Aerodynamics. Lecture 1: Introduction - Equations of Motion G. Dimitriadis
|
|
- Archibald Briggs
- 5 years ago
- Views:
Transcription
1 Aerodynamics Lecture 1: Introduction - Equations of Motion G. Dimitriadis
2 Definition Aerodynamics is the science that analyses the flow of air around solid bodies The basis of aerodynamics is fluid dynamics Aerodynamics only came of age after the first aircraft flight by the Wright brothers The primary driver of aerodynamics progress is aerospace and more particularly aeronautics
3 Applications (1) Basic phenomena: Flow around a cylinder Shock wave Flow around an airfoil
4 Applications (2) Low speed aerodynamics Trailing vortices High lift devices
5 Applications (3) High speed aerodynamics F14 shock wave causes condensation F14 shock wave visualized on water s surface
6 New concepts: Blended wing body Micro-air vehicles Forward-swept wings Applications (4)
7 Applications (5) Space: Rockets, spaceplanes, reentry, Airship 1
8 Applications (6) Non-aerospace applications: cars, buildings, birds, insects
9 Categories of aerodynamics Aerodynamics is an all-encompassing term It is usually sub-divided according to the speed of the flow regime under investigation: Subsonic aerodynamics: The flow is subsonic over the entire body Transonic aerodynamics: The flow is sonic or supersonic over some parts of the body but subsonic over other parts Supersonic aerodynamics: The flow is supersonic over all of the body Hypersonic aerodynamics: The flow is faster than four times the speed of sound over all of the body
10 Flow type applications Subsonic aerodynamics: Low speed aircraft, high-speed aircraft flying at low speeds, wind turbines, environmental flows etc Transonic aerodynamics: Aircraft flying at nearly the speed of sound, helicopter rotor blades, turbine engine blades etc Supersonic aerodynamics: Aircraft flying at supersonic speeds, turbine engine blades etc Hypersonic aerodynamics: Atmospheric re-entry vehicles, experimental hypersonic aircraft, bullets, ballistic missiles, space launch vehicles etc
11 Content of this course (1) This course will address mostly subsonic and supersonic aerodynamics Transonic aerodynamics is very difficult and highly nonlinear Small perturbation linearized solutions exist but their accuracy is debatable Hypersonic aerodynamics is beyond the scope of this course
12 Content of this course (2) Subsonic aerodynamics Incompressible aerodynamics Ideal flow 2D flow 3D flow Viscous flow Viscous-inviscid matching Compressibility corrections Supersonic aerodynamics 2D flow 3D flow
13 Simplifications The different categories of aerodynamics exist because of the different amount of simplifications that can be applied to particular flows Air molecules always obey the same laws, irrespective of the size or speed of the object that is passing through them However, the way we analyze flows changes with flow regime because we apply simplifications Without simplifications very few useful results can be obtained
14 Full Navier Stokes Equations The most complete model we have of the flow of air is the Navier Stokes equations These equations are nevertheless a model: they are not the physical truth They represent three conservation laws: mass, momentum and energy They are not the physical truth because they involve a number of statistical quantities such as viscosity and density
15 t + u t u t v t w t E + u2 Navier Stokes for Aerodynamicists + v + w y z + uv + uw + ue + uv y + v 2 y + vw y + ve y + u xx + v xy + w xz = 0 + uw z + vw z + w 2 z + we z = xx + xy y + xz z = xy + yy y + yz z = xz + yz y + zz z = q t + uq + vq y + wq z + ( y u xy + v yy + w yz ) + ( z u xz + v yz + w zz )
16 Nomenclature The lengths x, y, z are used to define position with respect to a global frame of reference, while time is defined by t. u, v, w are the local airspeeds. They are functions of position and time. p,, are the pressure, density and viscosity of the fluid and they are functions of position and time E is the total energy in the flow. q is the external heat flux
17 The stress tensor Consider a small fluid element. In a general flow, each face of the element experiences normal stresses and shear stresses The three normal and six shear stress components make up the stress tensor
18 More nomenclature The components of the stress tensor: xx = p + 2μ u, yy = p + 2μ v y, zz = p + 2μ w z xy = yx = μ v + u, y yz = zy = μ w y + v, z zx = xz = μ u z + w The total energy E is given by: E = e + 1 ( 2 u2 + v 2 + w 2 ) where e is the internal energy of the flow and depends on the temperature and volume.
19 Gas properties Do not forget that gases are also governed by the state equation: p = RT Where T is the temperature and R is Blotzmann s constant. For a calorically perfect gas: e=c v T, where c v is the specific heat at constant volume.
20 Comments on Navier-Stokes equations Notice that aerodynamicists always include the mass and energy equations in the Navier- Stokes equations Notice also that compressibility is always allowed for, unless specifically ignored This is the most complete form of the airflow equations, although turbulence has not been explicitly defined Explicit definition of turbulence further complicates the equations by introducing new unknowns, the Reynolds stresses.
21 Constant viscosity Under the assumption that the fluid has constant viscosity, the momentum equations can be written as u t v t t w + u2 + uv + uw + uv y + v 2 y + vw y + uw z + vw z + w 2 z = p + μ 2 u + 2 u 2 y + 2 u 2 z 2 = p y + μ 2 v + 2 v 2 y + 2 v 2 z 2 = p z + μ 2 w w y w z 2
22 Compact expressions There are several compact expressions for the Navier-Stokes equations: Tensor notation: Du i Dt p = + μ 2 u i 2 i i Vector notation: u t u u + ( u) u = p + μ2 u Matrix notation: u t + T uu T T = p + μ2 u
23 Non-dimensional form The momentum equations can also be written in non-dimensional form as u t v t t w + u2 + uv where + uw + uv y + v 2 y + vw y + uw z + vw z + w 2 z = p + = p y + = p z u Re + 2 u 2 y + 2 u 2 z v Re + 2 v 2 y + 2 v 2 z w Re w y w z 2 =, u = u, v = v, w = w, x = x U U U L, y = y L, z = z L, t = tl U, p = p 2 U
24 Solvability of the Navier- Stokes equations There exist no solutions of the complete Navier-Stokes equations The equations are: Unsteady Nonlinear Viscous Compressible The major problem is the nonlinearity
25 Flow unsteadiness Flow unsteadiness in the real world arises from two possible phenomena: The solid body accelerates There are areas of separated flows This course will only consider solid bodies that do not accelerate Attached flows will generally be considered Therefore, unsteady terms will be neglected All time derivatives in the Navier-Stokes equations are equal to zero
26 Unsteadiness Examples Flow past a circular cylinder visualized in a water tunnel. The airspeed is accelerating. The flow is always separated and unsteady. It becomes steadier at high airspeeds Flow past an airfoil visualized in a water tunnel. The angle of attack is increasing. The flow attached and steady at low angles of attack and vice versa.
27 Viscosity Viscosity is a property of fluids All fluids are viscous to different degrees However, there are some aerodynamic flow cases where viscosity can be modeled in a simplified manner In those cases, all viscous terms are neglected.
28 Cases where viscosity is Shock wave important Boundary layer Wake
29 Euler equations Neglecting the viscous terms, we obtain the unsteady Euler equations: t + u t u t v t w t E + u2 + v + w y z + uv + uw + ue + uv y + v 2 y + vw y + ve y = 0 + uw z + vw z + w 2 z + we z = p = p y = p z up = ( vp ) y wp z
30 Classic form of the Euler equations The Euler equations are usually written in the form: where U t + F + G y + H z = 0 u v w u p + u 2 uv uw U = v, F = uv, G = p + v 2, H = vw w uw uw p + w 2 E u( E + p ) v( E + p ) w( E + p )
31 Steady Euler Equations Neglecting unsteady terms we obtain the steady Euler equations: u u 2 uv uw + v y + uv y + v 2 y + vw y + w z = 0 + uw z + vw z + w 2 z = p = p y = p z
32 Example 1 Notice that in the steady Euler equations, the energy equation has disappeared. Show that neglecting unsteady and viscous terms turns the energy equation into an identity if the air s internal energy is constant in space.
33 Compressibility The compressibility of most liquids is negligible for the forces encountered in engineering applications. Many fluid dynamicists always write the Navier- Stokes equations in incompressible form. This cannot be done for gases, as they are very compressible. However, for low enough airspeeds, the compressibility of gases also becomes negligible. In this case, compressibility can be ignored.
34 Compressibility examples Hypersonic flow over blunt wedge Transonic flow over airfoil Supersonic flow over sharp wedge
35 Incompressible, steady Euler Equations The incompressible, steady Euler equations become u + v y + w z = 0 u u + v u y + w u z = 1 p u v + v v y + w v z = 1 p y u w + v w y + w w z = 1 p z
36 Comment on the Euler equations The Euler equations are much more solvable than the Navier-Stokes equations They are most commonly solved using numerical methods, such as finite differences There are very few analytical solutions of the Euler equations and they are not particularly useful In order to obtain analytical solutions, the equations must be simplified even further
37 Rotational flow: Flow rotationality Fluid rotation Fluid particle, time t 1 Irrotational flow: No fluid rotation Fluid particle, time t 2 Fluid particle, time t 3 Fluid particle, time t 1 Fluid particle, time t 2 Fluid particle, time t 3
38 Irrotationality (1) Some flows can be idealized as irrotational In general, attached, incompressible, inviscid flows are also irrotational Irrotationality requires that the curl of the local velocity vector vanishes: u = 0 where u=ui+vj+wk and = i + y j + z k
39 Irrotationality (2) This leads to the simultaneous equations: w y v z = 0, w u z = 0, v u y = 0 Integrating the momentum equations using these conditions leads to the wellknown Bernoulli equation 1 2 u2 + v 2 + w 2 + P = constant
40 Example 2 Integrate the incompressible, steady momentum equations to obtain Bernoulli s equation for irrotational flow You can start with the 2D equations
41 Velocity potential Irrotationality allows the definition of the velocity potential, such that It can be seen that all three irrotationality conditions are satisfied by this function Substituting these definitions in the mass equation leads to u = -, v = - y, w = - z y z 2 = 0
42 Laplace s equation The irrotational form of the Euler equations is Laplace s equation. This is an equation that has many analytical solutions. It is the basis of most subsonic, attached flow aerodynamic assumptions. The equation is linear, therefore its solutions can be superimposed The complete flow problem has been reduced to a single, linear partial differential equation with a single unknown, the velocity potential.
43 Potential flow Incompressible, inviscid and irrotational flow is also called potential flow because it is fully described by the velocity potential. The first part of this course will look at potential flow solutions: First in two dimensions Then in three dimensions Potential flow solutions have provided us with the most useful and trustworthy aerodynamic results we have to date. Their limitations must be kept in mind at all times.
44 Potential flow solutions We now have a basis for modelling the flow over 2D or 3D bodies. All we need to do is: Solve Laplace s equation With two boundary conditions (2 nd order problem): Impermeability: Flow cannot enter or exit a solid body Far field: The flow far from the body is undisturbed.
45 Boundary conditions (1) Neumann boundary condition n: unit vector normal to the surface q n : normal flow velocity component q t : tangential flow velocity component Impermeability: The normal flow velocity component must be equal to zero. q n = n surface =0 q n n qt
46 Boundary conditions (1bis) Dirichlet boundary condition An alternative form of the impermeability condition states that the potential inside the body must be a constant: (x,y,z) i (x,y,z) i (x,y,z)=constant
47 Boundary conditions (2) Far field: Flow far from the body is undisturbed. This usually is expressed as: * 0, as r r r 2 =x 2 +y 2 +z 2 r
48 2D Potential Flow Two-dimensional flows don t exist in reality but they are a useful simplification Two-dimensionality implies that the body being investigated: Has an infinite span Does not vary geometrically with spanwise position As examples, consider an infinitely long circular cylinder or an infinitely long rectangular wing
49 2D Potential equations Laplace s equation in two dimensions is simply y = 0 2 While the irrotationality condition is v u y = 0 We still need to find solutions to this equation.
50 Streamlines A streamline is a curve that is instantaneously tangent to the velocity vector of the flow x is the position vector of a point on a streamline, u is the velocity vector at that point and s is the distance on the streamline of the point from the origin x s u
51 Streamline definition A streamline is defined mathematically as: dx ds = u Where u has components u, v, w and x has components x, y, z. It can be easily seen that the definition leads to: dx ds = u, dy ds dz = v, ds = w, and therefore dx u = dy v = dz w
52 The stream function The stream function is defined at right angles to the flow plane, i.e. u = Where u=[u v 0] and =[0 0 ]. It can be seen that u = y, v = - The stream function is only defined for 2D or axisymmetric flows.
53 Properties of the stream function The stream function automatically satisfies the continuity equation. u + v y = + y y = 2 y 2 y = 0 The stream is constant on a flow streamline d = But, on a streamline Therefore dx + y dy = vdx + udy dx u = dy v d = udy + udy = 0
54 Elementary solutions There are several elementary solutions of Laplace s equation: The free stream: rectilinear motion of the airflow The source: a singularity that creates a radial velocity field around it The sink: the opposite of a source The doublet: a combined source and sink The vortex: a singularity that creates a circular velocity field around it.
55 Historical perspective 1738: Daniel Bernoulli developed Bernoulli s principle, which leads to Bernoulli s equation. 1740: Jean le Rond d'alembert studied inviscid, incompressible flow and formulated his paradox. 1755: Leonhard Euler derived the Euler equations. 1743: Alexis Clairaut first suggested the idea of a scalar potential. 1783: Pierre-Simon Laplace generalized the idea of the scalar potential and showed that all potential functions satisfy the same equation: Laplace s equation. 1822: Louis Marie Henri Navier first derived the Navier-Stokes equations from a molecular standpoint. 1828: Augustin Louis Cauchy also derived the Navier-Stokes equations 1829: Siméon Denis Poisson also derived the Navier-Stokes equations 1843: Adhémar Jean Claude Barré de Saint-Venant derived the Navier- Stokes equations for both laminar and turbulent flow. He also was the first to realize the importance of the coefficient of viscosity. 1845: George Gabriel Stokes published one more derivation of the Navier- Stokes equations.
V (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 informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
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 informationChapter 6: Incompressible Inviscid Flow
Chapter 6: Incompressible Inviscid Flow 6-1 Introduction 6-2 Nondimensionalization of the NSE 6-3 Creeping Flow 6-4 Inviscid Regions of Flow 6-5 Irrotational Flow Approximation 6-6 Elementary Planar Irrotational
More informationIntroduction to Aerodynamics. Dr. Guven Aerospace Engineer (P.hD)
Introduction to Aerodynamics Dr. Guven Aerospace Engineer (P.hD) Aerodynamic Forces All aerodynamic forces are generated wither through pressure distribution or a shear stress distribution on a body. The
More informationThin airfoil theory. Chapter Compressible potential flow The full potential equation
hapter 4 Thin airfoil theory 4. ompressible potential flow 4.. The full potential equation In compressible flow, both the lift and drag of a thin airfoil can be determined to a reasonable level of accuracy
More informationChapter 5. The Differential Forms of the Fundamental Laws
Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations
More informationAE/ME 339. K. M. Isaac Professor of Aerospace Engineering. 12/21/01 topic7_ns_equations 1
AE/ME 339 Professor of Aerospace Engineering 12/21/01 topic7_ns_equations 1 Continuity equation Governing equation summary Non-conservation form D Dt. V 0.(2.29) Conservation form ( V ) 0...(2.33) t 12/21/01
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 informationIn this section, mathematical description of the motion of fluid elements moving in a flow field is
Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small
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 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 informationFUNDAMENTALS OF AERODYNAMICS
*A \ FUNDAMENTALS OF AERODYNAMICS Second Edition John D. Anderson, Jr. Professor of Aerospace Engineering University of Maryland H ' McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas
More informationPerformance. 5. More Aerodynamic Considerations
Performance 5. More Aerodynamic Considerations There is an alternative way of looking at aerodynamic flow problems that is useful for understanding certain phenomena. Rather than tracking a particle of
More informationCompressible Potential Flow: The Full Potential Equation. Copyright 2009 Narayanan Komerath
Compressible Potential Flow: The Full Potential Equation 1 Introduction Recall that for incompressible flow conditions, velocity is not large enough to cause density changes, so density is known. Thus
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Hierarchy of Mathematical Models 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 29
More informationChapter 9: Differential Analysis
9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control
More informationSeveral forms of the equations of motion
Chapter 6 Several forms of the equations of motion 6.1 The Navier-Stokes equations Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationChapter 9: Differential Analysis of Fluid Flow
of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known
More informationChapter 4: Fluid Kinematics
Overview Fluid kinematics deals with the motion of fluids without considering the forces and moments which create the motion. Items discussed in this Chapter. Material derivative and its relationship to
More informationContinuity Equation for Compressible Flow
Continuity Equation for Compressible Flow Velocity potential irrotational steady compressible Momentum (Euler) Equation for Compressible Flow Euler's equation isentropic velocity potential equation for
More informationAE301 Aerodynamics I UNIT B: Theory of Aerodynamics
AE301 Aerodynamics I UNIT B: Theory of Aerodynamics ROAD MAP... B-1: Mathematics for Aerodynamics B-: Flow Field Representations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis
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 Navier-Stokes equations (unsteady, viscous
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. Second-order tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic
More informationNotes 4: Differential Form of the Conservation Equations
Low Speed Aerodynamics Notes 4: Differential Form of the Conservation Equations Deriving Conservation Equations From the Laws of Physics Physical Laws Fluids, being matter, must obey the laws of Physics.
More informationBLUFF-BODY AERODYNAMICS
International Advanced School on WIND-EXCITED AND AEROELASTIC VIBRATIONS OF STRUCTURES Genoa, Italy, June 12-16, 2000 BLUFF-BODY AERODYNAMICS Lecture Notes by Guido Buresti Department of Aerospace Engineering
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 informationJ. Szantyr Lecture No. 4 Principles of the Turbulent Flow Theory The phenomenon of two markedly different types of flow, namely laminar and
J. Szantyr Lecture No. 4 Principles of the Turbulent Flow Theory The phenomenon of two markedly different types of flow, namely laminar and turbulent, was discovered by Osborne Reynolds (184 191) in 1883
More informationChapter 2: Basic Governing Equations
-1 Reynolds Transport Theorem (RTT) - Continuity Equation -3 The Linear Momentum Equation -4 The First Law of Thermodynamics -5 General Equation in Conservative Form -6 General Equation in Non-Conservative
More informationConcept: AERODYNAMICS
1 Concept: AERODYNAMICS 2 Narayanan Komerath 3 4 Keywords: Flow Potential Flow Lift, Drag, Dynamic Pressure, Irrotational, Mach Number, Reynolds Number, Incompressible 5 6 7 1. Definition When objects
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) Counter-clockwise 3) Not enough information F y U 0 U F x V=0 V=0
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 informationAE 2020: Low Speed Aerodynamics. I. Introductory Remarks Read chapter 1 of Fundamentals of Aerodynamics by John D. Anderson
AE 2020: Low Speed Aerodynamics I. Introductory Remarks Read chapter 1 of Fundamentals of Aerodynamics by John D. Anderson Text Book Anderson, Fundamentals of Aerodynamics, 4th Edition, McGraw-Hill, Inc.
More information1. Introduction Some Basic Concepts
1. Introduction Some Basic Concepts 1.What is a fluid? A substance that will go on deforming in the presence of a deforming force, however small 2. What Properties Do Fluids Have? Density ( ) Pressure
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 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 informationIntroduction to Fluid Mechanics
Introduction to Fluid Mechanics Tien-Tsan Shieh April 16, 2009 What is a Fluid? The key distinction between a fluid and a solid lies in the mode of resistance to change of shape. The fluid, unlike the
More informationAE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says
More information2. Getting Ready for Computational Aerodynamics: Fluid Mechanics Foundations
. Getting Ready for Computational Aerodynamics: Fluid Mechanics Foundations We need to review the governing equations of fluid mechanics before examining the methods of computational aerodynamics in detail.
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 information3. FORMS OF GOVERNING EQUATIONS IN CFD
3. FORMS OF GOVERNING EQUATIONS IN CFD 3.1. Governing and model equations in CFD Fluid flows are governed by the Navier-Stokes equations (N-S), which simpler, inviscid, form is the Euler equations. For
More information1. Fluid Dynamics Around Airfoils
1. Fluid Dynamics Around Airfoils Two-dimensional 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 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 informationIntroduction to Flight
l_ Introduction to Flight Fifth Edition John D. Anderson, Jr. Curator for Aerodynamics, National Air and Space Museum Smithsonian Institution Professor Emeritus University of Maryland Me Graw Higher Education
More informationMAE 3130: Fluid Mechanics Lecture 7: Differential Analysis/Part 1 Spring Dr. Jason Roney Mechanical and Aerospace Engineering
MAE 3130: Fluid Mechanics Lecture 7: Differential Analysis/Part 1 Spring 2003 Dr. Jason Roney Mechanical and Aerospace Engineering Outline Introduction Kinematics Review Conservation of Mass Stream Function
More informationLecture-4. Flow Past Immersed Bodies
Lecture-4 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 informationIntroduction to Aerospace Engineering
Introduction to Aerospace Engineering Lecture slides Challenge the future 3-0-0 Introduction to Aerospace Engineering Aerodynamics 5 & 6 Prof. H. Bijl ir. N. Timmer Delft University of Technology 5. Compressibility
More informationIntroduction to Atmospheric Flight. Dr. Guven Aerospace Engineer (P.hD)
Introduction to Atmospheric Flight Dr. Guven Aerospace Engineer (P.hD) What is Atmospheric Flight? There are many different ways in which Aerospace engineering is associated with atmospheric flight concepts.
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 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 informationContents. I Introduction 1. Preface. xiii
Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics.........................
More informationIntroduction and Basic Concepts
Topic 1 Introduction and Basic Concepts 1 Flow Past a Circular Cylinder Re = 10,000 and Mach approximately zero Mach = 0.45 Mach = 0.64 Pictures are from An Album of Fluid Motion by Van Dyke Flow Past
More informationCOURSE NUMBER: ME 321 Fluid Mechanics I 3 credit hour. Basic Equations in fluid Dynamics
COURSE NUMBER: ME 321 Fluid Mechanics I 3 credit hour Basic Equations in fluid Dynamics Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET 1 Description of Fluid
More informationA Study of Transonic Flow and Airfoils. Presented by: Huiliang Lui 30 th April 2007
A Study of Transonic Flow and Airfoils Presented by: Huiliang Lui 3 th April 7 Contents Background Aims Theory Conservation Laws Irrotational Flow Self-Similarity Characteristics Numerical Modeling Conclusion
More informationME 425: Aerodynamics
ME 45: Aerodynamics Dr. A.B.M. Toufique Hasan Professor Department of Mechanical Engineering Bangladesh University of Engineering & Technology (BUET), Dhaka Lecture-0 Introduction toufiquehasan.buet.ac.bd
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 informationAerodynamics. High-Lift Devices
High-Lift Devices Devices to increase the lift coefficient by geometry changes (camber and/or chord) and/or boundary-layer control (avoid flow separation - Flaps, trailing edge devices - Slats, leading
More informationViscous flow along a wall
Chapter 8 Viscous flow along a wall 8. The no-slip condition All liquids and gases are viscous and, as a consequence, a fluid near a solid boundary sticks to the boundary. The tendency for a liquid or
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 informationAA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow
AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit
More informationAerodynamic force analysis in high Reynolds number flows by Lamb vector integration
Aerodynamic force analysis in high Reynolds number flows by Lamb vector integration Claudio Marongiu, Renato Tognaccini 2 CIRA, Italian Center for Aerospace Research, Capua (CE), Italy E-mail: c.marongiu@cira.it
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 informationNDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.
CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo
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 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 informationInviscid & 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. Low-speed wind tunnel
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 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 informationSupersonic Aerodynamics. Methods and Applications
Supersonic Aerodynamics Methods and Applications Outline Introduction to Supersonic Flow Governing Equations Numerical Methods Aerodynamic Design Applications Introduction to Supersonic Flow What does
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 informationThe E80 Wind Tunnel Experiment the experience will blow you away. by Professor Duron Spring 2012
The E80 Wind Tunnel Experiment the experience will blow you away by Professor Duron Spring 2012 Objectives To familiarize the student with the basic operation and instrumentation of the HMC wind tunnel
More informationLecture1: Characteristics of Hypersonic Atmosphere
Module 1: Hypersonic Atmosphere Lecture1: Characteristics of Hypersonic Atmosphere 1.1 Introduction Hypersonic flight has special traits, some of which are seen in every hypersonic flight. Presence of
More informationPDE Solvers for Fluid Flow
PDE Solvers for Fluid Flow issues and algorithms for the Streaming Supercomputer Eran Guendelman February 5, 2002 Topics Equations for incompressible fluid flow 3 model PDEs: Hyperbolic, Elliptic, Parabolic
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 informationFlight Vehicle Terminology
Flight Vehicle Terminology 1.0 Axes Systems There are 3 axes systems which can be used in Aeronautics, Aerodynamics & Flight Mechanics: Ground Axes G(x 0, y 0, z 0 ) Body Axes G(x, y, z) Aerodynamic Axes
More informationNumerical Heat and Mass Transfer
Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15-Convective Heat Transfer Fausto Arpino f.arpino@unicas.it Introduction In conduction problems the convection entered the analysis
More information( ) Notes. Fluid mechanics. Inviscid Euler model. Lagrangian viewpoint. " = " x,t,#, #
Notes Assignment 4 due today (when I check email tomorrow morning) Don t be afraid to make assumptions, approximate quantities, In particular, method for computing time step bound (look at max eigenvalue
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 informationVorticity Equation Marine Hydrodynamics Lecture 9. Return to viscous incompressible flow. N-S 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 informationMasters in Mechanical Engineering. Problems of incompressible viscous flow. 2µ dx y(y h)+ U h y 0 < y < h,
Masters in Mechanical Engineering Problems of incompressible viscous flow 1. Consider the laminar Couette flow between two infinite flat plates (lower plate (y = 0) with no velocity and top plate (y =
More informationChapter 1: Basic Concepts
What is a fluid? A fluid is a substance in the gaseous or liquid form Distinction between solid and fluid? Solid: can resist an applied shear by deforming. Stress is proportional to strain Fluid: deforms
More informationPart 3. Stability and Transition
Part 3 Stability and Transition 281 Overview T. Cebeci 1 Recent interest in the reduction of drag of underwater vehicles and aircraft components has rekindled research in the area of stability and transition.
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 Hunter-Jones) 1 2 BT Problem 14.2 (12 points: 3+3+3+3)
More informationPEMP ACD2505. M.S. Ramaiah School of Advanced Studies, Bengaluru
Two-Dimensional Potential Flow Session delivered by: Prof. M. D. Deshpande 1 Session Objectives -- At the end of this session the delegate would have understood PEMP The potential theory and its application
More informationi.e. the conservation of mass, the conservation of linear momentum, the conservation of energy.
04/04/2017 LECTURE 33 Geometric Interpretation of Stream Function: In the last class, you came to know about the different types of boundary conditions that needs to be applied to solve the governing equations
More informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
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 informationPrinciples of Convection
Principles of Convection Point Conduction & convection are similar both require the presence of a material medium. But convection requires the presence of fluid motion. Heat transfer through the: Solid
More informationGoverning Equations of Fluid Dynamics
Chapter 3 Governing Equations of Fluid Dynamics The starting point of any numerical simulation are the governing equations of the physics of the problem to be solved. In this chapter, we first present
More informationInvestigation potential flow about swept back wing using panel method
INTERNATIONAL JOURNAL OF ENERGY AND ENVIRONMENT Volume 7, Issue 4, 2016 pp.317-326 Journal homepage: www.ijee.ieefoundation.org Investigation potential flow about swept back wing using panel method Wakkas
More informationApplied Aerodynamics - I
Applied Aerodynamics - I o Course Contents (Tentative) Introductory Thoughts Historical Perspective Flow Similarity Aerodynamic Coefficients Sources of Aerodynamic Forces Fundamental Equations & Principles
More informationAerothermodynamics of high speed flows
Aerothermodynamics of high speed flows AERO 0033 1 Lecture 6: D potential flow, method of characteristics Thierry Magin, Greg Dimitriadis, and Johan Boutet Thierry.Magin@vki.ac.be Aeronautics and Aerospace
More informationρ Du i Dt = p x i together with the continuity equation = 0, x i
1 DIMENSIONAL ANALYSIS AND SCALING Observation 1: Consider the flow past a sphere: U a y x ρ, µ Figure 1: Flow past a sphere. Far away from the sphere of radius a, the fluid has a uniform velocity, u =
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 informationFluid Mechanics. Spring 2009
Instructor: Dr. Yang-Cheng Shih Department of Energy and Refrigerating Air-Conditioning Engineering National Taipei University of Technology Spring 2009 Chapter 1 Introduction 1-1 General Remarks 1-2 Scope
More informationWings and Bodies in Compressible Flows
Wings and Bodies in Compressible Flows Prandtl-Glauert-Goethert Transformation Potential equation: 1 If we choose and Laplace eqn. The transformation has stretched the x co-ordinate by 2 Values of at corresponding
More informationExperimental Aerodynamics. Experimental Aerodynamics
Lecture 6: Slender Body Aerodynamics G. Dimitriadis Slender bodies! Wings are only one of the types of body that can be tested in a wind tunnel.! Although wings play a crucial role in aeronautical applications
More informationMDTS 5734 : Aerodynamics & Propulsion Lecture 1 : Characteristics of high speed flight. G. Leng, MDTS, NUS
MDTS 5734 : Aerodynamics & Propulsion Lecture 1 : Characteristics of high speed flight References Jack N. Nielsen, Missile Aerodynamics, AIAA Progress in Astronautics and Aeronautics, v104, 1986 Michael
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 informationAerothermodynamics of High Speed Flows
Aerothermodynamics of High Speed Flows Lecture 1: Introduction G. Dimitriadis 1 The sound barrier Supersonic aerodynamics and aircraft design go hand in hand Aspects of supersonic flow theory were developed
More information