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 IIIIV LECTURES EXAMPLES OF CONCEPTUAL NATURE EXAMPLES IN STABILITY, TRANSITION AND IN 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 Turbulence
TRITTON 1988 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 LANDAU 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 NavierStokes 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 HOPF FOUNDING FATHERS OF FOUNDING 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 Among hydrodynamic (in)stability theory was seeking for insigts into the origins of turbulence.
der dynamischen Stabilität. W. Heisenberg 1923, Über Stabilität und Turbulenz von Flüssigkeitströmen, Ph.D. Thesis, p. 37.
* Das "Turbulenzproblem" der Hydrodynamik ist ein Problem der energetischen, nicht Das VICTOR YUDOVICH 19342006
Developed stability theory for Developed 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, Transl Providence – Rhode Island. Today Hydrodynamic Stability (HS)  both theory and experiment Today (physical and numerical)  is a vast field in its own far beyond the numerical 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 However 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 the sensitivity of the instability and transition phenomena to various details characterizing the basic flow and its environment. For example, the OrrSommerfeld equation governing the linear(ized) (in)stability contains the second derivative of the basic velocity profile. Many flows (some of the socalled 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 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. Primary instability followed by further (secondary, tertiary..) instabilities Primary (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 from one flow regime to another as manifestation of generic Transitions structural changes of the mathematical objects called phase flow and attractors in the phase space through bifurcations in a given flow geometry. Partly turbulent flows are not easily `fit' in this picture. A special feature of Partly 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 of the processes by which flows become turbulent as contrasted to Diversity (at least) qualitative universality of all turbulent flows.. The main difference between the transition to chaos and to turbulence is that The 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. It is a common view that the origin of turbulence is in the instability of some It 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 (or a similar parameter) is small, the flow remains laminar. As the Reynolds number increases, some instability sets in, which 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 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 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, 1948). However, partly turbulent flows (a special feature of these flows is the However, 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 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 by which flows become turbulent is in part due to the sensitivity of the instability and transition phenomena to various details characterizing the basic flow and its environment. For example, the OrrSommerfeld equation governing the linear(ized) (in)stability contains the second derivative of the basic velocity profile. Many flows (some of the socalled open flows, such as flows in pipes, boundary Many layers, jets, wakes, mixing 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, 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 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 regimes, such as developed turbulence (by tendency, it is meant that unversality occurrs on the qualitative, but not necessarily on the quantitative level. 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, whereas the second property is due to the production of strain. REYNOLDS’ EXPERIMENT
 1883 Puff at Re = 1900 Mullin, 2005 After reduction of Re`down to 1750 Faisst&Eckhardt, 2003 Wedin &Kerswell, 2004 Disordered signal Contains wavelength of 1.5 D Reynolds number dependence of Reynolds 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 46) is far from looking as purely laminar.
Adapted from Dubs (1939) Adapted Maloja Pass, 2001 Maloja COEXISTENCE OF COEXISTENCE LAMINAR AND TURBULENT REGIONS IN THE SAME FLOW SAME Vortex
breakdown DOUBLE DIFFUSIVE INSTABILITY PARTLY TURBULENT FLOWS I
Coexistence of laminar and turbulent regions in the same flow COEXISTENCE COEXISTENCE OF LAMINAR AND TURBULENT REGIONS IN THE SAME FLOW SAME
A turbulent jet turbulent from testing a Lockheed rocket engine in the Los Angeles hills COEXISTENCE COEXISTENCE OF LAMINAR AND TURBULENT REGIONS IN THE SAME FLOW SAME
Mount St. Helen Mount volcano on 18 May 1980 PARTLY TURBULENT FLOWS II
Coexistence of laminar and turbulent regions in the same flow
A turbulent boundary layer flow Flow past a bluff body COEXISTENCE OF COEXISTENCE LAMINAR AND TURBULENT REGIONS IN THE SAME FLOW SAME
Turbulent spots The front velocity is too small to explain the spot spreading. It is a common view that the origin of turbulence is in the instability of some basic laminar It 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 (or a similar parameter) is small, the flow remains laminar. As the Reynolds number increases, some instability sets in, which 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 field, but also at successive downstream locations of a single flow, such as the spatially developing 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 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, 1948). However, partly turbulent flows (a special feature of these flows 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 In of laminar flow into turbulent as result of the entrainment process occurring across the boundary between the two turbulent rotational ENTRAINMENT ABRUPT TRANSITION
Mount St. Helen volcano on 18 May 1980 The laminarturbulent “interface” is sharp so that fluid particles
laminar irrotational A turbulent jet from turbulent testing a Lockheed rocket engine in the Los Angeles hills found” abruptly in a turbulent environment (note the Lagrangian aspect !) “are A DNS in a box with random excitation at one wall DNS
102 ENTRAINMENT
R e num ber 101 100
10 10 10 10
0 L e n g t h s c a le
T o t a l s t r a in K in e t ic e n e r g y 2 A separate lecture later
3 1 0 4
6 1 V e lo c it y s c a le 10 10
10 10 10 8 2 0
1 0 1 E n s tro p h y 2 y  d ir e c t io n 1 2 Note the drasic drop of entsrophy across the interface as contrasted to strain
0 1 2 3 1 4 y  d ir e c t io n enstrophy Enstrophy production E N T R A I N M E N T enstrophy Enstrophy production INSTABILITY OF WHAT? INSTABILITY Oil slick past a WRECKED TANKER Re ~ 107 Flow past a 4 cm FLATPLATE Flow Re ~ 103 Re TANEDA 1963 TANEDA TANEDA 1963 TANEDA ABRUPT
TRANSITION ABRUPT
TRANSITION A vortex ring impinging a wall becomes turbulent in no time as it approaches the wall REYNOLDS 1883 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. and ROTTA, J. C.(1956) Experimenteller Beitrag zur Entstehung ROTTA, turbulenter Strömung im Rohr, Ing. Arch., 24, No. 4, 258–281. Ing. No. 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. and The transition is indeed frustratingly abrupt s l u g s Wygnanski & Champagne 1973 Durst & Unsal 2006 Durst p u f f s In a pipe flow which is held laminar at rather In large Reynolds number by special precautions, up to Re~105 (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 The ...the 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 law spectrum (and fractality). In other words significant variations down to very small scale can be produced by a single instability at much larger scale without any `cascade' of successive instabilities (Ott, 1999). An additional outcome is that nonlinearity in the Lagrangian representation cannot be interpreted in terms of some cascade. ENERGY OF DISTURBANCES ENERGY
INFINITESIMAL VERSUS FINAL The ReynoldsOrr equation for the total energy of a disturbance, ui, of an undisturbed shear flow Ui 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 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=  ∫uiuj(∂Ui)/(∂xj)dV  ∫εdV
This was interpreted (Henningson, 1996) in the sense that the disturbance energy produced by linear mechanisms is the only energy available. . In contrast the corresponding equation for enstrophy ω² In (d/dt)∫(1/2)ω²)dV = ∫{ωiuj(∂Ωi/∂xj) + ωiωjSij + ωisijΩj + ωiωjsij}dV  ∫ɛωdV.
does contain the cubic term, ωiωjsij corresponding to the selfamplification of vorticity. Hence the rate of change of the enstrophy of the disturbance Eω⁻¹dEω/dt does depend on the disturbance amplitude, and does is different for infinitesimal and finite amplitude disturbances (a similar
statement is true of the strain).
CONTRADICTION ? HOW MUCH STABILIZING HOW MUCH ARE STABILIZING FACTORS ? STABILIZING
Possible singularities Stable stratification Stable magnetic field magnetic Rotation Rotation Some other ? Some V I S C O S I T Y The mechanisms sustaining turbulence, at least some of them, are The believed to be closely related (but are not the same) to those by which laminar and transitional flows become turbulent. Apart from `natural‘ 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 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. MANY WAYS OF CREATING TURBULENT FLOWS A turbulent flow originates not necessarily out of a laminar flow with turbulent 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 NavierStokes 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.. ORIGIN OF TURBULENCE – MAIN POINTS
There is a great variety of ways/routes in which a laminar flow There becomes turbulent, just like there are many ways to establish the 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 Once origin. The reason is due to the chaotic nature and the irreversibility of turbulent flows. The main difference between the transition to chaos and to The 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. TURBULENCE VERSUS CHAOS I VERSUS
Chaotic behaviour as an intrinsic fundamental property of a wide Chaotic class of nonlinear physical systems (including turbulence) and not a and
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 The 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 Variety behaviour and a large diversity of such behaviours. Qualification of turbulence as a phenomenon characterized by a Qualification large number of strongly interacting degrees of freedom  a clear distinction between transition to turbulence and transition to chaotic behaviour (again Lagrangian chaos). TURBULENCE VERSUS CHAOS II VERSUS
As some parameter changes the number of degrees of freedom of As 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 The 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.’ TURBULENCE VERSUS CHAOS III
The idea that the essential feature of transition to turbulence is an increase of The 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 to be not precise (see Monin, 1986). However, Kolmogorov's ideas on the However, experimentalist's difficulties in distinguishing between quasiperiodic 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 in practice. There is an important difference between the number of degrees of There important number 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 dimension 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. TURBULENCE VERSUS CHAOS
A BIT OF MOCKERY BIT This was done by an ingenious This parametrization of subjectivity in the most objective way and by forcing the experiments to agree with our theory, which is universally correct INSTABILITY OF INSTABILITY PERIODIC/UNSTEADY FLOWS FLOWS
Out of scope. Just to mention an important issue Out ...
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 Fluid Dynamics, Aerodynamics, Turbulent Flows, Turbulent Flows Institute for Mathematical Sciences

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