In the Lagrangian representation, the problem of advection of a passive marker particle by a prescribed flow defines a dynamical system. For two-dimensional incompressible flow this system is Hamiltonian and has just one degree of freedom. For unsteady flow the system is non-autonomous and one must in general expect to observe chaotic particle motion. These ideas are developed and subsequently corroborated through the study of a very simple model which provides an idealization of a stirred tank. In the model the fluid is assumed incompressible and inviscid and its motion wholly two-dimensional. The agitator is modelled as a point vortex, which, together with its image(s) in the bounding contour, provides a source of unsteady potential flow. The motion of a particle in this model device is computed numerically. It is shown that the deciding factor for integrable or chaotic particle motion is the nature of the motion of the agitator. With the agitator held at a fixed position, integrable marker motion ensues, and the model device does not stir very efficiently. If, on the other hand, the agitator is moved in such a way that the potential flow is unsteady, chaotic marker motion can be produced. This leads to efficient stirring. A certain case of the general model, for which the differential equations can be integrated for a finite time to produce an explicitly given, invertible, area-preserving mapping, is used for the calculations. The paper contains discussion of several issues that put this regime of chaotic advection in perspective relative to both the subject of turbulent advection and to recent work on critical points in the advection patterns of steady laminar flows. Extensions of the model, and the notion of chaotic advection, to more realistic flow situations are commented upon.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112084001233

Stirring by chaotic advection

1. On the assumption that steady-state aerodynamics applies, simple analytical expressions are derived for the average lift coefficient, Reynolds number, the aerodynamic power, the moment of inertia of the wing mass and the dynamic efficiency in animals which perform normal hovering with horizontally beating wings. 2. The majority of hovering animals, including large lamellicorn beetles and sphingid moths, depend mainly on normal aerofoil action. However, in some groups with wing loading less than 10 N m -2 (1 kgf m -2 ), non-steady aerodynamics must play a major role, namely in very small insects at low Reynolds number, in true hover-flies (Syrphinae), in large dragonflies (Odonata) and in many butterflies (Lepidoptera Rhopalocera). 3. The specific aerodynamic power ranges between 1.3 and 4.7 WN -1 (11-40 cal h -1 gf -1 ) but power output does not vary systematically with size, inter alia because the lift/drag ratio deteriorates at low Reynolds number. 4. Comparisons between metabolic rate, aerodynamic power and dynamic efficiency show that the majority of insects require and depend upon an effective elastic system in the thorax which counteracts the bending moments caused by wing inertia. 5. The free flight of a very small chalcid wasp Encarsia formosa has been analysed by means of slow-motion films. At this low Reynolds number (10-20), the high lift co-efficient of 2 or 3 is not possible with steady-state aerodynamics and the wasp must depend almost entirely on non-steady flow patterns. 6. The wings of Encarsia are moved almost horizontally during hovering, the body being vertical, and there are three unusual phases in the wing stroke: the clap , the fling and the flip . In the clap the wings are brought together at the top of the morphological upstroke. In the fling, which is a pronation at the beginning of the morphological downstroke, the opposed wings are flung open like a book, hinging about their posterior margins. In the flip, which is a supination at the beginning of the morphological upstroke, the wings are rapidly twisted through about 180°. 7. The fling is a hitherto undescribed mechanism for creating lift and for setting up the appropriate circulation over the wing in anticipation of the downstroke. In the case of Encarsia the calculated and observed wing velocities at which lift equals body weight are in agreement, and lift is produced almost instantaneously from the beginning of the downstroke and without any Wagner effect. The fling mechanism seems to be involved in the normal flight of butterflies and possibly of Drosophila and other small insects. Dimensional and other considerations show that it could be a useful mechanism in birds and bats during take-off and in emergencies. 8. The flip is also believed to be a means of setting up an appropriate circulation around the wing, which has hitherto escaped attention; but its operation is less well understood. It is not confined to Encarsia but operates in other insects, not only at the beginning of the upstroke (supination) but also at the beginning of the downstroke where a flip (pronation) replaces the clap and fling of Encarsia . A study of freely flying hover-flies strongly indicates that the Syrphinae (and Odonata) depend almost entirely upon the flip mechanism when hovering. In the case of these insects a transient circulation is presumed to be set up before the translation of the wing through the air, by the rapid pronation (or supination) which affects the stiff anterior margin before the soft posterior portions of the wing. In the flip mechanism vortices of opposite sense must be shed, and a Wagner effect must be present. 9. In some hovering insects the wing twistings occur so rapidly that the speed of propagation of the elastic torsional wave from base to tip plays a significant role and appears to introduce beneficial effects. 10. Non-steady periods, particularly flip effects, are present in all flapping animals and they will modify and become superimposed upon the steady-state pattern as described by the mathematical model presented here. However, the accumulated evidence indicates that the majority of hovering animals conform reasonably well with that model. 11. Many new types of analysis are indicated in the text and are now open for future theoretical and experimental research.

Quick Estimates of Flight Fitness in Hovering Animals, Including Novel Mechanisms for Lift Production

The flight of insects has fascinated physicists and biologists for more than a century. Yet, until recently, researchers were unable to rigorously quantify the complex wing motions of flapping insects or measure the forces and flows around their wings. However, recent developments in high-speed videography and tools for computational and mechanical modeling have allowed researchers to make rapid progress in advancing our understanding of insect flight. These mechanical and computational fluid dynamic models, combined with modern flow visualization techniques, have revealed that the fluid dynamic phenomena underlying flapping flight are different from those of non-flapping, 2-D wings on which most previous models were based. In particular, even at high angles of attack, a prominent leading edge vortex remains stably attached on the insect wing and does not shed into an unsteady wake, as would be expected from non-flapping 2-D wings. Its presence greatly enhances the forces generated by the wing, thus enabling insects to hover or maneuver. In addition, flight forces are further enhanced by other mechanisms acting during changes in angle of attack, especially at stroke reversal, the mutual interaction of the two wings at dorsal stroke reversal or wing-wake interactions following stroke reversal. This progress has enabled the development of simple analytical and empirical models that allow us to calculate the instantaneous forces on flapping insect wings more accurately than was previously possible. It also promises to foster new and exciting multi-disciplinary collaborations between physicists who seek to explain the phenomenology, biologists who seek to understand its relevance to insect physiology and evolution, and engineers who are inspired to build micro-robotic insects using these principles. This review covers the basic physical principles underlying flapping flight in insects, results of recent experiments concerning the aerodynamics of insect flight, as well as the different approaches used to model these phenomena.

/pdf/the-aerodynamics-of-insect-flight-4ladwfpmp7.pdf

The aerodynamics of insect flight

Part I: Background and Fundamentals Introduction, Mohamed Gad-el-Hak, University of Notre Dame Scaling of Micromechanical Devices, William Trimmer, Standard MEMS, Inc., and Robert H. Stroud, Aerospace Corporation Mechanical Properties of MEMS Materials, William N. Sharpe, Jr., Johns Hopkins University Flow Physics, Mohamed Gad-el-Hak, University of Notre Dame Integrated Simulation for MEMS: Coupling Flow-Structure-Thermal-Electrical Domains, Robert M. Kirby and George Em Karniadakis, Brown University, and Oleg Mikulchenko and Kartikeya Mayaram, Oregon State University Liquid Flows in Microchannels, Kendra V. Sharp and Ronald J. Adrian, University of Illinois at Urbana-Champaign, Juan G. Santiago and Joshua I. Molho, Stanford University Burnett Simulations of Flows in Microdevices, Ramesh K. Agarwal and Keon-Young Yun, Wichita State University Molecular-Based Microfluidic Simulation Models, Ali Beskok, Texas A&M University Lubrication in MEMS, Kenneth S. Breuer, Brown University Physics of Thin Liquid Films, Alexander Oron, Technion, Israel Bubble/Drop Transport in Microchannels, Hsueh-Chia Chang, University of Notre Dame Fundamentals of Control Theory, Bill Goodwine, University of Notre Dame Model-Based Flow Control for Distributed Architectures, Thomas R. Bewley, University of California, San Diego Soft Computing in Control, Mihir Sen and Bill Goodwine, University of Notre Dame Part II: Design and Fabrication Materials for Microelectromechanical Systems Christian A. Zorman and Mehran Mehregany, Case Western Reserve University MEMS Fabrication, Marc J. Madou, Nanogen, Inc. LIGA and Other Replication Techniques, Marc J. Madou, Nanogen, Inc. X-Ray-Based Fabrication, Todd Christenson, Sandia National Laboratories Electrochemical Fabrication (EFAB), Adam L. Cohen, MEMGen Corporation Fabrication and Characterization of Single-Crystal Silicon Carbide MEMS, Robert S. Okojie, NASA Glenn Research Center Deep Reactive Ion Etching for Bulk Micromachining of Silicon Carbide, Glenn M. Beheim, NASA Glenn Research Center Microfabricated Chemical Sensors for Aerospace Applications, Gary W. Hunter, NASA Glenn Research Center, Chung-Chiun Liu, Case Western Reserve University, and Darby B. Makel, Makel Engineering, Inc. Packaging of Harsh-Environment MEMS Devices, Liang-Yu Chen and Jih-Fen Lei, NASA Glenn Research Center Part III: Applications of MEMS Inertial Sensors, Paul L. Bergstrom, Michigan Technological University, and Gary G. Li, OMM, Inc. Micromachined Pressure Sensors, Jae-Sung Park, Chester Wilson, and Yogesh B. Gianchandani, University of Wisconsin-Madison Sensors and Actuators for Turbulent Flows. Lennart Loefdahl, Chalmers University of Technology, and Mohamed Gad-el-Hak, University of Notre Dame Surface-Micromachined Mechanisms, Andrew D. Oliver and David W. Plummer, Sandia National Laboratories Microrobotics Thorbjoern Ebefors and Goeran Stemme, Royal Institute of Technology, Sweden Microscale Vacuum Pumps, E. Phillip Muntz, University of Southern California, and Stephen E. Vargo, SiWave, Inc. Microdroplet Generators. Fan-Gang Tseng, National Tsing Hua University, Taiwan Micro Heat Pipes and Micro Heat Spreaders, G. P. "Bud" Peterson, Rensselaer Polytechnic Institute Microchannel Heat Sinks, Yitshak Zohar, Hong Kong University of Science and Technology Flow Control, Mohamed Gad-el-Hak, University of Notre Dame) Part IV: The Future Reactive Control for Skin-Friction Reduction, Haecheon Choi, Seoul National University Towards MEMS Autonomous Control of Free-Shear Flows, Ahmed Naguib, Michigan State University Fabrication Technologies for Nanoelectromechanical Systems, Gary H. Bernstein, Holly V. Goodson, and Gregory L. Snider, University of Notre Dame Index

The MEMS Handbook

SUMMARY
Recent alluvial sediments are reviewed with respect to their geometrical, textural, structural and biological characteristics. These properties are related to the physiographic occurrence and hydraulic geometry of streams and to the dynamics of flowing water as controlling sediment transport-deposition and stream morphological activities. Based on this data, three-dimensional facies models are presented as an aid to the identification of ancient alluvial sediments, which are briefly reviewed also.

Oskar Karl Gustav Tietjens

Papers

Applied hydro- and aeromechanics

Fundamentals of hydro- and aeromechanics

Hydro- und Aeromechanik : nach Vorlesungen von L. Prandtl

Bewegung der Flüssigkeiten und Gase

Strömungslehre : physikalische Grundlagen vom technischen Standpunkt