{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

7875697 - FUNDAMENTAL AND CONCEPTUAL ASPECTS OF TURBULENT...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 fl...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern