takashi ishihara, kyo yoshida and yukio kaneda- anisotropic velocity correlation spectrum at small...

8/3/2019 Takashi Ishihara, Kyo Yoshida and Yukio Kaneda- Anisotropic Velocity Correlation Spectrum at Small Scales in a Hom

1/4

VOLUME 88, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 15 APRIL 2002

Anisotropic Velocity Correlation Spectrum at Small Scalesin a Homogeneous Turbulent Shear Flow

Takashi Ishihara,* Kyo Yoshida, and Yukio Kaneda

Department of Computational Science and Engineering, Graduate School of Engineering,

Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan(Received 4 September 2001; published 28 March 2002)

A simple theoretical analysis and direct numerical simulations on 5123 grid points suggest that thevelocity correlation spectrum tensor in the inertial subrange of homogeneous turbulent shear flow at high

Reynolds number is given by a simple form that is an anisotropic function of the wave vector. The tensor

is determined by the rate of the strain tensor of the mean flow, the rate of energy dissipation per unit

mass, the wave vector, and two nondimensional constants.

DOI: 10.1103/PhysRevLett .88.154501 PACS numbers: 47.27.Ak, 05.20.Jj, 47.27.Gs

Since the 1941 Kolmogorov theory, it has been widelyaccepted that the statistics of turbulence at very high Rey-

nolds numbers is universal and isotropic in the small scale

limit. Effect of anisotropy induced by forcing and/or

boundary conditions at large scales should be weaker at

smaller scales. However, recent quantitative studies sug-

gest that the anisotropy at small scales persists in experi-mental and numerical anisotropic turbulence at finite

Reynolds numbers with finite scale range (see, e.g., [1,2]

and the references cited therein). The small scale aniso-

tropy in turbulence is a theoretical challenge. The detailed

information on the anisotropic process of energy transfer,in particular that in the inertial subrange, should be useful

in turbulence modeling for practical problems. The pur-

pose of this Letter is to investigate theoretically and numeri-

cally the quantitative effects of the mean flow on the small

scale anisotropy in the inertial subrange of turbulence.In this paper, we consider the incompressible turbulent

shear flow that obeys the Navier-Stokes equations and has a

simple mean flow profile U; which is given by the simplest

but nontrivial one, i.e., by a linear function of the position

vector x as

Uix Sijxj , (1)

where Sij is a time-independent constant 3 3 3 matrixsatisfying Sii 0. The summation convention is usedfor repeated indices. This study is restricted to the small

scale anisotropy of the second order correlations of the

fluctuating velocity field, since much remains unknown

about them and they are important in practical applications.

The mean flow U given by (1) is compatible with thehomogeneity of single-time statistics; i.e., the statistics

remain homogeneous if they were so at some initial instant.

The single-time two-point correlation of the fluctuating

velocity at points x and x0 then depends on x and x0 only

through the difference r x 2 x0. It is convenient todefine the Fourier transform of this quantity as Qijk, t 2p23

Rd3r uix, tujx 1 r, te

2ik?r.The equation of motion governing the fluctuating field

u contains two types of terms: (i) those representing thedirect coupling ofu and the mean flow, which are bilinear

in U and u; and (ii) those representing the nonlinear cou-pling within the fluctuating field, which are quadratic inu. The mean field U has a spatially small gradient and istemporally coherent, whereas u may have a spatially largergradient and temporally shorter correlation. These can betaken into account and the relative importance of these cou-plings (i) and (ii) is evident when their characteristic timescales are compared. The time scale tS associated with thecoupling (i) is of order 1S, where S maxijjSijj, and in-dependent of the wave number k. However, the time scaletN of eddies with a length scale of that is associated with(ii) is of order u, where u is the characteristic veloc-ity of the eddies. If we assume Kolmogorov scaling u e13, then tN

2e13 1ek213, where e isthe energy dissipation rate per unit mass, and k 1.Thus, tN is dependent on the wave number, unlike tS, andit is much smaller than tS at large wave numbers. This sug-gests that in the inertial subrange, where k is much largerthan the characteristic wave number k

0of energy contain-

ing eddies in the fluctuating velocity field, the nonlinearcoupling between the eddies is more dominant than directinteractions with the mean flow.

The above consideration leads us to assume that thereexists a wave number range with k k0 in fully devel-oped turbulence at high Reynolds numbers such that theenergy spectrum is approximated by

Qijk Q0ij k 1 Q

1ij k , (2)

Q0ij k

Ko

4pe23k2113Pijk , (3)

where Q0ij

is the isotropic Kolmogorov spectrum in the

wave number range k0 k kd . Here, Pijk dij 2

ki kj, ki kik, Ko is the Kolmogorov constant, and kd

e14n234 is the Kolmogorov wave number. The term Q1ij

represents the modification due to the existence of themean flow, and is assumed to be small for small tNtS ,i.e., small dk Sek213.

Let us further assume that (A1) the two-point statistics inthis range are reflection invariant so thatQijkQij2k,which implies the symmetry condition Qijk Qjik;

154501-1 0031-900702 88(15)154501(4)$20.00 2002 The American Physical Society 154501-1


2/4


(A2) for small dk, only the linear terms in dk need tobe retained.

The latter implies that Q1ij is linear in Sab . Therefore,

there exists an isotropic fourth order tensor Cijab such that

Q1ij k CijabkSab . (4)

Since Qij and Q0ij

are divergence free and symmetric,

Q1ij

must also satisfy these conditions. The tensor

Cijabk must therefore satisfy the following constraints:(I) kiCijab k kjCijabk 0, (II) Cijab k

Cjiabk. Under these constraints, Q1ij

k may be

written, without loss of generality, in the form of (4) with

Cijab k ak PiakPjbk 1 PibkPja k

1 bkPijkkakb , (5)

for any traceless tensor Sab . Since Cijab k is indepen-dent ofSab , we assume that C depends only on e and k inthe inertial subrange, following Kolmogorov. The dimen-sional analysis then gives

ak Ae13k2133, bk Be13k2133, (6)

where A and B are universal constants.The anisotropic spectrum Q1

ijk is determined for any

Sab using (4) with (5). In particular, when Sab

Sda1db2, the anisotropic components E12k and E12ii k

are related to A and B according to

E12k 4p

157A 2 Bj ,

1

2E12ii k

4p

152A 1 Bj ,

(7)

where j e13k273S e23k253dk, Eijk Ppk Qijp, E

abij k

Ppk papbQijp, andP

pk denotes the integral or the summation withrespect to p over a spherical shell that satisfiesk 2 12, p # k 1 12.

The turbulence has been assumed above to be almost sta-tionary in the inertial subrange (not at the larger scales).It is not difficult to include the effects of small nonsta-tionarities in the dissipation rate e, te etdetdt.However, it does not contribute to the anisotropic spectrato be considered below and therefore it is neglected in thepresent analysis.

The k2133 dependence of Q1 is in agreement withprevious studies based on dimensional analysis, includingLumley [3]. However, there have been few investigationsof the tensorial dependence ofQ1 on the vector k com-pared to studies of the spectrum scaling. Leslie [4] andYoshizawa [5] did examine the tensorial dependence ofQ1, but their formulations of Q1 are different from

(4) with (5). The quantity Q1ij k that was obtained by

Leslie violates the symmetry condition Qijk Qjik,and does not satisfy the solenoidal condition; the one byYoshizawa is based on extra assumptions of Eulerian di-rect interaction approximation and scale separation, and

the bk term in (5) is missing. The form (4) with (5) isderived solely from assumptions (A1) and (A2) and is newto the authors knowledge. The form (4) with (5) may beFourier transformed with respect to k into real space rep-resentation which belongs to the j 2 sector of the SO(3)decomposition [6]. The present analysis shows that the lin-ear modification due to the existence of the mean shear iscompletely determined by two elements in the sector.

Direct numerical simulation (DNS) data of homoge-neous turbulence with an imposed simple shear mean flow(1) with Sij Sdi1dj2 was analyzed to test the predictionsof (4) with (5) on small scale anisotropy. The number ofgrid points is 5123. Prior to this study, we performed aDNS with 10243 grid points to simulate almost statisticallystationary forced turbulence without a mean flow field, us-ing periodic boundary conditions with periods of 2p ineach of the three Cartesian coordinate directions [7]. Theinitial conditions for our study were generated by cuttingoff the higher wave number components of the 10243 DNSfield. The characteristic parameters of the initial velocityfield are listed in Table I.

Under the presence of a simple shear mean flow, theinitially periodic field remains periodic in the coordinatesystem moving with the mean flow. The position vectorX is related to the position vector x in the fixed Eulerianframe, according to X1 x1 2 Stx2, X2 x2, and X3 x3. The wave vector K in the moving frame is relatedto the wave vector in the fixed frame according to K ?X k ? x. The present DNS fields are simulated using afourth order Runge-Kutta method to advance the time. Analias-free spectral method was used with the wave vectorK and a fast Fourier transform, in which the maximumwave number Kmax of the retained modes is 241. Theretained wave vector space is isotropic in K space, but not

in k space.Two simulations were performed with different shear

rates, S 0.5 and 1.0, up to time St 2.0. Although thesimulation domain in wave vector space is deformed, allthe modes whose wave numbers are smaller than 100 re-main in the domain throughout the simulation time. Thecharacteristic wave number kS e

212S32, defined as thewave number at which dkS 1, is 1.31.5 and 3.33.8for the runs with S 0.5 and S 1.0, respectively. Thedirect coupling to the mean flow is therefore smaller thanthe nonlinear coupling between the eddies of the fluctuat-ing field across almost the entire simulated wave numberrange.

TABLE I. The characteristic quantities of the initial velocityfield. Rl, Taylor microscale Reynolds number; E, total energy;e, energy dissipation rate; L0, integral length scale; l, Taylormicroscale; h, Kolmogorov microscale.

Rl E e L0 l h

284 0.500 7.12 3 1022 1.21 0.143 4.30 3 1023

154501-2 154501-2


3/4


The anisotropic energy spectrum components 2E12kand 212E12ii k obtained from the S 0.5 DNS withtime St 2.0 are shown in Fig. 1. The isotropic compo-nent 12Eiik calculated at the same time is also shownfor comparison. Both the anisotropic components E12kand E12ii k are negative for all but the highest wave num-bers, where the artificial effects of the simulation domaindeformation may have a direct influence on the spectra.

The Reynolds stress 2u1u2 is the integration of2E12kwith respect to k. The negative values ofE12k implythe positive Reynolds stress. Since 12 duiuidt 2Su1u2 . 0, where the viscous term is neglected, theenergy is supplied from the mean flow to the fluctuatingvelocities. The values ofE12k and E

12ii k in Fig. 1 show

an approximate k273 power-law dependence at k 10,which is in agreement with (6). Similar results were alsoobserved when S 1.0 (figure omitted). The spectra inthis range are quasistationary over the simulated time in-terval, except during the initial transient state. This im-plies that the anisotropy is almost stationary, unlike therapid-distortion limit (the linear limit ignoring the nonlin-

ear term) [8].Figure 2 shows the anisotropic components 2E12k

and 212E12ii k normalized by j e13k273S. The

plot suggests that the normalized spectrum E12kj maybe approximated by a constant in the vicinity of k 10for both runs between 1.0 # St # 2.0. Also, the normal-ized spectrum 12E

12ii kj may be approximated by a

constant over the same region, although a slight systematicdecay with time is observed. The mean values of the nor-malized spectra for S 0.5 and St 2.0 are computedby taking the average values over 4 # k # 16. In this re-gion, the energy spectrum Ek exhibits nearly the k253

power law, and the energy flux Pk is nearly constantwith relative deviation from e less than 10%. This gives

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1 10 100

k

1

2Eii(k)

E12(k)

1

2E12ii (k)

k5/3

k7/3

FIG. 1. The isotropic 12Eiik (solid line), anisotropicE12k (dashed line), and 212E

12ii (dotted line) components

of the energy spectrum at S 0.5 and St 2.0. The dot-dashed lines with a slope 273 are the line fits described by (8).

E12k 20.60 6 0.16j ,1

2E12ii k 20.20 6 0.04j ,

(8)

where the error is estimated from the variance. The coef-ficients A and B, estimated from (7) and (8), are

A 20.16 6 0.03, B 20.40 6 0.06. (9)

From (2), (4), and (5), some of the components of

E

ij

mni,j,m,n

1,2,3 are identically zero when Sab

da1db2, due to the symmetry in k space. Among the re-maining components, those that are zero for isotropic tur-bulence are expressed in terms ofA and B as

E1112k E2212k 4p105 13A 2 3Bj ,

E3312k 4p105 23A 2 Bj ,

E1211k E1222k 16p105 22A 1 Bj ,

(10)

E1233k 8p105 A 1 3Bj .

This theoretical prediction can be tested using DNS data,as shown in Fig. 3. There is good agreement between thepredictions and DNS.

Experiments have been performed to investigate theone-dimensional cross spectrum E12k1, which satisfies

0

0.2

0.4

0.6

0.8

1

1.2

1 10 100

E

12

(k)/

k

a) S=0.5,St=1.0S=0.5,St=2.0S=1.0,St=1.0S=1.0,St=2.0

0

0.1

0.2

0.3

0.4

0.5

0.6

1 10 100

1 2E

12

ii(k)/

k

b) S=0.5,St=1.0S=0.5,St=2.0S=1.0,St=1.0S=1.0,St=2.0

FIG. 2. The anisotropic components of the energy spectra (a)2E12k and (b) 212E

12ii k normalized by j e

13k273S.The straight dot-dashed lines indicate the values given by (8).

154501-3 154501-3


4/4


1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1 10 100

k

a) E1112(k)E2212(k)

E3312(k)

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1 10 100

k

b) E1211(k)E1222(k)

E1233(k)

FIG. 3. Simulated tensor components, (a) 2E1112k, 2E

2212k,

2E3312k and (b) 2E1211k, 2E

1222k, 2E

1233k, at S 0.5 and

St 2.0. Straight lines are the estimates using (9) and (10).

R`

0 dk1E12k1

u1u2. From (2),(4) (6), this can beexpressed as E12k1 2C0j1, C0 36p1729 3233A 1 7B in the scaling subrange, where j1 e13k273

1S. Substituting the values ofA and B in (9) to

the above equation gives C0 0.16 6 0.05. This is infairly good agreement with experimental values (0.14)obtained by Wyngaard and Cote [9] and by Saddoughi andVeeravalli [10]. On the other hand, in order to determinethe universal constants A and B completely from experi-ments, measurement of another anisotropic spectrumwhich is linearly independent ofE12k1 is needed.

In the present analysis, the linear assumption (A2) isused. The dimensional analysis used to derive (6) implic-

itly assumes that Cijab is independent of large scale flowstatistics, except e. These are not trivial assumptions. Re-cent studies of anomalous scaling of anisotropy show thatthe effect of the zero modes cannot be excluded a priori[11,12]. The agreement between the present theory and

DNS suggests that the zero modes may not be significantin the problem under consideration.

The assumptions of our theory are consistent with spec-tral closure approximations such as the Lagrangian renor-malized approximation (LRA) [13], which is free fromany ad hoc adjusting parameter and yields the Kolmogorovspectrum for isotropic turbulence. The universal constantsA and B may be estimated using the LRA by assuming

the formulas (2) (6) over the scaling range k0 , k , k1,and then substituting it into the closure equations. A pre-liminary analysis suggests that the estimate is sensitive tok0k for finite but small k0k0.2. Thus, the similarityrange must be quite large to obtain a reliable estimate of theasymptotic values ofA and B at high Reynolds numbersfrom experiments or DNS. The closure analysis based onthe LRA is now underway, and the details will be reportedelsewhere.

Since the gradient of the arbitrary mean flow is locallyconstant at sufficiently small scales, it is tempting to as-sume that the formulas (2) (6) can be applied to generalturbulent shear flows (which is at least formally possible).

The validity of such an assumption is left to be examinedin the future.

The DNSs were performed using a Fujitsu VPP5000/56computer at the Nagoya University Computation Center.This work was supported by the Research for the FutureProgram of the Japan Society for the Promotion of Scienceunder project JSPS-RFTF97P01101.

*Electronic address: [email protected][1] X. Shen and Z. Wahrhaft, Phys. Fluids 12, 2976 (2000).[2] L. Biferale and F. Toschi, Phys. Rev. Lett. 86, 4831 (2001).[3] J. L. Lumley, Phys. Fluids 10, 855 (1967).[4] D. C. Leslie, Developments in the Theory of Turbulence

(Clarendon Press, Oxford, 1973), Chap. 15.[5] A. Yoshizawa Hydrodynamic and Magnetohydrodynamic

Turbulent Flows (Kluwer Academic, Dordrecht, 1998),Chap. 6.

[6] I. Arad, V. S. Lvov, and I. Procaccia, Phys. Rev. E 59,6753 (1999).

[7] T. Ishihara and Y. Kaneda (to be published).[8] A. Townsend, The Structure of Turbulent Shear Flow

(Cambridge University Press, Cambridge, UK, 1976).[9] J. C. Wyngaard and O. R. Cote, Q. J. R. Meteorol. Soc. 98,

590 (1972).[10] S. G. Saddoughi and S. V. Veeravalli, J. Fluid Mech. 268,

333 (1994).[11] I. Arad, L. Biferale, and I. Procaccia, Phys. Rev. E 61,

2654 (2000).[12] K. Yoshida and Y. Kaneda, Phys. Rev. E 63, 016308 (2001).[13] Y. Kaneda, Phys. Fluids 29, 701 (1986).

154501-4 154501-4

takashi ishihara, kyo yoshida and yukio kaneda- anisotropic velocity correlation spectrum at small...

Documents