FMIA. Fluid Mechanics and Its Applications 113 Series Editor: A. Thess. Moukalled Mangani Darwish. F. Moukalled L. Mangani M.

Size: px
Start display at page:

Download "FMIA. Fluid Mechanics and Its Applications 113 Series Editor: A. Thess. Moukalled Mangani Darwish. F. Moukalled L. Mangani M."

Transcription

1 FMIA F. Moukalled L. Mangani M. Darwish An Advanced Introduction with OpenFOAM and Matlab This textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in Computational Fluid Dynamics (CFD). Readers will discover a thorough explanation of the FVM numerics and algorithms used in the simulation of incompressible and compressible fluid flows, along with a detailed examination of the components needed for the development of a collocated unstructured pressure-based CFD solver. Two particular CFD codes are explored. The first is ufvm, a three-dimensional unstructured pressure-based finite volume academic CFD code, implemented within Matlab. The second is OpenFOAM, an open source framework used in the development of a range of CFD programs for the simulation of industrial scale flow problems. Moukalled Mangani Darwish Fluid Mechanics and Its Applications 113 Series Editor: A. Thess The Finite Volume Method in Computational Fluid Dynamics With over 220 figures, numerous examples and more than one hundred exercises on FVM numerics, programming, and applications, this textbook is suitable for use in an introductory course on the FVM, in an advanced course on CFD algorithms, and as a reference for CFD programmers and researchers. Fluid Mechanics and Its Applications F. Moukalled L. Mangani M. Darwish The Finite Volume Method in Computational Fluid Dynamics The Finite Volume Method in Computational Fluid Dynamics An Advanced Introduction with OpenFOAM and Matlab Engineering ISBN Mathematical Review Chapter 02

2 Vector Operations z v = ui + vj+ wk v = u v w v T = u v w v = u 1 w 1 = v = u + v + w + = u w 1 + x u v w y sv = s( ui + vj+ wk) = sui + svj+ swk = su sv sw 2

3 Dot Product = cos(, ) i i = j j = k k = 1 i j = i k = j i = j k = k i = k j = 0 α = cos( α) = ( u 1 i + j+ w 1 k) ( i + j+ k) = u w 1 = cos α ( ) vector magnitude unit vector α v = v v e v = v v 3

4 Cross Product v 3 = = sin(, ) = ( u 1 i + j+ w 1 k ) ( i + j+ k ) = u 1 i i + u 1 i j+ u 1 i k + j i + j j+ j k +w 1 k i + w 1 k j+ w 1 k k ( ) ( ) i ( ) + w 1 0 = u u 1 k + u 1 j + k +w 1 j+ w 1 i determinant notation = v 3 = α i j k u 1 w 1 = area v 3 = sin α i i = j j = k k = 0 i j = k = j i j k = i = k j k i = j = i k w 1 w 1 u 1 u 1 ( ) 4

5 Scalar Triple Product volume ( ) v 3 u 1 w 1 ( ) = v 3 u 3 v 3 w 3 β v 3 α The scalar triple product represent the volume defined by the paralleled formed by the base vector 5

6 Gradients Nabla operator Gradient = i x + j y + k z s C e l l directional derivative s = s( x, y,z) = ds dl = s e l φ = i φ x + j φ y + k φ i k j x y Directional Derivative ds dl = s e = s cos s,e l ( l ) 6

7 Divergence divergence v = x x + y y + z divergence of the gradient of a scalar is the Laplacian of the scalar curl = 0 curl ( s) = 2 s = 2 s x + 2 s 2 y + 2 s 2 2 φ = ( φ) = 2 φ == 2 v x x v y y v z 2 v = x i + y j+ k = i j k x y u v w ( ui + vj+ wk) = y i + u x j+ x u y k divergence = 0 7

8 Tensors τ yy τ yx τ = τ xx τ xy τ xz τ yx τ yy τ yz τ zx τ zy τ zz τ xx τ xy τ xz τ yz τ zy τ zx τ zz τ = iiτ xx + ijτ xy + ikτ xz + jiτ yx + jjτ yy + jkτ yz + kiτ zx + kjτ zy + kkτ zz 8

9 Other Tensors { vv} = ( ui + vj+ wk) ( ui + vj+ wk) = iiuu + ijuv + ikuw + jivu + jjvv + jkvw + kiwu + kjwv + kkww { vv} = uu uv uw vu vv vw wu wv ww Velocity gradient { v} = x i + y j+ k = ii u + ij + ik x x x + ji u + jj + jk y y y + ki u + kj + kk ( ui + vj+ wk) 9 { v} = u x u y u x y x y

10 Tensor Operations Σ = σ + τ = σ xx + τ xx σ xy + τ xy σ xz + τ xz σ yx + τ yx σ yy + τ yy σ yz + τ yz σ zx + τ zx σ zy + τ zy σ zz + τ zz { sτ } = sτ xx sτ xy sτ xz sτ yx sτ yy sτ yz sτ zx sτ zy sτ zz τ v = iiτ + ijτ + ikτ + jiτ + xx xy xz yx ui + vj+ wk jjτ yy + jkτ yz + kiτ zx + kjτ zy + kkτ zz ( ) τ v = τ xx τ xy τ xz τ yx τ yy τ yz τ zx τ zy τ zz u v w 10 = τ xx u + τ xy v + τ xz w τ yx u + τ yy v + τ yz w τ zx u + τ zy v + τ zz w

11 Tensor Operations divergence of a tensor τ = τ xx x + τ yx y + τ zx i + τ xy x + τ yy y + τ zy j+ τ xz x + τ yz y + τ zz k double dot product of two tensors ( τ : v) = ii u iiτ xx + ijτ xy + ikτ xz + x jiτ yx + jjτ yy + jkτ yz + : ji u kiτ zx + kjτ zy + kkτ y zz ki u u ( τ : v) = τ xx x + τ xy τ yy y + τ yz u y + τ xz + τ zx u + τ yx 11 x + x + τ zy + ij x + jj y + kj y + τ zz + jk y + + kk + ik x +

12 Tensor Operations Deformation tensor D( v) = 1 2 Spin tensor ( v + vt ) = ) * + 1 # 2 $ % 1 2 # $ % x x x y + y x x + z x & ' ( & ' ( 1 # 2 $ % 1 # 2 $ % y x + x y y y y + z y & ' ( 1 2 # $ % 1 # 2 $ % z x + x z y + y & z ' ( symmetric part of &, ' (... &. ' ( v S ( v) = v D( v) = 1 2 ( v vt ) Skew-symmetric part of v trace( v) = v 12

13 Gradients Theorem for Line Integrals C z r(t) C s dr ( ) s( r( a) ) = s r( b) r(a) r(b) y x The gradient theorem for line integrals relates a line integral to the values of a function at its endpoints 13

14 Green s Theorem Convert a volume integral to a surface integral and reduce the order of equations by one ( udx + vdy) = x u! C R y dxdy C dr = dxi + dyj v = ui + vj ds = dxdyk! v dr = v ds C R R dr ds 14

15 Stoke s Theorem v ds =! v dr S C z S ds v y x C 15

16 Divergence Theorem ( v)dv =! v nds V S z ds S ds = nds y x v V 16

17 Leibniz Integral Rule d dt b( t) φ( x,t)dx = a( t) b( t) φ t dx a( t)!" # $# Term I ( ) b + φ b( t),t!#"# $ t Term II ( ) a φ a( t),t! #" ## $ t Term III φ φ( x,t + Δt) Term I φ( x,t) Term III Term II a( t) a( t + Δt) b( t) b( t + Δt) t 17

Transient, Source Terms and Relaxation

Transient, Source Terms and Relaxation F. Moukalled L. Mangani M. Darwish An Advanced Introduction with OpenFOAM and Matlab This textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in Computational

More information

The Finite Volume Mesh

The Finite Volume Mesh FMIA F Moukalled L Mangani M Darwish An Advanced Introduction with OpenFOAM and Matlab This textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in Computational

More information

Solving the Navier-Stokes Equations

Solving the Navier-Stokes Equations FMIA F. Moukalled L. Mangani M. Darwish An Advanced Introduction with OpenFOAM and Matlab This textbook explores both the theoretical oundation o the Finite Volume Method (FVM) and its applications in

More information

Getting started: CFD notation

Getting started: CFD notation PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =

More information

The Finite Volume Method in

The Finite Volume Method in FMIA F Moukalled L Mangani M Darwish An Advanced Introduction with OpenFOAM and Matlab This textbook explores both the theoretical oundation o the Finite Volume Method (FVM) and its applications in omputational

More information

AE/ME 339. K. M. Isaac Professor of Aerospace Engineering. 12/21/01 topic7_ns_equations 1

AE/ME 339. K. M. Isaac Professor of Aerospace Engineering. 12/21/01 topic7_ns_equations 1 AE/ME 339 Professor of Aerospace Engineering 12/21/01 topic7_ns_equations 1 Continuity equation Governing equation summary Non-conservation form D Dt. V 0.(2.29) Conservation form ( V ) 0...(2.33) t 12/21/01

More information

AE/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) 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 information

In this section, mathematical description of the motion of fluid elements moving in a flow field is

In 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 information

M E 320 Professor John M. Cimbala Lecture 10

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 information

Chapter 5. The Differential Forms of the Fundamental Laws

Chapter 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 information

Chapter 2 Review of Vector Calculus

Chapter 2 Review of Vector Calculus Chapter Review of Vector Calculus Abstract This chapter sets the ground for the derivation of the conservation equations by providing a brief review of the continuum mechanics tools needed for that purpose

More information

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

Numerical Modelling in Geosciences. Lecture 6 Deformation

Numerical Modelling in Geosciences. Lecture 6 Deformation Numerical Modelling in Geosciences Lecture 6 Deformation Tensor Second-rank tensor stress ), strain ), strain rate ) Invariants quantities independent of the coordinate system): - First invariant trace:!!

More information

PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT 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 information

THE I Establiifrad June, 1893

THE I Establiifrad June, 1893 89 : 8 Y Y 2 96 6 - - : - 2 q - 26 6 - - q 2 2 2 4 6 4«4 ' V () 8 () 6 64-4 '2" () 6 ( ) * 'V ( 4 ) 94-4 q ( / ) K ( x- 6% j 9*V 2'%" 222 27 q - - K 79-29 - K x 2 2 j - -% K 4% 2% 6% ' K - 2 47 x - - j

More information

Cartesian Tensors. e 2. e 1. General vector (formal definition to follow) denoted by components

Cartesian Tensors. e 2. e 1. General vector (formal definition to follow) denoted by components Cartesian Tensors Reference: Jeffreys Cartesian Tensors 1 Coordinates and Vectors z x 3 e 3 y x 2 e 2 e 1 x x 1 Coordinates x i, i 123,, Unit vectors: e i, i 123,, General vector (formal definition to

More information

Chapter 4: Fluid Kinematics

Chapter 4: Fluid Kinematics Overview Fluid kinematics deals with the motion of fluids without considering the forces and moments which create the motion. Items discussed in this Chapter. Material derivative and its relationship to

More information

EKC314: 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) EKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009) Dr. Mohamad Hekarl Uzir-chhekarl@eng.usm.my School of Chemical Engineering Engineering Campus, Universiti

More information

Closed-Form Solution Of Absolute Orientation Using Unit Quaternions

Closed-Form Solution Of Absolute Orientation Using Unit Quaternions Closed-Form Solution Of Absolute Orientation Using Unit Berthold K. P. Horn Department of Computer and Information Sciences November 11, 2004 Outline 1 Introduction 2 3 The Problem Given: two sets of corresponding

More information

VECTORS, TENSORS AND INDEX NOTATION

VECTORS, TENSORS AND INDEX NOTATION VECTORS, TENSORS AND INDEX NOTATION Enrico Nobile Dipartimento di Ingegneria e Architettura Università degli Studi di Trieste, 34127 TRIESTE March 5, 2018 Vectors & Tensors, E. Nobile March 5, 2018 1 /

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium

More information

7a3 2. (c) πa 3 (d) πa 3 (e) πa3

7a3 2. (c) πa 3 (d) πa 3 (e) πa3 1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin

More information

Course 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations

Course 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.

More information

A Study on Numerical Solution to the Incompressible Navier-Stokes Equation

A Study on Numerical Solution to the Incompressible Navier-Stokes Equation A Study on Numerical Solution to the Incompressible Navier-Stokes Equation Zipeng Zhao May 2014 1 Introduction 1.1 Motivation One of the most important applications of finite differences lies in the field

More information

Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI

Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI Lecture 8 Analyzing the diffusion weighted signal Room CSB 272 this week! Please install AFNI http://afni.nimh.nih.gov/afni/ Next lecture, DTI For this lecture, think in terms of a single voxel We re still

More information

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort - 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [

More information

Chapter 9: Differential Analysis

Chapter 9: Differential Analysis 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control

More information

CHAPTER 7 DIV, GRAD, AND CURL

CHAPTER 7 DIV, GRAD, AND CURL CHAPTER 7 DIV, GRAD, AND CURL 1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: (1 ϕ = ( ϕ, ϕ,, ϕ x 1 x 2 x n

More information

Stress, Strain, Mohr s Circle

Stress, Strain, Mohr s Circle Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected

More information

Candidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request.

Candidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request. UNIVERSITY OF EAST ANGLIA School of Mathematics Spring Semester Examination 2004 FLUID DYNAMICS Time allowed: 3 hours Attempt Question 1 and FOUR other questions. Candidates must show on each answer book

More information

Rigid body simulation. Once we consider an object with spatial extent, particle system simulation is no longer sufficient

Rigid body simulation. Once we consider an object with spatial extent, particle system simulation is no longer sufficient Rigid body dynamics Rigid body simulation Once we consider an object with spatial extent, particle system simulation is no longer sufficient Rigid body simulation Unconstrained system no contact Constrained

More information

The Discretization Process

The Discretization Process FMIA F Moukalled L Mangan M Darwsh An Advanced Introducton wth OpenFOAM and Matlab Ths textbook explores both the theoretcal foundaton of the Fnte Volume Method (FVM) and ts applcatons n Computatonal Flud

More information

Multiple Integrals and Vector Calculus: Synopsis

Multiple 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 information

Math review. Math review

Math review. Math review Math review 1 Math review 3 1 series approximations 3 Taylor s Theorem 3 Binomial approximation 3 sin(x), for x in radians and x close to zero 4 cos(x), for x in radians and x close to zero 5 2 some geometry

More information

Chapter 9: Differential Analysis of Fluid Flow

Chapter 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 information

MULTIVARIABLE CALCULUS

MULTIVARIABLE CALCULUS MULTIVARIABLE CALCULUS JOHN QUIGG Contents 13.1 Three-Dimensional Coordinate Systems 2 13.2 Vectors 3 13.3 The Dot Product 5 13.4 The Cross Product 6 13.5 Equations of Lines and Planes 7 13.6 Cylinders

More information

By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ.

By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ. Mohr s Circle By drawing Mohr s circle, the stress transformation in -D can be done graphically. σ = σ x + σ y τ = σ x σ y + σ x σ y cos θ + τ xy sin θ, 1 sin θ + τ xy cos θ. Note that the angle of rotation,

More information

CH.3. COMPATIBILITY EQUATIONS. Multimedia Course on Continuum Mechanics

CH.3. COMPATIBILITY EQUATIONS. Multimedia Course on Continuum Mechanics CH.3. COMPATIBILITY EQUATIONS Multimedia Course on Continuum Mechanics Overview Introduction Lecture 1 Compatibility Conditions Lecture Compatibility Equations of a Potential Vector Field Lecture 3 Compatibility

More information

PEAT SEISMOLOGY Lecture 12: Earthquake source mechanisms and radiation patterns II

PEAT SEISMOLOGY Lecture 12: Earthquake source mechanisms and radiation patterns II PEAT8002 - SEISMOLOGY Lecture 12: Earthquake source mechanisms and radiation patterns II Nick Rawlinson Research School of Earth Sciences Australian National University Waveform modelling P-wave first-motions

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y -» $ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - $ { Q» / X x»»- 3 q $ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q

More information

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t) IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common

More information

Properties of the stress tensor

Properties of the stress tensor Appendix C Properties of the stress tensor Some of the basic properties of the stress tensor and traction vector are reviewed in the following. C.1 The traction vector Let us assume that the state of stress

More information

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS I Main Topics A Why deal with tensors? B Order of scalars, vectors, and tensors C Linear transformation of scalars and vectors (and tensors) II Why

More information

Chapter 2: Basic Governing Equations

Chapter 2: Basic Governing Equations -1 Reynolds Transport Theorem (RTT) - Continuity Equation -3 The Linear Momentum Equation -4 The First Law of Thermodynamics -5 General Equation in Conservative Form -6 General Equation in Non-Conservative

More information

Rigid body dynamics. Basilio Bona. DAUIN - Politecnico di Torino. October 2013

Rigid body dynamics. Basilio Bona. DAUIN - Politecnico di Torino. October 2013 Rigid body dynamics Basilio Bona DAUIN - Politecnico di Torino October 2013 Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October 2013 1 / 16 Multiple point-mass bodies Each mass is

More information

Continuum Mechanics. Continuum Mechanics and Constitutive Equations

Continuum Mechanics. Continuum Mechanics and Constitutive Equations Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform

More information

Module 2: Governing Equations and Hypersonic Relations

Module 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 information

Math Review: Vectors and Tensors for Rheological Applications

Math Review: Vectors and Tensors for Rheological Applications Math Review: Vectors and Tensors for Rheological Applications Presented by Randy H. Ewoldt University of Illinois at Urbana-Champaign U. of Minnesota Rheological Measurements Short Course June 2016 MathReview-1

More information

Name: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8

Name: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8 Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is

More information

A Simple Turbulence Closure Model. Atmospheric Sciences 6150

A Simple Turbulence Closure Model. Atmospheric Sciences 6150 A Simple Turbulence Closure Model Atmospheric Sciences 6150 1 Cartesian Tensor Notation Reynolds decomposition of velocity: V = V + v V = U i + u i Mean velocity: V = Ui + V j + W k =(U, V, W ) U i =(U

More information

n i,j+1/2 q i,j * qi+1,j * S i+1/2,j

n i,j+1/2 q i,j * qi+1,j * S i+1/2,j Helsinki University of Technology CFD-group/ The Laboratory of Applied Thermodynamics MEMO No CFD/TERMO-5-97 DATE: December 9,997 TITLE A comparison of complete vs. simplied viscous terms in boundary layer

More information

INTEGRALSATSER VEKTORANALYS. (indexräkning) Kursvecka 4. and CARTESIAN TENSORS. Kapitel 8 9. Sidor NABLA OPERATOR,

INTEGRALSATSER VEKTORANALYS. (indexräkning) Kursvecka 4. and CARTESIAN TENSORS. Kapitel 8 9. Sidor NABLA OPERATOR, VEKTORANALYS Kursvecka 4 NABLA OPERATOR, INTEGRALSATSER and CARTESIAN TENSORS (indexräkning) Kapitel 8 9 Sidor 83 98 TARGET PROBLEM In the plasma there are many particles (10 19, 10 20 per m 3 ), strong

More information

A Simple Turbulence Closure Model

A Simple Turbulence Closure Model A Simple Turbulence Closure Model Atmospheric Sciences 6150 1 Cartesian Tensor Notation Reynolds decomposition of velocity: Mean velocity: Turbulent velocity: Gradient operator: Advection operator: V =

More information

Governing Equations of Fluid Dynamics

Governing Equations of Fluid Dynamics Chapter 3 Governing Equations of Fluid Dynamics The starting point of any numerical simulation are the governing equations of the physics of the problem to be solved. In this chapter, we first present

More information

Chapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)

Chapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a) Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J.. 7. 8 5. 6 6. 7. y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5

More information

Summary for Vector Calculus and Complex Calculus (Math 321) By Lei Li

Summary for Vector Calculus and Complex Calculus (Math 321) By Lei Li Summary for Vector alculus and omplex alculus (Math 321) By Lei Li 1 Vector alculus 1.1 Parametrization urves, surfaces, or volumes can be parametrized. Below, I ll talk about 3D case. Suppose we use e

More information

JJMIE Jordan Journal of Mechanical and Industrial Engineering

JJMIE Jordan Journal of Mechanical and Industrial Engineering JJMIE Jordan Journal of Mechanical and Industrial Engineering Volume Number, June.6 ISSN 995-6665 Pages 99-4 Computational Fluid Dynamics of Plate Fin and Circular Pin Fin Heat Sins Mohammad Saraireh *

More information

Inner Product Spaces 5.2 Inner product spaces

Inner Product Spaces 5.2 Inner product spaces Inner Product Spaces 5.2 Inner product spaces November 15 Goals Concept of length, distance, and angle in R 2 or R n is extended to abstract vector spaces V. Sucn a vector space will be called an Inner

More information

Lecture 3: Vectors. Any set of numbers that transform under a rotation the same way that a point in space does is called a vector.

Lecture 3: Vectors. Any set of numbers that transform under a rotation the same way that a point in space does is called a vector. Lecture 3: Vectors Any set of numbers that transform under a rotation the same way that a point in space does is called a vector i.e., A = λ A i ij j j In earlier courses, you may have learned that a vector

More information

Numerical Modelling in Geosciences. Lecture 1 Introduction and basic mathematics for PDEs

Numerical Modelling in Geosciences. Lecture 1 Introduction and basic mathematics for PDEs Numerical Modelling in Geosciences Lecture 1 Introduction and basic mathematics for PDEs Useful information Slides and exercises: download at my homepage: http://www.geoscienze.unipd.it/users/faccenda-manuele

More information

Vector Calculus, Maths II

Vector Calculus, Maths II Section A Vector Calculus, Maths II REVISION (VECTORS) 1. Position vector of a point P(x, y, z) is given as + y and its magnitude by 2. The scalar components of a vector are its direction ratios, and represent

More information

Lesson Rigid Body Dynamics

Lesson Rigid Body Dynamics Lesson 8 Rigid Body Dynamics Lesson 8 Outline Problem definition and motivations Dynamics of rigid bodies The equation of unconstrained motion (ODE) User and time control Demos / tools / libs Rigid Body

More information

Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles.

Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles. » ~ $ ) 7 x X ) / ( 8 2 X 39 ««x» ««! «! / x? \» «({? «» q «(? (?? x! «? 8? ( z x x q? ) «q q q ) x z x 69 7( X X ( 3»«! ( ~«x ««x ) (» «8 4 X «4 «4 «8 X «x «(» X) ()»» «X «97 X X X 4 ( 86) x) ( ) z z

More information

Math 11 Fall 2018 Practice Final Exam

Math 11 Fall 2018 Practice Final Exam Math 11 Fall 218 Practice Final Exam Disclaimer: This practice exam should give you an idea of the sort of questions we may ask on the actual exam. Since the practice exam (like the real exam) is not long

More information

Study on Numerical Simulation Method of Gust Response in Time Domain Jun-Li WANG

Study on Numerical Simulation Method of Gust Response in Time Domain Jun-Li WANG International Conference on Mechanics and Civil Engineering (ICMCE 4) Study on Numerical Simulation Method of Gust Response in Time Domain Jun-Li WANG School of Mechanical Engineering, Shaanxi University

More information

ACCEPTS HUGE FLORAL KEY TO LOWELL. Mrs, Walter Laid to Rest Yesterday

ACCEPTS HUGE FLORAL KEY TO LOWELL. Mrs, Walter Laid to Rest Yesterday $ j < < < > XXX Y 928 23 Y Y 4% Y 6 -- Q 5 9 2 5 Z 48 25 )»-- [ Y Y Y & 4 j q - Y & Y 7 - -- - j \ -2 -- j j -2 - - - - [ - - / - ) ) - - / j Y 72 - ) 85 88 - / X - j ) \ 7 9 Y Y 2 3» - ««> Y 2 5 35 Y

More information

M273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3

M273Q Multivariable Calculus Spring 2017 Review Problems for Exam 3 M7Q Multivariable alculus Spring 7 Review Problems for Exam Exam covers material from Sections 5.-5.4 and 6.-6. and 7.. As you prepare, note well that the Fall 6 Exam posted online did not cover exactly

More information

Solving Einstein s Equation Numerically III

Solving Einstein s Equation Numerically III Solving Einstein s Equation Numerically III Lee Lindblom Center for Astrophysics and Space Sciences University of California at San Diego Mathematical Sciences Center Lecture Series Tsinghua University

More information

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16. CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo

More information

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives. PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x

More information

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics 3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)

More information

12. Stresses and Strains

12. 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 1-D T(x) u(x)

More information

Multivariable Calculus

Multivariable Calculus Multivariable alculus Jaron Kent-Dobias May 17, 2011 1 Lines in Space By space, we mean R 3. First, conventions. Always draw right-handed axes. You can define a L line precisely in 3-space with 2 points,

More information

Continuum mechanism: Stress and strain

Continuum mechanism: Stress and strain Continuum mechanics deals with the relation between forces (stress, σ) and deformation (strain, ε), or deformation rate (strain rate, ε). Solid materials, rigid, usually deform elastically, that is the

More information

Uniformity of the Universe

Uniformity of the Universe Outline Universe is homogenous and isotropic Spacetime metrics Friedmann-Walker-Robertson metric Number of numbers needed to specify a physical quantity. Energy-momentum tensor Energy-momentum tensor of

More information

Lagrange Multipliers

Lagrange Multipliers Optimization with Constraints As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each

More information

CH.2. DEFORMATION AND STRAIN. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

CH.2. DEFORMATION AND STRAIN. Continuum Mechanics Course (MMC) - ETSECCPB - UPC CH.. DEFORMATION AND STRAIN Continuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Introduction Deformation Gradient Tensor Material Deformation Gradient Tensor Inverse (Spatial) Deformation Gradient

More information

Multiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015

Multiple 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 information

MATH 332: Vector Analysis Summer 2005 Homework

MATH 332: Vector Analysis Summer 2005 Homework MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,

More information

Two Posts to Fill On School Board

Two Posts to Fill On School Board Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83

More information

Topics. CS Advanced Computer Graphics. Differential Geometry Basics. James F. O Brien. Vector and Tensor Fields.

Topics. CS Advanced Computer Graphics. Differential Geometry Basics. James F. O Brien. Vector and Tensor Fields. CS 94-3 Advanced Computer Graphics Differential Geometry Basics James F. O Brien Associate Professor U.C. Berkeley Topics Vector and Tensor Fields Divergence, curl, etc. Parametric Curves Tangents, curvature,

More information

i.e. the conservation of mass, the conservation of linear momentum, the conservation of energy.

i.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 information

Group, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S,

Group, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, Group, Rings, and Fields Rahul Pandharipande I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ

More information

OCN/ATM/ESS 587. The wind-driven ocean circulation. Friction and stress. The Ekman layer, top and bottom. Ekman pumping, Ekman suction

OCN/ATM/ESS 587. The wind-driven ocean circulation. Friction and stress. The Ekman layer, top and bottom. Ekman pumping, Ekman suction OCN/ATM/ESS 587 The wind-driven ocean circulation. Friction and stress The Ekman layer, top and bottom Ekman pumping, Ekman suction Westward intensification The wind-driven ocean. The major ocean gyres

More information

Exercises for Multivariable Differential Calculus XM521

Exercises for Multivariable Differential Calculus XM521 This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done

More information

3D Elasticity Theory

3D 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 information

Chapter 6: Vector Analysis

Chapter 6: Vector Analysis Chapter 6: Vector Analysis We use derivatives and various products of vectors in all areas of physics. For example, Newton s 2nd law is F = m d2 r. In electricity dt 2 and magnetism, we need surface and

More information

A Brief Revision of Vector Calculus and Maxwell s Equations

A Brief Revision of Vector Calculus and Maxwell s Equations A Brief Revision of Vector Calculus and Maxwell s Equations Debapratim Ghosh Electronic Systems Group Department of Electrical Engineering Indian Institute of Technology Bombay e-mail: dghosh@ee.iitb.ac.in

More information

16.3 Conservative Vector Fields

16.3 Conservative Vector Fields 16.3 Conservative Vector Fields Lukas Geyer Montana State University M273, Fall 2011 Lukas Geyer (MSU) 16.3 Conservative Vector Fields M273, Fall 2011 1 / 23 Fundamental Theorem for Conservative Vector

More information

Divergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem

Divergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem Divergence Theorem 17.3 11 December 213 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:

More information

Mechanics of materials Lecture 4 Strain and deformation

Mechanics of materials Lecture 4 Strain and deformation Mechanics of materials Lecture 4 Strain and deformation Reijo Kouhia Tampere University of Technology Department of Mechanical Engineering and Industrial Design Wednesday 3 rd February, 206 of a continuum

More information

Math 212. Practice Problems for the Midterm 3

Math 212. Practice Problems for the Midterm 3 Math 1 Practice Problems for the Midterm 3 Ivan Matic 1. Evaluate the surface integral x + y + z)ds, where is the part of the paraboloid z 7 x y that lies above the xy-plane.. Let γ be the curve in the

More information

f. D that is, F dr = c c = [2"' (-a sin t)( -a sin t) + (a cos t)(a cost) dt = f2"' dt = 2

f. D that is, F dr = c c = [2' (-a sin t)( -a sin t) + (a cos t)(a cost) dt = f2' dt = 2 SECTION 16.4 GREEN'S THEOREM 1089 X with center the origin and radius a, where a is chosen to be small enough that C' lies inside C. (See Figure 11.) Let be the region bounded by C and C'. Then its positively

More information

INHOMOGENEOUS, NON-LINEAR AND ANISOTROPIC SYSTEMS WITH RANDOM MAGNETISATION MAIN DIRECTIONS

INHOMOGENEOUS, NON-LINEAR AND ANISOTROPIC SYSTEMS WITH RANDOM MAGNETISATION MAIN DIRECTIONS dx dt DIFFERENTIAL EQUATIONS AND CONTROL PROCESSES N 1 2002 Electronic Journal reg. N P23275 at 07.03.97 http://www.neva.ru/journal e-mail: diff@osipenko.stu.neva.ru Applications to physics electrotechnics

More information

α 2 )(k x ũ(t, η) + k z W (t, η)

α 2 )(k x ũ(t, η) + k z W (t, η) 1 3D Poiseuille Flow Over the next two lectures we will be going over the stabilization of the 3-D Poiseuille flow. For those of you who havent had fluids, that means we have a channel of fluid that is

More information

Review Sheet for the Final

Review Sheet for the Final Review Sheet for the Final Math 6-4 4 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And the absence

More information

Sections minutes. 5 to 10 problems, similar to homework problems. No calculators, no notes, no books, no phones. No green book needed.

Sections 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 information

1 Stress and Strain. Introduction

1 Stress and Strain. Introduction 1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may

More information

x + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the

x + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the 1.(8pts) Find F ds where F = x + ye z + ze y, y + xe z + ze x, z and where T is the T surface in the pictures. (The two pictures are two views of the same surface.) The boundary of T is the unit circle

More information

Introduction to Vector Calculus (29) SOLVED EXAMPLES. (d) B. C A. (f) a unit vector perpendicular to both B. = ˆ 2k = = 8 = = 8

Introduction to Vector Calculus (29) SOLVED EXAMPLES. (d) B. C A. (f) a unit vector perpendicular to both B. = ˆ 2k = = 8 = = 8 Introduction to Vector Calculus (9) SOLVED EXAMPLES Q. If vector A i ˆ ˆj k, ˆ B i ˆ ˆj, C i ˆ 3j ˆ kˆ (a) A B (e) A B C (g) Solution: (b) A B (c) A. B C (d) B. C A then find (f) a unit vector perpendicular

More information