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Unformatted text preview: FUNDAMENTAL AND CONCEPTUAL ASPECTS OF TURBULENT FLOWS Arkady Tsinober Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows Institute for Mathematical Sciences and Department of Aeronautics, Imperial College London Lectures series as a part of the activity within the frame Lectures as of the Marie Curie Chair “Fundamental and Conceptual Marie Chair Fundamental Aspects of Turbulent Flows”. Aspects We absolutely must leave room for doubt or there is no progress and no learning. There is no learning without posing a question. And a question requires doubt...Now the freedom of doubt, which is absolutely essential for the development of science, was born from a struggle with constituted authorities... FEYNMANN, 1964 FEYNMANN LECTURES IX-X PHENOMENOLOGY AND RELATED What is phenomenology? Cascade. Is cascade a dynamical process? Is it a kind of trivial consequence of nonlinearity and decompositions? Is there cascade in physical space? Is "cascade" in genuine turbulence well defined? Counter-examples. Is "cascade" of passive objects meaningful? Nature of dissipation - is it (un)important? I soon understood that there was little hope of developing a pure, closed theory, and because of the absence of such a theory the investigation must be based on hypotheses obtained in processing of experimental data... KOLMOGOROV 1985 KOLMOGOROV Correlations after experiments done is bloody bad. Correlations Only prediction is science, FRED HOYLE, 1957, The Black FRED Cloud, Harper, N-Y. Phenomenology - The branch of a science that Phenomenology classifies and describes its phenomena without any attempt at explanation, WEBSTER'S NEW WORLD DICTIONARY, 1962 WEBSTER'S …even wrong theories may help in designing machines, FEYNMAN, 1996 (i.e. the right results for the wrong reasons, who cares?) In our present state of understanding, these simple In models will be based, in part on good physics, in part on bad physics, and in part on shameless phenomenology, LUMLEY, 1992 LUMLEY, Our present understanding of anything turbulent is Our at best phenomenological, SIGGIA, 1994 does such a thing exist ? SIGGIA does There is no definition of what is phenomenology of turbulent There flows. In a broad sense, it can be defined by a statement of impotence: it is almost everything except the direct experimental results (numerical, laboratory and field) and/or results (a very small set indeed), which can be obtained from the first principles, e.g. NSE. Phenomenology involves use of dimensional analysis, variety of scaling arguments, symmetries, invariant properties and various assumptions, some of which are of unknown validity and obscured physical and mathematical justification. Thus in the broad sense phenomenology includes also Thus most of the semi-empirical approaches and turbulence modeling. Doing all this requires insight into the basic (!) physics of turbulence, hard experimentation and painful efforts of interpretation. The latter may be quite problematic, especially in models having enough free parameters to guarantee the right results not necessarily for the right reasons. CONVENTIONAL PHENOMENOLOGY AS IT APPEARS IN THE BOOK by ENRICO FERMI, Notes on Thermodynamics and Statistics, The University of Chicago Press, Chicago and London Midway Reprint Edition, 1988; pp. 181-182. (1951 lectures) CASCADE WHAT IS IT ? IS THERE CASCADE IN PHYSICAL SPACE ? 66 THE FUNDAMENTAL EQUATIONS oh. 4/8/0 On the other hand we find that convectional motions are hindered by the formation of small eddies resembling those due to dynamical instability. Thus 0. K. M. Douglas writing of observations from aeroplanes remarks : "The upward currents of large cumuli give rise to much turbulence within, below, and around the clouds, and the structure of the clouds is often very complex." One gets a similar impression when making a drawing of a rising cumulus from a fixed point; the details change before the sketch can be completed. We realize thus that: big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity— in the molecular sense… Thus, because it is not possible to separate eddies into clearly defined classes according to the source of their energy; and as there is no object, for present purposes, in making a distinction based on size between cumulus eddies and eddies a few metres in diameter (since both are small compared with our coordinate chequer), therefore a single coefficient is used to represent the effect produced by eddies of all sizes and descriptions. KOLMOGOROV 1941A KOLMOGOROV Do you see here a cascade? Do …. even wrong theories may help in designing machines. (FEYNMAN, 1996) Feynmann R., 1996 Lectures on Computation, Addison-Wesley. (FEYNMAN, Wesley. Or here? Or …. even wrong theories may help in designing machines. (FEYNMAN, 1996) Feynmann R., 1996 Lectures on Computation, Addison-Wesley. (FEYNMAN, One gets an impression of little, randomly structured and One distributed whirls in the fluid, with the cascade process consisting of the fission of the whirls into smaller ones, after the fashion of the Richardson poem. This picture seems to be drastically in conflict with what can be inferred about the qualitative structure of high-Reynolds-number turbulence from laboratory visualization techniques and from plausible application of Kelvin circulation theorem ... How a theoretical attack on the inertial-range problem should proceed is far from clear. KRAICHNAN, 1974. On Kolmogorov's inertial-range theories, J.. Fluid Mech., 62, 305-On J 330. ... the idea of conservative inertial cascade local in scale size is consistent prima facie, provided that the actual statistics do prima not differ strongly from gaussian (!). It is another, and unsettled, matter to establish that K41 or a related theory actually describes what happens in NS flows, KRAICHNAN, 1991. Turbulent cascade and intermittency growth, Proc. R. Soc. Lond., 434, 65--78 Proc. Lond The notion that turbulent flows are hierarchical and involve The entities... of varying sizes is a common idea... This common notion underlies the concept of cascade, the third key element of turbulence theory FRISCH AND ORSZAG, 1990. All this cascade in Fourier space is a dream of linearized physicists, BETCHOV, 1993. BETCHOV The three examples (jet, boundary layer, and wake)... show that there is something wrong with this idea (the Richardson poem. In each case turbulence begins at small scales and grows larger: not the other way around, GIBSON, 1996. The conceptual picture is that of a cascade organized by wall distance The and by eddy size, where energy is transferred to smaller scales at any given location, and to larger ones away from the wall, JIMENEZ, 1999. JIMENEZ, This suggests that the Komogorov cascade process is basically incorrect, albeit an excellent approximation, SHEN AND WARHAFT, 2000 SHEN Much more troubling is the idea that there might not exist an energy cascade. TENNEKES 2004 (private communication) The Richardson-Kolmogorov cascade picture was formulated in physical The space and is used frequently without much distinction both in physical and Fourier space, as well as some others. It was von NEUMANN 1949 NEUMANN (see also ONSAGER 1949 who introduced the term cacasde in 1945) who stressed that who this process occurs not in physical space, but in Fourier space: ... the the system is "open" at both ends, energy is being supplied as well dissipated. The two "ends" do not, however, lie in ordinary space, but in its Fourier-transform...The supply of energy occurs at the macroscopic end --- it originates in the forced motions of macroscopic (bounding) bodies, or in the forced maintenance of (again macroscopic) pressure gradients. The dissipation, on the other hand, occurs mainly at the microscopic end, since it is ultimately due to molecular friction, and this is most effective in flow-patterns with high velocity gradients, that is, in small eddies... Thus the statistical aspect of turbulence is essentially that of transport phenomenon (of energy) --- transport in the Fourier-transform space. That is, the nonlinear term in the Navier-Stokes equation That redistributes energy among the Fourier modes not ‘scales’ as is frequently claimed, unless the ‘scale’ is defined just as an inverse of the magnitude of the wave-number of a Fourier mode, which is not easy for everybody to swallow. A natural question is then what does the nonlinear term in physical space do? Is energy (and not only energy) transferred from large to small scales in physical space? The answer to the last question depends on the definition of what is a ‘scale’ in physical space. First, we recall that that there is no contribution from the nonlinear First, term in the total energy balance equation (and in a homogeneous/periodic flow it's contribution is null in both the total and the mean), since the nonlinear term in the energy equation has the form of a spatial flux, ∂{...}/∂xj. In other words the nonlinear term redistributes the energy in physical space, but does it do more than that? It is straightforward to see that in a statistically homogeneous turbulent flow the mean energy of volume of any scale (Lagrangian and/or Eulerian) is changing due to external forcing and dissipation only -- there is no contribution in the mean of the nonlinear term, which includes the term with the pressure. That is, if one chooses to define a ‘scale’, l, in physical space as a fluid That (or a fixed) volume, say, of order l³, then in a statistically homogeneous flow there is no cascade in physical space in the sense that, in the mean, there is no energy exchange between different scales. This happens because the nonlinear term in the energy equation has the form of a spatial flux, ∂{...}/∂xj, i.e. there is conservation of energy by non-linear terms. In other words, the nonlinear term redistributes the energy in physical space if the flow is statistically nonhomogeneous. So, generally, it is a misconception to misconception interpret this or any other process involving spatial fluxes, ∂{...}/∂xj (e.g. momentum flux), as a ‘cascade' in physical space. FRISCH 1995 (pp. 77,78) defines a quantity which he calls (!) physical-space energy flux At page 79 he ends up with the statement This is correct, but this is not an energy flux relation in physical space as just using the term physical-space energy flux for the quantity does not turn it into such. Neither does help the (very useful fact ) that this quantity is related the energy flux ΠK through the wave-number K appearing in the scale by scale energy budget relation in Fourier space On the other hand, in a statistically stationary state ∫F·udτ = ∫εdτ, On i.e. energy input, which is associated with large scales, equals dissipation, which occurs mostly in the small scales. Is there a contradiction? Does the equality ∫F·udτ = ∫εdτ mean that the energy should be somehow ‘transferred' from large to small scales via some multi-step process? Not necessarily - for example, two big neighboring eddies can dissipate energy directly through encounters with each other at small scale - much smaller than their own scales. Such a process still will look in Fourier space as continuous energy transfer from modes with small to modes with large wave numbers. CASCADE VERSUS DECOMPOSITIONS/REPRESENTATIONS Fourier transform ambiguity in turbulence… TENNEKES 1976 I think that the k -space decomposition does actually obscure the physics. MOFFATT, 1990 MOFFATT See also LIEPMANN, 1962; LOHSE AND MÜLLER-GROELING, 1996 LIEPMANN LOHSE The resolution of the apparent contradiction lies in clarifying the meaning of the The term `scale', which mostly is understood as an inverse of the wavenumber magnitude in a Fourier representation, and what is the meaning of ‘transfer of energy (or whatever) from large to small scales’ in physical space. This issue is directly related to the decomposition/representation of the turbulent flow field. Indeed, the reason for the above result on the absence of energy exchange between different scales in physical space is because no decomposition is involved in the above definition of ‘scale'. Any decomposition (be it in physical space, Fourier or any other) brings the ‘cascade' back to life. For example, there are various ways of filtering the flow field widely used in large eddy simulations. However, one of the problems with decompositions is that the nonlinear term redistributes the energy among the components of a particular decomposition in a different way for different decompositions, i.e. the energy exchange/transfer (and not just energy), generally is decomposition dependent. Therefore one may ask whether quantities like energy flux are well defined. In Therefore other words the term ‘cascade’ corresponds to a process of interaction/exchange of (not necessarily only) energy between components of some particular decomposition/representation of a turbulent field associated with the nonlinearity and the nonlocality of the turbulence phenomenon, two of the three N's: nonlinearity, nonlocality and nonintegrability, which make the problem so impossibly difficult . On the other hand, energy transfer, just like any physical process, should be invariant of particular decompositions/representations of a turbulent field. In this sense Kolmogorov's choice of dissipation (and energy input) are well defined and decomposition independent quantities, whereas the energy flux is (generally) not, since it is decomposition dependent. After all Nature may and likely does not know/care about our decompositions. our decompositions. ‘Cascade' arising from a decomposition of the flow field viewed as a process of exchange of energy, momentum, etc. between the components of this decomposition is a dynamical process. This should be distinguished from ‘cascading processes’ resulting from a decomposition of some quantity, e.g. dissipation, usually of its surrogate (∂u1/∂x1)², obtained from experimental signals. The former is a dynamical process, The whereas the latter is a representation characterizing some aspects of the spatial and/or temporal structure of some flow characteristics. ‘In other words, ‘structure’ is not synonymous of ‘process': it is the result of a process. Therefore, generally it is impossible to draw conclusions about the former from the information about the latter, though this is done quite frequently. For example, simple chaotic systems with few degrees of freedom only produce also ‘fine structure', possessing a continuous spectrum with a multitude of interacting modes. Of course, such a signal can be also cast in a multiplicative representation, but there is no ‘cascade' whatsoever. COUNTER-EXAMPLES COUNTER-EXAMPLES IN TRANSITIONAL FLOWS Abrupt transition Pipe. Entrainment. Vortex breakdown. Turbulent spots. Blow up instabilities. Bypass instabilities and transitions. In all laminar fluid becomes turbulent in ‘no time’ without any cascade whasoever. ABRUPT TRANSITION The transition between laminar and turbulent flows at the beginning ing and end of the turbulent region is abrupt relative to its duration. and ROTTA, J. C.(1956) Experimenteller Beitrag zur Entstehung ROTTA, turbulenter Strömung im Rohr, Ing. Arch., 24, No. 4, 258–281. Ing. No. ABRUPT TRANSITION The transition between laminar and turbulent flows at the beginning ing and end of the turbulent region is abrupt relative to its duration. and The transition is indeed frustratingly abrupt s l u g s Wygnanski & Champagne 1973 Durst & Unsal 2006 Durst p u f f s ABRUPT TRANSITION A vortex ring impinging a wall becomes turbulent in no time as it approaches the wall turbulent rotational PTF - ENTRAINMENT Mount St. Helen volcano on 18 May 1980 The laminarturbulent “interface” is sharp so that fluid particles laminar irrotational A turbulent jet from testing a Lockheed rocket engine in the Los Angeles hills (note the Lagrangian aspect !) “are found” abruptly in a turbulent environment COEXISTENCE OF COEXISTENCE LAMINAR AND TURBULENT REGIONS IN THE SAME FLOW SAME Vortex breakdown Small scales are not necessarily created from large scales via a stepwise turbulent ‘cascade’: it can be bypassed, and most probably is so in turbulent flows, for example via broad-band instabilities with highest growth rate at short wavelengths (PIERREHUMBERT AND WIDNALL, 1982) or some other approximately single step process (BETCHOV 1976, DOUADY ET AL. 1991, OTT 1999; SHEN AND WARHAFT 2000, VINCENT AND MENEGUZZI 1994). The problem goes back to TOWNSEND 1951: ...the postulated process differs from the ordinary TOWNSEND ...the type of turbulent energy transfer being fundamentally a single process An important example, is the complicated structure of vorticity (and passive An objects) with power law spectrum, (multi)fractality and significant variations down to very small scale that can be produced by a single instability at much larger scale without any ‘cascade’ of successive instabilities arising in a simple fluid flow via a single instability only(!), OTT 1999. ANTICASCADE ARGUMENT BY ANTICASCADE CHORIN 1994 Vorticity and Turbulence, Springer, pp. 55-57 CHORIN Do you see anything wrong with Chorin’s Chorin argument ? So you have a kind of home work: have a So look and we can discuss this later. look IS CASCADE LOCAL? IS # If cascade picture makes sense, one probably must have a complex interplay between distant shells CHORIN 1994 # One of the key ideas in turbulence theory is that of locality of cascades. The essential idea is that only modes near a given scale contribute to transfer across that scale. The concept has a central importance in the subject because locality is one of the conditions invoked to justify universal statistics at small scales. If excitations are transferred by a chain of chaotic steps from scale to scale, then the conditions at the large scales may be forgotten and only the interactions at a sequence of adjacent scales will determine the characteristics at small scales. EYINK 2005. EYINK In view of the existing evidence (part of which was presented in previous lectures) the latter (part statement looks somewhat outdated and seems to contradict the evidence. As the paper in question (Eyink, G.L. (2005) Locality of turbulent cascades, Physica, D207, 91-116) is pretty recent Physica and claims mathematical rigor some comments are in place. The paper is based on The The regularity that we shall assume for Euler solutions in the high The Reynolds number limit is the Hölder type that was conjectured theoretically [1,2] (ONSAGER 1949 AND PARISI & FRISH 1983) and is observed (in a space-mean sense) experimentally [3] (ANSELMET ET AL 1984). In other words the whole outcome is based at best on a conjecture. More serious is that it is not at all clear how the properties of Euler solutions can be observed experimentally (if at all), in general, and at large Reynolds numbers, in particular. All the experiments the author refer to (and all the other) are done at pretty low Reynolds numbers at which there are no very long inertial ranges to allow for local transfer to dominate (a crude estimate is that one needs about six decades for this, i.e. Taylor microscale (!) Reynolds numbers exceeding 10⁸, which cannot be reached in any experiment in the forseeable future). Thus the scaling exponents obtained from experiment as in ANSELMENT ET AL 1984 and similar ones (and, of ANSELMENT course, numerical) later are not necessarily those which can be expected at very large Reynolds numbers. Nobody seems to be sure at all what can be expected at very large Reynolds numbers. For example, there is a possibility (and a number of publications clearly For indicating that) in the limit a pure Kolmogorov scaling will recover (it happens pretty slowly) and that the present experimental and, of course, numerical observations are just a finite Reynolds number (and possibly finite size) effect*. This casts serious doubts about the multifractality hypothesis** * But this does not necessarily mean that nonlocal effects - even if they are not dominant in the sense used in the paper in question - may not spoil the universality of small scales at any whatever large Reynolds numbers. ** In fact, the main feature of the multifractal hypothesis (assumption, see Frisch 144) is that it's authors ARE back with universality postulating two universal exponents (i.e. a whole range) and a universal function postulated to be independent of the mechanism of production of the flow). Unfortunately, in contrast with K41, the multifractal model (like many other intermittency models) is an arbitrary construction in the sense that it lacks dynamical motivation (so far) in general and, with respect to the postulated multifractal universality in particular. (Recall the question Where is the physics? by L. P. Kadanoff 1986, Physics today, 39, 6.). Since structure functions exhibit some scaling this is all necessary to make the multifractal hypthesis "valid". indeed: Of course, having that much assumed (i.e. a whole range of exponents and a function) it is really easy to fit to this frame almost any experimental/DNS data and whatever. In other words, though multifractality was designed to ‘explain' the anomalous scaling, intermittency, etc., it is in fact another description and way of looking at the data as some people do at finite Reynolds numbers. The assumption (not made in the paper) that at low Reynolds numbers the large scales The (whatever this means) behave the same way as at very large Reynolds numbers does not help much: on the contrary this assumption meets lots of difficulties in view of the existing (pretty massive) evidence that at least at all achievable (so far) Reynolds numbers the nonlocal effects are essential. There is a problem concerning the relation of the exponents defined in the paper (based on Lp norm on the space domain of the flow) in claiming that these are related to the usual these (absolute) structure--function exponents by αp = ζp/p. The αp and ζp/p are pretty different objects. A related (technical, but not only) question in the definitions of exponents like αp is that first the limit ν → 0 is taken (assuming that it exists) and then r→0 whereas ζp’s are determined not really in this way. The above is related to the claim: It is important that we have established It locality for individual flow realizations without statistical averaging... This brings two questions. First, the exponents ζp are obtained essentially by statistical This averaging. Second, it is it is not clear in what sense the power laws exist for individual flow realizations. The paper ends up with the statement: The On the other hand, our estimates, like those from closures, show that the cascade is, as Kraichnan expressed it, only "asymptotically local" and "diffuse". The rate of vanishing of the nonlocal contributions with increasing scale-separation are quite slow power-law decays. Thus, very long inertial ranges might be required for local transfer to dominate. As mentioned very long inertial ranges (if the notion of inertial range is meaningful at all) are not achievable in the forseeable future. Thus even if asymptotically (as ν→0 ) the claims of the author are correct the cascade (whatever this means) is not local in any realistic situation. This brings us to the most popular and difficult question about the limit ν→0. NATURE OF DISSIPATION NATURE IS IT (UN) IMPORTANT ? It is quite a common view that the precise nature of dissipation is mostly unimportant in high Reynolds number turbulence except for the smallest scales. This This forms, e.g. the basis for what is called inertial range. In view if various aspects of nonlocality the natural question is what does it mean, what kind of quantities do not really depend on the nature of dissipation, why and and in what sense as well as many similar closely related questions We therefore conclude that, for the large eddies which We are the basis of any turbulent flow, the viscosity is unimportant and may be equated to zero, so that the motion of these eddies obeys Euler’s equation. ... The viscosity of the fluid becomes important only for the smallest eddies, whose Reynolds number is comparable with unity. ... we may say that none of the quantities pertaining to the eddies of sizes r >> η can >> depend on ν (more exactly, these quantities cannot be changed if ν varies but other conditions of the motion are unchanged). LANDAU AND LIFSHITZ 1954 The natural question is, therefore: Is it at all important that this subsidiary agent be viscosity? Might other dissipative, perturbing forces not do equally well? In planning for a test of this question, one might first think of investigating other forms of the law of viscosity, i.e. other equations of flow instead of those of Navier-Stokes, where viscosity might be described by a term other than ν∇2u or by entirely different, non-linear changes in the equations. … In any event, it would be interesting to determine, whether such modifications could lead to different forms of turbulence (in the pure limiting, i.e. ν→0 form) than the Navier-Stokes equations. The whole character of the KolmogoroffOnsager-Weizsa}cker theory would make one inclined to surmise that this is not the case. JOHN VON NEUMAN (1949) νΔu …there is nothing ``irrelevant'' in the (NSE) equation (except, may be, as ν→0, the precise nature of dissipative term) . FRISCH (1983)… FRISCH … it may be that turbulence is not dependent on the Navier-Stokes equations. There may be other equations, and not even necessarily diffrential equations, whose properties have the same kind of structure as the turbulent structure of the NavierStokes equations. And these other equations may be easier to solve... one should be prepared to consider systems of equations other than just the Navier-Stokes equations. SAFFMAN (1991) SAFFMAN In fact, turbulence is an inertial phenomenon. That is, turbulence is statistically indisdinguishable on energy-containing scales in gases, liquids, slurries, foams, and many non-Newtonian media. These media have markedly different fine structures, and their mechanisms for dissipation of energy are quire different. This observation suggests that turbulence is an essentially inviscid, inertial phenomenon, and is uninfluenced by the the precise nature of the viscous mechanism.. HOLMES, BERKOOZ AND LUMLEY (1996) HOLMES, Causality is from large to small scale, and how the energy is dissipated in the latter does not influence the former, as long as the amount is correct.. JIMENEZ JIMENEZ (2000) In any case, the dissipation processes, independently of their nature, serve only as energy sinks, which cut off the spectrum of turbulent fluctuations at small scales but do not affect the main turbulence scales. BISKAMP (2003) BISKAMP Iff the precise nature of dissipation is unimportant in high Reynolds I number turbulence and if the nature of dissipation is not important either, why to work hard specifically on NSE instead of, e.g. taking some modified version of NSE (LERAY 1934; LIONS, 1969; LADYZHENSKAYA, 1969, 1970; FRIEDLANDER AND PAVLOVIć, 2004) or lattice gas hydrodynamics approximation (CHEN AND DOOLEN, 1998; FRISCH ET AL., 1987; HARDY ET AL., 1976; WOLFRAM, 1986), all of which have regular solution for any time and at any Reynolds numbers ? The modification of NSE introduced by consists in mollifying the nonlinearity rather than changing the dissipative term as did the other authors. And why Clay Mathematics Institute (with $1M prize) insists specifically on NSE ? Is this really the case that the precise nature of dissipation is unimportant? It is possible that, after all, the It investigation in which viscosity is ignored altogether is inappropriate to the limiting case of a viscous fluid when the viscosity is small... certain features of the motion which could not enter into solutions were the viscosity ignored from the first are independent of the magnitude of the viscosity .. RAYLEIGH 1892 Rayleigh, (1892) On the question of the stability of the flow of fluids, Phil. Mag. 34, 59-70 Phil. SOME PROPERTIES OF SOME TURBULENT FLOWS DO NOT DEPEND OF THE NATURE OF DISSIPATION Turbulence is so rich that it can afford it The precise nature of dissipation is unimportant only in respect with a number of The manifestations of turbulence which are really weakly sensitive to the nature of dissipation at large Re (and even more generally to specific properties of the system as long as it is dissipative), e.g. things like 2/3, 4/5 and 4/15 laws, k-5/3 spectrum and some others. The most convincing example is the 4/5 law showing that the third order structure function is universal, i.e. it depends on the mean energy injection rate only in the so called inertial range. However, the 4/5 law is a direct consequence of Euler equations (Duchon and However, Robert 2000, Eyink 2003) and it is not obvious that the same result should hold for structure functions of higher orders and/or other objects. The argument that this may be not the case goes as follows. ONLY SOME PROPERTIES OF ONLY TURBULENT FLOWS DO NOT DEPEND OF THE NATURE OF DISSIPATION The starting point is that viscosity/dissipation modifies substantially The and qualitatively the nonlinearity, i.e. it is not a passive sink of energy. Indeed, this point is clearly seen from looking at the equations for vorticity and enstrophy, e.g. enstrophy production is approximately balanced by its viscous destruction 〈ωiωjsij〉 ~ 〈νωi∆ωi 〉 That is turbulence is an essential interaction between nonlinear and linear processes and not just a simple cascade of energy or whatever down to smaller scales. For example, in case of modified equations such as those with hyperviscosity the enstrophy balance is quite different from that for NSE, i.e. the nonlinearity is quite different either. Thus viscosity exerts direct influence on the vorticity field (and similarly Thus strain) and thereby should do the same with the velocity field either (indeed, we remind that the velocity filed is fully defined by the field of vorticity). Along with nonlocality (direct interaction of large and small scales irrespective of their separation, see references in Tsinober ( 2001, 2003) and broken scale invariance this means that the nature of dissipation should be felt in large scales as well. In particilar, it is not obvious why some statistical properties of velocity increments u(x+r) u(x) (both are functionals of vorticity and/or strain) do not depend on viscosity. This is true even of the third order structure function: it is interesting to explain how the viscous effects cancel out. The pure inertial range is not well defined due to nonlocality The of turbulence; because of scale invariance breaking, because the notion of inertial range is not well defined (ARNEODO ET AL., 1999). We wish to reiterate that independence of some (statistical) parameters or properties of viscosity at large Reynolds numbers does not mean that viscosity is unimportant. It means only that the effect of viscosity is Reynolds number independent. RELATED EXAMPLES SEE ALSO EXAMPLES IN THE LECTURES ON NONLOCALITY SEE NONLOCALITY Same dissipation – different flows: Dissipation (energy input) or drag Same only are not sufficient to define the properties of a turbulent flow. For example, BEVILAQUA AND LYKOUDIS (1978) performed experiments on flows past a sphere and a porous disc with the same drag. However, other properties of these flows even on the level of velocity fluctuations were quite different; see also WYGNANSKI ET AL. (1986) who performed similar experiments with a larger variety of bodies with the same drag. Similarly, many properties of turbulent flows with rough boundaries are not defined uniquely by their friction factor (KROGSTAD AND ANTONIA, 1999). Turbulent LS dynamo of magnetic fields is strongly dependent (sensitive) Turbulent on diffusivity. POSSIBLE CONSEQUENCES POSSIBLE FOR THE "INVISCID" LIMIT The differences in the behavior of the systems with different dissipation at The finite Reynolds numbers points to a possibility that the limiting solution will depend on the kind of dissipation we have at finite Reynolds number (recall the above quotation by NEUMANN, 1949) even if their "inviscid dissipation“ manifested in D(x,t) (DUCHON AND ROBERT 2000) would be the same. Of special interest is what happens in this limit with enstrophy/strain production and similar things. For example, assume that for a hyperviscous case that the mean disipation ε→const as some viscosity νh goes to zero, then velocity derivatives (both vorticity and strain) grow as νh-1/2h, which is pretty slow, if say, h= 8 as used in many simulations. We remind again that the requirement that D(x,t) ≥ 0 (DUCHON AND We ROBERT 2000) is nonnegative is not sufficient to determine the "right inviscid" solution (distribution) of Euler, and the whole issue of Redependence and finite Re-effects adds to its importance (see KRAICHNAN, 1991). Is it possible that the Re-dependence is different (which seems to be the case) but the limit is the same? More generally, does it make sense to speak about the inviscid limit without referring to some specific situation/properties? It is naturally to ask a more general question: Is turbulence "slightly viscous" at whatever small viscosity? (Isn't it similar to being slightly pregnant?) The answer seems to be negative even for regions far from boundaries. A BIT MORE FOR BIT MATHEMATICALLY MINDED EULER EQUATIONS AND WEAK SOLUTIONS OF EULER NAVIER-STOKES EQUATIONS NAVIER Att least since KOLMOGOROV (1941) an enormous effort is invetsed in A attempts to study asymptotic properties of turbulent flows at vanishingly small viscosity. Considerable evidence shows that these flows at whatever small viscosity possess finite dissipation, i.e. such flows are very much unlike classical solutions of the Euler equations, which have the property of energy conservation. One of natural conjectures in the mathematical community was that turbulent flows may be described asymptotically correctly by some sort of specially selected weak solutions of the Euler equations which are called "disspative" - an approach which goes back to ONSAGER (1949). Indeed, examples of weak (or distributional) solutions have been Indeed, constructed without energy conservation, see LIONS (1996), SHNIRELMAN (2003) and references therein. It appears that there exist very different classes of weak solutions, having little in common, and some of them are physically meaningless (with negative dissipation, i.e. energy creation), at least, in the context of turbulent flows. Moreover, there is no uniqueness of a weak solutions, SHNIRELMAN (2003). In other words one needs additional conditions to ensure physical meaning and uniqueness of solution. Simply stated the solution has to be dissipative in the first place. DUCHON AND ROBERT (2000) addressed this issue by considering local energy balance for any weak solution ∂u²/2/∂t +∂/∂xk {(u²/2+p)uk}+D(x,t) = 0 where D(x;t) is shown to be a distribution defined in terms of the local D(x;t smoothness of velocity field u. DISSIPATIVE SOLUTIONS DISSIPATIVE OF EULER EQUATIONS found an explicit expression for D(x;t) which makes the above equation identity in the sense of distributions. Thus D(x;t) measures a possible dissipation (or production) of energy caused by a lack of smoothness in the velocity field u in the spirit of ONSAGER (1949). For smooth solutions D(x;t)≡0. The next step is to impose a condition that locally D(x;t) ≥0 as physically acceptable, since for a general weak solution of Euler equation, there is no connection of D(x;t) with viscous dissipation, nor should D(x,t)≥0. However, as shown by DUCHON AND ROBERT D(x,t)≥0 for a weak solution of Euler which is the strong limit of a sequence of dissipative weak solutions of Navier--Stokes as viscosity goes to zero. In this they used the fact that the condition D(x,t)≥0 is satisfied by every In DUCHON AND ROBERT (2000) weak solution of the Navier- Stokes equation obtained as a limit of a subsequence of solutions uɛ of the regularized equation introduced by LERAY (1933, 1934). However, the condition D(x,t)≥0 does not garantee uniqueness However, - the phase space of Euler weak solutions seems to be too rich. Moreover nonuniqueness is not the only problem. To quote Moreover SHNIRELMAN (2003): as for today, we have no weak solution (of Euler equation) at hand which really (of describes a turbulent flow. In fact, having a "good candidate" it would be an extremely difficult (if not impossible) task to decide whether it really describes a turbulent flow. Moreover, to find such a candidate seems to be as difficult as the "solution of the problem of turbulence" itself. WEAK SOLUTIONS OF WEAK NAVIER-STOKES EQUATIONS In fact, DUCHON AND ROBERT (2000) started with weak solutions of In NSE and wrote a local energy balance for Navier-Stokes equations ∂u²/2/∂t+∂/∂xk {(u²/2+p)uk} -ν∆ u²/2+ν(∂ui/∂xk)(∂ui/∂xk)+ D(x;t)=0 with D(x;t) defined in the same way as for the Euler equation. Thus - they write - the non-conservation of energy originates from two sources: viscous dissipation and a possible lack of smoothness in the solution.; and stress that D(x;t) measures a possible dissipation (or production) of energy caused by a lack of smoothness in the velocity field u, this term is by no means related to the presence or absence of viscosity. The latter statement being formally/mathematically nice The is problematic from the physical point of view. As long as one is speaking about NSE this looks definitely unphysical: so far no physical process is known that can bring an additonal dissipation into operation which is formally described by the distribution formally D(x;t). The questions about the existence of a weak solution of Navier--Stokes The with D(x;t)≠0 as well as the uniqueness of such a solution with D(x;t) D(x;t)>0 remained unanswered. DUCHON AND ROBERT note also that DUCHON There is still some doubt as to whether weak solutions of the Navier--Stokes equation, the uniqueness of which is unknown, or hypothetical weak solutions of the Euler equation, are relevant to the description of turbulent flows at high Reynolds numbers. One may add that, moreover, if we look at real turbulence at finite One Reynolds numbers (whatever large) there seems to be no need for weak solutions at all. HOW MEANINGFUL IS "CASCADE" OF PASSIVE OBJECTS ass described by linear equations? a Is there ‘enough’ analogy (more on analogies in a separate lecture) between genuine and ‘passive’ turbulence or the differences are essential? Nonlinear versus linear. Is extension of Kolmogorov phenomenology justified for systems governed by linear equations? Itt is rather common, since Obukhov (1949) and Corrsin (1951), to speak about cascade in I case of a passive scalar and more recently passive . The main argument is from some analogy. Indeed, for instance in any random isotropic flow the rate of production of ‘dissipation' (i.e. corresponding field of derivatives) of both passive scalars and passive vectors is essentially positive, which can be interpreted as a sort of `cascade'. However, the equations describing the behavour of passive objects are linear. Hence, there is no interaction between modes of whatever decomposition of the field of a passive object: the princilple of superposition is valid in case of passive objects*. *Here by ‘mode' is meant as a solution of the appropriate equation, e.g. of the advection-diffusion equation . Of course, there are many ways to use ‘modes’ that are not solutions of this equation, such as Fourier modes. In this case the Fourier modes do interact, since one of the coefficients of the advection-diffusion equation, the velocity field, is not constant. This interaction is interpreted frequently as a 'cascade' of passive objects. But, as mentioned, this interaction is decomposition dependent, and therefore is not appropriate for description of physical processes, which are invariant of our decompositions. There is a point concerning the behavior of an individual solution. Namely, the evolution of its energy spectrum is expected to exhibit positive energy transfer to higher wave numbers as a consequence of production of the field of derivatives of the passive field. Can one see this as a kind of ‘cascade’? Even if the answer were affirmative it is a very different kind of cascade, if at all. Therefore, it seems more appropriate to describe the process in terms of production Therefore, of the field of derivatives of the passive object, which is performed by the velocity straining field, just like it is proposed above for the velocity ffield. ield. Hence the extension of Kolmogorov arguments and phenomenology to passive Hence objects seems to be much less justified. No wonder that the phenomenological paradigms for the velocity field failed in most cases when applied to passive objects*. We are reminded that the ‘analogy’ between the passive objects and the active variables is, at best, very limited for several reasons, the main of which are the linear nature of ‘passive’ turbulence, Lagrangian chaos, the irreversible effect of the randomness of the velocity field on passive objects independently of the nature of this randomness, e.g. even a Gaussian one, and the one-way interaction between the velocity field and the field of a passive object . *E.g. experiments by Villermaux et al. (2001) clearly show that this is the case. The behaviour of passive scalar in their experiments is distinctly nonlocal in the sense that the main mechanism responsible for mixiing involves direct interaction between large and small scales ‘bypassing' the (nonexistent) cascade. IS BELOW ANYTHING WRONG? Bogucki, D., Domaradzki, J.A. and Yeung. P.K. (1997) Direct numerical simulations of passive scalars with Pr>1 advected by turbulent flow, J. Fluid Mech., 343, 111-130. AT P. 117 THE MIXED DERIVATIVE SKEWNESS IS RELATED TO THE NONLINEAR TRANSFER OF SCALAR VARIANCE TO SMALL SCALES AND TAKES A VALUE ZERO WHEN THERE IS NO NET CASCADE TO HIGHER WAVENUMBERS. CONCLUDING CONCLUDING CHOOSE WHAT YOU LIKE MORE? Big whirls have little whirls, Big Which feed on their velocity. And little whirls have lesser whirls And so on to viscosity – In the molecular sense RICHARDSON (1922) Big whirls lack smaller whirls, Big To feed on their velocity. They crash and form the finest curls Permitted by viscosity BETCHOV (1976) The notion that turbulent flows are hierarchical, which underlies the The concept of cascade, though convenient, is more a reflection of the unavoidable (due to the nonlinear nature of the problem) hierarchical structure of models of turbulence and/or decompositions rather than reality. This is emphasized in the case of passive objects, whose evolution is governed by linear equations, with the velocity field entering multiplicatively in these equations, thus making them ‘statistically nonlinear’. AND SO WE ARE BACK WITH THE QUESTIONS Is the inertial range a conceptually well defined concept or Is is it "mostly a pedagogical imagery"? Are its properties really independent of the nature of dissipation? Is the Komlogorov 4/5 law an unequivocal evidence of such a cascade? Is "casade" Eulerian, Lagrangian or both? Are decompositions aiding understanding or obscuring the physics of turbulence? AND MANY OTHER ...
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