(c)2001 american institute of aeronautics & astronautics...

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

A01-31172

AIAA 2001-2645Numerical Simulations of the LagrangianAveraged Navier-Stokes (LANS-oc)Equations for Forced HomogeneousIsotropic Turbulence

Kamran MohseniDivision of Engineering and Applied ScienceCalifornia Institute of TechnologyPasadena, CA 91125

Steve Shkoller and Branko KosovicDepartment of Mathematics,University of California, Davis, CA 95616-8633

Jerrold E. MarsdenDivision of Engineering and Applied ScienceCalifornia Institute of TechnologyPasadena, CA91125

15th AIAA ComputationalFluid Dynamics Conference

11-14 June 2000 / Anaheim, CAFor permission to copy or to republish, contact the American Institute of Aeronautics and Astronautics,

1801 Alexander Bell Drive, Suite 500, Reston, VA, 20191-4344.


AIAA 2001-2645

NUMERICAL SIMULATIONS OF THE LAGRANGIAN AVERAGEDNAVIER-STOKES (LANS-a) EQUATIONS FOR FORCED

HOMOGENEOUS ISOTROPIC TURBULENCEKamran Mohseni

Mail-code 107-81, Division of Engineering and Applied ScienceCalifornia Institute of Technology

Pasadena, CA 91125

Steve Shkoller and Branko KosovicDepartment of Mathematics,

University of California, Davis, CA 95616-8633

Jerrold E. MarsdenMail-code 107-81, Division of Engineering and Applied Science

California Institute of TechnologyPasadena, CA 91125

ABSTRACTThe modeling capabilities of the Lagrangian Aver-aged Navier-Stokes-a equations (LANS-a) is inves-tigated in statistically stationary three-dimensionalhomogeneous and isotropic turbulence. The predic-tive abilities of the LANS-a equations are analyzedby comparison with DNS data. Two different forcingtechniques were implemented to model the energet-ics of the energy containing scales. The resolved flowis examined by comparison of the energy spectra ofthe LANS-a and the DNS computations; further-more, the correlation between the vorticity and theeigenvectors of the rate of the resolved strain tensoris studied. We find that the LANS-a equations cap-tures the gross features of the flow while the waveactivity below a given scale a is filtered by the non-linear dispersion.

1 INTRODUCTIONIn the last three decades computers made highly ac-curate numerical simulations of the Navier-Stokesequations feasible for moderate Reynolds numberflows. The capability of reproducing experimentallyobtained data for various flows has been demon-strated by these simulations; see Rogallo and Moin1

and Canuto et al.2 for an earlier account on theprogress in this area.

Copyright © 2001 by the authors. Published by theAmerican Institute of Aeronautics and Astronautics, Inc. withpermission.

The behavior of small scales in turbulent flows isusually characterized by statistical isotropy, homo-geneity, and universality. Consequently, by inves-tigating the simplest turbulent flow, i.e. isotropichomogeneous turbulence, we hope to understandsmall scale turbulence. Isotropic homogeneous tur-bulence can be categorized as either decaying orforced. While the forced isotropic turbulence is usu-ally designed to be statistically stationary, decayingturbulence is always statistically non-stationary.

Forced isotropic turbulence in a periodic box canbe considered as one of the most basic numericallysimulated turbulent flows. Forced isotropic turbu-lence is achieved by applying isotropic forcing tothe low wave number modes so that the turbulentcascade develops as the statistical equilibrium isreached. Statistical equilibrium is signified by thebalance between the input of kinetic energy throughforcing and its output through the viscous dissipa-tion. Isotropic forcing cannot be produced in a lab-oratory and therefore forced isotropic turbulence isan idealized flow configuration that can be achievedonly via a controlled numerical experiment; never-theless, forced isotropic turbulence represents an im-portant test case for studying basic properties of tur-bulence in a statistical equilibrium.

It is generally believed that the Kolmogorov cas-cade theory3 provides an approximate description ofhomogeneous isotropic turbulence. The almost uni-versal scaling of the dissipation range in Kolmogorovvariables and the fc~5/3 energy spectrum are among

1American Institute of Aeronautics and Astronautics


the most successful predictions in turbulent flows.Direct Numerical Simulation (DNS) of homoge-

neous isotropic turbulence is well established and hasbeen extensively studied in the last three decades.Such computations can be used to probe features,examine statistics of quantities normally not avail-able in conventional experiments, and test new tur-bulence models. The main difficulty in the turbu-lence engineering community is that performing theDNS of typical engineering problems (usually at highReynolds numbers) is very expensive, and thereforeunlikely to happen in the foreseeable future. Thereare methods for simulating turbulent flows whereone does not need to use the brute-force approachof DNS to resolve all scales of motion. Two popularalternatives are the Large Eddy Simulation (LES)models and the Reynolds averaging of the Navier-Stokes equations, known as the (RANS) equations.

In this paper, we shall numerically investigate theLANS-a equations in the subgrid-stress form in-troduced by Marsden and Shkoller (2001); the in-viscid form of these equations on all of E3 wereintroduced in Holm et ai (1998) and on compactmanifolds in Shkoller (1998). Unlike the tradi-tional averaging or filtering approach used for boththe RANS and the LES, wherein the Navier-Stokesequations are averaged, the LANS-a approach isbased on averaging at the level of the variationalprinciple from which the Euler equations are de-rived. Namely, a new averaged action principleis defined. The least action principle then yieldsthe so-called Lagrangian-Averaged Euler (LAE-a)equations. The Lagrangian-Averaged Navier-Stokesequations (LANS-a) are obtained by allowing theLagrangian flow to be stochastic and replacing de-terministic time derivatives with stochastic back-in-time mean derivatives; the viscous diffusion termnaturally arises in this fashion. An anisotropic ver-sion of the LANS-a equations was recently devel-oped by Marsden and Shkoller (2001). The decayinghomogeneous isotropic turbulence was considered inour previous works.4'5 In this study we will focuson modeling the forced homogeneous isotropic tur-bulence using (LANS-a) equations.

This paper is organized as follows. In the nextsection, a short review of the Lagrangian averagingtechnique is presented. The numerical techniqueand issues relating to forcing schemes are brieflydescribed in Section 3. Results of our numericalexperiments are presented in Section 4. A summaryof of our conclusions is given in Section 5.

2 THE LAGRANGIAN-AVERAGEDNAVIER-STOKES EQUATIONS

A detailed derivation of the isotropic and anisotropicLAE-a/LANS-a equations can be found in the ar-ticle by Marsden and Shkoller (2001). However, abrief summary of the relevant background materialon the Lagrangian averaging approach and the re-sulting LAE-a and LANS-a equations is given inMohseni et al. (2001).

The isotropic LANS-a equations in a periodic boxare

dtu + (u - V)u = -gradp + z/Au + Divra(u),divu = 0, (1)

where u is the macroscopic velocity and v is thekinematic viscosity. <Sa(u) is the subgrid stress ten-sor defined by

ra(u) = -<+Vu • Vu

- VuT - VuT - VuVuT - VuT] . (2)

In this equation, a is a length scale of the rapid fluc-tuations in the flow map, below which wave activityis filtered by the nonlinear dispersion. Notice thatthe divergence of the last term on the right handside can be expressed as a gradient of a scalar

Div (VuT - VuT) = grad (VuT : VuT) . (3)

Therefore this term can be absorbed into the modi-fied pressure - p. As with the usual Euler equations,the function p is determined from the incompressibil-ity condition, the pressure satisfies a Poisson equa-tion that is obtained by taking the divergence of themomentum equation.

Our Lagrangian averaging methodology permitsus to use a Lagrangian generalization of G.I. Tay-lor's frozen fluid hypothesis Taylor (1938) as a tur-bulence closure up to O(a3) in our averaging pa-rameter a. This is made possible by averaging andasymptotically expanding at the level of the varia-tional principle, instead of at the level of the momen-tum conservation equations as in the LES and RANSapproaches, and considering the Lagrangian fluctu-ations to be frozen into the mean flow on those timescales which are required to compute temporal ratesof change (time derivatives). The Taylor hypothesisis commonly invoked to compute spatial gradientsin fluid turbulence experiments, and provides a nat-ural turbulent closure in our Lagrangian averagingframework.

American Institute of Aeronautics and Astronautics


3 NUMERICAL METHOD

In this study we focus on the numerical solution ofhomogeneous turbulence. Our computational do-main is a periodic cubic box of side 27T. Given thenumber of grid points and the size of the compu-tational domain, the smallest resolved length scaleor equivalently the largest wave number, kmax, isprescribed. In a three-dimensional turbulent flowthe kinetic energy cascades in time to smaller, moredissipative scales. The scale at which viscous dissi-pation becomes dominant, and which represents thesmallest scales of turbulence, is characterized by theKolmogorov length scale 77. In a fully resolved DNS,the condition fcmoz?7 ^ 1 is necessary for the smallscales to be adequately represented. Consequently,kmax limits the highest achievable Reynolds numberin a direct numerical simulation for a given compu-tational box.

The full range of scales in a turbulent flow for evena modest Reynolds number spans many orders ofmagnitude, and it is not generally feasible to capturethem all in a numerical simulation. On the otherhand, in turbulence modeling, empirical or theoret-ical models are used to account for the net effect ofsmall scales on large energy- containing scales.

Direct Numerical Simulations (DNS) of isotropicflows using pseudospectral methods was pioneeredby the work of Orszag and Patterson6 and Ro-gallo.7 The accuracy in the calculation of the spatialderivatives that appear in the Navier-Stokes equa-tions is improved substantially by using pseudospec-tral technique as compared with the finite differencemethod. The core of the numerical method usedin this study is based on a standard parallel pseu-dospectral scheme with periodic boundary condi-tions similar to the one described in Rogallo (1981).The spatial derivatives are calculated in the Fourierdomain, while the nonlinear convective terms arecomputed in the physical space. The flow fields areadvanced in time in physical space using a fourth or-der Runge-Kutta scheme. The time step was chosenappropriately to ensure numerical stability. To elim-inate the aliasing errors in this procedure the twothirds rule is used, so that the upper one third ofwave modes is discarded at each stage of the fourthorder Runge-Kutta scheme. The resolutions of all ofour DNS computations are 1703 (2563 before dealias-ing). In the next section, the numerical simulationsof forced isotropic homogeneous turbulence based onthe full DNS and LANS-a modeling are presented.

3.1 Forcing schemes

The numerical forcing of a turbulent flow is usuallyreferred to the artificial addition of energy to thelarge scale (low wavenumber) velocity componentsin the numerical simulation. Forcing of the largescales of the flow is often used to generate a statis-tically stationary velocity field, in which the energycascades to the small scales and is dissipated by vis-cous effects. In the statistically stationary state, theaverage rate of energy addition to the velocity fieldis equal to the average energy-dissipation rate.

In using the low wavenumber forcing, we assumethat the time averaged small scale quantities arenot influenced by the details of the energy produc-tion mechanisms at large scales. This assumptionis closer to the truth at high Reynolds numbers,where the energy containing scales are widely sep-arated from the dissipation scales.

The forcing parameters influence both the smallscale quantities and the flow quantity variables suchas the Reynolds number and the Kolmogorov lengthscale; however, a physically plausible forcing schemeshould not influence the small scales independentlyof the flow quantity variables.

In order to achieve isotropic turbulence in sta-tistical equilibrium, we use two well-studied forcingmethods.

3.1.1 Forcing Scheme A

The first method consists of applying forcing overa spherical shell with shell walls of unit width cen-tered at wave number one, such that the total energyinjection rate is constant in time. This forcing pro-cedure was used by Misra and Pullin.8 The forcingamplitude is adjustable via the parameter 8 whilethe phase of forcing is identical to that of the veloc-ity components at the corresponding wave vectors.Th Fourier coefficient of the forcing term is writtenas

(4)

for all the modes in the specified shell. Here, f^ andUi are Fourier transforms of the forcing vector £$ andvelocity, u^ in the momentum conservation equa-tion, N is the number of wave modes that are forced.The above form of forcing ensures that the energyinjection rate, £] £$ • Ui, is a constant which is equalto 5. We chose 8 = 0,1 for all of our runs.



3.1.2 Forcing Scheme BThe second method corresponds to the forcing usedby Chen and Shan9 where wave modes in a sphericalshell |k| = &o of a certain width are forced in such away that the forcing spectrum follows Kolmogorov's-5/3 scaling law

(5)

This type of forcing ensures that the energy spec-trum assumes inertial range scaling starting fromthe lowest wave modes and thus an extended iner-tial range is artificially created. Imposing inertialrange scaling is particularly useful for studying iner-tial range transfer as well as scaling laws at higherwave numbers. We have chosen k$ = 2 and 8 = 0.03in all the runs.

4 THE NUMERICAL EXPERIMENTSThere are three main length scales characterizing anisotropic turbulent flow: the integral scale character-izing the energy containing scales is defined as

dk(6)

where E(k) is the energy spectrum function at thescalar wavenumber k = (k • &)1/2; the Kolmogorovmicroscale representative of dissipative scales is

(7)

where e is the volume averaged energy-dissipationrate; the Taylor microscale characterizing the mixedenergy-dissipation scales is defined as

\(8)

where u is the root mean square value of each com-ponent of velocity defined as

E(k)dk. (9)

The large eddy turnover time T = l/u is the timescale of the energy containing eddies.10 The Tay-lor Reynolds number (defined based on the Taylormicroscale)

(10)

10"

10'2

io-3

io-4

io-5

lO'6

io-7

io-8

io-9

10° 102

Figure 1: The energy spectra at the nondimensionaltime t = 25/0.82 = 30.5 for forcing scheme A.

We expect that simulations which share the samevalue of both ReA and the nondimensional Kol-mogorov length scale k^r] should be identical in astatistical sense.

4.1 Results for Forcing Scheme A

To present the results in a non-dimensional form,we use the integral length scale, /, and the rootmean square of velocity, u. Throughout the forcedsimulations, these two quantities vary significantly;however, as equilibrium is approached, the integrallength scale for the simulation with forcing schemeA approaches the value of approximately / « 0.2,and the root mean square of velocity is u « 0.25. Itfollows, that the corresponding eddy turnover timeis T = l/u = 0.8. The computations are continuedfor more than 30 eddy turnover times. The equilib-rium Taylor Reynolds number in these simulationsis approximately 80.

The energy spectra at the nondimensional timet — 30.5 is depicted in Figure 1. The total kinetic en-ergy energy (1/2(u)) for the DNS and LANS-a com-putations are shown in Figure 2. In these runs theflow is initialized from a fully developed turbulentflow at a higher Reynolds number. Consequently,the total kinetic energy drops as time evolves. Theplot of flatness and skewness of the DNS run (shownin Figure 3) and the plot of the total kinetic energyindicate that after 10 eddy turn over times the flowis essentially in statistical equilibrium.



0.4 r

0.3

0.2

0.1DNS, 1703

10 20 30

Figure 2: Total kinetic energy for forcing scheme A.

I 5

skewness

flatness

-0.2

-0.4

-0.611

-0.8

10 20 30

Figure 3: Flatness and skewness for forcing schemeA in the DNS run.

101

10°

10'1

io-2

io-3

io-4

io-5

lO'6

io-7

10° 102

Figure 4: The energy spectra at the nondimensionaltime t = 3.25/0.23 = 14.1 for forcing scheme B.

4-2 Results for Forcing Scheme BThe integral length scale for this run is approxi-mately I « 0.32, and the root mean square of veloc-ity is u w 1.4. Consequently the eddy turnover timecan be calculated as T — l/u = 0.23. All of the com-putations for the forcing scheme B are continued foralmost 25 eddy turn over time. The Taylor Reynoldsnumber, R\, is approximately 115 for these com-putations. The energy spectra at the nondimen-sional time 14.1 are shown in Figure 4. There isa well developed A:"5/3 region over one decade in thewavenumbers. The LANS-a calculations at the samefull resolution follows the same behavior up to thespatial scale of order a, where the inverse Helmholtzoperator in the LANS-a computations sharply steep-ens the energy cascades to smaller scales. While ittakes a relatively long time for the total kinetic en-ergy to settle (see Figure 5), the values of the skew-ness and the flatness reach an almost statistical equi-librium at around time t w 5, as shown in figure 6.

4-3 Subgrid Transfer and Alignment of Vorticityand Subgrid Stress Eigenvectors

In order to evaluate the performance of any SGSmodel, it is (in principle) possible to compare themodeled SGS stress ra with the corresponding stressobtained by filtering DNS and computing correlationcoefficients between SGS stress components. How-ever, Piomelli et a/.11 demonstrated that such an a



5 r

10 15 20 25

Figure 5: Total kinetic energy for forcing scheme B.

flatness-0.2

-0.4

-0.6 1

-0.8

10 20

Figure 6: Flatness and skewness for forcing schemeB in the DNS run.

priori technique does not account for the dynamiceffects of the subgrid models and therefore cannotrepresent a definitive test of a subgrid model.

With the development of novel experimental tech-niques and the availability of high resolution DNSdata, it is now possible to study basic structuralproperties of subgrid stresses with respect to thelarge scale characteristics of the flow field. A three-dimensional measurement technique such as holo-graphic particle velocimetry (HPIV) was used byTao et a/.12"14 to study the alignment of the eigen-vectors of actual and modeled components of thesubgrid stresses as well as the alignments betweeneigenvectors of the rate of strain tensor and vortic-ity vector. They confirmed the preferred local align-ment of the eigenvector of the rate of strain tensorcorresponding to the intermediate eigenvalue withthe vorticity vector, which was previously observedusing pointwise DNS data.15'16 They also found apreferred relative angle between the most compres-sive eigendirection of the rate of strain tensor andthe most extensive eigendirection of the SGS tensor.

Following the work of Tao et a/., we will discussthe statistics of alignment between the eigenvectorsof the SGS tensor ra, and both the eigenvectors ofthe rate of strain tensor, D, and the unit vorticityvector, u, respectively defined as

D = - (Vu 4- VuT) ;

and

V x u|Vxu |

(11)

(12)

The actual SGS tensor, r, can be computed by fil-tering the DNS data and using the definition

r = (13)

Here, overline denotes filtered quantities. We filterthe DNS data by applying a wave cut-off filter withcut-off wavenumber kc = 21, corresponding to thelargest wavenumber resolved in LANS-a simulationwith 483 grid points.

We denote the eigenvectors of D by [61,62,63],ordered according to the corresponding eigenvalues(Ai, A2, AS), with A3 < A2 < AI- The eigenvectors ofra are [ti,t2,ts] with eigenvalues (71,72,73), suchthat 73 < 72 < 71. Thus, for example, we refer toei as the most extensional eigendirection of D, andto t3 as the most compressive eigendirection of ra.Also, e2, t2 are intermediate eigendirections with



0.5

0.4

0.3

0.2

0.1

DNS -LANS a = 1/8 -

LANS a = 1/16 -

0.5

0.4

0.3

0.2

0.1

DNS •LANS a = l/4-

LANS a = 1/16 -

-15 -10 -5 10 15 -15 -10 10 15

Figure 7: Probability distribution of €sgs = r : D;solid line - Forced DNS 1703 - forcing scheme A,Rex = 80; dotted line - Forced LANS-a 483 witha = 1/8 and a = 1/16 - forcing scheme A. Thescalings are done based on the energy transfer T,sample size AT, and the bin size Aesps.

Figure 8: Probability distribution of esgs = r : D;solid line - Forced DNS 1703 - forcing scheme B,Rex = 80; dotted line - Forced LANS-a 643 witha = 1/4 and a = 1/16 - forcing scheme B. Thescalings are done based on the energy transfer T,sample size JV, and the bin size Aesps.

corresponding intermediate eigenvalues A2, 72, re-spectively.

We remark that commonly used linear,Smagorinsky-type, SGS models implicitly as-sume that the eigenvectors of the SGS stress arelocally aligned with the eigenvectors the eigenvec-tors of the rate of strain tensor, which is contraryto the experimental evidence.

We first used 1703 DNS data obtained using forc-ing scheme A and the data from 483 LANS-a sim-ulation to compute the distribution of the energytransfer, esgs = r : D, due to the contribution ofthe subgrid term. The distributions are presentedin Figure 7. In Figure 8, the same energy transferis computed for the cases with the forcing scheme B.Using the same data sets we computed the distribu-tions of the alignment (cosine of the angles) betweenthe eigenvectors of the rate of strain tensor and vor-ticity vector as well as the SGS tensor.

We first used forcing scheme A and LANS-a sim-ulations with 643 grid points and a = 1/8. Figures 9and 11 indicate that in the LANS-a simulations, thepreferential alignment between the vorticity vectorand the eigendirection of the rate of strain tensor

corresponding to the intermediate eigenvalue is notas pronounced as in DNS. As can be seen in Figures10 and 12 the eigenvectors of the modeled subgridtensor ra display qualitatively similar alignment asthat computed from DNS data. In particular, ex-perimentally observed preferred angle between 63and ti is captured by the LANS-a model. Similarobservations can be made base on the simulationswhere the forcing scheme B was used. For thesecases, probability distributions of the cosine of theangles between eigenvectors and the vorticity vectorare obtained by filtering the DNS data using a wavecut-off filter with cut-off wavenumber kc = 32 andis given in Figures 13 and 14. These results arecompared to LANS-a results corresponding to sim-ulations with 643 grid points and a = 1/16 whichare presented in Figures 15 and 16.



0.4 0.6

u • 62 •w • 63 •

0.8

p(u • ea)

0.2 0.4 0.8

Figure 9: Probability distribution of LJ • ea] ForcedDNS 1703 - forcing scheme A; Re\ = 80.

Figure 11: Probability distribution of u - ea', ForcedLANS-a 483 with a = 1/8 - forcing scheme A.

• 4)

0.2 0.4 0.6

e i •

0.8 0.2 0.4 0.6 0.8

Figure 10: Probability distribution of e\ - £3, e^ • ti, Figure 12: Probability distribution of e\ - T3, ef2 • t^and e3 • t~i; Forced DNS 1703 - forcing scheme A; and e3 • fi; Forced LANS-a 483 with a = 1/8 -

= 80. forcing scheme A.



p(uj •

• e-2 -----

I______I

0.2 0.4 0.6 0.8 1i______I______i______i

0 0.2 0.4 0.6 0.8

Figure 13: Probability distribution of (2 • ea] ForcedDNS 1703 - forcing scheme B; Rex = 80.

Figure 15: Probability distribution of Co • ea; ForcedLANS-a 643 with a = 1/16 - forcing scheme B.

e-2 • tiel - f 3

_j______i i______i0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

Figure 14: Probability distribution of Si • FS, 62 • *2»and 6*3 • ti; Forced DNS 1703 - forcing scheme B;Rex = 80.

Figure 16: Probability distribution of e\ • TS, e*2 • ̂ 2,and e3 - ti; Forced LANS-a 643 with a = 1/16 -forcing scheme B.



5 DISCUSSION AND SUMMARYWe performed a set of forced isotropic turbulencesimulations of the LANS-a equations and studiedtheir equilibrium states. The results of the LANS-asimulations were compared to the results obtainedfrom high-resolution DNS simulations, character-ized by the equilibrium Taylor Reynolds numbersRex = 80 and 115 for two different forcing schemes.We studied the effect of varying the model length-scale a. The key feature of the Lagrangian averagingtechnique is the dispersive (not dissipative) fashionin which it decays energy in inviscid Lagrangian-Averaged Euler (LAE-a) equations. In the LANS-aequations, the dispersive decay is accompanied bythe same viscous dissipation as appears in the orig-inal Navier-Stokes equations. Therefore, as can beseen from equation (1), the LANS-a equations on aperiodic box are given by a subgrid stress additionto the usual Navier-Stokes equations, and this sub-grid stress ra affects the energy transfer via non-linear dispersion. ra depends on the length scalea, and limits the effect of vortex stretching, caus-ing the energy spectrum to fall-off rapidly for scalessmaller than a. The slope of this drop at higherwave-modes can be predicted by an argument simi-lar to Kolmogorov's /c~5/3 in the inertial range.

The equilibrium energy spectrum in our LANS-asimulations were in good agreement with the DNSdata. Our simulations, in agreement with previousresults by Chen et a/.,17 show dependence of theequilibrium turbulence spectra on the model lengthscale a. At the wavenumber fc w I/a the slope ofthe spectra changes from the inertial range slope toa much steeper slope. Even with 1703 grid-pointLANS-a simulations due to the insufficient resolu-tion we were not able to determine the actual slopeand confirm or falsify its existing estimates. Ourstudy of the alignment of the vorticity vector andthe eigenvectors of the rate of strain tensor with theeigenvectors of the subgrid stress tensor showed abasic qualitative agreement of the LANS-a resultswith the filtered DNS data. We observed that theprobability densities of the energy transfer due tothe LANS-a subgrid model is in a good quantita-tive agreement with the corresponding probabilitydensity obtained from the DNS.

REFERENCES[1] R.S. Rogallo and P. Moin. Numerical simula-

tions of turbulent flows. Ann. Rev. Fluid Mech.,16:99-137, 1984.

[2] C. Canuto, M.Y. Hussaini ans A. Quarteroni,and T.A. Zang. Spectral Methods in FluidDynamics. Springer Series in ComputationalPhysics. Springer-Verlag, Berlin, 1988.

[3] A.N. Kolmogorov. The local structure of tur-bulence in incompressibel viscous fluids at verylarge Reynolds numbers. Dokl. Nauk. SSSR.,30:301-305, 1941.

[4] K. Mohseni, B. Kosovic, J. Marsden,S. Shkoller, D. Carati, A. Wray, and B. Ro-gallo. Numerical simulations of homogeneousturbulence using Lagrangian averaged Navier-Stokes equations. In Proceedings of the 2000Summer Program, pages 271-283. NASAAmes/Stanford Univ., 2000.

[5] K. Mohseni, S. Shkoller, B. Kosovic, andJ. Marsden. Numerical simulations of theLagrangian Averaged Navier-Stokes (LANS)equations. In Second International Sympo-sium on Turbulence and Shear Flow Phenom-ena, Stockholm, Sweden, July 2001. KTH,.

[6] S.A. Orszag and G.S. Patterson. StatisticalModels and Turbulence, volume 12 of LectureNotes in Physics, chapter Numerical simulationof turbulence. Springer, New York, 1972.

[7] R.S. Rogallo. Numerical experiments in ho-mogeneous turbulence. In NASA Tech. Memo81315, 1981.

[8] A. Misra and D.I. Pullin. A vortex-basedsubgrid stress model for large-eddy simulation.Phys. Fluids, 9(8):2443-2454, 1997.

[9] S. Chen and X. Shan. High-resolution turbu-lent simulations using the connection machine-2. Comp. Phys., 6(6):2443-2454, 1992.

[10] G.K. Batchelor. The Theory of HomogeneousTurbulence. Cambridge University Press, Cam-bridge, 1953.

[11] U. Piomelli, P. Moin, and J. Ferziger. Modelconsistency in large eddy simulation of turbu-lent channel flows. Phys. Fluids, 31:1884-1891,1988.

[12] B. Tao, J. Katz, and C. Meneveau. Applica-tion of HPIV data of turbulent duct flow forturbulence modelling. In Proceedings of 3rdASME/JSME Joint Fluids Engineering Confer-ence. ASME, June 18-22 1999.



[13] B. Tao, J. Katz, and C. Meneveau. Effectsof strain-rate and subgrid dissipation rate onalignment trends between large and small scalesin turbulent duct flow. In Proceedings of 3rdASME FEDSM'OO, Fluids Engineering Divi-sion Summer Meeting. ASME, June 11-15 2000.

[14] B. Tao, J. Katz, and C. Meneveau. Geometryand scale relationships in high reynolds numberturbulence determined from three-dimensionalholographic velocimetry. Phys. Fluids, 12:941-944, 2000.

[15] R.M. Kerr. Higher-order derivative correlationsand the alignment of small-scale structures inisotropic numerical turbulence. J. Fluid Mech.,153:31-58, 1985.

[16] W.T. Ashurst, A.R. Kerstein, R.M. Kerr, andC.H. Gibson. Alignment of vorticity and scalar-gradient in simulated Navier-Stokes turbulence.Phys. Fluids, 30:2343-2353, 1987.

[17] S.Y. Chen, D.D. Holm, L.G. Margoin, andR. Zhang. Direct numerical simulations of thenavier-stokes alpha model. Physica D, 133:66-83, 1999.


(c)2001 american institute of aeronautics & astronautics...

Documents