Chapter 6 2009_2

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    Chapter 6

    Flow Analysis Using Differential Methods(Differential Analysis of Fluid Flow)

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    In the previous chapter-- Focused on the use of finite control volume for the

    solution of a variety of fluid mechanics problems. The approach is very practical and useful since it

    doesnt generally require a detailed knowledge of thepressure and velocity variations within the control

    volume. Typically, only conditions on the surface of the

    control volume entered the problem.

    There are many situations that arise in which thedetails of the flow are important and the finitecontrol volume approach will not yield thedesired information

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    For example --

    We may need to know how the velocity varies over the cross

    section of a pipe, or how the pressure and shear stress vary

    along the surface of an airplane wing.

    we need to develop relationship that apply at a point,

    or at least in a very small region ( infinitesimal volume)within a given flow field.

    involve infinitesimal control volume (instead of finite

    control volume)

    differential analysis (the governing equations are

    differential equation)

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    In this chapter

    (1) We will provide an introduction to the differential

    equation that describe (in detail) the motion of fluids.

    (2) These equation are rather complicated, partial differential

    equations, that cannot be solved exactly except in a few

    cases.

    (3) Although differential analysis has the potential for

    supplying very detailed information about flow fields, the

    information is not easily extracted.

    (4) Nevertheless, this approach provides a fundamental basis

    for the study of fluid mechanics.

    (5) We do not want to be too discouraging at this point,

    since there are some exact solutions for laminar flow that

    can be obtained, and these have proved to very useful.

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    (6) By making some simplifying assumptions, many otheranalytical solutions can be obtained.

    for example , small 0 neglected

    inviscid flow.

    (7) For certain types of flows, the flow field can be conceptuallydivided into two regions

    (a) A very thin region near the boundaries of the system inwhich viscous effects are important.

    (b) A region away from the boundaries in which the flow isessentially inviscid.

    (8) By making certain assumptions about the behavior of the fluidin the thin layer near the boundaries, and

    using the assumption of inviscid flow outside this layer, a large

    class of problems can be solved using differential analysis .

    the boundary problem is discussed in chapter 9.

    Computational fluid dynamics (CFD) to solve differential eq.

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    )(.)((.)(.)(.)(.)(.)(.)

    .

    .

    problem.particularafort,z,y,on x,dependlyspecificalcomponents

    velocitythesehowdeterminetoisanalysisaldifferentiofgoalstheofOne

    vtz

    wy

    vx

    utt

    Da

    a

    z

    uw

    y

    uv

    x

    uu

    t

    ua

    dt

    vd

    z

    vw

    y

    vv

    x

    vu

    t

    va

    z

    y

    x

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    elementtheofndeformatioangularx

    v

    y

    u

    elementtheofndeformatiolinear

    z

    w

    y

    v

    x

    uNote

    v

    ratedilationvolumetricvz

    w

    y

    v

    x

    u

    dt

    d

    ,

    ,,:

    fluid,ibleincompressanfor0

    )(1

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    flowalIrrotation0

    zeroisaxis-zthearoundRotation

    only when)(i.e.blockundeformedanasaxis-zaboutRotation

    )(2

    1assuch(6.12)Eq.From

    2Define

    vor

    y

    u

    x

    v

    x

    v

    y

    uww

    y

    u

    x

    vw

    Vcurlvwvorticity

    OBOA

    z

    Vcurlv

    wvuzyx

    kji

    k

    y

    u

    x

    vj

    x

    w

    z

    ui

    z

    v

    y

    w

    kwjwiw

    vectorrotationtheW

    zyx

    2

    1)(

    2

    1

    2

    1

    })()(){(

    2

    1

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    6.2.1 Differential Form of Continuity Equation

    inout

    cv

    vAvA

    dAnv

    zyx

    t

    d

    t

    d

    t

    zyxd

    )()(

    elementtheofsurfacesthethroughflowmassofrateThe)(

    )(

    0

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    s

    yx

    xx

    xxx

    x

    uuu

    y

    x

    uuu

    )(,)(assuch,terms

    orderhighneglecting--expansionseriesTaylor

    2

    )(|

    2

    )(|

    direction-xtheinflowThe

    2

    2

    2

    zyxz

    w

    zyxyv

    zyxx

    uzy

    x

    x

    uuzy

    x

    x

    uu

    )(direction-zinrateNet

    )24.6()(direction-yinrateNet

    similarly

    )23.6()(

    ]2

    []2

    [

    direction-in xoutflowmassofrateNet

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    .formaldifferentiinequationcontinuityThe

    )27.6(0

    outflowmassofrateNet:

    0][

    0)(Since

    z

    w

    y

    v

    x

    u

    t

    zyxz

    wzyx

    y

    vzyx

    x

    uNote

    zyxz

    wzyx

    y

    vzyx

    x

    uzyx

    t

    dAnvd

    t cscv

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    )31.6(0

    )30.6(0

    0

    flowibleincompressFor--

    )29.6(0

    0)(

    fluidlecompressibofflowsteadyFor--

    )28.6(0

    formIn vector

    mechanicsfluidofequationslfundamentatheofOne--

    z

    w

    y

    v

    x

    uor

    vt

    const

    z

    w

    y

    v

    x

    uor

    v

    vt

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    equation.continuityhesatisfy ttorequired,w:Determine

    ?

    flowibleincompressanFor

    26.Example

    222

    w

    zyzxyv

    zyxu

    ),(2

    3nIntegratio

    3)(2

    0)()(

    0

    continuityofequationthefrom:Solution

    2

    222

    yxczxzw

    zxzxxz

    w

    zwzyzxy

    yzyx

    x

    z

    w

    y

    v

    x

    u

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    6.2.2 Cylindrical Polar Coordinates

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    01)(1

    )flowunsteadyorsteady(flowibleincompressFor

    0)()(1

    )(1

    flowlecompressibsteady,For

    scoordinatelcylindricain

    equationycontinualltheofformaldifferentitheisThis

    )33.6(0

    )()(1)(1

    z

    vv

    rr

    rv

    r

    vz

    vr

    vrrr

    z

    vv

    rr

    vr

    rt

    zr

    zr

    zr

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    6.2.3 The Stream Function

    )36.6(0

    )2(0)(0

    flowD-2&plane,ible,incompresssteady,ofequationcontinuityFor the

    0

    equationContinuity

    y

    v

    x

    u

    flowDz

    wcte

    twhere

    z

    w

    y

    v

    x

    u

    t

    0)()(

    eq.continuitythesatisfiesitthatso;where

    function,streamthe),(functionaDefine

    xyyxy

    v

    x

    u

    xv

    yu

    yx

    satisfiedbewillmassofonconservati

    unknowoneunknowstwofunction

    streamusing

    v

    u

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    6.3.Example

    )42.6(1

    01)(1

    0)()(1)(1

    flow.D-2place,,ibleIncompress

    forequationycontinuallthe,scoordinatelcylindricaIn

    rv

    rv

    v

    rr

    rv

    rz

    vv

    rr

    vr

    rt

    r

    rzr

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    Examp le 6.3 Stream Func t ion

    The velocity component in a steady, incompressible, twodimensional flow field are

    Determine the corresponding stream function and show on asketch several streamlines. Indicate the direction of glow alongthe streamlines.

    4xv2yu

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    Example 6.3 Solution

    (y)fx2(x)fy 2212

    Cyx222

    From the definition of the stream function

    x4x

    vy2y

    u

    For simplicity, we set C=0

    22yx2

    =0

    01

    2/

    xy22

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    6.3 Conservation of Linear Momentum

    amF

    amFordt

    vdmFEq

    VCsmallFor

    FdAnvvdvtdt

    vmD

    dmvdvP

    Pdt

    Ddv

    dt

    D

    dt

    vmD

    sys

    cvcv

    csvc

    cv

    sys

    syssys

    sys

    sys

    systemaforlaw2ndNewtonsThe

    )44.6(

    ..

    )44.6()()(

    momentumlinearfor thet theoremtransporReynoldstheFrom

    where

    )(

    momentumlinearFor the

    ..

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    Figure 6.9 (p. 287)Components of force acting on an arbitrary differential area.

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    Figure 6.10 (p. 287)Double subscript notation for stresses.

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    Figure 6.11 (p. 288)Surface forces in the x direction acting on a fluid element.

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    6.3.2 Equation of Motion

    Velocitiesstresses-----Unknowns

    .restatormotio