Multi-fractal formulation of compressible fully developed turbulence: Parabolic-profile approximation for the singularity spectrum

Annals of Physics 322 (2007) 967–976 www.elsevier.com/locate/aop

Multi-fractal formulation of compressible fully developed turbulence: Parabolic-profile approximation for the singularity spectrum Bhimsen K. Shivamoggi

*

International Centre for Theoretical Physics, Trieste, Italy Received 19 April 2006; accepted 31 July 2006 Available online 15 September 2006

Abstract A parabolic-profile approximation (PPA) for the singularity spectrum D (h) in the multi-fractal model for compressible fully developed turbulence (FDT) is considered and is then extrapolated to the Kolmogorov microscale regime. The generalization of Kolmogorov’s ‘‘4/5th law’’ relating the third-order velocity structure function to the mean energy dissipation rate e to compressible FDT is considered. The PPA is also shown to afford, unlike the generic multi-fractal model, an analytical calculation of probability distribution function (PDF) of velocity gradients and to describe intermittency corrections for this PDF that complement those provided by homogeneous-fractal model.  2006 Elsevier Inc. All rights reserved. PACS: 47.27.GS

1. Introduction Spatial intermittency is a common feature of fully developed turbulence (FDT) and implies that turbulence activity at small scales is not distributed uniformly throughout space. This leads to a violation of an assumption (Landau and Lifshitz [1]) in the *

Present address: University of Central Florida, Orlando, FL 32816-1364, USA. Fax: +1 4078236253. E-mail address: [email protected].

0003-4916/$ - see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2006.07.012

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Kolmogorov [2] theory that the statistical quantities show no dependence in the inertial range L  ‘  g on the large scale L (where the external stirring mechanisms are influential) and the Kolmogorov microscale g = (m3/e)1/4 (where the viscous effects become important), e being the mean energy dissipation rate. Spatial intermittency effects can be very conveniently imagined to be related to the multi-fractal aspects of the geometry of FDT (Mandelbrot [3]). The energy dissipation field may then be assumed to be a multi-fractal (Parisi and Frisch [4] and Mandelbrot [5]). The latter idea has received experimental support (Meneveau and Sreenivasan [6]). In the multi-fractal model one stipulates that the fine-scale regime of FDT possesses a range of scaling exponents h 2 I ” [hmin, hmax]. Each h 2 I has the support set SðhÞ  R3 of fractal dimension (also called the singularity spectrum) f(h) such that, as ‘ ) 0, the velocity increment has the scaling behavior dv(‘)  ‘h. The sets SðhÞ are nested so that Sðh0 Þ  SðhÞ for h 0 < h. Experimental data on three-dimensional (3D) incompressible FDT (Meneveau and Sreenivasan [7]) suggested that the singularity spectrum function D (h) around its maximum may be expanded up to second-order via the PPA [7] DðhÞ ¼ Dðh0 Þ þ 12D00 ðh0 Þðh  h0 Þ

2

ð1aÞ

where Dðh0 Þ ¼ 3:

ð1bÞ

(1) turns out to be equivalent to the log-normal model (Monin and Yaglom [8]). Recent analysis [9] of Voyager 1 data on solar wind turbulence over distances 40–85 AU confirmed (1) to be an excellent fit even over this enormous range of distances. On the other hand, Kolmogorov’s [10] celebrated ‘‘4/5th law’’ relating the third-order longitudinal velocity structure function to the mean energy dissipation rate e 3

h½dvð‘Þ i ¼ 45e‘

ð2Þ

is of enormous fundamental significance and serves as an anchor point for theoretical work on inertial-range dynamics as well as intermittency in FDT. (2) is essentially a statement of energy conservation in the inertial range. Compressibility effects on FDT are of importance in modern technological flow processes like waves of supersonic projectiles, hypersonic re-entry vehicles and high-speed internal flows in gas-turbine engines. Astrophysical processes like star formation in selfgravitating dense interstellar gas clouds via Jeans’ stability (Chandrasekhar [11] and Shu [12]) are other cases in point. So, considerable effort has been directed on this problem in the literature (Moiseev et al. [13], Passot and Pouquet [14], Porter et al. [15], Erlebacher et al. [16], Shivamoggi [17–23]). Generalization of the Kolmogorov exact result to compressible FDT has not been given. The previous work on compressible FDT via the multi-fractal model [18] which gave for the characteristic exponent np of the velocity structure function p¼

3c  1 : np ¼ 1 c1

ð3Þ

(c being the ratio of specific heats of the fluid which is assumed to be barotropic—c ) 1 corresponds to the incompressible fluid limit) has indicated (3) to be the possible generalization of the Kolmogorov exact result for compressible FDT. Application of (3) in the

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non-extensive statistical mechanics approach1 to compressible FDT [23] showed that the qualitative aspects of the spatial intermittency effects, in the weak-intermittency limit, are invariant to the inclusion of compressibility effects, as to be expected, hence lending credence to (3). The purpose of this paper is to use the PPA given by (1a) and (1b) for the singularity spectrum D (h) in the multi-fractal model for compressible FDT extrapolated to the Kolmogorov microscale to make a further compelling case for this generalization. The PPA also turns out also to be fruitful in providing an analytical calculation of probability distribution functions (PDF) of velocity gradients (the generic multi-fractal model is known not to afford such a procedure (Shivamoggi [19], Benzi et al. [32] and Shivamoggi [33])). 2. Generalization of the kolmogorov exact result for compressible FDT Consider first the inertial regime in the kinetic energy cascade of compressible FDT. According to the multi-fractal model for the pth order velocity structure function [18], we have Z p Ap  hjdvj i  ‘½phþ3DðhÞ dlðhÞ  ‘np ð4Þ where np ¼ inf ½ph þ 3  DðhÞ

ð5Þ

h

and this minimum h = h*, according to the saddle-point method, corresponds to D0 ðh Þ ¼ p:

ð6Þ

Writing (1a) and (1b) in the form 2

3  Dðh Þ ¼ aðh  h0 Þ ;

ð7Þ

a>0

(6) yields h ðpÞ ¼ h0 

p : 2a

ð8Þ

Using (8), (7) becomes 3  Dðh Þ ¼

p2 4a

ð9Þ

so the zero-intermittency limit corresponds to a ) 1. Using (7) and (8), (5) becomes np ¼ ph0 

p2 : 4a

ð10Þ

1 In dealing with the multi-fractal measures of systems at critical points (the critical point for FDT corresponds to the infinite Reynolds-number limit Nelkin [24], Yakhot and Orszag [25], Eyink and Goldenfeld [26] and Esser and Grossmann [27]), it has been proven useful to extend the Boltzmann–Gibbs thermodynamics by generalizing the concept of entropy to non-extensive regimes (Tsallis [28], see [29] for a recent extensive bibliography on the theory and applications). An alternative useful perspective to spatial intermittency aspects in FDT becomes available on assuming that the underlying statistics follows a non-extensive prescription (Beck [30], see also Shivamoggi and Beck [31]).

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The parameter h0 may now be determined by imposing (3) on (10)     c1 1 3c  1 h0 ¼ þ : 3c  1 4a c  1 Using (11), (10) becomes      c1 1 3c  1 np ¼ pþ p p 3c  1 4a c1 while (8) becomes      c1 1 3c  1 h ðpÞ ¼ þ  2p : 3c  1 4a c1 On rewriting (13a) as      1 2=3 1 2  ð3  2pÞ þ h ðpÞ ¼ þ 3 3c  1 4a c1

ð11Þ

ð12Þ

ð13aÞ

ð13bÞ

we observe that • the velocity-field singularities are stronger in compressible FDT; • the intermittency corrections (which act to make the velocity-field singularities stronger) are, however, smaller in compressible FDT; • in the incompressible fluid limit, c ) 1, (12) and (13) reduce to those given previously (Benzi and Biferale [34]). The first result appears to explain why the kinetic energy spectrum is found to be steeper in compressible FDT (group-theoretical arguments applied to a Hopf-type functional equation formulation [13], direct numerical simulations [14] and [16], scaling arguments applied directly to the Navier–Stokes equations [18]) than that for incompressible FDT. (Equilibrium statistical mechanics of compressible FDT [21] also indicated reduction of the mean kinetic energy of a Fourier mode in compressible FDT.) This result is also in accord with the reduction in the number of degrees of freedom of turbulence caused by compressibility effects ([19] and [33]). On the other hand, on comparing (12) with the multi-fractal result [18]       c1 c1 np ¼ p p  1 3  Dð c1 Þp ð14Þ 3c1 3c  1 3c  1 we obtain

  p 3c  1 Dð c1 Þp ¼ 3  3c1 4a c  1

ð15aÞ

(15a) implies D0 ¼ 3

ð15bÞ

which is also confirmed by (9) that yields Dðh ð0ÞÞ ¼ 3

ð15cÞ

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D (h*(0)) being the fractal dimension of the support of the measure, namely, D0. Thus, in the PPA, the support of the measure is not a fractal; consequently, in the PPA the multifractality manifests itself via the way the measure is distributed rather than the geometrical properties like the support of the set.2 In this sense, the PPA is complementary to the homogeneous-fractal model in describing the intermittency aspects of FDT. We now follow the previous formulations for the incompressible case (Paladin and Vulpiani [35] and Nelkin [36]) and extend the multi-fractal scaling in the compressible case to the Kolmogorov microscale g, where " #1=4 q30 m30 g : ð16Þ fqðgÞg2^eðgÞ Here, m0 ” l/q0, l being the dynamic viscosity, q being the fluid density and q0 being a reference density. We have ([19] and [33])  p Z  1 ½phpþ3DðhÞ ov cþ1 Bp     R 1þðc1Þh dlðhÞ ð17Þ ox where R is the Reynolds number R

ðh^eiL4 =q0 Þ m0

1=3

and ^ is the mean kinetic energy dissipation rate. Saddle-point evaluation of the integral in (17) yields       cþ1 cþ1 1þ ð18Þ h ½p  f 0 ðh Þ ¼ ½ph  p þ 3  Dðh Þ: c1 c1 Using (7), (18) leads to        cþ1 2c cþ1 ah2 þ 2ah þ p  2ah0  ah20 ¼ 0 c1 cþ1 c1

ð19Þ

from which,   "  #1=2 2  c1 c1 2c p þ h0  h ðpÞ ¼  : cþ1 cþ1 cþ1 a

ð20Þ

Imposing the condition B0 ¼ 1

ð21Þ

which, from (17), implies h ð0Þ ¼ h0 we see, from (20), that the negative root is to be discarded and obtain   "  #1=2 2  c1 c1 2c p þ h0  h ðpÞ ¼  : þ cþ1 cþ1 cþ1 a 2

A multi-fractal generalizes, as Mandelbrot [5] clarified, the self-similarity from sets to measures.

ð22Þ

ð23Þ

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On the other hand, using (7) and (18), (17) yields Bp  Rcp where cp  



ð24Þ  c1 ½p þ 2afh ðpÞ  h0 g: cþ1

ð25Þ

In order to determine the parameter h0, the most pertinent framework for the Kolmogorov microscale regime of compressible FDT appears to be imposing the physical condition of the inviscid dissipation of kinetic energy (IDE). This implies from (24) lB2  Rc2 1  const:

ð26Þ

We have from (26) c2  1 ¼ 0: Using (23) and (25), (27) yields     c1 1 3c  1 h0 ¼ þ 3c  1 4a c  1

ð27Þ

ð28Þ

which is identical to (11) that was obtained by imposing the compressible FDT multi-fractal result (3) for the inertial regime. The above result appears to indicate that the IDE has been incorporated into the result (3) in a manner similar to the case with the Kolmogorov exact result in the incompressible case, and hence makes a further compelling case for (3) to be a valid generalization of the Kolmogorov exact result to the compressible barotropic case. The ultimate validity of (3) is, of course, exact derivation from the Navier–Stokes equations for a compressible barotropic fluid a´ la Kolmogorov [10]. But, this is a formidable task which is still some time in the future. Using (28), (23) yields   "   2   #1=2 c1 4cðc  1Þ 1 3c  1 2c p h ðpÞ ¼  þ  : ð29Þ þ cþ1 ð3c  1Þðc þ 1Þ 4a c  1 cþ1 a For large a, (29) gives the following asymptotic result        c1 3c  1 1  p 1 h ðpÞ ¼ þ þO 2 : 3c  1 c1 4a a

ð30Þ

(29) is sketched in Figs. 1a and b for different values of the intermittency parameter a. Figs. 1a and b show that the velocity-field singularities are strengthened in the Kolmogorov-microscale regime (as in the inertial regime—see Eq. (13b)) by the compressibility effects (and of course the intermittency effects). 3. Probability distribution function for the velocity gradient The physical principle underlying the calculation of the intermittency correction to the PDF of the velocity gradient in the PPA turns out to be, however, the same as the one (namely, IDE) underlying the homogeneous-fractal model used in [32]. Noting the scaling behavior of the velocity gradient ([19] and [33])

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a 0.4

0.3

h 0.2

0.1

0.0 0

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

p 0.1

0.2

b 0.4

0.3

h 0.2

0.1

0.0 0

1

2

3

4

5

p 0.1

0.2

Fig. 1. (a) Scaling exponent h*(p) vs. p for the intermittency parameter a = 10 (—, compressible case c = 1.4; - - -, incompressible case). (b) Scaling exponent h* (p) vs. p for the intermittency parameter a = 100 (—, compressible case c = 1.4; - - -, incompressible case).

h 2ð1þc1 Þ

h 1

1þðcþ1Þh 1þðcþ1Þh v s   v0 c1 m0 c1 g

ð31Þ

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B.K. Shivamoggi / Annals of Physics 322 (2007) 967–976

and assuming v0 to be gaussian distributed, i.e. 

v2 0 2 v2 0

P ðv0 Þ  e h i

ð32Þ

we observe 1þðcþ1 c1Þh

v20  s

h 1þc1

ð33Þ

:

So, h*(p) corresponds to h ð~ pÞ where ~ p is now the solution of the equation   1 þ cþ1 pÞ h ð~ c1 ~ p¼ : h ð~ pÞ 1 þ c1 Using (34), and assuming again a to be large, we have from (30),       c1 1 3c  1 1 h ð~ pÞ ¼  þO 2 : 3c  1 12a c  1 a Using (35), the PDF of the velocity gradient ([19] and [33])    99 8 8  cþ1 1þð h ð~ p Þ >> 1h ð~pÞ > > 1h ð~pÞ c1Þ  > > > > h i h ð~pÞ > > > > h ð~ pÞ > >   h ð~pÞ = < > =>
2 P ðs; h ð~ exp  0 pÞÞ  > > > > jsj 2 v0 > > > > > > > > > > ; : > ;> :

ð34Þ

ð35Þ

ð36Þ

becomes   m0 P ðs; h ð~ pÞÞ  jsj



1þ 1 3c1 2 3 72a c1

ð

Þ



i 99 8 8   h 3 4 1 ð3c1Þ 1 3c1 2 > >> ð c1 Þ 3 108a cðc1Þ2 == < >
2 : exp  0 > > 2 v0 ; : > ;> :

ð37Þ

Using (35), (34) gives ~ p¼

  4 1 ð3c  1Þ3 1  þ O 3 108a cðc  1Þ2 a2

ð38Þ

which is the exponent of jsj in the argument of the exponential in (37), as to be expected from (33). Note the accentuation of the non-gaussianity of the PDF due to intermittency. On the other hand, writing (38) as    4 1 3c þ 1 2 ~ 1þ 1þ ð39Þ p¼  3 4a 9cðc  1Þ 3ðc  1Þ note, in the intermittent case (a ; 1), further accentuation of the non-gaussianity of the PDF due to compressibility effects (c ; 1), as also indicated by the homogeneous-fractal model ([19], [33]) which, as pointed out before, is, however, complementary to the PPA in describing the intermittency aspects of FDT.

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4. Discussion The PPA for D (h) appears to have the capacity to provide considerable insight into the qualitative aspects (quantitative aspects could be a different story as was the case with incompressible FDT—see Castaing [37]) of the effects of compressibility in the intermittency problem. This approach appears to be able to point toward a valid generalization of the Kolmogorov exact result to compressible FDT. Further, this approach appears to be able to complement even the generic multi-fractal model in some aspects like an analytical calculation of the PDF of the velocity gradient. Acknowledgments This work was carried out during the author’s visit to the International Centre for Theoretical Physics, Trieste, Italy. The author is thankful to Professor Katepalli Sreenivasan for his valuable remarks and suggestions. References [1] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, second ed., Pergamon Press, 1987. [2] A.N. Kolmogorov, Dokl. Akad. Nauk. SSSR 31 (1941) 19. [3] B. Mandelbrot, in: R. Temam (Ed.), Turbulence and Navier–Stokes Equations, Lecture Notes in Mathematics, vol. 565, Springer-Verlag, 1975. [4] G. Parisi, U. Frisch, in: M. Ghil, R. Benzi, G. Parisi (Eds.), Turbulence and Predictability in Geophysical Fluid Dynamics and Climatic Dynamics, North-Holland, 1985. [5] B. Mandelbrot, in: C.H. Scholz, B. Mandelbrot (Eds.), Fractals in Geophysics, Birkha¨user, 1989. [6] C. Meneveau, K.R. Sreenivasan, J. Fluid Mech. 224 (1991) 429. [7] C. Meneveau, K.R. Sreenivasan, Nucl. Phys. B Proc. Suppl. 2 (1989) 49. [8] A.S. Monin, A.M. YaglomStatistical Fluid Mechanics, vol. 2, MIT Press, 1975. [9] L.F. Burlaga, A.F. Vinas, Physica A 356 (2005) 375. [10] A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 32 (1941) 16. [11] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, 1961. [12] F.H. Shu, The Physics of Astrophysics, vol. II, Gas Dynamics, University Science Books, 1992. [13] S.S. Moiseev, V.I. Petviashvili, A.V. Toor, V.V. Yanovskii, Physica D 2 (1981) 218. [14] T. Passot, A. Pouquet, Eur. J. Mech. B 10 (1991) 377. [15] D.H. Porter, P.R. Woodword, A. Pouquet, Phys. Rev. Lett. 68 (1992) 3156. [16] G. Erlebacher, M.Y. Hussaini, C.G. Speziale, T.A. Zang, J. Fluid Mech. 256 (1993) 443. [17] B.K. Shivamoggi, Phys. Lett. A 166 (1992) 243. [18] B.K. Shivamoggi, Ann. Phys. 243 (1995) 169. [19] B.K. Shivamoggi, Ann. Phys. 243 (1995) 177. [20] B.K. Shivamoggi, Ann. Phys. 253 (1997) 265. [21] B.K. Shivamoggi, Europhys. Lett. 38 (1997) 657. [22] B.K. Shivamoggi, Ann. Phys. 283 (2000) 1. [23] B.K. Shivamoggi, Physica A 318 (2003) 358. [24] M. Nelkin, Phys. Rev. A 9 (1974) 388. [25] V. Yakhot, S.A. Orszag, Phys. Rev. Lett. 57 (1986) 1722. [26] G. Eyink, N. Goldenfeld, Phys. Rev. E 50 (1994) 4679. [27] A. Esser, S. Grossmann, Eur. Phys. J. B 7 (1999) 467. [28] C. Tsallis, J. Stat. Phys. 52 (1998) 479. [29] C. Tsallis, in: M. Gell-Mann, C. Tsallis (Eds.), Non-extensive Entropy—Interdisciplinary Applications, Oxford University Press, 2004. [30] C. Beck, Physica A 277 (2000) 115. [31] B.K. Shivamoggi, C. Beck, J. Phys. A 34 (2001) 4003.

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