Chapter 3 - Aerodynamics I would re-name this chapter Unpowered flight The drag polar, the relationship between lift and drag. Aerodynamically: = and = So the lift and drag forces are functions of the square of the speed. is the density of the air. S is a reference area (often the area of the wing). The chapter introduces the drag polar curve. This is a plot of CD vs. CL, though the independent variable (C L ) is plotted on the vertical axis and the dependent variable (C D ) on the horizontal axis. In Figure 9 of the Skript on page there are some errors, and the figure is misleading. Here s a revision of that figure: C L and C D are related to each other. As can be seen from the above curve, it looks something like a parabolic relationship. A polar curve for an aircraft is made by plotting C D vs. C L :
As can be seen, this curve is plotted backwards (see below). This curve is a simplified version of this relationship, a sort of prototype. This is called a square polar. Polar because it s a plot of C L vs. C D. Square because it s a parabolic relationship. The equation is = + So actually, what we have here is C D = C D(C L) not C L = C L(C D) ; it s just plotted backwards, with the independent variable on the vertical axis and the dependent variable on the horizontal axis. What s important, however, is that they are related. Why it s plotted backwards is anybody s guess. Aeronautical engineers are perverse. It was probably first done by the same guy who decided that down is up. Obviously what we want with aircraft is as much lift as possible and as little drag as possible. As we see from the equation at the start of this section of notes, both and have the same dependencies (, S, v 2 ). The place on the polar where L/D is maximum is the tangent to this curve drawn from the origin with the angle min shown. Turning our attention once again back to the polar equation = + we see that the curve has an offset C D0. This gives us the minimum drag. We have it regardless of how much lift we produce. The other term, is lift-related drag or induced drag. k is a measure of how fat the parabola will be. The smaller k is, the fatter the parabola will be. In fact, if k = 0, the curve would be a vertical line at C D0, and we could get more and more lift without increased drag. The higher k is, the more drag we will get for a given amount of lift. So we want the parabola to be as fat as possible; we want k to be a small as possible, if we consider the above equation. If it were very fat, say a straight vertical line, the drag would not increase with additional lift. k is often estimated by where = Λ e = 1 for elliptical wings e > 1 for non-elliptical wings
Λ = where b is the wing-span of the aircraft and A is some area, usually the area of the wing. For a fat parabola, a small k ), needs to be large. Thus we d want a lot of area for a given wing-span. Put another way, we want long wings with little area. As far as k goes, note that it depends just on properties of the aircraft and wing. So it is design conditions that determine the fatness of the parabola. Again, the polar equation here is a square polar, which is the most common. Other types of polar relationships have C L to other powers than 2. Still, here with this equation giving C D = C D(C L), we haven t said what C L depends on. Section 3.1 The relationship of lift and drag along the (square) polar Since the expressions for L and D are identical except for the coefficients of lift and drag, for any flight condition, these two forces are related as = 2 2 = This relationship will change, depending upon where you are operating on the polar. The figure above shows an aircraft with the throttles pulled back (T=0) in descent. Because of the kinematics, note that tan = = What happens if T 0? What happens, say, if the plane is flying horizontal, with = 0? We know that there is lift and drag, with D L/10. So the relationship between L, D, and does not apply to that case. In fact, only when there is no thrust is the L, D, and relationship valid, i.e. the lift and drag forces together are vertical. A pilot has the option of controlling the flight path of the aircraft. For example, if he/she turns the nose straight down to the ground, there will be no lift force. This will not make v 0, quite the contrary, nor
will any of the other terms be 0, so in such a case, with no lift, C L must be = 0. Since the aircraft must fly on the polar, it will be operating at the nose of the parabola, and C D = C D0. Thus, there no matter what the velocity, and = 0 2 = 0 = 2 The drag force will continue to increase as the velocity builds until it becomes equal to the gravity force, at which point we shall have reached terminal velocity, and the aircraft ceases to build speed in this balanced-force state. The other possibility is that the aircraft will exceed its structural speed and come apart. Also note that if the aircraft is headed straight down without lift, D/L =. arctan( ) = 90 ; i.e. the descent angle is 90 from the horizontal, i.e. straight down. Let s consider the other, opposite end of the flying spectrum on the polar. Rather than lowering the nose, we raise it. The lift force increases, but so does the drag force. Take again the case of unpowered flight, T = 0. If we pull the nose up, we get more get more lift, but because the parabola is mostly horizontal, we also increase the drag force inordinately. We move out along the polar, and increases. The descent rate increases again, even though we have more lift. Also because of the increased drag, the velocity drops off. Eventually the airflow cannot stay attached to the wing, so the aircraft stalls. Thus, there is a sweet spot. This is the point on the polar in between these two extremes where D/L = C D/C L is minimum. Since the tan increases as increases, this point is where a line from the origin is tangent to the parabola. This gives min, i.e. the minimum descent angle for unpowered flight. This is where a glider pilot wants to fly when he/she is hunting for the next source of lift. This is the speed at which a pilot of a powered aircraft wants to fly if his/her engine quits. This is the best glide or minimum descent speed. The moral of the story is that the airplane has a polar along which it flies. It cannot leave the polar, but it can fly at various points along it, depending upon the pitch attitude. Section 3.2 Minimum glide angle To find the sweet spot, take the derivative of with respect to C L and set this equal to 0. See Skript. This leads to = = 2 = arctan = arctan 2 = arctan 2
Notice what we would need to do to the square polar curve to get a small angle of descent. We need to move the parabola as close as possible to the origin (make C D0 small) and then make the parabola as fat as possible (make k small). Another way to look at this is to look at Figure 12. In this figure, if must be vertical, if increase, for whatever L we have, D will get greater and greater. In fact at = 90, L must be 0 to make vertical, and D is enormous. Minimum and maximum velocity We start with the expression for R shown in the Skript. Then we impose the condition that the resulting force needs to be equal to the weight. This would give us a descent rate that leads to no vertical acceleration. From this we can get an expression for the velocity in terms of parameters that don t change with the flight condition (m, g,, S) and those that do (C L and C D, as we move along the polar). Why are we limited to having = +? I assume it is for the reason shown in the three figures above. If we are flying along at our best L/D as in 1 above, for example, and we change the pitch angle downward, the airplane will then have and reorient as shown in 2. Now s vertical component is less than. The aircraft accelerates then downward. At the same time, now has a horizontal component, so the aircraft accelerates to the right too. The result is that the aircraft now has a steeper angle of descent and a greater speed. Thus the drag vector will be greater. We have moved away from the optimum unpowered descent angle, so we have moved to a new point on the drag polar, where there is less lift for a given drag. This is shown in 3 above. This point is then down the drag polar toward its nose. Also note that though this scenario started with us at the minimum descent angle, but there is actually no reason that that need be. We could have started at any point on the drag polar and moved to any other point on the drag polar. We could have started at 3, for example, pulled back on the stick or yoke, and wound up at 1. Then we would have moved up the drag polar away from the origin. On page 20 of the Skript, equations (3.9) and (3.10) give the values for minimum and maximum velocity along the drag polar. Since = 2 1 +
v max occurs when + is minimum, and that occurs at the nose of the parabola, when C D = C D0 and C L = 0. I m not sure about the assertion for v min. v min seems to me to be the stall speed, and that is not modelled on the drag polar. The condition given for i his expression is C L >> C D. The point on the curve where C L/C D is maximum is at the tangent point, as already discussed above. So this seems rather to be an approximation for the speed at minimum-descent flight-path angle. 3.3 Minimum rate of descent w is the vertical speed. So If is small, and Also So = sin sin tan = = 2 1 + + (Note that this equation is incorrect in the Skript.) etc To get the minimum rate of descent, minimize w with respect to C L. Figure 14 is a plot of w = w(v), i.e. of the equation above it. Note that v min is really v w-min, i.e. the velocity at minimum descent angle.