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(a) Development of fully-developed laminar flow

(b) Formation of laminar boundary layer

Flow is parallel to the surface in the x-direction with velocity u. Any movement of particles perpendicular to the surface move with velocity v in the y-direction. Boundary layer thickness, 8, is perpendicular to the flat surface. The free-stream velocity, U, is outside of the boundary layer.

46 eve U n CJ u r y y _ '. .,.-1 _ x

rest and a high velocity gradient is developed In the boundary layer. The high velocity gradients are associated with large frictional stresses in the boundary layer which cause the slowing down of the flow further downstream in the successive fluid elements. Thus the boundary layer steadily thickens downstream along the channel . The flow in the boundary layer can be either laminar or turbulent (Foust et aI, 1 980; Vennard and Street, 1 984, Lienhard, 1 987).

The Reynolds number is also used to characterise the flow in the boundary layer. The characteristic dimension used in Equation (2.20) is the boundary layer thickness (0) or the distance along the flow channel (x). Therefore, for the boundary layer the Re is written as,

u o p

. _

u x p

-_ or

11 11

(2.54) The critical values for the Re in the boundary layer is different to that for the free flowing fluid in the centre of the pipe, it is approximately 4,000 when 0 is used and approximately 500,000 when x is used. Below these critical values the boundary layer is l am inar (Vennard and Street, 1 984).

A schematic diagram of a boundary layer in l am inar flow is shown in F igure 2.4(b) Typically the boundary layer thickness (0) is arbitrarily defined as the distance from the wall at which the flow velocity approaches to within 1 % of the free-stream velocity ( U) (Foust et ai, 1 980; Lienhard, 1 987) In a laminar flow regime the boundary layer thickness can be estimated at a point x along the channel using the fol lowing equation which uses Re for the free flowing fluid in the centre of the pipe [(Re)x1 (Foust et ai ,

1 980),

4.64

(Re)� 5 (2 5 5 )

This equation implies that if the velocity is high o r the viscosity is low, when the Re i s large, then

olx

_{will b e relatively small and the boundary layer wil l be thin. If the} velocity is low the boundary layer will be relatively thick (Lienhard, 1 987).

For flow in a pipe the boundary l ayers can steadily thicken until they meet in the m iddle and envelop the entire flow. At that point the flow is " establ ished" or "fu lly-developed" and there is no further change in the velocity profil e (Foust et aI, 1 980; Vennard and Street, 1 984; Incropera and DeW itt, 1 98 5). Fully-developed flow in a cyl indrical pipe is shown in Figure 2.4(a) If the Reynolds number for ful ly-developed flow is less than

48 2 , 1 00, then it is inferred that the fully-developed flow has resulted from the growth of laminar flow boundary layers (Foust et ai, 1 980; Vennard and Street, 1 984) .

The distance from the entry of a pipe before fully-developed lam inar flow is given by the following equation, using Re for the entire flow channel

L '

-

entry length, m

L I _{;::; a}_Re

-

constant ranging from 0.05 to 0.0575 for laminar flow

(Foust et ai, 1 980; Vennard and Street, 1 984; Incropera and DeWitt, 1 985).

(2 . 56 )

These boundary layers are rel ated to fluid velocity and are called velocity boundary layers. Thermal boundary layers

(o()

_{may also develop if the fluid free-stream and} surface temperatures differ. Concentration boundary layers

(oe)

_{m ay develop when the} concentration at the surface (wall ) differs from the concentration in the free-stream It is the region of fluid i n which concentration gradients exits and its thickness is defined by

oe

_{(Incropera and DeWitt,}1 98 5 ) When the three boundary layers coexist, they rarely develop at the same rate and the values of

0, 0(, oe

_{at a given}_x_{location are not usually} the same.

For two-dimensional, steady-flow conditions, equations defining boundary layer conditions have been developed. A continuity equation for conservation of mass has been derived for a velocity boundary layer,

o (pu)

-

velocity in the x direction, m S - 1 V

-

velocity in the y direction, m S -l P

-

density of fluid, kg m -3

o (pv)

-

distance along the solid surface, m

-

distance perpendicular to the sol id surface, m

o (2 5 7)

Equating the rate of change i n the x momentum of the fluid to the sum of forces in the x direction (Newton's second law of motion), the fol lowing equation was derived for

momentum fluxes in the x direction

( au

p u-:::

ox and in the y direction

+- - p ) _+- y The equations for the associated stresses are

2 /l - - - /l -eu 2

(Cu

+ _

)

ax 3 ax 2 _{/l - - - /l - + -}ev 2

(eu Ov)

ey 3 ax

cy

l yX _{11 - + -}

(ex

ell

)

(Jy

- normal stress, kg m -I

S-2

- shear stress, kg m -I

S-2

(2 . 59) (2 60) l

subscript - first subscript indicates the orientation, second subscript indicates the direction of the force

)( y - components of the body force per unit volume, N m -3 (Incropera and DeWitt, 1 98 5 )

Equations (2. 5 7) to (2.60) can be solved to determine the velocity field in the boundary layer for two dimensional flow.

In a thermal boundary layer the equation for conservation of energy is ej

pu

+ pv _

In document Los millennials y sus procesos emocionales en red Un reto para la educación colombiana (página 62-75)

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