Flat plate boundary.docx

1.0 TITLE Flat plate boundary layer 2.0 INTRODUCTION Fluid flow is often confined by solid surfaces, and it is important to understand how the presence of solid surfaces affects fluid flow. Consider the flow of a fluid over a solid surface and the fluid is in direct contact with the surface with no slip. This is known as no-slip condition. By looking at the velocity gradient, the layer becomes slowed down from one to another. This is because of viscous forces between fluid layers. A consequence of the no-slip condition is that all velocity profiles must have zero values with respect to the surface at the points of contact between a fluid and a solid surface. The flow region adjacent to the wall in which the viscous effects are significant is called the boundary layer. 3.0 OBJECTIVES 3.1. To measured the boundary layer velocity layer and observed the growth of the boundary layer for the flat plate with smooth and rough surface 3.2. To measured the boundary layer properties for the measured velocity profile 3.3. To studied the effect of surface roughness on the development of the boundary layer 4.0 THEORY BACKGROUND Classical theory of real fluid flow that has been tested by experimental has shown that when a fluid flows over a surface there is no slip at the surface. The fluid in contact with the surface stays with it. The relative velocity increases from zero at the surface to that of the free stream some little distance away from the surface. The fluid in this small distance is called Boundary Layer. Consider a steady stream of fluid moving from left to right over a smooth plate. The free stream velocity, U, is constant over the entire plate. It is found that the boundary layer grows in thickness the further we travel downstream.

U

U U

Turbulence

Laminar Transition

Figure 1: Boundary Layer growth The initial motion is laminar with a gradual increase in thickness. If the plate is sufficiently long a transition to turbulence occurs.

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Laminar Boundary Layer In laminar boundary layer the flow is steady and smooth. Consequently the layer is thin. This give rise to drag. The velocity gradient is moderate and although significant viscous stresses exist is too small, so that skin friction is very small. Turbulence Boundary Layer In turbulence boundary layer the flow is unsteady and not smooth, but eddying. When specifying velocities, we must consider mean values over a small time interval and not instantaneous values as before. The distribution of mean velocity in any one time interval is the same as in another. Thus we can still draw velocity profiles, which have meaning. Due to the eddying nature of the flow there is a lot of movement of fluids between inner and outer layers of the regions. Thus the velocity near the wall will be higher than in a laminar boundary layer where the movement and energy transfer do not occur. The velocity gradient at the wall is consequently much higher so the skin friction and drag are also higher. Some measures of boundary layers are described in Figure 2 below.

U 0.99U x

area =

u ( U - u) dy 0

u y *

area = 0

( U - u) dy

Figure 2: Boundary Layer thickness definitions

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The boundary layer thickness, , is defined as the distance from the surface to the point where the velocity is within 1 percent of the stream velocity. uǀy=

The displacement thickness, *, is the distance by which the solid boundary would have to be displaced in a frictionless flow to give the same mass deficit as exists in the boundary layer.

∫ (

)

The momentum thickness, , is define as the thickness of a layer of fluid of velocity, U (free stream velocity), for which the momentum flux is equal to the deficit of momentum flux through the boundary layer.

∫

(

)

The equation for velocity measured by pitot tube is given as

√

The Blasius’s exact solutions to the laminar boundary yield the following equations for the above properties.



5.0 x Re x

 



1.72 x Re x

0.664 x Re x

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Due to the complexity of the flow, there is no exact solution to the turbulent boundary layer. The velocity profile within the boundary layer commonly approximated using the 1/7 power law.

u  y   U  

1

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The properties of boundary layer are approximated using the momentum integral equation, which result in the following expression.



0.370 x 1

Re x 5

 

0.0463 x 1

Re x 5



0.036 x 1

Re x 5

Another measure of the boundary layer is the shape factor, H, which is the ratio of the displacement thickness to the momentum thickness, H = */. For laminar flow, H increases from 2.6 to 3.5 at separation. For turbulent boundary layer, H increases from 1.3 to approximately 2.5 at separation.

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5.0 EXPERIMENTAL APPARATUS 5.1. Airflow bench The bench provides adjustable air stream enables a series of experiments to conduct when used with matching experimental equipment. The airflow is controlled by a damper linked to a control rod, which can be pulled in and out from the front panel of the bench. 5.2. Test apparatus It consists of rectangular duct with a flat plate in the middle of the duct. One side of the plate is smooth and other rough. 5.3. Total and static tube pressure probes and multi-tube manometer. The velocity is measured using total and static probes which are connected to multi-tube manometer.

6.0 EXPERIMENTAL PROCEDURES In this experiment the behaviour of boundary layer on flat plate with different surfaces are going to be observe. There are two surfaces being taken counted in this experiment which are smooth and rough surfaces. The boundary layer is to be observed at two different distances from the leading edge which are 50 mm and 200 mm. Therefore transformation of the boundary layer from laminar to turbulent can be studied. The boundary layer thickness supposed to be calculated by using theory before conducting the experiment. With the value, estimation can be made to decide the increment to be used in the experiment. As the increment begins to approaches the estimation value from theoretical calculation, the pressure falls should be observed. The pressure reading will not fall to zero as the Pitot tube has a finite thickness. A further indication that the wall has been reach is that the pressure reading will be zero. There should be a different in behaviour between boundary layer adjusted at 50 mm from trailing edge and adjusted at 200 mm with its different types of surfaces

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7.0 RESULT AND ANALYSIS Test 1 (Smooth Plate) Distance from leading edge, x = 50 mm Micrometer reading, y (mm)

Pitot tube Differential Differential Static pressure manometer manometer pressure manometer height height, Δh, (mBar) (mBar) (mBar) (mm)

0 5.3 7.4 0.25 5.3 7.8 0.50 5.3 8.0 0.75 5.3 8.1 1.00 5.3 8.2 1.25 5.3 8.3 1.50 5.3 8.4 1.75 5.3 8.5 2.00 5.3 8.5 2.25 5.3 8.5 2.50 5.3 8.5 Free stream velocity, U = 23.07 m/s Reynold number, Re = 73,864

2.1 2.5 2.7 2.8 2.9 3.0 3.1 3.2 3.2 3.2 3.2

26.88 32.00 34.56 35.84 37.12 38.40 39.68 40.96 40.96 40.96 40.96

u (m/s)

u/U

u/U (1- u/U)

18.69 20.39 21.19 21.58 21.96 22.34 22.70 23.07 23.07 23.07 23.07

0.8101 0.8839 0.9186 0.9354 0.9520 0.9682 0.9843 1 1 1 1

0.1538 0.1026 0.0748 0.0604 0.0457 0.0307 0.0155 0 0 0 0

u (m/s)

u/U

u/U (1- u/U)

17.78 19.98 20.39 21.19 21.58 21.96 22.34 22.70 22.70 23.07 23.07 23.07 23.07

0.7706 0.8660 0.8839 0.9186 0.9354 0.9520 0.9682 0.9843 0.9843 1 1 1 1

0.1768 0.1160 0.1026 0.0748 0.0604 0.0457 0.0307 0.0155 0.0155 0 0 0 0

Test 2 (Smooth Plate) Distance from leading edge, x = 200 mm Micrometer reading, y (mm)

Pitot tube Differential Differential Static pressure manometer manometer pressure manometer height height, Δh, (mBar) (mBar) (mBar) (mm)

0 5.3 7.2 0.25 5.3 7.7 0.50 5.3 7.8 0.75 5.3 8.0 1.00 5.3 8.1 1.25 5.3 8.2 1.50 5.3 8.3 1.75 5.3 8.4 2.00 5.3 8.4 2.25 5.3 8.5 2.50 5.3 8.5 2.75 5.3 8.5 3.00 5.3 8.5 Free stream velocity, U = 23.07 m/s Reynold number, Re = 295,456

1.9 2.4 2.5 2.7 2.8 2.9 3.0 3.1 3.1 3.2 3.2 3.2 3.2

24.32 30.72 32.00 34.56 35.84 37.12 38.40 39.68 39.68 40.96 40.96 40.96 40.96

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Test 3 (Rough Plate) Distance from leading edge, x = 50 mm Micrometer reading, y (mm)

Pitot tube Differential Differential Static pressure manometer manometer pressure manometer height height, Δh, (mBar) (mBar) (mBar) (mm)

0 5.3 7.2 0.25 5.3 7.5 0.50 5.3 7.7 0.75 5.3 8.0 1.00 5.3 8.1 1.25 5.3 8.2 1.50 5.3 8.3 1.75 5.3 8.4 2.00 5.3 8.5 2.25 5.3 8.5 2.50 5.3 8.5 Free stream velocity, U = 23.07 m/s Reynold number, Re =73,864

u (m/s)

u/U

u/U (1- u/U)

1.9 2.2 2.4 2.7 2.8 2.9 3.0 3.1 3.2 3.2 3.2

24.32 28.16 30.72 34.56 35.84 37.12 38.40 39.68 40.96 40.96 40.96

17.78 19.13 19.98 21.19 21.58 21.96 22.34 22.70 23.07 23.07 23.07

0.7706 0.8292 0.8660 0.9186 0.9354 0.9520 0.9682 0.9843 1 1 1

0.1768 0.1417 0.1160 0.0748 0.0604 0.0457 0.0307 0.0155 0 0 0

Differential manometer height (mBar)

Differential manometer height, Δh, (mm)

u (m/s)

u/U

u/U (1- u/U)

1.7 2.1 2.3 2.5 2.6 2.8 2.9 3.0 3.1 3.2 3.2 3.2 3.2

21.76 26.88 29.44 32.00 33.28 35.84 37.12 38.40 39.68 40.96 40.96 40.96 40.96

16.81 18.69 19.56 20.39 20.79 21.58 21.96 22.34 22.70 23.07 23.07 23.07 23.07

0.7289 0.8101 0.8478 0.8839 0.9014 0.9354 0.9520 0.9682 0.9843 1 1 1 1

0.1976 0.1538 0.1290 0.1026 0.0889 0.0604 0.0457 0.0307 0.0155 0 0 0 0

Test 4 (Rough Plate) Distance from leading edge, x = 200 mm Micrometer reading, y (mm)

Pitot tube Static pressure pressure manometer (mBar) (mBar)

0 5.3 7.0 0.25 5.3 7.4 0.50 5.3 7.6 0.75 5.3 7.8 1.00 5.3 7.9 1.25 5.3 8.1 1.50 5.3 8.2 1.75 5.3 8.3 2.00 5.3 8.4 2.25 5.3 8.5 2.50 5.3 8.5 2.75 5.3 8.5 3.00 5.3 8.5 Free stream velocity, U = 23.07 m/s Reynold number, Re =295,456

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Graph y vs. u/U for comparison the smooth and rough surfaces with distance from leading edge is 50 mm

Graph y vs. u/U 1.2 1 0.8 u/U 0.6

Smooth 50 mm Rough 50 mm

0.4 0.2 0 0

0.5

1

1.5

2

2.5

3

3.5

y

Graph y vs. u/U for comparison the smooth and rough surfaces with distance from leading edge is 200 mm

Graph y vs. u/U 1.2 1 0.8 u/U 0.6

Smooth 200 mm Rough 200 mm

0.4 0.2 0 0

1

2

3

4

y

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Graph y vs. u/U ( 1- u/U ) for comparison the smooth and rough surfaces with distance from leading edge is 50 mm

Graph y vs. u/U ( 1- u/U ) 0.20 0.18 0.16 0.14 0.12 u/U ( 1- u/U ) 0.10

Smooth 50 mm

0.08

Rough 50 mm

0.06 0.04 0.02 0.00 0

1

2

3

4

y

Graph y vs. u/U ( 1- u/U ) for comparison the smooth and rough surfaces with distance from leading edge is 200 mm

Graph y vs. u/U ( 1- u/U ) 0.22 0.20 0.18 0.16 0.14 0.12 u/U ( 1- u/U )

0.10

Smooth 200 mm

0.08

Rough 200 mm

0.06 0.04 0.02 0.00 0

1

2

3

4

y

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8.0 SAMPLE CALCULATIONS

,

√

√

(

) (

)

10

√

√

√

√

√

√

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9.0 DISCUSSIONS From the graph that we plotted, we see that its exactly a same between smooth and rough surface flat plate due to the errors occurs during experiments but from the theory, those smooth and rough surface flat plate graph at same distance from edge x, are suppose to be different. There are two types of flow in fluid that been showed in this experiment, laminar and turbulent flow. The differences between laminar and turbulence flow of fluid on the flat surface can be seen on the graph that have been plotted. Greater value of δ was obtained when the plane is rough while the δ value becomes lesser when the distance from the edge of the plate is further. Other than that, the roughness of the surfaces were effected the values of the pressures. The appearing of laminar and turbulent are depending on the smooth or rough of the flat plate, if the surface is smooth, the transition of laminar to be turbulent will delay, while when the surface is rough, the transition of laminar to become turbulent will be quick as there are small disturbance in the velocity profiles that make the flow easily pass through it. The differences of the velocity profiles showed on the graph plotted and the free stream velocity calculated was based on the smoothness and roughness of the surface. There are a few errors occurred in this experiment, such as parallax error during taking data from the experiment. Other than that, the error occurred when we calculated the U at 50mm and 200mm. Then at the same time we also measure the pitot tube in the same level. Unfortunately, our instrument is not capable on measuring the U at 50mm, is because the tube that measure at 50mm does not fit at the hole. So all the calculation involve U will be taken at 200mm only.

Figure 3(a)

Figure 3(b)

Figure 3: (a) boundary layer growth on a smooth surface; (b) boundary layer growth on a rough surface

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10.0 CONCLUSIONS The boundary layer velocities for the flat plate with smooth and rough surface have been obtained where the data can be seen from the table. The velocity profiles of the flat plate have been obtained through data read and the graphs have been plotted. The roughness of the flat plate gives the variety of the velocity profile. It can be concluded that the surface roughness of the flat plate influence the velocity profiles where the smooth surface will delay the transition while the rough surface will make the transition become faster.

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11.0 REFERENCES 11.1. Cengel Y.A., Cimbala J.M., Fluid Mechanic Fundamentals and Applications: Second Edition In SI Unit. McGraw Hill, New York, USA,2010 11.2. John F. Douglas, Janusz M. Gasiorek, John A. Swaffield, Fluid Mechanics, 4th Edition, Pearson Prentice Hall, Scotland, 2001 11.3. Bruce R. Munson, Donald F. Young, Theodore H. Okiishi, Fundementals of Fluid Mechanics, 5th Edition, John Wiley & Sons, Asia,2006

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