spectral modeling of turbulent ﬂows and the role of...

Spectral modeling of turbulent flows and the role of helicity

J. Baerenzung, H. Politano, and Y. PontyLaboratoire Cassiopée, UMR 6202, Observatoire de la Côte d’Azur, B.P. 4229, 06304 Nice Cedex 4, France

A. PouquetTNT/NCAR, P.O. Box 3000, Boulder, Colorado 80307-3000, U.S.A

Received 4 July 2007; revised manuscript received 6 November 2007; published 8 April 2008

We present a version of a dynamical spectral model for large eddy simulation based on the eddy dampedquasinormal Markovian approximation S. A. Orszag, in Fluid Dynamics, edited by R. Balian, Proceedings ofLes Houches Summer School, 1973 Gordon and Breach, New York, 1977, p. 237; J. P. Chollet and M.Lesievr, J. Atmos. Sci. 38, 2747 1981. Three distinct modifications are implemented and tested. On the onehand, whereas in current approaches, a Kolmogorov-like energy spectrum is usually assumed in order toevaluate the non-local transfer, in our method the energy spectrum of the subgrid scales adapts itself dynami-cally to the large-scale resolved spectrum; this first modification allows in particular for a better treatment oftransient phases and instabilities, as shown on one specific example. Moreover, the model takes into accountthe phase relationships of the small scales, embodied, for example, in strongly localized structures such asvortex filaments. To that effect, phase information is implemented in the treatment of the socalled eddy noisein the closure model. Finally, we also consider the role that helical small scales may play in the evaluation ofthe transfer of energy and helicity, the two invariants of the primitive equations in the inviscid case; this leadsas well to intrinsic variations in the development of helicity spectra. Therefore, our model allows for simula-tions of flows for a variety of circumstances and a priori at any given Reynolds number. Comparisons withdirect numerical simulations of the three-dimensional Navier-Stokes equation are performed on fluids drivenby an Arnold-Beltrami-Childress ABC flow which is a prototype of fully helical flows velocity and vorticityfields are parallel. Good agreements are obtained for physical and spectral behavior of the large scales.

DOI: 10.1103/PhysRevE.77.046303 PACS numbers: 47.27.E, 47.27.em, 47.27.ep, 47.27.er

I. INTRODUCTION

Turbulent flows are ubiquitous, and they are linked tomany issues in the geosciences, as in meteorology, oceanog-raphy, climatology, ecology, solar-terrestrial interactions, andfusion, as well as the generation and ensuing dynamics ofmagnetic fields in planets, stars and galaxies due to, e.g.,convective fluid motions. As manifestations of one of the lastoutstanding unsolved problems of classical physics, suchflows form today the focus of numerous investigations.

Natural flows are often in a turbulent state driven by largescale forcing novas explosion in the interstellar medium orby instabilities convection in the sun. Such flows involve ahuge number of coupled modes at different scales leading togreat complexity both in their temporal dynamics and in theiremerging physical structures. Nonlinearities prevail when theReynolds number Rv—which measures the amount of activetemporal or spatial scales in the problem—is large. In theKolmogorov framework 1, the number of degrees of free-dom increases as Rv9/4 for Rv1; for example in geophys-ical flows, Rv is often larger than 108. The ability to probelarge Rv, and to examine in details the large-scale behaviorof turbulent flows depends critically on the ability to resolvesuch a large number of spatial and temporal scales, or else tomodel them adequately.

Only modest Reynolds numbers can be achieved by directnumerical simulation DNS with nowadays computers. Oneway around this difficulty is to resort to large eddy simula-tions or LES, see, e.g., 2–5 and references therein. Suchtechniques are widely used in engineering contexts, as wellas in atmospheric sciences and, to a lesser extent, in geo-

physics see 6 and astrophysics. A specific class of LESmodels is based on two-point closures for turbulent flows,such as the eddy damped quasi normal Markovian approxi-mation, or EDQNM 7. These models, developed in themidseventies, gave rise to successful LES when implement-ing their eddy viscosity formulation 8,9. Such LES tech-niques have been used mostly in conjunction with pseu-dospectral methods, since they are best expressed in Fourierspace in terms of energy spectra; note that these numericalmethods allow for using global energy transfer quantities in-stead of using only local mesh measurement statistics as sub-grid models do in configuration space.

In this paper, we propose a new LES formulation thatgeneralizes the usual EDQNM approach which is based on aKolmogorov k−5/3 spectrum K41 hereafter, by allowing fora priori any kind of energy spectrum as may occur in thecomplex dynamical evolution of various turbulent flows,since our method as will be shown later is based on theevaluation of the transfer terms for energy and helicity. Forexample, there are small intermittency corrections to the K41spectrum due to the presence of strongly localized vortexfilament structures in fluid turbulence; similarly, the presenceof waves may alter the energy spectrum see, e.g., 10. Themethod proposed here may also be particularly importantwhen dealing with magnetohydrodynamics MHD flows,i.e., when coupling the velocity to the magnetic induction. Inthat case, the energy spectra can be either shallower 11 orsteeper than k−5/3, because of anisotropy induced by a uni-form magnetic field leading to Alfvén wave propagation andto weak turbulence for strong magnetic background 12.Similarly, in the case of strong correlations between velocity

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and magnetic fields 13, spectra that differ from the classicalK41 phenomenology may emerge. Note that, although wefocus here on neutral flows, the extension of our model toconductive MHD flows presents no particular difficulties14.

In general, the traditional formulation of turbulent energytransfers only takes into account the energy of the flow butnot its helicity. However, the closure transfer terms for he-licity are well known 9, including in MHD 15. The ki-netic helicity H=1 /2v ·w where w=v is the vorticityrepresents the lack of invariance of the flow by plane sym-metry w is an axial vector. This global invariant of theEuler equation 16 has been little studied until recently see,however, 9,17–19 and references therein. Furthermore, theintermittent structures that populate a turbulent flow at smallscales, namely the vortex filaments, are known to be helicalsee 16,18,19; this implies that the nonlinear transfer termsinvolving small scales are weakened. This is consistent withseveral recent findings, namely i helical vortex tubes, in awavelet decomposition of a turbulent flow into a Gaussiancomponent and a structure component, represent close to99% of the energy and corresponds to the strong tails of theprobability distribution function of the velocity gradients20; ii in a decomposition of the velocity field into large Vand small v components, dropping artificially the nonlocalin scale nonlinear interactions vV leads to less intermit-tency 21, indicating that intermittency involves interactionsbetween structures such as vortex tubes that incorporatesmall scales and large integral scales through a large aspectratio; and iii the spectrum of helicity is close to k−5/3 in theK41 range for energy, but not quite: the relative helicity =Hk /kEk decreases more slowly than 1/k 22,23, indi-cating that the return to full isotropy is not as fast as one mayhave conjectured in the small scales. Finally, helicity is alsoinvoked as possibly responsible for the so-called bottleneckeffect, i.e., the accumulation of energy at the onset of thedissipation range 24, although it is not clear whether thiseffect is or not an inertial range phenomenon 23.

Our dynamical spectral LES model, based on theEDQNM closure, is described in Sec. II. In Sec. III, numeri-cal tests of the model are performed by comparisons withthree-dimensional direct numerical simulations DNS forstrong helical ABC flows 25, as well as with the classicalChollet-Lesieur approach 8. We also intercompare our he-lical and nonhelical models, similarly to other studies per-formed with different helical subgrid-scale models 26, andfollow up with predictions for high Reynolds number flows.Section IV is the conclusion. Finally, details on closure ex-pressions of the nonlinear transfers for energy and helicity,and on the numerical implementation of the model are re-spectively given in Appendixes A and B.

II. MODEL DESCRIPTION

A. Equations

Let us consider the Fourier transform of the velocityvx , t and the vorticity wx , t=vx , t fields at wavevector k,

vk,t = −

vx,te−ik·xdx , 1

wk,t = −

wx,te−ik·xdx . 2

In terms of the Fourier coefficients of the velocity compo-nents, the Navier-Stokes equation for an incompressibleflow, with constant unit density, reads

t + k2vk,t = tvk,t + F

vk,t , 3

where Fv is the driving force, is the kinematic viscosity,and tvk , t is a bilinear operator written as

tvk,t = − iPkk

p+q=kvp,tvq,t; 4

Pk=−kk /k2 is the projector on solenoidal vectors,enforcing incompressibility. In the absence of viscosity, boththe total kinetic energy E=1 /2v2 and the total helicity H=1 /2v ·w are conserved. Such conservation laws constrainthe temporal evolution of the turbulent fluid: the direct cas-cade of energy to the small scales, and the related cascade ofhelicity 27 both stem from these laws. Furthermore, be-cause helicity may be playing an important role in the smallscales see, e.g., 16, and references therein and more re-cently 18,20,24, we are taking in this paper the approachof following the time evolution of both the energy and helic-ity spectra see below. Taking the rotational of Eq. 3 inFourier space leads to

t + k2wk,t = twk,t + F

wk,t 5

with

twk,t = kk

p+q=kvp,tvq,t , 6

Fwk,t = ikF

vk,t . 7

We respectively define the modal spectra of energy Ek , tand helicity Hk , t in the usual way as

Ek,t =1

2vk,t · v*k,t , 8

Hk,t =1

2vk,t · w*k,t , 9

where the asterisks stand for complex conjugates. Note thatw is a pseudoaxial vector, and correspondingly the helicityis a pseudoscalar. The integration of Ek , t and Hk , t overshells of radius k= k respectively gives the isotropic energyEk , t and helicity Hk , t spectra. Their spatiotemporal evo-lutions obey the following equations:

t + 2k2Ek,t = TEk,t + FEk,t , 10

t + 2k2Hk,t = THk,t + FHk,t , 11

where TEk , t and THk , t denote energy and helicity non-linear transfers at wave number k. They are functionals of the

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tensors involving triple correlations between vk , t, vp , t,and vq , t with the constraint that p+q=k due to the con-volution term in Fourier space emanating from the nonlin-earities of the primitive Navier-Stokes equation.

Finally, under the closure hypothesis customary to theEDQNM approach see 9 and references therein, the timeevolution of Ek , t and Hk , t can be described by theEDQNM equations where the exact transfer terms TEk , tand THk , t in the equations above are replaced by the clo-

sure evaluations denoted as TEk , t and THk , t. The closureis done at the level of fourth-order correlators which areexpressed in terms of third-order ones, with a proportionalitycoefficient—dimensionally, the inverse of a time—taken asthe sum of all characteristic rates appearing in a given prob-lem, namely the linear dispersive, nonlinear, and dissipa-tive rates. Tested against DNS 9, these closures allow forexponential discretization in Fourier space and hence for ex-ploration of high Reynolds number regimes. Their drawbackis that all information on moments of the stochastic velocityfield above second order is lost, and phase informationamong Fourier modes is lost as well so that, for example,intermittency is not present in this approach, nor is there anyinformation about spatial structures.

The full formulation of the EDQNM closure leads to a setof coupled integrodifferential equations for the energy andhelicity spectra Ek , t and Hk , t, with the nonlinear trans-fers decomposed into emission terms SE1

, SE3and SH1

, SH3,

and absorption terms SE2, SE4

and SH2, SH4

; note that we useSEi

and SHi, with i 1, 4,, as short-hand notations for the

full spectral functions SEik , p ,q , t and SHi

k , p ,q , t. The ex-pressions of these closure transfer terms are given in Appen-dix A. Note that absorption terms are linear in the spectra,the dynamical evolution of which we are seeking, whereasemission terms are inhomogeneous terms involving the p ,qwave numbers on which the double sum is taken with p+q=k. The absorption term, SE2

, leads to the classical con-cept of eddy viscosity, whereas the emission term, SE1

, is ingeneral modeled as an eddy noise, although it is knownthrough both experiments and DNS that the small scales arefar from following a Gaussian distribution, with substantialwings corresponding to strong localized structures. Here, wepresent a different method to treat the emission term seeSec. II E.

B. Spectral filtering

When dealing with an LES method, as a complement inthe unresolved small scales to the dynamical evolution of thelarge scales following the Navier-Stokes equation, we needto partition Fourier space into three regions; specifically, weneed to introduce a buffer region between the scales that arecompletely resolved above kc

−1, where kc is a cutoff wavenumber depending on the resolution of the LES run, and thescales that are completely unresolved, say beyond akc with aof O1. Following 28 and according to the test field modelclosure, the contribution of subgrid scales, to the explicitlyresolved inertial scales, leads to an eddy viscosity dependingboth on the wave number and the energy spectrum at thatwave number. It is also shown that beyond 2kc, 85% of the

transfer is covered by the eddy viscosity, while beyond 3kc,close to 100% is covered; we thus choose to take a=3.

The truncation of Eqs. 3, 5, 10, and 11 at two dif-ferent wave numbers k=kc and k=3kc gives rise to threetypes of transfer terms, corresponding, respectively, to local,nonlocal, and highly nonlocal interactions where locality re-fers to Fourier space, i.e., to interactions between modes ofcomparable wave number:

i The fully resolved transfer terms TE and TH

involvetriadic interactions such that k, p, and q are all three smallerthan kc; this interval is denoted .

ii The intermediate nonlocal tranfer terms TE,H , in which

p and/or q are contained in the buffer zone between kc and3kc hereafter denoted .

iii The highly nonlocal tranfer terms TE,H , in which p

and/or q are larger than 3kc hereafter denoted .We choose to model TE,H

and TE,H in Eqs. 10 and 11

by appropriately modified EDQNM transfer terms. We there-fore need to know the behavior of both energy and helicityspectra after the cutoff wave number kc=N /2−1, where N isthe linear grid resolution of the LES numerical simulation.Whereas it is customary to assume a k−5/3 Kolmogorov spec-trum in this intermediate range, here we choose a differentapproach: between k=kc and k=3kc, both spectra are as-sumed to behave as power laws with unspecified spectralindices followed by an exponential decrease, viz.

Ek,t = E0k−Ee−Ek, kc k 3kc, 12

Hk,t = H0k−He−Hk, kc k 3kc, 13

where E, E, E0, and H, H, H0 are evaluated, at each timestep, by a mean square fit of the energy and helicity spectra,respectively. Note that it is understood that the Schwarz in-equality HkkEk is fulfilled at all times. When eitherE or H is close to zero, we consider that the energy orhelicity spectrum has an infinite inertial range with a k−E,H

spower law see Eq. 20, so we can write

Ek,t = E0k−E, 3kc k , 14

Hk,t = H0k−H, 3kc k . 15

C. Eddy viscosity

In the context of spectral models for the Navier-Stokesequation, the concept of eddy viscosity was introduced byKraichnan 28. This transport coefficient, denoted k kc , t,allows one to model the nonlinear transfer through a dissipa-tive mechanism, as first hypothesized by Heisenberg. Withthe TE

and TE terms defined above, and where the hat sym-

bol denotes the fact that the EDQNM formulation of thesepartial transfers is taken, the eddy viscosity reads

kkc,t = −TE

k,t + TEk,t

2k2Ek,t= kkc,t + kkc,t ,

16

thus separating the contribution stemming from the bufferzone from those of the outer zone . Note that only

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the part of the transfer proportional to the energy spectrum atwave number k i.e., the SE2

k , p ,q , t term defined in Appen-dix A is taken into account in the derivation of k kc , t.Indeed, in our model, the closure transfer term TE

k , t isintegrated at each time step, but an eddy viscosity from thiswhole transfer term cannot be extracted; only the part thatexplicitely contains Ek , t the linear part of the transfer inEk,t enables the derivation of an eddy viscosity, namely,

kkc,t = −

kpqSE2k,p,q,t

2k2Ek,tdpdq

=

kpqp2

2k2qxy + z3Eq,tdpdq . 17

Let us now evaluate the eddy viscosity in the outer region,k kc , t, coming from the highly nonlocal EDQNM trans-fer terms. Since the k , p ,q triangles are very elongated inthe zone, with k p ,q, an algebraical simplification oc-curs leading to an explicit expression for k kc , t. Indeed,

it has been shown 29 that a Taylor expansion of TEk , t

with respect to k /q leads, at first order, to the followingasymptotic transfer term:

TEk = −

2

15k2Ek

3kc

kpp5Ep + pEp

pdp ,

18

where time dependency is omitted for simplicity. Since now

TEk explicitely depends on Ek, it is straightforward to

formulate the corresponding eddy viscosity k kc , t thusdefined as

kkc,t =1

15

3kc

kpp5Ep + pEp

pdp . 19

When Ep is replaced by its power law exponential decayapproximation see Eq. 14, we recover the so-called“plateau-peak” model 9,

kkc,t 0.315 − E

1 − E

3 − ECK−3/2E3kc,t

3kc1/2

.

20

Finally, in the energy equation, Eq. 10, the total eddy vis-cosity derived from the nonlocal and highly nonlocal transferterms is simply obtained by adding the two contributions, asstated before, k kc , t=k kc , t+k kc , t.

Note that in the helicity equation, Eq. 11, the transportcoefficient stemming from the helicity transfer term THk , tcan be similarly evaluated in the buffer zone and the outerzone, and written as

Hkkc,t = −TH

k,t + THk,t

2k2Hk,t= H

kkc,t + Hkkc,t ,

21

with

Hkkc,t = −

kpqSH2k,p,q,t

2k2Hk,tdpdq . 22

It is straightforward to show that this eddy viscosity term hasthe same formulation as k kc , t see Appendix A for thedefinition of SH2

k , p ,q , t. For the helicity transfer term

THk , t, simple algebraic calculations lead to

THk = −

2

15k2Hk

3kc

kpp5Ep + pEp

pdp .

23

The integrands in Eqs. 18 and 23 are thus identical; thisin turn provides the same eddy viscosity in the outer domainthan for the energy, namely H

k kc , t=k kc , t. Alto-gether, the same total eddy viscosity appears in both the en-ergy and helicity equations. This is expected from the formu-lation of the spectral closure, in which the temporaldynamics of the second-order velocity correlation function isseparated into its symmetric energetic and antisymmetrichelical parts.

D. Helical eddy diffusivity

At wave number k, the energy transfer obtained from theuse of the EDQNM closure involves a linear term in thehelicity spectrum Hk , t specifically, SE4

k , p ,q , t definedin Appendix A, Eq. A5; from this term, a new transportcoefficient, similar to the k kc , t eddy viscosity, can bebuilt. In the buffer zone, this new coefficient, hereafternamed “helical eddy diffusivity,” reads

kkc,t =

kpqSE4k,p,q,t

2k2Hk,tdpdq

=

kpq1

2k2qz1 − y2Hq,tdpdq . 24

Note that, dimensionally, this helical diffusivity scales as /k. As before a total helical eddy diffusivity can be definedas k kc , t= k kc , t+ k kc , t, where k kc , t repre-sents the contribution of the small-scale helicity spectrum tothe kinetic energy dissipation. Recall that, in the outer zone,

the Taylor expansion of the highly nonlocal transfer, TEk , t,

with respect to k /q1, leads at first order to Eq. 18, withno linear contribution from Hk , t. We therefore assume thatthe transfer part associated with helical motions in the outerzone is negligible, so that k kc , t=0. The total helicaleddy diffusivity thus reduces to k kc , t= k kc , t.

E. Emission transfer terms

The parts of the EDQNM transfer terms which are notincluded either in the eddy viscosity or in the helical eddydiffusivity, involve energy and helicity interactions at wavenumbers p and q both larger than kc. Respectively denoted

TEpqk , t and TH

pqk , t, they read

TEpqk,t =

kc

3kc k−p

k+p

kpqtSE1+ SE3

dpdq ,


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THpqk,t =

kc

3kc k−p

k+p

kpqtSH1+ SH3

dpdq , 25

where SEi,Histands for SEi,Hi

k , p ,q , t.Note that, on the one hand, the established eddy viscosity

and helical eddy diffusivity can be directly used in theNavier-Stokes equation for the modal energy and helicityspectra, Ek , t and Hk , t, respectively. On the other hand,in order to implement in these modal equations, the isotropic

transfers TEpqk , t and TH

pqk , t, we assume that they are uni-formly distributed among all k wave vectors belonging to thesame k shell. This means that the nonlocal modal energy and

helicity transfers, respectively, TEpqk , t and TH

pqk , t, can be

expressed as TEpqk , t= TE

pqk , t /4k2 and THpqk , t

= THpqk , t /4k2.

F. Numerical field reconstruction

To compute our LES model for all kkc, we proceed intwo steps. At a given time, the Navier-Stokes equation is firstsolved using the eddy viscosity and the helical eddy diffu-sivity, namely

t + k2vk,t = − iPkk p+q=k

k,p,qkc

vp,tvq

− kkc,tk2vk,t − kkc,tk2wk,t

+ Fvk,t . 26

Then, the effects of the emission terms, TEpqk , t and

THpqk , t, are introduced in the numerical scheme. In most

previous studies, these terms are taken into account througha random force, uncorrelated in time see, e.g., 30; thiscorresponds to the vision that they represent an eddy noiseoriginating from the small scales. However, the small scalesare all but uncorrelated noise; the phase relationships withinthe small-scale structures play an important role, albeit notfully understood, in the flow dynamics. It is well-known thata random field with a k−5/3 Kolomogorov energy spectrum,but otherwise random phases of the Fourier coefficients, isvery different from an actual turbulent flow, lacking, in par-ticular, the strong vortex tubes so prevalent in highly turbu-lent flows. Similarly, it has been recently shown 31 that,upon phase randomization, the ratio of nonlocal energytransfer i.e., the transfer involving widely separated scalesto total energy transfer reduces to a negligible amount,whereas this ratio is close to 20% at the resolutions of theperformed numerical experiments, corresponding to a TaylorReynolds number of about 103. These considerations lead usto directly incorporate the emission terms in the second stepof our numerical procedure. The modal spectra of the energyand the helicity, associated to the vk , t field computed fromEq. 26, now has to verify the following equations wherethe emission transfer terms are taken into account;

t + 2k2Ek,t = − 2kkc,tk2Ek,t − 2kkc,tk2Hk,t

+ TEk,t +

TEpqk,t4k2 + FEk,t , 27

t + 2k2Hk,t = − 2kkc,tk2Hk,t − 2kkc,tk4Ek,t

+ THk,t +

THpqk,t4k2 + FHk,t , 28

where FEk , t and FHk , t denote the spectral terms stem-ming from the driving force. Recall that TE

k , t and THk , t

are the resolved transfer terms based on triadic velocity in-teractions with k, p, and q all smaller than kc. Once theupdated Ek , t and Hk , t modal spectra are obtained, thevelocity field is updated. However, a difficulty immediatelyarises: the phase relationships between the three componentsof the velocity field in the EDQNM and other closures is ofcourse a priori lost. We thus proceed to the reconstruction ofthe three spectral velocity components, written as vk , t=k , teik,t, and rebuild the different velocity phases byusing the incompressibility and realizability Hk , tkEk , t conditions, as explained in Appendix B.

III. NUMERICAL TESTS OF THE MODEL

A. Numerical setup

In order to assess the model accuracy to reproduce thephysics involved in fluid flows, we performed direct numeri-cal simulations DNS of the Navier-Stokes equation andcomputations using our LES formulation. We denote LES Pthe code with partial recovery of phases and without helical

effects i.e., with 0 and THpq0, and LES PH the code

with helical effects incorporated. In our LES description, theenergy spectra—and helicity spectra when considered—ofthe subgrid scales self-adapt to the large scale resolved spec-tra i.e., no spectral scaling laws are prescribed. We cantherefore study a variety of flows, such as either low or highReynolds number flows, or the early phases of the temporaldevelopment of flows when Kolmogorov spectra are not yetestablished. Indeed, the cutoff wave number, kc, can as welllie in the dissipation range instead of the inertial range whichis the case of standard LES approaches based on closurestogether with a k−5/3 Kolmogorov spectrum. This enables ac-curate comparisons with DNS at a given viscosity. We alsocompared our numerical approach to a Chollet-Lesieur LESmodel CL 8, where a =2e−3 kinematic viscosity is addedto the turbulent viscosity for comparison purpose see TableI. The codes use a pseudospectral Fourier method in a 0−23 periodic box and an Adams-Bashforth second-orderscheme in time.

To test the ability of our LES models to reproduce helicalflow features, we focus on flows driven by a prototype Bel-trami flow v= w, namely the ABC flow see, e.g., 25,


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FABCk0 = B cosk0y + C sink0zC cosk0z + A sink0xA cosk0x + B sink0y

, 29

with k0=2, and A=B=C=1.The run parameters are summarized in Table I. The defi-

nitions used for the integral scale, L, and the Taylor micro-scale, , are based on the kinetic energy spectrum Ek; re-spectively, L=2k−1Ekdk /Ekdk and =2Ekdk /k2Ekdk1/2. Note that the characteristicflow quantities—L and scales, rms velocity, Urms, nonlin-ear time scale, NL, and Reynolds number, Rv—are time-averaged quantities once the steady state is achieved for thecomputed flow.

B. Spectral features

We first investigate the flow spectral behavior on one-dimensional energy, enstrophy, and helicity spectra obtainedfrom the different models. These spectra are averaged over67 nonlinear turnover times spanning the flow steady phasefrom t=10.0 up to t=60.0, the final time reached in the simu-lation. Figure 1 shows the time-averaged energy spectraEk , t for LES and DNS computations runs I, II, and III.Both LES results show good agreement with the correspond-ing DNS ones, up to kc=31, the maximum wave number ofthe LES calculations. Small differences are observed in bothLES P and LES PH models at the largest wave numbers.When looking at the vorticity density spectra, these differ-ences are amplified as small scales are emphasized see Fig.2. However, the mean characteristic wave number of thevelocity gradients, defined as the maximum of the time-averaged vorticity density spectrum, corresponds to k=9 inall runs.

Note that for the different flows computed in this paper,the time-averaged Reynolds number and characteristics inte-

gral scale are almost the same see Table I, while the Taylormicroscale, , and its associated Reynolds number, R, differdue to non-negligible intensities of the enstrophy spectrumnamely, k2Ek used to compute , with isotropy assumedafter the cutoff wave number kc. When downsizing the 2563

DNS data to 643 grid points by filtering all wave vectors ksuch as kkc case Ir in Table I, the Taylor small scalequantities obtained from Ir, LES PH II and LES P III runsbecome closer. Thus, our LES models can estimate the meancorrelation length scale of the vorticity when the smallestscales are properly filtered out.

The time-averaged helicity spectra are plotted in Fig. 3.At large scale, the LES P and LES PH models provide aclose approximation of the DNS helicity spectrum up to k20. For k20, the LES PH model, designed to take intoaccount helical effects, slightly overestimates the Hk , tmagnitudes of DNS data, while the LES P model dissipatestoo much helicity.

TABLE I. Parameters of the simulations I to IX. Linear grid resolution N, kinematic viscosity , and time-averaged quantities: Taylormicroscale and integral scale L; rms velocity Urms= v21/2; integral Reynolds number Rv=UrmsL /; eddy turnover time NL=L /Urms; tM

is the final time of integration. Note that the Ir label stands for 643 reduced data obtained from the 2563 DNS computation. The LES P vsPH label stands for our model computations without respectively with incorporating the helicity transport coefficients and emission

transfer terms THpq. The LES CL label stands for a Chollet-Lesieur scheme where the kinematic viscosity is added to the scheme eddy

viscosity see text.

N L Urms Rv NL tM

I DNS 256 5e−3 0.81 2.38 3.19 1525 0.75 60

Ir Reduced DNS 64 5e−3 0.92 2.38 3.19 1530 0.75 60

II LES PH 64 5e−3 0.93 2.37 3.20 1519 0.74 60

III LES P 64 5e−3 0.93 2.40 3.22 1544 0.74 60

IV DNS 512 2e−3 0.49 2.32 3.34 3881 0.70 7.0

V LES PH 128 2e−3 0.59 2.30 3.36 3877 0.69 7.0

VI LES P 128 2e−3 0.59 2.33 3.37 3925 0.69 7.0

VII LES CL 128 2e−3 0.66 2.38 3.29 3928 0.72 7.0

VIII LES CL 160 2e−3 0.61 2.36 3.31 3907 0.71 7.0

IX LES PH 256 5e−4 0.36 2.47 3.38 16693 0.73 10

X LES P 256 5e−4 0.36 2.35 3.31 15565 0.71 10

100

101

10−4

10−3

10−2

10−1

100

k

E(k

)

FIG. 1. Time-averaged energy spectra Ek , t for data I 2563

DNS, solid line, II 643 LES PH, dashed line, and III 643 LES P,dotted line. See Table I.


046303-6

C. Temporal evolution

In this section, to study the temporal behavior of theflows, in the spirit of the analysis performed for freely evolv-ing fluids, we focus on the temporal phase before the steady-state regime. Figure 4 shows the evolution of the global ki-netic energy Et and helicity Ht for DNS together withLES computations, namely runs IV 5123 DNS, V 1283

LES PH, and VI 1283 LES P. We first observe that forboth LES models, with and without helical effects, energiesclosely follow the growth phase of the DNS energy. Indeed,during the inviscid phase, t1.0, the small scales are gener-ated with negligible effects on large scales, since their inten-sities are very weak. Thus, at these times, the LES modelinghas only a reduced action. Later on, during the followinggrowth phase, up to t2.3, the effect of the subgrid scalesonto the resolved ones becomes important, and the LESmodels correctly reproduce the DNS dynamics again. Differ-ences then start to appear between the LES energy approxi-mations and the DNS energy, as all scale intensities increase,and therefore so does the influence of the intermediate andhighly nonlocal transfer terms see Sec. II B. However, theirmean values stay close to the DNS energy see Urms in TableI. The same remarks hold for the temporal evolution of thekinetic helicity. However, in this latter phase, t5, one can

note that the LES PH model provides a slightly better ap-proximation than the LES P one, for both energy and helic-ity. When computing the temporal mean of the relative errorbetween the LES and DNS data, we obtain for the energy1.28% for the helical model versus 1.36% for LES P. Forthe helicity, these errors are respectively 2.20% for LES PHand 2.27% for LES P. Considering that the cost of comput-ing the additional helical term is rather small the LES PHneeds 6% CPU time more than the LES P simulation, theslight improvement when using the helical model is worthconsidering; in particular, note that it reproduces better thetemporal oscillatory variation of the total energy, although ata higher intensity. On the other hand, the fact that the non-helical model performs almost as well shows that helicitydoes not play a significant role in the small scales, in agree-ment with the statistical argument of return to isotropy in thesmall scales, and with the fact that the relative helicity de-cays faster than k−1 in the small scales which are being trun-cated in an LES computation.

The temporal behavior of the LES flows can be under-stood when looking at kinetic energy spectra at early times,as plotted in Fig. 5. Instantaneous DNS energy spectra arewell fitted by LES spectra up to t3.0, including in the

100

101

100

101

k

Ω(k

)

FIG. 2. Time-averaged enstrophy spectra for runs I solid line,II dashed line, and III dotted line. See also Fig. 1.

100

101

10−3

10−2

10−1

100

k

H(k

)

FIG. 3. Time-averaged helicity spectra Hk , t for runs I solidline, II dash, and III dot. See also Fig. 1.

FIG. 4. Evolution of kinetic energy Et, lower curves, and he-licity Ht, upper curves, for runs IV 5123 DNS, solid line, V1283 LES PH, plus signs, and VI 1283 LES P, triangles.

100

101

102

10−10

10−5

100

k

E(k

)

FIG. 5. Temporal development of energy spectra Ek , t shownat t=0.5,1.0,2.0 and t=3.0, from bottom to top, for runs IV DNS,solid line and V LES PH, plus.


046303-7

phase of development toward a Kolmogorov spectrum. Atlarger times, the instantaneous modeled spectra slightly di-verge from the DNS ones, as seen in the steady state in theprevious section, with small scales slightly underestimatedand large scales slightly overestimated.

D. Statistical analysis

Flow statistics are now investigated. Probability densityfunctions hereafter, PDF, are computed from 20 velocitysnapshots extracted each 3NL and spanning the flow steadystates, between t=10.0 and t=60.0. We compare data setsobtained from runs I 2563 DNS, Ir 643 reduced DNS dataand II 643 LES PH, for which we have large velocitysamples. For clarity purpose, since differences between heli-cal and nonhelical models are not visible on the PDF, weonly represent LES PH results versus DNS ones. Figure 6adisplays the statistical distribution of the vx velocity compo-nent, after being normalized so that 2= vx

2=1 being thestandard deviation, together with a Gaussian distribution.The obtained distributions for all data runs are close toGaussian, typical of large scale velocity behavior. The LESmodels being designed to recover correctly the large-scaleflow, the LES PDF are identical to those of the truncatedDNS data set Ir where the smallest DNS scales have beenfiltered out for kkc. Note that they are also very close tothe velocity distribution of the full DNS data set.

Examples of spatial distributions of longitudinal and lat-eral velocity derivatives, vx /x and vy /x, respectively, areshown in Fig. 6b and Fig. 6c. The distributions are closerto an exponential than to a Gaussian. This behavior is evenmore pronounced for lateral derivatives with a slight depar-ture from an exponential law. The wings of the PDF aremainly due to small-scale velocity gradients. Since the DNSflow has more excited small scales, the wings of the associ-ated velocity derivatives distributions are more extendedthan for LES data. The LES distributions correctly repro-duced the DNS ones up to 3, however there is almost nodifferences with the PDF of the Ir data set.

In order to quantify the distributions of the velocity fluc-tuations, and their differences among DNS, LES PH, andLES P data, we compute low-order moments, namely theskewness S3 and flatness S4 factors of the velocity deriva-tives, defined as Sn= fn / f2n/2 where f stands for any ve-locity derivatives. Their temporal means and error bars aregiven in Table II and Table III, respectively. For a fair com-parison with the results of our LES simulations, we alsocompute the skewness and flatness factors based on theIr-reduced DNS velocity fields. The PDF of the longitudinalvelocity derivatives present an asymmetry that yields theirwell-known negative skewness, with S3−0.45 for DNS ve-locities, a value comparable to other simulations at R

500 e.g., 32. The skewness for the reduced DNS datafrom 2563 to 643 is about 22% lower, a reduction simplydue to the truncation of velocity fields in Fourier space asnoted in 33. The LES models give almost identical results,with, in most cases, slightly closer values for LES PH datasets. It remains an open problem to know whether this typeof agreement persists for higher Reynolds numbers. For all

runs, the lateral velocity derivatives are much more symmet-ric, S30 with fluctuations varying from 10−2 to 10−4, as

FIG. 6. Color Mean probability distributions of velocity andits derivative; a vx; b vx /x; c vy /x, normalized so that =1, for data I 2563 DNS, blue line, Ir 643 reduced DNS data,black, and II 643 LES PH, red, shown together with a Gaussiandistribution dotted line.

TABLE II. Temporal mean skewness of velocity derivatives forruns I–III and Ir data see Table I. Error bars are computed frominstantaneous data.

I DNS Ir II LES PH III LES P

vx /x −0.450.05 −0.350.05 −0.350.04 −0.330.05

vy /y −0.450.06 −0.340.04 −0.340.06 −0.340.06

vz /z −0.460.06 −0.350.05 −0.340.07 −0.330.05


046303-8

expected for fields that are almost statiscally isotropic 34.The longitudinal and lateral velocity derivatives do not havethe same S4 flatness factors. For the DNS data, the flatnessvalues for vx /x, vy /y, and vz /z are close to 5, whilethe flatness for the lateral velocity derivatives for example,vx /y and vx /z are larger, with values around 7. Thereduced Ir data present a loss of 20% and 30% for lon-gitudinal and lateral derivatives, respectively. Once again,both LES data provide similar values with a better approxi-mation of the lateral derivatives for the LES PH computa-tion.

E. Visualization in physical space

The topological properties of the different flows are nowinvestigated: comparisons between DNS and LES PH com-putations are carried out on either instantaneous and or meanvelocity fields.

For runs IV 5123 DNS and V 1283 LES PH, Fig. 7displays contour plots of the velocity intensity at time t=2.8, shown on three sides of the periodic box. Although ourLES model cannot exactly reproduce the DNS flow, one cannotice that the main flow structures, and their intensities, arewell represented at times before the statisticaly stationaryregime. More precisely, the mean spatial correlation of thepointwise LES PH velocity field with the DNS one is84.37%, while it is 84.32% in the case with the LES P point-wise velocity at the same time not shown.

For runs I 2563 DNS and II 643 LES PH, Fig. 8 showsone isosurface of mean velocity intensities at a level of 2,i.e., roughly at 63% of the maximum intensity for DNS data,and at 60% for LES PH data. The mean velocity fields aretime averaged each 3NL during the steady states that corre-spond to 67NL for the DNS flow and 68NL for the LES PHrun. With the longest stationary phases, we observe fromthese runs that the mean velocity field, and thus the large andintermediate flow scales, are not alterated by the modeling ofthe transfers linked to the subgrid scales. Indeed, the meanspatial correlation between the pointwise LES and DNS timeaveraged velocity fields is 91.96% for completness, it is92.38% when the LES P mean field is considered.

Finally, the flow isotropy is estimated by means of coef-ficients computed as in 35: for each wave vector k, an

orthonormal reference frame is defined as k / k,e1k / e1k, e2k / e2k, with e1k=kz and e2k=ke1k, where z is the vertical unit wave vector. In thatframe, since the incompressibility condition, say for the ve-locity field, yields k ·vk=0, vk is only determined by itstwo components v1k and v2k. The isotropy coefficient isthen defined as Ciso

v = v12 / v22, with thus a unit valuefor fully isotropic flows. A similar coefficient can be basedon the vorticity field, Ciso

w , characterizing the small scale isot-ropy. Isotropy coefficient values are given in Table IV forvelocity and vorticity fields of the flows visualized in Fig. 7and Fig. 8. Instantaneous values are computed at t=2.8 fordata IV 5123 DNS and V 1283 LES PH Fig. 7, while forthe mean flow shown in Fig. 8, the coefficients are basedon the time-averaged fields of runs I 2563 DNS and II 643

LES PH. For comparison, the isotropy coefficients are alsocomputed with the data of the LES P runs VI 2563 and III643. One can see that, altogether, the isotropic properties ofthe flow are correctly restored by the LES models.

TABLE III. Temporal mean flatness of the velocity gradients forthe same runs as in Table II.

I DNS Ir II LES PH III LES P

vx /x 5.00.2 4.00.2 4.00.3 4.00.3

vy /x 7.20.6 5.00.4 4.90.3 4.80.4

vz /x 7.30.4 5.10.4 4.90.6 4.80.5

vx /y 7.40.4 5.00.3 4.90.5 4.80.5

vy /y 5.10.2 3.90.1 3.90.2 3.90.2

vz /y 7.30.6 5.00.3 4.80.4 4.70.5

vx /z 7.30.6 5.00.5 4.90.5 4.70.4

vy /z 7.30.9 5.00.4 4.80.7 4.80.8

vz /z 5.10.4 4.00.2 3.90.3 3.90.3

FIG. 7. Color Contour plots of the velocity intensity vx , t attime t=2.8 on the boundaries of the periodic box, for run IV 5123

DNS, top and run V 1283 LES PH, bottom.


046303-9

F. Comparison with a Chollet-Lesieur approach

It may be instructive to test our model against anotherspectral LES approach, also based on the EDQNM closure;we have thus performed two simulations using a Chollet-Lesieur scheme 2 run VII and VIII LES CL in Table I.The CL model allows energy tranfer from subgrid to re-solved scales through a dissipation mechanism, with the helpof a dynamical eddy viscosity CLk , t defined as

CLk,t = C+k,t Ekcut,t/kcut, 30

kcut=N /2–3 is the cutoff wave number, N being the numberof grid points per direction, and +k , t is the so-called cuspfunction evaluated as +k , t= 1+3.58k /kcut8. We recallthat C Ekcut , t /kcut is the asymptotic expression of the non-

local tranfers from subgrid to resolved scales, and it assumesa k−5/3 Kolmogorov spectrum extending to infinity. The con-stant C is adjusted with the Kolmogorov constant computedfrom the ABC flow resolved by the DNS run using 5123 gridpoints, this leads to C=0.14. To be able to compare our DNSand LES simulations, runs IV to VI see Table I, to a CLsimulation with 1283 grid points, the kinematic viscosityused for the former runs, =2e−3, is added to CLk , t. Theasymptotic value of the eddy viscosity can be obtained as thetemporal mean of CL0, t in the time interval 3.5,7.0 andis here estimated to be 6e−4.

Both CL and LES PH runs provide a close agreementwith DNS temporal evolutions of kinetic energy and helicityduring the growth phase, i.e., up to t2.3 see Fig. 9. Insteady states, although noticeable deviations occur betweenDNS and LES approximations, a slightly better assessment isvisible for the LES PH model; indeed, the peak of oscillationat t5.4 is weaker in run VII. Note that comparisons of LESPH versus LES P data, runs V and VI respectively, are al-ready presented in Sec. III C.

With our approach, and in particular because of our nu-merical field reconstruction, we have increased the computa-tional cost roughly by a factor 2 when comparing with theclassic LES CL scheme. In order to evaluate the performanceof our model for a given numerical cost, we have also per-

FIG. 8. Color online Isosurface of the mean flow intensityvx , t for a run I 2563 DNS and b run II 643 LES PHplotted at a level of 2, with maximum intensity values of 3.18 forrun I and 3.31 for run II. Averages are taken over 50 computationaltimes spanning the steady states.

TABLE IV. Isotropy coefficients of velocity, Cisov , and vorticity

fields, Cisow , for flows shown in Fig. 7 and Fig. 8. For runs I to III,

values are computed from the mean flows. For runs IV to VI, theyare given at t=2.8. For completeness, the coefficient values are alsogiven for LES P runs.

Cisov Ciso

w

I DNS 0.990 1.010

II LES PH 1.026 0.979

III LES P 0.988 1.009

IV DNS 0.961 1.033

V LES PH 0.955 1.035

VI LES P 0.955 1.029

FIG. 9. Evolution of energy Et, lower curves, and helicityHt, upper curves, for data IV 5123 DNS, solid line, V 1283 LESPH, plus, and VII 1283 LES CL, square.


046303-10

formed a 1603 LES CL simulation VIII and compared themean energy and helicity spectra plotted in Fig. 10 withthose stemming from the runs IV, V and VII see Table I.The relative difference of energy respectively helicity be-tween the DNS and LES runs are emphasized in the inset ofFig. 10a respectively 10b. There is a better energy shellcorrespondence of our LES PH model with the DNS simu-lation than the LES CL displays, whatever the resolutionused Fig. 10a. Our spectral helical model produces onaverage a relative energy deviation not above 15%, whereasin the LES CL model, this deviation can reach the 50% level.Results are less clear for the relative deviation of the helicityspectra, with comparable errors in the large scales, smallererrors at intermediate scales, and a strong overshoot in thesmallest scales for the LES PH model; this may be related tothe slight overshoot of the energy spectrum at the smallestresolved scales of our model.

Energy spectra only exhibit a short k−5/3 inertial range forall runs, including the CL run, due to the procedure we choseavoiding a nonzero asymptotic viscosity. With =0 in the CLscheme, the results might be different. However, for bothenergy and helicity spectra, LES PH data give closer resultswhen compared to DNS data, as our CL simulation seems tooverestimate positive energy transfer from resolved scalesbetween k15 to kcut=61 to subgrid scales.

G. Predictions for high Reynolds number flow

In this last section, we present model computations forflows at high Reynolds number. Recently, Kurien et al. 24showed that for flows with maximum helicity, both energyand helicity spectra exhibit a k−4/3 scaling range followingthe k−5/3 Kolmogorov range and preceding the dissipationrange, a result also found in 36. This change in the energyspectrum is estimated from energy flux based on a character-istic time scale, denoted H, of distortion or shear of eddieswith wave number k submitted to out-of-plane velocity cor-relations, corresponding to helicity transfer. We recall thatthe two dynamical times in competition are estimated byH

2 kHkk2 /2−1 and NL2 kEkk3−1. Moreover,

from DNS of the forced Navier-Stokes equation, these au-thors associate the well-known bottleneck effect with thisscaling change when H becomes physically relevant. TheABC flow being known for the presence of strong helicalstructures, we performed two simulations using our LES PHand LES P models at kinematic viscosities =5e−4, with2563 grid points respectively run IX and run X in Table I.For these flows, the total helicity Ht=1 /2vt ·wt, av-eraged over 8NL in the steady state, is equal to 9.26 fordata set IX and to 8.65 for run X. Temporal means of thetotal relative helicity Ht /Et are also close within 1% forboth computations, namely 0.195 for the LES PH run versus0.186 for the LES P one.

More precisely, in the range 10k100, the relative he-licity Hk /kEk, viewed as an estimation of the ratioH

2 k /NL2 k, lies in between 13.5% and 3.5% for run IX,

and falls from about 14% to 2% for run X see Fig. 11, atypical ratio for strong helical flows 24. Note also that therelative helicity obtained by both models scales closely to ak−1 power law, as already observed in previous DNS experi-ments 36.

Figure 12 displays mean energy spectra for the two mod-eled flows compensated by k5/3 and k4/3, respectively. A k−5/3

scaling appears in the range 4k10, followed by a k−4/3

regime for 10k40, and with no appearance of a bottle-neck effect, although, in the latter wave number interval theestimated time ratio H /NL ranges from 37% to 20%. Simi-

100

101

102

10−4

10−3

10−2

10−1

100

k

E(k

)

DNS: 5123

LES PH: 1283

LES CL: 1283

LES CL: 1603

100

101

102

10−4

10−3

10−2

10−1

100

k

H(k

)

DNS: 5123

LES PH: 1283

LES CL: 1283

LES CL: 1603

FIG. 10. Mean spectra for the energy top and helicity bot-tom. The two insets show the differences between the LES andDNS spectra, normalized by the DNS. Runs IV 5123 DNS, solidline, V 1283 LES PH, dash, VII 1283 LES CL, dot, and VIII1603 LES CL, dashed dotted line.

100

101

102

10−2

10−1

100

k

<|H

(k)|

/kE

(k)

>

FIG. 11. Time-averaged relative helicity Hk /kEk for datasets, IX 2563 LES PH, solid line, and X 2563 LES P, dashedline. A k−1 slope is plotted for comparison dotted line.


046303-11

larly, time-averaged helicity spectra compensated by k5/3 andk4/3 are plotted in Fig. 13. A k−5/3 behavior is seen in approxi-mately the same range than for energy spectra, while thek−4/3 scaling occurs from k10 to k60 for the LES P flowand up to kc, the maximum computational wave number, forthe LES PH flow. Recall that at high wave numbers our LESPH model slightly overestimates helicity spectra, while ourLES P model underestimates them. However, both LESmodels reproduce well the observed spectral behaviors ob-tained from DNS of flows at lower kinematic viscosities =1e−4 and =0.35e−4 in 24.

IV. CONCLUSION

In this paper, we derive a consistent numerical method,based on the EDQNM closure, to model energy interactionsbetween large and small scales for the Navier-Stokes equa-tion. As no spectral behavior is a priori given, our dynamicalLES method allows for the modeling of various flows,whether turbulent or not, compared to former spectral mod-

els which can, in principle, simulate only infinite Reynoldsnumber flows. The phase relationships of the small scales aretaken into account through a numerical reconstructionscheme for the spectral velocity field. Helical effects in tur-bulent flows, such as in vortex filaments, are also consideredthrough the evaluation of the energy and helicity transfers.For this purpose, an “helical eddy diffusivity,” similar to aneddy viscosity, and the emission transfer terms in which thehelicity spectrum appears, are incorporated in a secondmodel. Numerical tests of our two methods, with and withouthelical effects included, are performed against DNS compu-tations. The spectral, statistical and spatial behaviors at largeand intermediate scales of DNS flows are well resaturated inboth modeled flows. We notice some advantages for themodel including helical effects, in particular, concerning theevaluation of the helicity spectra and the probability distri-butions of the lateral velocity field gradients. LES PH alsopredicts a 4 /3 spectrum for the helicity all the way to thecutoff, a point that will need further study.

However, in our approach, we need to calculate at eachtime step the nonlocal energy and helicity transfers, with anincreased computational cost roughly by a factor 2 whencompared with other spectral LES models 8,30. Neverthe-less, our LES model, assuming isotropy and within theframework of the EDQNM closure, takes carefully into ac-count all the energy and helical transfers present in the fluidsystem. Even if the spectral LES techniques are difficult toextend to more realistic geometries and boundaries configu-rations, they are working in spectral space and have access toglobal quantities at each scale, which are the natural math-ematical framework of the turbulence theory. In opposition,local mesh measurement subgrid models could simulatecomplex geometries, but they do not have access to thoseglobal energy transfer informations, present inside any self-similar turbulent flow. Pseudospectral simulations in periodicboxes are intensively used to study the incompressibleNavier-Stokes turbulent dynamics with the highest numericalresolution and accuracy reachable nowadays 37. In order tounderstand and modelize the impact of the small scales onthe large scales, spectral LES techniques are adequate to becompared with those direct numerical simulations at thehighest resolution.

Whereas the role of helicity in fluids is not necessarilydynamically dominant, such is not the case in MHD flowswhere both kinetic and magnetic helicity play a prominentrole; the former in the kinematic dynamo process and thelatter in its undergoing an inverse cascade to large scales.Furthermore, in MHD, energy spectra are not necessarilyKolmogorovian 38 and the models presented in this papermay thus be of some use in MHD as well. The extension toMHD turbulent flows, with coupled velocity and magneticfields, presents no major difficulties and is under study.

ACKNOWLEDGMENTS

We thank P.D. Mininni for useful discussions. This workis supported by INSU/PNST and PCMI Programs andCNRS/GdR Dynamo. Computation time was provided byIDRIS CNRS Grant No. 070597, and SIGAMM meso-center OCA/University Nice-Sophia.

100

101

102

1

2

3

4

5

6

7

8

k

LES PH: k5/3E(k)

LES P: k5/3E(k)

LES PH: k4/3E(k)

LES P: k4/3E(k)

100

101

102

1

2

3

4

5

6

7

8

k

LES PH: k5/3E(k)

LES P: k5/3E(k)

LES PH: k4/3E(k)

LES P: k4/3E(k)

FIG. 12. Temporal mean of energy spectra compensated by k5/3

two solid lines and k4/3 two dash lines for runs IX 2563 LESPH and X 2563 LES P see insert. Horizontal segments indicateranges of k−5/3 and k−4/3 scaling regime.

100

101

102

2

4

6

8

10

12

14

16

k

LES PH: k5/3H(k)

LES P: k5/3H(k)

LES PH: k4/3H(k)

LES P: k4/3H(k)

FIG. 13. Temporal mean helicity spectra compensated by k5/3

and k4/3 for the same runs as in Fig. 12.


046303-12

APPENDIX A: CLOSURE EXPRESSIONS OF TRANSFERTERMS

For completeness, we recall here the expressions of thenonlinear transfer terms for the energy and the helicity,SEk , p ,q , t and SHk , p ,q , t, respectively, under theEDQNM closure assumption 15,

TEk,t =

kpqtSEk,p,q,tdpdq , A1

THk,t =

kpqtSHk,p,q,tdpdq , A2

where is the integration domain with p and q such thatk , p ,q form a triangle, and kpqt is the relaxation time ofthe triple velocity correlations. As usual 9, kpqt is definedas

kpqt =1 − e−k+q+pt

k + q + p, A3

where k expresses the rate at which the triple correlationsevolve, i.e., under viscous dissipation and nonlinear shear. Itcan be written as

k = k2 + 0

k

q2Eq,tdq1/2

. A4

Note that is the only open parameter of the problem, takenequal to 0.36 to recover the Kolmogorov constant CK=1.4.The expressions of SEk , p ,q , t and SHk , p ,q , t can be fur-ther explicited with the time dependency of energy and he-licity spectra omitted here as

SEk,p,q,t =k

pqbk2EqEp − p2EqEk

−k

p3qck2HqHp − p2HqHk

= SE1k,p,q,t + SE2

k,p,q,t + SE3k,p,q,t

+ SE4k,p,q,t . A5

Here, SE1k , p ,q , t, SE2

k , p ,q , t, SE3k , p ,q , t, and

SE4k , p ,q , t are respectively used to denote the four terms

of the extensive expression of SEk , p ,q , t. Note thatSE3

k , p ,q , t and SE4k , p ,q , t are absent in the fully isotro-

pic case without helicity, and, of course, all SHik , p ,q , t

terms below are absent as well,

SHk,p,q,t =k

pqbk2EqHp − p2EqHk

−k3

pqcEpHq − HqEk = SH1

k,p,q,t

+ SH2k,p,q,t + SH3

k,p,q,t + SH4k,p,q,t ,

A6

with short notations analogous to what was used before.

In Eqs. A5 and A6, the geometric coefficientsbk , p ,q and ck , p ,q in short, b and c are defined as

b =p

kxy + z3, c =

p

kz1 − y2 , A7

where x, y, z are the cosines of the interior angles opposite tok ,p ,q.

Let us now introduce a cutoff wave number kc, and definethe following three zones for the integration domain ofEqs. A1 and A2: the inner zone with k, p, and q allsmaller than kc, the cutoff wave number, the buffer zone

with p and/or q between kc and 3kc, and the outer zone

with p and/or q larger than 3kc; then, the boundaries of thetransfer term integrals have to be adapted.

In the inner zone corresponding to the fully resolved flow,the resolved transfers write

TEk,t =

kk

− iPkk0

kkc

vp,tvk − p,tv

− k,tdpdk , A8

THk,t =

kk

kk0

kkc

vp,tvk − p,tv

− k,tdpdk , A9

where Pk=−kk /k2 is the projector on solenoidalvectors, as stated before.

The transfers of energy and helicity between the bufferzone and the inner zone become

TEk,t =

kc

3kc k−p

k+p


THk,t =

kc

3kc k−p

k+p

kpqtSHk,p,q,tdpdq , A11

and the transfers of energy and helicity between the outerzone and the inner zone read

TEk,t =

3kc

k−p

k+p


THk,t =

3kc

k−p

k+p

kpqtSHk,p,q,tdpdq . A13

APPENDIX B: NUMERICAL IMPLEMENTATION OF THEMODEL

As a first step, the Navier-Stokes equation is solved withthe eddy viscosity and the helical eddy diffusivity k kc , tand k kc , t see Eq. 26. At this intermediate stage, weobtain a partial estimation of the time updated velocity field,since the emission transfer terms are not yet taken into ac-count. We compute the corresponding energy and helicitydensity fields from this intermediate velocity, blue v, say at


046303-13

wave vector k. They are then corrected with the appropriateEDQNM emission terms according to Eqs. 27 and 28. Inthe time advance, the next step consists in reconstructing thethree velocity components from these updated energy andhelicity, Ek , t and Hk , t respectively. When the velocitycomponents are expressed as vk , t=k , teik,t, the in-compressibility condition leads to the following system ofequations for the velocity phases and amplitudes in short

and :

23 cos23 =k1

22Ek − 22k1

2 + k22 − 3

2k12 + k3

22k2k3

,

13 cos31 =k2

22Ek − 12k1

2 + k22 − 3

2k22 + k3

22k1k3

,

12 cos12 =k3

22Ek − 12k1

2 + k32 − 2

2k22 + k3

22k1k2

,

23 sin23 =k1

2Hkk2 ,

13 sin31 =k2

2Hkk2 ,

12 sin12 =k3

2Hkk2 , B1

with phase differences k , t=k , t−k , t, and standing for the component indices. Note that when onecomponent of the vector k is equal to zero, this case istreated separately in the code. Since only four of these equa-tions are independent because of the incompressibility con-dition, we are led to give an arbitrary value for two of the

variables. However, the choice of these arbitrary values isconstrained. Indeed, from the set of equations Eqs. B1, wecan derive an existence condition on the amplitudes de-pending on the realizability condition HkkEk andwhich reads with k dependency omitted

1 −k

2

k2E1 − 2 1 −

k2

k2E1 + , B2

with = 1−H2 /k2E2. Thus, the amplitudes can be ex-pressed as

2 =

i2 − 1 −k

2

k2Ei 2

i2+ 1 −

k2

k2E , B3

where the i superscript denotes quantities based on the inter-mediate velocity field, which is a solution of the modifiedNavier-Stokes equation, Eq. 26, with eddy viscosity andhelical eddy diffusivity incorporated, and with i

= 1−Hi2 /k2Ei2. Equation B3 represents a projection of i

computed from the intermediate velocity field and whichdepends on Ei and Hi to obtain the amplitude k , t de-pending now on E and H at the updated time step. This

allows one not to modify the velocity field when TEpq=0 and

THpq=0.

If one or two components of the k wave vector are equalto zero, the set of equations B1 is rewritten from thedivergence-free condition. Apart from this, the reconstructionprocedure is similar.

Finally, to rebuild the different velocity phases, 1 is as-sumed to be fixed to its value given by the intermediatecomponent v1k , t. The set of equations B1 is then solvedand we obtain the updated Fourier velocity field. Note that adifferent choice for the fixed phase leads to no significantchanges in our numerical tests.

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spectral modeling of turbulent ﬂows and the role of...

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