Continuum Mechanics Lecture 7 Theory of 2D potential flows
|
|
- Maurice Greene
- 5 years ago
- Views:
Transcription
1 Continuum Mechanics ecture 7 Theory of 2D potential flows Prof.
2 Outline - velocity potential and stream function - complex potential - elementary solutions - flow past a cylinder - lift force: Blasius formulae - Joukowsky transform: flow past a wing - Kutta condition - Kutta-Joukowski theorem
3 2D potential flows From the Helmholtz decomposition, we have 2D flows are defined by z ( ) =0 and v z =0. We have therefore A = ψez We consider in this chapter incompressible and irrotational flows. v =0 φ =0 + B. C. v =0 ψ =0 v n =0 We have two alternative but equivalent approaches. v = φ + A v x = x φ v x = y ψ v y = y φ v y = x ψ where the velocity potential satisfies the aplace equation. where the stream function satisfies the aplace equation. In the potential case, the irrotational condition is satisfied automatically. In the stream function approach, this is the divergence free condition. Since both conditions are satisfied, both velocity fields are equal.
4 Isopotential curves and stream lines The velocity field is defined equivalently by two scalar fields v x = x φ = y ψ v y = y φ = x ψ They are conjugate functions that satisfy the Cauchy-Riemann relations. They are also harmonic functions (aplace equation), with however different B. C. φ =0 in S with φ n =0 on the boundary ψ =0 in S with ψ t =0on the boundary or ψ = constant along Isopotential curves are defined by Stream lines are defined by dφ =dx x φ +dy y φ =0 dψ =dx x ψ +dy y ψ =0 or or dy dx = v x v y dy dx = v y v x Isopotential curves and stream lines are orthogonal to each other. x ψ x φ + y ψ y φ =( v y )v x +(+v x )v y =0
5 Complex potential and complex derivative We define the complex potential where z = x + iy and i 2 = 1. F (z) =φ(x, y)+iψ(x, y) From complex derivation theory, we know that any complex function F is differentiable if and only if the two functions Φ and ψ satisfy the Cauchy-Riemann relations. Such complex functions are called analytic. uckily, since the velocity potential and the stream function are conjugate, the complex velocity potential is differentiable. We define the complex velocity w(z) = df dz where the complex derivative is defined as df dz = F x = 1 i F y We obtain w(z) = x φ + i x ψ = y ψ i y φ = v x iv y In cylindrical coordinates, we have v r = r φ = 1 r θψ v θ = 1 r θφ = r ψ and the complex velocity writes w(z) =v x iv y =(v r iv θ )exp iθ
6 Uniform flow Complex potential F (z) =U exp iα z Complex velocity w(z) =U exp iα Velocity field v x = U cos(α) v y = U sin(α) Potential lines Streamlines Velocity potential Stream function φ = U cos(α)x + U sin(α)y ψ = U cos(α)y U sin(α)x
7 Stagnation flow F (z) =Cz 2 Streamlines are hyperbolae. φ = C(x 2 y 2 ) ψ =2Cxy v x =2Cx v y = 2Cy In polar coordinates φ = Cr 2 cos(2θ) ψ = Cr 2 sin(2θ) This potential can also be used to describe a flow past a corner.
8 Flow past an edge Complex potential F (z) =c z = cr 1/2 exp iθ/2 Velocity potential φ = c r cos θ 2 Stream function ψ = c r sin θ 2 Velocity field v r = c 1 cos θ 2 r 2 v θ = c 1 sin θ 2 r 2 The velocity field becomes infinite at the tip of the edge.
9 Flow around a source or a sink Complex potential F (z) = m 2π log (z z 0) Complex velocity w(z) = m 1 2π z z 0 φ = m 2π log r ψ = mθ 2π Velocity field v x = m x x 0 2π r 2 v y = m 2π v r = y y 0 r 2 In polar coordinates m 2πr v = r v r + v r The velocity divergence is zero everywhere r =0 for r>0. We apply the divergence theorem to a circle centered on the singularity: v ndl = v r 2πr = m 1 Rmax m 2 dr S 2 v2 dxdy = R min 4π r = m2 4π log Rmax The kinetic energy in the flow is R min In a real flow, the singularity is usually embedded inside the boundary condition.
10 Flow around a point vortex Complex potential F (z) = i Ω 2π log (z z 0) Complex velocity w(z) = i Ω 1 2π z z 0 φ = Ωθ 2π ψ = Ω 2π log r Velocity field v x = Ω y y 0 2π r 2 v y = Ω x x 0 2π r 2 In polar coordinates v θ = Ω 2πr e z =0 The velocity curl is zero everywhere v = r v θ + v θ r We apply the curl theorem on a circle centered on the singularity: v tdl = v θ 2πr = Ω 1 Rmax m 2 The kinetic energy in the flow is S 2 v2 dxdy = R min 4π There is a direct analogy with the energy of dislocations in a solid. for r>0. dr r = m2 4π log Rmax R min
11 Superposition principle and boundary conditions ike for the Navier equation for thermoelastic equilibrium problems, the aplace equation for the potential and/or the stream function is a linear boundary value problem. When proper boundary conditions are imposed (no vorticity), the solution always exists and is unique.two different solutions can be added linearly and the sum represent also a solution with the corresponding boundary conditions. The previous elementary solutions form a library that you can combine to build up more complex curl-free and divergence-free flows. Streamlines are perpendicular to potential curves. The velocity component normal to a streamline is always zero. Therefore, each streamline can be used to define a posteriori the boundary condition. You can therefore add up randomly complex potential to get any kind of analytical complex function. Then, you compute the streamlines. Then, you define the embedded body by picking any streamline. You finally get yourself a valid potential flow!
12 Flow around a doublet We superpose a source and a sink F (z) = m 2π (log (z z 0) log (z + z 0 )) very close to each other Taylor expanding, we find F (z) m exp iα = µ expiα π z z Parameter µ is called the doublet strength. For α =0, we find φ = µx r 2 ψ = µy r 2 cos θ = µ r = µ sin θ r The velocity field is given by It is a dipole field v = µ r r 3 z 0 = exp iα v r = r φ = µ cos θ r 2 Potential and streamlines are circles. and v θ = 1 r θφ = µ sin θ r 2 The general form around an arbitrary center z0 is: F (z) = µ expiα z z 0
13 Flow past a cylinder We superpose a uniform flow and a doublet. F (z) =U z + µ z We find φ = U x + µ x r 2 ψ = U y µ y r 2 µ The streamline ψ =0 is the circle r = We reverse engineer the process. For a cylinder a radius a, if we define U µ = U a 2 then the potential flow around the cylinder is F (z) =U z + a2 z The velocity field is given by v r = U 1 a2 cos θ v θ = U 1+ a2 sin θ r 2 S S The flow has 2 stagnation points S and S given by r=a and θ=0 and π. The doublet is inside the embedded body, so there is no singularity in the flow. r 2
14 Force acting on the cylinder Using the second Bernoulli theorem (curl-free, incompressible, no gravity), we know that the quantity H = P is uniform. ρ v2 We thus have p ρv2 = p ρu 2 The force acting on the cylinder is given by F = pndl On the cylinder, we have v r =0 and v θ = 2U sin θ p = p ρu 2 The pressure field on the cylinder is thus 1 4 sin 2 θ Using n = (cos θ, sin θ) F x = 1 2 ρu 2 a F y = 1 2 ρu 2 a 2π 0 2π 0 we find: 1 4 sin 2 θ cos θdθ 1 4 sin 2 θ sin θdθ sin θ 4 2π 3 sin3 θ 3 cos θ 4 3 cos3 θ 0 2π 0 =0 =0 Exercise: compute the torque on the cylinder (use the cylinder axis). It is also zero!
15 We superpose a uniform flow, a doublet and a vortex. F (z) =U z + a2 iω z z 2π log a Streamlines are given by ψ = U 1 a2 r 2 y Ω r 2π log a Flow past a cylinder with vorticity The cylinder r=a is still a proper boundary condition. v r = U 1 a2 r 2 cos θ v θ = U 1+ a2 r 2 sin θ + Ω 2πr On the cylinder, we have to stagnation point given by or one stagnation point away from the cylinder if At the boundary, we have v θ = 2U (sin θ sin θ s ) Using the Bernoulli theorem and integrating the pressure field on the boundary, we can compute the force on the cylinder (exercise) sin θ s = Ω < 4πU a F x =0 Ω 4πU a F y = ρu Ω
16 The Magnus effect Topspin tennis ball trajectory curves down. Rotating pipes induce a force perpendicular to the wind direction Warning: viscosity effects can t be ignored!
17 F = pndl We use curvilinear coordinates along the body t = (cos θ, sin θ) n =(sinθ, cos θ) The force components are F x = p sin θdl F y = The complex force: Blasius formulae p cos θdl In Cartesian coordinates, we have dx = cos θdl dy =sinθdl The complex force is defined as F x if y = F x n t p (dy + idx) = i pdz F y Bernoulli theorem: p = p ρu ρv2 with v 2 = w(z)w(z) Boundary condition: v n =0 v x dy v y dx =0 and w dz = wdz We finally get the force for an arbitrary shaped body: F x if y = i 2 ρ w 2 (z)dz
18 We consider an arbitrary closed contour in the complex plane. We define the complex circulation as C = w(z)dz Using the same definition as before along the contour, we have C = (v x dx + v y dy)+i (v x dy v y dx) where the Cartesian coordinates are related to the curvilinear ones by We finally get C = The complex circulation dx = dl cos θ v tdl + i Γ is the physical circulation and Q is the physical mass flux. dy = dl sin θ v ndl = Γ + iq On the contour defining the body shape, the mass flux is zero and we have C = Γ = v tdl
19 A conformal mapping is a differentiable complex function M that maps the complex plane z into another complex plane Z. We have Z = M(z) and z = m(z) with m = M 1 If a flow is defined by a potential function Conformal mapping We need to build more complex profile than just a cylinder. We use for that a mathematical trick called conformal mapping. f(z) F (Z) =f(m(z)) in the z plane, then the function is also analytic (it satisfies the Cauchy-Riemann relations). It is therefore a valid vector potential. The new streamlines and potential curves are the transform of the old one. The new complex velocity writes W (Z) = df dz = df dm dz dz = w(z)m (Z) The complex circulation is conserved by conformal mapping C = W (Z)dZ = w(z)m (Z)dZ = w(z)dz l
20 The Joukowski transform Definition: z = Z + c2 Z The circle of radius c becomes the line segment [-2c, 2c] c Z 2c 2c z Z = c exp iθ z =2c cos θ Z z The circle of radius a>c becomes an ellipse. Z = a exp iθ z =(a + c2 a ) cos θ + i(a c2 a )sinθ The inverse transform is Z = z 2 + z 2 2 c 2 The derivative M (z) = z 2 z 2 2 c 2 has 2 singular points at z = ±2c
21 Acyclic flow past an ellipse We use the Joukowski transform from a flow past a circular cylinder. The flow is acyclic: no circulation and no vortex component. We assume that the flow at infinity is at an angle with the x-axis. The complex potential and velocity of the original flow are F (Z) =U Z exp iα +a 2 expiα W (Z) =U Z exp iα 1 a2 Z 2 expi2α Using the Joukowski mapping ellipsoidal cylinder. Z = M(z) with a>c, we get the potential around an Using c 2, we get Z = z Z The original stagnation points f(z) =U Z s = ±a exp iα z a2 c 2 expiα +M(z) exp iα a2 c 2 expiα become z s = ± a exp iα + c2a exp iα
22 Acyclic flow past a plate eading edge Z s Trailing edge z s We use the previous results, taking c a f(z) =U z exp iα z z 2 2i sin α 2 + a 2 2 The stagnation points are on the x-axis The complex velocity is given by z s = ±2a cos α w(z) =W (Z)M (z) The velocity at the leading and trailing edges is: Z = ±a W (±a) = 2iU sin α w(±2a) (see flow past an edge). This is unphysical!
23 Flow past a plate with circulation from P. Huerre s lectures
24 Flow past a plate with circulation On the original circular cylinder, we have: F (Z) =U Z exp iα +a 2 expiα Z W (Z) =U exp iα 1 a2 Z 2 expi2α The stagnation points are now defined by iω 2π log iω 2ΠZ sin (θ s α) = Z a Ω 4πU a We still have an infinite number of solution, depending on the value of the point vorticity. For a particular value of the circulation, the stagnation point will coincide with the trailing edge, therefore removing the singularity. Ω c = sin α4πu a For a given body shape, we always choose the critical circulation as defining the unique physical solution.
25 The Kutta condition Initially, we have zero circulation Starting vortex produces vorticity Kelvin s theorem «A body with a sharp trailing edge which is moving through a fluid will create about itself a circulation of sufficient strength to hold the rear stagnation point at the trailing edge.»
26 The Joukowski profiles We consider now the more general case of a circular cylinder for which the center has been offset from the origin. F (Z) =U (Z b)exp iα +a 2 expiα iω Z b Z b 2π log a W (Z) =U exp 1 iα a 2 iω (Z b) 2 expi2α 2π(Z b) Recipe: using the Kutta condition, we impose the singular trailing edge to be a stagnation point. By adjusting b, we remove the singularity at the leading edge.
27 Critical circulation for Joukowski profiles The trailing edge Z =+c U exp iα 1 a2 (c b) 2 expi2α is imposed to be a stagnation point. Ω c = 4πU a sin (α + β) iω 2π(c b) =0 Since b is the center of the cylinder, we can define the angle c b = a exp iβ
28 Flow past an arbitrarily shaped cylinder We now consider the inverse problem: we know the shape of the cylinder and we would like to find the conformal mapping to a circular cylinder. Any analytic complex function can be expanded in its aurent series around the origin. We restrict ourselves to mapping for which points at infinity are invariants. a n M(z) =z + z n a n = 1 where M(z)z n+1 dz 2πi n=0 The general flow around the circular cylinder is given by the potential F (Z) =U Z exp iα +a 2 expiα iω Z Z 2π log a Injecting the mapping for Z and Taylor expanding around infinity, we get: f(z) =U z exp iα iω z 2π log b n + a z n and n=0 w(z) =U exp iα iω 2πz nb n z n+1 n=1 The general flow is uniform to leading order, then a vortex flow to next order, then a doublet flow to higher order, and so on... The circulation on the new body is C = w(z)dz = W (Z)dZ = Ω l l
29 We now compute the force acting on the arbitrarily shaped body. We have the Clausius formula F x if y = i 2 ρ w 2 (z)dz The kinetic energy is expanded as We have (residue theorem) The force is for any profile and We recover the force acting on the circular cylinder. General results: - no drag F x =0 The Kutta-Joukowski theorem - without circulation F y =0 (d Alembert s paradox). The force on a general Joukowski profile is w 2 (z) =U 2 exp 2iα iα iω U exp πz + dz z =2iπ F x if y = iρu exp iα Ω l dz z n = 0 for n 2 F y =4πρU 2 a sin (α + β) n=2 c n z n
30 ift coefficient The lift coefficient is a dimensionless number that measures the performance of a wing profile ( is the length of the wing section). C y = F y 1 2 ρu 2 For a Joukowski profile with small attack angle and small bending angle, C y =8π a sin (α + β) 2π(α + β) The theory disagrees more and more with the experiment: we have neglected viscous effects. It breaks down completely above 10 degrees. This is because the zero streamline is detaching from the wing.
Exercise 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 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 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 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 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 informationAn Internet Book on Fluid Dynamics. Joukowski Airfoils
An Internet Book on Fluid Dynamics Joukowski Airfoils One of the more important potential flow results obtained using conformal mapping are the solutions of the potential flows past a family of airfoil
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 informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
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 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 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 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 4-digit accuracy. Air density : Water density
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 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 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 informationII. Ideal fluid flow
II. Ideal fluid flow Ideal fluids are Inviscid Incompressible The only ones decently understood mathematically Governing equations u=0 Continuity u 1 +( u ) u= ρ p+ f t Euler Boundary conditions Normal
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 informationPart A Fluid Dynamics & Waves Draft date: 17 February Conformal mapping
Part A Fluid Dynamics & Waves Draft date: 17 February 4 3 1 3 Conformal mapping 3.1 Wedges and channels 3.1.1 The basic idea Suppose we wish to find the flow due to some given singularities (sources, vortices,
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 MATH-GA 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 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 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 informationMath 575-Lecture Failure of ideal fluid; Vanishing viscosity. 1.1 Drawbacks of ideal fluids. 1.2 vanishing viscosity
Math 575-Lecture 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 informationComplex Analysis MATH 6300 Fall 2013 Homework 4
Complex Analysis MATH 6300 Fall 2013 Homework 4 Due Wednesday, December 11 at 5 PM Note that to get full credit on any problem in this class, you must solve the problems in an efficient and elegant manner,
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 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 3-D characteristics) How
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 two-dimensional, nonviscous, incompressible flow field is given by the expression ψ = 2(x y) where the stream function has
More informationWhy airplanes fly, and ships sail
Why airplanes fly, and ships sail A. Eremenko April 23, 2013 And windmills rotate and propellers pull, etc.... Denote z = x + iy and let v(z) = v 1 (z) + iv 2 (z) be the velocity field of a stationary
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 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 informationChapter II. Complex Variables
hapter II. omplex Variables Dates: October 2, 4, 7, 2002. These three lectures will cover the following sections of the text book by Keener. 6.1. omplex valued functions and branch cuts; 6.2.1. Differentiation
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 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 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 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 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 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 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 informationF11AE1 1. C = ρν r r. r u z r
F11AE1 1 Question 1 20 Marks) Consider an infinite horizontal pipe with circular cross-section of radius a, whose centre line is aligned along the z-axis; see Figure 1. Assume no-slip boundary conditions
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 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 informationFluid Dynamics Problems M.Sc Mathematics-Second Semester Dr. Dinesh Khattar-K.M.College
Fluid Dynamics Problems M.Sc Mathematics-Second Semester Dr. Dinesh Khattar-K.M.College 1. (Example, p.74, Chorlton) At the point in an incompressible fluid having spherical polar coordinates,,, the velocity
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 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 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 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 informationSimplifications to Conservation Equations
Chater 5 Simlifications to Conservation Equations 5.1 Steady Flow If fluid roerties at a oint in a field do not change with time, then they are a function of sace only. They are reresented by: ϕ = ϕq 1,
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 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 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 informationFluid mechanics, topology, and complex analysis
Fluid mechanics, topology, and complex analysis Takehito Yokoyama Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan (Dated: April 30, 2013 OMPLEX POTENTIAL
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 information26.2. Cauchy-Riemann Equations and Conformal Mapping. Introduction. Prerequisites. Learning Outcomes
Cauchy-Riemann Equations and Conformal Mapping 26.2 Introduction In this Section we consider two important features of complex functions. The Cauchy-Riemann equations provide a necessary and sufficient
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 informationControl Volume. Dynamics and Kinematics. Basic Conservation Laws. Lecture 1: Introduction and Review 1/24/2017
Lecture 1: Introduction and Review Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study of motion without regard for the cause. Dynamics: On the other hand, dynamics
More informationLecture 1: Introduction and Review
Lecture 1: Introduction and Review Review of fundamental mathematical tools Fundamental and apparent forces Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study
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 informationVortex motion. Wasilij Barsukow, July 1, 2016
The concept of vorticity We call Vortex motion Wasilij Barsukow, mail@sturzhang.de July, 206 ω = v vorticity. It is a measure of the swirlyness of the flow, but is also present in shear flows where the
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 information18.325: Vortex Dynamics
8.35: Vortex Dynamics Problem Sheet. Fluid is barotropic which means p = p(. The Euler equation, in presence of a conservative body force, is Du Dt = p χ. This can be written, on use of a vector identity,
More informationSynopsis of Complex Analysis. Ryan D. Reece
Synopsis of Complex Analysis Ryan D. Reece December 7, 2006 Chapter Complex Numbers. The Parts of a Complex Number A complex number, z, is an ordered pair of real numbers similar to the points in the real
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 informationHW6. 1. Book problems 8.5, 8.6, 8.9, 8.23, 8.31
HW6 1. Book problems 8.5, 8.6, 8.9, 8.3, 8.31. Add an equal strength sink and a source separated by a small distance, dx, and take the limit of dx approaching zero to obtain the following equations for
More informationThe Aharonov-Bohm Effect: Mathematical Aspects of the Quantum Flow
Applied athematical Sciences, Vol. 1, 2007, no. 8, 383-394 The Aharonov-Bohm Effect: athematical Aspects of the Quantum Flow Luis Fernando ello Instituto de Ciências Exatas, Universidade Federal de Itajubá
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 information2.25 Advanced Fluid Mechanics Fall 2013
.5 Advanced Fluid Mechanics Fall 013 Solution to Problem 1-Final Exam- Fall 013 r j g u j ρ, µ,σ,u j u r 1 h(r) r = R Figure 1: Viscous Savart Sheet. Image courtesy: Villermaux et. al. [1]. This kind of
More information3 Generation and diffusion of vorticity
Version date: March 22, 21 1 3 Generation and diffusion of vorticity 3.1 The vorticity equation We start from Navier Stokes: u t + u u = 1 ρ p + ν 2 u 1) where we have not included a term describing a
More information1/3/2011. This course discusses the physical laws that govern atmosphere/ocean motions.
Lecture 1: Introduction and Review Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study of motion without regard for the cause. Dynamics: On the other hand, dynamics
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 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 informationOUTLINE FOR Chapter 3
013/4/ OUTLINE FOR Chapter 3 AERODYNAMICS (W-1-1 BERNOULLI S EQUATION & integration BERNOULLI S EQUATION AERODYNAMICS (W-1-1 013/4/ BERNOULLI S EQUATION FOR AN IRROTATION FLOW AERODYNAMICS (W-1-.1 VENTURI
More informationMaxwell s equations for electrostatics
Maxwell s equations for electrostatics October 6, 5 The differential form of Gauss s law Starting from the integral form of Gauss s law, we treat the charge as a continuous distribution, ρ x. Then, letting
More informationASTR 320: Solutions to Problem Set 2
ASTR 320: Solutions to Problem Set 2 Problem 1: Streamlines A streamline is defined as a curve that is instantaneously tangent to the velocity vector of a flow. Streamlines show the direction a massless
More informationAero III/IV Conformal Mapping
Aero III/IV Conformal Mapping View complex function as a mapping Unlike a real function, a complex function w = f(z) cannot be represented by a curve. Instead it is useful to view it as a mapping. Write
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
More information26.4. Basic Complex Integration. Introduction. Prerequisites. Learning Outcomes
Basic omplex Integration 6.4 Introduction omplex variable techniques have been used in a wide variety of areas of engineering. This has been particularly true in areas such as electromagnetic field theory,
More information2.6 Oseen s improvement for slow flow past a cylinder
Lecture Notes on Fluid Dynamics.63J/.J) by Chiang C. Mei, MIT -6oseen.tex [ef] Lamb : Hydrodynamics.6 Oseen s improvement for slow flow past a cylinder.6. Oseen s criticism of Stokes approximation Is Stokes
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 informationComplex Analysis Math 185A, Winter 2010 Final: Solutions
Complex Analysis Math 85A, Winter 200 Final: Solutions. [25 pts] The Jacobian of two real-valued functions u(x, y), v(x, y) of (x, y) is defined by the determinant (u, v) J = (x, y) = u x u y v x v y.
More informationMath 185 Fall 2015, Sample Final Exam Solutions
Math 185 Fall 2015, Sample Final Exam Solutions Nikhil Srivastava December 12, 2015 1. True or false: (a) If f is analytic in the annulus A = {z : 1 < z < 2} then there exist functions g and h such that
More informationUNIVERSITY OF EAST ANGLIA
UNIVERSITY OF EAST ANGLIA School of Mathematics May/June UG Examination 2007 2008 FLUIDS DYNAMICS WITH ADVANCED TOPICS Time allowed: 3 hours Attempt question ONE and FOUR other questions. Candidates must
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 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 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 informationLecture 10 - Moment of Inertia
Lecture 10 - oment of Inertia A Puzzle... Question For any object, there are typically many ways to calculate the moment of inertia I = r 2 dm, usually by doing the integration by considering different
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 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 informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2014 15 FLUID DYNAMICS - THEORY AND COMPUTATION MTHA5002Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
More informationDivergence Theorem and Its Application in Characterizing
Divergence Theorem and Its Application in Characterizing Fluid Flow Let v be the velocity of flow of a fluid element and ρ(x, y, z, t) be the mass density of fluid at a point (x, y, z) at time t. Thus,
More information65 Fluid Flows (6/1/2018)
65 Fluid Flows 6//08 Consider a two dimensional fluid flow which we describe by its velocity field, V x, y = p x, y, q x, y = p + iq R. We are only going to consider flows which are incompressible, i.e.
More informationMathematical Concepts & Notation
Mathematical Concepts & Notation Appendix A: Notation x, δx: a small change in x t : the partial derivative with respect to t holding the other variables fixed d : the time derivative of a quantity that
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 informationHere are brief notes about topics covered in class on complex numbers, focusing on what is not covered in the textbook.
Phys374, Spring 2008, Prof. Ted Jacobson Department of Physics, University of Maryland Complex numbers version 5/21/08 Here are brief notes about topics covered in class on complex numbers, focusing on
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2012-13 MODULE TITLE: Introduction to Fluids DURATION OF
More informationPotential flow of a second-order fluid over a sphere or an ellipse
J. Fluid Mech. (4), vol. 5, pp. 5. c 4 Cambridge niversity Press DOI:.7/S4954 Printed in the nited Kingdom Potential flow of a second-order fluid over a sphere or an ellipse By J. W A N G AND D. D. J O
More informationMath Final Exam.
Math 106 - Final Exam. This is a closed book exam. No calculators are allowed. The exam consists of 8 questions worth 100 points. Good luck! Name: Acknowledgment and acceptance of honor code: Signature:
More informationTHE VORTEX PANEL METHOD
THE VORTEX PANEL METHOD y j m α V 4 3 2 panel 1 a) Approimate the contour of the airfoil by an inscribed polygon with m sides, called panels. Number the panels clockwise with panel #1 starting on the lower
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 informationlim = F F = F x x + F y y + F z
Physics 361 Summary of Results from Lecture Physics 361 Derivatives of Scalar and Vector Fields The gradient of a scalar field f( r) is given by g = f. coordinates f g = ê x x + ê f y y + ê f z z Expressed
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 informationMA 412 Complex Analysis Final Exam
MA 4 Complex Analysis Final Exam Summer II Session, August 9, 00.. Find all the values of ( 8i) /3. Sketch the solutions. Answer: We start by writing 8i in polar form and then we ll compute the cubic root:
More information