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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 VII-VIII NONLOCALITY NONLOCALITY ‘KINEMATIC’ AND DYNAMIC A SET/VARIETY OF MANIFESTATIONS/ PHENOMENA UNDER THE COMMON NAME ‘DIRECT AND BIDIRECTIONAL INTERACTION OF LARGE AND SMALL SCALES’. This, however, contains two ambiguous notions: SCALES and SCALES COUPLING/INTERACTION. COUPLING/INTERACTION SCOPE. Nonlocality – kinematic and dynamic. Examples. SCOPE Manifestations of nonlocality. Small scale anisotropy at large Small Reynolds numbers. Substantial lack of reflexional symmetry in small scales in helically forced flows. Dependence of SS statistics (and accelerations) on LS. Memory effects and predictability. Modification of turbulence in multiphase flows. Drag reduction in turbulent flows of flows dilute polymer solutions and other drag reducing additives. Turbulence induced secondary mean flows. Related phenomena: Anistropic flows (MHD, stratified, rotating), steady streaming in a turbulent oscillating flows, ‘negative eddy viscosity’, longrange correlations in Lagrangian setting, predictability and data assimiliation, nature of forcing including acoustic; difference between quasi-2D and pure-2D turbulent fows. From the formal mathematical point of view a process is called local if all From the terms in the governing equations are differential. If the governing equations contain integral terms or other nonlocal operators then the process is nonlocal. The Navier-Stokes equations are integro-differential for the velocity field in both physical and Fourier space (and any other). Therefore, generally, the Navier-Stokes equations describe nonlocal processes. From the physical point of view (the above meaning of) nonlocality is associated with or long range or action at a distance. This again is closely related to what is called direct and bidirectional coupling of large and small scale ON SCALES AND THEIR COUPLING The ambiquity in ON definition of scales can be avoided by not using any decomposition (D): the small scales (whatever this means) are associated with the field of velocity derivatives (vorticity and strain). Therefore, it is naturally to look at this field as the one objectively (i.e. D-independent) representing the small scales. The velocity fluctuations represent the large scales.Taking the above position one can state that in homogeneous (not necessarily isotropic) turbulence the large and the small scales are uncorrelated. For example, in a homogeneous turbulent flow the mean Lamb vector 〈ω×u〉 = 0 (and 〈(u·∇)u〉 = 0 ). ON SCALES AND THEIR COUPLING. If the flow is statistically ON reflexionally symmetric then the mean helicity 〈ω·u〉 vanishes too. Similarly, in a homogeneous turbulent flow velocity, ui, and strain, sij do not correlate either, 〈uisij〉 = 0. However, vanishing correlations do not necessarily mean absence of dynamically important relations (vorticity and the rate of strain tensor are precisely uncorrelated too 〈ωisij〉 ≡ 0, but their interaction is in the heart of the physics of any turbulent flow). Indeed, the quantities (u·∇)u ≡ ω×u+∇(u²/2) and ω×u, are the main `guilty' for all we call turbulence. Both contain the large scales (velocity) and small scales (velocity derivatives). Therefore some kind of dynamically essential coupling between the two is unavoidable. A REMINDING REMINDING ON WHAT CAN BE CALLED KINEMATIC NONOLCALITY The whole flow field is defined by the vorticity in the whole flow domain with appropriate boundary conditions. This with is true of strain too, e.g. there is a relation between the fields of velocity and strain similar to that of Biot-Savart. It is essential that the above argument is based on vorticity and/or strain – as invariant physical quantities – rather than any ‘surrogate’ derivatives. This can be qualified as most elementary aspect of This ‘kinematic’ nonlocality as vortcity and starin can be seen as (mostly) representing ‘small scales’. Vorticity and strain are not just velocity derivatives. They Vorticity are special for several reasons as mentioned (and will be discussed at length below). The one to be stressed here is that the whole flow field is determined entirely by the field of vorticity or strain *with appropriate boundary conditions: ∇²u = -curl ω; ∇²ui=2∂sij/∂xj, i.e. the velocity field is∇ a linear functional of vorticity u i.e. = F{ω}, or strain ui = G{sij},*i.e. alteration of the field of velocity derivatives reflects on the velocity field, vorticity and strain are not passive – they react back and not only for the above (kinematic) reason. In the whole space the first functional is the well known Bio-Savart law, and the second has a similar form (see eq. C.14’ in Tsinober 2001) (see ui(x,t) = ∫µjjω(r)ωj(y,t)dy, µjjω (r)= - (4π)-1 εijk rj/r³, ri = xi-yi, ui(x,t) = ∫µjs(r)sij(y,t)dy, µjs(r)= - (2π)-1 rj/r³, ri = xi-yi, sij(x,t) = ∫P.V.αij(r)ωk (y,t)dy, αij (r)= - 3(8π)-1 {εijlrlrk+ εkjlrlri}/r5, ri = xi-yi, ∫P.V. stands for the Cauchy principal value. IN OTHER WORDS IN Alteration of the field of velocity derivatives reflects on the velocity field: vorticity and strain are not passive – they velocity react back and not only for the above (kinematic) reason react (likewise the field of a passive scalar is defined by its gradient G=∇θ and the field of a vector potential A off a solenoidal passive vector B (rot A=B) iss defined by B). A=B) i o . The above statement is not that trivial as may seem. The Three examples: FIRST, the purely kinematic reaction back, e.g. of vorticity in FIRST, the statistical sense, is seen from the following example. Taking a Helmholz decomposition of the most significant part of the nonlinear term in NSE, the Lamb vector ω×u = ∇α + ∇×β it can be shown that 〈(∇α)²〉 ~ 〈 ∇×β)²〉 if ω and u are random Gaussian and not related, i.e. ω ≠ rot u. However, if ω = rot u and u is random Gaussian, then rot is 〈(∇α)²〉 ~ 2〈∇×β)²〉 , i.e. ω ‘reacts back’ for purely kinematic reasons. SECOND, kinematic coupling plus self-amplification of vorticity and strain ‘react back' in creating the corresponding velocity field . (since as mentioned the whole flow field - including velocity, which is mostly a large scale object - is determined entirely by the field of vorticity or strain). In other words the small scales cannot be considered as a kind of passive objects swept by the large scales or just ‘slaved’ to them at any (whatever large) Reynolds numbers. This looks as a stong indication (and warning) that the nature of dissipative processes (viscosity, hyperviscosity, etc.) are important at any (whatever large) Reynolds numbers (this issue will be addressed in the next leture). It is noteworthy (this that due to nonlocality mostly small scale vorticity and strain are, generally, creating also some large scale velocity. (u·∇)u = ωxu + ∇(u2/2) In a plane channel flow (or any flow with d〈…〉/dy=0) d〈uv〉/dy = <ωxu>1 WEI AND WILMARTH 1989 ∇ Hence we have a clear indication of a dynamically important statistical dependence between the large (u) and small (ω) scales. and Since in these turbulent flows d〈uv〉/dy is essentially different from zero at any arbitrarily large Reynolds number, one can see from above that at least some correlations between velocity and vorticity are essentially different from zero too at any arbitrarily large Reynolds number. The above and other aspects of nonlocality (to be addressed The below) contradict the idea of cascade (in physical space), which is (in local by definition . This means that the common view that there exists a range of scales (the inertial range) in which the effects of viscosity, boundary conditions, and largescale structures are unimportant is suspicous as basically/conceptually incorrect (more in the next lecture on (more phenomenology and related). Nonlocality – dynamic. Long range forces due to pressure: Nonlocality Navier-Stokes equations are integro-differential. This is a property of nonlocality of Navier-Stokes equations is in physical space : pressure is a functional of the field of velocity derivatives p = P{Q}, where where Q=(1/4)(ω²-2sijsij) = -(1/4)(∂²uiuj/∂xi∂xj) is the second Q=(1/4)( invariant of the velocity gradient tensor ∂ui/∂xj. Again (it looks that) (it pressure too is completely determined by the field of velocity derivatives (not just the field of velocity) in a nonlocal way (The pressure field can be directly expressed in terms of the field of velocity itself. However, since the latter is defined by the field of velocity derivatives, so does the pressure). Nonlocality – dynamic. The nonlocality due to pressure is strongly The Nonlocality associated with essentially non Lagrangian nature of pressure. For example, replacing in the Euler equations the pressure Hessian ∂²p/∂xi∂xj ≡ Πij, which is both nonlocal and non Lagrangian, by a local quantity (1/3)δij∇²p=(1/6)ρ{ω²-2sijsij} turns the problem into a local and (1/3) integrable one and allows to integrate the equations for the invariants of the tensor of velocity derivatives ∂ui/∂xj in terms of a Lagrangian system of coordinates moving with a particle. The reason for disappearance of turbulence (and formation of singularity in finite time) in such models, called restricted Euler models, is that the eigenframe of sij in these models is fixed in space (same is true for Burgers vortex), (same whereas in a real turbulent flow it is oriented randomly. This means that nonlocality due to presure is essential for (self-) sustaining turbulence: no pressure Hessian no turbulence. Accelerations. A related aspect is that the Lagrangian Accelerations. acceleration a ≡ Du/Dt - a kind of small scale quantity – which is dominated by the pressure gradient, ∇p. Hence the scaling properties of the acceleration variance do not obey Kolmogorov-like scaling (e.g. Hill, 2002) , most probably due to dominating contribution of nonlocality. As Du/Dt = ∂u/∂t + u(∇⋅u) = ∂u/∂t + ωxu +∇(u2/2), i.e. it is a ‘mixed’ quantity due to i.e. presence of both velocity and velocity derivatives, this could be seen also as a reason for the impact of nonlocality on the behaviour of acceleration. However, the first one (i.e. ∇p) is a dynamical reason, whereas the second one is of purely kinematic nature. ACCELERATION VARIANCE GULITSKII ET AL., 2007 Acceleration variance does not scale as expected if it would be a small scale quantity The figure is based on Fig. 8c from GYLFASON ET AL. 2004 Loitsyanskii invariant. This is another aspect related to the Loitsyanskii issue of the long range forces due to pressure (see ISHIDA 2006 ET AL and (see references therein, also YAKHOT 2004). YAKHOT assuming that that the integral above is converging (if second and third order correlations (if assuming decay faster than r-4, but not exponentially). Contrary to many objections based on the importance of long range forces due to paressure at very large distances ISHIDA 2006 ET AL gave convinsing argumnents and numerical evidence that Loitsyanskii/Komogorov/Lanadau were right, i.e. the long-range interactions between remote eddies, as measured by the triple correlations, are very weak. This, however, does not mean that long range forces due to pressure at NOT very large distances are unimportant. * In 1970 Loytsianskii told me about some astropysical observations which were in conformity with the invariance of I. Taking curl of the NSE and getting rid of the pressure does not remove the Taking nonlocality. Indeed, the equations for vorticity and enstrophy are nonlocal in vorticity, ωi, since they contain the rate of strain tensor, sij, due to nonlocal relation between vorticity, ω, and the rate of strain tensor, sij (‘kinematic' nonlocality)*,** Both aspects of nonlocality are reflected in the equations for the rate of strain tensor and total strain/dissipation, s² ≡ sijsij, and the equations for the third order quantities ωiωjsij and sijsjkski. An important aspect is that the latter equations contain invariant (nonlocal) quantities ωiωk ∂²p/∂xi∂xj and sikskj∂²p/∂xi∂xj , reflecting the nonlocal dynamical effects due to pressure and can be interpreted as interaction between vorticity and pressure and between vorticity and and strain. These quantities are among possible candidates preventing formation of finite time singularity in NSE. ________________________ * Nonlocality of the same kind is encountered in problems dealing with the behaviour of vortex filaments in an ideal fluid. Its importance is manifested in the breakdown of the so called localized induction approximation as compared with the full Bio-Savart induction law). ** The two aspects of nonlocality are related, but are not the same. For example, in compressible flows there is no such relatively simple relation between pressure and velocity gradient tensor as above, but the vorticity-strain relation remains the same. MANIFESTATIONS OF NONLOCALITY MANIFESTATIONS There exist massive evidence not only on direct interaction/coupling between large and small scales but also that this interaction is bidirectional such as the example of turbulent flows in a channel. We mention also the well known effective use of fine honeycombs and screens in reducing large scale turbulence in various experimental facilities. One can substantially increase the dissipation and the rate of mixing in a turbulent flow by directly exciting the small scales. The experimentally observed phenomenon of strong drag reduction in and change of the structure of turbulent flows of dilute polymer solutions and other drag reducing additives is another example of such a ‘reacting back' effect of small scales on the large scales. SS Anisotropy I. Lack of rotational symmetry. Lack SS One of the manifestations of direct interaction between large and small scales is the One anisotropy in the small scales. Though local isotropy is believed to be one of the universal properties of high Re turbulent flows it appears that it is not so universal: in many situations the small scales do not forget the anisotropy of the large ones. There exist considerable evidence for this which has a long history starting somewhere in the 50-ies. Recently similar observations were made for the velocity increments and velocity derivatives in the direction of the mean shear both numerical and laboratory, see references in TSINOBER 2001, BIFERALE, L. AND TSINOBER PROCACCIA 2005, OULETTE ET AL 2006 . It was found that the stastistical properties of velocity increments and velocity derivatives in the direction of the mean shear do not conform with and do not confirm the hypothesis of local isotropy. SS Anisotropy I. Lack of rotational symmetry. Lack SS So it is not surprizing that in the ‘simlpe' example of turbulent channel flow the So flow is neither homogeneous nor isotropic even in the proximity of the midplane, y ≈ 0 , where dU/dy ≈ 0. Indeed, though 〈u1u2〉 ≈ 0 in this region too, dU/dy d 〈u1u2〉 /dy is essentially nonzero and is finite independently of Reynolds number as far as the data allow to make such a claim. This is also a clear indication of nonlocality, since in the bulk of the flow, i.e. far from the boundaries, dU/dy ≈ 0. Therefore the assumption that in the proximity in dU/dy of the centerline of the channel flow, the local isotropy assumption seem reasonable is incorrect. SS Anisotropy II. Lack of reflexional symmetry. Helicity. SS The hypothesis of local isotropy (K41) includes restoring of all the symmetries in small scales, i.e. the expectation is for restoring also of of reflection-invariance at small length scales and that reflection symmetry which is broken at large scales will tend to be restored asymptotically at small scales (CHEN ET AL 2003, Phys. Fluids, 15, 361) . These authors claimed that this is Phys. 15 These the case: that there is a tendency toward equalization of energy in the + and that components due to the nonlinear transfer between them. Hence, reflection symmetry which is broken at large scales will tend to be restored asymptotically at small scales. However, GALANTI AND TSINOBER 2004 showed However, showed that the opposite is true: to maintain finite helicity dissipation to balance the finite to helicity input (in a statistically stationary turbulence) the tendency to restore reflection symmetry at small scales can not be perfect, since helicity dissipation is associated with broken reflection symmetry at small scales, because helicity dissipation is just proportional to the superhelicity H_{s}, showing the lack of reflection symmetry of the small scales. Moreover, this lack of reflectional symmetry should increase as the Reynolds number increases. SS Anisotropy II. Lack of reflexional symmetry. Helicity. Lack SS Helicity – H = ∫u·ωdx; superhelicity - Hs = ∫ω·curlωdx, hyperhelicity Hh = ∫curlω·curlcurlωdx. The Equation for the helicity DH/Dt = -2νHs + FH FH = 2∫f·ωdx is the term associated with forcing f in the right hand side of NSE. The important point is that helicity dissipation -2νHs is The vanishing if reflexional symmetry holds. The Equation for the superhelicity DHs/Dt = PHs - 2νHh+FHs The term FHs = ∫curlf·curlωdx is associated with forcing f in the RHS of NSE, and the term PHs =2∫curlω·curl(u×ω)dx is the production term of the superhelicity Hs. There is a similar phenomenon of self-production of Hs and approximate balance of PHs and 2νHh and irrelevance of forcing at this level. SS Anisotropy II. Lack of reflexional symmetry. Helicity. SS GALANTI AND TSINOBER 2004 Note the tendency of DHl²u-3 to a to constant with increasing Reλ Re [10] Chen et al 2003, Phys. Phys. Fluids, 15, 361 . [16] Kurien et al. (2004) J.. Fluid [16] J Mech., 515 . The ABC case. Dependence of normalized dissipation of helicity DHl²u-3 and energy DElu-3 on the Taylor microscale Reynolds number Reλ. ◦ - corresponds to the data from Ref. [10], □ - correspond Ref. to the data from Ref. [16] and references therein. Along with other manifestations of direct interaction between large and small scales Along the deviations from local isotropy seem to occur due to various external constraints like boundaries, initial conditions, forcing (e.g. as in DNS), mean shear/strain, centrifugal forces (rotation), buoyancy, magnetic field, etc., which usually act as an organizing factor, favoring the formation of coherent structures of different kinds (quasi-two-dimensional, helical, hairpins, etc.). These are as a rule large scale features which depend on the particularities of a given flow and thus are not universal. These structures, especially their edges seem to be responsible for the contamination of the small scales. This ‘contamination' is unavoidable even in homogeneous and isotropic turbulence, since there are many ways to produce such a flow, i.e. many ways to produce the large scales. It is the difference in the mechanisms of large scales production which `contaminates' the small scales. Hence, nonuniversality. This brings us to the next issue. Anisotropy III. Anisotropy Statistical dependence of small and large scales. Statistical J. Fluid Mech., 5, 497—543 (1959). PRASKOVSKY ET AL 1993 J. Fluid Mech., 248, 493--511. 248 Anisotropy III. Anisotropy Statistical dependence of small and large scales. Statistical Enstrophy ω2, total strain s2 and squared acceleration a2 conditioned on magnitude of the Enstrophy velocity fluctuation vector, Field experiment, Sils-Maria, Switzerland, 2004, Reλ= 6800 (hopefully JFM, 2007) Closures and constitutive relations I. Closures The nonlocality due to the coupling between large and small scales is also manifested (and is a The concern) in problems related to various decompositions of turbulent flows and in the so called closure problem. For example, in the Reynolds decomposition of the flow field into the mean and the fluctuations and in similar decompositions associated with large eddy simulations (LES) the relation between the fluctuations and the mean flow (or resolved and unresolved scales in LES, etc.) is a nonlinear functional. That is the field of fluctuations at each time/space point depends on the mean (resolved) field in the whole time/space domain. Vice versa the mean (resolved) flow at each time/space point depends on the field of fluctuations (unresolved scales) in the whole time/space domain. This means that in turbulent flows point-wise flow independent `constitutive' relations analogous to real material constitutive relations for fluids (such as stress/strain relations) can not exist, though the `eddy viscosity' and `eddy diffusivity' are used frequently as a crude approximation for taking into account the reaction back of fluctuations (unresolved scales) on the mean flows (resolved scales). The fact that the `eddy viscosity' and `eddy diffusivity' are flow (and space/time) dependent is just another expression of the strong coupling between the large and the small scales. Closures and constitutive relations II. Closures The simplest version of this approach with a scalar eddy viscosity leads always to a positive The subgrid dissipation (positive energy flux from the resolved to the unresolved scales), whereas a priory tests of data from real flows (experiments and DNS) show that there exist considerable regions in the flow with negative subgrid-scale dissipation (called backscatter). The exchange of `information' between the resolved and unresolved scales is pretty reach and is not limited by energy. For example, BOS ET AL., 2002 (see also references therein) report that the subgridBOS scales have a variety of significant effects on the evolution of field of filtered velocity gradients. So it is too optimistic to claim, for examle, that LES of wall bounded flows ... resolve all the resolve important eddies... has received increased attention, in recent years, as a tool to study the physics of turbulence in flows at higher Reynolds number, or in more complex geometries, than DNS (PIOMELLI AND BALARAS 2002). The qualification of large scale (resolved) eddies as the most important ones is too subjective: all eddies are important in view of direct and bidirectional coupling of essentially all eddies. It is doubtful that LES or any other similar approach can be used as a tool to study the physics of turbulence, since a vitally important part of physics of turbulence resides in the unresolved scales. Memory effects. The far field statistical properties of free shear turbulent Memory flows (mixing layers, wakes, jets) and also boundary layers (Journal of Turbulence, 5, 015) are (Journal known to possess strong memory (`nonlocality in time'): they are sensitive to the conditions at their ‘start' with some properties not universal in Reynolds number, BEVILAQUA AND LYKOUDIS 1978, DIMOTAKIS 2001, GEORGE & DAVIDSON BEVILAQUA 2004. These flows develop in space beginning with small scales into the large ones, in apparent contradiction to the Richadrson-Kolmogorov cascade ideas. It is noteworthy that passive tracers in such flows possess even stronger memory, CIMBALA ET AL. 1988, due to importance of Lagrangian aspects of their CIMBALA evolution. The problem of predictability of turbulent flows involves nonlocality in time either: a small scale perturbation (both in time and space) perturb substantially the whole flow including the largest scales within time of the order of integfral time scale. In this sense instability can be seen as nonlocality in time. SAME FLOW - NOT THE SAME PATTERN CIMBALA ET AL. 1988 Particulate flows. The basic interaction of the carrier fluid flow with Particulate particulates occurs at the scale of the particle size, i.e. at small scales. However, a number of essential phenomena emerge at much larger scales in a variety of particualate flows: sedimenting suspensions, fluidized beds, formation of bedforms and their interaction with the carrier fluid, preferential concentration of particles/bubles (clustering) in and modification of turbulent flows. These phenomena are treated in terms of large scale instabilities, intrinsic convection in sedimenting suspensions, collective phenomena, long-range multibody hydrodynamic interactions/correlations, clusters. All these are essentially fluid mediated phenomena/interactions as contrasted with direct particle/particle interactions. Therefore, nonlocality is expected to be significant in these phenomena (see TSINOBER 2003 for more and references). Particulate flows. For example, an important process in the For Particulate interaction of the carrier fluid flow with particles (or any other additives) is the production (or more generally modification) of velocity derivatives, i.e. vorticity and strain. The modified field of velocity derivatives reacts back in changing the large scales of the flow (both velocity and pressure). It is tempting to see this process as the one underlying the formation of the mentioned large scale features, though the details in each case are different and in most cases are poorly understood. These processes are modified by specific features such as inertial bias, i.e. inertial response of particles to fluid accelerations and preferential concentration of particles(bubbles) in strain (vorticity) dominated regions. The latter may lead to enhanced bias of strain dominated regions (heavy particles), i.e. regions with large dissipation, or regions with strong enstrophy (bubbles). Clusters as manifestion of nonlocality. Both kinematic Clusters . (Lagrangian) aspects and dynamic and are important. Clustering of inertial particles in kinematic simulations, Lu Chen, Ph. D. Thesis 2007 Inertial Particles and stream-function Inertial particles and stagnation points The effect is considerable at The much larger scales than of the order of Kolmogorov scale. For example, see from figure 3b in KOSTINSKI AND SHAW KOSTINSKI 2001 in which the maximum occurrs at scales ~ 1cm, but the effect is considerable at 1 m and is seen even up to 10 m (see (see also figure 5 in ROUSON AND EATON ROUSON 2001, in which significant effect is seen at scales of half-width channel). CLUSTERS – CLUSTERS SMALL AND LARGE KOSTINSKI AND SHAW 2001 figure 3b CLUSTERS – CLUSTERS SMALL AND LARGE Dependence of particle spatial distribution for various Stokes numbers. Each panel represents a “slice” through the computational domain from a direct numerical simulation of homogeneous, isotropic turbulence containing particles. Figure courtesy of L. R. Collins. CLUSTERS – SMALL AND LARGE CLUSTERS FALLON AND ROGERS 2002 Turbulence-induced preferential concentration of solid particles in microgravity conditions ALISEDA ET AL 2002 Settling velocity of heavy partricles in a wind tunnel experiment Dilute polymer solutions. Turbulent flows can be strongly Dilute modified by additives in even extremely small concentrations. The most spectacular changes occur with only few parts per million of flexible polymers added to the solvent. These changes are exhibited in a number of flow parameters both large scales and small scales, though the direct action of the dissolved polymers is obviously in the small scales. Hence again nonlocality. The large scale manifestations are represented in the first place by strong reduction of drag (up to 80%) in turbulent shear flows. Along with this other global/ large scale effects on turbulence structure are observed both experimentally and in simulations. Other related issues I. The nonlocality in the sense of concern here is Other especially strongly manifested in the atmospheric convective boundary layers in which the common downgradient approximation is not satisfactory due to countergradient heat fluxes, ZILITINKEVICH ET AL. 1998 and references therein. We mention also a ZILITINKEVICH similar phenomenon in stably stratified turbulent flows - the so called PCG, persistent countregradient fluxes. The essence of PCG is the countergradient transport of momentum and active scalar. It is observed at large scales when stratification is strong, but in small scales it is present with weak stratification as well. on There is a class of flows with the so-called phenomenon of ‘negative eddy viscosity', There see references in TSINOBER 2001. It occurs in the presence of energy supply other TSINOBER than the mean velocity gradient. In such flows the turbulent transport of momentum occurs against the mean velocity gradient, i.e. from regions with low momentum to regions with high momentum (i.e. the Reynolds stresses as one of the agents of coupling the fluctuations with (i.e. the mean flow act in such flows in the `opposite' direction as compared to the usual turbulent shear flows). Concomitantly kinetic energy moves in the ‘opposite' direction too - from fluctuations to the mean flow.. Other related issues II . The flows with negative eddy viscosity are akin Other to nonturbulent but nonstationary flows in a fluid dominated by its fluctuating components and known (since RAYLEIGH 1883) under the name (acoustic) RAYLEIGH steady streaming, RILEY 2001, in the sense that in these flows a mean (time RILEY averaged) flow is induced and driven by the fluctuations. Recently turbulent flow of this were observed too, SCANDURA 2007. SCANDURA There are examples of ‘usual' turbulent flows with turbulence induced mean flows. The best known ones are flows in pipes with noncircular crossection. In such flows a mean secondary flow is induced which is absent in purely laminar flow. For example, see PATTERSON REIF AND ANDERSSON 2002 for references on such PATTERSON for flows in a square duct. Other related issues III . Predicatbility, data assimilation and Predicatbility Other ‘determining modes’. ARNOLD, 1991 SO WHAT ARE POSSIBLE SO CONSEQUENCES AND QUESTIONS ? CONSEQUENCES # NATURE OF DISSIPATION - IS IT (UN)IMPORTANT GENERALLY (is it just stupid passive sink of energy?) # AND FOR THE SO (is CALLED INERTIAL RANGE (IR)? # ARE (ALL) ITS PROPERTIES REALLY INDEPENDENT OF VISCOSITY/NATURE OF DISSIPATION? # PROBLEMS WITH VARIOUS ASPECTS OF PHENOMENOLOGY AND DECOMPOSTIONS SUCH AS: IS "CASCADE" IN GENUINE TURBULENCE WELL DEFINED? * IS THERE CASCADE IN PHYSICAL SPACE? *HOW MEANINGFUL IS CASCADE OF PASSIVE OBJECTS? These and similar in the next lecture on phenomenology and related at February 21 at ...
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