From the last time, we ended with an expression for the energy equation. u = ρg u + (τ u) q (9.1)
|
|
- Peter Hall
- 5 years ago
- Views:
Transcription
1 Lecture 9 9. Administration None. 9. Continuation of energy equation From the last time, we ended with an expression for the energy equation ρ D (e + ) u = ρg u + (τ u) q (9.) Where ρg u changes in potential energy q changes due to heat flux (τ u) a kinetic energy part and an internal heating part Clearly, the stress term is the painful one: (τ u) = u ( τ)+τ : u (9.) he first term, u ( τ), is the rate of work done by the surface forces. Imbalances in stress accelerate the fluid blob and change its kinetic energy. he second term, τ : u, is the work done by deformation. Not all stress moves the blob some of it deforms and causes heat change. Note the double contraction, A : B scalar. he u ( τ) terms is the easiest of the two since we came up with an expression for τ when we looked at momentum conservation. τ = (µe p 3 ) µ u = p + µ u + µ 3 ( u) u ( τ) = u ( p + µ u + µ ) 3 ( u) (9.3) We re advecting τ through our fixed blob of fluid. It is bringing in kinetic energy associated with the stress on the blob. (Jim You have a note on the side here saying think about this more ; I am not sure if this is directed at you or the students. In the least, advecting τ is a bit confusing.) Now the second term, τ : u double dot product
2 Notice, there is no between the and the u. u is the gradient of the velocity... he velocity gradient tensor! hus, this term looks like u = G (9.4) τ : G (9.5) But G = e + r, τ : (e + ) r (9.6) Recall that r is an antisymmetric tensor, and τ is a symmetric tensor. he double dot product of asymmetric and antisymmetric matrix is zero A : B = A ij B ij (9.7) i j In D [ ] τ τ : τ τ [ 0 ω 0 ω ] ( ω ) ( = τ (0) + τ + τ ω ) + τ (0) ω = τ τ ω = 0 (9.8) his last line is due to the fact that since τ is symmetric, τ = τ. hus, you can get rid of the r term and end up with τ : e (9.9) Substitute in our expression for τ, ( µe [p + 3 ] ) µ u I : e Lets start with an easy one µe : e pi : e µ ui : e (9.0) 3 pi : e = p u x + p v + p w z = p u (9.) hat is, change in energy due to compression or expansion. he last term is also pretty easy 3 µ ui : e = ( u 3 µ u x + v + w ) z = 3 µ u( u) = 3 µ( u) (9.)
3 Figure 9.: (fig:lec9muparameter) µ parameterizes work associated in pulling apart molecules inabloboffluid. Finally, the first term µe : e e : e = e e e 3 e e e 3 : e e e 3 e e e 3 e 3 e 3 e 33 e 3 e 3 e 33 e e + e e + e 3 e 3 + = e e + e e + e 3 e 3 + (9.3) e 3 e 3 + e 3 e 3 + e 33 e 33 Note e e = e e = = 4 ( ) u (9.4) x [ ( u + v )] x ( ( u ) + u ( ) ) v v x + x = 4 etc. ( u ) + u v x + 4 ( v x ) (9.5) Consider just the e e term. Remember, ( ) u µe : e µ = ( µ u ) u (9.6) he µ u u bit is associated with normal stress, and with the velocity gradient. hus, this term has something to do with the work it takes to pull apart molecules in a blob of fluid (see figure 9.). he other terms, such as ( u ) and u v x, say something similar about shear and twisting. Clearly this is a bit of a mess. hankfully, what is typically done is to sweep it all under the rug and define φ = (µe : e 3 ) µ( u) (9.7) his allows us to write viscous component of the deformation work rate τ : u = p( u) +φ (9.8) racking this all the way back to the energy equation gives: ρ D (e + ) u = ρg u + u [ p + µ u + µ ] 3 ( u) p( u)+φ q (9.9) 3
4 9.3 hermal energy (or heat) equation What a mess. What games can we play to make this more comprehensible? Let s go back to a version of the momentum equation: Multiply (dot) through by u ρ D ρ Du ( ) u = ρg + u ( τ) (9.0) = ρu g + u ( τ) (9.) his looks a little familiar. It is the mechanical energy equation. Subtract this from the total energy equation ρ D (e + ) u = ρg u + u ( τ) p( u)+φ q ρ D ( ) u = ρu g + u ( τ) and we get the thermal energy (or heat) equation: ρ De = p( u)+φ q (9.) OR = p( u)+φ q (9.3) he second equation is obtained using de = C v d.here 9.4 Approximations p( u) change in thermal energy due to compression φ change in thermal energy due to viscosity q change in thermal energy due to heat transfer Almost always it is safe to say that φ, and it is ignored (Jim Comparing φ to what other term that is O()?). + p( u) (9.4) Here, is temperature changes at constant volume, and p( u) is temperature changes due to changing volume. Remember the Boussinesq approximation? I stated it as And it allowed us to simplify the continuity equation to ρ = ρ o + ρ (x, y, z, t) (9.5) It would be tempting to simply remove the p( u) term, leaving u = 0 (9.6) (9.7) 4
5 But this isn t correct, because here u is similar in scale to, so we ll keep it. If we can t say u = 0, let s substitute in the continuity equation instead Dρ + ρ u =0 u = Dρ (9.8) ρ Substitution yields p Dρ (9.9) ρ Rewrite as ( + ρp ) Dρ ρ (9.30) Using the fact that ρ Dρ = D ρ + ρpdα where α = ρ specific volume. he equation of state for a perfect gas is (9.3) p = ρr or pα = R (9.3) Differentiate and rearrange: p Dα + αdp p Dα = R = R αdp (9.33) (9.34) Substitute: + ρr ραdp ρ(c v + R) Dp ρ Dp (9.35) Where C v + R =. Here, ρ is temperature change from heating at constant pressure, and Dp is the correction term for constant pressure. hus the approximations made are φ small and an ideal gas (though structurally similar to case with any equation of state). he Boussinesq approximation says that Dp is small. Hence ρ (9.36) Further, using Fourier s Law of conduction q = k (9.37) which comes from measuring the heat flow through a block with different temperatures on each side, we get the following result ρ = (k ) (9.38) 5
6 Here k thermal conductivity, and if constant we have ρ = k (9.39) (Jim Here we have used an atmospheric approximation, ideal gas, and an oceanic one, Boussinesq; perhaps less than ideal). his assumption that Dp is small is okay for the ocean, but poor for the atmosphere. What can we do about that? Go back to ρ Dp (9.40) Divide through by ρ Dp ρ = q ρ (9.4) hesourceontherhshasgonefromapervolumetoapermass.letsgiveitanewname, Q. Dp ρ = Q (9.4) Divide through by ρ Dp = Q (9.43) Noting that p = ρr p = R ρ R Dp p = Q (9.44) Since dx x dt = d dt ln x D ln R D Q ln p = Dη (9.45) Where η entropy per mass, and δη δq where Q heat per mass. How on earth could that possibly be useful? Let me introduce a new variable θ = ( ) R/Cp ps potential temperature (9.46) p Here p s surface pressure. he potential temperature is the temperature a parcel would have if you moved it adiabatically (no heat enters or leaves the parcel) from whatever height it sits at, down to the surface. ake the ln ln θ = ln + R ( ) ps ln p ln θ = ln R ln p + R ln p s (9.47) ake the time derivative D ln θ = C D p ln R D ln p (9.48) 6
7 Compare with our expression for rate of change of entropy: So, Dη D = ln R D ln p (9.49) D Dη ln θ = (9.50) his tells us that a parcel that conserves entropy travels along a surface of constant θ. Why do we care? In a compressible atmosphere it isn t the density gradient with height that determines vertical stability, it is the entropy gradient that determine stability. he potential temperature analog in the ocean is potential density, defined as the density obtained if a parcel is taken to a reference pressure at constant salinity and adiabatically. Back to the heat equation D Dη ln θ = = Q θ Dθ Dθ ρ ρ = θ q (9.5) ρ No assumptions here except ideal gas and φ small. Dp retained. 9.5 Reading for class 0 Noneassignedinclass. 7
(Jim You have a note for yourself here that reads Fill in full derivation, this is a sloppy treatment ).
Lecture. dministration Collect problem set. Distribute problem set due October 3, 004.. nd law of thermodynamics (Jim You have a note for yourself here that reads Fill in full derivation, this is a sloppy
More informationLecture Administration. 7.2 Continuity equation. 7.3 Boussinesq approximation
Lecture 7 7.1 Administration Hand back Q3, PS3. No class next Tuesday (October 7th). Class a week from Thursday (October 9th) will be a guest lecturer. Last question in PS4: Only take body force to τ stage.
More information1 Introduction to Governing Equations 2 1a Methodology... 2
Contents 1 Introduction to Governing Equations 2 1a Methodology............................ 2 2 Equation of State 2 2a Mean and Turbulent Parts...................... 3 2b Reynolds Averaging.........................
More informationChapter 1. Governing Equations of GFD. 1.1 Mass continuity
Chapter 1 Governing Equations of GFD The fluid dynamical governing equations consist of an equation for mass continuity, one for the momentum budget, and one or more additional equations to account for
More informationRational derivation of the Boussinesq approximation
Rational derivation of the Boussinesq approximation Kiyoshi Maruyama Department of Earth and Ocean Sciences, National Defense Academy, Yokosuka, Kanagawa 239-8686, Japan February 22, 2019 Abstract This
More information2 Equations of Motion
2 Equations of Motion system. In this section, we will derive the six full equations of motion in a non-rotating, Cartesian coordinate 2.1 Six equations of motion (non-rotating, Cartesian coordinates)
More informationCHAPTER 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 informationSummary of the Equations of Fluid Dynamics
Reference: Summary of the Equations of Fluid Dynamics Fluid Mechanics, L.D. Landau & E.M. Lifshitz 1 Introduction Emission processes give us diagnostics with which to estimate important parameters, such
More informationEntropy 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 informationExercise 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 informationConsider a volume Ω enclosing a mass M and bounded by a surface δω. d dt. q n ds. The Work done by the body on the surroundings is.
The Energy Balance Consider a volume Ω enclosing a mass M and bounded by a surface δω. δω At a point x, the density is ρ, the local velocity is v, and the local Energy density is U. U v The rate of change
More informationConservation 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 informationMath 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 informationChapter 2. Quasi-Geostrophic Theory: Formulation (review) ε =U f o L <<1, β = 2Ω cosθ o R. 2.1 Introduction
Chapter 2. Quasi-Geostrophic Theory: Formulation (review) 2.1 Introduction For most of the course we will be concerned with instabilities that an be analyzed by the quasi-geostrophic equations. These are
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 informationP = 1 3 (σ xx + σ yy + σ zz ) = F A. It is created by the bombardment of the surface by molecules of fluid.
CEE 3310 Thermodynamic Properties, Aug. 27, 2010 11 1.4 Review A fluid is a substance that can not support a shear stress. Liquids differ from gasses in that liquids that do not completely fill a container
More informationConvection and buoyancy oscillation
Convection and buoyancy oscillation Recap: We analyzed the static stability of a vertical profile by the "parcel method"; For a given environmental profile (of T 0, p 0, θ 0, etc.), if the density of an
More informationQuick 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( ) 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 informationBefore we consider two canonical turbulent flows we need a general description of turbulence.
Chapter 2 Canonical Turbulent Flows Before we consider two canonical turbulent flows we need a general description of turbulence. 2.1 A Brief Introduction to Turbulence One way of looking at turbulent
More informationwavelength (nm)
Blackbody radiation Everything with a temperature above absolute zero emits electromagnetic radiation. This phenomenon is called blackbody radiation. The intensity and the peak wavelength of the radiation
More informationTurbomachinery & Turbulence. Lecture 2: One dimensional thermodynamics.
Turbomachinery & Turbulence. Lecture 2: One dimensional thermodynamics. F. Ravelet Laboratoire DynFluid, Arts et Metiers-ParisTech February 3, 2016 Control volume Global balance equations in open systems
More informationMeteorology 6150 Cloud System Modeling
Meteorology 6150 Cloud System Modeling Steve Krueger Spring 2009 1 Fundamental Equations 1.1 The Basic Equations 1.1.1 Equation of motion The movement of air in the atmosphere is governed by Newton s Second
More informationESCI 485 Air/Sea Interaction Lesson 1 Stresses and Fluxes Dr. DeCaria
ESCI 485 Air/Sea Interaction Lesson 1 Stresses and Fluxes Dr DeCaria References: An Introduction to Dynamic Meteorology, Holton MOMENTUM EQUATIONS The momentum equations governing the ocean or atmosphere
More informationContinuum 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 informationmeters, we can re-arrange this expression to give
Turbulence When the Reynolds number becomes sufficiently large, the non-linear term (u ) u in the momentum equation inevitably becomes comparable to other important terms and the flow becomes more complicated.
More informationNumber-Flux Vector and Stress-Energy Tensor
Massachusetts Institute of Technology Department of Physics Physics 8.962 Spring 2002 Number-Flux Vector and Stress-Energy Tensor c 2000, 2002 Edmund Bertschinger. All rights reserved. 1 Introduction These
More informationρ x + fv f 'w + F x ρ y fu + F y Fundamental Equation in z coordinate p = ρrt or pα = RT Du uv tanφ Dv Dt + u2 tanφ + vw a a = 1 p Dw Dt u2 + v 2
Fundamental Equation in z coordinate p = ρrt or pα = RT Du uv tanφ + uw Dt a a = 1 p ρ x + fv f 'w + F x Dv Dt + u2 tanφ + vw a a = 1 p ρ y fu + F y Dw Dt u2 + v 2 = 1 p a ρ z g + f 'u + F z Dρ Dt + ρ
More informationPHYS 643 Week 4: Compressible fluids Sound waves and shocks
PHYS 643 Week 4: Compressible fluids Sound waves and shocks Sound waves Compressions in a gas propagate as sound waves. The simplest case to consider is a gas at uniform density and at rest. Small perturbations
More informationTotal energy in volume
General Heat Transfer Equations (Set #3) ChE 1B Fundamental Energy Postulate time rate of change of internal +kinetic energy = rate of heat transfer + surface work transfer (viscous & other deformations)
More informationFundamentals 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 informationASTR 320: Solutions to Problem Set 2
ASTR 320: Solutions to Problem Set 2 Problem 1: Streamlines A streamline is defined as a curve that is instantaneously tangent to the velocity vector of a flow. Streamlines show the direction a massless
More informationNavier-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 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 informationCH.9. CONSTITUTIVE EQUATIONS IN FLUIDS. Multimedia Course on Continuum Mechanics
CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS Multimedia Course on Continuum Mechanics Overview Introduction Fluid Mechanics What is a Fluid? Pressure and Pascal s Law Constitutive Equations in Fluids Fluid Models
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 informationATMOS 5130 Lecture 9. Enthalpy Conservation Property The Second Law and Its Consequences Entropy
ATMOS 5130 Lecture 9 Enthalpy Conservation Property The Second Law and Its Consequences Entropy CLASS Presentation Form group of 2 students Present ~20 minute presentation (~ 10 minute each person) Focus
More information6.2 Governing Equations for Natural Convection
6. Governing Equations for Natural Convection 6..1 Generalized Governing Equations The governing equations for natural convection are special cases of the generalized governing equations that were discussed
More informationIntroduction to Partial Differential Equations
Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 1 / 19 Introduction The derivation of the heat
More informationScale analysis of the vertical equation of motion:
Scale analysis of the vertical equation of motion: As we did with the hz eqns, we do for the vertical to estimate the order of magnitude of Dw/ we take the largest of the terms, Dw -- W/T h UW/L W 2 /H
More informationAA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow
AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit
More informationExercise 1: Vertical structure of the lower troposphere
EARTH SCIENCES SCIENTIFIC BACKGROUND ASSESSMENT Exercise 1: Vertical structure of the lower troposphere In this exercise we will describe the vertical thermal structure of the Earth atmosphere in its lower
More informationFirst Law of Thermodynamics
First Law of Thermodynamics September 11, 2013 The first law of thermodynamics is the conservation of energy applied to thermal systems. Here, we develop the principles of thermodynamics for a discrete
More information2 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 informationMath 234 Exam 3 Review Sheet
Math 234 Exam 3 Review Sheet Jim Brunner LIST OF TOPIS TO KNOW Vector Fields lairaut s Theorem & onservative Vector Fields url Divergence Area & Volume Integrals Using oordinate Transforms hanging the
More informationFluid Dynamics and Balance Equations for Reacting Flows
Fluid Dynamics and Balance Equations for Reacting Flows Combustion Summer School 2018 Prof. Dr.-Ing. Heinz Pitsch Balance Equations Basics: equations of continuum mechanics balance equations for mass and
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 informationLecture 1: Introduction to Linear and Non-Linear Waves
Lecture 1: Introduction to Linear and Non-Linear Waves Lecturer: Harvey Segur. Write-up: Michael Bates June 15, 2009 1 Introduction to Water Waves 1.1 Motivation and Basic Properties There are many types
More information5.1 Fluid momentum equation Hydrostatics Archimedes theorem The vorticity equation... 42
Chapter 5 Euler s equation Contents 5.1 Fluid momentum equation........................ 39 5. Hydrostatics................................ 40 5.3 Archimedes theorem........................... 41 5.4 The
More 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 informationComputer 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 information15. LECTURE 15. I can calculate the dot product of two vectors and interpret its meaning. I can find the projection of one vector onto another one.
5. LECTURE 5 Objectives I can calculate the dot product of two vectors and interpret its meaning. I can find the projection of one vector onto another one. In the last few lectures, we ve learned that
More informationFluid Animation. Christopher Batty November 17, 2011
Fluid Animation Christopher Batty November 17, 2011 What distinguishes fluids? What distinguishes fluids? No preferred shape Always flows when force is applied Deforms to fit its container Internal forces
More informationEULERIAN 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 informationGetting 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 informationLecture 2. Lecture 1. Forces on a rotating planet. We will describe the atmosphere and ocean in terms of their:
Lecture 2 Lecture 1 Forces on a rotating planet We will describe the atmosphere and ocean in terms of their: velocity u = (u,v,w) pressure P density ρ temperature T salinity S up For convenience, we will
More informationFundamentals of Atmospheric Modelling
M.Sc. in Computational Science Fundamentals of Atmospheric Modelling Peter Lynch, Met Éireann Mathematical Computation Laboratory (Opp. Room 30) Dept. of Maths. Physics, UCD, Belfield. January April, 2004.
More informationFundamentals of compressible and viscous flow analysis - Part II
Fundamentals of compressible and viscous flow analysis - Part II Lectures 3, 4, 5 Instantaneous and averaged temperature contours in a shock-boundary layer interaction. Taken from (Pasquariello et al.,
More informationn v molecules will pass per unit time through the area from left to
3 iscosity and Heat Conduction in Gas Dynamics Equations of One-Dimensional Gas Flow The dissipative processes - viscosity (internal friction) and heat conduction - are connected with existence of molecular
More information0.2. 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 informationLecture 23. Vertical coordinates General vertical coordinate
Lecture 23 Vertical coordinates We have exclusively used height as the vertical coordinate but there are alternative vertical coordinates in use in ocean models, most notably the terrainfollowing coordinate
More information- Marine Hydrodynamics. Lecture 14. F, M = [linear function of m ij ] [function of instantaneous U, U, Ω] not of motion history.
2.20 - Marine Hydrodynamics, Spring 2005 ecture 14 2.20 - Marine Hydrodynamics ecture 14 3.20 Some Properties of Added-Mass Coefficients 1. m ij = ρ [function of geometry only] F, M = [linear function
More informationAccretion Disks I. High Energy Astrophysics: Accretion Disks I 1/60
Accretion Disks I References: Accretion Power in Astrophysics, J. Frank, A. King and D. Raine. High Energy Astrophysics, Vol. 2, M.S. Longair, Cambridge University Press Active Galactic Nuclei, J.H. Krolik,
More informationATMO 551a Fall The Carnot Cycle
What is a arnot ycle and Why do we care The arnot ycle arnot was a French engineer who was trying to understand how to extract usable mechanical work from a heat engine, that is an engine where a gas or
More informationConsider a volume Ω enclosing a mass M and bounded by a surface δω. d dt. q n ds. The Work done by the body on the surroundings is
The Energy Balance Consider a volume enclosing a mass M and bounded by a surface δ. δ At a point x, the density is ρ, the local velocity is v, and the local Energy density is U. U v The rate of change
More information!t + U " #Q. Solving Physical Problems: Pitfalls, Guidelines, and Advice
Solving Physical Problems: Pitfalls, Guidelines, and Advice Some Relations in METR 420 There are a relatively small number of mathematical and physical relations that we ve used so far in METR 420 or that
More informationMath 575-Lecture Viscous Newtonian fluid and the Navier-Stokes equations
Math 575-Lecture 13 In 1845, tokes extended Newton s original idea to find a constitutive law which relates the Cauchy stress tensor to the velocity gradient, and then derived a system of equations. The
More informationTides in Higher-Dimensional Newtonian Gravity
Tides in Higher-Dimensional Newtonian Gravity Philippe Landry Department of Physics University of Guelph 23 rd Midwest Relativity Meeting October 25, 2013 Tides: A Familiar Example Gravitational interactions
More informationAstrophysical Fluid Dynamics (2)
Astrophysical Fluid Dynamics (2) (Draine Chap 35;;Clarke & Carswell Chaps 1-4) Last class: fluid dynamics equations apply in astrophysical situations fluid approximation holds Maxwellian velocity distribution
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 informationPHYS 633: Introduction to Stellar Astrophysics
PHYS 6: Introduction to Stellar Astrophysics Spring Semester 2006 Rich Townsend (rhdt@bartol.udel.edu Convective nergy Transport From our derivation of the radiative diffusion equation, we find that in
More informationTutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace
Tutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace Adapted from Publisher: John S. Wiley & Sons 2002 Center for Scientific Computation and
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 information2013/5/22. ( + ) ( ) = = momentum outflow rate. ( x) FPressure. 9.3 Nozzles. δ q= heat added into the fluid per unit mass
9.3 Nozzles (b) omentum conservation : (i) Governing Equations Consider: nonadiabatic ternal (body) force ists variable flow area continuously varying flows δq f ternal force per unit volume +d δffdx dx
More informationLecturer, Department t of Mechanical Engineering, SVMIT, Bharuch
Fluid Mechanics By Ashish J. Modi Lecturer, Department t of Mechanical Engineering, i SVMIT, Bharuch Review of fundamentals Properties of Fluids Introduction Any characteristic of a system is called a
More informationModule 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 informationES265 Order of Magnitude Phys & Chem Convection
ES265 Order of Magnitude Phys & Chem Convection Convection deals with moving fluids in which there are spatial variations in temperature or chemical concentration. In forced convection, these variations
More informationGeneral relativistic computation of shocks in accretion disc and bipolar jets
General relativistic computation of shocks in accretion disc and bipolar jets Rajiv Kumar Co-author Dr. Indranil Chattopadhyay Aryabhatta Research Institute of observational sciences (ARIES) rajiv.k@aries.res.in
More information7 The Navier-Stokes Equations
18.354/12.27 Spring 214 7 The Navier-Stokes Equations In the previous section, we have seen how one can deduce the general structure of hydrodynamic equations from purely macroscopic considerations and
More informationNumerical methods for the Navier- Stokes equations
Numerical methods for the Navier- Stokes equations Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Dec 6, 2012 Note:
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 informationNeeds work : define boundary conditions and fluxes before, change slides Useful definitions and conservation equations
Needs work : define boundary conditions and fluxes before, change slides 1-2-3 Useful definitions and conservation equations Turbulent Kinetic energy The fluxes are crucial to define our boundary conditions,
More informationFigure 1: Doing work on a block by pushing it across the floor.
Work Let s imagine I have a block which I m pushing across the floor, shown in Figure 1. If I m moving the block at constant velocity, then I know that I have to apply a force to compensate the effects
More informationConservation 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 informationHomework #4 Solution. μ 1. μ 2
Homework #4 Solution 4.20 in Middleman We have two viscous liquids that are immiscible (e.g. water and oil), layered between two solid surfaces, where the top boundary is translating: y = B y = kb y =
More informationGEFD SUMMER SCHOOL Some basic equations and boundary conditions (This is mostly a summary of standard items from fluids textbooks.
GEFD SUMMER SCHOOL Some basic equations and boundary conditions (This is mostly a summary of standard items from fluids textbooks.) The Eulerian description is used; so the material derivative D/Dt = /
More informationAdvection, 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 informationVectors Part 1: Two Dimensions
Vectors Part 1: Two Dimensions Last modified: 20/02/2018 Links Scalars Vectors Definition Notation Polar Form Compass Directions Basic Vector Maths Multiply a Vector by a Scalar Unit Vectors Example Vectors
More informationz g + F w (2.56) p(x, y, z, t) = p(z) + p (x, y, z, t) (2.120) ρ(x, y, z, t) = ρ(z) + ρ (x, y, z, t), (2.121)
= + dw dt = 1 ρ p z g + F w (.56) Let us describe the total pressure p and density ρ as the sum of a horizontally homogeneous base state pressure and density, and a deviation from this base state, that
More informationExercise: concepts from chapter 10
Reading:, Ch 10 1) The flow of magma with a viscosity as great as 10 10 Pa s, let alone that of rock with a viscosity of 10 20 Pa s, is difficult to comprehend because our common eperience is with s like
More informationOCN/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 informationSoft 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 informationFluid dynamics and moist thermodynamics
Chapter 1 Fluid dynamics and moist thermodynamics In this chapter we review the equations that represent the physical laws governing atmospheric motion. Rigorous derivations of the equations may be found
More informationGeneralizing the Boussinesq Approximation to Stratified Compressible Flow
Generalizing the Boussinesq Approximation to Stratified Compressible Flow Dale R. Durran a Akio Arakawa b a University of Washington, Seattle, USA b University of California, Los Angeles, USA Abstract
More informationCalculus II. Calculus II tends to be a very difficult course for many students. There are many reasons for this.
Preface Here are my online notes for my Calculus II course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus
More informationChapter 4: Fundamental Forces
Chapter 4: Fundamental Forces Newton s Second Law: F=ma In atmospheric science it is typical to consider the force per unit mass acting on the atmosphere: Force mass = a In order to understand atmospheric
More informationCONSERVATION 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 informationCoordinate systems and vectors in three spatial dimensions
PHYS2796 Introduction to Modern Physics (Spring 2015) Notes on Mathematics Prerequisites Jim Napolitano, Department of Physics, Temple University January 7, 2015 This is a brief summary of material on
More informationPhysics Dec Time Independent Solutions of the Diffusion Equation
Physics 301 10-Dec-2004 33-1 Time Independent Solutions of the Diffusion Equation In some cases we ll be interested in the time independent solution of the diffusion equation Why would be interested in
More informationShell/Integral Balances (SIB)
Shell/Integral Balances (SIB) Shell/Integral Balances Shell or integral (macroscopic) balances are often relatively simple to solve, both conceptually and mechanically, as only limited data is necessary.
More information