IMPUESTO SOBRE LA RENTA (ISR)

The Kolmogorov theory of homogeneous isotropic turbulence-HIT- has been dominant for over half a century among the scientists seek- ing fundamental understanding of turbulence. The Kolmogorov the- ory is an important subject of geophysical studies. Many geophysi- cists view turbulence from a fundamental view point may be because they are best exposed to its many grandiose manifestations. On the other hand practitioners of turbulence in civil engineering and aero- nautics who are mainly concerned with turbulence near to the solid boundaries, highly anisotropic boundary layer (BL) turbulence, usu- ally have very limited use for the Kolomogorov theory, some of them viewing it as may be correct but for a highly idealized model that is not relevant for real life applications. Turbulent BL flows are full of structures. Although these structures were not given real mathe- matical description they are clearly visualized, so that the engineers are convinced that they are all important in their contribution to the physical parameters important for applications, e.g., heat and mass transfer, turbulent friction (drag) at the boundaries, etc. There are no such structures in the Kolmogorov model of turbulence. Thereby follows the usual and liely false argument that CS is the intrinsic feature of specifically BL turbulence.

On the other hand the celebrated prediction of Kolmpogorov the- ory, the−5/3 power law for the energy spectrum is observed in lab- oratory experiments modeling HIT, e.g., decaying turbulence past a grid and in part in geophysical phenomena, albeit with modifications and degree of uncertainty.l This law is to some extent confirmed in DNS, the Reynolds numbers are not high enough for confidence. The most surprising thing about this spectrum is that it is observed even in conditions remote from HIT such as for instance in large scale and highly anisotropic atmospheric flows.

The Kolmogorov theory has been so dominant and for such a long time among so many thinking deeply on the nature of turbulence that

it has become almost an Aristotelian must. It is very difficult to make even a slightest dent in its basic postulates. Because of this and for immediate reference by readers it is useful to remind here briefly the main tenets of the theory even though it has been done in thousands of papers before in dozens of different ways. From the outset it is necessary to state that although some of the basic principles of Kol- mogorov theory will be asserted wrong here the Kolmogorov energy spectrum will remain intact. However the physics behind the spec- trum is quite different from the one postulated in the Kolmogorov theory or implied by it. The Kolmogorov theory is often portrayed as a simple scaling relation for the flow energy spectrum, i.e., the distri- bution of energy of turbulent motion among different scales of motion, the velocity harmonics. But in actual fact the theory is very com- plex logically, although the mathematics is indeed simple. It relies on deep intuition into the nature of the Navier-Stokes equations which while cannot be solved still can be analyzed to some, admittedly lim- ited, degree and furnish indications to at least certain features and properties of turbulent motion. Such analysis of the Navier-Stokes equation had been carried out by generations of scientists and engi- neers for over a half century prior to Kolmogorov contribution and surely played role in the formation of the Kolmogorov theory of 1941. The Kolmogorov basic concept of 1941, often called K41, starts from the preceding poetic observation of Richardson made quite ear- lier in 1922 that rhymes ostensibly like this:

Big whorls have little whorls, Which feed on their velocity; And little whorls have lesser whorls, And so on to viscosity (In the molecular sense)18

Laminar flows are driven by extraneous forces, like pressure gra- dient in pipe flows. The mechanism for the laminar flow to become turbulent is, as was emphasized above, a distinct scientific problem beyond the scope of this paper. The fact is that laminar flows are almost always unstable. The simplest way to obtain turbulence is to stir fluid in a semi- random way. Imagine a paddle stirring fluid in

18_{This rhyme is quoted and misquoted by many. It carries such clear and}

simple picture of hierarchy of whorls that it is impossible to resist the temptation to quote it once again

a large pail. Obviously this paddle can create a whorl or a vortex of the size comparable or bigger than the size of the paddle. This prime vortex size will be denoted as the integral scaleL. So far the motion is laminar. But the prime vortex or vortices are totally unstable and decompose into smaller vortices customary called eddies. And these eddies will decompose into smaller eddies and then into even smaller eddies and so forth till such time when eddies become small enough so that the viscous term in the Navier-Stokes equations becomes dom- inant, it is recalled that it is the second order space derivative of the velocity field, and these smallest viscous scale ld << L eddies

dissipate their motion into heat. Obviously the number of ld scale

eddies is much larger than the integral scale L eddies, since the to- tal incompressible fluid volume remains constant. The whole process can be likened to a cascade of water through the many steps in some waterfalls before sinking in a water reservoir below. But in this case the energy cascades through the eddies not in physical space but in the space of scales filling the same fluid domain.

The external forces are opposed by the internal viscosity friction of fluids and if there are boundaries by their friction at the boundaries In order for a steady state to set it should be that the same amount of energy as is injected into fluid by the stirring paddle to generate integral scale eddies is dissipated into heat by the many more small viscous scale eddies: the viscous forces balance the external forces so that the work done by the external forces is equal to the energy dissipated by the viscous forces.

The basic postulate of Kolmogorov theory is that the instabilities of integral scale laminar flow give rise to progressively smaller scales motion. As in the picture of Richardson the big eddy generates pro- gressively smaller and smaller eddies. The reason and possibility for eddies splitting into ever smaller eddies lies in the nonlinear nature of the Navier-Stokes equations for the velocity field. The simple mathe- matical details are discussed below in the next Section, but it is quite clear that when the Eqs. (1.1) - (1.3) are rewritten in Fourier space for the velocity field harmonics v(k,t) as a function of wave vectors

k:

v(k,t) = (2π)−1

Z

v(r,t)e−ik·rdV. (2.1) The quadratic nonlinear term couples triads of velocity harmon-

ics, v(k), v(q) and v(k−q) or the triads of eddies in the space of scales and therefore generates in general different triadic scales of fluid motion. But still it is a non-trivial assumption in the spirit of Richardson that the generated scales of the velocity harmonics are predominantly smaller than the prime characteristic scale of fluid motion that is called the integral scaleL. The integral scale can be associated with an external body force acting on the flow, or with a leading scale of natural instabilities of the incipient laminar flow, e.g., caused by interaction with boundaries. But in principle the non-linear interactions could have created the opposite process to cascade, the growth of eddies, or what is usual to call the inverse cascade. How- ever, most probably as a rule the inverse cascade does not realize in 3Dworld.19

The next fundamental postulate of Kolmogorov is that in the limit of highReand after many cascade steps the small scale eddies of tur- bulence are universal, isotropic and homogeneous. That is to say that a turbulent motion that ensues from the initial integral scale motion, at the scales l << Lbecome self similar and universal. Therefore it

19_{In 2Dworld on the contrary the inverse cascade would be a reality and eddies}

would predominantly grow in size rather than become smaller (e.g., Kraichman, 1962). The expectations of many including myself had been that the inverse cascade in one way or another would be realized in the flows like large scale atmospheric flows where the horizontal scales of motion are very large by com- parison with the vertical depth of Earth atmosphere; the latter is limited by gravity. The inverse cascade would have been an easy and very tempting mecha- nism for formation of large scale atmospheric structures. These expectations are almost certain to be unfulfilled. The flows anisotropy does not make turbulence 2D and the inverse cascade does not occur. On the contrary, most probably the general properties of the turbuleny flows remain very much the same despite the strong disparity between the scales of fluid motions in different dimensions. If inverse cascade realizes it is only as some sort of instabilities, or fluctuation in the otherwise unidirectional flow of energy from large scales to small ones. In this one might invoke a philosophical wisdom, akin to the second law of thermody- namics in equilibrium systems of which turbulence is not the one to be sure, that the real world all non - equlibrium systems tend to have a dominant propensity for decomposition and dissipation. And that the growth in size and extent of organization as is sometimes observed is a temporal fluctuation that should be strongly driven by appropriately arranged external forces. But this may be wrong or incomplete wisdom since we know that in turbulence the dissipation is possible only ifBCCare formed. In anisotropic flowsBCCmay stretch indefinitely in the flow direction since there are no apparent scales limiting the size ofBCCin unbounded direction.

should be independent of the prime flow integral scale Land ofRe. The independence of Land ofRe signifies that the motion at these scales is totally dominated by the local in wavenumbers nonlinear couplings. Let us comment that the assumption that only local non- linear coupling remains would indicate that somehow the non-linear term in Eqs. (1.1) - (1.3), which generally describes all the inter- actions would reduce in some rigorous mathematical analysis of the Navier-Stokes equation, if such was possible. Some of the interac- tions, the non-local ones, would cancel out and only the harmonics with the local triads, such that all the three wavenumbers |k|,|q|

and|k−q|are the same order of magnitude, would remain. This is an important postulate of locality of interactions in Fourier space to bear in mind.

On the other hand by the nature of Eq. (1.1) there are some relatively small viscous scales of order ofld at which the linear vis-

cous term in Eq. (1.1), since it is the second order velocity field space derivative, would necessarily become dominant for any given arbitrary large value of Re. When eddies become so small the vis- cous friction just dissipates their motion into heat.20 _{Also, the uni-}

versality naturally implies isotropy and homogeneity of small scale eddies in the inertial range after many steps of cascade, since there is no singled out scale in the inertial range to associate with either anisotropy, or inhomogeneity. This is the essence of the Kolmogorov HIT model. For consistency let us remind briefly the statistical math- ematical frame of this model.

The values of all turbulent fields are extremely variable in space and time and therefore it is natural to consider v(r,t) as a random function superimposed on the mean flow velocity, including the case of zero mean velocity. If the turbulent velocity were, indeed, a random field it would be fully characterized by the probability density:

W{v}Dv=W{v(N)}d(N)v(N)=

=limN→∞W{v(r1,t1),v(r2,t2), ...,v(rN,tN)}dv1v2....dvN,

(2.2)

20_{The energy spectrum is usually assuemd to be exponentially decaying in the}

viscous subrange of wavenumbers (Monin and Yaglom, 1975). The situation may be more complex than this in reality

Z

W{v}Dv= 1,

where the functional W{v} gives the probability of having certain velocity field space/time 4D = (3 + 1) realization, at all the points in space and at all times, in the multidimensional functional space of all possible realizations.

If homogeneity and steady state of the velocity field are assumed, as befits HIT model, then the turbulent flow can be described by means of the velocity correlation functions of different orders at all points and all times, i.e., by all correlation functions of the type:

<v(r1,t1), ...v(rN,tN)>=<v(0,0), ...v(rN−1,tN−1)>=

=

Z

ΠN,M_i=1 v(ri, ti)W{v}Dv. (2.3)

In principle the knowledge of all the 1,2, ...N,1,2, ...M space/time correlation functions, if they exist and are finite in the limit ofN, M→ ∞would be equivalent to the full knowledge ofv(r,t).

In real experiments the ergodic assumption is applied and the ensemble averaging is understood in reduced sense as the time aver- aging over the values ofv(r,t) at a point. This is because practically all measurements can be made only at one, or at best few points in order to calculate the velocity derivatives, in a fluid flow. In numer- ical simulations even further approximation is usually accepted. If the space-time lattice on which the discretizedv(r,t) is calculated is dense enough, than the space and time averaging over the grid points is assumed to be sufficient for accurate calculations of the real en- semble averaged correlation functions. Even if one space realization of turbulence is represented by large enough N it is enough to cal- culate the space average for certain quantities. This is what is done usually in DNS for the models of decaying turbulence. In this model turbulence is triggered att0 by injection of energy into flow and then

allowed to evolve in time as it follows the Navier-Stokes dynamics. In real DNS even now the number of grid points is still too limited for calculating high order correlation functions. Moreover since some of the basic quantities, enstrophy and its generation rate for instance, are determined by relatively small subsets of grid points.

The experiment in decaying turbulence shows that for one space point, one time velocityv(r,t), the probability functionalW{v}Dv

reduces to a nearly Gaussian distribution function. ButW{v}Dvis a strongly non-Gaussian functional for space-time variations of velocity at two or more points or its space derivatives related quantities, such as stress tensoreik, vorticityω, etc. This deviation from the normal

Gaussian law is associated with the phenomenon of intermittency.21

But let us go back to the Kolmogorov theory that does not concern itself with intermittency.

The velocity of the largest eddies vL is comparable to the total

variation of velocity ∆vLseparated by the distanceLthat is the scale

of the whole flow. ActuallyvL is smaller by a factor of two or three

but it does not matter really because ifRe= ∆VLL/v >>1 for the

whole flow it follows that:

v0l0>>1, (2.4)

where l0 is the size of the biggest eddies and byv0 ≈vL we rather

understand the root mean square of the fluctuating velocity typical; for the ensemble of proto-eddies, that ispv2

0.

The pro-eddies are the energy containing in the sense that they hold the major part of the energy associated with the total fluctuating part of the velocity, i.e., E ≈ v20. Successively the smaller eddies

are characterized by successively smaller, but still large Reynolds numbers and contain less energy,

Re>>Rei =v(i)li/v >Rei+1>>1, (2.5)

for each generation of eddies i. To avoid confusion it should be stressed that the notion of eddies is not really well defined. It would be safe to see the velocities v(i) as a characteristic fluctuating ve-

locity variance over a distance li, rather than the velocities of well

delineated vortices. In fact the latter can be seen as the definition of eddies. Eventually the size and the corresponding eddies velocity become small enough so that:

Red=vdld/v≈1, (2.6)

So that at these scales the dissipative term in the Navier-Stokes equa- tions becomes of the order as the nonlinear coupling. The scales of

21_{in Laboratory conditions the decaying turbulence is realized in turbulent flows}

order ofldare called thedissipation range, while the scales from the

interval:

L≈l0>> ld, (2.7)

this range is called theinertial range. For each scalelilet us introduce

the time scale:

∆ti =li/v(i), (2.8)

during which the eddy is likely to decompose into smaller ones. The characteristic time ∆t0 >>∆ti is called (large)eddy turnover time.

During the time comparable with the eddy turnover time the proto- eddies will decompose enough, as the result of nonlinear interactions, so that the inertial range is formed and the dissipation range is reached.22

It should be taken into account that in real conditions there is always present a mean inhomogeneous flow with velocity|U(r,t)|>

∆vL. It is quite clear that for the assumption of homogeneity and

isotropy of eddies (in the statistical sense) it is necessary that the following condition is fulfilled:

|∂kUi|−1>>∆t0>>∆ti, (2.9)

so that it is clear that the assumptions of HIT can be understood only locally in globally inhomogeneous flow and are applicable only to eddies from the inertial range.

Assume that turbulence in the inertial range is in a steady state. It is a safe assumption provided that the considered time interval sat- isfies (2.9). Then it should be from the balance of energy requirement that the average energy fluxper unit volume per secpassing through the successive generations of eddies with velocities vi remains con-

stant. Or more correctly the ensemble averaged energy flux density

< >= const. Clearly the flux is in the space of scales li dimin-