Unformatted text preview: FUNDAMENTAL AND CONCEPTUAL ASPECTS OF TURBULENT FLOWS
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 Arkady Tsinober 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 XIIIXIV IS TURBULENCE ERGODIC?
TAKING A STRICTLY RIGOROUS (MATHEMATICALLY) POSITION THE ANSWER SEEMS TO BE NEGATIVE! HOWEVER, THE ARE SEVERAL IMPORTANT “HOWEVERS”, WHICH ARE THE MAIN THEME OF THIS LECTURE. For statistically stationary flows ERGODICITY is ERGODICITY For (roughly) equivalence of `true' statistical properties (not only means/averages, but `almost' all statistical properties) of an ensemble to those obtained using time series in one very long realization. A similar property is defined in space by similar replacing time by space coordinate(s) in which the flow domain has an infinite extension, at least in one direction. Turbulent flows are believed/known Turbulent (empirically) to be ergodic.
There is no way to confirm that those turbulence There data used in analysis represent typical properties of turbulence. VAN VEEN, KIDA, & KAWAHARA
motion representing isotropic turbulence, Fluid Dynamics Research, 38, 19–46.
(2006) Periodic The ergodicity of turbulence sounds to me as an assumption which is hard to avoid or test Is it possible to do anything or should we stay with the belief ?? ? FOUNDING FATHERS FOUNDING OF STATISTICAL MECHANICS LUDWIG BOLTZMANN JAMES MAXWELL JOSIA GIBBS ALBERT EINSTEIN Introduced concepts such as ensembles, ergodicity and coarse graining . BOLTZMANN introduced the ergodic hypothesis in 1871
Raised serious mathematical problems ⇒ ergodic theory : There is much more to the mathematical study of Gibbs ensembles than that the question whether or not time averages and ensemble averages are equal. ON ‘ANALOGY’ BETWEEN STATISTICAL BETWEEN
MECHANICS AND TURBULENCE MECHANICS
The approach, which treats the fields of hydrodynamic variables of a turbulent flow as random fields, was initiated by the works of Kolmogorov and his school [see, e.g., Millionshchikov (1939)] and the work of Kampé de Fériet (1939), see Monin and Yaglom, 1971. Adopting the assumption of the existence of probability distributions for all fluid dynamic fields, we may further make wide use of the mathematical techniques of modern probability theory; the operation of averaging is then defined uniquely and has all the properties naturally required of it. MONIN AND YAGLOM 1971 RECENT REFERENCES – (S)NSE
DA PRATO, G. AND DEBUSSCHE, A. (2003) Ergodicity for the 3D stochasitc NavierStokes equations, J. de Math. Pures et Appl., 82, 877947.4. ROMITO, M. (2003) Ergodicity of the Finite Dimensional Approximation of the ROMITO 3D NavierStokes Equations Forced by a Degenerate Noise, J. Stat. Phys., 114, J. 114 155177. MATTINGLY, J. C. On Recent Progress for the Stochastic Navier–Stokes MATTINGLY, Equations. Journ´ees “Equations aux D´eriv´ees Partielles” (ForgeslesEaux, Journ 2003), XV, Summer 2003, Art. No. 11, 52 p. XV WAYMIREY, E.C. (2005) Probability & incompressible NavierStokes WAYMIREY equations: An overview of some recent developments, Probability Surveys, 2, 1Probability 32. FLANDOLI, F.(2005) An Introduction to 3D Stochastic Fluid Dynamics, CIME CIME lectures, 108 pp. 108 KUKSIN S.B. (2002) Ergodic theorems for 2D statistical hydrodynamics Ergodic Reviews in Mathematical Physics, 14, No. 6 , 585600. Note that this is all SNSE, i. e. stochastic forcing both in 3D and 2D. In the latter case it is unlikely that with a deterministic forcing one can expect anything like ergodicity AN IMPORTANT ASPECT
Note that the above is all SNSE, i. e. stochastic forcing. A typical statement is as follows
One of the oldest open problems in theoretical physics is that of describing fully developed turbulence on the basis of a mi(a)croscopic model. The latter is usually taken to be the stochastic Navier  Stokes (NS) equation subject to an external random force that models the energy injection by the largescale modes
ADZHEMYAN ET AL 2003 AN IMPORTANT ASPECT
However, what about a great variety of turbulent flows in which the ‘forcing’ is not random and in many cases is even not time dependent juts constant in time? Such flows at large enough Reynolds numbers become turbulent due to what can be called intrinsic stochasticity (nobody seems to know what it is), and, e.g. statistically stationary turbulent flows are massively studied using temporal statistics instead of the ‘true’ one based on ensembles or probability measures (which are anyhow not accessible). All All observed so far statistical (not only average) properties of many such turbulent flows (but not all) are remarkably reproducible (statistical stabilty) (statistical and – as mentioned – are believed to be ergodic in spite of the fact that, say, a constant in time large scale, i.e. deterministic forcing breaks the ergodicity ‘on large scales’ In statistically stationary situations the time statistics In obtained in experiments is believed to correspond to a probability measure invariant under time evolution. This comprises the essence of the ergodic hypothesis, which is usually expressed in terms of ensemble and is widely used in experiments. A similar statement is made for situations with at least one similar homogeneous spatial coordinate. ! In dynamical systems the equivalence of two is used as a definiton of egrodicity : Defintion 7.1 : An abstract
dynamical system is ergodic if for every complexvalued μsummable function the time mean is equal to the space mean. ARNOLD AND AVEZ (1968) ARNOLD WIGGINS, S. AND OTTINO J.M. (2004) Foundations of chaotic mixing, Phil. Trans. R. Soc. Lond. A 362, 937–970 Phil. 362 FLANDOLI, F.(2005) An Introduction to 3D Stochastic Fluid Dynamics, CIME lectures, 108 pp. FLANDOLI CIME 108 pp Turbulent flows are believed/known Turbulent (empirically) to be ergodic.
There is no way to confirm that those turbulence There data used in analysis represent typical properties of turbulence. VAN VEEN, KIDA, & KAWAHARA
(2006) Periodic motion representing isotropic turbulence, Fluid Dynamics Research, 38, 19–46. Fluid The ergodicity of turbulence sounds to me as an assumption which is hard to avoid or test Is it possible to do anything or should we stay with the belief ?? ? measures on function space that are time invariant. However, invariance under time evolution is not enough to specify a unique measure which would describe turbulence. Another problem is that it is not clear how the objects that the authors have constructed and used in their proofs are relevant/related or even have anything to do with turbulence.
We remind again that this is all SNSE, i. e. We stochastic forcing both in the RHS of NSE FOIAS ET AL. (2001) have shown that there are FOIAS OUR REFEREE “ The main objection that I am OUR raising is that the comparison between space and time averages is not at all what ergodicity is about. The authors should have compared the timeaveraged value of a given observable against the *ensemble*averaged value at a given time: the latter ones can be obtained by performing a large number of experiments with different initial conditions. This has not been done in the present manuscript. The confusion between space and ensemble averages …” REALLY? In statistically stationary situations the time statistics In obtained in experiments is believed to correspond to a probability measure invariant under time evolution. This comprises the essence of the ergodic hypothesis, which is usually expressed in terms of ensemble and is widely used in experiments. A similar statement is made for situations with at least one similar homogeneous spatial coordinate. ! In dynamical systems the equivalence of two is used as a definiton of egrodicity : Defintion 7.1 : An abstract
dynamical system is ergodic if for every complexvalued μsummable function the time mean is equal to the space mean. ARNOLD AND AVEZ (1968) ARNOLD Thus, one deals with two different aspects: one statistical Thus, analysis over the entire flow field at a certain moment in time, and another one for one position in space over a very long period of time. The first one may not be representative for a longer period of time, while the second one may not be representative for all the points in space. The point is that if the flow is ergodic the two types of statistics should give the same result.
Many ensembles, (like the human populations), are not ergodic. A NUMERICAL EXPERIMENT
Galanti and Tsinober (2004) Is turbulence ergodic?, Physics Letters A 330 , 173–180 There seems to exist no direct evidence regarding the validity of the ergodicity There hypothesis in turbulent flows. We made an attempt to obtain such evidence via direct numerical simulations of We the Navier–Stokes equations without (!) performing a large number of simulations at different initial conditions representing the members of an ensemble. The main idea is simple and is based on the fact that if a turbulent flow is both The statistically stationary in time and homogeneous in space than its temporal and spatial statistical properties should be the same if the ergodic hypothesis is correct. An important consequence is that it is not necessary to perform a large number of time/labor consuming “brutal force” experiments with different initial conditions in order to compare the timeaveraged value of a given observable against the “ensemble” averaged value at a given time (as suggested by our referee). (as Galanti and Tsinober (2004) Is turbulence ergodic?, Phys. Lett., A 330 , 173–180 Galanti Phys. V E L O C I T Y F L U C T U A T I O N S ENSTROPHY PRODUCTION AND ITS RATE ENSTROPHY PDFS OF cos{ω,W} PDF
Wi = ωisij – vortex stretching vector PDFS OF cos{ω,λi} PDF
λi – eigenvectors of the rate of strain tensor PDFS OF cos{s,S} PDF
s = sij,S = sikskj JOINT STATISTICS I JOINT
Temporal Spatial Q = (1/4){ω2 2s2} , R =  (1/3){sijsjkski+(3/4)ωiωjsij} Q = (1/4){ω2 2s2}
T e m p o r a l R =  (1/3){sijsjkski+(3/4)ωiωjsij} Q
S p a t i a l Third axis ω2 Third R Third axis s2 JOINT STATISTICS II JOINT
EIGEVALUES Λi OF THE RATE OF STRAIN TENSOR
IN THE PLANE Λ1 +Λ2+ Λ3= 0 Temporal Spatial JOINT STATISTICS III
Joint PDFS of cos{ω,λi} in HummerAitoff projection Joint cos
cos2{ω,λ1}+cos2{ω,λ2}+cos2{ω,λ3}=1 Temporal Spatial JOINT STATISTICS IV
Temporal Spatial T e m p o r a l S p a t i a l TWOPOINT STATISTICS I TWO TWOPOINT STATISTICS II TWO THREEPOINT STATISTICS THREE IN LIEU OF CONCLUSION The reported results from a long enough in time numerical The simulation provides clear evidence that if a turbulent flow is both statistically stationary in time and homogeneous in space than its temporal and spatial statistical properties are the same. This can be seen as evidence in favor of validity of the ergodic hypothesis in turbulence. Is this really the case for all turbulent flows. Can one claim more than that ? A natural question concerns the inhomogeneous flows. One can expect similar results as obtained above for flows with homogeneous coordinates, such as the flow in a plane channel. An obvious conjecture is that the temporal and spatial statistical properties of such a flow will be the same for fixed values of the distance from the wall. A positive addition to the answer on the question (when) do positive (when) simulations reproduce statistics? Att least in some cases one A time snapshot is pretty representative
IS THIS ALL? A ‘CONFESSION’ Whereas it is naturally to expect that nonlinear systems Whereas driven by a random force should be ergodic, our simulation was made with purely deterministic and with constant in time nonhelical forcing. Nevertheless, the flow clearly exhibited strong similarity between its temporal and spatial statistical properties with the exception of the largest scales. A possible explanation is that this happens due to the property of selfrandomization of fluiddynamical turbulence (intrinsic stochasticity) Chaotic behaviour versus ergodicity One of our referees wrote "The point in the conclusions stressing One
that the forcing is deterministic is very weak as it is very wellknown that deterministic forcing can yield a random dynamics even for a few degrees of freedom, let alone for a turbulent flow". Our referee is right, but seems to be not aware that most of lowOur dimensional chaotic systems are not ergodic!!! Moreover, the issue is broader and is a part of that on differences between ergodicity and randomness. The story goes back to the general belief that any kind of nonlinearity in a system with large number of degrees of freedom would give rise to ergodicity, see, e.g. FERMI, E. (1923), Beweis dass ein mechanisches normal
system im allgemeinen quasiperiodisch ist, Phys. Z., 24, 261 sd. Phys. There is another important and very difficult issue. Since the large There scale deterministic forcing breaks the ergodicity on large scales the authors removed the mean velocity before comparing the temporal and spatial statistics of the velocity field. So one may put forward an objection that ergodicity is a global property of the dynamical system represented by the NavierStokes equations and there cannot be smallscale ergodicity. Another question is about the impact of nonlocality, i.e direct and bidirectional coupling of large and small scales, especially in case of purely deterministic forcing. cing. Is it possible to speak about ‘approximate’ ergodicity or ‘modified’ Is ergodicity? ARE THERE NONERGODIC STATISTICALLY STATIONARY TURBULENT FLOWS?
There are many flows that cannot be easily qualified as ‘cleanly’ ergodic: Flows in diffusers with separation. All partly turbulent flows (mixing layers, jets, wakes past , bodies, boundary layers) which properties depend strongly on the conditions at the entrance (small oscillations of the body, acoustic excitation, etc.) and on the level of disturbances in the quasipotential flows outside. Minute changes in the latter (i.e. in the entrance conditions in the quasipotential flows outside) often result in dramatic ones in flows like mentioned above. Sometimes this is considered as ‘long memory’ of such flows, but there seems to be much more than that as minute changes produce dramatic ones in the statistical properties of the flows. Flows in Flows axisymmetric geometries, e.g. (spontaneously swirling) (spontaneously turbulent flows in jets and pipes ‘Wind’ in turbulent convection ARE THERE NONERGODIC STATISTICALLY STATIONARY TURBULENT FLOWS? RELATED ISSUES
Memory of turbulence: role of Memory initial conditions, conditions at the entrance etc. Passive objects, Lagrangian Passive issues ...
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