MECH 221 FLUID MECHANICS (Fall 06/07) Tutorial 9

1

MECH 221 FLUID MECHANICS(Fall 06/07)Tutorial 9

2

Example 1

A viscous fluid flows past a flat plate such that the boundary layer thickness at a distance 1.3m from the leading edge is 12mm. Determine the boundary layer thickness at distances of 0.2m, 2.0m and 20m from the leading edge. Assume laminar flow.

If the upstream velocity of the flow is 1.5m/s, determine the kinematic viscosity of the fluid.

3

Example 1

From similarity solution and normalization analysis,

21

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21

21

5247.10

5247.10

)3.1(12

12,3.1

,

~

x

C

C

xgiven

Cxor

x

X (m) δ (mm)

0.2 4.707

2.0 14.884

20 47.068

4

Example 1

From Eq. 9.15,

smx

U

mmmmxm/s.UGiven

x

U

U

x

/10646.6)3.1(25

)012.0)(5.1(

25

012.012,3.1 ,51

25

5

2622

2

5

Example 2

Water flows past a flat plate with an upstream velocity of U=0.02m/s. Determine the water velocity a distance of 10mm from the plate a distances of x=1.5m and x=15m from the leading edge.

6

Example 2

flowlaminar ReRe

10510975.1)10519.1(

)15)(02.0(Re ,15

Re ,

)( where),('

plate,flat aon flowlayer boundary for Solution Blasius From

crx

556x

x

21

mxfor

UxSince

x

Uyf

U

u

7

Example 2

smuu

U

uf

f

f

f

/006178.03089.002.0

)('

3089.0)(' ,937.0

ion,interpolatBy

3938.0)(' ,2.1

2647.0)(' ,8.0

9.1 TableIn

937.0))5.1)(10519.1(

)02.0()(01.0()

x

Uy(

0.01m,10mmy 1.5m,At x

applicabe. is 9.1 Table

plate,flat aon flowlayer boundary laminar for Solution Blasius From

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smuu

U

uf

f

f

f

/001965.00983.002.0

)('

0983.0)(' ,296.0

ion,interpolatBy

1328.0)(' ,4.0

0)(' ,0

9.1 TableIn

296.0))15)(10519.1(

)02.0()(01.0()

x

Uy(

0.01m,10mmy 15m,At x

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21

6

8

Example 3

Because of the velocity deficit, U-u, in the boundary layer, the streamlines for flow past a flat plate are not exactly parallel to the plate. This deviation can be determined by use of the displacement thickness, δ*. For air blowing past the flat plate shown in the figure, plot the streamline A-B that passes through the edge of the boundary layer (y=δB at x=l ) at point B. That is, plot y=y(x) for streamline A-B. Assume laminar boundary layer flow.

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Example 3

mU

x

UL

B

B

0382.0)1(

)4)(101.46(55

i.e. L,at x hicknessBoundary t

flowlaminar 1074.2)101.46(

)4)(1(Re

/sm101.46 4m;L 1m/s;given U

Re

flow, theof Re

5-

5-5-L

25-

L

x

10

Example 3

my

UyU

m

U

xwhere

UyU

BA

BA

BA

0251.001315.00382.0

)())((

01315.0)1(

)4)(1046.1(721.1

k)in textboo 9.16 (Eq. 721.1

)())((

,QQ

B,-A streamlineor plate he through tflow no is thereSince

.amount an by displaced plate with the velocity uniform aby carried

that toequal definitionby islayer boundary actual by the carried flowrate The

*

*

5*

*

*

BA

*

11

Example 3

xy

x

U

x

3

5

A*

A

*A

*A

A

1058.60251.0

)1(

)1046.1(721.10251.0y

721.1yyy

-yy

)-U(y)(U)(y

QQ

direction,-any xFor

12

Example 4

Fluid flows past a triangular flat plate oriented parallel to the free stream as shown in the figure. Integrate the wall shear stress over the plate to determine the friction drag on one side of the plate. Assume laminar boundary layer flow.

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Example 4

xU

dA

xy

w

w

23

332.0 where

2 D dragFriction

5.00 ;5.0x0 :region Area

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Example 4

N

smUNs

xxU

dxx

xU

dydxx

U

dydxx

U

dA

x

x

x

x

xy

y

x

x

xy

y

w

0296.0D

))5.0(3

2()5.0)(2(5.0)1012.1)(999()2.0(664.0 D

2.0;m

1012.1;m

kg999 1.6, Table From

)3

2())(2(5.0664.0 D

5.0664.0 D

1664.0 D

332.02 D

2 D dragFriction

32

21

23

32

21

23

23

23

23

3

23

3

5.0

0

5.0

0

5.0

0

5.0

0

5.0

0

5.0

0