A Comparison of Euler Finite Volume and Supersonic Vortex Lattice Methods Used During the Conceptual Design Phase of Supersonic Delta Wings
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Variational Formulation of Fluid and Geophysical Fluid Dynamics
Advances in Geophysical and Environmental Mechanics and Mathematics Gualtiero Badin Fulvio Crisciani Variational Formulation of Fluid and Geophysical Fluid Dynamics Mechanics, Symmetries and Conservation Laws Advances in Geophysical and Environmental Mechanics and Mathematics Series editor Holger Steeb, Institute of Applied Mechanics (CE), University of Stuttgart, Stuttgart, Germany More information about this series at http://www.springer.com/series/7540 Gualtiero Badin • Fulvio Crisciani Variational Formulation of Fluid and Geophysical Fluid Dynamics Mechanics, Symmetries and Conservation Laws 123 Gualtiero Badin Fulvio Crisciani Universität Hamburg University of Trieste Hamburg Trieste Germany Italy ISSN 1866-8348 ISSN 1866-8356 (electronic) Advances in Geophysical and Environmental Mechanics and Mathematics ISBN 978-3-319-59694-5 ISBN 978-3-319-59695-2 (eBook) DOI 10.1007/978-3-319-59695-2 Library of Congress Control Number: 2017949166 © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. -
The Vortex Lattice Method . for the Rotor-Vortex Interaction Problem
NASA CONTRACTOR NASA CR-2421 REPORT - THE VORTEX LATTICE METHOD . FOR THE ROTOR-VORTEX INTERACTION PROBLEM by Raghuveera Padakannaya Prepared by THE PENNSYL VANIA ST ATE UNIVERSITY University Park, Pa. 16802 for Langley Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • JULY 1974 1. Report No. , I 2. Government Accession No. 3. Recipient's Catalog No. CR-Z4Z1 4. TItle and Subtitle 5. Report Date THE VORTEX LATTICE METHOD FOR THE ROTOR-VORTEX JULY 1974 INTERACTION PROBLEM 6. Performing Organization Code 7. Author(s) 8. Performing Organization Report No. Raghuveera Padakannaya 1-------------------------------1 10. Work Unit No. g. Performing Organization Name and Address The Pennsylvania State University 11. Contract or Grant No. University Park, PA 16802 NGR - 39 - 009 - 111 ~------------------------------I13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address Contractor Report National Aeronautics and Space Administration 14. Sponsoring Agency Code Washington, D.C. 20546 15. Supplementary Notes Topical report 16. Abstract This study is concerned with the rotor blade-vortex interaction problem and the resulting impul sive airloads which generate undesirable noi se level s. A numerical 1ifting surface method to predict unsteady aerodynamic forces induced on a finite aspect ratio rectangular wing by a straight, free vortex placed at an arbitrary angle in a subsonic incompressible free stream is developed first. Using a rigid wake assumption, the wake vortices are assumed to move down steam with the free steam velocity. Unsteady load.distributions are obtained which compare favorably with the results of planar lifting surface theory. The vortex lattice method has been extended to a single bladed rotor operating at high advance 'ratios aodeocc)untering a ft:eevottex from afixed.wing upstream of the rotor. -
Potential Flow Theory
2.016 Hydrodynamics Reading #4 2.016 Hydrodynamics Prof. A.H. Techet Potential Flow Theory “When a flow is both frictionless and irrotational, pleasant things happen.” –F.M. White, Fluid Mechanics 4th ed. We can treat external flows around bodies as invicid (i.e. frictionless) and irrotational (i.e. the fluid particles are not rotating). This is because the viscous effects are limited to a thin layer next to the body called the boundary layer. In graduate classes like 2.25, you’ll learn how to solve for the invicid flow and then correct this within the boundary layer by considering viscosity. For now, let’s just learn how to solve for the invicid flow. We can define a potential function,!(x, z,t) , as a continuous function that satisfies the basic laws of fluid mechanics: conservation of mass and momentum, assuming incompressible, inviscid and irrotational flow. There is a vector identity (prove it for yourself!) that states for any scalar, ", " # "$ = 0 By definition, for irrotational flow, r ! " #V = 0 Therefore ! r V = "# ! where ! = !(x, y, z,t) is the velocity potential function. Such that the components of velocity in Cartesian coordinates, as functions of space and time, are ! "! "! "! u = , v = and w = (4.1) dx dy dz version 1.0 updated 9/22/2005 -1- ©2005 A. Techet 2.016 Hydrodynamics Reading #4 Laplace Equation The velocity must still satisfy the conservation of mass equation. We can substitute in the relationship between potential and velocity and arrive at the Laplace Equation, which we will revisit in our discussion on linear waves. -
Vortex-Lattice Utilization
NASA SP-405 VORTEX-LATTICE UTILIZATION (NRSA-5P-405) V3BTZX- LATTIZZ UTILIZF.TION N76-28163 (?iASA) 409 p HC 611.Ci0 CSCL 31A THRU N76-28186 rn Unclas .- H1/02 47615 A workshop held at LANGLEY RESEARCH CENTER Hampton, Virginia May 17-18, 1976 I NATIONAL AERONAUTICS AND SPAC ERRATA NASA SP-405 VORTEX-LATTICE UTILIZATION Page 231: Figure 6 is in error. Replace figure 6 with the following corrected version. Figure 6.- Span load and section suction distributions on a swept and skewed wing. h = 45O; A = 1; M = 0. NASA SP-405 VORTEX-LATTICE UTILIZATION A workshop held at Langley Research Center, Hampton, Virginia, on May 17-18, 1976 Prepared by Lungley Research Center Srietrlifir and Terhnirnl lrlformdtion Ofice !9'h u 5. b. NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Washington, D.C. CONTENTS PREFACE ..................................iii o)tl/7- . 1. HISTORICAL EVOLUTION OF VORTEX-LATTICE METHODS ............ 1 j. John DeYoung, Vought Corporation Hampton Technical Center - SESSION I - CONFIGURATION DESIGN AND ANALYSIS AND WALL EFFECTS Chairman: Percy J. Bobbitt, NASA Langley Research Center A. WING-BODY COMBINATIONS 2. SUBSONIC FINITC ELEMENTS FOR WING-BODY COMBINATIONS ......... 11 James L. Thomas, NASA Langley Research Center 3. EXTENDED APPLICATIONS OF THE VORTEX LATTICE METHOD ......... 27 Luis R. Miranda, Lockheed-California Company B. NONPLANAR CONFIGURATIONS 4. WMERICAL METHOD TO CALCULATE THE INDUCED DRAG OR OPTIMUM LOADING FOR ARBITRARY NON-PLANAR AIRCRAFT ................. 49 James A. Blackwell, Jr., Lockheed-Georgia Company 5. OPTIMIZATION AND DESIGN OF THREE-DIMEKSIONAL AERODYNAMIC CONFIGURATIONS OF ARBITRARY SHAPE BY A VORTEX LATTICE METHOD .............................. 71 Wlnfried M. Feifel, The Boeing Company 6. -
Far Field Velocity Potential Induced by a Rapidly Decaying Vorticity Distribution
Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/90/030395-24$ 1.50 + 0.20 Vol. 41, May 1990 1990 Birkh~iuser Verlag, Basel Far field velocity potential induced by a rapidly decaying vorticity distribution By Rupert Klein, Dept. of Mathematics, Princeton University, Princeton, NJ and Lu Ting, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA 1. Introduction The results to be obtained in this paper are applicable to a general problem regarding the far field behavior of the solution A(x) of the vector Poisson equation in space, zXA = -f~, (1.1) subjected to the boundary condition, IA]--*0 as r = Ixi- 00. (1.2) Here x denotes the position vector with Cartesian coordinates xi, i = 1, 2 and 3. In this paper, boldface symbols are used to denote vector quantities and the subscripts i = 1, 2 and 3 denote their components. Thus A; and coi denote the three components of A and 12 respectively. The inhomogeneous term, 12(x), is assumed to be divergence free, V.12 =0, (1.3) and to decay exponentially in r, that is 1121= o(r-"), for all n as r ~ 00. (1.4) Note that (1.4) is a realistic condition for vortical flows but is not necessary- for the results to be derived here. It suffices to require that (1.4) holds for n less than a sufficiently large integer instead of for all n. The above problem originated from our analysis of a three dimensional unsteady incompressible flow induced by a vorticity field 12. -
Unsteady Lift Prediction with a Higher-Order Potential Flow Method
aerospace Article Unsteady Lift Prediction with a Higher-Order Potential Flow Method Julia A. Cole 1,* , Mark D. Maughmer 2, Goetz Bramesfeld 3, Michael Melville 3 and Michael Kinzel 4 1 Mechanical Engineering, Bucknell University, 1 Dent Dr., Lewisburg, PA 17837, USA 2 Aerospace Engineering, The Pennsylvania State University, 229 Hammond Bldg, University Park, PA 16802, USA; [email protected] 3 Aerospace Engineering, Ryerson University, 350 Victoria St., Toronto, ON M5B 2K3, Canada; [email protected] (G.B.); [email protected] (M.M.) 4 Mechanical and Aerospace Engineering, University of Central Florida, P.O. Box 162450, Orlando, FL 32816, USA; [email protected] * Correspondence: [email protected] Received: 24 April 2020; Accepted: 18 May 2020; Published: 21 May 2020 Abstract: An unsteady formulation of the Kutta–Joukowski theorem has been used with a higher-order potential flow method for the prediction of three-dimensional unsteady lift. This study describes the implementation and verification of the approach in detail sufficient for reproduction by future developers. Verification was conducted using the classical responses to a two-dimensional airfoil entering a sharp-edged gust and a sinusoidal gust with errors of less than 1% for both. The method was then compared with the three-dimensional unsteady lift response of a wing as modeled in two unsteady vortex-lattice methods. Results showed agreement in peak lift coefficient prediction to within 1% and 7%, respectively, and mean agreement within 0.25% for the full response. Keywords: unsteady aerodynamics; potential flow; applied aerodynamics 1. Introduction 1.1. Motivation In previous work [1–4], the authors developed a time-stepping potential flow method that uses higher-order vorticity elements to represent lifting surfaces and wakes. -
A Generalized Vortex Lattice Method for Subsonic and Supersonic Flow Applications
https://ntrs.nasa.gov/search.jsp?R=19780008059 2020-03-22T05:30:34+00:00Z NASA Contractor Report 2865 A Generalized Vortex Lattice Method for Subsonic and Supersonic Flow Applications Luis R. Miranda, Robert D. Elliott, and William M. Baker ,8 J CONTRACT NAS 1-12972 DECEMBER 1977 N/ /X NASA Contractor Report 2865 A Generalized Vortex Lattice Method for Subsonic and Supersonic Flow Applications Luis R. Miranda, Robert D. Elliott, and William M. Baker Lockheed-California Company Burbank, California Prepared for Langley Research Center under Contract NAS1-12972 National Aeronautics and Space Administration Scientific end Technical Information Office 1977 TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS ......................... iv SUMMARY ............................... i INTRODUCTION ............................. i THEOP_TICAL DISCUSSION ........................ 3 The Basic Equations ....................... 3 Extension to Supersonic Flow ................. 5 The Skewed-Horseshoe Vortex ................. 9 Modeling of Lifting Surfaces with Thickness .......... 14 Modeling of Fusiform Bodies ................... 15 Computation of Sideslip Effects ................. 17 THE GENERALIZED VORTEX LATTICE METHOD ................. 18 Description of Computational Method ............... 18 Numeric 8.1 Considerations .................... 2O COMPARISON WITH OTHER THEORIES AND EXPERIMENTAL RESULTS ........ 21 CONCLUDING REMARKS ......................... 22 APPENDIX A USER'S MANUAL FOR A GENERALIZED VORTEX LATTICE METHOD FOR SUBSONIC AND SUPERSONIC FLOW APPLICATIONS -
Hydrodynamic Analysis of Ocean Current Turbines Using Vortex Lattice Method
HYDRODYNAMIC ANALYSIS OF OCEAN CURRENT TURBINES USING VORTEX LATTICE METHOD by Aneesh Goly A Thesis Submitted to the Faculty of The College of Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of Master of Science Florida Atlantic University Boca Raton, Florida August 2010 ACKNOWLEDGEMENTS I would like to express my gratitude to my advisor, Palaniswamy Ananthakrishnan, for his support, patience, and encouragement throughout my graduate studies. It is not often that one finds an advisor who always finds the time for listening to the little problems that unavoidably crop up in the course of performing research. His technical and editorial advice was essential to the completion of this master’s thesis and has taught me innumerable lessons and insights. I would like to sincerely thank my committee members Dr. Dhanak, Dr. Hanson and Dr. Xiros for their valuable comments, suggestions and input to the thesis. My special thanks go to Ranjith for his encouragement and numerous fruitful discussions related to the research. I feel privileged to have been able to work with Zaqie, Waazim, Lakitosh and Baishali who were encouraging me all through my thesis. My deepest gratitude goes to my family for their unflagging love and support throughout my life. In the first place to my parents, Srinivas and Rajani for giving me everything in my life. Second, to my grandparents and aunt, Jaishanker, Pushpavathy and Rekha for their unparalleled love and care. Next, to uncle Ravi for his motivation throughout my thesis. Last but not the least, to my uncle Kumar Allady, who directed me in choosing this field and together with aunt, Vishala, has given me unparalleled love, affection and guidance. -
An Introduction to Acoustics
An Introduction to Acoustics S.W. Rienstra & A. Hirschberg Eindhoven University of Technology 2 Oct 2021 This is an extended and revised edition of IWDE 92-06. Comments and corrections are gratefully accepted. This file may be used and printed, but for personal or educational purposes only. © S.W. Rienstra & A. Hirschberg 2004. Contents Page Preface 1 Some fluid dynamics 1 1.1 Conservation laws and constitutive equations ..................... 1 1.2 Approximations and alternative forms of the conservation laws for ideal fluids ..... 4 2 Wave equation, speed of sound, and acoustic energy 9 2.1 Order of magnitude estimates ............................. 9 2.2 Wave equation for a uniform stagnant fluid and compactness ............. 13 2.2.1 Linearization and wave equation ........................ 13 2.2.2 Simple solutions ................................. 14 2.2.3 Compactness .................................. 16 2.3 Speed of sound ..................................... 17 2.3.1 Ideal gas ..................................... 17 2.3.2 Water ...................................... 19 2.3.3 Bubbly liquid at low frequencies ........................ 19 2.4 Influence of temperature gradient ........................... 20 2.5 Influence of mean flow ................................. 22 2.6 Sources of sound .................................... 22 2.6.1 Inverse problem and uniqueness of sources ................... 22 2.6.2 Mass and momentum injection ......................... 23 2.6.3 Lighthill’s analogy ............................... 24 2.6.4 Vortex sound .................................. 27 2.7 Acoustic energy .................................... 29 2.7.1 Introduction ................................... 29 2.7.2 Kirchhoff’s equation for quiescent fluids .................... 30 2.7.3 Acoustic energy in a non-uniform flow ..................... 33 2.7.4 Acoustic energy and vortex sound ........................ 35 ii Contents 3 Green’s functions, impedance, and evanescent waves 38 3.1 Green’s functions ................................... -
Potential Flow Theory
2.016 Hydrodynamics Reading #4 2.016 Hydrodynamics Prof. A.H. Techet Potential Flow Theory “When a flow is both frictionless and irrotational, pleasant things happen.” –F.M. White, Fluid Mechanics 4th ed. We can treat external flows around bodies as invicid (i.e. frictionless) and irrotational (i.e. the fluid particles are not rotating). This is because the viscous effects are limited to a thin layer next to the body called the boundary layer. In graduate classes like 2.25, you’ll learn how to solve for the invicid flow and then correct this within the boundary layer by considering viscosity. For now, let’s just learn how to solve for the invicid flow. We can define a potential function, (xz,,t ) , as a continuous function that satisfies the basic laws of fluid mechanics: conservation of mass and momentum, assuming incompressible, inviscid and irrotational flow. There is a vector identity (prove it for yourself!) that states for any scalar, , = 0 By definition, for irrotational flow, r V = 0 Therefore r V = where = (,xy , z,t ) is the velocity potential function. Such that the components of velocity in Cartesian coordinates, as functions of space and time, are u = , v = and w = (4.1) dx dy dz version 1.0 updated 9/22/2005 -1- 2005 A. Techet 2.016 Hydrodynamics Reading #4 Laplace Equation The velocity must still satisfy the conservation of mass equation. We can substitute in the relationship between potential and velocity and arrive at the Laplace Equation,which we will revisit in our discussion on linear waves. -
Aerodynamics of 3D Lifting Surfaces Through Vortex Lattice Methods
6. Aerodynamics of 3D Lifting Surfaces through Vortex Lattice Methods 6.1 An Introduction There is a method that is similar to panel methods but very easy to use and capable of providing remarkable insight into wing aerodynamics and component interaction. It is the vortex lattice method (vlm), and was among the earliest methods utilizing computers to actually assist aerodynamicists in estimating aircraft aerodynamics. Vortex lattice methods are based on solutions to Laplace’s Equation, and are subject to the same basic theoretical restrictions that apply to panel methods. As a comparison, vortex lattice methods are: Similar to Panel methods: • singularities are placed on a surface • the non-penetration condition is satisfied at a number of control points • a system of linear algebraic equations is solved to determine singularity strengths Different from Panel methods: • Oriented toward lifting effects, and classical formulations ignore thickness • Boundary conditions (BCs) are applied on a mean surface, not the actual surface (not an exact solution of Laplace’s equation over a body, but embodies some additional approximations, i.e., together with the first item, we find DCp, not Cpupper and Cplower) • Singularities are not distributed over the entire surface • Oriented toward combinations of thin lifting surfaces (recall Panel methods had no limitations on thickness). Vortex lattice methods were first formulated in the late ’30s, and the method was first called “Vortex Lattice” in 1943 by Faulkner. The concept is extremely simple, but because of its purely numerical approach (i.e., no answers are available at all without finding the numerical solution of a matrix too large for routine hand calculation) practical applications awaited sufficient development of computers—the early ’60s saw widespread adoption of the method. -
A Review of Aerodynamic Flow Models, Solution Methods and Solvers – and Their Applicability to Aircraft Conceptual Design Literature Study Report
A review of aerodynamic flow models, solution methods and solvers – and their applicability to aircraft conceptual design Literature study report B. Peerlings November 1, 2018 Delft University of Technology A review of aerodynamic flow models, solution methods and solvers – and their applicability to aircraft conceptual design Literature study report by B. (Bram) Peerlings in partial fulfilment of the requirements for the degrees of Master of Science in Aerospace Engineering Student number 4079388 Project duration June 4, 2018 – September 5, 2018 October 8, 2018 – November 1, 2018 Department Flight Performance & Propulsion, faculty of Aerospace Engineering Supervisor dr. ir. R. Vos Electronic versions of this document suitable for screen and print are available at bram.peerlings.me/en/literature-study/, using the password ‘AE4020-2018’. Contents Preface v Summary vii List of figures ix List of tables xi List of abbreviations xiv List of symbols xviii 1 Introduction 1 2 Overview of flow models 3 2.1 Navier-Stokes equations . 5 2.1.1 Direct numerical simulation (DNS). 8 2.1.2 Large eddy simulation (LES). 8 2.1.3 Unsteady Reynolds-averaged Navier-Stokes equations (URANS) . 8 2.1.4 Reynolds-averaged Navier-Stokes equations (RANS) . 9 2.1.5 Thin-layer Navier-Stokes equations (TLNS). 9 2.1.6 Turbulence modelling . 10 2.2 Boundary layer equations . 10 2.3 Euler equations . 11 2.4 Full potential equation . 12 2.5 Linearised potential equation . 13 2.6 Laplace’s equation . 14 2.7 Empirical methods . 14 3 Overview of solution methods 15 3.1 Non-linear solution methods . 15 3.1.1 Finite difference methods (FDM).