Development of Boundary layer - Prof. K. M. 2-2.pdfarbitrarily defined as that distance from the surface boundary in which the velocity reaches 99 ... Thermal Boundary Layer.com

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    Near the leading edge of a flat plate, the boundary layer is wholly laminar. For a

    laminar boundary layer the velocity distribution is parabolic.

    The thickness of the boundary layer () increases with distance from the leading edge

    x, as more and more fluid is slowed down by the viscous boundary, becomes unstable and

    breaks into turbulent boundary layer over a transition region.

    For a turbulent boundary layer, if the surface is smooth, the roughness projections are

    covered by a very thin layer which remains laminar, called laminar sub layer. The

    velocity distribution in the turbulent boundary layer is given by Log law of Prandtls one-

    seventh power law.

    Development of Boundary layer

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    increases as distance x increases.

    decreases as U increases.

    increases as kinematic viscosity () increases.

    When U decreases in the downward direction, boundary layer growth is reduced.

    When U decreases in the downward direction, flow near the boundary is further

    retarded, boundary layer growth is faster and is susceptible to separation.

    0 decreases as x increases as

    Boundary layer thickness ( ):

    The velocity within the boundary layer increases from zero at the surface to the velocity

    of the main stream asymptotically. Therefore, the thickness of the boundary layer is

    arbitrarily defined as that distance from the surface boundary in which the velocity

    reaches 99 per cent of the free stream velocity.( i.e. u = 0.99U).

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    10:10:42

    V(x, y, z) in cartesian or V(r, , z) in cylindrical coordinates

    r z

    V(r)

    velocity profile

    (remains unchanged)

    One-dimensional flow in a circular pipe

    One, Two and Three Dimensional Flows

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    DIFFERENTIAL ANALYSIS OF FLUID FLOW

    Finite control volume approach is very practical and useful, since it does not generally require a detailed knowledge of the pressure and velocity variations within the control volume

    Problems could be solved without a detailed knowledge of the flow field

    Unfortunately, there are many situations that arise in which details of the flow are important and the finite control volume approach will not yield the desired information

    How the velocity varies over the cross section of a pipe, how the pressure and shear stress vary along the surface of an airplane wing

    In these circumstances we need to develop relationships that apply at a point, or at least in a very small region infinitesimal volume within a given flow field. This approach - DIFFERENTIAL ANALYSIS

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    DIFFERENTIAL ANALYSIS PROVIDES VERY DETAILED KNOWLEDGE OF A FLOW FIELD

    Control volume analysis

    Interior of the CV is

    BLACK BOX

    Differential analysis

    All the details of the flow are solved at every point within the flow domain

    Flow out

    Flow in

    Flow out

    Control volume

    F

    Flow out

    Flow out

    Flow domain

    F

    Flow in

  • x

    y

    z

    zyu

    zyxx

    uzyu

    zxv

    zyx

    y

    vzxv

    zyx

    z

    wyxw

    yxw

    Equation of Continuity

  • Let

    u, v and w are the velocities in x, y and z direction.

    Mass flow rate entering in the x direction

    Mx =

    And mass flow out of element Mx+x =

    zyx

    x

    uzyu

    x

    x

    MxMx

    zyu

    =

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    10:10:43

    Similar the Mass flow rate entering and leaving

    the element in y and z - direction are shown in

    figure.

  • Now, balanced equation for element can be written

    as.

    Mass inflow Mass outflow = Rate of change

    of mass inside the element

    zyxt

    zyxz

    wyxw

    zyxy

    vzxv

    zyxx

    uzyu

    yxwzxvzyu

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    zyxt

    zyxz

    wzyx

    y

    vzyx

    x

    uzyx

    t

    This is the differential equation of continuity in

    Cartesian Coordinate for unsteady and

    compressible fluid flow.

    For Steady and compressible flow,

    0

    z

    w

    y

    v

    x

    u

    0

    z

    w

    y

    v

    x

    u

    t

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    10:10:44

    For Steady and incompressible flow,

    0

    z

    w

    y

    v

    x

    u

    In polar coordinate, the general form of continuity

    equation for compressible and unsteady flow,

    0

    1

    z

    VV

    rr

    V

    r

    V

    t

    zrr

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    Momentum Equation Consider an element, two dimensional control volume ( dx . dy . Unit depth) within the boundary layer region.

    10:10:44

  • Assumption

    Let u = velocity of fluid flow at left hand face AB,

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    Since we are concerned only with the momentum in x-direction,

    momentum of fluid moving in y direction is obtain by, multiplying mass

    flow rate in y direction (my) by x-direction of velocity (u).

    Velocity of fluid flow at face AD =

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    For study and incompressible flow in,

    Blasius Exact Solution for Laminar Boundary Flows

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  • LINEAR DEFORMATION

    Because of the presence of velocity gradients, the element will generally be deformed and rotated as it moves. For example, consider the effect of a single

    velocity gradient , On a small cube having sides x

    u

    zandy,x

    A O

    B C

    O

    B C

    A

    uu x

    x

    uu x

    x

    x

    y

    u

    u

    ux t

    x

    C

    A

    y

    x

  • x component of velocity of O and B = u x component of velocity of A and C = This difference in the velocity causes a STRETCHING of the volume element by a volume Rate at which the volume V is changing per unit volume due the gradient

    xx

    uu

    tzyxx

    u

    x

    u

    x

    u

    t

    tx

    u

    0tLim

    td

    Vd

    V

    1

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    10:10:47

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    10:10:49

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  • Blasius Exact Solution for Laminar Boundary Flows

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  • Thermal Boundary Layer

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    Whenever a flow of fluid takes place past a heated or cold surface, a

    temperature field is set up in the field next to the surface.

    [If the surface of the plate is hotter than fluid, the temperature distribution

    will be as shown in the Figure.]

    The zone or this layer wherein the temperature field exists is called the

    thermal boundary layer.

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    10:10:49

    The thermal boundary layer thickness (t) is arbitrarily defined as the

    distance y from the plate surface at which

    [The shape of thermal boundary layer when ]

    The thermal boundary layer (t) depends on the fluid properties like

    specific heat, thermal conductivity, flow velocity, and viscosity.

  • The relative magnitude of and t affected by the thermo-physical