Mach Number

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  Speed of Light and Speed of Sound (Concepts and Applications)

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  This is somewhat reminiscent of the early modern ocean sounding unit mark (a synonym for fathom), which was also unit-first, and may have influenced the use of the term Mach. In the decade preceding faster-than-sound human flight, aeronautical engineers referred to the speed of sound as Mach's number , never Mach 1. Overview The Mach Number is commonly used both with objects traveling at high speed in a fluid, and with high-speed fluid flows inside channels such as nozzles, diffusers or wind tunnels. As it is defined as a ratio of two speeds, it is a dimensionless number. At a temperature of 15 degrees Celsius the speed of sound is 340.3 m/s (1225 km/h, or 761.2 mph, or 661.5 knots, or 1116 ft/s) in the Earth's atmosphere. The speed represented by Mach 1 is not a constant; for example, it is mostly dependent on temperature and atmospheric composition and largely independent of pressure. In the stratosphere, where the temperatures are constant, it does not vary with altitude even though the air pressure changes significantly with altitude. Since the speed of sound increases as the temperature increases, the actual speed of an object traveling at Mach 1 will depend on the fluid temperature around it. Mach Number ________________________ WORLD TECHNOLOGIES ________________________ is useful because the fluid behaves in a similar way at the same Mach Number. So, an aircraft traveling at Mach 1 at 20°C or 68°F will experience shock waves in much the same manner as when it is traveling at Mach 1 at 11,000 m (36,000 ft) at -50°C or -58F, even though it is traveling at only 86% of its speed at higher temperature like 20°C or 68°F. High-speed flow around objects Flight can be roughly classified in six categories: Regime Subsonic Transonic Sonic Supers onic Hypersonic High-hypersonic Mach <0.75 0.75–1.2 1.0 1.2–5.0 5.0–10.0 >10.0 For comparison: the required speed for low Earth orbit is approximately 7.5 km/s = Mach 25.4 in air at high altitudes.

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  Aviation Maintenance Technician: Airframe, Volume 1

  Structures

  • Dale Crane(Author)
  • 2008(Publication Date)
  • Aviation Supplies & Academics, Inc.

    (Publisher)

  Figure 1-41. The speed of sound in the air varies with the air temperature. Mach Number. A measurement of speed based on the ratio of the speed of the aircraft to the speed of sound under the same atmospheric conditions. An airplane flying at Mach 1 is flying at the speed of sound. 39 B ASIC A ERODYNAMICS Chapter 1 Flight Speed Ranges High-speed flight can be divided into four speed ranges: Subsonic — Below Mach 0.75 All airflow is below the speed of sound. Transonic — Mach 0.75 to Mach 1.20 Most of the airflow is subsonic, but in some areas, it is supersonic. Supersonic — Mach 1.20 to Mach 5.00 All of the airflow is faster than the speed of sound. Hypersonic — Greater than Mach 5.00 Subsonic Flight In low-speed flight, air is considered to be incompressible, and acts in much the same way as a liquid. It can undergo changes in pressure without any appreciable change in its density. But in high-speed flight the air acts as a compressible fluid, and its density changes with changes in its pressure and velocity. An airplane passing through the air creates pressure disturbances that surround it. When the airplane is flying at a speed below the speed of sound, these disturbances move out in all directions and the air immediately ahead of the airplane is affected and its direction changes before the air reaches the surface. This subsonic airflow pattern is shown in Figure 1-42. At speeds greater than the speed of sound, the disturbances do not spread out ahead of the airplane, and there is no change in flow direction ahead of the leading edge. subsonic flight. Flight at an airspeed in which all air flowing over the aircraft is moving at a speed below the speed of sound. transonic flight. Flight at an airspeed in which some air flowing over the aircraft is moving at a speed below the speed of sound, and other air is moving at a speed greater than the speed of sound. hypersonic speed. Speed of greater than Mach 5 (5 times the speed of sound).

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  Applied Gas Dynamics

  • Ethirajan Rathakrishnan(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  (1.5) reduces to Δ  i =  i 2 V 2 i E = 1 2 ( V a ) 2 (1.6) The ratio V/a is called the Mach Number M. Therefore, the condition of incompressibility for gases becomes M 2 ∕2 ≪ 1 Thus, the criterion determining the effect of compressibility for gases is that the magnitude of the Mach Number M should be negligibly small. Indeed, mathematics would stipulate this limit as M → 0. But Mach Number zero corresponds to stagnation state. Therefore, in engineering sciences flows with very small Mach Numbers are treated as incompressible. To have a quan- tification of this limiting value of the Mach Number to treat a flow as incompressible, a Mach Number corresponding to a 5% change in flow density is usually taken as the limit. It is widely accepted that compressibility can be neglected when Δ  i ≤ 0.05 or 5% that is when M ≤ 0.3. In other words, the flow may be treated as incompressible when V ≤ 100 m s −1 , that is when V ≤ 360 kmph under standard sea level conditions. The above values of M and V are widely accepted values and they may be re-fixed at different levels, depending upon the flow situation and the degree of accuracy desired. 1.4 Supersonic Flow – What Is it? The Mach Number M is defined as the ratio of the local flow speed V to the local speed of sound a M = V a (1.7) Thus, M is a dimensionless quantity. In general, both V and a are functions of position and time. Therefore, the Mach Number is not just the flow speed made nondimensional by dividing Basic Facts 5 by a constant. In other words, the flow Mach Number is the ratio of V to a and this relation should not be viewed as M proportional to V , or inversely proportional to a, in isolation. That is, we cannot write M ∝ V or M ∝ 1/a in isolation. However, it is almost always true that M increases monotonically with V . A flow with a Mach Number greater than unity is termed supersonic flow. In a supersonic flow V > a and the flow upstream of a given point remains unaffected by changes in conditions at that point.

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  Velocity Physics

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  Thus helium molecules (monatomic molecules) travel faster in a sound wave and transmit sound faster. (Sound generally travels at about 70% of the mean molecular speed in gases). Note that in this example we have assumed that temperature is low enough that heat capacities are not influenced by molecular vibration. However, vibrational modes simply cause gammas which decrease toward 1, since vibration modes in a polyatomic gas gives the gas additional ways to store heat which do not affect temperature, and thus do not affect molecular velocity and sound velocity. Thus, the effect of higher temperatures and ________________________ WORLD TECHNOLOGIES ________________________ vibrational heat capacity acts to increase the difference between sound speed in mona-tomic vs. polyatomic molecules, with the speed remaining greater in monatomics. Mach Number Mach Number, a useful quantity in aerodynamics, is the ratio of air speed to the local speed of sound. At altitude, for reasons explained, Mach Number is a function of temperature. Aircraft flight instruments, however, operate using pressure differential to compute Mach Number, not temperature. The assumption is that a particular pressure represents a particular altitude and, therefore, a standard temperature. Aircraft flight instruments need to operate this way because the stagnation pressure sensed by a Pitot tube is dependent on altitude as well as speed. Assuming air to be an ideal gas, the formula to compute Mach Number in a subsonic compressible flow is derived from Bernoulli's equation for M <1: where M is Mach Number q c is dynamic pressure and P is static pressure. The formula to compute Mach Number in a supersonic compressible flow is derived from the Rayleigh Supersonic Pitot equation: where M is Mach Number q c is dynamic pressure measured behind a normal shock P is static pressure. As can be seen, M appears on both sides of the equation.

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  Principles of Flight for Pilots

  • Peter J. Swatton, Peter Belobaba, Jonathan Cooper, Roy Langton, Allan Seabridge(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  (M FS). If the speed of the aeroplane is above the critical Mach Number (M CRIT) a shockwave may well form, even if the M FS is below Mach 1. b. Local Mach Number (M L). This is the ratio of the speed of the airflow at a point on the aeroplane to the speed of sound at the same point. c. Critical Mach Number (M CRIT). As M FS increases so do the local Mach Numbers. M CRIT is that M FS at which any M L has reached unity. It is the lowest speed in the transonic range. d. Critical Drag Rise Mach Number (M CDR). This is the M FS at which because of shockwaves, the C D for a given angle of attack increases significantly. e. Detachment Mach Number (M DET). This is the speed at which the bow shockwave of an accelerating aeroplane attaches to the leading edge of the wing or detaches if the aeroplane is decelerating. f. Indicated Mach Number. TAS is the difference between pitot pressure and static pressure. LSS is a function of static pressure and air density. Because air density is common to both TAS and LSS, both can be expressed as pressure ratios; this is what the Machmeter measures. Indicated Mach Number is therefore the ratio of the dynamic pressure to the static pressure. g. Shock Stall. The airflow over an aeroplane’s wings is disturbed when flying at or near the critical Mach Number. This causes the separation of the boundary layer from the upper surface of the wing behind the shockwave and is the shock stall, which is described in greater detail later in this chapter. h. The Speed of Sound (a). The speed of propagation of a very small pressure disturbance in a fluid in specified conditions. The local speed of sound through the air is equal to 38.94 multiplied by the square root of the absolute temperature (A)

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  Units of Speed and Time

  • (Author)
  • 2014(Publication Date)
  • White Word Publications

    (Publisher)

  Assuming air to be an ideal gas, the formula to compute Mach Number in a subsonic compressible flow is derived from Bernoulli's equation for M <1: where ________________________ WORLD TECHNOLOGIES ________________________ M is Mach Number q c is dynamic pressure and P is static pressure. The formula to compute Mach Number in a supersonic compressible flow is derived from the Rayleigh Supersonic Pitot equation: where M is Mach Number q c is dynamic pressure measured behind a normal shock P is static pressure. As can be seen, M appears on both sides of the equation. The easiest method to solve the supersonic M calculation is to enter both the subsonic and supersonic equations into a computer spreadsheet such as Microsoft Excel, OpenOffice.org Calc, or some equivalent program. First determine if M is indeed greater than 1.0 by calculating M from the subsonic equation. If M is greater than 1.0 at that point, then use the value of M from the subsonic equation as the initial condition in the supersonic equation. Then perform a simple iteration of the supersonic equation, each time using the last computed value of M , until M converges to a value—usually in just a few iterations. Experimental methods A range of different methods exist for the measurement of sound in air. The earliest reasonably accurate estimate of the speed of sound in air was made by William Derham, and acknowledged by Isaac Newton. Derham had a telescope at the top of the tower of the Church of St Laurence in Upminster, England. On a calm day, a synchronized pocket watch would be given to an assistant who would fire a shotgun at a pre-determined time from a conspicuous point some miles away, across the countryside. This could be confirmed by telescope. He then measured the interval between seeing gunsmoke and arrival of the noise using a half-second pendulum. The distance from where the gun was fired was found by triangulation, and simple division (time / distance) provided velocity.

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  Optimal Trajectories in Atmospheric Flight

  • Nguyen Vinh(Author)
  • 2012(Publication Date)
  • Elsevier

    (Publisher)

  CHAPTER 8 SUPERSONIC CRUISE 8. 1 INTRODUCTION The solutions to the different problems analyzed in Chapters 6 and 7 have made explicit the importance in the relationship between the altitude and the speed in optimal performance. Beginning in this chapter, an additional complexity is injected into that relationship. This is in the form of the influence of the Mach Number on aerodynamics and engine characteristics. As has been discussed in Chapter 4, in the range of the speed near and beyond the speed of sound, the aerodynamics and engine characteristics are strongly influenced by the Mach Number defined as M = — (8. 1) a where V is the actual speed and a is the speed of sound. On the other hand, the speed of sound is function of the altitude. Up to the altitude of 11 km , in the troposphere, it decreases as the altitude increases. Between 11km and 20 km, in the stratosphere, it becomes constant. In terms of the density , p , and pressure, p , of the ambient air, the speed of sound is expressed as a 2 = iE (8.2) P where k is the ratio of the specific heats, considered as constant with approxi-mate value equal to 1.4. It is assumed that both the density and the pressure decay exponentially as the altitude increases. But they do not have the same slope in the variation. We recall the equation of state written in differential form 165 166 d£ _ dp dT p p T (8.3) where T is the temperature of the air. Since T varies with the altitude in the troposphere, p is proportional to p only in the stratosphere. If a standard atmosphere is considered, we can use the simple law b (8.4) t-(t) where subscript * denotes the condition at the tropopause and the value of the constant exponent is b = 1.235 in the troposphere and b = 1 in the stratosphere [ 1 ] . Combining Eqs. (8. 2) and (8. 4) we obtain it) ( b -l ) / 2 b (8.5) The equations will be used to relate the different variables V, a,M,p and p in problems where the influence of the Mach Number is encountered.

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  Flight Theory and Aerodynamics

  A Practical Guide for Operational Safety

  • Joseph R. Badick, Brian A. Johnson(Authors)
  • 2021(Publication Date)
  • Wiley-Interscience

    (Publisher)

  The speed of sound is an important factor in the study of high‐speed flight. Small pressure disturbances are caused by all parts of an aircraft as it moves through the air. These disturbances move outward from their source through the air at the speed of sound. A two‐dimensional analogy is that of the ripples on a pond that result when a stone is thrown in the water. The behavior of these disturbances changes at airspeeds near, at, and above the speed of sound, and these changes have profound effects on the flow of air over the wings of aircraft operating at these speeds.

  The speed of sound in air is a function of temperature alone, as can be shown in Eq. 14.1 . The speed of sound varies from 661 kts. at sea level on a standard day (15 °C) to 574 kts. at the tropopause (−56.5 °C); see Table 2.1 for the speed of sound at various altitudes on a standard day.

  (14.1)

  where

  • a = speed of sound in air
  • a0 = speed of sound in air at sea level, standard day (661 kts.)
  • θ = temperature ratio, T/T0

  EXAMPLE

  Calculate the speed of sound at a pressure altitude of 25 000 ft with an outside air temperature of −36 °C.

  Since Eq. 14.1 shows that the speed of sound is only a function of temperature, we can see that the higher in altitude the lower the speed of sound. Because the aircraft’s speed in relation to the speed of sound is so important in high‐speed flight, airspeeds are usually measured as Mach Number. Mach Number is the aircraft’s true airspeed divided by the speed of sound (in the same atmospheric conditions, i.e. local speed of sound):

  (14.2)

  where

  • M = Mach Number
  • V = true airspeed (kts.)
  • a = speed of sound (kts. local speed of sound)

  EXAMPLE

  Using the local speed of sound found in the last example, calculate the Mach Number if that same airplane has a velocity of 375 kts. TAS.

  When an aircraft is flying below the speed of sound, the pressure disturbances will be moving faster than the airplane (ripples on a pond), and those disturbances that travel ahead of the aircraft influence or “warn” the approaching airflow. This “pressure warning” can be observed in a smoke wind tunnel as it causes the upwash well ahead of the wing. This is shown in Figure 14.1

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  Performance of the Jet Transport Airplane

  Analysis Methods, Flight Operations, and Regulations

  • Trevor M. Young, Peter Belobaba, Jonathan Cooper, Allan Seabridge, Peter Belobaba, Jonathan Cooper, Allan Seabridge(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  4 Satellite navigation is based on global networks of satellites in medium-Earth orbit, which transmit radio signals to onboard receivers. The best-known systems are GPS (Global Positioning System), GLONASS (Global Navigation Satellite System), and the more recently developed Galileo system. 6 Distance and Speed  6.4.2 Mach Number Mach Number was introduced in Section 2.5.9; the defining equation (i.e., Equation 2.78) can be used to describe the speed of an airplane: M = V a (6.7) where M is the flight Mach Number (dimensionless); V is the true airspeed of the airplane (typical unit: kt); and a is the speed of sound in the ambient air mass (typical unit: kt). For example, a cruise Mach Number of 0.84 (which, incidentally, is the design cruise speed of the B777) means that the airplane is cruising at 84% of the speed of sound determined for the atmospheric conditions of the air surrounding the airplane (specifically, the outside air temperature). Combining Equations 6.7 and 6.6 enables the true airspeed to be written in a convenient form: V = Ma = Ma 0 √ 𝜃 (6.8) This equation leads to an interesting deduction regarding the velocity–Mach Number relation-ship during a climb. As the air temperature decreases linearly with increasing altitude from sea level to the tropopause in the ISA, it is evident from Equation 2.77 that the speed of sound will also decrease, although not linearly but as a function of √ 𝜃 . In the lower ISA stratosphere, temperature is constant and hence the speed of sound is also constant. A hypothetical constant TAS climb from sea level would thus mean that the Mach Number would progressively increase as the airplane climbs to the tropopause, and thereafter, for flight in the stratosphere, the Mach Number would be constant (climb speeds are discussed later in Section 17.2). . Dynamic Pressure and Equivalent Airspeed 6.5.1 Dynamic Pressure The dynamic pressure of moving fluids is described in Section 2.5.5.

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  Aerothermodynamics and Jet Propulsion

  • Paul G. A. Cizmas(Author)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  In this region, Mach Number varies primarily due to changes in velocity V . In the transonic flow region, the differ- ences between V and a are smaller than either value. In the supersonic flow region, the velocity V and the speed of sound are of comparable magnitudes, although the former is larger than the latter. In this region, the Mach Number varies due to variations in both V and a. In the hypersonic flow region, velocity V is much larger than the speed of sound. Changes of velocity are small while changes of the speed of sound are large. The variation of the Mach Number is due almost exclusively to changes in the speed of sound. 3.3.5 λ Number The Mach Number has two limitations: (1) it is not proportional to the velocity alone because, for a given stagnation temperature, the speed of sound varies with velocity, as shown in (3.21), and (2) at high speeds, the Mach Number tends to infinity, as shown in Fig. 3.5. To avoid these limitations, a new dimensionless velocity, called the λ number (or M ∗ number), is defined as λ := V a cr . The λ number is proportional to velocity alone because the critical speed does not vary with velocity. To find out the bounds of the λ number, let us derive the relation between M and λ. λ can be rewritten as λ 2 = V 2 a 2 cr = V 2 a 2 · a 2 a 2 cr = M 2 a 2 a 2 cr . (3.28) To determine the expression of the a 2 /a 2 cr term, let us use (3.26) in (3.27) such that V 2 + 2 γ − 1 a 2 = 2 γ − 1 a 0 2 = V 2 max = γ + 1 γ − 1 a 2 cr . Dividing this equation by a 2 cr yields V 2 a 2 cr + 2 γ − 1 a 2 a 2 cr = γ + 1 γ − 1 , (3.29) Eliminating the a 2 /a 2 cr term between (3.28) and (3.29) yields λ 2 = γ + 1 2 M 2 1 + γ − 1 2 M 2 (3.30) 68 3 Steady One-Dimensional Gas Dynamics Table 3.1 Comparison between the values of Mach and λ numbers. Mach Number λ number 0 0 <1 <1 1 1 >1 >1 ∞  γ +1 γ −1 and M 2 = 2 γ + 1 λ 2 1 − γ − 1 γ + 1 λ 2 . It is apparent from (3.30) that when M goes to infinity, λ goes to  (γ + 1)/(γ − 1).

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