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Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM)
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Active Flow Control Papers contributed to the Conference “Active Flow Control 2006”, Berlin, Germany, September 27 to 29, 2006 Rudibert King (Editor)
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Library of Congress Control Number: 2007922725 ISBN-10 3-540-71438-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-71438-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 Printed in Germany
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[email protected] Preface
The dramatically increasing requirements of mobility through road-, rail- and airborne transport systems in the future necessitate non-evolutionary improvements of transportation systems. Without severe implications concerning the environment or restrictions concerning the performance, these requirements will only be met by a concerted action of many disciplines. It is believed that with ACTIVE FLOW CONTROL a key technology exists to supply an important block in the mosaic to be laid in the pursuit of best and sustainable solutions. Manipulation of fluid flows is highly advantageous in many cases. Aerodynamic or fluid flows around or inside bodies impose drag, lift and moments on the body, remove or supply energy by convection. Flow-induced noise may be produced by the interaction of a body with the surrounding air. Moreover, the interaction with the body changes the state of the flow drastically. A neatly aligned laminar flow around a wing of an aircraft giving enough lift, can become highly irregular and separated from the surface, with the result of a loss of lift. For cooling of engines of transport and other systems highly irregular turbulent fluid flows across the components are needed to guarantee a large heat transfer. In future engines of airplanes complying for example with the EU Vision 2020 an increased heat transfer, on the other hand, has to be avoided by all means in some parts of the engine. Turbine stages may be exposed here to extremely hot gases, needed for high efficiency, which would destroy the blades. In this application, more laminar flow regimes would be advantageous yielding a poorer heat transfer. The irregular flow in a combustor e.g., of an aero engine, determines to a great extent the thermodynamic efficiency of and the reactions occurring inside the system. Complying with environmental constraints concerning the production of exhaust gases like NOx will necessitate a low mixture ratio of fuel to air. However, this together with the requirements of light engines will result in thermo-acoustic instabilities in burners. Noise emission will only be one problem related with this situation. More severe consequences will result from the massive mechanical burden the burner has to sustain. Improving the efficiency of airplanes by flow control will significantly reduce the production of greenhouse gases and save money for the airlines and for the customers. It is not just the increased lift by flow control, which will allow smaller, lighter engines which consume less fuel. It is of course as well the reduced flow-induced drag exerted by the different components of an aircraft e.g., engines nacelles, fuselage, landing gear and so forth, which will allow for smaller engines. When steeper climbing and landing paths are possible thanks to higher lift obtained by flow control, the noise exposure on ground would be reduced quantitatively. Moreover, it is the improved design of the engines: If an engine can be run with fewer compressor stages, its size and weight will be
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reduced. Additionally, the drag induced by this component might be lowered. More flowrelated challenges for air-transport systems could easily be listed. From this selection of examples it can be concluded that manipulation of fluid flows is of paramount importance. Flows can be manipulated either by passive or by active means. Passive methods comprise the shaping of the bodies, the inclusion of devices such as spoilers, riblets or vortex generators. It goes without saying that manipulation of flows with these passive means has a very long, well established tradition in fluid mechanics. In contrast to passive means, active concepts of flow control are still in their infancy. Much basic and applied research will be needed to fully exploit the benefits associated with it. The most obvious advantage of active over passive manipulation is the possibility to adapt the kind and size of manipulation in a desired, if possible optimal way to the actual operating conditions of a process. Active methods are needed when passive ones reach their limits to extend regions of operation, when passive methods have positive and negative effects depending on the operating conditions, or when due to active methods conventional design constraints can be relaxed. Most importantly, active methods of flow control can result in a nongradual, drastic improvement of a system, as desired at the beginning of this section. Active flow control can be done by blowing, suction, acoustic actuation, and in some cases by magneto-hydrodynamic forces. It can be further subdivided in steady and in unsteady actuation. The simplest form is steady actuation. Fluid is blown into or sucked out of a flow over the contour of a body. It can be energized so that it can withstand an adverse pressure gradient which otherwise would lead to an undesired separation of the flow from the surface. A desired degree of mixing between different flow regimes can be influenced as well, just to name one more example. Unsteady flow control is known to offer the same or even larger control authority as steady control, but with much smaller control inputs. The art of active, unsteady flow control is to play around, to exploit instabilities present in the flow system. Due to their very nature as an unstable process, only a rather small amount of energy i.e., periodic mass flow is necessary to trigger the instability. The frequency and the size of actuation have to be matched exactly, to meet the demands of the control and to synchronize it with the operating point dependent character of the flow. Moreover, as the onset of instability itself is not the control goal, it has to be implemented such that the lift of a wing is increased, the drag reduced, the mixing in a burner improved, etc. Most of the work done so far in the area of active flow control is devoted to open-loop control. For open-loop control concepts extensive real and/or numerical experiments have to be performed to determine for example, the size, frequency or shape of the control signal necessary for every operating condition. This information is stored in the controller and retrieved afterwards when a specific situation is met. Various experiments in wind tunnels and for a real aircraft have shown the power of this dynamic active flow control. However, when the well- defined conditions of a wind tunnel are left and abundant disturbances act
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on the wing, car, or other flow systems, this open-loop control is bound to fail. It will fail also when operating conditions are chosen for which no information is available in the open-loop controller. Hence, to exploit the enormous potentials of active flow control, a control loop has to be established. By measuring the effect of the control with appropriate sensors, any disturbance or deterioration due to changing operating conditions is observed and a closed-loop controller can react accordingly. Besides a much better disturbance rejection and robustness with respect to process uncertainties, closed-loop control often increase the achievable goals, such as maximal lift. From a control engineering point of view, flow control systems are counted among the most difficult ones due to their nonlinearity and infinite parameter dimension. A modeling based on physics leads to the Navier-Stokes together with the governing mass conservation equations for which huge computer resources are needed to find numerical solutions. This complexity necessitates complementary approaches of controller synthesis, such as robust, adaptive, or nonlinear methods based on low-dimensional models if these controllers have to be brought into a real system. These challenges and perspectives in active flow control laid the ground at Technische Universit¨ at Berlin (TUB) almost 10 years ago to build up a Collaborative Research Center (SFB) 557 on CONTROL OF COMPLEX TURBULENT SHEAR FLOWS. It is financed by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) with significant contributions by TUB, Deutsches Zentrum f¨ ur Luft und Raumfahrt e.V. (DLR, German Aerospace Center), ZuseInstitut Berlin (ZIB) and Freie Universit¨at Berlin. This SFB 557 initiated the Conference ACTIVE FLOW CONTROL 2006, held September 27-29, 2006, at the TUB, Germany. The goal was to attract leading experts in the area of active flow control from around the world discussing the state-of-the-art of the aforementioned flow control aspects. Plenary and invited papers were dedicated to all important fields of active flow control, such as sensing, actuators, aerodynamics, acoustics, turbo-machinery, numerics or control. Preparing this book which was published shortly after the conference was challenging. First, due to size limitations the INTERNATIONAL PROGRAM COMMITTEE and the REVIEWERS had the difficult task to select a subset of the excellent papers presented at the conference to be included in this book1 . Reviewers were: A. Banaszuk, East Hardford; P. Bonnet, Poitiers; H. Choi, Seoul; H. Fernholz, Berlin; M. Gad-el-Hak, Richmond; K. Kunisch, Graz; V. Mehrmann, Berlin; M. M¨ oser, Berlin; W. Neise, Berlin; W. Nitsche, Berlin; B.R. Noack, Berlin; C.O. Paschereit, Berlin; W. Schr¨oder, Aachen; A. Seifert, Tel Aviv; S. Siegel, Colorado Springs; G. Tadmor, Boston; F. Thiele, Berlin. Their help is highly acknowledged. Second, deciding on the sequence of chapters was not straightforward. Due to the interdisciplinary character of flow control many
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More papers can be found in the conference CD.
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contributions comprise results from different areas concerning methods and/or applications. Neither an ordering according to sensors, actuators, controllers, nor to the various areas of applications, such as airfoils, bluff bodies, cavities etc., was possible without arbitrarily grouping some papers which cover different aspects at the same time. As a compromise, chapter headings were chosen according to the session headings of the conference. The international conference on ACTIVE FLOW CONTROL 2006 was used at the same time to highlight recent achievements obtained in the Collaborative Research Center in Berlin. The SFB 557 concentrates in its third period in the years 2004-2007 on four flow configurations, namely (1) a high-lift wing, (2) turbo-machines, (3) a generic car model, and (4) a burner. This concentration on very few, but theoretically and technically demanding problems has led to an increased cooperation and has improved drastically the insights into the benefits of synergy and mutual understanding. By this very simple instrument, mathematicians, theoreticians and experimenters from fluid and aerodynamics, control engineers, computational fluid dynamicists, acousticians, etc. are working together on the very same technical problems. Some of the results obtained are included in this volume as well. A main focus of the SFB 557 for all four flow configurations is dedicated to closed-loop flow control. As a result, the SFB 557 offers today a vast experience in experimental closed-loop flow control, not only for the above mentioned flow configurations but for other configurations, too, such as the backward-facing step, diffusers, etc. Again some related work is described in this volume as well. The ORGANIZING COMMITTEE is indebted to the DFG. Without their substantial financial support the making of this book would not have been possible. Furthermore, the editor is grateful to Mrs. Steffi Stehr for the significant help in the organization of the conference and the compilation of this volume.
Berlin, September 2006
Rudibert King
Table of Contents
The Taming of the Shrew: Why Is It so Difficult to Control Turbulence? Mohamed Gad-el-Hak …..……………………………………………………........
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Part I: Actuators Electromagnetic Control of Separated Flows Using Periodic Excitation with Different Wave Forms Christian Cierpka, Tom Weier, Gunter Gerbeth …………………………………
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Pulsed Plasma Actuators for Active Flow Control at MAV Reynolds Numbers B. Göksel, D. Greenblatt, I. Rechenberg, Y. Kastantin, C.N. Nayeri, C.O. Paschereit …………………………………………………………………...
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Experimental and Numerical Investigations of Boundary-Layer Influence Using Plasma-Actuators S. Grundmann, S. Klumpp, C. Tropea …………………………………………… 56 Designing Actuators for Active Separation Control Experiments on High-Lift Configurations* Ralf Petz, Wolfgang Nitsche ……………………………………………………... 69 Closed-Loop Active Flow Control Systems: Actuators A. Seifert ………………………………………………………………………….
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Part II: State Estimation and Feature Extraction State Estimation of Transient Flow Fields Using Double Proper Orthogonal Decomposition (DPOD) Stefan Siegel, Kelly Cohen, Jürgen Seidel, Thomas Mclaughlin ……………...... 105 A Unified Feature Extraction Architecture* Tino Weinkauf, Jan Sahner, Holger Theisel, Hans-Christian Hege, Hans-Peter Seidel …………………………………………………….…………
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Part III: Air Foils Control of Wing Vortices I. Gursul, E. Vardaki, P. Margaris, Z. Wang …………………………………… 137 Towards Active Control of Leading Edge Stall by Means of Pneumatic Actuators C.J. Kähler, P. Scholz, J. Ortmanns, R. Radespiel …………………………….... 152 Computational Investigation of Separation Control for High-Lift Airfoil Flows* Markus Schatz, Bert Günther, Frank Thiele …………………………………… 173 Steady and Oscillatory Flow Control Tests for Tilt Rotor Aircraft M. Schmalzel, P. Varghese, I. Wygnanski ………………………………………
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Part IV: Cavities Reduced-Order Model-Based Feedback Control of Subsonic Cavity Flows – An Experimental Approach M. Samimy, M. Debiasi, E. Caraballo, A. Serrani, X. Yuan, J. Little, J.H. Myatt ………………………………………………………………………. 211 Supersonic Cavity Response to Open-Loop Forcing David R. Williams, Daniel Cornelius, Clarence W. Rowley ……………………
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Part V: Bluff Bodies Active Drag Control for a Generic Car Model* A. Brunn, E. Wassen, D. Sperber, W. Nitsche, F. Thiele ………………………..
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Continuous Mode Interpolation for Control-Oriented Models of Fluid Flow* Marek MorzyĔski, Witold Stankiewicz, Bernd R. Noack, Rudibert King, Frank Thiele, Gilead Tadmor ………………………….………………………………………. 260 Part VI: Turbomachines and Combustors Active Management of Entrainment and Streamwise Vortices in an Incompressible Jet D. Greenblatt, Y. Singh, Y. Kastantin, C.N. Nayeri, C.O. Paschereit …………..
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Active Control to Improve the Aerodynamic Performance and Reduce the Tip Clearance Noise of Axial Turbomachines with Steady Air Injection into the Tip Clearance Gap L. Neuhaus, W. Neise …………………………………………………………... 293
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Part VII: Optimal Flow Control and Numerical Studies Drag Minimization of the Cylinder Wake by Trust-Region Proper Orthogonal Decomposition Michel Bergmann, Laurent Cordier, Jean-Pierre Brancher ……………………. 309 Flow Control on the Basis of a FEATFLOW-MATLAB Coupling* Lars Henning, Dmitri Kuzmin, Volker Mehrmann, Michael Schmidt, Andriy Sokolov, Stefan Turek ……………………………………..…………….
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On the Choice of the Cost Functional for Optimal Vortex Reduction for Instationary Flows Karl Kunisch, Boris Vexler ……………………………………………………..
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Flow Control with Regularized State Constraints* J.C. de los Reyes, F. Tröltzsch ………………………………………………......
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Part VIII: Closed-Loop Flow Control Feedback Control Applied to the Bluff Body Wake* Lars Henning, Mark Pastoor, Rudibert King, Bernd R. Noack, Gilead Tadmor …………………………………………………….…………….. 369 Active Blade Tone Control in Axial Turbomachines by Flow Induced Secondary Sources in the Blade Tip Regime* O. Lemke, R. Becker, G. Feuerbach, W. Neise, R. King, M. Möser …………….. 391 Phase-Shift Control of Combustion Instability Using (Combined) Secondary Fuel Injection and Acoustic Forcing* Jonas P. Moeck, Mirko R. Bothien, Daniel Guyot, Christian Oliver Paschereit …………………………………………………….. 408 Vortex Models for Feedback Stabilization of Wake Flows Bartosz Protas …………………………………………………………………… 422 Keyword Index …………………………………………………………………
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* These papers are results of the Collaborative Research Center 557 “Control of complex turbulent shear flows” (SFB 557, TU Berlin) and are printed according to the regulations of and funded by the DFG (German Research Foundation).
The Taming of the Shrew: Why Is It so Difficult to Control Turbulence? Mohamed Gad-el-Hak Virginia Commonwealth University, Richmond, VA 23284-3015, USA
[email protected] http://www.people.vcu.edu/∼gadelhak
Summary In the present chapter I shall emphasize the frontiers of the field of flow control, pondering mostly the control of turbulent flows. I shall review the important advances in the field that took place during the past few years and are anticipated to dominate progress in the future. By comparison with laminar flow control or separation prevention, the control of turbulent flow remains a very challenging problem. Flow instabilities magnify quickly near critical flow regimes, and therefore delaying transition or separation are relatively easier tasks. In contrast, classical control strategies are often ineffective for fully turbulent flows. Newer ideas for turbulent flow control to achieve, for example, skin-friction drag reduction focus on the direct onslaught on coherent structures. Spurred by the recent developments in chaos control, microfabrication and soft computing tools, reactive control of turbulent flows, where sensors detect oncoming coherent structures and actuators attempt to favorably modulate those quasi-periodic events, is now in the realm of the possible for future practical devices. In this chapter, I shall provide estimates for the number, size, frequency and energy consumption of the sensor/actuator arrays needed to control the turbulent boundary layer on a full-scale aircraft or submarine.
1 Introduction 1.1 The Taming of the Shrew Considering the extreme complexity of the turbulence problem in general and the unattainability of first-principles analytical solutions in particular, it is not surprising that controlling a turbulent flow remains a challenging task, mired in empiricism and unfulfilled promises and aspirations. Brute force suppression, or taming, of turbulence via active, energy-consuming control strategies is always possible, but the penalty for doing so often exceeds any potential benefits. The artifice is to achieve a desired effect with minimum energy expenditure. This is of course easier said than done. Indeed, suppressing turbulence is as arduous as the taming of the shrew. R. King (Ed.): Active Flow Control, NNFM 95, pp. 1–24, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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1.2 Control of Turbulence Numerous methods of flow control have already been successfully implemented in practical engineering devices. Delaying laminar-to-turbulence transition to reasonable Reynolds numbers and preventing separation can readily be accomplished using a myriad of passive and predetermined active control strategies. Such classical techniques have been reviewed by, among others, Bushnell (1983; 1994), Wilkinson et al. (1988), Bushnell and McGinley (1989), Gad-el-Hak (1989), Bushnell and Hefner (1990), Fiedler and Fernholz (1990), Gad-el-Hak and Bushnell (1991), Barnwell and Hussaini (1992), Viswanath (1995), and Joslin et al. (1996). Yet, very few of the classical strategies are effective in controlling free-shear or wall-bounded turbulent flows. Serious limitations exist for some familiar control techniques when applied to certain turbulent flow situations. For example, in attempting to reduce the skin-friction drag of a body having a turbulent boundary layer using global suction, the penalty associated with the control device often exceeds the saving derived from its use. What is needed is a way to reduce this penalty to achieve a more effective control. Flow control is most effective when applied near the transition or separation points; in other words, near the critical flow regimes where flow instabilities magnify quickly. Therefore, delaying/advancing laminar-to-turbulence transition and preventing/ provoking separation are relatively easier tasks to accomplish. To reduce the skinfriction drag in a non-separating turbulent boundary layer, where the mean flow is quite stable, is a more challenging problem. Yet, even a modest reduction in the fluid resistance to the motion of, for example, the worldwide commercial airplane fleet is translated into fuel savings estimated to be in the billions of dollars. Newer ideas for turbulent flow control focus on the direct onslaught on coherent structures. Spurred by the recent developments in chaos control, microfabrication and soft computing tools, reactive control of turbulent flows is now in the realm of the possible for future practical devices. The primary objective of the present chapter is to advance possible scenarios by which viable control strategies of turbulent flows could be realized. As will be argued in the following presentation, future systems for control of turbulent flows in general and turbulent boundary layers in particular could greatly benefit from the merging of the science of chaos control, the technology of microfabrication, and the newest computational tools collectively termed soft computing. Control of chaotic, nonlinear dynamical systems has been demonstrated theoretically as well as experimentally, even for multidegree-of-freedom systems. Microfabrication is an emerging technology which has the potential for producing inexpensive, programmable sensor/actuator chips that have dimensions of the order of a few microns. Soft computing tools include neural networks, fuzzy logic and genetic algorithms and are now more advanced as well as more widely used as compared to just few years ago. These tools could be very useful in constructing effective adaptive controllers. Such futuristic systems are envisaged as consisting of a large number of intelligent, interactive, microfabricated wall sensors and actuators arranged in a checkerboard pattern and targeted toward specific organized structures that occur quasi-randomly within a turbulent flow. Sensors detect oncoming coherent structures, and adaptive controllers process the sensors information and provide control signals to the actuators which in
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turn attempt to favorably modulate the quasi-periodic events. Finite number of wall sensors perceive only partial information about the entire flow field above. However, a low-dimensional dynamical model of the near-wall region used in a Kalman filter can make the most of the partial information from the sensors. Conceptually all of that is not too difficult, but in practice the complexity of such a control system is daunting and much research and development work still remain. 1.3 Outline In the present chapter, I shall review the important developments in the field of flow control that took place during the past few years and suggest avenues for future research. The emphasis will be on reactive flow control for future vehicles and other industrial devices. The present chapter is organized into four sections. Reactive flow control and the selective suction concept are described in Section 2. The number, size, frequency and energy consumption of the sensor/actuator units required to tame the turbulence on a full-scale air or water vehicle are estimated in that same section. Section 3 considers the emerging area of chaos control, particularly as it relates to reactive control strategies. Finally, brief concluding remarks are given in Section 4.
2 Reactive Control 2.1 Introductory Remarks Targeted control implies sensing and reacting to a particular quasi-periodic structure in the boundary layer. The wall seems to be the logical place for such reactive control, because of the relative ease of placing something in there, the sensitivity of the flow in general to surface perturbations, and the proximity and therefore accessibility to the dynamically all important near-wall coherent events. According to Wilkinson (1990), there are very few actual experiments that use embedded wall sensors to initiate a surface actuator response (Alshamani et al., 1982; Wilkinson and Balasubramanian, 1985; Nosenchuck and Lynch, 1985; Breuer et al., 1989). This decade-old assessment is fast changing, however, with the introduction of microfabrication technology that has the potential for producing small, inexpensive, programmable sensor/actuator chips. Witness the more recent reactive control attempts by Kwong and Dowling (1993), Reynolds (1993), Jacobs et al. (1993), Jacobson and Reynolds (1993a; 1993b; 1994; 1995; 1998), Fan et al. (1993), James et al. (1994), and Keefe (1996). Fan et al. and Jacobson and Reynolds even consider the use of self-learning neural networks for increased computational speeds and efficiency. Recent reviews of reactive flow control include those by Gad-el-Hak (1994; 1996), Lumley (1996), McMichael (1996), Mehregany et al. (1996), and Ho and Tai (1996). Numerous methods of flow control have already been successfully implemented in practical engineering devices. Yet, limitations exist for some familiar control techniques when applied to specific situations. For example, in attempting to reduce the drag or enhance the lift of a body having a turbulent boundary layer using global suction, global heating/cooling or global application of electromagnetic body forces, the actuator’s energy expenditure often exceeds the saving derived from the predetermined active
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control strategy. What is needed is a way to reduce this penalty to achieve a more efficient control. Reactive control geared specifically toward manipulating the coherent structures in turbulent shear flows, though considerably more complicated than passive control or even predetermined active control, has the potential to do just that. As will be argued in this and the following sections, future systems for control of turbulent flows in general and turbulent boundary layers in particular could greatly benefit from the merging of the science of chaos control, the technology of microfabrication, and the newest computational tools collectively termed soft computing. Such systems are envisaged as consisting of a large number of intelligent, communicative wall sensors and actuators arranged in a checkerboard pattern and targeted toward controlling certain quasi-periodic, dynamically significant coherent structures present in the near-wall region. 2.2 Targeted Control Successful techniques to reduce the skin friction in a turbulent flow, such as polymers, particles or riblets, appear to act indirectly through local interaction with discrete turbulent structures, particularly small-scale eddies, within the flow. Common characteristics of all these methods are increased losses in the near-wall region, thickening of the buffer layer and lowered production of Reynolds shear stress (Bandyopadhyay, 1986). Methods that act directly on the mean flow, such as suction or lowering of near-wall viscosity, also lead to inhibition of Reynolds stress. However, skin friction is increased when any of these velocity-profile modifiers is applied globally. Could these seemingly inefficient techniques, e.g. global suction, be used more sparingly and be optimized to reduce their associated penalty? It appears that the more successful drag-reducing methods, e.g. polymers, act selectively on particular scales of motion and are thought to be associated with stabilization of the secondary instabilities. It is also clear that energy is wasted when suction or heating/cooling is used to suppress the turbulence throughout the boundary layer when the main interest is to affect a near-wall phenomenon. One ponders, what would become of wall turbulence if specific coherent structures are to be targeted, by the operator through a reactive control scheme, for modification? The myriad of organized structures present in all shear flows are instantaneously identifiable, quasi-periodic motions (Cantwell, 1981; Robinson, 1991). Bursting events in wall-bounded flows, for example, are both intermittent and random in space as well as time. The random aspects of these events reduce the effectiveness of a predetermined active control strategy. If such structures are nonintrusively detected and altered, on the other hand, net performance gain might be achieved. It seems clear, however, that temporal phasing as well as spatial selectivity would be required to achieve proper control targeted toward random events. A nonreactive version of the above idea is the selective suction technique which combines suction to achieve an asymptotic turbulent boundary layer and longitudinal riblets to fix the location of low-speed streaks. Although far from indicating net drag reduction, the available results are encouraging and further optimization is needed. When implemented via an array of reactive control loops, the selective suction method is potentially capable of skin-friction reduction that approaches 60%.
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The genesis of the selective suction concept can be found in the papers by Gadel-Hak and Blackwelder (1987; 1989) and the patent by Blackwelder and Gad-el-Hak (1990). These researchers suggest that one possible means of optimizing the suction rate is to be able to identify where a low-speed streak is presently located and apply a small amount of suction under it. Assuming that the production of turbulence kinetic energy is due to the instability of an inflectional U (y) velocity profile, one needs to remove only enough fluid so that the inflectional nature of the profile is alleviated. An alternative technique that could conceivably reduce the Reynolds stress is to inject fluid selectively under the high-speed regions. The immediate effect of normal injection would be to decrease the viscous shear at the wall resulting in less drag. In addition, the velocity profiles in the spanwise direction, U (z), would have a smaller shear, ∂U/∂z, because the suction/injection would create a more uniform flow. Since Swearingen and Blackwelder (1984) and Blackwelder and Swearingen (1990) have found that inflectional U (z) profiles occur as often as inflection points are observed in U (y) profiles, suction under the low-speed streaks and/or injection under the high-speed regions would decrease this shear and hence the resulting instability. The combination of selective suction and injection is sketched in Figure 1. In Figure 1a, the vortices are idealized by a periodic distribution in the spanwise direction. The instantaneous velocity profiles without transpiration at constant y and z locations are shown by the dashed lines in Figures 1b and 1c, respectively. Clearly, the U (yo , z) profile is inflectional, having two inflection points per wavelength. At z1 and z3 , an inflectional U (y) profile is also evident. The same profiles with suction at z1 and z3 and injection at z2 are shown by the solid lines. In all cases, the shear associated with the inflection points would have been reduced. Since the inflectional profiles are all inviscidly unstable with growth rates proportional to the shear, the resulting instabilities would be weakened by the suction/injection process. The feasibility of the selective suction as a drag-reducing concept has been demonstrated by Gad-el-Hak and Blackwelder (1989) and is indicated in Figure 2. Low-speed streaks were artificially generated in a laminar boundary layer using three spanwise suction holes as per the method proposed by Gad-el-Hak and Hussain (1986), and a hot-film probe was used to record the near-wall signature of the streaks. An open, feedforward control loop with a phase lag was used to activate a predetermined suction from a longitudinal slot located in between the spanwise holes and the downstream hot-film probe. An equivalent suction coefficient of Cq = 0.0006 was sufficient to eliminate the artificial events and prevent bursting. This rate is five times smaller than the asymptotic suction coefficient for a corresponding turbulent boundary layer. If this result is sustained in a naturally developing turbulent boundary layer, a skin-friction reduction of close to 60% would be attained. Gad-el-Hak and Blackwelder (1989) propose to combine suction with non-planar surface modifications. Minute longitudinal roughness elements if properly spaced in the spanwise direction greatly reduce the spatial randomness of the low-speed streaks (Johansen and Smith, 1986). By withdrawing the streaks forming near the peaks of the roughness elements, less suction should be required to achieve an asymptotic boundary layer. Experiments by Wilkinson and Lazos (1987) and Wilkinson (1988) combine suction/blowing with thin-element riblets. Although no net drag reduction is yet attained in these experiments, their results indicate some
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Fig. 1. Effects of suction/injection on velocity profiles. Broken lines: reference profiles. Solid lines: profiles with transpiration applied. (a) Streamwise vortices in the y-z plane, suction/injection applied at z1 , z2 and z3 . (b) Resulting spanwise velocity distribution at y = yo . (c) Velocity profiles normal to the surface.
advantage of combining suction with riblets as proposed by Gad-el-Hak and Blackwelder (1987; 1989). The recent numerical experiments of Choi et al. (1994) also validate the concept of targeting suction/injection to specific near-wall events in a turbulent channel flow. Based on complete interior flow information and using the rather simple, heuristic control law proposed earlier by Gad-el-Hak and Blackwelder (1987), Choi et al.’s direct numerical simulations indicate a 20% net drag reduction accompanied by significant suppression of the near-wall structures and the Reynolds stress throughout the entire
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Fig. 2. Effects of suction from a streamwise slot on five artificially induced burstlike events in a laminar boundary layer (from Gad-el-Hak and Blackwelder, 1989). (a) Cq = 0.0. (b) Cq = 0.0006.
wall-bounded flow. When only wall information was used, a drag reduction of 6% was observed; a rather disappointing result considering that sensing and actuation took place at every grid point along the computational wall. In a practical implementation of this technique, even fewer wall sensors would perhaps be available, measuring only a small subset of the accessible information and thus requiring even more sophisticated control algorithms to achieve the same degree of success. Low-dimensional models of the nearwall flow (Section 3) and soft computing tools can help in constructing more effective control algorithms. Time sequences of the numerical flow field of Choi et al. (1994) indicate the presence of two distinct drag-reducing mechanisms when selective suction/injection is used. First, deterring the sweep motion, without modifying the primary streamwise vortices above the wall, and consequently moving the high-shear regions from the surface to the interior of the channel, thus directly reducing the skin friction. Secondly, changing the evolution of the wall vorticity layer by stabilizing and preventing lifting of the near-wall spanwise vorticity, thus suppressing a potential source of new streamwise vortices above the surface and interrupting a very important regeneration mechanism of turbulence. Three modern developments have relevance to the issue at hand. Firstly, the recently demonstrated ability to revert a chaotic system to a periodic one may provide
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optimal nonlinear control strategies for further reduction in the amount of suction (or the energy expenditure of any other active wall-modulation technique) needed to attain a given degree of flow stabilization. This is important since net drag reduction achieved in a turbulent boundary layer increases as the suction coefficient decreases. Secondly, to selectively remove the randomly occurring low-speed streaks, for example, would ultimately require reactive control. In that case, an event is targeted, sensed and subsequently modulated. Microfabrication technology provides opportunities for practical implementation of the required large array of inexpensive, programmable sensor/actuator chips. Thirdly, newly introduced soft computing tools include neural networks, fuzzy logic and genetic algorithms and are now more advanced as well as more widely used as compared to just few years ago. These tools could be very useful in constructing effective adaptive controllers. 2.3 Reactive Feedback Control A control device can be passive, requiring no auxiliary power, or active, requiring energy expenditure. Active control is further divided into predetermined or reactive. Predetermined control includes the application of steady or unsteady energy input without regard to the particular state of the flow. The control loop in this case is open, and no sensors are required. Because no sensed information are being fed forward, this open control loop is not a feedforward one. Reactive control is a special class of active control where the control input is continuously adjusted based on measurements of some kind. The control loop in this case can either be an open, feedforward one or a closed, feedback loop. The distinction between feedforward and feedback is particularly important when dealing with the control of flow structures which convect over stationary sensors and actuators. In feedforward control, the measured variable and the controlled variable differ. For example, the pressure or velocity can be sensed at an upstream location, and the resulting signal is used together with an appropriate control law to trigger an actuator which in turn influences the velocity at a downstream position. Feedback control, on the other hand, necessitates that the controlled variable be measured, fed back and compared with a reference input. Moin and Bewley (1994) categorize reactive feedback control strategies by examining the extent to which they are based on the governing flow equations. Four categories are discerned: adaptive, physical model-based, dynamical systems-based, and optimal control. Note that except for adaptive control, the other three categories of reactive feedback control can also be used in the feedforward mode or the combined feedforwardfeedback mode. Also, in a convective environment such as that for a boundary layer, a controller would perhaps combine feedforward and feedback information and may include elements from each of the four classifications. Each of the four categories is briefly described below. Adaptive schemes attempt to develop models and controllers via some learning algorithm without regard to the details of the flow physics. System identification is performed independently of the flow dynamics or the Navier–Stokes equations which govern this dynamics. An adaptive controller tries to optimize a specified performance index by providing a control signal to an actuator. In order to update its parameters,
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the controller thus requires feedback information relating to the effects of its control. The most recent innovation in adaptive flow control schemes involves the use of neural networks which relate the sensor outputs to the actuator inputs through functions with variable coefficients and nonlinear, sigmoid saturation functions. The coefficients are updated using the so-called back-propagation algorithm, and complex control laws can be represented with a sufficient number of terms. Hand tuning is required, however, to achieve good convergence properties. The nonlinear adaptive technique has been used with different degrees of success by Fan et al. (1993) and Jacobson and Reynolds (1993b; 1995; 1998) to control, respectively, the transition process and the bursting events in turbulent boundary layers. Heuristic physical arguments can instead be used to establish effective control laws. That approach obviously will work only in situations in which the dominant physics are well understood. An example of this strategy is the active cancellation scheme, used by Gad-el-Hak and Blackwelder (1989) in a physical experiment and by Choi et al. (1994) in a numerical experiment, to reduce the drag by mitigating the effect of nearwall vortices. As mentioned earlier, the idea is to oppose the near-wall motion of the fluid, caused by the streamwise vortices, with an opposing wall control, thus lifting the high-shear region away from the surface and interrupting the turbulence regeneration mechanism. Nonlinear dynamical systems theory allows turbulence to be decomposed into a small number of representative modes whose dynamics are examined to determine the best control law. The task is to stabilize the attractors of a low-dimensional approximation of a turbulent chaotic system. The best known strategy is the OGY method which, when applied to simpler, small-number of degrees of freedom systems, achieves stabilization with minute expenditure of energy. This and other chaos control strategies, especially as applied to the more complex turbulent flows, will be revisited in Section 3.2. Finally, optimal control theory applied directly to the Navier–Stokes equations can, in principle, be used to minimize a cost function in the space of the control. This strategy provides perhaps the most rigorous theoretical framework for flow control. As compared to other reactive control strategies, optimal control applied to the full Navier– Stokes equations is also the most computer-time intensive. In this method, feedback control laws are derived systematically for the most efficient distribution of control effort to achieve a desired goal. Abergel and Temam (1990) developed such optimal control theory for suppressing turbulence in a numerically simulated, two-dimensional Navier–Stokes flow, but their method requires an impractical full flow-field information. Choi et al. (1993) developed a more practical, wall-information-only, sub-optimal control strategy which they applied to the one-dimensional stochastic Burgers equation. Later application of the sub-optimal control theory to a numerically simulated turbulent channel flow has been reported by Moin and Bewley (1994) and Bewley et al. (1997; 1998). The recent book edited by Sritharan (1998) provides eight articles that focus on the mathematical aspects of optimal control of viscous flows. 2.4 Required Characteristics The randomness of the bursting events necessitates temporal phasing as well as spatial selectivity to effect selective control. Practical applications of methods targeted at
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controlling a particular turbulent structure to achieve a prescribed goal would therefore require implementing a large number of surface sensors/actuators together with appropriate control algorithms. That strategy for controlling wall-bounded turbulent flows has been advocated by, among others and in chronological order, Gad-el-Hak and Blackwelder (1987; 1989), Lumley (1991; 1996), Choi et al. (1992), Reynolds (1993), Jacobson and Reynolds (1993b; 1995), Moin and Bewley (1994), Gad-el-Hak (1994; 1996; 1998), McMichael (1996), Mehregany et al. (1996), Blackwelder (1998), Delville et al. (1998), and Perrier (1998). It is clear that the spatial and temporal resolutions for any probe to be used to resolve high-Reynolds-number turbulent flows are extremely tight. For example, both the Kolmogorov scale and the viscous length-scale change from few microns at the typical field Reynolds number—based on the momentum thickness—of 106 , to a couple of hundred microns at the typical laboratory Reynolds number of 103 . MEMS sensors for pressure, velocity, temperature and shear stress are at least one order of magnitude smaller than conventional sensors (Ho and Tai, 1996; 1998; L¨ofdahl et al., 1996; L¨ofdahl and Gad-elHak, 1999). Their small size improves both the spatial and temporal resolutions of the measurements, typically few microns and few microseconds, respectively. For example, a micro-hot-wire (called hot-point) has very small thermal inertia and the diaphragm of a micro-pressure-transducer has correspondingly fast dynamic response. Moreover, the microsensors’ extreme miniaturization and low energy consumption make them ideal for monitoring the flow state without appreciably affecting it. Lastly, literally hundreds of microsensors can be fabricated on the same silicon chip at a reasonable cost, making them well suited for distributed measurements and control. The UCLA/Caltech team (see, for example, Ho and Tai, 1996; 1998, and references therein) has been very effective in developing many MEMS-based sensors and actuators for turbulence diagnosis and control. The handbook edited by Gad-el-Hak (2006) offers a comprehensive coverage of the broad field of microelectromechanical systems. It is instructive to estimate some representative characteristics of the required array of sensors/actuators. Consider a typical commercial aircraft cruising at a speed of U∞ = 300 m/s and at an altitude of 10 km. The density and kinematic viscosity of air and the unit Reynolds number in this case are, respectively, ρ = 0.4 kg/m3 , ν = 3 × 10−5 m2 /s, and Re = 107 /m. Assume further that the portion of fuselage to be controlled has a turbulent boundary layer characteristics which are identical to those for a zero-pressure-gradient flat plate at a distance of 1 m from the leading edge. In this case, the skin-friction coefficient1 and the friction velocity are, respectively, Cf = 0.003 and uτ = 11.62 m/s. At this location, one viscous wall unit is only ν/uτ = 2.6 microns. In order for the surface array of sensors/actuators to be hydraulically smooth, it should not protrude beyond the viscous sublayer, or 5ν/uτ = 13 µm. Wall-speed streaks are the most visible, reliable and detectable indicators of the preburst turbulence production process. The detection criterion is simply low velocity near the wall, and the actuator response should be to accelerate (or to remove) the lowspeed region before it breaks down. Local wall motion, tangential injection, suction, 1
Note that the skin friction decreases as the distance from the leading increases. It is also strongly affected by such things as the externally imposed pressure gradient. Therefore, the estimates provided in here are for illustration purposes only.
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heating or electromagnetic body force, all triggered on sensed wall-pressure or wallshear stress, could be used to cause local acceleration of near-wall fluid. The recent numerical experiments of Berkooz et al. (1993) indicate that effective control of bursting pair of rolls may be achieved by using the equivalent of two wall-mounted shear sensors. If the goal is to stabilize or to eliminate all low-speed streaks in the boundary layer, a reasonable estimate for the spanwise and streamwise distances between individual elements of a checkerboard array is, respectively, 100 and 1000 wall units,2 or 260 µm and 2600 µm, for our particular example. A reasonable size for each element is probably one-tenth of the spanwise separation, or 26 µm. A (1 m × 1 m) portion of the surface would have to be covered with about n = 1.5 million elements. This is a colossal number, but the density of sensors/actuators could be considerably reduced if we moderate our goal of targeting every single bursting event (and also if less conservative assumptions are used). It is well known that not every low-speed streak leads to a burst. On the average, a particular sensor would detect an incipient bursting event every wall-unit interval of P + = P u2τ /ν = 250, or P = 56 µs. The corresponding dimensionless and dimensional frequencies are f + = 0.004 and f = 18 kHz, respectively. At different distances from the leading edge and in the presence of nonzero-pressure gradient, the sensors/actuators array would have different characteristics, but the corresponding numbers would still be in the same ballpark as estimated in here. As a second example, consider an underwater vehicle moving at a speed of U∞ = 10 m/s. Despite the relatively low speed, the unit Reynolds number is still the same as estimated above for the air case, Re = 107 /m, due to the much lower kinematic viscosity of water. At one meter from the leading edge of an imaginary flat plate towed in water at the same speed, the friction velocity is only uτ = 0.39 m/s, but the wall unit is still the same as in the aircraft example, ν/uτ = 2.6 µm. The density of required sensors/actuators array is the same as computed for the aircraft example, n = 1.5 × 106 elements/m2 . The anticipated average frequency of sensing a bursting event is, however, much lower at f = 600 Hz . Similar calculations have been recently made by Gad-el-Hak (1993; 1994; 1998), Reynolds (1993) and Wadsworth et al. (1993). Their results agree closely with the estimates made here for typical field requirements. In either the airplane or the submarine case, the actuator’s response need not be too large. According to Gad-el-Hak (2000), wall displacement on the order of 10 wall units (26 µm in both examples), suction coefficient of about 0.0006, or surface cooling/heating on the order of 40◦ C/2◦ C (in the first/second example, respectively) should be sufficient to stabilize the turbulent flow. As computed in the two examples above, both the required size for a sensor/actuator element and the average frequency at which an element would be activated are within the presently known capabilities of microfabrication technology. The number of 2
These are equal to, respectively, the average spanwise wavelength between two adjacent streaks and the average streamwise extent for a typical low-speed region. One can argue that those estimates are too conservative: once a region is relaminarized, it would perhaps stay as such for quite a while as the flow convects downstream. The next row of sensors/actuators may therefore be relegated to a downstream location well beyond 1000 wall units. Relatively simple physical or numerical experiments could settle this issue.
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elements needed per unit area is, however, alarmingly large. The unit cost of manufacturing a programmable sensor/actuator element would have to come down dramatically, perhaps matching the unit cost of a conventional transistor,3 before the idea advocated in here would become practical. An additional consideration to the size, amplitude and frequency response is the energy consumed by each sensor/actuator element. Total energy consumption by the entire control system obviously has to be low enough to achieve net savings. Consider the following calculations for the aircraft example. One meter from the leading edge, the skin-friction drag to be reduced is approximately 54 N/m2 . Engine power needed to overcome this retarding force per unit area is 16 kW/m2 , or 104 µW/sensor. If a 60% drag-reduction is achieved,4 this energy consumption is reduced to 4320 µW/sensor. This number will increase by the amount of energy consumption of a sensor/actuator unit, but hopefully not back to the uncontrolled levels. The voltage across a sensor is typically in the range of V = 0.1–1 V, and its resistance in the range of R = 0.1–1 MΩ. This means a power consumption by a typical sensor in the range of P = V 2 /R = 0.1–10 µW, well below the anticipated power savings due to reduced drag. For a single actuator in the form of a spring-loaded diaphragm with a spring constant of k = 100 N/m and oscillating up and down at the bursting frequency of f = 18 kHz with an amplitude of y = 26 microns, the power consumption is P = (1/2) k y 2 f = 600 µW/actuator. If suction is used instead, Cq = 0.0006, and assuming a pressure difference of ∆p = 104 N/m2 across the suction holes/slots, the corresponding power consumption for a single actuator is P = Cq U∞ ∆p/n = 1200 µW/actuator. It is clear then that when the power penalty for the sensor/actuator is added to the lower-level drag, a net saving is still achievable. The corresponding actuator power penalties for the submarine example are even smaller (P = 20 µW/actuator for the wall motion actuator, and P = 40 µW/actuator for the suction actuator), and larger savings are therefore possible.
3 Chaos Control 3.1 Nonlinear Dynamical Systems Theory In the theory of dynamical systems, the so-called butterfly effect denotes sensitive dependence of nonlinear differential equations on initial conditions, with phase-space solutions initially very close together separating exponentially. The solution of nonlinear dynamical systems of three or more degrees of freedom may be in the form of a strange attractor whose intrinsic structure contains a well-defined mechanism to produce a chaotic behavior without requiring random forcing. Chaotic behavior is complex, aperiodic and, though deterministic, appears to be random. A question arises naturally: just as small disturbances can radically grow within a deterministic system to yield rich, unpredictable behavior, can minute adjustments to 3
4
The transistor was invented in 1947. In the mid 1960s, a single transistor sold for around $70. In 1997, Intel’s Pentium II processor (microchip) contained 7.5 × 106 transistors and cost around $500, that is less than $0.00007 per transistor! A not-too-farfetched goal according to the selective suction results discussed earlier.
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a system parameter be used to reverse the process and control, i.e. regularize, the behavior of a chaotic system? Recently, that question was answered in the affirmative theoretically as well as experimentally, at least for system orbits which reside on lowdimensional strange attractors (see the review by Lindner and Ditto, 1995). Before describing such strategies for controlling chaotic systems, we first summarize the recent attempts to construct a low-dimensional dynamical systems representation of turbulent boundary layers. Such construction is a necessary first step to be able to use chaos control strategies for turbulent flows. Additionally, as argued by Lumley (1996), a lowdimensional dynamical model of the near-wall region used in a Kalman filter (Banks, 1986; Petersen and Savkin, 1999) can make the most of the partial information assembled from a finite number of wall sensors. Such filter minimizes in a least square sense the errors caused by incomplete information, and thus globally optimizes the performance of the control system. Boundary layer turbulence is described by a set of nonlinear partial differential equations and is characterized by an infinite number of degrees of freedom. This makes it rather difficult to model the turbulence using a dynamical systems approximation. The notion that a complex, infinite-dimensional flow can be decomposed into several low-dimensional subunits is, however, a natural consequence of the realization that quasi-periodic coherent structures dominate the dynamics of seemingly random turbulent shear flows. This implies that low-dimensional, localized dynamics can exist in formally infinite-dimensional extended systems—such as open turbulent flows. Reducing the flow physics to finite-dimensional dynamical systems enables a study of its behavior through an examination of the fixed points and the topology of their stable and unstable manifolds. From the dynamical systems theory viewpoint, the meandering of low-speed streaks is interpreted as hovering of the flow state near an unstable fixed point in the low-dimensional state space. An intermittent event that produces high wall stress—a burst—is interpreted as a jump along a heteroclinic cycle to different unstable fixed point that occurs when the state has wandered too far from the first unstable fixed point. Delaying this jump by holding the system near the first fixed point should lead to lower momentum transport in the wall region and, therefore, to lower skin-friction drag. Reactive control means sensing the current local state and through appropriate manipulation keeping the state close to a given unstable fixed point, thereby preventing further production of turbulence. Reducing the bursting frequency by say 50%, may lead to a comparable reduction in skin-friction drag. For a jet, relaminarization may lead to a quiet flow and very significant noise reduction. In one significant attempt the proper orthogonal, or Karhunen-Lo`eve, decomposition method has been used to extract a low-dimensional dynamical system from experimental data of the wall region (Aubry et al. 1988; Aubry, 1990). Aubry et al. (1988) expanded the instantaneous velocity field of a turbulent boundary layer using experimentally determined eigenfunctions which are in the form of streamwise rolls. They expanded the Navier–Stokes equations using these optimally chosen, divergence-free, orthogonal functions, applied a Galerkin projection, and then truncated the infinitedimensional representation to obtain a ten-dimensional set of ordinary differential equations. These equations represent the dynamical behavior of the rolls, and are shown to exhibit a chaotic regime as well as an intermittency due to a burst-like phenomenon.
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However, Aubry et al.’s ten-mode dynamical system displays a regular intermittency, in contrast both to that in actual turbulence as well as to the chaotic intermittency encountered by Pomeau and Manneville (1980) in which event durations are distributed stochastically. Nevertheless, the major conclusion of Aubry et al.’s study is that the bursts appear to be produced autonomously by the wall region even without turbulence, but are triggered by turbulent pressure signals from the outer layer. More recently, Berkooz et al. (1991) generalized the class of wall-layer models developed by Aubry et al. (1988) to permit uncoupled evolution of streamwise and cross-stream disturbances. Berkooz et al.’s results suggest that the intermittent events observed in Aubry et al.’s representation do not arise solely because of the effective closure assumption incorporated, but are rather rooted deeper in the dynamical phenomena of the wall region. The book by Holmes et al. (1996) details the Cornell research group attempts at describing turbulence as a low-dimensional dynamical system. In addition to the reductionist viewpoint exemplified by the work of Aubry et al. (1988) and Berkooz et al. (1991), attempts have been made to determine directly the dimension of the attractors underlying specific turbulent flows. Again, the central issue here is whether or not turbulent solutions to the infinite-dimensional Navier–Stokes equations can be asymptotically described by a finite number of degrees of freedom. Grappin and L´eorat (1991) computed the Lyapunov exponents and the attractor dimensions of two- and three-dimensional periodic turbulent flows without shear. They found that the number of degrees of freedom contained in the large scales establishes an upper bound for the dimension of the attractor. Deane and Sirovich (1991) and Sirovich and Deane (1991) numerically determined the number of dimensions needed to specify chaotic Rayleigh-B´enard convection over a moderate range numbers, Ra. of Rayleigh They suggested that the intrinsic attractor dimension is O Ra2/3 . The corresponding dimension in wall-bounded flows appears to be dauntingly high. Keefe et al. (1992) determined the dimension of the attractor underlying turbulent Poiseuille flows with spatially periodic boundary conditions. Using a coarse-grained numerical simulation, they computed a lower bound on the Lyapunov dimension of the attractor to be approximately 352 at a pressure-gradient Reynolds number of 3200. Keefe et al. (1992) argue that the attractor dimension in fully-resolved turbulence is unlikely to be much larger than 780. This suggests that periodic turbulent shear flows are deterministic chaos and that a strange attractor does underlie solutions to the Navier– Stokes equations. Temporal unpredictability in the turbulent Poiseuille flow is thus due to the exponential spreading property of such attractors. Although finite, the computed dimension invalidates the notion that the global turbulence can be attributed to the interaction of a few degrees of freedom. Moreover, in a physical channel or boundary layer, the flow is not periodic and is open. The attractor dimension in such case is not known but is believed to be even higher than the estimate provided by Keefe et al. for the periodic (quasi-closed) flow. In contrast to closed, absolutely unstable flows, such as Taylor-Couette systems, where the number of degrees of freedom can be small, local measurements in open, convectively unstable flows, such as boundary layers, do not express the global dynamics, and the attractor dimension in that case may inevitably be too large to be determined experimentally. According to the estimate provided by Keefe et al. (1992), the colossal
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data required (about 10D , where D is the attractor dimension) for measuring the dimension simply exceeds current computer capabilities. Turbulence near transition or near a wall is an exception to that bleak picture. In those special cases, a relatively small number of modes are excited and the resulting simple turbulence can therefore be described by a dynamical system of a reasonable number of degrees of freedom. 3.2 Chaos Control There is another question of greater relevance here. Given a dynamical system in the chaotic regime, is it possible to stabilize its behavior through some kind of active control? While other alternatives have been devised (e.g., Fowler, 1989; H¨ubler and L¨uscher, 1989; Huberman, 1990; Huberman and Lumer, 1990), the recent method proposed by workers at the University of Maryland (Ott et al., 1990a; 1990b; Shinbrot et al., 1990; 1992a; 1992b; 1992c; 1998; Romeiras et al., 1992) promises to be a significant breakthrough. Comprehensive reviews and bibliographies of the emerging field of chaos control can be found in the articles by Shinbrot et al. (1993), Shinbrot (1993; 1995; 1998), and Lindner and Ditto (1995). Ott et al. (1990a) demonstrated, through numerical experiments with the Henon map, that it is possible to stabilize a chaotic motion about any pre-chosen, unstable orbit through the use of relatively small perturbations. The procedure consists of applying minute time-dependent perturbations to one of the system parameters to control the chaotic system around one of its many unstable periodic orbits. In this context, targeting refers to the process whereby an arbitrary initial condition on a chaotic attractor is steered toward a prescribed point (target) on this attractor. The goal is to reach the target as quickly as possible using a sequence of small perturbations (Kostelich et al., 1993a). The success of the Ott-Grebogi-Yorke’s (OGY) strategy for controlling chaos hinges on the fact that beneath the apparent unpredictability of a chaotic system lies an intricate but highly ordered structure. Left to its own recourse, such a system continually shifts from one periodic pattern to another, creating the appearance of randomness. An appropriately controlled system, on the other hand, is locked into one particular type of repeating motion. With such reactive control the dynamical system becomes one with a stable behavior. The OGY-method can be simply illustrated by the schematic in Figure 3. The state of the system is represented as the intersection of a stable manifold and an unstable one. The control is applied intermittently whenever the system departs from the stable manifold by a prescribed tolerance, otherwise the control is shut off. The control attempts to put the system back onto the stable manifold so that the state converges toward the desired trajectory. Unmodeled dynamics cause noise in the system and a tendency for the state to wander off in the unstable direction. The intermittent control prevents that and the desired trajectory is achieved. This efficient control is not unlike trying to balance a ball in the center of a horse saddle (Moin and Bewley, 1994). There is one stable direction (front/back) and one unstable direction (left/right). The restless horse is the unmodeled dynamics, intermittently causing the ball to move in the wrong direction. The OGY-control needs only be applied, in the most direct manner possible, whenever the ball wanders off in the left/right direction.
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Desired trajectory
Unstable manifold
Stable manifold
Fig. 3. The OGY method for controlling chaos
The OGY-method has been successfully applied in a relatively simple experiment by Ditto et al. (1990) and Ditto and Pecora (1993) at the Naval Surface Warfare Center, in which reverse chaos was obtained in a parametrically driven, gravitationally buckled, amorphous magnetoelastic ribbon. Garfinkel et al. (1992) applied the same control strategy to stabilize drug-induced cardiac arrhythmias in sections of a rabbit ventricle. Other extensions, improvements and applications of the OGY-strategy include higher-dimensional targeting (Auerbach et al., 1992; Kostelich et al., 1993b), controlling chaotic scattering in Hamiltonian (i.e., nondissipative, area conservative) systems (Lai et al., 1993a; 1993b), synchronization of identical chaotic systems that govern communication, neural or biological processes (Lai and Grebogi, 1993), use of chaos to transmit information (Hayes et al., 1994a; 1994b), control of transient chaos (Lai et al., 1994), and taming spatio-temporal chaos using a sparse array of controllers (Chen et al., 1993; Qin et al., 1994; Auerbach, 1994). In a more complex system, such as a turbulent boundary layer, there exist numerous interdependent modes and many stable as well as unstable manifolds (directions). The flow can then be modeled as coherent structures plus a parameterized turbulent background. The proper orthogonal decomposition (POD) is used to model the coherent part because POD guarantees the minimum number of degrees of freedom for a given model accuracy. Factors that make turbulence control a challenging task are the potentially quite large perturbations caused by the unmodeled dynamics of the flow, the non-stationary nature of the desired dynamics, and the complexity of the saddle shape describing the dynamics of the different modes. Nevertheless, the OGY-control strategy has several advantages that are of special interest in the control of turbulence: (1) the mathematical model for the dynamical system need not be known; (2) only small changes in the control parameter are required; and (3) noise can be tolerated (with appropriate penalty). Recently, Keefe (1993a; 1993b) made a useful comparison between two nonlinear control strategies as applied to fluid problems. Ott-Grebogi-Yorke’s feedback method
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described above and the model-based control strategy originated by H¨ubler (see, for example, H¨ubler and L¨uscher, 1989; L¨uscher and H¨ubler, 1989), the H-method. Both novel control methods are essentially generalizations of the classical perturbation cancellation technique: apply a prescribed forcing to subtract the undesired dynamics and impose the desired one. The OGY-strategy exploits the sensitivity of chaotic systems to stabilize existing periodic orbits and steady states. Some feedback is needed to steer the trajectories toward the chosen fixed point, but the required control signal is minuscule. In contrast, H¨ubler’s scheme does not explicitly make use of the system sensitivity. It produces general control response (periodic or aperiodic) and needs little or no feedback, but its control inputs are generally large. The OGY-strategy exploits the nonlinearity of a dynamical system; indeed the presence of a strange attractor and the extreme sensitivity of the dynamical system to initial conditions are essential to the success of the method. In contrast, the H-method works equally for both linear and nonlinear systems. Keefe (1993a) first examined numerically the two schemes as applied to fullydeveloped and transitional solutions of the Ginzburg-Landau equation, an evolution equation that governs the initially weakly nonlinear stages of transition in several flows and that possesses both transitional and fully-chaotic solutions. The Ginzburg-Landau equation has solutions that display either absolute or convective instabilities, and is thus a reasonable model for both closed and open flows. Keefe’s main conclusion is that control of nonlinear systems is best obtained by making maximum use possible of the underlying natural dynamics. If the goal dynamics is an unstable nonlinear solution of the equation and the flow is nearby at the instant control is applied, both methods perform reliably and at low-energy cost in reaching and maintaining this goal. Predictably, the performance of both control strategies degrades due to noise and the spatially discrete nature of realistic forcing. Subsequently, Keefe (1993b) extended the numerical experiment in an attempt to reduce the drag in a channel flow with spatially periodic boundary conditions. The OGY-method reduces the skin friction to 60–80% of the uncontrolled value at a mass-flux Reynolds number of 4408. The H-method fails to achieve any drag reduction when starting from a fully-turbulent initial condition but shows potential for suppressing or retarding laminar-to-turbulence transition. Keefe (1993a) suggests that the H-strategy might be more appropriate for boundary layer control, while the OGYmethod might best be used for channel flows. It is also relevant to note here the work of Bau and his colleagues at the University of Pennsylvania (Singer et al., 1991; Wang et al., 1992), who devised a feedback control to stabilize (relaminarize) the naturally occurring chaotic oscillations of a toroidal thermal convection loop heated from below and cooled from above. Based on a simple mathematical model for the thermosyphon, Bau and his colleagues constructed a reactive control system that was used to alter significantly the flow characteristics inside the convection loop. Their linear control strategy, perhaps a special version of the OGY’s chaos control method, consists simply of sensing the deviation of fluid temperatures from desired values at a number of locations inside the thermosyphon loop and then altering the wall heating either to suppress or to enhance such deviations. Wang et al. (1992) also suggested extending their theoretical and experimental method to more complex situations such as those involving B´enard convection (Tang and Bau, 1993a; 1993b). Hu and Bau (1994) used a similar feedback control strategy to demonstrate that
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the critical Reynolds number for the loss of stability of planar Poiseuille flow can be significantly increased or decreased. Other attempts to use low-dimensional dynamical systems representation for flow control include the work of Berkooz et al. (1993), Corke et al. (1994), and Coller et al. (1994a; 1994b). Berkooz et al. (1993) applied techniques of modern control theory to estimate the phase-space location of dynamical models of the wall-layer coherent structures, and used these estimates to control the model dynamics. Since discrete wallsensors provide incomplete knowledge of phase-space location, Berkooz et al. maintain that a nonlinear observer, which incorporates past information and the equations of motion into the estimation procedure, is required. Using an extended Kalman filter, they achieved effective control of a bursting pair of rolls with the equivalent of two wallmounted shear sensors. Corke et al. (1994) used a low-dimensional dynamical system based on the proper orthogonal decomposition to guide control experiments for an axisymmetric jet. By sensing the downstream velocity and actuating an array of miniature speakers located at the lip of the jet, their feedback control succeeded in converting the near-field instabilities from spatial-convective to temporal-global. Coller et al. (1994a; 1994b) developed a feedback control strategy for strongly nonlinear dynamical systems, such as turbulent flows, subject to small random perturbations that kick the system intermittently from one saddle point to another along heteroclinic cycles. In essence, their approach is to use local, weakly nonlinear feedback control to keep a solution near a saddle point as long as possible, but then to let the natural, global nonlinear dynamics run its course when bursting (in a low-dimensional model) does occur. Though conceptually related to the OGY-strategy, Coller et al.’s method does not actually stabilize the state but merely holds the system near the desired point longer than it would otherwise stay. Shinbrot and Ottino (1993a; 1993b) offer yet another strategy presumably most suited for controlling coherent structures in area-preserving turbulent flows. Their geometric method exploits the premise that the dynamical mechanisms which produce the organized structures can be remarkably simple. By repeated stretching and folding of “horseshoes” which are present in chaotic systems, Shinbrot and Ottino have demonstrated numerically as well as experimentally the ability to create, destroy and manipulate coherent structures in chaotic fluid systems. The key idea to create such structures is to intentionally place folds of horseshoes near low-order periodic points. In a dissipative dynamical system, volumes contract in state space and the co-location of a fold with a periodic point leads to an isolated region that contracts asymptotically to a point. Provided that the folding is done properly, it counteracts stretching. Shinbrot and Ottino (1993a) applied the technique to three prototypical problems: a one-dimensional chaotic map; a two-dimensional one; and a chaotically advected fluid. Shinbrot (1995; 1998) and Shinbrot et al. (1998) provide recent reviews of the stretching/folding as well as other chaos control strategies.
4 Conclusions In the present chapter, I have emphasized the frontiers of the field of flow control, reviewing the important advances that took place during the past few years and providing
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a blueprint for future progress. In two words, the future of flow control is in taming turbulence by targeting its coherent structures: reactive control. Recent developments in chaos control, microfabrication and soft computing tools are making it more feasible to perform reactive control of turbulent flows to achieve drag reduction, lift enhancement, mixing augmentation and noise suppression. Field applications, however, have to await further progress in those three modern areas. Other less complex control schemes, passive as well as active, are more market ready and are also witnessing resurgence of interest. The outlook for reactive control is quite optimistic. Soft computing tools and nonlinear dynamical systems theory are developing at fast pace. MEMS technology is improving even faster. The ability of Texas Instruments to produce an array of one million individually addressable mirrors for around 0.01 cent per actuator is a foreteller of the spectacular advances anticipated in the near future. Existing automotive applications of MEMS have already proven the ability of such devices to withstand the harsh environment under the hood. For the first time, targeted control of turbulent flows is now in the realm of the possible for future practical devices. What is needed now is a focused, well-funded research and development program to make it all come together for field application of reactive flow control systems. In parting, it may be worth recalling that a mere 10% reduction in the total drag of an aircraft translates into a saving of $3 billion in annual fuel cost (at 2005 prices) for the commercial fleet of aircraft in the United States alone. Contrast this benefit to the annual worldwide expenditure of perhaps a few million dollars for all basic research in the broad field of flow control. Taming turbulence, though arduous, will pay for itself in gold. Reactive control as difficult as it seems, is neither impossible nor a pie in the sky. Beside, lofty goals require strenuous efforts. Easy solutions to difficult problems are likely to be wrong as Henry Louis Mencken once lamented.
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Electromagnetic Control of Separated Flows Using Periodic Excitation with Different Wave Forms Christian Cierpka, Tom Weier, and Gunter Gerbeth Forschungszentrum Rossendorf, P.O. Box 510119, 01314 Dresden, Germany
[email protected] http://www.fz-rossendorf.de/pls/rois/Cms?pNid=226
Summary Time periodic Lorentz forces have been used to influence the separated flow on an inclined flat plate in deep stall at a Reynolds number of 104 . The influence of the control parameters effective momentum coefficient and excitation frequency as well as excitation wave form is discussed based on phase averaged PIV measurements. As expected, control authority depends strongly on momentum input and excitation frequency, but effects of the excitation wave form can be shown as well.
1 Introduction Owing to its technological importance, flow separation and its control is a persistent topic of fluid dynamic research. A comprehensive and recent review can be found in [1]. While steady blowing is a tool investigated for almost eight decades, separation control by periodic addition of momentum has been a subject of intense research only since the early 1990s. Its most striking feature is that a control goal, e.g. a specific lift increase, can typically be attained by orders of magnitude smaller momentum input compared to steady actuation [2]. Control authority depends mainly on time averaged momentum input and excitation frequency. Greenblatt and Wygnanski [3] and more recently Seifert et al. [4] reviewed the state of the art of “active flow control”, a term now commonly used for periodic excitation. If the fluid is electrically conducting, like seawater or ionized air, momentum addition, usually accomplished by imposing mass fluxes, can be achieved as well by electromagnetic, i.e. Lorentz forces. The Lorentz force density F appears as a body force term on the right hand side of the Navier–Stokes–Equation for incompressible flow p F ∂u + (u · ∇)u = − + ν∇2 u + , ∂t ρ ρ
(1)
where u denotes the velocity, p the pressure, and t the time, respectively. ρ is the density and ν the kinematic viscosity of the fluid. As can be seen from (1), the electromagnetic force density acts as a momentum source for the flow. The Lorentz force density itself is the vector product of a current density j and a magnetic induction B F = j × B. R. King (Ed.): Active Flow Control, NNFM 95, pp. 27–41, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
(2)
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Ohm’s law j = σ(E + u × B)
(3)
for moving conductors describes the current density. E denotes an electric field, and σ the electrical conductivity of the fluid. In liquid metal Magnetohydrodynamics (MHD), the Lorentz force density and the flow are usually strongly coupled, since the flow induces currents via the u × B term in (3), these currents generate Lorentz forces (2), and the Lorentz forces change the flow (1). The reason for this strong coupling is the very high electrical conductivity of liquid metals, typically σ = O(106 ) S/m. In the case of seawater or other electrolytes, σ is small (∼ 10 S/m). Therefore, the induced currents are very low for moderate applied magnetic fields (B0 ∼ 1 T). Accordingly, the Lorentz forces due to these currents are negligible and too weak to act on the flow. In order to generate forces large enough to influence the flow, an additional electric field has to be applied. The ratio of the applied E0 to the electric field induced by the free stream velocity U∞ in the presence of the applied magnetic field B0 is commonly termed load factor [5] φ=
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For seawater flow control with moderate magnetic fields, it follows from the above φ 1.
(5)
This implies on one hand that the force density distribution can be calculated independently of the flow field. On the other hand, a large load factor means a small efficiency of momentum generation since the ratio of mechanical (∼ jU∞ B0 ) to electrical (∼ jE0 ) power is the reciprocal of φ. Strictly speaking, the flow acts not only on the electric field distribution, but deforms the magnetic field as well. However, even in the case of most liquid metal MHD problems – apart from dynamo experiments – it is well justified to ignore the induced magnetic fields when determining the generated Lorentz force. Despite its low energetical efficiency, the Lorentz force has several appealing features qualifying it as an interesting actuator for basic research: momentum is directly generated in the fluid without associated mass flux, the frequency response of the actuation is practically unlimited, no moving parts are involved. First investigations of the electromagnetic control of electrolyte flows date back to the 1950’s. Already in 1954, Crausse and Cachon [6] gave experimental evidence of successful separation postponement as well as separation provocation on a half cylinder. Similar experiments were performed later by Lielausis [7]. After a few investigations on laminar flow control [8,9] and related topics [10,11], activities in that field declined with the beginning 1970’s. A renewed interest in electromagnetic flow control arose in the 1990’s. The majority of publications concentrated on the topic of turbulent skin friction reduction, e.g. [12,13,14,15,16]. Compared to skin friction reduction, the use of Lorentz forces to control flow separation received less attention. A circular cylinder equipped with electrodes as well as permanent magnets generating a wall
Electromagnetic Control of Separated Flows Using Periodic Excitation
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parallel force in streamwise direction was used in the experiments and numerical calculations of Weier et al. [17]. Similar configurations were investigated later by Kim and Lee [18], Posdziech and Grundmann [19], and Chen and Aubry [20]. Results on separation prevention on hydrofoils using steady Lorentz forces have been reported by Weier et al. [21]. Weier and Gerbeth [22] examined time periodic Lorentz forces to control the separated flow around a NACA 0015 at chord length Reynolds numbers 5.2 × 104 < Re < 1.5×105. Essential features like charakteristic efficient excitation frequencies, effective momentum coefficients and resulting lift gain compare well to that found with alternative methods for periodic addition of momentum. Of special interest for the present paper is the influence of different excitation wave forms on the resulting lift and drag of the hydrofoil. Force balance measurements from [22] demonstrate clearly that under otherwise identical conditions different excitation wave forms can change the attainable lift by up to 70%. A further finding of interest by Weier and Gerbeth [22] is the scaling of the lift increase with the peak instead of the effective momentum coefficient under certain conditions. These results motivated the present paper which aims by help of particle image velocimetry (PIV) measurements to further investigate excitation wave form effects. This topic is of potential interest as well for active flow control by means other than electromagnetic forces, since using similar nondimensionalized excitation frequencies and momentum coefficients, different excitation methods achieve comparable results [3,22]. Up to now, the influence of the excitation wave form has received comparably little attention. Bouras et al. [23] used an elaborate device to apply pulsations of sinusoidal, triangular and square wave forms to the leading edge slot of a lambda wing. However, the authors found no significant effect of the pulsation wave form. Piezoelectric actuators are often used in active flow control experiments. Their utility is yet limited to a narrow frequency band around their resonance frequency. To partly overcome this limitation, Wiltse and Glezer [24] applied amplitude modulated driving signals. This technique has later been used by a number of researchers to control separated flows (e.g. [25,26,27,28]). Margalit et al. [25] report a distinct effect of the modulation wave form. Traditionally, the rms value of the excitation is used in active flow control applications to characterize momentum input [3]. However, supporting the above mentioned lift gain scaling with the peak momentum coefficient [22], alternative approaches are discussed as well. In reference to Chang et al. [29], Wu et al. [30] used the peak excitation velocity in the definition of their momentum coefficient. Recently, Kiedaisch et al. [31] proposed the ratio of peak excitation velocity to local flow velocity as a dominant parameter.
2
Experimental Setup and Parameters
Gailitis and Lielausis [8] and Rice [32] proposed the arrangement of flush mounted electrodes and permanent magnets shown in Fig. 1 to generate a wall parallel Lorentz force. Magnets and electrodes are of the same width a, a condition maximizing the attainable integral force density [33]. Apart from end effects, both electric as well as magnetic
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fields have only components in wall normal (y) and spanwise (z) direction. From the vector product (2) follows that the Lorentz force possesses a streamwise (x) component Fx only. The force density distribution shows non–uniformities in z–direction in the range of 0 ≤ y a [34]. Averaged over z, the mean force density decreases exponentially with increasing wall distance Fx =
π π j0 M0 e− a y . 8
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M0 denotes the magnetization of the permanent magnets and j0 the applied current density, respectively. The magnetic induction at the surface of the magnetic poles can be calculated from the geometry of the magnets and their magnetization. For the experiments described here, the electrode/magnet–array sketched in Fig. 1 has been mounted to the leading edge at the suction side of an inclined flat plate as sketched in Fig. 2. The body of the plate consists of polyvinyl chloride (PVC). It has a circular leading edge, a span width of 140 mm, a chord length of 130 mm, and a thickness of 10 mm. Shape and material were chosen as to allow for ease of manufacturing and durability in the electrolyte solution (0.25 M NaOH). Both, magnets as well as electrodes, have a quadratic surface with an edge length of a = 5 mm. At the surface of the magnets, a mean magnetic induction of B0 = 0.35 T was determined. A high power amplifier FM 1295 from FM Elektronik Berlin has been used to feed the electrodes. It was driven by a frequency generator 33220A from Agilent. The applied Lorentz force strength is given as an effective momentum coefficient T 1 aB 1 l 0 cµ = · · · j(t)2 dt (7) 2 2 ρU∞ c T 0
relating the rms momentum added by the Lorentz force to that in the free stream. In (7) j(t) denotes the time–depended current density, T its period of oscillation, and l the length of the actuator, respectively. For the plate investigated here l equals a. As common practice [3], we use percentage terms for cµ for convenience. A reduced frequency
Electromagnetic Control of Separated Flows Using Periodic Excitation
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fe c U∞
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characterizes the time dependency of the momentum input. fe = 1/T denotes the excitation frequency. The chord length c was chosen as a characteristic length in (8) since in our case it is in a good approximation identical to the distance from the excitation location to the trailing edge. The PIV measurements reported in the following were done in a small electrolyte channel. This channel is driven by a centrifugal pump. A settling chamber with a free surface and equipped with a filter pad, two honeycombs and a set of four screens results nevertheless in a relatively small turbulence level in the test section. The latter, featuring a free surface as well, is 1 m long and has a 0.2×0.2 m cross section. For more details we refer to [35]. The channel has been operated with a mean velocity of U∞ = 8 cm s−1 resulting in a chord length Reynolds number of Re = 1.04 × 104 for the plate. It has been mounted between rectangular endplates made from PMMA extending from the bottom of the test section to the free surface in vertical and from 3 cm in front of the leading edge to 3 cm behind the trailing edge in horizontal direction. The angle of attack of the plate was kept constant at α = 16◦ throughout the measurements. The PIV setup consists of a Spectra Physics continuous wave Ar+ –Laser type 2020–5 as light source and a Photron Fastcam 1024PCI 100K for recording the images. A light sheet, formed by two cylindrical lenses, was placed at mid–span of the plate extending in the direction of the flow (x) and normal to the test section bottom wall (y). The flow was seeded with polyamide particles of 25 µm mean diameter. In the plane of the light sheet, x and y velocity components have been calculated from the images using PIVview–2C 2.3 from PivTec. The camera was operated at 60 Hz frame rate, while the single image exposure time was set to 2 ms by the camera shutter. For each configuration a total of 6400 single images of 1024×512 pixel2 have been recorded synchronized to the excitation signal. Each image was correlated with its successor using multigrid interrogation with a final window size of 32×32 pixel2 and 50% overlap, image deformation and sub pixel shifting.
3
Results
3.1 Influence of the Excitation Frequency As noted above, a crucial role for the characteristics of the flow plays the excitation frequency. Although different observations exist, e.g. [36], it is widely recognized that excitation with a dimensionless frequency F + = O(1) has the greatest effect in re– establishing the lift of a stalled airfoil, at least for low to moderate momentum input [3]. In Fig. 3 the mean values for the streamwise velocity for sinusoidal excitation at F + = 0.5, 1, 2 and 4 are shown. As an indicator for the size of the separated region, areas with streamwise back flow, depicted by the dashed lines, can be used. As clearly visible, F + = 1 results in the smallest separation region, whereas for frequencies F + > 1 and F + < 1 its area increases significantly. This effect can possibly be explained by the different vortex structures resulting from forcing at different frequencies. On the left hand side in Fig. 4
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the phased averaged vorticity distribution for F + = 0.5 and 4 is given. Phase averaging of the synchronized PIV data was performed based on the excitation frequency. One period has been split into 20 equally wide bins. In the first half period, the Lorentz force points downstream, while it changes sign in the second half period. For the lower excitation frequency F + = 0.5, no discernible vortex structures can be found in Fig. 3. Instead, a region of higher vorticity indicates the shear layer, which originates at the leading edge and, instead of developing straight in downstream direction, is bended by the upstream pointing Lorentz force.
Electromagnetic Control of Separated Flows Using Periodic Excitation
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For the excitation with higher frequency several distinct regions of high vorticity appear within the shear layer above the front part of the plate. As the plot of the λ2 criteria, see [37] for definition and discussion, indicates, these regions are small and compact vortices, which loose their identity rapidly during downstream advection. Phase averaged vorticity and λ2 plots for forcing at the “optimum” frequency F + = 1 are part of Fig. 9 and Fig. 10, respectively. During most of the forcing period, two pronounced vortices are present on the suction side. It may be inferred from the averaged velocity contours in Fig. 3 and the phase averaged measurements discussed above that the momentum exchange between the separated region and the outer flow is most intensive for forcing at F + = 1. For the lower excitation frequency F + = 0.5, no discernible vortices are formed, while at the higher excitation frequency of F + = 4 small and compact vortices originate in the actuator region. However, due to the high excitation frequency, these structures are small. During their downstream advection, they neither grow markedly, nor do they merge. On the contrary, they dissipate quickly and are not any more detectable above the downstream half of the plate in the λ2 plot of Fig. 3. Consequently, they are not able to transfer much momentum from the free stream to the separated region. 3.2 Influence of the Excitation Amplitude In Fig. 5 the phase averaged flow fields for sinusoidal excitation at F + = 1 are shown for cµ = 1.3% and cµ = 5.2%. For the larger momentum coefficient, significant effects on the flow field can be observed. Due to the strong acceleration in the actuator region during the phase with downstream Lorentz force, the fluid follows the contour of the plate beginning at the leading edge. The vortices produced are comparably strong, the flow inbetween them is attached to the plate. In case of the lower momentum coefficient, the structures forming at the leading edge do not penetrate the separated region down to the surface of the plate. Their effect is limited to the shear layer region. Fig. 6 displays vorticity and λ2 contours for F + = 1 at the two momentum coefficients discussed above. As can be seen from the vorticity plot for cµ = 1.3%, the initial shear layer extends relatively far downstream. It is rolled up in the phase with upstream Lorentz force to an elongated vortex structure which dissipates quickly. For the higher momentum coefficient of cµ = 5.2%, the initial shear layer is considerably shorter, its roll up happens faster and more close to the leading edge. The vortex structures contain more concentrated vorticity and therefore have a longer lifetime than these generated with lower momentum coefficients. Presumably, the strong and compact vortex structures remain discernible vortices in the wake of the plate. Due to their strength, they will cause an intensive momentum exchange between freestream and separated region. Therefore, and due to the partial reattachement in the phase with downstream Lorentz force, the mean flow reattaches completely to the plate for cµ = 5.2% (not shown). 3.3 Influence of the Excitation Wave Form Fig. 7 shows contours of the mean flow component in streamwise direction u for the baseline flow and excitation with F + = 1 and cµ = 2.6% and different wave forms. At the low Reynolds number of Re = 1.04 · 104 a flat plate will experience leading edge
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stall already at inclination angles of about α = 5◦ [38]. Consequently, the plate is in deep stall at α = 16◦ and the large separation zone on the suction side shown in Fig. 7 results. This recirculation region extends downstream far behind the trailing edge with the maximum of backflow situated in the aft position of the plate and rearward of the trailing edge. While in all cases with excitation, separation is obviously still present, the structure of the separated region is markedly modified. First, its size is strongly reduced, and second, the region with maximum backflow is shifted towards the leading edge. The thin area of attached flow directly at the plate is due to the electrolysis bubbles generated at the electrodes, which are moved upwards by buoyancy. They carry along with them the wall near fluid. This happens, since the plate is mounted top down in the channel, while it has been flipped over for the graphs. Different excitation wave forms under otherwise identical conditions result in separated regions of different extension and shape. Excitation with a rectangular wave form generates the smallest separation bubble. Stronger backflow is localized in a relatively small region downstream of the leading edge. For excitation with a sinusoidal wave form, the separation bubble grows in downstream direction. As well, the region with stronger backflow is lengthened. Finally, excitation with a triangular wave form shows the largest separation bubble and consequently the greatest extend of the region with stronger backflow. The latter extends now almost to the trailing edge. Fig. 8 displays power spectra of the v–component of the velocity versus frequency normalized by the excitation frequency of 0.6Hz. The position of the points A, B, C, and D is denoted in Fig. 7. To compute the power spectra, time series of the velocity signal have been extracted from the PIV measurements at the respective points. The initial spectrum of the baseline flow directly behind the leading edge is very noisy. Subsequently, it develops a broadband peak with a maximum at 2.2 Hz in point B. Further downstream, larger vortex structures corresponding to lower frequencies dominate the flow, consequently the peak frequency shifts to lower values of 1.2 Hz in point C and 1 Hz in point D. The
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spectrum in point A under excitation with a rectangular wave form shows a sharp peak at the excitation frequency and lower but not less sharp peaks at their odd harmonics. These odd harmonics correspond to the Fourier coefficients of the rectangular signal. Their peak height decreases rapidly with increasing distance from the point of excitation. Only the 3rd and 5th harmonic are still discernible at point B, while in points C and D the excitation frequency alone dominates the power spectrum. Excitation with a pure sine wave results in a pronounced peak at the excitation frequency at all points, the same holds true for excitation with a triangular wave form. There, higher harmonics as in the case of the square wave could be expected, but the Fourier coefficients are much smaller in case of the triangle wave and therefore probably not resolved by the measurements. Investigating the excitation by Lorentz forces includes a wide range of frequencies and momentum coefficient as well as the freedom of the excitation wave form. For the present study sinusoidal, triangular and rectangular excitation was investigated. Keeping the integral of the applied Lorentz force, defined √ with the rms of the current density (7), = 2 jrec for excitation by sinusoidal and to constant, the peak value changes to j sin √ jtri = 3 jrec for excitation by triangular wave forms. In Fig. 9 the phase averaged vorticity distributions and in Fig. 10 the λ2 -criteria for the different wave forms are shown.
Electromagnetic Control of Separated Flows Using Periodic Excitation
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Fig. 9. Phased averaged values of the vorticity ωz for F + = 1, cµ = 2.6% for excitation with rectangular (left), sinusoidal (middle), and triangular wave forms (right)
For the excitation with the rectangular wave form, the flow at the leading edge is accelerated constantly in streamwise direction during the first half period. The shear layer has already rolled into a vortex like structure and due to the change from an upstream force to a downstream one, a small vortex separates, moves downstream and merges with the previously shed vortex. The shear layer starts again to grow and is lifted from the surface of the plate during the second half period. During number 13, 14, and 15 of the 20 intervals used for phase averaging, another merging of small vortices near the leading edge takes place, which is not shown here due to space limitations. The square wave excitation produces large structures of relatively high vorticity. Their size and energy remains roughly constant while they are convected downstream towards the trailing edge. Due to the jump from a constant positive to a constant negative Lorentz force the formation of compact and long living vortex structures is favoured. The vortex formations resulting from triangular and sinusoidal excitation wave forms are relatively similar to each other, but different from the vortices produced by square wave excitation. Especially the roll up of the shear layer into larger vortex structures
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occurs differently. Due to the much more gentle time dependence of the force, the shear layer is gradually wind up into a large vortex during the period with upstream pointing Lorentz force. This large vortex leaves the region near the leading edge, if the Lorentz force changes sign and moves the structure downstream. Only one large vortex is shed per period. In contrast to square wave excitation, no vortex amalgamation takes place for sine and triangle waves. However, a remarkable difference between sine and triangle wave excitation remains: the vortex structures forming under triangular excitation contain more vorticity and have a longer lifetime than that produced by sine wave forcing.
4 Conclusions Time periodic Lorentz forces have proven to be a viable tool to control separated flows. The Lorentz force actuator provides a great flexibility to choose excitation frequencies,
Electromagnetic Control of Separated Flows Using Periodic Excitation
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amplitudes as well as wave forms. As expected, control authority of the excitation frequency and amplitude is most striking. However, the excitation wave form as well has a large influence on the mean flow and on the phase averaged structures. Even if the low energetical efficiency of the Lorentz force actuator will hamper its use in industrial applications, it is a valuable tool for basic research.
Acknowledgements Financial support from Deutsche Forschungsgemeinschaft (DFG) in frame of the Collaborative Research Centre (SFB) 609 is gratefully acknowledged.
References [1] Gad-el Hak, M.: Flow control: passive, active, and reactive flow management. Cambridge University Press (2000) [2] Wygnanski, I.: Boundary layer and flow control by periodic addition of momentum. AIAA–paper 97–2117 (1997) [3] Greenblatt, D., Wygnanski, I.: The control of flow separation by periodic excitation. Prog. Aero. Sci. 36 (2000) 487–545 [4] Seifert, A., Greenblatt, D., Wygnanski, I.: Active separation control: an overview of Reynolds and Mach number effects. Aerosp. Sci. Techn. 8 (2004) 569–582 [5] Sutton, G., Sherman, A.: Engineering Magnetohydrodynamics. McGraw Hill, New York (1965) ´ Cachon, P.: Actions e´ lectromagn´etiques sur les liquides en mouvement, [6] Crausse, E., notamment dans la couche limite d’ obstacles immerg´es. Comptes rendus hebdomadaires des s´eances de l’ Acad´emie des Sciences 238 (1954) 2488–2490 [7] Lielausis, O.: Effect of electromagnetic forces on the flow of liquid metals and electrolytes. PhD thesis, Academy of Sciences of the Latvian SSR, Institute of Physics, Riga (1961) in Russian. [8] Gailitis, A., Lielausis, O.: On a possibility to reduce the hydrodynamic resistance of a plate in an electrolyte. Appl. Magnetohydrodynamics, Rep. Phys. Inst. 12 (1961) 143–146 in Russian. [9] Tsinober, A.B., Shtern, A.G.: On the possibility to increase the stability of the flow in the boundary layer by means of crossed electric and magnetic fields. Magnitnaya Gidrodinamica 3 (1967) 152–154 (in Russian). [10] Meyer, R.: Magnetohydrodynamic method and apparatus. US Patent 3,360,220 (1967) [11] Shtern, A.: Feasibility of modifying the boundary layer by crossed electric and magnetic fields. Magnitnaya Gidrodinamika 6 (1970) 124–128 [12] Nosenchuck, D., Brown, G., Culver, H., Eng, T., Huang, I.: Spatial and temporal characteristics of boundary layers controlled with the lorentz force. In: 12th Australian Fluid Mechanics Conference, Sydney (1995) [13] Henoch, C., Stace, J.: Experimental investigation of a salt water turbulent boundary layer modified by an applied streamwise magnetohydrodynamic body force. Phys. Fluids 7 (1995) 1371–1383 [14] Crawford, C.H., Karniadakis, G.E.: Reynolds stress analysis of EMHD–controlled wall turbulence. Part I. Streamwise forcing. Phys. Fluids 9 (1997) 788–806 [15] Berger, T.W., Kim, J., Lee, C., Lim, J.: Turbulent boundary layer control utilizing the lorentz force. Phys. Fluids 12 (2000) 631–649
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[16] Du, Y., Symeonidis, V., Karniadakis, G.: Drag reduction in wall–bounded turbulence via a transverse travelling wave. J. Fluid Mech. 457 (2002) 1–34 [17] Weier, T., Gerbeth, G., Mutschke, G., Platacis, E., Lielausis, O.: Experiments on cylinder wake stabilization in an electrolyte solution by means of electromagnetic forces localized on the cylinder surface. Experimental Thermal and Fluid Science 16 (1998) 84–91 [18] Kim, S., Lee, C.: Investigation of the flow around a circular cylinder under the influence of an electromagnetic force. Exp. Fluids 28 (2000) 252–260 [19] Posdziech, O., Grundmann, R.: Electromagnetic control of seawater flow around circular cylinders. European Journal of Mechanics–B/Fluids 20 (2001) 255–274 [20] Chen, Z., Aubry, N.: Active control of cylinder wake. Communications in Nonlinear Science and Numerical Simulation 10 (2005) 205–216 [21] Weier, T., Gerbeth, G., Mutschke, G., Lielausis, O., Lammers, G.: Control of flow separation using electromagnetic forces. Flow, Turbulence and Combustion 71 (2003) 5–17 [22] Weier, T., Gerbeth, G.: Control of separated flows by time periodic Lorentz forces. Eur. J. Mech. B/Fluids 23 (2004) 835–849 [23] Bouras, C., Nagib, H., Durst, F., Heim, U.: Lift and drag control on a lambda wing using leading-edge slot pulsation of various wave forms. Bulletin of the American Physical Society 45 (2000) 30 [24] Wiltse, J., Glezer, A.: Manipulation of free shear flows using piezoelectric actuators. J. Fluid Mech. 249 (1993) 261–285 [25] Margalit, S., Greenblatt, D., Seifert, A., Wygnanski, I.: Active flow control of a delta wing at high incidence using segmented piezoelectric actuators. In: 1st Flow Control Conference, St. Louis, MO (2002) AIAA–paper 2002–3270. [26] Pack, L.G., Scheffler, N.W., Yao, C.S.: Active control of separation from the slat shoulder of a supercritical airfoil. In: 1st Flow Control Conference, St. Louis, MO (2002) AIAA–paper 2002–3156. [27] Washburn, A., Amitay, M.: Active flow control on the stingray UAV: Physical mechanism. In: 42nd Aerospace Sciences Meeting & Exhibit, Reno, NV (2004) AIAA–paper 2004–0745. [28] Pack Melton, L.G., Yao, C.S., Seifert, A.: Application of excitation from multiple locations on a simplified high–lift system. In: 2nd Flow Control Conference, Portland, OR (2004) AIAA–paper 2004–2324. [29] Chang, R., Hsiao, F.B., Shyu, R.N.: Forcing level effects of internal acoustic excitation on the improvement of airfoil performance. J. Aircraft 29 (1992) 823–829 [30] Wu, J.Z., Lu, X.Y., Denny, A., Fan, M., Wu, J.M.: Post–stall flow control on an airfoil by local unsteady forcing. J. Fluid Mech. 371 (1998) 21–58 [31] Kiedaisch, J., Demanett, B., Nagib, H.: Active flow control of large separation: A new look at scaling parameters. In: 58th Annual Meeting of the APS Division of Fluid Dynamics. (2005) [32] Rice, W.: Propulsion system. US Patent 2,997,013 (1961) [33] Grienberg, E.: On determination of properties of some potential fields. Applied Magnetohydrodynamics. Reports of the Physics Institute 12 (1961) 147–154 (in Russian). [34] Weier, T., Fey, U., Gerbeth, G., Mutschke, G., Lielausis, O., Platacis, E.: Boundary layer control by means of wall parallel Lorentz forces. Magnetohydrodynamics 37 (2001) 177–186 [35] Weier, T., Gerbeth, G., Fey, U., Mutschke, G., Posdziech, O., Platacis, E., Lielausis, O.: Some results on electromagnetic control of flow around bodies. In: Int. Symp. on Seawater Drag Reduction. (1998) 229–235
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[36] Amitay, M., Glezer, A.: Role of actuation frequency in controlled flow reattachement over a stalled airfoil. AIAA J. 40 (2002) 209–216 [37] Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285 (1995) 69–94 [38] Schmitz, F.: Aerodynamik des Flugmodells. Tragfl¨ugelmessungen I. C.J.E. Volckmann Nachf. E. Wette, Berlin–Charlottenburg (1942)
Pulsed Plasma Actuators for Active Flow Control at MAV Reynolds Numbers B. Göksel1, D. Greenblatt2, I. Rechenberg1, Y. Kastantin2, C.N. Nayeri2, and C.O. Paschereit2 1
Technical University Berlin, Institute of Process Engineering Ackerstr. 71-76, Secr. ACK1, D-13355 Berlin, Germany
[email protected] 2 Technical University Berlin, Hermann-Föttinger-Institute of Fluid Mechanics MüllerBreslau-Strasse 8, D-10623 Berlin, Germany
Summary An experimental investigation of separation control using steady and pulsed plasma actuators was carried out on an Eppler E338 airfoil at typical micro air vehicle Reynolds numbers (20,000Re140,000). Pulsing was achieved by modulating the high frequency plasma excitation voltage. The actuators were calibrated directly using a laser doppler anemometer, with and without free-stream velocity, and this allowed the quantification of both steady and unsteady momentum introduced into the flow. At conventional low Reynolds numbers (Re>100,000) asymmetric single phase plasma actuators can have a detrimental effect on airfoil performance due to the introduction of low momentum fluid into the boundary layer. The effect of modulation, particularly at frequencies corresponding to F+≈1, became more effective with decreasing Reynolds number resulting in significant improvements in CL,max. This was attributed to the increasing momentum coefficient, which increased as a consequence of the decreasing free-stream velocities. Particularly low duty cycles of 3% were sufficient for effective separation control, corresponding to power inputs on the order of 5 milliwatts per centimeter.
1 Introduction Achieving sustained flight of micro air vehicles (MAVs) bring significant challenges due to their small dimensions and low flight speeds. This combination results in very low flight Reynolds numbers (Re200,000, achieve loiter targets by deploying flaps. This is not considered practical for MAVs loitering at Re1 indicates that it is more energy efficient to introduce power to the actuators, in order to improve L/D, rather than to the powerplant, to overcome drag. Besides the important applicability aspect, the fundamental nature of the interaction between the multiple modes of excitation resulting from the complex 3D actuator motion indicates the importance and relevance of studying 3D instability modes to enhance AFC effectiveness and efficiency. Margalit et al. [10] also cite AFM1>1 data using cavity installed internal Piezo fluidic actuators applied to control the flow over a delta wing at high incidence. It is highly desirable that the research associated with actuator development and utilization will include comprehensive measurement and reporting of the energetic and environmental impact of the actuator under consideration to supplement fluidic output characterization.
Fig. 4. The application of 10 surface mounted Piezo "benders" for separation control on the IAI-Pr8 airfoil (Seifert et al, 1998 [9]). Bottom right figure shows AFM1>1
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Piezo fluidic actuators Cavity based fluidic actuators are a class of devices capable of producing oscillatory flow excitation [35]. These devices contain three elements; pressure fluctuations generating mechanism a cavity and a slot, communicating the cavity with the flow to be controlled. Though many concepts have been suggested for generating the cavity pressure oscillations (e.g., a loudspeaker, a piston, a diaphragm, oscillating materials, pulsed heat or mass injection and removal), most of these concepts are impractical, due to large size, weight, compatibility issues and energy consumption. In the current application, the cavity pressure fluctuations are generated by an oscillating Piezoelectric elements that form at least one of the cavity walls and are driven by an electric AC signal. Consequently, velocity oscillations result at the actuator’s exit slot. The Helmholtz resonance mechanism can be utilized to generate larger velocities than one could obtain only due to volumetric cavity changes, but over a narrow frequency range. Higher magnitude velocity fluctuations could be achieved at the mechanical resonance frequency of the actuators, though at a narrower frequency range. The operation at the mechanical resonance is undesirable also from damage tolerance aspect and uniformity of multiple actuator application point of views. The problem at hand (“system”) contains two major components leading to the following groups of parameters: The actuator geometry and the actuator operating conditions. The geometry of the cavity is of lesser importance for the current discussion, but the neck geometry is of significant importance [34], determining the shape of the slot exit velocity profile. This is because, the slot vorticity flux during the blowing stage is crucial to the evolution of the external flow field. An application dependent optimum should be found between peak ejection velocity, maximum momentum and the largest vorticity flux out of the actuator's slot.
Fig. 5. Schematic structure of a generic oscillatory vorticity generator (OVG) configuration with leading parameters [32,33]
The main geometrical slot-exit boundary conditions and operational parameters of the 2D generic OVG configuration are shown in Fig. 5. Note that 3D effects are not considered here and that the exterior lines are treated as semi-infinite boundaries. As
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mentioned before, the actuator contains three main components – oscillating element(s), a cavity and a slot. Here, the cavity and the oscillating components are excluded from the dimensional considerations, since the current emphasis is on the relationship between the periodic pressure oscillation and the velocity oscillation at the slot and especially the external flow, and not on the mechanism that creates the pressure oscillations inside the cavity. Given that the flow is periodic, and for a pure sine wave excitation, we can write the slot exit velocity, Vs(r, t), as
~
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~
time) and V (r) is a normalized velocity vector distribution, r is a location vector and t is time. For order of magnitude analysis, the velocity profile can be treated as a "tophat", while for slots and operational conditions dominated by viscosity, it could be considered parabolic. Therefore, the slot velocity, for a given geometry, is a function of the excitation frequency, f, the kinematic viscosity and time. If we apply Buckingham's Pi theorem on the parameters indicated in Fig.5 we obtain: (5) F (h, b1, H1, α1, β1, b2, H2, α2, β2, f, Up, ρ, µ) = 0 Since we have nine dimensional parameters, a group of six independent dimensionless parameters can be formed (angles are already dimensionless). A possible set is: (6)
§b b H H F¨ 1 , 2 , 1 , 2 , ¨h h h h ©
· fh 2 U P h , , α 1 , α 2 , β1 , β 2 ¸ = 0 ¸ v ν ¹
The above dimensionless parameters can be divided into two groups: The boundary conditions group:
§ b1 b2 H 1 H 2 · , , α 1 , α 2 , β1 , β 2 ¸ ¨ , , ©h h h h ¹
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root of the excitation frequency. The ratio S tk form:
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amplitude parameter. Since the slot “neck” length (b1=b2 in the current configuration) used in common devices is significantly smaller than the unsteady wavelength, this parameter is important only outside the slot. The Strouhal number could also be viewed as a ratio of length scales, the excitation wavelength and the ejected fluid typical convection distance, which would become important while considering the resulting external flow field. For S tk approaching zero, the unsteady effects are negligible, and for sufficiently low Re numbers, the flow at the actuator's neck would resemble a plane Poisille flow. As S tk increases, the unsteady effects become important, and the velocity profile at the actuator's neck would become more like a top-hat profile (with an overshoot and a phase lag at the unsteady boundary layers forming at the neck). The velocity slug Reynolds number is probably the least representative Reynolds number of the problem at hand; however it is used for lack of a better, conveniently measured Reynolds number. The boundary conditions group also plays a significant role in the resulting flow field, influencing, for instance, vortex pair circulation and asymmetry. The actuator's "lips" width to slot width ratio, Hi/h, would most likely play a significant role in the suction process, influencing the flow field at the vicinity of the slot, together with the angles setup of the actuator. It is well known that suction and blowing are different processes in viscous fluid flow, where flow separation could take place. The same fluid mass leaving the actuator has to be entrained back into its cavity during the suction stage. Since sink flow is less affected than source (jet) flow by the details of the boundary conditions at the slot, the difference between the suction and blowing stages of the excitation cycle is expected to be strongly dependent on the boundary condition [32,33]. Piezo fluidic actuators have been successfully developed and applied during the last decade. Flight demonstration has not been performed yet, to the best knowledge of the author. It is expected that with proper design, careful fabrication and effective implementation, Piezo fluidic actuators will become a valuable tool for the AFC practitioners. Current trends demonstrate peak exit velocities reaching M=0.3, deemed sufficient for many AFC applications. To date, the Piezo fluidic actuators were the only ones to demonstrate AFM>1, still in laboratory experiments. There are many AFC applications where Piezo fluidic actuators are not compatible or suitable, or lack sufficient control authority. This also holds where there is no internal volume to use or the system is already in existence. In these cases surface mounted actuators is the way to go. In compressible flow and especially in supersonic flight speeds it is not expected that peak excitation Mach number with Mpeak=0.3 would suffice. This calls for further development of more robust actuation technology. 2.4 Additional Actuator Concepts Many other types of actuators are currently at different levels of development and application stages. A partial list and brief discussion follows. This is by no means a comprehensive review of the state-of-the-art or current trends.
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Combustion jet actuators are based on subsonic wave combustion process, and operate at a time scale much greater then the time scale of the wave expansion within the device. Their operation is similar to a small pulsejet engines or pulsed detonation engines [18]. They consist of an internal combustion chamber, an inlet valve that periodically introduces mixed fuel and Oxygen into the combustion chamber, an igniter to initiate the combustion process and exit slot to communicate the hot combustion gases to the external flow. Combustion jets actuators are capable of providing high impulse at a relatively wide frequency range (about 100Hz demonstrated to date) and therefore they can provide good solution to boundary layer separation control at high speeds and large scales, where such low frequencies might be adequate [20]. The main disadvantage of such devices is the need to use high temperature, exotic materials, the internal combustion process that consumes fuel, requires a pre-mixing process and a mixing device, complexity, safety, installation, weight and compatibility concerns. The Hartmann tube actuator is a device that uses shock wave oscillations to produce high amplitude sound waves. The phenomenon was discovered By Julius Hartmann in 1919 while using a Pitot probe in a compression region of a jet. The use of high intensity sound waves for flow control application was the focus of many studies, and shown to provide a limited control authority at low-speeds [21]. The combination of high-frequency-high-intensity pressure waves, may be suitable for high-speed flowcontrol- applications where marginal control authority may be sufficient. However, the receptivity of a separating turbulent boundary layer to sound waves is rather low. These devices are usually large and heavy. The installation issues of this class of actuators in an aerodynamic surface have not been assessed yet. The fundamental operational principle of plasma actuators is imposing kinetic energy on a limited region inside the boundary layer using a body-force. Methods of plasma generation include DC, AC, RF, microwave, arc, corona and spark electric discharge. Labergue et al (2004, [22]) explained that, for the case of a DC corona discharge established between two electrodes flush mounted along a non-conducting surface in air at atmospheric pressure, ions are produced at the anode and electrons at the cathode (see Fig. 6a). In their drift motion between the two electrodes under electrostatic Coulomb forces, these ions exchange momentum with neutral particles and induce an airflow termed “ionic wind”. The common plasma actuators operate in a similar manner. The above described process creates steady or pulsed near-wall velocity perturbations, which can be used to modify the near-wall flow. The amplitude of the plasma actuators depends on the supplied current and voltage and the fluid properties. Temperature has a strong effect on the actuator performance. Plasma actuators were proven to be efficient in high-speed cold jet [23], where the receptivity to the excitation is almost singular, and in boundary layer control at low-speeds [24]. One of the shortcomings of plasma actuators is the need to generate high voltage; accompanied with energy losses and excessive weight. Another disadvantage of plasma actuators is the sensitivity to the controlled fluid properties (e.g., humidity can cause dramatic changes in the efficiency and lead to damage), emitted light and electromagnetic radiation.
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Fig. 6. (a)Sketch of the Plasma actuator operation principle. [31]. (b) Sketch of the Spark jet actuator. [30].
The Spark jet actuator [30] is an actuator concept, similar to the combustion jet actuator, only that the entire energy deposition into the actuators cavity is due to an electric spark generated between two electrodes situated inside a small cavity. This type of actuator is under study for quite some time. The energy deposited by the spark into the cavity increases its temperature and pressure, causing the heated compressed gas to exit the cavity through a hole. This process have been demonstrated to be capable of generating supersonic exit speeds and been numerically simulated as well. However, it suffers from similar disadvantages as the combustion based actuator, namely heating, exotic materials, rapid degradation and compatibility issues. It has not been demonstrated in repeated operation at any applicable rate and has not been applied yet for flow control applications. Suction and oscillatory blowing actuator A new actuator for active flow control (AFC) applications is under development at the Tel-Aviv University, Meadow Aerodynamics laboratory. The new fluidic device combines steady suction and oscillatory blowing, both proven to be very effective for AFC (US Patent 2006-0048829-A1). The New fluidic device is a combination of a bi-stable fluidic amplifier and an ejector (Fig. 7). The bi-stable device is based on the principle of wall attachment (“Coanda effect”) to one of the two walls of a diverging diffuser. When a fluid jet is flowing in the proximity of a wall, a low-pressure region is formed between the jet and the wall. The low pressure draws the jet towards the wall, the jet will deflect and eventually become attached to the wall. In the case of two close and symmetric walls, such as in the configuration shown in Figure 7, the jet will randomly re-attach to one wall. If a pressure pulse is introduced at one of the control ports (control R or control L in Fig. 7) the jet will detach from that wall and reattach to the opposite wall, due to the two marginally stable conditions it can be in. The bi-stable fluidic amplifier can self-oscillate by connecting the two control ports by a tube. In this configuration the oscillation frequency is related to the tube length, volume, and resistance, the speed of the sound wave in the tube and the flow rate through the actuator’s main path. A detailed model of these aspects is provided in an accompanying paper [28]. The frequency of the valve, when operating in selfoscillating mode, can be adjusted by the length, diameter and resistance of the tube connecting the two control ports (Fig, 7).
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For active flow control use, and especially during the actuator design and development stages, there is a need to control the flow rate (which would dictate the magnitude of the external excitation) and the oscillation frequency. Therefore, there is a need to produce controlled frequency oscillatory pulses in the control ports. This can be done in many ways; piezoelectric actuators, spinning valve, solenoid valves and other devices and methods. The ejector is a simple fluidic device based on Bernoulli’s principle. When a jet stream expands into larger cylindrical cavity, it creates a low pressure region around it. If the cavity behind the jet is open to the free atmosphere or to a higher pressure environment (such as the surface of an airfoil) the pressure gradient will cause the external air to be sucked into the cavity around the internal jet. The purpose of the ejector in the current device is to increase the flow rate. However, if the fluid would be extracted from an aerodynamic surface it would create suction flow across the aerodynamic surface through slots or holes. The proposed fluidic actuator is intended to function over a wide range of frequencies and flow rates. In the self-oscillating mode, there are no moving or active elements while the frequency and amplitudes are coupled, which is desired from an applicability point of view. Furthermore, the distribution of suction and oscillatory-blowing over the surface to be controlled offer many intriguing combinations for effective AFC.
Fig. 7. Schematic of the suction and oscillatory blowing actuator
A detailed model and performance characteristics are presented in an accompanying paper (Arwatz et al, 2006, [28]). It was demonstrated that the new actuator is capable of near sonic output velocities and frequencies of order 1kHz. It is insensitive to exit load variations and demonstrated extremely efficient operation. A small scale (~2cm) device is being tested and its application for separation control is due next.
3 Actuation Systems Performance Comparison A detailed comparison of different actuators and actuation schemes, based on the three criteria suggested above was attempted. However, very little data is available in the open literature that is required for the calculations of the performance criteria suggested above. Only through private communication with the IIT group were we
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able to obtain sufficient information for comparison of TAU developed actuators with the actuators used for the flight tests of the XV-15 (Nagib et al, [38]). Further information that is provided below is based on conservative evaluations by the author and can be adjusted in the future when more information becomes available. The Overall Figure of Merit (OFM) compares the actuator performance as it is operated in still air, taking its fluidic output, its weight and power consumption into account. For the actuators used during the XV-15 fight tests and at the flight test conditions, the OFM=0.014 with Up=80.4m/s. Comparing this number to those of the TAU applied Piezo fluidic actuators results in OFM=0.052 (Up=20m/s, Margalit et al, [10]) and OFM=0.107 (Up=40m/s, Timor et al, [11]). For the compact actuator used by Yehoshua and Seifert (Up=60m/s, [32,33]) the OFM=0.295. However, the latest actuator was tested in free air and was not restricted by installation considerations. Note also the different peak slot velocities in the above comparisons. An estimation of the OFM for plasma actuators such as those used in [24,31], results in OFM1, the energy optimality of the POD decomposition in that direction leads to modes that are the optimum decomposition of a given physical mode as it evolves throughout a transient data set. If K=2 then modes Φ11 and Φ12 are the mean flow and the “shift mode” or “mean flow mode” as
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described by Noack and Siegel, respectively. Thus the modes with indices k>1 can be referred to as first, second and higher order “shift” modes that allow the POD mode ensemble to adjust for changes in the spatial modes. We will refer to all of these additional modes obtained by the DPOD decomposition as shift modes, since they modify a given physical mode to match a new flow state due to either a formation length or recirculation zone length change. This may be due to effects of forcing, a different Reynolds number, feedback or open loop control or similar events. Thus, in the truncated DPOD mode ensemble for each physical mode one or more shift modes may be retained based on inspection of energy content or spatial structure of the mode. We will now demonstrate how this DPOD procedure can be used to create mode ensembles that cover the entire unforced transient startup of the D-shaped cylinder wake. This mode ensemble will thus cover not just the limit cycle, but also the steady flow state before the onset of the vortex shedding. The latter is of particular importance since vortex shedding suppression control targets the steady flow state as the control goal.
4 SPOD Modes of Transient Startup Figure 2 indicates the boundaries of the bins used to derive SPOD modes. 21 SPOD mode sets were derived, the individual bins are numbered in Figure 2. As may be expected, the POD modes derived from the bins closer to the start of the simulation are quite different than those derived from snapshot sets from the end of the simulation. Figure 3 compares the second POD mode from bins 4 and 18. The second POD mode is the first mode of a mode pair representing the von Karman vortex street, and contains about 45% of the fluctuating energy. The peak amplitude of this mode can be found about 6d downstream of the base for the mode derived from bin 4, and about 2 base heights downstream of the base for the mode derived from bin 18. This indicates that the vortices are forming further downstream at the beginning of the simulation, and move closer to the base as the limit cycle develops. SPOD Mode 2 Bin 18 2
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While Figure 3 demonstrates the change of spatial modes during this transient simulation for one mode only, similar effects can be seen for all other SPOD modes (not shown here). This illustrates the need to derive shift type modes not just for the mean flow, but also the fluctuating modes.
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5 DPOD Model of the D-Shaped Cylinder The DPOD procedure introduced in a previous section achieves just that: Derivation of shift type modes for all physical modes that are deemed of importance in a transient flow field. Figure 4 shows a DPOD mode set for the transient simulation introduced in Figure 2, retaining 5 physical modes and 3 shift modes. Modes (1,n) represent the mean flow, Modes (2,n) and (3,n) the von Karman vortex street, while Modes (4,n) and (5,n) represent a higher order vortex shedding mode with twice the spatial frequency of the von Karman vortex street. Inspecting the energy contents of these modes presented in Figure 5, it can be seen that most energy is contained in the mean flow, followed by the von Karman vortex street modes which appear as pairs of approximately equal energy. This behavior is identical to a conventional POD procedure performed on a time periodic flow field. The energy content of the entire mode basis is now a two dimensional energy plane, with steep energy dropoff both towards higher order physical and higher order shift modes. While the energy content dropoff of the shift modes of the mean flow mode (Modes (1,2) and (1,3)) is fairly steep, it can be seen that the dropoff for the shift modes of the von Karman modes (Modes (2,2), (2,3) and (3,2), (3,3)) is far less steep, demonstrating the importance of including these modes in a low order model. Since the dropoff in energy is not uniform for all shift modes, one might retain different number of shift modes for each physical mode based on energy considerations. The spatial distribution of the shift modes demonstrates how they manage to adjust the distribution of each physical mode to the changes in the vortex shedding pattern seen during the transient startup simultation: The first shift mode of the mean flow mode, for example, will extend the length of the recirculation zone which is what can be observed during the development of the vortex shedding limit cycle. This behavior is almost identical the artificially created shift modes suggested by Noack et al. (2003) or the so called mean flow mode introduced by Siegel (2003). A consequence of the much longer recirculation zone at the startup of the limit cycle is also that the vortices are forming further downstream of the body, which can be seen by inspecting the SPOD Modes shown in Figure 3. This effect is modeled mostly by the first shift mode of the two von Karman vortex shedding modes, i.e. Modes (2,2) and (3,2). These have a similar appearance than the von Karman vortex shedding modes themselves, but with their maximum mode amplitude shifted further downstream. The same effect can be seen for the higher order vortex shedding modes, where the maximum of modal activity of the shift modes (Modes (4,2) and (5,2)) is also further downstream than for the modes themselves (Modes (4,1) and (5,1)). Since all of the spatial modes have been normalized to unity in magnitude, the temporal coefficients provide insight into the energy content as time evolves during the simulation. Moreover, the energy exchange between modes and their shift modes is also captured in these time coefficients. Thus it is of interest to inspect the time coefficients in order to understand the dynamics of energy exchange between the modes during the transient simulation. For example, the time coefficients of the two mean flow mode shift modes, Mode (1,2) and (1,3), reveal that these only contribute
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at the beginning of the limit cycle development, before about 1.5 s into the simulation. For the time periodic portion of the simulation, their contribution approaches zero. The von Karman shift modes show a different behavior: Their activity has a peak around 0.9 seconds, and a minimum at about 1.5 seconds. It can be shown that their phase with respect to their physical mode (Mode (n,1)) changes at that time by about 180 degrees and as a result they add to the main mode’s activity for one portion of the simulation, while they decrease the activity for the other portion of the simulation. In combination with their different spatial distributions, the observed shift in the shedding location is thus modeled correctly by the vortex shedding and their shift modes. The physical mode together with its shift modes thus offer a possibility to develop for example a multi input single output controller that adjusts its feedback
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parameters in accordance with the vortex shedding location shift: By applying different gains to the mode and shift modes, the feedback can be tuned to the required phase at any time.
6 Sensor Based State Estimation The time histories of the temporal coefficients of the POD model are determined by introducing the spatial Eigenfunctions into the flow field data using the least squares technique. For closed-loop control, we are interested in a system that maps, in realtime, velocity measurements provided by the sensors onto the estimates of the fifteen temporal modes. The estimation scheme predicts the temporal amplitudes of the first fifteen POD modes from a finite set of velocity measurements obtained from the CFD solution of the “D” shaped cylinder wake. For each sensor configuration, 341 velocity measurements were used equally spaced at 0.01seconds apart. Of the 341 snapshots, the first 170 were used for training of the estimator, whereas, the final 171 snapshots were used for validation purposes. Only data concerning velocity components in the direction of the flow were used for the sensor placement and number studies reported in this effort. The main approach in this effort is the incorporation of a non-linear dynamic estimator. The decision was to look into universal approximators, such as artificial neural networks (ANN), for their inherent robustness and capability to approximate any non-linear function to any arbitrary degree of accuracy. The ANN employed in this effort, in conjunction with the ARX model is the mechanism with which the dynamic model is developed using the POD time-coefficients extracted from the high resolution CFD simulation. Non-linear optimization techniques, based on the back propagation method, are used to minimize the difference between the extracted POD time coefficients and the ANN while adjusting the weights of the model (Nørgaard et al., 2003). The main hypothesis is that the non-linearity and scaling characteristics of the temporal coefficients lead to numerical stability issues which undermine the development and analysis of effective estimation/control laws. In order to assure model stability, the ARX dynamic model structure is incorporated. This structure is widely used in the system identification community. A salient feature of the ARX predictor is that it is inherently stable even if the dynamic system to be modeled is unstable. This characteristic of ARX models often lends itself to successful modeling of unstable processes as described by Nelles (2001). The artificial neural network (ANN) has the following features: •
Input Layer: Eight sensor signals, n (n = 8), namely, U-Velocity at the given sensor locations. In addition to these readings, in order to obtain a strong representation of the dynamics of the system, the input layer includes 8 past inputs and 15 time delays. No past outputs were used. The eight sensors are placed on the maxima/minima of the first eight modes as described by the heuristic procedure developed by Cohen et al. (2006). The total number of inputs to the net is as follows: # inputs to ANN = # time delays *[ # past outputs + (# past inputs per sensor)] + bias
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By inserting the values chosen after a brief sensitivity study, we obtain: # inputs to ANN = 15*[0 + 8] + 1 = 121
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From the above relation, we can see that the number of inputs is 121 for all the cases irrespective of the number of sensors. Hidden Layer: One hidden layer consisting of 15 neurons. The activation function in the hidden layer is based on the non-linear tanh function. A single bias input has been added to the output from the hidden layer. Output Layer: Fifteen outputs, namely, the 15 states representing the temporal coefficients of the 15 mode POD reduced order model developed in the previous Section. The output layer has a linear activation function. Weighting Matrices: The weighting matrices between the input layer and the hidden layer (W1) and between the hidden layer and the output layer (W2) depend on the number of sensors. In this effort, W1 is of the order of [121*15] and W2 is of the order of [15*16]. These weighting matrices are initialized randomly. Training the ANN: Back propagation, based on the Levenberg-Marquardt algorithm, was used to train the ANN using Nørgaard et al’s (2003) toolbox. The training procedure converged near 350 iterations. Generally speaking, the training data fits well and this should not be very surprising. Results for the training data are enclosed in Fig. 7. Validating the ANN: The validation data is as described in Table 1. Results for the validation data are enclosed in Fig. 8. Table 1. RMS Errors [%] for ANNE Estimates
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7 Discussion We introduce Double Proper Orthogonal Decomposition (DPOD) as a means to derive POD spatial modes that span different flow conditions. We demonstrate its ability by applying it to a transient simulation of the development of the limit cycle of a D shaped cylinder wake.
8 Outlook The derivation of valid spatial modes and temporal coefficients constitutes the first step towards developing a set of low dimensional equations capturing the dynamic behavior of a flow field. Both linear and nonlinear techniques, such as system identification and Galerkin projection may be used to do so, and we will explore their efficiency for this task in the future. While we demonstrate the capabilities of DPOD by applying it to a transient flow field which in essence consitutes one parameter change, DPOD may also be used to model the effects of several different parameters on a flow field, like the impact of changes in actuation frequency and amplitude, or Reynolds number. Thus it is a generic technique that can be applied to problems of any degree of complexity, not just the relatively simple sample flow field that was used in this work.
Acknowledgments The authors would like to acknowledge funding by the Air Force Office of Scientific Research, LtCol Sharon Heise, Program Manager. We would also like to thank Dr. Jim Forsythe of Cobalt Solutions, LLC for their support. The authors appreciate the fruitful discussions with Dr. Young Sug-Shin on system identification using Artificial Neural Networks and for his very helpful insight.
References Noack, B. R.; Afanasiev, K.; Morzynski, M.; Tadmor, G., Thiele, F., 'A hierarchy of lowdimensional models for the transient and post-transient cylinder wake', J. Fluid Mechanics, 497, 2003 Luchtenburg, M., G. Tadmor, O. Lehmann, B.R. Noack, R. King, M. Morzyinski, 'Tuned POD Galerkin models for transient feedback regulation of the cylinder wake', 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, AIAA-2006-1407, 2006 M. Bergmann, L. Cordier, and J.-P. Brancher, 'Optimal rotary control of the cylinder wake using POD reduced order model', 2nd AIAA Flow Control Conference, AIAA 2004-2323, 2004 Siegel, S.; Cohen, K.; McLaughlin, T., 'Feedback Control Of A Circular Cylinder Wake In Experiment And Simulation (invited)', 33rd AIAA Fluid Dynamics Conference, Orlando, AIAA 2003-3569, 2003
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Siegel, S., Cohen, K., Seidel, J., McLaughlin, T., 'Short Time Proper Orthogonal Decomposition for State Estimation of Transient Flow Fields', 43rd AIAA Aerospace Sciences Meeting, Reno, AIAA2005-0296, 2005 Siegel, S., Cohen, K., Seidel, J., McLaughlin, T., 'Two Dimensional Simulations Of A Feedback Controlled D-Cylinder Wake', AIAA Fluid Dynamics Conf Toronto, ON, CA, AIAA 20055019, 2005 Holmes P., Lumley, J. L., Berkooz, G., 1996, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press Cohen, K., Siegel S., and McLaughlin T.,"A Heuristic Approach to Effective Sensor Placement for Modeling of a Cylinder Wake”, Computers and Fluids, Volume 35, Issue 1, January 2006, pp. 103-120. Nelles, O., Nonlinear System Identification, Springer-Verlag, Berlin, Germany, 2001, Chap. 11. Nørgaard, M., Ravn., O., Poulsen, N.K., Hansen, L.K., Neural Networks for Modeling and Control of Dynamic Systems, 3rd printing, Springer-Verlag, London, U.K., 2003, Chap. 2.
A Unified Feature Extraction Architecture Tino Weinkauf1 , Jan Sahner1 , Holger Theisel2 , Hans-Christian Hege1 , and Hans-Peter Seidel2 1
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Summary We present a unified feature extraction architecture consisting of only three core algorithms that allows to extract and track a rich variety of geometrically defined, local and global features evolving in scalar and vector fields. The architecture builds upon the concepts of Feature Flow Fields and Connectors, which can be implemented using the three core algorithms finding zeros, integrating and intersecting stream objects. We apply our methods to extract and track the topology and vortex core lines both in steady and unsteady flow fields.
1 Introduction As the resolution of numerical simulations as well as experimental measurements like PIV have evolved significantly in the last years, the challenge of understanding the intricate structures within their massive result data sets has made automatic feature extraction schemes popular. Exploratory techniques alone do not suffice to analyze massive result data sets. Due to the sheer size of the data they have to be complemented by automatic feature extraction schemes, which give a reliable basis for subsequent manual explorations. In this paper we focus on the treatment of flow fields. They play a vital role in many research areas. Examples are combustion chambers, turbomachinery and aircraft design in industry as well as visualization and control of blood flow in medicine. For this class of data, topological and vortical structures are among the features of interest. Extraction of those structures helps in understanding processes inherent to the flow. This knowledge is the basis for manipulating those processes in terms of flow control. While [13] gives an overview on flow visualization techniques focusing on feature extraction approaches, we give a short introduction here. Topological methods have become a standard tool to visualize 2D and 3D vector fields because they offer to represent a complex flow behavior by only a limited number of graphical primitives. [6] and [5] introduced them as a visualization tool by extracting critical points and classifying them into sources, sinks and saddles, and integrating certain stream lines called separatrices from the saddles in the directions of the eigenvectors of the Jacobian matrix. Later, topological methods have been extended to higher order critical points [16] [28], boundary switch points [3] and curves [27], closed separatrices [32] [23], and saddle R. King (Ed.): Active Flow Control, NNFM 95, pp. 119–133, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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connectors [22]. In addition, topological methods have been applied to simplify [3] [25] [30], smooth [31], compress [9] and design [19] vector fields. While they aim at the segmentation of a vector field into areas of different flow behavior, vortex oriented methods highlight turbulent regions of the flow. Recently some work has been done to link these different areas: [4] [26] employ topological methods to analyze the phenomenon of vortex breakdown. Vortices play a major role due to their wanted or unwanted effects on the flow. In turbomachinery design, vortices reduce efficiency, whereas in burning chambers, vortices have to be controlled to achieve optimal mixing of oxygen and fuel. In aircraft design, vortices can both increase and decrease lift. Algorithms for the treatment of vortical structures can be classified in two major categories: – Vortex region detection is based on scalar quantities that are used to define a vortex as a spatial region where the quantity exhibits a certain value range. We refer to them as vortex region quantities. Examples of this are regions of high magnitude of vorticity or negative λ2 -criterion [8]. In general, these measures are Galilean invariant, i.e., they are invariant under adding constant vector fields. This is due to the fact that their computation involves derivatives of the vector field only. Isosurfaces or volume rendering are common approaches for visualizing these quantities, which requires the choice of thresholds and appropriate isovalues or transfer functions. As shown in [14], this can become a difficult task for some settings. – Vortex core line extraction aims at extracting line type features that are regarded as centers of vortices. Different approaches exist. [18] [12] consider lines where the flow exhibits a swirling motion around it. [1] extracts vorticity lines seeded at critical points and corrected towards pressure minima. [15] considers stream lines of zero torsion. All of these approaches include a Galilean variant part, i.e., they depend on a certain frame of reference. In contrast to vortex region detection described above, the extraction of those lines is parameter free in the sense that their definition does not refer to a range of values. This eliminates the need of choosing certain thresholds. In this paper we present an unified approach to extracting and tracking a variety of flow features. Hereby we define the term feature as follows: – A feature is an n-dimensional geometrical structure embedded into a m-dimensional domain. – It is located inside the domain of the analyzed data. – It yields certain “insight” into the data. Finally, the actual definition of a feature depends on the application. In this paper, we mainly treat topological and vortical structures of flow fields. The paper is organized as follows. Section 2 explains the unified feature extraction architecture, while sections 3 and 4 treat the main concepts behind it, namely Feature Flow Fields and Connectors. We apply our method in section 5 to a number of data sets and feature definitions. Conclusions are drawn in section 6.
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2 Unified Feature Extraction Architecture Almost every feature can be extracted and tracked using a combination of the following core algorithms: – Finding zeros – Integrating stream objects – Intersecting stream objects We show in section 3 how the first two algorithms can be combined in the Feature Flow Field approach to extracting and tracking features that are defined locally. The intersection of stream objects becomes necessary, where the features we are interested in have a global nature, like closed stream lines. Here the Connectors approach can be applied (section 4). We now briefly comment on each of the above algorithms. 2.1 Finding Zeros We are interested in extracting isolated zeros of functions f : Rn → Rn . Several approaches exist, some depending on the interpolation scheme: – Newton-Raphson: use the first derivative to repeatedly predict a zero [12]. – In piecewise linear fields, e.g. tetrahedral grids, the zeros can be computed explicitely. – In piecewise trilinear fields, e.g. regular grids, a component wise change-of-sign test is a necessary condition for a zero inside the grid cell. Based on this test, a recursive domain decomposition can be applied to the cell that converges to a zero. This method extends to finding zeros of functions f : Rn → Rm , is quick, robust, and easy to implement. So we favor this method over the Newton-Raphsonapproach for trilinear fields. 2.2 Integration of Stream Objects Given a flow field f : Rn → Rn we aim at constructing m + 1-dimensional stream objects from m-dimensional seeding structures. – For m = 0 we obtain a stream line or integral curve. – For m = 1 we obtain a stream surface by triangulating stream lines seeded equidistantly on the seeding line, see [7] for a thorough treatment of this topic. – Starting from seeding surfaces, we obtain flow volumes, see [10] for implementation details. Naively integrating stream objects from a seeding structure might result in passing through the whole dataset for each stream object, a costly undertaking, when the dataset is too large to fit into main memory and the number of stream objects is high. Nevertheless Weinkauf et al. showed in [29] that it is possible for a huge subclass of features to do all stream object integrations by sequentially loading the dataset only once, keeping just two consecutive time steps in memory at a time. Where section 3 shows that finding zeros and integration of stream objects suffices for finding features that are locally defined, stream object integration becomes necessary if such a definition is not at hand.
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2.3 Intersection of Stream Objects Given a flow field f : Rn → Rn and two m-dimensional (m > 1) stream objects R (integrated in forward direction) and A (integrated in backward direction) we aim at extracting the intersection of R and A, i.e., the m − 1-dimensional stream object that both R and A share. – For m = 2 we obtain a stream line which lies in both intersecting stream surfaces. – For m = 3 we obtain a stream surface which lies in both flow volumes. Figure 2 shows an example for m = 2, where two stream surfaces share a common stream line.
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The concept of feature flow fields was first introduced in [21]. It follows a rather generic idea: Consider an arbitrary point x known to be part of a feature in a (scalar, vector, tensor) field v. A feature flow field f is a well-defined vector field at x pointing into the direction where the feature continues. Thus, starting a stream line integration of f at x yields a curve where all points on this curve are part of the same feature as x. FFF have been used for a number of applications, but mainly for tracking features in time-dependent fields. Here, f describes the dynamic behavior of the features of v: for a time-dependent field v with n spatial dimensions, f is a vector field IRn+1 → IRn+1 . The temporal evolution of the features of v is described by the stream lines of f . In fact, tracking features over time is now carried out by tracing stream lines. The location of a feature at a certain time ti can be obtained by intersecting the stream lines with the time plane ti . Figure 1a gives an illustration. Depending on the dimensionality of the feature at a certain time ti , the feature tracking corresponds to a stream line, stream surface or even higher-dimensional stream object integration. The stream lines of f can also be used to detect events of the features: – A birth event occurs at a time tb , if the feature at this time is only described by one stream line of f , and this stream line touches the plane t = tb “from above” (i.e., the stream line in a neighborhood of the touching point is in the half-space t ≥ tb ). – A split occurs at a time ts , if one of the stream lines of f describing the feature touches the plane t = ts “from above”. – An exit event occurs if all stream lines of f describing the feature leave the spatial domain. The conditions for the reverse events (death, merge, entry) can be formulated in a similar way. Figure 1b illustrates the different events. Integrating the stream lines of f in forward direction does not necessarily mean to move forward in time. In general, those directions are unrelated. The direction in time may even change along the same stream line as it is shown in figure 1b. This situation is always linked to either a birth and a split event, or a merge and a death event. Even though we treated the concept of FFF in a rather abstract way, we can already formulate the basics of an algorithm to track all occurrences of a certain feature in a time-dependent field:
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(a) Tracking features by tracing stream lines. (b) Events: at the time tb a new feature is born, Features at ti+1 can be observed by interat the time ts it splits into two features. secting these stream lines with the time plane t = ti+1 . Fig. 1. Feature tracking using feature flow fields
Algorithm 1. General FFF-based tracking 1. Get seeding points/lines/structures such that the stream object integration of f guarantees to cover all paths of all features of v. 2. From the seeding structures: apply a numerical stream object integration of f in both forward and/or backward direction until it leaves the space-time domain. 3. If necessary: remove multiply integrated stream objects. Algorithm 1 is more or less an abstract template for a specific FFF-based tracking algorithm. Before showing how this template can be used to track critical points and extract and track vortex core lines in flow fields in the applications section 5, we can already note, how the steps of algorithm 1 correlate to the core algorithms given in section 2: the seeding points are usually extracted as critical points of some fields. Then we use the stream object integration from section 2.2 to track the feature. But what can be done, if the feature of interest does not admit a local definition? Here the connectors approach comes into play.
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Given a flow field f : Rn → Rn and two m-dimensional (m > 1) stream objects R (integrated in forward direction) and A (integrated in backward direction) we aim at extracting the intersection of R and A, i.e., the m − 1-dimensional stream object that both R and A share. Figure 2 shows an example for m = 2, where two stream surfaces share a common stream line. Since the intersection of R and A always starts at the repelling seeding structure of R and ends at the attracting seeding structure of A, it is called a connector. A connector is a global feature, i.e., it can not be locally defined.
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(a) Two stream surfaces starting from saddle points.
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An algorithm for the extraction of line-type connectors has been treated in [22]. To find the intersection between a separation surface in forward integration and a separation surface in backward integration, we integrate both separation surfaces simultaneously until a first intersection point p1 is found. After refining this intersection point (see [22] for details), a stream line from p1 is integrated both forwards and backwards. This stream line is the connector. Figure 3 gives an illustration of this algorithm.
5 Applications In this section we apply the Feature Extraction Architecture to a variety of feature extraction settings. In section 5.1, topological features are focused while section 5.2 shows how to extract and track vortex core lines using the Feature Extraction Architecture.
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5.1 Topological Feature Extraction Critical points, i.e. isolated points at which the flow vanishes, are perhaps the most important topological feature of vector fields. For static fields, their extraction and classification is well-understood both in the 2D [6] and the 3D case [27]. Critical points also serve as the starting points of certain separatrices, i.e. stream lines or surfaces which divide the field into areas of different flow behavior. Where the direct visualization of those stream surfaces result in cluttered images, Theisel et al. showed in [22] how restricting the display to the intersection lines of those surfaces, called saddle connectors, increases the comprehensibility. This has been achieved by using the connectors approach. In [23] and [24] Theisel et al. showed how to extract and track closed stream lines using the connectors approach. Considering a stream line oriented topology of time-dependent vector fields, critical points smoothly change their location and orientation over time. In addition, certain bifurcations of critical points may occur. To capture the topological behavior of timedependent vector fields, it is necessary to capture the temporal behavior of the critical points. Theisel et al. introduced in [21] a FFF-based approach to track critical points, which matches algorithm 1. We now show, how the Feature Extraction Architecture can be applied to this setting.
Fig. 4. Tracking 2D critical points: all points on a stream line of f have the same value for v. Note that the depicted stream line is a tangent curve of the feature flow field f and not of the original velocity field v.
Critical Point Tracking. Let v be a 3D time-dependent vector field, which is given as ⎛ ⎞ u(x, y, z, t) v(x, y, z, t) = ⎝ v(x, y, z, t) ⎠ (1) w(x, y, z, t) in the 4D space-time domain D = [xmin , xmax ] × [ymin , ymax ] × [zmin , zmax ] × [tmin , tmax ]. We can construct a 4D vector field f in D with the following properties: for any two points x0 and x1 on a stream line of f , it holds v(x0 ) = v(x1 ). This means that a stream line of f connects locations with the same values of v. Figure 4 gives an illustration in 2D. In particular, if x0 is a critical point in v, then the stream line of f describes the path of the critical point over time. To get f , we search for the direction in space-time in which both components of v locally remain constant. This is the direction perpendicular to the gradients of the three components of v:
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f ⊥ grad(u) = (ux , uy , uz , ut )T , f ⊥ grad(v) , f ⊥ grad(w). This gives a unique solution for f (except for scaling) ⎛ ⎞ + det(vy , vz , vt ) ⎜ − det(vz , vt , vx ) ⎟ ⎟ f (x, y, z, t) = ⎜ ⎝ + det(vt , vx , vy ) ⎠ . − det(vx , vy , vz )
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Theisel et al. showed in [24] that two classes of seeding points guarantee that all paths of critical points are captured: the intersections of the paths with the domain boundaries, i.e. critical points on the boundaries of the space time domain and fold bifurcations, locations where a pair of critical point emerges or vanishes. Fold bifurcations can be characterized by [ v(x) = (0, 0, 0)T , det(Jv (x)) = 0 ] .
(3)
Applying the Feature Extraction Architecture, we do the following: – Extraction of seeding structures boils down to finding zeros in the following flow fields: for intersections with the domain boundary, find zeros of the 4 3D flow fields v(x, y, z, tmin ) = 0 and v(x, y, z, tmax ) = 0 for the unknowns x, y, z, v(x, y, zmin , t) = 0 and v(x, y, zmax , t) = 0 for the unknowns x, y, t, v(x, ymin , z, t) = 0 and v(x, ymax , z, t) = 0 for the unknowns x, z, t, v(xmin , y, z, t) = 0 and v(xmax , y, z, t) = 0 for the unknowns y, z, t.
As mentioned above, also the fold bifurcation serve as seeding points. To extract those, a 4D zero extraction has to be applied to formula (3). – Trace out f from each of the seeding points to obtain the evolution paths of the critical points. In a postprocessing step remove all lines that are integrated twice, e.g. resulting from stream lines that leave the domain at two different locations. Example: Out-of-core tracking of critical points. Weinkauf et al. showed in [29] how to track critical points in 2D and 3D time-dependent vector fields in an effective out-of-core manner: in one sweep and by loading only two slices at once. We applied this algorithm to a random 2D time-dependent data set. Random vector fields are useful tools for a proof-of-concept of topological methods, since they contain a maximal amount of topological information. Figure 5 shows the execution of the tracking algorithm between two consecutive time steps ti and ti+1 . Example: Cavity. Figure 6 shows the visualization of a vector field describing the flow over a 2D cavity. This data set was kindly provided by Mo Samimy and Edgar
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(a) At ti .
(b) Entries.
(c) Births.
(d) Integration.
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(e) At ti+1 .
Fig. 5. Critical points tracked in one sweep through the data by applying the Feature Flow Fields concept
Caraballo (both Ohio State University) [2] as well as Bernd R. Noack (TU Berlin). 1000 time steps have been simulated using the compressible Navier-Stokes equations; it exhibits a non-zero divergence inside the cavity, while outside the cavity the flow tends to have a quasi-divergence-free behavior. The topological structures of the full data set visualized in Figure 6a elucidate the quasi-periodic nature of the flow. Figures 6b-c show approximately one period – 100 time steps – of the full data set, while Figures 6d-e point out some topological details. Figures 6b-c both reveal the overall movement of the topological structures – the most dominating ones originating in or near the boundaries of the cavity itself. The quasi-divergence-free behavior outside the cavity is affirmed by the fact that a high number of Hopf bifurcations has been found in this area. The tracked closed stream line in Figure 6d starts in a Hopf bifurcation and ends in another one – thereby enclosing a third Hopf. Figure 6e shows a detailed view of time step 22, where a saddle connection has been detected. In the front of this figure a sink is going to join and disappear with a saddle, which just happened to enter at the domain boundary. 5.2 Vortex Core Line Extraction and Tracking We apply the Feature Extraction Architecture to Vortex Core Line Extraction. While [12] gives a good overview of existing vortex core line definitions, we use the most prominent technique by Sujudi and Haimes [18]. Using the notation of [12] and denoting w1 := v, w2 := ∇v · v, we define a vortex core line as locations where w 1 w2 ,
(4)
where denotes vector parallelity and v is a time dependent flow field as in (1). In this setting, the Feature Extraction Architecture can solve different tasks: 1. Extract vortex core lines at some time step t0 . 2. Track a given vortex core line in time, i.e., given a vortex core line at some time t0 , compute the evolution path of this vortex core line in the 4D-spacetime domain. This will assemble a surface. 3. Extract the complete vortex core line surface from 2 at once and use it for vortex core line display and tracking in time.
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(b) Stream line oriented topology of the first 100 time steps.
(c) Path line oriented topology of the first 100 time steps.
(d) Tracked closed stream line starting and ending in a Hopf bifurcation.
(e) Detail view with a saddle connection and a fold bifurcation.
Fig. 6. 2D time-dependent flow at a cavity. The datasets consists of 1000 time steps which have been visualized in (a). All other images show the first 100 time steps.
Spatial Extraction of Vortex Core Lines. By (4), a point x is on a vortex core line, whenever ⎛ ⎞ k(x, y, z, t) (5) s(x, y, z, t) := ⎝ m(x, y, z, t) ⎠ := v × ∇v · v = w1 × w2 = 0. n(x, y, z, t) Given a point x0 = (x0 , y0 , z0 , t0 )T ∈ D = [xmin , xmax ] × [ymin , ymax ] × [zmin , zmax ] × [tmin , tmax ] on a vortex core line (i.e. s(x0 ) = 0), we can trace stream lines of the following feature flow field f from [20] to extract vortex core lines from the seed point x0 at time t0 :
A Unified Feature Extraction Architecture
⎛ ⎞ ⎛ ⎞ e det(sy , sz , a) f (x, y, z, t0 ) = ⎝ f ⎠ = ⎝ det(sz , sx , a) ⎠ . det(sx , sy , a) g
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(6)
For the choice of a we refer to [20]. In the notation of the Feature Extraction Architecture, the complete skeleton of vortex core lines at t0 can be extracted as follows: – From [20] we know, that all vortex core lines are either closed or cross the boundary. Therefore, we extract as starting points all intersections of the vortex core lines with the 2D spatial domain boundary [xmin , y, z, t0 ]∪[xmax , y, z, t0 ]∪[x, ymin , z, t0 ]∪ [x, ymax , z, t0 ] ∪ [x, y, zmin , t0 ] ∪ [x, y, zmax , t0 ]∪ at timestep t0 , e.g. at xmin : s(xmin , y, z, t0 ) = (0, 0, 0)T .
(7)
This is a function R2 → R3 with isolated zeros due to the dependencies of the components in the cross product (5). Closed vortex core lines can be detected by finding isolated zeros in the field [ s(x) = (0, 0, 0)T , e(x) = 0 ],
(8)
a function R3 → R4 , see again [20] for details. – Given those seeding points, we can extract all vortex core lines at time step t0 by tracing stream lines of f . Tracking of Vortex Core Lines in Time. For any vortex core line at a given time t0 , [20] shows that stream lines of g seeded from the vortex core line tracks the temporal evolution of the vortex core line: h×f h×f g(x, y, z, t) = = (9) f 2 e2 + f 2 + g 2 with
⎞ det(sx , st , a) h(x, y, z, t) = ⎝ det(sy , st , a) ⎠ . det(sz , st , a) ⎛
(10)
A complete Vortex Core Line Skeleton. In the 4D space time domain D, the vortex core lines build surface structures. In [20] a detailed algorithm is given, how this surface structure can be extracted based on a bifurcation analysis of the above feature flow field. In the Feature Extraction Architecture notation, the algorithm reads as follows: – Compute the seeding structures: 1. Compute the intersection curves of the vortex core line surface with the spatial boundaries of D. This can be done by spatial extraction of vortex core lines as explained above. 2. Extract all local bifurcations introduced in [20] by finding zeros of some function R4 → R4 .
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(a) Shortly before.
(b) The event.
(c) Shortly after.
Fig. 7. Example of the visualization of vortex core line surfaces. Shown is a saddle bifurcation of vortex core lines. The surfaces are displayed bright for future, dark for past times.
3. Extract closed vortex core lines at the times t = tmin and t = tmax respectively, pick a point on each extracted closed line, and apply a stream line integration of g starting from them. 4. Start a stream line integration of g from all inflow boundary bifurcations (zeros of some function R3 → R3 ). – Extract and visualize the vortex core line surface for a time interval [t0 , t1 ] with tmin ≤ t0 ≤ t1 ≤ tmax ): 1. Load the seeding lines obtained above. 2. Identify all parts of the seeding lines with t-values between t0 and t1 . 3. Starting from these seeding lines, apply a stream surface integration of f until it leaves D or returns to its starting point. 4. Visualize the stream surfaces obtained in 3. As projecting the complete vortex core surface to space leads to self-intersections already in quite simple settings, we use the following approach to visualize the evolution of vortex core line structures: at a given time we draw the vortex core lines as solid tubes inside the vortex core surface that is displayed only for a certain time range for future and past. At the boundary of the space domain the corresponding seeding lines are given for a larger time interval. Both the surfaces and the seeding lines fade out away from the current time. We use color coding to indicate past (dark) and future (bright). Figure 7 shows the evolution of a specific inner bifurcation called saddle bifurcation. Note that the width of the surface in figures 7a and 7c confirms the intuition that the most drastic movements of the vortex core line over time takes place near the bifurcation points. Example: Flow behind a Circular Cylinder. Figures 8 and 9 demonstrate results of the Unified Feature Extraction Architecture of vortex core line extraction in a flow behind a circular cylinder. The data set was derived by Bernd R. Noack (TU Berlin) from a direct numerical Navier Stokes simulation by Gerd Mutschke (FZ Rossendorf). It resolves the so called ‘mode B’ of the 3D cylinder wake at a Reynolds number of 300 and a spanwise wavelength of 1 diameter. The data is provided on a 265 × 337 × 65 curvilinear grid as a low-dimensional Galerkin model [11] [33]. The examined time
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Fig. 8. Flow behind a circular cylinder. Shown are vortex core lines in a certain frame of reference. Their evolution over time is tracked by our algorithm and depicted using transparent surfaces. Dark color encodes the past while bright shows the future.
Fig. 9. Flow behind a circular cylinder. The extracted seeding lines elucidate the alternating evolution of the vortical structures in transverse direction.
range is [0, 2π]. The flow exhibits periodic vortex shedding leading to the well known von K´arm´an vortex street. This phenomenon plays an important role in many industrial applications, like mixing in heat exchangers or mass flow measurements with vortex counters. However, this vortex shedding can lead to undesirable periodic forces on obstacles, like chimneys, buildings, bridges and submarine towers.
6 Conclusions In this paper we exemplified that a rich variety of flow features can be extracted and tracked by a combination of only three core algorithms, namely finding zeros, integrating and intersecting stream objects. The so-defined Unified Feature Extraction Architecture builds upon the concepts of Feature Flow Fields and Connectors.
Acknowledgments We thank Bernd R. Noack for the fruitful discussions and providing the Galerkin model of the cylinder data set. All visualizations in this paper have been created using A MIRA – a system for advanced visual data analysis [17] (see http://amira.zib.de/).
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References [1] D.C. Banks and B.A. Singer. A predictor-corrector technique for visualizing unsteady flow. IEEE Transactions on Visualization and Computer Graphics, 1(2):151–163, 1995. [2] E. Caraballo, M. Samimy, and DeBonis J. Low dimensional modeling of flow for closedloop flow control. AIAA Paper 2003-0059. [3] W. de Leeuw and R. van Liere. Collapsing flow topology using area metrics. In Proc. IEEE Visualization ’99, pages 149–354, 1999. [4] C. Garth, X. Tricoche, and G. Scheuermann. Tracking of vector field singularities in unstructured 3D time-dependent datasets. In Proc. IEEE Visualization 2004, pages 329–336, 2004. [5] A. Globus, C. Levit, and T. Lasinski. A tool for visualizing the topology of threedimensional vector fields. In Proc. IEEE Visualization ’91, pages 33–40, 1991. [6] J. Helman and L. Hesselink. Representation and display of vector field topology in fluid flow data sets. IEEE Computer, 22(8):27–36, August 1989. [7] J. Hultquist. Constructing stream surfaces in steady 3D vector fields. In Proc. IEEE Visualization ’92, pages 171–177, 1992. [8] J. Jeong and F. Hussain. On the identification of a vortex. J. Fluid Mechanics, 285:69–94, 1995. [9] S.K. Lodha, J.C. Renteria, and K.M. Roskin. Topology preserving compression of 2D vector fields. In Proc. IEEE Visualization 2000, pages 343–350, 2000. [10] N. Max, B. Becker, and R. Crawfis. Flow volumes for interactive vector field visualization. In Proc. Visualization 93, pages 19–24, 1993. [11] B.R. Noack and H. Eckelmann. A low-dimensional galerkin method for the threedimensional flow around a circular cylinder. Phys. Fluids, 6:124–143, 1994. [12] R. Peikert and M. Roth. The parallel vectors operator - a vector field visualization primitive. In Proc. IEEE Visualization 99, pages 263–270, 1999. [13] F.H. Post, B. Vrolijk, H. Hauser, R.S. Laramee, and H. Doleisch. Feature extraction and visualisation of flow fields. In Proc. Eurographics 2002, State of the Art Reports, pages 69–100, 2002. [14] M. Roth and R. Peikert. Flow visualization for turbomachinery design. In Proc. Visualization 96, pages 381–384, 1996. [15] M. Roth and R. Peikert. A higher-order method for finding vortex core lines. In D. Ebert, H. Hagen, and H. Rushmeier, editors, Proc. IEEE Visualization ’98, pages 143–150, Los Alamitos, 1998. IEEE Computer Society Press. [16] G. Scheuermann, H. Kr¨uger, M. Menzel, and A. Rockwood. Visualizing non-linear vector field topology. IEEE Transactions on Visualization and Computer Graphics, 4(2):109–116, 1998. [17] D. Stalling, M. Westerhoff, and H.-C. Hege. Amira: A highly interactive system for visual data analysis. The Visualization Handbook, pages 749–767, 2005. [18] D. Sujudi and R. Haimes. Identification of swirling flow in 3D vector fields. Technical report, Department of Aeronautics and Astronautics, MIT, 1995. AIAA Paper 95-1715. [19] H. Theisel. Designing 2D vector fields of arbitrary topology. Computer Graphics Forum (Eurographics 2002), 21(3):595–604, 2002. [20] H. Theisel, J. Sahner, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Extraction of parallel vector surfaces in 3d time-dependent fields and application to vortex core line tracking. In Proc. IEEE Visualization 2005, pages 631–638, 2005. [21] H. Theisel and H.-P. Seidel. Feature flow fields. In Data Visualization 2003. Proc. VisSym 03, pages 141–148, 2003.
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[22] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Saddle connectors - an approach to visualizing the topological skeleton of complex 3D vector fields. In Proc. IEEE Visualization 2003, pages 225–232, 2003. [23] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Grid-independent detection of closed stream lines in 2D vector fields. In Proc. Vision, Modeling and Visualization 2004, 2004. [24] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Topological methods for 2D timedependent vector fields based on stream lines and path lines. IEEE Transactions on Visualization and Computer Graphics, 11(4):383–394, 2005. [25] X. Tricoche, G. Scheuermann, and H. Hagen. Continuous topology simplification of planar vector fields. In Proc. Visualization 01, pages 159 – 166, 2001. [26] Xavier Tricoche, Christoph Garth, Gordon Kindlmann, Eduard Deines, Gerik Scheuermann, Markus Ruetten, and Charles Hansen. Visualization of intricate flow structures for vortex breakdown analysis. In Proc. IEEE Visualization 2004, pages 187–194, 2004. [27] T. Weinkauf, H. Theisel, H.-C. Hege, and H.-P. Seidel. Boundary switch connectors for topological visualization of complex 3D vector fields. In Data Visualization 2004. Proc. VisSym 04, pages 183–192, 2004. [28] T. Weinkauf, H. Theisel, H.-C. Hege, and H.-P. Seidel. Topological construction and visualization of higher order 3D vector fields. Computer Graphics Forum (Eurographics 2004), 23(3):469–478, 2004. [29] T. Weinkauf, H. Theisel, H.-C. Hege, and H.-P. Seidel. Feature flow fields in out-of-core settings. In Proc. Topo-In-Vis 2005, Budmerice, Slovakia, 2005. [30] T. Weinkauf, H. Theisel, K. Shi, H.-C. Hege, and H.-P. Seidel. Topological simplification of 3d vector fields by extracting higher order critical points. In Proc. IEEE Visualization 2005, pages 559–566, 2005. [31] R. Westermann, C. Johnson, and T. Ertl. Topology-preserving smoothing of vector fields. IEEE Transactions on Visualization and Computer Graphics, 7(3):222–229, 2001. [32] T. Wischgoll and G. Scheuermann. Detection and visualization of closed streamlines in planar flows. IEEE Transactions on Visualization and Computer Graphics, 7(2):165–172, 2001. [33] H.-Q. Zhang, U. Fey, B.R. Noack, M. K¨onig, and H. Eckelmann. On the transition of the cylinder wake. Phys. Fluids, 7(4):779–795, 1995.
Control of Wing Vortices I. Gursul, E. Vardaki, P. Margaris, and Z. Wang Department of Mechanical Engineering University of Bath Bath, BA2 7AY, United Kingdom
Summary Vortex control concepts employed for slender, nonslender and high aspect ratio wings were reviewed. For slender delta wings, control of vortex breakdown has been the most important objective, which is achieved by modifications to swirl level and pressure gradient. Delay of vortex breakdown with the use of control surfaces, blowing, suction, high-frequency and low-frequency excitation, and feedback control was reviewed. For nonslender delta wings, flow reattachment is the most important aspect for flow control methods. For high aspect ratio wings, vortex control concepts are diverse, ranging from drag reduction to attenuation of wake hazard and noise, which can be achieved by modifications to the vortex location, strength, and structure, and generation of multiple vortices.
1 Introduction Controlling vortical flows over wings may have various benefits, such as enhancement of lift force, generation of forces and moments for flight control, attenuation of buffeting, reduction of drag, and attenuation of noise due to vortex/blade interaction. Control methods include manipulation of one or more of the following flow phenomena: flow separation from the wing, separated shear layer, vortex formation, flow reattachment on the wing surface, and vortex breakdown. The occurrence and relative importance of these phenomena strongly depend on the wing sweep angle. For delta wings, flow reattachment and vortex breakdown are two important phenomena, which determine the effective flow control strategies. The reattachment location on the wing surface moves inboard with increasing angle of attack, and reaches the wing centreline at a particular incidence. Beyond this limiting angle of attack αFR, flow reattachment to the wing surface is not possible. The variation of predictions [1] of αFR with wing sweep angle is shown in Figure 1. This prediction is only valid for slender wings, and the only experimental observation [2] for a nonslender wing (by visualization of the reattachment line for a sweep angle of Λ=50°) is also given in Figure 1. For slender wings, it is seen that the reattachment incidence decreases with increasing wing sweep angle. Therefore, on highly swept wings, reattachment does not occur beyond small angles of attack. Vortex breakdown appears on the wing with increasing angle of attack and crosses the trailing-edge at a particular incidence αBD. The variation of this angle of attack R. King (Ed.): Active Flow Control, NNFM 95, pp. 137–151, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007
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ANGLE OF ATTACK [deg]
[3] with wing sweep angle is also shown in Figure 1. It is seen that this angle of attack increases with increasing wing sweep angle. On nonslender wings, vortex breakdown appears at very small incidences. As the vortex control concepts become increasingly diverse, new actuators and closed-loop control strategies are being developed. It is useful to consider the flow physics and dominant mechanisms as these determine which flow control methods are effective. The main objective of this paper is to review the vortex control concepts, which mainly depend on the wing sweep.
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2 Slender Delta Wings 2.1 Overview of Flow Physics The flow over a delta wing is characterised by a pair of counter-rotating leading-edge vortices that are formed by the roll-up of vortex sheets. The time-averaged axial velocity is jet-like at low and moderate incidences. The large axial velocities in the vortex core are due to very low pressures, which also generate additional suction and lift force, known as vortex lift, on the delta wings. Vortex lift contribution increases with wing sweep angle [4]. At a sufficiently high angle of attack, the vortices undergo a sudden expansion known as vortex breakdown. The axial flow downstream becomes wake-like with very low velocities. For slender wings (defined as Λ ≥ 65° in this paper), vortex breakdown is the dominant flow mechanism that is responsible for decreased lift. It is also the dominant source of unsteadiness that causes wing and fin buffeting [5]. Hence control of vortex breakdown has been the subject of many investigations [6]. There are two important parameters affecting the occurrence and movement of vortex breakdown: swirl level and pressure gradient affecting the vortex core. An increase in the magnitude of either parameter promotes the earlier occurrence of breakdown. Very early experiments [7] demonstrated that vortex breakdown moves upstream over delta wings when the magnitude of either parameter is increased.
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More recently, it was shown [8] that the minimum swirl level required for breakdown decrease with increasing magnitude of adverse pressure gradient. Naturally, flow control methods for the delay of vortex breakdown rely on modification of these two parameters. Active flow control methods generally rely on unsteady excitation in flow control applications. Unsteady forcing has been used for the control of vortex and breakdown in some cases. There are various sources of unsteadiness [5]: shear layer instabilities, vortex wandering, helical mode instability of vortex breakdown, oscillations of breakdown, vortex interaction, and vortex shedding. The frequency spectrum of the unsteady flow phenomena that exist over stationary wings is very wide, which is one of the challenges in numerical simulations of these flows. Vortex breakdown, vortex interactions, and vortex shedding, either alone or in combination, play an important role in wing and fin buffeting, although vortex breakdown is the main source of buffeting over slender wings. Flow control approaches for vortex breakdown are reviewed next. 2.2 Control Surfaces Various control surfaces [6] have been investigated to control the formation, location, strength, and breakdown of the leading-edge vortices: canards, strakes, leading-edge flaps, apex flap, and variable-sweep wings. It is well known that canards can provide substantial delays [9] in vortex breakdown by affecting the external pressure gradient acting on the vortex core. Since all of the vorticity of leading-edge vortices originates from the separation point along the leading-edge, leading-edge flaps are particularly attractive tools that can be used to influence the strength and structure of these vortices. Leading-edge flaps that are deflected downward have been found to reduce drag and improve the lift-to-drag ratio [10]. On the other hand, the flap deflection in the upward direction causes an increase in lift as well as drag. This type of vortex management can be used for landing or aerodynamic manoeuvres [10]. It has been found that the flaps deflected upward generate a stronger vortex lift at low and moderate angles of attack. However, flaps may also induce vortex breakdown [11]. Leading-edge flaps modify the strength and location of the vortices, thereby affecting the parameters that control vortex breakdown. It was shown [11] that breakdown location and its sensitivity strongly depend on incidence and flap angle. For large angles of attack, the variation of breakdown location is not monotonic, and therefore, not suitable for control purposes. A variable leading-edge extension [12] that effectively varies the sweep angle has been used to control leading-edge vortices and breakdown. The advantage of this method is that the variation of breakdown with sweep angle is monotonic, hence suitable for active control purposes. Because most of the vorticity within the vortex core originates from a small region near the apex of the wing, an apex flap can be an effective control surface. It was shown [13] that a drooping apex flap could delay vortex breakdown.
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2.3 Blowing and Suction Blowing and suction applied at various locations are commonly used as flow control methods for leading-edge vortices and breakdown. The most widely used versions include: (a) leading-edge suction/blowing, (b) trailing-edge blowing, (c) along-thecore blowing. These are discussed in detail below. Since the vorticity of the leading-edge vortices originates from the separation line along the leading-edge, control of separation characteristics or shear layer can be used to influence the strength and location of the vortices as well as the location of vortex breakdown. Steady blowing [14, 15] and suction [15, 16] at the leading-edge has been employed, but these methods differ in terms of their effect on swirl level. While blowing, in particular tangential blowing, may increase the swirl level, suction reduces the strength and swirl level due to removal of some of the vorticity shed from the leading-edge. Figure 2(a) shows how the location of shear layer and vortex is modified upstream of breakdown when suction is applied. Figure 2(b) shows the axial velocity contours at the trailing-edge, which show the change from wake-like to jet-like velocity as a result of the delay of vortex breakdown. Detailed measurements [16] show that maximum swirl angle in the core and circulation decrease with suction, which causes the vortex breakdown location to move downstream. It is also worth noting that the leading-edge suction technique does not require thick rounded leading edges. Control of vortices can be achieved without the use of the Coanda jet effect.
(a)
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Fig. 2. Variation of (a) rms axial velocity upstream of breakdown location; (b) time-averaged axial velocity at x/c = 1.0
The effect of trailing-edge jets on wing vortices and vortex breakdown has been investigated in several studies [17-20]. Blowing at the trailing-edge also modifies the external pressure gradient and causes delay of vortex breakdown. Favourable effect of a trailing-edge jet [17] could be observed even in the presence of a fin, which produces a strong adverse pressure gradient for a leading-edge vortex. It was shown that fin-induced vortex breakdown can be delayed even for the head-on collision of the leading-edge vortex with the fin [19]. Hence the adverse pressure gradient caused by the presence of the fin could be overcome with a trailing-edge jet. The
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effectiveness of a trailing-edge depends on the wing sweep angle [20]. It appears that it becomes more difficult to delay vortex breakdown with decreasing sweep angle. Along-the-core blowing [21, 22] accelerates the axial flow in the core, and modifies the pressure gradient favourably. Figure 3 shows the effectiveness of various blowing/suction methods from various studies published in the literature. Here the effectiveness is defined as (ǻxbd/c)/Cȝ, where ǻxbd is the change in breakdown location (positive corresponding to delay), c is the chord length and Cµ is the momentum coefficient. Figure 3 shows that along-the-core blowing is the most effective method in terms of delaying vortex breakdown. This can be attributed to the importance of the pressure gradient affecting the vortex core. It is also seen that the effectiveness of blowing and suction is nearly the same. The effectiveness of trailingedge blowing is the lowest among all blowing methods considered. 2
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2.4 Unsteady Control There have been various attempts to control vortices and breakdown by using unsteady excitation. These include small and large amplitude oscillations of leadingedge flaps, periodic variations of sweep angle, periodic suction or blowing, and combined use of leading-edge flaps and intermittent trailing-edge blowing, which are summarized in Reference [5]. These studies fall into two categories: (i) highfrequency excitation, St = fc/U∞ = O(1), (where f is the frequency and U∞ is the free stream velocity) and (ii) low-frequency excitation, St = O(0.1). For the high-frequency excitation, Gad-el-Hak and Blackwelder [23] applied periodic perturbations of injection and suction along the leading-edge of a delta wing (Λ = 60°). They found maximum changes in the evolution of the shear layer when the frequency of perturbations (St = 5.5) is the subharmonic of the frequency of KelvinHelmholtz instability. However, no results were reported regarding the structure of the main vortex and the effect on vortex breakdown. Gu et al. [24] applied periodic suction-blowing in the tangential direction along the leading-edge of the wing (Λ = 75°) and reported a delay of vortex breakdown. The most effective period of the alternate suction-blowing corresponded to fc/U∞ = 1.3. For a less slender wing
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(Λ = 60°), it was shown that oscillatory blowing at the leading edge can enhance the lift at high angles of attack [25], and optimum reduced frequency varied in the range of fc/U∞ = 1 to 2. For small amplitude flap oscillations, the strength of the vortices was larger than that of the quasi-steady case [26], when the excitation frequency was St = 1.2. This range of effective frequencies presumably corresponds to subharmonics of the Kelvin-Helmholtz instability due to vortex pairing. For the low-frequency excitation, there have been reports [5] of increased vortex circulation and delay of vortex breakdown for oscillating flaps, delay of vortex breakdown for harmonic variations of sweep angle for a variable sweep wing, and delay of breakdown for combined flaps and unsteady trailing-edge blowing. When the spectra of unsteady flow phenomena [5] are considered, it is less clear which unsteady phenomena are exploited for the low-frequency excitation. However, it is suggested [27] that the variations in the external pressure gradient generated by unsteady excitation plays a major role. 2.5 Feedback Control The pressure fluctuations induced by the helical mode instability of vortex breakdown can be measured and used as a feedback signal for active control. In such a approach, the rms value of pressure is chosen as the control variable and a feedback control strategy is considered [28]. The monotonic variation of the amplitude of the pressure fluctuations with vortex breakdown location makes the feedback control possible. In identifying a suitable flow controller, several methods were considered, including blowing, suction, and flaps. However, the relationship between the vortex breakdown location and control parameter is unknown or undesirable (i.e., not monotonic) for these methods. A desirable controller should have a monotonic relationship between the control parameter and breakdown location. It was shown [28] that it was feasible to use a variable sweep angle mechanism as a means of controlling the breakdown location by influencing the circulation of the leading-edge vortex. The system was idealized as a first-order system, and integral control was used. The feedback control of breakdown was demonstrated for stationary as well as pitching delta wings.
3 Nonslender Delta Wings 3.1 Overview of Flow Physics Vortical flow over nonslender delta wings (Λ ≤ 55°) has recently become a topic of increased interest in the literature. While the flow topology over more slender wings, typically Λ ≥ 65°, has been extensively studied and is now reasonably well understood, the flow over lower sweep wings has only recently attracted more attention [29]. Vortical flows develop at very low angles of attack, and form close to the wing surface. One of the distinct features of nonslender wings is that reattachment of the separated flow is possible even after breakdown reaches the apex of the wing. However, at large angles of attack in the post-stall region, reattachment
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is not possible, and completely stalled flow occurs on the wing. Active and passive control of reattachment may be beneficial for lift enhancement in the post-stall region. According to Polhamus’ leading-edge suction analogy [4], the vortex lift contribution becomes a smaller portion of the total lift as the sweep angle decreases. Vortex breakdown occurs over the wing even at small incidences, and there is no obvious correlation between the onset of vortex breakdown over nonslender wings and the change of the lift coefficient. Hence, vortex breakdown is not a limiting phenomenon as far as the lift force is concerned for nonslender wings. On the contrary, flow reattachment is key to any flow control strategy as suggested in Figure 1. 3.2 Passive Control with Wing Flexibility Passive lift enhancement for flexible delta wings has been demonstrated as a potential method for the control of vortex-dominated wing flows [30]. Force measurements over a range of nonslender delta wings (with sweep angles Λ = 40° to 55°) have demonstrated the ability of a flexible wing to enhance lift and delay stall compared with a rigid wing of similar geometry. An example for Λ = 40° is shown in Figure 4a. This recently discovered phenomenon appears to be a feature of nonslender wings. Flow visualization, PIV and LDV measurements show that flow reattachment takes place on the flexible wings in the post-stall region of the rigid wings. The lift increase in the post-stall region is accompanied with large self-excited vibrations of the wings as shown in Figure 4b. The dominant frequency of the vibrations of various nonslender delta wings is St = O(1). These vibrations promote reattachment of the shear layer, which results in the lift enhancement. Various measurements including wing-tip accelerations, rms rolling moment, and hot-wire measurements confirm that the dominant mode of vibrations occur in the second antisymmetric structural mode in the lift enhancement region. The self-excited vibrations are not observed for a halfwing model, hence passive flow control for a flexible wing occurs only in the anti-symmetric mode. 1.5
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Spectral analysis of velocity fluctuations [31] along the shear layer showed large sharp peaks, corresponding to the wing vibrations. There are also broad dominant peaks in the spectra of velocity fluctuations in the range of St = 1 to 5 for the poststall incidences, and these correspond to the shear layer instabilities. The center frequency of these peaks decreases with streamwise distance as the shear layer vortices shed conically. There is also a decrease in the spanwise direction due to the vortex pairing process. The frequency of the structural vibrations is in the same range as these natural frequencies [31]. 3.3 Active Control of Reattachment Rigid delta wings undergoing small amplitude oscillations [32] in the post-stall region exhibit many similarities to flexible wings, including reattachment in the post-stall region (see Figure 5). For simple delta wings and cropped delta wings [31] with Λ = 50°, 40°, and 30°, an optimum frequency around St = 1 was identified for which the reattachment is observed. Note that these dramatic changes are observed with unsteady forcing in the post-stall region, whereas there is little effect in the pre-stall region. Hence active control methods in the form of leading-edge oscillations or blowing can be effective. An important parameter is the wing sweep angle. The effect of excitation on a swept wing is similar to the response of the flow over a backward-facing step [33] to the periodic excitation. However, for zero sweep angle, formation of a closed separation bubble at high angles of attack in the post-stall region is not possible. It seems that moderate sweep angles (around 50°) help the formation of semi-open separation bubbles, hence the wing sweep is beneficial in flow reattachment. However, there is a lower limit of sweep angle below which the beneficial effect of wing sweep will diminish. This lower limit of sweep angle is around Λ = 20°. Symmetric perturbations in the form of small amplitude pitching oscillations (1° amplitude) were studied for Λ = 50° simple delta wing. The results show that symmetric perturbations also promote reattachment and vortex re-formation. Hence, St=0
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for active control purposes, both symmetric and anti-symmetric excitations are effective. However, passive control for a flexible wing occurs only in the antisymmetric mode. 3.4 Vortex Re-formation Within the reattachment region, axial flow may develop, resulting in re-formation of the leading-edge vortices. Figure 6 shows an example of vortex re-formation [32], which occurs when the amplitude of forcing is sufficiently large and the frequency of excitation is near an optimum value. The mean breakdown location becomes a maximum at an optimum frequency as shown in Figure 7.
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The time-averaged vorticity flux increases due to the oscillating leading edge, which leads to increased circulation. Although the leading edge vortices become stronger due to the leading edge motion, vortex breakdown is delayed for the oscillating wing compared to the stationary wing for which breakdown is at the apex. This appears to be in contrast to the well-known studies of vortex breakdown, which indicate that increased strength of vortices should cause premature, rather than
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delayed, breakdown. This result suggests that streamwise pressure gradient might be modified favourably due to the wing motion.
4 High Aspect Ratio Wings 4.1 Vortex Control Concepts Control of formation and development of tip vortices in both near-wake and far-wake has potential benefits in a variety of aerodynamic problems. Tip vortices form in a similar way to the leading-edge vortices over low aspect ratio wings and the roll-up process becomes complete within a few chord-lengths downstream of the trailingedge. There appear to be three main applications of flow control approaches with regard to the tip vortices: 1) reduction of induced drag, 2) attenuation of vortex wake hazard on following aircraft, 3) reduction of helicopter noise due to the blade-vortex interaction. These will be briefly reviewed with regard to vortex control concepts. For reduction of induced drag, various methods were considered [34], including planform shape and span, tip shape, winglets, fences and tip sails. Wing tip devices are used to redistribute the vorticity near the wing tip. Similarly, any flow control method that can displace the tip vortices in the outboard direction can be used to reduce the drag as it is inversely proportional to the square of the spanwise distance between the vortices. In order to attenuate wake vortex hazard on following aircraft, the core size of the vortices can be increased by turbulence injection into the core. Various wing tip devices were tested, which were found to be not much effective in the far-wake. Those that seemed to decrease the vortex hazard (such as decelerating chutes and splines placed behind the trailing-edge) had an unacceptable drag penalty [35]. It was noted [36] that passive devices in the form of turbulence generators could increase the size of the vortex core and also promote cooperative instabilities as an additional benefit. A second and more promising method is to rely on the long–wave instabilities occurring in a system of several vortices [37]. Compared to the alleviation of a single vortex by turbulent diffusion, excitation of the long-wave cooperative instabilities of multiple vortices has the potential for much faster destruction of vortex wakes. Hence deliberate creation of multiple vortices and excitation by oscillating surfaces or oscillatory blowing may be a promising strategy. One of the sources of helicopter noise is the interaction of rotor blades with the vortices shed from preceding blades. Noise and vibration caused by this interaction can be reduced by i) increasing the distance between the rotor blade and tip vortex, and (ii) by increasing the size of the vortex core and decreasing the maximum tangential velocity. 4.2 Tip Blowing Most of the intended modifications to the tip region (such as wing tip devices), to the vortex location, strength, and structure can be achieved without the use of passive devices. Active flow control using wing tip blowing can achieve multiple tasks
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during different flight regimes. It was shown that the strength, location, core structure, and number of vortices can be effectively manipulated by tip blowing [38]. The effect of continuous blowing using high-aspect ratio jets is very sensitive to the blowing direction. The location and strength of the vortices generated by the jet and their interaction with the tip vortex lead to different flow configurations. Blowing in the upward direction produces additional co-rotating vortices in the nearwake as shown in Figure 8. These strong vortices forming on the wing surface can be used to increase the lift in certain applications such as hovering rotorcraft. Figure 9 shows that a diffused vortex is obtained with spanwise blowing near the pressure surface. Also, downward blowing produces diffused vortices. For these cases, blowing appears to add a substantial amount of turbulence in the vortex core, resulting in diffused trailing vortices. For downward blowing, it was possible to displace the vortices in the outboard direction, which should result in an induced drag
Fig. 8. Vorticity distribution for upward blowing
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reduction. Spanwise blowing also results in a lift augmentation [39] as the downwash decreases with blowing. Drag reduction and lift augmentation are essentially inviscid phenomena due to an effective increase in span and aspect ratio of the wing. Synthetic jets can be useful for wing tip blowing as the air supply problem is avoided. Figure 10 shows that a synthetic jet is beneficial in terms of diffusing the trailing vortex [40]. Synthetic jets produce large turbulence and much larger vortex wandering. For the blowing coefficients used, the effect of excitation frequency was minor. While there are substantial changes in the circulation of the trailing vortex for continuous jets, the circulation remains virtually the same for synthetic jets. The intermittent and diffused jet vortices are weak and do not appear to affect the total circulation.
5 Conclusions For slender delta wings, control of vortex breakdown has been the primary goal of many investigations. Delay of vortex breakdown is possible with the modifications to the swirl level and pressure gradient. The use of control surfaces such as leadingedge flaps makes it possible to control the location, strength, and structure of the vortices. Blowing and suction at the leading-edge, trailing-edge, or along the core have differences in terms of their effects on swirl level and pressure gradient affecting the vortex core. Along-the-core blowing is the most effective method for delaying vortex breakdown. Active flow control using high-frequency excitation targets the Kelvin-Helmholtz instability of the separated shear layers. In contrast, low-frequency excitation appears to modify the external pressure gradient only. For nonslender delta wings, flow reattachment is the most important aspect for flow control methods. Passive lift enhancement on flexible wings is due to the selfexcited wing vibrations, which promote flow reattachment in the post-stall region. The frequency of the wing vibrations (St = O(1)) is in the same range as the natural frequencies of the shear layer instabilities. Rigid delta wings undergoing small amplitude oscillations in the post-stall region exhibit many similarities to flexible wings, including reattachment and vortex re-formation. Moderate sweep angles help the formation of semi-open separation bubbles, hence the wing sweep is beneficial.
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For high aspect ratio wings, vortex control concepts are diverse, ranging from drag reduction to attenuation of wake hazard and noise. Modifications to the vortex location, strength, and structure are common ideas in these applications. Active flow control using wing tip blowing is shown to have potential for various objectives. Depending on the blowing configuration and direction, more diffused or stronger vortices, and also, multiple vortices, that move inboard or outboard can be generated. The deliberate creation of multiple vortices and excitation of vortex instabilities may be a promising strategy for effective control in the far-wake.
Acknowledgements The authors acknowledge the financial support of the Air Force Office of Scientific Research (AFOSR), Engineering and Physical Sciences Research Council (EPSRC) and the Ministry of Defence in the UK.
References [1] Mangler, K.W. and Smith, J.H.B., “A Theory of the Flow Past a Slender Delta Wing with Leading Edge Separation”, Proceedings of Royal Society, A, vol. 251, 1959, pp. 200-217. [2] Taylor, G. and Gursul, I., “Buffeting Flows over a Low Sweep Delta Wing”, AIAA Journal, vol. 42, no. 9, September 2004, pp. 1737-1745. [3] Erickson, G.E., “Water-Tunnel Studies of Leading-Edge Vortices”, Journal of Aircraft, vol. 19, No. 6, June 1982, pp. 442-448. [4] Polhamus, E. C., ‘Predictions of vortex lift characteristics by a leading edge suction analogy’, Journal of Aircraft, Vol. 8, No. 4, pp. 193-199, 1971. [5] Gursul, I., “Review of Unsteady Vortex Flows over Slender Delta Wings”, Journal of Aircraft, vol. 42, no. 2, March-April 2005, pp. 299-319. [6] Mitchell, A.M. and Delery, J., “Research into Vortex Breakdown Control”, Progress in Aerospace Sciences”, vol. 37, 2001, pp. 385-418. [7] Lambourne, N.C. and Bryer, D.W., “The Bursting of Leading Edge Vortices: Some Observation and Discussion of the Phenomenon”, Aeronautical Research Council, R&M 3282, 1962. [8] Delery, J., Horowitz, E., Leuchter, O., and Solignac, J.L., “Etudes Fondamentales Sur Les Ecoulements Tourbillonnaires”, La Recherche Aerospatiale, no. 2, 1984, pp. 81-104. [9] Myose, R.Y., Hayashibara, S., Yeong, P.C. and Miller, L.S., “Effects of Canards on Delta Wing Vortex Breakdown During Dynamic Pitching”, Journal of Aircraft, vol. 34, no. 2, March-April 1997, pp. 168-173. [10] Lamar, J.E. and Campbell, J.F., “Vortex Flaps – Advanced Control Devices for Supercruise Fighters”, Aerospace America, January 1984, pp. 95-99. [11] Deng, Q. and Gursul, I., “Effect of Leading-Edge Flaps on Vortices and Vortex Breakdown”, Journal of Aircraft, vol. 33, no. 6, November-December 1996, pp. 10791086. [12] Yang, H. and Gursul, I., “Vortex Breakdown over Unsteady Delta Wings and its Control”, AIAA Journal, vol. 35, no. 3, 1997, pp. 571-574. [13] Klute, S.M., Rediniotis, O.K., and Telionis, D.P., “Flow Control over a Maneuvering Delta Wing at High Angle of Attack”, AIAA Journal, vol. 34, no. 4, 1996, pp. 662-668.
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[14] Wood, N.J., Roberts, L. and Celik, Z., “Control of Asymmetric Vortical Flows over Delta Wings at High Angles of Attack”, Journal of Aircraft, vol. 27, no. 5, May 1990, pp. 429435. [15] Gu, W., Robinson, O. and Rockwell, D., “Control of Vortices on a Delta Wing by Leading-Edge Injection”, AIAA Journal, vol. 31, no. 7, July 1993, pp. 1177-1186. [16] McCormick, S. and Gursul, I., “Effect of Shear Layer Control on Leading Edge Vortices”, Journal of Aircraft, vol. 33, no. 6, November-December 1996, pp. 1087-1093. [17] Helin, H.E. and Watry, C.W., “Effects of Trailing-Edge Jet Entrainment on Delta Wing Vortices”, AIAA Journal, vol. 32, no. 4, 1994, pp. 802-804. [18] Shih, C. and Ding, Z., “Trailing-Edge Jet Control of Leading-Edge Vortices of a Delta Wing”, AIAA Journal, vol 34, no(7), 1996, 1447-1457. [19] Phillips, S., Lambert, C., and Gursul, I., “Effect of a Trailing-Edge Jet on Fin Buffeting”, Journal of Aircraft, vol. 40, no. 3, 2003, pp. 590-599. [20] Wang, Z. and Gursul, I., “Effects of Jet/Vortex Interaction on Delta Wing Aerodynamics”, 1st International Conference on Innovation and Integration in Aerospace Sciences, 4-5 August 2005, Queen’s University Belfast, UK. [21] Guillot, S., Gutmark, E.J., and Garrison, T.J., “Delay of Vortex Breakdown over a Delta Wing via Near-Core Blowing”, AIAA 98-0315, 36th Aerospace Sciences Meeting and Exhibit, January 12-15, 1998, Reno, NV. [22] Mitchell, A.M., Barberis, D., Molton, P., and Delery, J., “Oscillation of Vortex Breakdown Location and Blowing Control of Time-Averaged Location”, AIAA Journal, vol. 38, no. 5, May 2000, pp. 793-803. [23] Gad-el-Hak, M. and Blackwelder, R.F., “Control of the Discrete Vortices from a Delta Wing”, AIAA Journal, vol. 25, no. 8, 1987, pp. 1042-1049. [24] Gu, W., Robinson, O. and Rockwell, D., “Control of Vortices on a Delta Wing by Leading-Edge Injection”, AIAA Journal, vol. 31, no. 7, July 1993, pp. 1177-1186. [25] Margalit, S., Greenblatt, D., Seifert, A. and Wygnanski, I., “Delta Wing Stall and Roll Control Using Segmented Piezoelectric Fluidic Actuators”, Journal of Aircraft, vol. 42, no. 3, 2005, pp. 698-709. [26] Deng, Q. and Gursul, I., “Effect of Oscillating Leading-Edge Flaps on Vortices over a Delta Wing”, AIAA 97-1972, 28th AIAA Fluid Dynamics Conference, June 29 – July 2, 1997, Snowmass Village, CO. [27] Yang, H. and Gursul, I., “Vortex Breakdown over Unsteady Delta Wings and Its Control”, AIAA Journal, vol. 35, no. 3, 1997, pp. 571-574. [28] Gursul, I., Srinivas, S. and Batta, G., “Active Control of Vortex Breakdown over a Delta Wing”, AIAA Journal, vol. 33, no. 9, 1995, pp. 1743-1745. [29] Gursul, I., Gordnier, R., and Visbal, M., “Unsteady Aerodynamics of Nonslender Delta Wings”, Progress in Aerospace Sciences, vol. 41, 2005, pp. 515-557. [30] Taylor, G., Kroker, A. and Gursul, I., “Passive Flow Control over Flexible Nonslender Delta Wings”, AIAA-2005-0865,43rd Aerospace Sciences Meeting and Exhibit Conference,10-13 January 2005,Reno,NV. [31] Gursul, I., Vardaki, E. and Wang, Z., “Active and Passive Control of Reattachment on Various Low-Sweep Wings”, AIAA-2006-506, 44th AIAA Aerospace Sciences Meeting and Exhibit, 9-12 January 2006, Reno, NV. [32] Vardaki, E., Gursul, I. and Taylor, G., “Physical Mechanisms of Lift Enhancement for Flexible Delta Wings”, AIAA-2005-0867, 43rd Aerospace Sciences Meeting and Exhibit, 10-13 January 2005, Reno, NV.
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[33] Roos, F.W. and Kegelman, J.T., “Control of Coherent Structures in Reattaching Laminar and Turbulent Shear Layers”, AIAA Journal, vol. 24, no. 12, December 1986, pp. 19561963. [34] Kroo, I., “Drag Due to Lift: Concepts for Prediction and Reduction”, Annual Review of Fluid Mechanics, vol. 33, 2001, pp. 587-617. [35] Spalart, P.R., “Airplane Trailing Vortices”, Annual Review of Fluid Mechanics, vol. 30, 1998, pp. 107-138. [36] Coustols, E., Stumpf, E., Jacquin, L., Moens, F., Vollmers, H., Gerz, T., “Minimised Wake: a Collaborative Research Programme on Aircraft Wake Vortices”, AIAA 2003-0938, 41st Aerospace Sciences Meeting and Exhibit, 6-9 January 2003, Reno, NV. [37] Jacquin, L., Fabre, D., Sipp, D., Theofilis, V., and Vollmers, H., “Instability and Unsteadiness of Aircraft Wake Vortices”, Aerospace Science and Technology, vol. 7, 2003, pp. 577-593. [38] Margaris, P., Gursul, I., “Effect of Steady Blowing on Wing Tip Flowfield,” AIAA 20042619. 2nd Flow Control Conference, Portland, Oregon, USA. June-July 2004. [39] Tavella, D. A., Wood, N J., Lee, C. S., Roberts, L., “Lift Modulation with Lateral WingTip Blowing,” Journal of Aircraft, Vol. 25, No. 4, 1988, pp 311-316. [40] Margaris, P., Gursul, I., “Wing Tip Vortex Control Using Synthetic Jets”, CEAS/KATnet Conference on Key Aerodynamic Technologies, 20-22 June 2005, Bremen, Germany.
Towards Active Control of Leading Edge Stall by Means of Pneumatic Actuators C.J. K¨ ahler, P. Scholz, J. Ortmanns, and R. Radespiel Technische Universit¨ at Braunschweig, Institut f¨ ur Str¨ omungsmechanik, Bienroder Weg 3, 38106 Braunschweig, Germany
[email protected] Summary This contribution summarizes the flow control research results obtained at TU Braunschweig and their implication for control on high-lift devices. The superordinate aim of the examination is the control of leading-edge stall on a two-element airfoil by means of dynamic 3D actuators. This is of great practical interest in order to increase the maximum angle of attack and/or the lift coefficient and to alter the drag coefficient in takeoff and landing configuration of future aircrafts. To reach this aim, several pneumatic actuators were designed and systematically tested to determine their characteristics and impulse response on the input signal at first. Secondly, their potential for active flow control was investigated in a small wind tunnel. Thirdly, the interaction of promising actuator concepts was studied in detail to examine the benefit of actuator arrays. Finally, the experiences were combined to examine the potential of the actuators in delaying leading edge separation on a generic airfoil. Therefore, an appropriate airfoil was designed, built and equipped with the developed actuator technology and investigated in a large wind tunnel. The results illuminate the potential of dynamic actuators for high-lift applications and the effect of different actuator parameters (actuator design, orientation, spacing and position as well as amplitude, frequency and duty-cycle). Furthermore, practical actuator designs and operation rules are deduced from the examination. In the future they may be assistant to assign the results on real aircraft configurations.
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The active control of leading-edge stall by means of dynamic actuators is of great industrial interest because of the ability to increase the maximum angle of attack and thus the lift and to alter the drag in takeoff and landing configuration. Hence, it becomes possible to reduce the number of gaps in high-lift configuration (slat-less wing). This reduces the noise during takeoff and landing and decreases the drag during the climb flight period. In addition it becomes possible to establish new low noise takeoff and landing approach procedures. Furthermore, the slat-less wing may enable laminar flow conditions during cruise flight which is important to reduce fuel consumption. Thus the potential of active control by R. King (Ed.): Active Flow Control, NNFM 95, pp. 152–172, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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means of dynamic actuators can lead to transport aircrafts with reduced design, development, fabrication and maintenance costs and with less impact on the environment. The active control by means of pneumatic actuators has been examined extensively [5]. Basically there are two control strategies, namely control by stimulating natural instabilities[21] and control by enhancing turbulent mixing. The stimulation of instabilities is quite efficient, e.g. in preventing stall on the flap of a high-lift configuration [18]. Unfortunately, this approach fails when the flow state is stable or the excitation frequency unmatched to the frequency of the instable modes. More robust is the transfer high momentum fluid from the outer part of the boundary layer toward the wall by mixing. Another advantage is the fact, that for the stimulation of instabilities dynamic pulsing is an indispensable part of the mechanism, whereas the turbulent mixing does work with either dynamic or static blowing. The actuation can be done directly, by transferring momentum into the desired region by means of tangential blowing [1], or indirectly. The first method is not applicable at cruise flight conditions because the efficiency decreases with increasing base flow velocity. In the second case, the blockage of the actuator jet is exploited to enhance the mixing of the base flow, similar to mechanical vortex generators [11]. In the past many attempts have been made to replace the stationary blowing actuators in favour of dynamic ones [6,13]. The motivation for applying these actuators is the efficiency gain, due to the reduced mass flux, and the excitation of span-wise oriented vortices, similar to starting vortices, which promote the mixing. An increase of approximately 30% of peak vorticity and a penetration of 50 % farther into the boundary layer can be achieved according to [9]. However, due to the large number of control parameters the optimization is difficult. Compton and Johnston varied the skew angle of a 45◦ inclined round jet to demonstrate that a skew angle of 90◦ is an optimal configuration to produce a strong longitudinal vortex [2]. McManus et al. showed that the ratio between the the boundary layer thickness and the actuator diameter should be around four for an efficient control [13]. Nagib et al. concluded that the reduced frequency must be approximately one [15]. In addition they describe that the most effective location for unsteady forcing is near but upstream of the point of separation. Unfortunately, the mutual dependency of the influencing factors is an open question along with the universality and assignability of the findings because of the different boundary conditititons. This will be examined systematically by determining the 1. 2. 3. 4.
impulse response of dynamic actuators working principle of single actuators in boundary layers synergy effects resulting from the interaction of actuator arrays aerodynamic potential for separation control on high-lift configurations.
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To assess the potential of pneumatic actuators for flow control applications their impulse response in terms of amplitude pV , frequency f , duty cycle ∆, waveform,
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settling chamber volume V and feed line length lS was examined at first. This is important for characterizing the actuators and the experimental boundary conditions. The dynamical switching was realized with a FESTO MH2 electromechanical fast-switching valve. This device allows to vary frequency and amplitude independently and it can operate at reasonably high frequencies (fmax ≈ 150 Hz) and adequate feeding pressures (pmax ≈ 8.0 bar). The input-signal was provided by a 16 channel custom built frequency generator. Each channel phasing and duty-cycle can be adjusted independently from the other channels. The generator is based on a 16 MHz EEPROM and is configurable via a dot matrix display or a serial port (RS232). The frequency generator delivers square-wave signals whose amplitude can be adjusted to the specifications of the magnetic valves by using two custom-built 8-channel amplifiers. To characterize the impulse response of the actuators on the input signal a hot-wire probe and a phased-locked stereoscopic and time resolved PIV system were applied [16]. In a first set of experiments the configuration of the pressure supply system was analyzed, especially regarding the question how the temporal development of the actuator exit velocity relates to the signal from the frequency generator. Beside the geometry and the size of the exit area S, the volume of the settling chamber V and the length of the feed line lS were of primary importance. For the investigation a modular construction that supports inlets with different size, geometry and orientation and a continuous variation of the volume V and the tube length lS was designed [16]. A round actuator-exit with a 1 and 2 mm diameter drill and a rectangular one with a variable aspect ratio in the range b = 5 - 20 mm (length) and h = 0 - 2 mm (width) was examined. In summary it can be stated that the feed line length must be minimized to avoid undesired pressure losses. However, a decreasing feed line length is associated with an amplification of pressure oscillations, which travel in the feeding lines with the speed of sound [16]. These oscillations, which are clearly visible in the passive part of the actuator process (actuator off), result in a significant disturbance at the actuator exit and thus in a reduced performance of the dynamic actuation process. At low frequencies the amplitude of the oscillations can reach 20% of the primary actuator exit signal (actuator on). To damp these disturbances, the volume and geometry of the settling chamber between the valve and the actuator exit must be properly designed. The volume must be sufficiently large to avoid pressure oscillations and disturbances due to the piston characteristics, but small enough to avoid a significant damping or modification of the exit velocity signal due to additional losses or secondary flows in the settling chamber. In a second series of experiments an actuator array was designed, based on the results of the single actuator examination, and optimized to obtain a similar impulse response at each outlet port. Therefore, the settling chamber was used to distribute the air from a single valve to multiple actuator exits. The essential result of this examination is the fact that the generation of a homogeneous squared wave output signal of identical amplitude at each port requires that the feed line connection is orthogonal to the actuator axis. Otherwise the kinetic
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energy of the jet entering the settling chamber is partly superimposed to the exit velocity of the actuators located adjacent to the feed line outlet. This causes a large scale modulation of the actuator array in span-wise direction which would further complicate any systematic analysis.
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Investigation of Single Actuators
The geometry and orientation of the actuator exit is probably one of the most important part of the design process. One frequently applied variant are circular holes, which can be pitched and skewed according to figure 1. The angle between the surface and the blowing direction is called the pitch angle α, the skew angle β is defined as the angle between the blowing axis and the bulk flow direction. A second common variant are rectangular, skewed slots. Other possibilities, such as pitched slots or elliptic holes, are less popular, as problems related with their fabrication are believed to be disproportionate to their possible advantages. z
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3.1
Flow Field Analysis
To determine the most suitable actuator geometry and orientation various concepts were systematically studied in a low-speed wind tunnel by means of phasedlocked stereoscopic PIV [16,19]. The different actuators were installed 300 mm behind the elliptical leading edge of a flat plate. The boundary layer thickness was about δ ≈ 14 mm at the actuator position and the free stream velocity was set to U∞ = 14 m/s. The orientation of the light-sheet plane was perpendicular to the main flow direction (yz-plane) to visualize the effect of longitudinal vortices. The vector fields displayed in figure 2 illustrate the mean flow field generated 10 mm behind the actuator at the phase angle of φ = 180◦ (just before closing the valve) for the skew angle α = 60◦ , 30◦ and 90◦ (top to bottom). It can be seen that one important parameter regarding the effectiveness of the actuators is the strength and position of the longitudinal vortices, induced by the actuator, because these flow structures transfer high momentum fluid from the outer part of the boundary layer towards the wall. A second important parameter is the size and location of the blockage domain, which is represented by red color in the lower result. This low momentum region must be small or far away from the wall as any blockage near the wall will promote flow separation [20].
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Fig. 2. Effect of the orientation of a circular actuator on the flow field (the out-of-plane velocity component is color coded). Top: α = 60◦ ; Center: α = 30◦ ; Bottom: α = 90◦ .
To visualize the change of stream-wise momentum caused by the actuation, the undisturbed boundary layer was subtracted in figure 3. In case of the circular actuator (left column) it can be seen that without a blowing component in spanwise direction (upper left image) no high momentum fluid is transferred into the near-wall region. However, when the blowing direction is rotated around the z axis the flow field becomes asymmetric and high momentum fluid is transferred indirectly towards the wall. The right column of figure 3 shows the performance for a rectangular actuator with b = 10 mm and h = 0.314 mm. Far away from the wall the stream-wise momentum is enhanced but in the near-wall region, right
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Fig. 3. Momentum variation relative to the undisturbed flow for the round actuator with 1 mm exit diameter (left) and the rectangular one with b = 10 mm and h = 0.314 mm (right). Recording parameter: f = 100 Hz, ∆ = 50%, φ = 180◦ . Left: α = β = 90◦ ; α = 45◦ , β = 90◦ ; α = 45◦ , β = 60◦ (top to bottom). Right: α = β = 90◦ ; α = 90◦ , β = 60◦ ; α = 90◦ , β = 30◦ (top to bottom).
behind the actuator, the flow is strongly delayed when the actuator is symmetric (upper right image). This implies that this actuator orientation is not suited to prevent flow separation. However, when β is altered to 60◦ and 30◦ , as shown in the lower two images, an efficient transfer of stream-wise momentum toward the wall takes place. This comparison indicates the significance of the geometry and orientation of the actuator on the gain in momentum close to the wall. 3.2
Effect of Actuator Geometry and Orientation
To judge the efficiency of various configurations in downstream direction the following assessment factor was defined to characterize the gain in momentum closed to the wall.
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∆I = b
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This factor can be calculated from PIV velocity fields by integrating the signed square of the change in stream-wise velocity ∆u(x, y, z) along a span-wise line with constant distance to the surface. The upper result of figure 4 displays the spatio-temporal momentum contour generated by an inclined slot actuator with β = 45◦ and Λ = Uexit /U∞ = 5.0 and the lower image shows the corresponding assessment factor as a function of the wall distance z. The result implies that the wall distance of the integration line is not important for estimating the efficiency of the actuator, provided the position is reasonably closed to the wall and below the lowest vortex core. Thus the gain in momentum according to equation (1) is a well suited measure to quantify the performance of the actuators for active flow control.
Fig. 4. Top: Spatio-temporal distribution of the stream-wise velocity component relative to the undisturbed flow for Λ = 5.0. Blue color indicate regions with increased stream-wise velocity. Bottom: Spatio-temporal distribution of the assessment factor defined by equation (1). Red color indicate a gain in momentum.
To determine an appropriate actuator design for leading edge separation control the assessment factor was calculated for different actuator geometries in dependency of the orientation. Figure 5 indicates the superior performance of the circular actuator with α = 30◦ . However, due to the fact that this actuator design is very sensitive on the Reynolds number and the flow direction it can be
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concluded that the inclined slot with β = 45◦ − 55◦ is better suited for industrial applications with changing boundary conditions. 3.3
Effect of Amplitude, Frequency and Duty-Cycle
To access the significance of amplitude, frequency and duty-cycle of the actuation the assessment factor was determined from wall-parallel PIV results measured sufficiently close to the wall according to the forgoing discussion. The left result in figure 6 visualizes that the velocity ratio Λ = Uexit /U∞ of the actuator exit velocity and the free stream velocity has quite a strong effect on the actuator performance. For small Λ the mixing process is inefficient due to the small strength of the longitudinal vortices. However, with increasing Λ the effectiveness of the actuation increases gradually with increasing amplitude so that an effective control of the far field can be expected. Only for very large amplitudes the performance decreases again because of the increasing distance of the longitudinal vortex from the wall. In contrast the central result of figure 6 shows that the frequency has only a very little effect on the gain of momentum in the near wall region along the x coordinate. The duty-cycle on the other hand, shown in the right graph, has a more significant effect on the velocity distribution closed to the wall. For ∆ = 50% the curve progression is almost equal to the steady case (which results in a doubling of efficiency because only half the input energy is needed), ∆ = 75 % even raises up to higher values. Another important result that can be deduced from the results is the loss of momentum close behind the actuators. This is caused by the blockage of the actuator jet and the fact that the transfer of high-momentum fluid toward the wall by means of a vortex takes some time or domain. For this reason a certain spacing is required between the position of actuation and the position of separation. The exact distance depends on the strength and position of the stream-wise vortex and thus mainly on the actuator geometry and orientation.
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4
Investigation of Actuator Arrays
As the span-wise area that can be controlled by a single actuator is limited, arrays of actuators must be implemented to prevent separation globally. The interactions between actuators have rarely been studied [22]. However, preliminary investigations indicated a promising amplification behavior due to non-linear interaction [19,17]. Two methods of combination are possible: Simple staggering of the slots with identical orientation (co-rotating vortices) or alternating staggering, which results in inversely orientated vortex-pairs. The interactions of corotating vortices are very hard to utilize, because the vortices merge quickly or they stay unaffected by their neighbors according to [17]. Alternating staggering of two actuators can be structured into diverging or converging configurations. If two slots are diverging a counter-rotating vortex pair is generated with a downward motion in between. Following the nomenclature of [17] this configuration can be referred to as the common-flow-down configuration. Converging slots on the other hand are enforcing an up-wash in the middle and overspeed areas beside it, the so called common-flow-up configuration. The result displayed in figure 7 reveals the assessment factor for the commonflow-up and common-flow-down configuration for various spacings between the actuators. Due to the retarded area behind the actuators the efficiency is negative in the near field as already discussed. Approximately 25 mm behind the actuator the overspeed starts to dominate and the efficiency becomes positive. The isolated actuator has a broad peak at x = 65 mm with ∆I/b ≈ 0.1. Further downstream the peak gradually decreases due to a deterioration of the vortex’ circulation. In contrast the efficiency of the arrays is much greater, but it should be noted that equation (1) does not take the amount of input energy into account. Two actuators actually require twice the flow-rate in the supply system, therefore they should deliver at least twice the gain of momentum! However, they can deliver more: Two slots with ∆y = 20 mm create a very pronounced peak at x ≈ 40 mm due to the conflated and therefore understated retardation
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areas and the higher peak velocity. After that initial peak the efficiency decreases and at x > 100 mm it is always approximately twice the efficiency of the isolated actuator, which means that there is no more amplification, only a superposition of the effect of two vortices. For the array ∆y = 30 mm the initial peak is less pronounced, but further downstream the efficiency is a little more than twice the one of the isolated actuator. And finally the arrays with ∆y = 40 and 50 mm do have a pronounced initial peak, then start to decrease in efficiency and afterwards feature a constant rise of efficiency in the far downstream part of the flow field, although the circulation of the vortices should decrease due to viscosity. This behavior can be explained by a beneficial interaction of the vortices [20]. The lower result displayed in figure 7 displays the efficiency of the commonflow-down configurations. The small spacing ∆y = 20 mm is not very beneficial, because the outstanding high initial peak ∆I/b ≈ 0.5 is counteracted by a huge loss of efficiency. Much more convenient is a spacing of ∆y = 30 mm, where the vortices are sufficiently close to amplify the downwash, but the viscosity does not retard the circulation of them – although the efficiency of this configuration diminishes quicker than that of the isolated actuator. With greater spacings the distribution changes its character: The initial peak is less pronounced and it takes some time until this combination of actuators starts to be efficient. But
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in the far field the greater spacings are actually the best choice – they do not feature high velocities, but their overspeed areas spread out broad. Maximum interaction takes places approximately x = 175 mm behind the actuator, where the common-flow-down configuration features three times the efficiency with only twice the supply-energy. By comparing the efficiency of common-flow-down and common-flow-up configuration it can be noticed that in principle the common-flow-down arrangement delivers more efficiency – at least in the displayed part of the flow field. Anyway, a choice between both alternatives is somehow insignificant, because in order to manipulate a large area obviously more than two actuators will have to be used, which is only possible with alternate orientation. Thus the investigations rather focused on the question of the spacings for the individual configurations.
5
Airfoil Design, Actuator Integration and Experimental Setup
To validate the potential of the actuators under realistic conditions a singleelement airfoil was designed with the typical leading-edge separation characteristics of a modern slat-less two-element airfoil in high-lift configuration. The advantage of this approach is that a single-element airfoil is less complicated to build and, when testing, it has no uncertainties related with the correct flap angle, gap and overlap, which quickly superimpose the leading edge stall behavior. The disadvantage on the other hand are the different pressure distributions. The flap on a two-element airfoil lowers the static pressure at the trailing-edge. Hence the rise in static pressure from the suction spike to the trailing edge is reduced and therefore the tendency to separate. To simulate this effect with a one-element airfoil it was necessary to first copy the pressure distribution of the two-element airfoil, then subtract a constant ∆cp so that the rise in pressure between cp,min and the trailing edge equals that one of the two-element airfoil, but the trailing edge is at cp ≈ 0. Finally the pressure gradients had to be designed, so that the boundary layers of the two element airfoil and that one of the design become as comparable as possible. The design was performed for a Re number of 1.5 · 106 with the codes XFoil (for one-element configurations) and MSES (for multielement configurations), both developed by Drela et al [3,4]. Figure 8 indicates that the one-element design matches the reference pressure distribution and the gradients quite well, when a constant ∆cp is added - the symbols highlight the pressure distribution and the pressure gradients of the reference configuration. Even the value of cp,min at the suction spike and the small bump due to the laminar-turbulent transition at x/c ≈ 0.01 agree quite well. So the remaining difference between the two configurations is the fact, that of course the boundary layer for both elements starts in the stagnation point (cp = 1) and therefore has a slightly longer drop of pressure for the two-element configuration. Figure 9 highlights the main boundary layer parameters ue /Uinf , δ ,Θ,Hk and ReΘ . The first 20% of the chord length, which are important for the leading edge flow control devices, agree very well. Just the ue /Uinf is of
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course on a higher level for the two-element configuration. Since the state of the boundary layer dominates the formation of separation it can be expected that the one element airfoil stalls similarly to the two element airfoil. For the experiments a wind-tunnel model with a wingspan of 1.3 m and a chord length of 0.4 m was built using classical carbon fiber reinforced plastic manufacturing methods. The actuator system inside the model consists of the valves, which are feed by a master pressure supply line. 20 of these valves and the master supply line are integrated in the space between the spar, the shells and the trailing edge reinforcement. From the valves the air is guided through the spar into the nose of the model. This nose is CNC-milled from a solid aluminumblock and holds 20 settling chambers and the corresponding connections to the valve system. Each valve feeds one individual settling chamber. The chambers are closed with inlays, which hold the actuator orifices. Each settling chamber supports one inlet with four slots. The slots are oriented with a skew angle of 45◦ . The inlays are fabricated using stereo-lithography, they are simply glued into a pocket in the aluminum nose and grinded into the airfoils contour, so they are flush with the model surface. The airfoil model is equipped with 31 static ports which are connected to a PSI pressure system. Thin airfoils at small Reynolds numbers often feature a laminar separation bubble shortly behind the pressure spike. The typical leading edge stall is then dominated by the burst of this bubble, which is a common problem when analyzing airfoil stall behavior at lower Re-numbers. This laminar leading edge
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stall is not the kind of stall that is supposed to appear on the single-element airfoil, which is designed to exhibit leading edge stall even in the absence of the laminar separation bubble. To remove the laminar separation bubble a 0.1 mm thick and 2mm wide transition trip was properly placed just behind the pressure minimum. The experiments were performed in the low-speed wind tunnel MUB which is a closed return atmospheric tunnel with a closed 1.3 x 1.3 m test section. The maximum Reynolds number is Remax = 1.3·106 and the turbulence level is about 0.2 % at 40 m/s [8]. The facility is equipped with a total-pressure rake behind the trailing edge to capture the wake of the model. Forces and coefficients can be determined by integrating the pressure distributions following Jones’ approach. However, in the following only the normal force coefficient cnp will be highlighted, which is very closely connected to the lift coefficient cl .
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Control of Leading Edge Separation
Figure 10(a) displays the different characteristics of the cases with free and fixed transition. Both show a distinct hysteresis, as it is typical for a low-Re leading edge stall. In the case with free transition the laminar bubble bursts at α = 10.2◦ . With fixed transition the laminar bubble is destroyed and consequently can not burst, hence greater normal force coefficients can be achieved (cnp,max ≈ 1.24). 1.4
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The testing-matrix covered three excitation frequencies (f = 50 Hz, 75 Hz and 100 Hz), two amplitudes (specified by a certain supply pressure: pV = 1.0 bar and 1.5 bar) four different duty cycles (∆ =12 %, 25 %, 50 % and 75%) and additionally two static cases with different amplitudes (pV = 1.0 bar and 1,5 bar). Figure 10(b) revals the measured cnp - α-curves for f = 75 Hz and pV = 1.5 bar. The sensitivity on the duty cycle is highlighted by different linestyles. It can be concluded that short the duty cycles is more effective than the longer ones. Only at very high angles of attack the slightly longer duty cycle ∆ =25 % can outreach the ∆ =12 %-case. This behavior is in fact characteristic for all the measured cnp curves and independent on the frequency f or the amplitude pV . Secondly all the curves show a small local maximum near α = 12.5◦ and a local minimum near α = 14.0◦. In figure 11(a) six different, selected configurations are shown. The three dashed lines and the three solid lines each represent one configuration with just a change in frequency. Generally it can be stated that the sensitivity on the excitation frequency is weak compared to a variation of the duty-cycle. Only slight differences can be measured with changing frequencies. Figure 11 (b) highlights the effect of the amplitude for two different configurations. The results indicate
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that a small amplitude (dashed lines) is better in the beginning of the stall (α ≈ 12◦ ), but with increasing angle the higher amplitude cases (solid lines) overcome the smaller ones. This again is consistent throughout the measured cnp -curves and not dependent on the frequency. Figure 12 displays the pressure distributions for two cases in order to highlight the effect of a variation of the duty cycle ∆. The left side displays a case at α = 14.5◦ just before stall. The clean configuration develops a distinct pressure spike. When the actuators are active the pressure spike degrades - for some reason it degrades most for the shortest duty cycle ∆ = 12 %. Exactly this configuration on the other hand develops a lowered cp in the region x ≈ 40 mm· · · 200 mm. The case with higher angle of attack α = 16.0◦ does show quite a similar behavior. The clean airfoil is stalled completely, but the configurations with actuation are able to maintain a pressure spike. Comparing the different duty cycles again the shortest duty cycle shows the greatest cp,min , but again develops the region of lower cp further downstream. To access the physical mechanism of the actuation process extended PIV measurements have been performed for many different configurations. Three exemplified results at α = 19.0◦ are shown in figure 13, along with the measured pressure distributions. The left row shows the basic case without actuation, where the flow is separated from the very leading edge. The point of separation is at the nose at x ≈ 2 mm (which equals x/c ≈ 0.005). The turbulent kinetic energy level is low, since the shear layer has relative static strength and position. The other two rows display configurations with actuation (refer to above figures for the normal fore coefficients of these configurations). With actuation the airfoil produces a reasonably high cnp coefficient, however it appears to be
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separated as well. Nevertheless, with actuation the flow turning at the nose is noticeably greater in both actuated configurations. With dynamic actuation (figure 13(b)) separation occurs slightly later than in the clean configuration, hence a lower nose-pressure can be abided, which is coherent to the measured pressure distributions. The turbulent kinetic energy level is high. With static actuation the ability to turn the flow around the leading edge is even greater, resulting in a nose-pressure of cp = −4.0, which is quite remarkable at α = 19.0◦ . In the contour plots a noticeable smaller separation area is visible as well as a level of the turbulent kinetic energy, which roughly equals that of the dynamic actuation. This is quite remarkable, since the displayed velocity and energy fields are mean values and not phase locked. So although for the dynamic actuation the turbulence level actually contains the unsteadiness of the actuators, the static actuation results in the same amount of turbulent kinetic energy. The reason for this is not really understood, however, the static actuation results in lower cnp , since the dynamic one develops the lowered cp in the region further downstream (as discussed in Fig. 12). Although the actuators cannot prevent separation totally, they have a very advantageous influence on the separation, as the lift is maintained far beyond the reference αmax of the clean configuration. This behavior explains the local maximum at α = 12.5◦ and the local minimum at α = 14.0◦ in the cnp -curves discussed above. These local extrema are due to a slow transition from a nonseparated (for α < 12.5◦) into a completely separated state (for α > 14.0◦). Since the separation is influenced by the flow control system this “stall” is not associated with a great loss of cnp , but only with a small “bump”, after which cnp rises again. As a matter of fact the overall behavior that was found with the airfoil model is not very consistent to the sensitivities that were found during the optimization process—for example the system was expected to be most effective with long
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Fig. 13. Result of the PIV-measurements at high angle of attack α = 19.0◦ . Velocity magnitude |V | in m/s, turbulent kinetic energy Ekin in [m2 /s2 ]. (a) clean airfoil (no actuation). (b) dynamic actuation f = 75 Hz, ∆ = 25 %, pV = 1.5 bar. (c) static actuation pV = 1.5.
duty cycles and almost independent of the excitation frequency (see Fig. 6). It has to be concluded that whether the amplitude of excitation is too low or the actuators are positioned too far downstream, so that the separation line is in the unaffective area shortly behind the actuator, where ∆I/b is negative. Both facts can be circumvented by positioning the actuators further upstream. Since the stall is effectively at the leading edge this means, that the actuators have to be positioned at the lower side, in between the stagnation point and the leading edge. Due to the relatively slow flow velocity a much higher velocity ratio λ can be realized at the same supply pressure pV or exit velocity vj . This variation was performed by simply turning the aluminum nose. Although the nose is not perfectly symmetric the difference in the pressure distribution is negligible. Fig. 14 displays a variation of amplitude pV and a variation of the duty cycle ∆. As can be seen the cnp,max was again raised by about 0.1,
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but with this turned-nose the cnp -curve was extended almost linearly. With this configuration the dependencies, that were found during the optimization process, are again present, namely greater duty cycle or greater supply pressure leading to better influence. The excitation frequency on the other hand has no effect on the displayed cnp -curves. The new cnp,max ≈ 1.4 is a somehow natural limit of this configuration, since none of the actuations were able to go further, neither by increasing the duty cycle nor by increasing the supply pressure.
7
Conclusions
Various pneumatic actuators were systematically examined to examine their efficiency and potential for active flow control on high-lift devices. The investigation indicates the significance of actuator geometry and orientation and the effect of the amplitude, frequency and duty-cycle for active flow control. In addition the advantage of actuator arrays was outlined and their potential for leading edge separation control could be estimated. The following main results can be deduced from the analysis: 1. To optimize pneumatic actuators the strength and position of the longitudinal vortices and the size and location of the blockage domain has to be properly aligned. This can be done by means of the geometrical and dynamical actuator parameter. 2. Pneumatic actuators become most efficient when the orientation is not symmetrical to the main flow direction, because a strong large-scale secondary flow is generated that transfers high momentum fluid in the near wall region.
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3. The circular actuator with α = 30◦ and β = 55◦ shows a very good mixing characteristics. However, as this actuator design is very sensitive on the Reynolds number and the flow direction the inclined slot with β = 45◦ − 55◦ is a better choice for industrial applications with changing boundary conditions. 4. A certain spacing between the position of actuation and the position of separation is required for efficient flow control, because of the blockage of the actuator jet and the fact that the transfer of high-momentum fluid towards the wall by means of a vortex requires some spatial distance. The exact distance depends on the strength and position of the stream-wise vortex and thus mainly on the actuator geometry and orientation. 5. The ratio between the actuator exit velocity and the free stream velocity has a strong effect on the actuator performance. For small values the mixing process is inefficient due to the small strength of the longitudinal vortices. With increasing actuator amplitude the effectiveness of the actuation increases gradually and an effective control of the far field is possible. 6. The frequency has a minor effect on the gain of momentum close to the wall in downstream direction. 7. The duty cycle has a strong effect on the gain of momentum close to the wall in downstream direction. Generally, large duty cycles are well suited to control separation in the far field. For near field control small duty cycles should be applied. 8. The efficiency of actuator arrays relatively large compared to the single actuators when the actuator spacing and orientation is properly selected. 9. The efficiency of common-flow-down actuator is superior to common-flow-up array configurations – at least in the displayed part of the flow field. 10. For leading edge stall separation control small actuator amplitudes and small duty cycles are better in the beginning of the stall (α ≈ 12◦ ). With increasing angle the higher amplitude case overcomes the smaller one as trailing edge stall appears gradually when the leading edge stall is suppressed. 11. Generally the configurations with actuation are able to maintain a pressure spike. The shortest duty cycle shows the greatest cp,min , but develops the region of lower cp further downstream. 12. Although a single span-wise actuator array cannot fully prevent the separation on the entire airfoil, they maintain the lift far beyond the αmax without actuation. 13. By displacing the actuation position further upstream the efficiency scales with the introduced momentum and due to the relatively slow mean flow velocity Uedge at the upstream position a much higher velocity ratio Λ = Uexit /Uedge can be realized at the same supply pressure pV or exit velocity Uexit . 14. At the upstream actuator position the turning of the base flow on the suction side caused by the jets is avoided and thus the local dip of the lift. In effect the cnp -curve increases quite linearly with increasing angle of attack.
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Acknowledgements The project is part of BMWA’s Lufo III “Innovative Hochauftriebskonfigurationen” and is being dealt with as participant of the joint research project “Dynamische 3D Str¨ omungskontrolle”.
References [1] Chang P. K.; Control of Flow Separation; Hemisphere Publishing Corporation, McGraw-Hill Book Company, 1976 [2] Compton, D. A. and Johnston, J. P.; Streamwise Vortex Production by Pitched and Skewed Jets in a Turbulent Boundary Layer; AIAA Journal, Vol. 30, No. 3, pp. 640-647, 1992 [3] Drela, M.; Youngren, H.; XFoil 6.94 User Guide; MIT, Aero & Astro, 2001 [4] Drela, M.: A User’s Guide to MSES 2.95; MIT, Computational Aerospace Sciences Laboratory; 1996 [5] Fiedler, H.E.; Fernholz, H.-H.; On Management and Control of Turbulent Shear Flows; Prog. Aerospace Sci., Vol. 27, Pergamon Press, 1990 [6] Gad-el-Hak, M.; Flow Control: The Future; Journal of Aircraft, Vol. 38, No. 3, pp. 402-418, 2001 [7] http://www.tu-braunschweig.de/ism/institut/wkanlagen [8] http://www.tu-bs.de/ism/institut/wkanlagen/mub [9] Johari, H. and Rixon, G. S.; ffects of Pulsing on the Vortex Generator Jet; AIAA Journal, Vol. 41, No. 12, pp. 2309-2315, 2003 [10] Johari, H., Pacheco-Tougas M. and Hermanson, J. C.; Penetration and Mixing of Fully Modulated Turbulent Jets in Crossflow; AIAA Journal, Vol. 37, No. 7, pp. 842-850, 1999 [11] Johnston, J. P. and Nishi, M.; Vortex Generator Jets - Means for Flow Separation Control; AIAA Journal, Vol. 26, No. 6, pp. 989-994, 1990 [12] K¨ ahler, C.J. and Scholz, U.; Investigation of laser-induced flow structures with time-resolved PIV, BOS and IR technology; 5th International Symposium on Particle Image Velocimety, Busan, Korea, September 22-24, Paper 3223, 2003 [13] McManus, K., Ducharme, A., Goldey, C. and Magill, J.; Pulsed Jet Actuators for Suppressing Flow Separation; AIAA Paper 96-0442, 1996 [14] McManus, K. R., Joshi, P. B., Legner, H. H. and Davis S. J.; Active Control of Aerodynamic Stall using Pulsed Jet Actuators; AIAA Paper 95-2187; 1995 [15] Nagib, H., Kiedaisch, J., Greenblatt D., Wygnanski, I. and Hassan, A.; Effective Flow Control for Rotorcraft Applications at Flight Mach Number; AIAA Paper 2001-2974, 2001 [16] Ortmanns, J.; K¨ ahler, C.J.: Investigation of Pulsed Actuators for Active Flow Control Using Phase Locked Stereoscopic Particle Image Velocimetry; Proceedings of the 12th Int. Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, 2004 [17] Pauley, W.R.; Eaton, J.K.; Experimental Study of the Development of Longitudinal Vortex Pairs Embedded in a Turbulent Boundary Layer; AIAA Journal, Vol. 26, No. 7, American Institute of Aeronautics and Astronautics, 1988 [18] Schatz, M.; Thiele, F.; Petz, R.; Nitsche, W.; Separation Control by Periodic Excitation and its Application to a High Lift Configuration; AIAA-2000-42507, American Institute of Aeronautics and Astronautics, 2004
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[19] Scholz, P.; Ortmanns, J.; K¨ ahler, C.J.; Radespiel, R.: Influencing the Mixing Process in a Turbulent Boundary Layer by Pulsed Jet Actuators; 12.DGLRsymposium of AG STAB, Bremen, 2004; (to be published in: “Notes on Numerical Fluid Mechanics and Multidisciplinary Design”, Springer-Verlag, 2005) [20] Scholz, P.; Ortmanns, J.; K¨ ahler, C.J.; Radespiel, R.: Performance Optimization of Jet Actuator Arrays for Active Flow Control; Proceedings of the CEAS/KATnet Conference, Bremen, Germany, 2005 [21] Seifert, A.; Bachar, T.; Wygnanski, I.; Koss, D.; Shepshelovich, M.; Oscillatory Blowing, A Tool to Delay Boundary Layer Separation; AIAA Journal, Vol. 31, No. 11, pp. 2052-2060, 1999 [22] Watson, M.; Jaworski, A.J.; Wood, N.J.; Contribution to the Understanding of Flow Interactions Between Multiple Synthetic Jets; AIAA Journal, Vol. 41, No. 4, American Institute of Aeronautics and Astronautics, 2003
Computational Investigation of Separation Control for High-Lift Airfoil Flows Markus Schatz, Bert G¨unther, and Frank Thiele Hermann-F¨ottinger-Institute of Fluid Mechanics, M¨uller-Breslau-Str. 8, 10623 Berlin, Germany
[email protected] http://www.cfd.tu-berlin.de
Summary This paper gives an overview of numerical flow control investigations for high-lift airfoil flows carried out by the authors. Two configurations at stall conditions, a generic two-element setup with single flap and a second configuration with slat and flap of more practical relevance are investigated by simulations based on the Reynolds-averaged Navier-Stokes equations and eddy-viscosity turbulence models. For both cases flow separation can be delayed by periodic vertical suction and blowing through a slot close to the leading edge of the flap. By simulating different excitation modes, frequencies and intensities optimum control parameters could be identified. Comparison of aerodynamic forces computed and flow visualisations to experiments allows a detailed analysis of the dominant structures in the flow field and the effect of flow control on these. The mean aerodynamic lift can be significantly enhanced by the active flow control concepts suggested here.
Nomenclature c, cclean , ck c l , cd Cµ f, F + H Re F, St u0 , ua α, δf , δs ∆t
chord length of main airfoil, clean airfoil, flap lift coefficient, drag coefficient 2 momentum coefficient, Cµ = 2 Hc uua0 excitation frequency, non-dimensional frequency F + = f ck /u0 slot width (H = 0.004ck ) Reynolds number based on chord length vortex-shedding frequency, Strouhal number based on flap length inflow velocity, excitation velocity ua in the slot angle of attack of the main airfoil, flap and slat deflection angle time stepping
1 Introduction Modern commercial airplanes are equipped with sophisticated multi-element high-lift devices consisting of slat and single or multiple flaps that must generate a tremendous R. King (Ed.): Active Flow Control, NNFM 95, pp. 173–189, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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amount of lift during take-off and landing in order to reduce ground speeds and runway lengths. As such elements tend to be complex, heavy and expensive, aerodynamic research has traditionally aimed at simplification of such elements without loosing efficiency. Experimental and numerical investigations showed that effectivity can significantly be improved by delaying flow separation on the flap in the case of high flap deflections for airfoil configurations at low and high Reynolds numbers [1,2,3]. In the last decades a large number of experimental and numerical studies showed the general effectiveness of flow control for single airfoils. In most investigations, leading edge suction is applied for transition delay [4]; nonetheless, jet flaps are also employed for lift increase and manoeuvering. Surface suction/blowing can be used to rapidly change lift and drag on rotary wing aircraft [5]. However, most control techniques considered in the past showed low or negative effectiveness. In further investigations oscillatory suction and blowing was found to be much more efficient with respect to lift than steady blowing. The process becomes very efficient if the excitation frequencies correspond to the most unstable frequencies of the free shear layer, generating arrays of spanwise vortices that are convected downstream and continue to mix across the shear layer. Suction and blowing can be applied tangential to the airfoil surface [6], rectangular [7] or with cyclic vortical oscillation. In order to create an effective and efficient control method previous studies have been primarily focused on the parameters of the excitation apparatus itself. Overviews are given by Wygnanski and Gad-el-Hak [8,9].
NACA4412
SCCH
NA
CA
441
5
Fig. 1. Active flow control by means of vertical suction and blowing at the flap, applied to; left: generic tow-element high-lift configuration; right: three-element configuration
In the investigations surveyed in this paper, periodic excitation is applied to delay flow separation on the flap of multi-element high-lift configurations resulting in enhanced lift and reduced drag. The objective of this investigation is to better understand the functionality of the periodic excitation on pressure-driven separated flow over multielement high-lift configurations. All presented investigations are obtained numerically. In wide parts however, the study includes results from cooperations with experimentalists using Particle Image Velocimetry (PIV) measurements. Additional benefit can be obtained from the combination of those results and the presented relatively low-cost simulations that allow an easy overview of the entire flow field in order to capture the effect of flow control techniques.
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2 Computational Method All numerical investigations are based on the ELAN code that was developed at the Hermann-F¨ottinger-Institute of Fluid Mechanics at the Berlin University of Technology. The method is based on a Finite-Volume solver for the incompressible Navier-Stokes equations. The method is fully implicit and of second order in space and time. The SIMPLE pressure correction algorithm is applied based on a collocated storage arrangement for all quantities. Convective fluxes are approximated by a TVD-MUSCL-scheme. 2.1 Turbulence Modelling The simulation program can be run in URANS mode, solving the Unsteady Reynoldsaveraged Navier-Stokes equations using statistical turbulence models as well as in a mode for Large-Eddy Simulation (LES) or combinations of both. In previous URANS investigations with a large variety of different one- and two-equation turbulence models as well as Explicit Algebraic Reynolds-stress Models (EARSM) the SST k-ω model by Menter [10] and the LLR k-ω model by Rung [11] exhibited the best overall performance for steady and unsteady airfoil flows with large separation [12]. The latter represents an improved two-equation eddy-viscosity model formulated especially with respect to the realizability conditions. Beside these two, three other eddy-viscosity turbulence models are applied in comparison: the Spalart-Allmaras (SA) and SALSA oneequation models [13,14] and the Wilcox k-ω model [15]. 2.2 Boundary Conditions At the wind tunnel entry all flow quantities including the velocity components and turbulent properties are prescribed. The level of turbulence at the inflow is set to T u = 1 2 1/2 = 0.1% and the turbulent viscosity µt /µ = 0.1. At the outflow a convecu0 ( 3 k) tive boundary condition is used that allows unsteady flow structures to be transported outside the domain. All surfaces of airfoil, slat and flap are modeled as a non-slip boundary condition. As the resolution is very fine, a low-Re formulation is applied. The wind tunnel walls however, are considered as frictionless walls. To model the excitation apparatus, a suction/blowing type boundary condition is used. The perturbation to the flow field is introduced through the inlet velocity to the small chamber representing the excitation slot. In the case of zero-net-mass excitation it is given by: u0 + 1 c Cµ · cos t · 2π F (1) uexc (t) = u0 2H ck where Cµ is the non-dimensional steady momentum blowing coefficient and H is the slot width. The other excitation modes discussed later are based on similar formulation.
3 Two-Component High-Lift Configuration The first part of the study is related to experimental investigations by Tinapp [1] and extensions of Petz [16]. The test model is a generic two element high-lift configuration,
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Fig. 2. Sketch of the two-component configuration in the wind tunnel and in the simulation with flow control by periodic suction and blowing through a slot on the flap
consisting of a NACA 4412 main airfoil and a NACA 4415 flap with ck /c = 0.4 relative chord length. Both profiles have bluff trailing edges. A previous study [17] of a similar test case showed that due to strong blocking the effect of the tunnel walls is very important and needs to be considered. The main airfoil is mounted at 52% of the tunnel height (h = 7.8 c), whereas the flap is situated at a fixed position underneath the trailing edge of the main airfoil, thus forming a gap of F G = 0.078c with an overlap of F O = 0.027c (figure 2). In the numerical study, the angle of attack is fixed at α = 3◦ for the main airfoil and β = α + δf = 40◦ for the flap. Under this condition the flow is characterized by the onset of stall on the flap. According to the experiments the freestream velocity corresponds to a ReynoldsNumber of Re = 1.6 · 105 based on the main-airfoil chord. Transition is fixed at the positions of turbulator strips at 4.5% chord on the main airfoil and 2.8% chord on the flap according to the experimental setup. In the experiments [1] periodic oscillating pressure pulses are generated externally by an electrodynamic shaker driving a small piston. This results in an oscillating jet emanating perpendicular to the chord from the narrow slot 4% chord behind the flap leading edge. The slot width is given by H = 0.004 ck . Due to the experimental setup this excitation is presumed to be completely two-dimensional and therefore all computations are correspondingly 2d. The computational domain starts 4 chords upstream and ends 8 chords downstream of the configuration. The computational c-type mesh provides 202 chordwise cells around the main airfoil and 196 around the flap resulting in 37,000 cells in total. The non-dimensional wall-distance of the first cell center remains below Y + = 1 on the complete surface for an attached steady case. The excitation slot is resolved by 20 × 20 cells. Additional simulations use a 64,000 cells mesh to evaluate the measure of mesh dependency (for more details see [2]). 3.1 Unexcited Flow First the flow around the configuration is simulated without excitation. In an early stage of the investigations Franke et al. [17] could obtain convincing results by using the LLR
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k-ω model of Rung and Thiele [11] that are significantly better compared to standard kε and k-ω formulations for similar conditions. Their simulation results strongly depend on the transition position on the flap as well as on the turbulence model. In the fully turbulent case, the flow remains attached to the flap surface, whereas tripped transition leads to flow separation. 6 -1
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Fig. 3. left: Pressure distribution on main airfoil and flap for different turbulence models without excitation in comparison to experiments [18]; right: Spectrum of lift coefficient for different models without excitation
In the present investigation the flow is characterized by massive separation which is predicted well by all turbulence models. The prediction of the flow topology agrees qualitively with experimental results as can be seen in the pressure distribution in figure 3 (left) that is available for the upper surfaces of main airfoil and flap. Compared to the experimental results the suction peak is overpredicted in the numerical simulation. This is caused by the direction of the flow-vector behind the main airfoil trailing edge. In the experiments strong three-dimensional effects appear that are neglected in the simulation. The most promising results were obtained by using the k-ω models (figure 3 left). The separation point for this case is located in the laminar part of the flap boundary-layer slightly upstream of the excitation slot, such that a large reverse flow region forms downstream (figure 4). The two-equation models provide a correct prediction of the flap separation. In the case of the Spalart-Allmaras model however, the flap separation position is located too close to the leading edge, resulting in slightly underpredicted pressure in the reverse flow region. Here the flow is almost steady and compared to the lift coefficient spectra of the two-equation models, no dominant frequencies, that might indicate vortex-shedding, occur (figure 3 right). Results of the two-equation models, however, show a strong amplitude for a Strouhal number based k ≈ 0.5. These results show the typical behaviour of the on the flap chord St = F uc∞ SA-model for unsteady airfoil flows reporting a very low unsteadyness. In the majority of following investigations the LLR k-ω model is used. To correctly capture all unsteady flow features a proper time-resolution is required and the simulation has to last at least several periods to avoid launching effects. In the present case, all simulations consider an unsteady flow field with a non-dimensional
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Fig. 4. Iso-contour plot of the mean velocity obtained by hot wire measurements of Tinapp/Petz [16] (left figure) and a numerical simulation (right figure) for the unexcited flow at a low Reynolds number of Re = 160, 000 k time stepping of ∆t = 0.01 uc∞ . One period of vortex-shedding is resolved by 230 time steps in this case. Results of a finer time-stepping with 595 steps per period (∆t = k ) do not show significant differences to the present results. 0.004 uc∞ The results of the URANS simulation are plotted in figure 4 right, compared to mean velocities obtained by traversing hot wire measurements in figure 4 left. Flow patterns in both results are in good agreement and show that leading edge separation on the flap occurs coupled with a large recirculation region downstream. The flow passing through the slot between main airfoil and flap behaves similar to a free jet rather than following the flap surface. Operating the configuration under these conditions is not effective and its performance must be improved in order to become applicable for airplanes. The effect of the Reynolds number on the mean lift coefficient was investigated by Petz [3]. A significant increase in lift can be observed for higher Reynolds numbers, however, the overall structure of the mean flow remains more or less similar to the lowest Reynolds number in figure 4. Dominant structures with a single frequency have been identified in URANS computations as well as in experiments. These correspond to a Kelvin-Helmholtz type of fluctuation with a Strouhal number that increases for higher Reynolds numbers.
3.2 Active Flow Control Simulations In the following investigations flow control mechanisms are applied. All flow control computations start from the baseline case solution as the initial flow condition. The k is used. LLR k-ω model and a time stepping of ∆t = 0.01 uc∞ Excitation modes. Figure 5 gives an overview of the investigated excitation modes. Very simple approaches of flow control are steady suction and blowing like the modes c) and d) in figure 5. First this kind of excitation is investigated and results are summarised in figure 5 right. The blowing/suction intensity is set to Cµ = 100 · 10−5 . The results indicate, that only steady suction is effective. Flap separation can be delayed and the mean lift increases by 2% while the drag drops. It is also remarkable
excitation
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a)
Case b)
c)
d)
mean mean flow Strouhal lift drag sep. number c¯l c¯d xd /ck St
without excitation 2.69 0.20 4.4% steady blowing c) 2.63 0.20 3.6% steady suction d) 2.74 0.17 14.6%
0.44 0.42 0.53
time
Fig. 5. left: Time evolution of excitation for different modes; right: Resulting flow properties for steady suction and blowing
that the frequency of detaching vortices significantly increases from St = 0.44 in the baseline simulation to St = 0.53 for steady suction. Steady blowing does not give any positive effect. For the majority of investigations a zero-net-mass-flux kind of excitation like mode a) in figure 5 is applied. In the case of excitation mode b) which represents a kind of periodic suction an enhancement of the mean lift coefficient appears (figure 9 centre). However, the effect always remains below that of excitation mode a) for comparable amplitudes. If not particularly mentioned, all following investigations are based on mode a) excitation. Periodic excitation with different frequencies. Experimental investigations show that different excitation frequencies require different intensities to get the same kind of flow control in post-stall cases [1,19]. In order to find an optimum excitation, simulations with different frequencies and Cµ = 50 · 10−5 are performed. The largest lift can be found at a frequency of F + = 0.62 (figure 6 right). In this case the lift coefficient can be enhanced by 14% compared to the baseline simulation. The optimum frequency is slightly larger than the frequency of detaching vortices without excitation (F + ≈ 0.5). At the same time the mean separation position moves from less than 5% chord downstream to 15.5% (figure 7 left) and drag drops from cd = 0.2 in the baseline case down to cd = 0.11 for frequencies of 0.4 < F + < 1.0 (figure 7 right). The differences in drag and separation position, however, are very small over a wide range of frequencies. Visualizations of the flowfield indicate the relevance of the detaching vortices for the effectivity of periodic excitation. This structure is dominated by the excitation frequency and turns out to be more important than the instability frequency of the free shear layer in the baseline case. The periodic excitation behaves like a periodic suction that always moves the free shear layer close to the flap surface. At very low frequencies more than one vortex can detach during one excitation period (figure 8 left). In the case of high frequency excitation, however, the time between two suction events is too short to form a complete vortex (figure 8 right). Consequently only a part of a complete vortex can detach during one period and each vortex is devided into subvortices. Both cases are less
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Fig. 6. left: Unsteady lift coefficient over the non-dimensional time for three different excitation frequencies; right: Mean lift coefficient for different excitation intensities 0.20 -5
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Fig. 7. left: Mean separation position for different excitation frequencies and Cµ = 50 · 10−5 intensity; right: Mean drag coefficient for different excitation frequencies
effective than excitation with a frequency that allows one complete vortex after the other to detach (figure 8 centre). The same effect appears in the time evolution of the lift coefficient in figure 6 left: in the case of low frequency excitation higher harmonics occur whereas for high frequency excitation a low frequency signal is superimposed on the main frequency. Only at F + = 0.62, where a clear sinusoidal signal can be seen, the lift coefficient reaches a maximum. In the numerical study frequencies around F + ≈ 0.62 appear to form an optimum excitation. In the experiments the range of frequencies with most promising results is 0.25 < F + < 0.5 [3]. The discrepancy might result from the 2d modelling. Periodic excitation with different intensity. To assess the effectiveness of oscillatory blowing and suction, various blowing coefficients Cµ are tested at two different excitation frequencies of F + = 0.62 and F + = 1.03.
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Fig. 8. Flowfield around flap represented by vorticity distribution for different excitation frequencies left: F + = 0.26; centre: F + = 0.62; right: F + = 1.54
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Fig. 9. left: Mean separation position for different excitation intensities; centre: Mean lift coefficient for different intensities; right: Mean drag coefficient for different intensities
Compared to the separated flow in the baseline simulations at Cµ = 0, excitation with low intensity does not increase the lift or delay separation (figure 9 left and centre). If Cµ becomes larger than 25 · 10−5 , however, lift continously climbs. This effect is perfectly in line with the experiments, but the maximum achievable intensity there was limited to Cµ = 65 · 10−5 . The present study shows that further increasing intensity might bring another gain in lift. In former investigations [6] flow control by tangential suction and blowing with stronger excitation also results in further increase of lift. In the present case of vertical excitation, separation position and drag coefficient, however, do not improve for extremely strong suction and blowing. The drag coefficient however, has a minimum at Cµ = 50 · 10−5 (see figure 9 right). All effects are documented for both excitation frequencies F + = 0.62 and F + = 1.03.
4
Three-Component High-Lift Configuration
The second test model represents the SCCH (Swept Constant Chord Half - model) high-lift configuration with practical relevance that has already been used for several experimental studies targeting passive flow and noise control concepts [20,21,22]. The typical three-component setup consists of a main airfoil equipped with an extended slat with 0.158 cclean relative chord length and an extended flap which has a relative chord length of ck = 0.254 cclean (figure 10). As seen in the first test model
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SCCH δS = 26.5°
δ F = 32°
Fig. 10. Sketch of the SCCH high-lift-configuration
all profiles have blunt trailing edges. The flap is situated at a fixed position underneath the trailing edge, forming a gap of 0.0202 cclean and an overlap of 0.0075 cclean (figure 11). In the experiments, the three-dimensional wing has a sweep angle of Φ = 30◦ and a constant wing depth in spanwise direction. This three-dimensionality, however, is neglected in the first part of the numerical investigations in order to reduce the computational costs. In all investigations the freestream velocity corresponds to a Reynolds number of Re = 106 , based on the chord of the clean configuration (with retracted high-lift devices). The transition positions for each element have not been reported from the experimental measurements.
0.075 c k δs ; δf
δ F = 32°
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gap/cclean overlap/cclean
slat
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32.0◦
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Fig. 11. Details of the geometry between main airfoil and flap
4.1 2d-Simulation of the Profile The dimensions of the computational domain are 15 chords forward, above and below the configuration and 25 chords behind. Around the main airfoil 309 chordwise cells are located, 116 around the slat and 118 around the flap. The computational c-type mesh consists of 75,000 cells in total. The non-dimensional wall-distance of the first cell centre again remains below Y + = 1 on the complete surface. The position of an effective perturbation is defined by the detachment position on the upper surface of the flap without excitation. For this reason the perturbation is introduced through a slot located at 7.5% chord behind the flap leading edge (figure 11) with a width of H = 0.0016 ck , and it is directed perpendicular to the flap surface. This first excitation appears to be two-dimensional and therfore all computations with excitation are limited to 2d.
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4.2 Unexcited Flow In the beginning, two-dimensional, unsteady investigations without excitation have been carried out. The angle of attack was fixed at zero degrees for the whole configuration. Results of all URANS-simulations were obtained by using the LLR k-ω model and considering fully turbulent flow on all elements. The time-dependent resolution of k , resolving the flow field was realised by a non-dimensional time step of ∆ t = 0.005 uc∞ each period of non-excitated vortex-shedding by 458 time steps.
-1
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-5
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3.0
Fig. 12. left: Flow field around flap represented by vorticity distribution and iso-contours of the mean velocity with vectors for the unexcited flow at a Reynolds number of Re = 1, 000, 000; right: Spectrum of lift coefficient without excitation
Similar to the two-component configuration the flow field of the SCCH-configuration without excitation is characterized by massive separation above the upper surface of the flap. The mean separation point is located at 7.5% chord behind the flap leading edge, and downstream a large recirculation region occurs. The unsteady behaviour of separated flow is mainly governed by large vortices shedding from the flap trailing edge and interacting with the vortices generated in the shear-layer between the recirculation region and the flow passing through the slot between main airfoil and flap nose (figure 12 left). The spectrum of the lift coefficient in figure 12 (right) shows a dominant k = 0.44, amplitude for a Strouhal number formed with the flap chord of St = F uc∞ mainly produced by this vortex-shedding. The above described base flow configuration involving massive separation over the flap is not desirable and needs to be improved by active flow control. 4.3 Excited Flow In order to reduce the computational costs, all simulations with active flow control consider only a two-dimensional configuration neglecting the effect of the swept wing. Based on the case of the unexcited flow, computational investigations with periodic, active flow control followed. Therefore simulations with a fixed frequency F + = 0.44 and varied intensities Cµ have been carried out. The selection of this excitation frequency
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is based on the vortex shedding frequency in the unexcited flow, as it is supposed that the excitation of such flows is particularly effective at this characteristic frequency. Periodic excitation with different intensity. In the cases of excited flow periodic perturbations with an intensity of Cµ = 50 · 10−5 . . . 300 · 10−5 have been introduced. Compared to the detached, unexcited flow the excited lift could be continuously increased with growing intensity. No local maximum of lift can be identified, so that the largest lift occurred for the strongest excitation of Cµ = 300 · 10−5 . In this case the lift coefficient can be enhanced by 17% compared to the baseline simulation (figure 14 left). 3
2
Cµ=300 x 10
−cp
−5
1
0
−1 0.0
0.2
0.4
0.6
0.8
x/cclean
0.20 +
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k
detachment position xd /ck
without excitation −5 Cµ=100 x 10
0.18 0.16 0.14 0.12 0.10 0×10
1.0
-5
100×10
-5
200×10
-5
excitation intensity Cµ
300×10
-5
Fig. 13. left: Pressure distribution for unexcited flow and two different excitation intensities; right: Mean separation position on the flap for different excitation intensities 0.14
mean drag coefficient cd
mean lift coefficient cl
1.6 +
F = 0.44
1.55 1.5 1.45 1.4 1.35 1.3 -5 0×10
100×10
-5
-5
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-5
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+
F = 0.44
0.13 0.12 0.11 0.10 0.09 -5 0×10
100×10
-5
-5
200×10
excitation intensity Cµ
300×10
-5
Fig. 14. left: Mean lift coefficient for different excitation intensities; right: Mean drag coefficient for different excitation intensities
The gain in lift by the excitation is mainly based on a change of flow direction at the trailing edge of the main airfoil. With the excited condition of flow above the flap the trailing edge angle is increased and the pressure distribution above the main airfoil could be enhanced (figure 13 left). The excited flow field above the flap is characterised by a decreased recirculation region. With this effect the mean separation position moves from less than 10% chord downstream to more than 15% (figure 13 right). At the same time the displacement of the detachment position is saturated with continously increased intensity.
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The distribution of the mean drag coefficient shows a similar behaviour. The drag drops from cd = 0.13 in the unexcited case down to cd ≈ 0.10 for the excited cases. At the same time the drag coefficient, however, has a minimum at Cµ = 100 · 10−5 (see figure 14 right) and also shows a saturated behaviour for higher intensities. 4.4 Quasi-3d-Simulation of a Infinite Swept Wing The three-dimensional mesh is based on an expansion of the two-dimensional mesh into the third direction. For consideration of an infinite swept wing 8 layers of the two-dimensional mesh are combined to resolve a three-dimensional wing section by 530,000 cells in total. The infinite character is simulated by means of periodic boundary conditions. In order to provide separated flow conditions that can be controlled by active methods, different flap settings have been tested. For four different angles of deflection, starting from the baseline configuration (δf = 32◦ ), steady computational investigations have been performed. Figure 15 shows the effect of flap deflection on the flow field for δf = 32◦ and δf = 37◦ .
flap deflection angle δf
38 o 37
o
36
o
35
o
34 o 33 o
°
32
32
o
31
o
0.00
0.05
0.10
0.15
0.20
0.25
°
37
detachment position xd/ck
Fig. 15. left: Different flap angles δf and corresponding flow fields represented by iso-contours of u-velocity; right: Steady separation position for different flap deflection angle
Iso-contours in figure 15 (left) give an idea of the expanding separated region with recirculation above the flap by increased flap deflection. As a result of this expanded separation, the detachment position moves from 13% relative chord at δf = 32◦ to 6% relative chord at δf = 37◦ (see figure 15, right). The larger recirculation region behaves like a flap with less effective camber and produces a lower suction peak. The upwash-effect reduced the velocity near the wall of the main trailing edge. Thereby the flow above the main trailing edge is subject to separation. If the angle of deflection is increased beyond δf = 37◦ , the detachment position remains unchanged. The following investigations with active flow control will be conducted on an infinite swept airfoil with a sweep angle of Φ = 30◦ . The effect of sweep is demonstrated in figure 16 for a steady computation of the baseline configuration with α = 6◦ incidence. The streamlines on the surface of the main airfoil show the three-dimensional component of the flow produced by the local pressure gradient in spanwise direction. On the
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Fig. 16. Surface streamlines on an infinite swept airfoil segment
upper surface oft the flap flow separation becomes visible by the direction of surface streamlines. The huge recirculation region downstream of the detachment line is based on near-wall flow in upstream direction with a dominant component in spanwise direction. This component is also produced by the sweep and is strongly developed in the slow detached flow.
5 Discussion The important features of the flow field and the effects of excitation on its properties can be discussed for both configurations together. Flow control by periodic excitation is dominated by the development of spanwise vortices in these cases. One of the results from simulations with different excitation frequencies and intensities is the difference in the size of these detaching vortices. Compared to both baseline cases without excitation the size of structures in the wake becomes smaller with increasing excitation frequency and high intensity. An optimum lift combined with minimum drag corresponds to smaller vortices. As the predominant part of the lift of a high-lift configuration is generated by the main airfoil, the most important effect of periodic excitation is to change the flow direction at the main airfoil trailing edge. By delaying separation on the flap the mean flow direction behind the trailing edge is changed (see figure 8 and figure 17). Vortices are generated and transported downstream and interact with those vortices detaching from the main airfoil. The surface pressure and the lift coefficient oscillate with the excitation frequency. Flow control with lower intensity means smaller vortices, which are able to
Fig. 17. Flow field around flap represented by vorticity distribution for different excitation intensities left: Cµ = 50 × 10−5 ; centre: Cµ = 100 × 10−5 ; right: Cµ = 200 × 10−5
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penetrate the flap boundary layer and the shear layer between the freestream and the reverse flow. However, larger vortices move away from the flap surface and are less effective. This may explain the small effect of low intensity excitation. The flow separation is located in the turbulent part of the flow. However, turbulence intensity is very low at the separation point. One effect of periodic excitation is to transfer energy from the potential flow region into the boundary layer. Steady simulations of turbulent flow with high turbulence intensity predict attached flow for high flap angles [2] stressing that high turbulence intensity can avoid flow separation on the flap.
6 Conclusion Two different high-lift configuration were studied for the application of active flow control by means of periodic suction and blowing in the flap boundary layer. First a twoelement configuration and later a setup with main airfoil, slat and flat were simulated by unsteady RANS simulations. The effect of vertical zero-net-mass excitation was studied. The results computed were compared with available wind tunnel test results to determine the prediction capability of the computational method. For the baseline simulations without excitation, the influence of mesh resolution and time-stepping was studied and reasonable results could be obtained with the LLR and SST k-ω turbulence models and correct transition fixing. Both configurations were adjusted to provide separated flow conditions on the flap. This desired behaviour with a large recirculation region over the flaps was predicted and the results agree fairly well with the experiments. For the active control cases, steady suction and blowing do not show satisfactory effects. To study flow control by periodic excitation, various blowing coefficients were investigated at a given excitation frequency. In general, the results for both cases show that the lift increases if the intensity exceeds a certain limit that is about Cµ > 25 · 10−5 in the present cases. However, for very large intensities with Cµ > 50·10−5 the positive effects become increasingly smaller. By changing the excitation frequency at a constant intensity an optimum frequency can be identified for the two-component configuration. It corresponds to a sinusoidal lift and detaching vortices of a suitable size. The mean lift coefficient increases by up to 30% compared to the baseline simulation, and separation can be delayed. The effect of Reynolds number turned out not to be crucial for the range 105 < Re < 106 investigated, and the flow control concept could successfully be applied. The behaviour of two completely different configurations showed the same principle influence of the excitation parameters on the resulting flow field. In both cases lift, drag and the mean flap separation could be controlled by parameters that do not differ substantially.
7 Outlook In the near future the investigations will be extended to simulations of flow control for the SCCH configuration including the 3d effects of the swept wing. The threedimensional model also allows investigations of excitation modes that vary in the
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spanwise direction and that might have a beneficial effect. Due to the uncertainties of URANS for simulations at higher Reynolds numbers the upcoming computations will be based on a Detached-Eddy Simulation (DES). Further investigations will be focused on a coupling of the flow control apparatus to an advanced active feed-back control system.
Acknowledgement This research is funded by the German National Science Foundation (Deutsche Forschungsgemeinschaft, DFG) under the umbrella of the Collaborative Research Center (Sonderforschungsbereich, Sfb 557, ’Kontrolle komplexer turbulenter Scherstr¨omungen’) at the Berlin University of Technology. The simulations were performed on the IBM pSeries 690 supercomputer at the North German Cooperation for High-Performance Computing (HLRN). This support is gratefully acknowledged by the authors.
References [1] F.H. Tinapp. Aktive Kontrolle der Str¨omungsabl¨osung an einer HochauftriebsKonfiguration. PhD thesis, Technische Universit¨at Berlin, 2001. [2] M. Schatz and F. Thiele. Numerical study of high-lift flow with separation control by periodic excitation. AIAA Paper 2001-0296, 2001. [3] M. Schatz, F. Thiele, R. Petz, and W. Nitsche. Separation control by periodic excitation and its application to a high lift configuration. AIAA Paper 2004-2507, 2004. [4] D.V. Maddalon, F.S. Collier, L.C. Montoya, and C.K. Land. Transition flight experiments on a swept wing with suction. AIAA Paper 89-1893, 1989. [5] A.A. Hassan and R.D. Janakiram. Effects of zero-mass synthetic jets on the aerodynamics of the NACA-0012 airfoil. AIAA Paper 97-2326, 1997. [6] S.S. Ravindran. Active control of flow separation over an airfoil. TM-1999-209838, NASA, Langley, 1999. [7] J.F. Donovan, L.D. Kral, and A.W. Cary. Active flow control applied to an airfoil. AIAA Paper 98-0210, 1998. [8] I. Wygnanski. The variables affecting the control separation by periodic excitation. AIAA Paper 2004-2505, 2004. [9] M. Gad-el Hak. Flow control: The future. Journal of Aircraft, 38(3), 2001. [10] F.R. Menter. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 32(8):1598–1605, 1994. [11] T. Rung and F. Thiele. Computational modelling of complex boundary-layer flows. In 9th Int. Symp. on Transport Phenomena in Thermal-Fluid Engineering, Singapore, 1996. [12] M. Schatz. Numerische Simulation der Beeinflussung instation¨arer Str¨omungsabl¨osung durch frei bewegliche R¨uckstromklappen auf Tragfl¨ugeln. PhD thesis, Technische Universit¨at Berlin, 2003. [13] P.R. Spalart and S.R. Allmaras. A one-equation turbulence model for aerodynamic flows. AIAA Paper 92-0439, 1992. [14] T. Rung, U. Bunge, M. Schatz, and F. Thiele. Restatement of the spalart-allmaras eddyviscosity model in a strain-adaptive formulation. AIAA Journal, 41(7):1396–1399, 2003. [15] D.C. Wilcox. Turbulence Modeling for CFD. DCW Industries, Inc., La Ca˜nada, 1993.
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[16] R. Petz, W. Nitsche, M. Schatz, and F. Thiele. Increasing lift by means of active flow control on the flap of generic high-lift configuration. In Proc. 14th DGLR-Fach-Symposium der AG STAB, Bremen, Germany, 2004. [17] M. Franke, T. Rung, M. Schatz, and F. Thiele. Numerical simulation of high-lift flows employing improved turbulence modelling. In ECCOMAS 2000, Barcelona, September 11-14, 2000. [18] F. Tinapp and W. Nitsche. On active control of high-lift flow. In W. Rodi and D. Laurence, editors, Proc. 4th Int. Symposium on Engineering Turbulence Modelling and Measurements, Corsica. Elsevier Science, 1999. [19] A. Seifert and L.G. Pack. Oscillatory excitation of unsteady compressible flows over airfoils at flight reynolds numbers. AIAA Paper 99-0925, 1999. [20] R. Meyer and D.W. Bechert. Beeinflussung von Str¨omungsabl¨osungen an Tragfl¨ugeln. Abschlussbericht, Hermann-F¨ottinger-Institut, TU Berlin, 1998. [21] L. Koop. Aktive und Passive Str¨omungsbeeinflussung zur Reduzierung der Schallabstrahlung an Hinterkanntenklappen von Tragfl¨ugeln. PhD thesis, Technische Universit¨at Berlin, 2005. [22] K. Kaepernick, L. Koop, and K. Ehrenfried. Investigation of the unsteady flow field inside a leading edge slat cove. In 11th AIAA/CEAS Aeroacoustics Conference (26th Aeroacoustics Conference), Monterey, CA, USA, 2005.
Steady and Oscillatory Flow Control Tests for Tilt Rotor Aircraft M. Schmalzel, P. Varghese, and I. Wygnanski The Aerospace and Mechanical Engineering Department The University of Arizona, Tucson, AZ. 85721
Nomenclature AFC AF c CD CDo Cdp Cdf CF CL CP CQ cµ Cµ D DL f F+
Active Flow Control Active Flap chord length drag coefficient: D / q c Where CD = Cdp for a particular Cµ form drag coefficient: ³ (p − p∞) dy / q c skin friction coefficient: ³ τ dy / q c integrated force coefficient: (CL2 + Cdp2)1/2 lift coefficient: L / q c pressure coefficient: (p − p∞) / q steady volume flow coefficient: Q / SU∞ Combined oscillatory momentum coefficient: Cµ+ steady momentum coefficient: (2h/c)(USlot/U∞) 2 oscillatory momentum coefficient: (h/c)(USlotMax /U∞) 2 drag download force frequency of excitation non-dimensional frequency: (f xc / U∞)
h L NK PK q Re T xc
slot height lift No Kruger flap Passive Kruger flap dynamic pressure: ½ ρU2∞ Reynolds Number: U∞ c / ν Thrust distance from slot to trailing edge angle of attack flap angle of attack
ηD ηL
Figure of Merit for decrease in drag per unit of momentum: ∆CD/Cµ Figure of Merit for increase in lift per unit of momentum: ∆CL/Cµ
α δf
R. King (Ed.): Active Flow Control, NNFM 95, pp. 190–207, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007
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Abstract The purpose of this manuscript is to address one of the many questions plaguing the application of fluidic active flow control for performance enhancement over wings and airplanes. Specifically, what mode of Active Flow Control (AFC) is most effective; steady suction, steady blowing, or a periodic variation of both? The tilt rotor model is chosen because it represents very demanding requirements over a wide range of incidence angles, α, varying from –90o 0 and W Γ > 0 are positive definite weighting functions for the state vector and the control signal, respectively. Minimization of J c results in asymptotic stabilization of the origin, while the control energy is kept small. In our design, the weights have been chosen as W a~ = I 4× 4 and W Γ = 1 for all three models, and the corresponding control gains with respect to the three flows read as 1
=
K
2
=
K
3
=
K
[− 56 [52.5 [17.6
.2
3.9
− 12 . 8 ] ,
− 417 . 2
8 .8
- 168
208 . 8
- 102
],
(5.7)
− 146 . 8 ] .
11 . 6
Applying the state feedback control, equation (5.5), to the linearized system, equation (5.3), results in mirroring all the right-half plane eigenvalues of the matrix ~ G to the left half plane. Figure 8 shows the simulation results obtained by applying the state feedback control, equation (5.5), to the finite-dimensional nonlinear model, equation (3.2), which indicate that the closed-loop state trajectories a(t ) converge to the corresponding equilibrium points in each case, given by a 01 = [ − 0.5036 0.2788 − 0.1930 0.4980]T , a 0 2 = [ − 0 . 3081
a 0 3 = [ − 0 .3261
0 . 1483
− 0 . 2083
0 . 4895 ] T
0 .2158
− 0 .2598
T
0 .4753 ]
,
(5.8)
.
It can be concluded that, in principle, the LQ controller, equation (5.5), designed for the linear approximation, equation (5.3), succeeds in stabilizing the equilibrium of the four modes nonlinear Galerkin system, equation (3.2).
Reduced-Order Model-Based Feedback Control of Subsonic Cavity Flows
(a)
(b)
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(c)
Fig. 8. Time coefficient solutions of the closed-loop simulation results. (a) baseline flow model, (b) open-loop forced flow model with forcing frequency of 3920 Hz, (c) combined flow model of the above two cases.
6 Feedback Control Results and Discussion 6.1 Modeling Results
Before presenting the results of the experimental implementation of the controller, it is worth summarizing the structure of the model-based controller derived in Section 5. As depicted in Figure 9, the model-based controller includes a stochastic estimation subsystem and a feedback from the estimated states. The estimate a~ˆ of the deviation from the equilibrium of the mode amplitudes of the Galerkin model, required to
Fig. 9. Diagram of the closed loop system with LQ state feedback control
implement the feedback law, equation (5.5), may be in principle obtained by means of stochastic estimation by first estimating aˆ (t ) from raw pressure measurements using equation (3.3), and then subtracting the equilibrium value a 0 computed from the model data. However, in implementing the controller, subtracting the equilibrium values from the estimated ones is not required, since the DC values have been removed from the pressure measurements by means of high-pass filtering. That is, equation (3.3) naturally produces the values of a~ˆ from the experimental measurements. It is important to point out that, to prevent any damage to the actuator, the control input signal is limited to the range ±10V. Since the gains of the LQ
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control, equation (5.7), are quite large, constant saturations of the actuator were observed during closed-loop experiments for all cases under investigation. Therefore, it was necessary to introduce a scaling factor 0R adjacent to the flap, above and below, each have the same sign as the primary vortices generated on the same side of the flap. At r 0). We note that the indeterminacy of the function f in (1) reflects the nonuniqueness of solutions of the Euler equations in a given domain Ω. For instance, expressing the function f (ψ) on the RHS as a linear combination of 2K Dirac delta distributions 2K k=1 Γk δ(x − xk )δ(y − yk ) with weights Γk , where K is the total number of singularities and their images, leads to systems known from the classical potential flow theory corresponding to 2K point vortices located at the points {xk , yk }2K k=1 . An example of such a solution was found in closed form by F¨oppl in [5], where the potential flow was obtained by placing behind the obstacle two counter–rotating point vortices located symmetrically with respect to the centerline (Figure 1a). With points √ of the plane characterized by their complex coordinates z = x + iy, where i = −1, the complex potential of this flow W0 (z) = (ϕ + iψ)(z), where ϕ and ψ are, respectively, the potential and the streamfunction, can be expressed as W0 (z) = WC (z) + WF,0 (z),
(2)
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B. Protas y 8
U δΩ
Γc Ω
11 00 00000000000 11111111111 11111111111 00000000000 Γ1 00000000000 11111111111 00000000000 11111111111 −Γ 00000000000 11111111111 11 0 00000000000 11111111111 0 1 00000000000 11111111111 00000000000 11111111111 1 0 00000000000 11111111111 00000000000 11111111111 1 0 00000000000 11111111111 00000000000 11111111111 −Γ2 00000000000 11111111111 Γ2 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 11 00
(a)
β
1 0
γ
α x
β 1 γ 0
α
(b)
Fig. 1. Schematics showing (a) the location of the singularities in the classical F¨oppl system and (b) the three modes of motion characterizing the linearized F¨oppl system (7): the unstable mode α, the asymptotically stable mode β and the neutrally stable (oscillatory) mode γ. In Figure (a) the dashed line represents the separatrix streamline delimiting the recirculation region.
where R2 WC (z) = U∞ z + , (2a) z Γ0 R2 R2 − ln(z − z 0 ) + ln z − (2b) WF,0 (z) = ln(z − z0 ) − ln z − 2πi z0 z0 and Γ0 and z0 = x0 + iy0 represent, respectively, the circulation and position of the F¨oppl vortices. In Equation (2) WC (z) represents the base flow symmetric with respect to the OY axis and due to the cylinder only, whereas WF,0 (z) corresponds to the two F¨oppl vortices located at z0 and z 0 and their two images located inside the obstacle. Two one–parameter families of steady vortex configurations given by (2) were found by F¨oppl in [5]: the configuration characterized by the condition ⎧ 2 2 2 2 2 ⎪ ⎨ (|z0 | − R ) = 4|z0 | y0 , (3) (|z0 |2 − R2 )2 (|z0 |2 + R2 ) ⎪ , ⎩ Γ0 = −2π 5 |z0 | hereafter referred to as the “classical F¨oppl system” (Figure 1a), and the configuration characterized by the condition (z0 ) = 0, i.e., corresponding to the vortices located on the OY axis. The latter solution, however, does not correspond to any physical situation and will not be discussed further in this investigation. The classical F¨oppl system has recently been used as a reduced–order model in the development of a simple feedback stabilization strategy for the cylinder wake flow in [6]. The cylinder rotation ΓC = ΓC (t) was used the flow actuation (i.e., the control variable) and observations of the centerline velocity downstream of the obstacle as the system output. The advantage of this approach is that, due to simplicity of the F¨oppl model, the synthesis of the stabilization algorithm becomes a simple task with a significant part of the calculations carried out analytically. The performance of the
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stabilization strategy, while quite encouraging, also showed some limitations of such very simple point–vortex systems employed as reduced–order models. The purpose of this investigation is to identify sources of these limitations and propose possible improvements. The structure of the paper is as follows: first in Section 2 we review the formulation of the Linear–Quadratic–Gaussian (LQG) compensator designed based on the F¨oppl system as a reduced–order model, in Section 3 we introduce a family of higher–order F¨oppl systems characterized by more desirable properties as candidates for reduced–order models, then some computational results are presented in Section 4 and conclusions are deferred to Section 5.
2
Control Design Based on the F¨oppl System as a Reduced–Order Model
We begin this Section by analyzing the stability properties of the classical F¨oppl system linearized around the equilibrium solution. This analysis will motivate the design of an LQG compensator for feedback stabilization of the linearized F¨oppl system. Careful analysis of the linear stability of the F¨oppl system and its relevance to modeling the onset of vortex shedding in 2D wake flows was presented by Tang and Aubry in [7]. Our discussion of the control–theoretic aspects will be here necessarily concise and the reader is referred to the publication [6] for further details. Different flow control problems also based on the F¨oppl system as the reduced–order model were studied in [8,9]. We will assume that the cylinder has unit radius R = 1 and the free stream at infinity has unit magnitude U∞ = 1. In addition, we will also assume that all quantities are nondimensionalized using these values. The F¨oppl model can be regarded as a nonlinear dynamical system with evolution described by ⎡ ⎤ [V1 (z1 , z2 , Γ1 , Γ2 )] ⎢−[V1 (z1 , z2 , Γ1 , Γ2 )]⎥ d ⎥ X = F(X) + b(X)ΓC ⎢ (4) ⎣ [V2 (z1 , z2 , Γ1 , Γ2 )]⎦ + b(X)ΓC , dt −[V2 (z1 , z2 , Γ1 , Γ2 )] where X [x1 y1 x2 y2 ]T is the state vector and the control matrix b(X) is expressed as ⎡ ⎤ −y1 /|z1 |2 2⎥ 1 ⎢ ⎢ x1 /|z1 |2 ⎥ . (5) b(X) ⎣ y /|z | 2π 2 2 ⎦ x2 /|z2 |2 The expressions for V1 and V2 in (4) are given by the velocity field 1 Γ1 1 1 V (z) = 1 − 2 − − z 2πi z − z1 z − 1/z1 1 Γ2 1 ΓC , + − + 2πi z − z2 z − 1/z2 2πiz
(6)
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1 1 evaluated at z1 and z2 with the singular “self–induction” terms ( z−z and z−z , re1 2 spectively) omitted. For the moment we will fix attention on the properties of the F¨oppl system without control, hence we will assume that ΓC ≡ 0, which renders (4) autonomous. The linear stability analysis of the F¨oppl system is performed by adding the perturbations (x1 , y1 ) and (x2 , y2 ) to the coordinates of the upper and lower vortex of the stationary solution and then linearizing the system (4) around X0 [x0 y0 x0 − y0 ]T assuming small perturbations. Thus, evolution of the perturbations is governed by the system d X = AX , (7) dt
where X [x1 y1 x2 y2 ]T is the perturbation vector and the system matrix is given ∂F by the Jacobian of the nonlinear system at the equilibrium A = ∂X (X0 ). We remark that (7) is a linear time–invariant (LTI) system. Eigenvalue analysis of the matrix A reveals (see [7] for details) the presence of the following modes of motion (Figure 1b): – unstable (growing) mode α corresponding to a positive real eigenvalue λ1 = λr > 0, – stable (decaying) mode β corresponding to a negative real eigenvalue λ2 = −λr < 0, – neutrally stable oscillatory mode γ corresponding to a conjugate pair of purely imaginary eigenvalues λ3/4 = ±iλi . These qualitative properties are independent of the downstream coordinate x0 characterizing the equilibrium solution. The analysis of the orientation of the unstable eigenvectors carried out in [7] revealed that the initial stages of instability of the F¨oppl system closely resemble the onset of vortex shedding in an actual cylinder wake undergoing Hopf bifurcation. The free parameter characterizing the equilibrium solution (3) of the F¨oppl system (i.e., the downstream location of the singularities x0 ) is chosen here, so that the length of the recirculation zone in the F¨oppl system is the same as the recirculation length in the steady unstable solution of the Navier–Stokes system at a prescribed Reynolds number. Further justification as well as details of calculations are described in [6]. In the examples presented hereafter the downstream position of the vortices was chosen, so that the recirculation length is the same as in the steady unstable solution of the Navier–Stokes system at the Reynolds number Re = 75. After including the control term representing the cylinder rotation the linearized F¨oppl system becomes d X = AX + BΓC , (8) dt where ⎡ ⎤ −y0 ⎥ 1 ⎢ ⎢ x0 ⎥ . B b(X0 ) = (9) 2 ⎣ y 2πr0 0⎦ x0 As mentioned in Section 1 our control objective is attenuation of vortex shedding which can be quantified by measuring the velocity at a point on the flow centerline with the
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streamwise coordinate xm (note that in the stationary symmetric solution the transverse velocity component vanishes on the centerline). Choosing this quantity as an output of system (4) we obtain the following output equation [V (xm )] h(z1 , z2 ) (10) + DΓC , −[V (xm )] 1 T where the matrix D 2πx represents the control–to–measurements map. 2 [0 xm ] m Linearization of equation (10) yields
h(z0 + z1 , z 0 + z2 ) ∼ = h(z0 , z 0 ) + CX ,
(11)
where zk = xk + iyk , k = 1, 2, and the linearized observation operator C is given by ∂u(xm ) ∂u(xm ) ∂u(xm ) ∂u(xm ) ∂x1 ∂y1 ∂x2 ∂y2 (x ,y ) (x ,y ) (x ,y ) (x ,y ) 0 0 0 0 0 0 0 0 C = ∂v(xm ) . (12) ∂v(xm ) ∂v(xm ) ∂v(xm ) ∂x1
(x0 ,y0 )
∂y1
(x0 ,y0 )
∂x2
(x0 ,y0 )
∂y2
(x0 ,y0 )
Uncertainty of the reduced–order model is represented by the presence of noise w which affects the linearized system dynamics via a [4 × 1] matrix G and the linearized system output via a [2 × 1] matrix H. Moreover, we assume that the velocity measurements may be additionally contaminated with noise m [m1 m2 ]T , where m1 and m2 are stochastic processes. With these definitions we can now put the linearized reduced– order model in the standard state–space form (see [10]) d X = AX +BΓC +Gw, dt Y = CX +DΓC +Hw + m.
(13a) (13b)
Prior to designing a controller for system (13) we have to verify whether the system is controllable and observable which is done by studying the ranks Nc and No of the controllability and observability Grammians (14) Nc rank B AB A2 B A3 B = 2, T T T T 2 T T 3 T (15) No rank C A C (A ) C (A ) C = 4. We conclude that the matrix pair {A, B} is not controllable and only two out of four modes present in the system can be controlled. On the other hand, the matrix pair {A, C} is completely observable. Converting system (13) to the minimal representation which consists of those modes only which are both controllable and observable will allow us to determine which modes are in fact controllable. We accomplish this by introducing an orthogonal transformation matrix ⎡ ⎤ 1/2 0 −1/2 0 √ ⎢ 0 1/2 0 1/2⎥ ⎥ (16) Tc 2 ⎢ ⎣1/2 0 1/2 0 ⎦ 0 1/2 0 −1/2
428
B. Protas
and making the following change of variables Xab
Xa = Tc X . The correspondXb
ing form of system (13) is now d Xa Ga Aa 0 Xa Ba ΓC + w, = + X X 0 Gb 0 A dt b b b H1 0 Cb Xa D1 Yb = + Γ + w + m. Ya Ca 0 Xb D2 C H2
(17a) (17b)
Our minimal representation is thus given by the upper row in equation (17a) and the lower row in (17b), i.e., d X = Aa Xa +Ba ΓC +Ga w, dt a Ya = Ca Xa +D2 ΓC +H2 w + m2 .
(18a) (18b)
Eigenvalue analysis of the matrices Aa and Ab reveals that Aa has two real eigenvalues (positive and negative) corresponding to the growing and decaying modes α and β, whereas the matrix Ab has a conjugate pair of purely imaginary eigenvalues which correspond to the neurally stable mode γ (Figure 1b). Hence, the uncontrollable part of the model system dynamics is associated with the neutrally stable oscillatory mode γ and the original system (13) is thus stabilizable, but not controllable. Practical effectiveness of the proposed algorithm depends on the location of the velocity sensor xm . As argued in [6], the distance xm is chosen so as to maximize the observability residual of the unstable mode α. Our objective here is to find a feedback control law ΓC = −KX , where K is a [4 × 1] feedback matrix, that will 1. stabilize system (13) and 2. minimizing a performance criterion represented by the following cost functional ∞ T (Y QY + ΓC RΓC )dt , (19) J (ΓC ) E 0
where E denotes the expectation, Q is a symmetric positive semi–definite matrix and R > 0. Note that the cost functional (19) represents a sum of the linearized system output Y [i.e., the velocity at the sensor location (xm , 0)] and the control effort. The feedback control law determines the actuation (i.e., the circulation of the control vortex ΓC representing the cylinder rotation) based on the state of the reduced–order model (i.e., the perturbation X of the stationary solution). In practice, however, the state X of the ˜ = [Y˜b Y˜a ]T of the actual model (13) is not known. Instead, noisy measurements Y system [i.e., the nonlinear F¨oppl model (4) or an actual wake flow] are available and can be used in an estimation procedure to construct an estimate Xe of the model state X . The evolution of the state estimate Xe is governed by the estimator system d ˜ − Ye ), X = AXe +BΓC + L(Y dt e Ye = CXe +DΓC ,
(20a) (20b)
Vortex Models for Feedback Stabilization of Wake Flows
w (system noise)
429
~ Y (measurements)
PLANT Γc (control) COMPENSATOR CONTROLLER
X’e
ESTIMATOR
v (measurement noise)
Γc (control) Fig. 2. Schematic of a compensator composed of an estimator and a controller
where L is a feedback matrix that can be chosen in a manner ensuring that the estimation error vanishes in the infinite time horizon, i.e., that Xe → X as t → ∞. Thus, the estimator assimilates available observations into the system model, so as to produce an evolving estimate of the system state. Finally, the controller and the estimator can be combined to form a compensator in which the feedback control is determined based on the state estimate Xe as (21) ΓC = −KXe . The flow of information in a compensator is shown schematically in Figure 2. The design of a Linear–Quadratic–Gaussian (LQG) compensator can be accomplished using standard methods of Linear Control Theory (see, e.g., [10]). Here we only remark that, since system (8) is stabilizable, but not controllable, the controller can in fact be designed based on the minimal representation. On the other hand, since system (8) is observable, the estimator is designed based on the original representation. Given small dimensions of systems (7) and (18), solution of the Riccati equations at the heart of these problems does not pose any difficulties. The reader is referred to [6] for further details.
3 Higher–Order F¨oppl Systems In this Section we describe potential flow solutions generalizing F¨oppl’s classical point– vortex system. They can approximate with desired accuracy the velocity field of the steady–state solutions of the Euler equations characterized by arbitrary compact vorticity support. Such system have the same dimension as the original F¨oppl model, however, are characterized by an arbitrary number of adjustable parameters, hence are more flexible as reduced–order models. We consider again solutions of system (1), however, instead of a collection of Dirac delta functions now we allow for more general forms of the RHS function f (ψ). A family of interesting solutions was computed by Elcrat et al. in [11] by choosing f (ψ) as follows −ω, ψ ≤ σ, f (ψ) = (22) 0, ψ > σ,
430
B. Protas
y
z
U
8
P
δΩ
z’
zs
ω0
Q Fig. 3. Schematic showing EFHM solutions with two opposite–sign vortex patches P (solid line) and Q (dashed line) located symmetrically with respect to the flow centerline
where σ is an adjustable parameter. Such a vorticity distribution is sometimes referred to as the Rankine core. These solutions will play an important role in our development and we will hereafter refer to them as the “EFHM flows”. When σ < 0, such flows are characterized by compact regions of constant vorticity embedded in an otherwise irrotational flow and are therefore related to the so–called Sadovskii flows [12]. In addition to this solution, shown schematically in Figure 3, it was found in [11] that regions of opposite–sign vorticity, hereafter denoted P and Q, may also exist above and below the obstacle, as well as in front of it. Since these solutions do not correspond to any physical situation, they will not be considered hereafter. Consider a compact region P of vorticity embedded in an irrotational flow past a circular cylinder (Figure 3) and characterized by the vorticity distribution ω = ω(z). When ω = Const, the corresponding steady–state solutions of the Euler equations defined by (1) and (22) are given by families of the EFHM flows computed in [11]. Below we construct algebraic systems that are approximations of such solutions. Our approach is conceptually related to the method devised by [13] in which moments of vorticity distribution are used to characterize the evolution of a system of vortex patches. Using complex Green’s function for the Laplace equation in a 2D unbounded domain 1 ln(z − z ), the complex potential induced by a vortex patch in such a G(z, z ) = 2πi domain can be expressed for points outside the patch z ∈ / P as ˜ P (z) = (ϕ + iψ)(z) = 1 ln(z − z )ω(z ) dA(z ), (23) W 2πi P where dA(z ) = dx dy . Tilde (˜) indicates that this potential represents a flow in an unbounded domain (i.e., without the obstacle), whereas the subscript indicates that the potential is due to the patch P . We now choose a point zs ∈ P as the origin of the local coordinate system associated with the patch P and set ζ = z − zs (see Figure 3). The complex potential (23) can now be expressed as
Vortex Models for Feedback Stabilization of Wake Flows
ln 1 −
˜ P (z) = Γ0 ln(z − zs ) + 1 W 2πi 2πi
P
ζ z − zs
431
ω(zs + ζ) dA(ζ).
(24)
The second term in (24) can, for |z − zs | > |z − zs |, be expanded in a Taylor series which yields ∞ cn ˜ P (z) = Γ0 ln(z − zs ) − 1 (z − zs )−n , W 2πi 2πi n=1 n
where
|z − zs | > ζm ,
(25)
ω(zs + ζ)ζ n dA(ζ)
cn (zs ) =
(26)
P
and ζm = max(zs +ζ)∈P |ζ|. Thus, the point zs represents also the location of a singularity which, for the moment, remains unspecified. The quantities cn (zs ), n = 1, . . . , N are the moments of the vorticity distribution in the patch P with respect to the point zs and therefore are related to the eccentricity of the patch (c1 ), its ellipticity (c2 ), etc. (unless required for clarity, hereafter we will skip the argument of cn ). The zeroth moment c0 is equal to the total circulation Γ0 of the patch. The complex potential due to a finite– area vortex patch P can be approximated for points of the plane lying outside this patch by truncating expression (25), i.e., replacing it with a finite sum of singularities located at the point zs N cn ˜ P (z) ∼ ˜ P,N (z) = Γ0 ln(z −zs )− 1 W (z −zs )−n , |z −zs | > ζm . (27) =W 2πi 2πi n=1 n
˜ . The complex The order of truncation is represented by the second subscript on W ˜ Q,N (z) due to the patch Q with the opposite–sign vorticity and located potential W symmetrically below the flow centerline (Figure 3) can be represented using an analogous expression in which zs is replaced with z s and cn with −cn for n = 1, . . . , N . Below we use these expressions to construct potential flows approximating solutions of the steady–state Euler equations (1) in the sense that the velocity field of the potential flow will converge, for z ∈ / P and z ∈ / Q, to the velocity field of the Euler ˜ P,N (z) flow as N → ∞. These potential flows are constructed using the potentials W ˜ Q,N (z), and adding suitable “image singularities” located inside the obstacle in and W a way ensuring that the boundary conditions for the wall–normal velocity component are satisfied. In general, such flows can be constructed using the “Circle Theorem” [4] which states that if w(z) ˜ is the complex potential of a flow in an unbounded domain and with singularities at some points zk , such that ∀k, |zk | > R, then the complex potential of the corresponding flow past the cylinder with radius R is given by the expression 2 ˜ Rz ). Thus, using this construction to enforce the boundary condiw(z) = w(z) ˜ + w( tions and including also the base flow with the potential WC (z) [cf. Eq. (2)], we obtain the following expression for the complex potential
432
B. Protas
WN (z) = WC (z) + WF,N (z)
2 2 R R ˜ ˜ ˜ ˜ = WC (z) + WP,N (z) + WQ,N (z) + W P,N + W Q,N z z 2 2 R R Γ0 − = U∞ z + − ln(z − zs ) − ln z − z 2πi zs N cn R2 1 1 − ln(z − z s ) + ln z − zs 2πi n=1 n (z − zs )n n n z z cn cn cn n n n n , − (−1) − + (−1) n 2 2 zs (z − z s ) zs R z−R z − zs zs (28)
where WF,N (z) represents the truncated potential due to the finite–area vortex patches and their images. We notice that setting N = 0 in (28) we recover the complex potential (2) of the classical F¨oppl system discussed in Section 1. Therefore, the family of the complex potentials given in (28) represents N –th order corrections to the F¨oppl system regarded as approximations of the corresponding solution of the steady–state Euler equations and hereafter we will refer to them as the “higher–order (N –th order) F¨oppl systems”. By taking N large enough we can obtain an arbitrarily accurate representation of the velocity field in the Euler flow valid for points in the flow domain outside the vortex patches, thereby improving applicability of the F¨oppl system as a model for steady wake flows. In general, in an inviscid and incompressible fluid singularities (e.g., a point vortex s ˆ located at zs ) move according to the velocity field dz dt = VN (zs ), where complex conjugation is required to account for the fact that the complex velocity field is given by VˆN = u ˆ − iˆ v . We note that the advection velocity VˆN of a singularity is not affected by its self–induction which can be seen by regarding the singularity as a limit of a sequence of finite–area circular distributions of the corresponding quantity. The velocity induced by such distributions can be shown to vanish at their center. Thus, the advection velocity of a singularity is obtained as N c Γ 1 0 n VˆN (z) = VN (z) − − + , (29) 2πi z − zs n=1 (z − zs )n+1 N (z) where VN (z) = dWdz , i.e., the terms which become singular as z → zs are removed from the velocity field. Hereafter, hats (ˆ) will distinguish quantities with these self–inductions terms subtracted off. We are interested in steady–state solutions of the higher–order F¨oppl systems, therefore, for a given truncation order N , we need to find the equilibrium points of system (29), i.e., the points zN such that setting zs = zN the following condition is satisfied for given Γ0 and {cn }N n=1
VˆN (zN ) = 0.
(30)
Vortex Models for Feedback Stabilization of Wake Flows
433
This condition can be expanded to ⎡ 2 1 1 R Γ 1 0 ⎣− − VˆN (zN ) = U∞ 1 − 2 − + R2 zN 2πi (z − z ) N N zN − z N zN − ⎡ N n−1 R 2 cn zN 1 ⎢ n+1 + ⎣(−1) n+1 n+1 2 2πi n=1 zN zN − zRN ⎤ −
cn − (−1)n+1 (zN − z N )n+1
R 2 cn zN −
R2 zN
n+1
⎤ R2 zN
⎦
1 ⎥ 2 ⎦ =0 zN
(31) which is a complex–valued equation characterizing one complex unknown zN . In the case when N = 0, one solution of (31) is given by (3). When N ≥ 1, solutions must be found numerically, e.g. using Newton’s method. Furthermore, since the order of the equation increases with N , it can be anticipated that so does the number of roots. In fact, it can be proved that there is always one root of (31) that is in a neighborhood of the solution characterized by (3) and the size of this neighborhood can be bounded by the magnitudes of the coefficients cn . Thorough analysis of this and other analytical properties of the higher–order F¨oppl system is deferred to a forthcoming paper [14].
4 Computational Results In this Section we present some preliminary computational results concerning construction of a higher–order F¨oppl system for a given EFHM flow and application of such a system as a reduced–order model to the design of an LQG–based stabilization strategy. Because of space limitations, our discussion here is necessarily short and the reader is referred to the forthcoming papers [14,15] for further particulars regarding the computational procedure and detailed results. To fix attention, we will focus on the F¨oppl system with the singularities located at [x0 , y0 ] = [4.32, ±2.3596] and with the circulation of the vortices given by Γ0 = −29.6015 (this is the configuration investigated in [6]). We will also consider an EFHM flow with the area of the vortex patch A = 20.43 as a desingularization of the classical F¨oppl system and will take N = 10 as the truncation order in the construction of the higher–order system. As analyzed in detail in [14], such higher–order systems are characterized by multiple solutions, however, in our analysis below we will focus on the equilibrium [xN , yN ] in the neighborhood of [x0 , y0 ], both of which are shown in Figure 4a. As can be easily verified, the higher–order F¨oppl system linearized about the new equilibrium [xN , yN ] has the same properties in terms of controllability and observability as the classical F¨oppl system [cf. Eqns. (14) and (15)]. Assuming measurements of two velocity components on the flow centerline as the observations and thecylinder rotation as the actuation, all four modes are observable, but only two of them are controllable. Stability analysis of this higher–order F¨oppl system indicates that, in
434
B. Protas
4
y/R
3
2
1
0
1
2
3
4
5
6
7
8
x/R (a)
2.6 2.5 2.4
y/R
2.3 2.2 2.1 2.0 1.9 1.8 3.5
4.0
4.5
5.0
5.5
6.0
x/R (b) Fig. 4. (a) Location of the equilibria of (circle) the classical F¨oppl system (3) and (square) the higher–order F¨oppl system (31) with N = 10. The boundary of the vortex patch in the EFHM flows used to construct the higher–order F¨oppl system is represented by the dotted line and the cylinder boundary is represented by a thick solid line. (b) Trajectories of the state of (solid line) the classical and (dotted line) higher–order F¨oppl system stabilized with an LQG compensator in the neighborhood of the corresponding equilibrium solutions.
Vortex Models for Feedback Stabilization of Wake Flows
435
addition to a growing and decaying mode (corresponding to, respectively, the modes α and β, cf. Figure 1b) characterized by purely real eigenvalues, there exists also a mode characterized by pair of complex–conjugate eigenvalues (corresponding to the mode γ, cf. Figure 1b). However, in contrast to the classical F¨oppl system, these complex eigenvalues have negative real parts, hence the oscillatory mode in the higher–order F¨oppl system is in fact exponentially stable. This difference has important consequences when a linear stabilization strategy, such as LQG, is applied to the original nonlinear system. As illustrated in Figure 4b, when the LQG compensator is applied to the classical F¨oppl system, the state of the system does not return to the equilibrium, but lands instead on a closed orbit. One can prove rigorously using methods of dynamical systems that this orbit has in fact the structure of a center manifold and the trajectory of the system on this manifold is stable (see [15] for precise statements and proofs of these theorems). On the other hand, when the LQG compensator is applied to the higher–order F¨oppl system, the system trajectory returns to the equilibrium owing to the exponential stability of the uncontrollable modes.
5 Conclusions The dynamics of both the classical and higher–order F¨oppl systems in the neighborhood of an equilibrium point is characterized by four degrees of freedom. However, in contrast to the classical system which has just one parameter, the higher—order systems are characterized by an arbitrary number of adjustable parameters represented by the expansion coefficients in (28). The number of these parameters is determined by the truncation order N . Therefore, by introducing a larger number of adjustable parameters, one can incorporate much more flexibility into F¨oppl–type models, so that, while remaining four–dimensional, they can reproduce more accurately certain properties of realistic flows. Advantages of having this additional flexibility were illustrated by the computational results presented in Section 4. We showed that the state of the classical F¨oppl system with an LQG stabilization converges to a center manifold, whose persistence prevents this state from reaching the equilibrium and, as a result, the amplitude of the state oscillations does not decrease. We conjecture that this is a possible reason for the oscillations of the velocity field in the near wake region occurring when this strategy was applied to stabilize an actual cylinder wake flow at Re = 75 (see [6]). On the other hand, the flexibility of the higher–order F¨oppl system investigated here made it possible to alter the stability properties of the new equilibrium in such way that the uncontrollable mode became stable. As a result, the same LQG compensation strategy was now able to stabilize completely the equilibrium. We anticipate that this additional flexibility of higher–order F¨oppl systems will play a role when employing these systems as reduced–order models to stabilization of actual cylinder wake flows. Verification of performance of such approaches is underway.
Acknowledgments The author wishes to express his thanks to Profs. Alan Elcrat and Ken Miller for many interesting and helpful discussions regarding the EFHM flows and for providing him
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with a code to reproduce the results of the paper [11]. This research was supported by an NSERC Discovery Grant (Canada) and CNRS (France).
References [1] J. Lumley and P. Blossey: “Control of Turbulence”, Ann. Rev. Fluid Mech. 30, 311-327, (1998). [2] T. R. Bewley and S. Liu: “Optimal and robust control and estimation of linear paths to transition”, J. Fluid Mech. 365, 305-349, (1998). [3] J. Kim: “Control of turbulent boundary layers”, Phys. Fluids 15, 1093-1105, (2003). [4] L. M. Milne–Thompson: “Theoretical Hydrodynamics”, MacMillan, (1955). [5] L. F¨oppl: “Wirbelbewegung hinter einem Kreiscylinder”, Sitzb. d. k. Bayr. Akad. d. Wiss. 1-17, (1913). [6] B. Protas: “Linear Feedback Stabilization of Laminar Vortex Shedding Based on a Point Vortex Model”, Phys. Fluids 16, 4473-4488, 2004. [7] S. Tang and N. Aubry: “On the symmetry breaking instability leading to vortex shedding”, Phys. Fluids 9, 2550-2561, (1997). [8] S. Tang and N. Aubry: “Suppression of vortex shedding inspired by a low–dimensional model”, J. Fluids and Struct. 14, 443-468, (2000). [9] F. Li and N. Aubry: “Feedback control of a flow past a cylinder via transverse motion”, Phys. Fluids 15, 2163-2176, (2003). [10] R. F. Stengel: “Optimal Control and Estimation”, Dover Publications, (1994). [11] A. Elcrat, B. Fornberg, M. Horn and K. Miller: “Some steady vortex flows past a circular cylinder”, J. Fluid Mech. 409, 13-27, (2000). [12] V. S. Sadovskii: “Vortex regions in a potential stream with a jump of Bernoulli’s constant at the boundary”, Appl. Math. Mech. 35, 729, (1971). [13] M. .V. Melander, N. J. Zabusky and A. .M. Styczek: “A moment model for vortex interactions of two–dimensional Euler equations. Part 1 — Computational validation of a Hamiltonian elliptical representation”, J. Fluid Mech. 167, 95-115, (1986). [14] B. Protas: “Higher–order F¨oppl models of steady wake flows”, Phys. Fluids 18, 117109, (2006). [15] B. Protas: “Center Manifold Analysis of a Point–Vortex Model of Vortex Shedding with Control”, (submitted), 2006.
Keyword Index
acoustic forcing Phase-Shift Control … Closed-loop Active… Designing Actuators for … aerodynamic improvement Active Control to … air injection Active Control to… Active Blade Tone … Airfoil Steady and Oscillatory … alternating suction and blowing Designing Actuators for … artificial neural network State estimation of … axial turbomachine Active Control to … backward facing step Flow Control on … Control of Wing Vortices bluff body flow Feedback control applied… boundary layer Experimental and Numerical… boundary layer separation Closed-loop Active … cavity flow control Reduced-order Model-based … cavity resonance Supersonic Cavity Response… CFD Computational Investigation… Experimental and Numerical Investigations… Flow Control on… chaos control Turbulence: The Taming … circulation flow Pulsed Plasma Actuators …
408 85 69 293 293 391 190 69 105 293 325 137 369 56
coherent structures Turbulence: The Taming … combustion control Phase-Shift Control … combustion instabilities Phase-Shift Control of … computational fluid dynamics Flow Control on … control design Pulsed Plasma Actuators … Flow Control on … control of entrainment Active Management of … control of turbulence Turbulence: The Taming … critical flow regimes Turbulence: The Taming … cylinder wake State estimation of … delta wing Control of Wing Vortices… drag control Active Drag control … drag reduction Feedback control …
1 408 408 325 42 325 281 1 1 105 137 247 369
85 211
230 173 56 325 1 42
Electromagnetic flow control Electromagnetic control of … excitation wave form Electromagnetic control of … experimental-based model Reduced-order Model-based … extremum-seeking control Active Blade Tone … fan noise Active Control to Improve … FEATFLOW Flow Control on… feature extraction A Unified Feature …
27 27 211 391 293 325 119
438
Keyword Index
feedback control Closed-loop Active … Turbulence: The Taming … Feedback control applied … State estimation of … figure of merit Closed-loop Active … flat plate Experimental and Numerical... flow analysis A Unified Feature… flow separation Electromagnetic control … flow stability Continuous mode … flow topology A Unified Feature … flow visualization A Unified Feature … forward facing step Flow control with … Galerkin model Continuous mode interpolation… Galerkin projection Reduced-order Model-based Feedback … generic car model Active Drag control … high-lift Designing Actuators for … Computational Investigation … Hover Steady and Oscillatory …
85 1 369 105 85 56 119 27 260 119 119 353
260
211 247 69 173 190
interstage hotwire measurements Active Blade Tone …
391
jet control Active Management of …
281
Lavrentiev regularization Flow control with … leading edge stall Towards Active control …
353 152
LES Active Drag control … linear control theory Vortex Models for … linear-quadratic optimal control Reduced-order Model-based … longitudinal vortex Active Drag control … low dimensional modeling State estimation of … Continuous mode interpolation … low-dimensional vortex models Feedback control … Matlab-interface Flow Control on … micro aerial Pulsed Plasma Actuators … microactuators Turbulence: The Taming .. microfabrication Turbulence: The Taming... microsensors Turbulence: The Taming mode interpolation Continuous mode interpolation … model-based feedback control Reduced-order Modelbased … Navier-Stokes equations On the choice of … Newton method Flow control with … noise control Active Blade Tone … noise reduction Active Control to I … nonlinear dynamical systems theory Turbulence: The Taming … nonlinear system identification State estimation of …
247 422
211 247 105 260 369 325 42 1 1 1 260
211 339 353 391 293
1 105
Keyword Index
open-loop control Supersonic Cavity Response … optimal flow control On the choice … Drag Minimization … Flow control with … periodic excitation Electromagnetic control of .. Computational Investigation ... phase-shift control Phase-Shift Control … PIV Active Drag control … Towards Active control of… Electromagnetic control of… plasma actuator Experimental and Numerical … pneumatic actuators Towards Active control … POD State estimation of … point vortex models Vortex Models for … pointwise state constraints Flow control with … proper orthogonal decomposition State estimation … Drag Minimization of … pulsed blowing Designing Actuators for … pulsed plasma actuators Pulsed Plasma Actuators … pulsed-blowing actuator Supersonic Cavity Response… RANS Computational Investigation… Flow Control on … reactive control Turbulence: The Taming … reattachment Control of Wing Vortices reduced order model Drag Minimization of … Reduced-order Model-based…
230 339 309 353 27 173 408 247 152 27 56 152 105 422 353 105 309 69 42 230 173 325 1 137 309 211
Reynolds-averaged Navier-Stokes equations Flow Control on … rotating instability Active Control to … secondary fuel injection Phase-Shift Control … semi-smooth Flow control with … separation control Designing Actuators for .. Steady and Oscillatory … Active Drag control … Computational Investigation … Pulsed Plasma Actuators … shear flows Flow Control on … snapshot POD Reduced-order Model-based … soft computing Turbulence: The Taming … spanwise vortex Active Drag control … stochastic estimation Reduced-order Model-based … streamwise vortices Active Management of … supersonic cavity Supersonic Cavity Response … tip clearance noise Active Control to … tonal fan noise Active Blade Tone … transient flow estimation State estimation of … transition Experimental and Numerical … trust-region optimization Drag Minimization of …
439
325 293 408 353 69 190 247 173 42 325
211 1 247
211 281
230 293 391 105
56 309
440
Keyword Index
turbulent boundary layers Turbulence: The Taming … unsteady aerodynamics Pulsed Plasma Actuators … V-22 model Steady and Oscillatory … Vehicles Pulsed Plasma Actuators … vortex breakdown Control of Wing Vortices…
Printing: Mercedes-Druck, Berlin Binding: Stein + Lehmann, Berlin
1 42 190 42
137
vortex core lines A Unified Feature … vortex reduction On the choice … Vortices Control of Wing Vortices… wake flow Drag Minimization of … Vortex Models for …
119 339
137 309 422
