10733697 - FUNDAMENTAL AND CONCEPTUAL ASPECTS OF TURBULENT...

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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 XV-XVI HELICITY HELICITY SCHRAUBENSINN− SCHRAUBENSINN WHAT IS IT AND WHAT IS IT GOOD FOR? A bit of history. Definition and elementary properties of helicity for a vector field. Geometrical aspects, including topology of field (e.g. vortex, magnetic) lines. Invariance in non-dissipative media, for both inviscid flows and for ideally conducting media . Effects of helicity. Kinetic helicity and turbulent dynamo. Kinetic helicity and fluid turbulence. Applications : atmospheric physics, plasma physics, astro-physics. ‘ BETCHOV 1961 At least intuitively it is quite clear that a vortex having a component of velocity along its axis is characterized by nonzero helicity, i.e. is a helical structure. There is a great number of structures falling under this category, e.g. such diverse structures as Taylor-Gortler vortices, leading edge and trailing vortices shed from wings and slender bodies, streamwise vortices in boundary layers and free shear flows, Langmuir circulations in the ocean and analogous structures in the atmosphere, tornados and rotating storms. Similar structures are observed in space and laboratory plasmas. Structures of this type are expected to lack reflexional symmetry. The key quantity characterizing the reflexional symmetry (or its lack) of a fluid flow is the so called helicity. A BIT OF HISTORY The term HELICITY in pure fluid dynamics was introduced by BETCHOV 1961 in the context of turbulent flows A GALLERY OF HELICAL GALLERY STRUCTURES VORTEX BREAKDOWN Vortex breakdown visualized. (a) In a water tunnel over a slender delta wing as a result of the emission of colored dye near the apex. (b) Around NASA’s F-18 High Angle of Attack Research Vehicle (HARV) using smoke to seed the vortex emanating from the nose of the aircraft at 201 incidence. Courtesy T. Leweke (Aachen 1989) ROTATING STIORM It has to be stressed that (quite popular) Clebsch variables are ‘defective’ in the sense that flows which (presumably) can be represented in these variables have oversimplified geometry as their helicity H = ∫ω·u dV = ∫ ω·∇ϕ dV = ∫ ∇·(ϕω)dV, i.e. it vanishes in many cases (ω = ∇λx∇μ, i.e. ω ⊥ λ∇μ) Prototype configuration of linked field tubes with nonzero helicity, MOFFATT AND TSINOBER, 1992.. Here the field is assumed to be identical zero except in two closed tubes with axes C1 and C2 and with vanishingly small cross-section. The field lines are untwisted within each tube, i.e. each line is a closed curve passing once round the tube, and unlinked with its neighbors in the same tube. In such a case HB=±2L1,2,where L1,2 is the linking (or winding) number of the two tubes (in the figure L1,2=1), F1,2 are the fluxes (circulations κ1,2) associated with each tube, and the + or - is chosen depending on whether the linkage is right - of left-handed. in non-dissipative media (no reconnections) In cases when the fluid flow is influenced by the presence of the magnetic field, that is by the Lorenz force {the term curl(j ×B) added to the LHS of NSE} the helicity of vorticity H is not conserved whereas the magnetic helicity remains conserved. Another conserved quantity in this latter case is the cross-helicity HB,ω= ∫ u·BdV. MODIFIED HELICITY Helicity is a global quantity which in many cases is not well defined. It appears that one can choose the gauge ϕ in such a way that the helicity density is a Lagrangian (non-dissipative) invariant, i.e. it is conserved (pointwise) along the paths of fluid particles and therefore for any fluid volume. Such a choice is possible both for magnetic field (ELSASSER, 1956; CHILDRESS & GILBERT, 1995) and for nonconducting fluid flows (KUZMIN, 1983; OSELEDETS, 1989). It is possible to do so also for a viscous flow (OSELEDETS, 1989) chosing ϕ obeying the equation Dφ/Dt = p − u2/2 + ν∇2φ Then the modified helicity density hm = ω·v, with v = u+∇ϕ satisfies the equation Dhm/Dt = ν{∇²hm -2(∂ωi/∂xk)(∂vi/∂xk)} i.e. is a Lagrangian invariant if ν = 0. EFFECTS OF HELICITY Note that magnetic fields are generated Note pretty successfully by velocity fields without any helicity, by flows possessing reflexional symmetry and even random gaussian isotropic ones The reason is that this argument is too simple as the Lamb vector ω×u = ∇α + ∇×β has a large potential part ∇α (such that 〈(∇α)²〉 ≥ 2〈∇×β)²〉), which can be included into pressure just like the Bernoulli term ∇(u2/2). It is the solenoidal part ∇×β = curl (ω×u) which matters, e.g. for vorticity dynamics. Moreover, if one cares about enstrophy (and strain) production, i.e. ω·∇×∇×β = uk∂ω2/∂xk – ωiωksik, only the second tem is of importance. 4/5 AND 2/15 KURIEN ET AL 2004 # Note that ∂jui∂jωi = 〈ω·curlω〉 is just the mean superhelicity. The helicity dissipation h = 2ν 〈ω·curlω〉 is assumed to be finite in the mentioned above and similar publications (see references in GALANTI AND TSINOBER 2004). # The 2/15 law has a nonlocal character as it involves correlation of both velocity (i.e. large scale quantities) and its differences (i.e. small scale quantities). It is noteworthy that the Kolmogorov 4/5 law can be interpreted in the same way since 〈(ΔuL)3〉 = - 3〈ΔuL [(uL(x+r) + uL(x)]2〉 Helicity dissipation seems to tend to a finite limit Helicity 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 (helical forcing). Dependence of normalized dissipation of helicity DH l² u-3 and The energy DE l u-3 on the Taylor microscale Reynolds number Reλ. ◦ - corresponds to the data from Ref. [10], □ - correspond to the data from Ref. [16] and references therein. Ref. Cω,curlω= 0.1 0.1 Note the pretty small value of the correlation coefficient Cω,curlω= 0.1between ω and curlω 0.1 curl A natural question is how it is natural possible that both energy and helicity dissipation (presumably) remain finite with increasing Reynolds number. A possible explanation is seen from the form of helicity dissipation as used above h = 2ν〈ω·curlω〉: the imperfect alignment between ω and curlω (as shown in figure at the left) can make it possible that the ‘singularitites’ arising from the finiteness of both energy and helicity dissipation can be matched also aided by the fact that ω·curlω is not a positively defined quantity. Helicity dissipation seems to tend to a finite limit even with Helicity nonhelical forcing nonhelical GALANTI AND TSINOBER 2004 T+(T-) is the total length of the time intervals where both H and Hs are positive (negative), etc. The total helicity dissipation in this case is DH = D++ D- and the total helicity production DpH = Dp++ Dp- . Their difference, DH - DpH, which is the net dissipation of helicity, is still positive. ADDITIONAL ASPECTS OF HELICITY ‘DISSIPATION’ from figures shown below from SELF-PRODUCTION OF HELICITY DISSIPATION/SUPERHELICITY The Equation for the superhelicity Hs = ∫ω·curlωdx 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 selfproduction of Hs and approximate balance of PHs and 2νHh and irrelevance of forcing at this level. Hyperhelicity Hh = ∫curlω·curlcurlωdx Three main results closely related to finite dissipation of helicity : i) - the self-production property of superhelicity i) production superhelicity H s=∫ω·curlωdx (which is proportional to helicity dissipation) in the sense that forcing does not play any role in forcing production of superhelicity, i.e. the dissipation of helicity; ii) finiteness of helicity dissipation is the manifestation of i.e. ii) the lack of reflection symmetry of the small scales; iii) - this lack of reflexional symmetry should increase with the iii Reynolds number, following from our third main result that the normalized helicity dissipation tends to remain finite as the Reynolds number increases, just like the energy dissipation, in the spirit of Kolmogorov 41. HELICAL FORCING D I S S I P A T I O N P R O D U C T I O N PRODUCTION of PRODUCTION Hs FORCING FORCING Three main results closely related to finite dissipation of helicity : i) - the self-production property of superhelicity production superhelicity Three helicity i) H s=∫ω·curlωdx (which is proportional to helicity dissipation) in the sense that forcing does not play any role in forcing production of superhelicity, i.e. the dissipation of helicity; ii) finiteness of helicity dissipation is the manifestation of the ii) lack of reflection symmetry of the small scales; iii) - this lack of reflexional symmetry should increase with the iii) Reynolds number, following from our third main result that the normalized helicity dissipation tends to remain finite as the Reynolds number increases, just like the energy dissipation, in the spirit of Kolmogorov 41. NON-HELICAL FORCING NON D I S S I P A T I O N P R O D U C T I O N PRODUCTION of Hs FORCING This This JOINT PDFS OF HELICITY H AND SUPERHELICITY HS a) NH-case (non-helical foring), b)ABC-case (helical forcing). In the NH case the correlation coecient CH;Hs between H and Hs is 0:70. In case ABC it is equal to0:52 without removal of the means of H, Hs and 0.98 with the means of H, Hs removed. Itt should be stressed, however, that an important difference with I true dissipation is that -2Hs can have any sign, so that when H and Hs have opposite signs the viscous term -2νHs acts as production of helicity. It is also noteworthy that finite deviation from reflection symmetry (i.e. finite Hs and also Hh) in small scales should remain at whatever small viscosity (and should be large for small viscosity), e.g. to maintain the helicity ‘cascade’ with definite sign helicity input, though this appears to be true in some sence in case of purely nonhelical forcing as well. Three main results closely related to finite dissipation of helicity defined Three as the integral of the scalar product of velocity and vorticity, H=∫u·ωdx: dx i) - the self-production property of superhelicity Hs=∫ω·curlωdx i) (which is proportional to helicity dissipation) in the sense that forcing does not play any role in production of superhelicity, i.e. the dissipation of helicity; ii) finiteness of helicity dissipation is the manifestation of the ii) lack of reflection symmetry of the small scales; iii) - this lack of iii reflexional symmetry should increase with the Reynolds number, following from our third main result that the normalized helicity dissipation tends to remain finite as the Reynolds number increases, just like the energy dissipation, in the spirit of Kolmogorov 41. The results were obtained from DNS of Navier-Stokes equations in a simplest flow geometry with periodical boundary conditions. COMPLEMENTARY RESULTS LIMITATIONS OF CORRELATIONS Itt is instructive to compare the relative helicities defined as I with the correlation coefficients HELICAL FORCING t NONHELICAL FORCING Two observations can be made here. First, the relative higher order helicities are of the same order and not that small in both cases. Second, the correlation coefficient Cu,curlω reflecting the coupling between u and curlω is considerably larger than the relative helicity Hr reflecting the coupling between u and ω . This is in spite of the fact that the scales of u and ω are ‘closer’ than those of u and curlω . Similarly the correlation coefficient Cω,curlcurlω, reflecting the coupling between ω and curlcurlω is considerably larger than the relative helicity Hsr reflecting the coupling between ω and curlω. Moreover, in flows with reflectional symmetry both Hr and Hsr (and also Hhr)vanish, whereas Cu,curlω and Cω,curlcurlω remain practically unchanged. This illustrates the limited value of quantities like correlations and correlation coefficients. This is an important This example of sensitivity of passive scalar to te nature of velocity field. Usually it is not so sensitive to the nature of velocity field so that many properties are the same, e.g. for genuine turbulence and gaussian velocity field. We remind another example of such sensitivity: the difference between forwards and backwards dispersion The 4/3 law is an for genuine turbulence and artificial velocity fields. immediate example. CONCLUDING CONCLUDING There seems to be little doubt that helicity, i.e. H = ∫u·ωdx, There being a quadratic (inviscid) invariant ... has a status comparable with that of kinetic energy¹. Nevertheless, the issue of the role of helicity in turbulent flows is one of the most controversial ones and is very far from a consensus. On one hand, it is thought that helicity should be of dynamical significance in turbulent flows, and on the other hand, helicity was shown to be dynamically irrelevant in some cases. So it is not surprising that the role of helicity in three-dimensional turbulence is considered to be still somewhat ‘mysterious’. However, it seems conceptually misleading to look at helicity as a causal However, factor as it is an integral property (as any other) of turbulent flows. An immediate example is comprised by flows in rotating fluids. Such flows lack reflexional symmetry and – as a consequence – possess nonvanishing helicity. On the other hand flows of this kind are characterized by reduced nonlinearity and consequently drag and dissipation A recent example is the numerical study of turbulent flow in a rotating pipe by ORLANDI 1997. He observed a clear positive correlation between increase in helicity and decrease in dissipation. However, the cause of both is rotation, i.e. an observed relation between two quantities/properties is not (necessarily) synonymous to direct causal relation between them: causality is different from relation. ...
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