INTRODUCTION

It is rather usual in colloquial speach to talk ofevery flow with a complicated dynamics as beingturbulent. In this sense, turbulence is synonymouswith disorganized flow and is in contrast to the con-cept of laminar flow which is a state of ordereddynamics. Turbulence is the result of a subtle bal-ance of terms involving many different length-scalesand frequencies and, because a detailed descriptionof every possible state of equilibrium is not possible,it is usually acknowledged that the ultimate goal ofany turbulence theory is to find a way to get rid ofthe small-scale dynamics which, in a sense, pollutethe flow. This is what we mean by modeling a tur-bulent flow: to produce a smooth flow where highfrequencies have been filtered out but which still

behaves at large as the original flow. As an exampleof this procedure we can mention the so-called largeeddy simulations - see, for instance, the reviewpaper by Lesieur et al. (1995).

In the study of the interaction between turbulenceand plankton, it should be noticed that the smoothedscales are of sizes comparable to, or even much larg-er than, plankton individuals. Therefore, thesesmoothed descriptions of a turbulent flow may not bevery useful in order to describe the dynamics of theseindividuals. In contrast, they can provide very usefulmodels describing the interaction of a whole popula-tion of them with the fluid. These interactions, whichcan either be a consequence of the body forces exert-ed by the plankton in the bulk of the fluid or the resultof the heat released during their life cycle, may enterthe model through the smoothing of the small scales.If this is the case, the interaction between the fluidand the plankton may strongly influence the dynam-ics of the flow, giving rise, for instance, to patchiness- in that respect, the reader may refer to the review byPedley and Kessler (1992).

FROM PRE-TURBULENT FLOWS TO FULLY DEVELOPED TURBULENCE 63

SCI. MAR., 61 (Supl. 1): 63-73 SCIENTIA MARINA 1997

LECTURES ON PLANKTON AND TURBULENCE, C. MARRASÉ, E. SAIZ and J.M. REDONDO (eds.)

From pre-turbulent flows to fullydeveloped turbulence*

JOSEP M. MASSAGUER†

Departament de Fisica Aplicada. Universitat Politècnica de Catalunya, B5 Campus Nord, E-08034 Barcelona, Spain

SUMMARY: In colloquial speech the term “turbulence” means any flow with complicated temporal and spatial dynamics. Incontrast, the properties of these flows are supposed to be described by models which are based on the assumption of a pre-cise and very well defined setting. This paper is an attempt to review the fundamentals of turbulence theory, with empha-sis on non-fully developed turbulence.

Key words: Transition, turbulence, chaos, instability.

†This paper did not go through the full review process due tothe decease of the author. Reprints to be requested to Dr J.M.Redondo at the same address

*Received March 1, 1996.

A more technical approach to turbulence, is toregard a turbulent flow as a physical realization of arandom process in statistical equilibrium - see, forinstance, the book by Monin and Yaglom (1971) foran introduction to this point of view. This is what isusually called fully developed turbulence. In aheuristic way, fully developed turbulence can bedefined by saying that it is a state such that the small-scale average of any physical variable is a welldefined quantity, no matter whether it is a time or aspace average, the average being independent of theset of measurements to be averaged. The assumptionof randomness is powerful and provides a basis for astatistical description of the flow, with the averagingprocess providing a convenient tool to smooth outthe dynamics, but this assumption is not free fromcriticism, as we shall argue below. No matter howconvenient it may be for modeling purposes, it is tooidealized to describe real flows. It neglects, forinstance, the presence of coherent structures andintermittency which, even if showing a modest con-tribution to the global energetic balance, can be ofgreat importance for the detailed dynamics.

What do we mean by turbulence?

A flow is said to be turbulent when it has no sim-ple structure neither in time nor in space. This is anegative definition, and thus a source of confusion.

There is a widespread confidence about the exis-tence of a well defined regime to be reached byincreasing external stresses, say the pressure head inpipe flows, but this asymptotic limit would be a kindof universal state whose existence and uniquenessstill raises many doubts. As an introduction to thesubject, we have reproduced in Figure 1 a set ofmeasurements of the drag coefficient C

D against theReynolds number Re for Poiseuille flow in a circu-lar pipe. The coefficient C

D gives a dimensionlessmeasure of the stresses exerted on the pipe wall bythe flowing fluid. For a small Reynolds number themeasurements fit the law C

D = 64/Re given by curve

1. This is a well known result that can be obtainedfrom the Navier-Stokes equation on the assumptionthat the fluid is in a steady state and flows parallel tothe axis of a cylindrical pipe. For a larger Reynoldsnumber the measurements fit the so-called Blasius’empirical law, given by curve 2, C

D = 0.3164Re—l/4.In a naive way, we could say that either curvedefines a regime, with the former being unstable ata given Reynolds number, so that by increasing theReynolds number the flow jumps from the statedefined by curve 1, termed laminar regime for obvi-ous reasons, to curve 2. Because in the latter case theflow is time dependent and does not show any regu-larity, it is called a turbulent regime. For further use,we must also notice that at a given Reynolds num-ber the so-called turbulent regime shows a larger

64 J.M. MASSAGUER

FIG. 1. – Experimental measurements of drag coefficient CD against the Reynolds number, Re, for Poiseuilleflow in a circular pipe. A laminar regime CD = 64/Re and a turbulent regime can be easily realized. For mildReynolds number values, the latter regime is described by the law CD = 0.3164Re-1/4, but at larger values depar-

tures from that law are apparent.

drag than that expected from the laminar one. Suchan enhancement of the effective viscosity is also acommon feature in any turbulent flow.

Unfortunately, real life is not that simple. Bymeasuring the external stresses acting on a fluidwhat we get is a highly averaged description of theflow. To illustrate the point, we have reproduced inFigure 2 measurements of stresses in a Couette flow.This is the flow of a fluid contained in the gapbetween two rotating coaxial cylinders. In the quot-ed experiments the inner cylinder rotates at a con-stant speed while the outer one is kept at rest. Theplot displays a set of measurements of the torque rapplied to the inner cylinder as a function of theangular velocity ω, ploted as its reciprocal T=2π/ω.

The laminar regime is given by the horizontalline, with the Reynolds number increasing fromright to left, and the turbulent regime is given by the(almost) vertical line. However, detailed inspectionof the flow for different rotating velocities of theinner and outer cylinders shows a complex situation.Many different regimes can be easily identified bysimply using standard visualization techniques.Andereck et al. (1986), gave a summary of observedregimes as a function of the inner and the outerrotating velocities and for a large range of Reynoldsnumber values. Two illustrative examples of theseflows have been displayed in Figures 3 and 4. Theyillustrate two examples of flows which are some-what intermediate between a laminar regime and aregime of fully developed turbulence.

Figure 3 displays a wavy vortex flow. Thisregime has been obtained with the inner cylinder

FROM PRE-TURBULENT FLOWS TO FULLY DEVELOPED TURBULENCE 65

FIG. 2. – Experimental measurements of the torque exerted by thefluid on the lateral walls of a Taylor-Couette apparatus as a functionof the rotation period. The inner cylinder rotates and the outer oneis at rest. The period provides a measure of the Reynolds number of

the flow.

FIG. 3. – This plate displays a Taylor-Couette experiment, with theinner cylinder rotating and the outer one at rest. The regime shownis a wavy vortex flow. The flow is organized as a set of precessingtoroidal vortices, and is periodic in time. Courtesy of D. Crespo.

FIG. 4. – This plate displays a Taylor-Couette experiment, withcylinders rotating in opposite directions. The resulting vortices spi-ral along the axis of the cylinder. A turbulent spot in the upper halfof the cylinder can be easily seen. It is an example of what is calledspatial intermittency, meaning that turbulence shows up in spots

surrounded by quasi-laminar flow. Courtesy of D. Crespo.

rotating at constant speed and the outer one at rest.The flow is organized as a set of toroidal vortices,i.e., a set of piled doughnuts, which are undulatedand precess. This is a flow periodic in time and thegeometry is rather simple, as corresponds to a mildReynolds number regime.

This is one of the many gentle regimes whichmay qualify as laminar. In contrast, Figure 4 dis-plays a regime where external stresses are impor-tant. Both cylinders rotate in opposite directions andthe resulting vortices spiral along the axis of thecylinder - they describe a helix, to be precise. Thisflow might seem rather laminar, but a turbulent spotin the upper half of the cylinder prevents the readerfrom being too naive. In fact, this is an example ofwhat is called spatial intermittency, meaning thatturbulence shows up in spots surrounded by quasi-laminar flow.

Fully developed turbulence

The flows described above, alike any flow in afinite container, are thought to be extremely condi-tioned by the details of the external stresses imposedon the flow as well as by the geometry of the con-tainer. Thus, closed flow turbulence is in suspicionof not being genuine turbulence in a sense to be

made precise later. But this criticism can be madeextensive even to open flows. In Figure 5 we displaythe wake of an inclined flat plate. It has been pro-duced by pulling a ruler in a water tank, with theplane of the ruler inclined with respect to its veloci-ty. On the large scale the wake shows some welldefined vortices, whose size and shapes depend onthe size and orientation of the ruler, i.e., on the forc-ing by the plate. Therefore, any attempt to describethe turbulence of this flow can only refer to scaleswhich are much smaller than the sizes of these vor-tices which, in its turn, depend on the external forc-ing. It is in the context of these small scales that wecan talk about fully developed turbulence. Whethera universal regime deserving such a name exists andwhether it is unique are still open questions - see, forinstance, the discussions about soft and hard- turbu-lence by Castaing et al. (1988). There is also anexcellent discussion on the existence of an asymp-totic limit in the book by Chorin (1994, chapter 3).

At this point we can quote the following para-graph from Lesieur (1987, p. 9). “Fully developedturbulence is a turbulence which is free to developwithout imposed constraints. The possible con-straints are boundaries, external forces or viscosity:one can easily observe that the structures of a flowof scale comparable to the dimensions of the domain

66 J.M. MASSAGUER

FIG. 5. – This plate displays the wake of an inclined flat plate. It has been produced by pulling a ruler in a water tank, with the plane of theruler inclined with respect to its velocity. On the large scale the wake shows some well defined vortices, whose size and shapes depend on

the size and orientation of the ruler. Courtesy of J. M. Redondo.

where the fluid evolves cannot deserve to be catego-rized as “developed”. The same remark holds for thestructures directly created by the external forcing, ifany. [...]. At smaller scales, however, turbulence willbe fully developed if the viscosity does not play adirect role in the dynamics of these scales”. Thescales where the turbulence is fully developed con-stitute the so-called inertial range. In this range, theflow at different scales shows a kind of self-similar-ity. Therefore, the inertial range is bounded aboveby a large scale, imposed by the external forcing,and below by the small scales of the viscous range,where dissipation of energy takes place.

Fully developed turbulence is the result of thefree interaction of structures, say eddies, of differentsizes. Therefore, we can expect fully developed tur-bulence to be of random nature, so that velocityfields are well described by using statistical meth-ods. This is the realm of the so called Statistical Tur-bulence Theory. As far as the randomness of thesmall scales is preserved, this is a powerful tool.But, “ in the statistical averages much of the infor-mation that may be relevant to the understanding ofthe turbulent mechanisms may be lost, especiallyphase relationships. [...] However, in order to under-stand highly intermittent turbulence productionmechanisms for which intrincate phase relationshipsare likely to play an essential role, standard averag-ing techniques are insufficient ...” (Landhl andMollo-Christensen, 1986, pp. 1,2). Whether coher-ent structures, intermittency, etc. can (and must) beintroduced in a turbulent theory, or they have to beconsidered as part of the scenario for pre-turbulentflows, is still a matter of discussion. The real ques-tion is, however, are we forced to deal, individually,with every type of turbulence or can we ignore themby simply introducing their behaviour in a large-scale average.

SEMIEMPIRICAL THEORIES OFTURBULENCE

The larger scales of a flow are smooth and slow-ly varying in time. On the contrary, turbulence isconcentrated on the small scales, which show highfrequencies. If the small scales are random, as is usu-ally the case, mean values and fluctuating quantitiescan be easily separated by averaging on shorttimescales, so that we can split up the velocity field as

where we denote with an overline the small-scaletime averages. For an incompressible fluid we canwrite the averaged continuity and Navier-Stokesequations

(1a)

(1b)

where we have introduced the dyadic notation(uu)

ij= u

iu

j, the density ρ has been taken constant, p–

is the averaged pressure field, ν is the kinematic vis-cosity, and we have assumed that there are no exter-nal body forces.

The system (l) includes as unknowns, besides theaveraged velocity and pressure fields, the Reynoldsstress tensor,

and it cannot be solved unless an additional condi-tion is provided. The simplest assumption in order toclose the system is given by the condition

(2)

which treats the turbulence itself as a Newtonianfluid. For instance, a simple derivation of this clo-sure condition can be obtained by modeling theturbulent flow as a perfect gas where, instead ofmolecules, there are eddies. This is the essence ofthe so-called mixing-length theory derived byPrandtl, with the mixing-length itself being theanalog of the mean-free-path in the kinetic theoryof gases.

If νtis constant in space, as it is usually assumed

in mixing-length theory, (1), (2) can be written

(3a)

(3b)

with . Therefore, turbulence can bemodelled by a turbulent viscosity νt which adds upto the molecular viscosity ν.

Although in mixing length theory νt is assumedto be constant, other assumptions have been shownto be plausible. From the mathematical point ofview, it is perfectly consistent to assume any func-tional relationship

νt = F ui ,∂ jui( )

p:= p + 3π

∂tu + u ⋅∇u = −ρ−1∇p + ν + νt( )∇2u

∇⋅u = 0

τij = −πδ ij + νt ∂iuj + ∂ jui( )

τij = −u′i ⋅u′ j

∂tu +∇⋅ u u + u′u′( ) = −ρ−1∇p + ν∇2u

∇⋅u = 0

u = u + u′ ,

FROM PRE-TURBULENT FLOWS TO FULLY DEVELOPED TURBULENCE 67

As an example we can mention Smagorinski’srecipe, where the turbulent viscosity is assumed tobe shear dependent,

with , and summation overrepeated indices is implied.

A more sofisticated procedure aims at derivingequations for the Reynolds stress tensor from theNavier-Stokes itself. We can obtain an equation forthe velocity fluctuations u′ by substracting from thefull equation the averaged equation, and anotherequation for the moments u

i′uj′ by multiplication ofthe Navier-Stokes equation times u′ and subsequentaveraging. But the equation for u

i′uj′ introduces theunknown u

i′uj′uk′, and some other averages whichinvolve the fluctuations of the pressure field. Again,ad hoc closures are required. A well known exam-ple of this procedure is provided by the so-called k-ε models. In these models the turbulent viscosity isνt=ck2/ε where k is the kinetic energy of the flow,

is the energy dissipa-tion rate and c is a constant. The implication is thatthere is no way of closing the problem at any levelwithout imposing some phenomenological relation-ships between moments. This is why these theoriesare called semiempirical. A good reference for thederivations in this section and in the next one, is thebook by Tennekes and Lumley (1972).

About scales

In the previous sections we have systematicallydealt with scales of length and time. The Navier-Stokes equation involves many balances betweenterms, and most of them rely on scalings which aredifferent for different regimes and which maychange from point to point. The two leading termsin the equation are u

i∂

iu

jand ν∂

ii2u

jand their ratio

can be estimated to be the Reynolds numberR=ul/ν, where u and l are, respectively the localvelocity and a local scale for length. The implica-tion is that for large scales the contribution of thedissipation term is small. On the other hand, it canbe realized from the Navier-Stokes equation itselfthat if the viscous term is negligible there is noenergy dissipation, i.e.: an inviscid fluid is a con-servative system. Therefore for a given velocity andviscosity, only the scales smaller that a given valuewill dissipate a significant amount of energy. But,as shown above, the turbulent viscosity only

includes the contribution of the non-linear terms,which do not dissipate by themselves. Thus, we canask, why is the effective viscosity enhanced by tur-bulence? Please, notice that the turbulent viscositycan be negative, but only if turbulence is notisotropic (Starr, 1968).

The answer to this question is that the nonlin-ear terms increase the transport of energy towardsthe smaller scales, so that we can draw the follow-ing picture. Energy is injected in the flow at agiven scale, smaller than the size of the container.We shall call this scale the integral scale of theflow because it measures the coherence length ofthe flow itself. The energy cascades from thisscale towards the smaller ones without dissipation.The range of scales without dissipation defines theinertial range. Finally, at a scale η, the Kol-mogorov scale, the flow dissipates the energy byviscous dissipation. If we call ε the dissipation rate(energy dissipated per unit mass), then ε is theenergy injected to the flow by the source term, thisenergy cascades without being dissipated alongthe inertial range, and finally dissipates in the vis-cous regime.

At this point, it is important to notice that viscousdissipation is controlled only by the input of energyat the source. Therefore, the length and time scaleswhich are characteristic of the dissipation process,the Kolmogorov’s scales (also called microscales),can only depend on ε and ν, and their expressionscan be obtained by simple dimensional arguments as

(4)

Let us now evaluate the ratio between the inte-gral and the Kolmogorov scales of the flow. If u isthe velocity of the largest eddy and l its characteris-tic length, the content of energy per unit mass forthis eddy is 1/2u2. If we estimate that the eddy canloose all this energy in a turnover time l/u, we willobtain for the dissipation rate

(5)

And by combining the previous expression with(4) we will get

(6)

which measures the size of the inertial range as afunction of the Reynolds number. A numericalmodel, for instance, has to be able to describe both

η l ~ ul ν( )−3 4 = R−3 4

ε ~ u3 l

η = ν3 ε( )1 4, τ = ν ε( )1 2

k = 1 2u'i u'i , ε = 2νD'ij D'ij

Dij = 12 ∂ jui + ∂ jui( )

νt = c Dij Dij

68 J.M. MASSAGUER

these scales, thus implying that the resolution alongevery coordinate will increase with the Reynoldsnumber of the flow as R3/4. This gives an idea of thechallenge of modelling large Reynolds numberflows in three dimensions.

An additional scale of relevance is the Taylormicroscale, λ, which for shear flows is defined bythe expression

from which we can estimate the dissipation rate,, as

(7)

The meaning of λ can be appreciated if we eval-uate the turbulent viscosity as νt∼ul. Then, by usingexpressions (5), (7) we can write

showing that the ratio of the integral scale to theTaylor microscale is a measure of the ratio betweenthe turbulent and molecular viscosity.

MODAL THEORIES OF TURBULENCE

The presentation made in the previous sectionsrelies on the concept of local scale, either temporalor spatial. The concept of scale is deeply rooted inphysics, but its use requires a deep knowledge ofthe phenomena that have to be described. Forinstance, in the so-called mixing-length theories,the mixing-length, as it is called the integral scale,can be thought of as the size of the largest eddy, butit can also be thought of as the distance to be trav-elled by an eddy before decaying. In a more strictsense, it can be taken as a coherence length. But inpractice, none of these images are really useful,and are only useful in helping to guess. Closurerelationships, for instance, are based on educatedguesses, thus putting semiempirical theories of tur-bulence somewhere between a science and an art.As an additional criticism towards these theories, itis important to mention that time and space aver-ages are only meaningful if the dynamic shows twowell separated scales: for instance, if the time-scales for the mean values and the ones for thefluctuations are of different order of magnitude -say a factor ten apart.

The only support for semiempirical theoriescomes from their ability to fit experimental results.In contrast, there are some other approaches whichare based on more solid ground. Statistical theoriesare very popular and provide a well-defined mathe-matical setting. Unfortunately, any of these moresoundly based approaches requires a frameworkwhich is, by far, much less intuitive. In the follow-ing we shall introduce the reader to this less intuitivepoint of view.

Modes and eddies

The above description of turbulence relies on theconcept of scale. In physical space, a scale can bethought of as a coherence length and, as such, tur-bulent elements are often visualized as eddies. Butthis picture cannot be pushed much further. In con-trast, the mathematical equations by themselves pro-vide a more powerful description. Let us assumethat the flow is described by some dynamical equa-tions, say the Navier-Stokes and continuity equa-tions, which we shall write symbolically as

(8)

where u is the velocity field, R is the Reynolds num-ber, and by F we denote any functional relationship.We shall also assume that the flow is unbounded inthe x direction and, for simplicity, we shall neglectany other direction. In addition we shall assume thatfor R < R

c the system has only one possible solu-tion, u = u

o, for instance the laminar solution of thePoiseuille flow, such that ∂

tu0=0, thus givingF(u

0,∂

x;R)=0. Introductory books on Fluid Mechan-

ics are full of examples where one such uo

is explic-itly computed.

Let us now assume that the flow uo

becomesunstable at a given Reynolds number value R = R

c .

This means that for R > Rc

there will exist perturba-tions of the flow that will grow exponentially withtime. Let us call them u’, with u = u

o + u’, and sub-stitute this expression in (8). As far as u’ is smallenough, we can neglect quadratic terms in u’, and bytaking into account the equation for u

o we shallobtain a linear equation for u’, say

(9)

where L is a linear operator including partial deriva-tives in space. We shall also assume the coefficientsin L do not depend either on space or on time: the

∂tu' = Lu'

∂tu = F u,∂x ;R( )

νt

ν~ 15

l

λ⎛⎝

⎞⎠

2

ε = 15νu2 λ2

ε = 2νD'ij D'ij

∂u

∂x⎛⎝⎜

⎞⎠⎟

2

=u2

λ2

FROM PRE-TURBULENT FLOWS TO FULLY DEVELOPED TURBULENCE 69

laminar flow is homogeneous, meaning that we seethe same velocity field from any frame of reference.Then (8) is a linear partial differential equation withconstant coefficients. Under such conditions, ele-mentary calculus shows that the solution of (9) canbe written

(10)

with

(11)

where c, α and ω are constants, the two latter beingreal valued and the former being complex. In thegeneral case, A will depend on the transverse coor-dinates, say the radial coordinate if the fluid flowsalong a pipe, but this is not introducing any signifi-cant change in the derivation.

Equation (10) describes a periodic structure ofwavenumber λ=2π/k. This can be thought of as aneddy of length scale λ or as a row of them. Period-icity allows both points of view. In fact, because oflinearity, the superposition principle allows us totake as a solution of (9) any linear combination ofsolutions (10) with different wavenumbers k. To beprecise, if we restrict ourselves to periodic solutions,we can write instead of (10)

(12)

where N is any integer. Equation (12), which caninclude terms of every possible lengthscale, canalso be thought of as a Fourier decomposition foru’(x,t), and the function A

n is called the amplitude

of the n-mode. Now, the concept of eddy as a ref-erence for lengthscales has been turned into theconcept of Fourier mode. To be more precise, avortex as such in Figure 5, can be described by asuperposition of these modes but, because this isnot a periodic structure, the wavenumbers k have tobe allowed to take any real value. Then, it is saidthat the flow displays a continuum spectrum for thewavenumber. Such a description will emerge quitenaturally below.

Landau’s description of turbulence

In order to get an equation for finite values of u’,Landau noticed that the amplitude given by (11) ful-fills the linear equation d|A|2/dt=2α|A|2. Expansion(12) is only valid for small values of u’, and there-

fore for small values of the amplitude |A|, and Lan-dau conjectured that for larger values of the ampli-tude, then d|A|2/dt has to be a function of |A|2. Forsimplicity, he proposed to aproximate the equationby keeping only two terms in the Taylor expansionof this function and wrote what is now called theLandau equation (Landau, 1963)

(13)

with α~R-Rc. The Landau equation is far more

general than could be expected from the previousderivation, and is a good point of reference so asto introduce some basic ideas on turbulence (for agood discussion see Monin and Yaglom, 1971, p. 160).

Let us notice, first of all, that if β is positive, theamplitude in (13) reaches a steady regime at|A|2=2α/β for any R > Rc , while for R < Rc the coef-ficient α is negative, the amplitude decays to zeroand so does u’. An additional, and very importantpoint to notice is that the amplitude obtained from(13) is defined up to a phase, with A=|A|eiθ for anyvalue of θ, which, as shown in (11), can be timedependent.

By increasing R, the new regime can becomeunstable again. If the new regime can still bedescribed by a Landau equation (13), and this is farfrom being obvious, the process described aboverepeats, thus introducing a new arbitrary phase θ′.By increasing more the Reynolds number, theprocess repeats again and again, as many times aswe like, thus increasing the number of arbitraryphases at will. The implication is that the solutionwill depend on a large number of arbitrary phases.And because the phases can be randomly chosen,say from the initial conditions, the velocity field canbe treated as a random variable.

Landau’s theory of turbulence can be, and hasbeen, criticized in many respects, but it introduces,at least, two important concepts. One is the descrip-tion of turbulence in terms of modes, given by equa-tion (10), and the other is the idea that turbulencecan be described in terms of random field variables.However, Landau’s theory does not escape therequirement of two well separated time scales, thefast one, given by τ

f ~2π/ω and the slow one, given

by τS ~1/α. An additional source of criticism comesfrom the assumption, implicit in the theory, that tur-bulence can only be reached once the system hasevolved through an infinite number of differentregimes.

d A 2

dt= 2α A 2 −β A 4

u' x,t( ) = An t( )n=0,± N∑ einkx

A t( ) = ce α +iω( )t

u' = A t( )eikx

70 J.M. MASSAGUER

Deterministic chaos and turbulence

When a solution of a differential equationbecomes unstable and the system moves into a dif-ferent state, we say that the system undergoes abifurcation. This is current terminology in dynam-ical systems. It is well known that chaos can bereached after a finite number of bifurcations, sothat the system does not need to have a large num-ber of degrees of freedom to reach complicateddynamics. In Landau’s picture a finite number ofbifurcations implies a finite number of degrees offreedom, and the flow is uniquely determined byinitial conditions. Randomness must now bethought of in a completely different way. Ratherschematically, we can say that for a given experi-ment, even under well controlled physical condi-tions, a chaotic flow may wander erraticallybetween many different regimes. These many dif-ferent regimes do not exist by themselves as inde-pendent states: that is just an idealization. Theyare simply part of a path described by the system.But the system is always attracted by this path,where it tends asymptotically. It is for that reasonthat this path is called an attractor of the system,and because this attractor is neither a fixed point,nor a periodic orbit, it is often called an strangeatractor.

Many systems displaying chaotic flows havebeen found in recent decades. The Couette flowbetween concentric cylinders and the Bénard prob-lem for thermal convection are two well knownexamples. These systems can be described by lowdimensional systems, i.e: by a small number ofmodes, but they often reach states of time-intermit-tency and develop coherent strucutures by phaselocking and resonance processes. Some times theyeven show a complicated spatial structure. But, byconstruction, these low-dimensional models onlyinvolve a small number of independent spatialscales, and this is a real flaw in order to be useful todescribe real turbulence.

The Kolmogorov’s spectrum

Fully developed turbulence provides a good testfor every moel of turbulence. It can be thought of asa state of statistical equilibrium between modes oreddies. In that there were no non-linear terms in theNavier-Stokes equation, every set of amplitudes in(12) would describe a real flow. But it is because ofthe non-linear interaction between modes, which is

given by the inertial term u·∇u, that amplitudes canreach an equilibrium state. This is a conservativeterm, and the non-linear interactions may be thoughtto describe the elastic collisions between eddies,with energy cascading from the larger to the smallerones. Because the nonlinearity is quadratic, theinteraction requires three modes, which in terms oftheir (vector) wavenumbers can be written as kl + k2+ k3 = 0. By book keeping the terms in the interac-tion a convolution product of Fourier modes- it canbe shown that the number of modes involved isinfimte.

Whether for some given conditions the non-lin-ear interactions can be described in terms of afinite number of modes or not, is a rather technicalquestion. The answer is in the so-called centermanifold theorem (Guckenheimer and Holmes,1984) which tells under what conditions a dynam-ical system described by a given (finite) number ofmodes can be reduced to a smaller dimension. It isusually agreed that for large external stresses - say,large Reynolds number values - the answer is neg-ative, mostly because the spectra tends to be con-tinuous. The implication is that fully developedturbulence can hardly be described in terms ofdeterministic chaos. In order to be more precise,we shall now derive an expression for the spec-trum of homogeneous, isotropic, fully developedturbulence.

A necessary condition for a flow to be homoge-neous is for it to extend over an unbounded domain.In such a case, there is no preferred length to definethe periodicity of the domain. Every wavenumber ispossible, thus meaning that the k-spectrum is con-tinuous and the expansion (12) has to be changed to

(14)

Expression (14) describes a very large set offunctions. There is almost no other restriction for u′′than that being bounded at infinity. As an additionalremark, it must be noticed that An and u, as definedin (12) and (14), do not share the same units. Theratio of their dimensions is a volume.

If a flow is homogeneous and isotropic, we canexpect its distribution function to be given by somesimple law expressing a universal equilibrium or athe tendency to equilibrium. If such a state exists, itsenergy spectrum can be easily obtained from dimen-sional arguments. As discussed earlier, the extent ofthe inertial range is controlled by the amount ofenergy per unit mass, ε, injected to the flow by the

u′ x,t( ) = u k,t( )∫ eik⋅xd3k

FROM PRE-TURBULENT FLOWS TO FULLY DEVELOPED TURBULENCE 71

external forcing, i.e.: at the integral scale. Let ustake as a measure of the amplitude of a given modeits energy content. If we designate by E(k,t)dk thekinetic energy of the modes with wavenumbers inthe range between k and k + dk, then

(15)

An expression for the energy spectrum can beobtained by noticing that, by assumption, E(k, t) canonly depend on k and ε. Straightforward dimension-al arguments lead to

(16)

where Ko is the so-called Kolmogorov constant, withan experimental value Ko = 1.44. Expression (16)gives the so-called Kolmogorov’s spectrum. It givesthe amplitude of the modes in the inertial range,where energy cascades without dissipation. There-fore, ε is constant in this range, and the flow isexpected to display the E(k)∝k-5/3 Fourier spectrumwhich is usually taken as a signature for fully devel-oped turbulence. The inertial range ends at the Kol-mogorov’s microscale η, defined in (4), where thedissipation range begins.

The Kolmogorov spectrum (16) describes verywell the experimental results and, therefore, it is atest for every theory of turbulence. In particular, itshows the limits to any description of a turbulentflow in terms of deterministic chaos. Notwith-standing, the agreement of (16) with experimentalmeasurements is a challenge, because the previousderivation is based on the assumption that themode-mode interaction is a local process, which itis not. Non-local interactions between modes farapart can be very important. For a classical discus-sion on this subject, the reader is addressed toLeslie (1973). A very interesting and updated,albeit somewhat personal view on equilibriumstates and non-local processes can be found inChorin (1994).

Chaos in extended systems

Turbulence is a state where the flow showscomplicated dynamics in both space and time, andinvolves a continuum of scales. It is in this respectthat most experiments on closed flows, say thermalconvection, Taylor-Couette flow, etc. are suspectedof not being genuine models for turbulence.Indeed, it is an open question whether for high

enough Reynolds number values these flows maydisplay genuine turbulence. There are still ques-tions about the meaning of genuine in the presentcontext. Thus, it is not surprising that in recenttimes there has been increasing interest in systemswhere chaos occurs both in time and space. Oneexample of such a flow is provided by thermal con-vection in very large aspect ratio containers, i.e.:large width and thickness as compared to depth.Experiments show for these systems complicateddynamics since the onset of instability. Morris etal.(1993) have shown that under prescribed condi-tions, mild thermal convection can take place as arandom distribution of spiral-like vortices. Labora-tory and numerical experiments (Decker et al.,1994) show a remarkable agreement. Figure 6 is anexample of such a flow.

Extended systems, as may be called the sys-tems just described, show randomness in both,time and space, so that they cannot be describedby a small number of modes. At first sight itseems a good setting to describe turbulence. How-ever, energy is injected in these flows at the scaleof the depth of the layer, which is much smallerthan the horizontal scales. Thus, interactionsbetween vortices a long distance apart require thata fraction of the energy cascades backwards,towards the larger scales. The interaction is non-

E k( ) = Koε2 3k−5 3

E k,t( ) = 4 3πk2u k,t( )2

72 J.M. MASSAGUER

FIG. 6. – This plate is an example of the weak turbulence that hasbeen found for thermal convection in very large aspect ratio con-tainers. Spiral-like eddies are dominant. For this range of values,laboratory experiments and numerical simulations produce

undistinguishable patterns and dynamics. Courtesy of W. Pesch.

local, and the energy flows in reverse along thespectrum. Indeed, inverse cascades are wellknown non-isotropic flows such as stratified tur-bulence which, otherwise, display a Kol-mogorov’s spectrum. Because most of the theoryfor these flows has been done under conditions ofsmall external stresses, say near the onset of con-vection, they are often referred to as weak turbu-lence. A recent review on these topics can befound in Manneville (1990).

CONCLUSIONS

Flows with complicated dynamics are the rule,not the exception, in natural environments. Butmany of these flows may not fulfil the requirementsto qualify as fully developed turbulence, thoughthey often display a temporal and spatial structuremuch richer than could be expected from theoriesbased on (temporal) deterministic chaos. In contrast,the dimensional laws obtained from Kolmogorov’shypotheses are soundly established, buth they onlygive an idealized approach to turbulence. Thedetailed dynamics may be very different from thestatistical description that emerges from thesehypotheses. Coherent structures, intermittency, etc.are some examples of these differences. Therefore,the statistical theories of turbulence, which haveproven to be extremely powerful in understandinglarge scale dynamics, may not be that useful tounderstand dynamics at smaller scales. They aregood for describing patchiness, but they may not beso good to describe the dynamics of plankton indi-viduals. In the latter case, a detailed analysis of thedynamics is required.

ACKNOWLEDGEMENTS

Some of the pictures displayed in the presentpaper have been done by Drs. D. Crespo and J.M.Redondo in the laboratory of Fluid Dynamics of ourDepartment. My deep appreciation to them both.This work has received financial support from DGI-CYT, Spain, under grant PB94-1216.

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