“Isn’t it astonishing that all these secrets have been preserved for so many years just so that we could discover them!!”
3.1 DESIGN MOTIVATION The Physics of Aerodynamic Forces
Figure 3.1 shows a cross section view of an aircraft wing. A wing cross section like this is called an airfoil. Lines drawn above and below the airfoil indicate how the air flows around it. The shape of the airfoil and the pattern of airflow around it have profound effects on the lift and drag generated by the wing. Aircraft designers choose a particular airfoil shape for a wing in order to optimize its lift and drag characteristics to suite the requirements for a particular mission. It is essential that an aircraft designer understand how the changes that occur in air as it flows past a wing create lift and drag, and how airfoil shape influences this process.
Figure 3.1. Flowfield Around an Airfoil
The Basis for Airspeed Indication
The changes which occur in the properties of moving air as it encounters obstructions provide the basis for the airspeed indicating systems used on most aircraft. An understanding of how these systems work is essential to anyone who designs, builds, or operates aircraft.
3.2 BASIC AERODYNAMICS The Language
A number of terms must be defined to facilitate a discussion of aerodynamics. The lines in Figure 3.1 which indicate how the air flows are known as streamlines. Each streamline is drawn so that at every point along its length, the local velocity vector is tangent to it. A tube composed of streamlines is called a stream tube. In a steady flow, each streamline will also be the path taken by some particle of air as it moves through the flowfield (a region of air flow). A steady flow is defined as one in which the flow properties (pressure, temperature, density and velocity) at each point in the flowfield do not change with time. If, as in Figure 3.2, a streamline runs into an obstruction, the airflow along the streamline comes to a stop at the obstruction. The point where the flow stops is called a stagnation point, and the streamline leading to the stagnation point is called a stagnation streamline.
Figure 3.2. Stagnation Point and Stagnation Streamline
If, at each point along a streamline, there is no variation in the flow properties in a plane perpendicular to the flow direction, the flow is said to be one-dimensional. Figure 3.3 illustrates a flow that is one-dimensional at stations 1 and 2.
Figure 3.3. Flow Which is One-Dimensional at Station 1 and Station 2 The Continuity Equation
Figure 3.3 depicts a flow in a stream tube. Because the walls of the stream tube are composed of streamlines, the velocity vectors are everywhere tangent to the walls of the tube, so no air can pass through the tube walls. The rate at which mass is flowing through a plane perpendicular to a one-dimensional flow is given by:
where is the mass flow rate and A is the cross-sectional area of the stream tube. In nature, in the absence of nuclear reactions, matter is neither created nor destroyed. Therefore, mass which flows into the tube must either accumulate there or else flow out of the tube again. The case where matter is accumulating in the tube like air filling a balloon is an unsteady, time-varying flow. If the flow is a steady flow, then the rate at which mass is flowing into the tube at station 1 must just equal the rate at which mass is flowing out if the system at station 2:
Equation 3.2 is known as the continuity equation. It is a statement of the law of conservation of mass for fluid flows. Applying this equation to the flowfield shown in Figure 3.3 reveals a phenomenon which is very important to the production of aerodynamic forces. If we assume that the flow is incompressible ( density is constant everywhere in the flowfield ) or at least that the changes in air density are small, then (3.2) makes it obvious that the reduction in stream tube area at station 2 will produce an increase in the velocity there relative to the velocity at station 1. A simple demonstration of this effect occurs when an obstruction such as a person’s thumb is placed over the end of a garden hose which has water flowing out of it. The obstruction of the flow reduces the area of the stream tube and forces the fluid to accelerate in order to maintain the mass flow rate. Figure 3.4 shows a stream tube in a portion of the flowfield around an airfoil. The airfoil is an obstruction to the flow. It reduces the area of the stream tube and forces the flow to speed up as it flows around it. The changes which occur in the properties of the air as it flows past the airfoil produce aerodynamic forces.
Figure 3.4 A Stream Tube in Air Flowing Past an Airfoil
Air flows through a tube which changes cross-sectional area similar to the one illustrated in Figure 3.3. At a point in the tube (Station 1) where the cross-sectional area is 1 m2, the air density is 1.2 kg/m3 and the flow velocity is 120 m/s. At another point in the tube (Station 2) the cross sectional area is .5 m2 and the air density has decreased to 1.0 kg/m3. What is the mass flow rate through the tube and what is the flow velocity at station 2?
Solution: Using (3.1), the mass flow rate is:
Then, solving (3.2) for V2 :
To understand the changes which occur in the flow properties of a fluid as its velocity changes, consider an infinitesimally small particle of air moving along a streamline in a steady flow, as shown in Figure 3.5. A number of forces may act on this particle. Gravity and magnetic fields may exert body forces on it. Viscous shear forces may retard the particle’s motion. Pressure imbalances may also exert a net force. If we consider only flows of relatively lightweight gases which do not have large vertical components and no strong magnetic attractions, then the effects of body forces may be ignored. If we consider only inviscid (frictionless) flows, then viscous shear forces can also be ignored. For such a situation, the only significant forces remaining are due to pressure imbalances along the streamline.
Applying Newton’s second law to the motion of the particle along the streamline, the sum of the forces in the streamwise direction, , is equal to the mass of the fluid particle multiplied by the rate of change of its velocity:
now the volume of the fluid particle is the infinitesimal streamwise distance, ds, multiplied by the area of the perpendicular face, dA, so:
m = r ds dA Also, since the velocity vector is everywhere tangent to the streamline, the direction of ds is everywhere parallel to the local velocity, so:
Equation (3.3) is called Euler’s equation, after the eighteenth-century Swiss mathematician,
who first derived it. The differential equation is a statement of Newton’s second law for a weightless, inviscid fluid. It essentially states that for any increase in a fluid’s velocity, there must be a corresponding decrease in its pressure. Because it relates the rate of change of a fluid’s momentum to the forces acting on it, (3.3) is also known as the momentum equation.
For many purposes, the integral form of (3.3) will be more useful to us. For a compressible fluid, the integral of the right-hand side requires a relationship for density. However, many useful flow problems can be solved with reasonable accuracy by assuming density has a constant value throughout the flowfield. This is an extremely accurate assumption for liquids. It also gives reasonable results for air if the velocities throughout the flowfield remain below 100 m/s or 330 ft/s. With r assumed constant (incompressible flow,) integrate (3.3) from some arbitrary point along the streamline, station 1, to another point, station 2 to yield:
Equation (3.4) is known as Bernouilli’s Equation after another eighteenth-century Swiss mathematician, Daniel Bernouilli. The two terms on each side of Bernouilli’s equation are given descriptive names. The pressure term is called the static pressure. The velocity squared term is called the dynamic pressure, and is often identified by the symbol q.
The sum of static pressure and dynamic pressure is called total pressure. It is identified by the symbol Po. Total pressure in a flow governed by (3.4) is invariant along a streamline.
When using (3.4), it is important to remember that it is only valid for the steady flow along a streamline of an inviscid, incompressible fluid for which body forces are negligible. Together with the continuity equation, Bernouilli’s equation provides the key to understanding such diverse concepts as how wings generate lift and how airspeed indicating systems work.
One of the simpler applications of the aerodynamic equations developed to this point is the analysis and design of common airspeed indicating systems. These systems function by using the relationship between pressure and velocity described by Bernouilli’s equation. Figure 3.6 shows a schematic
Figure 3.6. A Pitot-Static Tube and Manometer The system consists of a Pitot tube, one or more static ports, and a device for indicating differential pressure (a manometer in Figure 3.6.). The Pitot tube is named for its inventor, Henri Pitot, an eighteenth-century French scientist. It is placed in a flowfield with its opening perpendicular to the flow velocity so that if its opposite end were open, air would flow directly through it. Since the opposite end of the Pitot tube is blocked by the differential pressure indicator, the air in the tube cannot flow, and a stagnation point exists at the entrance to the tube. We assume that if we look far enough upstream in the flowfield, the flow becomes essentially undisturbed by the Pitot tube and any shape to which it is attached. The undisturbed flow is called the free stream, and the properties of this undisturbed state are called the free stream conditions. Free stream conditions are usually identified by a subscript infinity, e.g. . Since total pressure is constant along a streamline, the total pressure for the stagnation streamline leading to the stagnation point at the entrance to the Pitot tube is:
Velocity is zero at the stagnation point, so (3.4) requires that the static pressure there is equal to the total pressure. The Pitot tube therefore measures the total pressure of the flow and transmits it to one side of the manometer.
The static ports are oriented parallel to the flow velocity so that no stagnation point develops and the pressure they measure is as close to the free stream static pressure as possible. Aircraft designers use great care in placing static ports, and they often use multiple ports in order to get good approximations to the free stream static pressure. The static ports in Figure 3.6 are placed on the sides of the Pitot tube to form a Pitot-static tube. The static pressure is transmitted through the connecting tube to the other side of the manometer. Solving (3.6) for yields:
A manometer connected to a Pitot-static tube as in Figure 3.7 has a difference in the height of the two collumns of water of 10 cm when the Pitot-static tube is placed in a flow of air at standard sea level conditions. What is the velocity of the airflow?
Solution: In a normally functioning Pitot-static tube, the pressure measured at the static port will always be lower than or equal to the total pressure measured at the stagnation point, so the collumn of water connected to the static port will be higher than the other. Using the manometry equation, with the subscript o identifying total pressure and the subscriptidentifying the freestream static pressure approximated at the static port:
Then, substituting the required values into (3.7):
The manometer or other differential pressure device measures the difference between the total pressure and the static pressure of the free stream. According to (3.6), this difference is the dynamic pressure. If the air density is known, then the dynamic pressure is a direct indication of the free stream velocity. In aircraft, a differential pressure gauge is normally used instead of a manometer. In the differential pressure gauge, the static and total pressure lines are connected to opposite sides of a metal diaphragm. The pressure difference causes the diaphragm to deflect. A linkage connected to the diaphragm moves a needle on the gauge dial when the diaphragm moves. By calibrating the dial scale in terms of velocity instead of pressure, the differential pressure gauge becomes an airspeed indicator. Figure 3.7 shows a schematic of an airspeed indicator connected to a Pitot-static tube.
The airspeed which the needle on the airspeed indicator points at for a given set of flight conditions is called the indicated airspeed. If the airspeed indicator is geared and calibrated based on (3.6), then it is accurate only at speeds below 100 m/s or 330 ft/s where the flow is incompressible. Aircraft built prior to around 1925 operated exclusively at incompressible airspeeds and had incompressible airspeed indicators. Incompressible flow indicators are inaccurate for high speed flight and are no longer used. The Euler equation may be integrated without assuming incompressible flow. The details of this integration go beyond the scope of this text, but the result is a compressible form of Bernouilli’s equation. Virtually all modern airspeed indicators are geared and calibrated to represent the compressible analog of (3.7) which is:
Note that (3.8) is not a simple equation to engineer into a mechanical instrument. In addition, values of rare difficult to measure accurately in flight. For these reasons, it is difficult to build a simple and reliable airspeed indicator based on (3.8). Engineers surmounted this problem, however, by simplifying the equation. Airspeed indicators are manufactured with gears calibrated to use sea level standard atmospheric values of P and r. In effect, an airspeed indicator is calibrated to solve the expression:
where Vc is called the calibrated airspeed. Yet, this is still not what is indicated on the airspeed indicator. The static ports on the aircraft may be located such that they do not accurately measure the freestream static pressure. This is referred to as position or installation error. Additionally, there may be small inaccuracies in the machining of the instrument. To account for these discrepancies, errors are quantified during flight testing and equated to a velocity change ( Vp ) called position error. The relationship between what is displayed on the airspeed indicator (indicated airspeed - Vi) and the calibrated airspeed is given as:
On a perfect airspeed indicator, with zero position error, a pilot reading indicated airspeed would also be reading calibrated airspeed. However, in most cases DVp does not equal zero, and indicated airspeed will be slightly greater or less than calibrated airspeed.
In order to obtain true airspeed (3.8) from calibrated airspeed (3.9), two corrections must be made, one for the actual existing pressure and the other for the actual existing density. Making the pressure correction yields equivalent airspeed which is defined as:
Note that the actual static pressure is used in (3.11), as opposed to the sea level values in (3.9). The ratio between Ve and Vcis generally called the compressibilitycorrection factor and is given the symbol f :
Note that f varies only with (Po -) and. All other variables in (3.13) are constant. can be obtained by setting a standard sea level reference pressure in the aircraft altimeter, and (Po -) can be obtained from knowing the calibrated airspeed. In this manner, a table of f factors such as Table 3.1 can be produced which apply for any aircraft. It is normally more convenient to find a value for f from the table than to evaluate (3.13).
Table 3.1. Compressibility Correction f Factors
Calibrated Airspeed (knots)
For the density correction, observe that:
Since the density ratio is usually less than or equal to 1, is usually Ve. Notice that when flying at sea level on a standard day = 1, and = Ve. Recall that dynamic pressure is given by
Equivalent airspeed may be alternately defined as that airspeed that would produce the same dynamic pressure at sea level as is measured for the given flight conditions. It will become apparent later on in this chapter and in Chapter 4 that, in the absence of compressibility effects, aircraft with identical configurations and orientation to the flow will produce the same aerodynamic forces if the dynamic pressures they are exposed to are the same. Since Ve is a direct measure of dynamic pressure, it is a very useful indicator of an aircraft’s force generating capabilities. This fact is very useful to both engineers and pilots.
It is worthwhile at this point to recapitulate the process for correcting an indicated airspeed. The steps are as follows:
The result, , is called true airspeed. The series of corrections from indicated to calibrated to equivalent to true airspeed is often called an ICeT (“ice tee”) problem, with the lower case e being used as a reminder that equivalent airspeed is usually less than the other airspeeds. However, true airspeed is frequently not very useful until another correction is made. is the magnitude of the aircraft’s true velocity relative to the air mass. However, the air mass itself may be moving relative to the ground. The velocity of the air mass relative to the ground is the wind velocity. This must be added vectorially to the true velocity relative to the air mass in order to determine the aircraft’s ground speed, Vg. Ground speed is the magnitude of the aircraft’s velocity relative to the earth’s surface. To help distinguish between true airspeed and groundspeed, consider the following example: