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    ME 3250Fluid Dynamics I

    Spring 2014

    Prof. Mike Renfro

    AUST 110

    TuTh 2:00-3:15 PM

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    Handouts

    Course Policy

    Syllabus

    Academic Conductfill out and returnNews articleMars Orbiter

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    Course Information (1)

    Text: Munson, B.R., Rothmayer, A.P., Okiishi,

    T.H., and Huebsch, W.W., Fundamentals of

    Fluid Mechanics,7thedition, Wiley (2013).

    Office: UTEB 472

    Phone: 486-2239

    E-mail: [email protected] Office Hours: M 10-11, Tu 10-11, Th 3:30-

    4:30 (or by appointment)

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    Course Information (2)

    HuskyCT Site

    Updated homework assignments, announcements

    Link to publisher websitevideos, sample quizzes

    Lecture notes

    Old test solutions

    Grades

    Pre-req: ME 2233 (Thermo I) and its pre-reqs

    (calculus, physics, etc.)

    Pre-req quiz (5%) on Tu 1/28, 30 minutes in class

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    Grade Distribution (1)

    Homework (10%)

    2-3 problems assigned each lecture due followingThursday

    2-3 problems graded per setall must be turned infor full credit

    No late homework accepteddue at start of class

    Format/units (see Mars Lander handout)

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    Grade Distribution (2)

    i>clicker 2 questions (15%)

    3-4 clicker questions each class will be used to:

    Review material from previous lecture

    Test new material presented during lecture See if concepts need additional attention

    Grading for clicker questions is credit foranswering + credit for answering correctly

    You may usually talk through your answers withother students unless I specifically say otherwise

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    Attendance

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    Grade Distribution (3)

    Exams (35%)

    Exam 1Tu March 4

    Exam 2Tu April 22

    1 crib sheet can be brought to each exam

    Final Exam (25%)

    Tu May 6, 1-3pm (tentative)

    Comprehensive

    2 crib sheets can be brought to exam

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    Grade Distribution (4)

    FLUENT Projects (10%)

    One or two projects using FLUENT software

    2ndfloor ME computer lab in E-II

    Project includes brief 1 page report and data

    analysis

    Final grades are relative to class performance

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    Academic Honesty

    Anything you turn in for a grade must be

    entirely your own work

    Cheating includes obvious copying of exams

    or homework but also listing the books or a

    friends answer or working backwards from this

    answer

    As long as you turn in your own work, I

    encourage working with others on homework

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    Intro to Fluid Dynamics (1)

    What is a fluid? (Chap. 1)

    When a small force is applied to a solid, the solid

    strains (displacement) until the stresses in the

    solid balance the forceat equilibrium a solid is atrest and if the force is removed the solid recovers

    stress

    F

    strain

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    Intro to Fluid Dynamics (2)

    When a force is applied to a fluid (liquid or gas),

    the stresses lead to continuous straining (a rate of

    strain) via fluid motion (velocity)at equilibrium

    a fluid flows Fluids flow even for infinitesimally small stresses

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    Intro to Fluid Dynamics (3)

    Fluid Statics = A study of forces caused by

    stationary fluids

    Fluid Dynamics = A study of fluid (gas or

    liquid) motion due to applied forces

    Fluid Mechanics = A study of forces caused byfluid motion

    These are often used synonymously

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    Applications of Fluid Mechanics (1)

    Study the behavior of fluids at rest

    Fluid statics analysis (Chap. 2)

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    Applications of Fluid Mechanics (2)

    Study the forces that fluids impart on systems

    Study the global behavior of fluid in a system

    Integral (control volume) analysis (Chap. 5)

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    Applications of Fluid Mechanics (3)

    Study the local behavior of fluid flow

    Differential analysis (Chap. 6)

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    Properties of Fluids (1)

    Pressure

    Density

    Ideal gas

    Most liquids

    331

    f t

    lbm

    m

    kg

    vV

    m

    ][)(22

    atmpsiinlbfPaor

    mN

    AFp

    RT

    pRTp

    const

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    Properties of Fluids (2)

    Specific weight

    Specific gravity

    33 ft

    lbf

    m

    Ngg

    water

    SG

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    Viscosity (1)

    You should be familiar with concepts of

    density, temperature, pressure, and velocity

    Viscosity is one of the most important

    properties (with density) that make fluids

    behave differently

    ViscosityMovie

    Viscosity of the oil is 104larger than water

    same density

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V1_1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_1.mov
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    Viscosity (2)

    What is viscosity?

    All fluids strain when a stress is applied

    The viscosity, m, is the stress, , required to

    achieve a given rate of strain,(we will discuss fluid strain and stress

    more in Chap. 6)

    g

    dy

    du

    tt

    g

    0lim

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    Viscosity (3)Newtonian Fluids A velocity gradient is a rate

    of shearing strain

    As the stress increases (bypulling the upper platefaster), the shearing strainincreases (the fluid flows

    with a steeper velocitygradient)

    Newtonian fluids have alinear relationship between and strain rate (du/dy)

    This linear coefficient is theviscosity, m

    dy

    dum

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    Viscosity (4)Non-Newtonian

    Fluids

    Non-Newtonian fluidsdo not have a linearrelationship betweenstress and strain rate

    Non-Newtonian Movie Corn starch forms a

    shear thickening fluidviscosity is larger for

    high shearing strain In this course we will

    deal with mostlyNewtonian fluids

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V1_4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_4.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_4.mov
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    Viscosity (5) - Units

    Rate of shearing strain

    Stress is a force per unit area

    Viscosity

    Also called dynamic viscosity

    Kinematic viscosity

    (other units exist for viscositybe careful with

    unit conversions)

    ][ 1 sdy

    dug

    22

    in

    lbfor

    m

    N

    22in

    slbfor

    m

    sNm

    s

    ftor

    s

    m 22

    m

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    Viscosity (6)Temperature Effects

    Viscosity stronglydepends on T

    For liquids viscosity

    decreases with T (effectsof molecular interactionsdecrease)

    For gases viscosity

    increases with T (effectsof molecular collisionsincrease)

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    Viscosity (7)Walls and No-slip

    At walls, viscosity causes the fluid to stick to

    the wall the wall and fluid have the same

    velocity (otherwise there would be infinite

    strain and stress)

    This is called the No-slip condition (movie)

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V1_2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_2.mov
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    30 minutes

    Pre-req Quiz

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    Additional Fluid Properties (1)

    Bulk modulus

    If p is increased, how much does decrease (as a percentage)?

    since

    Evis the Bulk Modulus with [N/m2]

    Change in pressure required for a givenpercentage change in density or volume

    ddp

    ddpEv

    /

    m

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    Bulk Modulus

    For ideal gases undergoing an isothermal

    process

    Thus, bulk modulus for air is of the order 105

    N/m2

    For liquid water Ev=2x109N/m2

    CRTp

    Cddp

    pCd

    CdEv

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    Additional Fluid Properties (2)

    Liquid Surfaces Gases characterized by weak intermolecular forces

    (flying through space unaffected by neighbors)

    Liquids characterized by significant intermolecularforces

    In liquid neighboring molecules pull

    equally in all directions

    At surface, net force is down(balanced by pressure)

    Surface also stretches molecules

    (surface tension)

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    Surface Tension (1)

    Droplet Formation

    Liquids in free space form spherical drops due to

    surface tension

    Surface tension is property of fluid, [N/m] =

    force per unit length of surface

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    Surface Tension (2)

    Pressure in a drop

    Balance of forces on cross

    section of drop

    22 RpAppRL ambdrop

    Rp

    2

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    Surface Tension (3)

    Bending of surface around an object provides

    tension that must be broken

    Surface acts like membrane

    Movie

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V1_5.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V1_5.mov
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    Surface Tension (4)

    At solid surface, attractive force between the

    solid and liquid molecules can be large enough

    to overcome surface tensionthe solid surface

    wets (fig A)

    Or, too weak and the surface does not wet (fig

    c)

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    Surface Wetting

    Wetting pulls liquid up small capillary tubes

    Angle of liquid surface causes net upward

    force to balance weight of liquid

    g cos22 RhRgmg

    g

    Rh

    cos2

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    Additional Fluid Properties (3)

    Vapor Pressure

    Fast molecules from liquidwith sufficient velocity toescape surface evaporate

    Slow molecules from gasare captured by surface forcesand condense

    Equilibrium occurs whensufficient vapor exists(evaporation=condensation)

    This equilibrium pressure isVapor pressure, pv

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    Vapor Pressure

    Vapor pressure, pv[N/m2] or [atm] is a

    property of a liquid and is strongly T

    dependent

    At boiling T, vapor pressure = 1 atm

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    Fluid Statics

    If fluid has no velocity (left), or is movingtogether as a system (right) then there is nostraining motion

    Each fluid element moves rigidly with itsneighbor

    No shear force on fluid00 m

    dy

    duand

    dy

    du

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    Balance of Forces (1)

    If there is no shear force, only forces on fluid

    element are pressure and weight

    Can balance forces on any control volume

    (fluid element) using Newtons Second Law

    (F=ma)

    Wedge chosen

    as an example

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    Balance of Forces (2)

    Consider case where acceleration is zero

    Regardless of angle, pressure acts equally in all directions,even with acceleration (see further analysis on text p. 39)

    However, pressure can vary from point to point

    sxpzxp

    maF

    sy

    yy

    sin0

    0

    zs sin

    sy

    sy

    pp

    zxpzxp

    0

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    Variation of pressure in a fluid (1) Now consider a finite sized

    cubic fluid element with

    pressure variations

    y

    y

    yy

    y

    ay

    p

    zyxazyxy

    p

    zyxama

    zxyyppzxy

    yppF

    22

    xax

    p

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    Variation of pressure in a fluid (2) In z-direction, gravity also acts

    on fluid element

    gaz

    p

    zyxgazyxz

    p

    mamgyx

    z

    z

    p

    pyx

    z

    z

    p

    pF

    z

    z

    z

    22

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    Variation of pressure in a fluid (3)

    gaz

    pa

    y

    pa

    x

    pzyx

    ,,

    kzpj

    ypi

    xpp

    kajaiaa zyx

    gap

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    Fluids at Rest (1)

    If a=0

    For an

    incompressible fluid(density=constant)

    g

    gzp

    y

    p

    x

    p

    gp

    0

    hp

    zzgpp

    gdzdp

    BABA

    g

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    Fluids at Rest (2)

    For a compressible fluid

    (e.g., ideal gas)

    If T, R, and g areconstant with height then

    dzRT

    g

    p

    dp

    RT

    pgg

    dz

    dp

    RTp

    )0(

    exp

    ln

    pCwhere

    RT

    gzCp

    C

    RT

    gzp

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    Standard Atmosphere

    Considerable variations in atmospheric

    properties with height

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    Pressure Measurements

    Barometers measure

    atmospheric pressure

    through balance of

    pressure and liquidmercury weight

    vapatm php g

    0105.1000023.0 6, atmpsip Hgvap

    Hgmmatm 7601

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    Manometers (1)

    Manometers use fluid

    statics to measure

    relative pressure

    between two points Balance of forces

    If fluid A is a gas, then

    pressure rise at h1is

    negligible (gA

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    Gage Pressure

    Manometers do not determine absolutepressure (since patmis not measured)

    Gage pressure is pressure relative to

    atmospheric Gage pressure can be negative for absolute

    pressures below atmospheric

    Usually, if no information is given on apressure it is assumed to be a gage pressure

    gatmA ppph g

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    Manometers (2)

    Inclined manometers increase the sensitivity of

    the measurement

    g sin2221 lpp

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    Bourdon Pressure Gages

    Metal tube straightens as pressure increases movingneedle

    Set to zero for atmospheric pressure Gage pressuremeasurement

    Most common pressure sensor

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    Forces on Surfaces (1)

    For fluids in motion both pressure and shear

    forces act on surfaces

    For static fluids (including systems in motion

    as a whole) there is no shear force

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    Forces on Surfaces (2)

    Net force is for isobaric

    surfaces (normal to gravity, horizontal)

    Atmospheric pressure cancels since acting on

    both sides of surface (usually)

    Gage pressure

    ghApAF

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    Inclined Surfaces (1)

    For an inclined surface, the pressure varies

    across the surface

    Defining

    pdAF

    hdApdAFR g

    sinyh

    ydAdAyFR gg sinsin

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    Area Centroid

    Integral is areas Centroid

    Thus, inclined surfaces can be treated as flat if

    area centroid location is known The resulting force acts through another point,

    yR

    AyydA c

    AhAyF ccR gg sin

    Inclined Surfaces (2)

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    Inclined Surfaces (2)

    For moment of

    resulting forceto act like actual

    distributed force

    dAypydAyF RR g sin2

    g sincR AyF

    Ay

    I

    Ay

    dAyy

    c

    x

    c

    R

    2

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    Tabulated Moments of Inertia

    See p. 59-60for

    transformations

    to use tabulated

    Ixy data

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    Example: Plane surface (wall)

    Centroid is located at h/2

    Thus, force is

    Force actsthrough 2/3 point

    2/hAFR g

    hhhhbh

    bdyyAydAyy

    h

    c

    R32

    32

    ))(2/(

    3

    20

    22

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    Example: Curved surface

    Direct integration difficult for arbitrary surface

    Free body diagram used to compute resultant

    force

    F1 is gh acting through center of surface AB

    F2 acts through 2/3 point of surface AC

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    Buoyancy (1)

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    Buoyancy (1) . Submerged surfaces are subject to

    an upward force due to higherpressure at greater depth

    Force balance on only fluid in

    ABCD

    Buoyancy force only depends on

    volume (Movie)

    AB

    AhF11

    g CD

    AhF22

    g

    g

    ggg

    B

    objB

    fluidB

    F

    AhhAhFAh

    WFFF

    1212

    12

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V2_5.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V2_5.mov
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    Buoyancy (2)

    Since buoyancy only depends on volume and

    not density of the object, its force acts through

    the centroid of the object (see p. 70 for formal

    proof) However, the weight of the object acts through

    the center of mass which is different from the

    centroid if the object is not homogeneous

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    Stability

    If center of mass is below centroid, object will be stable

    If center of mass is above centroid, object will be unstable

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    Rigid body rotation (1)

    Last time we derived for accelerating systems

    For a rotating system

    gap

    gazz

    pp

    rr

    r

    pp

    1

    rra 2

    2rr

    p

    g

    z

    p

    dzdrrdp g 2

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    Rigid body rotation (2)

    Along an isobar dp=0 dzdrrdp g 2

    g

    r

    dr

    dz 2

    cg

    rz

    2

    22

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    Fl Vi li i (1)

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    Flow Visualization (1)

    Dye injectionin liquid canbe used tosee flow dye followsthe flow

    Photographsshow streaksrepresenting

    path of fluidparticles

    Fl Vi li i (2)

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    Flow Visualization (2)

    Streaklines, Pathlines, and Streamlines

    Streaklines = instantaneous location of fluid

    particles that once passed through a specified point

    inject dye continuously at fixed points and take snapshotat later time

    Pathlines = path that particles follow

    inject dye briefly at fixed points and take time-lapsed

    photo for a period of time

    Moviedifferences in pathlines and streaklines

    St li

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V4_6.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V4_6.mov
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    Streamlines

    Streamlines = lines in the flow that are locally

    tangent to the velocity of the fluid

    St li (2)

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    Streamline (2)

    Streamlinesdetermined bymeasuringinstantaneousvelocity and

    integrating tofind tangent lines

    Harder tomeasure than

    streaklines Most useful to

    mathematicallydescribe flow

    St li (3)

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    Streamlines (3)

    For steady flowspathlines, streaklines, andstreamlines are identical - Movie

    I i id Fl (1)

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V4_5.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V4_5.mov
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    Inviscid Flow (1)

    In real fluids, if there is fluid motion with non-

    uniform velocity then there will be strain and

    shear forces

    However, it is often true that these shear forcesare much smaller than forces due to pressure

    gradients or gravity

    In these cases the fluid is assumed to beinviscid (m=0)

    I i id Fl (2)

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    Inviscid Flow (2)

    Inviscid flows are not strongly affected by drag atsurfaces and can flow around sharp corners

    Viscid flows are slowed by drag at the surface much

    more strongly

    I i id Fl (3)

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    Inviscid Flow (3)

    Changes in overallvelocity or geometryof a problem canchange the

    importance ofviscous forces

    Some regions of a

    flow may be inviscidwhile others showstrong viscouseffects

    St li A l i

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    Streamline Analysis

    Fluid particlessubject to

    Since streamline is

    tangent to V

    amF

    nsVV 0

    dt

    ds

    s

    V

    dt

    dn

    n

    V

    sanadt

    Vda sn

    2Van V

    s

    Vas

    F Al St li (1)

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    F=ma Along Streamline (1)

    Note: we are applying F=ma to a fluid particlethe fluid particle follows a

    pathline but we are using derivatives along the streamline to represent the

    fluid acceleration the flow must be steady

    Force due to gravity along streamline is

    s

    VVamF ss

    sinsin,

    gmgFg

    s

    F Al St li (2)

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    F=ma Along Streamline (2)

    Force due to pressure:

    2

    s

    s

    pppp s

    2

    s

    s

    pppp s

    2

    s

    s

    pps

    s

    pynp

    ynppynppF

    s

    ssps

    2

    )()(,

    F Al St li (3)

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    F=ma Along Streamline (3)

    Net Force = ma

    Note: we have not included shear forceswe are assuming the flow is

    inviscid (pressure forces are more important than viscous forces)

    In static fluids the pressure gradient was balanced by gravity

    In moving fluids, any imbalance in pressure and gravity (LHS) causes fluidparticle acceleration (RHS)

    s

    VV

    s

    pgF sin

    s

    VV

    s

    pg

    sin

    F ma Along Streamline (4)

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    F=ma Along Streamline (4)

    Along streamline

    Since along a streamline dn=0, for any derivative

    partial and ordinary derivates in s are the same

    (Note: analysis is limited to along a streamline)

    Finally,

    s

    VV

    s

    pg

    sin

    s

    z

    sin

    0

    szg

    sVV

    sp

    dss

    pdn

    n

    pds

    s

    pdp

    s

    V

    s

    VV

    2

    2

    F ma Along Streamline (5)

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    F=ma Along Streamline (5)

    Integrating along a streamline

    If we assume fluid is incompressible (negligible density change)

    02

    2

    ds

    dzg

    ds

    dV

    ds

    dp

    02

    1 2 gdzdVdp

    .2

    1 2 ConstgdzdVdp

    .2

    1 2 ConstgzVdp

    .2

    1 2 ConstgzVp

    Bernoulli Equation

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    Bernoulli Equation

    Can only be applied to

    Steady flow

    Inviscid flow

    Incompressible flow

    Flow along a streamline

    .21 2 ConstgzVp

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    Stagnation Flow

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    Stagnation Flow

    Stagnation point occurswhere flow is diverted around

    two sides of an object

    Dividing streamline includesstagnation point

    Movie

    Flow decelerates toward

    stagnation point (higherpressure)

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V3_3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V3_3.mov
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    Pressure Along Dividing Streamline

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    Pressure Along Dividing Streamline

    F=ma Normal to Streamline (1)

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    F=ma Normal to Streamline (1)

    Force due to gravity across streamline is

    Force due to pressure

    coscos, gmgF gn

    2V

    amF nn

    n

    p

    ysp

    ysppysppF

    n

    nnpn

    2

    )()(,

    2

    n

    n

    ppn

    F=ma Normal to Streamline (2)

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    F=ma Normal to Streamline (2)

    Across streamline

    Since normal to streamline ds=0, for any derivative

    partial and ordinary derivates in n are the same

    (Note: analysis is limited to normal to a streamline)

    n

    z

    cos

    dnn

    p

    dnn

    p

    dss

    p

    dp

    2

    cos V

    n

    pg

    02

    nzgV

    np

    02

    gdzdnV

    dp

    F=ma Normal to Streamline (3)

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    F=ma Normal to Streamline (3)

    Integrating normal to streamline

    If we assume fluid is incompressible

    .

    2

    Constgzdn

    V

    p

    0

    2

    gdzdnV

    dp

    .2

    ConstgdzdnVdp

    Steady Inviscid Incompressible Flow

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    Steady, Inviscid, Incompressible Flow

    Along streamline:

    Across streamline:

    Pressure changes along streamline accelerates fluid

    particles Pressure changes normal to streamline turns fluid

    particles (changes streamline direction)

    .

    2

    ConstgzdnV

    p

    .21 2 ConstgzVp

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    Movie

    Bernoulli Equation

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V3_1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V3_1.mov
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    Meaning of Terms

    Along the streamline:

    Static pressure dynamic pressure hydrostatic pressure total pressure

    Each term has units of pressure (N/m2)

    Static pressure = actual local pressure in the flow (thermodynamic pressure)

    Dynamic pressure = pressure change due to velocity

    Hydrostatic pressure = pressure change due to height

    Total pressure = sum of all parts = constant

    The Bernoulli Equation conserves pressurepressure is only converted from one

    type to another (Static/Dynamic/Hydrostatic)

    TpConstgzVp .2

    1 2

    Bernoulli Equation

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    Bernoulli Equation

    Dividing by specific weight:

    Pressure head velocity head elevation head total head

    Each term has units of height (m)

    Instead if we multiply by specific volume:

    Elevation head is equivalent to potential energy per unit mass

    Velocity head is equivalent to kinetic energy per unit mass

    Pressure head is equivalent to flow work (pv) per unit mass

    The Bernoulli Equation is a form of energy conservation (with no thermal orviscous losses or work or heat additions)

    Hzg

    Vp

    2

    2

    g

    .2

    2

    ConstgzV

    pv

    Example Stagnation Streamline

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    ExampleStagnation Streamline

    Free stream: high velocity (high dynamic pressure),low static pressure

    Stagnation point: zero dynamic pressure, high staticpressure

    E.g., sticking your hand out the window of a movingcar

    z

    Stagnation Pressure

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    Stagnation Pressure

    Stagnation pressure = pressure that will beachieved if a fluid is brought to rest

    Neglecting hydrostatic pressure, stagnation

    pressure is simply static + dynamic pressure

    2

    22

    2

    1

    2

    1.

    2

    1

    Vpp

    VpConstVp

    stag

    stagstag

    Measuring Static/Stagnation Pressure

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    Measuring Static/Stagnation Pressure

    A side wall tap measures thestatic pressure by convertingstatic pressure tohydrodynamic pressure(pressure head)dynamic

    pressure at point 3 is zero (noslip)

    Tap facing into flow convertsstatic and dynamic pressureto hydrostatic pressure -dynamic pressure at point 2 is

    NOT zero

    Pitot-static Tube

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    Pitot static Tube

    Two concentrictubesone with a

    forward facing tap

    and the other with aside tap

    43

    4

    2

    3

    2

    2

    1

    ppV

    pp

    Vpp

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    Why does the pressure drop between 2 and 1?

    Hydraulic grade line

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    Hydraulic grade line

    Energy line istotal head offluid(measured by

    pitot tube)

    Hydraulicgrade line is

    pressure andelevation

    head only(measured bystatic tube)

    Why does HGL decrease in this system?

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    Modified Bernoulli Equation

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    Modified Bernoulli Equation

    Bernoulli equation can only be used for steady,inviscid, incompressible flow

    Modified forms of equation can be used

    carefully for special cases of compressible,viscous, or unsteady flows

    E.g., unsteady Bernoullis (see Sect. 3.8)

    2

    2

    22

    1

    2

    1

    2

    112

    1

    2

    1gzVpds

    t

    VgzVp

    s

    s

    Application of Bernoulli Equation

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    Application of Bernoulli Equation

    Free Jets

    Confined Flows

    Venturi and orifice flowmeters

    Sluice gates

    Free Jets

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    Free Jets

    If streamlines are straight at jet exit (free jet,

    R=) then no pressure gradient across jet,

    p2=p1 V1=0

    2

    2

    221

    2

    112

    1

    2

    1

    gzVpgzVp

    )(2 212 zzgV

    ghV 22

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    Vena Contracta (1)

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    Vena Contracta (1)

    If a jet exit is too sharp,the streamlines cannotturn and flow is smallerthan hole diameter

    90turn would require

    infinite pressure gradient At a-a, pressure is

    atmospheric across jet(straight streamlines)

    p2>p1to cause streamline

    curvature Contraction coefficient =

    Aj/Ah

    Vena Contracta (2)

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    Vena Contracta (2)

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    Confined Flows

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    Co ed ows

    If a flow is completely enclosed, then no pointmust be at atmospheric pressure (or zero

    velocity)less information known

    Must generally use VAQ VAmassflow

    Repeat for static taps on

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    Repeat for static taps on

    both ends of the manometer

    Cavitation

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    High velocity causesa locally lowpressure

    If static pressuredrops below vapor

    pressure of liquidboiling occurs

    This cavitationleads to gas bubbles

    flowing in liquiduntil pressureincreases

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    Orifice and Venturi Meters

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    Restriction in flow byorifice or Venturi tube

    (specially shaped nozzle)

    causes increase invelocity and decrease in

    pressure

    For horizontal flow:222

    2

    112

    1

    2

    1VpVp

    2211 AVAVQ

    Flow Meters

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    Flow rate is directly related to static pressure

    drop across orifice and known geometry

    In practice, these must be calibrated to account

    for non-uniform flow, viscous effects, etc.

    2

    1

    2

    212

    2

    1

    2

    2

    2

    22

    1

    2

    2

    21

    1

    2

    122

    1)(

    A

    A

    ppAQ

    AA

    AQ

    AQ

    AQpp

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    Sluice Gates

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    2

    2

    221

    2

    11

    2

    1

    2

    1gzVpgzVp 2211 AVAVQ

    2

    221

    2

    12

    1)(

    2

    1VzzgV

    2

    12

    21

    2

    2

    1

    222

    2

    2

    21

    2

    1

    2

    2

    21

    1

    )(2

    1)(2

    2

    1)(

    zz

    zzgbzQ

    z

    zQzbzzg

    bz

    Q

    bz

    Qzzg

    12 2gzbzQ If z1>>z2

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    ***ADD**

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    Add physical interpretation of acrossstreamline equation and a couple of examples

    E.g. Pressure in a vortex

    Chapter 4 - Fluid Kinematics

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    p

    Fluid kinematics = study of fluid motionwithout concern for forces driving flow

    What is the velocity, acceleration?

    In reality, a point in space does not have avelocity (it is unlikely there is a moleculeexactly at that point)

    When we talk velocity of a fluid particle wemean the average over all molecules in a smallregionContinuum Hypothesis

    Eulerian - Velocity Field

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    Eulerian description of flow describes the velocity at

    all points at all times Velocity field is a 3 component vector in 4

    dimensions Movie

    Some simpler cases often occur: Steady 3-D flow

    Unsteady, 2-D flow

    Movie Steady, 2-D flow

    Movie

    Steady, 1-D flow

    ktzyxwjtzyxvitzyxuV ),,,(),,,(),,,(

    kzyxwjzyxvizyxuV ),,(),,(),,(

    jyxviyxuV ),(),(

    ixuV )(

    jtyxvityxuV ),,(),,(

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V4_2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V4_1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V4_3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V4_3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V4_1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V4_2.mov
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    LagrangianParticle Velocity

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    Lagrangian description offlow follows individualparticles

    The position of a fluid particleis

    The velocity of that fluidparticle is

    This does not tell us what thevelocity will be at a pointaway from the fluid particle

    ktzjtyitxr )()()(

    dt

    rdV

    Acceleration - Eulerian

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    In Eulerian representation of velocity

    Since x, y, z do not depend on tEulerian =

    fixed coordinates

    dt

    Vd

    a

    ktzyxwjtzyxvitzyxuV ),,,(),,,(),,,(

    t

    Vk

    t

    wj

    t

    vi

    t

    u

    dt

    Vd

    AccelerationLagrangian (1)

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    In Lagrangian representation of velocity, x, y,

    and z depend on t as particle follows flow

    dt

    Vd

    a

    kttztytxwjttztytxvittztytxuV )),(),(),(()),(),(),(()),(),(),((

    kdt

    dz

    z

    w

    dt

    dy

    y

    w

    dt

    dx

    x

    w

    t

    w

    jdtdz

    zv

    dtdy

    yv

    dtdx

    xv

    tv

    idt

    dz

    z

    u

    dt

    dy

    y

    u

    dt

    dx

    x

    u

    t

    u

    dt

    Vd

    u

    dt

    dx

    vdtdy

    wdt

    dz

    AccelerationLagrangian (2)

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    Lagrangian acceleration includes local acceleration

    and change in velocity due to fluid particle motion

    kzww

    ywv

    xwu

    jz

    vw

    y

    vv

    x

    vu

    iz

    uw

    y

    uv

    x

    uu

    t

    V

    dt

    Vd

    VV

    t

    V

    dt

    Vd

    kwjviuV

    k

    z

    j

    y

    i

    x

    Material Derivative

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    Material or Substantial derivative includeslocal and spatial variation in a quantity

    VVt

    V

    Dt

    VD

    termconvective

    termunsteady

    VtDt

    D

    z

    Tw

    y

    Tv

    x

    Tu

    t

    T

    Dt

    DT

    Steady/Unsteady Flow

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    In steady flow, partial derivatives with respectto time are zero

    However, spatial variations can still cause a

    derivative for a fluid particle to be non-zero

    z

    Tw

    y

    Tv

    x

    Tu

    z

    Tw

    y

    Tv

    x

    Tu

    t

    T

    Dt

    DT

    0Dt

    DT0

    t

    T

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    Control Volumes / Systems

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    Just like in Thermo we sometimes consider systems (controlmass, Lagrangian) and sometimes control volumes (Eulerian)

    We will start with relationships for systems

    m=constant

    F=ma

    and derive from them relationships for control volumes

    Steady Control Volume Unsteady Control VolumeControl Mass

    Extensive/Intensive

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    Consider an intensive property b (units/kg) B=mb (units)

    systemsystem

    sys bdbdmB cvvolumecontrol

    cv bdbdmB

    dt

    bdd

    dt

    dB syssys

    dt

    bdd

    dtdB cvcv

    dt

    dB

    dt

    dB syscv

    Example: Mass, b=1 (B=m)

    0dt

    dmsys mdt

    dmcv

    Relating CV and CM (1)

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    Consider a control mass that moves from (1) to (2) int

    Define control volume as common region plus regionI

    Relating CV and CM (2)

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    III

    cvsys

    BBt

    B

    Dt

    DB

    22221111 bVAbVA

    t

    B

    Dt

    DBcvsys

    AVbbmB

    Rate change of property B for a system equals

    change for a control volume plusinflow/outflow to control volume

    Relating CV and CM (3)

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    In general: velocity may not be normal to surface

    property and velocity may vary across surface

    csout dAnVbbmB

    cv

    cv bdtt

    B

    Reynolds Transport Theorem

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    Formal relationship between changes in a control

    mass and control volume This is similar to a material derivative except it

    applies to a finite sized control volume and materialderivative applies at a point

    RTT becomes material derivative as volume goes tozero

    cscvsys dAnVbbd

    tDt

    DB

    bVt

    b

    Dt

    Db

    Deforming Control Volumes

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    Reynolds Transport Theorem can be applied to

    any CV even if it moves

    V is relative velocity

    = V1-V0

    cscv

    sysdAnVbbd

    tDt

    DB

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    3:57 (19thOlympics) versus 4:02 (20th

    Olympics) speed skating (3000 m)

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    19th- Salt Lake City (4200 ft): p = 650 mm Hg

    20th- Turin, Italy (810 ft): p = 739 mm Hg

    Density atm = 1.23 kg/m3

    Find difference in air resistance

    Conservation of Mass (1)

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    Mass is conserved for a system, which bydefinition is a control surface that follows a

    specific set of mass

    Apply Reynolds Transport Theorem with B=mb=B/m=1

    cscv

    sysdAnVbbd

    tDt

    DB

    cscv

    sysdAnVd

    tDt

    Dm

    Conservation of Mass (2)

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    Since mass is conserved, msys=constant For a closed control surface (a system)

    Lagrangian description

    For an open control volume (Eulerian

    description), this becomes

    0

    cscv

    dAnVdt

    dDt

    D

    Dt

    Dmsys

    0

    Integral Continuity Equation

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    For steady flows this becomes

    The term

    since mass flow is out if V is in same direction as n

    0

    cscv dAnVdt

    0 cs

    dAnV

    inoutnetcs

    mmmdAnV

    Uniform Flow

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    Recall from Thermo 1, for uniform flow

    We define average velocity at an inlet or outlet so that

    We usually mean average velocitywhen we speak about the velocity at

    inlet or exit

    VAm

    AVm

    A

    dAnV

    V cs

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    Moving Control Volumes

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    If the controlvolume is moving,the coordinatesystem is fixed tothe control volume

    The velocity offlow across thecontrol surface isevaluated relativeto this coordinate

    system

    0

    cscv

    dAnWdt

    cvVVW

    Deforming Control Volumes

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    Deforming control volumes are both movingand unsteady

    Local relative velocities used at all surfaces

    0

    cscv

    dAnWdt

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    Newtons Second Law

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    but since mass is conserved for a system

    Sum of forces on a system in any direction

    equals change in momentum for the system in

    that direction

    DtVDmamF syssyssys

    Dt

    VmDF

    sys

    sys

    Linear Momentum Equations (1)

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    Applying Reynolds Transport Theorem withB=mV, b=V

    This is a vector equation (it is really 3equations in x, y, z directions)

    cscv

    sysdAnVVdV

    tF

    cscv

    sysdAnVbbd

    tDt

    DB

    Dt

    VmDF

    sys

    sys

    Integral Linear Momentum

    Equations

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    cscv

    sysdAnVVdV

    tF

    cscv

    z

    cscv

    y

    cscv

    x

    dAnVwwdtF

    dAnVvvdt

    F

    dAnVuud

    t

    F

    zyx nwnvnunV

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    Moving Reference Frame

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    Moving control volumescan be used

    Inertial reference frame(not accelerating)

    Non-inertial referenceframe (accelerating)

    Accelerating referenceframes require caution

    (relative velocities cannotsimply be used)

    Steady Inertial Control Volume

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    cscvsys

    dAnWVdV

    t

    F

    cvVVW

    cs

    cv

    steady

    cv

    cvsysdAnWVWdVW

    tF

    )(0

    cs

    cv

    cs

    sysdAnWVdAnWWF

    )(0

    onconservatimass

    cs

    cv

    cssys

    dAnWVdAnWWF

    cs

    sysdAnWWF

    Relative velocities used for

    inertial control volumes

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    Rotating Systems (1)

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    For rotating systems, the moment of themomentum equation (angular momentum) is

    often convenient

    At a point, F=ma becomes Taking cross product of location of fluid

    particle and F=ma

    Dt

    dVD

    Dt

    maDF

    )(

    dVDt

    rD

    Dt

    dVrD

    Dt

    dVDrFr

    )0(0

    VV

    dVVDt

    dVrDFr

    Rotating Systems (2)

    I t ti t l l

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    Integrating over a control volume

    Applying Reynolds Transport Theorem(b=rxV)

    Moment of momentum equation

    syssyssys

    dVrDt

    D

    Dt

    dVrDFr

    sys

    dVrDt

    DFr

    cscvsys

    sysdAnVbbd

    tbd

    Dt

    D

    Dt

    DB

    cscv

    dAnVVrdVrt

    Fr

    Moment of momentum equation

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    For a steady system

    Sign of rxV and T terms determined by right-

    hand-rule

    cs fluxmomentumangular

    momentumangularunsteady

    cvsystemontorquesofSum

    dAnVVrdVr

    t

    T

    cs dAnVVrT

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    Energy Equation (1)

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    From Thermo 1, energy of a system follows

    where u is the internal energy from thermodynamics

    innetinnet

    sysWQ

    Dt

    DE,,

    syssys edE

    gzV

    upekeue 2

    2

    Energy Equation (2)

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    Applying Reynolds Transport Theorem withB=E, b=e

    cscv

    innetinnet

    sysdAnVeed

    tWQ

    DtDE

    ,,

    gzV

    upekeue 2

    2

    cscv

    sysdAnVbbd

    tDt

    DB

    Work Work = force through a distance (s)

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    Work force through a distance (s)

    Can occur internal to system (as shaft work) or work at controlsurface (flow work)

    Force at flow surface is pressure times area acting inndirection

    Rate change of distance is velocity

    cs

    shaftinnet sFWW

    ,

    dAnpVWWcs

    shaftinnet

    )(,

    inshaftinnetcscv

    WQdAnVpeedt

    ,,

    Energy Equation (3)

    VV 22

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    For steady, uniform flow

    inshaftinnetcscv WQdAnVgz

    Vp

    udgz

    V

    ut ,,

    22

    22

    hpvup

    u

    inshaftinnet

    cscv

    WQdAnVgzV

    hdgzV

    ut

    ,,

    22

    22

    inshaftinnetinout WQgzV

    hAVgzV

    hAV ,,

    22

    22

    Steady, Uniform FlowVV 22

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    For a 1 inlet, 1 outlet control volume

    If flow is inviscid, shaft work will be zero

    inshaftinnetinout

    inout

    inout

    WQgzgzVV

    hhm,,

    22

    22

    inshaftinnetinout WQgzV

    hmgzV

    hm ,,22

    qgzgz

    VV

    pvpvuu inoutinout

    inoutinout 22

    22

    lossBernoulliquugzgzVV

    pvpv inoutinoutinout

    inout 22

    22

    Extended Bernoulli Equation

    VpVp22

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    In units of length (head)

    hL= head loss

    hs= shaft work head (pump head)

    inshaftin

    in

    in

    in

    out

    out

    out

    out wlossgzVp

    gzVp

    ,22

    slininin

    outoutout hhz

    g

    V

    g

    pz

    g

    V

    g

    p

    22

    22

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    Why Differential Analysis?

    I l l i ll

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    Integral analysis allows us tocompute overall (global) flow

    behavior without concern for thedetailed flow inside a device

    Integral analysis requires careful

    integration at system boundaries(velocity profiles at exits must

    be given or assumed)

    Differential analysis is required

    when we need to know thedetailed flow behavior at pointsinside a system (velocity profilesare computed directly) Recirculation zone will not show up in

    integral analysis

    Fluid Element Motion

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    Differential equations for fluid flow can bederived by considering the motions and forces

    of small fluid elements

    Movierotation/translation/angulardeformation

    Translation

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V6_1.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V6_1.mov
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    Translationfluid elements translate at localfluid velocity zwyvxuV

    tux

    tvy

    twz

    Linear Deformation

    zyx

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    Volume of differential fluid element: Change in volume of fluid element in x direction

    In general for 3-D:

    zdtyxx

    ud

    zyx

    x

    u

    dt

    d

    1

    V

    z

    w

    y

    v

    x

    u

    dt

    d

    1

    Rotation/Angular Deformation (1)

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    Define angles aand as rotation of x and yaxis

    aa

    t

    x

    vtan

    t

    y

    utan

    Rotation/Angular Deformation (2)

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    Rate of rotation of x and y axis

    Note different sign convention for aand

    If OA=-OBthen the fluid element will only rotate andnot deform

    If OA=+OBthen the fluid element will only deformand not rotate

    x

    v

    ttOA

    a

    0lim

    yu

    ttOB

    0lim

    Rotation/Angular Deformation (3)

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    Rate of rotation of fluid elementdefined as average of OA and -OB

    Likewise

    y

    u

    x

    vOBOAz

    2

    1

    2

    zv

    yw

    x21

    xw

    zu

    y21

    Rotation and Vorticity

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    Rotation rate is a vector:

    Vorticity is defined as twice the rotation rate - Movie

    y

    u

    x

    vz

    2

    1

    z

    v

    y

    wx

    2

    1

    x

    w

    z

    uy

    2

    1

    zyx zyx

    wvu

    zyx

    zyx

    V

    2

    1

    2

    1

    V

    2

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V6_2.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V6_2.mov
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    Angular Deformation

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    Rate of angular deformation (rate of shearing strain) offluid element defined as twice the average of OA and

    +OB

    Likewise

    y

    u

    x

    vOBOAz

    22

    g

    z

    v

    y

    wxg

    x

    w

    z

    uyg

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    Mass Conservation (1)

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    Previously we derived mass conservation for acontrol volume

    For a differential element:

    We apply control

    volume equation to

    element and let

    dV0

    0

    cscv

    sysdAnVd

    tDt

    Dm

    Mass Conservation (2)

    0

    sysdAnVd

    Dm

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    As volume0, all flows become uniform and

    volume integral becomes homogeneous

    cscv

    tDt

    0

    22

    22

    22

    xyz

    z

    wwxy

    z

    z

    ww

    zxy

    y

    vvzx

    y

    y

    vv

    zyx

    x

    uuzy

    x

    x

    uuzyx

    t

    cs

    cs

    cscv

    Mass Conservation (3)

    0

    wvu

    t

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    For steady flow:

    For incompressible fluids:

    Incompressible flows have zero deformation(zero dilatation)Velocity field is solenoidal

    zyxt

    0

    V

    t

    0 V

    0 V

    Cylindrical Coordinates

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    Mass conservation can be applied in anycoordinate system:

    For cylindrical coordinates

    Can look up form for gradient in cylindrical

    coordinates:

    0

    V

    t

    0

    11

    z

    vv

    rr

    vr

    rt

    zr

    zvvrvV zr

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    Stream Function (1)

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    For plane (2-D) steady incompressible flow,mass conservation becomes:

    Define the stream function, y(x,y) such that

    Thus, if ycan be found it automaticallysatisfies mass conservation

    0

    V

    t

    y

    v

    x

    uV

    0

    uy

    yv

    x

    y0

    xyyxy

    v

    x

    u yy

    Stream Function (2)

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    What is the stream function?

    For constant y

    Line of constant stream function is a

    streamline

    udyvdxdyy

    dxx

    d

    yyy

    0yduv

    dxdy

    Stream Function (3)

    Since flow cant cross a streamline, flow between two

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    differentially close streamlines: Difference in stream

    function is flow rate

    between streamlines

    yyy

    ddxx

    dyy

    vdxudydq

    12 yyy dvdxudydqQ

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    Potential Flow

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    Potential Flow = Inviscid, Incompressible,Irrotational

    Under these conditions the velocity potential,f, exists and satisfies:

    Laplace equation is linear, so superposition ofsolutions can be used (this is generally not true

    for fluids since full momentum equation isnonlinear)

    02

    f

    gpVVt

    V

    What is the Potential Function?

    vdyudxdydxd

    ff

    f

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    Thus, if then

    Compared to stream function:

    Lines of constant f(equipotential lines) are

    orthogonal to lines of constant y(streamlines)

    yx 0fd

    v

    u

    dx

    dy

    0yduv

    dxdy

    Streamlines/Potential Lines

    Streamlines and

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    Streamlines andequipotential linesform grid describingflow

    High velocities inregions wherestreamlines arecompressed

    Low velocities wherestreamlines expand

    How is Potential Flow Used? First, we find the potential function for several simple flows

    (next slides)

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    (next slides)

    If we can then describe a new flow as a sum of simple flows,the total potential function = sum of simple potential functions

    MovieExample: flow over a surface can be approximated bya point source and uniform flow

    Simple Flow 1: Uniform Flow

    http://localhost/var/www/apps/conversion/tmp/scratch_7/ME250-Fluids/Lectures/V6_3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/ME250-Fluids/Lectures/V6_3.mov
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    Straight flow in one direction:

    If a=0:

    xUu

    facos

    yUv

    fasin

    )(cos

    )(sin

    yfUx

    xfUy

    af

    af

    aaf cossin xyU

    Uxf

    Simple Flow 2: Source/Sink

    Flow originating from a point with equal velocity in

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    all directions and a volume flow rate, m

    Source (m>0), Sink (m

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    Half-body can be approximated as uniformflow + source (Movie)

    Stagnation point occurs at r=b (angle

    f cosUrUx

    rm

    ln2

    f

    f

    f

    f

    sin1

    2cos

    ln2

    cos

    Ur

    v

    rmU

    rv

    rm

    Ur

    r

    U

    mb

    b

    mUvr

    2

    2cos0

    Flow over a Half-body (2)

    http://localhost/var/www/apps/conversion/tmp/scratch_7/ME250-Fluids/Lectures/V6_3.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/ME250-Fluids/Lectures/V6_3.mov
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    Can also find streamfunction for halfbody (see text):

    Value of stream function at stagnation point:

    Streamlines from stagnation point:

    yyy2

    sin m

    Ursourceuniform

    22

    sin

    2

    mm

    U

    mU

    y

    y

    sin2

    2sin

    2sin

    2

    U

    mr

    m

    Ur

    mUr

    m

    Simple Flow 3: Vortex

    Flow circulating about a point

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    Free vortex:

    Free vortex is irrotational

    Other vortices can be rotational No potential function

    f K

    r

    K

    rv

    f

    1

    Circulation

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    Define the circulation for a closed curve in anyflow as:

    For vortex: sdVccw

    rdvsdV

    Kdrr

    K 2

    f2

    Simple Flow 4: Doublet (1)

    A doublet is a combination of a source and sink

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    As a0

    1

    212 ln

    2ln

    2ln

    2 rrmrmrm

    f

    2/1222 cos2 ararrrsource

    2/1221 cos2 ararrrsink

    f

    cos2

    cos41ln

    4 22

    arar

    arm

    r

    am

    arar

    arm

    aa

    f

    cos

    cos2

    cos4

    4limlim

    2200

    Simple Flow 4: Doublet (2)

    A doublet is a combination of a source and sink

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    Streamlines:

    Why is this useful?

    Next slide

    rK

    rma

    f coscos

    fcosKr

    Flow over a Circular Cylinder (1)

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    Lets combine uniform flow and a doublet tosee what flow that represents:

    Choose: where a is the radius of acylinder

    r

    K f

    cosUxf

    f coscoscos2 r

    r

    KU

    r

    KUr

    UaK 2

    f cos12

    2

    Urr

    a

    Flow over a Circular Cylinder (2)

    f cos12

    2

    Ura

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    Thus

    This potential functionhappens to yield flowover a cylinder

    r

    f

    cos12

    2

    U

    r

    a

    r

    vr

    f sin1

    12

    2

    Ur

    a

    rv

    Pressure Distribution over Cylinder

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    Since potential flow analysis assumes inviscidincompressible flow, Bernoullis equation can

    be used

    Pressure on cylinder surface (r=a):22

    2

    1

    2

    1 vpUp surface

    )sin41(2

    1sin4

    2

    1

    2

    1 22222 UpUUppsurface

    sin2sin1 2

    2

    UUr

    av

    Pressure Distribution over Cylinder

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    Comparison ofinviscid and

    experimental

    pressure

    distributions:

    )sin41(2

    1 22 Uppsurface

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    Momentum Equation (1)

    Integral momentum equation for a control

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    g qvolume:

    Apply this to differential element:

    Forces:

    Normal stress,

    (N/m2)

    Shear stress, (N/m2)

    Gravity

    cscv

    sysdAnVVdV

    tF

    Momentum Equation (2)

    In x- directionzyxg

    yzxF x

    yxzxxx

    x

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    Momentum Equation (3)

    In x- direction

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    VAu

    t

    udAnVVdV

    t cscv

    2222

    2222

    2222

    z

    z

    uuyx

    z

    z

    ww

    z

    z

    uuyx

    z

    z

    ww

    y

    y

    uuzxy

    y

    vvy

    y

    uuzxy

    y

    vv

    x

    x

    uuzy

    x

    x

    uu

    x

    x

    uuzy

    x

    x

    uuzyx

    t

    u

    u

    A

    V

    Momentum Equation (4)

    Dividing by volume and equating to force:

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    Subtracting mass conservation times u:

    22

    1

    22

    1

    22

    1

    22

    1

    221

    221

    z

    z

    u

    uz

    w

    z

    wz

    z

    u

    uz

    w

    z

    w

    y

    y

    uu

    y

    v

    y

    vy

    y

    uu

    y

    v

    y

    v

    xxuu

    xu

    xux

    xuu

    xu

    xu

    tug

    yzx x

    yxzxxx

    u

    z

    w

    z

    uwu

    y

    v

    y

    uvu

    x

    u

    x

    uu

    tu

    t

    ug

    yzx x

    yxzxxx

    0

    u

    z

    wu

    y

    vu

    x

    uu

    t

    Momentum Equation (5)

    z

    uw

    y

    uv

    x

    uu

    t

    ug

    yzx x

    yxzxxx

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    The normal stress is due to both viscous stress() and pressure

    z

    uw

    y

    uv

    x

    uu

    t

    ug

    yzxx

    px

    yxzxxx

    z

    vw

    y

    vv

    x

    vu

    t

    vg

    xzyy

    py

    xyzyyy

    xxxx p

    zww

    ywv

    xwu

    twg

    xyzzp

    zxzyzzz

    gpVVt

    V

    Momentum Equation gpVV

    t

    V

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    z

    uw

    y

    uv

    x

    uu

    t

    ug

    yzxx

    px

    yxzxxx

    z

    vw

    y

    vv

    x

    vu

    t

    vg

    xzyy

    py

    xyzyyy

    zww

    ywv

    xwu

    twg

    xyzzp z

    xzyzzz

    Inviscid Flow

    gpVVt

    V

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    In inviscid flow there is no shear stress (=0)

    Euler equation:

    Steady inviscid flow:

    (see text for proof that this reduces to Bernoulliequation)

    Neglecting gravity:

    gpVVt

    V

    1

    gpVV

    1

    pVV

    1

    Irrotational Flows (1)

    Define the velocity potential

    ff f f

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    Vorticity is:

    Only irrotational flows have a velocity potential

    vy

    fu

    x

    f w

    z

    f fV

    0

    k

    xyyx

    j

    zxxz

    i

    yzzy

    ky

    u

    x

    vj

    x

    w

    z

    ui

    z

    v

    y

    w

    ffffff

    Irrotational Flows (2) For incompressible, irrotational flow, mass

    i i

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    conservation is:

    This is a much easier equation to solve thanmomentum and is useful if flow can beapproximated as irrotational and incompressible

    We will see, only inviscid flows can beirrotational

    2

    2

    2

    2

    2

    220

    zyxV

    fffff

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    Full Momentum Equation

    gpVVt

    V

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    Need model for viscous forces () dydu

    m

    Newtonian Fluids (1)

    id

    dum

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    isjust part of the

    full viscous

    stress The complete

    model:

    dy

    yxxyx

    v

    y

    um

    zyyz y

    w

    z

    vm

    zxxzz

    u

    x

    wm

    x

    uxx

    m 2

    y

    vyy

    m 2

    z

    wzz

    m 2

    Newtonian Fluids (2) The viscous stress tensor for Newtonian fluids:

    uwvuu2

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    If viscosity is constant:

    z

    w

    y

    w

    z

    v

    z

    u

    x

    w

    y

    w

    z

    v

    y

    v

    x

    v

    y

    u

    zxxyx

    2

    2m

    VV

    Vz

    w

    Vy

    v

    V

    x

    u

    z

    w

    y

    w

    yz

    v

    xz

    u

    x

    w

    zy

    w

    z

    v

    y

    v

    x

    v

    xy

    u

    z

    u

    zx

    w

    yx

    v

    y

    u

    x

    u

    mmmm 2

    2

    2

    2

    2

    2

    2

    222

    2

    2

    2

    2

    2

    2

    2

    2

    22

    2

    222

    2

    2

    2

    2

    2

    2

    2

    Navier-Stokes Equations

    gpVVt

    V

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    Substituting viscous stress tensor yields NSE

    gVVpVVt

    V

    mm

    2

    xgz

    w

    y

    v

    x

    u

    xz

    u

    y

    u

    x

    u

    x

    p

    z

    uw

    y

    uv

    x

    uu

    t

    ummmm

    2

    2

    2

    2

    2

    2

    ygz

    w

    y

    v

    x

    u

    yz

    v

    y

    v

    x

    v

    y

    p

    z

    vw

    y

    vv

    x

    vu

    t

    vmmmm

    2

    2

    2

    2

    2

    2

    zgz

    w

    y

    v

    x

    u

    zz

    w

    y

    w

    x

    w

    z

    p

    z

    ww

    y

    wv

    x

    wu

    t

    wmmmm

    2

    2

    2

    2

    2

    2

    NSE Example: Flow between plates

    Infinite plates in x and z, steady flow, gy=-g,

    1 D fl ( 0) 0)(

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    1-D flow (v=w=0) 0)( zxutu

    2

    2

    0y

    u

    x

    p

    m g

    y

    p

    0 00

    Flow between plates (2)

    2

    0 up

    gy

    p

    0 )(xfgyp

    uxgyp )(1

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    Flow is parabolic

    Flow rate per unit width

    20

    yu

    xp

    m

    yuxgy

    xp

    )(1

    m

    )(21 2 xhy

    xpu

    m

    2

    21)(0)( h

    xpxhhu

    m

    0)(0)0(

    xg

    y

    u

    222

    1hy

    x

    pu

    m

    x

    phh

    h

    x

    pdyhy

    x

    pudyq

    hh

    mmm 332

    1

    2

    12

    33

    3

    0

    22

    0

    x

    phu

    m2

    2

    m ax

    Keys to Success

    St t ith f ll NSE

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    Start with full NSE Make reasonable assumptions and eliminate

    terms

    Solve simplified differential equations Apply boundary conditions

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    Navier-Stokes Equations

    R i

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    Review

    gVVpVVt

    V

    mm

    2

    xgz

    w

    y

    v

    x

    u

    xz

    u

    y

    u

    x

    u

    x

    p

    z

    uw

    y

    uv

    x

    uu

    t

    ummmm

    2

    2

    2

    2

    2

    2

    ygz

    w

    y

    v

    x

    u

    yz

    v

    y

    v

    x

    v

    y

    p

    z

    vw

    y

    vv

    x

    vu

    t

    vmmmm

    2

    2

    2

    2

    2

    2

    zgz

    w

    y

    v

    x

    u

    zz

    w

    y

    w

    x

    w

    z

    p

    z

    ww

    y

    wv

    x

    wu

    t

    wmmmm

    2

    2

    2

    2

    2

    2

    Couette Flow (1)

    Flow driven by moving plate

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    1-D, steady, infinite in z and x

    v=w=0

    Same equations at pressure driven flow boundary conditions are different

    2

    2

    0 y

    u

    x

    p

    mgy

    p

    0

    Ubuu )(,0)0(

    )(xfgyp

    )()(2

    1 2 xhyxgyx

    pu

    m

    Couette Flow (2)

    Ubuu )(0)0(

    )()(2

    1 2 xhyxgyx

    pu

    m

    b

    Uybyyx

    pu 2

    2

    1

    m

    b

    y

    b

    yp

    U

    b

    b

    y

    U

    u

    12

    2

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    Ubuu )(,0)0(

    0)( xh

    )(2 xgx

    pb

    b

    U

    m

    bbxUbU 12m

    Pipe Flow (1) 1-D steady flow (vr=0, v=0), axisymmetric

    (see p. 321 for full cylindrical NSE)21

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    No gradients in z (besides pressure)

    2

    2

    21

    z

    v

    r

    vr

    rrz

    p

    z

    vv zzzz

    mm

    r

    vr

    rrz

    p z1m

    Pipe Flow (2)

    Boundary conditions:

    r

    vr

    rrz

    p z1m

    0)0(

    r

    vz 0)( Rvz

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    r

    vrzfr

    z

    p z)(2

    1 2

    m0)( zf

    )(4

    1 2 zgrz

    pvz

    m

    2

    4

    1)( R

    z

    pzg

    m 22

    4

    1Rr

    z

    pvz

    m

    Pipe Flow (3)

    224

    1Rr

    z

    pvz

    m

    http://localhost/var/www/apps/conversion/tmp/scratch_7/ME250-Fluids/Lectures/V6_6.mov
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    MovieParabolic velocity profiles in pipe

    flow

    z

    pRv

    m4

    2

    m ax

    z

    pRRR

    z

    prdrRr

    z

    pQ

    R

    m

    m

    m 82422

    4

    1 444

    0

    22

    28

    max2 v

    z

    pR

    A

    QV

    m

    http://localhost/var/www/apps/conversion/tmp/scratch_7/ME250-Fluids/Lectures/V6_6.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/ME250-Fluids/Lectures/V6_6.mov
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    Unsteady Flow (1)

    Infinite plate, infinite fluid (in z and x), fluid

    i i i ll 1 D fl )()(

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    initially at rest, 1-D flow

    At t=0, plate moves at velocity U

    2

    2

    y

    u

    t

    u

    m

    gy

    p

    0

    00

    0)()(

    zx

    0 wv

    Cgyp

    Unsteady Flow (2)

    Similarity solution: 22

    y

    u

    t

    u

    m

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    Assume solution is of form:

    2

    2

    2

    2

    2

    y

    u

    y

    u

    y

    u

    yt

    u h

    h

    h

    h

    h

    h

    h

    h

    tff

    tt

    yf

    4

    1)0(

    4

    t

    yfu

    hh

    2:)(

    ff h2

    21 )( CerfCf h

    A simple problem

    Consider flow out of an infinitely long slot

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    Problem is 2-D (w=0, d/dz=0), steady, gravity

    negligible, incompressible

    x

    y

    V

    Governing Equations (1)

    xgz

    w

    y

    v

    x

    u

    xz

    u

    y

    u

    x

    u

    x

    p

    z

    u

    wy

    u

    vx

    u

    ut

    u

    mmmm

    2

    2

    2

    2

    2

    2

    0

    z

    w

    y

    v

    x

    u

    t

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    Simplify to

    gzyxxzyxxzwyvxut mmmm

    ygz

    w

    y

    v

    x

    u

    yz

    v

    y

    v

    x

    v

    y

    p

    z

    vw

    y

    vv

    x

    vu

    t

    vmmmm

    2

    2

    2

    2

    2

    2

    zgz

    w

    y

    v

    x

    u

    zz

    w

    y

    w

    x

    w

    z

    p

    z

    w

    wy

    w

    vx

    w

    ut

    w

    mmmm

    2

    2

    2

    2

    2

    2

    y

    v

    x

    u

    xy

    u

    x

    u

    x

    p

    y

    uv

    x

    uu mmm

    2

    2

    2

    2

    y

    v

    x

    u

    yy

    v

    x

    v

    y

    p

    y

    vv

    x

    vu mmm

    2

    2

    2

    2

    00

    0

    y

    v

    x

    u

    Governing Equations (2)

    2

    2

    2

    2

    1 uupuvuu

    0

    y

    v

    x

    u

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    Three equations, three unknowns (u, v, p) u=u(x,y,,m,v=v(x,y,,m), p=p(x,y,,m)

    Problem has a solution, but cannot simply

    integrate equations CFD is an approach to estimate an answer to

    governing equations

    22 yxxp

    yx

    2

    2

    2

    21

    y

    v

    x

    v

    y

    p

    y

    vv

    x

    vu

    Computational Mesh

    Computational fluid dynamics (CFD) solves

    these differential equations on a grid

    ( ( ) ( )

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    u=u(xi,yi,,m,v=v(xi,yi,,m), p=p(xi,yi,,m)

    xiand yiare discretespacing x and y

    NxN number of nodes

    Examples of Meshes

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    Numerical Methods Finite Difference

    Differential form of governing equations are discretized

    and solved

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    and solved

    Finite Volume

    On each cell, conservation laws are applied at a discrete

    point of the cell [node].

    Integral Control Volume Form of

    Governing Equations

    Taylor Series Expansion

    !3!2!1

    3

    ,

    3

    32

    ,

    2

    2

    ,

    ,,1

    x

    x

    ux

    x

    ux

    x

    uuu

    jijiji

    jiji

    3322 xuxuxuuu

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    Differential is converted to discrete algebraicexpression:

    Error of order (x)2

    !3

    2!1

    2

    3

    ,

    3

    3

    ,

    ,1,1

    x

    x

    ux

    x

    uuu

    jiji

    jiji

    !3!2!1

    ,

    3

    ,

    2

    ,

    ,,1xxx

    uu

    jijiji

    jiji

    62

    2

    ,

    3

    3

    ,1,1

    ,

    x

    x

    u

    x

    uu

    x

    u

    ji

    jiji

    ji

    x

    uu

    x

    u jiji

    ji

    2

    ,1,1

    ,

    Second Derivatives

    !3!2!1

    3

    ,

    3

    32

    ,

    2

    2

    ,

    ,,1

    x

    x

    ux

    x

    ux

    x

    uuu

    jijiji

    jiji

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    With an error of order (x)2

    !42

    !222

    4

    ,4

    42

    ,2

    2

    ,,1,1

    x

    x

    ux

    x

    uuuu

    jiji

    jijiji

    !3!2!1

    3

    ,

    3

    32

    ,

    2

    2

    ,

    ,,1

    x

    x

    ux

    x

    ux

    x

    uuu

    jijiji

    jiji

    12

    2 2

    ,

    4

    4

    2

    ,1,,1

    ,

    2

    2 x

    x

    u

    x

    uuu

    x

    u

    ji

    jijiji

    ji

    2,1,,1

    ,

    2

    2 2

    x

    uuu

    x

    u jijiji

    ji

    Discretized Equations

    Apply the derivative estimates to the

    governing equations: 22

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    governing equations:

    Similar additional equation for v momentum

    2

    2

    2

    21

    y

    u

    x

    u

    x

    p

    y

    uv

    x

    uu

    0

    y

    v

    x

    u

    x

    uu

    x

    u jiji

    ji

    ,,1

    , 2

    ,1,,1

    ,

    2

    2 2

    x

    uuu

    x

    u jijiji

    ji

    21,,1,

    2,1,,1,1,11,1,

    ,,1,1

    , 222

    122 y

    uuux

    uuuxpp

    yuuv

    xuuu jijijijijijijijijijijijijiji

    01,1,,1,1

    y

    vv

    x

    uujijijiji

    System of Algebraic Equations

    Discretization turns 3 partial differential

    equations into thousands of algebraic

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    equations into thousands of algebraic

    equations (3 for each mesh point i,j)

    E.g.:

    (2,2)

    (4,2)

    01,1,,1,1

    y

    vv

    x

    uujijijiji

    01,23,22,12,3

    y

    vv

    x

    uu

    01,43,42,32,5

    y

    vv

    x

    uu

    NxN number of algebraic

    equations

    Boundary Conditions

    Since the mesh is finite, derivatives must be

    computed differently at the edge of the mesh

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    computed differently at the edge of the mesh

    (u-1,-1doesnt exist)

    Boundary conditions must be specified at all

    mesh edges

    Boundary conditions

    are always approximations

    of realistic conditions

    u,v=0

    d/dx=0

    d/dy=0

    u=U, v=0

    CFD OutputInfinite slot (1)

    Solution is only available at grid points

    interpolation used for values between grid points

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    CFD OutputInfinite slot (2)

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    CFD OutputVortex Generation

    Many CFD simulations are unsteady to capture

    transient features of flow

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    transient features of flow

    Cylinders in cross flow shed vortices that can

    be visualized by streamlines

    MovieCFDcomputed streamlines over

    bluff-body

    Errors in CFD

    Numerical errorthe iterative solution did not

    find the correct answer to the algebraici

    http://localhost/var/www/apps/conversion/tmp/scratch_7/V6_7.movhttp://localhost/var/www/apps/conversion/tmp/scratch_7/V6_7.mov
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    find the correct answer to the algebraicequations

    Discretization errorthe derivative estimates

    were not accurate enough (x too large) Governing equation errora term was

    removed that shouldnt have been (gravity?)

    Boundary condition errorthe boundaryconditions do not reflect reality

    CFD Validation/Accuracy (1) Numerical error can be

    assessed by examiningresiduals

    Residualy

    vv

    x

    uu jijijiji

    1,1,,1,1

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    Residuals would ideallybe zero, but actuallyconverge to a small

    constant value Once the residuals are

    sufficiently small thealgebraic equationshave been solved butthey still may beinnaccurate

    CFD Validation/Accuracy (1) Discretization error can be assessed by

    recomputing a solution on a finer/different grid

    The difference in the solutions is an estimate of

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    The difference in the solutions is an estimate of

    effect of grid (grid independence)

    Errors in governing equations and boundary

    conditions can be assessed by turning on/off

    physical terms or by adding perturbations to

    boundary conditions (sensitivity studies)

    Finally, simulations for some cases should becompared to experiments (validation)

    Post-Processing

    Solution from CFD must be post-processed to

    extract information of interest

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    extract information of interest

    E.g.: Flowlab assignment you will integrate

    velocity profiles to compute mass and

    momentum flux

    i

    iii rruurdrudAndAVm 22

    CFD Overview

    Simplify equations as far as possible

    Discretize equations using a finite grid (derivativesbecome differences)

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    q g g (become differences)

    A few partial differential equations become

    thousands+ of algebraic equations

    Solve using numerical methods (ME elective course)

    Validate solution to insure accuracyEven with

    commercial CFD codes this step MUST be done by

    user CFD always gives a pretty answeryou must work

    to make sure that answer is useful

    Why Dimensional Analysis?

    Imagine we are interested in the solution to

    flow in a round pipe for 1000 differentcombinations of velocity fluid type (density

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    ow a ou d p pe o 000 d e e tcombinations of velocity, fluid type (density,viscosity), and pipe diameter

    Do we have to perform 1000 experiments?

    1000 calculations?

    Dimensional analysis determines how these

    1000 cases are related to minimize the numberof independent calculations/experiments thatmust be performed

    Problem Variables

    For the pipe flow example, assuming the pipe

    is smooth and the velocity profile does not

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    y pchange with position in the pipe, the variablesare (experience required to generate this list)

    Pipe diameter, D Centerline velocity, V

    Viscosity, m

    Density,

    Pressure drop, p

    Pipe length, L

    ),,,,( LDVfp m

    Buckingham Pi Theorem (1)

    If an equation involving k variables is

    dimensionally homogeneous it can be reduced

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    dimensionally homogeneous, it can be reduced

    to a relationship among k-r independent

    dimensionless groups, where r is the minimum

    number of dimensions required to describe thevariables

    Buckingham Pi Theorem (2)

    k=6

    p (N/m2), (kg/m3), m (N-s/m2), V (m/s), D

    ),,,,( LDVfp m

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    p (N/m ), (kg/m ), m(N s/m ), V (m/s), D(m), L (m)

    Both kg and N are not independent dimensions

    since 1 N = 1 kg-m/s2; thus (N-s2/m4) r=3 (N, m, s)

    p relationship can be expressed with k-r=3dimensionless Pi groups

    Dimensionless Pi Groups (1)

    To determine Pi groups

    Select one variable for each of the independent

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    Select one variable for each of the independent

    dimensions (N, s, m)these are called repeating

    variables

    E.g.: D (for m), V (for s), (for N) For the other non-repeating k-r variables construct

    dimensionless parameters using only the non-

    repeating variable and the repeating variables as

    needed

    Dimensionless Pi Groups (2)

    p (N/m2) zyx

    mDs

    mV

    m

    sN

    m

    Np

    4

    2

    2

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    This is dimensionless for x=-1, y=-2, z=0

    m(N-s/m2)

    x=-1, y=-1, z=-1

    L (m)

    By inspection

    12

    V

    p

    zyx

    mDs

    mV

    m

    sN

    m

    sN