COMPUTATIONAL FLUID DYNAMIC SIMULATION OF AIRFOILS IN UNSTEADY LOW REYNOLDS NUMBER FLOWS by Mohammadreza Amiralaei BSc, Sharif University of Technology, Tehran, Iran, 2004 MSc, Sharif University of Technology, Tehran, Iran, 2007 A Dissertation Presented to the School of Graduate Studies at Ryerson University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Program of Aerospace Engineering Toronto, Ontario, Canada, 2012 (cid:13)c Mohammadreza Amiralaei, 2012 Author’s Declaration I hereby declare that I am the sole author of this dissertation. This is a true copy of the dissertation, including any required final revisions, as accepted by my examiners. I authorize Ryerson University to lend this dissertation to other institutions or individu- als for the purpose of scholarly research. I further authorize Ryerson University to reproduce this dissertation by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. I understand that my dissertation may be made electronically available to the public. ii Abstract Computational Fluid Dynamic Simulation of Airfoils in Unsteady Low Reynolds Number Flows (cid:13)c Mohammadreza Amiralaei, 2012 Doctor of Philosophy Aerospace Engineering Ryerson University The inherent complexity of low Reynolds number (LRN) flows and their respective viscous vortical patterns demand an accurate solution method to achieve the desired accuracy. This complicated flow field needs even more robust methods when the flow is unsteady. The flow field of unsteady airfoils and wings in LRN regime is challenging to solve and Computational Fluid Dynamics (CFD) simulations stand out as solid solution techniques in this area. This thesis is motivated by an existing rotating-flapping mechanism, whose kinematics compo- nents can be broken into pitching, plunging and a novel figure-of-eight-like flapping motion of its blades and each blade’s cross section. The focus is on two-dimensional low Reynolds number (LRN) flows using Computational Fluid Dynamics (CFD) and a Finite Volume Method (FVM). As one of the targets is to simulate a pair of blades, and consequently a pair of airfoils, a mesh motion library is developed to perform rotational and translational motions of multi-body configurations. The library and its sub-routines are tested on pairs of pitching, plunging and flapping airfoils, where the moving mesh problem is performed with a significant gain in the computational time compared to other moving mesh techniques such as Laplacian smoothing algorithm. The simulations of a single airfoil under harmonic and the novel figure-of-eight-like flapping motions, respectively, are conducted within 67% and 80% time it took to obtain a steady solution using the Laplace smoothing mesh motion algorithm, while the calculated force coefficients were in reasonably close agreement. Flow fields of single unsteady airfoils under pitching, plunging and figure-of-eight flapping mo- tions are also simulated in this thesis accompanied with extensive parametric studies. The iii simulations of the considered figure-of-eight flapping pattern shows that its highly inclined asymmetrical kinematics results in higher vertical lift coefficients than the existing flapping patterns in the literature, useful for stable hovering flight. The studies over paired-airfoils arrangements under pitching and plunging and the figure- of-eight flapping motion show that the airfoil-airfoil interaction affects the fluid forces notice- ably. The multi-plunging analysis, for example, reveals that the maximum lift coefficient is higher than that of a single plunging airfoil, while minimum drag coefficient is lower, showing the favorable effect of airfoil-airfoil interaction in the studied multi-plunging cases. iv Acknowledgments I would like to extend my gratitude to my supervisors professors H. Alighanbari and S. M. Hashemi and thank them for their guidance, support and constant encouragement during the course of this research. I would also like to thank Dr. D. Poirel of Royal Military College for serving as my external examiner. The positive feedback, suggestion and advice of my internal committee members professors B. Jubran, F. Sharifi, P. Walsh and J. Yokota from Ryerson University are greatly appreciated. I would also thank my friends: Noushin, Robin and Masroor for their support, encour- agement and advise, and the last but certainly not the least to my mother for her love and support and my father who always believed in me and encouraged me at all the times. v Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv 1 Introduction 1 1.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 Modeling airfoils under different LRN unsteady conditions . . . . . . 4 1.3.2 Extensive parametric studies . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.3 Implementation, validation and modification of the fast and robust mesh motion algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.1 Analysis of flow field characteristics and force signatures of airfoils under unsteady conditions . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.2 Development of mesh motion methodologies for simulating multi-body problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Background and literature review 9 vi 2.1 LRN flow physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Dynamic stall and leading edge vortex . . . . . . . . . . . . . . . . . 10 2.1.2 Rotational circulation (rotational forces) . . . . . . . . . . . . . . . . 10 2.1.3 Wake-capturing mechanism (wing-wake interaction) . . . . . . . . . . 11 2.1.4 Weis-Fogh clap and fling mechanism . . . . . . . . . . . . . . . . . . 12 2.2 LRN solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Analytical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Numerical Simulation Method 23 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Discretization of N-S equations . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.1 Discretizationoftemporalandspatialtermsinincompressiblemomen- tum equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.5 Solution of the governing equations . . . . . . . . . . . . . . . . . . . . . . . 32 3.6 N-S equations for moving grid problems . . . . . . . . . . . . . . . . . . . . . 34 3.7 Moving mesh algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4 On the LRN flow characteristics of a pitching airfoil 43 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Theory and numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2.1 Governing equations and flow solver . . . . . . . . . . . . . . . . . . . 45 4.2.2 Mesh generation and boundary conditions . . . . . . . . . . . . . . . 46 4.2.3 Validation studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 vii 4.3.1 Effects of amplitude of oscillation . . . . . . . . . . . . . . . . . . . . 53 4.3.2 Effects of reduced frequency . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.3 Effects of Reynolds number . . . . . . . . . . . . . . . . . . . . . . . 59 4.4 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5 Numerical modeling of the flow field characteristics of a LRN plunging airfoil 63 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Reference velocity, force and moment coefficients calculations . . . . . . . . . 66 5.4 Evaluation of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.4.1 Mesh and time step sensitivity analyses . . . . . . . . . . . . . . . . . 67 5.4.2 Validation studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.5 Evaluation of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.5.1 Effect of the Strouhal number without freestream flow . . . . . . . . 71 5.5.2 Effect of the Strouhal number with freestream flow . . . . . . . . . . 73 5.5.3 Effect of the Reynolds number . . . . . . . . . . . . . . . . . . . . . . 77 5.6 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6 Flow field characteristics of a figure-of-eight-like flapping airfoil 85 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.2 Numerical simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3 Kinematics and equations of motion . . . . . . . . . . . . . . . . . . . . . . . 90 6.4 Reference velocity, force and moment coefficients calculations . . . . . . . . . 92 6.5 Evaluation of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.5.1 Effect of the amplitude of pitching oscillations . . . . . . . . . . . . . 97 6.5.2 Effect of the phase angle . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.5.3 Effect of the mean angle of attack . . . . . . . . . . . . . . . . . . . . 106 viii 6.5.4 Effect of the Reynolds number . . . . . . . . . . . . . . . . . . . . . . 108 6.5.5 Effect of Strouhal Number . . . . . . . . . . . . . . . . . . . . . . . . 109 6.5.6 Effect of the pitching axis location . . . . . . . . . . . . . . . . . . . 112 6.6 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7 The unsteady aerodynamics of a pair of pitching airfoils 115 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 Numerical simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.3 Evaluation of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.4.1 Effect of amplitude of pitching oscillations . . . . . . . . . . . . . . . 120 7.4.2 Effect of Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . 125 7.5 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8 On the force signatures of a pair of plunging airfoils 129 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8.2 Numerical simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.3 Evaluation of the numerical approach . . . . . . . . . . . . . . . . . . . . . . 133 8.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.4.1 Effect of the Reynolds number . . . . . . . . . . . . . . . . . . . . . . 137 8.4.2 Effect of the frequency of oscillations . . . . . . . . . . . . . . . . . . 142 8.4.3 Effect of the amplitude of oscillations . . . . . . . . . . . . . . . . . . 146 8.4.4 Effect of the airfoil spacing . . . . . . . . . . . . . . . . . . . . . . . . 151 8.5 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9 Multi figure-of-eight flapping airfoils 154 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.2 Numerical simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . 156 9.3 Evaluation of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 ix 9.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.5 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 10 Concluding Remarks 169 10.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 10.2 Thesis Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 10.2.1 Published Chapters in Books . . . . . . . . . . . . . . . . . . . . . . 172 10.2.2 Refereed Journal Publications . . . . . . . . . . . . . . . . . . . . . . 172 10.2.3 Submitted Journal Publications . . . . . . . . . . . . . . . . . . . . . 172 10.2.4 Refereed Conference Publications . . . . . . . . . . . . . . . . . . . . 173 A Motion Methodology for Multi Figure-of-Eight Flapping Motions 176 Bibliography 210 x