Drag of Cylinders & Cones

Drag of Cylinders & Cones


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This subject shares some commonality with previous articles in which we discussed the drag coefficient of a sphere and using that of a circular disk to calculate the terminal velocity of a falling penny. In fact, a flat circular disk oriented perpendicular to the airflow is commonly used for calibrating wind tunnel equipment. An example of this application can be found in NACA Technical Note 253, for instance. This shape is so popular because its drag coefficient (CD) is nearly constant across all operating conditions like Reynolds number. Above a Reynolds number of 1000, the drag coefficient of the circular disk becomes a constant that has been consistently measured as 1.17. This coefficient is based upon a reference area equal to the frontal area of the disk, or  r² where r is the radius of the disk.



Drag coefficient of a circular disk and a sphere versus Reynolds number

It sounds like the shape you have is not a flat disk but a cylinder. Estimates for the drag coefficient of a cylinder oriented so that the blunt end is perpendicular to the flow can be found in the classic book Fluid Dynamic Drag by Dr. Sighard Hoerner. According to the following graph, the coefficient of drag for a cylinder in this orientation is about 0.81 so long as the l/d (length-to-diameter ratio or fineness ratio) is greater than 2. As the fineness ratio shrinks to zero, the cylinder collapses to a flat circular disk. It is therefore not surprising that the drag coefficient for a cylinder with l/d=0 is about 1.17, the same as a flat disk. The reference area for a cylinder is also the same as that used for the disk except the radius is that of the cylinder's circular cross-section.



Drag coefficient of blunt nose and rounded nose cylinders versus fineness ratio l/d

The drag coefficient for a cone pointed into the airflow is a bit more complex since it depends on the cone's shape. In particular, the drag will vary depending on how steep the angle of the cone is. The angle of interest is called the half-vertex angle, , measured from the centerline of the cone to one of its walls. The larger this angle becomes, the higher the drag of the cone is.



Drag coefficient of wedges and cones versus half-vertex angle

Based on the above graph provided by Hoerner, it appears that the drag coefficient is nearly linear with the half-vertex angle from 0° up to 90°. We can use this information to derive the following equation that closely approximates the experimental data for drag coefficient versus the half-vertex angle of the cone. Simply use the appropriate angle (in degrees) for the cone in your wind tunnel experiment to calculate the drag coefficient.



A half-vertex angle of 90° causes the cone to turn into a flat circular disk. As we have already seen, the drag coefficient of this shape is about 1.17. Plugging 90° into the above equation results in a drag coefficient of 1.17, exactly what we should expect at this particular angle. Again, the reference area for a cone is equal to the cross-sectional area of its base and can be calculated as  r².

Above 90°, the cone folds backwards and becomes hollow like a cup. The drag of this cupped shape remains fairly constant as the half-vertex angle of this cone increases to 180°. Typical drag coefficients of these and other basic shapes at Reynolds numbers between 10,000 and 1 million are compared in the following diagram.



Drag coefficients of several simple 3D and 2D shapes

The table on the left compares three-dimensional shapes like disks, cones, and spheres while the table on the right is for two-dimensional shapes like plates, wedges, and cylinders. On the 3D side, note the flat circular disk in shape #7. The drag coefficient for this shape is given as 1.17, as we have discussed. The shape just above this one is a 60° cone, or a cone with a half-vertex angle of 30°. The drag coefficient of this shape is listed as 0.5. The simple equation we derived earlier predicts 0.498, a very close approximation.

Also note the two-dimensional cylinder in shape #12. This cylinder is oriented with its axis perpendicular to the flow rather than into the flow as described earlier. If a cylinder is mounted in this orientation in a wind tunnel, the drag coefficient should be about 1.17 using the same circular reference area assumed for all the shapes we have discussed.

This explanation has probably gone into much greater detail than was required, but understanding the drag characteristics of these simple shapes can often be very useful in predicting how more complicated objects behave. We have also explained the similarities between these different shapes, such as how they all collapse into flat disks and produce the same drag. Simple rules of thumb like these are often very useful as a method of quickly evaluating the accuracy of experimental data compared to theoretical predictions. Judgment skills of this kind can prove indispensable in the day-to-day work of an engineer.

- answer by Jeff Scott, 5 June 2005

What is meant by the term finite aspect ratio (2D) or infinite aspect ratio (3D) on an airfoil's coefficient of lift? Is this difference related to the air coming around the wing at the tip?



You are partially correct. As you surmised, the difference between a finite wing and an infinite wing is in that a finite wing has tips. As a result, the higher pressure air from beneath the wing tries to move around the tips towards the lower pressure above the wing. This motion creates a swirling vortex of air from each tip that trails behind the wing. For that reason, we call these vortices trailing vortices. You can read more about this phenomenon in a previous question about ground effect.



Creation of trailing vortices due to a difference in pressure above and below a lifting surface

However, you have the terms 2D and 3D reversed. A 2D wing is the same as an infinite wing while a 3D wing is a finite wing. We call a finite wing "3D" because the air is able to travel up and around the wingtip to produce trailing vortices. The flow around a 2D wing is not able to move in this third dimension. This situation is not possible on a real aircraft since one cannot build an infinite wing. However, an airfoil section tested in a wind tunnel is a 2D wing because the walls of the tunnel prevent the flow from being able to travel around the tips. An example of a 2D wing being tested in a wind tunnel is shown below. In this case, the wing is mounted vertically so that the floor and ceiling prevent the air from being able to flow around the tips.



2D infinite wing being tested in a wind tunnel

Aerodynamically, the effect of trailing vortices reduces the slope of the coefficient of lift vs. angle of attack curve. The lower the aspect ratio of the wing, the more the lift-curve slope is reduced. This behavior results from the fact that the trailing vortices are able to influence a larger portion of the wing the smaller the wingspan becomes. The ideal lift curve slope of any 2D wing is 2. If you look at wind tunnel data for any airfoil shape, you'll see that the slope of the lift curve is indeed very close to this value. As aspect ratio decreases, however, the lift curve slope becomes less than 2 which reduces the overall lift that the wing can produce. You can learn more about this behavior in an article about estimating the lift coefficient.

The reasoning above explains why commercial airliners like the Boeing 747 and other long-range aircraft like the B-52 Stratofortress bomber have very long, slender wings. These wings have a high aspect ratio that reduces the effect of trailing vortices and maintains a high lift curve slope. Such a wing is more aerodynamically efficient and allows the plane to maximize its range.



Overhead view of a B-52 illustrating its high aspect ratio wing

By contrast, most fighters like the F-16 Fighting Falcon or MiG-21 have very short, stubby wings. These aircraft need to be very fast and maneuverable, which requires low aspect ratio wings. The drawback of this design is that such planes typically have a very short range.



MiG-21 fighter showing its low aspect ratio wing

You can see the effect of aspect ratio on the lift produced by a wing quite clearly in the following graph.



The data compares the lift coefficient of the Cessna 172, which has a high aspect ratio wing, against the lift coefficient of the Lightning, a supersonic fighter with a low aspect ratio wing. The slope of the Cessna 172 curve is clearly much higher than that of the Lightning up to the stall angle. At any angle of attack below stall, the Cessna will have a higher lift coefficient and be a more efficient wing.

F-14 Wing Sweep & Aspect Ratio

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I read that airplanes with a wing of high aspect ratio can still fly at high angles of attack and a wing of a small aspect ration usually stalls at about 16° AoA. If this is true, then why can the F-14 land on a carrier at high AoA with its wings fully extended? Doesn't fully extending the wings make a small aspect ratio?

- question from Jeremy Ashley

I believe you have your definitions a bit confused. Aspect ratio (AR) is defined as the square of the wingspan (b) divided by the surface area (S) of the wing, as shown in the following equation.



In general, a "skinny" wing with a large span will be considered a high aspect ratio wing while a "stubby" wing with a short span is a low aspect ratio wing. A Boeing 747, for example, has a high aspect ratio wing compared to the F-16 Fighting Falcon.



F-14 with wings fully extended during takeoff

An aircraft like the F-14 Tomcat is a special case since it can sweep its wings backward and then forward again to change its wingspan. When swept fully forward, the Tomcat's wingspan is about 64.08 ft (19.54 m) and its wing area is 565 ft² (52.49 m²). The variable-sweep mechanism, however, allows the wings to sweep backward during flight reducing the wingspan to as little as 38.17 ft (11.65 m). There is also some small change is wing area as the wings move but this can be neglected for the sake of our discussion.

During takeoff and landing, the wings of the F-14 are swept fully forward. This position maximizes wingspan for the same wing area and results is an aspect ratio of approximately 6.95. When fully swept back for high speed flight, however, the wingspan is significantly reduced and the aspect ratio is only 2.58. While there is no definitive cutoff point, a wing with an AR less than 4 or 5 is typically considered to be a low aspect ratio surface while one higher is considered a high aspect ratio wing.



F-14 with wings fully swept back during high speed flight

Your definitions of wing stall are also reversed. A high aspect ratio wing will generally stall at lower angles of attack than a low aspect ratio wing. These trends become apparent in the following graph that compares the lift of a Cessna 172 with a high aspect ratio wing to a Lightning supersonic fighter with a low aspect ratio wing. The Cessna wing stalls at an angle of attack of about 16° while the Lightning does not stall until 30°.



Comparison of lift coefficients for high and low aspect ratio wings

Although both your definitions of aspect ratio and wing stall are incorrect, the two mistakes cancel each other out leaving the same fundamental question: why does the F-14 fly at high angles of attack at landing when the wing is more likely to stall? As shown in the above graph, a high aspect ratio wing produces much more lift than a low aspect ratio wing. At its stall angle, the Cessna 172 generates a lift coefficient of about 1.7 whereas the Lightning only generates a coefficient of about 1.3 at its stall angle. The lift equation tells us that the higher the lift coefficient is, the slower the plane needs to fly. It is much safer to fly as slow as possible during landing, so a high aspect ratio wing is far more desirable during this stage of flight.

This topic was also explored in greater detail in a previous article about takeoff and landing flap settings.

Reference Area and Lift

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The question you are referring to explains the terms used in the basic equation that describes how the lift of an airplane or helicopter is computed. Among these terms is the reference area, denoted by Sref. Now the important thing to realize about reference area is that it is really only an arbitrary value. It exists because aeronautical engineers need some area by which to nondimensionalize the lift coefficient, as well as other aerodynamic coefficients (such as drag).

This is particularly true when it comes to collecting wind tunnel data. In the wind tunnel, balances are used to measure the actual lift or drag force as well as aerodynamic moments that are generated by some body in a given set of conditions (such as velocity and angle of attack). Knowing the forces and moments themselves is useful, but it is more convenient to express these values in terms of some nondimesional form that can be more easily compared for different configurations at different conditions.

Let us consider an example and compare the lift produced by a jet fighter to that produced by a sailplane. If you were to compare them simply of the amount of lift force they produce, the fighter would appear to be several orders of magnitude "better" at producing lift. But this is an unfair comparison simply because the fighter weighs orders of magnitude more than the sailplane and needs to generate that much more lift to remain airborne. But when we nondimensionalize the lift force and compare the two vehicles based on their lift coefficients, we'd see that in actuality the sailplane generates a higher lift coefficient making it the more effective lifting surface.

These nondimensional forms of the aerodynamic coefficient are expressed as CL for lift, CD for drag, Cm for pitching moment, and so on. So at this point, what we need is some way to convert the dimensional forces and moments expressed in pounds, Newtons, ounces, foot-pounds, Newton-meters, ounce-inches, or some other unit system into the nondimensional coefficient form. We can understand how this is done by turning back to our good old friend the lift equation:



Here we see the dimensional form of lift (L, in pounds or Newtons) is computed as a function of operating conditions (density and velocity), reference area, and the lift coefficient. But in our case, we have specified the operating conditions in the wind tunnel and measured the lift force, so we can compute the lift coefficient in this way:



Here it is that we see how the term Sref is needed simply as an area term to nondimensionalize the coefficient. And it really makes no difference what value we use so long as that same value is used consistently in any future work with that coefficient data. It is simply a rule of thumb that the wing area is used as the reference area for aircraft and the rotor area is used for helicopters. For rockets or missiles, we typically use the maximum cross-sectional area of the body. However, this is not always the case as for some missile configurations it makes more sense to use a wing or fin area or even a total planform area. For airfoils, we need a reference length rather than an area, so we use the chord, or length, of the airfoil section.

The key point to remember from this discussion is that the reference area itself has nothing to do with how much lift a vehicle produces, but it is important to know what area was used in a given application to make correct conclusions about the aerodynamic behavior of that body. If one were to conduct two wind tunnel tests of the same missile and in one case used the fin area as the reference area and in the other used the body cross-sectional area, one set of lift coefficients would likely appear to be much greater than the other, so a comparison of the two sets of data would be useless. Therefore, while it is not necessary to use wing area for planes, rotor area for helicopters, or cross-sectional area for missiles, it is useful to do so because it provides a common ground for comparison between the lift coefficient produced by two different planes, two different helicopters, or two different missiles.

As for number of blades and the blade airfoil section, of course these variables factor in to how much lift a helicopter rotor will generate. But the same can be said of an airplane where the wing airfoil section, the use of flaps or slats, wing incidence angle, and a variety of other factors will increase or decrease the amount of lift the plane generates. While these more complex considerations are not addressed directly in the basic equation for lift, where they do factor in is in the lift coefficient itself. For example, CL increases with increasing angle of attack for both an aircraft wing and a helicopter rotor. The latter is accomplished by increasing the angle at which an individual rotor blade or all of the rotor blades collectively meets the oncoming air. Similarly, the increase in lift associated with a plane dropping flaps or slats is reflected in the lift equation through an increase in CL. Keep in mind that this equation for lift is only a very basic expression used to analyze the overall aerodynamic behavior of an entire vehicle. Engineers must use much more complex methods to do detailed design and analysis work.

- answer by Jeff Scott, 19 May 2002

Lift Equation

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The equation for lift is:



where

Variable Units Description

L English:

lb

Metric:

N lift force

English:

slugs/ft3

Metric:

kg/m3 air density

Air density changes as a function of altitude, so the value of this variable depends on the height you want to find the lift at. You can find tables of the air properties at different altitudes in the appendices of any elementary aerodynamics book, and you can also compute these properties at the Aerospaceweb.org Atmospheric Properties Calculator. All you need to do is enter the altitude you want and the script will compute the temperature, density, pressure, speed of sound, and other properties at that height. The other input values, like velocity and reference length, are not necessary unless you want to perform the rest of the calculations available.

V English:

ft/s

Metric:

m/s aircraft velocity

The form of velocity applicable to the lift equation is the true airspeed. True airspeed is defined as the actual speed of the aircraft through the air and includes corrections for density, compresibility, and instrumentation error.

Sref English:

ft2

Metric:

m2 reference area

For an airplane, the reference area is the area of the wing when viewed from overhead. This can be somewhat confusing because it includes the area of the wing if it were extended through the fuselage:



For a helicopter, reference area is the rotor disk area, or the area of the circle through which the rotor blades turn:



These values are provided for many aircraft at The Aircraft Museum.


CL - coefficient of lift

This variable is a non-dimensional value that changes with speed as well as angle of attack and is dependent on the aircraft. Although CL is usually determined from wind tunnel experiments (or from computational methods that are beyond the scope of this question), the lift coefficient can be estimated fairly accurately for most aircraft. The following graph compares wind tunnel data for two actual aircraft. One set is based on the Cessna 172, a single-engine piston-powered plane that flies at low speeds. The other set of data is for the BAC Lightning, a Mach 2 jet-powered British fighter with a wing sweep of 60°. Planforms of these aircraft are shown along with the lift coefficient graphs.



Most aircraft will be behave similarly to the Cessna 172 while high-speed planes with short wingspans, like fighters, will more closely resemble the Lightning data. Unfortunately, I haven't been able to find any comparable results for helicopters.

What is the difference between airspeed, true airspeed, equivalent airspeed, calibrated airspeed, indicated airspeed, and ground speed? Which of these speeds does a pilot see in the cockpit?



Speed is probably one of the most misunderstood concepts in aviation because of the confusion over its many forms. We previously addressed this subject in part in an article explaining the difference between airspeed and ground speed. Although it is not explicitly stated in that article, the form of airspeed it addresses is the true airspeed. The other types of airspeed mentioned in the question are related to how the speed is measured and corrected by instruments aboard the aircraft. These speeds are typically expressed in the unit of knots.



Example of a pitot tube mounted on the wing of a Cessna 172

A key device you will see on the nose or wing of most aircraft is called a pitot tube. The purpose of this instrument is to measure the dynamic pressure, sometimes called the ram or impact pressure, the plane experiences as it moves through the air. Aircraft also take another measurement of the atmospheric pressure that is called the static pressure. The difference between the dynamic and static pressures is used to determine the indicated airspeed (IAS) that is displayed to the pilot on the airspeed indicator in the cockpit. This display is usually shown in KIAS which stands for Knots Indicated Air Speed.



Diagram of a pitot-static airspeed measurement system

However, the indicated airspeed is not always completely accurate. Errors are often introduced by the design of the measuring instruments, lag in the time it takes for the system to update, or the location of the pressure probes on the aircraft. Although these errors are typically small, they can introduce discrepancies as large as several knots. The most significant source of instrumentation error is that due to the position of the pitot-static probes. This type of error tends to vary primarily with speed and angle of attack since it is difficult to find a location on an aircraft that will always measure static pressure correctly at all combinations of these variables. The position error is usually greatest at low speeds and high angles of attack, like those encountered during takeoff and landing, but smallest during cruise flight.



Example of a position error calibration chart to convert IAS to CAS for a specific aircraft

To correct for these errors, manufacturers provide an airspeed calibration chart for each aircraft. This chart allows a pilot to correct for the discrepancies and calculate the calibrated airspeed (CAS). Calibrated airspeed is defined as the indicated airspeed corrected for instrumentation errors in the pitot-static pressure measurement system. Many modern aircraft correct for these errors internally and automatically display the CAS, instead of IAS, on the airspeed indicator gauge as in the example pictured below. Regardless, the instrumentation errors are typically small such that IAS and CAS are very close.



Typical airspeed indicator gauge from an aircraft cockpit

Another source of error in airspeed measurement is independent of the measuring instruments but due to an aerodynamic effect called compressibility. When flying faster than about 200 knots, the air being rammed into the pitot tube becomes compressed or squeezed to a higher pressure than it would if the fluid were an ideal incompressible substance. This compressibility error increases the faster the aircraft flies and grows particularly large near Mach 1. It is because of this effect that many World War II pilots like Hans Mutke believed they had flown faster than the speed of sound even though their planes were not actually capable of doing so.

Like instrumentation error, the compressibility error can also be accounted for using an airspeed correction chart. The result of this correction is the equivalent airspeed (EAS). The faster and higher an aircraft flies, the larger the correction becomes and the greater the difference between CAS and EAS. Equivalent airspeed is defined as the speed at sea level that would produce the same dynamic pressure as the true airspeed at the altitude the vehicle is flying at. For example, if a plane is flying at 500 KTAS (Knots True Air Speed) at 20,000 ft, the true dynamic pressure is 451 lb/ft² and the equivalent airspeed is 365 KEAS (Knots Equivalent Air Speed). Conversely, a true airspeed of 365 KTAS at an altitude of 0 ft results in a dynamic pressure of 451 lb/ft².



Compressibility correction chart to convert CAS to EAS at different altitudes

The final source of error that must be accounted for is the decrease in air density that a plane experiences the higher it flies. At very high altitudes, the density eventually drops to near zero. A vehicle in orbit, for example, typically travels at a speed of about 17,000 mph (27,355 km/h). At this extreme altitude, however, there is very little air to ram into a pitot tube so the indicated, calibrated, and equivalent airspeeds will be virtually zero.

This density error can be corrected for if the pilot knows the atmospheric density at the plane's current altitude. This piece of information is typically calculated using the static pressure provided by the pitot-static system and the atmospheric temperature. Once density is known, the equivalent airspeed can be converted to true airspeed (TAS). Another approach to estimate the true airspeed is a rule of thumb technique based on the indicated airspeed and altitude. In this approach, the pilot simply increases the IAS by two percent for every thousand feet of altitude to approximate the TAS. Many airspeed indicators also include a sliding scale to perform this conversion for the pilot and provide the TAS. You can perform many of these airspeed conversions yourself using theAtmospheric Properties Calculator on this site.

True airspeed is the actual speed of an aircraft with respect to the air through which it flies. This speed is what determines the aerodynamic behaviors of an aircraft such as its Mach number, lift, and drag. True airspeed is also important for navigation purposes since it is related to how quickly the aircraft will reach its destination. The other critical factor that plays a role in navigation is the wind speed. The winds add to or subtract from the true airspeed to determine the ground speed, which is ultimately the speed that a pilot wants to know for navigation. Ground speed is the speed of an aircraft with respect to a fixed point on the Earth's surface.

However, there is no direct method for an aircraft to measure the speed of the winds in the air mass through which it flies. The determination of ground speed instead relies on some external source like radio beacons or the Global Positioning System (GPS). These systems allow the position of an aircraft to be determined at any point in time during its flight. By calculating the rate of change in its position, it is possible to determine the speed of the plane along the ground. The difference between airspeed and ground speed can then be used to estimate the wind speed.

The speeds a pilot typically sees in the cockpit are either the indicated or calibrated airspeeds, depending on what is displayed on the airspeed indicator gauge, and the ground speed if the aircraft is equipped with some kind of GPS receiver. While a pilot has the means to calculate equivalent and true airspeeds, these are primarily intermediate results that are not typically needed in and of themselves.



Explanations of the markings on a typical airspeed indiactor

The airspeed indicator in the cockpit is also covered in color-coded markings that provide the pilot with important information to operate the plane safely. These markings denote regions where the plane can be safely operated and limits that should not be exceeded. The purpose of the markings used on a typical single-engine light plane is described below.

• Flap Operating Range - The white arc specifies the speed range over which the wing flaps can be extended.

• Power-Off Stall Speed with Flaps Extended and Landing Gear Deployed (VS0) - The lower limit of the white arc denotes the minimum speed at which the plane can be flown in landing configuration without stalling. Once stalled, the wing rapidly loses lift and the aircraft becomes difficult or impossible for the pilot to control.

• Maximum Flaps Extended Speed (VFE) - The upper limit of the white arc dictates the maximum airspeed at which the flaps can be fully extended. Higher speeds will induce loads on the flaps that may exceed their structural limits resulting in damage to the aircraft.

• Normal Operating Range - The green arc specifies the range of airspeeds over which the plane can be safely flown in cruise configuration with flaps and landing gear retracted.

• Power-Off Stall Speed with Flaps and Landing Gear Retracted (VS1) - The lower limit of the green arc denotes the minimum speed at which the plane can be flown in cruise configuration without stalling.

• Maximum Normal Operation Speed (VNO) - The upper limit of the green arc dictates the maximum cruise speed in a clean configuration.

• Caution Range - The yellow arc specifies a speed range the pilot should avoid unless flying in very smooth air.

• Never Exceed Speed (VNE) - The red line dictates the maximum speed at which the plane can be operated in smooth air. The pilot should never fly faster than this airspeed or risk structural damage.