a) Airfoil Terminology;
b) Pressure Gradient across a Curved Streamline;
c) Streamlines over an Airfoil
12
Figure 2
Airfoil in the Wing Tunnel
13
Figure 3
Schematic Diagram (Side View) of Pressure Measurement
13
Figure 4
Forces and Pressure on an Airfoil
14
Figure 5
Pressure Distribution around an Airfoil; at a=10° and Re=2.33x105
15
Figure 6
Area between Two Curves using Trapezoidal Method
16
Table 1
Coordinates of Pressure Tappings
16
Table 2
Manometer Readings
17
Table 3
Pressure Coefficients
18
Appendix A
Location of Engineering Workshop 2
Nomenclature
A’
axial component of force per unit span (Fig 4)
c
airfoil chord (Fig 4)
CD
2D drag coefficient, =
CL
2D lift coefficient, =
CP
D′
1
2 ρU ∞ c
2
L′
1
2
ρU ∞ c
2
pressure coefficient, =
P − P∞
1
2 ρU ∞
2
3
19
D’
drag force per unit span
h
manometer reading (see Fig 3)
L’
lift force per unit span
N’
normal component of force per unit span (Fig 4)
P
pressure
Re
Reynolds number, =
t
airfoil thickness
U∞
free stream velocity, =
ρU ∞ c µ PT − P∞
1
ρ
2
Greek Symbols α angle of attack
µ
dynamic viscosity of air, = 1.84 x 10-5 Ns/m2
ρ
density of air, = 1.18 kg/m3
ρw
density of water, = 103 kg/m3
θ
inclination of manometer with horizontal
Subscripts
T
stagnation value
f
front surface (upstream of maximum thickness)
}
ℓ
lower surface (below chord line)
}
r
rear surface (downstream of maximum thickness)
} see Fig 4
u
upper surface (above chord line)
}
∞
free-stream value
}
4
1. Introduction
1.1
Background
An airfoil (Figure 1a) is a two dimensional cross-section of an airplane wing. It may be thought of as a wing of infinite span with constant cross-sectional shape. With a forward speed, wings can generate a lift force which enables the airplane to stay airborne. Airfoil shapes are designed to provide high lift values at low drags, for given flight conditions.
Airfoil studies are not only relevant for airplanes, but also applicable to wings on F1 cars and blades of a helicopter, propeller, hydrofoil, and wind turbine.
A typical subsonic airfoil has a streamline profile with a fairly rounded nose (leading edge) and a sharp tail (trailing edge). A chord line is a straight line joining the leading to trailing edges, the length of which is called the chord c. The acute angle between the free stream velocity direction and the chord line is called the angle of attack α (Figure 1a).
1.2
Lift Generation
The pressure in a curved flow increases radially outwards (or reduces radially inwards), in order to accelerate the flow (Figure 1b). The curved streamlines around an airfoil result in the pressure at the upper surface being lower than P∞, and the pressure at the lower surface being higher P∞, as illustrated in Figure 1c. The pressure difference between the lower and upper surfaces of an airfoil generates a lift force, which is the component of force perpendicular to the direction of motion. The component parallel to the direction of motion is called drag.
1.3
Objectives and Scope
The objectives of this experiment were to investigate the pressure distribution around the airfoil and to calculate the lift and drag forces. The experiment was conducted at a specified angle of incidence relative to the wind direction and at a specified wind speed.
5
2. Experimental Set up
2.1
Wind Tunnel
The air flow was generated by the blower of a subsonic wind tunnel (Figure 2), which was of the open-circuit type. It has a working section of square cross-section, 0.30 m x 0.30 m.
.
2.2
Airfoil
The airfoil used in this experiment was a NACA 0015 section of chord length 101.6 mm. It has a symmetrical profile with a maximum thickness 15% of the chord. The airfoil spanned the test section of the wind tunnel, and was supported by two end plates. One of the end plate was graduated in degree for determining the angle of attack.
2.3
Pressure Measurement
The airfoil has 11 static pressure taps at the mid-section (mid-span) on the upper surface.
The same tappings can be used to measure pressures on the lower surface at negative incidence, due to symmetry of the airfoil. The pressure tappings were connected to a multitube manometer to measure the static pressure distribution around the airfoil (Figure 3). The manometer was inclined at an angle θ to increase the sensitivity.
2.4
Velocity Measurement
The flow speed in the wind tunnel was measured by using a standard Pitot-static tube
(Figure 3) and by applying Bernoulli's Equation.
3. Analysis
3.1
Free Stream Velocity
Apply Bernouli’s Equation from free stream to the nose of the pitot tube (Figure 3):
1
1
2
P∞ + ρU ∞ =
PT + ρU T2
2
2
= 0 at pitot nose
6
(1)
The free stream velocity may be expressed as
=
U∞
ρ w g ( hT − h∞ ) sin θ
PT − P∞
=
1 ρ 2
The Reynolds number, Re =
3.2
(2)
1 ρ 2
ρU ∞ c µ (3)
Pressure Coefficient
P - P∞
1
2 rU ∞
2
P - P∞
Substitute from Equation (1), CP =
PT - P∞
Pressure coefficient is defined as CP =
(4)
Pressure difference is measured by the manometer: P - P∞ =r w g ( h-h ∞ ) sinq where r w is the density of water in the manometer
g is gravitational acceleration, 9.81 m/s 2 and h is the reading of the manometer column (Figure 3)
Thus, pressure coefficient may be expressed in terms of the manometer readings
=
CP
3.3
rq g ( h - h∞ ) sin h - h∞
=
rq hT h∞ g ( hT -h∞ ) sin
(5)
Force Coefficients
The resultant aerodynamic force that acts on an airfoil can be resolved into a pair of orthogonal forces. The two most commonly used pairs are those which are perpendicular and parallel to the free stream direction ( i.e. lift L’ and drag D’ forces respectively) and those which are perpendicular and parallel to the chord line (i.e. normal
N’ and axial A’ forces respectively). These components of force are shown in Figure 4.
Referring to Figure 4, the normal component of force is
=
N′
c
∫ ( P − P ) dx
( 6)
u
0
where c is the chord length, Pℓ and Pu are the pressures on the lower and upper surfaces.
7
−
The axial component of force = is A′
t
2
∫ (P
f
− Pr ) dy
(7)
t
−
2
where Pf is the pressure upstream of the maximum thickness, Pr is the pressure downstream of the maximum thickness, and t is the maximum thickness.
From the resolution of forces, the lift and drag forces are
( 8a )
(8b )
L′
= N ′ cos a − A′ sin a
D′
= N ′ sin a + A′ cos a
1
2
Substitute Equations (6) and (7) into Equation (8a) and divide throughout by ρU ∞ c , the lift
2
coefficient is given by t +
c
2
1
CL =
− ∫ ( CPf − CPr ) dy sin
( CP − CPu ) dx cos αα c ∫ t 0
−
2
+0.075
1
0
x
y
y
= cos αα ∫ ( CPf − CPr ) d + ∫ ( CPf − CPr ) d
∫ ( CP − CPu ) d c − sin −0.075
u
c
c
0
0
(9)
Similarly, the drag coefficient is given by t +
c
2
1
+ ∫ ( CPf − CPr ) dy cos
CD =
∫ ( CP − CPu ) dx sin αα c 0 t −
2
+0.075
1
0 x y
y
= sin αα ∫ ( CPf − CPr ) d + ∫ ( CPf − CPr ) d
( CP − CPu ) d + cos
∫
u c
c
c
0
0
−0.075
The integrals of the above equations are areas between pressure distribution curves in
Figure 5.
3.4
Trapezoidal Method
Each area may be approximated by several trapezoids, the width of which may be nonuniform. The trapezoidal method, as illustrated in Figure 6, is a simple geometric approximation to a strip between the curves y=f(x) and y=g(x) by assuming the change
8
(10)
between any two points x=a and x=b is linear:
f (a ) + f (b) g (a ) + g (b)
−
(b − a )
2
2
b
∫ [ f ( x) − g ( x)] dx ≈
a
(11)
By summing up the areas of several trapezoids, the area enclosed by the pressure curves or loops may be approximated.
3.5
Thin Airfoil Theory
The thin airfoil theory is an inviscid theory which is used to predict the lift acting on an airfoil. It predicts that the lift coefficient is directly proportional to the angle of attack in radian. Analytically, the above statement can be stated as
CL = 2πα
(12)
This prediction is quite accurate when the angle of attack is smaller than the stall angle.
When the angle of attack is small, the flow over it is attached to the upper and lower surfaces since an airfoil is a streamlined body However, as the angle of attack increases, a certain critical angle will be reached at which the flow can no longer stay attached to the upper side of the airfoil. When this happens, flow separation is said to have occurred and the phenomenon is known as stall. The angle of attack at which stall first occurs is called the stall angle.
4. Procedure
The Experiment and Tables may be done as a group effort.
4.1
a.
Experiment
Check that there is no air bubble in the manometer tubes. Level the manometer base, and record the inclination θ of the manometer tubes to the horizontal.
b.
Start the wind tunnel motor and run it to give a specified speed in the test section, as given by the Instructor. Each group will do the experiment at only one speed. For the purpose of comparing results, one group will work at the lower speed around 7.5 m/s and the other, at the higher speed around 15 m/s, as
9
assigned by the Instructor. Measure the exact speed with a pitot-static tube at a location upstream of the airfoil.
c.
Note the atmospheric temperature.
d.
Check t h a t the zero angle of incidence corresponds to that on the end plate, by observing the pressure reading at the leading edge (tube 1). Zero incidence occurs when the leading edge pressure is a maximum (i.e. a stagnation point)
e.
Take manometer readings with the airfoil at a small angle of incidence specified by the Instructor. Check that the reference of the manometer readings is connected to the free-stream pressure tapping, as shown in Figure 3. For the purpose of comparing results at different speeds, both groups will do the experiment at the same incidence. Pressure on the other surface may be obtained from negative incidence.
f.
4.2
Repeat the measurement of the wind speed in the test section.
Tables
a.
Table 1 gives the non-dimensional coordinates of the pressure tappings.
b.
Record the manometer readings, at the specified angle of incidence, in Table 2.
c.
Compute and tabulate the pressure coefficients in Table 3.
5. Results and Discussion
The Results (including calculations & graphs) and Discussion should be done individually.
5.1
Results
a.
Calculate the free-stream velocity and the Reynolds number.
b.
Plot CPℓ and CPu against x/c as illustrated in Figure 5. Extrapolate your curves to the trailing edge x/c = 1.
c.
Plot CPf and CPr against y/c as illustrated in Figure 5. Clearly indicate whether it corresponds to the lower or upper surfaces. Make sure that the pressure distributions are continuous at the leading and trailing edges, y/c=0; that is, (CPf)u can only be joined to (CPf)ℓ; and (CPr)u can only be joined to (CPr)ℓ. In this plot, two pressure loops should be obtained. Check carefully to ascertain whether each loop is contributing to a positive or negative axial force
10
d.
The lift and drag coefficients, CL and CD, may be obtained from the integration of the pressure coefficient curves as indicated by Equations (9) and (10). The definite integral is the area between the pressure curves or net area enclosed by the pressure loops, as shown in Fig 5. Be careful with the signs of the areas!
e.
5.2
a.
Calculate the lift-drag ratio CL/CD.
Discussion
State the value of the maximum CP and its location (x/c, y/c). Do you expect the value of maximum CP to be higher, if you increased the angle of incidence?
b.
Was the shear stress along the airfoil included in the pressure measurement by the manometer? Hence, state whether your experimental CL and CD corresponded to the total lift and drag on the airfoil. Suggest another method to measure lift and drag other than pressure distribution.
c.
Compare your experimentally measured CL with the Thin Airfoil Theory prediction of
CL = 2πα.
d.
Compare your experimental lift coefficient with the other group, which was obtained at a different speed. Do you expect the CL to be higher if the speed was higher?
6. Conclusions and Recommendations
6.1
Conclusions
This experiment investigated the pressure distribution over an airfoil at a small angle of attack with the flow. The pressure distribution curves were integrated to obtain the lift and drag forces on the airfoil. The experiment illustrated the concept of lift and drag forces generated by a streamlined body moving through air. It demonstrated the significance of the non-dimensional lift and drag coefficients.
6.2
Recommendations
Further studies include experiments at other angles of attack to investigate the variation of lift with incidence and determine the stall angle. Studies at higher Reynolds number will be interesting to show the effect of boundary layer transition on the surface from laminar to turbulent flow.
11
References
1.
Anderson J.D. Fundamentals of Aerodynamics, McGraw Hill.
2.
Bertin J.J. and Smith M.L. Aerodynamics for Engineers, Prentice Hall.
3.
Kermode, A.C. Mechanics of Flight, Pitman.
a)
b)
P∞
c)
P∞
Figure 1. a) Airfoil Terminology; b) Pressure Gradient across a Curved Streamline;
c) Streamlines over an Airfoil
12
airfoil flow flow
end plate multi manometer
Figure 2. Airfoil in the Wind Tunnel
Figure 3. Schematic Diagram (Side View) of Pressure Measurement
13
Pℓ
(Pf)u
(Pr)u
Figure 4. Forces and Pressure on an Airfoil
14
1.5
Cp
1
trailing edge
0.5
CPℓ
0
0.000
-0.5
0.200
x/c
0.400
-1
0.600
0.800
1.000
1.200
CPu
-1.5
leading edge
-2
-2.5
-3
1
∫ (C
= area between curves
0
(Cpf)ℓ
+ve
1
-0.050
(cPr)ℓ
lower surface
trailing edge
0
0.000
-0.5
y/c
(cPr)u
0.050
-1
0.100
upper surface
-1.5
-ve
x
− CPu ) d
c
cP
0.5
-y/c
-0.100
1.5
P
-ve
-2
(cPf)u
leading edge
-2.5
-3
+0.075
y
y net area within loops = ∫ ( CPf − CPr ) d + ∫ ( CPf − CPr ) d
u
c
c
−0.075
0
0
horizontal shading
vertical shading
Figure 5. Pressure Distribution around an Airfoil; at a=10° and Re=2.33x105
15
Figure 6. Area between Two Curves using Trapezoidal Method
Table 1. Coordinates of Pressure Tappings y/c y/c
upper
lower
0
0
0
3.268
0.025
0.032
-0.032
5
4.443
0.049
0.044
-0.044
4
10
5.853
0.098
0.058
-0.058
5
20
7.172
0.197
0.071
-0.071
6
30
7.502
0.295
0.074
-0.074
7
40
7.254
0.394
0.071
-0.071
8
50
6.617
0.492
0.065
-0.065
9
60
5.704
0.591
0.056
-0.056
10
70
4.58
0.689
0.045
-0.045
11
80
3.279
0.787
0.032
-0.032
12 (extrapolated)
101.6
0
1
0
0
Tapping No.
x mm
|y| mm
x/c
1
0
0
2
2.5
3
Note: c = 101.6 mm
16
Table 2. Manometer Readings
Atmospheric temperature (at beginning of experiment):
(at end of experiment) :
Airfoil angle of incidence, a:
Manometer inclination to horizontal, θ:
Manometer reading of dynamic pressure, hT-h∞, (at beginning of experiment):
(at end of experiment) :
Manometer readings of pressure distribution, h- h∞ (see Fig 3)
