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Über dieses Buch

The German Aerospace Research Establishment (DFVLR) has initiated a new series of seminars concerning fundamental problems in applied engineering sciences. These seminars will be devoted to interdisciplinary topics related to the vast variety of DFVLR activities in the fields of fluid mechanics, flight mechanics, guidance and control, materials and struc­ tures, non-nuclear energetics, communication technology, and remote sensing. The purpose of the series is twofold, namely, to bring modern ideas and techniques to the attention of the DFVLR in order to stimulate internal activi ties, and secondly, to promulgate DFVLR achievements wi thin the international scientific/technical community. To this end, prominent speakers from Germany and other countries will be invited to join in a series of lectures and discussions on certain topics of mutual interest. The first colloquium of this series dealt with the dynamics of nonlinear systems, especially in relation'to its application to fluid mechanics, particularly in transcritical flows. Of special interest are questions concerning the formation of nonlinear three-dimensional structures in classical fluid mechanical stability problems, the physical process of transition to turbulence, and the appearance of chaotic solutions. The scope of lectures reaches from self-organization in physical systems to structural stability of three-dimensional vortex patterns, the treatment of dissipative and conservative systems, the formation of nonlinear structures in the region of laminar-turbulent transition, and numerical simulation of cumulus cloud convection in meteorology. The seminar should provide an insight into the extent to which theoretical findings in Non­ linear Dynamics apply to the comprehension of fluid-mechanical problems.




Nonlinear Dynamics

Nonlinear Dynamics Temporal and Spatial Structures in Fluid Mechanics

Fluid mechanical instabilities are the primary cause of well-organized structures of different characteristic wavelengths in a flow field. The key to a physical understanding of the laminar-turbulent transition process lies in understanding the temporal and spatial development and decay of these flow structures. In this volume, we will be concerned with selected examples of the formation of structures. In the fluid mechanical part, we will deal only with transcritical flows, i.e., flows across and beyond a critical state and therefore beyond one or more fluid mechanical instabilities. Figs. 1 and 2 portray transcritical flows. The first photograph of Fig. 2 shows a carbon dioxide jet in air, which is coming out of a round nozzle. The jet flow is laminar at first; however, after a characteristic length, it becomes unstable, forming vortex rings, and the transition to turbulence occurs, at which point a characteristic microscale structure becomes visible. A mixing layer allows us to observe a plane transcritical flow. Again it starts with a laminar shear flow, which becomes unstable and causes vortices with a spatially increasing flow amplitude. With increasing relative velocity of the upper and lower portions of the flow forming the shear layer, the transition to a turbulent flow takes place.
H. Oertel


Self-Organization in Physics

In my contribution, I shall present a small cross-section of a rather new field of interdisciplinary research called “synergetics”, HAKEN [1, 2, 3]. Emphasis will be laid on problems of fluid dynamics. In synergetics, we study the co-operation of individual parts of a system, which makes possible a self-organized formation of spatial, temporal, or functional structures on macroscopic scales. Systems studied so far belong to the fields of physics, electronics, mechanical and electrical engineering, chemistry, biology, and some “soft sciences”. We shall ask whether or not there are general principles that govern self-organization irrespective of the nature of the individual subsystems, which may be electrons, molecules, photons, biological cells, or animals. In particular, we wish to develop an operational approach that will allow us to actually calculate the evolving structures (or “patterns”). The price we have to pay for the general validity of our approach is that we are limited to considering only such situations in which the state of a system undergoes qualitative macroscopic changes. Figures 1 and 2 provide us with some simple examples.
H. Haken

Nonlinear Dynamical Systems

Tori and Chaos in a Simple C1-System

The world is a partial differential equation (P.D.E.) — to some approximation at. least. The qualitative behavior of P.D.E.s, however, is potentially infinitely complex. It is therefore fortunate that some P.D.E.s behave precisely like ordinary differential equations (O.D.E.s) in some of their regimes. Two recent cases in point are the surprising discovery of silent turbulence by DALLMANN [1], and the successful analytic demonstration of constant-shape traveling chaotic waves in a boundary value problem of reaction-diffusion type, RÖSSLER & KAHLERT [2]. A third example is the new observation by SREENIVASAN [3] that turbulent vortex streets involve, with comparable probabilities, either hypertori of up to 50 dimensions or related hyperchaos of the same dimensionality (cf. RÖSSLER [4] for these notions).
O. E. Rössler, C. Kahlert, B. Uehleke

Transcritical Flows

Transitional and Turbulent Wakes and Chaotic Dynamical Systems

Recent studies of the dynamics of simple nonlinear systems with chaotic solutions have produced very interesting and (perhaps) profound results with several implications in many disciplines. However, the interest of fluid dynamicists in these studies stems primarily from the expectation that they will help us better understand the process of transition and turbulence in fluid flows. At this time, much of this expectation remains untested, especially in ‘open’ or unconfined fluid systems. This work is aimed at filling some of this gap.
K. R. Sreenivasan

Fluid Mechanical Structures

Structural Stability of Three-Dimensional Vortex Flows

In fluid dynamics, we intuitively use the notion structure in connection with features of flow fields that are qualitatively quite different. For instance, we recognize an evolution of well-organized flow structures on various scales caused by various kinds of flow instabilities. We also talk about large-scale separated vortex structures. Coherent structures are all of a sudden being identified in a variety of flows. On the other hand, we use terms like vortex, eddy, and turbulent spot as synonyms for some unknown, i.e., undefined structures that appear in velocity or pressure signals. Finally, loosely speaking, the strange attractors exhibit some strange structures of trajectories in phase space, which we would like to relate to small-scale turbulent flow structures in real physical space and time.
U. Dallmann

Self Similarity, Critical Points, and Hill’s Spherical Vortex

Hill’s spherical vortex, a classical solution dating from 1894, is examined in a new perspective. The axisymmetric vorticity transport equation is taken to its low Reynolds number limit (no convective term) and cast into a form that is self similar in time. The subsequent equation is solved by separation of variables and shown to have two linearly independent solutions. One of these two solutions satisfies the Navier-Stokes equation exactly for all Reynolds numbers. However, to satisfy boundary conditions for the unbounded problem, both of these solutions as well as irrotational components are required. The results are represented by self similar particle paths. The flow topology is examined in the context of critical points of the self similar particle paths. It is shown that this result undergoes two transitions in topological structure with changing Reynolds number. Also shown is that the creeping flow (low Reynolds number) solution has three possible topological states in the axisymmetric case.
G. A. Allen

Transition and Turbulence

Three-Dimensional Processes in Laminar-Turbulent Transition

The transition of a laminar flow into a turbulent state is a classical problem of fluid mechanics that has been investigated for more than a hundred years. Numerous types of flows have been considered, and a great variety of phenomena and of factors affecting transition have been discovered. Despite the enormous progress achieved, transition to turbulence remains a challenge for many disciplines from mathematics to practical engineering design. Many technologically important properties of flows, such as drag or heat and mass transfer, change drastically during transition. There is therefore an urgent need for improved knowledge, prediction, and possible control of transition. In addition to this practical aspect, there has always been a fundamental interest in the transition phenomenon as a clue to understanding the origin of turbulence. Generally, transition to turbulence occurs via a sequence of increasingly complex but still laminar intermediate stages. These are usually connected with flow instabilities. Flow patterns generated during transition often remain visible far into the turbulent region. The past ten years have seen an explosive growth in a novel direction of research in this field, namely the investigation of low-dimensional dynamical systems and their possible relation to transition and turbulence (OTT [52], ECKMANN [12], TATSUMI [61]).
L. Kleiser

Renormalization Group Formulation of Large Eddy Simulation

Perhaps the most distinguishing characteristic of high Reynolds number turbulent flows is their large range of excited space and time scales. In homogeneous turbulence, dissipation-scale eddies are of order R3/4 times smaller than energy-containing eddies, where R is the Reynolds number. In order to solve the Navier-Stokes equations accurately for such a turbulent flow, it is necessary to retain order (R3/4)3 spatial degrees of freedom. Also, since the time scale for significant evolution of homogeneous turbulence is of the order of the turnover time of an energy containing eddy, it is necessary to perform order R3/4 time steps to calculate for a significant evolution time of the flow. Even if these calculations require only O(1) arithmetic operations per degree of freedom per time step, the total computational work involved would be order R3, while the computer storage requirement would be R9/4. In this case, doubling the Reynolds number would require an order of magnitude improvement in computer capability. With this kind of operation and storage count, it is unlikely that forseeable advances in computers will allow the full numerical simulation of turbulent flows at Reynolds numbers much larger than Rλ = 0(100) already achieved (see BRACHET et al. [2]).
V. Yakhot, S. A. Orszag

Meteorological Dynamics

Three-Dimensional Cumulus Cloud Convection

A cloud can be defined as a visible ensemble or aggregate of so-called hydrometers, i.e., minute particles consisting of liquid or frozen water in a wide variety of forms. The processes involved in the formation of clouds range from the very small-scale processes responsible for the nucleation and growth of cloud particles (cloud microphysics) up to the very large-scale dynamical processes that are associated with synoptic weather systems. A large portion of the study of clouds has been focused on their microphysical processes, evidenced by the voluminous material on cloud microphysics, with less attention paid to the dynamics of cloud formation, COTTON, LILLY [1,2]. Much has been learned in recent years about the structure of strong convection, with the help of observational tools, particularly doppler-radar, BROWNING, SCHROTH [3,4]. With the advent of powerful scientific computers such as the CRAY-1, substantial advances in numerical modelling of cumulus convection has been achieved up to now, and even more can be expected in the near future, SCHLESINGER [5].
U. Schumann


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