Offshore Hydromechanics Module 1
|
|
- Gabriel Robinson
- 5 years ago
- Views:
Transcription
1 Offshore Hydromechanics Module 1 Dr. ir. Pepijn de Jong 4. Potential Flows part 2
2 Introduction Topics of Module 1 Problems of interest Chapter 1 Hydrostatics Chapter 2 Floating stability Chapter 2 Constant potential flows Chapter 3 Constant real flows Chapter 4 Waves Chapter 5 2
3 Learning Objectives Chapter 3 Understand the basic principles behind potential flow To schematically model flows applying basic potential flow elements and the superposition principle To perform basic flow computations applying potential flow theory 3
4 Fluid Mechanics Laws Summarizing: Conservation of mass (continuity): Incompressible flow: ρ t + ρv = 0 u x + v y + w z = 0 V = 0 Conservation of momentum (inviscid flow): ρ DV = ρg p Dt ω x = 1 2 Rotation of a fluid element: w y v z ω y = 1 2 ω z = 1 u z w 2 x v x u y Vorticity: ζ = 2ω = V 4
5 Fluid Mechanics Laws Velocity Potential Assumptions Homogeneous Continuous Incompressible Non-viscous (inviscid) Irrotational flow The velocity potential is a function of time and position: Φ x, y, z, t The spatial derivatives of the velocity potential equal the velocity components at a time and position: Φ x = u Φ y = v Φ z = w 5
6 Fluid Mechanics Laws Velocity Potential Continuity equation for potential flow: 2 Φ x Φ y Φ z 2 = 0 2 Φ = 0 Laplace equation Irrotationality v x = u y in the (x,y) plane ω x = 1 2 w y v z w y = v z in the (y,z) plane ω y = 1 2 u z w x u z = w x in the (x,z) plane ω z = 1 2 v x u y 6
7 Fluid Mechanics Laws Bernoulli Equation Potential flow: V = Φ Φ t Φ 2 gz + p ρ = constant Bernoulli equation More general: also valid on a streamline 7
8 Potential Flow Stream funtion (2D) Ψ is the (2D) stream function, with: dψ dy = u dψ dx = v Difference of Ψ streamlines Orthogonality: between neighboring stream lines: rate of flow between dψ dy = dφ dx = u dψ dx = dφ dy = v Impervious boundaries equals streamline: dφ dn = 0 Ψ = constant 8
9 Potential flow elements Introduction Using the previous we can define 'flow elements' We can add these elements up to construct realistic flow patterns Modeling of submerged bodies by matching streamlines to body outline Using the velocity potential, stream function and Bernoulli equation to find velocities, pressures and eventually fluid forces on bodies Discussed: Uniform flow element Source/sink element Doublet element Vortex element 9
10 Potential flow elements Uniform flow y y U U Φ = U x x Φ = U x x Ψ = U y Ψ = U y u = dφ dx = dψ dy = U u = dφ dx = dψ dy = U 10
11 Potential flow elements Source and sink flow y y so r θ x si r θ x Φ = + Q 2π lnr = + Q 2π ln x2 + y 2 Φ = Q 2π lnr = Q 2π ln x2 + y 2 Ψ = + Q 2π θ = + Q 2π arctan y x Ψ = Q 2π θ = Q 2π arctan y x 11
12 Potential flow elements Source and sink flow y Φ = + Q 2π lnr Ψ = + Q 2π θ v r = Φ r = 1 r Ψ θ = Q 2πr so r θ x v θ = 1 r Φ θ = Ψ r = 0 12
13 Potential flow elements Circulation or vortex elements y Φ = + Γ 2π θ Ψ = + Γ 2π lnr v r = Φ r = 1 r Ψ θ = 0 Γ r θ x v θ = 1 r Φ θ = Ψ r = Γ 2πr 13
14 Potential flow elements Circulation or vortex elements Φ = + Γ 2π θ v r = Φ r = 1 r Ψ θ = 0 v θ = 1 r Φ θ = Ψ r = Ψ = + Γ 2π lnr Γ 2πr Circulation strenght constant: y Γ r θ x Γ = v θ ds = 2πr v θ = constant Therefore: no rotation, origin singular point: velocity infinite 14
15 Methodology (source in positive uniform flow) The resulting velocity fields, potential fields or stream function fields may be simply superposed to find the combined flow patterns (Using stream function values) Uniform flow field Source flow field 15
16 Methodology (source in positive uniform flow) The resulting velocity fields, potential fields or stream function fields may be simply superposed to find the combined flow patterns (Using stream function values) 16
17 Methodology (source in positive uniform flow) The resulting velocity fields, potential fields or stream function fields may be simply superposed to find the combined flow patterns (Using stream function values) 17
18 Sink in negative uniform flow Besides graphically this works also with formulas: Ψ = Q 2π arctan y x U y Φ = Q 2π ln x2 + y 2 U x For instance: Find location stagnation point (Blackboard...) 18
19 Separated source and sink Ψ source = + Q 2π θ 1 = + Q 2π arctan y x 1 Ψ sink = Q 2π θ 2 = Q 2π arctan y x 2 Ψ = Q 2π arctan 2ys x 2 + y 2 s 2 x 1 x 2 2s 19
20 Separated source and sink in uniform flow s s Ψ = Q 2π arctan 2ys x 2 + y 2 s 2 + U y 20
21 Separated source and sink in uniform flow s s Streamline resembles fixed boundary (Rankine oval) The flow outside this streamline resembles flow around solid boundary with this shape Shape can be changed by using more source-sinks along x-axis with different strenghts 21
22 Separated source and sink in uniform flow s s This approach can be extended to form ship forms in 2D or 3D: Rankine ship forms Useful for simple flow computations 22
23 Potential flow elements Doublet or dipole When distance 2s becomes zero a new basic flow element is produced: Doublet or dipole producing flow in positive x-direction Q Ψ = lim s 0 2π arctan 2ys x 2 + y 2 s 2 Ψ = lim s 0 Q π s y x 2 + y 2 s 2 s s Note: in book errors wrt to doublet and its orientation! 23
24 Potential flow elements Doublet or dipole When distance 2s becomes zero a new basic flow element is produced: Doublet or dipole producing flow in positive x-direction Q Ψ = lim s 0 2π arctan 2ys x 2 + y 2 s 2 Ψ = lim s 0 Q π s y x 2 + y 2 s 2 Set constant: μ = Q π s 24
25 Potential flow elements Doublet or dipole When distance 2s becomes zero a new basic flow element is produced: Doublet or dipole producing flow in positive x-direction Q Ψ = lim s 0 2π arctan 2ys x 2 + y 2 s 2 Ψ = lim s 0 Q π s y x 2 + y 2 s 2 Disappears when: s 0 Set constant: μ = Q π s 25
26 Potential flow elements Doublet or dipole When distance 2s becomes zero a new basic flow element is produced: Doublet or dipole producing flow in positive x-direction Ψ = lim s 0 Q 2π arctan 2ys x 2 + y 2 s 2 Ψ = μ Φ = μ y x 2 + y 2 = μ sinθ r x cosθ x 2 + y2 = μ r 26
27 Doublet in a uniform flow Φ = μ x x 2 + y 2 U x Ψ = μ y x 2 + y 2 U y Φ = μ cosθ r U rcosθ Wrong in book! Ψ = μ sinθ r U rsinθ Doublet pointing in positive x-direction, uniform flow in negative x- direction: + 27
28 Doublet in a uniform flow Φ = μ Φ = μ cosθ r x x 2 + y 2 U x U rcosθ Ψ = μ Ψ = μ sinθ r y x 2 + y 2 U y U rsinθ Set Ψ = 0 then: True when: Ψ = y μ x 2 + y 2 U = 0 y = 0 μ x 2 + y 2 U = 0 x 2 + y 2 = μ U 28
29 Doublet in a uniform flow: flow around a circle The radius of the circle: R = μ U Doublet strength needed for radius R: Φ = μ cosθ r Ψ = μ sinθ r U rcosθ U rsinθ μ = U R 2 This yields the following: Φ = U R 2 cosθ r U rcosθ = RU R r + r R cosθ Ψ = U R 2 sinθ r U rsinθ = RU R r r R sinθ 29
30 Doublet in a uniform flow: flow around a circle Φ = RU Φ = U R 2 R r + r R cosθ x x 2 + y 2 U x u = dφ dx = U R 2 x2 y 2 x 2 + y 2 2 U 30
31 Doublet in a uniform flow: flow around a circle Φ = RU Φ = U R 2 R r + r R cosθ x x 2 + y 2 U x u = dφ dx = U R 2 x 2 y 2 x 2 + y 2 2 U x = ±R, y = 0 u = dφ dx = U R 2 R2 0 2 R U = U R 2 R2 R 4 U = 0 Stagnation points! 31
32 Doublet in a uniform flow: flow around a circle Φ = RU Φ = U R 2 R r + r R cosθ x x 2 + y 2 U x u = dφ dx = U R 2 x 2 y 2 x 2 + y 2 2 U x = 0, y = ±R u = dφ dx = U R R R 2 2 U = U R 2 R2 R 4 U = 2U 32
33 Evaluate velocities on cylinder wall Generally, velocity on cylinder wall: Ψ = μ sinθ r U rsinθ v θ r = R = Ψ r r=r = r U R 2 sinθ r U rsinθ =... = 2U sinθ 33
34 Evaluate pressures on cylinder wall Use the Bernoulli equation: 1 2 ρu = p ρv θ 2 v θ = 2U sinθ Pressure at stagnation points: Pressure at cylinder boundary: v = 0 v r = 0 Assuming constant elevation Result: p = 1 2 ρu 2 1 4sin 2 θ 34
35 Evaluate pressures on cylinder wall Velocity profile: Pressure profile: v θ = 2U sinθ p = 1 2 ρu 2 1 4sin 2 θ stagn.pt stagn.pt stagn.pt 35
36 Evaluate pressures on cylinder wall Velocity profile: Pressure profile: v θ = 2U sinθ p = 1 2 ρu 2 1 4sin 2 θ No net resulting force!!! D'Alembert's Paradox stagn.pt stagn.pt stagn.pt 36
37 Pipeline near seabed How to calculate the flow around a pipeline near the seabed? y U y 0 37
38 Pipeline near seabed Mirror flow in seabed! Superpose flows: undisturbed cylinder mirrored cylinder y U y 0 y 0 38
39 Pipeline near seabed Mirror flow in seabed! Superpose flows: undisturbed cylinder mirrored cylinder y U Result is zero normal flow on seabed y 0 y 0 39
40 Pipeline near seabed Mirror flow in seabed! Superpose flows: undisturbed cylinder mirrored cylinder y U Result is zero normal flow on seabed y 0 Is flow exactly right? y 0 40
41 Pipeline near seabed Mirror flow in seabed! Superpose flows: undisturbed cylinder mirrored cylinder y U Result is zero normal flow on seabed y 0 Is flow exactly right? NO: interaction cylinders not modeled! y 0 41
42 Circulation Add circulation (or vortex flow element) to doublet in uniform flow Resulting velocity field: U r + Γ θ + 42
43 Circulation Add circulation (or vortex flow element) Resulting velocity field: 43
44 Circulation Add circulation (or vortex flow element) Resulting velocity field: 44
45 Circulation Add circulation (or vortex flow element) Resulting velocity field:
46 Circulation Now integration of pressure yields a net force perpendicular to the undisturbed flow direction: the lift force However: still no net force in the flow direction: no drag 46
47 Sources images All images are from the book Offshore Hydromechanics by Journée and Massie. 47
Offshore 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationMethod of Images
. - Marine Hdrodnamics, Spring 5 Lecture 11. - Marine Hdrodnamics Lecture 11 3.11 - Method of Images m Potential for single source: φ = ln + π m ( ) Potential for source near a wall: φ = m ln +( ) +ln
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 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 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 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 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 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 information(Jim You have a note for yourself here that reads Fill in full derivation, this is a sloppy treatment ).
Lecture. dministration Collect problem set. Distribute problem set due October 3, 004.. nd law of thermodynamics (Jim You have a note for yourself here that reads Fill in full derivation, this is a sloppy
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 information3.1 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 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 informationChapter 6 Equations of Continuity and Motion
Chapter 6 Equations of Continuity and Motion Derivation of 3-D Eq. conservation of mass Continuity Eq. conservation of momentum Eq. of motion Navier-Strokes Eq. 6.1 Continuity Equation Consider differential
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 informationxy 2 e 2z dx dy dz = 8 3 (1 e 4 ) = 2.62 mc. 12 x2 y 3 e 2z 2 m 2 m 2 m Figure P4.1: Cube of Problem 4.1.
Problem 4.1 A cube m on a side is located in the first octant in a Cartesian coordinate system, with one of its corners at the origin. Find the total charge contained in the cube if the charge density
More informationPressure in stationary and moving fluid Lab- Lab On- On Chip: Lecture 2
Pressure in stationary and moving fluid Lab-On-Chip: Lecture Lecture plan what is pressure e and how it s distributed in static fluid water pressure in engineering problems buoyancy y and archimedes law;
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 informationMath 3435 Homework Set 11 Solutions 10 Points. x= 1,, is in the disk of radius 1 centered at origin
Math 45 Homework et olutions Points. ( pts) The integral is, x + z y d = x + + z da 8 6 6 where is = x + z 8 x + z = 4 o, is the disk of radius centered on the origin. onverting to polar coordinates then
More information(You may need to make a sin / cos-type trigonometric substitution.) Solution.
MTHE 7 Problem Set Solutions. As a reminder, a torus with radii a and b is the surface of revolution of the circle (x b) + z = a in the xz-plane about the z-axis (a and b are positive real numbers, with
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 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 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 informationMAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.
MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant
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 informationPressure in stationary and moving fluid. Lab-On-Chip: Lecture 2
Pressure in stationary and moving fluid Lab-On-Chip: Lecture Fluid Statics No shearing stress.no relative movement between adjacent fluid particles, i.e. static or moving as a single block Pressure at
More informationStream Tube. When density do not depend explicitly on time then from continuity equation, we have V 2 V 1. δa 2. δa 1 PH6L24 1
Stream Tube A region of the moving fluid bounded on the all sides by streamlines is called a tube of flow or stream tube. As streamline does not intersect each other, no fluid enters or leaves across the
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 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 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 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 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 informationNotes 19 Gradient and Laplacian
ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can
More informationLecture 4. Differential Analysis of Fluid Flow Navier-Stockes equation
Lecture 4 Differential Analysis of Fluid Flow Navier-Stockes equation Newton second law and conservation of momentum & momentum-of-momentum A jet of fluid deflected by an object puts a force on the object.
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 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 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 informationWhere does Bernoulli's Equation come from?
Where does Bernoulli's Equation come from? Introduction By now, you have seen the following equation many times, using it to solve simple fluid problems. P ρ + v + gz = constant (along a streamline) This
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 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 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 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 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 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 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 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 information5.1 Fluid momentum equation Hydrostatics Archimedes theorem The vorticity equation... 42
Chapter 5 Euler s equation Contents 5.1 Fluid momentum equation........................ 39 5. Hydrostatics................................ 40 5.3 Archimedes theorem........................... 41 5.4 The
More informationThe Bernoulli Equation
The Bernoulli Equation The most used and the most abused equation in fluid mechanics. Newton s Second Law: F = ma In general, most real flows are 3-D, unsteady (x, y, z, t; r,θ, z, t; etc) Let consider
More informationLecture notes on fluid mechanics. Greifswald University 2007
Lecture notes on fluid mechanics Greifswald University 2007 Per Helander and Thomas Klinger Max-Planck-Institut für Plasmaphysik per.helander@ipp.mpg.de thomas.klinger@ipp.mpg.de 1 1 The equations of fluid
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 informationexcept assume the parachute has diameter of 3.5 meters and calculate how long it takes to stop. (Must solve differential equation)
Homework 5 Due date: Thursday, Mar. 3 hapter 7 Problems 1. 7.88. 7.9 except assume the parachute has diameter of 3.5 meters and calculate how long it takes to stop. (Must solve differential equation) 3.
More information0 = p. 2 x + 2 w. z +ν w
Solution (Elliptical pipe flow (a Using the Navier Stokes equations in three dimensional cartesian coordinates, given that u =, v = and w = w(x,y only, and assuming no body force, we are left with = p
More information!! +! 2!! +!"!! =!! +! 2!! +!"!! +!!"!"!"
Homework 4 Solutions 1. (15 points) Bernoulli s equation can be adapted for use in evaluating unsteady flow conditions, such as those encountered during start- up processes. For example, consider the large
More informationCreated by T. Madas LINE INTEGRALS. Created by T. Madas
LINE INTEGRALS LINE INTEGRALS IN 2 DIMENSIONAL CARTESIAN COORDINATES Question 1 Evaluate the integral ( x + 2y) dx, C where C is the path along the curve with equation y 2 = x + 1, from ( ) 0,1 to ( )
More informationChapter 5 Trigonometric Functions of Angles
Chapter 5 Trigonometric Functions of Angles Section 3 Points on Circles Using Sine and Cosine Signs Signs I Signs (+, +) I Signs II (+, +) I Signs II (, +) (+, +) I Signs II (, +) (+, +) I III Signs II
More informationMATHS 267 Answers to Stokes Practice Dr. Jones
MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the
More informationOffshore Hydromechanics Module 1
Offshore Hydromechanics Module 1 Dr. ir. Pepijn de Jong 1. Intro, Hydrostatics and Stability Introduction OE4630d1 Offshore Hydromechanics Module 1 dr.ir. Pepijn de Jong Assistant Prof. at Ship Hydromechanics
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 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 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 information1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem
1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem Objective of the Chapter: Formulation of the potential flow around the hull of a ship advancing and oscillationg in waves Results of the Chapter:
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 informationFinal Review of AerE 243 Class
Final Review of AeE 4 Class Content of Aeodynamics I I Chapte : Review of Multivaiable Calculus Chapte : Review of Vectos Chapte : Review of Fluid Mechanics Chapte 4: Consevation Equations Chapte 5: Simplifications
More informationFluid Mechanics Answer Key of Objective & Conventional Questions
019 MPROVEMENT Mechanical Engineering Fluid Mechanics Answer Key of Objective & Conventional Questions 1 Fluid Properties 1. (c). (b) 3. (c) 4. (576) 5. (3.61)(3.50 to 3.75) 6. (0.058)(0.05 to 0.06) 7.
More informationProf. Scalo Prof. Vlachos Prof. Ardekani Prof. Dabiri 08:30 09:20 A.M 10:30 11:20 A.M. 1:30 2:20 P.M. 3:30 4:20 P.M.
Page 1 Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.) Circle your lecture section (-1 point if not circled, or circled incorrectly): Prof. Scalo Prof. Vlachos
More informationIYGB Mathematical Methods 1
IYGB Mathematical Methods Practice Paper B Time: 3 hours Candidates may use any non programmable, non graphical calculator which does not have the capability of storing data or manipulating algebraic expressions
More informationPractice Final Solutions
Practice Final Solutions Math 1, Fall 17 Problem 1. Find a parameterization for the given curve, including bounds on the parameter t. Part a) The ellipse in R whose major axis has endpoints, ) and 6, )
More informationMATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS
MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,
More informationMAY THE FORCE BE WITH YOU, YOUNG JEDIS!!!
Final Exam Math 222 Spring 2011 May 11, 2011 Name: Recitation Instructor s Initials: You may not use any type of calculator whatsoever. (Cell phones off and away!) You are not allowed to have any other
More information6.641, Electromagnetic Fields, Forces, and Motion Prof. Markus Zahn Lecture 5: Method of Images Φ Φ Φ. x y z
6.641, Electromagnetic Fields, Forces, and Motion Prof. Markus Zahn Lecture 5: Method of Images I. Point Charge Above Ground Plane 1. Potential and Electric Field q 1 1 Φ p 4π x + y + z d x + y + z + d
More informationφ(r, θ, t) = a 2 U(t) cos θ. (7.1)
BioFluids Lectures 7-8: Slender Fish Added Mass for Lateral Motion At high Reynolds number, most of the effort required in swimming is pushing water out of the way, that is our energy goes in providing
More informationSections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.
MTH 34 Review for Exam 4 ections 16.1-16.8. 5 minutes. 5 to 1 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed. Review for Exam 4 (16.1) Line
More informationF1.9AB2 1. r 2 θ2 + sin 2 α. and. p θ = mr 2 θ. p2 θ. (d) In light of the information in part (c) above, we can express the Hamiltonian in the form
F1.9AB2 1 Question 1 (20 Marks) A cone of semi-angle α has its axis vertical and vertex downwards, as in Figure 1 (overleaf). A point mass m slides without friction on the inside of the cone under the
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 Fluid Dynamics: Waves in Fluids
Fundamentals of Fluid Dynamics: Waves in Fluids Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/ tzielins/ Institute
More informationPart A: 1 pts each, 10 pts total, no partial credit.
Part A: 1 pts each, 10 pts total, no partial credit. 1) (Correct: 1 pt/ Wrong: -3 pts). The sum of static, dynamic, and hydrostatic pressures is constant when flow is steady, irrotational, incompressible,
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 informationCE Final Exam. December 12, Name. Student I.D.
CE 100 - December 12, 2009 Name Student I.D. This exam is closed book. You are allowed three sheets of paper (8.5 x 11, both sides) of your own notes. You will be given three hours to complete four problems.
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 informationMATH 52 FINAL EXAM SOLUTIONS
MAH 5 FINAL EXAM OLUION. (a) ketch the region R of integration in the following double integral. x xe y5 dy dx R = {(x, y) x, x y }. (b) Express the region R as an x-simple region. R = {(x, y) y, x y }
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 information