Kitabı oku: «Introduction to Flight Testing», sayfa 4

McCormick, B.W. (2011). Introduction to Flight Testing and Applied Aerodynamics. Reston, VA: American Institute of Aeronautics and Astronautics.

McCrink, M. H. and Gregory, J. W. (2021). Design and development of a high‐speed UAS for beyond visual line‐of‐sight operations. Journal of Intelligent & Robotic Systems 101: 31. https://doi.org/10.1007/s10846-020-01300-2.

Miller, J. (2001). The X‐planes – X‐1 to X‐45, 3e. Hinckley, UK: Midland Publishing.

Mondt, M.J. (2014). The Tao of Flight Test: Principles to Live By. Boone, IA: J. I. Lord.

Olson, W.M. (2003). Aircraft Performance Flight Testing, AFFTC‐TIH‐99‐01. Edwards AFB, CA: Air Force Flight Test Center.

Peebles, C. (2014). Probing the Sky: Selected NACA Research Airplanes and their Contributions to Flight. Washington, DC: National Aeronautics and Space Administration.

Reader, K. R. and Wilkerson, J. B. (1977). Circulation Control Applied to a High Speed Helicopter Rotor. Report 77‐0024, David W. Taylor Naval Ship Research and Development Center, Bethesda, MD: DTIC accession number ADA146674.

Smith, H.C. (1981). Introduction to Aircraft Flight Test Engineering. Basin, WY: Aviation Maintenance Publishers.

Stinton, D. (1998). Flying Qualities and Flight Testing of the Airplane. Reston, VA: American Institute of Aeronautics and Astronautics.

Stoliker, F.N. (1995). Introduction to Flight Test Engineering, AGARD Flight Test Techniques Series, AGARD‐AG‐300, vol. 14. Neuilly‐sur‐Seine, France: Advisory Group for Aerospace Research and Development.

Stoliker, F., Hoey, B., and Armstrong, J. (1996). Flight Testing at Edwards: Flight Test Engineers' Stories 1946–1975. Lancaster, CA: Flight Test Historical Foundation.

Tavoularis, S. (2005). Measurement in Fluid Mechanics. Cambridge, UK: Cambridge University Press.

Tischler, M.B. and Remple, R.K. (2012). Aircraft and Rotorcraft System Identification, 2e. Reston, VA: American Institute of Aeronautics and Astronautics.

U.S. Code of Federal Regulations. (2021). Airworthiness Standards: Normal Category Airplanes. Title 14, Chapter I, Subchapter C, Part 23 (14 CFR §23). http://www.ecfr.gov (accessed 01 January 2021).

U.S. Air Force Test Pilot School (1986). Performance Phase Textbook, vol. 1, USAF‐TPS‐CUR‐86‐01. Edwards AFB, CA: US Air Force Test Pilot School.

U.S. Naval Test Pilot School. (1977). Fixed Wing Performance: Theory and Flight Test Techniques. USNTPS‐FTM‐No. 104, Patuxent River, MD: Naval Air Test Center.

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Ward, D., Strganac, T., and Niewoehner, R. (2006). Introduction to Flight Test Engineering, 3e, vol. 1. Dubuque, IA: Kendall/Hunt.

Ward, D., Strganac, T., and Niewoehner, R. (2007). Introduction to Flight Test Engineering, vol. 2. Dubuque, IA: Kendall/Hunt.

Warwick, G. (2017). Ohio State pushes speed envelope in UAS Research. Aviation Week and Space Technology 18 (2017): 44.

Wolfe, T. (1979). The Right Stuff. New York: Farrar, Straus and Giroux.

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Young, J.O. (1997). Meeting the Challenge of Supersonic Flight. Edwards AFB, CA: Air Force Flight Test Center.

2
The Flight Environment: Standard Atmosphere

In this chapter, we will discuss the properties of the environment for flight testing – Earth's atmosphere. It is critical to understand the nature of the atmosphere, since aircraft performance depends significantly on the properties of air. For example, the lift produced by the aircraft is proportional to the air density, and the amount of power produced by an internal combustion engine also varies with density. For these two reasons, aircraft performance decreases as density decreases. We will see in this chapter that density decreases with altitude, so key aircraft performance metrics such as takeoff distance, rate of climb, acceleration, etc. all degrade with altitude. Since aircraft performance depends significantly on the local properties of air, we need some way to factor out altitude effects. We also need to be able to predict the performance of an aircraft as a function of altitude, once its baseline performance is known. Thus, we need an agreed‐upon standard definition of the properties of the atmosphere – this is the standard atmosphere. Definition of the standard atmosphere allows us to evaluate and compare aircraft performance in a consistent manner, no matter what the altitude is.

The important atmospheric parameters are the atmospheric temperature, pressure, density, and viscosity, which depend on the distance from the earth surface, geographic location, and time. In order to describe the atmosphere in a universal way, a standard atmosphere model has been developed, where the atmospheric parameters are determined as the univariate functions of altitude from sea level. Temperature exhibits strong variations with time of year, geographic location, and altitude. And, on a daily basis, temperature depends on current weather conditions in a stochastic manner. It is impossible to develop a first‐principles model that will capture all of these parameters that influence the temperature profile; thus, the standard temperature profile is determined from an average of a large ensemble of atmospheric measurements. The variation of pressure with altitude, however, is rigorously described by some basic physical principles – we will derive these here. In fact, pressure is so intricately and reliably linked to altitude that aircraft altimeters measure pressure and convert the measurement to an indicated altitude through the definition of the standard atmosphere. Density is related to the estimated value of temperature and the derived value of pressure via the ideal gas law. Finally, we will provide a relationship that determines the viscosity of air as a function of temperature. Based on these developments, we will define a standard atmosphere that can be expressed in tabular form, or equations coded for computational analysis. This chapter will start with a physical description of the atmosphere and then present a detailed development of the standard atmosphere. Most of the development of the standard atmosphere presented in this chapter will rely on SI units, since this was the unit system used to define the standard atmosphere and the boundaries of atmospheric regions. The input and output of the standard atmosphere can be easily converted from SI units to English units as needed.

2.1 Earth's Atmosphere

Earth's atmosphere is an envelope of air surrounding the planet Earth, where dry air consists of 78.08% nitrogen, 20.95% oxygen, 0.93% argon, 0.031% carbon dioxide, and small amounts of other gases (NOAA et al. 1976). In addition, air contains a small amount of water vapor (about 1%). The entire atmosphere has an air mass of about 5.15 × 10¹⁸ kg (1.13 × 10¹⁹ lb), and three quarters of the total air mass are contained within a layer of about 11 km (∼36,000 ft) from the Earth's surface.

There is a general stratification of Earth's atmosphere, which leads to the definition of distinct regions of the atmosphere: the troposphere (0–11 km), stratosphere (11–50 km), mesosphere (50–85 km), and thermosphere (85–600 km). The atmosphere becomes thinner as the altitude increases, and there is no clear boundary between the atmosphere and outer space. However, the Kármán line has been defined at 100 km and is often used as the border between the atmosphere and outer space. Atmospheric effects become noticeable during atmospheric reentry of spacecraft at an altitude of around 120 km. Aircraft propelled by internal combustion engines and propellers are generally limited to operating in the troposphere, while jet‐propelled aircraft routinely operate in the stratosphere.

Figure 2.1 illustrates the bottom three layers of Earth's atmosphere, which is where all atmospheric flight vehicles conduct flight. The delineation between the various regions of the atmosphere is based on historical measurements of temperature profiles, which lead to distinct regions with different temperature lapse rates. In the troposphere (the layer of the atmosphere nearest the surface), the air temperature generally decreases linearly with the altitude. This temperature reduction with altitude is due to the increasing distance from Earth and a concomitant reduction in heating from Earth's surface. Weather phenomena are directly dependent on this temperature reduction with altitude, causing most storms and other weather phenomena to develop and reside within the troposphere. The dividing boundary between the troposphere and the next layer (the stratosphere) is called the tropopause, at 11 km. Within the lower portion of the stratosphere (11–20 km), the air temperature remains constant; it then increases with altitude in the upper stratosphere (20–50 km), due to absorption of the sun's ultraviolet radiation by ozone in this region of the atmosphere.

Figure 2.1 The layers of Earth's atmosphere.

In contrast with the temperature–altitude profile, the variation of pressure with altitude is highly repeatable and deterministic. Air pressure continually decreases with altitude from Earth's surface all the way to the edge of the atmosphere. The primary reason for this is the action of Earth's gravitational acceleration on air, causing a given mass of air to exert a force on the air below it. Air at a given altitude must support the weight of all of the air mass above it, and it balances this force by pressure. As altitude increases, there is less air mass above that altitude, so there is less force (weight) acting on the air at that point and the pressure decreases. Thus, pressure decreases as altitude increases. We will discuss this physical mechanism in greater detail in Section 2.2, when we derive an expression for the variation of pressure with altitude.

2.2 Standard Atmosphere Model

A standardized model of the atmosphere allows scientists, engineers, and pilots in the flight testing community to have a commonly agreed‐upon definition of the properties of the atmosphere. The definition of the standard atmosphere includes the variation of gravitational acceleration, temperature, pressure, density, and viscosity as a function of altitude. There is actually more than one definition of the standard atmosphere: the U.S. Standard Atmosphere (NOAA et al. 1976) and the International Civil Aviation Organization (ICAO) Standard Atmosphere (ICAO 1993). Thankfully, the two definitions are identical at lower altitudes where aircraft fly – the only differences are in the upper stratosphere and beyond. Our discussion here will generally follow the development of the U.S. Standard Atmosphere (NOAA et al. 1976).¹

2.2.1 Hydrostatics

The development of the standard atmosphere directly results from the hydrostatic equation, which is derived here based on a control volume analysis. Figure 2.2 illustrates an arbitrary control volume, measuring dx × dy × dh_G, and the forces acting upon it (here, h_G is the geometric altitude, or height above mean sea level (MSL)). The forces due to pressure acting on all of the side walls balance one another out in this static equilibrium condition, and we will consider only the forces acting in the vertical direction. The force acting upward on the bottom surface of the control volume is the pressure, p, times the cross‐sectional area dx dy. Similarly, on the top surface, we have a force of (p + dp)dx dy acting downward. (Here, the differential pressure dp accounts for pressure changes in the vertical direction.) Finally, we have the weight of the air inside the control volume acting downward, W = mg, where g is the local gravitational acceleration and the mass of the air inside the control volume can be found from the product of density and the volume,

(2.1)

Figure 2.2 Forces acting on a hydrostatic control volume.

Summing all the forces in the vertical direction and setting equal to zero (from Newton's second law applied to a stationary control volume), we obtain

(2.2)

Canceling terms leads to

(2.3)

which is the hydrostatic equation as a function of geometric altitude. This expression mathematically expresses the physical explanation that we presented earlier for the variation of pressure with altitude. As altitude increases (positive dh_G), the minus sign indicates that the pressure decreases (negative dp). The ρg term is an expression of the weight of the air inside the control volume, which is the reason for the pressure difference.

2.2.2 Gravitational Acceleration and Altitude Definitions

As we proceed with the development of the standard atmosphere, we must consider how gravitational acceleration varies with altitude. From Newton's law of universal gravitation, we know that gravitational acceleration varies inversely with the square of the distance to the center of the earth. Thus, we have

(2.4)

where g is the local gravitational acceleration (varies with altitude), g₀ is the gravitational acceleration at sea level (9.806 65 m/s² or 32.174 ft/s²), h_A is the distance from the center of the earth (defined here as the absolute altitude²), and r_Earth is Earth's mean radius, which is 6356.766 km (NOAA et al. 1976).

Despite the fact that gravity varies with altitude, it is convenient to derive the standard atmosphere based on the assumption of constant gravitational acceleration. In order to do so, we must define a new altitude, the geopotential altitude, h, which we will use in the hydrostatic equation with the assumption of constant gravity. Referring to Eq. (2.3), we can also write the hydrostatic equation as a function of geopotential altitude and constant gravitational acceleration,

(2.5)

Taking the ratio of (2.5) and (2.3), we have

(2.6)

since the differential pressure and density terms cancel out for a given change of pressure. The small difference between g₀ and g then leads to a small difference between the geopotential and geometric altitudes. Combining Eqs. (2.4) and (2.6) produces

(2.7)

which can be integrated between sea level and an arbitrary altitude to find

(2.8)

This expression defines the relationship between geopotential altitude, h, and geometric altitude, h_G, which can also be solved for geometric altitude,

(2.9)

In our derivation of the standard atmosphere, we will use geopotential altitude, h, and assume constant g₀. Properties of the standard atmosphere such as temperature, pressure, and density, i.e., (T, p, ρ), will be found as a function of geopotential altitude, h, and then mapped back to geometric altitude, h_G, by Eq. (2.9). In this work, we are focused on the lower portions of the atmosphere where most aircraft fly (h ≤ 20 km or 65, 617 ft). At that upper altitude limit, Eq. (2.9) predicts a maximum difference of 0.31% between the geometric and geopotential altitude. Thus, in many cases related to flight testing, this difference between geopotential and geometric altitudes can be neglected.

2.2.3 Temperature

Temperature at any given point in the Earth's atmosphere will depend not only on the altitude but also on time of year, latitude, and local weather conditions. Since the variation of temperature has spatial, temporal, and stochastic input, the development of the standard atmosphere as a function of only altitude inherently involves many approximations. Thus, we might anticipate that the actual temperature at a given location can deviate significantly from the standard value.

The standard temperature profile has been determined through an average of significant amounts of data from sounding balloons launched multiple times a day over a period of many years, at locations around the globe. The resulting temperature profile is a function of geopotential altitude, with the lapse rate, a = dT/dh, representing the linear variation of temperature with altitude for each region (see Table 2.1 and Figure 2.3). In the troposphere (0 ≤ h ≤ 11 km), the standard temperature lapse rate is defined as −6.5 K/km, starting at T_SL = 288.15 K. In the lower portion of the stratosphere (11 < h ≤ 20 km), the temperature is presumed to be constant at 216.65 K. Starting at 20 km, the temperature then increases at a rate of 1 K/km due to ozone heating of the upper stratosphere. Based on the data in Table 2.1, we can write expressions for the temperature profile throughout the standard atmosphere as

(2.10)

where “ref” refers to the base of the layer (defined by either sea level conditions, or the top of the prior atmospheric layer, working upwards). Output from Eq. (2.10) can be stacked for each altitude layer, one on top of another, to define the entire standard temperature profile. Since most flight testing, especially for light aircraft and drones, occurs at altitudes below 20 km, we will focus our attention on the troposphere and lower portion of the stratosphere.

Table 2.1 Definition of temperature lapse rates in various regions of the atmosphere.

Source: Data from NOAA et al. 1976.

h₁ and h₂ are the beginning and ending altitudes of each region, respectively.

Figure 2.3 Standard temperature profile.

2.2.4 Viscosity

We also need to define an expression for dynamic viscosity, μ, which depends on temperature. The most significant impact of viscosity is in the definition of Reynolds number,

(2.11)

which is an expression of the ratio of inertial to viscous forces (here, U_∞ is the freestream velocity or airspeed, and c is the mean aerodynamic chord of the wing). Reynolds number has a significant impact on boundary layer development and aerodynamic stall, as we will see in Chapter 12.

The viscosity of air is related to the rate of molecular diffusion, which is a function of temperature (Sutherland 1893). This relationship has been distilled down to Sutherland's Law,

(2.12)

where T is the temperature in absolute units, and β and S_visc are empirical constants, provided in Table 2.2 for both English and SI units (NOAA et al. 1976). Based on Eq. (2.12), the viscosity of a gas increases with increasing temperature. Thus, the dynamic viscosity decreases gradually through the troposphere, starting with the standard sea level value of μ_SL = 1.7894 × 10⁻⁵ kg/(s m) = 3.7372 × 10⁻⁷ slug/(s ft) at a temperature of T_SL = 288.15 K = 518.67 ° R. If kinematic viscosity (ν) is desired instead of dynamic viscosity, it can be found based on its definition,

(2.13)

Table 2.2 Constants used in Sutherland's Law.

Source: Based on NOAA et al. 1976.

2.2.5 Pressure and Density

To derive an expression for the variation of pressure with altitude, we need to integrate the hydrostatic equation. Since density and gravitational acceleration also vary with altitude, we need to cast the hydrostatic equation in terms of only pressure and altitude, with all other variables being constant. We will work with the hydrostatic equation shown in Eq. (2.5), based on geopotential altitude and constant gravity. Density can be expressed as a function of pressure and temperature via the equation of state for a perfect gas,

(2.14)

where R = 287.05 J/(kg K) is the gas constant for air. Taking a ratio of Eqs. (2.5) and (2.14), we have

(2.15)

We will work with Eq. (2.15) for two different cases: first, where the temperature is constant with altitude, and then when temperature varies linearly with altitude.

Equation (2.15) can be directly integrated to find pressure as a function of altitude for the isothermal regions of the atmosphere (11 < h ≤ 20 km and 47 < h ≤ 51 km) since all terms in the equation are constant except pressure and altitude. Performing this integration between the base (h_ref) and an arbitrary altitude within that region (h) yields

(2.16)

where “ref” indicates the base of that particular region of the atmosphere. When the ideal gas law, Eq. (2.14), is applied to isothermal regions of the atmosphere, we see that density is directly proportional to pressure. Thus, we can also write an expression for density in the isothermal regions as

(2.17)

Equations  then form a complete definition of temperature, viscosity, pressure, and density in the isothermal regions of the standard atmosphere.

Portions of the atmosphere with a linear lapse rate, however, require a different approach to integrating Eq. (2.15). In this case, T is no longer constant with respect to altitude, so we must substitute it in the temperature lapse rate. Combining a = dT/dh with Eq. (2.15) yields

(2.18)

Integration of Eq. (2.18) gives the pressure ratio as a function of the temperature ratio, i.e.,

(2.19)

where p_ref and T_ref are the pressure and temperature at a reference altitude, respectively. Again applying the ideal gas law, Eq. (2.14), the density ratio is given by

(2.20)

Thus, for regions of the atmosphere with linear temperature lapse rates, Eqs. (2.10), (2.12), (2.19), and (2.20) form a complete description of the temperature, viscosity, pressure, and density variation with altitude. The reference condition for the base of each region is simply the values corresponding to the top of the previous (lower) region.

In the flight testing community and elsewhere, we often express the above ratios as specific variables referenced to sea level conditions. Temperature ratio, pressure ratio, and density ratio are defined as

(2.21)

In the standard atmosphere, sea level conditions are defined as T_SL = 288.15 K, p_SL = 101.325 kPa, and ρ_SL = 1.225 kg/m³, where the subscript “SL” denotes sea level. The ratios defined in Eq. (2.21) still satisfy the ideal gas law, giving

(2.22)

It is important to bear in mind that these equations are a function of geopotential altitude, which presumes constant gravitational acceleration. If properties are desired as a function of geometric altitude, then the corresponding geometric altitudes can be found by solving for h_G in Eq. (2.9).