In the previous lecture we started the discussion
about the forces acting on an aircraft during flight. For that purpose we took a very simple situation:
a horizontal flight at constant speed. In this case there are four forces:
the Weight, Lift, Drag and Thrust. The equilibrium equations are also very simple:
Lift equals weight and Drag equals Thrust. In this lecture we will focus on the Lift force. The lift force is given by a rather simple equation. The lift is equal to a constant CL times the
product half rho V squared times S. The meaning of these parameters are as follows:
CL is the so-called lift coefficient. This parameter will be discussed
in more detail on the next slides. The rho stand for the air density,
the V for the air speed and and S is the parameter for the wing area,
which is the surface area projected on the X-Y plane, when we take a Cartesian coordinate
system and where Z is the vertical direction. For all parameters, except for the coefficient,
the dimensions of the parameters are given. The question is therefore:
what is the dimension of CL? How do we determine its dimension? In this case we use a dimension analysis. Such analysis can also be used to check whether
a formula or equation is correct. If not, the dimensions on both sides of the
equal sign are different. In our case we can fill in the known dimensions
of the parameters. We use a question mark for the lift coefficient. On the left hand side we see Newtons, on the
right hand side we see kilos per cubic meter times meters per second squared
times meters squared. First we replace Newtons by kilos
times meters per seconds squared. Next we eliminate as much as possible
on the right hand side. It happens to be that in the end that the
lift coefficient has no dimension. The lift coefficient, which is dimensionless,
includes a number of aspects of a particular wing. like the geometry, the airflow around the wing, etc. Key parameters influencing the lift coefficient
are the angle of attack and the geometry of the airfoil, which is the cross section
of the wing in flying direction. The airfoil is usually not symmetric. If we draw a straight line between the front,
the leading edge, and the rear, the trailing edge, it becomes visible that the profile has a camber. A camber means that the line precisely in
the middle between the upper and lower surface has a curve, with respect to the straight line. The straight line itself, the chord line,
may have an angle with respect to the airflow at some distance in front of the airfoil. This angle is the so-called "angle of attack"
and has a significant influence on the Lift. Because of the importance of the airfoil,
a lot of experimentation has been done in the early ages of aviation,
searching for the ultimate airfoil. This resulted in a wide variety of airfoils
as can be seen on this slide. What is remarkable is the fact that many airfoils
have their own code. Later, in the nineteen twenties
a more systematic nomenclature was introduced, which is still used today, the NACA profiles. NACA stands for National Advisory Committee
for Aeronautics, a committee founded in the US. NACA airfoils are identified
by a specific 4-digit number. In the figure you can see a number of features
of an airfoil like the positions of the leading edge and trailing edge, the chord line and its
length, the profile thickness and the camber. Some of the dimensions of these features are
represented in the NACA number. Note that all dimensions are expressed as
percentage or fraction of the chord length. As example, NACA 2412: The first 2 means that the profile has a camber of 2%. So it has a maximum distance of 2% times
the chord length between camber and chord line. The second digit in the number indicates
the position of that camber with reference to the leading edge:
0.4 times or 40% of the chord length. The last two digits indicate the thickness of the profile,
in this case 12% of the chord length. Let us return to the formula for lift. The other parameters in the formula are the
Wing surface area and the velocity or air speed. Both are typical design parameters. During the design of an aircraft you need
to optimize these values with many others, in order to obtain the best ratios between
performance, weight and costs. The last variable in this equation is the air density,
which depends on altitude and temperature. As designer, by choosing a flying altitude,
you can have some, although limited, influence on this parameter as well. We have looked at the formula for Lift,
but how is the Lift force generated? One way of explaining is by using
the law of Bernoulli. He postulated his formula as: The sum of static
and dynamic pressure is constant. In formula: p plus half rho V squared is constant. This means that when the speed increases,
the local pressure drops, or decreases. So in a tube with a throat, the pressure in
the narrow section is the lowest, but the air speed is the highest. This explanation tells that pressure differences
can be created by differences in air speed. These pressure differences
can be used to create lift. Note that Bernoulli's law is applicable
for particular situations only, for example for low airspeeds
and incompressible media. So for an airfoil the air speed over the airfoil is
usually higher than the airspeed below the airfoil. This is due to the camber of the airfoil and
the angle of attack as can be seen in the first picture. This results in a pressure distribution
as can be seen in the second picture. There is some over pressure under the airfoil
but significant under pressure at the upper surface of the airfoil. This pressure distribution results
in a lift force and a drag force. With the program NASA Foil-Sim,
and I recommend you to play with it, you can play with many different parameters
of the airfoil and the angle of attack and see what the effect is on the lift force. Can we use Bernoulli's law to measure the
airspeed? If we look at the formula, we see that the total pressure
is the pressure for zero air speed. In a pitot tube we can make several small holes. One hole is in front of the tube,
head-on in the air flow. The air enters the tube from the left end side
and comes to a standstill. So here we measure the total pressure. Further we make several holes
at the side of the tube. They measure the local pressure,
a pressure in an air flow with velocity V. The difference between the total and static
pressure can be used to calculate the velocity as demonstrated by the equation. When we return to the lift coefficient, we said that the
coefficient depends on the angle of attack of the wing. This is expressed in a Lift curve. For such curve, we plot the value of CL
as function of the angle of attack, alpha. Most wing profiles have a camber, this means
that at a negative alpha, or alpha = 0, there is already a Lift force available. When increasing the angle of attack, the Lift
coefficient increases more or less linear with alpha. Look also at the simulations,
when the flow around the airfoil changes, and the Lift force increases steadily. Just before it's maximum
the curve becomes less linear. and ultimately the maximum coefficient is obtained. A further increase in angle of attack
results in a fast decrease of the Lift coefficient. This is what we call the stall of the wing. In this case the Lift force drops, and look also again
at the airflow around the airfoil, which separates. At landing and take-off we fly at low air speeds. As you can see in the formula,
this has a large impact on the lift force. Question is: What to do? Because the weight
of the aircraft does not change. In order to compensate for a lower value of the
airspeed, the other parameters should increase. The air density is beyond our control,
although most airfields are at low altitude and therefore the density has
rather high values, which helps. The other two parameters, the lift coefficient and the wing surface area can be changed by using all kinds of high lift devices. In this picture you see an Airbus A380
with high lift devices. First I show you the flaps,
which are used during take-off and landing. Second, the slats or leading edge devices,
which are mainly used during landing. Both devices increase the wing surface area
and to some extent they increase the camber of the wing profile and thereby
increase the lift coefficient. This is visible in this plot. Slats increase the maximum angle of attack
and as result also the maximum lift coefficient. The flaps at the trailing mainly increase
the camber of the airfoil and shift the curve upwards. In addition to these effects both devices
also increase the wing surface area. To conclude this lecture I will briefly mention
a number of parameters for the wing geometry. Most of these are optimized during the design
of the wing and will have an impact on the lift coefficient and thereby on the lift force. First there is the wing span,
which is the length from wingtip to wingtip. Then we have the wing surface area; it should
be noted that there might be some small differences between several definitions of this area.
Next we have the root chord and the tip chord, which represent the chord length
at the wing root and wing tip respectively. The taper is the ratio between these two. The last parameter to mention is the sweep angle,
which is the angle between the leading edge and the Y-axis of our coordinate system. This concludes the lecture about the lift force. Next time I will talk about the drag force.