Getting started: CFD notation

Size: px
Start display at page:

Download "Getting started: CFD notation"

Transcription

1 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 = 1,2,... Nabla operator = i x + j y + k x = (x, y, z), v = (v x, v y, v z ) [ u = i u x + j u y + k u = v = v x v = det x + v y y + v z i j k x y u x, u y, u = ] T gradient v z y v y v x v z x divergence curl i z k j v y v x v y v z v y x v x y x u = ( u) = 2 u = 2 u x + 2 u 2 y + 2 u 2 2 Laplacian

2 Tensorial quantities in fluid dynamics elocity gradient v = [ v x, v y, v z ] = v x x v x y v x v y x v y y v y v z x v z y v z v Remark. The trace (sum of diagonal elements) of v equals v. Deformation rate tensor (symmetric part of v) D(v) = 1 2 ( v + vt ) = v x x ( vx y + v y x ( vx + v z x ) ) ( ) vy x + v x y v y y ( vy + v z y ) ( vz x + v x ( vz y + v y v z ) ) Spin tensor S(v) = v D(v) (skew-symmetric part of v)

3 ector multiplication rules Scalar product of two vectors a,b R 3, a b = a T b = [a 1 a 2 a 3 ] b 1 b 2 b 3 = a 1b 1 + a 2 b 2 + a 3 b 3 R Example. u v u = v x x + v u y y + v u z convective derivative Dyadic product of two vectors a,b R 3, a b = ab T = a 1 a 2 [b 1 b 2 b 3 ] = a 1 b 1 a 1 b 2 a 1 b 3 a 2 b 1 a 2 b 2 a 2 b 3 R3 3 a 3 a 3 b 1 a 3 b 2 a 3 b 3

4 Elementary tensor calculus 1. αt = {αt ij }, T = {t ij } R 3 3, α R 2. T 1 + T 2 = {t 1 ij + t2 ij }, T 1, T 2 R 3 3, a R 3 t 11 t 12 t a T = [a 1, a 2, a 3 ] t 21 t 22 t 23 = 3 a i [t i1, t i2, t i3 ] i=1 }{{} t 31 t 32 t 33 i-th row t 11 t 12 t 13 a 1 4. T a = t 21 t 22 t 23 a 2 = 3 t 1j t 2j a j (j-th column) j=1 t 31 t 32 t 33 a 3 t 3j t 5. T 1 T t 1 12 t 1 13 t = t 1 21 t 1 22 t t 2 12 t 2 { t 2 t 1 31 t 1 32 t 1 21 t 2 22 t 2 3 } 23 = t 1 ik t2 kj 33 t 2 31 t 2 32 t 2 k= T 1 : T 2 = tr (T 1 (T 2 ) T ) = 3 3 t 1 ik t2 ik i=1 k=1

5 Divergence theorem of Gauß Let Ω R 3 and n be the outward unit normal to the boundary Γ = Ω\Ω. Then Ω f dx = Γ f n ds for any differentiable function f(x) Example. A sphere: Ω = {x R 3 : x < 1}, Γ = {x R 3 : x = 1} where x = x x = x 2 + y 2 + z 2 is the Euclidean norm of x Consider f(x) = x so that f 3 in Ω and n = x x volume integral: surface integral: Ω Γ f dx = 3 f n ds = Γ Ω dx = 3 Ω = 3 x x x ds = Γ [ ] 4 3 π13 x ds = Γ on Γ = 4π ds = 4π

6 Governing equations of fluid dynamics Physical principles 1. Mass is conserved 2. Newton s second law 3. Energy is conserved Mathematical equations continuity equation momentum equations energy equation It is important to understand the meaning and significance of each equation in order to develop a good numerical method and properly interpret the results Description of fluid motion z v Eulerian monitor the flow characteristics Lagrangian in a fixed control volume track individual fluid particles as i k j ܼ Ý ¼ Þ ¼µ y they move through the flow field x ܽ Ý ½ Þ ½µ

7 Trajectory of a fluid particle Description of fluid motion x i z k j v ܽ Ý ½ Þ ½µ ¼µ Þ ¼ Ý Ü¼ y x = x(x 0, t) x = x(x 0, y 0, z 0, t) y = y(x 0, y 0, z 0, t) z = z(x 0, y 0, z 0, t) dx dt = v x(x, y, z, t), x t0 = x 0 dy dt = v y(x, y, z, t), y t0 = y 0 dz dt = v z(x, y, z, t), z t0 = z 0 Definition. A streamline is a curve which is tangent to the velocity vector v = (v x, v y, v z ) at every point. It is given by the relation dx v x = dy v y = dz v z y v y(x) dy dx = v y v x x Streamlines can be visualized by injecting tracer particles into the flow field.

8 Flow models and reference frames Eulerian Lagrangian S S integral fixed C of a finite size moving C of a finite size d fixed infinitesimal C ds d moving infinitesimal C ds differential Good news: all flow models lead to the same equations

9 Eulerian vs. Lagrangian viewpoint Definition. Substantial time derivative d dt is the rate of change for a moving fluid particle. Local time derivative is the rate of change at a fixed point. Let u = u(x, t), where x = x(x 0, t). The chain rule yields du dt = u + u dx x dt + u dy y dt + u dz dt = u + v u substantial derivative = local derivative + convective derivative Reynolds transport theorem d dt t u(x, t) d = u(x, t) t d + u(x, t)v n ds S S t rate of change in a moving volume = rate of change in a fixed volume + convective transfer through the surface

10 Derivation of the governing equations Modeling philosophy 1. Choose a physical principle conservation of mass conservation of momentum conservation of energy 2. Apply it to a suitable flow model Eulerian/Lagrangian approach for a finite/infinitesimal C 3. Extract integral relations or PDEs which embody the physical principle Generic conservation law Z Z Z u d + f n ds = S ds n f S q d f = vu d u flux function Divergence theorem yields Z Z Z u d + f d = Partial differential equation u + f = q in q d

11 Derivation of the continuity equation Physical principle: conservation of mass dm dt = d ρ ρ d = dt t t d + ρv n ds = 0 S S t accumulation of mass inside C = net influx through the surface Divergence theorem yields [ ] ρ + (ρv) Lagrangian representation d = 0 Continuity equation ρ + (ρv) = 0 (ρv) = v ρ + ρ v dρ dt + ρ v = 0 Incompressible flows: dρ dt = v = 0 (constant density)

12 Conservation of momentum Physical principle: f = ma (Newton s second law) d g ds h n total force f = ρg d + h ds, where h = σ n body forces g gravitational, electromagnetic,... surface forces h pressure + viscous stress Stress tensor σ = pi + τ momentum flux For a newtonian fluid viscous stress is proportional to velocity gradients: τ = (λ v)i + 2µD(v), where D(v) = 1 2 ( v + vt ), λ 2 3 µ Normal stress: stretching τ xx = λ v + 2µ v x x τ yy = λ v + 2µ v y y τ zz = λ v + 2µ v z Ý ÜÜ Ü Shear stress: deformation vy τ xy = τ yx = µ + v x x y τ xz = τ zx = µ ` v x + v z x v τ yz = τ zy = µ z + v y y Ý ÝÜ Ü

13 Derivation of the momentum equations Newton s law for a moving volume d (ρv) ρv d = d + (ρv v) n ds dt t t S S t = ρg d + σ n ds t S S t Transformation of surface integrals [ ] (ρv) + (ρv v) d = [ σ + ρg] d, σ = pi + τ (ρv) Momentum equations + (ρv v) = p + τ + ρg [ ] [ ] (ρv) v ρ + (ρv v) = ρ + v v + v + (ρv) = ρ dv dt }{{}}{{} substantial derivative continuity equation

14 Conservation of energy Physical principle: δe = s + w (first law of thermodynamics) d g ds δe accumulation of internal energy n h s heat transmitted to the fluid particle w rate of work done by external forces Heating: s = ρq d f q ds Fourier s law of heat conduction q f q T κ internal heat sources diffusive heat transfer absolute temperature thermal conductivity f q = κ T the heat flux is proportional to the local temperature gradient Work done per unit time = total force velocity w = f v = ρg v d + v (σ n) ds, σ = pi + τ

15 Derivation of the energy equation Total energy per unit mass: E = e + v 2 2 e specific internal energy due to random molecular motion v 2 2 specific kinetic energy due to translational motion Integral conservation law for a moving volume d (ρe) ρe d = d + ρe v n ds dt t t S S t = ρq d + κ T n ds t S S t + ρg v d + v (σ n) ds t S S t accumulation heating work done Transformation of surface integrals [ ] (ρe) + (ρev) d = [ (κ T) + ρq + (σ v) + ρg v] d, where (σ v) = (pv) + (τ v) = (pv) + v ( τ) + v : τ

16 Total energy equation (ρe) Different forms of the energy equation + (ρev) = (κ T) + ρq (pv) + v ( τ) + v : τ + ρg v (ρe) Momentum equations [ ] [ ] E ρ + (ρev) = ρ + v E + E + (ρv) }{{}}{{} substantial derivative continuity equation ρ dv dt ρ de dt = ρde dt + v ρdv dt = (ρe) Internal energy equation (ρe) = p + τ + ρg = ρ de dt (Lagrangian form) + (ρev) + v [ p + τ + ρg] + (ρev) = (κ T) + ρq p v + v : τ

17 Summary of the governing equations 1. Continuity equation / conservation of mass ρ + (ρv) = 0 2. Momentum equations / Newton s second law (ρv) + (ρv v) = p + τ + ρg 3. Energy equation / first law of thermodynamics (ρe) + (ρev) = (κ T) + ρq (pv) + v ( τ) + v : τ + ρg v E = e + v 2 2, (ρe) + (ρev) = (κ T) + ρq p v + v : τ This PDE system is referred to as the compressible Navier-Stokes equations

18 Conservation form of the governing equations Generic conservation law for a scalar quantity u + f = q, where f = f(u,x, t) is the flux function Conservative variables, fluxes and sources ρ ρv U = ρv, F = ρv v + pi τ ρe (ρe + p)v κ T τ v, Q = 0 ρg ρ(q + g v) Navier-Stokes equations in divergence form U + F = Q U R 5, F R 3 5, Q R 5 representing all equations in the same generic form simplifies the programming it suffices to develop discretization techniques for the generic conservation law

19 Constitutive relations ariables: ρ, v, e, p, τ, T Equations: continuity, momentum, energy The number of unknowns exceeds the number of equations. 1. Newtonian stress tensor τ = (λ v)i + 2µD(v), D(v) = 1 2 ( v + vt ), λ 2 3 µ 2. Thermodynamic relations, e.g. p = ρrt ideal gas law R specific gas constant e = c v T caloric equation of state c v specific heat at constant volume Now the system is closed: it contains five PDEs for five independent variables ρ, v, e and algebraic formulae for the computation of p, τ and T. It remains to specify appropriate initial and boundary conditions.

20 Initial and boundary conditions Initial conditions ρ t=0 = ρ 0 (x), v t=0 = v 0 (x), e t=0 = e 0 (x) in Ω Boundary conditions Inlet Γ in = {x Γ : v n < 0} Let Γ = Γ in Γ w Γ out Û ρ = ρ in, v = v in, e = e in Ò Å ÓÙØ prescribed density, energy and velocity Solid wall Γ w = {x Γ : v n = 0} v = 0 no-slip condition T = T w given temperature or ( T f n) = q κ prescribed heat flux Outlet Γ out = {x Γ : v n > 0} v n = v n or p + n τ n = 0 v s = v s or s τ n = 0 prescribed velocity vanishing stress The problem is well-posed if the solution exists, is unique and depends continuously on IC and BC. Insufficient or incorrect IC/BC may lead to wrong results (if any).

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

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

Convection Heat Transfer

Convection 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 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

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 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

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

Conservation of Mass. Computational Fluid Dynamics. The Equations Governing Fluid Motion

Conservation of Mass. Computational Fluid Dynamics. The Equations Governing Fluid Motion http://www.nd.edu/~gtryggva/cfd-course/ http://www.nd.edu/~gtryggva/cfd-course/ Computational Fluid Dynamics Lecture 4 January 30, 2017 The Equations Governing Fluid Motion Grétar Tryggvason Outline Derivation

More information

Fundamentals of Fluid Dynamics: Elementary Viscous Flow

Fundamentals of Fluid Dynamics: Elementary Viscous Flow Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research

More information

Numerical Heat and Mass Transfer

Numerical Heat and Mass Transfer Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15-Convective Heat Transfer Fausto Arpino f.arpino@unicas.it Introduction In conduction problems the convection entered the analysis

More information

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

Chapter 2: Fluid Dynamics Review

Chapter 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 information

Introduction to Fluid Mechanics

Introduction to Fluid Mechanics Introduction to Fluid Mechanics Tien-Tsan Shieh April 16, 2009 What is a Fluid? The key distinction between a fluid and a solid lies in the mode of resistance to change of shape. The fluid, unlike the

More 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 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

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

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

ENGR Heat Transfer II

ENGR Heat Transfer II ENGR 7901 - Heat Transfer II Convective Heat Transfer 1 Introduction In this portion of the course we will examine convection heat transfer principles. We are now interested in how to predict the value

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

AE/ME 339. K. M. Isaac. 9/22/2005 Topic 6 FluidFlowEquations_Introduction. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.

AE/ME 339. K. M. Isaac. 9/22/2005 Topic 6 FluidFlowEquations_Introduction. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept. AE/ME 339 Computational Fluid Dynamics (CFD) 1...in the phrase computational fluid dynamics the word computational is simply an adjective to fluid dynamics.... -John D. Anderson 2 1 Equations of Fluid

More information

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

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

More information

Numerical Solution of Partial Differential Equations governing compressible flows

Numerical Solution of Partial Differential Equations governing compressible flows Numerical Solution of Partial Differential Equations governing compressible flows Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore

More information

Mathematical Theory of Non-Newtonian Fluid

Mathematical Theory of Non-Newtonian Fluid Mathematical Theory of Non-Newtonian Fluid 1. Derivation of the Incompressible Fluid Dynamics 2. Existence of Non-Newtonian Flow and its Dynamics 3. Existence in the Domain with Boundary Hyeong Ohk Bae

More information

Chapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature

Chapter 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

AA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow

AA210A 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 information

Where does Bernoulli's Equation come from?

Where 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 information

Review of fluid dynamics

Review 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 information

Navier-Stokes Equation: Principle of Conservation of Momentum

Navier-Stokes Equation: Principle of Conservation of Momentum Navier-tokes Equation: Principle of Conservation of Momentum R. hankar ubramanian Department of Chemical and Biomolecular Engineering Clarkson University Newton formulated the principle of conservation

More information

Math background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids

Math background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids Fluid dynamics Math background Physics Simulation Related phenomena Frontiers in graphics Rigid fluids Fields Domain Ω R2 Scalar field f :Ω R Vector field f : Ω R2 Types of derivatives Derivatives measure

More information

UNIVERSITY of LIMERICK

UNIVERSITY 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 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

0.2. CONSERVATION LAW FOR FLUID 9

0.2. CONSERVATION LAW FOR FLUID 9 0.2. CONSERVATION LAW FOR FLUID 9 Consider x-component of Eq. (26), we have D(ρu) + ρu( v) dv t = ρg x dv t S pi ds, (27) where ρg x is the x-component of the bodily force, and the surface integral is

More information

Continuum Mechanics Fundamentals

Continuum Mechanics Fundamentals Continuum Mechanics Fundamentals James R. Rice, notes for ES 220, 12 November 2009; corrections 9 December 2009 Conserved Quantities Let a conseved quantity have amount F per unit volume. Examples are

More information

Dynamics of Ice Sheets and Glaciers

Dynamics of Ice Sheets and Glaciers Dynamics of Ice Sheets and Glaciers Ralf Greve Institute of Low Temperature Science Hokkaido University Lecture Notes Sapporo 2004/2005 Literature Ice dynamics Paterson, W. S. B. 1994. The Physics of

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

Chapter 6: Momentum Analysis

Chapter 6: Momentum Analysis 6-1 Introduction 6-2Newton s Law and Conservation of Momentum 6-3 Choosing a Control Volume 6-4 Forces Acting on a Control Volume 6-5Linear Momentum Equation 6-6 Angular Momentum 6-7 The Second Law of

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

KINEMATICS OF CONTINUA

KINEMATICS 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 information

2. Conservation Equations for Turbulent Flows

2. Conservation Equations for Turbulent Flows 2. Conservation Equations for Turbulent Flows Coverage of this section: Review of Tensor Notation Review of Navier-Stokes Equations for Incompressible and Compressible Flows Reynolds & Favre Averaging

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

2. FLUID-FLOW EQUATIONS SPRING 2019

2. FLUID-FLOW EQUATIONS SPRING 2019 2. FLUID-FLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid

More information

2. Getting Ready for Computational Aerodynamics: Fluid Mechanics Foundations

2. Getting Ready for Computational Aerodynamics: Fluid Mechanics Foundations . Getting Ready for Computational Aerodynamics: Fluid Mechanics Foundations We need to review the governing equations of fluid mechanics before examining the methods of computational aerodynamics in detail.

More information

2.20 Fall 2018 Math Review

2.20 Fall 2018 Math Review 2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more

More information

Differential relations for fluid flow

Differential 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 information

Lecture 3: 1. Lecture 3.

Lecture 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 information

Vector Calculus. A primer

Vector Calculus. A primer Vector Calculus A primer Functions of Several Variables A single function of several variables: f: R $ R, f x (, x ),, x $ = y. Partial derivative vector, or gradient, is a vector: f = y,, y x ( x $ Multi-Valued

More information

Computer Fluid Dynamics E181107

Computer Fluid Dynamics E181107 Computer Fluid Dynamics E181107 2181106 Transport equations, Navier Stokes equations Remark: foils with black background could be skipped, they are aimed to the more advanced courses Rudolf Žitný, Ústav

More information

CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION

CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,

More information

Computational Astrophysics

Computational Astrophysics Computational Astrophysics Lecture 1: Introduction to numerical methods Lecture 2:The SPH formulation Lecture 3: Construction of SPH smoothing functions Lecture 4: SPH for general dynamic flow Lecture

More information

Rotational & Rigid-Body Mechanics. Lectures 3+4

Rotational & Rigid-Body Mechanics. Lectures 3+4 Rotational & Rigid-Body 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 information

Dynamics of Glaciers

Dynamics of Glaciers Dynamics of Glaciers McCarthy Summer School 01 Andy Aschwanden Arctic Region Supercomputing Center University of Alaska Fairbanks, USA June 01 Note: This script is largely based on the Physics of Glaciers

More information

The Shallow Water Equations

The Shallow Water Equations The Shallow Water Equations Clint Dawson and Christopher M. Mirabito Institute for Computational Engineering and Sciences University of Texas at Austin clint@ices.utexas.edu September 29, 2008 The Shallow

More information

Exercise 5: Exact Solutions to the Navier-Stokes Equations I

Exercise 5: Exact Solutions to the Navier-Stokes Equations I Fluid Mechanics, SG4, HT009 September 5, 009 Exercise 5: Exact Solutions to the Navier-Stokes Equations I Example : Plane Couette Flow Consider the flow of a viscous Newtonian fluid between two parallel

More information

Chapter 6: Momentum Analysis of Flow Systems

Chapter 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 information

Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics

Fundamentals 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 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

Advanced Course in Theoretical Glaciology

Advanced Course in Theoretical Glaciology Advanced Course in Theoretical Glaciology Ralf Greve Institute of Low Temperature Science Hokkaido University Lecture Notes Sapporo 2015 These lecture notes are largely based on the textbook Greve, R.

More information

Stability of Shear Flow

Stability of Shear Flow Stability of Shear Flow notes by Zhan Wang and Sam Potter Revised by FW WHOI GFD Lecture 3 June, 011 A look at energy stability, valid for all amplitudes, and linear stability for shear flows. 1 Nonlinear

More information

1 Equations of motion

1 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 information

Basic concepts in viscous flow

Basic concepts in viscous flow Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic Adapted from Chapter 1 of Cambridge Texts in Applied Mathematics 1 The fluid dynamic equations Navier-Stokes equations Dimensionless

More information

EULERIAN DERIVATIONS OF NON-INERTIAL NAVIER-STOKES EQUATIONS

EULERIAN DERIVATIONS OF NON-INERTIAL NAVIER-STOKES EQUATIONS EULERIAN DERIVATIONS OF NON-INERTIAL NAVIER-STOKES EQUATIONS ML Combrinck, LN Dala Flamengro, a div of Armscor SOC Ltd & University of Pretoria, Council of Scientific and Industrial Research & University

More information

2 GOVERNING EQUATIONS

2 GOVERNING EQUATIONS 2 GOVERNING EQUATIONS 9 2 GOVERNING EQUATIONS For completeness we will take a brief moment to review the governing equations for a turbulent uid. We will present them both in physical space coordinates

More information

Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost

Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit

More information

Conservation Laws in Ideal MHD

Conservation Laws in Ideal MHD Conservation Laws in Ideal MHD Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 3, 2016 These lecture notes are largely based on Plasma Physics for Astrophysics

More information

Mechanics of solids and fluids -Introduction to continuum mechanics

Mechanics of solids and fluids -Introduction to continuum mechanics Mechanics of solids and fluids -Introduction to continuum mechanics by Magnus Ekh August 12, 2016 Introduction to continuum mechanics 1 Tensors............................. 3 1.1 Index notation 1.2 Vectors

More information

Modelling ice shelf basal melt with Glimmer-CISM coupled to a meltwater plume model

Modelling ice shelf basal melt with Glimmer-CISM coupled to a meltwater plume model Modelling ice shelf basal melt with Glimmer-CISM coupled to a meltwater plume model Carl Gladish NYU CIMS February 17, 2010 Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 1 / 24 Acknowledgements

More information

Quick Recapitulation of Fluid Mechanics

Quick Recapitulation of Fluid Mechanics Quick Recapitulation of Fluid Mechanics Amey Joshi 07-Feb-018 1 Equations of ideal fluids onsider a volume element of a fluid of density ρ. If there are no sources or sinks in, the mass in it will change

More information

Lecture: Wave-induced Momentum Fluxes: Radiation Stresses

Lecture: Wave-induced Momentum Fluxes: Radiation Stresses Chapter 4 Lecture: Wave-induced Momentum Fluxes: Radiation Stresses Here we derive the wave-induced depth-integrated momentum fluxes, otherwise known as the radiation stress tensor S. These are the 2nd-order

More information

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.

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. 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 information

AA210A Fundamentals of Compressible Flow. Chapter 5 -The conservation equations

AA210A Fundamentals of Compressible Flow. Chapter 5 -The conservation equations AA210A Fundamentals of Compressible Flow Chapter 5 -The conservation equations 1 5.1 Leibniz rule for differentiation of integrals Differentiation under the integral sign. According to the fundamental

More information

Physics 451/551 Theoretical Mechanics. G. A. Krafft Old Dominion University Jefferson Lab Lecture 18

Physics 451/551 Theoretical Mechanics. G. A. Krafft Old Dominion University Jefferson Lab Lecture 18 Physics 451/551 Theoretical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 18 Theoretical Mechanics Fall 018 Properties of Sound Sound Waves Requires medium for propagation Mainly

More information

Chapter 7. Kinematics. 7.1 Tensor fields

Chapter 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 information

The Navier-Stokes Equations

The Navier-Stokes Equations s University of New Hampshire February 22, 202 and equations describe the non-relativistic time evolution of mass and momentum in fluid substances. mass density field: ρ = ρ(t, x, y, z) velocity field:

More information

Dynamics of the Mantle and Lithosphere ETH Zürich Continuum Mechanics in Geodynamics: Equation cheat sheet

Dynamics of the Mantle and Lithosphere ETH Zürich Continuum Mechanics in Geodynamics: Equation cheat sheet Dynamics of the Mantle and Lithosphere ETH Zürich Continuum Mechanics in Geodynamics: Equation cheat sheet or all equations you will probably ever need Definitions 1. Coordinate system. x,y,z or x 1,x

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

Entropy generation and transport

Entropy generation and transport Chapter 7 Entropy generation and transport 7.1 Convective form of the Gibbs equation In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?

More information

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)

SOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours) SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please

More information

Mathematical Preliminaries

Mathematical Preliminaries Mathematical Preliminaries Introductory Course on Multiphysics Modelling TOMAZ G. ZIELIŃKI bluebox.ippt.pan.pl/ tzielins/ Table of Contents Vectors, tensors, and index notation. Generalization of the concept

More information

Chapter 2. General concepts. 2.1 The Navier-Stokes equations

Chapter 2. General concepts. 2.1 The Navier-Stokes equations Chapter 2 General concepts 2.1 The Navier-Stokes equations The Navier-Stokes equations model the fluid mechanics. This set of differential equations describes the motion of a fluid. In the present work

More information

Lecture 8: Tissue Mechanics

Lecture 8: Tissue Mechanics Computational Biology Group (CoBi), D-BSSE, ETHZ Lecture 8: Tissue Mechanics Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015/16 7. Mai 2016 2 / 57 Contents 1 Introduction to Elastic Materials

More information

Introduction to Continuum Mechanics

Introduction to Continuum Mechanics Introduction to Continuum Mechanics I-Shih Liu Instituto de Matemática Universidade Federal do Rio de Janeiro 2018 Contents 1 Notations and tensor algebra 1 1.1 Vector space, inner product........................

More information

Mathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length.

Mathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length. Mathematical Tripos Part IA Lent Term 205 ector Calculus Prof B C Allanach Example Sheet Sketch the curve in the plane given parametrically by r(u) = ( x(u), y(u) ) = ( a cos 3 u, a sin 3 u ) with 0 u

More information

CHAPTER 8 ENTROPY GENERATION AND TRANSPORT

CHAPTER 8 ENTROPY GENERATION AND TRANSPORT CHAPTER 8 ENTROPY GENERATION AND TRANSPORT 8.1 CONVECTIVE FORM OF THE GIBBS EQUATION In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?

More information

Soft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies

Soft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Soft-Body Physics Soft Bodies Realistic objects are not purely rigid. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Deformed

More information

CONSERVATION OF ENERGY FOR ACONTINUUM

CONSERVATION OF ENERGY FOR ACONTINUUM Chapter 6 CONSERVATION OF ENERGY FOR ACONTINUUM Figure 6.1: 6.1 Conservation of Energ In order to define conservation of energ, we will follow a derivation similar to those in previous chapters, using

More information

Tensor Visualization. CSC 7443: Scientific Information Visualization

Tensor 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 information

Sec. 1.1: Basics of Vectors

Sec. 1.1: Basics of Vectors Sec. 1.1: Basics of Vectors Notation for Euclidean space R n : all points (x 1, x 2,..., x n ) in n-dimensional space. Examples: 1. R 1 : all points on the real number line. 2. R 2 : all points (x 1, x

More information

Mathematical Concepts & Notation

Mathematical 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 information

Fluid Dynamics Exercises and questions for the course

Fluid 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 information

2 Law of conservation of energy

2 Law of conservation of energy 1 Newtonian viscous Fluid 1 Newtonian fluid For a Newtonian we already have shown that σ ij = pδ ij + λd k,k δ ij + 2µD ij where λ and µ are called viscosity coefficient. For a fluid under rigid body motion

More information

From Navier-Stokes to Saint-Venant

From Navier-Stokes to Saint-Venant From Navier-Stokes to Saint-Venant Course 1 E. Godlewski 1 & J. Sainte-Marie 1 1 ANGE team LJLL - January 2017 Euler Eulerian vs. Lagrangian description Lagrange x(t M(t = y(t, z(t flow of the trajectories

More information

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is

More information

Lecture 4. Differential Analysis of Fluid Flow Navier-Stockes equation

Lecture 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 information

Module 2 : Convection. Lecture 12 : Derivation of conservation of energy

Module 2 : Convection. Lecture 12 : Derivation of conservation of energy Module 2 : Convection Lecture 12 : Derivation of conservation of energy Objectives In this class: Start the derivation of conservation of energy. Utilize earlier derived mass and momentum equations for

More information

Microscopic Momentum Balance Equation (Navier-Stokes)

Microscopic Momentum Balance Equation (Navier-Stokes) CM3110 Transport I Part I: Fluid Mechanics Microscopic Momentum Balance Equation (Navier-Stokes) Professor Faith Morrison Department of Chemical Engineering Michigan Technological University 1 Microscopic

More information

Module 2 : Lecture 1 GOVERNING EQUATIONS OF FLUID MOTION (Fundamental Aspects)

Module 2 : Lecture 1 GOVERNING EQUATIONS OF FLUID MOTION (Fundamental Aspects) Module : Lecture 1 GOVERNING EQUATIONS OF FLUID MOTION (Fundamental Aspects) Descriptions of Fluid Motion A fluid is composed of different particles for which the properties may change with respect to

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

Advection, Conservation, Conserved Physical Quantities, Wave Equations

Advection, Conservation, Conserved Physical Quantities, Wave Equations EP711 Supplementary Material Thursday, September 4, 2014 Advection, Conservation, Conserved Physical Quantities, Wave Equations Jonathan B. Snively!Embry-Riddle Aeronautical University Contents EP711 Supplementary

More information

Week 6 Notes, Math 865, Tanveer

Week 6 Notes, Math 865, Tanveer Week 6 Notes, Math 865, Tanveer. Energy Methods for Euler and Navier-Stokes Equation We will consider this week basic energy estimates. These are estimates on the L 2 spatial norms of the solution u(x,

More information

CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM

CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM Summary of integral theorems Material time derivative Reynolds transport theorem Principle of conservation of mass Principle of balance of linear momentum

More information