Some Aspects of Strain Vorticity and Material Element Dynamics as Measured with 3D Particle Tracking Velocimetry in a Turbulent Flow

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Diss. ETH No.: 14893

Some Aspects of Strain, Vorticity and Material Element Dynamics as Measured with 3D Particle Tracking Velocimetry in a Turbulent Flow A dissertation submitted to the Swiss Federal Institute of Technology Zürich for the degree of Doctor of Technical Sciences

presented by

Beat Lüthi Dipl. Masch.-Ing. ETH Zürich born on April 30, 1971 citizen of Zürich (ZH) examination committee: Prof. Dr. Wolfgang Kinzelbach, examiner Prof. Dr. Thomas Rösgen, co-examiner Prof. Dr. Arkady Tsinober, co-examiner Dr. Ulrich Burr, co-examiner

2002

2 Abstract The full set of velocity derivatives, Uij = ∂ui /∂xj , is measured experimentally in a Lagrangian way along particle trajectories in a turbulent flow. This is achieved by producing a suitable turbulent flow that is continuously forced electromagnetically and by applying and further developing a 3D Particle Tracking Velocimetry (3D-PTV) technique. The critical steps that allowed to go from 3D-PTV measurements of velocity to measurements of velocity derivatives are (i) an increased rate of image recording, from 30Hz to 60Hz, (ii) an improved ’spatio-temporal’ particle tracking algorithm and (iii) a weighted polynomial fitting procedure, applied to interpolated velocity derivative signals along particle trajectories, using relative divergence obtained from ∂ui /∂xi as a criterion to choose proper weights for each particle’s contribution to the fit. The quality of interpolated ∂ui /∂xj is estimated according to relative divergence for three reasons: (i) divergence reflects incompressibility of water - a physical property which otherwise is not used in any of the procedures, (ii) it treats flow situations with small and large ∂ui /∂xi in an equally fair way and (iii) it has no singularities, contrary to some other kinematic measures. It is demonstrated that the use of weighted polynomials clearly enhances the quality of the measured and interpolated components of ∂ui /∂xj . The procedure to obtain the spatial velocity derivatives, ∂ui /∂xj , involves a chain consisting of measurements, image processing, linear interpolations and polynomial fitting. Therefore, a number of checks which are based on precise kinematic relations are performed. They show that on a qualitative level the 3D-PTV measurements are correct and that the statistical set of data that ’survived’ processing and the applied quality check, based on relative divergence, are representative of the actual physical flow. In a study on enstrophy and strain and on the self-amplifying nature of their dynamics, known characteristic properties of turbulent flows are reproduced; most of which are not observed in a Gaussian flow field, i.e. are specific of genuine fluid turbulence. These are the positive skewness of the intermediate eigenvalue of the rate of strain tensor, sij , hΛ2 i > 0, the predominance of vortex stretching over vortex compression, hω i ω j sij i > 0, a predominant alignment of vorticity, ω, with the intermediate principal axis of strain, λ2 and the fact that regions in a turbulent flow with intense strain, s2 = sij sij , and more so regions with intense enstrophy, ω 2 , occupy only a relatively small portion of the domain, which is reflecting the intermittent nature of strain and enstrophy. With the use of a new representation of ∂ui /∂xj in the space spanned by {Q, ω i ω j sij , sij sjk ski } a mean cyclic evolution is observed consisting of intense events of strain, enstrophy production, concentrated vorticity and enstrophy destruction, along with strain production. These findings are a first indication that turbulence is more likely to be understood if it is looked at not as a cascade but as a succession of strong stretching and folding processes of fluid blobs. Lagrangian measurements of Dω 2 /Dt reveal that there is no direct relation between ω i ω j sij and 1/2 Dω 2 /Dt. In the context of the concept of an ’energy cascade in physical space’, this finding questions the argument commonly brought forward that vortex stretching is the primary mechanism producing smaller scales, and 2 hence is responsible for the cascade. The role played by viscosity in enstrophy changes, 21 Dω Dt , is found to be just as important as vortex stretching, ω i ω j sij . These viscous effects might be related to vortex reconnection which has the capability to change the topology of a turbulent flow. We arrive at the conjecture that whereas viscosity in combination with strain is responsible for dissipation, viscosity in combination with vorticity is associated with continuous change of the topology of turbulent flow by means of vortex reconnection. After these investigations on enstrophy and strain dynamics, which are related to the dissipative nature of turbulence, the stretching and folding of material elements, which in turn are related to the diffusive characteristics of turbulent flows, are studied. Results on the evolution in time of material lines, l, as compared to vortex lines, ω, are presented. In agreement with literature it is found that the mean stretching rates of material lines are well above the mean stretching rates of vorticity. Material line growth is governed by stretching, whereas growth of enstrophy is a combined effect of vortex stretching and viscous effects. Further, material lines have a strong tendency to align with the most positive principal axis of strain, λ1 , in contrast to vorticity which tends to align with the intermediate principal axis, for which there are indications that viscosity is primarily responsible. Special material lines which at some time, t0 , are perfectly aligned with vorticity, l0 = ω 0 , appear ’reluctant’ to act like ’proper’ material lines, in the sense that their initial λ2 alignment is persistent over a considerable time of τ η ∼ 7 and the λ1 -alignment develops only very slowly. Two effects that might be partly responsible for this persistence are identified: The

3 ’Λ2 cos (l, λ2 ) > Λ1 cos (l, λ1 )’ effect which is found to be active in high strain regions and when l is in the proximity of the intermediate eigenvector, λ2 , and the effect of vorticity ’acting back’ on the flow which is observed to be correlated to enstrophy, ω 2 . Regarding material surfaces, it is found that material surface normals, N , after a period of transition of τ η ∼ 2 align with the most compressing principal strain axis, λ3 . The mean stretching rates for material surfaces are measured to be only slightly higher than those for material lines and we note that after a time of τ η ∼ 7 both, material lines and surfaces, have roughly doubled their magnitude. It is concluded that due to the observed behavior of material elements, diffusivity across material surfaces is significantly increased. To study the rotation of material surfaces, the rotation of their carriers, namely the rotation of material volumes, defined by the eigenvectors, wi , of the Cauchy-Green tensor W, is measured. The rotation, ω 2w , of material volumes is obtained by looking at the to the rotation of the eigen-frame, w1,2,3 . It is found that the magnitude of ω 2w is comparable ¡ ¢ magnitude of enstrophy, ω 2 , and that ω 2w can be related in some ways with Q = 14 ω 2 − 2s2 . When surface stretching is of low intensity, surfaces are re-oriented via rotation of their carriers - the infinitesimal material volumes. This presumably results in material volume folding. The two processes of stretching and folding are important for turbulent diffusivity. They are driven by the combined effects of enstrophy and strain and their processes of self-amplification.

4 Zusammenfassung Der volle Tensor der Geschwindigkeits-Ableitungen, Uij = ∂ui /∂xj , wird experimentel und auf Lagrange’sche Art entlang von Partikelbahnkurven in einer turbulenten Strömung gemessen. Dies gelingt durch Erzeugung einer geeigneten Strömung mittels kontinuierlicher, elektro-magnetischer Krafteinleitung in die Strömung und durch Anwendung und Weiterentwicklung einer 3D-Particle-Tracking-Velocimetry (3D-PTV) Messtechnik. Die entscheidenden Schritte, die es erlauben, von einer reinen Geschwindigkeitsmessung zu einer Geschwindigkeitsableitungs-Messung überzugehen, sind (i) eine erhöhte Bildaufzeichnungsrate von 30Hz auf 60Hz , (ii) ein verbesserter spatio-temporal particle tracking Algorithmus und (iii) eine gewichtete ’polynomial fitting’ Prozedur, welche auf die interpolierten Geschwindigkeitsableitungs-Signale entlang der Bahnkurven angewendet wird. Zur Bestimmung der einzelnen Partikelbeiträge zum fitting wird ein Kriterium der relativen Divergenz, gebildet aus ∂ui /∂xi , benutzt. Die Qualitätsbestimmung der interpolierten ∂ui /∂xj mit Hilfe der relativen Divergenz geschieht hauptsächlich aus drei Gründen: (i) Die Inkompressibilität des Wassers wird in der Divergenzfreiheit wiederspiegelt - eine physikalische Eigenschaft, die sonst in keiner der Prozeduren genutzt würde. (ii) Strömungssituation mit grossen und kleinen ∂ui /∂xi werden gleich fair behandelt, und (iii) im Gegensatz zu anderen kinematischen Qualitätsmassen, weist die relative Divergenz keine Singularitäten auf. Es wird gezeigt, dass die gewichtete ’polynomial fitting’ Prozedur die Qualität der gemessenen und interpolierten Komponenten von ∂ui /∂xj eindeutig verbessert. Die notwendige Prozedur, um Geschwindigkeitsableitungen, ∂ui /∂xj , mit der gewünschten Genauigkeit zu erhalten, ist eine lange Verkettung von Messung, Bildverarbeitung, linearer Interpolation und ’polynomial fitting’. Deshalb werden eine Reihe von Kontrollen durchgeführt, die auf exakten, kinematischen Gleichungen beruhen. Diese Kontrollen zeigen, dass die 3D-PTV Messungen qualitativ korrekt sind und weiter, dass das Set der statistischen Daten, welches alle Prozeduren und den Qualitätscheck überlebt hat, die eigentliche physikalische Strömung immer noch repräsentiert. In einer Studie über enstrophy und strain und über die Natur ihrer selbstverstärkenden Dynamik werden bekannte charakteristische Eigenschaften turbulenter Strömungen reproduziert. Es handelt sich dabei um Eigenschaften, die ausschliesslich in der Turbulenz auftreten und die in einer Gauss’schen Strömung nicht zu beobachten sind. Es sind dies (i) der positive Mittelwert des mittleren Eigenwerts, hΛ2 i > 0, des Spannungsraten-Tensors, sij = 21 (∂ui /∂xj + ∂uj /∂xi ), (ii) die Dominanz der Wirbelstreckung über Wirbelkompression, hω i ω j sij i > 0, (iii) eine Ausrichtung von vorticity, ω , vorwiegend mit dem mittleren Eigenvektor, λ2 , des Tensors sij und (iv) die Tatsache, dass in turbulenten Strömungen Gebiete mit intensivem strain, s2 = sij sij , und - sogar noch ausgeprägter - mit intensiver enstrophy, ω 2 , nur einen relativ kleinen Anteil der gesamten Domäne ausmachen. Dies wiederspiegelt die Tatsache, dass strain und enstrophy von intermittierender Natur sind. Mit Hilfe einer neuen Darstellung von ∂ui /∂xj in einem {Q, ω i ω j sij , sij sjk ski }-Raum kann eine im Mittel zyklische Abfolge von speziellen ∂ui /∂xj -Zuständen über verschiedene Stadien hinweg beobachtet werden. Diese Zustände sind der Reihe nach: intensiver strain, Produktion von enstrophy, konzentrierte vorticity, Zerstörung von enstrophy und Produktion von strain. Das ist ein erster Hinweis darauf, dass die Turbulenz wohl eher verstanden wird, wenn man sie nicht als eine Kaskade, sondern mehr als eine Abfolge von starken Streckungs- und Faltungs-Erscheinungen von Flüssigkeitsballen betrachtet. Lagrange’sche Messungen von Dω 2 /Dt zeigen, dass es keinen direkten Zusammenhang zwischen ω i ω j sij und 1/2 Dω2 /Dt gibt. Im Kontext des Konzepts einer Energie-Kaskade im physikalischen Raum stellen diese Resultate das gemeinhin vorgebrachte Argument in Frage, dass Wirbelstreckung der Hauptmechanismus zur Erzeugung von kleinen Skalen - und also auch einer Kaskade - sei. Es zeigt sich, dass die Rolle 2 der Viskosität in der Änderungen von enstrophy, 12 Dω Dt , mindestens so wichtig ist, wie diejenige der Wirbelstreckung. Diese viskosen Effekte können mit dem Phänomen der vortex reconnection zusammenhängen, welches die Möglichkeit besitzt, die Topologie einer turbulenten Strömung zu verändern. Wir gelangen zur Vermutung, dass Viskosität, die in Kombination mit strain für Dissipation verantwortlich ist, in Kombination mit vorticity dafür verantwortlich sein kann, der Turbulenz ihren drei-dimensionalen Rahmen zu geben; einen Rahmen, der die Möglichkeit besitzt durch vortex reconnection seine Gestalt fortlaufend zu verändern. Nach diesen Untersuchungen über die Dynamik von enstrophy and strain im Zusammenhang mit der dissipativen Eigenschaft der Turbulenz, werden das Strecken und Falten von Materialelementen untersucht, welche

5 mehr mit der diffusiven Natur turbulenter Strömungen im Zusammenhang stehen. Es werden Resultate über die zeitliche Entwicklung von Materiallinien, l, gezeigt und verglichen mit derjenigen von Wirbellinien, ω . In Übereinstimmung mit Resultaten aus der Literatur wird festgestellt, dass die mittleren Streckungsraten von Materiallinien leicht höher sind als diejenigen von Wirbellinien. Das Wachstum von Materiallinien ist bestimmt durch Streckung, im Gegensatz zu Wirbellinien, deren Veränderungen aus der Kombination von Wirbelstreckung und viskosen Effekten hervorgeht. Materiallinien weisen eine starke Tendenz auf, sich mit der Achse der grössten Hauptspannung, λ1 , des Tensors sij auszurichten, im Gegensatz zu vorticity, welche sich vornehmlich mit der mittlerern Hauptspannungsachse, λ2 , ausrichtet. Es gibt Hinweise dafür, dass dies hauptsächlich der Viskosität zuzuschreiben ist. Spezielle Materiallinien, welche zu einem bestimmten Zeitpunkt t0 perfekt mit vorticity ausgerichtet sind, l0 = ω 0 , scheinen sich nur sehr zögerlich wie ’richtige’ Materiallinien zu verhalten. Wir beobachten, wie sich die ursprüngliche λ2 -Ausrichtung ’hartnäckig’ über einen Zeitraum von τ η ∼ 7 zu halten vermag, und wie sich eine λ1 -Ausrichtung nur sehr langsam einstellt. Zwei Effekte werden identifiziert, die wohl zum Teil für diese ’Hartnäckigkeit’ verantwortlich sind. Es sind dies zum einen der ’Λ2 cos (l, λ2 ) > Λ1 cos (l, λ1 )’-Effekt, hauptsächlich aktiv in Gebieten mit starkem strain und starker l − λ2 -Ausrichtung, und - stark korreliert mit enstrophy - die Rückwirkung von vorticity auf die sie umgebende Strömung. Betreffend Materialflächen wird festgestellt, dass sich Flächennormalen, N , nach einer Übergangszeit von τ η ∼ 2 mit der grössten komprimierenden Hauptspannungsachse, λ3 , des Tensors sij ausrichten. Die mittlere Streckungsrate von Materialflächen ist nur leicht grösser als diejenige der Materiallinien und wir stellen fest, dass nach einer Zeit von τ η ∼ 7 sowohl Materiallinien als auch Materialflächen ihre Grösse ungefähr verdoppeln. Wir ziehen unter anderem daraus den Schluss, dass durch das beobachtete Verhalten von Materialelementen die Diffusion über Materialflächen entscheidend verstärkt wird. Um die Rotation von Materialflächen zu untersuchen wird die Rotation ihrer Träger, der Materialvolumen, gemessen. Materialvolumen sind definiert durch den Raum, den die Eigenvektoren, wi , des Cauchy-Green Tensors, W , aufspannen. Die Rotation, ω 2w , eines Materialvolumens wird bestimmt, indem die Rotation des Eigenrah2 mens, w1,2,3 , gemessen wird. Es wird festgestellt, dass Grösse und Verteilung von ¡ ω2w sehr2 ¢vergleichbar mit 1 2 denjenigen von enstrophy sind, und dass ω w zu einem gewissen Grad mit Q = 4 ω − 2s im Zusammenhang steht. Hauptsächlich dann, wenn die Streckung von Materialflächen gering ist, werden grosse Rotationen von Materialvolumen beobachtet. Diese Rotationen führen vermutlich zur Faltung von Materialvolumen. Die zwei beobachteten Prozesse, Streckung und Faltung, sind wichtig für die turbulente Diffusion und resultieren aus den kombinierten Effekten von enstrophy und strain und deren selbstverstärkenden Prozessen.

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People After I finished writing this thesis I felt it to be appropriate to put a list of those people at the beginning of the text, that are responsible - in one way or another - that I might be getting the degree of a Ph.D. sometime soon. I feel deeply thankful to all of them.

In order of probable appearance in the sequence of events: Margrit Lüthi Jürgen Lüthi Hans Asper Erdal Karamuk George Raeber Joël Schlinger Michael Hauser Matthias Machacek Roland Reber Rösgen Thomas Carlos Härtel Petros Koumoutsakos Kleiser Leonard Albert Gyr Wolfgang Kinzelbach Arkady Tsinober Ulrich Burr Barbara Tänzler ALL of IHW team Hannes Bühler Thomy Keller Toni Blunschi Xaver Studerus René Weber Jochen Willneff Schneuwly Bruno Tony Leonard

Contents 1 Introduction 1.1 Velocity derivatives . . . . . . . . . . . . . . . 1.2 Material elements and Lagrangian description 1.3 Overview of previous Lagrangian studies . . . 1.3.1 Experimental . . . . . . . . . . . . . . 1.3.2 Numerical . . . . . . . . . . . . . . . . 2 Method 2.1 Experimental setup . . . . . . . . . . . . . . . 2.1.1 Forcing . . . . . . . . . . . . . . . . . 2.1.2 First experiment . . . . . . . . . . . . 2.1.3 Second experiment . . . . . . . . . . . 2.2 3D Particle Tracking Velocimetry . . . . . . . 2.3 Trajectory processing . . . . . . . . . . . . . . 2.4 Velocity derivatives . . . . . . . . . . . . . . . 2.4.1 Spatial velocity derivatives . . . . . . 2.4.2 Temporal velocity derivatives . . . . . 2.4.3 Velocity derivatives along trajectories 2.5 Checks and verification of procedures . . . . . 2.5.1 Statistical checks . . . . . . . . . . . . 2.5.2 Single Point Checks . . . . . . . . . . 2.5.3 Multi Point Checks . . . . . . . . . . . 2.5.4 Checks along trajectories . . . . . . . 2.5.5 Influence of trajectory length . . . . . 2.5.6 Summary on checks . . . . . . . . . .

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3 Results 3.1 Governing equations and relevant scales . . . . . . 3.2 Overview on vorticity and strain . . . . . . . . . . 3.3 Some universal features of vorticity and strain . . . 3.4 Q − ωi ωj sij − sij sjk ski probability orbital . . . . . 3.4.1 Construction . . . . . . . . . . . . . . . . . 3.4.2 Description . . . . . . . . . . . . . . . . . . 3.4.3 Strain . . . . . . . . . . . . . . . . . . . . . 3.4.4 Enstrophy . . . . . . . . . . . . . . . . . . . 3.5 Cyclic mean process . . . . . . . . . . . . . . . . . 3.6 Enstrophy dynamics . . . . . . . . . . . . . . . . . 3.7 Strain dynamics . . . . . . . . . . . . . . . . . . . 3.8 Material Elements - Integrated rate of strain tensor

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CONTENTS 3.9 Randomly oriented material lines . . . . . . . 3.10 Differences between material and vortex lines 3.10.1 λ2 -alignment . . . . . . . . . . . . . . 3.10.2 Viscous change of enstrophy . . . . . . 3.11 Evolution of material surfaces and volumes . 3.11.1 Stretching . . . . . . . . . . . . . . . . 3.11.2 Folding . . . . . . . . . . . . . . . . .

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4 Conclusion

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5 Bibliography

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List of Figures 2.1

2.2

2.3

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2.5

2.6

2.7

2.8

Schematic of the experimental facility and coordinate system (x,y,z). Electromagnetic forcing from two opposite walls produces swirls over each magnet. At short distances of the walls the flow becomes three dimensional and towards the observation volume fully turbulent. The flow is recorded by four CCD cameras at recording rates of first 30Hz and - in a later stage - at 60Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Schematic of electromagnetic forcing. A current density field (arrows, top left) interacts over each magnet with the magnetic field (arrows, top right) producing Lorentz forces (gray scale, top left). This forcing results in torus-like swirls over each magnet.

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The first experimental facility is set up inside a 50` tank. For forcing two walls, each with a 4×4 permanent magnet array, are facing each other at a distance of 17cm. A DC electric current of 7A is a applied to two copper plates which cover the magnet arrays (not shown here), resulting in a current density of ∼70 A/m2 . As a calibration target a regular array of 7×9 points with a grid distance of 2mm is used. . . . . . .

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1/200

The flow is recorded with four JAI M10 CCD progressive scan monochrome cameras with 8bit/pixel resolution. The pixel resolution is 640×480. The cameras are positioned manually. Each camera is mounted on a linear stage with a travel range of 20mm. The stages are used to focus the macro lenses onto the observation volume. To align the camera axis the linear stages are mounted on ball bearings. . .

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The setup of the second and final experiment is shown. The flow is again produced with electromagnetic forcing as schematically shown in figure 2.1, but now in a much smaller volume of 120×120×140 mm3 . The observation volume is illuminated with a continuous Argon-Ion laser. Before entering the flow through the bottom of the tank the laser beam is expanded by two cylindrical lenses and reflected by a mirror. In the back the four recoding CCD cameras, mounted on linear stages and ball bearings, can be seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A schematic of the calibration block, where x, y are the positions and ω, ϕ the rotation angles of each camera, as determined from the calibration procedure for the second experiment, are shown. z , the distance of the cameras from the tank is between 59mm and 63mm. The rotations, κ, around the z-axis are between 2◦ to 4◦ . . . . . .

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At a recording rate of 60Hz the signal from camera to frame grabber is transmitted through two cables. As a negative effect this results in different signal magnitudes for even and odd image lines. This deficiency is corrected by mapping each grayscale value from odd lines with the corresponding mean of upper and lower even line values. Here the effect of grayscale-correction is shown for one examplary image, before and after the procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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To get a qualitative idea of the flow as obtained from 3D-PTV some trajectories of the second experiment are shown. The selected trajectories begin within the first second of recording and can be tracked over 0.3s or longer. . . . . . . . . . . . . . .

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LIST OF FIGURES 2.9

The frequency response of the ’moving’ cubic spline procedure - used to filter the position- and thus also the velocity- and acceleration-signals along a particle trajectory - is shown. The procedure effectively acts as a low pass filter, with a cut-off frequency for velocity and acceleration signals of ∼ 10Hz and ∼ 7Hz respectively. The response for signal frequencies up to 5Hz is above 0.95. Damping is maximal at the Nyquist frequency of 30Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.10 The effect of the ’moving-spline’ procedure is demonstrated on a qualitative level. Two renderings of an arbitrarily selected trajectory from the second experiment are shown. For this trajectory a particle is tracked through 80 time steps. The arrows represent Lagrangian acceleration, a. The trajectory on the left (blue) has been processed with the ’moving-spline’ procedure while a of the trajectory on the right (red) is obtained using a central approximation only. . . . . . . . . . . . . . . . . . .

39

2.11 Velocity and acceleration components ui , ubi (a) and ai , abi (b) of the same trajectories as shown in figure 2.10 are plotted over time. Black lines are used for the components as obtained from central approximations and red lines for the components as obtained from the low pass filtering ’moving-spline’ procedure. The necessity for filtering becomes especially clear in b) where the unprocessed acceleration fluctuations appear completely uncorrelated in time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.12 The qualitative effect of different procedures to obtain velocity derivatives along a particle trajectory is shown. In (a) for exemple the behavior of the component ∂u3 /∂x3 along the same trajectory that is used above for figures 2.10 and 2.11 is shown. The inset in (a) shows the weight, wi , as a function of relative divergence. Figure (b) shows the relative divergences as obtained from the different procedures along the corresponding trajectory. Only the procedure with ’moving-spline’ and weighted polynomial fitting produces qualitatively acceptable results. . . . . . . . . . . . . . .

45

2.13 The number of trajectories of a certain length, the number of points belonging to a certain trajectory length and the summation of points with a trajectory length equal to or greater than a specific trajectory length are shown for experiment 30Hz and 60Hz. The 60Hz set is always at least twice as large as the 30Hz set. With increasing trajectory length tracking becomes more difficult. . . . . . . . . . . . . . . . . . . . .

47

2.14 The mean kinetic energy, < k >= 21 ui ui (a), and the mean square of Lagrangian acceleration, < a2 >= ai ai (b), are conditioned on their trajectory length. The mean of both quantities decreases with increasing trajectories lengths, indicating that trackability is sensitive especially to a2 . The polynomial fitting and the selection according to relative divergence have a neglectable influence. . . . . . . . . . . . . .

49

2.15 The PDFs of kinetic enrgy, k (a), and squared Lagrangian acceleration, a2 (b), for trajectory lengths of τ η = 1 and τ η = 4 are shwon. Apparently mainly the pdf for a2 is affected by longer trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

∂u

∂uk j i 2.16 The joint PDFs of ∂u ∂xi versus ∂xj + ∂xk (here no summation over i, j, k) for three cases, 60Hz with full numerical processing, 60Hz, not using the polynomials and 30Hz with full numerical processing. The superior behaviour of the 60Hz experiment with full numerical processing can be seen both in amount of data - as compared to 30Hz and alignment to the diagonal - as compared to 60Hz, not using the polynomials. . .

52

∂ui ∂ui i 2.17 The expression Du Dt = ∂t + uj ∂xj is checked for each component with joint PDFs of ai versus al,i +ac,i for the three cases 60Hz, 60Hz no polynomial and 30Hz. It reveals the strong increase of accuracy due to the weighted polynomials. . . . . . . . . . . .

53

LIST OF FIGURES 2.18

2.19

2.20

2.21

2.22

2.23

2.24

2.25

3.1

3.2

3.3

3.4

3.5

11

dli dt

∂ui is plotted versus lj ∂x in joint PDFs for the 60Hz, 60Hz no polynomials and j 30Hz experiments and each component. The aspect ratios of the shapes are again highest for the 60Hz experiment with the full polynomial procedure, confirming the general findings from the single point checks. . . . . . . . . . . . . . . . . . . . . . . ∂uj dNi dt is plotted versus -Nj ∂xi in joint PDFs for the 60Hz, 60Hz no polynomials and 30Hz experiments and each component. The aspect ratios of the shapes are again highest for the 60Hz experiment with the full polynomial procedure. Generally the shapes are still aligned well with the diagonal but their aspect ratios are relatively low. The evolution of the ratios of volumes over initial volumes is shown by their PDFs for each point in time. For the 30Hz (a) the PTV volumes are conserved only up to 0.5τ η , for the 60Hz (b) up to 1τ η . For volumes derived locally through the integrated deformation matrix B (c) the volumes are conserved up to 7τ η . . . . . . . . . . . . . PDFs of three different sets of div l which had initial values of −1, 0 and 1. Only after a few τ η the peaks are shifted slightly towards their random distribution, which is centered around zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . For each set of points belonging to a trajectory of a certain length the corresponding mean of relative divergence is plotted. It shows the effectiveness of the use of weighted polynomial fits and how they reach their full impact only for trajectories wich are 2τ η of length or more. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . For each set of points ­ ® length the corresponding ­ ® belonging to a trajectory of a certain mean enstrophy, ω 2 , is plotted. It can be seen that ω 2 drops by 30% over lengths of trajectories of 2 − 7τ η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . For each set of points belonging to a trajectory of a certain length the corresponding mean strain, s2 , is plotted. Contrary to the expectation that, if anything,­polynomial ® fitting and the criteria of relative divergence would dampen the data, s2 is 30% higher for the fully processed data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . For each set of points belonging to a trajectory of a certain length the corresponding ratio of mean enstropy, ω 2 over mean strain, s2 , is plotted. Only the fully processed data converge to the theoretical value of 2. . . . . . . . . . . . . . . . . . . . . . . . .

PDFs of ω i ω j sij and − 43 sij sjk ski show how both enstrophy production and strain production are positively skewed. When conditioned on high strain or high enstrophy the skewness generally increases. Additionally, conditioning of ω i ω j sij on large ω 2 and − 43 sij sjk ski on large s2 results in a positive shift of the most probable events. . Logarithmic joint PDF of ω i ω j sij and − 34 sij sjk ski . Despite their similar PDFs in figure 3.1 the pointwise relation of ω i ω j sij and − 34 sij sjk ski is strongly nonlocal and they are only weakly correlated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PDFs of cos (ω, W ) conditioned on s2 (a) and ω 2 (b) are shown. In both cases skewness increases with increasing magnitude of s2 and ω 2 . Physically, a positive cos (ω, W ) reflects the situation where the projection of the vortex stretching vector W on ω points in the same direction as vorticity, if it is zero it will not change the magnitude but only the direction of ω and if it is negative vortex compression occurs. PDFs of cos (ω, λi ) conditioned on weak and strong strain (a) and on weak and strong enstrophy (b) are shwon. The main feature is the predominant alignment of ω with the intermediate eigenvector λ2 of the rate of strain tensor. The preferential alignments increase with both, higher strain and higher enstrophy. . . . . . . . . . . . . . . . . . PDFs of the eigenvalues, Λ1,2,3 , of the rate of strain tensor are shown. The positively skewed PDF of Λ2 comprises a fundamental property of turbulence. If hΛ2 i = 0 essentially no strain and therefore also no enstrophy could be produced by turbulence.

54

55

57

58

59

60

61

62

68

68

70

71

72

12

LIST OF FIGURES 3.6

3.7

Relative volume of the flow occupied by regions of u2 , ω 2 and s2 of different magnitude. As an example we take i.e. 70% of ω2 : From ® figure we read that 70% of the total ­ 2the amount of enstrophy is (i) larger than 1.6 ω and (ii) that it occupies only 20% of the entire volume of the flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The joint probability R − Q plot, with the second invariant of

∂ui ∂xj

73

Q = 14 (ω2 − 2s2 )

and the third invariant R = − 13 (sij sjk ski + 43 ω i ω j sij ), shows a typical ’tear drop’ shape. The most characteristic feature is the predominance of strain production over enstrophy production in regions with high strain (bottom right). Regions of different kinds of events can be allocated to distinct regions within the R − Q phase space. . .

74

As an extension of joint PDFs of ω i ω j sij and − 34 sij sjk ski and of R − Q plots we introduce here the Q − ω i ω j sij − sij sjk ski probability orbital. Represented in this manner, 58% of all events are inside the yellow core (inner region) and 92% of all events are within the envelopping surface of the blue slices (outer region). Only 8% of events are outside the orbital. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Two side views of the Q − ω i ω j sij − sij sjk ski probability orbital are shown. The orbital appears much ’leaner’ viewed in the Q − sij sjk ski plane (b) than viewed in the Q − ωi ωj sij plane (a). Apparently this has to do with the fact that the axis around which fluid particles evolve in the mean, is roughly perpendicular to the Q − ωi ωj sij plane (a). As yet another reflexion of the overall positiveness of h−sij sjk ski i and hω i ω j sij i we note that the entire orbital is skewed towards positive production of s2 and ω 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

3.10 PDFs of strain, s2 , as found in inner and outer regions (figures 3.8, 3.9) of the Q − ωi ωj sij −sij sjk ski probability orbital. The division made in the Q−ω i ω j sij −sij sjk ski probability orbital separates weaker from stronger strain events. . . . . . . . . . . . .

77

3.11 The mean strain distribution in the Q−ω i ω j sij −sij sjk ski probability orbital is shown. ­ ® Orbital regions which in the mean have values for strain that are than 1.25 s2 ­ higher ® are rendered blue and if in the mean strain is higher than 2.5 s2 they are rendered yellow. The blue region in real space occupies 35% of the flow and the yellow region 10% respectively. In terms of dynamical relevance this is equivalent to 70% and 30% of total strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.12 PDFs of enstrophy, ω 2 , as found in inner and outer regions (figure 3.8, 3.9) of the Q − ω i ω j sij − sij sjk ski probability orbital. The division made in the Q − ω i ω j sij − sij sjk ski probability orbital separates weaker from stronger enstrophy events. . . . .

79

3.13 The mean enstrophy distribution in the Q − ω i ω j sij − sij sjk ski probability orbital is shown. Orbital ­ 2 ® regions which in the mean have values for enstrophy that are higher ­ ® than 1.25 s are rendered blue and if in the mean enstrophy is higher than 2.5 s2 they are rendered yellow. The blue region in real space occupies 30% of the flow and the yellow region 12% respectively. In terms of dynamical relevance this is equivalent to 78% and 55% of total enstrophy! . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

˙ −sij s˙ jk ski , ωi ω˙ j sij which can be thought of as the components of a vector field 3.14 Q, are plotted. Arrows are blue when Q˙ < 0 and red when Q˙ > 0. The dominant feature is a cyclic mean evolution of the state (Q, −sij sjk ski , ω i ω j sij ). If we think of the vector pattern as a ’rigid-body-like-rotation’ then the ’axis of rotation’ is roughly perpendicular to the Q−ω i ω j sij plane and sligthly inclined to the −sij sjk ski −ωi ωj sij plane, just like the viewing axis of this figure. . . . . . . . . . . . . . . . . . . . . . .

81

3.8

3.9

13

LIST OF FIGURES 3.15 Orbital regions of different contributions­ to vortex stretching are shown. In ® (a) contributions to vortex stretching due to ω 2 Λ1 cos2 (ω, λ1 ) + ω2 Λ3 cos2 (ω, λ3 ) > 0 are rendered blue when in the mean they are above 3s−3 and yellow when in the mean are larger than 6s−3 . In (b) contributions to vortex stretching due to ­ 2 they ® 2 ω Λ2 cos (ω, λ2 ) > 0 are rendered using blue and yellow for the same mean values as in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

3.16 Regions of medium and strong mean viscous enstrophy destruction are shown. To al® ­ 2 < ∇ ω low comparison with figure 3.15 blue cross-sections represent regions where νω i i ­ ® 2 −3 −3 −3s and the yellow surface encloses regions where νωi ∇ ω i < −6s . In most regions mean viscous destruction is of the same order of magnitude as the mean enstrophy production. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.17 Orbital regions of different contributions to ® are shown. In (a) con­ vortex compression tributions to vortex compressing due to ω 2 Λ2 cos2 (ω, λ2 ) < 0 are rendered blue when in the mean they are below −1s−3 and yellow when in the ­ mean they are smaller ® than −3s−3 . In (b) contributions to vortex compression due to ω2 Λ1 cos2 (ω, λ1 ) + ω 2 Λ3 cos2 (ω, λ3 ) < 0 are rendered using blue and yellow for the same mean values as in (a). . . . . . . 86 3.18 PDFs of ω i ω j sij /ω 2 conditioned on weak, intermediate and strong values for Q are shown. They are all skewed positively. For Q < −2 the peak of most probable events is shifted towards more positive ω i ω j sij /ω 2 values and both, strong vortex compression and stretching rate events become more probable. . . . . . . . . . . . .

87

/ω 2

are conditioned 3.19 Means of the relative contribution, cΛ2 , for each value of ω i ω j sij on weak, intermediate and strong Q. cΛ2 is maximal at intermediate compression and stretching rates. The cases −2 < Q < 1 and Q > 1 are qualitatively similar. For events where Q < 2 in the mean there is no contribution from Λ2 cos2 (ω, λ2 ) to the rate of vortex compression and the influence on the rate of vortex stretching exceeds 60%! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Λ2 cos2 (ω, λ2 ),

defined as 3.20 Events that are dominated by the contribution of the term cΛ2 , are represented in blue color where the intermediate contributor is responsible for half or more of the rate of vortex compression or stretching and the yellow color represents the region where the contribution of cΛ2 is larger even than two thirds. . . 3.21 Regions where in the mean the value for cos (ω, λ2 ) is higher then 0.707 are rendered in colors according to mean value of cos (ω, λ2 ). Note how hcos (ω, λ2 )i is strongly correlated with the rate of enstrophy production that is associated with the term Λ2 cos2 (ω, λ2 ) or cΛ2 . The shape of the regions corresponding to high values of hcos (ω, λ2 )i ’follows’ the direction suggested by the mean evolution of fluid particles in the Q − sij sjk ski − ω i ω j sij probability orbital. . . . . . . . . . . . . . . . . . . . . 3.22 In (a) regions with values for −sij sjk ski that are above 3s−3 are rendered in blue color and such with values above 6s−3 in yellow. In (b) blue is used for regions where 2p 2p + νsij ∇2 sij > < −3s−3 and yellow where < −sij ∂x∂i ∂x + νsij ∇2 sij > < −sij ∂x∂i ∂x j j < −6s−3 . The combination of (a) and (b) demonstrates how mean strain production due to −sij sjk ski > 0 is approximately balanced by mean strain destruction due to the combined effect of pressure and viscous terms of expression (3.21). . . . . . . .

89

90

92

−1s−3

3.23 (a) strain destruction due to −sij sjk ski < 0 at levels of −sij sjk ski < (blue) and −sij sjk ski < −3s−3 (yellow) is shown. In (b) mean strain production due to combined effects of pressure and viscous terms of expression (3.6) are shown. Rendered in blue 2p color are regions where the mean of −sij ∂x∂i ∂x + νsij ∇2 sij is above 1s−3 and regions j 2

p where the mean of −sij ∂x∂i ∂x + νsij ∇2 sij is above 3s−3 are rendered in yellow. . . . j

94

14

LIST OF FIGURES 3.24 The evolution of each component of material elements li as obtained from 3D-PTV and from integrating B over time at three selected times are compared. Generally there is a good agreement between lP T V and lB . . . . . . . . . . . . . . . . . . . . .

96

3.25 The evolution of ∆l, ∆l = lt − lt=0 , for each component of a material elements l as obtained from 3D-PTV and from integrating B over time at three selected times is shown. Generally there is a good agreement between ∆lP T V and ∆lB . . . . . . . . .

97

3.26 The main features of the mean stretching rate of material lines are that (i) it is higher than the mean stretching rate of vorticity, (ii) it is higher than the intermediate principal strain rate and (iii) it is lower than the most positive principal strain rate.

98

3.27 The evolution of the PDFs of material line stretching, li lj sij /l2 , over time is shown. The large variance converges to a stable distribution within a time interval of 2τ η . A shift of the most probable events towards more positive rates of stretching and a significant persistence of compressing events is observed. . . . . . . . . . . . . . . . .

99

Wl

∂ui lj ∂x , j

= the 3.28 The evolution in time of the cosine of the angle between l and material line stretching vector, is shown. A predominance of stretching over tilting and compressing is reached after 2τ η . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.29 The evolution in time of the PDFs of cos (l, λi ) is shown. After a transition time of 2τ η a clear alignment of l’s with λ1 can be observed . The PDFs for cos (l, λ2 ) reveal a constantly flat distribution for all times and the pdf’s for cos (l, λ3 ) suggest that l and λ3 are oriented almost perpendicular to each other. . . . . . . . . . . . . . . . . 101 3.30 The evolution of the PDFs of the stretching rates of material lines with l0 = ω 0 are shown. The variance of li lj sij /l2 barely changes over time. The only notable change is a a slight decrease of compressing events. . . . . . . . . . . . . . . . . . . . . . . . 102 ¢ ¡ 3.31 The evolution of the PDFs of cos l, W l is shown. Only a slight reduction of the material line compressing and a slight increase of the material line stretching events can be observed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.32 The evolution in time of the PDFs of cos (l,λi ), with l0 = ω0 , is shown. The alignment of l with λ1 develops, but only relatively slowly. The alignment of l with λ2 is becoming weaker with time but it is surprisingly persistent over a time period of τ η = 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.33 PDFs of kcos (l, λ2 )k at τ η = 3 that are conditioned on different initial situations for l0 at time τ η = 0 are shown. (a) ls with l0 = ω 0 and l0 = λ2 conditioned on the sign of Λ2 at time τ η = 0. (b) ls with l0 = λ2 conditioned on the cosine of the angle between l0 and ω 0 at time τ η = 0. (c) ls with l0 = ω 0 and l0 = λ2 conditioned on the magnitude of enstrophy, ω 2 , at time τ η = 0. (d) ls with l0 = ω0 and l0 = λ2 conditioned on the magnitude of strain, s2 , at time τ η = 0. . . . . . . . . . . . . . . 106 3.34 Joint PDF plots of stretching of material elements versus 1/2 Dl2 /Dt (a) and the stretching of vorticity versus 1/2 Dω 2 /Dt (b) are shown. ω i ω j sij and 1/2 Dω 2 /Dt do not seem to be correlated implying that in enstrophy change viscosity plays a role just as important as vortex stretching. . . . . . . . . . . . . . . . . . . . . . . . . . . 108 ¡ ¢ 3.35 Joint PDF plots of stretching rates of material elements ¢versus 1/2 Dl2 /Dt /l2 (a) ¡ and the stretching rates of vorticity versus 1/2 Dω 2 /Dt /ω 2 (b) are shown. For the rate of enstrophy production the role of viscosity becomes even more important. . . 109 3.36 The joint PDF plots of ω i ω j sij versus 1/2 Dω 2 /Dt are shown when they are conditioned on (a) weak strain, (b) high strain, (c) weak enstrophy and (d) high enstrophy. Only when enstrophy is high and vortex stretching moderate, center of (d), there appears to be a slight correlation between ω i ω j sij and 1/2 Dω 2 /Dt. . . . . . . . . . 110

LIST OF FIGURES ¡ ¢ 3.37 Joint PDFs of ωi ω j sij /ω 2 versus 1/2 Dω2 /Dt /ω 2 are conditioned on (a) weak strain, (b) high strain, (c) weak enstrophy and (d) high enstrophy. Only when enstrophy is high and the vortex stretching rate is moderate, center ¢of (d), there appears ¡ to be a slight correlation between ω i ω j sij /ω 2 and 1/2 Dω 2 /Dt /ω 2 . . . . . . . . . . 3.38 PDFs of 1/2 Dω 2 /Dt conditioned on (a) weak and high strain and (b) weak and 2 2 high enstrophy. Conditioning on small ° ° and large events of s and ω reveals that the 2 probability for strong °1/2 Dω /Dt° events to occur depends very much on strain and even slightly more so on enstrophy. . . . . . . . . . . . . . . . . . . . . . . . . . ¡ ¢ 3.39 PDFs of 1/2 Dω 2 /Dt /ω 2 , the rate of enstrophy change, are conditioned on (a) weak and high strain and (b) weak and high enstrophy. Contrary to the results on 1/2 Dω2 /Dt, here its rate reacts most sensitive when conditioned on strain. High strain results in an increased probability for high rates of change of enstrophy (a). Conditioning on high enstrophy however, leads to less intense rates of change (b). . . 3.40 The PDFs of (a) strain and (b) enstrophy both conditioned magnitude of ° ° on the °1/2 Dω2 /Dt°and s2 (a) change of enstrophy. A weak positive correlation between ° ° and between °1/2 Dω2 /Dt°and ω 2 (b) can be observed. . . . . . . . . . . . . . . . . 3.41 The PDFs of strain and enstrophy are conditioned on the rates of change of enstrophy. (a) conditioned PDFs of strain and (b) those for enstrophy respectively. In (a) the probability of weak strain events decreases for high rates of enstrophy changes, whereas in (b) the ¡probability of¢ weak enstrophy events increases for high rates of enstrophy changes, 1/2 Dω2 /Dt /ω2 . This finding is comparable to the ’high strain, low enstrophy, bridging and vortex reconnection’ phenomena which Fernandez et al. (1995) report to be associated with higher than exponential rate of enstrophy change. ¡ ¢ 2 2 , (b) 1/2 Dω 2 /Dt −ω 2 , (c) 1/2 Dω 2 /Dt /ω 2 −s2 3.42 Joint PDFs ¡ of (a)2 1/2 ¢Dω 2/Dt −s and (d) 1/2 Dω /Dt /ω − ω 2 . In the cases (a), (b) and (c) it can be seen how the moduli of quantities of both axes¡are at least slightly correlated. Case (d) reveals a ¢ strong nonlocal relation between 1/2 Dω2 /Dt /ω 2 and ω 2 . . . . . . . . . . . . . . . 3.43 Evolutions in time of PDFs of geometrical invariants involving the normal, N , of material surfaces are shown. In (a) the evolution of the PDF of the cosine of the angle between N and the surface stretching vector W N shows strong preferential surface stretching after 2τ η . In (b)-(d) evolutions in time of PDFs of the cosine of the angle between N and the eigenvectors λi , i = 1 − 3, show how N is strongly preferentially aligned with λ3 after 2τ η . . . . . . . . . . . . . . . . . . . . . . . . . . 3.44 A fluid volume at some time τ η > 0 with side length ratios (w1 )1/2 : (w2 )1/2 : (w3 )1/2 defined by the eigenvalues of the Cauchy-Green tensor, W , is sketched. The axis along w1 is constantly stretched, the axis along w2 is also stretched in the mean and the axis along w3 is compressed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.45 The evolution of the mean eigenvalues, hwi i, of the Cauchy-Green tensor, W , is shown. If at τ η = 0 the volume is unity then, due to conservation of volume, w1 · w2 · w3 = 1 for finite times, τ η > 0. Hence also hln (w1 )i + hln (w2 )i + hln (w3 )i = 0 for finite times, τ η > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.46 Joint PDF plots of ln (w1 ) and ln (w2 ) for different time moments of evolution, τ η = 1, 2, 4, 6. ln (w2 ) / ln (w1 ) is skewed towards ’pancake-shapes’. . . . . . . . . . . . . . 3.47 Evolution in time of stretching rates (a) and absolute stretching (b) for material lines, l, material surfaces, N , and the surfaces of material volumes, S. . . . . . . . . . . . . 3.48 Conditional PDFs of the magnitude of the rotation of material volumes, ω 2w , are shown. In (a) the conditions are weak and high events of strain (light blue, blue) and enstrophy (orange, red). In (b) ω2w is conditioned on different values of Q, Q = 14 (ω 2 − 2s2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

111

112

113

115

116

117

118

119

120 121 123

125

16

LIST OF FIGURES ­ ® 3.49 Joint PDF plot of Q versus ω2w . The mean rotation of a material volume, ω 2w , conditioned on values of Q is plotted as a white line. . . . . . . . . . . . . . . . . . . 126 3.50 A³joint´PDF plot of material volume rotation, ω 2w , and its rate of surface stretching, 1 Ds2 2 2 Dt /S , reveals a strong nonlocal behavior. Strong rates of surface stretching inhibit strong rotations of material volumes and vice versa. . . . . . . . . . . . . . . 127

List of Tables 2.1 2.2

2.3

2.4

2.5

2.6

3.1 3.2

Characteristics of the four lenses Rodenstock macro lenses used for both experiments. After high pass filtering of the recorded images a particle detection algorithm, provided by IGP, determines particle image coordinates. In this table the specific detection parameters used for both experiments are given. . . . . . . . . . . . . . . . . . . A particle correspondence algorithm, provided by IGP, finds for particles of one image corresponding particles in the remaining three images. The intersection of three or four epipolar lines of corresponding particles is then used to determine their positions in space. In this table the parameters used for both experiments are given. . . . . . In this table the settings for the algorithm, IGP, used to find corresponding particles in image and object space of consequtive time steps are given. For effective assignment three criterias are used: (i) a 3D search volume is definded by minimum and maximum velocities in all three coordinate directions, (ii) the Lagrangian acceleration of a particle is limited, defining a conic search area, (iii) in case of ambiguities the particle leading to the smallest Lagrangian acceleration is chosen. Similarities in brightnes, width, height and sum of gray values of the pixel of a particle image prooved to be not as valuable as expected. . . . . . . . . . . . . . . . . . . . . . . . . Some characteristics of the statistical sets are compared for the 30Hz and the 60Hz experiments. In the 60Hz experiment the Kolmogorov time, τ η , decreased and both, frames per τ η and recording time per τ η increased. . . . . . . . . . . . . . . . . . . . Assuming that two sets of a a2 - with low and high intensities - have a probabilty of beeing tracked to their next frames of 0.99 and 0.98 - then, the ratio of trackable ’low’ a2 over ’high’ a2 more than doubles over a time period from 2τ η to 7τ η . . . . Characteristic flow properties as obtained for both experiments, 30Hz recording over 60s in the ’large faciltiy’ and 60Hz recording over 100s in the ’small faciltiy’. . . . . . Contributions of terms associated with Λi to mean enstrophy production, hω i ω j sij i, and to the magnitude of the vortex stretching vector, W 2 . The largest contribution to hωi ωj sij i is associated with the first term, Λ1 , despite the preferential alignment between ω and λ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

29

34

34

36

46

48 66

69

18

LIST OF TABLES

Chapter 1

Introduction The motivation for this thesis is to make use of the essentially Lagrangian method of three dimensional Particle Tracking Velocimetry (3D-PTV) to eludicate some mechanisms of turbulence that previous Eulerian approaches could not access. In particular, this means to further develop the non intrusive flow measurement technique of 3D-PTV such, that the full tensor of velocity derivatives can be measured along trajectories of fluid particles. This Lagrangian approach should give new insight into some aspects of strain and enstrophy dynamics. Moreover it will add to the understanding of the behavior of material volumes and surfaces and the differences between stretching of vortex lines and material lines.

1.1

Velocity derivatives

Velocity derivatives, Uij = ∂ui /∂xj 1 , play an outstanding role in the dynamics of turbulence for a number of reasons. Their importance became clear since the papers by Taylor (1937, 1938) and Kolmogorov (1941a,b). Taylor emphasized the role of vorticity, ω = curl u, i.e. the antisymmetric part of the velocity gradient tensor Uij = ∂ui /∂xj , with the stress on one of the most basic phenomena and distinctive features of three dimensional turbulence - the predominant vortex stretching, which is manifested in positive net enstrophy production, hω i ω j sij i > 0. Kolmogorov stressed the importance of dissipation, 2νsij sij or 2νs2 , and thereby of strain, sij = 12 (∂ui /∂xj + ∂uj /∂xi ), i.e. the symmetric part of the velocity gradient tensor. It is known that the production of the field of velocity derivatives is a spatially local selfamplification process (Galanti and Tsinober, 2000) where the type of forcing does not play any role. The local production terms, ω i ω j sij and sij sjk ski , for vorticity and strain are reported by Tsinober (2001) to be orders of magnitude higher than their corresponding terms associated with forcing. There are indications that the process of self-amplification has a universal character for a wide range of different flow types. During the whole history of turbulence research the main bulk of experimental results was associated with the velocity field itself and some of its simple consequences as spectra, structure functions and Probability Density Functions (PDFs). More frequently than otherwise one velocity component only was observed. The main reason for such a state of matters was the relatively 1 The indices, i, j, k, used throughout this text are applied according to the following rules: (i) summation over indices that in a term appear more than once, e.g. 1) ωj sij = ω1 si1 + ω2 si2 + ω3 si3 , e.g. 2) ωi ωj sij after summation over j and i becomes a scalar. (ii) If there is only one index or after summation there remains only one index, it is a vector, e.g. ui = {u1 , u2 , u3 } or ωj sij = Wi = {W1 , W2 , W3 }. (iii) if there are or - after summation - remain two indices, it is a tensor, e.g. sij = {(s11 , s12 , s13 ), (s21 , s22 , s23 ), (s31 , s32 , s33 )}. For vectors and tensors bold notations are used, e.g. ui = {u1 , u2 , u3 } = u.

19

20

CHAPTER 1. INTRODUCTION

simple accessibility of the streamwise velocity component. Even with the ability to measure more than one component either by hotwire anemometry, Laser Doppler Velocimetry (LDV) or other techniques, the field of velocity derivatives remained inaccessible until recently. The full set of velocity derivatives was measured by the group of Tsinober (2001) by employing a multi-hotwire probe both in laboratory and in field experiments (Kholmyansky et al., 2001a, 2001b). This approach is essentially Eulerian, one in which the measurements are performed in a fixed point in space. To our knowledge no Lagrangian measurements of Uij were conducted when this work started in 1999, until the two-dimensional experiment by Voth et al. (2002) published recently. With the advent of powerful computers, regions of strong vorticity were discovered in Direct Numerical Simulation (DNS) of homogeneous, isotropic turbulence (i.e. Siggia, 1981; Kerr, 1985; Ashurst et al. 1987 and Hosokawa and Yamamoto, 1989). They are usually referred to as ’worms’ and brought excitement to the part of the turbulence community concerned with its fundamental aspects, see i.e. Jiménez et al. (1993) or Vincent and Meneguzzi (1994). The idea was straight forward: Now, with the problem broken down to small regions of concentrated vorticity and large un-important regions with background turbulence, which is also referred to as ’random sea’, it would be only a matter of time for substantial progress in the understanding of the basic turbulent mechanisms and phenomena such as intermittency and anomalous scaling. While this progress did never really happen the opposition of the ’worm’ or concentrated vorticity theories argued that the ’background’ of turbulence is very structured and dynamically active and that ’worms’ were more the passive result than the active cause of anything. This argument was put in words somewhat pronounced by Brasseur (1999): ”..., describing turbulence evolution in terms of dynamical interactions among vortex tubes and sheets is akin to describing a happy chubby dog as being wagged by its skinny vibrating tale”. Though, formally all of the flow field is determined entirely by the field of vorticity or strain (Tsinober, 2001) there are several reasons why one should study both, the vorticity and strain together. Formally, the relation between strain and vorticity is strongly nonlocal, but the main reason is that vortex stretching is essentially a process of interaction of vorticity and strain. Strain in some respect is even more important since strain dominated regions appear to be the most active/nonlinear in a number of aspects. For example, the energy cascade and its final result dissipation, are associated with predominant self-amplification of the rate of strain production and vortex compression rather than with vortex stretching. Apart from vorticity and strain/dissipation, there are other reasons for a special interest in the characteristics of the field of velocity derivatives in turbulent flows. For example: • The field of velocity derivatives is much more sensitive to the non-Gaussian nature of turbulence or more generally to its structure, and hence reflects more of its physics, Tsinober (2001). • The possibility of singularities being generated by the Euler and the Navier-Stokes equations (NSE) and possible breakdown of NSE are intimately related to the field of velocity derivatives, Doering and Gibbon (1995) . • In the Lagrangian description of fluid flow in a frame following a fluid particle, each point is a critical one, i.e. the direction of velocity is not determined, Perry and Fairlie (1974). So everything happening in the proximity of a fluid particle is characterized by the velocity gradient tensor Uij = ∂ui /∂xj . For instance, local geometry/topology is naturally described in terms of critical points terminology, (see Cantwell, 1990, 1992; Chacin et al., 1996; Chertkov et al., 1999; Martin et al., 1998).

1.2. MATERIAL ELEMENTS AND LAGRANGIAN DESCRIPTION

1.2

21

Material elements and Lagrangian description

One of the common views is that the prevalence of vortex stretching is due to the predominance of stretching of material lines. This view originates with Taylor (1938): Turbulent motion is found to be diffusive, so that particles which were originally neighbors move apart as motion proceeds. In a diffusive motion the average value of d2 /d20 continually increases. It will be seen therefore from (1) ( i.e. from ω/ω0 = d/d0 , where d0 is the initial distance between two neighboring particles, and d is the distance at a subsequent time) that the average value of ω 2 /ω20 continually increases. This view is widely accepted, e.g. in Hunt (1973)... the interesting physical argument that hω i ω j sij i is positive because two particles on average move apart from each other and therefore vortex lines are on average stretched rather than compressed. However, this view is - at best - only partially true, since there exist several essential qualitative differences between the two processes. These were listed by Tsinober (2001) as follows: • The equation for a material line element l is a linear one and the vector l is passive, i.e. the fluid flow does not ‘know’ anything whatsoever about l: the vector l (as any passive vector) does not exert any influence on the fluid flow, expression (3.28). The material element is stretched (compressed) locally at an exponential rate proportional to the rate of strain along the direction of l, since the strain is independent of l.

• On the contrary, the equation for vorticity is a nonlinear partial differential equation and the vector ω is an active one -it ‘reacts back’ on the fluid flow. The strain does depend in a nonlocal manner on ω and vice versa, i.e. the rate of vortex stretching is a nonlocal quantity, whereas the rate of stretching of material lines is a local one. Therefore the rate of vortex stretching (compressing) is different from the exponential one and is unknown. There are many ‘fewer’ vorticity lines than material lines - at each point there is typically only one vortex line, but there are infinitely many material lines. This leads to differences in the statistical properties of the two fields. In the absence of viscosity vortex lines are material lines, but they are special in the sense that they are not passive as all the other passive material lines.

• Consequently while a material element l tends to be aligned with the largest (positive) eigenvector of sij , vorticity ω tends to be aligned with the intermediate (mostly positive) eigenvector of sij : the eigenframe of sij rotates with an angular velocity Ω of the order of vorticity ω. • For a Gaussian isotropic velocity field the mean enstrophy generation vanishes identically, hωi ωj sij i ≡ 0, whereas the mean rate of stretching of material lines is essentially positive. The same is true of the mean rate of vortex stretching hω i ω j sij /ω 2 i and for purely two-dimensional flows. This means that in turbulent flows the mean growth rate of material lines is larger than that of vorticity. The nature of the vortex stretching process is dynamical and not a kinematic one as the stretching of material lines is.

• The curvature of vortex lines increases with strain and positive rate of vortex stretching, whereas the curvature of material lines decreases with strain and positive rate of material line stretching.

• An additional difference due to viscosity becomes essential for regions with concentrated vorticity, in which there is an approximate balance between enstrophy generation and its reduction. Vortex reconnection is allowed by nonzero viscosity. No such phenomena exist for material lines.

• Comparing vorticity with a passive vector in the presence of the same diffusivity as viscosity, the analogy is partial not just because the equation for vorticity is nonlinear, but also because in the case of vorticity the process is due to self-amplification of the field of velocity derivatives, whereas in case of a passive vector it is not.

22

CHAPTER 1. INTRODUCTION • Vorticity is divergence-free, whereas material elements are not.

It should be stressed that the evolution of material line elements is essentially Lagrangian and that it is associated with the tensor of velocity derivatives Uij and the deformation tensor Bij , d dt B = (∂ui /∂xj )t · B(t), B(0) = I. Likewise, the dynamics of vorticity is intimately associated with the Lagrangian aspects of fluid flows, especially those related to the field of velocity derivatives. Therefore, in order to obtain further insights on the nature of both material line evolution and the dynamics of vorticity and the differences between the two, it is vital to study these processes in the Lagrangian setting. Hence, along with velocity the necessity for Lagrangian measurements of the tensor of velocity derivatives, i.e. their evolution along the particle trajectories, comprises the main goal of this thesis both in essence and from the technical point of view. The Lagrangian description of fluid flows is physically more natural than the Eulerian one, since it is related most directly to the motion of fluid elements as is done in Newtonian mechanics. However, it is much more difficult to come by. Until recently the technical difficulties both in laboratory and numerical approaches strongly hindered the use of the Lagrangian approach in most of fluid dynamical problems. The traditional particular problems for which Lagrangian description is especially appropriate are turbulent diffusion, transport and mixing in a great variety of applications, e.g. geophysical, cloud formation, atmospheric and environmental transport, tracers on the ocean surface, chemical technology, combustion systems and material processing, production of nano particles, sedimentation, bio-medical, delivery of drugs and many others, Monin and Yaglom (1971, ch. V, pp. 527-693), Tennekes and Lumley (1972), Kim and Stinger (1992), Babiano et al. (1987). Another aspect is associated with the dynamics of inviscid fluids, theoretical problems associated with Euler equations, vortex dynamics, dynamics of interfaces and surface waves, Saffman (1991), Moffatt (2000), and Lundgren and Koumoutsakos (1999). In all these the basic ingredient is the motion of fluid particles, which is essentially Lagrangian in its nature as contrasted to the Eulerian description in which the observation of the system is made in a fixed frame as the fluid goes by. Hence the importance of particle tracking either in real flows and/or in direct numerical simulations. There is also a view that from a purely theoretical point ‘the power of Lagrangian approach to fluid turbulence...allows the analytical description of most important statistics of particles and fields’, Falkovich et al. (2001). This is true of the so called ‘passive turbulence’ dealing with the behavior of passive objects in prescribed velocity fields (i.e. described by linear equations), which is not genuine dynamical turbulence. Nevertheless, it is thought that Lagrangian approaches can be also useful in a number of basic aspects of fluid dynamical problems beyond traditionally purely kinematical ones, Stuart and Tabor (1990).

1.3

Overview of previous Lagrangian studies

Most of the work done in connection with Lagrangian approaches was motivated by turbulent diffusion/dispersion since the work by Taylor (1921) with the main emphasis on Lagrangian correlations of velocity and later of accelerations.

1.3.1

Experimental

The first approach is based on determination of Lagrangian velocity correlation from dispersion of a scalar substance like heat. This work was summarized by Shlien and Corrsin (1974), who measured heat dispersion behind a heated wire in a wind tunnel for this purpose. The second approach consists of tracking neutral tracer particles that approximate the Lagrangian fluid motion. The first such tracking was made by Shyder and Lumley (1971). They used

1.3. OVERVIEW OF PREVIOUS LAGRANGIAN STUDIES

23

hollow spherical glass beads of 46.5µ in diameter to follow and estimate the Lagrangian fluid properties in an air flow past a grid in a wind tunnel. Along with light hollow glass beads experiments were made also with heavier 87µm glass beads, 87µm corn pollen and 46.5µm copper spherical beads. Sato and Yamamoto (1987) performed similar experiments in a water tunnel grid turbulence. A stereoscopic system for following simultaneously many particles in three dimensions was used by Virant and Dracos (1997) in a water boundary layer in a flow with free surface, whereas Mann et al. (1999) used a similar system in an oscillating grid turbulent flow. Voth et al. (1998) and La Porta et al. (2001) used silicon strip detectors as optical imaging elements originally developed to measure particle tracks in the vertex detector of the CLEO III experiment operating at the Cornell Electron Positron Collider. This (due to very high sampling rate - 60kHz) allowed them to approach the Taylor microscale Reynolds number Reλ ∼ 103 , which is more than an order of magnitude larger than in previous experiments. The only other large Reynolds number data come from balloon tracking in the atmospheric boundary layer, Morel and Larcheveque (1974), Hanna (1981). As mentioned most of the above work was concentrated on obtaining the Lagrangian velocity correlations and later accelerations, Virant and Dracos (1997), Mann et al. (1999), Voth et al. (1998), La Porta et al. (2001) and studying relative diffusion, Mann et al. (1999), Morel and Larcheveque (1974). An important recent development was made in the work by Voth et al. (2002). This is the only work in which along with the flow itself the field of velocity derivatives, and the Cauchy-Green tensor were obtained in a particle tracking experiment. This was achieved by following of about 800 fluorecsent latex particles 120µm in diameter with sampling rate 10Hz, measuring their velocities from the particle trajectories using polynomial fitting and interpolating the velocities onto a grid to obtain the velocity as a function of space and time. This is in principle the same procedure as used in our experiments, Lüthi et al. (2001), and which is described in more detail below along with later improvements and developments. The main difference - which from technical point is more than essential - is that the experiment made by Voth et al. (2002) was a two-dimensional one, whereas ours is three-dimensional.

1.3.2

Numerical

Modern computers allow to extract Lagrangian information from the Eulerian direct numerical simulations of the Navier-Stoke equations at moderate Reynolds numbers and were reviewed recently by Yeung (2002). These studies were made in simple geometries and as the experimental ones emphasized the Lagrangian velocity correlations and accelerations. Yeung (1994) studied two-particle relative diffusion in isotropic turbulence. The work of Girimaji and Pope (1990b) is of particular relevance to ours. They performed an extensive study of the behavior of material elements (lines, surfaces, volumes) in NSE isotropic turbulence, which along with conventional analysis includes the growth rates of material lines and surfaces, deformation of volume elements and alignments with T . Huang (1996) extensive use of the deformation matrix Bij and Cauchy-Green tensor Wij = Bik Bkj also addressed similar issues, but in addition made important comparisons between the properties of material lines and those of vorticity such as stretching rates and alignments. Similar comparison between vorticity and passive vector with the same diffusivity as the fluid viscosity was made by Ohkitani (2002) and Tsinober and Galanti (2001). Two closely related papers deal with the geometry and evolution of Lagrangian tetrahedrons in quasi-isotropic turbulence, Chertkov et al. (1999), Pumir et al. (2000).

24

CHAPTER 1. INTRODUCTION

Chapter 2

Method The experimental method designed to reach the goal of measuring the full set of velocity derivatives, Uij = ∂ui /∂xj , along particle trajectories in a turbulent flow can be divided into five parts: • Experimental setup designed to produce a turbulent flow which is (i) measurable via 3D-PTV and (ii) comparable to results from Direct Numerical Simulations (DNS) from literature. • Application of an existing 3D Particle Tracking Velocimetry (3D-PTV) algorithm. • Processing of obtained particle trajectories in order to interpolate from them velocity derivatives of acceptable accuracy. • Interpolation of spatial and temporal velocity derivatives along particle trajectories. • Verification of qualitative accuracy of the obtained velocity derivatives. They will be described in the following sections along with the evolution of individual experimental components.

2.1 2.1.1

Experimental setup Forcing

As stated above the experiment should produce a turbulent flow with velocity derivatives that are measurable through 3D-PTV. Besides the trivial requirement that the turbulence intensity should not exceed the capabilities of the 3D-PTV system - which are mainly defined by the recording rate of the cameras and particle seeding density, as explained below in the section on 3D-PTV- the flow should be such, that it allows for monitoring its properties along particle trajectories. Lagrangian studies of fluid turbulence are difficult however in open systems with a mean velocity transporting the fluid elements out of the observation region. Even more severe is the fact that in open systems a considerable part of the resolvable range of velocities would be occupied by the mean velocities and not by their turbulent fluctuations, responsible for the velocity derivatives. For these reasons it was clear that the flow has to be produced in a closed system. Thermal convections in enclosures belong to such systems. However, buoyant forcing is likely to be distributed in the entire flow on all relevant scales, which makes a direct comparison to homogeneous isotropic flows, obtained from direct numerical simulations problematic. Isothermal flows with mechanical forcing using oscillating grids, see i.e. Frenkiel et al. (1979), also belong to the class of closed systems. They have two major shortcomings. The 3D-PTV system can be corrupted by mechanical vibrations and the forcing is not sufficiently flexible, i.e. it is not confined to large scales and the oscillations may introduce time scales which are remembered throughout the flow’s history. 25

26

CHAPTER 2. METHOD

For these reasons it was decided to use electromagnetical forcing. The flow domain is a rectangular box (figure 2.1) with the flow forced from two opposite walls by Lorentz forces, fL , fL = j × B,

(2.1)

where j is the current density and B the magnetic field (figure 2.2). Simple systems of such kind were successfully used starting from the seventies by the group of A. Obukhov (e.g. Obukhov, 1983; Cardoso et al., 1996; Honji et al., 1997) with simple quais-two-dimensional flows and Yule (1975) who has used electromagnetic forcing of low Reynolds number flow in a pipe. To obtain a sufficiently high current density, j, between the two copper plates a well saturated aqueous copper sulphate (CuSo4 ) test fluid was used. Comparable conductivity could be reached using a NaCl solution.

Figure 2.1: Schematic of the experimental facility and coordinate system (x,y,z). Electromagnetic forcing from two opposite walls produces swirls over each magnet. At short distances of the walls the flow becomes three dimensional and towards the observation volume fully turbulent. The flow is recorded by four CCD cameras at recording rates of first 30Hz and - in a later stage - at 60Hz. This would have the negative effect of producing hydrogen bubbles at both electrodes which contaminate the flow both optically and physically. Since salt water would have advantages over the copper sulphate solution, such as better optical properties, easier availability and handling, a variety of different materials for the electrodes were tested. However, metals such as bronze and brass corroded and platinum plated electrodes produced gas.The copper sulphate solution used in the final experimental setup has a conductivity of 16.7mS/cm, a density, ρ, of 1050kg/m3 , and a dynamic viscosity, υ, of 1.2 · 10−6 m2 /s. For the magnetic field cylindrical rare earth sintered strong permanent magnets (RECOMA 28) are used. They have a diameter of 42mm, an axial length of

2.1. EXPERIMENTAL SETUP

27

Figure 2.2: Schematic of electromagnetic forcing. A current density field (arrows, top left) interacts over each magnet with the magnetic field (arrows, top right) producing Lorentz forces (gray scale, top left). This forcing results in torus-like swirls over each magnet. 40mm and 20mm, and a strength, B, of over 1T esla. As sketched in figure 2.2 volumetric Lorentz forces, fL , are produced through the interaction of the - in our case - wall normal current density field, j, and the magnetic field, B. It is clear from expression (2.1) and figure 2.2 that fL are highest when B is parallel to the wall. This is the case right above the rim of each cylindrical permanent magnet. This leads to tangential - with respect to the axis of the magnets - volumetric forcing, that is strongest in a torus-like region above the magnets with a diameter of ∼ 42mm reaching up to 5mm into the flow. The forcing produces a non-oscillating swirling motion in the proximity of each magnet. Within a few seconds these circulations cause a three dimensional time dependent flow region with a front that quickly propagates towards the center resulting in a turbulent velocity field with zero mean flow and fluctuations, ui , of order O(0.01m/s), occupying the entire volume of the tank.

2.1.2

First experiment

In the course of this thesis two different experiments were conducted of which the first will be described here. The flow is produced as schematically shown in figure 2.1 where the copper plates measure 320 × 320mm2 and are 170mm apart. This results in an active volume of 18`, which is situated in an even larger tank, namely 50`, as can be seen from figure 2.3.Behind each copper

28

CHAPTER 2. METHOD

Figure 2.3: The first experimental facility is set up inside a 50` tank. For forcing two walls, each with a 4×4 permanent magnet array, are facing each other at a distance of 17cm. A DC electric current of 7A is a applied to two copper plates which cover the magnet arrays (not shown here), resulting in a current density of ∼70 A/m2 . As a calibration target a regular array of 7×9 points with a grid distance of 2mm is used. plate a 4 by 4 array of the permanent magnets described above, here with axial length of 40mm, is mounted. The magnets are arranged in such a way that positive and negative magnetic fluxes alternate, forming a chessboard pattern. The flow is seeded with neutrally buoyant polystyrene particles of 40 − 60µm in diameter. The seeding density of the detected particles is around 50particles/cm3 which corresponds to a typical particle distance of ∼ 2.7mm. Since the particle size is comparable to debris occurring naturally the fluid is pumped through a 10µm filter several times. Before measuring the test fluid is at rest for several hours to allow air bubbles of size of order 1 − 100µm to surface. For illumination two 400W att metal halide lamps with reflector (Osram) are focused onto the observation volume with two cylindrical lenses (f = 150mm) for each lamp. The setup produces a light sheet of 25mm thickness. Due to the relatively high power of the metal halide lamps (2 × 400W att), a significant amount of heat is produced. This results in un-wanted, thermal forcing in the observation volume, producing velocities of order O(1mm/s).The flow is recorded with four JAI M 10 1/200 CCD progressive scan monochrome cameras with 8bit/pixel resolution, shown in figure 2.4. Their pixel resolution is 640 × 480. The cameras are positioned manually. Each camera is mounted on a linear stage with a travel range of 20mm. They stages are used to focus the macro lenses onto the observation volume. To align the camera axis the linear stages are mounted on ball bearings, as can be seen in figure 2.4. As lenses four Rodenstock macro lenses are used. Their characteristics are given in table 2.1. The camera orientations are calibrated with images of the calibration target in figure 2.3. The calibration target is a regular array of 7 × 9 points with a grid distance of 2mm. It is recorded at three different z − positions, according to figure 2.1, before and after the experiment. Calibration is

29

2.1. EXPERIMENTAL SETUP

Figure 2.4: The flow is recorded with four JAI M10 1/200 CCD progressive scan monochrome cameras with 8bit/pixel resolution. The pixel resolution is 640×480. The cameras are positioned manually. Each camera is mounted on a linear stage with a travel range of 20mm. The stages are used to focus the macro lenses onto the observation volume. To align the camera axis the linear stages are mounted on ball bearings. model mount magnification working distance F-stop (from m=2 numerical aperture) field coverage for 1/200 CCD distortion depth of field

M R03/8 C − M ount 0.3 90 mm 8 21 × 16 mm2 < 0.1 % ±3 mm

Table 2.1: Characteristics of the four lenses Rodenstock macro lenses used for both experiments. a crucial step of the procedure since particle position accuracy is determined mainly by the quality of the calibration. It is explained in great detail in Maas et al. (1991, 1992a, 1992b, 1993). From the calibration algorithm - discussed below in the section on 3D Particle Tracking Velocimetry - the particle position accuracy is estimated to be ∼ 3 microns. The synchronized images are stored on two PC’s, Pentium 200M hz, SCSI, at a frame rate of 30Hz. Each PC has a M atrox Genesis frame grabber card (GEN/F 64/8/ST D) from which a rate controller writes the images to three hard disks simultaneously. This system is capable of recording 1800frames which is equivalent to 60s, or in other words it is able to store 36M B/s over 60s. It has been developed by Maas (1992), Virant (1996) and Stüer (1999) and is described in great detail in their Ph.D. thesis. To force the flow a DC electric current of 7A is a applied. With the 0.32 × 0.32m2 copper plates

30

CHAPTER 2. METHOD

2 this is equivalent to a current density of ∼ 70A/m R . From a simple numerical model for the Lorentz force field (figure 2.2) the total Lorentz forces, V fL , over the N = 32 magnets is estimated to be ∼ 0.03N . With the active volume of the fluid, V , and the characteristic velocity huM i the mean rate of dissipation of energy, ε, at steady state can be estimated from expression (2.2) as · ¸ · 2¸ m N · huM i m · kg m 1 m3 ε∼ , · · 3· = . (2.2) 2 V ·ρ s s m kg s3

With huM i ∼ 0.1m/s, V = 0.0184m3 and ρ = 1050kg/m3 we obtain ε ∼ 1.5 · 10−4 m2 /s3 . From observation of the experiment it becomes clear that in the proximity of the copper plates the flow is more intense, while less isotropic, than in the observation volume which is in the center of the tank. The rate of dissipation, ε, can therefore be estimated to be of order O(1 · 10−6 − 1 · 10−5 m2 /s3 ). With η= and

µ

υ3 ε

¶ 14

(2.3)

³υ ´1 2

(2.4) ε the Kolmogorov scale, η, and the Kolmogorov time, τ η , can be estimated to be around 0.6 − 1mm and 0.3 − 1s respectively. These two scales are briefly discussed in 3.1 Governing equations and relevant scales, further below. Usually the observed smallest scales in turbulent flows are a factor 3 − 5 higher than η which is derived on a dimensional basis only. Together with the typical particle distance of ∼ 2.7mm mentioned above this indicates that the flow measurements could just be resolved high enough. τη =

2.1.3

Second experiment

After processing the data obtained in the first experiment the need of improvement of the measurement system was quickly felt. The second experiment differed from the first with respect to two items. First a 60Hz camera system was used, second the experiment was placed in a smaller water tank. The reasons for the development of a 60Hz camera system was the promise of an improved trackability of particles and an increased number of frames per Kolmogorov time, τ η , as compared to the first experiment. The new system is based on the existing 30Hz version, and has been custom developed by BS Engineering. Other components were also changed based on experience with the first experiment. As will be demonstrated below the overall performance improved not only in quality but also in the amount of statistical data available. The flow is again produced as schematically shown in figure 2.1 but now in a much smaller volume of 120 × 120 × 140mm3 (figure 2.5) for two reasons. First, the resulting total volume of 5` of the test fluid can be handled in a much easier fashion. Filtering can be done without the use of a pump and, more important, from the first experiment it became clear that the fluid needs to be de-gassed. The available 10` facility is used to vacuumize the fluid for several hours which effectively left a fluid with no visible oxygen bubbles that might contaminate the recorded images. Second, from the estimated and later measured Kolmogorov time, τ η , it became desirable to have a flow with comparable velocities but smaller time scales to increase recording time in terms of τ η , as compared to the first experiment. It can be seen from expression (2.2) and (2.4) that the overall τ η is proportional with the square root of the volume, τ η ∼ V 1/2 .Behind each copper plate now an array of only 2 by 2 permanent magnets is mounted, with an axial length of 20mm each. The magnets are arranged in such a way that the resulting swirls are analogous to a ’milling’ flow in the laminar situation.

31

2.1. EXPERIMENTAL SETUP

Figure 2.5: The setup of the second and final experiment is shown. The flow is again produced with electromagnetic forcing as schematically shown in figure 2.1, but now in a much smaller volume of 120×120×140 mm3 . The observation volume is illuminated with a continuous Argon-Ion laser. Before entering the flow through the bottom of the tank the laser beam is expanded by two cylindrical lenses and reflected by a mirror. In the back the four recoding CCD cameras, mounted on linear stages and ball bearings, can be seen. The flow is seeded with neutrally buoyant polystyrene particles but they are of 40µm in diameter only. The recorded images of the first experiment reveal that the 40 − 60µm particles occupied a pixel area slightly too large, which leads to many situations were particles overlap. The seeding density of the detected particles is only slightly increased to 75particles/cm3 which corresponds to a typical particle distance of ∼ 2.3mm for better spatial resolution of the velocity field. Increasing seeding density too much, however, would spoil the improved particle trackability. The velocities resolved by PTV can roughly be estimated as umin = errorposition · frame rate

(2.5)

umax = particle distance · frame rate.

(2.6)

and For the first experiment expressions (2.5) and (2.6) give umin = 0.1mm/s and umax = 80mm/s and for the second experiment umin = 0.2mm/s and umax = 140mm/s respectively. For illumination of the second experiment a continuous 5W att Argon Ion Laser was available. In the first experiment, the light of two metal halide lamps needs to be focused onto the observation volume, whereas now the laser beam only needs to be widened to a laser sheet which is spread to a sheet of thickness 20mm with two cylindrical lenses of f = 80mm. Besides improved ease of handling there is a physically relevant advantage of this light source. The observation volume is exposed to much less thermal energy and the flow is no longer influenced by buoyant forcing. Lenses, cameras and their mounting are left unchanged as compared to the first experiment. Calibration is done in an alternative way using a calibration block where 36 points are distributed

32

CHAPTER 2. METHOD

in a volume of 18 × 15 × 7mm3 . This is schematically sketched in figure 2.6. The obtained particle position accuracy is again ∼ 3microns. The 60Hz image recording system is based on the 30Hz version and is identical up to the stage of the M atrox Genesis frame grabber cards. The storing of the image streams from the frame grabbers onto the hard disks has been identified as the major bottle neck of the process in the first experiment. Even the elaborate technique with rate controllers produces internal overflows after a few seconds. Therefore, in collaboration with BS Engineering, the images are now stored in two PCs 4GB RAM memories. The new system now records at 72M B/s over 100s which corresponds to 6000 f rames, as compared to 1800 frames of the first experiment. The flow is forced with a DC electric current of 1A. With the smaller copper plates, of size 0.12 × 0.12m2 , this is equivalent to a current density of again ∼ 70A/m2 , which reduces the overall force by a factor four since only 8 magnets are used as compared to 32. With the Lorentz forces reduced by a factor four but the volume reduced by a factor nine, as compared to the first experiment, the overall rate of dissipation can be estimated to be twice as high, as ε = 3.4 · 10−4 m2 /s3 , again using expression (2.2). Kolmogorov scale, η, and Kolmogorov time, τ η , are estimated to be around 0.5 − 0.8mm and 0.2 − 0.6s respectively. In terms of recorded frames per τ η the second experiment can be expected to have a Kolmogorov time resolved with 11−36 frames whereas the first experiment has an estimated 9 − 30 frames per Kolmogorov time scale, τ η .

2.2

3D Particle Tracking Velocimetry

The 3D Particle Tracking Velocimetry (3D-PTV) technique is a flexible non intrusive image analysis based flow measurement technique. It measures particle trajectories from which - if the particles are neutrally buoyant and small enough to perfectly follow the flow - particle velocities and Lagrangian accelerations can be obtained directly. As a measure of just how perfectly a particle follows the flow the Stokes number, St, is used, as defined in expression (2.7). It compares a particle’s response time, τ p , with a characteristic time scale of the flow here the Kolmogorov time, τ η . τp = St = τη

(ρp /ρf )d2p 18ν

τη

=

ρp d2p ε1/2 18ρf ν 3/2

(2.7)

For both experiments the Stokes number is of order O(10−4 ) which means that the particles used indeed perfectly follow the flow. The range of minimal and maximal particle velocities resolved by this method is roughly given by the product of particle position accuracy and recording rate (expression 2.5) and by the product of particle separation and recording rate respectively (expression 2.6). The idea of 3D-PTV was introduced by Chang and Taterson (1983), and was further developed by Racca and Dewey (1988), Maas (1991, 1992a, 1992b, 1993, 1995, 1996) and Malik et al. (1993). The technique has a history of development for more than a decade at the Institute of Geodesy and Photogrammetry (IGP) and at the Institute for Hydromechanics and Water Resources Management (IHW) both at the Swiss Federal Institute (ETH) Zürich. In a cooperation of both institutes, hardware and software have been tested and further developed and the method has been applied to several flow problems during the course of three Ph.D. theses (Maas, 1992b, Virant, 1996, Stüer, 1999) and two ongoing Ph.D. theses, Willneff (to be submitted 2003) and the one presented here. 3D-PTV can be divided into two major parts, (i) determination of particle positions in spatial coordinates and (ii) tracking of individual detected particles through consecutive images. While in former implementations these two steps were completely separated Willneff and Gruen (2002) combined the two steps and developed a new spatio-temporal matching method which improves trackability of particles by 10 − 30%. The first experiment has been processed with the former version and the second experiment could benefit from the latter.

2.2. 3D PARTICLE TRACKING VELOCIMETRY

33

The 3D-PTV technique has reached a status of an operational and reliable measurement tool that is used in hydrodynamics and space applications. Instead of repeating all the technical details which are described in great detail in the publications by Chang and Taterson, Racca and Dewey, Maas, Malik et al., Virant, Stüer and Willneff mentioned above, only the principal ideas that lead to the new spatio-temporal matching method and some specific algorithm parameters, used for the processing of the two experiments, will be described in the following. Particle trajectories which are longer than the relevant Kolmogorov scales, η, and τ η , are the key prerequisite for a Lagrangian flow analysis. Due to interruptions of particle trajectories caused by unsolved ambiguities the number of long trajectories decreases exponentially with the trajectory length (Willneff and Gruen, 2002). So far, the probability of ambiguities could only be reduced by using low particle seeding densities with the negative effect of drastically reducing the spatial resolution, which in turn made determination of velocity derivatives impossible. Willneff’s new idea was to exploit the redundant information in image and object space more effectively. Predictions of particle motion, based on particle tracking in image and object space, are used to resolve ambiguous particle image positions and correspondences. In other words, ’temporal’ information of time, t, or frame, i, is used to solve ’spatial’ uncertainties regarding the existence and positions of particles in the next time step t + ∆t, or frame, i + 1.

Figure 2.6: A schematic of the calibration block, where x, y are the positions and ω, ϕ the rotation angles of each camera, as determined from the calibration procedure for the second experiment, are shown. z , the distance of the cameras from the tank is between 59mm and 63mm. The rotations, κ, around the z-axis are between 2◦ to 4◦ . The following calibration, particle detection, particle correspondence and particle linking algorithms and procedures were and are developed by IGP, ETH Zurich. The calibration procedure predicts a particle position accuracy of ∼ 3microns. Additionally, it gives the camera positions, xi , yi , zi , and their rotations ω i , ϕi , κi around the axes x, y and z. Since

34

CHAPTER 2. METHOD

grayvalue threshold tolerable discontinuity min #pixels max #pixels min #pixels in x direction max #pixels in x direction min #pixels in y direction max #pixels in y direction min sum of grayvalues

1st experiment 5 8 3 600 2 20 2 20 30

2nd experiment 7 10 4 400 2 20 2 20 50

Table 2.2: After high pass filtering of the recorded images a particle detection algorithm, provided by IGP, determines particle image coordinates. In this table the specific detection parameters used for both experiments are given.

min corr for ratio #x min corr f or ratio #y min corr for #pix min corr f or sum of grayvalues min f or weighted correlation tolerance to epipolar band

1st experiment 0.02 0.02 0.02 0.02 15 0.035 mm

2nd experiment 0.02 0.02 0.02 0.02 10 0.025 mm

Table 2.3: A particle correspondence algorithm, provided by IGP, finds for particles of one image corresponding particles in the remaining three images. The intersection of three or four epipolar lines of corresponding particles is then used to determine their positions in space. In this table the parameters used for both experiments are given. the cameras are positioned manually this gives useful information about the symmetry of the setup. As an example figure 2.6 shows x, y positions and ω, ϕ rotation of each camera as determined from the calibration procedure for the second experiment. z, the distance of the cameras from the tank is between 59mm and 63mm. The rotation angle, κ, around the z − axis is between 2◦ to 4◦ . For the second experiment the image signal from camera to frame grabber is transmitted through two cables. As a negative effect this results in different signal magnitudes for even and odd image lines. This deficiency is corrected by mapping each grayscale value from odd lines with the corresponding mean of upper and lower even line values.The resulting grayscale mapping and its effect is demonstrated in figure 2.7, where a typical particle image and a magnified area before and after the odd-even correction are shown. After high pass filtering the images a particle detection algorithm determines particle image coordinates. The specific detection parameters used for both experiments are given in the table 2.2. A particle correspondence algorithm finds for particles of one image corresponding particles in the remaining three images. The intersection of three or four epipolar lines of corresponding particles is then used to determine their positions in space. The correspondence parameters used for both experiments are given in table 2.3. To track particles, i.e. to find corresponding particles in image and object space of consecutive time steps mainly three criteria are used for effective assignment. First, a 3D search volume is defined by minimum and maximum velocities in all three coordinate directions. Second, the Lagrangian acceleration of a particle is limited, defining a conic search area. Third, in case of ambiguities the particle leading to the smallest Lagrangian acceleration is chosen. Similarities in brightness, width,

2.2. 3D PARTICLE TRACKING VELOCIMETRY

35

Figure 2.7: At a recording rate of 60Hz the signal from camera to frame grabber is transmitted through two cables. As a negative effect this results in different signal magnitudes for even and odd image lines. This deficiency is corrected by mapping each grayscale value from odd lines with the corresponding mean of upper and lower even line values. Here the effect of grayscale-correction is shown for one examplary image, before and after the procedure.

height and sum of gray values of the pixel of a particle image in two consecutive time steps could have been used as a fourth criteria. This proofed to be not as valuable as expected and the parameters defining this criteria were set very high. In table 2.4 the settings for the tracking algorithm used in

36

CHAPTER 2. METHOD

dV (mm/f rame) dAcc (mm/f rame2 ) angle ∆ sum of gray values ∆ size in pixels ∆ size in x ∆ size in y

1st experiment 0.5 0.5 − − − − −

2nd experiment 0.6 0.6 110◦ 1000 100 10 10

Table 2.4: In this table the settings for the algorithm, IGP, used to find corresponding particles in image and object space of consequtive time steps are given. For effective assignment three criterias are used: (i) a 3D search volume is definded by minimum and maximum velocities in all three coordinate directions, (ii) the Lagrangian acceleration of a particle is limited, defining a conic search area, (iii) in case of ambiguities the particle leading to the smallest Lagrangian acceleration is chosen. Similarities in brightnes, width, height and sum of gray values of the pixel of a particle image prooved to be not as valuable as expected. both experiments are given. For the first experiment typically 500 particles per frame are assigned a position in space. The ratio of triple correspondences to quadruplets is 25% to 75%. The tracking efficiency of 60% leads to a typical value of 300 links per frame and the final number of linked particles belonging to trajectories of length 5 frames or longer is around 250 particles per frame. For the second experiment an average of 800 particles per frame could be assigned a position in space with a triplets-quadruplets ratio of 30% to 70%. An increased tracking efficiency of 70% leads to typically 500 linked particles per frame which belong to trajectories over 5 frames or longer. For a qualitative illustration of the result of the procedures described above figure 2.8 shows trajectories of the second experiment that start within the first second of recording and that can be tracked over 0.3s or longer.

2.3

Trajectory processing

From the derived particle trajectories the flow velocity, u, and Lagrangian acceleration, a, can be derived directly trough expressions (2.8) and (2.9). 1 ui (t) = 2 1 ai (t) = 2∆t

µ

µ

xi (t + ∆t) xi (t − ∆t) − ∆t ∆t

¶

xi (t) xi (t − ∆t) xi (t + ∆t) −2 + ∆t ∆t ∆t

(2.8) ¶

(2.9)

For the first and the second experiment - according to the Nyquist criterium - only frequencies, α, of a position signal xi (t), of up to 15Hz and 30Hz respectively can be resolved properly. From the considerations above, a minimal Kolmogorov time, τ η , can be estimated to be ∼ 0.2s which corresponds to a maximal position signal frequency, α, of ∼ 5Hz. In the following the details of a procedure to derive filtered velocities, ubi , and accelerations, abi , along particle trajectories will be described. The procedure acts as a low pass filter with a cut-off frequency of ∼ 10Hz. This filter reduces noise stemming from particle position inaccuracies which would result in aliasing effects on the lower frequencies of the velocity and acceleration spectra. The results are compared on a qualitative level to those obtained directly through expressions (2.8) and (2.9) along an exemplary trajectory in figures 2.10 and 2.11.

37

2.3. TRAJECTORY PROCESSING

Figure 2.8: To get a qualitative idea of the flow as obtained from 3D-PTV some trajectories of the second experiment are shown. The selected trajectories begin within the first second of recording and can be tracked over 0.3s or longer. For each time step, t, and around each point of a measured trajectory a third order polynom is fitted from t − 10 · ∆t to t + 10 · ∆t for each component, x1 , x2 , x3 . This results in a fit to 21 measured trajectory points. The constants, ci , for the polynomial of type xi (t) = ci,o + ci,1 t + ci,2 t2 + ci,3 t3 are determined as where

and



¢−1 ¡ ¢¡ , ci = AT xi AT A

(2.10) (2.11)

 1 t − 10 · ∆t (t − 10 · ∆t)2 (t − 10 · ∆t)3  1 t − 9 · ∆t (t − 9 · ∆t)2 (t − 9 · ∆t)3   A=  ...  ... ... ... 2 3 1 t + 10 · ∆t (t + 10 · ∆t) (t + 10 · ∆t)

(2.12)

 xi (t − 10 · ∆t)  xi (t − 9 · ∆t)  . xi =    ... xi (t + 10 · ∆t)

(2.13)



Position, velocity and acceleration, xbi (t), ubi (t) and abi (t), after filtering are now obtained as ¡ ¢ ¡ ¢ xbi (t) = ci,o + ci,1 (t) + ci,2 t2 + ci,3 t3 , (2.14) ubi (t) = ci,1 + 2ci,2 t + 3ci,3 t2

(2.15)

38

CHAPTER 2. METHOD

and abi (t) = 2ci,2 + 6ci,3 t.

(2.16)

The frequency response of this ’moving’ cubic spline was computed for two recording rates of 30Hz and 60Hz by sampling and processing cos (αt) signals for α’s ranging from 0.1 − 60Hz with the procedure described above. Figure 2.9 shows the response of this procedure - that effectively acts as a low pass filter - for the 60Hz case. The cut-off frequency of velocities and accelerations is around 10Hz and 7Hz respectively and the response for frequencies up to 5Hz is above 0.95. Damping is maximal at the Nyquist frequency of 30Hz. This corresponds well to the low pass filter characteristics desired. For the 30Hz case, not shown here, the same characteristics are obtained if the cubic spline is fitted to 15 measured trajectory points.To demonstrate the effect and the

Figure 2.9: The frequency response of the ’moving’ cubic spline procedure - used to filter the positionand thus also the velocity- and acceleration-signals along a particle trajectory - is shown. The procedure effectively acts as a low pass filter, with a cut-off frequency for velocity and acceleration signals of ∼ 10Hz and ∼ 7Hz respectively. The response for signal frequencies up to 5Hz is above 0.95. Damping is maximal at the Nyquist frequency of 30Hz. necessity of this ’spline’ procedure on a qualitative level, figure 2.10 shows two renderings of an arbitrarily selected trajectory from the second experiment. For this trajectory a particle is tracked through 80 time steps. The arrows represent Lagrangian acceleration, a. The trajectory on the left has been processed with the ’spline’ procedure while a of the right trajectory is obtained with the central approximations of expressions (2.8) and (2.9).To see this effect in more detail in figure 2.11 the velocity and acceleration components ui and ai and ubi and abi of the same trajectories are plotted over time for both procedures. High frequency velocity fluctuations of order O(1mm/s) for the third component and of order O(< 1mm/s) for components 1 and 2 are strongly dampened (figure 2.11 a). The larger fluctuations of the third component are a first indication of the lower accuracy of measurements in z − axis. This can be explained with the relatively small camera angles of ∼ 20◦ as shown in figure 2.6. The low camera angles are the result of a trade-off between observable volume and accuracy along the z − axis. In addition, velocities along the z − axis are

2.4. VELOCITY DERIVATIVES

39

Figure 2.10: The effect of the ’moving-spline’ procedure is demonstrated on a qualitative level. Two renderings of an arbitrarily selected trajectory from the second experiment are shown. For this trajectory a particle is tracked through 80 time steps. The arrows represent Lagrangian acceleration, a. The trajectory on the left (blue) has been processed with the ’moving-spline’ procedure while a of the trajectory on the right (red) is obtained using a central approximation only. more difficult to measure due to the limited depth of field of ±3 mm. If a particle moves away from the focus plane its shape changes from a circle to an ellipsoid. Together with illumination effects the apparent center of gravity of a particle image becomes more difficult to determine. Figure 2.11 b) demonstrates the necessity of low pass filtering accelerations obtained from expression (2.9). The unprocessed fluctuations appear completely uncorrelated in time and the magnitude of their ¢ ¡ fluctuations of order O 0.1m/s2 is clearly too high.

2.4

Velocity derivatives

From the velocities, derived with the above described procedure, spatial and temporal velocity ∂ui derivatives are interpolated for every particle trajectory point. The interpolation procedures for ∂x j i and ∂u ∂t at position x0 are linear and use weighted contributions from particles according to their distances, kxi − x0 k, to x0 .

40

CHAPTER 2. METHOD

Figure 2.11: Velocity and acceleration components ui , ubi (a) and ai , abi (b) of the same trajectories as shown in figure 2.10 are plotted over time. Black lines are used for the components as obtained from central approximations and red lines for the components as obtained from the low pass filtering ’moving-spline’ procedure. The necessity for filtering becomes especially clear in b) where the unprocessed acceleration fluctuations appear completely uncorrelated in time.

2.4.1

Spatial velocity derivatives

For ubi (x0 ), i = 1, 2, 3, the linear Ansatz

ubi (x0 ) = ci,0 + ci,1 x1 + ci,2 x2 + ci,3 x3 ,

(2.17)

41

2.4. VELOCITY DERIVATIVES with ∂ui ∂x1 ∂ui ∂x2 ∂ui ∂x3

= ci,1 , = ci,2 , = ci,3 ,

can be made. Theoretically expression (2.17) for ui (x) with its four unknowns ci,0−3 could be solved for ci with the information of three points in the proximity of x0 , (x1 , x2 , x3 ) and the point at location x0 , only as (2.18) ci = vi A−1 , where



1  1 A=  1 1

and

x0,1 x1,1 x2,1 x3,1

x0,2 x1,2 x2,2 x3,2

 x0,3 x1,3   x2,3  x3,3

 ubi (x0 )  ubi (x1 )   vi =   ubi (x2 )  . ubi (x3 ) 

This however makes ci very sensitive to small errors of vi and it is therefore preferable to overdetermine the linear system, Aci = vi , by using the information of all available points, n, n À 3, that are within proximity of x0 , so that kx1 − x0 k < `. Clearly the linear Ansatz made in expression (2.17) is decreasingly valid for points with large r, r = k∆xk, ∆x = x1 − x0 . Their contribution should therefore be weighted less in an over-determined system of equations for ci . To choose proper weights according to r, the a priori knowledge of the spatial correlation functions ¶ µ ∂ui , ui (x + k∆xk) , (2.19) ρ ui (x) + ∆x ∂xj and ρ

µ

¶ ∂ui ∂ui (x) , (x + k∆xk) ∂xj ∂xj

would be desirable. However, a priori only the spatial correlation function of ubi is known. It is obtained as

ρ [ubi (x) , ubi (x + k∆xk)]

ubi (x) · ubi (x + ∆x) ρ [ubi (x) , ubi (x + k∆xk)] = ³ ´2 ubi (x) − ubi (x)

(2.20)

(2.21)

(2.22)

where (·) denotes averaging over all ∆x with equal k∆xk. It was found that the correlation function of ubi (expression 2.21) is a good conservative estimate of expression (2.19). It will therefore be used to determine the weights of individual equations of Aci = vi . With this procedure ci is obtained with the information of n points, n À 3, as ¡ ¢−1 ¢¡ , ci = AT vi AT A

(2.23)

42

CHAPTER 2. METHOD

where 

and

 (1 x0,1 x0,2 x0,3 )· 1  (1 x1,1 x1,2 x1,3 )· ρ [ubi (x) , ubi (x + kx1 − x0 k)]  , A=  ... ...  ... ... ... (1 xn,1 xn,2 xn,3 )· ρ [ubi (x) , ubi (x + kxn − x0 k)]  ubi (x0 ) · 1  ubi (x1 ) · ρ [ubi (x) , ubi (x + kx1 − x0 k)]  . vi =    ... ... ubi (xn ) · ρ [ubi (x) , ubi (x + kxn − x0 k)] 

For practical reasons only information of points with kxn − x0 k < 4 mm has been used, which resulted in a typical value for n of ∼ 20. For the second experiment at 4 mm the correlation coefficient ρ [ubi (x) , ubi (x + 4mm)] is about 0.5. The quality of the linear interpolation described above is strongly dependent on the local spatial correlation function of ∂ui /∂xj . In regions with ­ 2 ® 2 2 large ∇ u, ∇ u À ∇ u , the linear Ansatz made in expression (2.17) is valid for a much smaller region than is estimated from expression (2.21). This will result in a poor approximation of ∂ui /∂xj . Having no a priori knowledge of ∇2 u - and other effects that might influence the quality of the interpolation in a negative way- additional ’treatment’ of the ∂ui /∂xj tensor is necessary. This will be explained in the second next section: Velocity derivatives along trajectories.

2.4.2

Temporal velocity derivatives

For ubi (x) the linear Ansatz with

ubi (x) = ci,0 + ci,1 x1 + ci,2 x2 + ci,3 x3 + ci,4 t ∂ui ∂x1 ∂ui ∂x2 ∂ui ∂x3 ∂ui ∂t

(2.24)

= ci,1 , = ci,2 , = ci,3 , = ci,4 ,

can be made which is similar to expression (2.17) but has one additional term, namely ci,4 t. Again, a system of weighted linear equations is used to obtain ci (expression 2.25) with the weighted information of n points that are in the proximity of x0 . The points are spatially distributed in a sphere of radius r = 4mm and spread over 5 time steps which is roughly equivalent to 0.3τ η for both experiments. ¢−1 ¡ ¢¡ ci = AT vi AT A ,

with 

 (1 x0,1 x0,2 x0,3 x0,4 )· 1  (1 x1,1 x1,2 x1,3 x1,4 )· ρ [ubi (x) , ubi (x + kx1 − x0 k)]  , A=  ... ...  ... ... ... (1 xn,1 xn,2 xn,3 xn,4 )· ρ [ubi (x) , ubi (x + kxn − x0 k)]

(2.25)

43

2.4. VELOCITY DERIVATIVES and



 ubi (x0 , t0 ) · 1  ubi (x1 , t1 ) · ρ [ubi (x) , ubi (x + kx1 − x0 k)]  . vi =    ... ... ubi (xn , tn ) · ρ [ubi (x) , ubi (x + kxn − x0 k)]

It could be argued that the same procedure to derive ∂ui /∂t can be used to obtain the tensor of velocity derivatives, ∂ui /∂xj . The use of 5 time steps essentially smooths the information on ∂ui /∂xj over time. Therefore the spatial velocity derivatives, ∂ui /∂xj , are derived local in time with the separate procedure described above.

2.4.3

Velocity derivatives along trajectories

In order to compensate for effects that lead to inaccurate ­approximations of ∂ui /∂xj and ∂ui /∂t, such ® 2 2 2 as inaccurate or under-resolved ubi , large ∇ u, ∇ u À ∇ u , small space correlations of ∂ui /∂xj and possibly others, the fact that ∂ui /∂xj and ∂ui /∂t are - to some extent - auto-correlated in time along their trajectories is used in the following. The Lagrangian correlation of velocity derivatives is not known in advance but, as will be seen below, it does make sense to again apply a procedure that effectively acts as a low pass filter to velocity derivative signals, ∂ui /∂xj (t), along their trajectories. The desired cut-off frequency for the velocity derivative signal needs to be below 15Hz and 30Hz for the first and second experiment respectively. On the other hand, it should be high enough, in order not to dampen relevant fluctuations of ∂ui /∂xj and ∂ui /∂t. A variety of functional approximations, with which the desired low pass filter characteristics could be achieved, are thinkable, such as Fourier series, Legendre-polynomials or Taylor-polynomials. It was decided to use Taylor type polynoms (expression 2.27) of appropriate orders, n. The polynomial order, n, was chosen to be adequate for the specific trajectory lengths, `, and the quality of the approximations for each component of ∂ui /∂xj and ∂ui /∂t. As a measure for quality it was decided to define a relative divergence as ° ° ° ∂u1 ∂u2 ∂u3 ° ° ∂x1 + ∂x2 + ∂x3 ° ° ° ° ° ° (2.26) rel. div = ° ° ∂u1 ° ° ∂u2 ° ° ∂u3 ° . ° ∂x1 ° + ° ∂x2 ° + ° ∂x3 °

Estimation of the quality of ∂ui /∂xj (t) and ∂ui /∂t (t) according to relative divergence (expression 2.26) is done for several - mostly practical - reasons. First, divergence reflects incompressibility of water - a physical property which has not yet been used in any of the above procedures. Second, because of the way it is normalized, it treats flow situations with small and large ∂ui /∂xj equally fair. Third, it has no singularities, contrary to some other kinematic measures of interpolation quality. These exact kinematic relations are used as additional checks and are presented below in Checks and verification of procedures. And last, relative divergence is direction independent, i.e. it produces a scalar measure that is independent of the direction of the flow or vorticity or other quantities. To determine the order, n, of the polynomial n

X ∂ui ∂ui (t) , (t) = c0 + ck tk ∂xj ∂t

(2.27)

k=1

the following empirical expressions are defined as ¶ µ ` , n ∈ N, n = trunc 3.5 + 35

(2.28)

44

CHAPTER 2. METHOD

or, for weighted polynomials Ã

nw = trunc 3.5 +

P`

i=1 (1 − rel.

div (ti ))

35

!

, nw ∈ N.

(2.29)

Polynomials, as defined in expressions (2.27- 2.29) are fitted along every trajectory to each component of ∂ui /∂xj and ∂ui /∂t. Defining the order, n, of the polynomials according to expressions (2.28) and (2.29) is an arbitrary choice. It is based on the empirical observation that the order should be high enough to adequately resolve fluctuations of ∂ui /∂xj (t) and ∂ui /∂t¡(t) and ¢ that orders which are chosen higher result in poorly conditioned, close to singular matrices AT d of the linear system of equations, Ac = d, that needs to be solved to obtain the polynomial coefficients ci of expression (2.27). With ` as the trajectory length, c is obtained as ¡ ¢¡ ¢−1 c = AT d AT A

with 

(1  (1 A=  ... (1

t11 t12 ... t1` 

  d= 

t21 t22 ... t2`

∂ui ∂(·) ∂ui ∂(·)

(2.30)

 ... tn1 ) · w1 ... tn2 ) · w2  ,  ... ... n ... t` ) · w`

(t1 ) · w1 (t2 ) · w2 ... ∂ui ∂(·) (t` ) · w`



  , 

and wi , a quality function which is used to weight those ∂ui /∂xj (t) and ∂ui /∂t (t) with a low relative divergence highest, 1 . (2.31) wi = 1 − 1 + exp [−10 · (rel. div (ti ) − 0.25)] In the inset of figure 2.12 a) the weight wi as a function of relative divergence is shown. Figure 2.12 shows the behavior of an arbitrarily selected component of ∂ui /∂xj (t) and ∂ui /∂t (t), namely ∂u3 /∂x3 , along the same trajectory that was used above in figures 2.10 and 2.11, processed in five different ways. It demonstrates the positive impact on interpolation quality of ∂ui /∂xj of the ’moving spline’, ’polynomial’ and ’weighted polynomial’ procedures discussed above. From figure 2.12 a) in combination with figure 2.12 b) it becomes clear, that while all procedures that involve polynomial fitting lead to smooth velocity derivatives, only the procedures with ’moving-spline’ and weighted polynomial fitting produce qualitatively acceptable and correct results. The reduction of overall relative divergence that can be observed from the case central, polynomial to the case spline, polynomial, no weight, demonstrates the importance of filtering velocity signals before interpolating - from them - velocity derivatives in space. This is not clear a priori. It could have been argued that the information that is ’destroyed’ during the ’moving-spline’ filtering could be more valuable for interpolating velocity derivatives, than its high level of noise is of negative influence. The benefit of using weighted polynomial fits on interpolated velocity derivatives is confirmed by the significant decrease of relative divergence that can be observed in figure 2.12 b) passing from the case spline, polynomial, no weight to case spline, polynomial.

2.5. CHECKS AND VERIFICATION OF PROCEDURES

45

Figure 2.12: The qualitative effect of different procedures to obtain velocity derivatives along a particle trajectory is shown. In (a) for exemple the behavior of the component ∂u3 /∂x3 along the same trajectory that is used above for figures 2.10 and 2.11 is shown. The inset in (a) shows the weight, wi , as a function of relative divergence. Figure (b) shows the relative divergences as obtained from the different procedures along the corresponding trajectory. Only the procedure with ’moving-spline’ and weighted polynomial fitting produces qualitatively acceptable results.

2.5

Checks and verification of procedures

The procedure described above involves a long chain of measurements, image processing, linear approximations and polynomial fitting before it actually produces the desired temporal and spatial

46

CHAPTER 2. METHOD

30Hz 60Hz (·)60Hz (·)30Hz

recording time (s) 60 100 1.7

τ η (s) 0.31 0.23 0.75

f rames τη

10 14 1.4

¡1¢ s

recording time τη

(−)

194 435 2.2

Table 2.5: Some characteristics of the statistical sets are compared for the 30Hz and the 60Hz experiments. In the 60Hz experiment the Kolmogorov time, τ η , decreased and both, frames per τ η and recording time per τ η increased. velocity derivatives. To get an idea on how much the improved experiment actually helped to improve the quality of the data and whether the final accuracy of the data is sufficient for our purpose, a number of checks were performed. These checks are based on precise kinematic relations and can be divided into single and multi point checks. It is important to keep in mind that the experiment and the post processing was designed to produce statistically correct results along particle trajectories and not for highly resolved instantaneous spatial fields or high accuracy point measurements. The latter two could be obtained much easier via other experimental methods. Additionally it needs to be shown that the set of data that remains at the end of the procedure is still significantly representing the entire flow. In other words we have to make sure that no or only a tolerable amount of filtering occurs through PTV, linear interpolation, polynomial fitting and selection based on the divergence criterion. This comparison can of course only be performed for quantities which can be measured directly from the trajectories, such as velocity, u, and Lagrangian acceleration, a.

2.5.1

Statistical checks

As discussed above two experiments, one with a 30Hz recording system in a larger tank and one with a newly developed 60Hz recording system in a smaller tank, were conducted and processed. Before discussing precise kinematic checks of the derived velocity derivatives a quantitative analysis of the magnitude of the obtained statistical sets for the two experiments is performed. When processing the data obtained from the 30Hz experiment it is found that many statistics such as PDFs of various quantities and geometrical alignments converge only just. It appears that the set of independent data is slightly too small. Therefore part of the motivation for the 60Hz experiment is to increase the number of recorded frames per Kolmogorov time, τ η , in order to increase trackability, but also to reduce τ η to get from the 100s available a recording time as long as possible in terms of τ η . Table 2.5 shows how these numbers which characterize the statistical sets changed from the 30Hz to the 60Hz experiment, i.e. how much the quality of the statistical set of the 60Hz experiment improved. The increase of recording time over τ η of 2.2 is a first indication that the 60Hz experiment actually can improve the statistics. In figure 2.13 the two sets are compared in terms of lengths of trajectories that could be tracked. The number of trajectories of a certain length is higher by a factor 2 for the 60Hz compared to the 30Hz experiment. This factor increases with increasing lengths. The same trend can be observed for the number of points belonging to a specific class of trajectory length as for the sum of all available points belonging to a trajectory length equal to or greater than a certain length. At a summation starting from trajectory length 2τ η we find a total of 0.5 · 106 points for 30Hz and 2.2 · 106 points for 60Hz. Keeping the ratio of frames per τ η of 1.4 between the two experiments in mind this is still an increase of a factor of ∼ 3. This suggests that improved trackability, partly due to the increased rate of frames per τ η and to an equal part due to

2.5. CHECKS AND VERIFICATION OF PROCEDURES

47

the latest version of the tracking algorithm developed by Willneff and Gruen (2002), is responsible for the larger available data set than could be expected from the ratio of recording time per τ η of 2.2 between the 60Hz and 30Hz experiments.To get an idea if and how different sets of the flow

Figure 2.13: The number of trajectories of a certain length, the number of points belonging to a certain trajectory length and the summation of points with a trajectory length equal to or greater than a specific trajectory length are shown for experiment 30Hz and 60Hz. The 60Hz set is always at least twice as large as the 30Hz set. With increasing trajectory length tracking becomes more difficult. are representative of the actual physical flow, we look at two quantities from the 60Hz experiment that are obtained from 3D-PTV directly, namely the kinetic energy, k = 12 ui ui , and the square of Lagrangian acceleration, a2 = ai ai , from three different viewpoints: -Influence of 3D-PTV, i.e. how is trackability of particles affected by k or a2 ? -The polynomial fitting can act as a low pass filter if its order is chosen too low for the actual behavior of the particle trajectory. -The selected criterion for a quality estimate of the local velocity derivatives could also act as a low pass filter. ­ ® Figure 2.14 shows that in the ’no polynom’ case both the means of hki and a2 , conditioned on the length of the trajectory they belong to, decrease with increasing ­ ® trajectory length. hki drops by a factor two over a length difference of 4τ η (figure 2.14 a) and a2 even by an order of magnitude (figure 2.14 b). This is quite severe and has to be kept in mind when looking at results in the chapter further below. However the situation is not that bad because, as we will see below, the field of velocity derivatives is - while being a very subtle matter - not quite as sensitive to trajectory length. This is especially true if one looks not at the magnitudes of individual terms of the ∂ui /∂xj tensor, but more at its internal structure. By internal structure we mean the statistics on geometrical invariants of ∂ui /∂xj , such as alignments, production and stretching rates of various quantities. The decreasing trackability of particles especially with high a2 indicates that the intensity of the measured flow field is rather at the limit of the system capabilities. However, taking a much slower flow would not

48

CHAPTER 2. METHOD

low a2 high a2

probability to reach next frame 0.99 0.98

low a2 high a2

2τ η = 14 frames

7τ η = 100 frames

0.75 0.57 1.3

0.37 0.13 2.8

Table 2.6: Assuming that two sets of a a2 - with low and high intensities - have a probabilty of beeing tracked to their next frames of 0.99 and 0.98 - then, the ratio of trackable ’low’ a2 over ’high’ a2 more than doubles over a time period from 2τ η to 7τ η . solve anything, because then τ η would increase and produce a data set much smaller - in the sense of recording time per τ η - than the one obtained from the 60Hz experiment. To put the tracking sensitivity to a2 in the right perspective - that is, that even a very reliable tracking algorithm is very sensitive - a simple consideration is presented. Let’s assume that particles with a ’low’ value for a2 have a probability of 0.99 of being tracked to the next frame and particles with a ’high’ value for a2 have a slightly lower probability of only 0.98. A simple calculation shows that the ratio of tracked a2 that are weak over tracked a2 that are intense more than doubles over a time period from 2τ η to 7τ η , as can be seen from the numbers given in table 2.6. The polynomial fitting and the selection according ­ 2 ® to relative divergence has an only negligible influence on the conditional means of hki and a . However, the statistical behavior of the two cases using the polynomials is ’rougher’ (figure 2.14 b). Velocities and Lagrangian accelerations are obtained in a very straight-forward way from PTV and the ’moving-spline’ procedure. It appears that any further processing is not beneficial. In the case of polynomials not much can be gained - on the contrary, through a non unique selection of the order, n, of the polynomial a mismatch between these fits and the ’raw’ but more accurate PTV data - that is processed only with the moving spline procedure - is unavoidable. For further results it was decided to use for u and a only low-pass filtered PTV data, calculated from expressions (2.15) and (2.16).For the velocity derivatives however the picture is quite different. For them we do not have direct access through 3D-PTV but only from linear interpolation of varying quality along particle trajectories. To ensure their accuracy it is necessary to make use of the fact that ∂ui /∂xj and ∂ui /∂t are auto-correlated in time along their trajectories and to apply the weighted low pass filter procedure that was described above in Velocity derivatives along trajectories. The necessity and the positive effect of this procedure will be presented further below.The PDFs of kinetic energy, k, and squared Lagrangian acceleration, a2 , for trajectory lengths of τ η = 1 and τ η = 4 are shown in figure 2.15. As can be expected they show a 2 shift from strong ­ 2 ® events to weak events, especially for ­ 2 ®the case of a . The PDFs help to understand in part why a is­ so® much more sensitive than k . In the case of kinetic energy we see from figure 2.14 a) that k2 has a range from roughly 4 · 10−5 to 8 · 10−5 m2 /s2 . From the PDF of k in figure 2.15 a) it can be seen that events around this mean are only 0.2 − 0.5 times­ less ® likely than the most probable events. For accelerations the situation is quite different. Here a2 has a range from 8 · 10−4 to 40 · 10−4 m2 /s4 and from its corresponding PDF (figure 2.15 b) we see that this mean is governed by extremely rare events which have a probability to occur that is two to three orders of magnitude smaller than the most probable events. In other words, the current 3D-PTV system is able to capture a set of points for long trajectories with only half as many ’strong’ but highly unlikely events as for short trajectories.

2.5.2

Single Point Checks

To check the quality of velocity derivatives a number of precise kinematic relations can be used. It should be stressed that all the following checks are purely local. Due to incompressibility of water

49

2.5. CHECKS AND VERIFICATION OF PROCEDURES

Figure 2.14: The mean kinetic energy, < k >= 21 ui ui (a), and the mean square of Lagrangian acceleration, < a2 >= ai ai (b), are conditioned on their trajectory length. The mean of both quantities decreases with increasing trajectories lengths, indicating that trackability is sensitive especially to a2 . The polynomial fitting and the selection according to relative divergence have a neglectable influence. it seems natural to start with the check on divergence. Ideally the trace of ∂ui ∂uj ∂uk + + = 0. ∂xi ∂xj ∂xk

∂ui ∂xj

should be zero, i.e.: (2.32)

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CHAPTER 2. METHOD

Figure 2.15: The PDFs of kinetic enrgy, k (a), and squared Lagrangian acceleration, a2 (b), for trajectory lengths of τ η = 1 and τ η = 4 are shwon. Apparently mainly the pdf for a2 is affected by longer trajectories.

∂u

∂uk j i Figure 2.16 shows the joint PDFs of − ∂u ∂xi versus ∂xj + ∂xk (here no summation over i, j, k) for three cases: 60Hz, the final stage of the experimental procedure; 60Hz, not using the polynomials; and 30Hz using polynomials. Before looking at the shapes of the joint PDFs the much higher amount of data points close to the diagonal of the fully processed, 60Hz experiment as compared to the others

2.5. CHECKS AND VERIFICATION OF PROCEDURES

51

needs to be mentioned. From the shapes which are very similar from column to column we see, that the flow is more or less isotropic a least on a level of spatial velocity derivatives. The difference between the 60Hz and 30Hz experiment is not only revealed in the amount of data but also in its quality. The aspect ratio of the shapes for 60Hz is slightly higher. The most interesting feature of figure 2.16 however is the strong effect of the weighted polynomials. Particularly if we take a look 3 at the behavior of the velocity component, wz = ∂u ∂x3 , along the axis, z = x3 , we note that all the shapes which already have a relatively low aspect ratio are ’stretched’ along the axis where wz is involved. The way the 3D-PTV was set up, the determination of a particle’s z position was the most difficult. Estimates of position accuracies were around 0.014mm for x and y components and around 0.045mm for the z component. This noise is clearly filtered out by the use of the weighted i polynomials, without damping the overall magnitude of the components of ∂u ∂xi . In other words, the weighted polynomials clearly enhance the quality of the measured and interpolated components ∂ui .The next check involves everything that was measured, namely, Lagrangian accelerations, of ∂x j ∂ui ∂ui i ai = Du Dt , local accelerations, al,i = ∂t , and convective accelerations, ac,i = uj ∂xj . Since they all are derived in a different manner they make an excellent (very hard to pass) overall check of the applied procedures, i.e.: ∂ui ∂ui Dui = + uj (2.33) Dt ∂t ∂xj

In figure 2.17 expression (2.33) is checked for each component with joint PDFs of ai versus al,i + ac,i . The emerging picture is similar to the one obtained from the divergence check. Generally the aspect ratios are much lower, but still on a tolerable level, if we keep in mind that we are after statistical results for qualitative aspects of turbulence.What is seen much clearer here than above is the zcomponent inaccuracy. Through the use of weighted polynomials it can only be corrected up to a certain degree. From the 60Hz no polynomial PDFs we see a slight misalignment to the diagonal for the components x and y and a huge misalignment for the component z. This can be caused by either a systematic underestimation of a or a systematic overestimation of al . Because of the checks above, a systematic error in ac can be ruled out. It is more likely that the noise is caused by al since ∂ui their interpolation is very much dependent on a high quality ∂x tensor, not only for a precise point j in time, t, but for a small interval around this time. Again, it needs to be noted that the weighted polynomial procedure clearly enhances the quality of the measured and interpolated components of ∂ui ∂ui ∂xi and ∂t .

2.5.3

Multi Point Checks

Exact kinematic expressions exist for three types of infinitesimal material elements. Namely, evolution of material lines, evolution of material surfaces and conservation of material volumes. Infinitesimal material elements are comprised by arbitrarily selected infinitesimally separated fluid points which connected can be looked at as either lines (2 connected points), surfaces (3 connected points) or volumes (4 connected points). Here we use actually measured tracer particles as points. Their separation (2mm − 4mm) is not infinitesimal but finite and most importantly - larger than the Kolmogorov scale, η. In this sense the kinematic expressions used below loose their strict validity and the measurements can therefore not be expected to pass the checks, unless the Kolmogorov scale, η, underestimates the smallest relevant scale by a significant factor. As will be shown this is really the case - apparently the smallest relevant scale is ∼ 5η.When selecting material elements arbitrary choices have to be made on the maximum initial separation of the particle pairs, triplets or quadruplets and on how long they can be tracked in time. In order to compare the checks for lines, surfaces and volumes for both experiments, 30Hz and 60Hz with the same criteria, the initial separation was chosen to be no larger than 3.5mm and the tracking time had to be no shorter than 0.7τ η which corresponds to 10 time f rames. A shorter distance and a longer tracking time would

52

CHAPTER 2. METHOD

∂u

∂uk j i Figure 2.16: The joint PDFs of ∂u ∂xi versus ∂xj + ∂xk (here no summation over i, j, k) for three cases, 60Hz with full numerical processing, 60Hz, not using the polynomials and 30Hz with full numerical processing. The superior behaviour of the 60Hz experiment with full numerical processing can be seen both in amount of data - as compared to 30Hz - and alignment to the diagonal - as compared to 60Hz, not using the polynomials.

have resulted in much too small data sets especially for the 30Hz data and the material volumes. The Lagrangian change of an infinitesimal material line element, l, is described by: ∂ui dli = lj dt ∂xj

(2.34)

It allows for a precise check (under the assumption of infinitesimality) of how well the paths of two particles are consistent with the local velocity derivatives since a material line, l, can be measured directly with 3D-PTV by following two particle positions in time and the r.h.s. of expression ∂ui which is obtained through interpolation in space. Figure 2.18 shows for each (2.34) involves ∂x j component joint PDF plots of the l.h.s. versus the r.h.s. of expression (2.34) for the 60Hz and 30Hz experiments. All cases as well as all components appear to be free of any systematic error.

2.5. CHECKS AND VERIFICATION OF PROCEDURES

53

∂ui ∂ui i Figure 2.17: The expression Du Dt = ∂t + uj ∂xj is checked for each component with joint PDFs of ai versus al,i +ac,i for the three cases 60Hz, 60Hz no polynomial and 30Hz. It reveals the strong increase of accuracy due to the weighted polynomials.

The aspect ratios of the shapes are again highest for the 60Hz experiment with the full polynomial procedure, confirming the general findings from the single point checks. The good agrement with ∂ui dli dt = lj ∂xj comes as a slight surprise if one considers that the particle pairs were allowed a relatively large initial separation of 3.5mm, which is ∼ 5η scales, a distance at which it might very well be ∂ui that the local approximation for ∂x has already lost its validity. This is a first indication that the j Kolmogorov scale η underestimates the smallest relevant scale of a turbulent flow by a factor of at least 5. The Lagrangian change of an infinitesimal material surface element, N, is described by: ∂uj dNi = −Nj dt ∂xi

(2.35)

where N is the normal of a surface defined by two material lines l1 and l2 . N = l1 × l2

(2.36)

This check is much harder because through the involved cross product in expression (2.36) any error can be expected to be squared also. The joint PDFs shown in figure 2.19 indeed have a rather noisy

54

CHAPTER 2. METHOD

∂ui i Figure 2.18: dl dt is plotted versus lj ∂xj in joint PDFs for the 60Hz, 60Hz no polynomials and 30Hz experiments and each component. The aspect ratios of the shapes are again highest for the 60Hz experiment with the full polynomial procedure, confirming the general findings from the single point checks.

character. While the shapes are aligned well with the diagonal their aspect ratios are much lower than those of the previous checks. It is believed that these inaccuracies are introduced through the above mentioned cross product of l1 × l2 . While noise of the particle positions is compensated ∂u in the determination of ∂xji through the use of typically 20 particles for the linear interpolation, N is defined with only three particle positions and their position error is roughly squared. This suggests that material elements obtained directly from 3D-PTV can be used only as an additional ∂u confirmation that the derived local velocity derivative tensor, ∂xji , is qualitatively adequate. To get reliable statistics of material elements themselves, they have to be obtained in a purely local manner, which will be outlined below. In the next section the check on material volumes will be discussed. There the benefit of determining the material elements locally becomes obvious.

2.5.4

Checks along trajectories

Due to incompressibility infinitesimally small material volumes are conserved along particle trajectories, i.e.:

55

2.5. CHECKS AND VERIFICATION OF PROCEDURES

∂u

j i Figure 2.19: dN dt is plotted versus -Nj ∂xi in joint PDFs for the 60Hz, 60Hz no polynomials and 30Hz experiments and each component. The aspect ratios of the shapes are again highest for the 60Hz experiment with the full polynomial procedure. Generally the shapes are still aligned well with the diagonal but their aspect ratios are relatively low.

dl·N = 0. dt

(2.37)

Alternatively to defining l and N directly from 3D-PTV they can be derived in a purely local manner. Referring to Monin & Yaglom (1997) and Girimaji and Pope (1990b) a material line element which is initially l(0), is given at any later time t by (2.38)

l(t) = B(t) · l(0), where B, the deformation matrix, evolves according to the equation d B = h(t) · B(t), dt

h(t) =

µ

∂ui ∂xj

¶

,

(2.39)

t

with the initial condition B(0) = I. The tensor B(t) contains all of the local infinitesimally small material element information at time t. The equation (2.39) for B is integrated in time using a third

56

CHAPTER 2. METHOD

order Runge Kutta scheme. Defining arbitrarily or choosing from 3D-PTV three line elements l1 (0), l2 (0) and l3 (0) an initial volume is defined as: volinitial = l1 · (l2 × l3 )

(2.40)

Figure 2.20 shows how these initial volumes evolve along particle trajectories by plotting the PDFs of the ratios of volumes over initial volumes for each time moment. For the 30Hz experiment the 3D-PTV volumes are conserved only up to 0.5τ η , for the 60Hz up to 1τ η . It appears that volumes defined by particles which are separated ∼ 5η length scales of each other are either only a very poor approximation of infinitesimal volumes or that the cumulative error produced by the triple product of l1 , l2 and l3 in expression (2.40) is ruining whatever accuracy there might be initially at time, τ η = 0. Results on material lines and surfaces suggest that the latter is more likely to be true. Volumes derived locally through the integrated deformation matrix B are however conserved up ∂ui is of adequate quality but, more to 7τ η .This is a further confirmation that the derived tensor ∂x j ∂ui importantly, it is the first indication that ∂x is accurate if integrated along particle trajectories. j A stronger check of this feature can be made by looking at the conservation of divergence of the material field l. The expression Dl = (l · ∇)u (2.41) Dt

or expression (2.34) as above Dli ∂ui = lj Dt ∂xj

(2.42)

∂l = rot(u × l) − u div l. ∂t

(2.43)

can be written as

Taking div of this equation, we find that for any material field, l D (div l) = 0, Dt

(2.44)

which means that div l is a pointwise Lagrangian invariant. This is important since no diffusivity is involved here. Expression (2.44) does not contain any velocity derivatives. However, l is determined ∂ui by the matrix B which in turn is defined by the matrix ∂x integrated along particle trajectories.Since j there exists an infinite number of material fields l and hence as many values for divergences of l, three initial sets of div l = −1, 0, 1 were created randomly and then the evolution of their initial values was monitored along their trajectories. The arbitrary values −1, 0 and 1 were chosen based on the following consideration. If the initial line elements l defining div l are normalized to a value ∂li are bounded between −2 of 1 then, again normalized over a distance 1, the maximal gradients ∂x i and 2, resulting in a PDF of div l centered around zero with a standard deviation of order O (1). Therefore, this selection allows for a meaningfull test of the conservation of div l: for a tolerable l) l) of ∼ 10% the initial value would still be detectable and for a D(div of more than amount of D(div Dt Dt 50% after a few τ η the initial PDF would loose its three peaks around −1, 0 and 1 quickly and obtain a random Gaussian shape with a peak around zero. As can be seen from figure 2.21 div l is conserved and only after a few τ η , corresponding to 100 time frames, the peaks of the PDFs are shifted slightly towards their random state around zero. This is a last strong indication that the tensor of velocity derivatives can be derived qualitatively correct not only local in time but also along a particle trajectory from 3D-PTV through the described processing.

2.5. CHECKS AND VERIFICATION OF PROCEDURES

57

Figure 2.20: The evolution of the ratios of volumes over initial volumes is shown by their PDFs for each point in time. For the 30Hz (a) the PTV volumes are conserved only up to 0.5τ η , for the 60Hz (b) up to 1τ η . For volumes derived locally through the integrated deformation matrix B (c) the volumes are conserved up to 7τ η .

2.5.5

Influence of trajectory length

Above in figure 2.14 and 2.15 we saw how the trajectory length of individual particles biased the statistics of the PDFs of kinetic energy k but especially of Lagrangian acceleration a. In this section

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CHAPTER 2. METHOD

Figure 2.21: PDFs of three different sets of div l which had initial values of −1, 0 and 1. Only after a few τ η the peaks are shifted slightly towards their random distribution, which is centered around zero. it will be checked if the situation is as critical in the case of velocity derivatives. There is at least one good reasons to hope that this is not the case. The velocity derivatives are the result of a linear ∂ui tensor at a single location. Even if this point interpolation typically involving 20 particles for a ∂x j is part of a long trajectory it is likely that many of the 20 particles contributing to the interpolation stem from much shorter trajectories. Therefore they should be compensating for the negative effect of trackability. From the above it can be concluded that trackability becomes more difficult if a particle makes fast movements which let it travel further than a typical particle distance or if it makes sudden changes of velocity in magnitude or direction, that is if a is large. In terms of velocity derivatives this means that trackability becomes difficult if large ω 2 is associated with large angular velocity of the local fluid volume - leading to large direction changes of particles - and large s2 with strong stagnation points - resulting in large changes of velocity in magnitude and direction. ∂ui is affected by the trajectory length, First we will look at how a selected internal quantity of ∂x j namely the quantity ’relative divergence’: ° ° ° ∂u1 ∂u2 ∂u3 ° ° ∂x1 + ∂x2 + ∂x3 ° ° ° ° ° ° (2.45) rel. div = ° ° ∂u1 ° ° ∂u2 ° ° ∂u3 ° . ° ∂x1 ° + ° ∂x2 ° + ° ∂x3 ° The entire idea of using the weighted polynomials along a trajectory is to selectively use only those points with a low relative divergence. Figure 2.22 is the most direct measure of the effectiveness of this concept. The comparison between the 60Hz and 60Hz no polynomial curves shows that after a delay of 0.5τ η , which corresponds to 7 time frames, the mean relative divergence drops ∂ui asymptotically from a value of 0.4, almost equal to the value if the components of ∂x were complete j

2.5. CHECKS AND VERIFICATION OF PROCEDURES

59

random numbers, to an acceptable value of 0.1.The delay of 0.5τ η or 7 time frames can be explained

Figure 2.22: For each set of points belonging to a trajectory of a certain length the corresponding mean of relative divergence is plotted. It shows the effectiveness of the use of weighted polynomial fits and how they reach their full impact only for trajectories wich are 2τ η of length or more. when looking in more detail at how the weights, which contribute toP the polynomial fits (expression 2.30), are distributed. It was found that only when the total weight, wi of expression (2.31), is the sum of few but strong weights, rather than many small weights stemming from inaccurate points, a qualitatively good fit can be expected. If, e.g. only 20% to 50% of the interpolated points are of high quality a cubic spline fit requires at least 6 − 15 or even more points to result in a ’good’ fit, i.e. a fit with low relative divergence. Judging from figure 2.22, where the mean relative divergence is observed to be below 0.2 only for trajectories longer than ∼ 15 frames, it is more likely that only 20% of the interpolated points are helpful contributors to accurate polynomial fits. The comparison between the 30Hz and the 60Hz experiment shows that the 60Hz run not only produced larger statistical sets but also sets of 10 − 30% better quality, at least in terms of relative divergence. length. From Figure 2.23 shows if and how strongly enstrophy, ω 2 , is affected by the­trajectory ® 2 trajectories with lengths ranging from 2τ η to 7τ η a drop of mean enstrophy, ω , of 30% is observed. This is considerable but much less than in the case of Lagrangian acceleration, a. Even though the above considerations seem to hold, ω 2 also seems to influence trackability slightly. It will have to be kept in mind when using ω 2 as a condition for other statistics presented further below. For shorter trajectories than 2τ η it appears that polynomials are not capable of filtering data ­ 2erroneous ® sufficiently and only after applying the relative divergence criteria the behavior of ω is consistent with the behavior for trajectories of lengths ranging from 2 − 7τ η . No significant influence can be observed from the use of polynomials or relative divergence as a selective criteria at trajectory lengths ∂ui is still of 2τ η or longer. This is a first indication that our processed and selected set of data for ∂x j representative of the actual physical flow. The polynomial fitting and the selection based on relative divergence apparently do not eliminate physical information of ­the® flow.Figure 2.24 is analogous to figure 2.23, but it is showing the behavior of mean strain, s2 , instead. Again the relevant

60

CHAPTER 2. METHOD

Figure 2.23: For ­each® set of points belonging to a trajectory of a certain length the corresponding ­ 2® 2 mean enstrophy, ω , is plotted. It can be seen that ω drops by 30% over lengths of trajectories of 2 − 7τ η .

­ ® curve that shows s2 after the use of polynomial fitting and application of the relative divergence criterion drops by 30% for trajectories with lengths ranging from 2 − 7τ η . The most noteworthy feature of figure 2.24 however is, that we observe for the first time a significant difference of the mean of a quantity between processed and unprocessed data. Contrary to the expectation that - if anything - polynomial fitting and ­ 2 ®the relative divergence criteria would dampen the data, the mean for the fully processed strain, s , is 30% higher! A possible explanation is, that in the course of linear interpolation considerable ’strain damping’ occurs when the overall quality of ubi , used for the interpolation, is low and the linear Ansatz is met only poorly. Through polynomials the missing or ’spoiled’ information on strain is then ’recovered’ along the trajectories with the help of points that suited better to the linear Ansatz and could therefore keep their high values for s2 . If­ this ® is true 2 however, the question arises why the same effect could not be observed in the case of ω . Going even further into speculations this could be explained by the fact that strain is a symmetric quantity, ³ ´ ∂uj ∂ui 1 ∂ui tensor.In figure sij = 2 ∂xj + ∂xi , while vorticity is the result of differences of terms of the ∂x j ­ 2® ­ 2® 2.25 the means of enstrophy, ω from figure 2.23 are plotted over the means ­ 2s® , from ­ of2 ®strain, figure 2.24. From theory it is known that in a homogeneous turbulent flow ω = 2 s , see for example (Monin and Yaglom, 1997). Only the fully processed data involving the polynomial ­ ® ­ ®fitting and the relative divergence criteria converge to the theoretically correct ratio of ω 2 / s2 = 2. The results shown in figure 2.24 in combination with figure 2.25 can be interpreted as a further ∂ui indication that the processed and selected set of data for ∂x is representative of the actual physical j flow. Regarding strain, it appears that only the fully processed data is representative of the actual physical nature of a turbulent flow.

2.5. CHECKS AND VERIFICATION OF PROCEDURES

61

Figure 2.24: For each set of points belonging to a trajectory of a certain length the corresponding mean strain, s2 , is plotted. Contrary to the expectation that, ­ ® if anything, polynomial fitting and the criteria of relative divergence would dampen the data, s2 is 30% higher for the fully processed data.

2.5.6

Summary on checks

From 3D-PTV measurements in a turbulent flow qualitatively correct velocity derivatives can be obtained and they can be monitored in a Lagrangian way along particle trajectories. The applied checks show that the 60Hz experiment is superior to the 30Hz experiment, both in the amount of data it produces for adequate statistics and also in its quality. The weighted polynomials which are fitted along a particle trajectory using only the information of points with a low relative divergence always result in improved qualitative accuracies for temporal and spatial velocity derivatives. Relative divergence used as a criteria to select data for a statistical set has been shown to work satisfactorily. Material elements measured directly from 3D-PTV could only be used as additional checks. For further statistical use material elements are better computed locally using an integrated deformation matrix. The observed weak sensitivity of particle trackability to enstrophy, strain and kinetic energy ω 2 , s2 , and k and the strong sensitivity to Lagrangian acceleration a produce statistically biased sets of data especially for long particle trajectories. This will have to be kept in mind whenever these quantities are used as conditions for further statistics.

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CHAPTER 2. METHOD

Figure 2.25: For each set of points belonging to a trajectory of a certain length the corresponding ratio of mean enstropy, ω 2 over mean strain, s2 , is plotted. Only the fully processed data converge to the theoretical value of 2.

Chapter 3

Results 3.1

Governing equations and relevant scales

To clarify the terminology used below, the relevant transport equations for kinetic energy and hence the term for viscous dissipation, vorticity, enstrophy, enstrophy production, rate of strain and rate of strain production are given here. For the sake of completeness the forcing terms are included in the expressions, despite the fact that in the observation volume of the experiments presented they are irrelevant. A detailed derivation and discussion is given in Tennekes and Lumley (1972), Novikov (1967), Yanitski (1982) and Tsinober (2001, ch. 12). A brief introduction to some relevant turbulent scales will be given at the bottom of this section. Namely, to the Kolmogorov scales and the Taylor microscale. Equation for the kinetic energy, 12 u2 , µ ¶ ¶ ¾ ½ µ D 1 2 ∂ 1 2 u =− uj p/ρ + u − 2νui sij − 2νsij sij + ui Fi . Dt 2 ∂xj 2

(3.1)

Expression (3.1) contains the dissipation term 2νsij sij or 2νs2 for kinetic energy. Equation for vorticity, ωi , Vorticity, ω i , is the curl of the velocity vector, u, and is related only to the skew-symmetric part, ∂ui , rij , of the tensor ∂x j ω = curl u, (3.2) or with

1 ω i = εijk rkj , 2 µ ¶ ∂uj 1 ∂ui . − rij = 2 ∂xj ∂xi

The transport equation for vorticity reads:

∂Fk Dω i = ωj sij + ν∇2 ω i + εijk . Dt ∂xj

(3.3)

∂ui The vortex stretching vector, Wi ≡ ωj ∂x = ω j sij , reflects the interaction between vorticity and rate j of strain tensor and is responsible for stretching (compressing) and tilting of the vorticity vector, ω. ∂ui , The rate of strain tensor, sij , is the symmetric part of ∂x j µ ¶ ∂uj 1 ∂ui . + sij = 2 ∂xj ∂xi

63

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CHAPTER 3. RESULTS

∂ui , can be split up in its symmetric part, sij , and in its skewThe velocity derivative tensor, ∂x j symmetric part, rij , associated with vorticity,

∂ui = sij + rij . ∂xj Equation for enstrophy, 21 ω 2 1 Dω2 ∂Fk = ωi ωj sij + νω i ∇2 ω i + εijk ω i 2 Dt ∂xj

(3.4)

The term ω i ω j sij is responsible for the enstrophy production and is known to be positive in the mean. νω i ∇2 ω i is the viscous enstrophy destruction term. Since enstrophy is not an inviscid invariant νω i ∇2 ωi cannot be called dissipation of enstrophy. Equation for the rate of strain tensor, sij ,

where Fij =

³

∂Fi ∂xj

Dsij 1 ∂2p = −sik skj − (ω i ω j − ω 2 δ ij ) − + ν∇2 sij + Fij Dt 4 ∂xi ∂xj ´ ∂F + ∂xji .

(3.5)

Equation for the total strain, sij sij ≡ s2 ,

1 ∂2p 1 Ds2 = −sij sjk ski − ω i ωj sij − sij + νsij ∇2 sij + sij Fij . 2 Dt 4 ∂xi ∂xj

(3.6)

Just like the term ωi ω j sij in (3.4) is called enstrophy production the term −sij sjk ski − 41 ω i ω j sij 2p in (3.6) can be called (inviscid) production of total strain. Note that the term −sij sjk ski −sij ∂x∂i ∂x j of expression (3.6) is equivalent to −Λ31 − Λ32 − Λ33 , where Λi are the eigenvalues of the rate of strain tensor sij . νsij ∇2 sij is the viscous strain destruction term. The nonlinearities including the viscous terms do not contribute to the rate of change of energy but only to its redistribution in space. In homogeneous flows the relation 1 1 h−sij sjk ski − ωi ωj sij i = hω i ω j sij i 4 2

(3.7)

is valid and both terms of (3.7) are strictly positive and comprise the generation of strain and 2p enstrophy in incompressible flows. Due to incompressibility we have sij ∂x∂i ∂x = ∂x∂ k {· · · } and j hence the mean of the pressure term in homogeneous flows is zero, hsij

∂2p i = 0. ∂xi ∂xj

Invariants Writing such equations allows us to identify in a natural way the dynamically significant geometrical invariant quantities and relations between such invariants of different order via dynamical equations, all of which are the consequence of the Navier-Stokes equations. Namely the quantities of 2p the second order are ω 2 and s2 ; the invariants of the third order are ωi ωj sij , sij sjk ski and sij ∂x∂i ∂x j (Tsinober 2001). Geometrical invariants remain unchanged under the full group of rotations (i.e.

3.1. GOVERNING EQUATIONS AND RELEVANT SCALES

65

rotations plus reflections). The cosines cos2 (ω, λi ) and cos(ω, W) are examples of such invariant quantities. An invariant of the third order, the enstrophy production term, allows a useful geometrical interpretation, (3.8) ω i ω j sij = ω 2 Λi cos2 (ω, λi ) ≡ ω · W = ωW cos(ω, W), where λi is the eigenframe of the rate of strain tensor, sij , and Λi are its eigenvalues. The vector Wi = ω j sij is the vortex stretching vector. Besides the magnitude of ω 2 and s2 expression (3.8) depends only on geometrical properties of the velocity gradients: the mutual orientation of the vorticity vector, ω, in the eigenframe, λi , and of the shape of sij , e.g. the ratios Λ2 /Λ1 and Λ3 /Λ1 . The terms ω 2 Λi cos2 (ω, λi ) (i = 1, 2, 3) will be referred to as contribution terms to enstrophy production, associated with each eigenvalue and eigenvector of sij . Another way of choosing invariant quantities is to look directly at the invariants of the velocity ∂ui i : the first invariant — P = ∂u gradient tensor ∂x ∂xi , vanishing for incompressible flow; the second j invariant — Q = 14 (ω 2 − 2s2 ), and the third invariant R = − 13 (sij sjk ski + 43 ω i ω j sij ), both written for incompressible flows (Chacin et al. 1996, 2000, Martin et al., 1998 and Ooi et al., 1999). These invariants arise naturally as coefficients in expression (3.9) of the characteristic equation for the ∂ui ∂ui and comprise a tempting reduction of the dynamics of ∂x to only two quantities. eigenvalues of ∂x j j α3 + P α2 + Qα1 + R = 0

(3.9)

Kolmogorov scales, η, τ η . The Kolmogorov length scale is defined on dimensional grounds as η = (ν 3 /²)1/4 ,

(3.10)

where ν is the fluid’s kinematic viscosity and ² is the mean rate of energy being pumped into the system, which is equal to the mean dissipation in statistically stationary flows. Similarly, the corresponding time, velocity and acceleration scales are defined as τ η = (ν/²)1/2 , uη = (ν²)1/4 and a = ²3/2 ν −1/2 . The Kolmogorov scale, η, can be defined in various ways. For example, imposing u η the condition Re = νη ≈ 1 and writing ² ≈ ν(uη /η)2 one arrives again at (3.10) with uη ≈ ν/η. In Frisch (1995) an overview on the impact of Kolmogorov’s work is given. It is known that adequate space/time resolution (in laboratory, field or numerical experiments) is achieved if the smallest resolved scales are of the order of the space/time Kolmogorov scales as defined above. For instance, η is considered as the smallest relevant spatial scale in turbulence. This, however, is not obvious, since the instantaneous dissipation ε is not narrowly banded around its mean ², but it is distributed with a rather long tail, so that values as large as, 102 ² are not that rare. This corresponds to scales more than three times smaller than η. It is due to the rather weak dependence of η on ², η ∼ ²1/4 , that the resolution as mentioned above appears to be reasonably adequate. Taylor microscale, λ. The length scale λ is called Taylor microscale in honor of G.I. Taylor who defined µ

∂u1 ∂x1

¶2

≡

u21 , λ2

relating isotropic velocity fluctuations with isotropic velocity derivatives. The Taylor microscale is neither related to a characteristic length of the strain rate field nor does it represent any group of ’swirl’ or ’eddy’ sizes in which dissipative effects are strong. Its existence has historical reasons,

66

CHAPTER 3. RESULTS

30Hz 60Hz

u 7mm/s 10mm/s

η 0.6mm 0.5mm

τη 0.31s 0.23s

λ 7mm 6mm

Reλ 40 50

Table 3.1: Characteristic flow properties as obtained for both experiments, 30Hz recording over 60s in the ’large faciltiy’ and 60Hz recording over 100s in the ’small faciltiy’. 1 since for quite some time it was only possible to measure ∂u ∂x1 instead of the full tensor of velocity ∂ui . The derivation of λ is discussed in more detail in Tennekes & Lumley (1972). derivatives ∂x j With the Taylor microscale the Taylor microscale Reynolds number, Reλ , commonly used to characterize isotropic turbulent flows, is defined as

Reλ = where u=

r

uλ , ν

(3.11)

1 ui ui . 3

In table 3.1 the characteristic scales, as found for the first and second experiment, are given.

3.2

Overview on vorticity and strain

As already mentioned above, vorticity and strain are comprised by the antisymmetric and skew∂ui . From Galanti and Tsinober (2000) it is known symmetric parts of the velocity derivative tensor ∂x j that the production of the field of velocity derivatives is a spatially local self-amplification process where the type of forcing does not play any important role. The local production terms, ω i ω j sij and sij sjk ski , are reported to be orders of magnitude higher than their corresponding terms in expressions (3.4) and (3.6) associated with forcing. There are indications that the process of selfamplification has a universal character for a wide range of different flow types. Manifestations are ∂ui , R and Q, with their characteristic ’tear-drop’ e.g. maps of the second and third invariants of ∂x j feature (Cantwell, 1992, Chacin et al. 1996, Martin et al., 1998 and Ooi et al., 1999) or characteristic alignments between vorticity and the eigenframe of the rate of strain tensor sij (Ashurst et al., 1987, Saffmann, 1991, Constantin, 1994, Tsinober, 1992, 1995a, 1996, Kholmyansky et al., 2001a, 2001b and Lüthi et al., 2001). The self amplified field has two nonlocally interconnected weakly correlated processes of enstrophy and strain production. The nonlocality is discussed in e.g. Novikov (1967), Constantin (1994), Ohkitani (1995) and Tsinober (2001). These two processes are understood only to a certain degree. Understood is used here in the sense of how the underlying mechanisms work. Taylor (1938) was the first to show that the mean enstrophy production, hω i ω j sij i, is positive. He expressed the view that stretching of vortex filaments must be regarded as the principal mechanical cause of the higher rate of dissipation which is associated with turbulent motion. Kolmogorov (1941a, 1941b) stressed the importance of dissipation (strain). They were followed by people who, motivated by the energy flux in Fourier space from high to lower wave numbers, favored the concept of an energy cascade in physical space from large eddies down to small eddies (e.g. Tennekes and Lumley, 1972, Meneveau and Sreenivasan, 1991, Vincent and Meneguzzi, 1994, She and Waymire, 1995 and Jiménez, 2000), generally with the more conceptual than physical explanation that vortex stretching produces smaller and smaller scales down to the dissipation scale. With the advent of strong computer power, regions of strong vorticity were discovered in DNS of homogeneous isotropic turbulence (e.g. Siggia, 1981, Kerr, 1985, Ashurst et al. 1987, Hosokawa and Yamamoto, 1989). They are usually referred to as ’worms’ and brought excitement to some of the

3.3. SOME UNIVERSAL FEATURES OF VORTICITY AND STRAIN

67

turbulence community, who were concerned with its fundamental aspects. The idea was straightforward: Now with the problem broken down to small regions of concentrated vorticity and large un-important regions with background turbulence, referred to as ’random sea’, it would be only a matter of time for substantial progress in the understanding of the basic turbulent mechanisms. While this progress never really happened those people opposing cascade and ’worm’ theories argued that the ’background’ is very structured and dynamically active and that the ’worms’ were more the consequences than the cause of anything. Tsinober (1998) stresses the importance to devote an at least equal amount of attention to the process of production of strain. The arguments are (i) that dissipation and its self-amplification are directly related to strain and not to vorticity, (ii) vortex stretching is essentially a process of interaction of vorticity and strain, with high enstrophy, ω 2 , being more the result than the cause of the turbulent process and (iii) that strain dominated regions appear to be most active/nonlinear in a number of aspects. The second argument was put in words somewhat more pronounced by Brasseur (1999): ”..., describing turbulence evolution in terms of dynamical interactions among vortex tubes and sheets is akin to describing a happy chubby dog as being wagged by its skinny vibrating tale”. The minimal common ground on the issue of ’cascade’ is that velocity derivatives represent the small scales and that dissipation is associated precisely with the strain field, s2 . These matters are discussed in more detail in Yeung et al. (1995), Tsinober (2001) and Brasseur (1999). In either case velocity derivatives play the dominant role in the dynamics of turbulence, which has yet to be understood. The idea at the beginning of this thesis was that since the progress on understanding turbulence got stuck with approaches using the Eulerian frame of reference, the essentially Lagrangian method of 3D-PTV should be further developed so that it can be used to study ∂ui . It should bring more light to the relevant physical processes in the set of velocity derivatives ∂x j general and clarify the differences between stretching of vortex lines and material lines in particular.

3.3

Some universal features of vorticity and strain

The data set from which all of the following single point results are derived comprises a subset of all measured points along particle trajectories. Note that from here on only results from the second experiment, 60Hz, are presented. According to the checks discussed above single point results are chosen only if they belong to a particle trajectory that was tracked over 20 time frames or longer and if their relative divergence is 0.1 or smaller. This results in a total of 7.5 · 105 data points. The 5 total number of points that allowed for a Lagrangian time derivative D(·) Dt of any quantity is 6.9 · 10 . As an additional important verification of the experimental data some selected results on geometrical statistics with vorticity and on strain are shown first. The short selection reproduces well known features of turbulence most of which are not observed in a Gaussian flow field, i.e. are specific of genuine fluid turbulence. It is natural to start with the positive rates of enstrophy and strain production. It is the most basic distinction of turbulence to a random Gaussian velocity field. In figure 3.1 the PDFs of ω i ω j sij and − 43 sij sjk ski show how both enstrophy and strain production are positively skewed. If conditioned on high strain or high enstrophy the PDFs - while remaining positively skewed - become flatter in favor of higher probabilities of strong production or destruction events. In addition we note that if ω i ω j sij is conditioned on ω 2 or − 43 sij sjk ski is conditioned on s2 the peaks of the most probable events themselves are shifted towards to positive regime.To show the non-locality of enstrophy and strain production, despite their qualitative similar univariate PDFs, it has become common to show the joint PDF of ω i ω j sij and − 34 sij sjk ski . By ’non-locality’ here we mean that strong production of enstrophy does not happen at the same time or location as production of strain. In figure 3.2 the logarithmic joint PDF again shows the positive skewness of ω i ω j sij and − 34 sij sjk ski but in addition it shows how strong production of enstrophy inhibits strong production of strain and vice versa.An

68

CHAPTER 3. RESULTS

Figure 3.1: PDFs of ω i ω j sij and − 34 sij sjk ski show how both enstrophy production and strain production are positively skewed. When conditioned on high strain or high enstrophy the skewness generally increases. Additionally, conditioning of ω i ω j sij on large ω 2 and − 43 sij sjk ski on large s2 results in a positive shift of the most probable events.

Figure 3.2: Logarithmic joint PDF of ω i ω j sij and − 43 sij sjk ski . Despite their similar PDFs in figure 3.1 the pointwise relation of ωi ωj sij and − 34 sij sjk ski is strongly nonlocal and they are only weakly correlated. even stronger manifestation of the positiveness of mean enstrophy production is shown in figure 3.3. The physical interpretation of the geometrical invariant cos (ω, W) is straight-forward. If it is

3.3. SOME UNIVERSAL FEATURES OF VORTICITY AND STRAIN ­ 2 ® ω Λ1 cos2 (ω, λ1 ) ® ­ 2 2 1.79 ω Λ1 cos2 (ω, λ1 ) 0.50

­ 2 ® ω Λ2 cos2 (ω, λ2 ) ­ 2 2 0.50 ® ω Λ2 cos2 (ω, λ2 ) 0.11

69

­ 2 ® ω Λ3 cos2 (ω, λ3 ) ­ 2 2 -1.29 ® ω Λ3 cos2 (ω, λ3 ) 0.39

Table 3.2: Contributions of terms associated with Λi to mean enstrophy production, hωi ωj sij i, and to the magnitude of the vortex stretching vector, W 2 . The largest contribution to hω i ω j sij i is associated with the first term, Λ1 , despite the preferential alignment between ω and λ2 . positively skewed it reflects the situation where the projection of the vortex stretching vector, W, on ω tends to point in the same direction as vorticity. If cos (ω, W) is zero it will not change the magnitude but only the direction of ω and if it is negative vortex compression occurs. It could be argued that the cosine itself proofs nothing since the magnitude of ω could be anti-correlated with cos (ω, W) resulting in a net vortex compression despite the strong positive skewness. However, conditioning cos (ω, W) on s2 (figure 3.3a) and ω 2 (figure 3.3b) reveals that the contrary is the case: Stronger s2 and ω 2 are correlated with stronger positive skewness of cos (ω, W). The stronger reaction of cos (ω, W) on conditioning it on s2 than on ω 2 that is observed in figure 3.3 is a reflection of a broader characteristic feature of turbulence. Tsinober (2000) reports that ”...nonlinearities such as enstrophy production, production of strain and many others are an order of magnitude larger in the regions dominated by strain than in the enstrophy dominated regions.” The best known manifestation that turbulence has a dynamically relevant structure even in less intense regions is the orientation of ω relative to the eigenframe of the rate of strain tensor sij , cos (ω, λi ). Figure 3.4 shows the PDFs of cos (ω, λi ) conditioned on weak and strong strain and enstrophy respectively. The main feature is the predominant alignment of ω with the intermediate eigenvector λ2 of the rate of strain tensor, a feature recognized by Siggia (1981), first reported by Ashurst (1987) and confirmed in Tsinober (1998) and references therein.The preferential alignments increase with both, higher strain and higher enstrophy. Even for weak events the preferential alignments persist, suggesting that turbulence has structure also in less intense regimes. In figure 3.5 the PDFs of Λ1,2,3 are given. The main feature is the positively skewed PDF of Λ2 which comprises a fundamental property of turbulence, Tsinober et al. (1997). It is essentially responsible for ’everything’ since without it, turbulence would not be able to produce strain and therefore also no enstrophy. The strain production ­ ® ­ term, ® ­ −s®ij sjk ski , of expression (3.6) is equivalent to −Λ31 − Λ32 − Λ33 and h−sij sjk ski i = − Λ31 − Λ32 − Λ33 would be identically zero or negative if hΛ2 i ≤ 0. With no strain being constantly produced from expressions (3.4) and (3.8) it is clear that ultimatively also no enstrophy can be produced. The magnitude of hΛ2 i ∼ 0.2s−1 however remains relatively small compared to hΛ1 i ∼ 1s−1 and − hΛ3 i ∼ −1.2s−1 . The ratio of hΛ1 i : hΛ2 i : hΛ3 i is consistent with findings from numerical simulations reported in e.g. Girimaji and Pope (1990b), Dresselhaus and Tabor (1994), Huang (1996) and Tsinober (2001).The combined effect of the alignment between ω and λi and the behavior of Λi leads to peculiar contributions to the mean enstrophy production hωi ωj sij i from the three terms associated with each Λi , ω i ω j sij = ω 2 Λk cos2 (ω, λk ). In table 3.2 mean values for contributions to enstrophy production and the magnitude of the vortex stretching vector are given. The largest contribution to hω i ω j sij i is associated with the first term, Λ1 , despite the preferential alignment between ω and λ2 . This is explained by the facts that the magnitude of Λ2 is much smaller than the magnitude of Λ1 and that the eigenvalue Λ2 takes both positive and negative values ­(figure ® 3.5) (Kholmyansky et al. 2001a). Similarly the largest contribution to vortex stretching W2 comes also from the term associated with Λ1 (table 3.2). The dynamics of enstrophy production will be discussed in more detail below. Finally in figure 3.6 we show a last characteristic feature of turbulent flows regarding the volumes occupied by regions of u2 , ω 2 and s2 of different magnitude. Strain regions that represent, e.g. 50% of the total strain occupy

70

CHAPTER 3. RESULTS

Figure 3.3: PDFs of cos (ω, W ) conditioned on s2 (a) and ω 2 (b) are shown. In both cases skewness increases with increasing magnitude of s2 and ω 2 . Physically, a positive cos (ω, W ) reflects the situation where the projection of the vortex stretching vector W on ω points in the same direction as vorticity, if it is zero it will not change the magnitude but only the direction of ω and if it is negative vortex compression occurs.

almost twice as much volume as corresponding regions of strong vorticity. This is a reflection of the fact that strain has to be present both in regions of strain and enstrophy production whereas

3.3. SOME UNIVERSAL FEATURES OF VORTICITY AND STRAIN

71

Figure 3.4: PDFs of cos (ω, λi ) conditioned on weak and strong strain (a) and on weak and strong enstrophy (b) are shwon. The main feature is the predominant alignment of ω with the intermediate eigenvector λ2 of the rate of strain tensor. The preferential alignments increase with both, higher strain and higher enstrophy.

enstrophy is located more in small concentrated regions. In other words enstrophy is much more intermittent than strain. This is similar to the observations made by Boratav and Pelz (1997) in their DNS of NSE.

72

CHAPTER 3. RESULTS

Figure 3.5: PDFs of the eigenvalues, Λ1,2,3 , of the rate of strain tensor are shown. The positively skewed PDF of Λ2 comprises a fundamental property of turbulence. If hΛ2 i = 0 essentially no strain and therefore also no enstrophy could be produced by turbulence.

3.4

Q − ω i ωj sij − sij sjk ski probability orbital

The equations (3.4) and (3.6) indicate that along with enstrophy, ω 2 , and strain, s2 , the third moments, enstrophy and strain production, ω i ω j sij and sij sjk ski , are the key quantities of turbulence dynamics. It is a convenient and useful way to study some of the local flow properties in the so ∂ui , called R − Q plane. Q is the second invariant, Q = 14 (ω 2 − 2s2 ), of the velocity gradient tensor, ∂x j

and R is its third invariant, R = − 31 (sij sjk ski + 43 ω i ω j sij ). Both are written for incompressible flows, see Soria et al. (1994), Tsinober et al. (1997), Martin et al. (1998), Ooi et al. (1999) and Chacin et al. (1996). In joint PDF plots of R versus Q they all find a qualitatively identical ’tear drop’ shape for different kind of flows. The most characteristic feature is, that in regions with high strain the strain production term, −sij sjk ski , is much more dominant over ω i ω j sij than the enstrophy production term is dominant over the strain production term in regions with high vorticity. Regions of different kinds of events, such as large values of geometrical invariants, production rates and even Reynolds stresses can be allocated to distinct regions within the R − Q phase space. A qualitative summary of these local flow properties, such as alignments, enstrophy and total strain production as well as their rates, is given in Tsinober (2000). In figure 3.7 the joint probability R − Q plot, as obtained from our measurements, reveals for the first time in experiments the same qualitative features.Martin et al. (1998) were the first to bring evidence that ”...the resulting dynamical system in the (R,Q) phase space is a clockwise spiral....”. They illustrated that - in the mean - there is a cyclic sequence of topological evolution from enstrophy dominated regions to strain dominated regions and back, with a characteristic cycle time, τ 0 ∼ 30τ η . For this they introduced the concept of Conditional Mean Trajectories (CMT).

3.4. Q − ω I ωJ SIJ − SIJ SJK SKI PROBABILITY ORBITAL

73

Figure 3.6: Relative volume of the flow occupied by regions of u2 , ω2 and s2 of different magnitude. 2 As an example we take i.e. 70% ­ of ® ω : From the figure we read that 70% of the total amount of 2 enstrophy is (i) larger than 1.6 ω and (ii) that it occupies only 20% of the entire volume of the flow. Though R − Q invariants are useful in several respects they are insufficient since both quantities, R and Q, represent only differences between ω 2 and s2 and −sij sjk ski and ωi ωj sij respectively. Since all of these quantities are important by themselves it is necessary to use a representation in a four dimensional (4D) space spanned by, ω 2 , s2 , ωi ωj sij and sij sjk ski . With 4D representations being not very practical, we present here a three dimensional (3D) space instead, spanned by Q, ω i ω j sij and sij sjk ski . As will be shown below, in this space, henceforth referred to as Q − ωi ωj sij − sij sjk ski probability orbital, the quantities ω 2 and s2 are sufficiently well separated from each other and thus the representation allows to look almost directly at ω 2 , s2 , ω i ω j sij and sij sjk ski in parallel. The Q − ω i ωj sij − sij sjk ski probability orbital is more transparent to different contributors to physical processes such as enstrophy and strain production and their interactions as it expands single points of the R − Q or − 34 sij sjk ski − ω i ω j sij phase spaces to lines. Similar to the CMT, used by Martin et al. (1998), that revealed a mean cyclic sequence of topological evolution in the R − Q phase space, a mean cyclic evolution over regions with intense strain, enstrophy production, concentrated vorticity and enstrophy destruction along with strain production is observed here. The mean cyclic evolution found suggests that turbulence is more likely to be understood if it is looked at not as a cascade but as a succession of strong stretching and folding processes of fluid blobs.

3.4.1

Construction

Since the {Q, ω i ω j sij , sij sjk ski } representation is new, a somewhat detailed description of how the following results are obtained is given here. The concept however is simple, since it is nothing else but

74

CHAPTER 3. RESULTS

Figure 3.7: The joint probability R − Q plot, with the second invariant of

∂ui ∂xj

Q =

1 2 4 (ω

− 2s2 )

and the third invariant R = − 31 (sij sjk ski + 34 ωi ωj sij ), shows a typical ’tear drop’ shape. The most characteristic feature is the predominance of strain production over enstrophy production in regions with high strain (bottom right). Regions of different kinds of events can be allocated to distinct regions within the R − Q phase space. the natural expansion of the idea of univariate PDFs over joint PDFs to a three dimensional PDF. In univariate PDFs the second axis is available and therefore it is used to represent the probabilities. In joint PDFs the probabilities need to be represented either in color coding or by introducing a third axis. In a three dimensional PDF these representations are not possible anymore. Therefore it was decided to show either color coded slices through the Q − ω i ω j sij − sij sjk ski probability orbital, or iso-probability surfaces within the Q − ωi ωj sij − sij sjk ski probability orbital. A sub-region out of the entire {Q, ω i ω j sij , sij sjk ski } space is chosen such, that 92% of the total data is inside the rendered probability orbital, as shown in most of the figures ¡ −2 ¢3.8 through 3.23. For our experiment this is achieved¡ with axis limits chosen as: −10 < Q < 10 s , −20 < ω i ω j sij < 20 ¢ −3 and −10 < −sij sjk ski < 30 s . This sub-region is divided in 80 × 40 × 40 cells, ci,j,k . The probability, p, P of a measured fluid particle to be in a specific cell, ci,j,k , can be approximated as p = ni,j,k / ni,j,k where n is the number of particles counted in each cell during the entire experiment. In figure 3.8 the yellow core represents an iso-probability surface of p = 6 · 10−4 . The volume included by an iso-probability of p = 7 · 10−6 is rendered as blue slices. This is equivalent to 58% of all measured particles being inside the yellow core (inner region) and 34% of all particles being in-between the inner region and the enveloping surface of the blue slices with yellow rims (outer region). 8% of particles are outside of the orbital.

3.4.2

Description

The shape of the orbital, figure 3.8, resembles that of a - rather fat - boomerang or a very asymmetric propeller. From ’the top’ it would look identical to the contours of the PDFs of ωi ωj sij and

3.4. Q − ω I ωJ SIJ − SIJ SJK SKI PROBABILITY ORBITAL

75

Figure 3.8: As an extension of joint PDFs of ω i ω j sij and − 34 sij sjk ski and of R − Q plots we introduce here the Q − ω i ω j sij − sij sjk ski probability orbital. Represented in this manner, 58% of all events are inside the yellow core (inner region) and 92% of all events are within the envelopping surface of the blue slices (outer region). Only 8% of events are outside the orbital. − 43 sij sjk ski as shown in figure 3.2. Cross sections of low values for Q are ellipse like shapes which are elongated parallel to the axis −sij sjk ski . With increasing values for Q their orientation is twisted until at ’the top’ they are elongated along the axis ω i ω j sij . While the ’center of gravity’ of the rendered shape intersects the origin, the center of gravity of the cross sections is shifted increasingly into the positive quadrant of ω i ω j sij and −sij sjk ski as the modulus of Q increases.Two side views shown in figure 3.9 complete and aid the geometrical description of the orbital. Note how the orbital appears much ’leaner’ viewed in the Q − sij sjk ski plane, figure 3.9 b), than as viewed in the Q − ω i ω j sij plane, figure 3.9 a), where it appears to have a more ’circular’ shape. As will be discussed further below, apparently this has to do with the fact that the axis around which fluid particles evolve in the mean, is roughly perpendicular to the Q − ω i ω j sij plane (figure 3.9 a). It is noteworthy that both, at large values for Q (top of figure 3.9a) and much more so at small values for Q (bottom of figure 3.9b) the ’tips’ of the orbitals are offset towards the positive regime of ω i ω j sij and much stronger towards the regime of strain production, to positive −sij sjk ski . This already indicates that in the ’tips’ especially strong viscous destruction of enstrophy and strong viscous destruction or pressure re-distribution of strain is occurring, as will be explained in the following: It can be seen from expressions (3.4) and (3.6) that in order for enstrophy change or strain change to stay low, viscous destruction of enstrophy and strain (for strain also pressure re-distribution) need to neutralize enstrophy and strain production. If this were not so, fluid particles being in a state of very high or very low Q would experience continuous production of enstrophy or strain, which in turn would result in shapes of iso-probability surfaces that would be completely different compared

76

CHAPTER 3. RESULTS

Figure 3.9: Two side views of the Q − ωi ωj sij − sij sjk ski probability orbital are shown. The orbital appears much ’leaner’ viewed in the Q − sij sjk ski plane (b) than viewed in the Q − ω i ω j sij plane (a). Apparently this has to do with the fact that the axis around which fluid particles evolve in the mean, is roughly perpendicular to the Q − ω i ω j sij plane (a). As yet another reflexion of the overall positiveness of h−sij sjk ski i and hω i ωj sij i we note that the entire orbital is skewed towards positive production of s2 and ω2 .

to those measured. Not only the ’tips’ but the entire orbital is skewed towards positive production

3.4. Q − ω I ωJ SIJ − SIJ SJK SKI PROBABILITY ORBITAL

77

of s2 and ω 2 . This is yet another reflection of the overall positiveness of h−sij sjk ski i and hω i ω j sij i.

3.4.3

Strain

The inner region rendered yellow in figure 3.8 and 3.9 contains 32% of the total strain and the outer region represents 46%. From figure 3.10 which shows the PDFs of strain for both regions it can be seen that they separate weaker from stronger strain events. However the separation by means of Q − ω i ω j sij − sij sjk ski iso-probability surfaces is not sharp. Therefore in figure 3.11 the ® ­ 2volumes within the orbital which ­ 2 ® in the mean have values for strain that are higher than 1.25 s (blue) and higher than 2.5 s (yellow) are shown. ­ 2 ® Combined with the result from figure 3.6 above we 2 know that the region where s > 1.25 s (blue) ­in ®real space occupies 35% of the domain and the strong strain region (yellow), where s2 > 2.5 s2 , 10% respectively. In terms of dynamical relevance they represent much more of the flow - the regions are equivalent to 70% of total strain and 30% respectively. It is noteworthy that while strain increases along the ’central axis’ of the orbital towards negative Q - as can be expected from Q = 41 (ω 2 − 2s2 ) - the region in the proximity of the origin is excluded from this trend.

Figure 3.10: PDFs of strain, s2 , as found in inner and outer regions (figures 3.8, 3.9) of the Q − ω i ω j sij − sij sjk ski probability orbital. The division made in the Q − ωi ωj sij − sij sjk ski probability orbital separates weaker from stronger strain events.

3.4.4

Enstrophy

The emerging picture for enstrophy is similar to that of strain. The important qualitative difference is - as already mentioned - that the relevant amount of enstrophy is concentrated within much smaller regions in space. The inner regions rendered yellow in figures 3.8 and 3.9 contain only the weakest 22% of the total enstrophy whereas the outer region represents 46%. This means that where the orbital excludes only ∼ 20% of the strongest strain events, for enstrophy ∼ 30% are excluded. From the PDFs of enstrophy

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Figure 3.11: The mean strain distribution in the Q − ω i ωj sij − sij sjk ski probability­orbital is shown. ® 2 are rendered Orbital regions which in the mean have values for­strain that are higher than 1.25 s ® blue and if in the mean strain is higher than 2.5 s2 they are rendered yellow. The blue region in real space occupies 35% of the flow and the yellow region 10% respectively. In terms of dynamical relevance this is equivalent to 70% and 30% of total strain. in figure 3.12 it can be seen that the ’high-low’ separation through iso-probability surfaces is slightly stronger for enstrophy than for strain. Figure 3.13 shows ­ 2 ® which in ­ 2 ® the volumes within the orbital (blue) and higher than 2.5 ω (yellow). the mean have values for enstrophy higher than 1.25 ω ­ ® The region where ω 2 > 1.25 ω 2­ , in® real space occupies 30% of the domain and the strong vorticity region (yellow) where ω 2 > 2.5 ω 2 , 12% respectively. This is similar to the behavior observed for strain. Their dynamical relevance is much stronger though - the regions are equivalent to 78% and 55% of total enstrophy respectively. Here it has to be kept in mind that strong enstrophy does not necessarily imply strong dynamical relevance. Tsinober (2000) stresses the fact that regions with very strong ω 2 are associated with low curvature of vortex lines, strong alignment with λ2 and that they are at an equilibrium with viscous enstrophy destruction. These regions are usually referred to as ’worms’. The region in the proximity of the origin is again excluded from the trend regions of strain and of increasing values for ω 2 along the positive direction of­ the ® Q axis. ­Strong ® 2 2 enstrophy are completely separated 2.5 s­ ®and 2.5 ω ­ ® for values ­ above ® ­ .® However, intermediate events with values between 1.25 s2 and 2.5 s2 and 1.25 ω 2 and 2.5 ω 2 overlap, as can be seen from figures 3.11 and 3.13. This is especially true when Q ∼ 0 and ω i ω j sij > 0. Again this is a manifestation that in order to produce enstrophy both strain and enstrophy are required whereas strain production is less dependent on enstrophy.

3.5

Cyclic mean process

Similar to the approach described in Martin et al. (1998) the mean Lagrangian evolution of fluid particle states in the {Q, −sij sjk ski , ω i ω j sij } space is determined - where in our case the fluid

79

3.5. CYCLIC MEAN PROCESS

Figure 3.12: PDFs of enstrophy, ω 2 , as found in inner and outer regions (figure 3.8, 3.9) of the Q−ω i ω j sij −sij sjk ski probability orbital. The division made in the Q−ω i ω j sij −sij sjk ski probability orbital separates weaker from stronger enstrophy events. particles are tracer particles. Martin et al. essentially computed and studied the mean Lagrangian DQ evolution of fluid particle states in the R − Q space, DR Dt and Dt , and observed ...a mean cyclic sequence of topological evolution in the R − Q phase space. For such a computation an accurate particle displacement integration is necessary which requires the use of slow Fourier transforms. They overcame this computational drawback by employing an alternative approach based on the conditional average technique. Following Girimaji and Pope (1990a) they made use of the fact that ...for homogeneous isotropic turbulence Lagrangian and Eulerian formulations are statistically equivalent for one-point statistics. In simpler words: They computed the mean Lagrangian evolution as Eulerian time derivatives of R and Q as dR , R˙ ≡ dt

dQ Q˙ ≡ . dt

To obtain the mean evolution of R˙ and Q˙ as a function of R and Q they conditioned the evolution on R and Q as D E D E R˙ | R, Q , Q˙ | R, Q . As a major advantage of the 3D-PTV technique the Lagrangian derivatives can be obtained ˙ −sij s˙ jk ski and ω i ω˙ j sij over each of directly along the measured particle trajectories. Averaging Q, the 80 × 40 × 40 cells, ci,j,k , of our probability orbital and over the entire time of the experiment we obtain in the space of {Q, −sij sjk ski , ωi ωj sij } the mean evolution of fluid particles that is conditioned on their {Q, −sij sjk ski , ω i ω j sij } state as À ¿ DQ ˙ | Q, −sij sjk ski , ωi ωj sij , Q≡ Dt

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CHAPTER 3. RESULTS

Figure 3.13: The mean enstrophy distribution in the Q − ωi ω j sij − sij sjk ski probability orbital ­ 2is® shown. Orbital regions which in the mean have values for enstrophy­that are higher than 1.25 s ® 2 are rendered blue and if in the mean enstrophy is higher than 2.5 s they are rendered yellow. The blue region in real space occupies 30% of the flow and the yellow region 12% respectively. In terms of dynamical relevance this is equivalent to 78% and 55% of total enstrophy!

and

À ¿ D − sij sjk ski | Q, −sij sjk ski , ω i ω j sij , −sij s˙ jk ski ≡ Dt ¿ À Dω i ω j sij ω i ω˙ j sij ≡ | Q, −sij sjk ski , ω i ω j sij . Dt

What we call mean analogous to what Martin et al. call CMT. The o n evolution here is completely ˙ −sij s˙ jk ski , ω i ω˙ j sij can be thought of as the components of a vector field. In three quantities Q, figure 3.14 this vector field is represented by arrows that are plotted in blue color if Q˙ < 0 and in red color if Q˙ > 0 respectively.The dominant feature of figure 3.14 is a clear cyclic mean evolution of {Q, −sij sjk ski , ω i ω j sij }. If we think of the pattern formed by the arrows of a ’rigid-body-likerotation’ then we observe that the ’axis of rotation’ is roughly perpendicular to the Q − ω i ω j sij plane and slightly inclined to the −sij sjk ski − ω i ω j sij plane, just like the viewing axis of figure 3.14. At positive values for Q the evolution of {Q, −sij sjk ski , ω i ω j sij } reminds of the motion of a rotating rigid disk that would fit into the ellipse shaped cross-sections of the orbital parallel to Q − ωi ωj sij . At negative values for Q the mean evolution of particles resembles more to a ’sharp U-turn’. It is probably noteworthy - if not striking - that the observed shape of the Q − ω i ω j sij − sij sjk ski probability orbital (figure 3.8, figure 3.9) is matching the mean evolution of fluid particles: Viewed in the plane parallel to most of the observed evolution the shape of the orbital is circular and viewed in the Q − sij sjk ski plane the shape is relatively ’lean’. A mean cyclic evolution of a fluid particle in the Q − ωi ω j sij − sij sjk ski probability orbital is completed in a time of order O(10 − 100τ η ) and looks as follows: Starting at the high enstrophy region the dominant component of the evolution is ωi ω˙ j sij < 0. It brings particles into the

3.5. CYCLIC MEAN PROCESS

81

˙ −sij s˙ jk ski , ω i ω˙ j sij which can be thought of as the components of a vector field are Figure 3.14: Q, plotted. Arrows are blue when Q˙ < 0 and red when Q˙ > 0. The dominant feature is a cyclic mean evolution of the state (Q, −sij sjk ski , ω i ω j sij ). If we think of the vector pattern as a ’rigid-body-likerotation’ then the ’axis of rotation’ is roughly perpendicular to the Q − ω i ω j sij plane and sligthly inclined to the −sij sjk ski − ωi ωj sij plane, just like the viewing axis of this figure. negative regime of ω i ω j sij with −sij sjk ski centered around zero. The predominant component of ˙ Q˙ < 0. As enstrophy decreases, strain increases and at Q ∼ 0 the ˙ −sij s˙ jk ski , ωi ω˙ j sij ) now is Q, (Q, mean evolution dominated by the term Q˙ < 0 brings fluid particles more and more into the positive regime of −sij sjk ski . Concurrently the component ω i ω˙ j sij becomes positive. As the particles approach ωi ω j sij ∼ 0 mean strain increases to its maximum. After the ’U-turn’ at ωi ωj sij ∼ 0 strain decays only slowly while the values for enstrophy grow. Again at Q ∼ 0 the component ω i ω˙ j sij changes its sign and the distribution of −sij sjk ski looses much of its skewness as the particles return to the strong vorticity region to complete their cycle. A single fluid particle does off course not follow precisely a mean evolution as just described. Preliminary inspections of Lagrangian tracking of individual tracer particles reveal that typical {Q, −sij sjk ski , ωi ω j sij } evolutions follow paths that are much more complicated. Significant fluctuations parallel and normal to iso-probability levels of the orbital are observed. Parallel fluctuations alone would still result in a true mean cyclic evolution for most individual particles, true in the sense that a particle would successively spend time in strain and enstrophy regions of comparable strength. However, with large fluctuations normal to iso-probability surfaces the individual cycle is bound to become chaotic. For example, a particle may well ’leave’ a high strain region, complete its cycle in low strain and enstrophy - close to the origin {Q ∼ 0, −sij sjk ski ∼ 0, ωi ωj sij ∼ 0} - and return again to the high strain region. In other words, fluctuations normal to iso-probability surfaces shuffle the succession of comparable strain-enstrophy events of individual particle’s. As a result a

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particles sequence of strong strain and enstrophy events becomes apparently random. The following simplifying consideration tries to explain why this randomness makes sense even from a turbulence point of view. Roughly ∼ 60% of the flow is represented by the ’inner region’ of figures 3.8 and 3.9 where no strong events occur. Turbulence will hardly allow such fluid particles to stay in the ’inner region’ for ever. On the contrary, it will see to it that every particle takes part in a strong event once in while, i.e. intermittently, just as particles involved in strong events will eventually have a rest in the ’inner region’ of the Q − ω i ω j sij − sij sjk ski probability orbital. The only way this exchange between inner and outer regions can be accomplished, is through evolutions normal to iso-probability surfaces. From the point of view of the turbulent flow itself the history of an individual particle might be less relevant. If a particles evolution departs from the mean path in the probability orbital it will be replaced by another particle. In other words, the dynamics of a swarm of fluid particles (Pumir et al. 2002) - a fluid blob - might be more relevant, and at the same time also more associated with a true cyclic evolution.

3.6

Enstrophy dynamics

In this section and in the section on Strain dynamics a description of the self-amplification mechanisms will be given. Mostly they will be described in the context of the Q − ω i ω j sij − sij sjk ski probability orbital introduced above. Different contributions to mechanisms involved in enstrophy and strain dynamics are found to be predominantly active in different but distinct regions of the Q − ω i ω j sij − sij sjk ski probability orbital. Understanding enstrophy dynamics governed by expression (3.4) means among other things understanding the positiveness of hω i ω j sij i. So far, no theoretical arguments in favor of the positiveness of hωi ωj sij i have been given. There is evidence that hω i ω j sij i would grow without bounds if it wasn’t for viscosity, see Brachet et al. (1992). This evidence is based on DNS simulations of the Euler equations. Such Eulerian simulations however are always contaminated with some small numerical viscosity. In high enstrophy regions of viscous flows the mean generation due to predominant vortex stretching is approximately balanced by viscous destruction (Tsinober 2000). It was also shown for a Gaussian field - with hω i ω j sij i identically zero at time t = 0 - that within a D very short time interval the mean enstrophy production will become positive, Dt hω i ω j sij i > 0, see Proudman and Reid (1954). As already mentioned (table 3.2) the largest contribution to hωi ωj sij i is associated with the term Λ1 and alignment of the largest eigenvector λ1 with ω, in spite of the preferential alignment between ω and λ2 . Contrary to the expectation that all strong enstrophy production events are associated with regions of high vorticity Tsinober (1998, 2000, 2001) reports that in regions dominated by strain the rate of enstrophy production is an order of magnitude higher than both its viscous destruction rate and the enstrophy production rate in the enstrophy dominated regions. Using the frame of the Q − sij sjk ski − ω i ω j sij orbital the different mechanisms of enstrophy dynamics and their approximate balance will be discussed in the following. Namely, from expressions (3.4) and (3.8) it can be seen that the mechanisms are (i) mean enstrophy production due to predominant vortex stretching over vortex compression E D ω 2 Λ1 cos2 (ω, λ1 ) + ω 2 Λ3 cos2 (ω, λ3 ) >0,

(3.12)

or mean enstrophy destruction due to predominant vortex compression over vortex stretching E D ω 2 Λ1 cos2 (ω, λ1 ) + ω 2 Λ3 cos2 (ω, λ3 ) 0, (3.14)

or mean enstrophy destruction when the intermediate eigenvalue is negative ­ 2 ® ω Λ2 cos2 (ω, λ2 ) 5. The regions occupied by both contributors are of comparable size for medium and strong enstrophy production. Looked at in the Q−ω i ω j sij plane the regions covered almost coincide. It is noteworthy that contributions associated with the intermediate eigenvalue dominate enstrophy production at Q > 6 which is the region of strongest vorticity. Both contributions are strongest in low probability regions of the orbital, i.e. in its outer regions. The most distinct difference between the two contributions becomes visible along the axis −sij sjk ski or in the Q−sij sjk ski plane. The Λ1 , Λ3 contribution is strongest when −sij sjk ski is minimal and ω i ωj sij is maximal. On the other hand strong Λ2 contributions are located in a ’flat’ region normal to the Q − sij sjk ski plane when −sij sjk ski is maximal. The combined effect of these two mechanisms results in the observed mean enstrophy production.Figure 3.16 shows regions of medium and strong viscous enstrophy destruction (expression 3.16). To allow for a comparison with the competing production terms iso-surfaces of equal values as used in figure 3.15 are chosen. Blue ® ­ 2 −3 and the yellow surface encloses regions ∇ ω < −3s cross-sections represent regions where νω i i ® ­ 2 −3 where νω i ∇ ω i < −6s . The important feature is that - on an order of magnitude scale - viscous destruction ­approximately neutralizes enstrophy production terms everywhere. The shape of the re® −3 follows gion where νω i ∇2 ωi < −6s ® the shape of regions of strong enstrophy production ­ 2 roughly 2 (ω, λ ) > 0 - with two qualitative differences. First, the shape due to the contribution of ω Λ cos 2 ® 2 ­ of the region where νω i ∇2 ω i < −6s−3 is much less concentrated on high −sij sjk ski values only and that viscous second, in the high vorticity region it extends to ω i ω j sij ∼ 0. This suggests ® destruction ­ 2 2 predominantly counteracts enstrophy production contributions of ω Λ2 cos (ω, λ2 ) > 0. From the following analysis of the vortex compression terms it also appears that the mean < Q˙ > < 0 evolution at high vorticity values is dominated by the viscous term of expression (3.16).Next, the two contributors to vortex compression (expression 3.13 and 3.15) are ­compared. Figure® 3.17 a) shows vortex compression due to the mechanism associated with Λ2 , ω 2 Λ2 cos2 (ω, λ2 ) 0 are rendered blue −3 when in the mean they are above 3s−3 and yellow ­ 2 when2 in the ®mean they are larger than 6s . In (b) contributions to vortex stretching due to ω Λ2 cos (ω, λ2 ) > 0 are rendered using blue and yellow for the same mean values as in (a). the Q − ω i ω j sij plane the contributors to vortex compression (expression 3.13 and 3.15) are of much smaller magnitude than the corresponding vortex stretching terms. Blue cross-sections represent

3.6. ENSTROPHY DYNAMICS

85

Figure 3.16: Regions of medium and strong mean viscous enstrophy destruction are shown. To ­ ® 2 allow comparison with figure 3.15 blue cross-sections represent regions where νωi ∇ ω i < −3s−3 ® ­ 2 and the yellow surface encloses regions where νω i ∇ ωi < −6s−3 . In most regions mean viscous destruction is of the same order of magnitude as the mean enstrophy production. mean contributions of −1s−3 or smaller and the yellow surface encloses values of −3s−3 or smaller. It is also clear that negative Λ2 contributions are restricted to the small region where −sij sjk ski < 0 and ωi ωDj sij < 0. The entire rest of the orbitalEwhere < Q˙ > < 0 is dominated by vortex compression

due to ω 2 Λ1 cos2 (ω, λ1 ) + ω 2 Λ3 cos2 (ω, λ3 ) < 0.A close inspection of the highlighted regions of figure 3.17 b) shows that enstrophy strongly decreases along the negative QE axis. Therefore also the D 2 effect of vortex compression due to ω Λ1 cos2 (ω, λ1 ) + ω 2 Λ3 cos2 (ω, λ3 ) < 0 can be expected to decrease. The question arises how turbulence manages to sustain a mean < Q˙ > that is significantly negative. Two answer ® observations made can­ in part ® this question. (i) It is here that in the mean ­ 2 2 cos (ω, λ1 ) has its minimum and cos (ω, λ3 ) its maximum (not shown). Alignment effects appear to make the most out of the weak enstrophy situation in order to maximize vortex compres˙ ˙ is weak though ® a mean evolution, < Q >, < Q > < 0. The observed maximum ­sion 2and hence cos (ω, λ3 ) max ∼0.6 which corresponds to an alignment of ω and λ3 of only 40◦ . (ii) It is also here E D E D −Λ3 2 or is increased (since due to incompressibility Λ1 + Λ2 + Λ3 = 0, that the magnitude of Λ Λ1 Λ1 3 results also in an increased value for −Λ effectively enhances Λ1 ). This again E D 2 the vortex compression effect of the ω 2 Λ1 cos2 (ω, λ1 ) + ω Λ3 cos2 (ω, λ3 ) mechanism.

an increased value for

Λ2 Λ1

® ­ Mean conditional rates of enstrophy production and destruction, ω i ω j sij /ω 2 , were found to have their maxima in regions of low Q, Q < −2. Instead of showing the distribution of mean ® ­ ωi ωj sij /ω 2 in the frame of the Q − ωi ωj sij − sij sjk ski probability orbital, in figure 3.18 we show conditional, univariate PDFs of ωi ωj sij /ω2 . The PDFs of ω i ω j sij /ω 2 are conditioned on

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CHAPTER 3. RESULTS

Figure 3.17: Orbital regions of different contributions to vortex compression are shown. In (a) ® ­ contributions to vortex compressing due to ω2 Λ2 cos2 (ω, λ2 ) < 0 are rendered blue when in the mean they are smaller than −3s−3 . In (b) mean they are below −1s−3 and yellow when ­ 2 in the ® contributions to vortex compression due to ω Λ1 cos2 (ω, λ1 ) + ω 2 Λ3 cos2 (ω, λ3 ) < 0 are rendered using blue and yellow for the same mean values as in (a). large negative, intermediate and large positive values for Q. This is equivalent to high strain, weak and high enstrophy events. All of the PDFs are found to be positively skewed. The most important feature of figure 3.18 however is, that in the PDF conditioned on high strain, Q < −2, the peak of the most probable events is shifted towards more positive values of ωi ωj sij /ω 2 and both,

3.6. ENSTROPHY DYNAMICS

87

strong vortex compression and stretching rate events become more probable.An explanation for the

Figure 3.18: PDFs of ω i ω j sij /ω 2 conditioned on weak, intermediate and strong values for Q are shown. They are all skewed positively. For Q < −2 the peak of most probable events is shifted towards more positive ω i ω j sij /ω 2 values and both, strong vortex compression and stretching rate events become more probable. increased probability of strong rates of vortex compression is the observation mentioned above that D E D E −Λ3 Λ2 the magnitude of Λ1 or Λ1 is increased in regions with Q < −2, which effectively enhances the

3 rate of vortex compression of the Λ1 cos2 (ω, λ1 )+Λ3 cos2 (ω, λ3 ) mechanism, since when −Λ Λ1 À 0 then Λ1 + Λ3 1, i.e. high enstrophy. Qualitatively nothing changes while the extrema for cΛ2 are slightly higher, −0.4 and 0.5. For events where Q < −2, i.e. in high strain regions, figure 3.19 suggests that in the mean there is no contribution from Λ2 cos2 (ω, λ2 ) to the rate of vortex compression and the influence, cΛ2 , on the rate of vortex stretching exceeds 60%. In other words, the increased positiveness of the rate of enstrophy production in high strain regions is strongly associated with the term Λ2 cos2 (ω, λ2 ).Figure 3.20 shows in more detail where the vortex compression and stretching rate events that are dominated by the term Λ2 cos2 (ω, λ2 ), defined as orbital. cΛ2 with expression (3.17), are to be found within the Q − sij sjk­ski − ωi ωj sij probability ® Represented in blue color are the two regions where the mean of Λ2 cos2 (ω, λ2 ) is responsible for

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CHAPTER 3. RESULTS

Figure 3.19: Means of the relative contribution, cΛ2 , for each value of ωi ωj sij /ω2 are conditioned on weak, intermediate and strong Q. cΛ2 is maximal at intermediate compression and stretching rates. The cases −2 < Q < 1 and Q > 1 are qualitatively similar. For events where Q < 2 in the mean there is no contribution from Λ2 cos2 (ω, λ2 ) to the rate of vortex compression and the influence on the rate of vortex stretching exceeds 60%!

half or more of the rate of vortex compression or­ stretching, hkc®Λ2 ki > 0.5, and the yellow color represents the region where the contribution of Λ2 cos2 (ω, λ2 ) is larger even than two thirds, hcΛ2 i > 2/3.The section on enstrophy dynamics is completed with a representation of the mean of the geometrical invariant cos (ω, λ2 ) in the Q − sij sjk ski − ω i ω j sij probability orbital. As already mentioned the geometrical invariant, cos (ω, λ2 ), is found to have a universal behavior in the sense that for a number of flows - if not every flow - there exists a strong alignment tendency for ω and λ2 . The predominant alignment of ω to λ2 becomes stronger even in regions with high strain or high vorticity, as shown in figure 3.4 above and as discussed in Tsinober (1998, 2000 and 2001). Figure 3.21 shows regions in the Q − sij sjk ski − ω i ω j sij orbital where in the mean the value of the invariant hcos (ω, λ2 )i is higher then 0.707. It shows how cos (ω, λ2 ) is strongly correlated with the rate of enstrophy production associated with the term Λ2 cos2 (ω, λ2 ) and cΛ2 ,defined in expression (3.17). The correlation of hcos (ω, λ2 )i with hcΛ2 i is much stronger than just with high enstrophy or strain regions. It also needs mentioning that the shape of the regions corresponding to high mean values of cos (ω, λ2 ) ’follows’ the direction suggested by the mean evolution of fluid particles in the Q−sij sjk ski −ωi ωj sij probability orbital. This predominant alignment of ω to λ2 - as will be shown below in section 3.10 one of the essential differences of vortex lines to material lines - appears to be a key ingredient to the two nonlocal self-amplification processes of turbulence.

3.7. STRAIN DYNAMICS

89

Figure 3.20: Events that are dominated by the contribution of the term Λ2 cos2 (ω, λ2 ), defined as cΛ2 , are represented in blue color where the intermediate contributor is responsible for half or more of the rate of vortex compression or stretching and the yellow color represents the region where the contribution of cΛ2 is larger even than two thirds.

3.7

Strain dynamics

Strain dynamics governed by expression (3.6) are just as poorly understood as enstrophy dynamics discussed above. The problem that arises when trying to explain the positiveness of h−sij sjk ski i is usually shifted to the problem of the positiveness of hΛ2 i > 0, since Λ1 > Λ2 > Λ3 and in incompressible flows Λ1 + Λ2 + Λ3 = 0 and −sij sjk ski = −(Λ31 + Λ32 + Λ33 ). What is clear is ”...that the appropriate level of dissipation moderating the growth of turbulent kinetic energy is achieved by the build up of strain of sufficient magnitude” (Tsinober 2000). The only situation that produces s2 is when Λ2 > 0. With Λ1 +Λ2 +Λ3 = 0 a positive Λ2 results in larger values for kΛ3 k as compared to Λ1 , which in turn causes the strain production term, −sij sjk ski = −(Λ31 + Λ32 + Λ33 ), to be positive. From the positively skewed PDFs for Λ2 , −sij sjk ski , the joint PDF for −sij sjk ski and ω i ω j sij and the shape of the Q − sij sjk ski − ω i ω j sij probability orbital, looked at in the Q − sij sjk ski plane, it is clear that most fluid particles indeed have a Λ2 > 0. Large −Λ3 , positive Λ1 and Λ2 is what Betchov (1956) calls a jet collision situation. He argues that with the produced strain, vorticity is stretched along λ1 and λ2 and hence enstrophy is essentially produced through Λ3 . ”... . It is clear, therefore, that production of vorticity is associated essentially with Λ3 and production of ω 1 and ω 2 . This suggests that the most important processes associated with production of vorticity and energy transfer resemble a jet collision and not the swirling of a contracting jet.” In terms of an - in the mean cyclic - evolution of nonlocal self-amplifying processes this can be commented by stressing the fact that strain needs to be built up to a level sufficiently high, in order for it to significantly

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Figure 3.21: Regions where in the mean the value for cos (ω, λ2 ) is higher then 0.707 are rendered in colors according to mean value of cos (ω, λ2 ). Note how hcos (ω, λ2 )i is strongly correlated with the rate of enstrophy production that is associated with the term Λ2 cos2 (ω, λ2 ) or cΛ2 . The shape of the regions corresponding to high values of hcos (ω, λ2 )i ’follows’ the direction suggested by the mean evolution of fluid particles in the Q − sij sjk ski − ωi ωj sij probability orbital. contribute to the production of enstrophy. The different mechanisms involved in strain dynamics are again discussed in the context of the Q − sij sjk ski − ω i ω j sij probability orbital. Similarly to the section on Enstrophy dynamics here the different mechanisms of strain dynamics and their approximate balance will be discussed in the following. From expression (3.6) it follows that these mechanisms are (i) mean strain production due to (3.18) h−sij sjk ski i > 0 and mean strain destruction due to h−sij sjk ski i < 0,

(3.19)

and (ii) mean strain production due to the combined effect of the pressure and the viscous term of expression (3.6) as À ¿ ∂2p 2 (3.20) + νsij ∇ sij > 0 −sij ∂xi ∂xj and mean strain destruction due to the combined effect of the pressure and the viscous term of expression (3.6) as À ¿ ∂2p 2 + νsij ∇ sij < 0. (3.21) −sij ∂xi ∂xj

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3.7. STRAIN DYNAMICS

Using the technique of 3D-PTV does not allow for measurements of pressure Hessians and therefore 2p and νsij ∇2 sij can be obtained as only the sum of −sij ∂x∂i ∂x j −sij

∂2p 1 Ds2 1 + sij sjk ski + ω i ω j sij , + νsij ∇2 sij = ∂xi ∂xj 2 Dt 4

(3.22)

with the three terms of the r.h.s. of expression (3.22) being accessible by 3D-PTV. From results reported by Tsinober (2000) it is known that in expression (3.21) the magnitude of the pressure terms is of the same order as that of the viscous terms. To discuss the different mechanisms involved in strain dynamics we start with the region of the Q − sij sjk ski − ωi ωj sij probability orbital where −sij sjk ski is positive. The combination of figure 3.22 a) and figure 3.22 b) demonstrates how mean strain production due to −sij sjk ski > 0 is approximately balanced by mean strain destruction due to the combined effect of the pressure and the viscous term of expression (3.21). Figure 3.22 a) shows, just like in the figures for enstrophy dynamics, values for −sij sjk ski that are above 3s−3 in blue color and values above 6s−3 in yellow. Since −sij sjk ski is plotted on itself figure 3.22 a) is only interesting in combination with figure 3.22 b) which shows the corresponding sum Dof expression (3.21) using E the same color coding as ∂2p 2 figure 3.22 a), namely blue for regions where −sij ∂xi ∂xj + νsij ∇ sij < −3s−3 and yellow where E D 2p 2 < −6s−3 . The main observation that can be made from figure 3.22 is that −sij ∂x∂i ∂x + νs ∇ s ij ij j apparently in regions with high strain, Q / 0, the total of the terms −sij sjk ski − sij

∂2p + νsij ∇2 sij , ∂xi ∂xj

(3.23)

is close to zero. In regions of Q ' 0 however, in the mean the strain production term, −sij sjk ski , slightly exceeds the contributions from pressure and viscous destruction. From expression (3.6) we 2 1 know that in 12 Ds Dt vortex stretching plays the role of strain destruction, − 4 ω i ω j sij , and from figure 3.15 it is seen that at Q ' 0 vortex stretching starts to become large. Therefore, not only the total of the terms of expression (3.23) is close to zero but also the total of the terms 1 ∂2p + νsij ∇2 sij . −sij sjk ski − ω i ω j sij − sij 4 ∂xi ∂xj

(3.24)

The results presented above on enstrophy suggest that in the Q − sij sjk ski − ω i ω j sij probability orbital at ω i ω j sij < 0 vortex compression, contributing to a mean evolution of Q, < Q˙ > < 0, is mainly due to the negativeness of the contributions associated with Λ1 and Λ3 (expression 3.13). When ω i ω j sij > 0 the mean vortex stretching contribution associated with Λ2 (expression 3.14) dominates, especially its rate, which adds to a positive mean evolution of Q, < Q˙ > > 0. The observed approximate balance between strain production, −sij sjk ski , and the mean of the sum of pressure and viscous terms of expression (3.21) is in full agreement with Tsinober reporting ...that the interaction of strain and pressure Hessian is such that it is opposing the production of strain when it [strain] becomes large (Tsinober 2000). The region in the Q − sij sjk ski − ωi ω j sij probability orbital where −sij sjk ski is negative is much smaller. Figure 3.23 a) shows strain destruction due to −sij sjk ski < 0 at levels of −sij sjk ski < −1s−3 (blue) and −sij sjk ski < −3s−3 (yellow). It is interesting to observe how the shape of the Q − sij sjk ski − ω i ω j sij probability orbital makes strong strain destruction due to −sij sjk ski < 0 to occur mainly in regions where vorticity is compressed, ωi ωj sij < 0. In other words, the system of NSE, which after all defines the shape of the Q − sij sjk ski − ωi ωj sij probability orbital, sees to it that destruction of strain due to −sij sjk ski < 0 is partly compensated by vortex compression which in turn contributes to strain production as can be seen from expression (3.6).

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Figure 3.22: In (a) regions with values for −sij sjk ski that are above 3s−3 are rendered in blue color 2p + and such with values above 6s−3 in yellow. In (b) blue is used for regions where < −sij ∂x∂i ∂x j 2

p νsij ∇2 sij > < −3s−3 and yellow where < −sij ∂x∂i ∂x + νsij ∇2 sij > < −6s−3 . The combination j of (a) and (b) demonstrates how mean strain production due to −sij sjk ski > 0 is approximately balanced by mean strain destruction due to the combined effect of pressure and viscous terms of expression (3.21).

93

3.8. MATERIAL ELEMENTS - INTEGRATED RATE OF STRAIN TENSOR

Figure 3.23 b) shows the mean strain production (expression 3.20) caused by the combined effects of pressure and viscous terms of expression (3.6). Rendered in blue color are regions where the mean 2p 2p of −sij ∂x∂i ∂x + νsij ∇2 sij is above 1s−3 and regions where the mean of −sij ∂x∂i ∂x + νsij ∇2 sij is j j above 3s−3 are rendered in yellow. Since here, in the colored regions of figure 3.23 b), strain 3.23 b)Ein good and therefore also viscous strain destruction, νsij ∇2 sij - is relatively small, figure D 2

p , alone. approximation represents the mean strain production due to pressure effects, −sij ∂x∂i ∂x j E D 2 p > 3s−3 , in the mean significantly produce strain, The region, where pressure effects, −sij ∂x∂i ∂x j extends into regions with high values for enstrophy and enstrophy production. This region coincides with another region where ω is found to be predominantly aligned with λ1 and hence enstrophy is mainly produced by ω 2 Λ1 cos2 (ω, λ1 ) contributions. Summarizing the mean effect of the pressure 2p term, −sij ∂x∂i ∂x , it can be said that in the part of the Q − sij sjk ski − ω i ω j sij probability orbital j where −sij sjk ski is negative, it contributes to an evolution of Q, < Q˙ > < 0 by producing strain, except in regions where it neutralizes attempts of ω 2 Λ1 cos2 (ω, λ1 ) contributions to destroy strain. 2p plays a key role as a ’re-distributor’ in strain dynamics. It is ’only’ The pressure term −sij ∂x∂i ∂x j a re-distributor since for homogeneous and incompressible flows we have ¿ À ∂2p sij = 0. (3.25) ∂xi ∂xj

Without having direct access to pressure gradients through 3D-PTV to date the characteristics of the role played by pressure have to be accessed by other means.

3.8

Material Elements - Integrated rate of strain tensor

In the last sections of this chapter we turn our attention to the evolution of infinitesimal material elements for which Girimaji and Pope (1990b) give the following introduction: A material element is defined as any line, surface or volume that always consists of the same material points or fluid particles. The basic diffusive character of turbulence - which tends to move two fluid particles, however close initially, away from each other - renders the study of finite-sized lines and surfaces difficult. Batchelor (1952) was the first to simplify the general analysis of lines and surfaces to the analysis of infinitesimal line and surface elements. As Batchelor observed, subject to the assumption that velocity gradients in turbulence are bounded, an initially infinitesimal material element remains infinitesimal for a finitely long time, and during this period the velocity gradients can be considered uniform over the material element. So a one-point description of the velocity gradients following fluid particles suffices for the study of the evolution of infinitesimal elements. The study of enstrophy and strain dynamics is relevant for a deeper understanding of the dissipative nature of turbulence. A consequence of strain and enstrophy and the effect of their combined evolution, the stretching and folding of material elements, is related to the equally important characteristic of turbulent flows - its diffusivity. It causes rapid mixing of passive scalars, increased rates of momentum, heat and mass transfer. According to Tennekes and Lumley (1972) diffusivity is the single most important feature [of turbulence] as far as applications are concerned: it prevents boundary-layer separation on airfoils at large (but not too large) angles of attack, it increases heat transfer in machinery of all kinds, it is the source of the resistance of flow in pipelines, and it increases momentum transfer between winds and ocean currents. Infinitesimal material elements are by no means believed to be sufficient to capture the essentials of diffusivity but their analysis comprises a starting point for studies of more finite sized material elements. Additionally, material lines are important in at least two other respects. In the context of drag reduction and understanding of the stretching and alignment, evolutions of lines will help to pose

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Figure 3.23: (a) strain destruction due to −sij sjk ski < 0 at levels of −sij sjk ski < −1s−3 (blue) and −sij sjk ski < −3s−3 (yellow) is shown. In (b) mean strain production due to combined effects of pressure and viscous terms of expression (3.6) are shown. Rendered in blue color are regions where 2p ∂2p 2 −3 and regions where the mean of −s +νs ∇ s is above 1s +νsij ∇2 sij the mean of −sij ∂x∂i ∂x ij ij ij ∂x j i ∂xj is above 3s−3 are rendered in yellow. better questions for polymer dynamics. Second the evolution of material lines differs only in one term - the viscous term - from the evolution of vortex lines. For clarity the relevant expressions for

3.8. MATERIAL ELEMENTS - INTEGRATED RATE OF STRAIN TENSOR

95

vortex lines and material lines are shown again in this section even though they have been introduced already in sections 2.5.3, 2.5.4 and 3.1 above. Dω i ∂ui = ωj sij + ν∇2 ωi = ω j + ν∇2 ω i Dt ∂xj 1 Dω2 = ωi ωj sij + νω i ∇2 ωi 2 Dt Dli ∂ui = lj Dt ∂xj

(3.26) (3.27) (3.28)

1 Dl2 = li lj sij (3.29) 2 Dt Material lines, especially those which initially are perfectly aligned with vorticity allow to study to some degree the influence of viscosity on the evolution of vorticity. This section is organized as follows. First, some additional checks are presented to validate the use of integrated rate of strain tensors along particle trajectories. Second, results on randomly oriented material lines are presented. Third, differences between material and vortex lines are discussed with some emphasis on enstrophy dynamics. Finally, the evolutions of material surfaces and volumes will be discussed briefly. What is a material line? A material line, l, is a ’thought’ connection between any two points at locations x and x + l in a flow that always consists of the same fluid particles. If the vector li is i infinitesimal the evolution of the vector, l, follows the equation Dl Dt = ui (x + l)−ui (x) and can be written as ∂ui 1 Dli = lj = lj sij + lj rij = lj sij + εijk ω j lk , (3.30) Dt ∂xj 2 ´ ´ ³ ³ ∂uj ∂uj ∂ui 1 ∂ui where sij = 12 ∂x + is the symmetric part and r = − ij ∂xi 2 ∂xj ∂xi is the skew-symmetric part j of the velocity derivative tensor, D Dt

∂uk ∂ui 1 ∂xj , and rij = − 2 εijk ω k , where ω i = εijk ∂xj = ∂ ∂ ∂t + uj ∂xj . From 3D-PTV the evolution of l

εijk rkj , ω =curlu

is the vorticity vector and = can be monitored directly. However, if the set of l is restricted to those particles which are reasonably close together, i.e. are to a good approximation separated infinitesimally, the available set of data becomes very small. There are two simple reasons for this. First, with the seeding density, which for technical reasons had to be kept relatively low a typical particle distance is of order O (1 − 5η). Second, the probability of tracking n particles simultaneously over a given time decreases as a power of n. As already mentioned in a previous section on multipoint checks, there is an alternative way of defining l directly from 3D-PTV. Infinitesimal material elements can be derived in a purely local manner. Referring to Monin and Yaglom (1997) an infinitesimal material line element which is initially l(0), is given at any later time, t, by (3.31)

l(t) = B(t) · l(0), where B evolves according to the equation d B = h(t) · B(t), dt

h(t) =

µ

∂ui ∂xj

¶

,

(3.32)

t

with the initial condition B(0) = I and I as the identity matrix.To show that (i) even ls of order O (5η) behave like infinitesimal ls and (ii) that integration of B over time yields qualitatively correct results two kinds of joint PDF plots are shown. Figure 3.24 compares the evolution of each component li as obtained directly from 3D-PTV and as obtained be integrating B over time at three selected times. Generally there is a good agreement between lP T V and lB . The observed scatter

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Figure 3.24: The evolution of each component of material elements li as obtained from 3D-PTV and from integrating B over time at three selected times are compared. Generally there is a good agreement between lP T V and lB .

is a combined effect of line elements l which connect particles that are too far apart to satisfy the approximation of infinitesimal l, and errors in integration of B. This is especially true when along a particle trajectory the relative divergence is only piece-wise sufficiently low, i.e. when the approx∂ui tensor is insufficient. The data become very scarce, especially at time τ η = 4. imation to the ∂x j This reflects the simple fact that it is less likely to simultaneously track two particles as compared to one through a certain interval of time. Since it could be argued that the lP T V and lB of the above figure 3.24 are only in good agreement because l itself does not change much in direction and/or magnitude over time, in figure 3.25 also the differences between lt=0 and lt , ∆l, are shown. Again, the evolution of each component as obtained directly from 3D-PTV and using integration of B over time is compared for three different times. The correlation between ∆lP T V and ∆lB is still acceptable on a qualitative level. Two main features can be observed. The aspect ratios of the elliptical shapes of the joint PDF plots become higher as time increases and again the scarcity of data especially at time τ η = 4 can be observed. The first feature is somewhat counter intuitive since one would expect better correlation between ∆lP T V and ∆lB at very early times. As will be shown below however, material elements have a strong tendency to align with the principal stretching axis of the rate of strain tensor. This implies that the strongest changes of orientation - which are therefore most difficult to measure - occur in the early time interval from τ η = 0 to τ η = 4, when the typical material line orientation changes from completely random to aligned with the eigenvector λ1 of the rate of strain tensor sij .

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97

Figure 3.25: The evolution of ∆l, ∆l = lt − lt=0 , for each component of a material elements l as obtained from 3D-PTV and from integrating B over time at three selected times is shown. Generally there is a good agreement between ∆lP T V and ∆lB .

3.9

Randomly oriented material lines

The term randomly oriented material lines is used to distinguish them from those material lines studied below which initially are perfectly aligned with the vorticity vector. Each point along a particle trajectory is intersected by an infinite number of material lines. For convenience we selected only 5 material elements at each starting point of a trajectory that could be tracked for τ η = 2 or longer. The results below show that this is enough for the statistics to converge. We start with the mean rates at which material lines are being stretched. They have been studied ever since 1949 (Batchelor and Townsend, 1949; Batchelor, 1952; Girimaji and Pope, 1990b; Dresselhaus and Tabor, 1994; Huang, 1996). From these studies it is known that the mean stretching or growth rate of a material line is larger than the intermediate principal rate, but smaller than the most positive one. It is also known that the mean stretching rates of material lines are by a factor ∼ 2 − 3 larger than the mean stretching rates of vorticity. If the rates are normalized by τ η then the results suggest that there is very little (if at all) Reynolds number dependence on the stretching rate.Figure 3.26 summarizes the main features of material line stretching. It shows that roughly at τ η = 2 the mean stretching rate reaches a stable level of 0.45 s−1 which is well above the mean stretching rate for vorticity that is measured to be around 0.25 s−1 . With hΛ1 i ∼ 1 s−1 and hΛ2 i ∼ 0.2 s−1 the measured mean stretching rate of material lines of 0.45 s−1 is well in between, which is consistent with results from literature (Girimaji and Pope, 1990b; Dresselhaus and Tabor, 1994; Huang, 1996).

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Figure 3.26: The main features of the mean stretching rate of material lines are that (i) it is higher than the mean stretching rate of vorticity, (ii) it is higher than the intermediate principal strain rate and (iii) it is lower than the most positive principal strain rate. Looking for the mechanisms that explain how a material line in the mean is stretched it is instructive to look at the following expression (3.33) that decomposes li lj sij /l2 in terms that are associated with each eigenvector and eigenvalue of sij . li lj sij /l2 = Λ1 cos2 (l, λ1 ) + Λ2 cos2 (l, λ2 ) + Λ3 cos2 (l, λ3 )

(3.33)

Randomly selected material elements have no preferential alignment with λi . With Λ1 +Λ2 +Λ3 = 0 ∂ui ∂ui i it can be expected that initially li lj sij /l2 ∼ 0. From Dl Dt = lj ∂xj and the assumption that ∂xj changes

i sufficiently slow compared to Dl Dt (assumption of persistent straining) it follows that l should align with the most positive principal axis of strain (Batchelor, 1952).The - questionable - persistent ∂ui is more a symmetric straining tensor in the sense that any straining assumption implies that ∂x j rigid body rotation is compensated by the rotation of the eigenframe λi . In other words l is projected onto the eigenframe of sij which relatively to l is fixed. Then it would be clear that components aligned with λ1 are amplified and components aligned with λ3 are compressed, which would finally result in an almost perfect alignment with® the most positive principal axis of the rate of strain tensor ­ and a mean stretching rate of li lj sij /l2 = hΛ1 i. A slight misalignment of l and λ1 would remain caused by contributions associated with Λ2 and λ2 , which in the mean lead to stretching along the intermediate axis. This ­is more or® less the conclusion Batchelor (1952) arrived at. Results however show that the value of li lj sij /l2 is closer to hΛ2 i than to hΛ1 i. Figure 3.27 shows the evolution of the PDFs of material line stretching, li lj sij /l2 , over time. The most prominent feature is how the large variance converges quickly to a stable distribution within a time interval of 2τ η . Second, a shift of the most probable events towards more positive rates of stretching is observed. Third, a significant negative part of the PDF persists. The remaining variance is a combined effect of varying values for Λi and varying alignments of l with respect to the eigen-frame, λi . Trivially, compressing material line events can only be explained with alignments of l to negative principal strain axes.

3.9. RANDOMLY ORIENTED MATERIAL LINES

99

Figure 3.27: The evolution of the PDFs of material line stretching, li lj sij /l2 , over time is shown. The large variance converges to a stable distribution within a time interval of 2τ η . A shift of the most probable events towards more positive rates of stretching and a significant persistence of compressing events is observed. ∂ui the material line In figure 3.28 the evolution in time of the cosine between l and Wl , Wl = lj ∂x j ¢ ¡ l l stretching vector, °is shown. cos ¡ ¡ ¢° l, W indicates whether W is actually leading to a stretching ¢ or l l ° ° > 0, or if it mainly causes a change of direction of l, cos l, W ∼ compression of l, cos l, W 0.Again, it is observed that within the time interval of the first 2τ η the PDF undergoes a qualitative change. Starting from a practically flat distribution it converges to a strongly positively skewed distribution, which is another manifestation ¢of the mean positive stretching rate of material lines. ¡ Still, for roughly half of all events cos l, Wl < 0.5, i.e. Wl is outside an imaginary cone around l with an opening angle of ∼ 120◦ . This reflects a considerable non-perfect alignment with λ1 , a slight alignment with λ2 and a significant attempt of material line kinematics to tilt the orientation of l. It is likely that most of the tilting is trying to re-align any ls that are either λ3 - or λ2 -aligned back to λ1 : In the case of λ3 -alignment the λ3 component of l is compressed until the other two components become more dominant. When l is λ2 -aligned this orientation is, e.g. for a ratio of hΛ2 /Λ1 i ∼ 0.25, only stable when l is inside an imaginary ’cone’ around λ2 with an opening angle of no more than 28◦ . As soon as the angle between l and the intermediate principal axis is larger, the λ1 component of Wl becomes dominant, i.e. Λ1 cos (l, λ1 ) > Λ2 cos (l, λ2 ). This ’stability’ of orientation of ls - with respect to the eigen-frame of sij - very close to the intermediate principal strain axis is of course strongly dependent on the local value of Λ2 /Λ1 . Figure 3.29 finally shows the evolution in time of the PDFs of cos (l, λi ). After the above discussion it is not surprising that a clear, but non-perfect, alignment of ls with λ1 can be observed after a transition time of 2τ η . The PDFs for cos (l, λ2 ) reveal a constantly flat distribution for all times and the PDFs for cos (l, λ3 ) suggest that indeed any λ3 component of l is compressed resulting in an almost perpendicular orientation of l with respect to λ3 . Mainly two papers deal with the issue of persistency of straining of material lines. Girimaji and

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∂ui Figure 3.28: The evolution in time of the cosine of the angle between l and W l = lj ∂x , the material j line stretching vector, is shown. A predominance of stretching over tilting and compressing is reached after 2τ η .

Pope (1990b) attribute the smaller than expected stretching rates and cos (l, λ1 ) to the effects of vorticity and ’non-persistent straining’. They conclude that the misaligning effects are due to the rotation of the principal strain axes. Dresselhaus and Tabor (1994) point out that it is obvious that vorticity and strain rotation are dynamically dependent and come up with equations that identify the role played by the competing effects of vorticity and strain-basis rotation. However, they arrive at the strange conjecture that vortex line stretching should be greater than material line stretching, which is confirmed neither by DNS (Huang 1996) nor by our experiment.A few sections later in 3.11 Evolution of material surfaces and volumes infinitesimal material volumes and the stretching rate of their surfaces will be discussed. Any three infinitesimal material lines define an infinitesimal material volume, which in incompressible flows and finite times is conserved. In this sense material lines and their orientation with respect to the strain basis are restricted. As will be shown the time required for any material element to be stretched to twice its original length or size is observed to be of order O (10τ η ). The non-perfect alignment of ls with λ1 is thus partly necessary simply to conserve initial volumes spanned by any three material elements. Also in the context of material volumes, the issue of strain-basis rotation will be picked up again. Results will be shown for the integrated history of the deformation tensor, the Cauchy-Green tensor.

3.10

Differences between material and vortex lines

From the results presented above it is clear that between randomly selected material lines and vorticity lines there exist two major qualitative differences. First, the stretching rate of material lines is clearly larger than the rate of vortex stretching. Second, material lines have a strong tendency to align with the most positive principal axis of strain while vorticity tends to align with the intermediate principal axis. However, for both cases the alignment is non-perfect. In this section

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101

Figure 3.29: The evolution in time of the PDFs of cos (l, λi ) is shown. After a transition time of 2τ η a clear alignment of l’s with λ1 can be observed . The PDFs for cos (l, λ2 ) reveal a constantly flat distribution for all times and the pdf’s for cos (l, λ3 ) suggest that l and λ3 are oriented almost perpendicular to each other. we will look at a special kind of material elements, namely at such elements which at some initial time τ = 0 are perfectly aligned with the vorticity vector. This kind of experiment allows to study to some degree the role played by viscosity and the effect of the nonlinearity of the term ω i ω j sij as compared to li lj sij in λ2 -alignment, and the effect of viscosity on enstrophy change. This can

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be seen from the above expressions (3.26) and (3.28) for material and vortex line stretching which differ qualitatively in the sense that the expressions for l are linear and those for ω are not and by the viscous terms in the equations for vorticity. The latter issue - viscosity and enstrophy change will be discussed in some detail at the end of this section.

3.10.1

λ2 -alignment

Figure 3.30: The evolution of the PDFs of the stretching rates of material lines with l0 = ω 0 are shown. The variance of li lj sij /l2 barely changes over time. The only notable change is a a slight decrease of compressing events. Figure 3.30 shows the evolution of the PDFs of the stretching rates of these special material lines over time. Since l0 = ω 0 for the very first time interval it can be argued that li lj sij /l2 is equivalent to ω i ω j sij /ω 2 with ν = 0. The main feature of figure 3.30 is that the variance of li lj sij /l2 barely changes over time. The only notable change is a slight decrease of compressing events. This means that either not much change of alignment between l and ω occurs or that while the alignment of l changes this is not reflected much in the variance of li lj sij /l2 . ¡ ¢ In figure 3.31 the evolution of the PDFs of cos l, Wl is shown. The change is again not very dramatic. A slight reduction of the material line compressing and a slight increase of the material line stretching events can be observed. It is interesting that¡ after¢a time of τ η = 2 the transition appears to be completed even though the skewness of cos l, Wl remains less than for initially randomly distributed material elements l as shown in figure 3.28. This would imply that a material element l with l0 = ω 0 in the mean possesses some kind of memory that lasts longer than the typical transition time observed above and cited in literature of τ η = 2. The only way l can remember its initial state l0 = ω 0 for a certain time is when the orientation with the intermediate principal axis is stable under some conditions.Figure 3.32 shows the evolution in time of the PDFs of cos (l, λi ), with l0 = ω 0 . It exhibits a similar behavior as observed in figure 3.31, in the sense that material lines l with l0 = ω 0 appear ’reluctant’ to act like ’proper’ material lines l. Namely, figure 3.29 above

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103

¢ ¡ Figure 3.31: The evolution of the PDFs of cos l, W l is shown. Only a slight reduction of the material line compressing and a slight increase of the material line stretching events can be observed. suggests that a material line element seeks an orientation that is perpendicular to λ3 and almost parallel to λ1 . The orientation with λ2 remains as if it was completely random. Figure 3.32, with l0 = ω 0 , however shows how the alignment of l with λ1 develops only relatively slowly and the alignment with λ2 is surprisingly persistent over a time period of τ η = 6. In other words, while special material lines l with l0 = ω 0 seem to show the same characteristic behavior as randomly selected material lines l, their transient time is significantly larger.So what do these observations tell us about the role played by viscosity in λ2 -alignments of vorticity? The observations made in figures 3.30, 3.31 and 3.32 suggest that viscosity is necessary for λ2 -alignment of vorticity and primarily responsible for it. Because, when absent, the alignment slowly but steadily decreases. In addition, from the results presented above there appears to be a significant kinematic effect that assists viscosity in stabilizing the λ2 -alignment of vorticity. By kinematic effect we mean the simple mechanism, as already mentioned above, that Λ2 cos (l, λ2 ) > Λ1 cos (l, λ1 ) when the angle between l and λ2 is smaller than e.g. 28◦ and Λ2 /Λ1 > 0.25. As will be discussed below, it is likely that there is yet another effect, namely the ’acting-back’ of vorticity to the flow. In other words, viscosity takes care of bringing vorticity vectors in the preferential alignment with λ2 , from where on it is assisted by a dominant Λ2 cos (l, λ2 ) term of Wl and the just mentioned ’acting-back’ effect. Since the ’dominance’ of Λ2 cos (l, λ2 ) is not very strong, and it is limited to a narrow region around the intermediate principal axis, already small strain frame rotations suffice to disturb this weak balance. The above findings on viscosity being responsible for λ2 -alignments are consistent with numerical results reported by Ohkitani (2002). He finds passive vectors to align with λ2 when a diffusive term for material elements l is added to the transport equation of expression (3.28) as: ∂ui Dli = lj + ν∇2 li . Dt ∂xj

(3.34)

Sofar, one could speculate that the observed λ2 -alignment of vorticity or the persistent λ2 alignments

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Figure 3.32: The evolution in time of the PDFs of cos (l,λi ), with l0 = ω0 , is shown. The alignment of l with λ1 develops, but only relatively slowly. The alignment of l with λ2 is becoming weaker with time but it is surprisingly persistent over a time period of τ η = 6. of material elements are assisted - or in the latter case - caused solely by the Λ2 cos (l, λ2 ) > Λ1 cos (l, λ1 ) effect. However, it can be argued that the observed ’stabilization’ of λ2 orientation of l and ω is not solely due to effects associated with Λ2 and λ2 , but due to vorticity itself - or its ’acting back’ on the flow - which just happens to be preferentially aligned with the intermediate eigenvector. In the

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latter scenario passive ls aligned to ω would profit - in the sense of λ2 -alignment - from the active, nonlinear interaction of vorticity with its surrounding region. This interaction would be such, that it favors alignment of passive and dynamically active vectors with the axis of vorticity. To eludicate whether viscosity is mainly assisted either by Λ2 cos (l, λ2 ) > Λ1 cos (l, λ1 ) mechanisms, or ’acting back’ of vorticity on the flow, or if it is a combination of both, figure 3.33 compares PDFs of kcos (l, λ2 )k at τ η = 3 that are conditioned on different initial situations for l0 at time τ η = 0. The different initial conditions allow to some degree to distinguish between the importance of l0 being initially aligned to λ2 or ω, and hence to compare the importance of the two effects. The restriction to cos (l, λ2 ) and time τ η = 3 is justified by the results of previous figures. Namely, the statistical evolution of alignments of ls leaves its traces only in the PDFs of cos (l, λ1 ) and cos (l, λ2 ) while the PDFs of cos (l, λ3 ) remain practically unchanged. Since almost all of the action takes place in the plane of λ1 and λ2 , everything that can be observed in the PDFs for cos (l, λ1 ) is also reflected in the PDFs for cos (l, λ2 ). The evolution of ls with l0 = ω 0 , after sufficiently long time is gradually drifting towards a state that initially randomly oriented ls take. For the purpose of comparison of different initial conditions and the two effects, it therefore suffices to look at how much the corresponding PDFs of kcos (l, λ2 )k have already progressed towards their known final states - PDF with ’flat’ distributions - at an arbitrary selected time of τ η = 3. Figure 3.33 a) shows that ls which at τ η = 0 had a perfect alignment with the eigenvector λ2 (blue and light blue) are slightly more persistent in keeping their initial orientation than such ls which at time τ η = 0 were perfectly aligned with the vector of vorticity (red and orange). In addition, initial situations where Λ2 > 0 at τ η = 0 (blue and red), seem to increase the persistency of λ2 -alignment, while situations with initial Λ2 < 0 (light blue and orange) decrease it respectively. This indicates that the Λ2 cos (l, λ2 ) > Λ1 cos (l, λ1 ) effect actually exists and is relevant, i.e. it indicates that it is more than just caused by coinciding λ2 and ω orientations. Figure 3.33 b) and c) show the relevance of vorticity ’acting back’ on the flow. In figure 3.33 b) PDFs of ls that are initially perfectly aligned with λ2 - according to figure 3.33 a) seemingly preferable conditions for increased persistency of λ2 alignment - show a high sensitivity to their relative alignment with vorticity. ls that are aligned with vorticity which themselves are aligned to the intermediate principal axis of strain seem to be very persistent in respect to their original λ2 alignment. Such ls that initially are misaligned to ω loose their initially perfect λ2 -alignment quickly. This supports the significance of the effect of vorticity ’acting back’ on the flow.Formally vorticity is affected - in part - by the vortex stretching vector, Dω Dt = W+ν..., with Wi = ω i sij . Vorticity is changed by the interaction of vorticity with strain. Since the field of strain can be expressed as a nonlocal functional of vorticity it is also affected by changes of vorticity. The interaction of ω and sij changes vorticity, which acts back on the flow changing sij , which in turn acts on the evolution of vorticity. Apparently one effect of this nonlinearity is ’stabilizing’ λ2 -alignments of vectors which are close to vorticity. In figure 3.33 c) it can be seen that for such ls which initially are aligned with ω the magnitude of vorticity seems to decide just how much ’acting back’ occurs and hence assists significantly in ’stabilizing’ λ2 -alignments. ls which initially are aligned with λ2 are (i) much less affected by the magnitude of enstrophy and (ii) their reaction is just contrary to those with initial vorticity alignment. Low enstrophy seems to result in a slightly increased persistence of λ2 -alignments for ls with l0 = ω 0 . In figure 3.33 d) it can be seen that high strain is for both types of ls related to an increased persistence of λ2 alignment. Not much surprising the PDFs of figure 3.33 d) resemble strongly those shown in figure 3.33 a), which reflects the fact that the sign of Λ2 is correlated with the magnitude of s2 which could already be observed above in figure 3.11. The results presented in figure 3.33 suggest the following conclusion. At least two effects assist viscosity in ’stabilizing’ orientations of material and vortex lines that are aligned with the inter-

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Figure 3.33: PDFs of kcos (l, λ2 )k at τ η = 3 that are conditioned on different initial situations for l0 at time τ η = 0 are shown. (a) ls with l0 = ω 0 and l0 = λ2 conditioned on the sign of Λ2 at time τ η = 0. (b) ls with l0 = λ2 conditioned on the cosine of the angle between l0 and ω 0 at time τ η = 0. (c) ls with l0 = ω 0 and l0 = λ2 conditioned on the magnitude of enstrophy, ω2 , at time τ η = 0. (d) ls with l0 = ω0 and l0 = λ2 conditioned on the magnitude of strain, s2 , at time τ η = 0. mediate principal axis of the rate of strain tensor sij , namely, the ’Λ2 cos (l, λ2 ) > Λ1 cos (l, λ1 )’ or shorter, ’Λ2 ’ effect and vorticity ’acting back’ on the flow. Overall their influence is comparable - with the ’Λ2 ’ effect being just slightly larger than the effect of vorticity ’acting back’. More important, the latter is very sensitive to the magnitude of enstrophy and slightly sensitive to the magnitude of strain, whereas the ’Λ2 ’ effect is (i) generally less sensitive to enstrophy and strain and (ii) is strongest in regions with low enstrophy and or high strain. Again, it has to be kept in mind, that in order to really understand the process, the influences of vorticity, enstrophy production and rotation of strain frames on alignment kinematics and dynamics need to be investigated more deeply.

3.10.2

Viscous change of enstrophy

The effect of viscosity on the change of enstrophy is relevant not only because of its direct physical influence but also in the context of the ’energy cascade’. In physical space the concept of energy cascade or production of smaller scales from larger scales is often explained with vortex stretching (e.g. Tennekes and Lumley, 1972; Jiménez et al., 1993). It is argued that vortex stretching is the primary mechanism producing smaller scales. In the absence of viscosity the angular momentum of a material volume element is conserved. If vortex stretching leads to an increased angular velocity

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in the direction of the stretching, only a decreased cross-sectional area can conserve the angular momentum. The argument makes three assumptions. First, that it is possible to assign certain scales to vorticity, second, that for vorticity on scales of the inertial range viscous terms can be neglected, and third, that vortex stretching leads to an increased angular velocity. At least the second and third assumptions can be checked directly by comparing the rate of change of ω 2 and l2 , with initial conditions l0 = ω 0 . Figure 3.34 shows joint PDF plots of stretching of material elements versus 1/2 Dl2 /Dt (a) and the stretching of vorticity versus 1/2 Dω 2 /Dt (b). From expression (3.29) it follows that for material elements there is a direct relation and ideally all points should fall onto a diagonal. The data shown in figure 3.34 a) stems from material elements which are taken directly from 3D-PTV measurements. From figure 3.24 and 3.25 we know that these line elements, l, typically have a length klk ∼ 1 −4mm which is relatively large for the assumption of infinitesimally small material elements. Partly for these reasons we observe a considerable scattering of data. However, the scattering is precisely centered around the diagonal. In the case of vorticity we know from expression (3.27) that there is only a direct relation between ωi ωj sij and 1/2 Dω 2 /Dt if the term νω i ∇2 ω i is relatively small. Figure 3.34 b) suggests that at least in our case viscosity cannot be neglected and that there is no direct local relation between vortex stretching and change of enstrophy. What is even more striking is the fact that the variance of enstrophy change due to viscosity is larger than the variance of vortex stretching. This is true for both positive and negative changes of ¡enstrophy.Figure ¢ 3.35 shows joint PDF plots of the rates of stretching¡ of material line elements versus 1/2 Dl2 /Dt /l2 ¢ (a) and the rates of stretching of vorticity versus 1/2 Dω 2 /Dt /ω 2 (b). Figure 3.35 suggests that the¡ role of viscosity ¢ becomes even more important for the rate of enstrophy change. The variance of 1/2 Dω 2 /Dt /ω 2 is an order of magnitude larger than the variance of the vortex stretching rate ωi ωj sij /ω 2 . One possible candidate for a mechanism that could explain the importance of viscosity in enstrophy dynamics is the phenomena of vortex reconnection or collapsing of vortex configurations. Vortex reconnection is in fact only possible because of the existence of viscosity. Fernandez et al. (1995) observe in numerical experiments that during the time of a vortex reconnection phenomenon dramatic changes of enstrophy values can be observed. In the vicinity of vortex collapsing and reconnection in their numerical vortex filament simulation they observe a faster than exponential growth of vorticity on the filament and a maximum strain rate off the filament. They find that vorticity aligns - despite the faster than exponential growth - not with the largest, but with the intermediate principal axis of the strain rate tensor. They draw the conclusion that - despite the faster than exponential growth in a situation where two strong filament-like regions of high enstrophy are about to reconnect - vorticity will not pick up the largest available strain rate on the filaments themselves. They speculate that bridge formation - between the two strong regions - in vortex reconnection may be associated with low-level vorticity picking up the very large, off the filament strain rate growth. The mechanisms of vortex collapsing, vortex reconnection and of bridging are also discussed and summarized in Kida and Takaoka (1994). In order to study reconnection events directly, information on the history of spatial coherence of an entire field of velocity derivatives of a finite fluid volume is necessary. In other words, one needs to know the history of vortex lines of finite size in order to see which segments of different lines reconnect at a later time. From our experiment only information on infinitesimal vortex elements along particle path lines is available and therefore we can only observe events that might be correlated locally with reconnection events. It is not clear that large variations of 1/2 Dω 2 /Dt which are unrelated to ω i ω j sij , are caused solely by reconnection ¡mechanisms. ¢Therefore, in the following we restrict our attention to statistics on 1/2 Dω 2 /Dt and 1/2 Dω 2 /Dt /ω 2 . Results are presented for conditional joint PDF plots and conditional, univariate PDFs of change and rate of change of enstrophy, as well as PDFs of strain and enstrophy that are conditioned on change and rate of change of enstrophy.In figure 3.36 and figure 3.37 the joint PDF plots of ωi ωj sij versus 1/2 Dω 2 /Dt and their rates

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Figure 3.34: Joint PDF plots of stretching of material elements versus 1/2 Dl2 /Dt (a) and the stretching of vorticity versus 1/2 Dω2 /Dt (b) are shown. ωi ωj sij and 1/2 Dω 2 /Dt do not seem to be correlated implying that in enstrophy change viscosity plays a role just as important as vortex stretching. ¡ ¢ ω i ω j sij /ω 2 versus 1/2 Dω 2 /Dt /ω 2 respectively, are conditioned on weak and strong events of strain and enstrophy. The main result is that only for events with high enstrophy and moderate vortex stretching or rate of stretching a slight correlation can be observed, i.e. only for these events

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¡ ¢ Figure 3.35: Joint PDF plots of stretching material elements versus 1/2 Dl2 /Dt /l2 (a) and ¢ ¡ rates of the stretching rates of vorticity versus 1/2 Dω 2 /Dt /ω2 (b) are shown. For the rate of enstrophy production the role of viscosity becomes even more important. there seems to be a significant local connection of vortex stretching and change of enstrophy. In weak strain and enstrophy events the ratio between variance of enstrophy change and variance of vortex stretching is particularly high, mainly due to the fact that - as discussed above - weak stretching events are associated with weak strain and enstrophy. Especially in figures 3.37 (a-d) it can be

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Figure 3.36: The joint PDF plots of ω i ω j sij versus 1/2 Dω 2 /Dt are shown when they are conditioned on (a) weak strain, (b) high strain, (c) weak enstrophy and (d) high enstrophy. Only when enstrophy is high and vortex stretching moderate, center of (d), there appears to be a slight correlation between ω i ω j sij and 1/2 Dω 2 /Dt.

observed that only very moderate rates of enstrophy change are correlated to vortex stretching rates. Note that in figures 3.37 (a-d) only sub-windows with axes ranging from −2s−1 to +2s−1 are shown as compared to the corresponding figures 3.36 (a-d) that have axes ranging from −20s−1 to +20s−1 . A still somewhat blurred picture emerges that (i) large changes of 1/2 Dω 2 /Dt or rates of change of enstrophy are associated with strong strain and even more so with strong enstrophy events and (ii) that apparently only small changes of 1/2 Dω 2 /Dt can be directly related to vortex stretching.To see more clearly just how strain, enstrophy, change and rate of change of enstrophy are related to each other, in figures 3.38 (a) and ¡ ¢ 2(b) and figures 3.39 (a) and (b) conditional, univariate 2 2 on small and large events of PDFs of 1/2 Dω /Dt and 1/2 Dω /Dt /ω are° shown. Conditioning ° to occur depends s2 and ω 2 reveals that the probability for strong °1/2 Dω 2 /Dt° events °¡ ¢ 2 ° very much 2 ° on strain and slightly more so on enstrophy (figure 3.38). For strong 1/2 Dω /Dt /ω ° events the situation is slightly different. First, the rates of enstrophy change seem to°¡be less related¢to enstrophy ° and strain. Second, and much more important the probability of strong ° 1/2 Dω 2 /Dt /ω 2 ° events to occur is higher in high strain regions and lower in high enstrophy regions, see figures 3.39 (a) and (b).The results from figures 3.38 and 3.39 can be summarized as follows: The probability of ° ° 2 ° ° to enstrophy. Due to this proportionality strong 1/2 Dω /Dt events ° °¡ is more or ¢less proportional the probability for rates, ° 1/2 Dω 2 /Dt /ω 2 °, which are changes divided by enstrophy, to occur is only weakly sensitive to enstrophy. For strain the relation is less straight forward. Change and the

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¡ ¢ Figure 3.37: Joint PDFs of ω i ω j sij /ω 2 versus 1/2 Dω2 /Dt /ω 2 are conditioned on (a) weak strain, (b) high strain, (c) weak enstrophy and (d) high enstrophy. Only when enstrophy is high and the vortex stretching¡ rate is moderate, ¢ 2 center of (d), there appears to be a slight correlation between 2 2 ω i ω j sij /ω and 1/2 Dω /Dt /ω . rate of change of enstrophy depend on strain. This is consistent with previous findings above and results reported by Huang (1996) and Tsinober (1999b, 2001) where rates of nonlinearities such as sij sjk ski /s2 , ωi ω j sij /ω 2 and li lj sij /l2 are observed to be much stronger in regions with high strain than in regions with high enstrophy. In the context of vortex reconnection it could be speculated that in the tails of the PDF shown in figure 3.39 a), where an increased rate of enstrophy change is observed for high strain events, we might be seeing events that occur in regions with bridging between regions of concentrated vorticity. According to Fernandez et al. (1995) in such events the level of strain is high and the level of vorticity relatively low. To check our speculation, in figures 3.40 and 3.41 PDF plots of strain and enstrophy conditioned on change and rate of change of enstrophy are shown. They show just¢ how ¡ 2 strain and enstrophy are distributed in the tails of the PDFs of 1/2 Dω /Dt and 1/2 Dω 2 /Dt /ω 2 presented above. In figure 3.40 PDFs of strain and enstrophy are conditioned on negative, intermediate and positive 1/2 Dω 2 /Dt events. In figure 3.40 a) the PDFs for strain are given and in° figure 3.40 b) ° those for enstrophy respectively. Mainly three things can be observed. First, large °1/2 Dω 2 /Dt° events coincide with both larger strain and larger enstrophy events. Second, negative and positive 1/2 Dω 2 /Dt events are associated with almost identical distributions of strain and enstrophy. Third, 2 separates events of different intensity much stronger than the magnitude of 1/2­Dω ­ enstrophy ® ® /Dt −2 2 strain events. With s ∼ 4s and ω 2 ∼ 8s−2 it can be seen that small 1/2 Dω 2 /Dt events

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Figure 3.38: PDFs of 1/2 Dω 2 /Dt conditioned on (a) weak and high strain and (b) weak and high enstrophy. Conditioning on small and large events of s2 and ω 2 reveals that the probability for ° ° strong °1/2 Dω2 /Dt° events to occur depends very much on strain and even slightly more so on enstrophy. ­ ® practically inhibit enstrophy above 2 ω 2 ∼ 16s−2 (figure 3.40 b) whereas for strain the probability difference for ­very ® strong events to occur remains within one order of magnitude up to a strain intensity of 5 s2 ∼ 20s−2 (figure 3.40 a).

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¡ ¢ Figure 3.39: PDFs of 1/2 Dω2 /Dt /ω 2 , the rate of enstrophy change, are conditioned on (a) weak and high strain and (b) weak and high enstrophy. Contrary to the results on 1/2 Dω 2 /Dt, here its rate reacts most sensitive when conditioned on strain. High strain results in an increased probability for high rates of change of enstrophy (a). Conditioning on high enstrophy however, leads to less intense rates of change (b).

Figure 3.41 shows PDFs of strain and enstrophy that are¢ conditioned on the rates of negative, ¡ intermediate and positive changes of enstrophy, 1/2 Dω 2 /Dt /ω 2 . In figure 3.41 a) again the PDFs

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for strain are given and in figure 3.41 b) those for enstrophy. The emerging picture is similar to that of figure 3.40 - with one important difference. In fact, this difference is the only indication that the results from figure 3.39 a) can be associated with a bridging mechanism in vortex reconnection events. In figure 3.41 a) the probability of weak strain events decreases for high rates of enstrophy changes, whereas in figure the¢ probability of weak enstrophy events increases for high rates ¡ 3.41 b) 2 of enstrophy changes, 1/2 Dω /Dt /ω 2 . This finding is comparable with the ’high strain, low enstrophy, bridging and vortex reconnection’ phenomena which Fernandez et al. (1995) report to be associated with higher than exponential rate of enstrophy change.As a last figure on the issue of viscous change we show in figure 3.42 the joint PDF plots of 1/2 Dω 2 /Dt and ¡ ¢ enstrophy 2 2 1/2 Dω /Dt /ω versus strain and ° ° versus enstrophy. Figure 3.42 a) and b) suggest that strong 2 ° changes of enstrophy, 1/2 Dω /Dt° À 0, are slightly correlated to both, °¡ higher strain¢ and °higher enstrophy. Figure 3.42 c) shows that high rates of change of enstrophy, ° 1/2 Dω 2 /Dt /ω 2 ° À 0, are very weakly correlated to high strain and that the relation to enstrophy is purely nonlocal (figure 3.42 d). Strong rates of change of enstrophy inhibit high enstrophy and vice versa. Summarizing the differences between material and vortex lines it can be noted that (i) stretching rates of material lines are larger than the rates of vortex stretching, (ii) material lines have a strong tendency to align with the most positive principal axis of strain while vorticity tends to align with the intermediate principal axis, (iii) viscosity is very likely to be responsible for alignment of the vorticity vector with the intermediate principal axis, (iv) a ’Λ2 effect’ and vorticities ’acting-back’ on the flow assist viscosity in keeping passive vectors alined to the intermediate principal axis and (v) material line growth is governed by its stretching whereas growth of enstrophy is a combined effect of vortex stretching and viscous effects. Especially those viscous effects that might be related to vortex reconnection and collapse in part have the capability to change the topology of a turbulent flow. Kida and Takaoka (1994) report that two parallel vortex sections essentially produce vorticity that has components which are perpendicular to the original orientation. In other words, viscosity in combination with strain is responsible for dissipation and viscosity in combination with vorticity is associated with continuous change of topology of turbulent flow by means of vortex reconnection. Topological ideas in fluid mechanics are discussed in Ricca and Berger (1996). In order to get more insight into these mechanisms, Lagrangian measurements that can access the vector ∇2 ω i directly are needed.

3.11

Evolution of material surfaces and volumes

The results presented here are essentially about growth, orientation and rotation of infinitesimal material surfaces and of infinitesimal material volumes, with emphasis on their surfaces. The passive evolution of these types of material elements in a turbulent flow are the result of the combined dynamics of enstrophy and strain. It has to be noted, that positive mean material line stretching and positive mean surface stretching is also obtained for any isotropic random field, as was proven by Cocke (1969). In a flow it is trivially surfaces across which diffusive processes of various passive quantities are driven by their gradients. The question is, how these surfaces grow and how they evolve in space and how these kinematic processes are different in a turbulent flow than in an isotropic random flow. Answers would add to a better understanding of diffusivity in general and to a better understanding of turbulent diffusivity in particular. It should be mentioned once more that the study of these two types of infinitesimal surfaces is looked at as a starting point for a more extensive investigation on finite sized material elements.

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Figure 3.40: The PDFs of (a) strain and (b) enstrophy both conditioned ° on the magnitude of ° °1/2 Dω 2 /Dt°and s2 (a) and between change of enstrophy. A weak positive correlation between ° ° °1/2 Dω2 /Dt°and ω 2 (b) can be observed.

3.11.1

Stretching

Two randomly oriented material lines, l1 and l2 , are used to define a material surface with its vector normal, N, defined as the cross product of l1 and l2 as: N = l1 × l2 . The equation for the i li infinitesimal area, identified by Ni follows from the conservation of fluid volume, Ni li , i.e. DN Dt = 0,

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Figure 3.41: The PDFs of strain and enstrophy are conditioned on the rates of change of enstrophy. (a) conditioned PDFs of strain and (b) those for enstrophy respectively. In (a) the probability of weak strain events decreases for high rates of enstrophy changes, whereas in (b) the ¢ probability of weak ¡ enstrophy events increases for high rates of enstrophy changes, 1/2 Dω 2 /Dt /ω 2 . This finding is comparable to the ’high strain, low enstrophy, bridging and vortex reconnection’ phenomena which Fernandez et al. (1995) report to be associated with higher than exponential rate of enstrophy change.

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¡ ¢ 2 /Dt −s2 , (b) 1/2 Dω 2 /Dt −ω 2 , (c) 1/2 Dω 2 /Dt /ω 2 − s2 Figure 3.42: Joint PDFs of (a) 1/2 Dω ¡ ¢ and (d) 1/2 Dω 2 /Dt /ω2 − ω2 . In the cases (a), (b) and (c) it can be seen how the moduli of quantities¡ of both axes¢ are at least slightly correlated. Case (d) reveals a strong nonlocal relation between 1/2 Dω 2 /Dt /ω 2 and ω2 . which together with (3.28) gives ∂uk DNi = −Nk = WN . Dt ∂xi

(3.35)

In complete analogy to expression (3.29) which governs the growth of the length of a material line the growth of the area of surfaces is governed by expression 1 DN2 = −Ni Nj sij . 2 Dt

(3.36)

Hence, a good statistical indicator of if and how much such surfaces or compressed is ¡ are stretched ¢ again analogous to material lines - the geometrical invariant cos N, WN . To see what mechanism could be responsible for a surface stretching, using the decomposition of −Ni Nj sij as −Ni Nj sij = −N2 Λ1 cos2 (N, λ1 ) − N2 Λ2 cos2 (N, λ2 ) − N2 Λ3 cos2 (N, λ3 )

(3.37)

is helpful. From expression (3.37) it follows that substantial surface stretching can only stem from preferential alignments of N with the compressing principal axis of the rate of strain tensor sij , λ3 . From literature it is known that the mean rate of surface stretching behaves very similar to the mean rate of material line stretching, both, in terms of temporal evolution and magnitude (Girimaji and Pope, 1990b). The mean surface stretching rate was observed to be well below Batchelor’s estimate,

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hΛ1 + Λ2 i, (Batchelor, 1952) and just slightly higher than the mean material stretching rate. Also, it was found that after a transitional phase the mean rate reached a stable value suggesting that in the mean, material surfaces experience exponential growth. This exponential growth is confirmed by results presented by (Girimaji and Pope, 1990b) and Goto and Kida (2002). In figure 3.43 the evolution of PDFs of some geometrical invariants related to N are shown. All Ns ¡are at some ¢ time τ η = 0 defined by randomly oriented l1 s and l2 s. In figure 3.43 a) the PDFs of cos N, WN for different times are shown. The PDF for τ η = 0 has a relatively flat distribution that slightly favors surface compressing. In a period of transition of τ η ∼ 2 the PDFs become strongly positively skewed. In the time that follows only little changes can be observed in the distributions of the PDFs, suggesting that they have converged to a statistically stationary state of predominant surface stretching. This can be expected from what is known on the evolution of material lines, as discussed above. In figures 3.43 b), c) and d) the PDFs of cos (N, λi ) for the same times as in a) are shown. Again at time τ η = 0 all PDFs have very flat distributions - as can be expected from randomly selected ls and therefore also for Ns. Most of the change also occurs in the first period of transition of τ η ∼ 2. A strong preferential alignment of N with λ3 develops while the alignment of N with λ2 and stronger even with λ1 becomes less probable.This is another manifestation of the fact

Figure 3.43: Evolutions in time of PDFs of geometrical invariants involving the normal, N , of material surfaces are shown. In (a) the evolution of the PDF of the cosine of the angle between N and the surface stretching vector W N shows strong preferential surface stretching after 2τ η . In (b)-(d) evolutions in time of PDFs of the cosine of the angle between N and the eigenvectors λi , i = 1 − 3, show how N is strongly preferentially aligned with λ3 after 2τ η . that material lines are preferentially oriented close to the plane λ1 , λ2 resulting in surface normals N being aligned to λ3 , which leads to preferential surface stretching. The actual rates of surface

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stretching will be shown further below (figure 3.47) in comparison with the corresponding rates for material lines and surfaces defined by material volumes which are discussed next.An infinitesimal

Figure 3.44: A fluid volume at some time τ η > 0 with side length ratios (w1 )1/2 : (w2 )1/2 : (w3 )1/2 defined by the eigenvalues of the Cauchy-Green tensor, W , is sketched. The axis along w1 is constantly stretched, the axis along w2 is also stretched in the mean and the axis along w3 is compressed. material volume - for simplicity we think of a fluid block with initial side length ratios of 1 : 1 : 1 - deforms under the influence of straining into a volume with side length ratios of their axes as a : b : c. Following Girimaji and Pope (1990b) the Cauchy-Green tensor W, W = BBT ,

(3.38)

describes this deformation. B is the deformation tensor as described above and in expression (3.32). As it happens, a : b : c is equal to (w1 )1/2 : (w2 )1/2 : (w3 )1/2 , where w1 ≥ w2 ≥ w3 ≥ 0 are the eigenvalues of W. Given that B (0) = I, the initial values at time τ η = 0 of all eigenvalues are unity. Figure 3.44 illustrates the essence of what has just been described. A fluid volume at some time τ η > 0 with side length ratios defined by the Cauchy-Green tensor, W, is sketched. The axis along w1 is constantly stretched, the axis along w2 is also stretched in the mean, reflecting the fact that hΛ2 i of the rate of strain tensor, sij , is positive, and the axis along w3 is compressed, where wi are the eigenvectors of W. With initially equal side lengths of 1 the side lengths of such a fluid volume, or block, evolve as the squareroots of the eigenvalues of W as (w1 )1/2 , (w2 )1/2 and (w3 )1/2 .If at τ η = 0 the volume of a material element is unity then, due to conservation of volume, w1 · w2 · w3 = 1 for finite times, τ η > 0. Therefore it is also true that hln (w1 )i + hln (w2 )i + hln (w3 )i = 0

(3.39)

for finite times, τ η > 0. From figure 3.45 it can be seen that the measured and time integrated Cauchy-Green tensor in the mean is free of divergence and hence satisfies the conservation of volume

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Figure 3.45: The evolution of the mean eigenvalues, hwi i, of the Cauchy-Green tensor, W , is shown. If at τ η = 0 the volume is unity then, due to conservation of volume, w1 · w2 · w3 = 1 for finite times, τ η > 0. Hence also hln (w1 )i + hln (w2 )i + hln (w3 )i = 0 for finite times, τ η > 0. criterion, since for each time moment the sum of hln (wi )i is zero. The evolution of the mean eigenvalues, wi , of W, plotted as hln (wi )i over time, confirms the conclusion that hΛ2 i > 0 indeed results in a hln (w2 )i > 0. This is in very good agreement with numerical results presented by Girimaji and Pope (1990b) where, in addition, over a range of Reλ from 38 − 90 no Reynolds number dependence of the evolution of hln (wi )i was observed. Further, figure 3.45 reveals another noteworthy consequence of material volume deformation in the context of gradient amplification and increased diffusivity. In a simplified situation one could imagine two neighboring material volumes, V1 and V2 , each of which has a roughly homogeneously distributed concentration, c1 and c2 of some passive, diffusive quantity. At time τ η = 0 the diffusion over these two volumes would then be ∆c proportional to S ∆x , with ∆c = c2 − c1 , ∆x = 1 and S as the dividing surface between V1 and V2 . As time evolves the surface normal, N, of the dividing surface S, will - based on the result presented above in figure 3.43 d) - align with λ3 and hence ∆x ∼ (w3 )1/2 . The evolution of hln (w3 )i shown 1 at τ η ∼ 2 is already twice (exp (−1.3)−1/2 ∼ 2) and at τ η ∼ 6 four in figure 3.45 suggests that ∆x times (exp (−2.6)−1/2 ∼ 4) larger than initially. S, due to surface stretching increases and hence also amplifies diffusivity. The behavior of S will be discussed in more detail below. Since, as a result of this increased diffusivity, ∆c decreases the effect is slightly dampened. Most of the above results - material surface alignment, material surface stretching and material volume deformation - could also be observed in a Gaussian flow field which raises the question: what is so special about a turbulent flow field in this context? Essentially the answer is not known possibly with one exception. Namely, the properties of the eigenvalue Λ2 of sij , a positively skewed quantity with a mean value hΛ2 i close to 0.2 hΛ1 i for a wide range of turbulence measurements both experimental and numerical (Ashurst et al., 1987; She, 1991; Su and Dahm, 1996; Tsinober et al., 1992), have some important consequences for diffusivity. In a Gaussian flow field hΛ2 i is precisely zero. The ratios of the eigenvalues Λ2 /Λ1 of the rate of strain tensor or for the Cauchy-

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Green tensor the corresponding ratio ln (w2 ) / ln (w1 ) are often referred to as ’the shape’ of sij or W respectively. The term ’shape’ is used because, in an incompressible flow, the two theoretical extreme values of 1 and −0.5 the ratios Λ2 /Λ1 and ln (w2 ) / ln (w1 ) can take, lead to shapes of deformed material volumes that resemble those of pancakes or cigars. If the shape of W is ’pancakelike’ then - in the context of the above discussion - (w3 )1/2 ∼ ∆x is minimal and S is maximal for a given w1 . Hence, it can be said that pancake like shapes of W increase diffusivity between ∆c , whereas cigar like shapes decrease it. Figure two material volumes, which is proportional to S ∆x 3.46 shows joint PDF plots of ln (w1 ) and ln (w2 ) for different times of evolution, τ η = 1, 2, 4, 6. The ’shape’, ln (w2 ) / ln (w1 ), is clearly skewed towards ’pancake-shapes’ but the majority of events is closer to ln (w2 ) / ln (w1 ) ∼ 0 than to any of the two extrema. Over time little qualitative change can be observed. This experimentally obtained result is again in perfect agreement with the numerical findings in Girimaji and Pope (1990b).Therefore it can be conjectured that turbulent

Figure 3.46: Joint PDF plots of ln (w1 ) and ln (w2 ) for different time moments of evolution, τ η = 1, 2, 4, 6. ln (w2 ) / ln (w1 ) is skewed towards ’pancake-shapes’. flows as compared to Gaussian flows are special in that at least some of their properties, such as hΛ2 /Λ1 i > 0 and hln (w2 ) / ln (w1 )i > 0, lead to increased diffusivity. Now, the above question could be reformulated as: Why doesn’t turbulence go further and sees to it that hΛ2 i ∼ hΛ1 i instead of keeping the mean value of hΛ2 i only just slightly positive? Presently there is little hope for an answer. Among others such an answer would require an understanding of the reasons of the observed behavior of Λ2 and λ2 also in the context of vortex stretching. Galanti et al. (1996) suggest that well aligned and straight vortex lines, which are observed in regions with very high vorticity and thus are strongly aligned to λ2 , are a manifestation of a self-correcting mechanism that prevents singular behavior of simulations of NSE. They conclude that their observation ’...strengthens the connection

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between vortex dynamics and regularity’. We are tempted to speculate that since in enstrophy dynamics preferential alignments of vorticity with λ2 and hΛ2 i > 0, but hΛ2 /Λ1 i ¿ 1 play the role of moderating the growth of enstrophy, then the observation of hΛ2 i being only just slightly positive could be reflecting a trade-off between the two goals of maximizing diffusivity and keeping the overall magnitude of vorticity finite.1 Figure 3.47 shows how the surface S mentioned above actually evolves and how its growth compares to the growth of material lines, l, and material surface normals, N. We define S as the dividing surface not just between two material volumes V1 and V2 but, more generally, as the dividing surface between a material volume V1 and its surroundings, simply as the surface of the fluid ’block’ sketched in figure 3.44 as √ √ √ S = c · (2 w1 w2 + 2 w2 w3 + 2 w3 w1 ) .

(3.40)

With c = 1/6 the initial value of S is unity. The evolution of all mean stretching rates, shown in figure 3.47 a), reveal a characteristic change at τ η ∼ 2 − 3. After a transitional increase the rates apparently level off to some constant values. The mean stretching rates for material surface normals, N, are clearly higher than those for material lines, l, which is again consistent with findings in (Girimaji and Pope, 1990b). The mean stretching rate of S is lower than the corresponding mean rate of N and somewhat higher than the mean rate for l. From figure 3.47 b) it can be seen that after a time of τ η ∼ 7 both, material lines and surfaces, have roughly doubled their magnitude. The initial transitional period of the stretching rates reflects how material elements take some finite time to orient in the eigen-frame of the rate of strain tensor in the case of l and N and in the evolving field of deformation in the case of S respectively. After the transition the observed roughly constant stretching rates result in exponential growth. The higher rates for both types of surfaces, N and S as compared to l was already foreseen by Batchelor (1952). With the same assumptions he used to estimate the mean rate of material line stretching to be ∼ hΛ1 i, one arrives at a mean rate of surface stretching of hΛ1 + Λ2 i. As was discussed above this is overestimating the rates as compared 2 1 Dl2 to those obtained from numerical experiments and measurements, but the result that 12 DN Dt > 2 Dt remains correct. This is also reflected in figure 3.47 b) which shows a higher total stretching of N compared to l for every time instant. The finding that the mean stretching rate of S is lower than the corresponding rate of N (figure 3.47 a) and also that the total growth of S is significantly lower than the growth of N (figure 3.47 b) can be explained by the fact that S - being the surface of a material volume - also has areas that in the mean are compressed. Especially during the first period this dampens the overall mean rate of stretching and only when the last two terms of expression (3.40) are sufficiently small the rate for S approaches a similar value as the rate for N. Contrary to the numerical results reported in Girimaji and Pope (1990b) and Huang (1996) here no ’peaking’ of material line and surface stretching around time τ η = 2 is observed. This is noteworthy and asks for some explanation in the future. Girimaji and Pope make an attempt to at least explain the ’peaking’ of material line stretching. Since their simplified model, involving assumptions such as velocity gradients being frozen in time, vorticity perfectly aligned to λ2 , and material lines oriented only in the plane of λ1 , λ3 , in our view fails, a more extensive study is necessary to clarify the different observations, which at present is beyond our scope. 1 To claim that turbulence actually does what it does because it has goals, is of course questionable. Therefore, this sentence starts with ...tempted to speculate.. . To hope that turbulence has extremal properties, might be wishful thinking. This thinking is based on examples where nature obtimizes. Mostly these examples would stem from things that can evolve in time according to some selective criteria. Such things are associated with being alive. To speculate that the NSE also evolve selectively and thereby find a state that is optimal in some sense, is trying to make a living thing out of turbulence.

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Figure 3.47: Evolution in time of stretching rates (a) and absolute stretching (b) for material lines, l, material surfaces, N , and the surfaces of material volumes, S.

3.11.2

Folding

Finally, some results on the rotation of material volumes are presented. In section 3.5 Cyclic mean process above a mean cyclic evolution of fluid particles in the {Q − ωi ωj sij − sij sjk ski } space is observed. This mean evolution is suggesting the existence of some systematic succession of stretching and folding events, if one assumes, that mainly in strain regions material elements are being stretched

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and in regions dominated by high vorticity material elements, i.e. material volumes, experience high rotation, and further, that fluid rotation ultimately results in folding of material elements. If fluid rotation, i.e. angular velocity of a material volume, is auto-correlated over some distance, `r , then this will lead to folding of finite sized surfaces over a scale `f , with `f ≥ `r . The results of such a process is a new ’neighborhood’ for each of the folded material volumes. Where the assumption that high strain is correlated with high stretching of fluid ’blobs’ is relatively straight-forward the latter assumption - on the correlation between vorticity and rotation or folding - is not so clear, since high vorticity may also be associated with high shear events. The effect of vorticity on the angular velocity of a material volume may be compensated by the effect of strain or even by the rotation of the rate of strain frame (Dresselhaus and Tabor, 1994). This is essentially connected with the difficulty of giving a unique formal definition of a flow structure that by the eye easily is recognized as a ’swirl’ or ’eddy’. In the context of material elements as opposed to single point analysis - this difficulty can be partially avoided when we look at the rotation of a material volume, ω w , i.e. at the rotation of the frame of the principal axes defined by tensor, W. Technically, ω w is obtained by first solving the eigenvectors, wi , of the Cauchy-Green ¤ £− → expression (3.41) for the rotation tensor ω w . £→ ¤ ω w · ∆t = [(w1 ) (w2 ) (w3 )]τ +∆t (3.41) [(w1 ) (w2 ) (w3 )]τ · − Where w1 , w2 and w3 are the eigenvectors of the £Cauchy-Green tensor W at times t = τ and ¤ → 1 ω w , the rotation vector, ω w , is defined as s. From the rotation tensor, − t = τ + ∆t and ∆t is 60 ¤ £→ ¤    £−  → ω w 3,2 − − ω w 2,3 [ω w ]1 £− £− ¤ ¤  → →  [ω w ]2  =  − ω (3.42)  £ w ¤1,3 £ ω w ¤3,1  , → − → [ω w ]3 ω w 1,2 ω w 2,1 − −

in complete analogy to the definition of vorticity in expression (3.2) above. This procedure is only first order accurate and results should therefore be interpreted only for their qualitative character. ∂ui using weighted polynomial fitting - acting Future work will derive ω w in a similar fashion as ∂x j as a low pass filter with cut-off frequency of ∼ 30Hz - along particle trajectories. As can be seen however in the following, already this low order scheme allows for a preliminary glimpse on some characteristic properties of ω w . In figure 3.48 PDFs of the magnitude of the rotation of material volumes, ω 2w , are shown. First of all, it is noteworthy that the magnitude of ω 2w and its distribution is very comparable with the magnitude of enstrophy which is shown in figure 3.12 above. The fact that the PDF for ω 2w shows significant probabilities for events as high as 102 s−2 should be interpreted with care, keeping in mind that this could also be an artefact caused by the low order scheme defined by expression (3.41).In figure 3.48 a) the PDFs of ω 2w are conditioned on low and high events of strain and enstrophy. It is interesting that while the PDFs are apparently completely insensitive to the magnitude of strain (blue ­and® light-blue) there is a clear dependence on enstrophy (red and orange). ­Namely, ® for ω2 < 0.5 ω 2 the probability for weak events of ω 2w is almost doubled and for ω 2 > 2 ω 2 the probability for weak events of ω 2w is roughly reduced by a factor of ∼ 4 while strong events are observed to be slightly more probable. This can be interpreted as an indication that vorticity can be at least slightly associated material volume rotation. However, despite the observed dependence the correlation between the two types of events is found to be weak. In figure 3.48 b) the PDFs of ∂ui . ω 2w are conditioned on Q = 41 (ω2 − 2s2 ), the second invariant of the velocity derivative tensor ∂x j Here the effect is stronger. High values for Q apparently inhibit slow material volume rotation while strong ω 2w events become clearly more probable. The same dependence can be observed ­ 2 ® in the joint a material volume, ω w , conditioned PDF plot of Q versus ω 2w of figure 3.49. The mean rotation of ­ ® on values of Q is plotted in the same figure. According to ω 2w only, material volume rotation is

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Figure 3.48: Conditional PDFs of the magnitude of the rotation of material volumes, ω 2w , are shown. In (a) the conditions are weak and high events of strain (light blue, blue) and enstrophy (orange, red). In (b) ω 2w is conditioned on different values of Q, Q = 14 (ω 2 − 2s2 ). seemingly linearly dependent on Q, doubling its value from ∼ 25 to ∼ 50ω 2w over a range from −10 to 10Q. However, from the joint PDF plot it becomes clear that the situation is more complex. Namely, the most striking feature of figure 3.49 is, that most of the strong ω 2w events occur around Q ∼ 0. This cannot be seen from the univariate PDF in figure 3.48 b) since also most of the weak ω 2w events

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occur around Q ∼ 0.Going back to the above assumption on a simple correlation between vorticity

­ ® Figure 3.49: Joint PDF plot of Q versus ω2w . The mean rotation of a material volume, ω 2w , conditioned on values of Q is plotted as a white line. and rotation of material volumes the above results suggest the following conclusions: (i) There is a weak dependence of material volume rotation on vorticity (figure 3.48 (a)), (ii) the dependence on Q is stronger even (figure 3.48 (b)) and (iii) despite the observed dependence there is no direct linear correlation between Q and ω 2w , since most of the strong ω 2w events occur when the values for enstrophy are roughly balanced by those for strain, i.e. Q ∼ 0 (figure 3.49). In other words, despite some similarities - no simple analogy can be drawn from the evolution of single point fluid particles to the behavior of material volumes in the context of stretching and rotating or folding of fluid elements. More studies on the relationship between ω 2 , s2 and ω 2w , including their history, are necessary. Figure 3.50 shows - as a last result of this work - the nonlocal relation between the rotation of material and the stretching rate of their surfaces, S. From the joint PDF plot of ω 2w ³ volumes ´ 2 2 versus 12 Ds Dt /S it becomes clear that strong rates of surface stretching inhibit strong rotations of material volumes and vice versa. Apparently turbulence ’waits’ with re-orienting material surfaces, carried by their material volumes, for a suitable moment where no significant surface stretching occurs. This is consistent with the results from figure 3.49 above that surface stretching is associated with high strain, whereas intense material volume rotation is mostly associated with strain and enstrophy of equal magnitude and also, but less intense, with positive values of Q.In terms of diffusivity, one of the key properties of turbulent flows, from the above results the following picture can be drawn: The evolution of material volumes is such, that the normals, N, of its most stretched surfaces align with the ’compressing’ principal directions λ3 and w3 , the third eigenvectors of the rate of strain tensor, sij , and of the Cauchy-Green tensor, W = BBT . This and a positively skewed distribution of the intermediate eigenvalue of the rate of strain tensor, hΛ2 i > 0, result in increased surface stretching - with respect to ’available’ strain and to regularity of NSE (Galanti et al. 1996) or to moderating the growth of enstrophy maybe even maximal - and in increased diffusion across

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2 Figure ³ 2 ´3.50: A joint PDF plot of material volume rotation, ω w , and its rate of surface stretching, 1 Ds 2 2 Dt /S , reveals a strong nonlocal behavior. Strong rates of surface stretching inhibit strong rotations of material volumes and vice versa.

dividing surfaces of two material volumes. Preferentially when surface stretching is of low intensity, surfaces are re-oriented via rotation of their carriers - the infinitesimal material volumes - which results in material volume folding. The two processes of stretching and folding are driven by the combined effects of enstrophy and strain and their processes of self-amplification. While surface is stretching can be associated with strain, s2 , the relation to material volume rotation ¢ ¡ 2 or folding 1 2 not as straight-forward. It is observed to be associated in some ways with Q = 4 ω − 2s .

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Chapter 4

Conclusion To the best of our knowledge, in the course of this thesis it was possible for the first time to experimentally measure in a Lagrangian way the full set of velocity derivatives, Uij = ∂ui /∂xj , along particle trajectories in a turbulent flow. This is achieved by producing a suitable turbulent flow that is continuously forced electromagnetically and by applying and further developing a 3D Particle Tracking Velocimetry (3D-PTV) technique. The critical steps that allowed to go from 3DPTV measurements of velocity to measurements of velocity derivatives are (i) an increased rate of image recording, from 30Hz to 60Hz, (ii) an improved ’spatio-temporal’ particle tracking algorithm and (iii) a weighted polynomial fitting procedure, applied to interpolated velocity derivative signals along particle trajectories, using relative divergence associated with ∂ui /∂xi as a criterion to choose proper weights for each particle’s contribution to the fit. It is demonstrated that the use of weighted polynomials clearly enhances the quality of the measured and interpolated components of ∂ui /∂xj . The procedure to obtain the spatial velocity derivatives, ∂ui /∂xj , involves a chain of measurements, image processing, linear interpolations and polynomial fitting. Therefore, a number of checks which are based on precise kinematic relations are performed. They show that on a qualitative level the 3D-PTV measurements are correct and that the statistical set of data that ’survived’ processing and the applied quality check, based on relative divergence, are representative of the actual physical flow. In a study on enstrophy and strain and on the self-amplifying nature of their dynamics, known characteristic properties of turbulent flows, most of which are not observed in a Gaussian flow field, i.e. are specific of genuine fluid turbulence, are reproduced. These are the positive skewness of the intermediate eigenvalue of the rate of strain tensor, sij , hΛ2 i > 0, the predominance of vortex stretching over vortex compressing, hωi ωj sij i > 0, a predominant alignment of vorticity, ω, with the intermediate principal axis of strain, λ2 and the fact that regions in a turbulent flow with intense strain, s2 = sij sij , and more so regions with intense enstrophy, ω 2 , occupy only a relatively small portion of the domain which is reflecting the intermittent nature of strain and enstrophy. We speculate that in enstrophy dynamics preferential alignments of vorticity to λ2 , hΛ2 i > 0, but hΛ2 /Λ1 i ¿ 1 play the role of moderating the growth of enstrophy. With the use of a new representation of ∂ui /∂xj in the {Q, ω i ω j sij , sij sjk ski } space a mean cyclic evolution is observed consisting of intense events of strain, enstrophy production, concentrated vorticity and enstrophy destruction along with strain production. These findings are a first indication that turbulence is more likely to be understood if it is looked at not as a cascade but as a succession of strong stretching and folding processes of fluid blobs. In physical space the concept of energy cascade, or production of smaller scales from larger scales, is often explained with vortex stretching. It is argued that vortex stretching is the primary mechanism producing smaller scales. Lagrangian measurements of Dω 2 /Dt however reveal that there is no direct relation between ωi ω j sij and 1/2 Dω 2 /Dt. The role played by viscosity in 129

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enstrophy changes, 21 Dω Dt , is found to be just as important as vortex stretching. Viscous effects that might be related to vortex reconnection and collapse in part have the capability to change the topology of a turbulent flow. We arrive at the conjecture that where viscosity in combination with strain is responsible for dissipation, viscosity in combination with vorticity is associated with continuous change of the topology of turbulent flow by means of vortex reconnection. Enstrophy and strain dynamics are relevant for a deeper understanding of the dissipative nature of turbulence. A consequence of their combined evolution, the stretching and folding of material elements, is related to the equally important characteristic of turbulent flows - its diffusivity. The evolution in time of material lines, l, as compared to vortex lines, ω, is studied. In perfect agreement with results from literature it is found that the mean stretching rates of material lines are well above the mean stretching rates of vorticity. Material line growth is governed by its stretching whereas growth of enstrophy is a combined effect of vortex stretching and viscous effects. Further, material lines have a strong tendency to align with the most positive principal axis of strain, λ1 , in contrast to vorticity which tends to align with the intermediate principal axis, for which there are indications that viscosity is primarily responsible for it. Special material lines which at some time, t0 , are perfectly aligned with vorticity, l0 = ω 0 , appear ’reluctant’ to act like material elements, in the sense that their initial λ2 -alignment is persistent over a considerable time and the λ1 -alignment develops only very slowly. Two effects that might be partly responsible for this persistence are identified: The ’Λ2 cos (l, λ2 ) > Λ1 cos (l, λ1 )’ effect which is active in high strain regions and when l is in the proximity of the intermediate eigenvector, λ2 , and the effect of vorticity ’acting back’ on the flow, which is very sensitive to the magnitude of enstrophy, ω 2 . Regarding material surfaces, it is found that material surface normals, N, after a period of transition of τ η ∼ 2 align with the most compressing principal strain axis, λ3 . The mean stretching rates for material surfaces are found to be only slightly higher than those for material lines and we note that after a time of τ η ∼ 7 both, material lines and surfaces, have roughly doubled their magnitude. We conclude that due to the observed behavior of material elements which is driven by the dynamics of enstrophy and strain, diffusivity across material surfaces is significantly increased. To study the rotation of material surfaces we measure the rotation of their carriers, namely the rotation of material volumes. The rotation, ω 2w , of material volumes defined by the eigenvectors, wi , of the Cauchy-Green tensor, W, is measured by looking at the rotation of its eigen-frame. It is found that the magnitude of ω 2w is comparable to the magnitude of enstrophy, ω 2 , and that ω 2w ¢ ¡ can be related in some ways with Q = 14 ω 2 − 2s2 . Preferentially when surface stretching is of low intensity, surfaces are re-oriented via rotation of their carriers - the infinitesimal material volumes which results in material volume folding. The two processes of stretching and folding are driven by the combined effects of enstrophy and strain and their processes of self-amplification. We conclude that the application of 3D-PTV measurements on a turbulent flow did give new insights into some aspects of strain and enstrophy dynamics and further, to us it did add to the understanding of differences between stretching of vortex lines and material lines and the behavior of material surfaces and volumes.

Chapter 5

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