Hydrodynamic Analysis of Ocean Current Turbines Using Vortex Lattice Method
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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. -
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. -
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 ................................... -
A Comparison of Euler Finite Volume and Supersonic Vortex Lattice Methods Used During the Conceptual Design Phase of Supersonic Delta Wings
A Comparison of Euler Finite Volume and Supersonic Vortex Lattice Methods used during the Conceptual Design Phase of Supersonic Delta Wings THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Daniel Guillermo-Monedero, B.S. Graduate Program in Aeronautical and Astronautical Engineering The Ohio State University 2020 Master’s Examination Committee: Dr. Clifford A. Whitfield, Advisor Dr. Richard J. Freuler Dr. Matthew H. McCrink 1 Copyrighted by Daniel Guillermo-Monedero 2020 2 Abstract This thesis uses two different methods to analyze wings in supersonic flows with a focus on preliminary design. The primary goals of this study are to compare the Euler equations finite volume method and supersonic vortex lattice method in predicting surface pressure on wings, and to develop a low-order supersonic vortex lattice method as a baseline tool that can be extended for further wing design and analysis. The supersonic vortex lattice method uses vortical sources to model the flow on the boundary, which replicates the aerodynamic shape of interest, in an inviscid and irrotational flow field and obtain solutions. The Euler equations of flow can be discretized using finite volume methods and can be integrated over the volume of interest and solutions can be obtained over the surface. These mathematically similar methods have a lot of differences in their numerical formulations, which can be critical in the design and analysis of wings. Hence, it is important to understand the key differences of these in order to develop a reliable baseline low-order design tool. -
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. -
Chapter 10 Elements of Potential Flow ( )
CHAPTER 10 ELEMENTS OF POTENTIAL FLOW ___________________________________________________________________________________________________________ 10.1 INCOMPRESSIBLE FLOW Most of the problems we are interested in involve low speed flow about wings and bodies. The equations governing incompressible flow are Continuity !iU = 0 (10.1) Momentum !U $ P ' + "i UU + I = *"2U (10.2) t & ) ! % # ( The convective term can be rearranged using !iU = 0 and the identity # UiU & Ui!U = ! " U " U + ! (10.3) ( ) % 2 ( $ ' The viscous term in (10.2) can be rearranged using the identity ! " ! " U = ! !iU # !2U (10.4) ( ) ( ) Using these results and !iU = 0 the momentum equation can be written in terms of the vorticity. ! = " # U (10.5) in the form !U & P UiU ) + " # U + $ + + ,$ # " = 0 (10.6) t ( 2 + ! ' % * If the flow is irrotational, ! = 0 , and the velocity can be expressed in terms of a velocity potential. 1 U = !" (10.7) The continuity equation becomes Laplace’s equation 2 !iU = !i!" = ! " = 0 (10.8) and the momentum equation is fully integrable. % "# P UiU ( ! + + = 0 (10.9) ' t 2 * & " $ ) The quantity in parentheses is at most a function of time !" P UiU + + = f (t) (10.10) !t # 2 The expression (10.10) is called the Bernoulli integral and can be used to determine the pressure throughout the flow once the velocity potential is known from a solution of Laplace’s equation (10.7). Generally the flow is specified within a volume V surrounded by surface A (Figure 10.1). A solid body defined by the function G(x,t) = 0 may be imbedded inside V . Figure 10.1 Flow volume V with surface A and imbedded solid body G(x,t) = 0 . -
Unsteady Aerodynamic Vortex Lattice of Moving Aircraft
Unsteady Aerodynamic Vortex Lattice of Moving Aircraft September 30, 2011 Master thesis Supervisors: Author: Arthur Rizzi Enrique Mata Bueso Jesper Oppelstrup Aeronautical and Vehicle engineering department, KTH, Kungliga Tekniska H¨ogskolan SE-100 44 Stockholm, Sweden Contents 1 Introduction 1 1.1 CEASIOM............................................1 1.2 TORNADO, a brief overview..................................3 1.3 Linear Aerodynamics.......................................5 1.3.1 Steady Aerodynamics..................................6 1.3.2 Quasi-steady Aerodynamics...............................7 1.3.3 Unsteady Aerodynamics.................................7 2 Vortex Lattice Method9 2.1 Basics of fluid dynamics.....................................9 2.1.1 Kinematic concepts................................... 10 2.1.2 The Continuity Equation................................ 10 2.1.3 Bernoulli equation.................................... 11 2.2 Potential Flow Theory...................................... 11 2.2.1 Kelvin's circulation theorem............................... 11 2.2.2 Helmholtz theorems................................... 12 2.2.3 Kutta-Joukowsky theorem................................ 12 2.2.4 Boundary conditions................................... 13 2.2.5 Biot-Savart's Law.................................... 13 2.2.6 Horseshoe vortex..................................... 14 2.2.7 Vortex Rings....................................... 15 2.3 Classical Vortex Lattice Method................................ 15 2.4 Unsteady VLM, theoretical -
ME 306 Fluid Mechanics II Part 1 Potential Flow
Inviscid (Frictionless) Flow ME 306 Fluid Mechanics II • Continuity and Navier-Stokes equations govern the flow of fluids. • For incompressible flows of Newtonian fluids they are Part 1 훻 ⋅ 푉 = 0 퐷푉 휕푉 휌 = 휌 + 푉 ⋅ 훻 푉 = 휌푔 − 훻푝 + 휇훻2푉 Potential Flow 퐷푡 휕푡 • These equations can be solved analytically only for a few problems. • They can be simplified in various ways. These presentations are prepared by −3 Dr. Cüneyt Sert 휇푤푎푡푒푟 = 10 Pa s • Common fluids such as water and air have small viscosities. 휇 = 2 × 10−5 Pa s Department of Mechanical Engineering 푎푖푟 Middle East Technical University • Neglecting the viscous term (zero shear force) gives the Euler’s equation. Ankara, Turkey 퐷푉 [email protected] 휌 = 휌푔 − 훻푝 퐷푡 You can get the most recent version of this document from Dr. Sert’s web site. • Still difficult get a general analytical solution for the unknowns 푝 and 푉. Please ask for permission before using them to teach. You are not allowed to modify them. 1-1 1-2 Inviscid Flow (cont’d) Inviscid and Irrotational Flow • Bernoulli Equation (BE) is Euler’s equation written along a streamline. • To simplify further we can assume the flow to be irrotational. 푝 푉2 휉 = 2휔 = 훻 × 푉 = 0 (irrotational flow) + + 푧 = constant along a streamline 휌푔 2푔 Vorticity (ksi) Angular velocity Exercise: Starting from the Euler’s equation derive the BE. • Question: What’s the logic behind irrotationality assumption? Inviscid flow away from the object (negligible shear forces) • Irrotationality is about velocity gradients. 휕푤 휕푣 휕푢 휕푤 휕푣 휕푢 휉 = − 푖 + − 푗 + − 푘 = 0 휕푦 휕푧 휕푧 휕푥 휕푥 휕푦 Airfoil or 1 휕푉 휕푉 휕푉 휕푉 1 휕 푟푉 1 휕푉 휉 = 푧 − 휃 푖 + 푟 − 푧 푖 + 휃 − 푟 푖 푟 휕휃 휕푧 푟 휕푧 휕푟 휃 푟 휕푟 푟 휕휃 푧 Adapted from http://web.cecs.pdx.edu/~gerry/class/ME322 Viscous flow close to the object and in the wake of it (non negligible shear forces) 1-3 1-4 Inviscid and Irrotational Flow (cont’d) Inviscid and Irrotational Flow (cont’d) • One special irrotational flow is when all velocity gradients are zero. -
Stream Function and Velocity Potential)
NPTEL – Mechanical – Principle of Fluid Dynamics Module 3 : Lecture 1 INVISCID INCOMPRESSIBLE FLOW (Fundamental Aspects) In general, fluids have a well-known tendency to move or flow. The slight change in shear stress or appropriate imbalance in normal stresses will cause fluid motion. Fluid kinematics deals with various aspects of fluid motion without concerning the actual force that causes the fluid motion. In this particular section, we shall consider the ‘field’ concept to define velocity/ acceleration of fluid by virtue of its motion. In the later part, some ‘visualization’ concepts are introduced to define the motion of the fluid qualitatively as well as quantitatively. There are two general approaches in analyzing the fluid motion. In the first method (Lagrangian approach), the individual fluid particles are considered and their properties are studied as a function of time. In the second method (Eulerian approach), the ‘field’ concept is introduced and the properties are completely prescribed as the functions of space and time. In other words, the attention is focused at fixed points in space as the fluid passes those points. Velocity and Acceleration Field Since the ‘continuum’ assumption holds well for fluids, the description of any fluid property (such as density, pressure, velocity, acceleration etc.) can be expressed as a function of location. These representation as a function of spatial coordinates is called as “field representation” of the flow. One of the most important fluid variables is the velocity field. It is a vector function of position and time with components uv, and w as scalar variables i.e. V=++ u( xyzt,,,) iˆˆ v( xyzt ,,,) j w( xyzt ,,,) kˆ (3.1.1) The magnitude of the velocity vector i.e.