Solution
2.3.7 Mach Number and the Regimes of Flight
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Introductory Concepts 129
Examining the data for the trim shot, the most important few seconds, preceding the maneuver, are shown. The altitude is stable at 7000 ft and the airspeed is about 154 knots, about 1 knot less than the target of 155 knots. The pitch angle is stable at about 5∘. Since the test maneuver was an aileron roll, it was critical that the trim shot start at a zero roll angle and with zero roll rate. If these were non-zero at the start of the maneuver, it would be difficult to accurately determine the aircraft roll performance. This again emphasizes the importance of the trim shot. From the data, the wings are level, as indicated by a roll angle of zero and the roll rate is zero. Based on the data, the trim shot looks correct and stable, prior to initiating the roll maneuver.
The roll maneuver is completed in less than two seconds. The roll angle is seen to go from −180 to +180 degrees, indicating a 360-degree roll. The altitude remains constant throughout the roll, while the airspeed increases, indicating that the nose was dropping in the maneuver, as also verified by the decrease in pitch attitude. A parameter of particular interest in assessing roll performance is the roll rate. As is seen by the data, the maximum roll rate was about 250 degrees per second. This maximum roll rate was obtained for a little less than one second of the two second-duration roll, due to the finite amount of time to achieve the roll rate and to recover from the roll. This review of the aileron roll shows the degree of analysis detail that can be obtained from a few, properly selected measurement parameters.
k k same direction. The speed of sound is related to the random motion of the air molecules, which is
a function of the air temperature. The higher the temperature, the more “excited” the air molecules become, increasing their random motion. Therefore, the Mach number can be physically interpreted as the ratio of the directed motion to the random thermal motion of the air molecules.
So, why do we often use the Mach number to quantify how fast we are flying rather than just using the velocity? The answer is that by using the Mach number, we are not only quantifying the magnitude of the flow velocity, but we are also saying something about the physical characteristics of the flow. As we change the Mach number of a flow, there are distinct changes in the physical nature of the flow. Keep this in mind, as we define the various flow regimes based upon specific physical phenomena and characteristics of the flow.
To explore the various flight regimes, let’s think about what happened during your F-18 familiar-ization flight. When you were sitting in the cockpit on the runway at sea level, assume that the air temperature was 59∘F (519∘R, 288 K). The engines were running, so there were sound waves from the engine noise traveling away from the aircraft in all directions at the speed of sound, roughly 760 miles per hour (1200 km/h). This situation is shown in Figure 2.20 for M = 0, where the sound waves form concentric circles emanating from center.
When the F-18 leveled off at 30,000 ft (9100 m), you checked several of the cockpit instruments.
The airspeed indicator read 350 knots (403 mph, 644 km/h), the outside air temperature (OAT) was
−48∘F (412∘R, 229 K), and the Mach indication was about 0.6. As shown in Chapter 3, the speed of sound is proportional to the square root of the temperature, therefore the speed of sound at 30,000 ft, a30K, is given by
a30K aSL =
√ T30K
TSL =
√ 412∘R
519∘R = 0.891 (2.41)
a30K = 0.891 aSL = 0.891(661.6kt) = 589kt (2.42) where T30Kis the air temperature at 30,000 ft, TSLis the air temperature at sea level, and aSLis the speed of sound at sea level. Using Equation (2.40), the Mach number at 30,000 ft, M30K, was
M30K = V30K
a30K = 350 kt
589 kt= 0.594 (2.43)
which agrees with the Mach indication that you read. The aircraft was in the subsonic flow regime, where the velocity is less than the speed of sound and the Mach number is less than one. The sound waves from the engine were still moving at the speed of sound, but since the aircraft had a forward velocity, the sound waves “bunched up” in front of the aircraft and spread out behind the aircraft, as shown in Figure 2.20 for M< 1. In the subsonic flow regime, the flow properties, such
Zone of silence
M = 0 M < 1 M = 1 M > 1
Zone of silence
Zone of silence Zone of action
Zone of action
Figure 2.20 Sound waves patterns corresponding to different Mach numbers.
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Introductory Concepts 131
as the air temperature and pressure, change smoothly and continuously throughout the flow. Later, in Chapter 3, we will see that the air density of the flow remains constant, or nearly so, in much of the subsonic regime and this is called incompressible flow.
Returning to the F-18 flight, the aircraft was leveled off at 30,000 ft, stabilized for a moment, then a level acceleration was performed. Looking through the side of the canopy during the acceleration, you saw some light and dark shadowy lines “dancing” on the wing of the aircraft. Glancing at the airspeed indicator at this moment, you noted that the airspeed was about 530 knots (610 mph, 982 km/h). Let us calculate the Mach number corresponding to this airspeed and altitude. Since the acceleration was performed at a constant altitude of 30,000 ft, the speed of sound is still 589 knots, so that the Mach number is
M30K= V30K
a30K = 530 kt
589 kt = 0.900 (2.44)
During the level acceleration, the Mach number increased from about 0.6 to 0.9, from the sub-sonic to the transub-sonic flow regime. We can define the flow as subsub-sonic below about Mach 0.8 and transonic starting from about Mach 0.8 up to about Mach 1.2. In the transonic flow regime, there is both subsonic, and for the first time, supersonic flow present. Supersonic flow is defined as flow where the Mach number is greater than one. In the transonic mixed flow region, there are localized
“pockets” of the flow that accelerate from a subsonic Mach number to slightly beyond Mach 1, to supersonic flow. The supersonic flow, in these pockets of flow, has to readjust to the overall sub-sonic flow in an abrupt fashion, through what is called a shock wave. The Mach number decreases discontinuously from above Mach 1 to less than 1 through the shock wave. Other thermodynamic flow properties (pressure, temperature, density, etc.) also change discontinuously through the shock wave. In the other parts of the flow, where there are no supersonic pockets of flow, the flow proper-ties change continuously, since the flow is subsonic. The sound wave pattern around the aircraft in transonic flow is similar to that in subsonic flow, except for the small pockets of flow, as we have just described.
As the level acceleration continued in the F-18, the aircraft reached Mach 1, meaning that it was traveling at an airspeed equal to the speed of sound. The aircraft was moving at the same speed as the sound waves from the engine, so that these waves could not travel forward of the aircraft and remained stationary with respect to the aircraft, as shown in Figure 2.20 for M = 1.
The wavefronts of the sound waves overlap or coalesce to form a near perpendicular, dividing line between the upstream region, where the sound waves cannot travel, called the zone of silence, and the downstream region, where the sound can still be heard, called the zone of action.
As the F-18 accelerated past about Mach 1.2, it entered the supersonic flow regime. The aircraft was traveling faster than the sound waves emanating from the engine, so that these waves start to “fall behind”, as shown in Figure 2.20 for M> 1. The wavefronts from the sound waves start forming a conical shock wave around the aircraft, where again sound waves cannot travel upstream past this conical boundary. In the supersonic flow regime, the flow upstream of the shock wave is entirely supersonic, with a Mach number greater than one. The flow properties, such as the pressure and temperature, change discontinuously through shock waves. Unlike subsonic flow, the air density cannot be considered constant, rather the air is compressible in supersonic flow.
Table 2.6 summarizes the different flow regimes that have been discussed, as a function of the Mach number. If you could have flown at much higher Mach numbers in the F-18, you would have reached the hypersonic flow regime beyond about Mach 5. Here, the shock waves form at a steeper angle, with respect to the flow direction and there are larger jumps in the flow properties across these stronger shock waves. At such a high Mach number, the flow has a large amount of kinetic energy. Hypersonic flows are synonymous with high temperature flows, which are discussed further in Chapter 3.
k k Table 2.6 Classification of flight regimes based on Mach number.
Flight regime
Mach number range
Physical flow features
Subsonic M< 0.8 Smoothly changing flow properties Constant density flow (incompressible flow) Acoustic disturbances (sound waves) can propagate upstream
Transonic 0.8 < M < 1.2 Subsonic and supersonic flow present
Local pocket(s) of supersonic flow, terminating in a shock wave
Supersonic 1.2 < M < 5 Shock waves and expansion waves are present in flow Discontinuous flow properties across shock waves Flow density is not constant (compressible flow) Acoustic disturbances (sound waves) cannot propagate upstream
Hypersonic M> 5 Shock waves are closer to a body than for supersonic flow Very high heat transfer
High temperature, chemically reacting flows
Keep in mind that the Mach numbers that bound the different flow regimes are only approximate.
The Mach number where the effects of the different flow regimes are realized may vary, depending on factors such as the vehicle geometry. For instance, a slender body does not disturb the flow as much as a non-slender, thicker body, so that the onset of transonic shock waves on a slender body occurs at a slightly higher Mach number than for a non-slender body.
As a final note, we look at Figure 2.21, a photograph of a bullet in a supersonic flow, obtained using a flow visualization technique that makes the shock waves visible. This was the first photo-graph ever obtained of shock waves in a supersonic flow. The photophoto-graph was taken by the 19th century Austrian physicist Ernst Mach (1838–1916), after whom the Mach number is named. Ernst Mach pioneered many of the principles of supersonic flow and developed optical techniques to visu-alize these flows. The shock wave is clearly visible at the front of the bullet, trailing downstream at an angle. Weaker waves are seen trailing from the body of the bullet and from the turbulent wake behind the bullet.