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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 V-VI VORTICITY AND STRAIN IN TURBULENCE VORTICITY Why velocity derivatives? Vorticity versus strain. Self-amplification (SA) of both: who is responsible for enhanced disspation? Stretching versus compressing. Geometrical statistics. Concentrated vorticity or strain. Unrestanding the physics of SA of vorticity and strain is the heart of approaching the turbulence problem. Self-amplification of the strain-rate field is something I overlooked up to now ... TENNEKES 2003 TENNEKES One of the important common features of One processes resulting in turbulence is that all of them tend to enhance the rotational and dissipative properties of the flow in the process of transition to turbulence. The first property is associated with the production of vorticity (turbulence is highly rotational), whereas the second property is due to the production of strain (turbulence is strongly dissipative). The latter is a unique process of genuine turbulence and does not have any analogue in ‘passive’ turbulence. VORTICITY AND STRAIN LIVE TOGETHER CHEN, 2000 BUT HAVE A VERY DIFFERENT LIFE DNS of forced NSE in a periodic box Reλ=220, ω = 3ωmean, s = 3smean One of the important One common features of processes resulting in turbulence is that all of them tend to enhance the rotational and dissipative properties of the flow in the process of transition to turbulence. The first property is associated with the production of vorticity, whereas the second property is due to the production of strain. INTEREST TO Aik= ∂ui/∂xk VORTICITY AND RELATED Skew-symmetric part of Aik= ∂ui/∂xk Skew TAYLOR 1938 ω - obsession DISSIPATION/STRAIN DISSIPATION/STRAIN Symmetric part of Aik = ∂ui/∂xk, i.e. sik Symmetric KOLMOGOROV 1941 ε - religion Vorticity obsession since HELMHOLTZ and KELVIN HELMHOLTZ and 1867 TAYLOR 1937, CORRSIN 1953 The turbulence syndrome includes the following symptoms:…it is essentially nonlinear and rotational STEWART 1963 STEWART Turbulence is rotational and three dimensional. Turbulence is characterized by high levels of fluctuating vorticity. For this reason, vorticity dynamics plays an essential role in the description of turbulent flows. TENNEKES AND LUMLEY 1972 What is turbulence but a random chaotic field of vorticity, What whose strong nonlinear interactions makes the problem so nonlinear impossibly difficult? ... the concept of the coherent structure in turbulent shear flows has led to the picture of such flows as a superposition of organized, ‘deterministic ’ vortices whose evolution and interaction is the turbulence. SAFFMAN 1981 WHY SMALL SCALES? AND WHAT THEY ARE ? A great deal of interest is concentrated on the great Reynolds stress tensor 〈uiuj〉 (or similar quantities in LES) and the “closure problem”, i.e. how it is related to the mean flow (with the – not obvious - assumption that a relatively simple relation – if at all - does exist). Such an approach is Such based on the view that small scales (whatever this means) are “slaved” to the large scales and are mostly a kind of passive sink of energy. This is the so called “classical approach” with the view that in order to understand turbulence one needs only to ‘resolve’ the large scales and ‘model’ the small scales (can this be done without sufficient understanding and/or full resolution down to all physically relevant scales?). However, today it is high time to ask how However, much justified is such oversimplified treatment of small scales via methods like eddy viscosity, and similar. The small scales contain a great deal of essential physics of turbulent flows, much of which is not known or poorly understood, and which are intimately and bidirectionally related to the large scales (nonlocality). Apart of basic there are numerous problems in which one has to deal explicitly with the nature, structure and dynamics/evolution of small scales. For instance, special information on small scale structure(s) is For needed in problems concerning, e.g. combustion, disperse multiphase flow, mixing, cavitation, turbulent flows with chemical reactions, some environmental problems, generation and propagation of sound and light in turbulent environments, and some special problems in blood flow related to such phenomena as hemolysis and thrombosis. In such problems, not only special statistical properties are of importance like those describing the behaviour of smallest scales of turbulence, but also actual ‘nonstatistical' features like maximal concentrations in such systems as an explosive gas which should be held below the ignition threshold, some species in chemical reactions, concentrations of a gas with strong dependence of its molecular weight on concentration (such as hydrogen fluoride used in various industries e.g. in production of unleaded petrol) and toxic gases. THE ABOVE IS A CLEAR INDICATION THE WHY VELOCITY DERIVATIVES ARE SO IMPORTANT, BUT THERE IS MUCH MORE Velocity derivatives, Aij = ∂ui/∂xj, play an outstanding Velocity role in the dynamics of turbulence for a number of reasons. Their importance has become especially clear since TAYLOR, 1938 and KOLMOGOROV, 1941. TAYLOR, KOLMOGOROV Taylor emphasized the role of vorticity, i.e. the antisymmetric part of the velocity gradient tensor Aij = ∂ui/∂xj, whereas Kolmogorov stressed the importance of dissipation, and thereby of strain, i.e. the symmetric part of the velocity gradient tensor. It is important (see It (see below) that the whole (incompressible) flow field is fully determined by the fields of vorticity or strain with appropriate boundary conditions. Apart from vorticity and strain/dissipation, there are many other reasons for special interest in the characteristics of the field of velocity derivatives, Aij=∂ui/∂xj, in turbulent flows:. In the Lagrangian description of fluid flow in a frame In following a fluid particle, each point is a critical one, i.e. the direction of velocity is not determined. So everything happening in its proximity is characterized by the velocity gradient Aij=∂ui/∂xj. For instance, local geometry/topology is naturally described in terms of critical points terminology. The field of velocity derivatives is much more sensitive to the The non-Gaussian nature of turbulence or more generally to its structure, and hence reflects more of its physics. The possibility of singularities being generated by the Euler The and the Navier-Stokes equations (NSE) and possible breakdown of NSE are intimately related to the field of velocity derivatives. There is a generic ambiguity in defining the meaning of the term small There scales (or more generally scales) and consequently the meaning of the term cascade in turbulence research. The specific meaning of this term and associated interscale energy exchange/`cascade' (e.g. spectral energy transfer) is essentially decomposition/representation dependent. Perhaps, the only common element in all decompositions/representations (D/R) is that the small scales are associated with the field of velocity derivatives. Therefore, it is natural to look at this field as the one objectively (i.e. D/R independent) representing the small scales (one can – following CORRSINCORRSIN (one use higher order derivatives, e.g. curl ω, ∇²sij ). Indeed, the dissipation is Indeed, associated precisely with the strain field, sij, both in Newtonian and nonNewtonian fluids. There is a number of more specific reasons why studying the field of velocity derivatives is so important in the dynamics of turbulence. 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 ∇²ui = 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. WHY STRAIN TOO? WHY IS IT AN EQUAL PARTNER OF VORTICITY? SStrain controls the flow in the same way as does vorticity also in the sense of possible train vorticity also breakdown of smooth solutions for 3D flows, Ponce, G. (1985) Remarks on a paper J.T. Beale, T. Ponce, breakdown Kato and A. Majda, Commun. Math. Phys., 98, 345-3353. Majda Commun 98 53. STRAIN – AN EQUAL PARTNER STRAIN In some cases even more In ‘Dolphins in phosphorescent sea’. The inspiration for this woodcut, created by M. C. Escher in 1923, created was the flow-induced induced bioluminescence that occurs on dolphins when they swim through waters that contain high levels of waters bioluminescent plankton. Some reasons More below 1. Most important stresses in fluids (both Newtonian and non-Newtonian) flows are defined by strain 2. Energy dissipation is directly associated with strain (both Newtonian (both and non-Newtonian) and not with vorticity. 3.. The energy cascade (whatever this means) and its final result 3 dissipation, are associated with predominant self-amplification of the rate of strain and vortex compression rather than with vortex stretching. That is another nonzero odd moment - sijsjksjk responsible for the production of strain, is not less important than the enstrophy production ωiωjsij 4.. Vortex stretching is essentially a process of interaction of 4 vorticity and strain. “Vortices” iinteract via their strain fields. nteract 5.. Strain dominated regions appear to be the most active/nonlinear 5 in a number of aspects. 6. Interaction of the flow field with additives (particles, polymers, blood cells) is mostly via strain . 7. Though formally all the flow field is determined entirely by the field of vorticity the relation between the strain and vorticity is strongly nonlocal. In many cases, they are only weakly statistically correlated or not correlated at all. BASIC EQUATIONS Dω/Dt = (ω·∇)u + νΔω + ɛijk∂Fk/∂xj (½)Dω²/Dt = ωiωjsij + νωi∆ωi + ɛijkωi ∂Fk/∂xj Dsij/Dt = - sjkski – (1/4)(ωiωj- ω2δij) – ∂²p/∂xi∂xj + ν∆sij + Fij (½)Ds²/Dt = - sijsjkski – (1/4)ωiωjsij – sij∂²p/∂xi∂xj + νsij∆sij + sijFij In what sense vortex stretching plays a central role in the energy cascade to small scales and dissipation? OR WHO IS THE ‘GUILTY’ ? I One of the most basic phenomena and distinctive features of three-dimensional turbulence is the predominant vortex One stretching. This process occurs via interaction of vorticity and strain. strain (½)Dω²/Dt = ωiωjsij + νωi∆ωi + ɛijkωi ∂Fk/∂xj νω It is a very common view that this process is responsible for the enhanced dissiption in 3-D turbulent flows (TAYLOR 1938, ONSAGER 1949 and everybody on). It seems that the stretching of vortex filaments must be regarded as the principal mechanical cause of the high rate of dissipation which is associated with turbulent motion, TAYLOR 1938 Vortex-line stretching plays a central role in the energy cascade to small scales and dissipation. RHINES, 1997 (p.102) RHINES BUT, dissipation is BUT, 2= 2νs 2νsijsij !! !! The true physical causal relation is between The dissipation and strain both in Newtonian and nonNewtonian fluids. Therefore it is a misconception to associate dissipation directly with vorticity. However, the rotational nature of turbulence (i.e. vorticity) is crucial for dissipation. As a home work show that in irrotational flow (i.e. As u=∇ϕ) the dissipation is either zero or volume integral 2∫s2dV over the flow domain is equal to the surface integral ∫∇u2dV ∫∇ WHO IS THE ‘GUILTY’ ? II There exist two nonlocally connected and weakly correlated two processes. The ‘second’ is the self-amplification of strain. The (½)Ds²/Dt = - sijsjkski – (1/4)ωiωjsij – sij∂²p/∂xi∂xj + νsij∆sij + sijFij It is this ‘second’ process which is directly responsible for the It directly responsible enhanced dissipation in turbulent flows. Moreover the ‘first’ one (i.e. the enstrophy production) is opposing production of strain. Note that the production of strain is 1) much more ‘’self‘, 2) it is a specific (!!) feature of the dynamics of genuine turbulence having no counterpart in the behavior of passive objects, 3) Enstrophy 3) production ωiωjsij has an additional role in exchanging “energy” between enstrophy and strain. NONLOCALITY OF NONLOCALITY OF VORTICITY/STRAIN RELATION AND THE ISUE OF SURROGATES ISUE Field experiment 2000, Israeli field station, Reλ=104 ωiωjsij AND sjkskisij VERSUS SURROGATE 17.5∂u1/ ∂x1 17.5 Field experiment 2004, Sils-Maria, Switzerland, Reλ= 6800 THE MARIA SILS SITE, SWITZERLAND SILS FIELD EXPERIMENT SUMMER 2004 SILS MARIA, SWITZERLAND Height 1850 m Experiment was Experiment performed in collaboration of Institute of Hydromechanics and Water Resources Management, ETH Zurich Management, The Israeli team The calibration unit at 3 m in the field THE PROBE THE 3 mm hot wires cold wires Manganin is used as a material for the sensor prongs instead of tungsten because the temperature coefficient of the electrical resistance of manganin is 400 times smaller than that of tungsten. The tip of the probe with prongs made of manganin The manganin SELFAMPLIFICATION OF VORTICITY AND STRAIN (½)Dω²/Dt = ωiωjsij + νωi∆ωi + ɛijkωi ∂Fk/∂xj (½)Ds²/Dt = - sijsjkski – (1/4)ωiωjsij – sij∂²p/∂xi∂xj + νsij∆sij + sijFij The property of self amplification of vorticity and strain is responsible for the fact the neither enstrophy ω² nor the total strain s² are inviscid invariants as is the kinetic energy u² SELFSELF AMPLIFICATION OF VORTICITY AND STRAIN SELF-RANDOMIZATION/INTRINSIC STOCHASTICITY: NO SOURCE OF RANDOMNESS IS NEEDED, THE FORCING CAN BE CONSTANT IN TIME AT THE LEVEL OF VELOCITY DERIVATIVES: VORTICITY AND STRAIN (DISSIPATION) THE EXTERNAL FORCING IS IRRELEVANT Three cases: 1. DNS in a periodic box, Reλ=102 2. DNS in a channel flow, Re=5600 3. Atmospheric SL, Reλ=104; Re=108 0.5 F O R C I N G VELOCITY DERIVATIVES 5 0 0 -5 -0.5 -5000 -10 0 5000 Assume there is no production of enstrophy in the Assume mean 〈ωiωjsij〉 = 0. Is there turbulence? Is What about a similar assumption for strain production 〈sijsjkski〉 = 0 ? . Enstrophy production is approximately balanced by viscous terms at any - whatever large - Reynolds number?* TENNEKES AND LUMLEY (1972, P.91): P.91 (½)Dω²/Dt = ωiωjsij + νωi∆ωi + ɛijkωi ∂Fk/∂xj νω Three cases: flow, Re=5600 1. DNS in a periodic box, Reλ=102 2. DNS in a channel 3. Atmospheric surface layer, Reλ=104; Re=108 More: Spatial integrals, running averages Similar result holds for strain production *(In this sense - but not only in this - turbulence is not slightly viscous at whatever large Reynolds number. In this context the *(In but umber. question: what happens with enstrophy/strain production as ν→0 is of special interest) enstrophy/strain ν→ TEMPORAL EVOLUTION OF SPATIAL INTEGRALS IN THE ENSTROPHY BALANCE EQUATION Reλ = 2x102 ∫(½)Dω²/Dt = ωiωjsij + νωi∆ωi + ɛijkωi ∂Fk/∂xj νω dEω/dt Pω --Dω Fω Note i) approximate balance between Pω and ---Dω and ii) irrelevance of the forcing term Fω Similar behavour is observed with hyperviscosity ARE VORTEX LINES (OR VORTICITY) ARE VORTICITY APPROXIMATELY FROZEN IN FLUID FLOW AT LARGE REYNOLDS NUMBERS? AT a material line which is initially coinsides material with a vortex line continues to do so. It is thus possible and convenient to regard a vortexline as having a continuing identity and as moving with the fluid (In a viscous fluid it is, of course, possible to draw the pattern of vortex lines at any instant, but there is no way in which particular vortex-line can be identified at diffrent instants). BATCHELOR, 1967, p. 274 1967 ... STRETCHING VERSUS (AND/OR?) STRETCHING (AND/OR?) COMPRESSING In turbulent flows 〈ωiωjsij〉 > 0 due to predominant stretching. However, there is no stretching without compressing (div u=0), so what is the meaning of so predominant stretching? Also 〈- sijsjkski〉 > 0. Is it 0. Is too due predominat stretching? To clarify this matter a bit of geometrical statistics is needed GEOMETRY OF VORTEX STRETCHING VORTEX STRETCHING ωiωjsij = ω2{Λk ω,λk)} cos2( BETCHOV 1956 λk – eigenvectors of the rate of strain tensor sij, Λk - eigenvectors of the rate of strain tensor sij, Λ1 > Λ2 >Λ3; Λ1+Λ2 +Λ3 = 0 (div u=0); Λ1 > 0, Λ3<0; ω2{Λ1 cos2(ω,λ1)} = I ω2{Λ2cos2(ω,λ2)} = II II ω,λ3)} = III III ω2{Λ3 cos2( 〈I〉: 〈II〉: 〈III〉 = 3: 1: -1 3: EIGEN CONTRIBUTIONS EIGEN Λ α cos 2 (ω, λ α ) Λ2 cos 2 (ω, λ α ) α 1.2m 2.0m 3.0m 4.5m 7.0m 10m 1 1.44 1.60 1.36 1.31 1.53 1.04 1.37 2 0.44 0.62 0.67 0.46 0.46 0.58 0.49 -0.87 -1.22 -1.03 -0.77 -0.99 1 0.53 0.33 0.29 0.52 0.46 0.49 0.49 2 0.09 0.05 0.15 0.14 0.13 0.16 0.15 0.38 0.63 0.56 0.34 0.41 0.35 0.36 1 1.77 1.56 1.63 1.91 2.08 1.55 2.19 2 0.47 0.50 0.52 0.45 0.47 0.54 0.47 3 ω2 Λ2 cos 2 (ω, λ α ) α 0.8m 3 ω2 Λ α cos 2 (ω, λ α ) α 3 Value -1.24 -1.07 -1.15 -1.36 -1.55 1 0.51 0.50 0.50 0.51 0.49 0.50 0.49 2 0.08 0.09 0.10 0.10 0.10 0.11 0.10 3 0.41 0.41 0.41 0.40 0.41 0.40 0.41 -0.62 -0.85 -1.09 -1.66 ωiωjsij = ω2{Λ1 cos2(ω,λ1)} I + ω2{Λ2 cos2(ω,λ2)} II + ω2{Λ3 cos2(ω,λ3)} III Note that the dominating term is the first one (I) Note associated with the first eigenvector λ1 corresponding to the largest (purely positive) eigenvalue. This is This comprises the meaning of ‘predominant stretching’. Yet vorticity is preferentially aligned with the second eigenvector λ2 which is quite a bit counterintuitive. ALIGNMENT BETWEEN THE EIGENFRAME λI OF THE RATE OF STRAIN TENSOR Sij AND VORTICITY ω Note essentially the same behavior at large and small Reλ Field experiment; Reλ= 104 3D-PTV; Reλ=60 Same in DNS PDFS OF EIGENVALUES, Λk PDF OF EIGENVALUES OF THE RATE OF STRAIN TENSOR sij Field experiment 2004, Sils-Maria, Switzerland, Reλ= 6800 EIGENVALUES Λi EIGENVALUES OF RATE OF STRAIN, sij Value i 0.8m 2m 10m 1 0.53 0.51 0.47 2 0.09 0.09 0.06 -0.62 -0.60 -0.53 1/ 2 Λα / s2 3 3/2 0.40 0.40 0.41 0.04 0.04 0.06 3 Λ3 / s 2 α 1 2 Λ2 / s 2 α 0.56 0.56 0.55 1 0.48 0.54 0.60 2 0.01 0.02 0.02 -0.73 -0.83 -1.04 3 So what about 〈- sijsjkski〉 > 0 (in homogeneous So flows equal to - 4/3 〈ωiωjsij〉 ) is it due to predominant stretching? predominant sijsjkski = Λ13 +Λ23 +Λ33 = 3Λ1Λ2 Λ3 〈Λ13〉 : 〈Λ23〉 : 〈Λ33 〉 = 1.5 : 0.05: -2.5 That is 〈- sijsjkski〉 = -{〈Λ13〉 + 〈Λ23〉 +〈Λ33 〉} is That positive due to dominant contribution from - 〈Λ33 〉, i.e. i.e. predominant compressing rather than stretching ! compressing There are several other (see the examples below) important processes (see in which predominant compressing is the main player : there is no reason to push everywhere stretching, especially vortex stetching. MORE EXAMPLES WITH PREDOMINANT CONTRIBUTION OF COMPRESSING TKE PRODUCTION IN TURBULENT SHEAR FLOWS 〈uiuk〉Sik EVOLUTION OF DISTURBANCES IN GENUINE AND PASSIVE TURBULENCE PRODUCTION OF VORTICITY GRADIENTS IN TWO- DIMENSIONAL TURBULENCE PRODUCTION OF GRADIENTS/ DISSIPATION OF PASSIVE SCALAR TKE PRODUCTION IN SHEAR FLOWS TKE STRETCHING OR COMPRESSING? The turbulent energy production in a turbulent shear flow is known to The be represented by the term -<uiuk>Sik, with ui being the components with of velocity fluctuations, and Sik the mean rate of strain. In turbulent flows which are two-dimensional in the mean (i.e. such that ∂〈⋯ /∂x3 = 〉 0) the production term can be represented as - <uiuk>Sik = - 〈u²Λ1Scos²(u,λ1S)〉 - 〈u²Λ2Scos²(u,λ2S)〉 where u² = u1² + u2², ΛiS are the eigenvalues and λiS are the corresponding eigenvectors of the mean rate of strain tensor Sik, and Λ1S>0,Λ2S<0 (Λ3S=0). Since the term associated with the stretching of material elements is negative, - 〈u²Λ1Scos²(u,λ1S)〉 < 0, and the term associated with the compressing of material elements is positive, 〈u²Λ2Scos²(u,λ1S)〉 > 0, the production term -<uiuk>Sik can be (and usually is) positive due to positiveness of the term associated with the compressive (negative) eigenvalue/eigenvector {Λ2, λ2S}, of the mean strain Sik. In this sense the turbulent energy production is due to the predominant compressing of material elements rather than stretching. TKE PRODUCTION: STRETCHING OR TKE COMPRESSING? Turbulent channel flow (same in BL) Phys Fluids, 16, 16 2704 (2004) The main feature - tendency of alignment of the vector u with both λ1S and λ2 S. The difference is that the latter alignment is somewhat stronger, which results in the positive value of the TKE production TKE PRODUCTION: PRODUCTION: STRETCHING OR COMPRESSING? STRETCHING Tennekes & Lumley 1972 pp 40-41 TKE PRODUCTION: TKE PRODUCTION: STRETCHING OR COMPRESSING? STRETCHING TKE PRODUCTION: STRETCHING OR TKE PRODUCTION: COMPRESSING? SUBGRID STRESSES TKE PRODUCTION: STRETCHING OR TKE PRODUCTION: COMPRESSING? SUBGRID STRESSES GROWTH OF DISTURBANCES IN GENUINE (EΔu, EΔω, EΔs) AND GENUINE PASSIVE (EΔA , EΔB , EΔθ , EΔG) TURBULENCE Looking at the evolution of the disturbance Δu of some flow realization u in a statistically steady state and similarly for other quantities. For more details see Tsinober and Galanti 2003, Phys. For Phys. Fluids, 15, 3514-3531. Again compression Again rather than stretching GROWTH OF DISTURBANCES IN GENUINE, EΔu, EΔω, EΔs GROWTH AND PASSIVE, EΔA , EΔB , EΔθ , EΔG, TURBULENCE The process of The evolution and amplification of disturbances - both in genuine and `passive' turbulence - is dominated by the strain field of the basic flow. In all their energy production has the form ΔuiΔujsij Tsinober & Galanti, 2003 Galanti PRODUCTION OF VORTICITY GRADIENTS IN TWO-DIMENSIONAL TURBULENCE Dξ/Dt = - (ξ·∇)u + νΔξ ; ξ = ∇ω (1/2)Dξ2/Dt = - ξiξk sik + νξiΔξi The main contribution to the production (-〈ξiξksik〉 > 0) - 〈ξiξk〉sik = - ξ2Λkcos2(ξ,λk) is associated with compressive eigenvalue Λ3: it is due to -ξ2Λ3 is cos2(ξ; λ3). Note that ‘partner’ to ξi has the same partner as does vorticity (strain), but they (ξi and sik) are not equal partners as ξi lives are at much smaller scales than sij and there is no strain production (!) either. However, production of palinstrophy, i.e. (curl ω)2 is due to predominant contribution from the term associated with the stretching eigenvalue Λ1. PRODUCTION OF GRADIENTS/ DISSIPATION OF PASSIVE SCALAR DISSIPATION DΘ/Dt = - (Θ·∇)u + κΔΘ ; G = ∇Θ (1/2)DG2/Dt = - GiGk sik + κGiΔGi (1/2) The main contribution to the production (-〈GiGksik〉 > 0) - GiGksik = - G2Λkcos2(G;λk) is associated with the alignment of the temperature gradient and the eigenvector¸ λ3 corresponding to the compressive eigenvalue Λ3: it is due to -G2Λ3 cos2(G; λ3). Selected results PDF OF THE PRODUCTION -GiGjsij, PDF The PDF of the production, -GiGjsij, obtained in our experiments in a slightly heated jet is very similar to that obtained in DNS of NSE by Tsinober and Galanti (2003). The PDF of -GiGjsij is positively skewed positively and the mean <-GiGjsij > is positive. positive 1D 2 2 G = −Gi G j sij + κGi∇ Gi 2 Dt Alignments between the temperature gradient G and the eigenframe λi of the rate of strain tensor sij (Λk are the corresponding eigenvalues); Λ1> Λ2> Λ3; Λ1> 0, Λ3<0; Λ1+ Λ2+ Λ3 = 0 0, -GiGksik=-G2[Λ1cos2(G;λ1) + Λ2cos2(G;λ2) + Λ3cos2(G;λ3)] The main effect, the alignment between the temperature gradient and the eigenvector λ3 corresponding to the compressive eigenvalue Λ3< 0, is captured well in the measurements and is similar to that obtained from DNS BACK TO VORTICITY BACK VERSUS STRAIN STRONGER NON LINEARITY IN STRAIN DOMINATED REGIONS CONCENTRATED VORTICITY OR STRAIN ARE THEY THAT IMPORTANT? JOINT PDFS OF ENSTROPY (top) AND STRAIN (bottom) PDF ENSTROPY AND STRAIN PRODUCTION WITH ENSTROPHY (left) AND STRAIN (right) PRODUCTION FROM OUR FROM FIELD EXPERIMENT AT Reλ=104 AT Physics of Fluids, 13, 311 (2001) Note much larger positive shift with strain (ω ∇ ω / ω ) 2 i 2 i Rate of enstrophy Rate enstrophy production and its production viscous reduction conditioned on strain and vorticity vorticity ω ωi ∇ 2 ωi / ω 2 (ω ω s i k ik ) /ω 2 ω ωi ωk sik / ω 2 (ω ∇ ω / ω ) 2 i 2 i s ωi ∇ 2 ωi / ω 2 (ω ω s i k ik /ω 2 ωi ωk sik / ω ) NOTE THE LARGE RATE OF OF ENSTROPHY PRODUCTION IN STRAIN DOMINATED REGIONS STRAIN (RED CURVE) AS COMPARED TO AS REGIONS OF LARGE ENSTROPHY (BLUE CURVE) s 2 s / s ,ω/ ω ALL NONLINEAR ALL TERMS BEHAVE THIS WAY! THIS Eigen-contributions to ωiωjsij/ω2=Λkcos2(ω,λk) Eigen Non-linear terms are growing much slower in the enstrophy dominated regions than in the strain dominated regions. Q Summary of threeSummary dimensional, incompressible flow patterns/ local structure of the MR R flow field in the frame following a fluid particle (from Soria et al. 1994, after Perry et al. 1990). Q = 1/4(ω² - 2sijsij), R = - 1/3(sijsjkski + + 3/4ωiωjsij). . STRONGER NON LINEARITY IN STRONGER STRAIN DOMINATED REGIONS HOW THIS LOOKS ON THE R – Q PLANE PLANE Q = (1/4){ω2 -2s2}; R = - (1/3){sijsjkski+(3/4)ωiωjsij} – Second and third invariants of the velocity gradient tensor Aik= ∂ui/∂xk R-Q - PLOT Field experiment, Reλ= 6800 DNS of NSE, ReΘ= 690 PTV, Reλ= 80 velocity gradient ( ) Chacin et al 2000 12 ω − 2 sik sik - second invariant of the velocity gradient tensor 4 1⎛ 3 ⎞ R = − ⎜ sik skm smi + ω iω k sik ⎟ - third invariant of the velocity gradient tensor 3⎝ 4 ⎠ Q= The first invariant is vanishing as a consequence of incompressibility ∂ui ∂xk Q = (1/4){ω2 -2s2}; R = - (1/3){sijsjkski+(3/4)ωiωjsij} CONDITIONAL AVERAGES ON THE R-Q PLANE OF REYNOLDS STRESS (LEFT) AND REYNOLDS TKE `generating events’ (RIGHT), CHASIN AND CANTWELL , 2001 TKE D=0 D=0 Q = (1/4){ω2 -2s2}; R = - (1/3){sijsjkski+(3/4)ωiωjsij} CONDITIONAL AVERAGES ON THE R-Q PLANE OF REYNOLDS STRESS (LEFT) AND PLANE AND DISSIPATION (RIGHT), CHASIN AND CANTWELL , 2001 Q Q R R Ejections (-u+v) D=0 D=0 Sweeps (+u-v) -1.3 uv/uτ2 -0.3 0.13 ε/(uτ4) 0.36 One of the most interesting (from our point of view) One findings is that the main contribution to the shear stress, turbulent energy production, and dissipation comes from the regions with Q<0 with larger contribution from the lower right quadrant, i.e. Q<0 and R>0, not only dominated by strain, but also by production of strain, -sikskmsmi, see figure 5, also figures 8, 11 and 14 in Chacin and Cantwell (2000). It should be emphasized that these regions are mainly not the ones corresponding to vortices (hairpins or whatever), which are located mostly in the regions with Q>0, or a bit more precisely in regions with D>0, where D=((27)/4)Q³+R² is the discriminant of ∂ui/∂xj. That is the regions of major nonlinear activity are really associated with large strain (mainly corresponding to what Chacin and Cantwell call `blank' spaces) rather than with regions of concentrated vorticity with lower dissipation. In other words it seems that concentrated vorticity is not In that important also in turbulent shear flows and that structure(s) associated with turbulence (not only its energy) production are mainly due to the large strain rather than large vorticity. Structure(s) associated with the latter seem to be the consequence of the turbulent dynamics rather than its dominating factor. A final remark is that these results do not contradict the importance of vorticity in maintaining the Reylolds stress. First, these are relations for the mean quantities, and second, there is no turbulent flow without vorticity. However, important details of the relations between Reynolds stress, vorticity, strain and their production remain not clear. The interpretation of the results by The Chacin and Cantwell, 2000 (and similar) given here is not in full agreement with their conclusions, especially regarding the role of vortices and concentrated vorticity in turbulent flows (again vortex obsession). NICE VORTICES/WORMS/FIALMENTS OR WHATEVER vorticity velocity velocity SHE et al. 1991 At this stage, this alternative approach (i.e. the ‘structural’) has not led to a At i.e. has generally applicable quantitative model, neither – for better or worse – has it a for has major impact on the statistical approaches. Consequently the deterministic major viewpoint is neither emphasized nor systematically presented, POPE 2000. POPE nor This does not mean that there exists “generally applicable quantitative This generally model” based on statistical approaches. model REAL SHE et al. 1991 GAUSSIAN ‘STRUCTURES’ OF INTENSE VORTICITY AND STRAIN, Moisy & Jimenez, 2004 a b c (a)Big structure with |ω|>3ω; (b)Small structure with |ω|>3ω; (c)Big structure with |ω|>6ω SIMILAR TO THOSE IN SHE et al., 1991 and SHE BORATAV AND PELZ, 1997 CHEN, 2000 CHEN, a b (a) |s|>2.8s; (b) |s|>4.2s IS THE FLOW FIELD IN ‘WORMS’ SIMPLE? IS IT QUASI-2D? JIMENEZ ET AL 1993 JIMENEZ IS THE FIELD OF VORTICITY IN ‘WORMS’ SIMPLE? CONSEQUENCES FOR REPRESENTATION OF TURBULENT FIELDS JIMENEZ & WRAY 1996 JIMENEZ THE ‘AMOUNT’ OF COMPRESSING IN ‘WORMS’ IS THE SAME AS IN THE WHOLE FIELD ! HENCE INADEQUATE REPRESENTATION OF THE FLOW FILED BY A COLLECTION OF PURELY STRETCHED VORTICES (or other `simple' objects), ESPECIALLY THOSE WHICH DO NOT INTERACT WITH WITH STRAIN.. THESE LATTER ARE THESE DESCRIBED BY DIFFRENTIAL EQUATIONS. THE REAL ONES ARE DESCRIBED BY INTEGRO – DIFFERENTIAL EQUATIONS. SOME COMMENTS ON SOME WHAT IS A“ VORTEX” ? I DO NOT KNOW. WHO DOES? THERE ARE SOME DEFINTIONS (MOSTLY AD HOC). A COMMON AD DO FEATURE IS THAT (MORE OR LESS) CONCENTRATED VORTICITY IS SURROUNDED BY PLENTY VORTICITY OF STRAIN , E.. G. ‘POTENTIAL’ VORTEX: VORTICITY ONLY IS NOT REALLY ‘ONLY’ STRAIN E A popular method to look for structure(s) (and `vortices’) is to use a criterion based on one parameter only, e.g. enstrophy ω². Though such an approach is useful and `easy', it is inherently limited and reflects the simplest aspects of the problem. For example, even for characterization of some aspects of the local (i.e. in a sense `point'-wise) structure of the some aspects flow field in the frame following a fluid particle requires at least two parameters: the second and the third invariants of the velocity gradient tensor ∂ui/∂xk: Q = 1/4(ω² - 2sijsij) and R = - 1/3(sijsjkski + 3/4ωiωjsij). Therefore attempts to adequately identify/characterize finite size 3/4 structure(s) – and ‘vortices’ seem to be of this kind - by one parameter only are unlikely to be successful, and one needs something like pattern recognition based on some conditional sampling scheme involving definitely more (perhaps much more) than two parameters. But then all the ‘simplicity’ (and attraction) will be gone. So it is safer to use well defined quantities – vorticity and strain. Vorticity alone is not ‘enough’. Concentrated vorticity (very popular) is not that important and other regions (e.g. regions dominted by strain) play essential role in the evolution and dynamics of turbulent flows (Tsinober, 1998, 2001) IN LIEU OF IN CONCLUSION Looking in more essential details (both physics and the mathematical Looking ones) of the processes of self- amplificiation of the field of velocity derivatives seems to be the key issue for most of the problems of turbulence but also 3-D NSE and Euler. This includes subtle geometrical relations between vorticity and strain (which are likely to be the main guilty of almost happening in turbulence: after all there is no turbulence without vorticity production) and several others of dynamical significance . Understanding of these processes and thereby essential aspects of turbulence physics seems to form the basis for constructive approach to a great variety of problems including effective handling of nonlinearities which will allow to solve the standard 3D-NSE and Euler problems and not the other way round. ...
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This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.

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