Airspeed Definitions & Measurement

J. Gordon Leishman

doi:https://doi.org/10.15394/eaglepub.2022.1066.n29

35 Airspeed Definitions & Measurement

Introduction

Knowing how fast an aircraft travels through the air is critical to piloting and engineering. An aircraft’s performance characteristics are generally presented in terms of its airspeed and/or Mach number, as well as a function of its flight altitude and in-flight weight. However, determining an aircraft’s airspeed must be done with great care, and the process requires an acute knowledge of aerodynamics and other engineering principles.

Pilots are usually concerned more about the indicated airspeed, or IAS, which is read off the airspeed indicator (ASI) or the electronic flight display (EFIS) in the cockpit. When the IAS can be corrected for mechanical (if any) and static pressure reading errors (always present), it is called the equivalent airspeed or EAS. This particular airspeed is significant because it measures the actual dynamic pressure acting on the aircraft. However, engineers are also concerned with the true airspeed or TAS of the aircraft through the undisturbed air, which will generally not be the same as the IAS or the EAS. True airspeed, TAS, and ground speed (which include the effects of the winds) are also needed by pilots for navigation purposes. Therefore, to avoid confusion and potential misinterpretations of what is called “airspeed,” the proper basis of airspeed measurement and the various definitions of airspeed must be understood, as well as how such measurements are used in engineering and aviation practice.

Learning Objectives

Understand the aerodynamic principles associated with airspeed measurement.
Know the difference between indicated, equivalent, calibrated, and true airspeeds.
Be able to calculate the true airspeed of an aircraft.
Know how to calculate airspeed at higher Mach numbers.

Airspeed & Mach Number

When a flight vehicle’s performance is addressed, its “airspeed” capabilities are critical. This usually means the speed through the air or airspeed and specifically the true airspeed of the aircraft relative to the air, or what is referred to in engineering terms as the quantity “ $V_{\infty}$ .” It is often taken for granted that $V_{\infty}$ is known or can be measured by recording a physical distance covered in a given time as on a terrestrial vehicle, but this is not the case. Instead, the airspeed of an aircraft must be determined using pneumatic measurements such as Pitot probes, and the “true” speed of the aircraft through the air must be obtained by applying the principles of aerodynamics, hydrostatics, and thermodynamics.

A flight vehicle’s achievable airspeed (or airspeed range) is best classified by its corresponding flight Mach number, as shown in the figure below. Recall that the Mach number is the ratio of airspeed to the speed of sound, so the flight Mach number $M_{\infty}$ at any altitude will be given by

(1) $\begin{equation*} M_{\infty} = \frac{V_{\infty}}{a_{\infty}} \end{equation*}$

where $V_{\infty}$ will be the true airspeed and $a_{\infty}$ is the ambient speed of sound at that altitude and temperature.

The flight Mach number is one of the most useful measures of a flight vehicle’s aerodynamics. However, measuring ways to categorize airspeed and Mach numbers requires careful analysis.

The speed of sound depends on the outside air temperature $T_{\infty}$ at any altitude, i.e.,

(2) $\begin{equation*} a_{\infty} = \sqrt{ \gamma R T_{\infty}} \end{equation*}$

so for a given true airspeed, the flight Mach number increases with increasing altitude in the atmosphere. Again, engineers would generally default to using the ISA model for any atmospheric properties needed for calculations unless instructed otherwise. The ISA has the advantage of standardizing definitions, calculations, and measurements because it can be used as a universal basis for instrument calibration, performance standardization, or as another atmospheric reference.

A flight vehicle classified as being low subsonic speed capable will cruise in the range up to $M_{\infty} <$ 0.3 to 0.4, i.e., a predominantly incompressible flow region where the effects of compressibility about the vehicle are minor and would be expected to have a minimal impact on any aspect of the vehicle’s performance. Most general aviation aircraft, rotorcraft, and uncrewed aerial vehicles (UAVs) fly in this range. In this regime, Bernoulli’s equation is helpful for the calculation of airspeed based on measurements of dynamic pressure using a Pitot probe.

A flight vehicle that would be classified as a high subsonic vehicle will fly at Mach numbers up to $M_{\infty} <$ 0.7 to 0.85, which is where many commercial airliners fly, and here compressibility effects would be expected to manifest in some form. In subsonic flow, airspeeds are readily measurable, but the effects of compressibility mean that thermodynamic principles are needed to measure airspeed correctly. Recall that the transonic region occurs approximately when $M_{\infty}$ = 0.8 to 1.2, so most airliners begin to intrude into the transonic region of flight, i.e., $M_{\infty} = 0.8-0.85$ .

Vehicles capable of supersonic flight will fly from $M_{\infty}$ = 1 up to as much as 3, or perhaps higher. Under these conditions, shock waves affect the flow of Pitot probes or other devices used to measure pressure and the determination of airspeed. So again, thermodynamic principles are needed to calculate the aircraft’s speed relative to the air.

Definitions of Airspeeds

When dealing with airplanes (and aircraft, in general), several different speeds may be used, which must now be defined. For example, the aircraft’s speed over the ground differs from its true airspeed through the air. This is because the ground speed depends on the relative magnitude and direction of the winds in the air mass in the direction of flight. Also, the speed read by an airspeed indicator (or the EFIS equivalent) is not the actual or true speed of the aircraft or the ground speed. This may all sound confusing, and it certainly is to new engineers, so the various airspeeds used for aircraft must be understood.

What Does an ASI Measure?

It is best to understand precisely what an airspeed indicator (ASI) measures. An ASI, as shown in the figure below, is one of the most fundamental instruments used for flight, and the readings made by an ASI are used not only for piloting but also for engineering purposes. An ASI is a pressure-measuring (pneumatic) instrument that responds to dynamic pressure $q$ variations with a scale calibrated in speed units. The ASI is calibrated to read airspeed, usually in units of “knots” (kts), which means nautical miles per hour. Units of “knots” are used universally in aviation, but some ASIs may be calibrated in units of km/h (kph).

An airspeed indicator is calibrated in speed units but responds to dynamic pressure. The pressure source is either a pitot-static probe or a pitot probe and a static vent.

Recall that dynamic pressure is the difference between the total or stagnation pressure $p^*$ (the “ram” pressure) and the static pressure $p$ , i.e., $q = p^* - p_s$ . Sometimes, a Pitot-static probe will be used to measure dynamic pressure, but other times, a separate total pressure probe (which measures $q^*$ ) and a static pressure vent placed somewhere on the aircraft’s surface (which measures $p_s$ ) will be used. In either case, the measured pressure difference is the dynamic pressure, $q$ , which the ASI responds to by design.

Analysis

Consider flight at low Mach numbers where the flow can be assumed incompressible. If the total (stagnation) pressure is denoted as $p^*$ , then the use of Bernoulli’s equation (which is valid for incompressible flow) gives that

(3) $\begin{equation*} p_s + \frac{1}{2} \varrho V_{\infty}^2 = p^* \end{equation*}$

Rearranging this latter equation gives the true airspeed (TAS), i.e.,

(4) $\begin{equation*} V_{\rm {\tiny TAS}} = \sqrt{ \frac{p^* - p_s}{2 \varrho}} = \sqrt{ \frac{\Delta p}{2 \varrho_0} \left( \frac{1}{\sigma} \right) } \end{equation*}$

which is exact if the density of the air $\varrho$ is known and the measured value of $\Delta p$ .

Of course, this latter equation is hardly convenient to find TAS even if $p^*$ and $p_s$ are known because $\varrho$ depends on pressure altitude and temperature. Also, a speed scale on the ASI must be associated with any given $\Delta p$ , so a formal calibration is needed against some reference. It will also be apparent that the scale required on the ASI will be non-linear because of the relationship between the value of static pressure and the square of the airspeed (or dynamic pressure) as formalized by the Bernoulli equation.

Basis of Calibration

To resolve these apparent dilemmas, the airspeed measurement on the ASI is referenced to mean sea level (MSL) density conditions by calibrating the speed scale based on the assumption that $\varrho = \varrho_0$ , the corresponding measured airspeed then being known as the equivalent airspeed or EAS, i.e.,

(5) $\begin{equation*} V_{\rm {\tiny EAS}} = \sqrt{ \frac{p^* - p_s }{2 \varrho_0} } = \sqrt{ \frac{\Delta p}{2 \varrho_0} } \end{equation*}$

In other words, then

(6) $\begin{equation*} \frac{1}{2} \varrho_0 V_{\rm {\tiny EAS}}^2 = \frac{1}{2} \varrho V_{\rm \tiny{ TAS}}^2 \end{equation*}$

The true airspeed is then obtained from the calculation by using

(7) $\begin{equation*} V_{\rm {\tiny TAS}} = \frac{V_{\rm{\tiny{EAS}}}}{\sqrt{\displaystyle{\frac{\varrho}{\varrho_0}}}} = \frac{V_{\rm{\tiny{EAS}}}}{\sqrt{\sigma}} \end{equation*}$

which requires the value of $\sigma$ to be evaluated.

Finding Air Density

To find $\sigma$ first requires a measurement of pressure altitude and the outside air temperature and then using the equations of the ISA model. The value of $\sigma$ can be obtained from

(8) $\begin{eqnarray*} \sigma = \frac{\varrho}{\varrho_0} & = &\frac{T_0}{(T_{\rm {\tiny OAT}} +273.16)} \left( 1 - \frac{B h_p}{T_0} \right)^{5.252} \nonumber \\[8pt] & = & \frac{288.16}{(T_{\rm {\tiny OAT}} +273.16)} \left( 1 - \frac{0.001981~h_p}{288.16} \right)^{5.252} \end{eqnarray*}$

where the local pressure altitude, $h_p$ , is in feet and the local $T_{\rm {\tiny OAT}}$ is in units of Celsius ( $^{\circ}$ C) or using

(9) $\begin{equation*} \sigma = \frac{\varrho}{\varrho_0} = \frac{518.4}{(T_{\rm {\tiny OAT}}+459.4)} \left( 1 - \frac{0.001981~h_p}{288.16} \right)^{5.252} \end{equation*}$

where $h_p$ is in feet and $T_{\rm {\tiny OAT}}$ is in units of Fahrenheit ( $^{\circ}$ F). Notice the use of mixed units in some of these equations, which requires care in their proper evaluation. It is a convention in international aviation (per ICAO) that altitudes are measured and reported in feet, but temperatures are reported in units of $^{\circ}$ C.

Summary

The critical point to remember from the preceding is that TAS is not a directly measurable quantity and requires the measurement of EAS followed by a calculation using the value of $\sigma$ to obtain the TAS. Nevertheless, measuring the EAS on the ASI is very convenient from a piloting perspective because the actual dynamic pressure (to which the aircraft responds aerodynamically) is directionally proportional to the square of EAS at a given altitude and temperature. However, it does not solve the problem that engineers generally require TAS, for which a calculation is always needed. Notice that the EAS and TAS will only be equal if the aircraft is flying at MSL and that the EAS measurement is entirely error-free, which is an issue that must now be considered in more detail.

Why is Airspeed Measured in knots?

The “knot” or “knots” is a unit of speed equal to one nautical mile per hour. It is equivalent to 1.15078 mph or 1.852 km/h or 0.514 m/s. The knot’s standard symbol is “kn.” However, using “kt” or “kts” is common, especially in aviation. The symbol “kt” is the specific form recommended by the ICAO. The term knot or knots originally derives from pre-19th century nautical use when sailors would estimate the speed of a ship by counting the number of knots made on a rope that was unspooled behind the ship during a specific time.

Pitot-Static System

Pitot probes (or tubes) and Pitot-static probes are often used to measure flow velocities in general and airspeeds in particular. Consider point 1 upstream of the Pitot tube and point 2 at the entrance to the tube, as shown in the figure below. From Bernoulli’s equation it is known that $p + \dfrac{1}{2}\varrho V^{2} =$ constant along any given streamline, so that

(10) $\begin{equation*} p_{1} + \dfrac{1}{2} \varrho V_{1}^{2} = p_{2} + \dfrac{1}{2} \varrho V_{2}^{2} \end{equation*}$

The dynamic pressure is the difference between the total pressure measured using a Pitot tube and the static pressure measured using a pressure tap.

But at point 2 at the entrance to the Pitot tube, the fluid is brought to rest so $V_{2} = 0$ , i.e., it is a stagnation point, and

(11) $\begin{equation*} p_1 + \dfrac{1}{2} \varrho V_{1}^2 = p_{2} = \mbox{Total pressure at point 2} \end{equation*}$

The static pressure $p_1 = p_{s}$ must be measured to obtain the flow velocity from this latter expression. Like the venturi problem, this static pressure can be measured with a separate static vent or the ports on the outer side of a Pitot-static tube. Solving for $V_{1}$ gives

(12) $\begin{equation*} V_{1} = \sqrt{\frac{ 2\left(p_{2} - p_{s}\right)}{\varrho}} \end{equation*}$

Therefore, the upstream flow velocity can be measured by measuring the difference between a flow’s total and static pressure.

While sometimes a pitot-static probe is used to measure dynamic pressure, in practice, it is found that it can be measured somewhat more accurately by using a separate pitot probe and static vent, as shown in the schematic below. On an airplane, the static vent or source is usually located somewhere on the side of the fuselage. There may also be more than one static vent for redundancy. The pitot probe is set off from the aircraft’s surface into the flow; these pitot probes may be mounted on an airplane’s nose or, in some cases, under the wing. The total pressure is connected to the airspeed indicator (ASI), with the static pressures connected to the altimeter and vertical speed indicator. A heating element is used on the Pitot probe to prevent icing and avoid false ASI readings.

A typical pitot-static system on an airplane provides the reference pressures needed for the airspeed indicator and other pneumatic instruments. Notice, in this case, the use of a pitot probe to measure total pressure and a separate static vent to measure the static reference pressure.

Measuring Total Pressure

The total (ram) pressure can be measured without much error because the probe points directly into the airflow, and the resulting pressure is relatively high compared to the static pressure. The inlet shape of the Pitot tube must be clean and smooth so that there are no pressure losses or other pressure disturbances that could affect the reading. The only exception when a loss occurs might be in cases where the aircraft is significantly yawed with respect to the oncoming airstream, although this would not be a common flight condition.

The photograph below shows an example of a total pressure probe placed on an airplane’s nose. The probe must be well out of the surface boundary layer so it is displaced a short distance from the aircraft’s skin. Such probes must not be located where upstream flow disturbances, such as from propellers, antennas, air scoops, or other probes, are possible.

Pitot probe mounted on the right side of the nose of an airplane.

Measuring Static Pressure

The static pressure is always more challenging to measure accurately because its value is relatively smaller and measured on the airplane’s outer skin, usually at a location where the local static pressure is close to the ambient static pressure. However, there can be no point on the surface where the local static pressure exactly equals the static pressure in the free-stream flow. Therefore, the static pressure measurement available to the ASI is always in error to a lesser or greater degree, which is called the static position error or SPE.

Static vents on the side of the nose of an airplane. Notice the multiple separate vents for redundancy if one or more vents become blocked.

Static Position Error (SPE)

The “error-free” issue in airspeed measurement mentioned previously requires further elaboration because the preceding arguments are predicated on two major points:

The total and static pressure can both be measured accurately.
The ASI can be adequately calibrated against a suitable reference in terms of speed units.

One source of error is the Static Position Error or SPE, which is the error in measuring the local static pressure outside the aircraft relative to the actual static pressure at that altitude and airspeed. If the IAS reading is corrected for the SPE and mechanical errors, the resulting speed is called the calibrated airspeed or CAS. If the SPE and mechanical instrument errors are small, it is sufficient to state that IAS = CAS = EAS. However, it must be appreciated that, in general, the IAS differs from the EAS and the TAS.

The effects of SPE must be measured and, in most cases, corrected for engineering and piloting. This process uses a formal calibration during flight testing with the specific make and model of the airplane. The actual static pressure measurements are compared to the local static pressure measured at the pressure tap(s) on the outside skin of the aircraft. As shown in the figure below, there are a few points on the outside skin of the aircraft where the local static pressure is equal to the true static pressure in which the aircraft is flying. The static pressure port on a prototype airplane is usually repositioned after flight tests so that it is ultimately located where the SPE is minimized as much as possible, which is the location used on production versions of the aircraft. However, the SPE will never be precisely zero.

Several options exist for placing a static pressure port to minimize the SPE, although it will vary somewhat with airspeed and altitude.

In the SPE calibration process, which is performed during flight testing, the actual or true static pressure is measured well behind the aircraft using a trailing cone apparatus, as shown in the photograph below. The SPE will vary somewhat at different airspeeds and altitudes, so the SPE must be mapped out over the entire operating envelope of the aircraft.

The trailing cone method is used to calibrate the pitot-static system on a test aircraft to account for static position errors from the positioning of the static pressure tap(s) on the airframe.

Usually, the SPE is highest at lower airspeeds, lower at higher speeds, and higher at higher altitudes, an example being shown in the figure below. For higher-performance aircraft such as jets, the SPE is likely to be a function of flight Mach number. The process must also be conducted with the aircraft in the “clean” configuration (i.e., flaps and landing gear retracted) and in the “dirty” configuration (i.e., flaps and landing gear down, such as for landing).

An airspeed SPE correction chart. The indicated airspeed may read high or low compared to the equivalent airspeed.

The results obtained and the corrections subsequently derived as the SPE affects the airspeed readings will apply to all production aircraft of the same make and model. The SPE results (or the corrected airspeed values), by regulation, are always included (by FAA regulations) in the aircraft’s flight manual. In the case of a modern airliner, the correction is included in the EFIS as a software correction to airspeed. For piloting purposes, the SPE and mechanical errors are usually small enough to be ignored on many aircraft (perhaps a couple of kts). However, for engineering purposes, all such errors must be accurately accounted for in all aircraft performance analyses.

Mechanical Errors

The mechanical error in the ASI can be obtained in a laboratory setting using a calibration against a reference ASI, which results in a standard mechanical error calibration chart (or table), as shown in the example below. Each instrument error correction (IEC) calibration will apply to a specific ASI, and a different calibration will be needed if the ASI is replaced. The errors are usually small enough to be unimportant for piloting purposes, but they should be included in any data reduction process for engineering work.

A mechanical instrument error correction (IEC) chart for an ASI. The ASI may read high or low, depending on the specific ASI. Each ASI has its own IEC chart.

Check Your Understanding #1 – Correcting an ASI reading

A test aircraft is flying at a pressure altitude of 4,200 ft where the outside air temperature is 68.4 $^{\circ}$ F. The airspeed indicator (ASI) reads 134.5 kts. Calibrations available show the mechanical error of the ASI, $\Delta V_{E}$ , is -0.7 kts, and the static position error, $\Delta V_{\rm SPE}$ , is equivalent to +0.3 knots. Calculate the true airspeed of the aircraft.

Show solution/hide solution.

The ISA ambient temperature at this pressure altitude is

$T = T_0 - B h = 59 - 0.00357 \times 4,200 = 44.0~\mbox{$^{\circ}$F}$

Remember that an airspeed indicator (ASI) is calibrated based on sea-level density ( $\varrho_0$ = 0.002378 slugs ft ${^{-3}}$ ). The equivalent speed (EAS) is

$V_{\rm {\tiny TAS}} = \frac{V_{\rm {\tiny EAS}}}{\sqrt{\sigma}}$

which also requires $\sigma$ (= $\varrho/\varrho_0$ ) to be evaluated. The actual air temperature is 68.4 $^{\circ}$ F, much warmer than standard by 68.4 – 44.0 = 24.39 $^{\circ}$ F. Calculating the air density using the ISA model at this pressure altitude and temperature gives $\varrho$ = 0.0020 slugs ft ${^{-3}}$ .

In this case, first, we need to correct the $V_{\rm {\tiny EAS}}$ for both the instrument error correction (IEC) and for static position error (SPE) to get the calibrated airspeed $V_{\rm CAS}$ so

$V_{\rm CAS} = V_{\rm IAS} - \Delta V_{IEC} - \Delta V_{\rm SPE}$

and from the calibrations, then

$V_{\rm CAS} = 134.5 - (-0.7) - 0.3 = 134.9~\mbox{knots}$

Therefore, the true airspeed will be

$V_{\rm {\tiny TAS}} = \frac{V_{\rm CAS}}{\sqrt{ \sigma} } = V_{\rm CAS} \sqrt{\frac{\varrho_0}{\varrho}} = \sqrt{ \frac{0.002378}{0.0020}} \times 134.9 = 147.1~\mbox{kts} \equiv V_{\infty}$

Airspeed Measurement at Higher Mach Numbers

The issue of airspeed measurement in higher-speed flight where compressibility issues manifest must also be addressed. In this case, using the Bernoulli equation would not be appropriate. In practice, compressibility effects on airspeed measurement are negligible below about 10,000 feet and 200 knots; at higher altitudes, the lower air temperatures decrease the speed of sound, thereby increasing Mach numbers so that compressibility thresholds are met at lower airspeeds.

Remember that airspeed values on the ASI are referenced to MSL density conditions by calibrating the speed scale based on the assumption that $\varrho = \varrho_0$ , then the corresponding equivalent airspeed or EAS is then

(13) $\begin{equation*} V_{\rm {\tiny EAS}} = \sqrt{ \frac{p^* - p }{2 \varrho_0} } = \sqrt{ \frac{\Delta p}{2 \varrho_0}} \end{equation*}$

The ASI still responds as a dynamic pressure measuring instrument. Still, the compressibility of the air affects the reading of total pressure from the Pitot tube, which must be considered.

Subsonic Flight Speeds

For subsonic Mach numbers above 0.3, the total and static pressures can be related using the isentropic thermodynamic relationships. In this case

(14) $\begin{equation*} \frac{p^*}{p_s} = \bigg( 1 + \frac{\gamma - 1}{2} M_{\infty}^2 \bigg)^{\gamma/(\gamma -1)} \end{equation*}$

which can be used to solve for the Mach number, $M_{\infty}$ , i.e.,

(15) $\begin{equation*} M_{\infty} = \sqrt{\left( \frac{2}{\gamma - 1} \right) \bigg( \left( \frac{p^*}{p_s} \right)^{\gamma/(\gamma -1)} - 1 \bigg) } \end{equation*}$

Therefore, the Pitot system now measures flight Mach numbers.

Not only do the total pressure and static pressure need to be measured in higher-speed flight, but the total temperature must also be measured, from which true airspeed can again be obtained by calculation. To this end, the airspeed is then given by

(16) $\begin{equation*} V_{\rm {\tiny TAS}} = a M_{\infty} = \frac{a_s^2 M_{\infty}^2}{1 + \left( \displaystyle{\frac{\gamma-1}{2} } \right) M_{\infty}^2} \end{equation*}$

which requires either the value of the “static” speed of sound $a$ or the “stagnation” speed of sound $a_s$ at any given pressure altitude. As previously discussed, the stagnation temperature or total air temperature (TAT), i.e., $T_T$ , is usually measured during flight so that $a_s$ can be obtained by using

(17) $\begin{equation*} a_s = \sqrt{ \gamma R \, T_T} \end{equation*}$

The EAS and the TAS can then be determined by calculation.

Check Your Understanding #2 – Finding the flight Mach number & TAS

An experimental turboprop airplane is flying at a pressure altitude of 25,000 ft. The Pitot-static system monitored by the test engineers measures the total pressure, $p^*$ , as 30.65 kPa, and the corrected static pressure, $p_s$ , as 23.91 kPa. What are the aircraft’s flight Mach number and true airspeed (TAS)? Assume: 1. ISA standard conditions. 2. No mechanical or SPE errors.

Show solution/hide solution.

Using the compressible flow equations, then

$\frac{p^*}{p_s} = \bigg( 1 + \frac{\gamma - 1}{2} M_{\infty}^2 \bigg)^{\gamma/(\gamma -1)}$

Inserting the measured numerical values gives

$\frac{30,650}{23,910} = 1.282 = \bigg( 1 + \left( \frac{1.4 - 1}{2}\right) M_{\infty}^2 \bigg)^{3.5}$

giving

$\bigg( 1 + 0.3 M_{\infty}^2 \bigg)^{3.5} = 1.24$

Therefore, the flight Mach number of the airplane is

$M_{\infty} = \sqrt{ \frac{1.282^{({\tiny 0.2857})} - 1}{0.2}} = 0.606$

The speed of sound, $a$ , at a pressure altitude of 25,000 ft at ISA conditions is 1,016.1 ft/s. Therefore, the TAS is

$V_{\rm TAS} = a M_{\infty} = 1,016.1 \times 0.606 = 615.8~\mbox{ft/s}= 364~\mbox{kts}$

Note: Determining the actual speed of sound other than for assumed ISA conditions requires measuring outside air temperature.

Supersonic Flight Speeds

For supersonic Mach numbers, a bow shock wave forms ahead of the Pitot probe, as shown in the figure below, further complicating the airspeed measurement issue. Therefore, the flow does not decelerate isentropically because of this shock wave. However, the normal shock equations can be used across a shock wave to find the relationship between the total pressure after the shock, the static pressure, and the Mach number.

A Pitot tube placed in a supersonic flow will produce a shock wave ahead of the stagnation point in the tube, requiring shock wave relationships to obtain total pressure.

The total pressure is

(18) $\begin{equation*} p^* =\bigg( 1 + \frac{\gamma - 1}{2} M_{\infty}^2 \bigg)^{\gamma/(\gamma -1)} \end{equation*}$

and the static pressure is

(19) $\begin{equation*} p_s = \bigg(\frac{2 \gamma}{\gamma + 1} M_{\infty}^2 - \frac{\gamma -1}{\gamma +1}\bigg)^{1/(\gamma -1)} \end{equation*}$

Therefore, the ratio of total to static pressure is

(20) $\begin{equation*} \frac{p^*}{p_s} = \frac{\bigg( 1 + \displaystyle { \frac{\gamma - 1}{2}} M_{\infty}^2 \bigg)^{\gamma/(\gamma -1)} } {\bigg(\displaystyle {\frac{2 \gamma}{\gamma + 1}} M_{\infty}^2 - \frac{\gamma -1}{\gamma +1}\bigg)^{1/(\gamma -1)} } \end{equation*}$

which can be rearranged to give

(21) $\begin{equation*} \frac{p^*}{p_s} = M_{\infty}^2 \left( \frac{\gamma + 1}{2} \right) \bigg( \frac{ (\gamma + 1)^2 M_{\infty}^2}{4 \gamma M_{\infty}^2 - 2 (\gamma -1)} \bigg)^{1/(\gamma -1)} \end{equation*}$

This is called the Rayleigh pitot tube formula. Because the flight Mach number is an implicit function of the total and static pressures in the preceding equation, it must be calculated iteratively. Again, the value obtained is predicated on the assumption that the total and static pressures can be measured accurately and any corrections needed, such as SPE, can be applied.

Summary of Airspeed Definitions

The following definitions of airspeeds in aviation may be encountered:

1. TAS is referred to as “True airspeed,” which is the actual speed of an aircraft through the air relative to an undisturbed air mass. In engineering work, this is called “ $V_{\infty}$ .”

2. IAS is called “Indicated airspeed,” which is the speed shown on an airspeed indicator or ASI. IAS differs from TAS because the ASI is calibrated based on MSL conditions, i.e., $\varrho = \varrho_0$ .

3. CAS is “Calibrated airspeed,” which is the indicated airspeed corrected for errors such as static position error (i.e., the error in measuring accurate static pressure on the aircraft) and mechanical instrument errors.

4. EAS is called “Equivalent airspeed,” which is the airspeed reading on an error-free ASI corresponding to a given dynamic pressure. The EAS must also be corrected for compressibility effects on the dynamic pressure reading at flight Mach numbers greater than 0.3 to find the aircraft’s TAS.

Summary & Closure

Airspeed measurement is straightforward using a pitot-static system, but precisely what airspeed is used must be understood and carefully qualified, e.g., equivalent airspeed, calibrated airspeed, or true airspeed. Engineers usually require a measurement of the aircraft’s true airspeed, which must be obtained indirectly by measurement and calculation. True airspeed is not a direct measurement. Pilots, however, require the equivalent airspeed (to which the aircraft responds aerodynamically) and the true airspeed (for navigation). Static position errors and mechanical errors in airspeed measurement must also be addressed.

One must also remember that the ASI is calibrated based on mean sea-level (MSL) density, i.e., the calibration is performed by assuming that $\varrho = \varrho_0$ . For higher subsonic and supersonic airspeeds, the compressibility effects will affect the total and static pressure measured by the Pitot probe, so appropriate corrections (based on isentropic flow relations) are needed to determine the Mach number and true airspeed. At supersonic airspeeds, other issues arise, including the need to relate total and static pressure measurements to relationships for the flow across shock waves.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

An airplane is flying at a pressure altitude of 10,000 ft with an indicated airspeed of 200 kts. What is its true airspeed? Hint: Assume no SPE and an error-free airspeed indicator.
In practical aviation, why does a pilot need the indicated airspeed and the ground speed?
What might be the consequences if an aircraft is found to have a significant static position error? For the pilot? For engineering analysis?
Research the effects of compressibility on airspeed measurement. What happens to the reading from a pitot probe when an aircraft exceeds the speed of sound?
Can you explain the concept of airspeed limitations and how they are indicated on the instrument?
What are the limitations of airspeed indicators and factors that can affect their accuracy?
Can you describe the effects of airspeed on aircraft performance?
How are airspeed indicators calibrated and tested for accuracy?
Air France 447 Accident: Explain how a failed Pitot tube resulted in the loss of control of the aircraft.

Other Useful Online Resources

Great video on “The Airspeed Indicator & Types of Airspeed.”
This video explains how a barometric altimeter works and how the readings are interpreted.
To learn more about the evolution of aircraft instruments, read “70 Years of Flight Instruments and Displays.”
An excellent discussion on airdata measurement and calibration by NASA.

License

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2024 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n29