We are now going to explore some of the factors that affect the lift of a wing. We will be using the input sliders and boxes located at the lower left and will be watching the output bar graph marked "Lift" at the bottom on the left. We will also be using the Plotter Panel at the lower right. In this lesson we will try to answer the question "How does the lift change?" based on what we learned in the last lesson. In the next lesson we will explore a little more of "Why does the lift change?" by looking at the details of the pressure and velocity around an airfoil. For now, let's investigate the effects of each of the input variables separately.
Let's start out by considering the effects of changing the angle of attack. We've already learned that increasing the angle of attack will increase the lift. Now let's see by how much.
If you have been playing with the package, it will now be necessary to set the input to some initial conditions.
Let's set the following conditions:
Airspeed = 100 mph
Altitude = 0
feet
Angle = 2.0 degrees
Thickness = 0.5
Camber = 0
Area = 100 sq ft
This gives us a symmetrical airfoil generating 627 pounds of lift.
Now let's set up the plotter to give us some additional information. In the lower right panel, click on the button marked "Lift vs." Additional buttons will appear in the box and a plot will appear in the window above, with a line going from the upper right to the lower left. (Sorry about the goofy scales on the plot - but we won't really need them in this lesson). With the lift now at 627 pounds,
Let's first double the Angle of attack from 2 degrees to 4 degrees.
WHAT IS THE NEW VALUE OF LIFT AND HOW IS THIS RELATED TO THE OLD VALUE?
--> Doubling the angle of attack doubles the lift. <--
OK - Reset the Angle of attack to 2 degrees.
Now, cut this value in half to 1 degree.
WHAT IS THE NEW VALUE OF LIFT AND HOW IS THIS RELATED TO THE OLD VALUE?
--> Halving the angle of attack halves the lift. <--
In fact, tripling the angle will triple the lift, quadrupling, quadruples ... etc. Then:
--> Lift is directly (linearly) proportional to angle of attack. <--
Or mathematically: Lift = constant x angle of attack
The graph shown in the Plotter View Panel shows the linear relation between lift and angle of attack.
Lift = constant x angle of attack
Let's let the letter "L" be the lift, and "a" be the Angle of attack and "C" be the unknown constant. Then:
L = C a
************ EXTRA CREDIT #1***************
Strictly speaking, L = C a (ONLY for small values of "a")
Do a little experiment where you make a big change in angle of attack. Note the ratio of the two angles and the ratio of the two lifts. ARE THEY THE SAME?
Now, let's look at the variation of lift with wing area. We know that increasing the area will increase the lift.
Our initial conditions
will be:
Airspeed = 100 mph
Altitude = 0 feet
Angle = 5.0 degrees
Thickness = 0.5
Camber = 0
Area = 10 sq
ft
This gives a lift equal to 156.6 pounds.
As before, let's double the Area to 20 square feet.
WHAT IS THE NEW VALUE OF LIFT? HOW IS THIS RELATED TO THE OLD VALUE?
--> Doubling the area doubles the lift. <--
OK, set the Area back to 10
square feet.
Now, cut this value in half to 5 square
feet.
WHAT IS THE NEW VALUE OF LIFT? HOW IS THIS RELATED TO THE OLD VALUE?
--> Halving the area halves the lift. <--
This relationship is identical to what we've seen with the Angle of attack!
So we can see
that:
Note--the constant for the wing area relation is not the same constant as the Angle of attack.
Let's let the letter "L" be the lift, and "A" be the area, and "K" be the new constant. Then: L = K A
We can check our result by pushing the button labeled "Area" on the Plotter Control Panel below the lighted button labeled "Lift vs."
The graph shows a straight line relationship between Lift and Area.
Unlike the Angle of attack, this relationship is true for ANY increase in area. Check it out by making a big change in Area.
OK, let's try our luck with changing the Camber (curvature) of the airfoil. We know that increasing the camber increases the lift and that lift is very sensitive to Camber.
Our initial conditions will
be:
Airspeed = 100 mph
Altitude = 0
feet
Angle = 0 degrees
Thickness =
0.5
Camber = 0.2
Area = 10 sq
ft
This gives a lift equal to 158.6 pounds.
As before, let's double the Camber to 0.4.
WHAT IS THE NEW VALUE OF LIFT? HOW IS THIS RELATED TO THE OLD VALUE?
--> Doubling the camber doubles the lift. <--
OK, set the camber back to 0.2.
Now,
cut this value in half--to 0.1.
WHAT IS THE NEW VALUE OF LIFT? HOW IS THIS RELATED TO THE OLD VALUE?
--> Halving the camber halves the lift. <--
Looks like our old buddy the linear relationship!
So we can see that:
Again, the constant for the camber relation is not the same constant as either Area or Angle of Attack.
Let's let the letter "L" be the lift, and "c" be the camber and "K" be the new constant. Then: L = K c
We can check our result by pushing the button labeled "Camber" on the Plotter Control Panel below the lighted button labeled "Lift vs."
The graph shows a straight line relationship between Lift and Camber.
*************** EXTRA CREDIT #2 ******************
Notice that we can have an airfoil generate lift, using camber, without inclining the airfoil to the incoming flow. Angle = 0.
And we can generate lift using an angle of attack with no camber to the airfoil Camber = 0.
HOW CAN YOU MAKE A CAMBERED AIRFOIL PRODUCE NO LIFT??
This becomes very important if you wish to control the lift of your wing. Going up in the air is easy--but how do you kill the lift to get back down?
The answer is to combine the effects of Camber and Angle of attack.
Set the Angle of attack to the correct negative value and the wing quits lifting.
Of course, you can also take the Airspeed to 0!!
Which leads us to our next variable--how does lift
change with
airspeed?
We know that increasing the airspeed increases the lift.
Our initial conditions will be the same as before:
Airspeed = 100 mph
Altitude = 0 feet
Angle = 0 degrees
Thickness = 0.5
Camber =
0.2
Area = 10 sq ft
This gives a lift equal to 158.6 pounds.
As before, let's double the Airspeed to 200 mph.
WHAT IS THE NEW VALUE OF LIFT? HOW IS THIS RELATED TO THE OLD VALUE?
--> Doubling the airspeed quadruples the lift. <--
OK, set the
airspeed back to 100 mph.
Now, cut this value in half to 50 mph.
WHAT IS THE NEW VALUE OF LIFT? HOW IS THIS RELATED TO THE OLD VALUE?
What's going on here ? Half of 158.6 pounds would be 79.3
pounds.
This looks like one-fourth of 158.6 pounds.
--> Halving the airspeed cuts the lift by one-fourth. <--
This is not a linear relationship! In
fact, this is a quadratic relationship.
This constant is again different from the previous constants.
Letting the letter "L" be the lift and "V" be the airspeed and "K" be the new constant,
Then: L = K V^{2 }
We can check our result by pushing the button labeled "Airspeed" on the Plotter Control Panel below the lighted button labeled "Lift vs."
The graph shows a curved quadratic relationship between Lift and Airspeed. Not a straight line!
Well, let's move along and see how the lift changes with altitude. We know that increasing the altitude decreases the lift.
Our initial conditions will be:
Airspeed = 100 mph
Altitude = 1,000 feet
Angle = 0 degrees
Thickness =
0.5
Camber = 0.2
Area = 100 sq ft
This gives a lift equal to 1540 pounds.
As before, let's double the Altitude to 2000 ft.
WHAT IS THE NEW VALUE OF LIFT? HOW IS THIS RELATED TO THE OLD VALUE?
OK, set the Altitude back to 1000 ft.
Now, cut
this value in half to 500 ft.
WHAT IS THE NEW VALUE OF LIFT? HOW IS THIS RELATED TO THE OLD VALUE?
When in doubt .... Cheat!
Let's look at the plot by pushing the "Altitude" button on the Plotter Control Panel below the lighted button labeled "Lift vs."
Well, that's a weirdly shaped curve! It's not a straight line and it's not a quadratic either. It's just curved.
What's going
on here?
Well, actually, we're trying to plot against
the "wrong" variable. Lift changes with altitude,
but there is more to it than
just the height above sea level.
The physical quantity which actually affects the lift as we
change altitude is the air
density. Air density itself decreases
with altitude; and it is this decrease which causes the lift to
drop. We can use the
simulator to find out how the lift changes
with density, but it will take a little work.
At the lower left corner of the Airfoil View Panel you will see an output value of the air density. As you change the altitude, this value will change. Let's do another little experiment:
Set
the Altitude to 5800 ft. The Density is then
0.002 slugs/cubic ft.
Set the Airspeed to 100 mph,
Angle to 5 degrees, Thickness = 0.5, Camber =
0.0.
This produces 1317 pounds of lift force.
Now let's increase the altitude while watching the density output.
We're shooting for a value of Density equal to 0.001 slugs/cu ft--exactly half the previous value.
WHAT IS THE NEW VALUE OF ALTITUDE?
WHAT
IS THE NEW VALUE OF LIFT?
HOW IS THIS RELATED TO THE OLD
LIFT VALUE?
Looks like an Altitude of around
26,850 ft gives the desired Density = 0.001.
The
lift is then 658 pounds = almost exactly half of the
lift at 5,800 ft (Density = 0.002)
--> Halving the density cuts the lift in half. <--
Let's do
one more check--increase the Altitude until Density = 0.0005.
An Altitude of around 43,210 ft gives the desired
Density = 0.0005.
The lift is then 329.3 pounds--almost exactly one-fourth of the lift at 5800 ft. (Density = 0.002)
--> Quartering the density cuts the lift by one quarter. <--
This is the old linear relationship again. So we have seen that:
And the Density depends on the altitude in a more complicated way.
Well, that leaves only the airfoil thickness to look at. In the previous lesson we noticed that lift increases with increasing airfoil thickness. So let's set:
Airspeed = 100 mph
Altitude = 0 feet
Angle = 5.0 degrees
Thickness = 0.5
Camber = 0
Area = 100 sq ft
This gives a symmetric airfoil generating 1566 pounds of lift.
If we push the "Thickness" button on the Plotter Control Panel below the "Lift vs." button, we see that the plot of lift versus thickness is not linear, but that lift increases as thickness increases.
[Our previous technique for finding linear relations by doubling and halving the input will not work here because the plot does not go through the point (0,0). Near zero thickness (flat plate), there is still a lift value.] OK, one final check--let's create the lift on the airfoil through camber instead of through angle of attack.
Change the Angle to 0 and the Camber to 0.5 This gives a curved airfoil generating 3963 pounds.
WHAT HAS HAPPENED TO THE LIFT CURVE IN THE PLOTTER
WINDOW?
It's still fairly linear, BUT ... lift now
decreases with increasing thickness!
Maybe the plotter is wrong!
Let's check it by changing the value of thickness on the input panel.
HOW DOES THE LIFT VARY WITH CHANGES IN THICKNESS?
For the curved airfoil, lift does indeed decrease with increasing thickness. There is something much more complex going on here. If we take our curved airfoil and increase the angle of attack, watching the lift plot, we see that the variation can change from decreasing with thickness to increasing with thickness. The fact that both camber and thickness are involved together indicates that the explanation for this behavior will be found in more detailed studies of airfoil geometry. So what have we learned in this little exercise?
--> Good engineers must be thorough!
Let's summarize our results:
So, there are some factors which are rather easy to figure out and some that will require more investigation.
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