PEMP ACD2505. M.S. Ramaiah School of Advanced Studies, Bengaluru


 Brian Moody
 2 years ago
 Views:
Transcription
1 TwoDimensional Potential Flow Session delivered by: Prof. M. D. Deshpande 1
2 Session Objectives  At the end of this session the delegate would have understood PEMP The potential theory and its application i to 2D Dirrotational i flows The flow field in simple flows like uniform, source, vortex, doublet flows The principle of superposition and its application to simple cases The method of images Flow around a cylinder and forces on a general 2D cylinder Stokes Theorem, KuttaJoukowski theorem Kutta condition 2
3 Session Topics 1. Basic Potentail Theory 2. 2D Potential Flows 3. Simple Flows: Uniform, Source, Vortex, Doublet 4. Superposition of Flows 5. Method of Images 6. Flow around a Cylinder 7. Force on a General 2D Cylinder 8. Circulation, Vorticity, Stokes Theorem 9. KuttaJoukowski Theorem 10. Kutta Condition 3
4 2D Potential Flow This chapter starts the description of solution methods in detail. Beginning with the simplest flows: twodimensional, inviscid and irrotational, the chapter describes the basic theoretical results. These are applied to airfoil il problems in later chapters and then modified to include the effects of compressibility and viscosity. Basic Theory Sources and Vortices Interactive Calculations References 4
5 Basic 2D Potential Theory PEMP We outline here the way in which the "known" solutions used in panel methods can be generated and obtain some useful solutions to some fundamental fluid flow problems. Often the known solutions can be applied, but sometimes other approaches are possible. The simplest case, twodimensional potential flow illustrates this process. We shall discuss 2D incompressible potential flow and just mention the extension to linearized compressible flow. For this case the relevant equation is Laplace's equation: 2 2 φ φ 2 Φ = 0, OR in 2D + = x y 5
6 There are several ways of generating fundamental solutions to this linear, homogeneous, second order differential equation with constant coefficients. Two methods are particularly useful: Separation of variables and the use of complex variables. Complex variables are especially useful in solving Laplace's equation because of the following: We know, from the theory of complex variables, that in a region where a function of the complex variable z = x + iy is analytic, the derivative with respect to z is the same in any direction. This leads to the famous CauchyRiemann conditions for an analytic function in the complex plane. 6
7 Consider the complex function: W = φ +iψ ψ The CauchyRiemann conditions are: Differentiating the first equation with respect to x and the second with respect to y and adding gives: Thus, analytic function of a complex variable is a solution to Laplace's equation and may be used as part of a more general solution. 7
8 W=φ+iψ ψ is called the complex potential. It consists of the usual velocity potential as the real part and the stream function as its imaginary part. The flow velocities can then be written as a single complex number: dw/dz = u  iv (Try deriving this.) We consider some simple analytic functions for W that are of great use in applied aerodynamics: Uniform flow W(z) = U z Line Source or Vortex W(z) = [K / (2 π )] ln z Doublet W(z) = C / (2 π z) 8
9 Uniform Flow If U is real the flow is in the x direction with a speed U. The flow direction can be adjusted by changing real and imaginary parts. This is a good example of the fact that the potential is not defined apart from an arbitrary constant. Although the flow is uniform everywhere, the potential depends on our choice of the origin. Differences in the potential are physically meaningful, though and do not depend on the choice of the origin. 9
10 Line Source or Vortex PEMP The same expression describes a "point" source or vortex in 2D (which can be thought of as a vortex line or line of sources in 3D). When K is real the expression describes a source with radially directed induced velocity vectors; imaginary values lead to vortex flows with induced velocities in the tangential direction. Further discussion of these flows is given in the next section. 10
11 Line Source or Vortex (Contd) Φ = (S / 2 π)lnr R Φ = ( Γ / 2 π) ln R 11
12 Doublet PEMP A doublet is formed by superimposing a source and a sink along the xaxis. The doublet strength is given by (S 2x 0 ). The fundamental doublet singularity with the potential shown above is formed by taking the limit as 2x 0 goes to zero and S goes to infinity while keeping the product constant. The doublet is commonly used as one of the fundamental singularities in many panel methods. Sourcesink pair 12
13 Doublet (Contd) Streamlines of a sourcesink pair and Streamlines of a doublet 13
14 Sources and dvortices 14
15 Notice that many of the solutions to the 2D potential equation that we proposed are singular. In fact, the source solution seems the ultimate way of violating continuity while the vortex is the essence of rotational (not irrotational as we assumed) flow. These solutions are indeed singular at a point and do not satisfy the differential equation at that point. Away from the singularity, however, they are perfectly adequate solutions as can be seen by evaluating the integral forms of the continuity and irrotationality conditions. 15
16 Why the flow field near a source satisfies continuity: 16
17 Why the flow field near a vortex satisfies irrotationality: 17
18 The solutions are singular at a point, but even near the singular point strange things happen: the velocity gets very large. In real life, the large velocities in this region give rise to compressibility effects; viscous effects smear the discrete vortex into a distribution of vorticity in a viscous core. The actual velocity distribution near the core of a free vortex behaves more like a solid body with a velocity distribution V(R) = kr. (This is the result obtained by assuming a Gaussian distribution of distributed vorticity in the core region. The size of the viscous core depends on the Reynolds number, often taken as Γ/ν.). 18
19 Decay of a vortex filament in a viscous fluid. At t = 0, u θ = Γ /( 2 π r). Dashed lines correspond to the case of rigid body rotation corresponding roughly to core radii proportional to sqrt(ν t). From: Kuethe & Chow PEMP 19
20 This 1/r behavior of the vortex induced velocity is not just a mathematical result. It is essential for the flow to exist in equilibrium. We can easily see that the velocity must vary as 1/r for the pressure gradients to balance the centrifugal force acting on the fluid. The derivation is shown later. 20
21 Examples We can combine these singularities in different locations to produce the desired flow pattern. Since the solution to Laplace's equation is uniquely determined in regions without singularities when the solution on the boundaries is specified, we can use combinations of singularities to model many flows of interest. Ground Effects Cylinder Group of Vortices (Method of Images) (Source Doublets) (Stokes Theorem) 21
22 Streamlines for ψ = uniform flow (ψ 1 ) + Source (ψ 2 ) for V = 1 and h = Λ /2 V = L. Note, for instance, that the ψ = 0 streamline (OA & BAB ) is the locus of the intersections of the streamlines ψ 2 and ψ 1 ( =ψψ 2 ). Similarly, for the ψ = 0.1 streamline, ψ 1 + ψ 2 = 0.1, and so forth. (From: Kuethe and Chow) PEMP NOTE: ψ 1 = V y Λ = Source strength ψ 2 = (Λ /2 π) θ ψ = ψ 1 + ψ 2 = V [(h θ / π )  y] h = Λ /2 V = L =Characteristic length of the combined flow. Vel = 0, (h / π) ahead of the source location. 22
23 Flow pattern from distributing m identical line sources along the dashed line in the presence of a uniform flow. Only the upper half of the flow is shown. Circles indicate the locations of the line sources. (a) m = 5. (b) m = 11 (same total source strength). (c ) m = 101 (same total source strength). (d) m = 101 (but the source strength is reduced). (e) m infinity with λ /2 = V λ = Source density = flow rate /area. See that forward velocity is zero. Rear side velocity is λ /2 = V and normal to the panel. (f) Boundary condition at inclined panel. (From: Kuethe and Chow) 23
24 Derivation of Vortex Velocity Distribution ib ti The inward force on an element of fluid due to pressure gradients may be found by summing the contributions from the inner, outer, and side faces. 24
25 The result is: F = r dθ dp + higher order terms. Centrifugal force (outward): F = ρ r dr dθ V 2 / r For equilibrium: r dθ dp = ρ rdrdθ dθ V 2 /r So, dp = ρ V 2 dr / r, and from the Euler equation, dp = ρ V dv. Integrating: g ρ V 2 dr / r = ρ ρ V dv yields: V = constant / r 25
26 Mthd Method of fimages The flow field created by singularities in the presence of solid boundaries can be simulated by superimposing "image vortices". This works because the symmetry of the problem on the right ensures that there is no flow through the plane of symmetry. The boundary does the same thing for the problem on the left. Since both of these problems have the same boundary conditions and satisfy the same linear differential equation, the flow must be the same. 26
27 A source near a plane wall (From: Kuethe and Chow) 27
28 A vortex near a plane wall (From: Kuethe and Chow) 28
29 This technique is useful for simulating the effects of the ground on the aerodynamics of cars or airplanes at low altitude. It can also be used in more complex situations. Here, three images are required to simulate the boundary conditions associated with a corner. 29
30 This technique is used to predict the effects of wind tunnel walls on the flow field of models being tested. Imagine the system of image vortices that would be required to simulate wall effects on a 2D airfoil test. Yes, more than 2 images are required. The 3D situation cannot in general be solved with images. 30
31 Motion of a vortex pair near the ground (From: Kuethe and Chow) 31
32 Cylinders Streamlines for (a doublet + Uniform flow): Synthesis of flow around a circular cylinder in uniform flow PEMP 32
33 Cylinders (Contd) PEMP The flow around a circular cylinder may be computed from a uniform stream and a doublet. (See previous sections.) The potential of the combined flow is: Differentiating to find the velocities gives: NOTE: On the cylinder wall, only radial velocity is zero. 33
34 Some interesting conclusions and generalizations follow from the expressions for the velocity and the potential on a circular cylinder shown above. Note that on the surface of the cylinder, the tangential velocity is: V = 2U sin θ,, so the maximum velocity is twice the freestream value. 34
35 Continuous distribution of doublets in a uniform flow (From: Kuethe and Chow) 35
36 The more general forms of these results hold for all ellipsoids: V max = U (1 + t/c) and V at surface =  n (n V max ) Notice that this holds exactly in incompressible potential flow, even if the ellipse has a t/c much larger than 1. Of course, in such a case, the real flow will probably look quite different from the potential flow solution. 36
37 The force on a general 2D cylinder The force on a general 2D cylinder can be computed by calculating the velocities, using Bernoulli's law to compute pressures, then integrating the surface pressures. However, the total forces and moments can be derived directly from the complex potential. The result is called the Blasius theorem. It is not derived d here, but the result follows from the theory of residues, the complex potential, and the incompressible Bernoulli equation. (Or one might just use the momentum equation and compute the net force by far field integrals.). 37
38 The force on a general 2D cylinder (Contd) where Γ is the total circulation (measured counterclockwise) and dsi is the net source strength. th In the case of no net source strength, the net force exerted on a collection of sources and vortices in a flow with freestream velocity U is perpendicular to the freestream and proportional to U and the total circulation. 38
39 Circulation, Vorticity, it and Stokes Theorem Stokes' theorem is an integral identity that may be written: The first integration is done over volume V. (dv missing). When the vector function F is taken to be the velocity field, V, then this relation in 2D may be restated as: 39
40 This result implies that the circulation around a contour that contains a group of vortices is just equal to the sum of the enclosed vortex strengths. This allows application of the Blasius theorem to find the force acting on a group of vortices. The result is sometimes called the KuttaJoukowski law: 40
41 KuttaJoukowski Theorem As seen before this theorem states; L = ρ V Γ Note the direction of the lift force carefully. We can also treat the flow field far from a group of vortices as if it were created by a single vortex with a strength equal to the sum of the individual vortices. Such far field solutions can be especially simple and useful as a check of more complex results. Far field solutions can also be used as boundary conditions for the more complex near field solution, reducing the required extent of computational grids. 41
42 We should note here that just because we find a superposition of singularities that satisfies the boundary conditions and the differential equation, it does not mean that we have found the only solution to the problem. For example, we could add a vortex to the doublet that was used to model the circular cylinder, and we would still find that the flow went around the cylinder. These nonunique solutions are problemsome and we appeal to additional considerations to find the one(s) that actually will appear in nature. Just such an auxiliary condition, the Kutta condition, is provided by viscous effects which then determine the value of circulation. 42
43 Kutta condition We are considering steady, incompressible, irrotational 2D motion. KuttaJoukowski theorem states that force experienced by a body in a uniform stream acts perpendicular to the flow direction and is given by F = ρ V Γ. With the given boundary conditions on the boundary (normal vel = 0) and at infinity (uniform flow) we try to solve the Laplace equation. The flow is not unique. One way to make it unique is to specify the circulation. See that it is equivalent to specifying Lift itself! Then the formula above is not of practical use. In the figure below Γ is specified to be zero and hence we get zero lift. PEMP 43
44 Kutta condition (Contd) In the problem we are considering i if the circulation is specified the problem has a unique solution. The bodies we are considering are airfoils with a sharp trailing edge. They have the rear stagnation point at the trailing edge since the flow cannot take a sharp turn as shown in the last figure due to viscosity. Hence in real fluids the body with a sharp trailing edge will create enough circulation to hold the stagnation point at the trailing edge as shown in the figure below. This fixes the entire flow and the values of circulation and lift. Hence we are able to evaluate the value of lift by ideal fluid flow itself. PEMP 44
45 Free Vortices Singularities that are free to move in the flow do not behave in response to F = m a (what is m?). Rather they move with the local flow velocity. Thus, vortices and sources are convected downstream with the flow. And interacting singularities can produce complex motions due to their mutual induced velocities. 45
46 A pair of counterrotating vortices moves downward because of their mutual induced velocities. Corotating vortices orbit each other under the influence of their mutual induced velocities. 46
47 Streamlines Past Sources and Vortices (Interactive Program) Drag any of the singularities from the well on the right into the main computation area. Set the freestream speed (the flow is from left to right), then click Compute. The marks on the page simulate small tufts and indicate the direction of the local flow. Experiment with multiple singularities to simulate a pair of wing trailing vortices, a source/sink doublet, or a spinning baseball. If you do not see the results you may want to try an alternate version of this applet. Due to certain platformdependent java problems, this program may not work with some browsers on some platforms. If not, try here for a less cool, but simpler version of the program. 47
48 Summary The following topics were dealt in this session The Potential theory and its application to 2D irrotational flows The flow field in simple flows like uniform, source, vortex, doublet flows Application i of superposition i principle i to simple flows Method of images Flow around a cylinder and the forces involved Stokes theorem, KuttaJoukowski theorem Kutta condition 48
49 Thank you 49
Inviscid & Incompressible flow
< 3.1. Introduction and Road Map > Basic aspects of inviscid, incompressible flow Bernoulli s Equation Laplaces s Equation Some Elementary flows Some simple applications 1.Venturi 2. Lowspeed wind tunnel
More informationSome Basic Plane Potential Flows
Some Basic Plane Potential Flows Uniform Stream in the x Direction A uniform stream V = iu, as in the Fig. (Solid lines are streamlines and dashed lines are potential lines), possesses both a stream function
More informationLifting Airfoils in Incompressible Irrotational Flow. AA210b Lecture 3 January 13, AA210b  Fundamentals of Compressible Flow II 1
Lifting Airfoils in Incompressible Irrotational Flow AA21b Lecture 3 January 13, 28 AA21b  Fundamentals of Compressible Flow II 1 Governing Equations For an incompressible fluid, the continuity equation
More informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More information1. Fluid Dynamics Around Airfoils
1. Fluid Dynamics Around Airfoils Twodimensional flow around a streamlined shape Foces on an airfoil Distribution of pressue coefficient over an airfoil The variation of the lift coefficient with the
More informationCopyright 2007 N. Komerath. Other rights may be specified with individual items. All rights reserved.
Low Speed Aerodynamics Notes 5: Potential ti Flow Method Objective: Get a method to describe flow velocity fields and relate them to surface shapes consistently. Strategy: Describe the flow field as the
More informationGeneral Solution of the Incompressible, Potential Flow Equations
CHAPTER 3 General Solution of the Incompressible, Potential Flow Equations Developing the basic methodology for obtaining the elementary solutions to potential flow problem. Linear nature of the potential
More informationChapter 6: Incompressible Inviscid Flow
Chapter 6: Incompressible Inviscid Flow 61 Introduction 62 Nondimensionalization of the NSE 63 Creeping Flow 64 Inviscid Regions of Flow 65 Irrotational Flow Approximation 66 Elementary Planar Irrotational
More informationDetailed Outline, M E 521: Foundations of Fluid Mechanics I
Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Secondorder tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic
More informationOffshore Hydromechanics Module 1
Offshore Hydromechanics Module 1 Dr. ir. Pepijn de Jong 4. Potential Flows part 2 Introduction Topics of Module 1 Problems of interest Chapter 1 Hydrostatics Chapter 2 Floating stability Chapter 2 Constant
More informationAE301 Aerodynamics I UNIT B: Theory of Aerodynamics
AE301 Aerodynamics I UNIT B: Theory of Aerodynamics ROAD MAP... B1: Mathematics for Aerodynamics B: Flow Field Representations B3: Potential Flow Analysis B4: Applications of Potential Flow Analysis
More informationMarine Hydrodynamics Prof.TrilochanSahoo Department of Ocean Engineering and Naval Architecture Indian Institute of Technology, Kharagpur
Marine Hydrodynamics Prof.TrilochanSahoo Department of Ocean Engineering and Naval Architecture Indian Institute of Technology, Kharagpur Lecture  10 Source, Sink and Doublet Today is the tenth lecture
More informationMAE 101A. Homework 7  Solutions 3/12/2018
MAE 101A Homework 7  Solutions 3/12/2018 Munson 6.31: The stream function for a twodimensional, nonviscous, incompressible flow field is given by the expression ψ = 2(x y) where the stream function has
More information6.1 Momentum Equation for Frictionless Flow: Euler s Equation The equations of motion for frictionless flow, called Euler s
Chapter 6 INCOMPRESSIBLE INVISCID FLOW All real fluids possess viscosity. However in many flow cases it is reasonable to neglect the effects of viscosity. It is useful to investigate the dynamics of an
More informationPEMP ACD2505. M.S. Ramaiah School of Advanced Studies, Bengaluru
Governing Equations of Fluid Flow Session delivered by: M. Sivapragasam 1 Session Objectives  At the end of this session the delegate would have understood The principle of conservation laws Different
More informationFluid Dynamics Problems M.Sc MathematicsSecond Semester Dr. Dinesh KhattarK.M.College
Fluid Dynamics Problems M.Sc MathematicsSecond Semester Dr. Dinesh KhattarK.M.College 1. (Example, p.74, Chorlton) At the point in an incompressible fluid having spherical polar coordinates,,, the velocity
More informationAll that begins... peace be upon you
All that begins... peace be upon you Faculty of Mechanical Engineering Department of Thermo Fluids SKMM 2323 Mechanics of Fluids 2 «An excerpt (mostly) from White (2011)» ibn Abdullah May 2017 Outline
More informationSPC Aerodynamics Course Assignment Due Date Monday 28 May 2018 at 11:30
SPC 307  Aerodynamics Course Assignment Due Date Monday 28 May 2018 at 11:30 1. The maximum velocity at which an aircraft can cruise occurs when the thrust available with the engines operating with the
More informationu = (u, v) = y The velocity field described by ψ automatically satisfies the incompressibility condition, and it should be noted that
18.354J Nonlinear Dynamics II: Continuum Systems Lecture 1 9 Spring 2015 19 Stream functions and conformal maps There is a useful device for thinking about two dimensional flows, called the stream function
More informationFluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition
Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition C. Pozrikidis m Springer Contents Preface v 1 Introduction to Kinematics 1 1.1 Fluids and solids 1 1.2 Fluid parcels and flow
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationOUTLINE FOR Chapter 3
013/4/ OUTLINE FOR Chapter 3 AERODYNAMICS (W11 BERNOULLI S EQUATION & integration BERNOULLI S EQUATION AERODYNAMICS (W11 013/4/ BERNOULLI S EQUATION FOR AN IRROTATION FLOW AERODYNAMICS (W1.1 VENTURI
More informationGiven a stream function for a cylinder in a uniform flow with circulation: a) Sketch the flow pattern in terms of streamlines.
Question Given a stream function for a cylinder in a uniform flow with circulation: R Γ r ψ = U r sinθ + ln r π R a) Sketch the flow pattern in terms of streamlines. b) Derive an expression for the angular
More informationContinuum Mechanics Lecture 7 Theory of 2D potential flows
Continuum Mechanics ecture 7 Theory of 2D potential flows Prof. http://www.itp.uzh.ch/~teyssier Outline  velocity potential and stream function  complex potential  elementary solutions  flow past a
More informationFUNDAMENTALS OF AERODYNAMICS
*A \ FUNDAMENTALS OF AERODYNAMICS Second Edition John D. Anderson, Jr. Professor of Aerospace Engineering University of Maryland H ' McGrawHill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas
More informationMath 575Lecture Failure of ideal fluid; Vanishing viscosity. 1.1 Drawbacks of ideal fluids. 1.2 vanishing viscosity
Math 575Lecture 12 In this lecture, we investigate why the ideal fluid is not suitable sometimes; try to explain why the negative circulation appears in the airfoil and introduce the vortical wake to
More informationACD2503 Aircraft Aerodynamics
ACD2503 Aircraft Aerodynamics Session delivered by: Prof. M. D. Deshpande 1 Aims and Summary PEMP It is intended dto prepare students for participation i i in the design process of an aircraft and its
More informationFluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay
Fluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay Lecture No. # 35 Boundary Layer Theory and Applications Welcome back to the video course on fluid
More informationMATH 566: FINAL PROJECT
MATH 566: FINAL PROJECT December, 010 JAN E.M. FEYS Complex analysis is a standard part of any math curriculum. Less known is the intense connection between the pure complex analysis and fluid dynamics.
More informationGiven the water behaves as shown above, which direction will the cylinder rotate?
water stream fixed but free to rotate Given the water behaves as shown above, which direction will the cylinder rotate? ) Clockwise 2) Counterclockwise 3) Not enough information F y U 0 U F x V=0 V=0
More informationUNIT IV BOUNDARY LAYER AND FLOW THROUGH PIPES Definition of boundary layer Thickness and classification Displacement and momentum thickness Development of laminar and turbulent flows in circular pipes
More informationAerodynamics. HighLift Devices
HighLift Devices Devices to increase the lift coefficient by geometry changes (camber and/or chord) and/or boundarylayer control (avoid flow separation  Flaps, trailing edge devices  Slats, leading
More informationHigh Speed Aerodynamics. Copyright 2009 Narayanan Komerath
Welcome to High Speed Aerodynamics 1 Lift, drag and pitching moment? Linearized Potential Flow Transformations Compressible Boundary Layer WHAT IS HIGH SPEED AERODYNAMICS? Airfoil section? Thin airfoil
More informationLab Reports Due on Monday, 11/24/2014
AE 3610 Aerodynamics I Wind Tunnel Laboratory: Lab 4  Pressure distribution on the surface of a rotating circular cylinder Lab Reports Due on Monday, 11/24/2014 Objective In this lab, students will be
More informationExercise 9: Model of a Submarine
Fluid Mechanics, SG4, HT3 October 4, 3 Eample : Submarine Eercise 9: Model of a Submarine The flow around a submarine moving at a velocity V can be described by the flow caused by a source and a sink with
More informationWater is sloshing back and forth between two infinite vertical walls separated by a distance L: h(x,t) Water L
ME9a. SOLUTIONS. Nov., 29. Due Nov. 7 PROBLEM 2 Water is sloshing back and forth between two infinite vertical walls separated by a distance L: y Surface Water L h(x,t x Tank The flow is assumed to be
More informationComplex functions in the theory of 2D flow
Complex functions in the theory of D flow Martin Scholtz Institute of Theoretical Physics Charles University in Prague scholtz@utf.mff.cuni.cz Faculty of Transportation Sciences Czech Technical University
More informationFundamentals of Aerodynamics
Fundamentals of Aerodynamics Fourth Edition John D. Anderson, Jr. Curator of Aerodynamics National Air and Space Museum Smithsonian Institution and Professor Emeritus University of Maryland Me Graw Hill
More informationThe purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and engineering.
Lecture 16 Applications of Conformal Mapping MATHGA 451.001 Complex Variables The purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and
More informationDetailed Outline, M E 320 Fluid Flow, Spring Semester 2015
Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015 I. Introduction (Chapters 1 and 2) A. What is Fluid Mechanics? 1. What is a fluid? 2. What is mechanics? B. Classification of Fluid Flows 1. Viscous
More informationHomework Two. Abstract: Liu. Solutions for Homework Problems Two: (50 points total). Collected by Junyu
Homework Two Abstract: Liu. Solutions for Homework Problems Two: (50 points total). Collected by Junyu Contents 1 BT Problem 13.15 (8 points) (by Nick HunterJones) 1 2 BT Problem 14.2 (12 points: 3+3+3+3)
More informationIran University of Science & Technology School of Mechanical Engineering Advance Fluid Mechanics
1. Consider a sphere of radius R immersed in a uniform stream U0, as shown in 3 R Fig.1. The fluid velocity along streamline AB is given by V ui U i x 1. 0 3 Find (a) the position of maximum fluid acceleration
More informationIncompressible Flow Over Airfoils
Chapter 7 Incompressible Flow Over Airfoils Aerodynamics of wings: D sectional characteristics of the airfoil; Finite wing characteristics (How to relate D characteristics to 3D characteristics) How
More informationExercise 9, Ex. 6.3 ( submarine )
Exercise 9, Ex. 6.3 ( submarine The flow around a submarine moving at at velocity V can be described by the flow caused by a source and a sink with strength Q at a distance a from each other. V x Submarine
More informationINSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad
INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad  500 043 AERONAUTICAL ENGINEERING TUTORIAL QUESTION BANK Course Name : LOW SPEED AERODYNAMICS Course Code : AAE004 Regulation : IARE
More informationFundamentals of Aerodynamits
Fundamentals of Aerodynamits Fifth Edition in SI Units John D. Anderson, Jr. Curator of Aerodynamics National Air and Space Museum Smithsonian Institution and Professor Emeritus University of Maryland
More informationASTR 320: Solutions to Problem Set 3
ASTR 30: Solutions to Problem Set 3 Problem : The Venturi Meter The venturi meter is used to measure the flow speed in a pipe. An example is shown in Fig., where the venturi meter (indicated by the dashed
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the NavierStokes equation for incompressible constant density, i.e. homogeneous flows:
More information1. Introduction  Tutorials
1. Introduction  Tutorials 1.1 Physical properties of fluids Give the following fluid and physical properties(at 20 Celsius and standard pressure) with a 4digit accuracy. Air density : Water density
More informationPART II: 2D Potential Flow
AERO301:Spring2011 II(a):EulerEqn.& ω = 0 Page1 PART II: 2D Potential Flow II(a): Euler s Equation& Irrotational Flow We have now completed our tour through the fundamental conservation laws that apply
More informationAirfoils and Wings. Eugene M. Cliff
Airfoils and Wings Eugene M. Cliff 1 Introduction The primary purpose of these notes is to supplement the text material related to aerodynamic forces. We are mainly interested in the forces on wings and
More informationBLUFFBODY AERODYNAMICS
International Advanced School on WINDEXCITED AND AEROELASTIC VIBRATIONS OF STRUCTURES Genoa, Italy, June 1216, 2000 BLUFFBODY AERODYNAMICS Lecture Notes by Guido Buresti Department of Aerospace Engineering
More informationDepartment of Mechanical Engineering
Department of Mechanical Engineering AMEE401 / AUTO400 Aerodynamics Instructor: Marios M. Fyrillas Email: eng.fm@fit.ac.cy HOMEWORK ASSIGNMENT #2 QUESTION 1 Clearly there are two mechanisms responsible
More informationContinuum Mechanics Lecture 5 Ideal fluids
Continuum Mechanics Lecture 5 Ideal fluids Prof. http://www.itp.uzh.ch/~teyssier Outline  Helmholtz decomposition  Divergence and curl theorem  Kelvin s circulation theorem  The vorticity equation
More informationVorticity Equation Marine Hydrodynamics Lecture 9. Return to viscous incompressible flow. NS equation: v. Now: v = v + = 0 incompressible
13.01 Marine Hydrodynamics, Fall 004 Lecture 9 Copyright c 004 MIT  Department of Ocean Engineering, All rights reserved. Vorticity Equation 13.01  Marine Hydrodynamics Lecture 9 Return to viscous incompressible
More informationChapter 3 Bernoulli Equation
1 Bernoulli Equation 3.1 Flow Patterns: Streamlines, Pathlines, Streaklines 1) A streamline, is a line that is everywhere tangent to the velocity vector at a given instant. Examples of streamlines around
More information3.5 Vorticity Equation
.0  Marine Hydrodynamics, Spring 005 Lecture 9.0  Marine Hydrodynamics Lecture 9 Lecture 9 is structured as follows: In paragraph 3.5 we return to the full NavierStokes equations (unsteady, viscous
More information1. Introduction, tensors, kinematics
1. Introduction, tensors, kinematics Content: Introduction to fluids, Cartesian tensors, vector algebra using tensor notation, operators in tensor form, Eulerian and Lagrangian description of scalar and
More informationSyllabus for AE3610, Aerodynamics I
Syllabus for AE3610, Aerodynamics I Current Catalog Data: AE 3610 Aerodynamics I Credit: 4 hours A study of incompressible aerodynamics of flight vehicles with emphasis on combined application of theory
More informationNUMERICAL SIMULATION OF THE FLOW AROUND A SQUARE CYLINDER USING THE VORTEX METHOD
NUMERICAL SIMULATION OF THE FLOW AROUND A SQUARE CYLINDER USING THE VORTEX METHOD V. G. Guedes a, G. C. R. Bodstein b, and M. H. Hirata c a Centro de Pesquisas de Energia Elétrica Departamento de Tecnologias
More informationMestrado Integrado em Engenharia Mecânica Aerodynamics 1 st Semester 2012/13
Mestrado Integrado em Engenharia Mecânica Aerodynamics 1 st Semester 212/13 Exam 2ª época, 2 February 213 Name : Time : 8: Number: Duration : 3 hours 1 st Part : No textbooks/notes allowed 2 nd Part :
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIERSTOKES EQUATIONS Under the assumption of a Newtonian stressrateofstrain constitutive equation and a linear, thermally conductive medium,
More informationFluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur
Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture  15 Conservation Equations in Fluid Flow Part III Good afternoon. I welcome you all
More informationFluid Mechanics Qualifying Examination Sample Exam 2
Fluid Mechanics Qualifying Examination Sample Exam 2 Allotted Time: 3 Hours The exam is closed book and closed notes. Students are allowed one (doublesided) formula sheet. There are five questions on
More informationFluid Mechanics (3)  MEP 303A For THRID YEAR MECHANICS (POWER)
كلية الھندسة جامعة القاھرة قسم ھندسة القوى الميكانيكية معمل التحكم األوتوماتيكى Notes on the course Fluid Mechanics (3)  MEP 303A For THRID YEAR MECHANICS (POWER) Part (3) Frictionless Incompressible
More informationAERODYNAMICS STUDY NOTES UNIT I REVIEW OF BASIC FLUID MECHANICS. Continuity, Momentum and Energy Equations. Applications of Bernouli s theorem
AERODYNAMICS STUDY NOTES UNIT I REVIEW OF BASIC FLUID MECHANICS. Continuity, Momentum and Energy Equations. Applications of Bernouli s theorem UNIT II TWO DIMENSIONAL FLOWS Complex Potential, Point Source
More informationComputing potential flows around Joukowski airfoils using FFTs
AB CD EF GH Computing potential flows around Joukowski airfoils using FFTs Frank Brontsema Institute for Mathematics and Computing Science AB CD EF GH Bachelor thesis Computing potential flows around
More informationFigure 3: Problem 7. (a) 0.9 m (b) 1.8 m (c) 2.7 m (d) 3.6 m
1. For the manometer shown in figure 1, if the absolute pressure at point A is 1.013 10 5 Pa, the absolute pressure at point B is (ρ water =10 3 kg/m 3, ρ Hg =13.56 10 3 kg/m 3, ρ oil = 800kg/m 3 ): (a)
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 201213 MODULE TITLE: Introduction to Fluids DURATION OF
More informationAn alternative approach to integral equation method based on Treftz solution for inviscid incompressible flow
An alternative approach to integral equation method based on Treftz solution for inviscid incompressible flow Antonio C. Mendes, Jose C. Pascoa Universidade da Beira Interior, Laboratory of Fluid Mechanics,
More informationAerodynamics. Basic Aerodynamics. Continuity equation (mass conserved) Some thermodynamics. Energy equation (energy conserved)
Flow with no friction (inviscid) Aerodynamics Basic Aerodynamics Continuity equation (mass conserved) Flow with friction (viscous) Momentum equation (F = ma) 1. Euler s equation 2. Bernoulli s equation
More informationFLUID MECHANICS. Atmosphere, Ocean. Aerodynamics. Energy conversion. Transport of heat/other. Numerous industrial processes
SG2214 Anders Dahlkild Luca Brandt FLUID MECHANICS : SG2214 Course requirements (7.5 cr.) INL 1 (3 cr.) 3 sets of home work problems (for 10 p. on written exam) 1 laboration TEN1 (4.5 cr.) 1 written exam
More informationPEMP ACD2505. Finite Wing Theory. M.S. Ramaiah School of Advanced Studies, Bengaluru
Finite Wing Theory Session delivered by: Prof. M. D. Deshpande 1 Session Objectives  At the end of this session the delegate would have understood The finite wing theory Lifting line theory Elliptic
More informationNumerical study of the steady state uniform flow past a rotating cylinder
Numerical study of the steady state uniform flow past a rotating cylinder J. C. Padrino and D. D. Joseph December 17, 24 1 Introduction A rapidly rotating circular cylinder immersed in a free stream generates
More informationFluid Mechanics. Chapter 9 Surface Resistance. Dr. Amer Khalil Ababneh
Fluid Mechanics Chapter 9 Surface Resistance Dr. Amer Khalil Ababneh Wind tunnel used for testing flow over models. Introduction Resistances exerted by surfaces are a result of viscous stresses which create
More informationChapter 2 Dynamics of Perfect Fluids
hapter 2 Dynamics of Perfect Fluids As discussed in the previous chapter, the viscosity of fluids induces tangential stresses in relatively moving fluids. A familiar example is water being poured into
More informationAerodynamics. Lecture 1: Introduction  Equations of Motion G. Dimitriadis
Aerodynamics Lecture 1: Introduction  Equations of Motion G. Dimitriadis Definition Aerodynamics is the science that analyses the flow of air around solid bodies The basis of aerodynamics is fluid dynamics
More information1.060 Engineering Mechanics II Spring Problem Set 3
1.060 Engineering Mechanics II Spring 2006 Due on Monday, March 6th Problem Set 3 Important note: Please start a new sheet of paper for each problem in the problem set. Write the names of the group members
More information7 EQUATIONS OF MOTION FOR AN INVISCID FLUID
7 EQUATIONS OF MOTION FOR AN INISCID FLUID iscosity is a measure of the thickness of a fluid, and its resistance to shearing motions. Honey is difficult to stir because of its high viscosity, whereas water
More informationANALYSIS OF HORIZONTAL AXIS WIND TURBINES WITH LIFTING LINE THEORY
ANALYSIS OF HORIZONTAL AXIS WIND TURBINES WITH LIFTING LINE THEORY Daniela Brito Melo daniela.brito.melo@tecnico.ulisboa.pt Instituto Superior Técnico, Universidade de Lisboa, Portugal December, 2016 ABSTRACT
More informationAEROSPACE ENGINEERING
AEROSPACE ENGINEERING Subject Code: AE Course Structure Sections/Units Topics Section A Engineering Mathematics Topics (Core) 1 Linear Algebra 2 Calculus 3 Differential Equations 1 Fourier Series Topics
More informationTURBULENT FLOW ACROSS A ROTATING CYLINDER WITH SURFACE ROUGHNESS
HEFAT2014 10 th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics 14 16 July 2014 Orlando, Florida TURBULENT FLOW ACROSS A ROTATING CYLINDER WITH SURFACE ROUGHNESS Everts, M.,
More informationMAE 222 Mechanics of Fluids Final Exam with Answers January 13, Give succinct answers to the following word questions.
MAE 222 Mechanics of Fluids Final Exam with Answers January 13, 1994 Closed Book Only, three hours: 1:30PM to 4:30PM 1. Give succinct answers to the following word questions. (a) Why is dimensional analysis
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationFLUID MECHANICS. Chapter 9 Flow over Immersed Bodies
FLUID MECHANICS Chapter 9 Flow over Immersed Bodies CHAP 9. FLOW OVER IMMERSED BODIES CONTENTS 9.1 General External Flow Characteristics 9.3 Drag 9.4 Lift 9.1 General External Flow Characteristics 9.1.1
More informationIntroduction to Aerospace Engineering
4. Basic Fluid (Aero) Dynamics Introduction to Aerospace Engineering Here, we will try and look at a few basic ideas from the complicated field of fluid dynamics. The general area includes studies of incompressible,
More informationF11AE1 1. C = ρν r r. r u z r
F11AE1 1 Question 1 20 Marks) Consider an infinite horizontal pipe with circular crosssection of radius a, whose centre line is aligned along the zaxis; see Figure 1. Assume noslip boundary conditions
More informationNumerical Investigation of Laminar Flow over a Rotating Circular Cylinder
International Journal of Mechanical & Mechatronics Engineering IJMMEIJENS Vol:13 No:3 32 Numerical Investigation of Laminar Flow over a Rotating Circular Cylinder Ressan Faris AlMaliky Department of
More informationFLUID MECHANICS. ! Atmosphere, Ocean. ! Aerodynamics. ! Energy conversion. ! Transport of heat/other. ! Numerous industrial processes
SG2214 Anders Dahlkild Luca Brandt FLUID MECHANICS : SG2214 Course requirements (7.5 cr.)! INL 1 (3 cr.)! 3 sets of home work problems (for 10 p. on written exam)! 1 laboration! TEN1 (4.5 cr.)! 1 written
More informationThe vorticity field. A dust devil
The vorticity field The vector ω = u curl u is twice the local angular velocity in the flow, and is called the vorticity of the flow (from Latin for a whirlpool). Vortex lines are everywhere in the direction
More information3.1 Definition Physical meaning Streamfunction and vorticity The Rankine vortex Circulation...
Chapter 3 Vorticity Contents 3.1 Definition.................................. 19 3.2 Physical meaning............................. 19 3.3 Streamfunction and vorticity...................... 21 3.4 The Rankine
More informationThe Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)
Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green
More informationAerodynamics of Spinning Sphere in Ideal Flow R. C. Mehta Department of Aeronautical Engineering, Noorul Islam University, Kumaracoil , India
Scholars Journal of Engineering and Technology (SJET) Sch. J. Eng. Tech., 06; 4(5):59 Scholars Academic and Scientific Publisher (An International Publisher for Academic and Scientific Resources) www.saspublisher.com
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
More informationHow Do Wings Generate Lift?
How Do Wings Generate Lift? 1. Popular Myths, What They Mean and Why They Work M D Deshpande and M Sivapragasam How lift is generated by a moving wing is understood satisfactorily. But the rigorous explanation
More informationSteady waves in compressible flow
Chapter Steady waves in compressible flow. Oblique shock waves Figure. shows an oblique shock wave produced when a supersonic flow is deflected by an angle. Figure.: Flow geometry near a plane oblique
More informationFundamentals of Fluid Mechanics
Sixth Edition Fundamentals of Fluid Mechanics International Student Version BRUCE R. MUNSON DONALD F. YOUNG Department of Aerospace Engineering and Engineering Mechanics THEODORE H. OKIISHI Department
More informationFLUID DYNAMICS, THEORY AND COMPUTATION MTHA5002Y
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 FLUID DYNAMICS, THEORY AND COMPUTATION MTHA5002Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
More informationOffshore Hydromechanics
Offshore Hydromechanics Module 1 : Hydrostatics Constant Flows Surface Waves OE4620 Offshore Hydromechanics Ir. W.E. de Vries Offshore Engineering Today First hour: Schedule for remainder of hydromechanics
More informationLecture4. Flow Past Immersed Bodies
Lecture4 Flow Past Immersed Bodies Learning objectives After completing this lecture, you should be able to: Identify and discuss the features of external flow Explain the fundamental characteristics
More information