Aerodynamics (Chapter 1)

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    Aerodynamics

    )(

    3

    rd

    year Mechanical Engineering-

    2012/2011

    ..

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    Syllabus )(

    :/683

    ::5

    ::2

    :1

    :1

    Subject Number: ME\683

    Subject: AerodynamicsUnits: 5

    Weekly Hours: Theoretical: 2

    Experimental: 1

    Tutorial : 1ContentsWeek

    1

    2

    3

    4

    -

    -

    -

    --

    -

    -

    Navier-Stokes equations

    - Introduction

    - Derivation

    - Laminar flow between parallel plates

    - Couette flow- Hydrodynamic lubrication

    - Sliding bearing

    - Laminar flow between coaxial

    rotating cylinders

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    -

    -

    -

    -

    -

    -

    -

    -

    Boundary layer theory

    - Introduction

    - Displacement, Momentum, and

    Energy thicknesses

    - Momentum equation for the

    boundary layer

    - Laminar boundary layer

    - Turbulent boundary layer

    - Transition from laminar to turbulent

    flow

    - Effect of pressure gradient

    - Separation and pressure drag

    5

    6

    7

    8

    9

    10

    11

    )(

    --

    -

    Potential flow theory

    (Ideal fluid)

    - Introduction- Continuity equation

    - Vorticity equation

    11

    12

    -

    -

    -

    Basic concepts in potential flow

    - Stream function

    - Potential function

    - Circulation

    12

    13

    -

    -

    -

    -

    Basic flow patterns

    - Uniform flow

    - Source , Sink

    - Doublet

    - Free vortex

    13

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    14

    15

    -

    -

    -

    -

    Combination of basic flows- Flow past a half body

    - Flow past a Rankine oval

    - Flow past a cylinder

    - Flow past a cylinder with circulation

    14

    15

    16

    17

    -

    -

    -

    -

    Incompressible flow over airfoils

    - Introduction

    - The Kutta condition

    - Kelvins circulation theorem

    - Thin airfoil theory

    16

    17

    18

    -

    -

    -

    -

    Airfoil characteristics

    - Wind tunnel tests

    - Estimation of aerodynamic coefficients

    from pressure distribution

    - Compressibility effects

    - Reynolds number effects

    18

    19

    -

    -

    -

    Airfoil maximum lift characteristics

    - Geometric factors effects

    - Effect of Reynolds number

    - Effect of leading and trailing edges

    devices

    19

    20

    21

    -

    -

    -

    Incompressible flow over wings

    - Introduction

    - Circulation, downwash, lift and

    induced drag

    - Finite wing theory

    20

    21

    22

    -

    -

    -

    Wing stall

    - Stall characteristics

    - Effect of planform and twist

    - Stall control devices

    22

    23

    -

    -

    Lift control devices

    - High lift devices

    - Spoilers

    23

    24

    -

    -

    Flow control devices

    - Boundary layer control

    - Reduction of drag24

    25

    26

    27

    28

    -

    -

    -

    -

    Propellers- Momentum theory

    - Simple blade element theory

    - Combined blade element theory and

    momentum theory

    - Propeller performance

    25

    26

    27

    28

    29

    30

    -

    -

    Computational methods

    - Introduction to panel methods

    for airfoils

    - Introduction to panel methods

    for wings

    29

    30

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    References

    1- Fluid Mechanics By Streeter & Wylie

    2- Introduction to Fluid Mechanics

    By Fox & McDonald

    3- Aerodynamics for Engineering Students

    By Houghton & Carpenter

    4- Airplane Aerodynamics and Performance

    By Roskam & Edward Lan

    :

    .

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    Chapter One

    Navier Stokes EquationsContents

    1- Navier-Stokes equations.

    2-Steady laminar flow between parallel flat plates.

    3- Hydrodynamic lubrication.4- Laminar flow between concentric rotating cylinders.5- Example.

    6- Problems; sheet No. 1

    1- Navier-Stokes equations:

    The general equations of motion for viscous incompressible, Newtonian fluids may

    be written in the following form:

    x- direction:

    -(1)-----2

    2

    2

    2

    2

    2

    +

    +

    +

    =

    +

    +

    +

    z

    u

    y

    u

    x

    u

    x

    pg

    z

    uw

    y

    uv

    x

    uu

    t

    ux

    y- direction:

    -(2)-----2

    2

    2

    2

    2

    2

    +

    +

    +

    =

    +

    +

    +

    z

    v

    y

    v

    x

    v

    y

    pg

    z

    vw

    y

    vv

    x

    vu

    t

    vy

    Equations (1) and (2) are called: Navier Stokes equations.

    2- Steady laminar flow between parallel flat plates:

    The fluid moves in the x- direction without acceleration.

    v= 0 , w= 0 , 0=t

    the Navier-Stokes equation in the x- direction (eq. 1) reduces to:

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    (3)---------2

    2

    dy

    ud

    dx

    dpg

    x =+

    dx

    dhgggx == sin.

    eq. 3 will be

    ( )-(4)---------

    12

    2

    dx

    hpd

    dy

    ud

    +=

    Integration of eq. 4:

    ( )Ay

    dx

    hpd

    dy

    du+

    +=

    1

    ( )(5)-----------

    2

    1 2BAyy

    dx

    hpdu ++

    +=

    B.C (Two fixed parallel plates)B

    ( )dx

    hpdaAuay

    uy

    +===

    ===

    20

    000

    eq. 5 will be

    ( ) ( ) -(6)-----------2

    1 2ayy

    dx

    hpdu

    +=

    B.C (One plate is fixed and the other plate moves with a constant

    velocity U) (Couette flow)

    ( )dx

    hpda

    a

    UAUuay

    Buy

    +===

    ===

    2

    000

    eq. 5 will be

    ( ) ( ) (7)-----------a

    Uy21 2 ++= ayy

    dxhpdu

    For the case of horizontal parallel plates:

    0=dx

    dh

    eq. 7 will be

    ( ) (8)-----------aUy212 += ayydx

    dpu

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    The location of maximum velocity umax may be found by evaluatingdy

    duand setting it to

    zero.

    The volume flow rate is

    =

    a

    dyuwQ0

    -(9)-----------.

    3- Hydrodynamic lubrication:

    Sliding bearing

    Large forces are developed in small clearance when the surfaces are slightly inclined

    and one is in motion so that fluid is wedged into the decreasing space. Usually the oils

    employed for lubrication are highly viscous and the flow is of laminar nature.

    Assumptions:

    The acceleration is zero.

    The body force is small and can be neglected.

    Also2

    2

    2

    2

    2

    2

    2

    2

    and

    z

    u

    y

    u

    x

    u

    y

    u

    The Navier-Stokes equation in the x-direction (eq. 1) reduces to:

    dx

    dp

    dy

    ud

    12

    2

    =

    Integration:

    BAyydx

    dpu ++

    = 22

    1

    B.C

    x

    xx

    h

    U

    dx

    dphAuhy

    UBUuy

    ===

    ===

    20

    0

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    ( )

    +=

    x

    xh

    yUyhy

    dx

    dpu 1

    2

    1 2

    The volume flow rate in every section will be constant.

    1wassume.

    0

    == xh

    dyuwQ

    dx

    dphUhQ xx

    122

    3

    = --------(*)

    ** For a constant taper bearing:

    l

    hh 21 =

    ( )xhhx = 1

    Sub in eq.(*) and solving for

    dx

    dpproduces:

    ( ) ( )312

    1

    126

    xh

    Q

    xh

    U

    dx

    dp

    =

    Integration gives:

    ( ) ( )C

    xh

    Q

    xh

    Uxp +

    =

    2

    11

    66)(

    ---------(**)

    B.C

    0

    00

    ===

    ===

    o

    o

    pplx

    ppx

    ( )2121

    21 6andhh

    UC

    hh

    hUhQ

    +

    =+

    =

    With these values inserted in eq.(**) we obtain the pressure distribution inside the bearing.( )

    ( )212

    26)(

    hhh

    hhUxxp

    x

    x

    +

    =

    The load that the bearing will support per unit width is:

    =l

    dxxpF0

    ).(

    ( )

    +

    =

    1

    )1(2ln

    62

    21

    2

    k

    kk

    hh

    UlF

    where

    2

    1

    h

    hk =

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    4- Laminar flow between concentric rotating cylinders:

    Consider the purely circulatory flow of a fluid contained between two long

    concentric rotating cylinders of radius R1 and R2 at angular velocities 1 and 2.

    In this case the Navier-Stokes equations in cylindrical coordinates are used.

    r- direction:

    rrrrrrrr

    rr g

    z

    uu

    r

    u

    rr

    u

    r

    ur

    rrr

    p

    z

    uw

    r

    uu

    r

    u

    r

    uu

    t

    u+

    +

    +

    +

    =

    +

    +

    +

    2

    2

    22

    2

    22

    22111

    - direction:

    g

    z

    uu

    r

    u

    rr

    u

    r

    ur

    rr

    p

    rz

    uw

    r

    uuu

    r

    u

    r

    uu

    t

    urr

    r +

    +

    +

    +

    +

    =

    ++