AY2014 CE2134 Hydraulics 4P Boundary layer flow over a flat plate II

8/10/2019 AY2014 CE2134 Hydraulics 4P Boundary layer flow over a flat plate II

1/15


2/15

LAMINAR BOUNDARY LAYER OVER A SMOOTH FLAT

PLATE

At any x

" #w =d

dx0

$

%& u2 "Umu( )dz'(

)

*+

,

-".u

.z

/

01

2

34w

=

d

dx&u2dz

0

$

%/

01

2

34"Um

d

dx&udz

0

$

%/

01

2

34

For similar velocity profilesu

Um= f

z

$

/

01

2

34= f 5( )

-Um

$

df 5( )

d5

'(

)

*+

,5=0=&

.

.x

Um2$ 1" f 5( )[ ]f 5( )d5

0

1

%'(

)

*+

,


3/15

LAMINAR BOUNDARY LAYER OVER A SMOOTH FLAT

PLATE

Let

"= 1# f $( )[ ]f $( )d$0

1

%

&=df $( )d$

'()

*+,$= 0

-Um&

.=/Um

2"0.

0x

Integrating

&x =1

2

/Um".2+ const

With .= 0 atx = 0

.=2&

"

x

Rex

'()

*+,


4/15

LAMINAR BOUNDARY LAYER OVER A

SMOOTH FLAT PLATE

"W

=

d

dx#U

m$u( )udz

0

1

%&

'(

)

*+

=#Um

2,d-

dx

=#Um

2, 2.

#Um,

&

'(

)

*+1

2x

$1/ 2

=#Um

2 ,.

2Rex

F = "Wdx

0

L

#

=$Um

2%&0

L

= 2%'$Um

3L[ ]

The total friction force on one side of

the smooth flat plate between x=0

and x=L for a unit width of the plate


5/15

EXAMPLE: LAMINAR BOUNDARY LAYER

Ftotaldragforce

! = 22 sides

! 2"#$Um3

L[ ] where "= 1% f&( )[ ]

0

1

' f&( )d& and #=

df&( )d&

(

)*

+

,-& =0

" = 1 # 2$# $2( ){ }0

1

% 2$# $2( )d$ = 2$# 5$2 + 4$3 #$4{ }0

1

% d$

= $2 #

5

3$

3+$

4 #1

5$

5&

'(

)

*+0

1

=1 #5

3+1 #

1

5=

2

15 ,

" =df#( )

d#

$

%

&'

(

)

# =0

= 2 * 2#[ ]# =0

= 2 +


6/15

EXAMPLE : LAMINAR BOUNDARY LAYER

F = 2 2"#$U3L

[ ] = 2 2

2

15

%

&'

(

)*2( )$U3L

+

,-

.

/0=1.46 $U

3L

By definition,

F =CD

1

2$ 2L(

)U

2

1CD =1.46 $U3L

$LU2=

1.46

$UL

=

1.46

ReL

2


7/15

TURBULENT BOUNDARY LAYER OVER A FLAT

PLATE

In the case of the turbulent boundary layer, the shearing stress atthe plate may be expressed by

"w= +#

t( ) $u

$z

%

&'

(

)*z= 0

"tthe eddy viscosity is a property of the flow; i.e. it depends onthe character of the flow. The eddy viscosity can be different

all over the flow and it is not readily determined


8/15

BLASIUS

FORMULA FOR TURBULENT SHEAR

STRESS OVER SMOOTH SURFACE (Re


9/15

(b) Blasius formula for f for smooth pipes with diameter D and

mean velocity U. This appears to fit the experimental results forReynolds number between 3000 and 105.

f = 0.32UD

"

#

$%

&

'(

)1/ 4

* f = 0.32 0.8UmD"

#

$% &

'(

)1/ 4

= 0.32 1.6UmR"

#

$% &

'(

)1/ 4

= 0.321.6Um+

"

#

$%

&

'(

)1/ 4

,w =

-fU2

8=

-

8 0.8Um( )

2

0.32( ) "

1.6Um+

#

$%

&

'(

1/ 4

= 0.0227-Um2 "

Um+

#

$%

&

'(

1/ 4

.

BLASIUS

FORMULA FOR TURBULENT SHEAR

STRESS OVER SMOOTH SURFACE (Re


10/15

TURBULENT BOUNDARY LAYER OVER A FLAT PLATE

Use the momentum equation of von Karman for the mean timequantities, assume similarity in velocity profiles along the plate andaccount for the wall shear stress by using an experimentally derivedvalue.

If thenand Umis constant along the direction of x outside the boundarylayer.

"p*

"x=0

"Um

"x=0

Simple velocity profile suggested by Prandtl in the turbulent

boundary layer

u

Um

= z

"

#

$%

&

'(

n

, n =1

7

(c) Assume a velocity profile


11/15


12/15

TURBULENT BOUNDARY LAYER OVER A FLAT PLATE

"

Um

#

$%

&

'(

1/ 4

x = 3.45)5 / 4

+ constant

*)

x= 0.376

Umx

"

#

$%

&

'(

+1/ 5

,

Turbulent boundary layer starts at the transition location and has some

thickness at this position. Location of the transition is difficult todetermine.

The length of the laminar boundary layer is generally small and one canapproximately imagine that the turbulent boundary layer commences atthe leading edge x = 0.

In the turbulent boundary layer, the thickness varies as

In the laminar boundary layer, the thickness varies as

The turbulent boundary layer grows faster along x than the laminar case.

5/4x

x1/ 2


13/15

TURBULENT BOUNDARY LAYER DRAG OVER A FLAT PLATE

The drag for unit width on both sides of the smooth plate

Drag = 20

L

! w! dx= 0.072"Um2L #

UmL

"#$ %

&'

1/5

=0.072 "Um2LR

eL

(1/5

The drag coefficient CDof an object immersed in a fluid stream ofvelocity U and mass density #is defined by

CD =Drag force

1

2"Um

2Area( )

CD

=

0.072!Um

2L.1( )ReL

!1/5

0.5!Um

22L

two sides

!

= 0.072ReL

!1/5


14/15

DRAG ON FLAT PLATE


15/15

BLANK

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AY2014 CE2134 Hydraulics 4P Boundary layer flow over a flat plate II