Lift, Thrust, Drag, and the Inevitable Consequences of Gravity
This is the traditional starting point for almost any discussion of flight aerodynamics I have ever seen. If we are to accept Sir Isaac Newton’s contributions to modern physics, then there are several immutable truths. An aircraft in level, unaccelerated flight (constant speed, constant direction, constant altitude) must have an amount of lift exactly equal to its weight and must produce thrust exactly equal to its drag:
Now, of course, if any of these forces are varied, acceleration occurs. For example, if the lift exceeds the weight of the aircraft, it will climb, or if the thrust exceeds the drag, it will increase its forward velocity. It is left as an exercise to the reader to determine the effect of the inverses of those conditions.
| Here is a climb: | This is a dive: |
 |  |
Now that we can climb, dive, and vary speed, what about turning? Unlike ground vehicles which derive their motive forces via the contact of wheels with road or rail, an aircraft must depend on its wing (and tail) surfaces to develop aerodynamic forces to achieve the desired result. In order to turn, an aircraft must roll into a bank to develop a component of its lift force in the direction of the desired turn. The lift of a wing is developed at right angles to its surface, so banking the aircraft tilts the lift "vector" away from the vertical by the amount of the bank angle. The following head-on illustration of my little purple airplane will illustrate this:
Forgive a small amount of math, but if this turn is made at constant altitude and at constant forward speed we can calculate that the lift required to support the weight of the aircraft is:
L = W / Cos x , where W is the weight and x is the bank angle.
Also, the centripetal force, which is the horizontal component providing the turning force is:
T = W / Sin x , where W is the weight and x is the bank angle.
We could also express these relationships in terms of acceleration units, or "Gs". A "G" is simply the accleration due to gravity. Force is mass x acceleration, so if the mass of the aircraft is m, and the acceleration due to gravity is G, then we can restate the first relationship as:
m LA = mG / Cos x, where LA is the net acceleration perpendicular to the surface of the wing.
Note that the mass term may be divided out of both sides of the expression which leaves us with:
LA = G / Cos x
The acceleration experienced by the aircraft (and its pilot) in the direction perpendicular to the wing is thus independent of aircraft mass and velocity and is solely determined by the bank angle. That one bears repeating. Regardless of the aircraft type, its altitude, the density of the air, the temperature, or any other factor, the G-load (or Load Factor) experienced by the aircraft and pilot is related only to the bank angle. A Cessna 150 at 100 knots in a 60° bank will experience 2 G as will an F-16 at 500 knots at the same bank angle.
Here is a little table for various bank angles:
| Bank Angle | Acceleration |
| 10° | 1.02 G |
| 20° | 1.06 G |
| 30° | 1.15 G |
| 40° | 1.31 G |
| 50° | 1.56 G |
| 60° | 2.00 G |
| 70° | 2.92 G |
| 80° | 5.76 G |
| 85° | 11.47 G |
| 87° | 19.11 G |
From this table, it can easily be seen why normal passenger aircraft generally limit bank angle to 30° - 35°. The structural limit for symmetrical G loading in most large aircraft (Boeing 727, C-141) is 2.5 G, a T-38 is rated for 7.33 G, while an F-16 can sustain 12 G. The big problem at the higher G loads is the pilot. In a normal, erect seating position, 6 -7 Gs can be tolerated for only limited amounts of time. In order to get to 12 Gs, the pilot must be in a recumbent position (as is the case for an F-16). In many high performance aircraft it is possible to exceed the structural design limits over a wide range of the flight "envelope" and these machines have G meters as one of the cockpit instruments. The flight envelope of an aircraft is simply the region of altitude and airspeed in which it can be operated.
Airfoils and Lift
An airfoil, the cross-sectional shape of an aircraft wing, owes its design to the work of Daniel Bernoulli, another physicist, who discovered and described certain principles of fluid flow. The "fluid", of course, is air, the medium through which aircraft move and depend upon for their lift. Mr. Bernoulli discovered some fairly simple relationships between fluid velocity and pressure which are the fundamental basis for the design of wings. Note that the discussion here really only applies to subsonic flow, not the compressible flow found in the transonic and supersonic cases. If you remember the picture of the pipe with a narrowed center section through which air is flowing from your high school or college physics class, you may also remember that as the air flows through the center narrow section, its velocity increases and its (static) pressure decreases. In the illustration below, V2 > V1 and P1 > P2.
Without getting into a lot of math, Bernoulli found that the Total Pressure was a constant. The total pressure is defined as the static pressure plus the kinetic, or dynamic, pressure. This is really just another statement of the conservation of energy. This principle forms the basis for a wing to develop lift and also is the basis for the operation of an airspeed indicator.
Compare the illustration above to a simple wing cross-section:
The airfoil is just an inside-out, asymmetrical version of the pipe used to demonstrate Bernoulli's Principle. The lift an airfoil generates is a reflection of the fact that the static pressure on the top surface of the wing is lower than that on the bottom surface. With a difference in static pressure, there is a net force in the direction of bottom to top.
The dynamic pressure of a gas flow is q = 1/2 d V2, where d is the density and V is the velocity. The Total Pressure, Pt ,can thus be expressed as:
Pt = Ps + q (where Ps is the static pressure)
We could also express the relationship as Ps = Pt - q
(I will have to be excused for the substitution of Latin characters where, traditionally, Greek characters would be used, because Hypertext Markup Language simply has no Greek characters in its repertoire. Density would normally be represented with the Greek rho.)
We can make some very broad simplifying assumptions and gain some insight into the operation of a wing, although even from the simple illustration above, it can be seen that the mathematics of the problem is already getting pretty messy. Let's just assume that the average flow velocity on the bottom surface of the wing is V1 and the average velocity on the top surface is V2. That being the case, using the second expression, above, the difference in static pressure across the wing would be:
Pdiff = (Pt - 1/2 d V12) - (Pt - 1/2 d V22)
Since Pt is a constant, the above expression reduces to:
Pdiff = 1/2 d (V22 - V12)
Since pressure is a force per unit area, the total lift is directly proportional to the wing area and is directly proportional to the air density. Once again, skipping a lot of the math, an airfoil can be characterized by an artifice called its Coefficient of Lift or CL. The Coefficient of Lift is not a constant, but varies with the Angle of Attack. Once in possession of the Coefficient of Lift, however, we can state the total lift of a wing as:
L = 1/2 d V2 S CL , where d is the air density, V is the velocity, and S is the total projected area of the wing
Let's define some terms before moving forward. First, Angle of Attack, is the angle between the wing chord and the relative wind. Oh great, define a new term with two more new terms! The chord of an airfoil is the straight line joining the ends of the mean camber line. OK, let's start at the beginning:
Mean Camber Line - The locus of points equidistant from the upper and lower surfaces of an airfoil.
Chord - The straight line joining the ends of the mean camber line.
Flight Path Velocity - The speed and direction of the airfoil.
Relative Wind - Equal and opposite to the Flight Path Velocity
Angle of Attack - The angle between the relative wind and the chord of the airfoil.
Some illustrations are in order.
The Mean Camber Line and Chord:
In this sketch, the Mean Camber Line is exaggerated in order to show the chord clearly.
The Chord, Relative Wind, and Angle of Attack:
Now that the terms have been a little better described, a look at the Coefficient of Lift is in order. This is best done with another illustration:
These Coefficient of Lift curves are shown for both a subsonic asymmetric airfoil and a swept wing symmetrical airfoil. What do these curves mean? As the angle of attack of a wing is increased, its coefficient of lift increases - up to a point. If the angle of attack is increased further, the rate of increase of the coefficient of lift is less, until finally it starts to actually decrease. Physically, the hook at the top of the curve is the onset of an aerodynamic stall - the point at which the airflow over the wings ceases to be a well-behaved laminar flow and the wing starts to lose lift. The basic lesson is that, although pilots often talk about stall "speed", a wing stalls at a predetermined angle of attack. That stall angle of attack is manufactured into the wing and nothing (well, almost nothing) changes it. A given wing can be stalled at any airspeed (if it is strong enough!). The reason I say "if it is strong enough" is that the way to stall a wing at high airspeed is simply to try to make it produce more lift than its coefficient of lift can provide. This is done by "pulling Gs"; for example, in a level turn at a bank angle approaching 90°. Practically speaking, most normal aircraft reach their structural limit before stalling the wing at high airspeed - this is not true of high performance aircraft, however. Referring again to the curves above, the line representing a subsonic asymmetrical airfoil is characteristic of a light aircraft like a Cessna 172. The Cessna 172 has a very abrupt and well-defined stall in which the nose of the aircraft drops, decreasing the wing's angle of attack and increasing its speed. The stall of a high performance aircraft, represented by the lower curve which is characteristic of an aircraft like the T-38, is much more insidious. It is possible to get a T-38 into a fully-developed stall in a level attitude with a very high sink rate (on the order of 20,000 ft/min).
Many aircraft provide tactile feedback to the pilot of an impending stall. As the airflow on the top surface of the wing becomes turbulent and that turbulence flows past the horizontal stabilizer, the pilot can feel the buffeting in the controls. Pilots refer to this as the "burble" or a "nibble". Some aircraft, particularly those with "T" tails, such as the C-141 or the Boeing 727 have their horizontal stabilizers deliberately positioned to avoid wing turbulence and provide no tactile feedback at all. These aircraft depend upon an angle of attack sensor and a "stick shaker" to warn the pilot of an impending stall condition. Ironically, neither of those aircraft provide a cockpit instrument which would allow the pilot to monitor the angle of attack.
Drag
Drag comes is two basic flavors. The first, parasitic drag, is very easy to understand as it is simply the resistance of the aircraft to the air through which it moves. Although very difficult to describe in precise mathematical terms, parasitic drag is easily visualized in the difference between a barn door placed at right angles to the wind and an arrow moving through the air or the difference between a person who jumps out of an airplane with and without a parachute. Parasitic drag increases with the square of the speed through the air. The second basic type of drag is less obvious, but very important. This type is called induced drag and is related to the wing's production of lift. The net lift, or aerodynamic force of a wing is produced at right angles to the chord of the wing. Since there must be a positive angle of attack at low airspeed (the angle between the chord and the relative wind) to produce lift, there is a component of the total aerodynamic force which resolves to a vector in the drag direction. This is exactly what is observed in practice - at low airspeed and high angles of attack, induced drag increases and becomes a large factor. The following graph shows the effects of the two types of drag.
This curve, although its precise shape was dictated more by the tool I used to produce it than reality, nonetheless shows the basic effect of the two types of drag. It depicts the condition of an aircraft in level unaccelerated flight over a range of airspeeds. Since the lift in this case is a constant (the weight of the aircraft), the minimum point in the drag curve is the point at which the lift to drag ratio is at its maximum. This point is termed L/D max. Below L/D max, the increase in drag is due to induced drag and above the increase in drag is due to parasitic drag. Airspeeds below L/D max are area known as the region of "reverse command"; that is, as you go slower, you must add more power to overcome the drag. This is somewhat counterintuitive, but an essential part of the aviator's knowledge base. The point labeled "Buffet Limit" is the airspeed at which you have reached the maximum angle of attack and are starting to get the stall buffet. At the high end of the curve, you may actually be limited either by reaching the maximum power output of the engine(s) or reaching the point at which transonic speeds are achieved. As Mach 1 is approached, drag increases dramatically, due to the formation of shock waves. In fact, this is another distinct type of drag known as "wave drag".
As the above discussion indicates, a supersonic aircraft also has to deal with the drag due to the formation of shock waves, the onset of which is influenced by the specific airframe design, but generally starts around .75 Mach. Aircraft designed for supersonic flight power through the high drag of the transonic region and then experience a reduction in drag once established in the supersonic realm. In the transition from subsonic (< .75 Mach), through the transonic region (.75 Mach to 1.3 Mach), and into the supersonic region (> 1.3 Mach), Bernoulli's rules no longer apply and the very nature of lift production changes.
Yet another form of drag is called interference drag. If you take a clean aircraft and determine its drag at a given airspeed and then take an external store like a fuel tank or bomb and measure its drag at the same airspeed, the total drag of the aircraft with the external store attached exceeds the sum of the individual drag values. Interference drag also occurs as a result of external engine nacelles and at the junction of the wing and tail surfaces with the fuselage.
Many of these three dimensional fluid flow problems only recently became candidates for computer analysis as the power of supercomputers has reached a level that make it practical to attempt their analysis. Many of the current high performance aircraft were designed by people gifted with a sense of what would work and what wouldn't. The classic example which comes to mind is Kelley Johnson and his design of the SR-71 at Lockheed's "Skunk Works". Kelley was a very gifted aircraft designer whose mind would be a match for the best Cray has to offer.
Thrust - The Powerplant(s)
Modern flight was made possible largely through the development of engines light enough and powerful enough to be able to overcome the drag forces discussed in the previous section. From the Wright brothers through most of WWII, the internal combustion engine was refined as an aircraft powerplant. Probably the most highly developed of all the internal combustion engine types was the air-cooled radial which reached its pinnacle with the Pratt and Whitney R-4360 Wasp Major, a turbo-supercharged 28 cylinder engine with 4 rows of seven cylinders each, displacing a total of 4,360 cubic inches. As installed in the Convair B-36, it produced 3800 horsepower and represented the crowning achievement of the engine designers. No discussion, however, of aviation powerplants would be complete without mentioning the another engine type which was also highly successful in its application in the P-51D. That engine is the Rolls-Royce-designed Merlin, a liquid-cooled narrow V-12 with a displacement of 1649 cubic inches and a power output of 1695 horsepower.
Well before the end of WWII, though, a German aircraft, the Messerschmitt Me-262 Schwalbe (Swallow) heralded the end of an era. With two Junkers Jumo 004B-1 engines, the Me-262 was the first operational jet aircraft. Even though it was not brought on line early enough to have a significant effect on the outcome of the war and had notoriously unreliable engines, the Me-262 showed the promise of turbine propulsion. Even the Convair B-36 was a hybrid, with 6 Pratt and Whitney R-4360s and 4 J47-GE-19 jet engines.
The military, of course, pioneered the jet engine revolution as only the government had the money required for the necessary research and development. The six engine B-47 flew years before Boeing developed its famous 707. In fact, the Pratt & Whitney J-57 engine which, in May of 1953, enabled the Air Force F-100 to be the first production aircraft to break the speed of sound, was used just about a year later in July of 1954 to power the first prototype of the B707. That commercial version of the military J-57 was designated the JT-3. The Douglas DC-8 later used the same engine. The commercial switch to turbine powerplants lagged the military by nearly a decade. In their early days, jet engines were pretty unreliable, wheezy affairs that only really provided advantage at higher altitudes and airspeeds unattainable by piston-powered aircraft. The engineering challenges were overcome, though, and the jet engine became the mainstream for both heavy aircraft and high performance aircraft.
What are the differences between the powerplant types? The first glance provides some insight. All recips have a big fan mounted on the business end of the crankshaft (the "business" end varies among aircraft - on most airplanes it is the front, but the historical turning point, the B-36, had its R-4360s mounted as "pushers" with the props aft of the wing trailing edge). The propeller consists of a number of blades (I have seen anywhere from 2 to 6). Each blade is, in effect, a long narrow "wing"; that is, an airfoil. As the prop is turned by the engine, the blades produce lift, which is manifested as forward thrust. The design of the blades, however, is complicated by the fact that their tangential velocity increases with increasing distance from the hub. Without delving into the math, in order to maintain a more or less constant angle of attack along the length of a blade, it is necessary to impart to its airfoil a significant amount of twist. The problem encountered with a prop is that the tip velocity must be kept in the subsonic realm. Consider a 15 foot diameter prop turning at 1000 RPM. I'll leave the arithmetic to you, but I calculate a tip speed of 535.50 mph. At sea level in a standard atmosphere that corresponds to Mach .70, while at 20,000 feet it is .76 Mach - already encroaching into the transonic region. You might ask why not make the blades shorter and allow them to turn faster? The issue here is one of tip losses. The most efficient airfoils are those with a high aspect ratio; that is large wingspan compared to chord width. As you approach the end of a wing, a significant amount of spanwise flow develops, spilling the lower pressure air from above the wing off the end and mixing it with the higher pressure air from below. This creates a vortex at the tip which increases drag and generally reduces the effective lift of the wing. This is a significant effect and a large waste of engine horsepower. The next problem is that as aircraft speed increases, the effective angle of attack of the blade increases and since we can't increase the blade's tangential speed to compensate, we can only increase the pitch. These effects end up in a stalemate - you just get up against a wall at some speed.
Another problem endemic to the recip engine is that with increasing altitude comes decreasing air density. As any hot rodder knows, the way to get more power out of a recip engine is to cram more fuel and air into the cylinders to increase its Brake Mean Effective Pressure (BMEP). Decreasing the air density is destructive to this end. Recip engines, therefore, need superchargers to maintain efficiency at altitude. Relatively simple radial engines, like the P & W R-2000, have a single speed supercharger which provides about 15 psi of boost at takeoff and, typically, 10 psi of boost at cruise. More sophisticated engines use gearboxes in the supercharger drives to allow for higher speed operation at higher altitudes. Once again, however, it is a losing battle. The rule of thumb aviators learn quickly is that at 18,000' altitude, half of the atmosphere is below you. That is the point at which pressure has dropped to half that of sea level. At sea level, 10 psi of boost gives you a net absolute manifold pressure of almost 25 psi, while at 18,000', you only have a bit over 17 psi. A high performance recip aircraft, such as the P-51D, could only achieve a maximum speed of about 380 knots at 25,000 feet. Virtually any modern jet passenger aircraft can easily exceed that.
In an engine/propeller system, the propeller is the speed governor for the engine. The propellers are deceptively complicated devices which can vary the blade angle over a wide range. The props are literally set up as governors for engine speed through a complex mechanical linkage which varies the blade pitch angle to regulate engine speed. The pilot can control engine speed by setting it with his prop levers. He then sets the desired power output with engine's throttles. Power output is determined by directly measuring the intake manifold pressure or even BMEP. In the C-7A, a typical cruise setting was 2000 RPM (engine speed - prop speed was half that through a planetary gearset) and 20 inches (of mercury) manifold pressure boost.
It is not coincidental that recip engines are measured by their horsepower and jet engines are measured by their thrust. To a first order approximation, a jet engine does, indeed, produce a fixed amount of thrust, independent of its speed through the air. Thrust is nothing more than the total force the engine produces and if you remember from that physics class of long ago, power is Force x Velocity. Thus the "horsepower" of a jet engine increases in direct proportion to its speed, whereas the net "thrust" of a recip engine decreases with increased speed. To put some numbers on that, a C-141A with P & W TF-33 P7 engines (21,000 lbs. of thrust at takeoff, and perhaps 15,000 lbs. in cruise) in a 500 mph cruise produces somewhere on the order of 20,000 horsepower per engine! Those are only medium-sized engines by today's standards. Large, wide-body commercial aircraft now have engines in the 50,000 - 90,000 pound thrust class. What is ironic is that modern jet engines bypass varying amounts of air past the high pressure compressors and combustion chambers and directly out the rear of the engine. In a TF-33 P7, the bypass ratio was about 50%, while in the new large commercial engines, the ratio may be 6 or 7 to one. The low pressure compressor, in effect, is nothing more than a ducted propeller!
Several views of the Pratt & Whitney PW4000 Series engines
which clearly show the large first-stage fan and bypass duct
That duct makes all the difference. By maintaining very close clearances between the end of the compressor blades and the compressor shroud, blade tip effects are almost eliminated and thrust is produced very efficiently by relatively short, wide chord blades. This concept is being explored with great interest now, since we have both the supercomputers to analyze three-dimensional compressible flow and the precise numerically controlled machine tools to create extremely complex fan blade shapes.
Another difference between the recip and the jet is that of reliability. A C-7A had a high time engine at 1000 hours of total time. At 1200 hours, you literally held your breath when you made power changes (the most likely time for failure) and I never saw one with 1300 hours on it. In something like 700 hours of flying time in the C-7A, I had two catastrophic engine failures. A modern jet engine routinely goes 10,000 hours or more between overhauls and in thousands of flying hours, I have never seen a catastrophic failure. The jet, unlike the recip, has no reciprocating parts which generate very high periodic bearing loadings and has almost no direct contact parts like the piston/piston rings and cylinder wall. The rotating parts in a jet engine are very precisely balanced, using machines similar to the dynamic wheel balancers we have all seen when we buy a new set of tires.
Since we all are familiar with automobiles and the engines which power them, understanding the basic principle of a recip engine is almost second nature. Jet engines, on the other hand, are much more mysterious. A jet engine inhales air at the front and compresses it through a number of axial compressor stages. The high pressure air exiting the compressor is introduced into a combustion chamber where it is mixed with fuel and ignited. The now very hot gases exiting the combustion chamber flow through a set of axial flow turbine stages which extract enough energy to run the compressor and to power accessories such as the hydraulic pump, the generator, and the high pressure fuel pump. The whole idea, though, is to take a in mass of air and accelerate it before expelling it out the back. If Sit Isaac Newton can be believed, then Force = Mass x Acceleration. To put some rough numbers on things, the compressor increases the inlet pressure by about a factor of about 30 and increases inlet air temperature by over 1000° F. Temperature rises to around 4000° F in the combustor and has dropped to about 750° F at the turbine exit.
In modern fan jet engines, like those pictured above, it is desirable to run the fan (first compressor) stage at a much slower speed than the high pressure end of the compressor section, so the engine will have two, or even three separate "spools". In a two spool engine, the outermost compressor and turbine stages are mounted on a shaft. A second hollow shaft, through which the first spool's shaft runs coaxially, connects the inner compressor and turbine stages. Turbine and compressor blade geometries are carefully designed the achieve the correct rotational speeds on each spool. The first spool, or N1 section is completely mechanically independent of the second, or N2 spool. In fact, you can reach up and spin the fan section with your finger - it rotates freely. Since the accessory drive must be taken off the outermost shaft, it is the N2 section which drives the accessories and provides a connection point for a starter. A three spool engine, such as the Rolls-Royce RB-211, adds another spool in an attempt to better match compressor speed with the very short compressor blade lengths in the highest pressure section. In this case, the N1 and N2 sections are both free-rotating and the N3 section drives the accessories.
The accessories themselves are quite impressive. Generators are driven by hydraulic devices called constant speed drives or CSDs. A CSD is basically a variable volume hydraulic pump which is closely connected to a hydraulic motor. The gear-driven engine power takeoff runs the pump and the generator is connected to the motor. Over a reasonable range of input speeds (engine idle to takeoff thrust) the output motor is able to maintain an almost constant speed. For most aircraft, 115 volt, three phase, 400 Hz. power is standard. These generators are large - a B727 has three 40 KVA units and a C-141 has four 50 KVA units. The generators are sized generously to allow for engine failure situations. Both of those aircraft can maintain essential electrical power in a two engine out situation. The hydraulic pump is also a variable volume pump which can produce high volumes of hydraulic fluid at 3000 psi. For most modern aircraft, 3000 psi is about the standard hydraulic system pressure. In addition to operating the flight controls, hydraulic cylinders operate the landing gear, hydraulic motors operate the trailing edge flaps, and even the brakes are powered by the hydraulic system through control valves operated by the rudder pedals and moderated by the anti-skid system. Unlike the electrical system, in which the generators operate in parallel on a common bus system with a hierarchical load-shedding system, the hydraulic systems are carefully isolated. The flight controls, which have the highest priority, have completely independent dual hydraulic systems, operated from separate pumps. In the B727, system A is provided by the pumps on engines 1 and 2, while system B is powered by two electrical pumps. In the C-141, the two primary systems are powered by engine pumps and the tertiary system is powered by electric pumps.
All in all, jet engines are more powerful, retain their performance at high speed and altitude, and are more reliable. They certainly have removed some of the romance of flying, however. Starting a jet engine is pretty boring. Some aircraft like the T-38 and C-141 have completely automatic engine start sequencers - you just push the start button and make sure you get a clean light off. Others, like the Boeing 727 allow the pilot to actually turn on the fuel at the appropriate N2 RPM. Nothing can compare to the art of starting a cantankerous P & W radial, though. With the radial there is absolutely no science involved - it is a black art which requires a complex vocabulary of curses and encouragements, appropriate body English, and a highly skilled left hand (at least on a Caribou). There is no finer sound than 2000 cubic inches of Pratt and Whitney coming to life in a billowing cloud of blue oil smoke and settling down to a contented idle.
