Implicite coherent fluid systems

Lagrange Coherent Systems

Induction effectiveness

Resume

Bibliograhy

Implicite coherent fluid systems Fluid within a Fluid

Michel Felgenhauer, Berlin

Lagrange coherent structures (LCS) are separated surfaces of trajectories in a dynamic system, which exert a great influence on neighboring trajectories over a relevant time interval. The nature of this influence can vary, but invariably creates a coherent trajectory pattern for which the underlying LCS serves as the theoretical core. When observing tracer patterns in nature, one can easily identify coherent features, but it is often the underlying structure that creates these features of interest.

Lagrange-Kohärenzstrukturen (LCS) sind separierte Oberflächen von Trajektorien in einem dynamischen System, die über ein relevantes Zeitintervall großen Einfluss auf benachbarte Trajektorien ausüben. Die Art dieses Einflusses kann variieren, erzeugt jedoch ausnahmslos ein kohärentes Trajektorienmuster, für das das zugrundeliegende LCS als theoretischer Kern dient. Bei der Beobachtung von Tracermustern in der Natur kann man leicht kohärente Merkmale identifizieren, aber es ist häufig die zugrundeliegende Struktur, die diese Merkmale erzeugt, von Interesse.

Lagrange Coherent Systems

In September 1988 the Gesellschaft Deutscher Naturwisssenschaftler und Ärzte GDNĹ, takes place to discuss new developments in natural sciences, medicine and technology. At the end of the 1980s we are in the high phase of chaos theory. In the explanations on the general topic of the meeting it is said that the conference will show how order structures can arise from disordered, chaotic conditions and a reversal of order to a disordered state can take place². The transition to the non-linearity of dynamic systems is linked to the loss and resolution of superposable probability amplitudes (Großmann 1989). As an example of the non-linearity of the equations of motion for flows, called the Karman vortex street. The sum of all positive Lyapunoff exponents K in a region with At=F[(1/K) ln(Ax/df)] is a feature of finding them in otherwise homogeneous flows and, as it were, a measure of the nonlinearity in a law of motion. A decade later, the acronym "LCS (Lagrange Coherent Structures)" appears for the first time. At this time, the Lyapunoff exponent also played a decisive role in the localization of coherence structures from image data. Today (2021) we have a closed definition:

Lagrangian coherence structures (LCS) are separated surfaces of trajectories in a dynamic system, which exert a great influence on neighboring trajectories over an interesting time interval. The nature of this influence can vary, but invariably creates a coherent trajectory pattern for which the underlying LCS serves as the theoretical core. When observing tracer patterns in nature, one can easily identify coherent features, but it is often the underlying structure that creates these features of interest.

Physical phenomena controlled by LCS include floating debris, oil spills, surface drifts, and patterns of chlorophyll in the ocean; Clouds of volcanic ash and spores in the atmosphere; and coherent mass patterns formed by humans and animals.

The established theory of Lagrange coherent systems was developed in the early 2010s at the Lefschetz Center for Dynamical Systems at Brown University, later at the ETH Zurich, there at the Department of Mechanical and Process

Engineering. The acronym "Lagrange Coherent Structures, LCS" comes from Haller & Yuan. Haller was looking for an approach to describe the repulsive and attractive fluid movements in real shear layers. Accelerated shear layers can only be reliably generated in the laboratory in special multi-fan wind tunnels. In Zurich this was known from the experimental past and had observed that system boundaries exist within fluidic regimes in the sense of "material surfaces" that separate coherent structures from the rest of the flow (Lyapunoff). These extraordinary systems develop complex body and directional dynamics within a flow. When research started in Zurich, the scientists were unable to access any solid theoretical models for the quantitative description of these strange physical events. It soon became clear that in the future large-volume, numerical models and complex simulations would be required to isolate these strange system surfaces of the systems now called Lagrange Coherent Structures, LCS, in flow scenarios from experimental and numerical data and, if necessary, to describe them with a simplifying approach do. For to understand. Haller's research initially addressed the question of whether it could be possible to predict or even influence mixing, demixing and mass transport in and around Lagrange coherent systems in complex fluidic systems. In the course of theorising, nonlinear dynamic system methods were developed to solve complex problems in applied science and technology, such as the analysis of transport processes and coherence in an ocean and in the atmosphere, the real-time recording of air turbulence near airports, theory and Control of the unsteady, aerodynamic separation, the dynamics of inertial particles under memory effects and the theory of the dynamic transition state in chemical reactions.

Good flow simulation models are "physics-based interaction generators left to their own devices". It is typical and the real motive for the use of flow simulation models that an experimenter often, but not always, knows how the synthetic fluid will behave in the flow space before a simulation. The advantage of a simulation over observations in nature is that there is an informational control of the physical interaction parameters at any point in time and at any place. Only in a subsequent process does the data material of physical interactions become analytically usable image material. Gradual eddies and velocity distributions from and in LCO in flows can be represented as coherent structures of different colors in a fluid. Pressure or velocity gradients can therefore be easily isolated from the data set of a flow simulation and Lagrange coherent structures can be extracted from a flow in this way.

In the case of a (flow) simulation, the necessary pre-processes for the analysis of LCS-affected flows focus on the viewing of the data material and the detection and isolation of Lagrangian coherent structures based on physical effects.

This information is not directly contained in the images from the observation of natural currents. Around 2010, the Zurich group around G. Haller developed image processing methods for measurement data from the real world that target the special contour properties of Lagrange coherent systems. Finding the interrelated Lagrangian coherent structures succeeds in this way with the so- called "Finite-Time-Lyapunov-Exponent³ (FTLE)" discussed above. The Lyapunov exponent (adapted to the digital image processing) describes the speed with which two neighboring points in the flow space move away from or approach each other. A field from FTLE thus contains characteristics of the coherence of structures and surfaces in dynamic scenarios. In principle, this gradient analysis also works in the artificial world of CFD simulations. A research group led by Sun and Colagrossi formulated (2016) the FTLE method in the context of a smoothed particle hydrodynamics simulation. Methods of the finite-time Lyapunov exponent are now part of the general instrumental repertoire in the run-up to an investigation of coherent flow quantities. FTLE can also precede a digital analysis of complexity parameters (e.g. the fractal dimension) of a substructure contained in the flow (fluid within a fluid) whenever the information about the flow is available as time-based data; vulgar: as a film.

However, this is precisely what is rarely the case. As a rule, and in the vast majority of studies, the data on Lagrange coherent structures is limited to a series of representations or even consists of a single image of the flow. Processing visual data is complicated and the data power of images is inherently high. Formerly an isolated specialty, digital image processing and the use of professional computer programs have developed into a standard instrument for digital pattern and signal analysis over the past twenty years.

But how is the physicality of a fluid in a fluid, motivated by system boundaries, explained in terms of enveloping surfaces of Lagrange coherent objects? We identify an opaque, contiguous structure within a fluid and call it homeomorphic. A feature of coherent structures should be the (system) boundary to the rest of the flow space. The surface of a Lagrange coherent vortex filament is a separated accelerated shear layer. It coincides with a velocity gradient and thus represents a system limit compared to the rest of the flow space. With all the consequences for a future formulation of the Navier-Stokes equations for Lagrange coherent objects⁴.

The novel approach about the inner milieu of Lagrange coherent objects interprets the "idea of an artificial viscosity" as it is used today in the theories of particle-based flow simulation (smoothed-particle hydrody-namics, SPH)⁵ in a radical way. The Navier-Stokes equation does not fit an artificial viscosity, but the circulation rimplicit in the Lagrange coherent object. The search for the theoretical foundations of an implicit path-dependent approach is the subject of recent efforts (Lagrange Implicite Vorticity Theory, LIV). Cassey and Naghdi⁶ worked out first considerations for an approach to fluidic objects of implicit circulation as early as 1991. In the light of modern, particle-based flow models, the idea of implicit circulation appears much less theoretical, but with a useful reference to simulation practice. With the mass balance and the momentum in Lagrange notation, the definition of a Lagrange implicit circulation follows:

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Induction effectiveness

As Helmholtz's theorems of vortices state, vortex filaments subject to circulation have coherence, topology maintenance and freedom from overlapping. The model of Lagrange coherent systems proposed in this article is based on the fluid mechanics of macroscopic vortex structures, their traditional teaching and their inherent physics. The inner milieu of the systems examined follows the vortex statements that Hermann von Helmholtz formulated around 1859:

First Helmholtz's vortex law:

In the absence of vortex-stimulating external forces, vortex-free flow areas remain vortex-free.

Second Helmholtz's vortex law:

Fluid elements that lie on a vortex line remain on this vortex line. Vortex lines are therefore material lines.

Third Helmholtz's vortex law:

The circulation along a vortex tube is constant. A vortex line cannot therefore end in the fluid. Vortex lines are closed, literally infinite, or run towards the edge.

The first vortex law means that both the circulation along the edge curve of a surface which lies entirely on the mantle of a vortex tube vanishes and that the circulation of different cross sections of a vortex tube is the same. The second vortex law means that vortex tubes are also flow tubes, vortices adhere to matter (the fluid) and thirdly, that particles that have once formed a vortex line continue to do so (coherence). The third vortex law demands the temporal constancy of the circulation in a vortex tube. The Helmholtz vortex theorems are the basis of the vortex model proposed here based on Lagrange coherent systems.

In contrast to the theory (Haller's), a phenomenology of Lagrange coherent vortex systems is allowed to speculate about their nature. The phenomenology discussed here interprets the (Helmholtz) vortex filaments as Lagrange coherent systems (LCS), which are subject to circulation, appear separately in a flow field and interact with it. From the point of view of this phenomenology, they are by nature vortex filaments in the sense of Helmholtz's vortex theory and fluidic trajectories as it were, as Haller describes them. This should result in (1) that without an external force stimulating the vortex, the vortex-free flow areacontinues to remain free of vortices and (2) fluid elements that lie on an LCS vortex line remain on this vortex line and (3) that the circulation along an LCS- Vortex tube remains constant. And (4) the most significant, most important element in our context: Lagrange coherent systems do not end (anywhere) in the fluid; they are closed, literally infinite; Only a small section of your system is considered within the given system limits.

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The potential theoretical approach for the induction effect is described in detail [Fel-20]. The stationary observer is aware of the induction effectiveness of Lagrangian coherent objects, for example as (indicated) speed c. Preferably as the local components of the velocity induced in the fluid from a location, the source point Q of the Lagrangian coherent vortex filament, at a reference point P there. Position vectors r from source points to reference points, as I like to use them in the sketch above, again simplify the notation. The components of the velocity c {u,v,w} induced by the Lagrangian coherent vortex filament thus follow⁷:

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Every vector direction-dependent induction effect of Lagrange coherent objects emerges in nature from observable causal phenomena, which in turn are direction-dependent. In the case discussed here as a simulation model, the circulations r or the vortex strength Q in the cross-sections along the central axis of the Lagrange coherent system are induction-relevant and cause those abstract vectorial induction effects of the generating system, which are path- dependent and with a direction vector Y=ye¡ correlated. The local directions can be written into the Lagrange coherent system in such a way that y varies locally, with the direction vector {yi eii,y2 ei2,y3 e¡3>. The induction effect is therefore a tensor⁸. In the case of the induction effects distributed in the flow space, which originate from the existence of a Lagrange coherent system, we use local Cartesian coordinate systems with the unit vector (ei,e2,e3). The induction effect of the Lagrange coherent generating system in a flow field accumulates the induced speed that already exists there. So c is the induced velocity in the reference points P and this in the sense of an induction effect s and x the direction vector of coherent objects that together form a Lagrange coherent system (LCS) and q the position vector with the coordinates of all source points Qk, at which the Lagrange system exists as a coherent, coherent entity. This is already shown in the schematic sketch above.

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Resume

The phenomenology discussed here interprets the Helmholtz vortex filaments as Lagrange Coherent Systems (LCS). They appear in a flow field and have a separable surface, they are opaque. Since LCO are objects subject to circulation, they have the ability to organize their fluidic environment by generating induction effects. These induction effects are perceived by the Eulerian observer as (induced) velocities or as pressure gradients in the fluidic field. At the same time, Lagrangian coherent objects are self-referential "autopoietic systems". The induction effects add shape changes to one's own shape. This may be one reason why structures of Lagrangian coherent objects are of astonishing shape stability.

Mi. D., Berlin 2021

Bibliography, sources and further reading

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¹ Die Gesellschaft Deutscher Naturforscher und Ärzte e. V. (GDNÄ) ist die älteste und größte interdisziplinäre wissenschaftliche Gesellschaft in Deutschland. Sie wurde 1822 von dem Naturphilosophen und Arzt Lorenz Oken in Leipzig gegründet. Berühmte Forscherpersönlichkeiten wie Alexander von Humboldt, Albert Einstein, Max Planck und Christiane Nüsslein-Volhard präsentierten ihre Forschungsergebnisse auf den Versammlungen der GDNÄ und stellten sich der fachübergreifenden Diskussion.

² Tagungsband der 115. Verhandlungen der Gesellschaft Deutscher Naturforscher und Ärzte e. V. (1988). Der Autor war von 1988 bis 2012 Md GDNÄ.

³ At = F[(1/K) ln (Ax/öf) ]3. Messgrößen Ax werden in einem Zeitintervall At mit der Messungenauigkeit öf untersucht. K ist der Lyapunoff-Exponent. Lagrange kohärente Struktur, nach wikipedia - https://de.qaz.wiki/wiki/Lagrangian coherent structure

⁴ Die Rede greift der Ausformulierung einer Theorie der Lagrange Implicite Vorticity voraus, dies bitte ich zu entschuldigen. Aber es eilt. Wir Alle begreifen dieser Tage, dass wir sterblich sind.

⁵ Smoothed-particle hydrodynamics (SPH; deutsch: geglättete Teilchen-Hydrodynamik) ist eine numerische Methode, um die Hydrodynamischen Gleichungen zu lösen. SPH ist eine Lagrange-Methode, d. h. die benutzten Koordinaten bewegen sich mit dem Fluid mit. SPH ist eine besonders robuste Methode (nach wikipedia).

⁶ J. CASEY & P. M. NAGHDI (1991) On the LagrangianDescriptionof Vorticity, in: Arch. Rational Mech. Anal. 115 (1991) 1-14. Springer-Verlag.

⁷ Quellpunkt Q {Qx,Qy,Qz}, Aufpunkt P {Px,Py,Pz}. Ortsvektoren r von Quellpunkten zu Aufpunkten, vereinfachen erneut die Schreibweise. Mit r2=(xP-xQ)2+(yP -yQ)2+(zP -zQ)2

⁸ Um einen Tensor zahlenmäßig darzustellen (Euler), benötigt man ein lokales, körperfestes Koordinatensystem (Lagrange).