Lift on a Wing
Prepared by Professor J. M. Cimbala, Penn State University
Latest revision: 11 January 2012
Nomenclature
Symbols
a
A
AR
c
CD
CL
FD
FL
g
Ma
P
R
Rec
s
T
Tamb
V
x





speed of sound (NOTE: we use a instead of c in this lab so as not to confuse between chord length
and speed of sound.) Depending on the reference, both a and c are used for the speed of sound.
planform area (area looking at a wing at zero angle of attack from the top): for a rectangular wing, A
= sc
aspect ratio of a wing: for a rectangular wing, AR = s/c
chord length of an airfoil or wing
drag coefficient: CD = 2FD /(V2A)
lift coefficient: CL = 2FL /(V2A)
drag force
lift force
gravitational constant (9.81 m/s2)
Mach number: Ma = V/a
static pressure
specific ideal gas constant: for air, R = 287. m2/(s2K)
Reynolds number based on chord length c: Rec = Vc/ (other Reynolds numbers defined similarly)
span of an airfoil or wing
temperature
ambient air temperature
mean freestream velocity
flow direction
angle of attack of an airfoil or wing
average surface roughness height in a pipe or on the surface of a body
coefficient of dynamic viscosity (also called simply the viscosity)
coefficient of kinematic viscosity (for water,   1.0  10-6 m2/s at room temperature)
density of the fluid (for water,   998 kg/m3 at room temperature)
Educational Objectives
1.
2.
3.
Develop familiarity with operation of the lift balance in a wind tunnel.
Measure lift, and plot lift coefficient as a function of angle of attack and Reynolds number.
Reinforce the concept of airfoil stall.
Equipment
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
high speed undergraduate wind tunnel
United Sensors Pitot-static probe, with traverse and stand
Validyne electronic pressure transducer and digital display unit, Model CD23
Meriam micro manometer, Model SC-2897, with hand pressure pump and thermometer
Measurements Group, Inc. strain indicator, Model 3800
TSI integrating voltmeter, Model 1076
barometric pressure gauge (mounted on the wind tunnel to the left of the test section)
strain gage drag and lift balance with rotational traverse system
lift and drag balance calibration pulley system, with Ohaus 10-1000 gram weight set
personal computer with data acquisition system and data acquisition software
NACA 0012 two-dimensional airfoil, with 5.8 inch chord length, 9.7 inch span, and 56.3 square inch planform area
Background
The purpose of an airfoil-shaped wing is to generate lift with as little drag as possible. When an airfoil is pitched to an
angle of attack , the fluid travels faster over the upper wing surface (the “suction side”) than over the lower surface (the
“pressure side”). By Bernoulli's principle, the pressure on the upper surface is therefore less than that on the lower; a
relatively large lifting force accompanied by a relatively small drag force results. As angle of attack is increased, the lift also
increases. Elementary airfoil theory predicts that lift coefficient CL increases as the sine of angle of attack . Specifically, for
a two-dimensional, thin, symmetric airfoil, theory predicts that CL = 2 sinα. For small angles of attack, CL can be
approximated as 2α since sin   when  is small. (Note:  must be in radians, not degrees, for this approximation to be
valid.) Experiments show that actual symmetric airfoils follow theory fairly well at low to moderate angles of attack (Fig. 1).
However, as  is increased further, the adverse (increasing) pressure gradient on the upper surface becomes stronger and
stronger, and soon the upper surface boundary layer can no longer remain attached. The boundary layer separates off the
upper surface, usually at first near the trailing edge of the airfoil. The point of separation moves forward rapidly as angle of
attack is further increased. Beyond some angle (typically between 10o and 20o, depending on airfoil shape and Reynolds
number), the flow separates completely from the upper surface, with the separation point occurring very near the leading
edge of the airfoil. The airfoil is then said to be stalled. Not only does lift force decrease with stall, but drag force increases
significantly as well. Stall is therefore generally undesirable in aircraft flight, and has been the culprit of numerous airplane
accidents. Further increases in angle of attack beyond the stall angle lead to even less lift and more drag – the wing cannot
recover from stall unless its angle of attack is decreased to sub-stall angles.
3.5
3
Theory: CL = 2 sin(α)
2.5
CL
2
1.5
1
Stall
0.5
Typical experimental data
0
0
5
10
15
 (degrees)
20
25
Figure 1. Comparison of lift coefficient for a thin airfoil as a function of angle of attack ; theory is for a two-dimensional
(infinite span) rectangular wing, and experimental data are for a high-aspect-ratio, finite span rectangular wing.
A real wing, however, is not two-dimensional (infinite span), but rather has a finite span. We define a rectangular
wing’s aspect ratio as AR = s/c. It turns out that a finite aspect ratio wing does not have the same lift and drag characteristics
as do two-dimensional airfoils. In general, the lift produced by a finite-span wing is significantly less than that of a twodimensional wing, and its drag is also significantly higher. Because fluid can flow from the pressure side to the suction side
at the tip of the fin, strong tip vortices are generated, which severely increase the airfoil’s drag. This type of drag, induced by
the tip vortices, is appropriately called induced drag. On the other hand, stubby (short aspect ratio) fins can typically reach
higher angles of attack before stall occurs.
Airplanes are not the only machines that employ airfoils. “Wings” and fins of various shapes and sizes can also be
found on racing cars, sailboats, rockets, missiles, and submarines. Of particular interest here are the fins mounted near the
rear of a submarine. These fins are used as control surfaces to steer the vehicle. Typically four such fins are utilized  the two
vertical fins are used to steer to the right or left, while the two horizontal fins are used to pitch the vehicle (nose up or nose
down), and to provide stability. Some submarines and torpedoes also have horizontal fins near the front that control the pitch
of the vehicle. The fins are either rectangular (same chord length across the entire span), or trapezoidal (larger chord at the
root and smaller chord at the tip, and with a span typically about the same length as the average chord length). These shapes
are sketched in Fig. 2.
Some years ago, Penn State students designed a human powered submarine called “The Sea Lion,” which could travel
at a speed of around 3.0 knots (1 knot = 1.152 miles/hour = 0.515 m/s) through sea water. One preliminary design of this
submarine included four control surfaces (fins) near the rear of the model. The fins were initially rectangular in shape, with a
chord length of 11.6 inches, and a span of 19.4 inches. A one-half scale model of one of the fins was constructed for testing
in the undergraduate wind tunnel.
The wind tunnel has both a drag balance that measures the force parallel to the freestream flow direction (to the right
of an observer looking into the test section of the wind tunnel) and a lift balance that measures the horizontal side force,
perpendicular to the flow direction. In this lab, lift is defined as positive towards an observer looking into the test section of
the wind tunnel when the wing is mounted vertically as in Fig. 2. In other words, lift is positive when there is a force towards
the door of the wind tunnel test section.
cavg
c
s
s
s/2

V
(a)

V
(b)
Figure 2. Two common fin designs: (a) rectangular and (b) trapezoidal, with definitions of chord length c, span s, and
angle of attack  relative to freestream velocity V. Note that for the trapezoidal fin, the average chord length is
that at mid span.
In this lab experiment you will measure the lift produced by this one-half scale model fin, which is a rectangular,
short-aspect ratio NACA 0012 airfoil (a symmetric airfoil with 12% thickness-to-chord ratio) at various angles of attack and
at various Reynolds numbers. At high enough angles of attack, airfoil stall should be observable in our wind tunnel facility.
The wind tunnel model can be used to predict the performance of the prototype fins on the actual human powered submarine,
provided of course that proper dimensional scaling is applied. This experiment complements the water channel experiment in
which stall on an airfoil is investigated visually.
References
1.
Çengel, Y. A. and Cimbala, J. M., Fluid Mechanics – Fundamentals and Applications, McGraw-Hill, NY, 2006.
2.
White, F. M., Fluid Mechanics, Ed. 5, McGraw-Hill, NY, 2003.