular viscosity and external length scale (in the case of flow in pipes, the diameter d) from the list of arguments leads to the relation

ayu=

-

KY

[3.3]

This derivation of Eq. 3.3 is based on the implicit assumption that the limit of the function D in Eq. 3.1 when both arguments tend to infinity, exists and is finite and different from zero: K = 1/FD(oo, ox). This relation leads automatically to the universal logarithmic law u

1

-= U * K ln

u*y

I)

+ constant,

[3.4]

where the constant should also be universal. The assumption of incomplete similarity in the parameter u *y/ v, on the other hand, leads to the general scaling law

U -

=B(Re)

U*

(1996)

/U *y 3(Re)

[3.5]

V

Equation 3.2 is the expansion of Eq. 3.5 at large Re around the limiting state corresponding to Re = oc. Scaling laws similar to Eq. 3.5 were popular among engineers, especially before the papers of von KaIrmaln (16) and Prandtl (17). Recently (18, 19), arguments in favor of the incomplete similarity Eqs. 3.5-3.2 were proposed and the coefficients Bo, B1, and 13i were determined from the Nikuradze data (20):

Bo

15

-32

B1=4, p=

3 2.

[3.6]

We emphasize that neither the scaling law (Eq. 3.5) nor the universal logarithmic law (Eq. 3.4) can be considered as merely convenient representations of experimental data. They have precise and equally justified theoretical foundations; both are based on the assumption of self-similarity, but the logarithmic law is based on the assumption of complete similarity, whereas the scaling law is based on the assumption of incomplete similarity. The question is, which of these assumptions is correct. This question can be answered in full only by further advances in the theory of Navier-Stokes equations and/or by further experimental studies, as is the case in the study of the inertial range in the local structure. The issue here is analogous to the question whether the power laws (Eq. 3.5) have a nonsingular limit as v -- 0 and allow a statement analogous to the statement Ao # 0 in the theory of the inertial range.

4. Further Asymptotic Relationships for Wall-Bounded Shear Flow: Discussion of the Experimental Data

The difference between the cases of complete and incomplete similarity is essential in practice. In the first case the experimental data should concentrate in the 4 = u/u*, lnqj plane (,q = u *y/ v) on a universal curve, the straight line of the logarithmic law. We emphasize that both the slope 1/K and the additive constant entering the logarithmic law (Eq. 3.4) should, by the logic of the derivation, be universal, i.e., Reynolds number independent. (In particular, it is not legitimate to say that in a certain range of Reynolds numbers there is one best fit for the constants in the logarithmic law but that in a different range the constants are different.) In the second case the experimental points may occupy an area in the 4, lnqj plane bounded by the envelope of the family of scaling law curves, having the Reynolds number as parameter. According to previous suggestions (18) the power law has the form 5

4= ( 13nR1

3

U

U*y

U*

V

[4.1] that can be represented also as -

(1TInRe)

exp

[4.2]

The difficulty in distinguishing between complete and incomplete similarity on the basis of the experimental data was due until recently to a simple reason: All the available experimental data were concentrated near the envelope. This was specifically mentioned in ref. 19. The envelope corresponding to Eq. 4.1 is represented by a smooth curve, which can be approximated by piecewise linear functions of lnq. Asymptotically, at very large Re never achieved in the experiments and -at very large lnq, the envelope is represented by the straight line (18)

Applied Mathematics: Barenblatt and Chorin

23e lnq + 6.79.

=

Proc. Natl. Acad. Sci. USA 93

[4.3]

2

Moreover, the relationships (Eqs. 4.1 and 4.2) show that at such very large Re the members of the family of scaling law (Eq. 4.2) have the property of self-similarity: In the plane v = (/lnRe, ; = lInr/lnRe, the family is represented by a single curve

1

v=

+n-y]1 In

lU[v

A]

According to ref. 18, at small v, i.e., large Re, ui/u* [(1/V3)lnRe + 5/2], so that the term ln(u/u*) in the denominator of the right-hand side of Eq. 4.6 is asymptotically small, of the order of lnlnRe, and can be neglected at large Re. The crucial point is that due to the small value of the viscosity v the first term ln(u *A/ v) in both the numerator and denominator of Eq. 3.4 should be dominant, so that 3 lnq/2 lnRe is close to 3/2 (y is obviously less than d/2). Therefore the quantity 1

-

[4.9]

and

15

n (+ 4lnRe,Je 3l 2 aln,nP = (2

[4.10]

[4.5]

q.

u*/A 3 [ ln

InRe- 4e3/2 4 lnR

Note that this law has a finite limit independent of Re. At the same time it can be easily shown that for the envelope of the power-law curves the asymptotic relation is

(Eq. 4.3 gives a next-term correction to Eq. 4.5.) Regrettably, the Reynolds numbers that would allow such self-similar asymptotics had never been achieved in the experiments. Let us analyze further the curves of the family (Eqs. 4.1-4.2), at large but finite Reynolds numbers. We notice that in experimental measurements using any probe, including the Pitot tube used by Nikuradze (20), it is impossible to approach the wall closer than a certain distance 8, say the diameter of the Pitot tube. Consider the experimental possibilities for a certain member of the family (Eqs. 4.1-4.2). It was shown in ref. 19 that the experimental points presented by Nikuradze (20) are close to the envelope. So we assume that up to some distance A > 8 the experimental points are close to the envelope. What happens farther? Consider the combination 31n-q/2lnRe. It can be represented in the following form 3ln71

(p = e3/2( 2 + 4 )lnTl 2 4 InRe 2

[4.4]

The envelope of the family is the curve

2=2 eln

6751

A, but at the same timey is slightly less than d/2, the following asymptotic relations should hold with accuracy O(1/ln2Re):

3

3

exp-;.

(1996)

a

=

(

2

+

O(ln2Re))

[4.11]

The difference in slopes between Eq. 4.10 and Eq. 4.11 is essential. It means that the individual members of the family Eq. 4.1-4.2 should have at large Re an intermediate part, represented in the plane O,lnqj by straight lines, having a slope essentially different from the slope of the envelope by a factor N/e 1.65. Therefore, the graph of the individual members of the family Eq. 4.1-4.2 should have the form presented schematically in Fig. 1. Recently an experimental paper by Zagarola et al. (21) presented new data obtained in a high-pressure pipe flow. High pressure creates a large density and therefore a low kinematic viscosity. The experimentalists were thus able to enlarge the range of Reynolds numbers by an order of magnitude in comparison with Nikuradze's (20). Zagarola and colleagues write (page 1); "The friction data show von Kairman's constant to be 0.44. If the velocity profile is normalized using scaling variables, a log-law with this slope and an additive constant of 6.3 is in excellent agreement with the data." Later (page 5) the authors clarify: "The data in the rangey+ (our q) 2 200, andy/R < 0.15 closely follow a log-law with values of K and B of 0.44 and 6.3, respectively." The published graph of a part of the data in the range of Re from 32 103 to 35.106 (see -

I

III

I

II

lnq/lnRe

can be considered in the intermediate region A < y < d/2 as a small parameter, so that the factor exp(31nr/2lnRe) is approximately equal to

II

I

[2

2(

InRe)]

e/[

3/2 3

- Envelope

2(- 77n)]

ln1

[2 lnRe

1

2-

'

[4.7]

According to Eq. 3.1 we have also

,qano = ainO =

( 2 +

415)exp(2

lne)

[4.8]

and the approximation (Eq. 4.7) can also be used in (Eq. 4.8). Thus, in the intermediate asymptotic range of distancesy: y >

FIG. 1. The individual members of the family of scaling laws (Eqs. 4.1-4.2) near the envelope in the 4,lniq plane have a straight intermediate interval with a slope essentially larger than that of the envelope. I, A part close to the envelope; II, straight intermediate part; III, the fast growing ultimate part having no physical meaning; IV, the region near the maximum where the scaling law is not valid. The horizontal scale is logarithmic.

Proc. Natl. Acad. Sci. USA 93

Applied Mathematics: Barenblatt and Chorin

6752

" "" " I I" I" '11

40

Re = 35X106 Re - 11Re = 1Ox1o6-

Re = .x106_ 30

R

Re-= 1.0x106_ vZ~u+= 1/.44 1ny++6.3\

' ' 25 Re = 99x103 _ 25

100

104

1000

lent flow has a well-defined, experimentally demonstrable limit as v -> 0. By analogy, one may conclude thatA0 0 0 in the case of the inertial range in free turbulence, a conclusion that lends support to recent work as typified by ref. 14; in particular, it lends support to the idea that turbulence is well-defined in the limit of vanishing viscosity. We acknowledge with gratitude the kind permission of Dr. M. V. Zagarola to reproduce Fig. 2 from ref. 21, as well as the pleasant discussion with Prof. N. Goldenfeld. G.I.B. thanks the Miller Institute for Basic Research at Berkeley for its generous hospitality. A.J.C. is supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract DE-AC03-76-SF00098, and in part by the National Science Foundation under Grants DMS94-14631 and DMS89-19074.

20 Re= 2x03-103 20-

10

(1996)

105

106

(our il) FIG. 2. A graph of the velocity profiles normalized using inner scaling variables for 13 different Reynolds numbers between 32 X 103 and 35 X 106. Also shown is a log-law with K and B equal to 0.44 and 6.3, respectively. [Reproduced with permission from Zagarola et al. (21) AIAA Paper 96-0654, Reno, NV.]

Fig. 2, reproducing figure 4 of ref. 21) obviously contradicts this statement. The restriction from above by y/R < 0.15 explains what happens: At such smally the deviation from the envelope is too small to be noticed (part I of the curve in Fig. 1). However, for larger y, up to a very close vicinity of the maximum of 4 achieved in the experiments, the data are split. To each Reynolds number corresponds its own curve with a pronounced linear part having a slope clearly larger than the slope of the envelope; the ratio of the slope of the curves of the family to that of the envelope is always larger than 1.5. Contrary to the opinion of Zagarola et al. (21), we consider this graph to be a clear confirmation of the scaling law, and a strong argument against the logarithmic law according to which all the points up to a close vicinity of maxima should lie on the universal logarithmic straight line. 5. Conclusions

Our analysis leads to a specific prediction of the asymptotic form of the scaling laws for wall-bounded turbulence; this form is distinct from the usual law of the wall, by a difference of \/e in the slope. This conclusion is apparently wellverified experimentally; it shows that wall-bounded turbu-

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York). 15. Chorin, A. J. (1996) Report CPAM-665 (Univ. of California, Berkeley Math. Dept., Berkeley, CA). 16. Von Karman, T. (1930) Nachr. Ges. Wiss. Goettingen Math-Phys. Kl., 58-76. 17. Prandtl, L. (1932) Ergebnisse Aerodyn. Versuch., Series 4 (Goettingen). 18. Barenblatt, G. I. (1993) J. Fluid Mech. 248, 513-520. 19. Barenblatt, G. I. & Prostokishin, V. N. (1993) J. Fluid Mech. 248, 521-529. 20. Nikuradze, J. (1932) VDI-Forschungs. 356. 21. Zagarola, M. V., Smits, A. J., Orszag, S. A. & Yakhot, V. (1996) AIAA Paper 96-0654.