Pharos UniversityME 352 Fluid Mechanics II

Problems on Boundary Layer

2

Boundary Layer Thicknesses

Problem 1: Determine the displacement and momentum thickness if the velocity profile over flat plate is given by

Solution: Displacement thickness (*)

my

U

u/1

0

/1

0

* 11 dyy

dyU

um

3

Velocity Profile

Similarly momentum thickness ()

1

1*

m

0

/1/1

011 dy

yydy

U

u

U

umm

21

mm

m

4

Shape Factor

Shape factor (H) is defined as ratio of displacement thickness to momentum thickness

So, H = */ = (m+2)/m For m=7

*= /8, =7/72 and H = 9/7=1.286

5

Problem 2

Model airplane wings are tested in a wind tunnel under atmospheric pressure and with a wind velocity of 2 m/s. If the aerofoil wing section can be treated as a plate of length 15 cm and width 50 cm, calculate The boundary layer thickness at trailing edge Displacement thickness at trailing edge The drag force exerted by the wind

Take kinematic viscosity of air as 1.5x10-5 m2/s and density as 1.22 kg/m3.

6

Solution

Reynolds number at the trailing edge

Since Re is less than critical Re, i.e. 5x105, so boundary layer is wholly laminar and so we can use Blasius solution

45 102

105.1

15.02

Ux

Re

7

Blasius Solution

Boundary layer thickness ()

Displacement thickness (*)

Average skin friction coefficient (Cf)

mmmR

x

x

2.50052.0102

15.091.491.44

mmmR

x

x

8.10018.0102

15.0729.1729.14

*

0094.0102

328.1328.14

L

fR

C

8

Drag force

Drag force Ff from one side is given by

Since drag force would act on both up and down face so total drag force = 0.0034 N

NF

BLUCF

f

ff

0017.015.05.0222.12

10094.0

2

1

2

2

9

Problem 3

A streamline train is designed to run between two cities at an average speed of 172 km/h. The train is 3 m wide, 3 m high and 100 m long. Compute the power required to overcome the skin friction drag.

Assume that the entire friction drag is offered only by sides and the top of the train and can be approximated as a flat plate.

10

Solution

Free stream velocity U = 172 kmph=172x103/3600 m/s = 47.7 m/s

Taking kinematic viscosity of air as 1.5x10-5 m2/s, Reynolds number at the trailing edge

Assuming wholly turbulent boundary layer

00147.0102.3

074.0074.05/185/1

L

fR

C

85 102.3

105.1

1007.47

UL

RL

11

Drag and Power Required

For = 1.22 kg/m

Power required to overcome skin friction is P = FfxU=1836x47.7 watts P = 87.6 kW

NF

AUCF

f

ff

183610097.4722.12

100147.0

2

1

2

2

12

Problem 4

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.

13 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

21

21

21

21

5247.10

5247.10

)3.1(12

12,3.1

,

~

x

C

C

xgiven

Cxor

x

From similarity solution and normalization analysis,

X (m) δ (mm)

0.2 4.707

2.0 14.884

20 47.068

14

Problem 5

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.

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

15smuu

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

21

21

6

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

21

21

6

16

Problem 6

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.

17

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

18

Problem 7

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

*

19

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

20

Problem 8

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.

21

xU

dA

xy

w

w

23

332.0 where

2 D dragFriction

5.00 ;5.0x0 :region Area

22

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

23

Problem 9 The boundary layer over a thin aircraft wing can be treated

as that over a flat plate. The speed of the aircraft is 100m/s and the chord length of the wing is 0.5m. At an altitude of 4000m, the density of air is 0.819kg/m3 and the kinematic viscosity is 20x10-6m2/s, Assuming the flow over the wing is 2D and incompressible, Calculate the boundary-layer thickness at the trailing

edge Estimate the surface friction stress at the trailing edge Will the boundary layer thickness and friction stress

upstream be higher or lower compared to those at the trailing edge?

What will be the boundary-layer thickness and the surface friction stress at the same chord location if the speed of the aircraft is doubled?

24

SOLUTIONS

The Reynolds number at x=0.5m:

The boundary layer thickness:

Friction stress at trailing edge of the wing

mx

Ux

x

x

0016.0105.2

5.05

Re

5

105.21020

5.0100Re

6

66

2

6

2

2

2

/72.1105.2

664.0100819.05.0

Re

664.05.0

5.0

mN

u

Cu

x

fw

25

SOLUTIONS

Will the boundary layer thickness and friction stress upstream higher or lower compared to those at the trailing edge? is smaller upstream as is proportional to x1/2

w is higher upstream as is thinner.

If the speed of the aircraft is doubled, Rex will be doubled since

will be smaller as Re increases.

w will be higher as the increase in U∞ has a greater effect than the increase in Rex.

RexU x

x

w uRe

664.05.0 2

x

x

Re

5