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Open Nicholaslabarbera.Pdf The Pennsylvania State University The Graduate School College of Engineering A COMPARISON OF MODELING APPROACHES OF A HIGH ASPECT CYLINDER IN AXIAL FLOW A Thesis in Engineering Science and Mechanics by Nicholas LaBarbera © 2015 Nicholas LaBarbera Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science December 2015 The thesis of Nicholas LaBarbera was reviewed and approved∗ by the following: Jonathan S. Pitt Research Associate, Applied Research Laboratory Assistant Professor of Engineering Science and Mechanics Thesis Advisor Robert L. Campbell Research Associate, Applied Research Laboratory Assistant Professor of Mechanical Engineering Judith A. Todd P.B. Breneman Department Chair Department of Engineering Science and Mechanics ∗Signatures are on file in the Graduate School. ii Abstract Simulating a fully-coupled fluid-structure interaction system from first-principles can be very computationally expensive especially for use in design-level analyses; therefore, it is advantageous to explore less computationally expensive methods. By making assumptions about the relevant physics of the problem, simplifications to the governing equations can be applied. These simplifications result in a reduced- order model that can significantly decrease the computational cost; however, the governing equations simplifications result in the reduced-order model neglecting to take into account all the physics of the system. Ideally, the neglected physics would have little to no impact on the dynamics of the system; however, this is not always the case for all input parameters. Therefore, it is important to determine the parameter space for which a reduced order model is valid. In this thesis, numerical simulations of a slender cylinder in axial flow were performed using two different methods. The first is a strongly-coupled fluid- structure interaction simulation based on first-principles. First-principles for this case is solving the incompressible Navier-Stokes equations for the fluid and the full equations of elasticity for the solid domain. The second is a reduced-order model for a cylinder in axial flow that was devised by Paidoussis. It consists of a single linear partial differential equation that requires significantly less computational resources to solve than the non-linear governing equations used in the first-principles model. For certain input parameters, the reduced-order model is capable of predicting stability with comparable accuracy to the first-principles model, but is obtained in a fraction of the time. The goal of this thesis is to investigate the parameter space for which the less computational expensive reduced order model can be utilized and for which the more computationally expensive first-principles solver is required. To accomplish the goal, the two models were compared through the variation of input parameters to determine the parameter ranges corresponding to agreement between models. Results are presented and conclusions discussed. iii Table of Contents List of Figures vi List of Tables viii List of Symbols ix Acknowledgments xi Chapter 1 Introduction 1 1.1 Introduction . 1 Chapter 2 Literature Review 5 2.1 Introduction . 5 2.2 Existing Models . 9 2.2.1 String Approximation . 10 2.2.2 Beam Modelling . 11 2.2.3 Coupled FSI Simulations . 13 2.2.4 Recap by Decade . 14 2.3 State of the Art . 16 2.3.1 Plan to Advance State of the Art . 16 Chapter 3 First Principle solver 17 3.1 Introduction . 17 3.2 Governing equations . 18 3.3 Solvers . 18 3.3.1 Solid Solver - FEANL . 19 3.3.2 Fluid Solver - OpenFOAM . 19 iv 3.3.2.1 Derivation of pressure equation . 20 3.3.2.2 IcoFoam Algorithm . 21 3.4 Verification and Validation . 22 Chapter 4 Reduced-Order Model 25 4.1 Equation of Motion . 25 4.2 Implementation . 27 4.2.1 Spatial Discretization . 29 4.2.2 Temporal Discretization . 30 4.3 Verification . 31 4.3.1 Review of Method of Manufactured Solutions . 31 4.3.2 Verification . 31 4.4 Best Practices . 36 Chapter 5 Model Comparison 39 5.1 Introduction . 39 5.2 Comparison case . 39 5.2.1 Reduced-Order Model Setup . 40 5.2.2 First Principles Simulation Setup . 40 5.3 Clamped-free with a blunt end comparison . 42 5.4 Clamped-free with a streamlined end comparison . 44 5.4.1 U=1 . 46 5.4.2 U=2 . 46 5.4.3 U=4 . 49 5.4.4 U=6 . 52 Chapter 6 Conclusions 56 6.1 Summary of Work Performed . 56 6.2 Future Work . 57 Appendix A Non Technical Abstract 59 Bibliography 61 v List of Figures 1.1 Flow chart outlining the thesis . 4 2.1 A typical towed array . 6 2.2 A towed array close up . 6 2.3 A picture to show the difference between the continuous distribution of mass in the beam model(top) as opposed to the lumped masses found in the cable model (bottom). 13 3.1 FSI flow chart . 19 3.2 PISO flow chart . 22 3.3 Long Turek setup . 23 3.4 Vertical tip displacement of Turek cases with different tail lengths . 24 4.1 Physical diagram for the Paidoussis Model . 27 4.2 L2 error for displacement for grid refinement study . 35 4.3 L∞ error for displacement for grid refinement study. 35 4.4 Schematic of vibrating string . 36 4.5 Backward-Euler Energy Graph . 38 4.6 θ = 0.5 Energy Graph . 38 5.1 Paidoussis Experimental Pictures . 42 5.2 Schematic of first principle simulation with a blunt tail . 43 5.3 Grid used for a first principle simulation with a blunt tail . 43 5.4 Tip displacement for full order model with blunt tail and U=4 m/s 44 5.5 Schematic of first principle simulation with a streamlined tail . 45 5.6 Grid used for a first principle simulation with a streamlined tail . 45 5.7 Tip displacement for full order model with streamlined tail at U=1 m/s . 46 5.8 Full order simulation with streamlined tail at U=2 m/s at various times . 48 vi 5.9 Tip displacement for full order model with streamlined tail at U=2 m/s . 49 5.10 Full order simulation with streamlined tail at U=4 m/s at various times . 50 5.11 Tip displacement for full order model with streamlined tail at U=4 m/s . 51 5.12 Tip displacement for both models with streamlined tail at U=4 m/s 51 5.13 Full order simulation with streamlined tail at U=6 m/s at various times . 52 5.14 Tip displacement for full order model with streamlined tail at U=6 m/s . 53 5.15 Tip displacement for both models with streamlined tail at U=6 m/s 53 5.16 Tip displacement for full order model with streamlined tail at U=1,2,4,6 m/s . 55 vii List of Tables 2.1 Prakla Array Information . 7 3.1 Dimensions for the standard and long Turek cases . 23 4.1 Refinement table for verification of y . 33 4.2 Refinement table for verification of w . 33 4.3 Refinement table for verification of u . 34 4.4 Refinement table for verification of v . 34 5.1 A recap of the results of all the comparison cases performed. 55 viii List of Symbols E Young’s Modulus I Moment of Inertia U Free-stream speed of the fluid D Diameter of the cylinder L Length of the cylinder m mass per unit length of the cylinder M Mass of the displaced fluid ct Longitudinal drag coefficient cn Normal drag coefficient c2 Base drag coefficient f Measure of tail piece slenderness l Length of tail piece t time ρ density p pressure µ viscosity u fluid velocity vector ix um mesh velocity S Second Piola-Kirchhoff stress E Lagrangian Green strain λ Lame Constant y Displacement of the cylinder w Slope of the cylinder u Curvature of the cylinder v velocity of the cylinder x Acknowledgments I would like to acknowledge Naval Sea Systems Command Advanced Submarines Systems Development Office (SEA 073R) and the Penn State Applied Research Laboratory for funding to conduct this research. I also owe a major thanks to Dr. Jonathan Pitt & Dr. Scott Miller for advising during this research. Without their help and guidance, this thesis wouldn’t have been possible. Finally, I would like to thank my family for encouraging me to go back to school to pursue a graduate degree. xi Chapter 1 | Introduction 1.1 Introduction Computational mechanics has played an important role in the engineering design process over the past few decades by providing the necessary tools to go from a mathematical model of a complex system to its predicted behavior. The role of computational mechanics is likely to only expand as advancements continue to be made in both software and hardware. Increases in computational resource availability have made it possible to simulate problems and phenomenon that were once too computationally expensive. In particular, the demand for coupled multi-physics simulations has been increasing over the years. A multi-physics simulation couples different physical models into a single model. An example would be coupling thermal effects with a structure’s mechanical response. This thesis focuses on the specific multi-physics problem of fluid-structure interaction (FSI). A fluid-structure interaction simulation couples the governing equations of solid mechanics with the governing equations of fluid mechanics. Fluid- structure interaction is characterized by the interaction of a deformable solid whose deformation is dependent on the fluid’s flow field and a fluid whose flow field is dependent on the solid’s deformation. It is this two-way coupling that exemplifies the multi-physics nature of a fluid-structure interaction problem.
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