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 of the Marie Curie Chair "Fundamental and Conceptual Aspects of Turbulent Flows". 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 SCOPE: The lectures are devoted to a rather informal overview of basic features of Please, note that there will be no lectures at February 7 and 14.
PLACE: Seminar Room, Institute for Mathematical Sciences, Imperial College turbulent flows (TF). The main emphasis is on conceptual and problematic aspects, physical phenomena, observations, misconceptions and unresolved issues rather than on conventional formalistic aspects, models, etc.. The lectures are based on the book by the author An informal introduction to turbulence , Kluwer , 2001 and its revision A conceptual introduction to turbulence , Kluwer , 2007 and recent developments. PDF files of the lectures will be accessible from internet. AUDIENCE AND PREREQUISITES: Ph.D students and higher. Fluid dynamics including turbulence. Knowledge of physics and mathematics at graduate level. TIME: Wednesdays, 16001700 and/or 16001800. London, 53 Princes Gate, Exhibition Road, South Kensington, London SW7 2PG GENERAL NOTES ON THE NATURE OF LECTURES
This is a selection of lectures/topics rather than a course and contains as much questions and similar things (or even more) as answers, (hopefully) unbiased discussion of the unresolved issues, controversies major problems and misconceptions. In particular there are no lapses into brevity at difficult places. Naturally, this involves critical remarks, which are intended to be as constructive as possible. The specific sequence of themes depends on the audience response as the lectures will be made as interactive as possible.
It is much easier to present nice rational linear analysis than it is to wade into the morass that is our understanding of turbulence dynamics. With the analysis, professor and students feel more comfortable; even the reputation of turbulence may be improved, since the students will find it not as bad as they had expected. A discussion of turbulence dynamics would create only anxiety and a perception that the field is put together out of folklore and arm waving (LUMLEY, 1987). In selecting the references I used the (genuinely small) parameter introduced by Saffman (1978), which he called information density, I, and defined as the ratio, S/N, in the literature, with S = signal (understanding), and N = noise (mountains of publications). In order to increase the value of I I did all my best to concentrate on the numerator, S, and to reduce the denominator, N, to the best of my knowledge, ability and judgment/understanding (this includes scanning about 9000 references since the publication of my book at 2001) . However, absence of references does not necessarily mean that  in my view  they belong to N, but is due to my ignorance or the lack of space needed to discuss them here. I used extensively quotations which  to my view  are essential in any field of science. SONNET TO TURBULENCE by S. Corrsin
(For Hans Liepmann on the occasion of his 70th birthday, with apologies to Bill S. and Liz B.B.) Shall we compare you to a laminar flow? You are more lovely and more sinuous. Rough winter winds shake branches free of snow, And summer's plumes churn up in cumulus. How do we perceive you? Let me count the ways. A random vortex fieldwith strain entwined. Fractal? Dig and small swirls in the maze May give us paradigms of flows to, find. Orthonormal forms nonlinearly renew Intricate flows with many free degrees Or, in the latest fashion, merely few As strange attractor. In fact, we need Cray 3 's. Experiment and theory, unforgiving For serious searcher, fun ... and it's a living! The following is an exemplifying list of some possible questions to be discussed and/or mentioned:  Reynolds number dependence or where is the ? Is it (always) necessary to have large parameters to study the basic physics of turbulence? Is the nature of dissipation unimportant in the "inviscid" limit? Is the inertial range a conceptually well defined concept? Are its properties really independent of the nature of dissipation?  Is "cascade" in genuine turbulence conceptually well defined notion or is it "mostly a pedagogical imagery"? Is there cascade in physical space? Is the Komlogorov 4/5 law an unequivocal evidence of such a cascade? How meaningful is "cascade" of passive objects as described by linear equations? Is "casade" Eulerian, Lagrangian or both? Are decompositions aiding understanding or obscuring the physics of turbulence?  Is the physics of vortex stretching well understood? Is it a result of the kinematics of turbulence: are vortex lines on average stretched rather than compressed because two particles on average move apart from each other? Is the physics of vortex stretching well understood? Is it a result of the kinematics of turbulence: are vortex lines on average stretched rather than compressed because two particles on average move apart from each other?  Is enhanced dissipation in turbulence due to vortex stretching? Are vorticity and strain equal partners?  What is (are) the meaning(s) of non locality of turbulence? What are its manifestations? Is non locality important? Is there screening in turbulence?  How (much) statitsical is turbulence? Is turbulence a part of statistical physics? Will nonequilibrium statistical mechanics play an increasingly important role in further progress of turbulence? Is turbulence ergodic?  How rigorous is turbulence modelling? Modelling versus physics of turbulence.  How analogous are quasitwodimensional flows? Diversity of strongly anisotropic flows (and the corresponding limiting states) as compared to pure twodimensional ones.  How analogous are the genuine and passive turbulence? What can be learned about genuine turbulence from its signature on the evolution of passive objects? What is the importance (if any) of statistical conservation laws in genuine turbulence?
 How prospective is Lagrangian description of turbulence? Is it different conceptually (not only technically) from the Eulerian one? Is it possible in this approach to separate the Lagrangian (kinematic) chaos from the genuinly dynamical intrinsic (Eulerian) stochasticity? Is this a conceptual difficulty?  What is unviersality of turbulence (if such exists)? What is the role of (the nature of) forcing and intial/upstream conditions in this and other issues? What are the situations and what are the properties which are (approximately) invariant of IC and BC? Is there qulitative universality?  What are the main reasons for slow progress in handling the physics of turbulence (which is the key for the progress in any aspect of the problem). Is it due to i) inadequate tools to handle both the problem and the phenomenon of turbulence, ii) lack of fresh ideas, which is directly related to the (in)ability/skill/art to ask the right and correctly posed questions, iii) insufficient conceptual progress and dominance of some misconceptions or ill defined conceptions? LECTURES III INTRODUCTORY
Introductory remarks and premises. Main qualitative (universal) features. Examples of real turbulent flows. Definitions of turbulence and of the problem of turbulence? The nature of the problem, why turbulence is so impossibly difficult. THE RISE AND FALL OF IDEAS IN TURBULENCE LIEPMANN 1979
1500 Recognition of two states of fluid motion by Leonardo da Vinci and use of the term la turbolenza. 1839 `Rediscovery' of two states of fluid motion by G. Hagen. 1883 Osborne Reynolds' experiments on pipe flow. Concept of critical Reynolds number  transition from laminar to turbulent flow regime. 1887 Introduction of the term `turbulence' by Lord Kelvin. 1895 Reynolds decomposition. Beginning of statistical approach. 1909 D. Riabuchinsky invents the constantcurrent hotwire anemometer. 1912 J.T. Morris invents the constanttemperature hotwire anemometer. 1921, 1935 Statistical approach by G.I. Taylor. 1922 L.F. Richardson's hierarchy of eddies. 1924 L.V. Keller and A.A. Friedman formulate the hierarchy of moments. 1938 G.I. Taylor discovers the prevalence of vortex stretching. 1941 A.N. Kolmogorov local isotropy, 2/3 and 4/5 laws. 1943 S. Corrsin establishes the existence of the sharp laminar/turbulent interface in shear flows. 1949 Discovery of intrinsic intermittence by G. Batchelor and A. Townsend 1951 Turbulent spot of H.W. Emmons 1952 E. Hopf functional equation. 1967 Bursting phenomenon by S.J. Kline et al. 1976 Recapitulation of large scale coherent structures by A. Roshko. A BIT OF HISTORY
, PREMISES
I soon understood that there was little hope of developing a pure, closed theory, and because of the absence of such a theory the investigation must be based on hypotheses obtained in processing of experimental data... KOLMOGOROV 1985 In searching for a theory of turbulence, perhaps we are looking for a chimera...
One of the things that I always found troubling in the study of the problem of turbulence is that I am not quite sure what the theoretical turbulence problem actually isAA I just cannot think of anything where a genuine prediction for the dynamics of turbulent flow has been confirmed by an experiment. So we have a big vast empty field SAFFMAN 1978, 1991 Nothing can be proven in the theory of turbulence, BATCHELOR, NOV.1998 ... it is amazing how many different, nearly orthogonal, points of view there are about a phenomenon which is governed by Newton's innocentlooking, linear second law of motion, with a little help or hindrance from viscosity. BRADSHAW, 2003 Every aspect of turbulence is controversial,
SALMON, 2003 Sometimes experiments provide us with so beautiful and clear results that it is a shame on theorists that they cannot interpret them., YUDOVICH, 2003
... and so we can hold strong opinions either way FEYNMANN 1963. BATCHELOR'S FRUSTRATION OF TURBULENCE
Formal mathematical investigations have produced remarkably little value... A number of general procedures for calculation of various dynamical aspects of homogeneous turbulence have been devised, but none of them impresses me as being likely either to advance our understanding of turbulence or to achieve results on which we can place reliance...The universal similarity theory of the smallscale components of the motion stands out in this rather grey picture as a valuable contribution... BATCHELOR 1962 ... that sense of frustration that afflicted Batchelor (and many others) from 1960 onward...These frustrations came to the surface at the now legendary meeting held in Marseille (1961) to mark the opening of the former Institut de Mecanique Statistique de la Turbulence (Favre 1962). MOFFATT (2002) The story started before
My overall impression of the Symposium is that no really new and important ideas have been presented... I think we must admit that little new theory has been put before us. BATCHELOR (1959) Some reflections on the theoretical problems raised at the Symposium, Proceedings of a Symposium on Atmospheric diffusion and air pollution, editors F.N. Frenkiel and P.A. Sheppard, Advances in Geophysics, 6, 449452. Formal mathematical investigations have produced remarkably little value...
BATCHELOR (1962). THREE MAJOR DIFFICULTIES
Inadequate tools to handle both the problem and the phenomenon of turbulence are not developed enough Lack of fresh ideas, which is directly related to the (in) ability/skill/art to ask the right and correctly posed questions Insufficient conceptual progress and dominance of some misconceptions or ill defined conceptions . The Rise and Fall of Ideas in Turbulence , LIEPMANN 1979 INADEQUATE TOOLS
The entire experience with the subject indicates that the purely analytical approach is beset with difficulties, which at this moment are still prohibitive....our intuitive relationship to the subject is still too loose not having succeeded at anything like deep mathematical penetration in any part of the subject, we are still quite disoriented as to the relevant factors, and as to the proper analytical machinery to be used, VON NEUMANN,1949. THE TOOLS
THEORY: TOO MANY "THEORIES" (MOSTLY PRETTY USELESS, BUT ALL AGREEING WELL WITH EXPERIMENT)
I just cannot think of anything where a genuine prediction for the dynamics of turbulent flow has been confirmed by an experiment. So we have a big vast empty field. SAFFMAN, 1991 Nothing can be proven in the theory of turbulence, BATCHELOR, NOV.1998
There is no way and tools so far, if ever, to treat turbulence analytically turbulence is beyond analytics (TBA). Unfortunately, this is true of other theoretical approaches such as attempts to construct statistical and/or other theories. EXPERIMENTS: PHYSICAL AND NUMERICAL
The essential mathematical complications of the subject were only disclosed by actual experience with the physical counterparts of these equations , VON NEUMANN, 1949 REMAIN A MAJOR EXPLORATORY TOOL BOTH in
Elucidating the properties of turbulence as a physical phenomenon Host of applications THESE ARE THE REASONS WHY THE DISCUSSION OF conceptual and problematic aspects, physical phenomena, observations, misconceptions and unresolved issues (rather than conventional formalistic aspects, models and similar)
IS SO IMPORTANT CAN ONE DEFINE TURBULENCE?
Turbulence is a phenomenon which sets in in a viscous fluid for small values of the viscosity coefficient ... hence its purest, limiting form may be interpreted as the asymptotic, limiting behavior of a viscous fluid for 0 NEUMANN, 1949. Turbulence can be defined by a statement of impotence reminiscent of the second law of thermodynamics: flow at a sufficiently high Reynolds number cannot be decelerated to rest in a steady fashion. The deceleration always produces vorticity, and the resulting vortex interactions are apparently so sensitive to the initial conditions that the resulting flow pattern changes in time and usually in stochastic fashion. LIEPMANN 1979, The rise and fall of ideas in turbulence, American Scientist, 67. 221.
See Appendix A in my book for a collection of attempts to give a definition and snags. ARE ATTEMPTS TO GIVE A DEFINITION OF WHAT IS TURBULENCE CONCEPTUALLY CORRECT?
In mathematics the definition of the main object of study precedes the results. In physics it is vice versa. Usually it happens when one studies a new phenomenon and only at a later stage, after understanding (!) it sufficiently (!), classifying it, etc. a most reasonable definition is found. Turbulence in not a new field but this time has yet to come.
Can one define what is mathematics? One has to learn it first Following Stewart and Tennekes & Lumley IN LIEU OF DEFINITION: MAJOR QUALITATIVE UNIVERSAL FEATURES
Intrinsic spatiotemporal stochasticity, both E and Lof predictability turbulent, in contrast to pure Lagrangian chaos, which is Elaminar but Lturulent . Loss Huge range of strongly interacting `scales'/degrees of freedom. Highly dissipative Threedimensional and rotational. Strongly diffusive Generally, a single property (or part) as above is not synonymous to turbulence. For example, randomness alone as was believed for a long time. Today too there is a tendency not to make a clear distinction between genuine turbulence (the big Tproblem) and a set of phenomena in evolution of passive objects (PS, PV, diffusion/dispersion) in random velocity fields (small tproblems) prescribed a priory (more in lectures on analogies). As a latest example see Falkovich, G. and Sreenivasan, K.R. (2006) Lessons from Hydrodynamic Turbulence, Physics Today, 59(4), 4349. More in lectures on analogies. Mount GALUNGGUNG ,west Java in , August 1982, M.A. del Marmol Mount St. Helen volcano on 18 May 1980, US Geological Survey Re~20,000 Re~30,000 Re~300,000 Courtesy H. Werl, ONERA Flow patterns of a square jet of cold flow (nitrogen, top) and combusting gas (propane, bottom) exhibiting strong dependence on forcing (4 Hz, left column) at the jet exit by piezoelectric actuators. Courtesy A. Glezer (1992), Phys. Fluids, A4, p.1877. BLOOD FLOWS Where turbulence is present and where it is essential for flow regimes (e.g. stenotic vessels, aneurismas; artificial devices), transport processes (PO) and on microscales (hemolysis, thrombosis...) ? Partly Turbulent Flows
Flow past a 4 cm FLAT PLATE Re ~ 1000 Flow past a bluff body Oil slick past a WRECKED TANKER Re ~ 100 million A turbulent boundary layer flow A turbulent jet from testing a Lockheed rocket engine in the Los Angeles hills Mount St. Helen volcano on 18 May 1980, US Geological Survey It seems more meaningful to look at a turbulent flow (TF) as a whole and `indecomposable': separate components are not so meaningful, e.g. it is impossible to say that the flow is completely laminar or not at smallest scales It is a matter of principle whether one looks at TF as a whole or via various decompositions. The latter is pretty problematic (more later), just like the following statements: ...for the very smallest eddies the motion is entirely laminar BATCHELOR, G.K. 1947, Kolmogoroff's theory of locally isotropic turbulence, Proc. Camb. Phil. Soc., 43, 53359, see p. 535. .. at the small scales it becomes more difficult to argue for fundamental differenecs between these two types of flows.
SOUTHERLAND, K.B., FREDERIKSEN, R.D. AND DAHM, W.J.A. 1994 Comparisons of mixing in chaotic and turbulent flows, Chaos, Solitons and Fractals, 4(6), 10571089. see p 1065: so many random phenomena having nothing to do with turbulence TURBULENCE IS `RANDOM' , but there are W.Feller (1964) An introduction to probability theory and its applications, vol 1., p. 82, fig. 5: The record of 10,000 tosses of an ideal coin. IN LIEU OF DEFINITION OF TURBULENCE:
DEFINITION OF TURBULENCE PROBLEM It has been realized since the beginning that the problem of turbulence is a statistical problem; that is a problem in which we study instead of the motion of a given system, the distribution of motions in a family of system...It has not, however, been adequately realized just what has to be assumed in a statistical theory of turbulence, WIENER, 1939. There is probably no such thing as a most favored or most relevant, turbulent solution. Instead, the turbulent solutions represent an ensemble of statistical properties, which they share, and which alone constitute the essential and physically reproducible traits of turbiulence. VON NEUMANN, 1949 Juts like in statistical physics, the statistical approach should be adopted in turbulence theories from the outset/start due to the extreme complexity of turbulence phenomenon(a). In both cases certain statistical hypotheses are made. But the former was quite successful in making a number of important predictions, whereas the latter, with few exceptions, such as the Kolmogorov four fifths law (Kolmogorov, 1941b), was unable to produce genuine predictions based on the first principles. All the rest  in the words of P.G. Saffman  are postdictions. Apart from the abovementioned reasons for such a failure it should be mentioned that, unlike statistical physics, in turbulence neither `simple objects' (such that a collection of these objects would adequately represent turbulent flows) `to do statistical mechanics' with them, nor `right' statistical hypotheses have so far been found. The question about the very existence of both remains open....
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 Fluid Dynamics, Batchelor, Saffman, G. Batchelor

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