Boundary Layer Theory_P2

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Chapter presentation on Boundary Layer Theorems

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  • CVEN 311 Fluid Dynamics 1

    Further the equation can be re-written as

    MOMENTUM INTEGRAL EQUATION

    ( ) 00 0

    h hd dU dUu u U dy udy h U

    dx dx dx

    =

    ( ) ( ) 00 0

    h hd dUu U u dy U u dy

    dx dx

    + =

    or

    ( ) ( )2 * 0d dUU Udx dx + =

    For a flat plate at zero incidence (i.e. no imposed

    pressure, dp/dx = 0, then

    Hence,

    ( ) ( )2 * 0d dUU Udx dx + =

    MOMENTUM INTEGRAL EQUATIONMOMENTUM INTEGRAL EQUATION of

    the boundary layer which forms the

    basis for approximate methods of

    solving boundary layer problems

    0=dxdUU

    02

    dU dx

    =

  • CVEN 311 Fluid Dynamics 2

    In order to use this equation to estimate the boundarylayer thickness as a function of x, we must first: Obtain a first approximation to the freestream velocity

    distribution, U(x). This is determined from inviscid flow theory(the velocity that would exist in the absence of boundarylayer) and depends on body shape.

    Assume a reasonable velocity-profile shape inside theboundary layer

    Derive an expression for 0 using the result obtained from theprevious item.

    Reviewing the assumptions made in the derivation, it can be seen thatthe equation is:

    Restricted to steady, incompressible, two-dimensional flow with nobody forces parallel to the surface.

    Valid for either a laminar or turbulent boundary layer flow.

    Step 1. Assume a suitable velocity profile for u(y)

    inside the BL

    For example, assume a linear velocity

    distribution inside a BL

    Determine constants A and B with the boundary

    conditions as

    u(0) = 0 B = 0

    u() = U A = U/

    VON KARMANS MOMENTUM INTEGRAL APPROACH

    BAyyu +=)(

    y

    Uu

    =

  • CVEN 311 Fluid Dynamics 3

    Thus U

    yu

    y

    =

    =

    =00

    VON KARMANS MOMENTUM INTEGRAL APPROACH

    *

    0 0

    1 12

    u ydy dyU

    = = =

    0 0

    1 16

    u u y ydy dyU U

    = = =

    dxd

    U

    =20 1

    6ddx U U

    = =

    2( ) 12d

    dx U

    =

    Integrating

    Local drag coefficient

    Skin friction drag

    VON KARMANS MOMENTUM INTEGRAL APPROACH

    2 12 x CU = +

    At x=0, = 0, hence C = 0

    2 12 xU =

    2

    212 12

    Rexx Ux

    = =12 3.464Re Rex xx

    = =

    02

    2 0.5774Re

    2

    fx

    cU U

    = = =

    3 2 1 2 1 20

    0

    0.5774L

    DF Bdx U L B= =

  • CVEN 311 Fluid Dynamics 4

    VON KARMANS MOMENTUM INTEGRAL APPROACH Average drag coeffcient

    Displacement thickness

    Momentum thickness

    21.1548

    2 ReD

    fL

    F BLCU

    = =

    where UL

    L =Re

    * 1.7322 Re x

    x = =

    0.57746 Re x

    x = =

    This problem dealt with linear velocity profile as an approximate solution. The results

    obtained are rough. However the exercise illustrates the use of the momentum

    integral method. Practice this method with other types of approximated velocity

    profile, such as parabolic, sinusoidal, etc.

    One key point to remember

    Be careful not to confuse the calculation for cf and Cf.

    cf is a local calculation at a particular x location

    (including x=L) and can only be used to calculate local

    shear stress, NOT drag force. Cf is an integrated

    average over a specified length (including any x L)

    and can only be used to calculate average shear

    stress and the integrated force over the length

  • CVEN 311 Fluid Dynamics 5

    EXAMPLE PROBLEM

    Water at 15C flows over a flat plate at a speed of 1 m/s. Theplate is 0.4 m long and 1 m wide. The boundary layer on eachsurface of the plate is laminar. Assume that the velocity profilemay be approximated as linear. Determine the drag force on theplate.

    Given

    Working fluid is water at T = 15 C = 999 kg/m3 & = 1.14 10-3Ns/m2

    U = 1 m/s

    L = 0.4 m

    W = 1 m

    The boundary layer on each surface of the plate is laminar

    Velocity profile is linear (assuming approximately)

    Assumptions

    Steady state condition

    Incompressible fluid flow

    Laminar boundary layer

    System diagramU = 1 m/s

    L = 0.4 m

  • CVEN 311 Fluid Dynamics 6

    Governing Equations

    Skin friction coefficient definition:

    Reynolds number definition for a flat plate:

    2

    21 U

    C wf

    =

    Ux

    x =Re

    LAMINAR BOUNDARY LAYER ON A FLAT PLATE:

    APPROXIMATE SOLUTION USING PARABOLIC

    VELOCITY PROFILE

    Consider two-dimensional laminar boundary

    layer flow along a flat plate. Assume the

    boundary layer as parabolic.

    Find expressions for:

    The rate of growth of as a function of x.

    The displacement thickness, *, as a function of x.

    The total friction force on a plate of length L and width b.

  • CVEN 311 Fluid Dynamics 7

    In 1908 Blasius, a student of Prandtl, obtained an exact solution of thefollowing BL equations for a flat plate and demonstrated the shape ofthe boundary layer profile.

    With the following boundary conditions u(y = 0) = 0

    v(y = 0) = 0

    u U as y

    Blasius exact solution is valid only for laminar BL flow with no pressuregradient.

    LAMINAR BOUNDARY LAYER (BLASIUS EQN.)

    Parallel flow along a

    plate with zero

    pressure gradient

    LAMINAR BOUNDARY LAYER Inside the boundary layer since the viscous forces

    are predominant Reasonable to assume: inertial and viscous forces are of

    the same order of magnitude in a laminar boundary layer

    Inertial forces/unit volumex

    Ux

    uu

    2

    For a flat plate

    Viscous forces/unit volume 22

    2

    U

    yu

    yu

    yy

    =

    =

    If these forces are proportional to each other, then

    x

    kUx

    kx Re2

    2

    =

    =

    2

    2

    Uk

    x

    U=

    x

    Uk

    = A non-dimensional parameter

    Rex

    kkx Ux

    = =

  • CVEN 311 Fluid Dynamics 8

    RESULTS OF BLASIUSEXACT SOLUTION

    u approaches to 99 % of U at k = 5.

    In other words when k becomes 5, y becomes 5.

    Therefore using the definition of k, BL thickness atany x becomes

    Using local Reynolds number definition in theabove equation we get

    Ux 5=

    grows with

    Uxx =Re

    x

    x

    Re5

    =

    Shear stress

    Since in the boundary layer

    Replacing in the above equation

    RESULTS OF BLASIUSEXACT SOLUTION

    U

    yuU

    yu

    yy

    =

    == 00

    0

    ;

    x

    U

    3

    0 constant =

    xx

    f Uxc

    U Re0.664

    Reconstantconstant

    2

    20

    ====

    Blasiusexact analytical solution

    cf = local drag coefficient

  • CVEN 311 Fluid Dynamics 9

    Total horizontal force (or skin friction drag)

    Average drag coeffcient

    Displacement thickness

    Momentum thickness

    RESULTS OF BLASIUSEXACT SOLUTION

    BLUBdxFL

    D21

    0

    21230 664.0 ==

    L

    Df U

    BLFCRe328.1

    22==

    where UL

    L =Re

    x

    x

    Re729.1*

    =

    x

    x

    Re664.0

    =

    VON KARMANS MOMENTUM INTEGRAL APPROACH

    Blasiusexact solution

    laminar BL

    over a flat plate

    With zero pressure gradient (dp/dx=0)

    Momentum Integral Approach (MIA)

    both laminar and turbulent BLs

    over flat and curved surfaces

    for any known U(x) and poutside (x) distributions

  • CVEN 311 Fluid Dynamics 10

    COMPARISON OF DIFFERENT SOLUTIONS

    Source: Munson, Yong, Okiishi, Fundamentals of Fluid Mechanics, 3rd ed. Willey, 1998

    PROBLEM

    Air flows over a sharp edged flat plate 1 m long,

    3 m width with a velocity of 2 m/s. For one side

    of the plate, determine at the end of the plate,

    0 at the middle of the plate, FD. [ = 1.23 kg/m3;

    =1.46 10-5 m2/s]

  • CVEN 311 Fluid Dynamics 11

    TURBULENT BOUNDARY LAYER

    Turbulent Boundary layers are usually thicker

    than laminar ones.

    Velocity distribution in a turbulent boundary

    layer is much more uniform than that in a

    laminar boundary layer

    Large velocity change occur in a relatively small

    vertical distance

    Velocity gradient (dv/dy) is steeper in a turbulent

    boundary layer than in laminar boundary layer

  • CVEN 311 Fluid Dynamics 12

    From experiments,

    Velocity distribution in a turbulent boundary profilefollows 1/7th power law i.e.

    Satisfactorily describes velocity distribution for mostof the region of turbulent boundary layer but givesinfinite slope at the wall,

    Therefore it can not be used to predict 0

    TURBULENT BOUNDARY LAYER

    17u y

    U

    =

    ( ) 617 7u 1 7 U at y = 0yy

    = =

    Instead experimentally obtained measurements of the

    shear profile are used such as

    Putting the expression for the 1/7 power law into the

    equations for displacement and momentum thickness

    14

    20 0.0225 U U

    =

    * 7,

    8 72 = =

    =99%

    dm

    TURBULENT BOUNDARY LAYER

  • CVEN 311 Fluid Dynamics 13

    Using momentum thickness obtained and the above 0relation in the integral momentum equation we can

    obtain

    Equating this to the experimental value of shear stress:

    20

    772

    dUdx

    =

    147 0.0225

    72ddx U

    =

    Integrating gives:

    The turbulent boundary grows as x4/5, faster than the lamin