Organisation chaotischer Dynamik in der Taylor-Couette Strömung

Obwohl die hydrodynamischen Grundgleichungen schon seit langem bekannt sind, sind die grundlegenden Mechanismen, die zum Auftreten der phänomenologisch beobachteten Strömungszustände führen, nur teilweise verstanden. Insbesondere das Auftreten von Instabilitäten und raum-zeitlichen Strukturen, sowie der Übergang ins Chaos und in die Turbulenz sind Gegenstand intensiver Forschung. In dieser Arbeit wurden experimentelle Untersuchungen an einer Taylor-Couette Strömung - der Strömung einer viskosen Flüssigkeit zwischen zwei konzentrischen, rotierenden Zylindern - durchgeführt. Es konnte gezeigt werden, daß beim Übergang von der laminaren Couette- zur Taylor-Wirbelströmung aufgrund der Ränder kein "critical-slowing-down" auftritt, so daß sich das Verhalten des endlichen von dem eines unendlichen Systems prinzipiell unterscheidet, was in Übereinstimmung mit früheren Stabilitätsuntersuchungen ist. Ferner konnte im Rahmen dieser Arbeit erstmals ein Szenario ins Chaos ausgehend von der wellenförmigen Taylor-Wirbelströmung eindeutig identifiziert und in Verbindung mit theoretisch bekannten niederdimensionalen Mechanismen gebracht werden. Dabei zeigte sich, daß Chaos in dem untersuchten Bereich durch eine Wechselwirkung von Symmetriebrechung und einer oszillatorischen Mode durch das Auftreten von homoklinen Orbits chaotisch im Sinne Shil'nikovs wird. Außerdem wurden in dieser Arbeit Zustände mit lokalisierten Anregungen untersucht

Real‐time capable multiple‐input–multiple‐output SONAR systems—An algorithmic approach

In recent years, significant effort has been allocated to research on multiple‐input–multiple‐ output (MIMO) sound navigation and ranging (SONAR) and RADAR systems. Most work has been conducted on the general theoretical functionality of such systems. Less effort has been applied to considerations of the real‐time MIMO capability, although this is an important factor for the application of these new algorithms in real SONAR systems. To account for this, the following work focusses, after introducing the used methodology and revisiting the general MIMO idea and considered system, on more effective permutations of the involved algorithms in the reduction of floating ‐point operations. In this context, the general necessity of the algorithms utilized is shown. Furthermore, it is proven that the reduction in computational load does not affect the performance of the system. In addition, the main algorithmic parts of MIMO systems can be exchanged almost arbitrarily under given restrictions without changing the result. Therefore, the performance differences in floating ‐point operations are depicted to give an estimate of the achievable degree of complexity reduction. The results for the investigated systems and algorithms are obtained by applying a system simulation of a simple underwater channel. The obtained results were also verified using a real MIMO SONAR system operating in real time

Imperfect Homoclinic Bifurcations

Experimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations. These results are discussed in the light of a theory of imperfect global bifurcations. It is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system.Comment: 8 pages, 11 figures, submitted to PR

An Organizing Center for Thermohaline Excitability

The bifurcation behavior of a conceptual heat–salt oscillator model is analyzed by means of numerical continuation methods. A global (homoclinic) bifurcation acts as an organizing center for the dynamics of the simplified convective model. It originates from a codimension-2 bifurcation in an extended parameter space. Comparison with earlier work by Cessi shows that the intriguing stochastic thermohaline excitability can be understood from the bifurcation structure of the model. It is argued that global bifurcations may play a crucial role in determining long-term variability of the thermohaline circulation

Spatio-temporal behavior of spiral vortex flow

Experimental realizations of Taylor-Couette flow often include rigid end plates at bottom and top of the system. As a consequence of such end plates the bifurcation behavior of the basic laminar flow as well as the spatio-temporal properties of the emerging pattern, such as e.g. spiral vortex flow, can change. The latter point is in the focus of our present experimental study. The spatio-temporal behavior of spiral vortex flow in a Taylor-Couette system with rigid end plates is analyzed by a measurement technique based on Doppler-shift. This enables us to determine the spatial amplitude profile of up- and downward propagating spiral vortices within oscillatory flow states. Our study confirms experimentally recent numerical results of Hoffmann et al. [1] on the spatio-temporal properties of the spiral vortex state in finite systems with rigid end plates

Bifurcation behavior of standing waves

Two different types of standing waves (SW0 and SWπ) can appear instead of spiral vortices from a supercritical Hopf bifurcation in counter-rotating Taylor-Couette flow for sufficiently small aspect ratios [1,2]. The bifurcation sequence from basic flow to spiral vortices via SW0 can include modulated waves, homoclinic bifurcations, and hysteresis as a consequence of broken translational invariance [3]. Here we show that the same kind of sequence can also occur for the other type of standing wave, i.e., SWπ. Furthermore we show that SWπ can exist also up to much larger inner Reynolds numbers than is has been found for SW0. Far from onset SWπ can undergo bifurcation sequences that differs qualitatively from those close to onset. These sequences involve a supercritical symmetry breaking as well as a supercritical Hopf bifurcation towards a new type of modulated wave

Standing waves in flow between finite counter-rotations cylinders

Experimental evidence for standing waves resulting from a supercritical Hopf bifurcation that appears as the first pattern-forming instability in counterrotating Taylor-Couette flow is presented. Depending on the aspect ratio two different types of standing waves, denoted as SW0 and SWp, could be observed. Both modes have an azimuthal wave number m51 but differ in symmetry. While for SWp , a spatiotemporal glide-reflection symmetry could be found, SW0 is purely spatial reflection symmetric. The transition between the two modes is found to be organized in a cusp bifurcation unfolded by variations of the aspect ratio. The ‘‘classical’’ spiral vortex flow appears in this control parameter regime only as a result of a secondary steady bifurcation from SW0. This transition is found to be either subcritical or supercritical. The experimentally observed bifurcation structure has been predicted by theory of Hopf bifurcation to spiral vortex flow in finite counterrotating Taylor-Couette systems

Chaos from Hopf Bifurcation in a fluid flow experiment

Results of an experimental study of a Hopf bifurcation with broken translation symmetry that organizes chaotic homoclinic dynamics from a T2 torus in a fluid flow as a direct consequence of physical boundaries are presented. It is shown that the central features of the theory of Hopf bifurcation in O(2)-symmetric systems where the translation symmetry is broken are robust and are appropriate to describe the appearance of modulated waves, homoclinic bifurcation, Takens-Bogdanov point, and chaotic dynamics in a fluid flow experiment

The effects of physical boundaries on oscillatory bifurcation in counterrotating Taylor-Couette flow

The results of an experimental study on the bifurcation structure of oscillatory modes in counterrotating Taylor–Couette flow with stationary end plates are presented. It is shown that the cylinder length L acts as an important geometric control parameter of the system. As a result of a supercritical Hopf bifurcation it is found that for an aspect ratio Γ=L/d>16 (d gap width) only spiral vortices appear in basic laminar flow. For Γ<10.5 spiral vortices are almost entirely replaced by two types of standing waves called SW0 and SWπ as supercritical oscillatory flow. Experimental evidence is presented that the mode exchange between standing waves SW0 and SWπ is governed by underlying Ekman induced vortices which appear as a result of stationary end plates in the flow. In this regime spiral vortices appear only from a sub- or supercritical symmetry breaking bifurcation of the standing waves. Within an “intermediate regime” between 10.5⩽Γ⩽16 spiral vortices are found to be the predominant primary oscillatory flow but small stability intervals of standing waves are also observed. Surprisingly, the experimentally determined critical Reynolds number is found to deviate not more than 2% from the numerical values for all values of aspect ratio even though they are calculated under the assumption of infinite axial length. Moreover, the critical oscillation frequency is also in agreement with the numerical values and is independent from Γ