Vortex-Based Aero- and Hydrodynamic Estimation

176 Pages·2013·11.16 MB·English

University of California
Los Angeles
Vortex-Based
Aero- and Hydrodynamic Estimation
A dissertation submitted in partial satisfaction
of the requirements for the degree
Doctor of Philosophy in Mechanical Engineering
by
Maziar Sam Hemati
2013
(cid:13)c Copyright by
Maziar Sam Hemati
2013
Abstract of the Dissertation
Vortex-Based
Aero- and Hydrodynamic Estimation
by
Maziar Sam Hemati
Doctor of Philosophy in Mechanical Engineering
University of California, Los Angeles, 2013
Professor Jeff D. Eldredge, Co-chair
Professor Jason L. Speyer, Co-chair
Flow control strategies often require knowledge of unmeasurable quantities, thus presenting
a need to reconstruct flow states from measurable ones. In the present work, the model-
ing, simulation, and estimator design aspects of flow reconstruction are considered. First,
a vortex-based aero- and hydrodynamic estimation paradigm is developed to design a wake
sensing algorithm for aircraft formation flight missions. The method assimilates wing dis-
tributed pressure measurements with a vortex-based wake model to better predict the state
of the flow. The study compares Kalman-type algorithms with particle filtering algorithms,
demonstrating that the vortex nonlinearities require particle filters to yield adequate per-
formance. Furthermore, the observability structure of the wake is shown to have a negative
impact on filter performance regardless of the algorithm applied. It is demonstrated that
relative motions can alleviate the filter divergence issues associated with this observability
structure. In addition to estimator development, the current work addresses the need for
an efficient unsteady multi-body aerodynamics testbed for estimator and controller valida-
tion studies. A pure vortex particle implementation of a vortex panel-particle method is
developed to satisfy this need. The numerical method is demonstrated on the impulsive
startup of a flat plate as well as the impulsive startup of a multi-wing formation. It is clear,
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from these validation studies, that the method is able to accommodate the unsteady wake
effects that arise in formation flight missions. Lastly, successful vortex-based estimation is
highly dependent on the reliability of the low-order vortex model used to represent the flow
of interest. The present work establishes a systematic framework for vortex model improve-
ment, grounded in optimal control theory and the calculus of variations. By minimizing
model predicted errors with respect to empirical data, the shortcomings of the baseline vor-
tex model can be revealed and reconciled. Here, the method is demonstrated on an impulse
matching model for canonical unsteady wing maneuvers and reveals the shortcomings of the
Kutta condition in such flows. The resulting analysis sheds light on the governing physi-
cal processes and provides guidance for model improvement for the unsteady aerodynamics
associated with these canonical wing maneuvers.
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The dissertation of Maziar Sam Hemati is approved.
Andrea L. Bertozzi
John Kim
Jason L. Speyer, Committee Co-chair
Jeff D. Eldredge, Committee Co-chair
University of California, Los Angeles
2013
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“Vortices of pure energy can exist and, if my theories are right,
can compose the bodily form of an intelligent species.”
—Sir William Thomas, 1st Baron Kelvin
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Table of Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Flow Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Reduced Order Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Mathematical ROM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Physics-Based ROM . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Vortex-Based Aero- and Hydrodynamic Estimation . . . . . . . . . . . . . . 7
1.4 Overview of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Aero- and Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 The Navier-Stokes Equations and the Euler Limit . . . . . . . . . . . . . . . 12
2.2 Flow Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Helmholtz Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Biot-Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.3 Line Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Flow Kinetics and Vorticity Transport . . . . . . . . . . . . . . . . . . . . . 16
2.4 Kelvin-Helmholtz Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Vortex Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Kutta Condition and Vorticity Shedding . . . . . . . . . . . . . . . . . . . . 22
3 Vortex Model Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Vortex Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Complex Potential: System of Vortices in the Presence of a Flat Plate 27
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3.2.2 Force on the Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.3 Impulse Matching Model . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 A Variational Approach to Vortex Model Improvement . . . . . . . . . . . . 31
3.3.1 Constrained Optimization Formulation . . . . . . . . . . . . . . . . . 31
3.3.2 Method of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.3 True Force Histories: High-Fidelity Viscous Vortex Particle Simulation 34
3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Pitching Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.2 Impulsive Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Remaining Challenges and Paths to Enhancement . . . . . . . . . . . . . . . 53
3.5.1 Convergence for Large Time-Windows and Early-Times . . . . . . . . 53
3.5.2 Extending Time Windows via Stitching . . . . . . . . . . . . . . . . . 57
3.5.3 A Framework for Subsequent Vortex Shedding . . . . . . . . . . . . . 58
3.5.4 Distilling a Modified Kutta Condition . . . . . . . . . . . . . . . . . . 64
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Simulation Testbed: A Vortex Panel-Particle Method . . . . . . . . . . . 72
4.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 The Mathematical Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Discretization of Vortex Sheets . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Vortex Particle Representation of Line Vortex Segments . . . . . . . 78
4.4 Unsteady Trailing Edge Kutta Condition . . . . . . . . . . . . . . . . . . . . 79
4.5 Wake Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6 Aerodynamic Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 Code Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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4.7.1 Simulation Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7.2 Impulsive Start of a Flat Plate . . . . . . . . . . . . . . . . . . . . . 84
4.7.3 Impulsive Start of a Two Flat Plate Formation . . . . . . . . . . . . 84
5 Nonlinear Estimation and Filtering . . . . . . . . . . . . . . . . . . . . . . . 90
5.1 Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1.1 Linear Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1.2 Extended Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . 94
5.1.3 Variations of Extended Kalman Filtering . . . . . . . . . . . . . . . . 96
5.2 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.1 Particle Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 Bias and Divergence Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3.1 Modeling Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3.2 Observability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6 Wake Estimation for Formation Flight . . . . . . . . . . . . . . . . . . . . . 104
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Aerodynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2.1 Lead Aircraft and Wake Representation . . . . . . . . . . . . . . . . 108
6.2.2 Trailing Aircraft Representation . . . . . . . . . . . . . . . . . . . . . 110
6.3 Wake Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.1 System Observability and Conditioning . . . . . . . . . . . . . . . . . 116
6.3.2 Lifting Line Based Measurement Jacobian . . . . . . . . . . . . . . . 117
6.3.3 Wake Observability and Conditioning Comparison . . . . . . . . . . . 121
6.4 Wake Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
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6.4.1 States and Measurements . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4.2 Measurement and Process Noise . . . . . . . . . . . . . . . . . . . . . 125
6.4.3 Kalman-Type Filtering: Measurement Function Linearization . . . . 125
6.4.4 Particle Filtering Procedure . . . . . . . . . . . . . . . . . . . . . . . 126
6.4.5 Offline Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4.6 System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5 Performance Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 128
6.5.1 Two-Aircraft Static Configuration . . . . . . . . . . . . . . . . . . . . 129
6.5.2 Two-Aircraft with Relative Motions . . . . . . . . . . . . . . . . . . . 132
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7 Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . 148
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
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