FUNDAMENTAL AND CONCEPTUAL ASPECTS OF TURBULENT FLOWS

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FUNDAMENTAL AND CONCEPTUAL ASPECTS OF TURBULENT FLOWS Arkady Tsinober Professor and Marie Curie Chair in Fundamental and Conceptual Aspects of Turbulent Flows Institute for

Conceptual Aspects of Turbulent Flows. We absolutely must leave room for doubt or there is no progress and no learning. There is no learning without posing a question.

1 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 of the Marie Curie Chair Fundamental and Conceptual Aspects of Turbulent Flows. 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

2 LECTURES III-IV IV EXAMPLES OF CONCEPTUAL NATURE IN STABILITY, TRANSITION AND ORIGINS OF TURBULENCE Origin(s) ) of turbulence. Overview of instability and transition in various typical flows. Partly turbulent flows and abrupt transition. Forcing. TF versus chaos (onset of turbulence versus onset of chaos, routes to chaos and turbulence). Common features and qualitative differences between the two. Turbulence is a state of continuous instability TRITTON 1988

Yet not every solution of the equation of motion, even if it is exact, can actually occur in Nature.

LIFSHITS,, 1959 Kolmogorov's scenario was based on the complexity of the dynamics along the attractor rather than on its

there correspond certain solutions of the Navier-Stokes equations.

3 Yet not every solution of the equation of motion, even if it is exact, can actually occur in Nature. The flow that occur in Nature must not only obey the equations of fluid dynamics but also be stable, LANDAU AND LIFSHITS,, 1959 Kolmogorov's scenario was based on the complexity of the dynamics along the attractor rather than on its stability, ARNOLD, 1991 To the flows observed in the long run after the influence of the initial conditions has died down there correspond certain solutions of the Navier-Stokes equations. These solutions constitute a certain manifold in phase space invariant under phase flow... The notion of stability here refers to the whole manifold and not to the single motions contained in it. HOPF,, 1948

F O U N D I N G F A T H E R S O F HYDRODYNAMIC STABILITY THEORY LORD KELVIN William

Heisenberg Among the motivations of the father founders of hydrodynamic (in) in)stability

* Das "Turbulenzproblem" der Hydrodynamik ist ein Problem der energetischen, nicht der

4 F O U N D I N G F A T H E R S O F HYDRODYNAMIC STABILITY THEORY LORD KELVIN William Thomson LORD RAYLEIGH John Strutt Hermann Von Helmholtz Arnold Sommerfeld Werner Heisenberg Among the motivations of the father founders of hydrodynamic (in) in)stability theory was seeking for insigts into the origins of turbulence. * Das "Turbulenzproblem" der Hydrodynamik ist ein Problem der energetischen, nicht der dynamischen Stabilität. W. Heisenberg 1923, Über Stabilität t und Turbulenz von Flüssigkeitstr ssigkeitströmen, Ph.D. Thesis, p. 37.

VICTOR YUDOVICH 1934-2006 Developed stability theory for infinite dimensional systems in the early seventies.

5 VICTOR YUDOVICH Developed stability theory for infinite dimensional systems in the early seventies. His theory includes a justification of the linearization method in the study of the stability of solutions of Navier Stokes equations (and of abstract parabolic equations as well). V. I. Yudovich,, The linearization method in hydrodynamical stability theory, Transl.. of math. monographs 74,, AMS, Providence Rhode Island.

6 Today Hydrodynamic Stability (HS) - both theory and experiment (physical and numerical) - is a vast field in its own far beyond the field of turbulence, though the origins of turbulence are not much more clear than a century ago. However HS did produce an incredible amount of information about the bewildering variety of routes to turbulence and transitional behaviour (MORKOVIN 1969) The diversity of the processes by which flows become turbulent is in part due to the sensitivity of the instability and transition phenomena to various v details characterizing the basic flow and its environment. For example, the Orr- Sommerfeld equation governing the linear(ized) ) (in)stability( contains the second derivative of the basic velocity profile. Many flows (some of the e so-called open flows, such as flows in pipes, boundary layers, jets, wakes, mixing ing layers) are very sensitive to external noise and excitation. There are essential differences in the instability features of turbulent shear flows of different kinds (wall bounded - pipes/channels, boundary layers, and free - jets, wakes and mixing layers), thermal, multidiffusive and compositional convection, vortex breakdown and other vortex instabilities, breaking of surface and internal waves and many others. It is important that such differences occur also for the same flow geometry metry.

7 Primary Primary instability followed by further (secondary, tertiary..) instabilities (bifurcations), transition and a fully developed turbulent state either throughout the whole flow field or at successive downstream locations of a single flow. Sudden transition. Transitions Transitions from one flow regime to another as manifestation of generic structural changes of the mathematical objects called phase flow and attractors in the phase space through bifurcations in a given flow geometry. Partly Partly turbulent flows are not easily `fit' in this picture. A special s feature of these flows is the coexistence of regions with laminar and turbulent states of flow and continuous transition of laminar flow into turbulent as result of the entrainment process occurring across the boundary between the two. Diversity Diversity of the processes by which flows become turbulent as contrasted to (at least) qualitative universality of all turbulent flows.. The The main difference between the transition to chaos and to turbulence is that in the former the number of degrees of freedom remains fixed (typically small), whereas in the latter the number of degrees of freedom increases strongly with increases in the Reynolds number and/or other similar parameters.

8 It is a common view that the origin of turbulence is in the instability of some basic laminar flow(s). This is understood in the sense that any flow is started at some moment in time from rest, and as long as the Reynolds number r (or a similar parameter) is small, the flow remains laminar. As the Reynolds number n increases, some instability sets in, which is followed by further (secondary, tertiary..) instabilities (bifurcations), transition and a fully developed turbulent t state. Such sequences of events occur not only throughout the whole flow field, but also at successive downstream locations of a single flow. However, it is important to stress that transition to turbulent regime may be quite sudden. From the mathematical point of view the transitions from one flow w regime to another with increasing Reynolds number -- as we observe them in physical space -- are believed to be a manifestation of generic structural changes s of the mathematical objects called phase flow and attractors in the phase space through bifurcations in a given flow geometry (Hopf( Hopf,, 1948).

9 However, partly turbulent flows (a special feature of these flows s is the coexistence of regions with laminar and turbulent states of flow) ) are not easily `fit' in this picture. Note that in partly turbulent flows there is a continuous transition of laminar flow into turbulent as result of the entrainment process occurring across the boundary between the two. Whereas the processes by which flows become turbulent are quite diverse, all known quantitative properties of many (but not all) turbulent flows do not depend either on the initial conditions or on the history and particular r way of their creation, e.g. whether the flows were started from rest or from some other flow and/or how fast the Reynolds number was changed. The qualitative properties of all turbulent flows are the same. The diversity of the processes s by which flows become turbulent is in part due to the sensitivity of the instability ility and transition phenomena to various details characterizing the basic flow and its i environment. For example, the Orr-Sommerfeld equation governing the linear(ized) ) (in)stability( contains the second derivative of the basic velocity profile.

10 Many flows (some of the so-called open flows, such as flows in pipes, boundary layers, jets, wakes, mixing layers) are very sensitive to external noise and excitation. There are essential differences in the instability features f of turbulent shear flows of different kinds (wall bounded - pipes/channels, boundary layers, and free - jets, wakes and mixing layers), thermal, multidiffusive and compositional convection, vortex breakdown, breaking of surface and internal waves and many others. Such differences occur also for the same flow geometry, which display in words of M..V. Morkovin bewildering variety of transitional behaviour. The specific route may depend on initial conditions, level of external disturbances (receptivity), forcing, time history and other details in most of the flows. This diversity is especially distinct for the very initial l stage - the (quasi)linear(ized)) instability. Later nonlinear stages are less sensitive to such details. Hence there is a tendency to universality in strongly nonlinear n regimes, such as developed turbulence (by tendency, it is meant that unversality occurrs on the qualitative, but not necessarily on the quantitative level. l.

11 One of the important 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.

12 REYNOLDS EXPERIMENT Puff at Re = 1900 After reduction of Re`down to Faisst&Eckhardt,, 2003 Wedin &Kerswell,, 2004 Mullin, 2005 Disordered signal Contains wavelength of 1.5 D

Reynolds number dependence of the friction

flow visualization at particular values of

follows the laminar law, 64/Re, but the flow

13 Reynolds number dependence of the friction factor in a circular pipe with corresponding flow visualization at particular values of Reynolds number. Note that there is a range of values of Reynolds number in which the friction factor follows the laminar law, 64/Re, but the flow pattern (pictures 4-6) 4 is far from looking as purely laminar. Adapted from Dubs (1939)

14 Maloja Pass, 2001

16 COEXISTENCE OF LAMINAR AND TURBULENT REGIONS IN THE SAME FLOW Vortex breakdown

18 DOUBLE DIFFUSIVE INSTABILITY

19 PARTLY TURBULENT FLOWS I Coexistence of laminar and turbulent regions in the same flow

20 COEXISTENCE OF LAMINAR AND TURBULENT REGIONS IN THE SAME FLOW A turbulent jet from testing a Lockheed rocket engine in the Los Angeles hills

21 COEXISTENCE OF LAMINAR AND TURBULENT REGIONS IN THE SAME FLOW Mount St. Helen volcano on 18 May 1980

22 PARTLY TURBULENT FLOWS II Coexistence of laminar and turbulent regions in the same flow A turbulent boundary layer flow Flow past a bluff body

23 COEXISTENCE OF LAMINAR AND TURBULENT REGIONS IN THE SAME FLOW Turbulent spots The front velocity is too small to explain the spot spreading.

24 It is a common view that the origin of turbulence is in the instability of some basic laminar flow(s). This is understood in the sense that any flow is started at some s moment in time from rest, and as long as the Reynolds number (or a similar parameter) is small, s the flow remains laminar. As the Reynolds number increases, some instability sets in, which h is followed by further (secondary, tertiary..) instabilities (bifurcations), transition and a fully developed turbulent state Such sequences of events occur not only throughout the whole flow w field, but also at successive downstream locations of a single flow, such as the spatially developing eloping flows as shown in the figures 1.4 and 1.5 of chapter 1. However, it is important to stress that transition to turbulent regime may be quite abrupt. For example, this may happen in pipe flows under certain conditions or in the process involving the impingement of a laminar vortex ring upon a rigid wall From the mathematical point of view the transitions from one flow w regime to another with increasing Reynolds number -- as we observe them in physical space -- are believed to be a manifestation of generic structural changes of the mathematical objects called phase flow and attractors in the phase space through bifurcations in a given flow geometry (Hopf( Hopf,, 1948). However, partly turbulent flows (a special feature of these flows s is the coexistence of regions with laminar and turbulent states of flow) are not easily `fit' in this picture. Note that In partly turbulent flows there is a continuous transition of laminar flow into turbulent as result of the entrainment process occurring across the boundary between the two

laminar irrotational turbulent rotational

laminar- turbulent interface is sharp so

25 laminar irrotational turbulent rotational A turbulent jet from testing a Lockheed rocket engine in the Los Angeles hills ENTRAINMENT ABRUPT TRANSITION The laminar- turbulent interface is sharp so that fluid particles (note the Lagrangian aspect!) are found abruptly in a turbulent environment Mount St. Helen volcano on 18 May 1980

26 ENTRAINMENT A DNS in a box with random excitation at one wall 10 2 Re number A separate 10 1 lecture Kinetic energy Length scale Totalstrain later V elocity scale E n s tro p h y 2 3 y-direction Note the drasic drop of entsrophy across the interface as contrasted to strain y-direction

27 enstrophy Enstrophy production E N T R A I N M E N T enstrophy Enstrophy production

28 INSTABILITY OF WHAT?

29 Oil slick past a WRECKED TANKER Re ~ 10 7 Flow past a 4 cm FLATPLATE Re ~ 10 3

30 TANEDA 1963

31 TANEDA 1963

32 ABRUPT TRANSITION

33 ABRUPT TRANSITION A vortex ring impinging a wall becomes turbulent in no time as it approaches the wall

34 REYNOLDS 1883

35 ABRUPT TRANSITION The transition between laminar and turbulent flows at the beginning ing and end of the turbulent region is abrupt relative to its duration. ROTTA, J. C.(1956) Experimenteller Beitrag zur Entstehung turbulenter Strömung im Rohr, Ing. Arch., 24, No. 4,

36 ABRUPT TRANSITION The transition between laminar and turbulent flows at the beginning ing and end of the turbulent region is abrupt relative to its duration. The transition is indeed frustratingly abrupt s l u g s Wygnanski & Champagne 1973 Durst & Unsal 2006 p u f f s

37 In a pipe flow which is held laminar at rather large Reynolds number by special precautions, up to Re~10 5 (PFENNIGER 1961), and then subject to disturbance of finite amplitude the transition to turbulent regime is quite (frustratingly) abrupt: the flow becomes turbulent extremely fast ("in no time ) ) That is a large range of scales is created in one shot without any cascade. The problem goes back to Townsend (1951):...the postulated process differs from the ordinary type of turbulent energy transfer being fundamentally a single process. There are examples of flows for which is was shown that a single (!) linear instability results in a power p law spectrum (and fractality). In other words significant variations down to very small scale can be produced by a single instability at much m larger scale without any `cascade' of successive instabilities (Ott( Ott, 1999). An additional outcome is that nonlinearity in the Lagrangian representation cannot be interpreted in terms of some cascade.

39 ENERGY OF DISTURBANCES INFINITESIMAL VERSUS FINAL

40 The Reynolds-Orr equation for the total energy of a disturbance, u i, of an undisturbed shear flow U i does not contain cubic terms in the disturbance (corresponding to the nonlinear terms in NSE). This means that the rate of change of the energy of the disturbance E ¹dE/dt E does not depend on the disturbance amplitude, i.e. in some sense,, is the same for infinitesimal and finite amplitude disturbances. A (d/dt) (1/2)u²dV= - u i u ( U j i )/( x j )dv - εdv This was interpreted (Henningson( Henningson,, 1996) in the sense that the disturbance energy produced by linear mechanisms is the only energy available..

41 In contrast the corresponding equation for enstrophy ω² (d/dt) (1/2)ω²)dV = {ω i u ( Ω j i / x j ) + ω i ωs j ij + ω i s ij ijω j + ω i ω j s ij does contain the cubic term, ω i ω j sij }dv - ɛ ω dv. ij }d corresponding to the self- amplification of vorticity.. Hence the rate of change of the enstrophy of the disturbance E ω ¹dE ω /dt does depend on the disturbance amplitude, and is different for infinitesimal and finite amplitude disturbances (a similar statement is true of the strain). CONTRADICTION?

42 HOW MUCH STABILIZING ARE STABILIZING FACTORS? Possible singularities Stable stratification magnetic field Rotation Some other?

43 V I S C O S I T Y

44 The mechanisms sustaining turbulence, at least some of them, are believed to be closely related (but are not the same) to those by which laminar and transitional flows become turbulent. Apart from `natural ural ways resulting from instabilities, turbulent flows can be produced by `brute force which can be random or deterministic. However, random forcing of integrable systems (Burgers, Korteveg de Vries,, restricted Euler) does not produce what is called genuine (i.e. NSE turbulence in Euler setting). At small enough Reynolds numbers, the flow produced by deterministic forcing of NSE is not random, it is laminar, but t a flow produced by random forcing, though random, is in many respects trivial (it is linear/laminar), e.g. there is no interaction between its degrees of freedom/modes (at Re=0 the NSE are integrable). However, such a flow is not trivial (and not integrable) ) in Lagrangian contexts and possess complex phenomena in evolotion of passive objects.

45 MANY WAYS OF CREATING TURBULENT FLOWS A turbulent flow originates not necessarily out of a laminar flow w with the same geometry. It can arise from any initial state including a `turbulent' one, such as random initial conditions in direct numerical simuations of the Navier-Stokes equations. That is, the transition from laminar to turbulent regime is not the only causal relation. This problem is related to a somewhat `philosophical' question on whether flows become or whether they just are turbulent, and to the unknown properties of the phase flow, attractors and related matters..

46 ORIGIN OF TURBULENCE MAIN POINTS There is a great variety of ways/routes in which a laminar flow becomes turbulent, just like there are many ways to establish the e same turbulent flow. In other words, the view that turbulent flows always develop from the laminar ones is too narrow. Once a flow becomes turbulent, it seems impossible to find out its i origin. The reason is due to the chaotic nature and the irreversibility ibility of turbulent flows. The main difference between the transition to chaos and to turbulence is that in the former the number of degrees of freedom remains fixed, whereas in the latter the number of degrees of freedom increases strongly with increases in the Reynolds number and/or other similar parameters.

47 TURBULENCE VERSUS CHAOS I Chaotic behaviour as an intrinsic fundamental property of a wide class of nonlinear physical systems (including turbulence) and not a result of external random forcing or errors in the input of the numerical simulation on the computer or the physical realization in the laboratory. The nonlinear systems and the equations describing them produce an apparently random output `on their own', `out of nothing' -- it is their very nature. Variety of qualitatively different systems exhibiting such a behaviour and a large diversity of such behaviours. Qualification of turbulence as a phenomenon characterized by a large number of strongly interacting degrees of freedom - a clear distinction between transition to turbulence and transition to chaotic c behaviour (again Lagrangian chaos).

48 TURBULENCE VERSUS CHAOS II As some parameter changes the number of degrees of freedom of low dimensional chaotic systems remains the same, only the character of the interaction of these degrees of freedom changes. Their dynamics is essentially temporal: it is chaotic but rather `simple' - the chaos is temporal only. The number of excited degrees of freedom in fluid flows increases es rapidly with the Reynolds number. This steep increase in the number of excited degrees of freedom results in a qualitative change in the behaviour of the flow - it is chaotic as well, but qualitativey different, much more complicated kind of chaos - it is both temporal and spatial and high dimensional: more is different.

49 TURBULENCE VERSUS CHAOS III The idea that the essential feature of transition to turbulence is an increase of the number of excited degrees of freedom dates back to Landau (1944) and Hopf (1948) and is correct, though the details of their scenario appeared ared to be not precise (see Monin,, 1986). However, Kolmogorov's ideas on the experimentalist's difficulties in distinguishing between quasi- periodic systems with many basic frequencies and genuinely chaotic systems have not yet been formalized (Arnold, 1991). In other words it is very difficult if not impossible to make such a distinction inction in practice. There is an important difference between the number of degrees of freedom rougly proportional to the number of ordinary differential equations necessary to adequately represent a system described by partial differential equations (NSE) and the dimension of the attractor of the system (if such exists). In a particular dynamical system, the former is obviously fixed and is independent of the parameters of the system, whereas the latter is changing with the parameters but is bounded. In turbulence both are essentially increasing with the Reynolds number and become very large at large Reynolds number.

50 TURBULENCE VERSUS CHAOS A BIT OF MOCKERY This was done by an ingenious parametrization of subjectivity in the most objective way and by forcing the experiments to agree with our theory, which is universally correct

51 INSTABILITY OF PERIODIC/UNSTEADY FLOWS Out of scope. Just to mention an important issue

FUNDAMENTAL AND CONCEPTUAL ASPECTS OF TURBULENT FLOWS

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