16holm_pp172-180

8/13/2019 16holm_pp172-180

1/9

Darryl D. Holm

G. I. Taylors Contributions to Lagrangian vs EulerianThinking about Turbulence

G. I. Taylors Dispersion Law. An understanding of Lagrangian statistics isof great importance in the ongoing effort to develop both fundamental and practi-cal descriptions of turbulence. For example, Prandtls turbulent mixing lengthcame from a Lagrangian viewpoint: It was envisioned as the turbulent analog of the mean free path of molecules in a gas. In fact, until the famous paperDiffusion by Continuous Movements by G. I. Taylor (1921), most turbulencetheory was discussed exclusively from the Lagrangian viewpoint. However,

despite the obvious importance of the Lagrangian viewpoint in turbulent combus-tion, reacting flows, and pollutant transport, until recently, very few measure-ments of Lagrangian statistics were performed at large Reynolds numbers.Instead, experimentalists performed Eulerian measurements and tried to link thesemeasurements as best they could to the Lagrangian statistics. For example, G. I.Taylor (1921) pursued the idea originating with Prandtl and others that, by anal-ogy with the kinetic theory of gases, one should attempt to find ways of predict-ing statistical properties of the flow by taking measurements at a given point inspace. One of his most influential contributions in this regard was the formula

(1)

This formula links the Lagrangian and Eulerian statistics of turbulence. In thisformula, denotes an appropriate statistical average and the velocity u(t ) withassumed zero mean u(t ) = 0 is defined by the fundamental formula X

.(t , X(0)) =

u(X(t ), t ), as a composition of functions. This is the Eulerian velocity evaluatedalong the Lagrangian trajectory x = X(t , X(0)) whose initial position is X(t = 0,X(0)) = X(0).

Taylors formula is actually a definition, and it is independent of the dynamicsof how a real fluid moves. For example, it does not refer to the Navier-Stokesequations. However, the formula is important because it relates two differenttypes of experimental measurements: Its left side represents the dispersion of Lagrangian traces in the types of flows that can be measuredfor example, byobserving how dye spreads in a turbulent flow or how a bunch of balloons dis-

perses in the wind.1

In contrast, the right side of Taylors formula can be measured by sampling theEulerian velocity field at a single spatial location, then averaging over time, andthereby measuring its velocity correlations.

Taylor argued that the correlation function on the right (Eulerian) side of thisformula specifies the statistical properties of a stationary random function, anidea which had great influence in the subsequent development of statistical treat-ments in turbulence theory and elsewhere. In general, the properties of the(Lagrangian) displacement would depend on the specific trajectory under consid-

172 Los Alamos Science Number 29 2005

Taylors Hypothesis, Hamiltons Principle, andthe LANS- Model for Computing Turbulence

8/13/2019 16holm_pp172-180

2/9

eration. However, Taylor argued for assuming statistical homogeneity of theEulerian velocities, which assumes that the stochastic process generating u(t )does not depend on the initial position X(0) of the trajectory. If, in addition, thestochastic process is statistically stationary, then so are the Eulerian velocity sta-tistics. Thus, one reason for Taylors formula to have been influential was that itmade experimental measurements of Eulerian velocity at a single point seem rele-vant to turbulence. Eulerian measurements are much easier than Lagrangianmeasurements. Averaging the velocity at a fixed location, or comparing velocitiesat two fixed points in space at the same instant is much easier to perform thanmeasuring the motion of fluid parcel trajectories carried in a chaotic flow thenapplying averaging techniques to them. However, Eulerian statistics are notequivalent to Lagrangian statistics, in general, and turbulence modeling musteventually deal with Lagrangian statistics.

G. I. Taylors Microscale and Its Scaling Laws. G. I. Taylor (1921) intro-duced the length scale now called Taylors microscale, which is intermediatebetween the integral scale L and the Kolmogorov dissipation scale . The integralscale L contains the most energy on the average. Due to the nonlinearity of fluid

dynamics, energy cascades from the integral scale down through the inertialrange of smaller scales, until it reaches the Kolmogorov scale, = ( v3 / )1/4 ,where viscous dissipation finally balances nonlinearity in the Navier-Stokes equa-tions. Thus, Kolmogorovs dissipation scale signals the end of the inertial range,and it determines the average size of the smallest eddies, which are responsiblefor the energy dissipation rate effected by the viscosity v. In contrast, Taylorsmicroscale is an intermediate length scale associated with energy dissipationrate, the viscosity and the Eulerian time-mean kinetic energy of the circulationsu2

by Taylors formula

(2)

G. I. Taylor (1921) argued that, dimensionally,

(3)

and if one assumes that viscous energy dissipation may be estimated as

(4)

Number 29 2005 Los Alamos Science 173

Hamiltons Principle and the LANS- Model

1 The Lagrangian statistics for the spread of such passive tracers was first studied quan-titatively by Lewis F. Richardson (1926), in his observation of the spread of ten thousandballoons released simultaneously at the London Expo on a windy day. Each balloon con-tained a note asking the finder to call and tell him the location and time when the ballooncame to Earth. On collecting these observations Richardson obtained the formula,

which implies the Lagrangian dispersion increases with time as < X(t ) 2 > t 3. Thisfamous Richardson Dispersion Law still challenges researchers in turbulence for manyreasons, not least because it shows that the dispersion properties of turbulence are anom-alous (non-Gaussian). This is one indication of the intermittency of turbulence. (In con-trast, ordinary diffusion due to Gaussian random motion would yield the linear timedependence < X(t ) 2 > t for the dispersion of particles.)

8/13/2019 16holm_pp172-180

3/9

(5)

where Re = L4/3 1/3 / v is the Reynolds number based on the integral scale.A similar estimate yields the well-known formula

(6)

for the ratio of Kolmogorovs dissipation scale to the integral scale. Thus, at agiven Reynolds number Re (at the integral scale), Taylors microscale exceedsKolmogorovs dissipation scale by the factor

(7)

A physical interpretation of Taylors microscale has recently emerged in the con-text of Lagrangian-averaged computational turbulence models. In particular, theLANS- model is parameterized by the length scale , which is the mean corre-lation length of a Lagrangian trajectory with its own running time average.Remarkably, the Lagrangian-averaged dynamics of the LANS- model achieves

a balance between its modified nonlinearity and its viscous dissipation, occurringat a length scale that has precisely the same Reynolds scaling as Taylorsmicroscale. Before explaining this result, we need to review another of Taylorscontributions linking Lagrangian statistics to the experimental interpretation of Eulerian measurements in turbulence.

G. I. Taylors 1938 Frozen-in Turbulence Hypothesis. G. I. Taylor (1938)made the hypothesis that, because turbulence has high power at large lengthscales, the advection contributed by the turbulent circulations themselves must besmall, compared with the advection produced by the larger integral scales, whichcontain most of the energy. Therefore, in such a situation, the advection of a fieldof turbulence past a fixed point can be taken as being mainly due to the larger,energy containing scales. This is the frozen-in turbulence hypothesis of G. I.Taylor. Although only valid when the integral scales have sufficiently high powercompared with the smaller scales, this hypothesis delivered another very conven-ient linkage between the Eulerian and Lagrangian viewpoints of turbulence.Taylors hypothesis holds, provided u2

8/13/2019 16holm_pp172-180

4/9

data taken from single-point spatial measurements into their corresponding inter-pretation as temporal data, and vice versa. (Other approaches, such as two-pointspatial measurements, must be used when the assumptions of Taylors hypothesisbreak down.)

Using the Frozen-in Turbulence Hypothesisin a Turbulence Closure

Lagrangian averaging and the corresponding adaptation of Taylors hypothesisof frozen-in turbulence circulations was used in Chen et al. (1998) to derive theclosed system of Lagrangian-averaged Navier-Stokes- (LANS- ) equations.This work treated the Lagrangian average of the exact flow as the large scaleflow into which the turbulence circulations are frozen. Thus, Lagrangian averag-ing was first used to find a decomposition of the exact Navier-Stokes flow into itsLagrangian mean and rapidly circulating parts. Then Taylors hypothesis wasused as a closure approximation.

Lagrangian averaging of fluid equations is a standard technique, which is

reviewed, for example, in Andrews and McIntyre (1978). However, Lagrangianaveraging does not give closed equations. That is, it does not give equationsexpressed only in terms of Lagrangian-averaged evolutionary quantities.Something is always left over, which must be modeled when averaging nonlineardynamics. This is because the average of a product is not equal to the product of the averages, regardless of how one computes the averages. This difficulty is theLagrangian-average version of the famous closure problem in turbulence.

The approach used in Chen et al. (1999) for deriving the closed Eulerian formof the inviscid convection nonlinearity in the LANS- equations was based oncombining two other earlier results. First, the Lagrangian-averaged variationalprinciple of Gjaja and Holm (1996) was applied for deriving the inviscid aver-aged nonlinear fluid equations, which had been obtained by averaging Hamiltonsprinciple for fluids over the rapid phase of their small turbulent circulations at afixed Lagrangian coordinate. Second, the Euler-Poincar theory for continuummechanics of Holm, Marsden, and Ratiu (1998) was used for handling theEulerian form of the resulting Lagrangian-averaged fluid variational principle.Next, Taylors hypothesis of frozen-in turbulence circulations was invoked forclosing the Eulerian system of Lagrangian-averaged fluid equations. Finally, theNavier-Stokes Eulerian viscous dissipation term was added, so that viscositywould cause diffusion of the newly defined Lagrangian-average momentum andproper dissipation of its total Lagrangian-averaged energy.

Gjaja and Holm had earlier derived (1996) a Lagrangian-average wave, mean-flow turbulent description, which allowed the turbulent circulations to propagaterelative to the fluid. However, this Lagrangian-mean description was accom-plished at the cost of adding complication in the form of self-consistent additional

dynamical equations for the Lagrangian statistics of this type of turbulence. Theuse in Chen et al. (1998) of Taylors hypothesis of frozen-in turbulence circula-tions simplified the description of the Lagrangian statistics, by assuming it isswept along by the Eulerian mean flow. Following the assumption that theseLagrangian statistics are homogeneous and isotropic, we and colleagues derivedthe new LANS- turbulence equations with only one additional (constant) param-eter, which is the length scale alpha.

According to the theory, alpha is the mean correlation length of a Lagrangiantrajectory with its own running time average, at fixed Lagrangian label.



8/13/2019 16holm_pp172-180

5/9

Practically speaking, the quantity alpha is the length scale in isotropic homoge-neous turbulence at which the sweeping of the smaller scales by the larger onesfirst begins according to Taylors hypothesis. That is, circulations at length scalessmaller than alpha do not interact nonlinearly to create yet smaller ones in theprocess of their advection. However, these smaller circulations are fully present.In particular, their Lagrangian statistics contribute to the stress tensor, the inertialterms in the nonlinearity and the circulation theorem for the resulting LANS- model.

Deriving the LANS- Model

The motion equation for the LANS- model is

(9)

with Eulerian mean velocity u satisfying

(10)

The inviscid part of this nonlinear motion equation (its left side) emerges fromthe Lagrangian-averaged Hamiltons principle for ideal fluids, upon usingTaylors hypothesis of frozen-in turbulence circulations. A sketch of its derivationis given below. For full details, see Holm (1999).

Hamiltons Principle for the Euler Equations. One begins with the Lagrangianl [u , D ] in Hamiltons principle S = 0 with S = l [u , D ]dt for the Eulerequations of incompressible fluid motion.

(11)

This Lagrangian is the kinetic energy, constrained by the pressure p to preservethe volume element D d 3 x. Conservation of the volume element D d 3 x, in turn,summons the continuity equation

(12)

The constraint D = 1 then implies incompressibility, u = 0, and preservationof incompressibility will determine the pressure as a Lagrange multiplier.Varying the action yields

(13)

As expected, stationarity of S under the variation of pressure p imposes preser-



8/13/2019 16holm_pp172-180

6/9

vation of volume, D 1 = 0. The variations D and u are given in terms of arbi-trary variations of the Lagrangian trajectory X = (x, t ) as

(14)

Integration by parts and use of the continuity equation yield

(15)

Cancellation between the third and fourth terms finally implies Eulers equations,

(16)

by vanishing of the coefficient of the arbitrary vector function . This is the stan-dard derivation of Eulers equations in the Euler-Poincar theory of Holm,Marsden, and Ratiu (1998).

Hamiltons Principle for the Lagrangian-Averaged Euler Equations. Thederivation of the Lagrangian-averaged Euler-alpha (LAE- ) equations proceedsalong the same lines, except one first decomposes the fluid velocity and volumeelement into their Eulerian mean and fluctuating parts, as

(17)

The fluctuating parts D and u of the Eulerian quantities D and u at a fixed pointin space x are associated with fluctuations of the fluid parcel trajectory X = X

~+

(X~, t ) around its Lagrangian mean trajectory X~(t , X0). (For example, the runningtime average of X is taken at a fixed Lagrangian coordinate X0.) The relationsbetween the D and u and the Lagrangian fluctuation , all expressed as func-tions of Eulerian position and time ( x, t ) are

(18)

These are linearized relations, which apply for sufficiently small fluctuations.Having used these linearized relations, we need not distinguish between Eulerianand Lagrangian averaging because the difference is only relevant at higher orderin the relative amplitudes of the fluctuations. The simplest variant of theLagrangian-averaged Euler equations is derived by substituting Taylors hypothe-

sis in the form

(19)

Thus, Taylors hypothesis drastically simplifies the velocity decomposition. Wenow substitute this form of Taylors hypothesis into the decomposition of fluidvelocity on the Lagrangian for Eulers equations, perform the Eulerian average(in time) using the projection property u= = u and then constrain the Eulerian-mean volume to be preserved ( D 1). Following these steps yields the averaged



8/13/2019 16holm_pp172-180

7/9

Lagrangian

(20)

By Taylors hypothesis, the Lagrangian statistic ( j k ) in this expression satisfies

(21)

upon using the projection property u= = uu again. Consequently, homogeneousisotropic initial conditions satisfying ( j k ) = 2 jk with constant 2 are preservedby the dynamics, and the averaged Lagrangian l[u, D] in Hamiltons principle S

= 0 with S = l[u, D]dt for these initial conditions simplifies to

(22)

Note that the constant 2

appears in the relative kinetic energy much the sameway as Taylor argued dimensionally for his microscale. That is, 2 encodes therelative specific kinetic energies of the Eulerian mean fluid velocity u 2 and theturbulent circulations, which satisfy u 2 = 2 u 2 because of Taylors hypoth-esis of frozen-in turbulence. However, as we shall see, is not Taylorsmicroscale.

Reapplying Hamiltons variational principle with this averaged Lagrangian byfollowing the Euler-Poincar theory, as we did before for the Euler equations,now yields the motion equation for the Lagrangian-averaged Euler- (LAE- )model.

(23)

with

(24)

Finally, adding viscosity in Navier-Stokes form and forcing on the right side of the LAE- model recover the LANS- equation of motion:

(25)

Relation of LANS- Inertial Subrange to Taylors MicroscaleThe LANS- system of equations has a variety of properties, only one of

which we shall discuss here; that is, its inertial regime has two different scalings,depending on whether the circulations are either larger or smaller than alpha. Infact, its Krmn-Howarth theorem discussed in Holm (2002) implies that itskinetic energy spectrum changes from k 5/3 for large scales, corresponding towave numbers k > 1. For a dimensional argument justifying this change of scaling in the iner-



8/13/2019 16holm_pp172-180

8/9

tial regime for the LANS- model, see Foias, Holm, and Titi (2001).Because of this change of scaling in the LANS- model for circulations that

are larger or smaller than alpha, the inertial range is shortened for the LANS- model. With fixed, the wave number at the end of the second, steeper k

3

LANS- inertial range is determined in Foias, Holm, and Titi (2001) to be

(26)

Since the Kolmogorov dissipation wave number ( Ko) scales with integralscale Reynolds number as Ko Re3/4 , one finds that dissipation balances nonlin-earity for the LANS- model at Re1/2 , which is precisely the Reynolds scal-ing for the Taylor microscale. Thus, there is a relationship among the three pro-gressively larger wave numbers

(27)

Shortening the inertial range for the LANS- model to k < Re1/2 rather thank < Ko Re3/4 implies fewer active degrees of freedom in the solution for theLANS- model, which clearly makes it much more computable than Navier-Stokes at high Reynolds numbers.

Counting Degrees of Freedom. If one expects turbulence to be extensive inthe thermodynamic sense, then one may expect that the number of activedegrees of freedom N dof for LANS- model turbulence should scale as

(28)

where L is the integral scale (or domain size), is the end of the LANS- iner-tial range, and Re = L4/3 1/3 / v is the integral-scale Reynolds number (with totalenergy dissipation rate and viscosity v). The corresponding number of degreesof freedom for Navier Stokes with the same parameters is

(29)

and one sees a possible trade-off in the relative Reynolds number scaling of thetwo models, provided one resolves down to the Taylor microscale. (In practice,users of the LANS- model often find acceptable results by setting its resolution

scale to be just a factor of 2 smaller than .)Should these estimates of the number of degrees of freedom needed fornumerical simulations that use the LANS- model relative to Navier-Stokes notbe overly optimistic, the implication would be a two-thirds power scaling advan-tage for using the LANS- model. That is, in needing to resolve only the Taylormicroscale, the LANS- model could compute accurate results at scales largerthan by using two decades of resolution in situations that would require threedecades of resolution for the Navier-Stokes equations at sufficiently high Re .

The argument for this advantage is as follows: One factor of ( N NSdof / N

dof )1/3



8/13/2019 16holm_pp172-180

9/9

in relative increased computational speed is gained by the LANS- model foreach spatial dimension and yet another factor (at least) for the accompanyinglessened Courant-Friedrichs-Levy (CFL) time step restriction. Altogether, thiswould be a gain in speed of

(30)

Since / L > 1, the two factors in the last expressions do com-pete, but the Reynolds number should win out, because Re can keep increasingwhile the number / L is expected to tend to a constant value, say / L = 1/100, athigh (but experimentally attainable) Reynolds numbers, at least for simple flowgeometrics. Empirical indications for this tendency were found in Chen, Foias etal. (1998, 1999a, 1999b) by comparing steady LANS- solutions with experi-mental mean-velocity-profile data for turbulent flows in pipes and channels.

Thus, according to this scaling argument, a factor of 10 4 in increased speed foraccurate computation of scales greater than could occur, by using the LANS-

model at the Reynolds number for which the ratio Ko / = 10. An early indica-tion of the feasibility of obtaining such factors in increased computational speedwas realized in the direct numerical simulations of homogeneous turbulencereported in Chen, Holm et al. (1999), in which Ko / 4 and the full factorof 4 4 = 256 in computational speed was obtained using spectral methods in aperiodic domain at little or no cost of accuracy in the statistics of the resolvedscales.



Andrews, D. G., and M. E. McIntyre. 1978. AnExact Theory of Nonlinear Waves on aLagrangian-Mean Flow. J. Fluid Mech. 89 :609.

Chen, S. Y., D. D. Holm, L. G. Margolin, andR. Zhang. 1999. Direct NumericalSimulations of the Navier-Stokes AlphaModel. Physica D 133 : 66.

Chen, S., C. Foias, D. D. Holm, E. J. Olson, E.S. Titi, and S. Wayne. 1999a. A Connectionbetween the Camassa-Holm Equations andTurbulence in Pipes and Channels. Phys.Fluids 11 : 2343.

. 1999b. The Camassa-Holm Equationsand Turbulence in Pipes and Channels.Physica D 133 : 49.

. 1998. The Camassa-Holm Equations asa Closure Model for Turbulent Channel andPipe Flows. Phys. Rev. Lett. 81 : 5338.

Foias, C., D. D. Holm, and E. S. Titi. 2001. TheNavier-Stokes-Alpha Model of FluidTurbulence. Physica D 152 : 505.

Gjaja, I., and D. D. Holm. 1996. Self-Consistent Wave-Mean Flow InteractionDynamics and Its Hamiltonian Formulationfor a Rotating Stratified IncompressibleFluid. Physica D 98 : 343.

Holm, D. D. 1999. Fluctuation Effects on 3DLagrangian Mean and Eulerian Mean FluidMotion. Physica D 133 : 215.

. 2002. Krmn-Howarth Theorem forthe Lagrangian-Averaged Navier-Stokes-Alpha Model of Turbulence. J. Fluid Mech.467 : 205.

Holm, D. D., J. E. Marsden, and T. S. Ratiu.1998. The Euler-Poincre Equations andSemidirect Products with Applications toContinuum Theories. Adv . Math . 137 : 1.

Kolmogorov, A. N. 1941. Dissipation of Energyin a Locally Isotropic Turbulence. Dokl.

Akad. Nauk SSSR 32 : 141. (English transla-tion in Proc . R. Soc . London A 434 : 15,1991).

Richardson, L. F. 1926. Atmospheric DiffusionShown on a Distance-Neighbour Graph.Proc . R. Soc . London A 110 : 709 .

Taylor, G. I. 1921. Diffusion by ContinuousMovements. Proc . London Math . Soc. 20 :196.

. 1938. The Spectrum of Turbulence.Proc. R. Soc. London A 164 : 476.

Further Reading

Documents

16holm_pp172-180