Boundary Layer Theory_2

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Transcript of Boundary Layer Theory_2

  • CE F312 Hydraulics Engineering 1

    STANDARD BOUNDARY LAYER THICKNESS

    That distance from the boundary surface (or

    plate) upto which the velocity reaches 99% of the

    velocity of the main stream.

    y = where u = 0.99U

    Somewhat an easy to understand

    but arbitrary definition as the limit

    of boundary layer is not easily

    defined13

    99% THICKNESS

    U

    U is the free-stream velocity

    (x)

    x

    y

    (x) is the boundary layer thickness when u(y) ==0.99U

    14

  • CE F312 Hydraulics Engineering 2

    For greater accuracy the boundary layer

    thickness is defined in terms of certain

    mathematical expressions which are the

    measures of the effect of boundary layer on the

    flow.

    Displacement thickness (*)

    Momentum thickness ()

    Energy thickness (E)

    THICKNESS OF BOUNDARY LAYER

    15

    DISPLACEMENT THICKNESS (*)

    The boundary layer retards the fluid, so that the mass flux is less

    than it would be in the absence of the boundary layer.

    Thus the displacement thickness is the distance by which the

    boundary surface would have to be moved so that the actual

    discharge deficit would be same as that of an ideal (or

    frictionless) fluid past the displaced boundary.

    Amount of fluid

    being displaced

    outward

    *

    Equal

    Areas

    U-u

    16

  • CE F312 Hydraulics Engineering 3

    DISPLACEMENT THICKNESS (*)

    =

    0

    *dy

    U

    u1

    Uq *=

    The areas under each curve are defined as being equal:

    ( )

    =0

    dyuUq

    and

    Equating these gives the equation for the displacement

    thickness:

    The loss due to boundary layer

    17

    MOMENTUM THICKNESS ()

    The distance from the actual boundary surface

    such that the momentum flux corresponding to

    the main stream velocity V through this distance

    is equal to the deficiency or loss in momentum

    due to the boundary layer formation.

    18

  • CE F312 Hydraulics Engineering 4

    MOMENTUM THICKNESS ()

    Um 2=

    =

    0

    dyU

    u1

    U

    u

    In the boundary layer, the fluid loses momentum, so imagining an

    equivalent layer of lost momentum:

    ( )

    =0

    dyuUum and

    Equating these gives the equation for the momentum thickness:

    19

    ENERGY THICKNESS (E)

    The distance from the actual boundary surface

    such that the energy flux corresponding to the

    main stream velocity U through this distance (E)is equal to the deficiency or loss of energy due to

    the boundary layer formation

    20

  • CE F312 Hydraulics Engineering 5

    ENERGY THICKNESS (E)

    3 2 2

    0

    1 1( )

    2 2EU U u udy

    =

    2

    2

    0

    1Eu u

    dyU U

    =

    21

    SOLVE! Air flowing into a 2-ft-square duct with a uniform velocity of 10 ft/s

    forms a boundary layer on the walls as shown in Fig. The fluid withinthe core region (outside the boundary layers) flows as if it were inviscid.From advanced calculations it is determined that for this flow theboundary layer displacement thickness is given by

    where * and x are in feet. Determine the velocity U = U(x) of the air within the duct but outside of

    the boundary layer.

    ( ) 2/1* 0070.0 x=

    22

  • CE F312 Hydraulics Engineering 6

    SOLUTION

    The volume flow rate across any section of the duct is equal to

    that at the entrance (i.e., Q1=Q2). That is

    According to the definition of the displacement thickness, the

    flowrate across section (2) is the same as that for a uniform flow

    with velocity U through a duct whose walls have been moved

    inward by *

    ( ) ===)(

    3 /ft ft ft/s 2

    2

    1140210 udAsAU

    ( ) ( )

    ( )ft/s

    .

    .*/ft

    /

    /

    )(

    3

    221

    2212

    2

    007001

    10

    00700142240

    xU

    xUUudAs

    =

    ===

    23

    PROBLEM

    Assuming that the shear stress distribution in a

    laminar boundary layer is such that

    Calculate the displacement and momentum

    thickness of this boundary layer in terms of .

    =

    y

    10

    24

  • CE F312 Hydraulics Engineering 7

    BOUNDARY LAYER EQUATIONS

    Equations of continuity and motion for the

    steady flow of an incompressible, inviscid fluid in

    2-D without body forces are:

    1v v pu vx y y

    + =

    CONTINUITY EQUATION

    EULERS EQUATION IN X DIR.

    EULERS EQUATION IN Y DIR.

    0u v

    x y

    + =

    1u u pu vx y x

    + =

    Chapter 6, Sec 6.6

    Chapter 7, Sec 7.3

    25

    If a viscous fluid is considered, eqn. of continuity

    remains unchanged

    Additional terms due to viscous stresses will be

    introduced in the eqns. of motion

    BOUNDARY LAYER EQUATIONS

    + d

    y

    x

    y

    x

    Edge of

    boundary layer

    ( ) x y xy

    + = +

    Net force = x yy

    26

  • CE F312 Hydraulics Engineering 8

    1 1u u pu vx y x y

    + = +

    BOUNDARY LAYER EQUATIONS

    u

    y

    =

    Simplification.. Since the boundary layer is thin, it is expected that the component

    of velocity normal to the plate is much smaller than the parallel tothe plate and that the rate of change of any parameter across theboundary layer should be much greater than that along the flowdirection. That is

    Again2

    2

    1u u p uu vx y x y

    + = +

    v