Chapter 4: Fluid Kinematics


 Griffin Sims
 2 years ago
 Views:
Transcription
1
2 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 Lagrangian and Eulerian descriptions of fluid flow. Fundamental kinematic properties of fluid motion and deformation. Reynolds Transport Theorem. Meccanica dei Fluidi I 2
3 Lagrangian Description Lagrangian description of fluid flow tracks the position and velocity of individual particles. Based upon Newton's laws of motion. Difficult to use for practical flow analysis. Fluids are composed of billions of molecules. Interaction between molecules hard to describe/model. However, useful for specialized applications Sprays, particles, bubble dynamics, rarefied gases. Coupled EulerianLagrangian methods. Named after Italian mathematician Joseph Louis Lagrange ( ). Meccanica dei Fluidi I 3
4 Eulerian Description Eulerian description of fluid flow: a flow domain or control volume is defined by which fluid flows in and out. We define field variables which are functions of space and time. Pressure field, P = P(x,y,z,t) Velocity field, V V x, y, z, t Acceleration field,,,,,,,,,, V u x y z t i v x y z t j w x y z t k a a x, y, z, t,,,,,,,,, a a x y z t i a x y z t j a x y z t k x y z These (and other) field variables define the flow field. Well suited for formulation of initial boundaryvalue problems (PDE's). Named after Swiss mathematician Leonhard Euler ( ). Meccanica dei Fluidi I 4
5 Example: Coupled EulerianLagrangian Method Forensic analysis of Columbia accident: simulation of shuttle debris trajectory using Eulerian CFD for flow field and Lagrangian method for the debris. Meccanica dei Fluidi I 5
6 Acceleration Field Consider a fluid particle and Newton's second law, Fparticle mparticlea particle The acceleration of the particle is the time derivative of the particle's velocity: dvparticle aparticle dt However, particle velocity at a point is the same as the fluid velocity, V V x t, y t, z t particle particle particle particle To take the time derivative, chain rule must be used. a particle V dt V dx V dy V dz t dt x dt y dt z dt particle particle particle, t) Meccanica dei Fluidi I 6
7 Acceleration Field Since dxparticle dy particle dz particle u, v, w dt dt dt V V V V aparticle u v w t x y z In vector form, the acceleration can be written as dv V ax, y, z, t V. V dt t First term is called the local acceleration and is nonzero only for unsteady flows. Second term is called the advective (or convective) acceleration and accounts for the effect of the fluid particle moving to a new location in the flow, where the velocity is different (it can thus be nonzero even for steady flows). Meccanica dei Fluidi I 7
8 Material Derivative The total derivative operator d/dt is call the material derivative and is often given special notation, D/Dt. DV dv V V. Dt dt t Advective acceleration is nonlinear: source of many phenomena and primary challenge in solving fluid flow problems. Provides transformation ' between Lagrangian and Eulerian frames. Other names for the material derivative include: total, particle, Lagrangian, Eulerian, and substantial derivative. V Meccanica dei Fluidi I 8
9 Flow Visualization Flow visualization is the visual examination of flowfield features. Important for both physical experiments and numerical (CFD) solutions. Numerous methods Streamlines and streamtubes Pathlines Streaklines Timelines Refractive flow visualization techniques Surface flow visualization techniques Meccanica dei Fluidi I 9
10 Streamlines and streamtubes A streamline is a curve that is everywhere tangent to the instantaneous local velocity vector. Consider an infinitesimal arc length along a streamline: dr dxi dyj dzk dr By definition must be parallel to the local velocity vector V ui vj wk Geometric arguments result in the equation for a streamline dr dx dy dz V u v w Meccanica dei Fluidi I 10
11 Streamlines and streamtubes NASCAR surface pressure contours and streamlines Airplane surface pressure contours, volume streamlines, and surface streamlines Meccanica dei Fluidi I 11
12 Streamlines and streamtubes A streamtube consists of a bundle of individual streamlines. Since fluid cannot cross a streamline (by definition), fluid within a streamtube must remain there. Streamtubes are, obviously, instantaneous quantities and they may change significantly with time. In the converging portion of an incompressible flow field, the diameter of the streamtube must decrease as the velocity increases, so as to conserve mass. Meccanica dei Fluidi I 12
13 Pathlines A pathline is the actual path traveled by an individual fluid particle over some time period. Same as the fluid particle's material position vector,, x t y t z t particle particle particle Particle location at time t: x xstart Vdt start Particle Image Velocimetry (PIV) is a modern experimental technique to measure velocity field over a plane in the flow field. t t Meccanica dei Fluidi I 13
14 Streaklines A streakline is the locus of fluid particles that have passed sequentially through a prescribed point in the flow. Easy to generate in experiments: continuous introduction of dye (in a water flow) or smoke (in an airflow) from a point. Meccanica dei Fluidi I 14
15 Comparisons If the flow is steady, streamlines, pathlines and streaklines are identical. For unsteady flows, they can be very different. Streamlines provide an instantaneous picture of the flow field Pathlines and streaklines are flow patterns that have a time history associated with them. Meccanica dei Fluidi I 15
16 Timelines A timeline is a set of adjacent fluid particles that were marked at the same (earlier) instant in time. Experimentally, timelines can be generated using a hydrogen bubble wire: a line is marked and its movement/deformation is followed in time. Meccanica dei Fluidi I 16
17 Plots of Data A Profile plot indicates how the value of a scalar property varies along some desired direction in the flow field. A Vector plot is an array of arrows indicating the magnitude and direction of a vector property at an instant in time. A Contour plot shows curves of constant values of a scalar property (or magnitude for a vector property) at an instant in time. Meccanica dei Fluidi I 17
18 Kinematic Description In fluid mechanics (as in solid mechanics), an element may undergo four fundamental types of motion. a) Translation b) Rotation c) Linear strain d) Shear strain Because fluids are in constant motion, motion and deformation is best described in terms of rates a) velocity: rate of translation b) angular velocity: rate of rotation c) linear strain rate: rate of linear strain d) shear strain rate: rate of shear strain Meccanica dei Fluidi I 18
19 Rate of Translation and Rotation To be useful, these deformation rates must be expressed in terms of velocity and derivatives of velocity The rate of translation vector is described mathematically as the velocity vector. In Cartesian coordinates: V ui vj wk Rate of rotation (angular velocity) at a point is defined as the average rotation rate of two lines which are initially perpendicular and that intersect at that point. Meccanica dei Fluidi I 19
20 Rate of Rotation In 2D the average rotation angle of the fluid element about the point P is a w = (a a + a b )/2 The rate of rotation of the fluid element about P is 1 u w 1 v u i j k 2 z x 2 x y w In 3D the angular velocity vector is: 1 w v 1 u w 1 v u w i j k 2 y z 2 z x 2 x y Meccanica dei Fluidi I 20
21 Linear Strain Rate Linear Strain Rate is defined as the rate of increase in length per unit length. In Cartesian coordinates u xx, v yy, w zz x y z The rate of increase of volume of a fluid element per unit volume is the volumetric strain rate, in Cartesian coordinates: 1 DV u v w xx yy zz V Dt x y z (we are talking about a material volume, hence the D) Since the volume of a fluid element is constant for an incompressible flow, the volumetric strain rate must be zero. Meccanica dei Fluidi I 21
22 Shear Strain Rate Shear Strain Rate at a point is defined as half the rate of decrease of the angle between two initially perpendicular lines that intersect at a point. positive shear strain negative shear strain Meccanica dei Fluidi I 22
23 Shear Strain Rate The shear strain at point P 1 is xy =  d aab dt 2 Shear strain rate can be expressed in Cartesian coordinates as: 1 u v 1 1 xy, w u zx, v w yz 2 y x 2 x z 2 z y Meccanica dei Fluidi I 23
24 Shear Strain Rate We can combine linear strain rate and shear strain rate into one symmetric secondorder tensor called E: strainrate tensor. In Cartesian coordinates: u 1u v 1u w x 2 y x 2z x xx xy xz 1 v u v 1 v w ij yx yy yz 2 x y y 2 z y zx zy zz 1w u 1w v w 2 x z 2 y z z E Meccanica dei Fluidi I 24
25 State of Motion Particle moves from O to P in time D t Taylor series around O (for a small displacement) yields: The tensor u can be split into a symmetric part (E, the strain tensor) and an antisymmetric part W, the rotation tensor part as so that Meccanica dei Fluidi I 25
26 Shear Strain Rate Purpose of our discussion of fluid element kinematics: Better appreciation of the inherent complexity of fluid dynamics Mathematical sophistication required to fully describe fluid motion Strainrate tensor is important for numerous reasons. For example, Develop relationships between fluid stress and strain rate. Feature extraction and flow visualization in CFD simulations. Meccanica dei Fluidi I 26
27 Translation, Rotation, Linear Strain, Shear Strain, and Volumetric Strain Deformation of fluid elements (made visible with a tracer) during their compressible motion through a convergent channel; shear strain is more evident near the walls because of larger velocity gradients (a boundary layer is present there). Meccanica dei Fluidi I 27
28 Strain Rate Tensor Example: Visualization of trailingedge turbulent eddies for a hydrofoil with a beveled trailing edge Feature extraction method is based upon eigenanalysis of the strainrate tensor. Meccanica dei Fluidi I 28
29 Vorticity and Rotationality The vorticity vector is defined as the curl of the velocity vector z V Vorticity is equal to twice the angular velocity of a fluid particle: z 2w Cartesian coordinates w v u w v u z i j k y z z x x y Cylindrical coordinates 1 uz u ru ur uz u r z er e e r z z r r In regions where z = 0, the flow is called irrotational. Elsewhere, the flow is called rotational. z Meccanica dei Fluidi I 29
30 Vorticity and Rotationality Meccanica dei Fluidi I 30
31 Comparison of Two Circular Flows Special case: consider two flows with circular streamlines u 0, u wr r solidbody rotation 2 ru u wr 1 r 1 z e 0 e 2we r r r r z z z K line vortex ur 0, u r 1ru u r 1K z e 0e 0e r r r r z z z Meccanica dei Fluidi I 31
32 Circulation and vorticity Circulation: Meccanica dei Fluidi I 32
33 Circulation and vorticity Stokes theorem: If the flow is irrotational everywhere within the contour of integration C then G = 0. Meccanica dei Fluidi I 33
34 Reynolds Transport Theorem (RTT) A system is a quantity of matter of fixed identity. No mass can cross a system boundary. A control volume is a region in space chosen for study. Mass can cross a control surface. The fundamental conservation laws (conservation of mass, energy, and momentum) apply directly to systems. However, in most fluid mechanics problems, control volume analysis is preferred over system analysis (for the same reason that the Eulerian description is usually preferred over the Lagrangian description). Therefore, we need to transform the conservation laws from a system to a control volume. This is accomplished with the Reynolds transport theorem (RTT). Meccanica dei Fluidi I 34
35 Reynolds Transport Theorem (RTT) There is a direct analogy between the transformation from Lagrangian to Eulerian descriptions (for differential analysis using infinitesimally small fluid elements) and the transformation from systems to control volumes (for integral analysis using large, finite flow fields). Meccanica dei Fluidi I 35
36 Reynolds Transport Theorem (RTT) Material derivative (differential analysis): Db Dt b V. t RTT, moving or deformable CV (integral analysis): b V r = V  V cs Mass Momentum Energy Angular momentum B, Extensive properties m mv E b, Intensive properties 1 V e In Chaps 5 and 6, we will apply RTT to conservation of mass, energy, linear momentum, and angular momentum. r H V Meccanica dei Fluidi I 36
37 Reynolds Transport Theorem (RTT) RTT, fixed CV: db dt sys CV t b dvv CS. bv nda Time rate of change of the property B of the closed system is equal to (Term 1) + (Term 2) Term 1: time rate of change of B of the control volume Term 2: net flux of B out of the control volume by mass crossing the control surface Meccanica dei Fluidi I 37
M E 320 Professor John M. Cimbala Lecture 10
M E 320 Professor John M. Cimbala Lecture 10 Today, we will: Finish our example problem rates of motion and deformation of fluid particles Discuss the Reynolds Transport Theorem (RTT) Show how the RTT
More informationME 3560 Fluid Mechanics
ME 3560 Fluid Mechanics 1 4.1 The Velocity Field One of the most important parameters that need to be monitored when a fluid is flowing is the velocity. In general the flow parameters are described in
More informationChapter 6: Momentum Analysis of Flow Systems
Chapter 6: Momentum Analysis of Flow Systems Introduction Fluid flow problems can be analyzed using one of three basic approaches: differential, experimental, and integral (or control volume). In Chap.
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 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 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 informationGetting started: CFD notation
PDE of pth 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 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 informationChapter 9: Differential Analysis
91 Introduction 92 Conservation of Mass 93 The Stream Function 94 Conservation of Linear Momentum 95 Navier Stokes Equation 96 Differential Analysis Problems Recall 91 Introduction (1) Chap 5: Control
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 informationChapter 6: Momentum Analysis
61 Introduction 62Newton s Law and Conservation of Momentum 63 Choosing a Control Volume 64 Forces Acting on a Control Volume 65Linear Momentum Equation 66 Angular Momentum 67 The Second Law of
More informationCLASS Fourth Units (First part)
CLASS Fourth Units (First part) 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
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 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 informationAE301 Aerodynamics I UNIT B: Theory of Aerodynamics
AE301 Aerodynamics I UNIT B: Theory of Aerodynamics ROAD MAP... B1: Mathematics for Aerodynamics B: Flow Field Representations B3: Potential Flow Analysis B4: Applications of Potential Flow Analysis
More informationReview of fluid dynamics
Chapter 2 Review of fluid dynamics 2.1 Preliminaries ome basic concepts: A fluid is a substance that deforms continuously under stress. A Material olume is a tagged region that moves with the fluid. Hence
More informationKINEMATICS OF CONTINUA
KINEMATICS OF CONTINUA Introduction Deformation of a continuum Configurations of a continuum Deformation mapping Descriptions of motion Material time derivative Velocity and acceleration Transformation
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 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 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 NonConservative
More informationDetailed Outline, M E 521: Foundations of Fluid Mechanics I
Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Secondorder tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic
More 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 informationConvection Heat Transfer
Convection Heat Transfer Department of Chemical Eng., Isfahan University of Technology, Isfahan, Iran Seyed Gholamreza Etemad Winter 2013 Heat convection: Introduction Difference between the temperature
More informationIntroduction to Fluid Mechanics
Introduction to Fluid Mechanics TienTsan 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 informationLecture 2 Flow classifications and continuity
Lecture 2 Flow classifications and continuity Dr Tim Gough: t.gough@bradford.ac.uk General information 1 No tutorial week 3 3 rd October 2013 this Thursday. Attempt tutorial based on examples from today
More informationLECTURE NOTES  III. Prof. Dr. Atıl BULU
LECTURE NOTES  III «FLUID MECHANICS» Istanbul Technical University College of Civil Engineering Civil Engineering Department Hydraulics Division CHAPTER KINEMATICS OF FLUIDS.. FLUID IN MOTION Fluid motion
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 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 ElastoDynamics 6.2. Principles of Measurement 6.3. The PulseEcho
More information2. Conservation of Mass
2 Conservation of Mass The equation of mass conservation expresses a budget for the addition and removal of mass from a defined region of fluid Consider a fixed, nondeforming volume of fluid, V, called
More informationNumerical Heat and Mass Transfer
Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15Convective Heat Transfer Fausto Arpino f.arpino@unicas.it Introduction In conduction problems the convection entered the analysis
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 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 informationFLUID MECHANICS. Gaza. Chapter CHAPTER 44. Motion of Fluid Particles and Streams. Dr. Khalil Mahmoud ALASTAL
FLUID MECHANICS Gaza Chapter CHAPTER 44 Motion of Fluid Particles and Streams Dr. Khalil Mahmoud ALASTAL Objectives of this Chapter: Introduce concepts necessary to analyze fluids in motion. Identify differences
More informationROAD MAP... D0: Reynolds Transport Theorem D1: Conservation of Mass D2: Conservation of Momentum D3: Conservation of Energy
ES06 Fluid Mechani UNIT D: Flow Field Analysis ROAD MAP... D0: Reynolds Transport Theorem D1: Conservation of Mass D: Conservation of Momentum D3: Conservation of Energy ES06 Fluid Mechani Unit D0:
More informationFLUID MECHANICS FLUID KINEMATICS VELOCITY FIELD FLOW PATTERNS
www.getmyuni.com FLUID MECHANICS FLUID KINEMATICS Fluid Kinematics gives the geometry of fluid motion. It is a branch of fluid mechanics, which describes the fluid motion, and it s consequences without
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 xdirection [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit
More information2. FLUIDFLOW EQUATIONS SPRING 2019
2. FLUIDFLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Nonconservative differential equations 2.4 Nondimensionalisation Summary Examples 2.1 Introduction Fluid
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 informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIERSTOKES EQUATIONS Under the assumption of a Newtonian stressrateofstrain constitutive equation and a linear, thermally conductive medium,
More information12. Stresses and Strains
12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM  Formulation Classification of Problems Scalar Vector 1D T(x) u(x)
More information6. Basic basic equations I ( )
6. Basic basic equations I (4.24.4) Steady and uniform flows, streamline, streamtube One, two, and threedimensional flow Laminar and turbulent flow Reynolds number System and control volume Continuity
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 informationRotational & RigidBody Mechanics. Lectures 3+4
Rotational & RigidBody Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion  Definitions
More informationChapter 4 Continuity Equation and Reynolds Transport Theorem
Chapter 4 Continuity Equation and Reynolds Transport Theorem 4.1 Control Volume 4. The Continuity Equation for OneDimensional Steady Flow 4.3 The Continuity Equation for TwoDimensional Steady Flow 4.4
More informationDifferential relations for fluid flow
Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow
More informationFluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur
Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture  11 Kinematics of Fluid Part  II Good afternoon, I welcome you all to this session
More informationApplications in Fluid Mechanics
CHAPTER 8 Applications in Fluid 8.1 INTRODUCTION The general topic of fluid mechanics encompasses a wide range of problems of interest in engineering applications. The most basic definition of a fluid
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 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 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 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 Nonconservation form D Dt. V 0.(2.29) Conservation form ( V ) 0...(2.33) t 12/21/01
More informationEKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009)
EKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009) Dr. Mohamad Hekarl Uzirchhekarl@eng.usm.my School of Chemical Engineering Engineering Campus, Universiti
More informationCHAPTER 4. Basics of Fluid Dynamics
CHAPTER 4 Basics of Fluid Dynamics What is a fluid? A fluid is a substance that can flow, has no fixed shape, and offers little resistance to an external stress In a fluid the constituent particles (atoms,
More informationChapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature
Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte
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 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 informationA Study on Numerical Solution to the Incompressible NavierStokes Equation
A Study on Numerical Solution to the Incompressible NavierStokes Equation Zipeng Zhao May 2014 1 Introduction 1.1 Motivation One of the most important applications of finite differences lies in the field
More informationMultiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015
Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction
More informationLecture 3: 1. Lecture 3.
Lecture 3: 1 Lecture 3. Lecture 3: 2 Plan for today Summary of the key points of the last lecture. Review of vector and tensor products : the dot product (or inner product ) and the cross product (or vector
More informationChapter 1 Fluid Characteristics
Chapter 1 Fluid Characteristics 1.1 Introduction 1.1.1 Phases Solid increasing increasing spacing and intermolecular liquid latitude of cohesive Fluid gas (vapor) molecular force plasma motion 1.1.2 Fluidity
More informationTensor Visualization. CSC 7443: Scientific Information Visualization
Tensor Visualization Tensor data A tensor is a multivariate quantity Scalar is a tensor of rank zero s = s(x,y,z) Vector is a tensor of rank one v = (v x,v y,v z ) For a symmetric tensor of rank 2, its
More informationMultiple Integrals and Vector Calculus: Synopsis
Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors  recap of basic principles. Things which are (and are not) vectors. Differentiation and integration
More informationThe most common methods to identify velocity of flow are pathlines, streaklines and streamlines.
4 FLUID FLOW 4.1 Introduction Many civil engineering problems in fluid mechanics are concerned with fluids in motion. The distribution of potable water, the collection of domestic sewage and storm water,
More informationLecture 8. Stress Strain in Multidimension
Lecture 8. Stress Strain in Multidimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]
More informationKinematics of fluid motion
Chapter 4 Kinematics of fluid motion 4.1 Elementary flow patterns Recall the discussion of flow patterns in Chapter 1. The equations for particle paths in a threedimensional, steady fluid flow are dx
More informationClassical Mechanics. Luis Anchordoqui
1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body
More informationHigher Education. Mc Grauu FUNDAMENTALS AND APPLICATIONS SECOND EDITION
FLUID MECHANICS FUNDAMENTALS AND APPLICATIONS SECOND EDITION Mc Grauu Higher Education Boston Burr Ridge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur
More informationFluid Dynamics Problems M.Sc MathematicsSecond Semester Dr. Dinesh KhattarK.M.College
Fluid Dynamics Problems M.Sc MathematicsSecond Semester Dr. Dinesh KhattarK.M.College 1. (Example, p.74, Chorlton) At the point in an incompressible fluid having spherical polar coordinates,,, the velocity
More informationSeveral forms of the equations of motion
Chapter 6 Several forms of the equations of motion 6.1 The NavierStokes equations Under the assumption of a Newtonian stressrateofstrain constitutive equation and a linear, thermally conductive medium,
More informationFluid Mechanics. du dy
FLUID MECHANICS Technical English  I 1 th week Fluid Mechanics FLUID STATICS FLUID DYNAMICS Fluid Statics or Hydrostatics is the study of fluids at rest. The main equation required for this is Newton's
More informationAstrophysical Fluid Dynamics
Astrophysical Fluid Dynamics What is a Fluid? I. What is a fluid? I.1 The Fluid approximation: The fluid is an idealized concept in which the matter is described as a continuous medium with certain macroscopic
More informationChapter 7. Kinematics. 7.1 Tensor fields
Chapter 7 Kinematics 7.1 Tensor fields In fluid mechanics, the fluid flow is described in terms of vector fields or tensor fields such as velocity, stress, pressure, etc. It is important, at the outset,
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 informationCourse Syllabus: Continuum Mechanics  ME 212A
Course Syllabus: Continuum Mechanics  ME 212A Division Course Number Course Title Academic Semester Physical Science and Engineering Division ME 212A Continuum Mechanics Fall Academic Year 2017/2018 Semester
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 informationLecture 4. Differential Analysis of Fluid Flow NavierStockes equation
Lecture 4 Differential Analysis of Fluid Flow NavierStockes equation Newton second law and conservation of momentum & momentumofmomentum A jet of fluid deflected by an object puts a force on the object.
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 informationPEAT SEISMOLOGY Lecture 2: Continuum mechanics
PEAT8002  SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a
More informationModule 2: Governing Equations and Hypersonic Relations
Module 2: Governing Equations and Hypersonic Relations Lecture 2: Mass Conservation Equation 2.1 The Differential Equation for mass conservation: Let consider an infinitely small elemental control volume
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 informationMOMENTUM PRINCIPLE. Review: Last time, we derived the Reynolds Transport Theorem: Chapter 6. where B is any extensive property (proportional to mass),
Chapter 6 MOMENTUM PRINCIPLE Review: Last time, we derived the Reynolds Transport Theorem: where B is any extensive property (proportional to mass), and b is the corresponding intensive property (B / m
More informationAn Overview of Fluid Animation. Christopher Batty March 11, 2014
An Overview of Fluid Animation Christopher Batty March 11, 2014 What distinguishes fluids? What distinguishes fluids? No preferred shape. Always flows when force is applied. Deforms to fit its container.
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 informationCE 204 FLUID MECHANICS
CE 204 FLUID MECHANICS Onur AKAY Assistant Professor Okan University Department of Civil Engineering Akfırat Campus 34959 TuzlaIstanbul/TURKEY Phone: +902166771630 ext.1974 Fax: +902166771486 Email:
More informationVortex stretching in incompressible and compressible fluids
Vortex stretching in incompressible and compressible fluids Esteban G. Tabak, Fluid Dynamics II, Spring 00 1 Introduction The primitive form of the incompressible Euler equations is given by ( ) du P =
More informationCH.1. DESCRIPTION OF MOTION. Continuum Mechanics Course (MMC)
CH.1. DESCRIPTION OF MOTION Continuum Mechanics Course (MMC) Overview 1.1. Definition of the Continuous Medium 1.1.1. Concept of Continuum 1.1.. Continuous Medium or Continuum 1.. Equations of Motion 1..1
More informationArtificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.Ing. John Nassour J.
Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik Robot Dynamics Dr.Ing. John Nassour 25.1.218 J.Nassour 1 Introduction Dynamics concerns the motion of bodies Includes Kinematics
More informationLecture 3: The NavierStokes Equations: Topological aspects
Lecture 3: The NavierStokes Equations: Topological aspects September 9, 2015 1 Goal Topology is the branch of math wich studies shapechanging objects; objects which can transform one into another without
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 informationIntroduction to Fluid Dynamics
Introduction to Fluid Dynamics Roger K. Smith Skript  auf englisch! Umsonst im Internet http://www.meteo.physik.unimuenchen.de Wählen: Lehre Manuskripte Download User Name: meteo Password: download Aim
More information2.29 Numerical Fluid Mechanics Fall 2011 Lecture 5
.9 Numerical Fluid Mechanics Fall 011 Lecture 5 REVIEW Lecture 4 Roots of nonlinear equations: Open Methods Fixedpoint Iteration (General method or Picard Iteration), with examples Iteration rule: x g(
More informationCURRENT MATERIAL: Vector Calculus.
Math 275, section 002 (Ultman) Spring 2012 FINAL EXAM REVIEW The final exam will be held on Wednesday 9 May from 8:00 10:00am in our regular classroom. You will be allowed both sides of two 8.5 11 sheets
More informationNumerical Modelling in Geosciences. Lecture 6 Deformation
Numerical Modelling in Geosciences Lecture 6 Deformation Tensor Secondrank tensor stress ), strain ), strain rate ) Invariants quantities independent of the coordinate system):  First invariant trace:!!
More information1 Equations of motion
Part A Fluid Dynamics & Waves Draft date: 21 January 2014 1 1 1 Equations of motion 1.1 Introduction In this section we will derive the equations of motion for an inviscid fluid, that is a fluid with zero
More informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationOCN/ATM/ESS 587. The winddriven ocean circulation. Friction and stress. The Ekman layer, top and bottom. Ekman pumping, Ekman suction
OCN/ATM/ESS 587 The winddriven ocean circulation. Friction and stress The Ekman layer, top and bottom Ekman pumping, Ekman suction Westward intensification The winddriven ocean. The major ocean gyres
More information3D Elasticity Theory
3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.
More informationThe conservation equations
Chapter 5 The conservation equations 5.1 Leibniz rule for di erentiation of integrals 5.1.1 Di erentiation under the integral sign According to the fundamental theorem of calculus if f is a smooth function
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 information