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Title: Adaptive Robust Fixed Structure Control of Turbulence Submitted to: Air Force Office of Scientific Research Program Manager: Dr. Mark Glauser AFOSR/NA 110Duncan Avenue, Suite B115 Bolling AFB, DC 203320001 Submitted by: Polytechnic University Department of Mechanical, Aerospace & Manufacturing Engineering Six Metrotech Center Brooklyn, NY 11201 Principal Investigators: Dr. M. Volkan Otugen, Dr. Anthony Tzes and Dr. Vikram Kapila Proposed Duration: 3 years Funding Level: $ 305,542 Starting Date: November 1, 1998 Table of Contents Page I. Abstract 2 II. Introduction 3 2.1 Background 2.2 Review of Previous Work in Flow Control III. Proposed Work 3.1 Objective 3.2 Approach IV. Proper Orthogonal Decomposition V. Adaptive Identification Strategy 5 VI. Robust LowOrder Controller Synthesis 7 VII. Study of Controller Performance via DNS 8 VIII. References 12 IX. Work Plan 9 X. Key Personnel XI. Proposed Budget 10 XII. Curriculum Vitae of Senior Personnel I. Abstract A research effort is proposed focusing on the systematic modeling and active control of nearwall turbulence using advanced control theory. Adaptive set membership identification and robust fixedstructure control schemes will be developed and tested for the control of turbulence in the wall region of a turbulent channel flow. The proper orthogonal decomposition (POD) approach in conjunction with a Galerkin procedure will be used to obtain the finitedimensional dynamic model of the nearwall turbulent flow. In order to capture the true dynamics of turbulence and to obtain a high model accuracy, a large number of POD modes will be retained in the truncation procedure. The computational complexity and the associated throughput constraints due to the large number of modes used in the system will be overcome by designing high performance loworder controllers based on fixedstructure control theory. The robust control scheme will accommodate uncertainties caused by: (i) the truncation of the model dynamics and (ii) the actuator nonlinearities. The adaptive control scheme which relies on the set membership identification techniques, will serve to mitigate the effects of actuatorinduced flow perturbations. The research effort will deal with controlturbulence interaction phenomenon with equal emphasis on both the flow physics and control. It will take advantage of the new developments in POD modeling of turbulence as well as the modern advances in adaptive and robust control theory. Specifically, it will exploit the recent POD mode calculation results in the nearwall region of a turbulent flow. It is well known that turbulence is produced primarily in the nearwall layer and organized structures in this region are the principle contributors to wall shear stress. Therefore, control strategies targeting this flow region have the best overall chance of turbulence control and drag reduction. The research will be carried out in two phases. In the first phase, the control schemes using the POD model of Aubry et. al. (1988) and Sanghi (1992) for nearwall turbulence will be developed. The suitability of the schemes and the fundamental questions of controllability and observability will be evaluated by performing numerical tests based on the POD model. The second phase of the research will be dedicated to the comprehensive and independent evaluation of the performance of the control schemes. This will be carried out in the form of numerical experiments via direct numerical simulation (DNS) of a planar channel flow together with feedback control which will include sensing and actuation in the nearwall region. This phase will allow for the "fine tuning" of the proposed control laws. Complex issues such as the optimal sensor/actuator placement will also be addressed in this phase. II. Introduction 2.1. Background Recent technological advancements in microelectromechanical systems (MEMS), coupled with the several successful applications of proper orthogonal decomposition (POD) to the dynamical system modeling of turbulence during the last decade, have raised the possibility of active robust control of turbulence in the nearwall region of bounded flows. The cost effective application of micro actuators and sensors to the practical problem of drag reduction over large vehicles may yet be some years away (McMichael, 1996). However, it is feasible today to sense and control random turbulent flow structures in a boundary layer and hence reduce skin friction at least in a small section of the surface (Ho & Tai, 1996; Rathnasingham et. al., 1997). Therefore, the important task of developing and demonstrating robust feedback control strategies for turbulent flows can now be initiated. Efforts in this area are critical in achieving a fundamental understanding of the basic issues related to controllability and observability of turbulence. Research in this area can also play a pivotal role in the future advances of MEMS devices for turbulence control. The successful design of robust feedback controllers for turbulence can help the future evolution of autonomous sensor/actuator devices as well as large arrays of such devices with integrated controllers. The MEMS technology integrates sensors, actuators, and electronics on a single substrate using integrated circuit design and micromachining techniques. Thus, these devices can be made very small (down to about a micron in physical dimensions) and in large batches with uniform performance characteristics. They tend to be inexpensive while capable of implementing high performance and sophisticated functions. From flow control point of view, perhaps the most attractive feature of MEMS devices is their small size. For example, it is well known that in boundary layers organized structures in the nearwall region carry the bulk of turbulent energy, and are the primary contributors to turbulent skin friction. In flows with Reynolds numbers relevant to most engineering applications, the typical dimensions of these randomly spaced structures range from hundreds of microns to a few millimeters with life cycles ranging between tens of microseconds to milliseconds. MEMS can match these flow structures in size and response time thus, as pointed out by Ho & Tai (1996), these devices have the best chance of directly manipulating turbulence. The small sensor size provides structureresolved information and, with an appropriate control strategy, actuators of proper size (for example, of the order of a boundary layer displacement thickness for external flows) can accomplish turbulence reduction with high energy efficiency. However, it should be emphasized here that the success in developing efficient turbulence control techniques depends equally strongly on the appropriateness of the control scheme. Various types of MEMS sensors are currently being developed that are applicable to fluid mechanics. These include pressure sensors, wall shearstress sensors, and micro hotwires (Jiang et. al., 1994; Ho & Tai, 1996; Mehregany, et. al., 1996). These sensors, again, have the advantage of being smaller (up to two orders of magnitude) and having faster response times (about an order of magnitude) than the conventional researchtype sensors. As for actuators, several types have been developed including “synthetic” jets (Smith & Glezer, 1997), micro flaps (Ho & Tai, 1996), and electromagnetic microtiles (Singh & Bandyopadhyay, 1997). These devices are larger than the MEMS sensors due to the power and strength requirements demanded by practical flows. As will be apparent in the survey presented in Section 2.2, unfortunately, little work has been accomplished to date in the application of modern control theory to turbulence management. Yet, successful control schemes are a critical ingredient in flow control. Clearly, there is a need for advanced system identification and control strategies complementing the MEMS technology if tangible progress is to be made towards the management of turbulence and drag reduction in flows with Reynolds numbers relevant to air vehicles. In view of the above, a research effort is proposed here which aims to develop adaptive identification and robust control schemes for nearwall turbulence in a channel flow. As a first step towards successful control, an accurate dynamical system representation of turbulence will be developed. We propose to use the POD approach (Lumley, 1967) to obtain a finite dimensional system model which provides a highaccuracy representation of the flow dynamics. For this purpose, the mode calculations of Sanghi (1992) for a nearwall turbulent flow will be used. These calculations are based on the experimental results obtained by Herzog (1986) in the nearwall region of a turbulent pipe flow. In order to obtain an accurate ordinary differential equation (ODE) model which properly represents the dynamic structure of the flow and captures most of the turbulent energy, a large number of modes will be retained in the POD truncation process. The computational loads resulting from the high dimensionality of the system will be addressed by designing high performance loworder controllers derived from fixedstructure control theory as described in Section VI. In addition, an adaptation mechanism will be developed in order to minimize the effects of actuatorinduced flow perturbations on system model. The adaptation mechanism will rely on set membership identification techniques and will provide a family of transfer functions consistent with sensor outputs and the structure of the problem (see Section V). Therefore, as time progresses, the uncertainty of the identified transfer function parameters will be reduced, thereby allowing for the “fine tuning” of the controller. 2.2. Review of Previous Work in Flow Control Most previous appraches to control of turbulence over surfaces have been based on passive techniques. In these techniques, no energy is added to the flow and, in most cases, the flow dynamics is controlled by modifying the surface characteristics such as adding riblets or using compliant materials (Bushnell & McGinley, 1989; GadelHak, 1989). The more recent approaches include active control methodologies, where energy/momentum is added to the flow in order to reduce turbulence. Active control can be implemeted as either openloop or closedloop (feedback). The openloop approach involves the addition of momentum into the flow in advance, either continuously or periodically (for example, blowing or suction). In the feedback control approach, on the other hand, the addition of energy/momentum is in response to the occurance of a turbulent event detected by sensors. Therefore the feedback approach to turbulence control can be more effective and efficient since the controller allows for a more judicious use of the added energy. Although many strategies have been proposed varying from simple feedforward approaches to formal optimal control (a review of these can be found in Moin & Bewley, 1994), few actual attempts have been made for closedloop control of turbulence. As pointed out earlier, some of the earlier studies concentrated on computer simulations of transition delay in laminar boundary layers using suction and blowing. The basis for these studies was essentially wave cancellation (Metcalfe et al., 1986; Laurien & Kleiser, 1989; Danabasanoglu et al., 1991; Joslin et al., 1995). Disturbances were superimposed inphase and, outofphase onto the instability waves to augment and suppress transition, respectively. Jackobsen & Reynolds (1995) carried out simple experiments to test the applicability of neural networks to active control of a turbulent boundary layer. Neural networkbased control schemes require substantial realtime computations and long training periods. A feedforward control approach to a turbulent boundary layer was used by Rathnasingham & Breuer (1997). The identification and control method used in this experimental work is essentially based on qualitative physical arguments. Wall sensors placed upstream edge of a control region detect large energy containing (coherent) structures near the wall. A linear transfer function is then used along with a single “synthetic jet” as actuator. Although simple in scope, this study nevertheless showed the possibility of reducing turbulence intensities and wall shear stress in a boundary layer using wall sensors and miniature jet actuators. A 30% reduction in the local streamwise turbulence intensity along with about 7% reduction in wall shear stress was achieved in this study. In another study with a similar approach, Choi et al. (1994) explored the use of a simple active control strategy to reduce drag (or average pressure gradient to drive a fixed mass flow rate) in a channel flow using DNS rather than physical experiments. Several sensor locations and actuator concepts were explored for the largest drag reduction in the channel. These numerical experiments indicated that drag can indeed be reduced by detecting and counteracting certain structures in the nearwall region. Most importantly, the study showed that DNS can be a powerful tool in the initial development and testing stage of advanced control concepts prior to actual physical experiments. Attempts towards active control of turbulence discussed above are essentially based on flow physics arguments. In these studies, the focus has been on sensing the correct “significant” turbulent structure in the near wall region and counteracting it, again, in a predetermined fashion (by either pushing the structure away from the wall or adding enery to cancel it) with little use of the modern control theory. In order to take advantage of advanced control strategies, a dynamical system modeling of nearwall turbulence has to be used. In this approach, the NavierStokes equations are reduced to a set of ordinary differential equations (ODEs). In one of the few attempts using a dynamical systembased approach, Singh & Bandyopadhyay (1997) used linear feedback control approach to control the nearwall zone of a turbulent salt water flow over a flat plate. Electromagnetic tiles were used as actuators in this numerical study which aimed to demonstrate the viability of both microtiles and the feedback control approach used. The scope of the work was limited to the control of small perturbations in the viscous sublayer and the use of linear control theory was hence justified. In order to apply the feedback control, the NavierStokes equations were also linearized citing the limitation of the work to the viscous sublayer. A dynamic system model of this simplified flow was obtained by Galerkin projection. Chebyschev polynomials were used to form the basis functions for velocity in the wallnormal direction. The results showed that the stabilization of the viscous sublayer can be achieved with this approach. Turbulent flows even at moderate Reynolds numbers typically pose a large range of scales with significant energy. This requires the use of large dimensions in order to obtain an accurate dynamic model representation of turbulence. Severe model truncations leave out significant turbulent energy from the model and lead to noisy systems that render control difficult in any realistic flow. However, recent studies have shown that higher accuracies with lower dimensions can be obtained in a dynamic model using the proper orthogonal decomposition (POD) approach (Aubry et al., 1988; Berkooz et al., 1993). Therefore, control strategies using the PODbased ODE models are more likely to succeed in turbulence mitigation. Following up on this, Coller et al. (1994) laid the ground work for future studies using this approach. They carried out a mathematical analysis exploring the use of PODbased low dimensional models for feedback control of turbulent events in the nearwall region of a boundary layer. They used POD models with a limited number of modes; starting with as low as 2 modes and progressing to 5 and then 10 modes (model earlier obtained by Aubry et al.; 1988). The objective was to determine whether or not such a low number of modes can be used, in principle, to affect the burst/sweep events in the boundary layer. The assumed actuation mechanism was a crossstream flow that could be created, for example, by a streamwise vortex pair. The study indicated that although a control design based on such small number of modes cannot completely stabilize the flow under realistic conditions including noise, it can still affect the flow by increasing the interburst interval. Thus, in essence, Coller et al. (1994) showed the feasibilty of the use of PODbased models in the feedback control of nearwall turbulent events in a fully turbulent flow. 