The simplest decomposition of a turbulent field

Physica D 241 (2012) 284–287

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The simplest decomposition of a turbulent field Massimo Germano ∗ Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torimo, Italy

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Article history: Available online 23 July 2011 Keywords: Turbulence structure Scale interaction

abstract In this paper we will present some recent results concerning the simplest decomposition of a turbulent field. The large scale filtering operator is simply given by the two-point sum in space and the associated fluctuation is given by the two-point difference. In the paper we will present the general properties of this simple decomposition and their particular relations with the subgrid stresses and the generalized central moments associated to a generic filtering operator. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Every approach to turbulence is based on a decomposition of the original turbulent field in two or more contributions. Due to the nonlinearity of the physical phenomena the different modes interact among themselves and exchange energy. The associated nonlinearity of the equations produces coupling terms between the different levels, and the study of such interactions is fundamental both to theory and to the practical computation of turbulent flows. The Reynolds decomposition, the spectral and the wavelet decompositions, the proper orthogonal decomposition, the large eddy decomposition based on hierarchies of filters are different formulations of such strategy. Multilevel and multiscale methods are more and more applied to the study of turbulence, see [1], a general framework for the multiresolution representation of data has been produced by Harten [2] and multilevel closures have been studied and applied [3,4] to the large eddy simulation of turbulent flows. As remarked by Sagaut [5], a multilevel representation of a turbulent field is essentially based on a decomposition yielding to a multiscale representation of the original field. In conclusion the interest for simple decompositions, particularly in the physical space, is presently very high in order to extend the study of turbulent flows to nonhomogeneous situations. It is interesting to notice that a strong impulse to these studies has been produced in the framework of the Large Eddy Simulation of turbulent flows. We remark that the relationship between the velocity increments and the subgrid stresses has been studied in the past by many researchers following different points of view, and we cite in particular the paper by Vreman et al. [6] devoted to the realizability conditions for the subgrid stress tensor, the paper of Eyink [7] where a multiscale gradient expansion of the

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subgrid stress has been developed and the paper of Brun et al. [8] concerning the use of the velocity increments for subgrid scale modeling. We refer to the paper [9] for more details on that and in particular we remark that the simplest Large Eddy Simulation, LES, of a turbulent field is that produced by the two-point average; see [10]. This simple average is a useful tool to understand some peculiar aspects of LES and the duality between the two-point sum and the two-point difference is important as regards the elementary interaction between large and small scales; see [11]. Strictly speaking, LES is a technique to solve under resolved numerical equations. At large it is related to the general problem of the decomposition of scales, and as such is a typical multiscale approach to the study of turbulence. Given a quantity a(x, t ) we define a large scale operator G, that we assume linear and constant preserving, such that

G[a(x, t )] = a¯ (x, 1, t )

(1)

where 1 is a new variable, the length scale introduced by the filter G. A general theory of LES in the physical and the scale space is at present missing, and that could be very important in order to better understand the interaction between them. The simple twopoint sum operator also represents the simplest LES operator, and could be the starting point for such exploration. Another point to remark is that since the fundamental paper of Kolmogorov [12], the statistical properties of the two-point difference have been explored in great detail both in the case of a velocity field and in the case of a passive scalar, but strangely enough the study of the associated two-point sum has not received a similar attention. It is clear that in order to better understand the dependence and the interaction between large and small scales we need more information on the two-point sum, in particular as regards its relations with the two-point difference. The interest for the universal properties of turbulence dates probably to the first studies on it, and the great step on is undoubtedly due to the Kolmogorov assumptions, [12,13]; see [14] for a recent presentation. Obviously the main interest of a researcher is in the

M. Germano / Physica D 241 (2012) 284–287

more general properties, and the idea that the small scales of a turbulent flow are asymptotically provided with some geometric and dynamic universality has been very fruitful and has given plenty of useful results. During the years however some doubts on these assumptions have arisen and a more detailed exploration based on the exact Navier–Stokes equations, conducted in particular by Hill [15–17], have paved the way to a reexamination of many assumptions concerning the isotropy, the homogeneity and the universality of the small turbulent scales; see [18,19]. One point in particular is the object of an extended study, the assumed statistical independence of the large and the small turbulent scales that recent papers have vigorously put on discussion. A simple exact relation equivalent to the Kolmogorov 4/5 law pointed out by Hosokawa [20] and experimentally verified by Kholmyansky and Tsinober [21] put a lot of doubts on the assumed independence of large and small scales under the assumption of isotropic turbulent flows. The same happens for another exact relation derived by the present author, see [22], that is equivalent to the Kolmogorov law for homogeneous fields. In the following we will explore in detail the properties of the two-point sum and the twopoint difference decomposition, the elementary dual faces of the turbulence representation, and in particular we will relate the two point differences with the subgrid stresses and the generalized central moments associated to a generic filtering operator. 2. General properties of the two-point sum and difference operators Let us start with a very simple consideration. It is well known, see for example Papoulis [23], that given two random variables a1 and a2 their sum a¯ and their difference a′ , here normalized for convenience as a¯ =

a1 + a2

a′ =

;

a1 − a2

(2) 2 2 are provided with interesting properties. Let us indicate with σ12 , σ22 and r respectively their variances and their cross correlation

σ12 = ⟨a1 a1 ⟩;

σ22 = ⟨a2 a2 ⟩;

r = ⟨a1 a2 ⟩

(3)

where the angle brackets stand for the statistical average. It is easy to see that

⟨¯aa¯ ⟩ =

σ12 + σ22 4

+

⟨a′ a′ ⟩ =

σ12 + σ22

⟨¯aa′ ⟩ =

σ12 − σ22

4

r 2

−

a(x + 1, t ) = a¯ (x, 1, t ) + a′ (x, 1, t ) = a1 a(x − 1, t ) = a¯ (x, 1, t ) − a′ (x, 1, t ) = a2

a¯ =

a1 + a2 2

a′ =

;

a1 − a2

2 a1 = a¯ + a′ ; a2 = a¯ − a′ a1 b 1 + a2 b 2 = a¯ b¯ + a′ b′ ab = 2 a1 b 1 − a2 b 2 ¯ ′ (ab)′ = = a¯ b′ + ba 2 a1 b1 c1 + a2 b2 c2 abc = 2 = a¯ b¯ c¯ + a′ b′ c¯ + b′ c ′ a¯ + c ′ a′ b¯

(abc )′ =

a1 b1 c1 − a2 b2 c2 2

¯ ′ + b¯ c¯ a′ + c¯ a¯ b′ . = a′ b′ c ′ + a¯ bc

∂a ∂ a¯ ∂a ∂ a¯ = ; = ∂t ∂t ∂ xk ∂ xk  ′   ∂a ∂ a′ ∂a ′ ∂ a′ = ; = ∂t ∂t ∂ xk ∂ xk

(5)

and let us introduce the two-point sum operator S and the twopoint difference operator D defined as

S [a(x, t )] = a¯ (x, 1, t ) =

a(x + 1, t ) + a(x − 1, t )

D [a(x, t )] = a (x, 1, t ) =

2 a(x + 1, t ) − a(x − 1, t )

⟨¯a⟩ = ⟨a⟩;

⟨a′ ⟩ = ⟨a⟩′

2

(10)

where the angle brackets stand for the statistical average. Other important properties related to the derivatives in the physical space x and the scale space 1 are the following

∂ a¯ ∂ a′ = ; ∂ xi ∂ ∆i

∂ a′ ∂ a¯ = ∂ xi ∂ ∆i

∂ 2 a¯ ∂ 2 a¯ = ; ∂ xi ∂ xj ∂ ∆i ∂ ∆j

∂ 2 a′ ∂ 2 a′ = ∂ xi ∂ xj ∂ ∆i ∂ ∆j

(11)

(12)

we have

⟨¯a⟩ = ⟨a⟩ ⟨ab⟩ = ⟨ab⟩ ⟨abc ⟩ = ⟨abc ⟩

(13)

and

⟨a′ ⟩ = 0 ¯ ′⟩ = 0 ⟨(ab)′ ⟩ = ⟨¯ab′ ⟩ + ⟨ba ¯ ′ ⟩ + ⟨b¯ c¯ a′ ⟩ + ⟨¯c a¯ b′ ⟩ = 0. ⟨(abc )′ ⟩ = ⟨a′ b′ c ′ ⟩ + ⟨¯abc

(14)

It is worth noting that the two-point sum and the two-point difference are uncorrelated in the case of homogeneous turbulence

⟨¯aa′ ⟩ = 0 (6)

(9)

and we remark the two-point average and the two-point difference also commute with the statistical operator

⟨a1 ⟩ = ⟨a2 ⟩ ⟨a1 b1 ⟩ = ⟨a2 b2 ⟩ ⟨a1 b1 c1 ⟩ = ⟨a2 b2 c2 ⟩

a2 = a(x − 1, t )

(8)

Let us now consider the commutation of the two-point sum and difference operators with the time and space derivatives. We can write

2 (4)

(7)

and important properties of these operators are the following

and in the particular case of homogeneous turbulence

so that when σ12 = σ22 the sum and the difference are uncorrelated. Moreover, it is easy to see that if a1 and a2 are jointly normal, their sum and their difference in the homogeneous case are not only uncorrelated, but also independent, so that in some sense they are more ‘‘universal’’ than the given variables. Let us now assume that the random variables a1 and a2 are the values that a random turbulent quantity a(x), a component of the velocity field, or the pressure or the temperature assumes at two different points distant 21

′

where the overline and the apex stand for abbreviated writings. We have the reconstruction rules

r

4

a1 = a(x + 1, t );

285

but that does not mean that they are independent. We have

(15)

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M. Germano / Physica D 241 (2012) 284–287

⟨a′ a′ a′ ⟩ = −3⟨¯aa¯ a′ ⟩

(16)

and as remarked by Hosokawa [20] that poses a lot of questions when applied to the longitudinal velocity difference in the inertial range as regards the refined similarity hypothesis of Kolmogorov [13]. The same considerations can be extended to higher order moments, and here we will remark that in the case of homogeneous turbulence many other statistical relations between the two-point sum and the two point difference can be derived. As remarked by Hill [17] the ensemble average commutes with differential operators, and for incompressible flows

∂ un ∂ u′ ∂ u′n ∂ u¯ n ∂ u¯ n = n = = = =0 ∂ xn ∂ xn ∂ ∆n ∂ xn ∂ ∆n

(17)

owing to the fact that in the case of homogeneous turbulence the statistical averages are space independent, we also have

∂⟨aun ⟩ ∂⟨a un ⟩ = =0 ∂ ∆n ∂ ∆n ′

′ ′

∂⟨abu′n ⟩ ∂⟨(ab)′ u′n ⟩ = = 0. ∂ ∆n ∂ ∆n

(18)

In particular from the relation

∂⟨ui ui u′n ⟩ =0 ∂ ∆n

(19)

τ (a, b) = ⟨a′′ b′′ ⟩ τ (a, b, c ) = ⟨a′′ b′′ c ′′ ⟩ τ (a, b, c , d) = ⟨a′′ b′′ c ′′ d′′ ⟩

where the angle brackets stand for the statistical average and the double apex for the statistical fluctuation, a = ⟨a⟩ + a′′ ;

τ (a, b, c , d) = a′ b′ c ′ d′ .

The statistical central moments are very important in the statistical study of turbulence, and we recall that they have been recently generalized for a generic filtering operator in the framework of the Large Eddy Simulation approach [24]. The first three generalized central moments, see [25], are defined as follows for a generic filtering operator

τ (a, b) = ab − a¯ b¯ τ (a, b, c ) = abc − a¯ τ (b, c ) − b¯ τ (c , a) − c¯ τ (a, b) − a¯ b¯ c¯ τ (a, b, c , d) = abcd − a¯ τ (b, c , d) − b¯ τ (c , d, a) − c¯ τ (d, a, b) − d¯ τ (a, b, c ) − a¯ b¯ τ (c , d) − a¯ c¯ τ (b, d) − a¯ d¯ τ (b, c ) − b¯ c¯ τ (a, d) − b¯ d¯ τ (a, c ) − c¯ d¯ τ (a, b) − a¯ b¯ c¯ d¯ .

2

6

p(a) =

2π

∞

ϕ(α)e

−iaα

−

=

α

gα δ(x + rα − ξ)

gα

3 ∏

δ(xk + rαk − ξk )

(26)

k=1

where α

gα = 1.

(27)

In this case we have u¯ i (x) =

∫

G(x − ξ)ui (ξ)dξ =

− α

gα uiα

(28)

where uiα = ui (x + rα )

(29)

and it is easy to derive the following simple relation for the subgrid stress τij 1− 2 α,β

gα gβ dijαβ

(30)

where (21)

(22)

and the generalized probability density function, see [26], associated to a is given by 1

−

τij = ui uj − u¯ i u¯ j =

They spontaneously arise in the generalized formulation of the probability density function and are associated to the generalized centered characteristic function ϕ(α) of the filtered quantity a¯ . In one dimension we can write

  α2 iα 3 ϕ(α) = eiaα = eia¯ α 1 − τaa − τaaa + · · ·

(25)

We remark again that the subgrid stresses can be read as the second order generalized central moments [24,25] associated to a generic filtering operator for the velocity components ui . As such they represent for a large eddy simulation what the Reynolds stresses represent in the case of the Reynolds Averaged Navier Stokes equations, RANS. Let us consider a generic discrete filter given by

− 3. The generalized central moments and their relations with two point differences

(24)

4. The subgrid stresses and their relation with the two-point differences

(20)

that results in a generalization of the Kolmogorov inertial law; see [22].

···

τ ( a, b ) = a′ b ′ τ ( a, b , c ) = 0

α

∂⟨u′i u′i u′n ⟩ ∂⟨¯ui u¯ i u′n ⟩ =− ∂ ∆n ∂ ∆n

∫

b = ⟨b⟩ + b′′ ;

while for the two-point average we have

G(x − ξ) =

we can derive

(23)

dijαβ = (uiα − uiβ )(ujα − ujβ ).

We remark that this expression relates the subgrid stress to the two point differences between different points, and in the case of the simplest discrete filter, the two point sum, we obtain

τij = 0.25(ui1 − ui2 )(uj1 − uj2 )

It is easy to see that owing to the operational properties of the statistical operator we have

(32)

owing to the fact that in this case we have simply g1 = g2 = 0.5. We remark that we can extend this new formulation of the subgrid stress in terms of the velocity increments to all filtering operators. The subgrid stress τij (x) can be explicitly written as

τij (x) =

∫

dα.

−∞

(31)

G(x − ξ)ui (ξ)uj (ξ)dξ

∫ −

G(x − ξ)ui (ξ)dξ

∫

G(x − ξ′ )uj (ξ′ )dξ′

and it is interesting to notice that we can equivalently write

(33)

M. Germano / Physica D 241 (2012) 284–287

τij (x) =

1

∫∫

2

G(x − ξ)G(x − ξ′ )dij (ξ, ξ′ )dξdξ′

(34)

where dij (ξ, ξ′ ) = (ui (ξ) − ui (ξ′ ))(uj (ξ) − uj (ξ′ ))

(35)

due to the fact that

∫

G(x − ξ)dξ = 1.

(36)

If we now introduce the coordinates r and s defined as r = ξ − ξ′ ;

s=

ξ + ξ′

(37)

2

we can write

τij (x) =

1

∫∫



G x−s−

2

r 2



G x−s+

r 2

dij (r, s)drds

(38)

and finally it is easy to verify that the subgrid force fi fi =

∂τij ∂ xj

(39)

is given by fi =

1 2

∫∫



G x−s−

r 2



G x−s+

r  ∂ dij 2

∂ sj

drds.

(40)

5. Conclusions The separation of scales is a central problem in turbulence. Consistently with Kolmogorov the simplest small scale is the two-point difference of a turbulent quantity. Dually the simplest large scale associated to the two-point difference seems to be the two-point sum for a lot of reasons. First, it is well known that, given two random variables, their sum and difference are statistically orthogonal, and in particular if they are jointly normal they are independent. Second, these particular linear combinations represent the simplest decomposition of a turbulent field. We remark that every approach to turbulence is based on a decomposition of the original turbulent field in two or more contributions. Due to the nonlinearity of the physical phenomena the different modes interact among themselves and exchange energy. The associated nonlinearity of the equations produces coupling terms between the different levels, and the study of such interactions is fundamental both to theory and to the practical computation of turbulent flows. In this paper we have explored the properties of such simple decomposition by defining two joint operators, the two-point sum and the two point difference operators. The first one represents the elementary averaging operator, and in the paper we have explored the related algebraic properties and their connection to the elementary fluctuation operator, the two-point difference. We have shown that the

287

generalized central moments, that formally extend to a generic large scale filtering operator the statistical central moments, are strictly related to the two-point differences. In particular the generalized central moments of the second order, the subgrid stresses in the language of the Large Eddy Simulation approach, can be read in terms of these differences and are intimately related to the second order structure functions. References [1] P. Sagaut, S. Deck, M. Terracol, Multiscale and Multiresolution Approaches in Turbulence, Imperial College Press, London, 2006. [2] A. Harten, Multiresolution representation of data: a general framework, SIAM J. Numer. Anal. 33 (1996) 1205. [3] M. Terracol, P. Sagaut, C. Basdevant, A multilevel algorithm for large eddy simulation of turbulent compressible flows, J. Comput. Phys. 167 (2001) 439. [4] T.J.R. Hughes, L. Mazzei, A.A. Oberai, A.A. Wray, The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence, Phys. Fluids 13 (2001) 505. [5] P. Sagaut, Large Eddy Simulation for Incompressible Flows, third ed., Springer, 2005. [6] B. Vreman, B. Geurts, H. Kuerten, Realizability conditions for the turbulent stress tensor in large eddy simulation, J. Fluid Mech. 278 (1994) 351. [7] G.L. Eyink, Multi-scale gradient expansion of the turbulent stress tensor, J. Fluid Mech. 549 (2006) 159. [8] C. Brun, R. Friedrich, C.B. da Silva, A non-linear SGS model based on the spatial velocity increment, Theor. Comput. Fluid Dyn. 20 (2006) 1. [9] M. Germano, A direct relation between the filtered subgrid stress and the second order structure function, Phys. Fluids 19 (2007) 038102/2. [10] M. Germano, The simplest LES, in: Ercoftac Workshop on Direct and Large Eddy Simulation 7, Conference Proceedings, Springer, 2008. [11] M. Germano, The two point average and the related subgrid model, in: Turbulence and Shear Flow Phenomena TSFP-5, Conference Proceedings 1, 2007, pp. 309–314. [12] A.N. Kolmogorov, Dissipation of energy in locally isotropic turbulence, Proc. R. Soc. Lond. Ser. A 434 (1941) 15–17 (English translation). [13] A.N. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech. 13 (1962) 82–85. [14] U. Frisch, Turbulence. The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995. [15] R.J. Hill, Applicability of Kolmogorov’s and Monin’s equations of turbulence, J. Fluid Mech. 353 (1997) 67–81. [16] R.J. Hill, Equations relating structure functions of all orders, J. Fluid Mech. 434 (2001) 379–388. [17] R.J. Hill, Exact second order structure function relationships, J. Fluid Mech. 468 (2002) 317–326. [18] K.R. Sreenivasan, B. Dhruva, Is there scaling in high Reynolds-number turbulence? Progr. Theoret. Phys. Suppl. 130 (1998) 103. [19] U. Frisch, J. Bec, E. Aurell, Locally homogeneous turbulence: is it an inconsistent framework? Phys. Fluids 17 (2005) 081706/4. [20] I. Hosokawa, A paradox concerning the refined similarity hypothesis of Kolmorogov for isotropic turbulence, Progr. Theoret. Phys. 118 (2007) 169. [21] M. Kholmyansky, A. Tsinober, Kolmogorov 4/5 law, nonlocality, and sweeping decorrelation hypothesis, Phys. Fluids 20 (2008) 041704/4. [22] M. Germano, The elementary energy transfer between the two-point velocity mean and difference, Phys. Fluids 19 (2007) 085105/5. [23] A. Papoulis, Probability, Random Variables and Stochastic Processes, McGrawHill, 1965. [24] M. Germano, Turbulence: the filtering approach, J. Fluid Mech. 238 (1992) 325–336. [25] M. Germano, Fundamentals of large eddy simulation, in: Advanced Turbulent Flows Computations, in: R. Peyret, E. Krause (Eds.), CISM Courses and Lectures, vol. 395, Springer, 2000, pp. 81–130. [26] S.B. Pope, Computations of turbulent combustion: progress and challenges, in: Twenty Third International Symposium on Combustion, Plenary Lecture, Orleans, France, July 22–27, 1990.