flat plate boundary layer 7over10 complete

8/10/2019 Flat Plate Boundary Layer 7over10 Complete

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Abstract

The following experiment will give the proper procedure to determining and measuring the

boundary layer using the momentum integral equation. During this experiment, Airflow

bench, Test apparatus, Micrometer scale and Velocity measurement were used as apparatus to

study the boundary layer behaviour when using smooth and rough plate. By measuring the

boundary layer thickness, () displacement thickness, ( ) momentum thickness,

(

)thereforevelocity profile can be measured.


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TABLE OF CONTENTS

CONTENTS PAGE

1. Introduction 1

2. Objectives 5

3. Apparatus 6

4. Experimental Procedures 8

5. Data Collected 12

6. Sample of calculations 16

7. Results

8.

Discussions 17

9. Conclusions 28

10.References 39

Appendices and Raw Data -


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List of figures, graphs and tables

Figure 1.1: Boundary layer on a flat plate 1

Figure 1.2: Boundary layer thickness 3

Figure 4.1: 6

Figure 4.2: 6

Figure 4.3: 7

Figure 5.1: 8

Graph 6.1: Graph y of vs u/U for test 1.

Graph 6.2: Graph of y vs (u/U)(1-u/U) for test 1

Graph 6.3: Graph y of vs u/U for test 2


Graph 6.5: Graph y of vs u/U for test 3.


Graph 6.7: Graph y of vs u/U for test 4



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1. INTRODUCTION

As for flow in a duct, flow in boundary layer may be laminar or turbulent.

There is no unique value of the Reynold number at which transition from laminar to

turbulent flow occur in a boundary layer. Among the factors that affect boundary

layer transition are pressure, gradient, surface, roughness, heat transfer, body forces

and free stream disturbances.

In many real life flow situations, a boundary layer develops over a long,

essentially flat surface. A qualitative picture of the boundary layer growth over a flat

plate is shown in Figure 1.1 below.

Figure 1.1: Boundary layer on a flat plate.

Some measures of boundary layers are describe in Figure 1.2 below.

Figure 1.2: Boundary layer thickness.


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The boundary layer thickness,

, is used for a thickness beyond which the velocity is

essentially the free-stream velocity, U. This is customarily defined as the distance from the

wall to the point where

u|y= =0.99

U

The displacement thickness, *, is the distance by the solid boundary would have to be

displaced in a frictionless flow the same mass deficit exist in the boundary layer. The

mathematcal definiton of the displacement thickness for incompressible flow is given by

*= ( )

The momentum thickness, , is defined as the thickness of the layer fluid of velocity, U(free stream velocity), for which the momentum flux is equal to the deficit of momentum flux

through the boundary layer. Mathematically it is defines as

= ( )

The equation for velocity measured by pitot tube is given as

u =

The Blasiuss exact solution to the laminar boundary layer yields the following equations for

the above properties.

=

* =

=


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Due to the complexity of the flow, there is no exact solution to the turbulent boundary layer.

The properties of the boundary layer are approximated using the momentum integral equation

which results in the following expression

=

* =

=


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2. OBJECTIVES

1. To measured the boundary layer velocity layer and observed the growth of the

boundary layer for the flat plate with smooth and rough surface.

2. To measured the boundary layer properties for the measured velocity profile.

3. To studied the effect of surface roughness on the development of the boundary

layer.


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3. APPARATUS

1. Airflow benchprovie adjustable air stream which enables a series of

experiment to be connected

2. Test apparatusconsists of rectangular duct with flat plate. One side of the

plate is smooth and other rough. Pitot tube is set in zero plane of scale. By

moving plate up and down, the leading edge can be set to given

distance from pitot tube tip.

3. Micrometer scaleto measure the displacement of pitot tube from wall.

4. Velocity measurementvelocity is measured using total and static probes

which connected to multi-tube manometer.


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4. EXPERIMENTAL PROCEDURES

1. The apparatus had been set up on the bench. Smooth surface of the plat was

used for the first part of experiment.

2. To set the measurement playe at the required distance from leading edge, the

position of the central plate was adjusted to 50mm.

3. The fan and the air flow speeds was switched on to set the air stream velocity

at medium speed. The total pressure of the pitot tube was read for a range of

several points as the tube traverse toward the plate at an interval of 0.25mm.

4. The increment of the advanced reduced as the pressure begins to fall to clearly

define the velocity profile. The pressure reading was not fall to zero as the

pitot tube has a finite thickness.

5. Step 2 to 4 were repeated for set up of measurement plate at 200mm.

6. The entire experiments was repeated for the rough surface plate.


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5. DATA COLLECTED

Test 1: Smooth Surface Plate

Distance from leading edge, x: 0.05 m

Free stream velocity, U: 22.37 m/sRe: 0.738 X 105

Micrometer

reading, y

(mm)

Static

pressure

manometer,

(mbar)

Total

pressure

Manometer

(mbar)

Differential

manometer

height

(mbar)

Differential

manometer

height

h, (mm)

u

(m/s)

0 10.4 12.4 2.0 25.60 18.08 0.81 0.15

0.25 10.4 12.8 2.4 30.72 19.81 0.89 0.10

0.50 10.4 13.0 2.6 33.28 20.62 0.92 0.07

0.75 10.4 13.2 2.8 35.84 21.40 0.96 0.04

1.00 10.4 13.4 3.0 38.40 22.15 0.99 0.01

1.25 10.4 13.4 3.0 38.40 22.15 0.99 0.01

1.50 10.4 13.4 3.0 38.40 22.15 0.99 0.01

1.75 10.4 13.4 3.0 38.40 22.15 0.99 0.01

2.00 10.4 13.4 3.0 38.40 22.15 0.99 0.01

2.25 10.4 13.4 3.0 38.40 22.15 0.99 0.01

2.50 10.4 13.4 3.0 38.40 22.15 0.99 0.01


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Test 2: Smooth Surface Plate


Free stream velocity, U: 23.82 m/s

Re: 3.14 X 105

Micrometer

reading, y

(mm)

Static

pressure

manometer,

(mbar)

Total

pressure

Manometer

(mbar)

Differential

manometer

height

(mbar)

Differential

manometer

height

h, (mm)

u

(m/s)

0 10.2 12.4 2.2 28.16 18.97 0.80 0.16

0.25 10.2 12.8 2.6 33.28 20.62 0.87 0.11

0.50 10.2 13.0 2.6 33.28 20.62 0.87 0.11

0.75 10.2 13.2 3.0 38.40 22.15 0.93 0.07

1.00 10.2 13.2 3.0 38.40 22.15 0.93 0.07

1.25 10.2 13.4 3.2 40.96 22.88 0.96 0.04

1.50 10.2 13.4 3.2 40.96 22.88 0.96 0.04

1.75 10.2 13.4 3.2 40.96 22.88 0.96 0.04

2.00 10.2 13.6 3.4 43.52 23.58 0.99 0.01

2.25 10.2 13.6 3.4 43.52 23.58 0.99 0.01

2.50 10.2 13.6 3.4 43.52 23.58 0.99 0.01

2.75 10.2 13.6 3.4 43.52 23.58 0.99 0.01

3.0 10.2 18.6 3.4 43.52 23.58 0.99 0.01

3.25 10.2 13.6 3.4 43.52 23.58 0.99 0.01

3.5 10.2 13.6 3.4 43.52 23.58 0.99 0.01


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Test 3: Rough Surface Plate



Re: 0.762 X 105

Micrometer

reading, y

(mm)

Static

pressure

manometer,

(mbar)

Total

pressure

Manometer

(mbar)

Differential

manometer

height

(mbar)

Differential

manometer

height

h, (mm)

u

(m/s)

0 10.4 12.4 2.0 25.60 18.08 0.78 0.17

0.25 10.4 12.8 2.4 30.72 19.81 0.86 0.12

0.50 10.4 13.0 2.6 33.28 20.62 0.89 0.10

0.75 10.4 13.2 2.8 35.84 21.40 0.93 0.07

1.00 10.4 13.4 3.0 38.40 22.15 0.96 0.04

1.25 10.4 13.4 3.0 38.40 22.15 0.96 0.04

1.50 10.4 13.4 3.0 38.40 22.15 0.96 0.04

1.75 10.4 13.4 3.0 38.40 22.15 0.96 0.04

2.00 10.4 13.6 3.2 40.96 22.88 0.99 0.01

2.25 10.4 13.6 3.2 40.96 22.88 0.99 0.01

2.50 10.4 13.6 3.2 40.96 22.88 0.99 0.01

2.75 10.4 13.6 3.2 40.96 22.88 0.99 0.01

3.0 10.4 18.6 3.2 40.96 22.88 0.99 0.01

3.25 10.4 13.6 3.2 40.96 22.88 0.99 0.01

3.5 10.4 13.6 3.2 40.96 22.88 0.99 0.01


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Test 4: Rough Surface Plate



Re: 3.23 X 105

Micrometer

reading, y

(mm)

Staticpressure

manometer,

(mbar)

Totalpressure

Manometer

(mbar)

Differentialmanometer

height

(mbar)

Differentialmanometer

height

h, (mm)

u

(m/s)

0 10.2 12.2 2.0 25.60 18.08 0.74 0.19

0.25 10.2 12.4 2.2 28.16 18.97 0.77 0.18

0.50 10.2 12.8 2.6 33.28 20.62 0.84 0.13

0.75 10.2 13.0 2.8 35.84 21.40 0.87 0.11

1.00 10.2 13.0 2.8 35.84 21.40 0.87 0.11

1.25 10.2 13.2 3.0 38.40 22.15 0.90 0.09

1.50 10.2 13.4 3.2 40.96 22.88 0.93 0.07

1.75 10.2 13.4 3.2 40.96 22.88 0.93 0.07

2.00 10.2 13.6 3.4 43.52 23.58 0.93 0.07

2.25 10.2 13.6 3.4 43.52 23.58 0.93 0.07

2.50 10.2 13.6 3.4 43.52 23.58 0.93 0.07

2.75 10.2 13.8 3.6 46.08 24.26 0.99 0.01

3.0 10.2 13.8 3.6 46.08 24.26 0.99 0.01

3.25 10.2 13.8 3.6 46.08 24.26 0.99 0.01

3.5 10.2 13.8 3.6 46.08 24.26 0.99 0.01

3.75 10.2 13.8 3.6 46.08 24.26 0.99 0.01

4.0 10.2 13.8 3.6 46.08 24.26 0.99 0.01

4.25 10.2 13.8 3.6 46.08 24.26 0.99 0.01


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6. SAMPLE OF CALCULATIONS

Smooth surface with distance from the leading edge, x = 0.05m

air = 1.204 kg/m3

air= 1.825 x 10-5

m2

/soil = 784 kg/m

3

velocity, u =a

o hg

2m/s

=204.1

106.2581.97842 3

m/s

= 18.08 m/s

Free stream velocity, U = highest value of u / 0.99

= 22.15 = 22.37

Reynolds number, Rex = airUXair

=5

10825.1

05.037.22204.1

= 0.735 X 105< 500000 (laminar)


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Sample calculation for boundary layer thickness,(

)displacement thickness, (

) momentum

thickness, (

)by usingtheoretical

For smooth surface with x = 50mm

x

x

Re

0.5

=

= 0.9203mm

x

x

Re

72.1

=

= 0.3166mm

x

x

Re

664.0

=

= 0.1222mm


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Sample calculation for boundary layer thickness,(

)displacement thickness, (

) momentum

thickness, (

) for experimental values

Boundary layer thickness, = 1.0 mm

Displacement thickness, * =- u/U)dy= - y/ 1/7)dy=/

= 1/8(1.0)

= 0.125 mm

Momentum thickness, = u/U - u/U)

= y/ 1/7)(1- y/ 1/7)=/

= 7/72(1.0)

= 0.0972 mm


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7. RESULTS

Graph 6.1: Graph of y vs u/U for test 1.

Graph 6.2: Graph of y vs (u/U)(1-u/U) for test 1.

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2

m

icrometerreading,y

(mm)

u/U

test1 graph y versus u/U

0

0.5

1

1.5

2

2.5

3

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

heigh

t,y(mm)

(u/U)(1-u/U)

test1 graph y vs (u/U)(1-u/U)


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0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1 1.2

height,y(mm)

u/U

test2, y vs u/U

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

height,y(mm)

(u/U)(1-u/U)

test2 y vs (u/U)(1-u/U)


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0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1 1.2

height,y(mm)

u/U

test3 y vs u/U

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

height,y(mm)

(u/U)(1-u/U)



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0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.2 0.4 0.6 0.8 1 1.2

height,y(mm)

u/U

test4 y vs u/U

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

height,y(mm)

(u/U)(1-u/U)



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Boundary layer

thickness,() (mm)

Displacement

thickness, (

) (mm)

Momentum

thickness, ()

Test1 theoretical 0.9203 0.3166 0.1222

Test1 experimental 1.0 0.125 0.0972








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9. REFERENCES

i.

P.M Gerhart, R.J. Gross and J.I. Hochstein, Fundamentals of fluid Mechanics, 2nd

Edition, 1992.

ii.

F.M White, Fluid Mechanics, McGraw-Hill, 4th Edition, 1999.

iii. Fundamental of Fluid Mechanics, B.r. Munson, D.F. Young and T.H. Okiishi, John

Wiley and Sons, 3rd Edition, 1998

iv.

Yunus A. engel and John M. Cimbala, 2010, Fluid Mechanics Fundamentals And

Applications Second Edition in SI Unit, Published By McGraw Hill International

Edition. In Singapore.

v. A.Cengel, Y., & John M. Cimbala. (N.D.), Fluid Mechanics Fundamental and

Applications, (2nd Edition).

TABLE OF CONTENTS

CONTENTS PAGE

11.OBJECTIVES 2

12.INTRODUCTION 2

13.

THEORY 3

14.APPARATUS 5

15.PROCEDURES 7

16.RESULTS 8

17.DISCUSSIONS 10


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18.CONCLUSIONS 11

19.

RECOMMENDATIONS 12

20.REFERENCES 39

flat plate boundary layer 7over10 complete

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