Introduction to Boundary Layer

Faculty of ACES

Wind tunnel test

Boundary layer development and separation on a flat plate

Coursework 2

Module Leader: Xinjun CUI

Leo Hamad

B1050081

Date: 18/03/2014

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ContentsTable of Figures and tables:........................................................................................2

Introduction:................................................................................................................3

Apparatus:...................................................................................................................3

Procedure:...................................................................................................................4

Introduction to boundary layer and background theory:..............................................4

Laminar Flow over a Flat Plate:..................................................................................6

Boundary layer equations by Blasius:......................................................................6

Sample calculations:...................................................................................................8

Results:.......................................................................................................................9

Discussion and conclusion:.......................................................................................12

Bibliography..............................................................................................................13

Table of Figures and tables:Figure 1The main parts of the flat plate boundary layer model AF106........................3Figure 2: Fluid flow over flat plate (y-axis enlarged)....................................................5Figure 3 Development of boundary layer for 20 m/s at 0 AoA..................................10Figure 4 Development of boundary layer for 25 m/s at 0 AoA...................................10Figure 5 Development of boundary layer for 30 m/s at 0 AoA...................................11Figure 6 Development of boundary layer for 35 m/s at 0 AoA...................................11Figure 7 Development of boundary layer for 20 m/s at AoA.....................................12Figure 8 Development of boundary layer for 25 m/s at AoA.....................................12Figure 9 Development of boundary layer for 30 m/s at AoA.....................................12Figure 10 Development of boundary layer for 35 m/s at AoA...................................13

Table 1 Table of Results for velocity of 20 m/s at 0 AoA.............................................8

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Introduction:Boundary layer has been an important subject to investigate since 1904, as its theory is applied in the development on hydraulics and aerospace sciences. In this laboratory, we investigated the development of the boundary layer and its separation on a flat plate model (AF106) using the wind tunnel (AF100).

Apparatus:As mentioned above, the apparatus used in this lab were a flat plate model (AF106) and the wind tunnel (AF100) in addition to the wind tunnel computers that were used to record specific data such as pressure and a manometer.

The flat plate is made of two parts, they are both hinged stainless steel flat plates where one of the, is flexible to be adjusted vertically to a decided angle to set an ideal condition as wished for the experiment. And are both fitted inside the working section of the wind tunnel.

The upper surface of the fixed flat plate has five small aerofoils set at right angle to the plate with the leading edge facing the incoming air. Each of these aerofoils has five tapping on the leading edge where each is connected to a pressure tube that is connected to a manometer outside the wind tunnel to measure the pressure of the air. The aerofoils are set in a way where their wakes do not interfere with each other.

The adjustable plate is connected to an adjuster that is located outside the wind tunnel to make it easier to adjust without the need of taking it out of the working section of the tunnel as shown in the figure below.

Figure 1The main parts of the flat plate boundary layer model AF106

Aerofoils are located at a horizontal distance of 40, 90, 150, 220 and 300 mm from the leading edge. And each Aerofoils has the tapings at 0.5, 1.5, 3, 6 and10mm as the vertical distance from the base of the flat plate.

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Procedure:Firstly, the room pressure of and temperature were inserted to the computer before the experiment was started.

After that, the flat plate model was adjusted carefully to give the ideal conditions to achieve a satisfactory boundary layer by having a horizontal reference line at 0 degrees to the x-axis on the windshield of the wind tunnel with the trailing edge on the same level. this was done to prevent the separation of the boundary layer off the flat plate before it reaching to the aerofoil fitted closer to the trailing edge, and this is because if the boundary layer separates, there will be high turbulence which will give the pressure tapings wrong readings.

Following that, the pressure reading of the pressure tapings on the wind tunnel was zeroed.

After that was done, the experiment was conducted using four different airflow velocities varying from 10 m/s to 35 m/s. Data was collected after that.

Following that, the angle of the trailing edge was changed to a negative value and the same procedure was repeated.

Finally local velocity at each location was calculated using the Bernoulli's principle

V=√2× pT−p0ρ where ρ=

paRT a

and graphs were plotted.

Introduction to boundary layer and background theory:

In 1904, a German engineer named Ludwig Prandtl first introduced the revolutionary concept of a boundary layer in a paper "on the motion of a fluid and very small viscosity". Prandtl stated that when an ideal fluid (incompressible and nonviscous fluid) flows past a stationery solid boundary, the flow will then be divided into two parts. A part where a boundary layer attached to the surface has a viscous force that cannot be neglected and is of main significance. And the other part which is a flow in the outer region. This separation of the two fluid flows when considering the theory was responsible for bridging the gap between the classic hydrodynamic theory and the experimentally based branch of aeronautics known as hydraulics.

Boundary layers appear on surfaces as the fluid passes over them because it seems to "stick" to them. This is known as the zero slip condition. At the surface, the flow

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has no relative speed and through the action of viscosity, the flow transfers momentum to the neighbouring layers. Therefore, a layer must exist between the stationery molecules at the surface of the object and the dynamic fluid flow, this layer is known as the boundary layer. As the distance increases from the surface, the velocity of the boundary layer increases until it reaches the velocity of the mainstream flow.

As the boundary layer develop along the surface, its character change. It starts initially with an entirely laminar flow, and then gradually thickens and the flow becomes disturbed and develops to a turbulent flow; this short region of developing is a mixture of both flows and is known as the transition region. After that, the boundary layer then continues to thickens and develop until it reaches a point where it separates from the surface under certain conditions. This can be shown in figure 2. (Messey and Ward-Smith 2012)

Figure 2: Fluid flow over flat plate (y-axis enlarged)

As the boundary layer thickens and goes through the mentioned above transitions, it slowly increases it's velocity until it is caught up with the speed of the outer main flow stream. The thickness of the boundary layer, δ, can be described as the distance required for the flow to about 99% of the velocity of the free stream,u∞. (Messey andWard-Smith 2012) Boundary layers can be measured by more significant parameters (Kundu and Cohen 2008). The main parameters are as followed:

1) The displacement thickness, which is described as the distance by which the main streamlines shifted due to the existence of the boundary layer:

δ ¿=∫(1− uu∞ )dy

2) The momentum thickness, which is the height of the free stream flow which would be needed to make up the momentum loss due to the presence of the boundary layer:

θ=∫ uu∞ (1− u

u∞ )dy

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3) The Reynolds number: This is a dimensionless number that indicates whether the fluid flow past the surface is laminar or turbulent:

ℜ∞=ρu∞Lμ

Laminar Flow over a Flat Plate:One of the most fundamental problems in the sector of fluid mechanics is the laminar flow over a flat plate. Boundary layer equations for flow along a flat plate have small viscosity or large Reynolds numbers. With larger Reynolds numbers and small viscosity, boundary layer thickness decreases. Navier-Stokes equations approximate solutions of viscosity at high Reynolds numbers by finding u, v and p. Blasius derived a simplified version of the equations that uses only u and v. These equations cannot be solved analytically and a numerical method should be introduced to be able to be solved such as the Runge-Kutta Numerical Method.

Boundary layer equations by Blasius:The boundary-layer flow across a flat plate can be expressed as:

∂u∂ x

+ ∂ v∂ y

=0

u ∂u∂ x

+v ∂ v∂ y

=v ∂2u∂ y2

The boundary conditions for these equations are: y=0, u=v=0, y=∞ , and u=U∞ , assuming that the leading edge of the plate is x=0 and the plate is infinity long. This system of equations can be simplified further to an ordinary differential equation. To do this, the following equation is used:

δ ~√ vxU∞

To make this quantity dimensionless it can be divided by y to become:

η= y √U∞

vx

The stream function can be found from the equation of continuity as:

ϕ=√vxU∞ f (η)

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Where f(n) is the dimensionless stream function. The velocity component u which is

equal to

∂ϕ∂ y can be expressed as follows:

u=∂ ϕ∂ y

=∂ϕ∂η

∂η∂ y

Since

∂ ϕ∂ η

=√ vxU∞ f' (η )

, and

∂η∂ y

=√U ∞

vx . So,

u=U∞ f' (η )

Also the transverse velocity component can be expressed as:

v=−∂ϕ∂ x

=12 √ vU∞

x(ηf '−f )

Now inserting these equations into the second boundary layer flow equation:

''')'(2

'''2

222

fxvUvff

xU

ffx

U

and with furthur simplification we have the Blasius ordinary differential equation.

ff ''+2 f '''=0

Where the boundary conditions are: whenη=0 , 0f andf '=0 ; and whenη=∞ , f '=1

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Sample calculations:The table below shows one of the results obtained and calculated for the boundary layer thickness. Only one table was attached as a sample due to the high number of data sheets. All results obtained were calculated as will be mentioned below.

No Po (Pa) Pt (Pa) v(m/s) x(mm) y(mm) ReBoundary layer Thickness (mm)

0 0 40 0 01 0.13 238.2 13.54215 40 0.5 52737.43 0.8552281782 0.096 238.2 15.52472 40 1.5 52737.43 0.8552281783 0.05 238.2 17.8601 40 3 52737.43 0.8552281784 0.013 238.2 19.53701 40 6 52737.43 0.8552281785 0.009 238.2 19.70976 40 10 52737.43 0.8552281786 0.096 238.2 15.52472 90 0.5 118659.2 1.2828422687 0.059 238.2 17.42782 90 1.5 118659.2 1.2828422688 0.027 238.2 18.91999 90 3 118659.2 1.2828422689 0.013 238.2 19.53701 90 6 118659.2 1.28284226810 0.01 238.2 19.66671 90 10 118659.2 1.28284226811 0.097 238.2 15.47004 150 0.5 197765.4 1.65614224612 0.056 238.2 17.57309 150 1.5 197765.4 1.65614224613 0.028 238.2 18.87515 150 3 197765.4 1.65614224614 0.015 238.2 19.45007 150 6 197765.4 1.65614224615 0.023 238.2 19.09832 150 10 197765.4 1.65614224616 0.161 238.2 11.43886 220 0.5 290055.9 2.00568786317 0.142 238.2 12.76914 220 1.5 290055.9 2.00568786318 0.109 238.2 14.79808 220 3 290055.9 2.00568786319 0.061 238.2 17.33029 220 6 290055.9 2.00568786320 0.04 238.2 18.32845 220 10 290055.9 2.00568786321 0.11 238.2 14.7407 300 0.5 395530.7 2.34213882622 0.078 238.2 16.47803 300 1.5 395530.7 2.34213882623 0.045 238.2 18.09579 300 3 395530.7 2.34213882624 0.018 238.2 19.31891 300 6 395530.7 2.34213882625 0.011 238.2 19.62358 300 10 395530.7 2.342138826

Table 1 Table of Results for velocity of 20 m/s at 0 AoA

Pt was calculated by taking an average of the taping pressure and P0 was calculated by taking the average of upstream and downstream pressures respectively.

The local velocity at each location was measured using V=√2× pT−p0ρ by

considering the density of the fluid (Air) as 1.18 kg/m3. For Example :

v=√2× 238.2−1301.18=13.54215m/ s

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Furthermore, Reynolds number was calculated as it is important to determine the boundary layer thickness. And it was calculated using:

ℜ= ρ×V ×dμ

Where density was considered as the density of air at sea level which is 1.18 Kg/m3 and Viscosity is considered as the viscosity of air at sea level which equals to 1.79*10^-5 Pa.s .

For Example:

ℜ=1.18×20×0.041.79×10−5

=52737.43

Note: Reynolds number is dimensionless.

Finally, Boundary layer thickness was obtained using:

δ=4.91×x√ℜ

For Example:

δ= 4.91×40√52737.43

=0.8552mm

The mentioned above calculation was repeated for every set of data in the same way.

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Results:

0 50 100 150 200 250 300 3500

2

4

6

8

10

12

Velocity profile for 20 m/s at 0 AoA

horizontal distance from the leading edge (mm)

verti

cal d

ista

nce

from

the

base

of t

he fl

at p

late

(m

m)

Figure 3 Development of boundary layer for 20 m/s at 0 AoA

0 50 100 150 200 250 300 3500

2

4

6

8

10

12



verti

cal d

ista

nce

from

the

base

of t

he fl

at p

late

(m

m)


10

0 50 100 150 200 250 300 3500

2

4

6

8

10

12


horizontal distance from the leading edge (mm)verti

cal d

ista

nce

from

the

base

of t

he fl

at p

late

(m

m)


0 50 100 150 200 250 300 350 4000

2

4

6

8

10

12

velocity profile for 35 m/s at 0 AoA


cal d

ista

nce

from

the

base

of t

he fl

at

plat

e (m

m)


0 50 100 150 200 250 300 3500

2

4

6

8

10

12

Velocity profile for 20 m/s at -5 AoA


verti

cal d

ista

nce

from

the

base

of t

he fl

at p

late

(m

m)

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Figure 7 Development of boundary layer for 20 m/s at AoA

0 50 100 150 200 250 300 3500

2

4

6

8

10

12



cal d

ista

nce

from

the

base

of t

he fl

at

plat

e (m

m)


0 50 100 150 200 250 300 3500

2

4

6

8

10

12



verti

cal d

ista

nce

from

the

base

of t

he fl

at p

late

(m

m)


12

0 50 100 150 200 250 300 3500

2

4

6

8

10

12



cal d

istan

ce fr

om th

e ba

se o

f the

fla

t pla

te (m

m)


Discussion and conclusion:The results above represent eight different set of data. The first four, are the set of data of four different velocities from 20 m/s to 35 m/s at 0 AoA. The second four set of data are of the same set of velocities but at -5 AoA.

By looking at the first four graphs, as speed increases, boundary layer thickness decreases and the velocity profile decreases and there is higher sheer stress. This means the faster the flow, the lower the Reynolds number, the later the separation

and the greater the laminar flow. This indeed matches the theory. By looking at

boundary layer thickness equation δ=4.91×x√ℜ

It clearly shows that the relationship between boundary layer thickness and Reynolds number is a positive correlation. Whereas the smaller the Reynolds number, the narrower the thickness of the boundary layer, the laminar the flow.

Also, this is dependent on the velocity of the flow and is stated in the following

equation: ℜ= ρ×V ×dμ

The above equation of Reynolds number states the negative correlation between Reynolds number itself and the velocity. Where the higher the velocity, the lower the Reynolds number is and in that case the thinner the thickness of the boundary layer.

By comparing the set of data between a 0 AoA and a -5 AoA, we can notice that velocity profile on the aerofoil is narrower at -5 AoA. This means that the laminar flow separates of the aerofoil at -5 AoA earlier than it separates of an aerofoil at 0 AoA.

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In conclusion, it can be said that the experiment conducted was successful with a minimum amount of errors and it also approved with the theory:

* As speed increases, Reynolds number decreases.

* The smaller the Reynolds number, the greater the laminar flow.

* The greater the laminar flow, the later the separation.

BibliographyKUNDU, Pijush and COHEN, Ira (2008). Fluid Mechanics. 4th ed., Oxford, Elsevier Inc.

MESSEY, Bernard and WARD-SMITH, John (2012). Mechanics Of Fluids. 9th ed., Oxon, Spon Press.

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Documents

Introduction to Boundary Layer