Atmosphere, aerodynamics, and propulsion

Atmosphere, Aerodynamics, and Propulsion
In the previous chapter, functional relations were presented for the aerodynamic and propulsion terms appearing in the equations of motion used for trajectory analysis. It is the intention of this chapter to verify these relations as well as to present a procedure for estimating the aerodynamic characteristics of a subsonic jet airplane. Engine data isassumed to be provided by the manufacturer, so that no estimation of propulsion terms is attempted. First, a standard atmosphere is defined and approximated by an exponential atmosphere. Then, aerodynamics is discussed functionally, and an algorithm for estimating the angle of attack and the drag polar of a subsonic airplane at moderate values of the lift coefficient is presented. Because the graphs ofthese quantities have simple geometric forms, approximate aerodynamic formulas are developed. All of the aerodynamics figures are for the SBJ of App. A. Finally, data for a subsonic turbojet and turbofan are presented, and the propulsion terms are discussed functionally. Because of the behavior of these engines, approximate formulas can be developed.

3.1

Standard Atmosphere

The realatmosphere is in motion with respect to the earth, and its properties are a function of position (longitude, latitude, and altitude)

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Chapter 3. Atmosphere, Aerodynamics, and Propulsion

and time. From an operational point of view, it is necessary to have this information, at least in the region of operation. However, from a design point of view, that is, when comparing the performance oftwo aircraft, it is only necessary that the atmospheric conditions be characteristic of the real atmosphere and be the same for the two airplanes. Hence, it is not important to consider the motion of the atmosphere or to vary its characteristics with respect to longitude and latitude. A simple model in which atmospheric properties vary with altitude is sufficient. There are two basic equations whichmust be satisfied by air at rest: the aerostatic equation dp = −ρg dh and the equation of state for a perfect gas p = ρRτ (3.2) (3.1)

where p is the pressure, ρ the density, R the gas constant for air, and τ the absolute temperature. For the region of the atmosphere where airplanes normally operate, the acceleration of gravity and the composition of air can be assumed constant (g = 32.174 ft/s2and R = 1716.5 ft2 /s2 ◦ R). To complete the system of equations defining the standard atmosphere, it is assumed that the temperature is a known function of the altitude. Actual measurements of atmospheric properties using balloons and sounding rockets have shown that the atmosphere can be approximated by a number of layers in which the temperature varies linearly with the altitude, that is, thetemperature gradient β = dτ /dh is constant. The assumed temperature profile for the first three layers of the 1962 U.S. Standard Atmosphere (Ref. An) is shown in Fig. 3.1. Note that the layer of the atmosphere closest to the earth (0 ≤ h ≤ 36089 ft) is called the troposphere; the next two layers (36089 ≤ h ≤ 104,990 ft) are part of the stratosphere; and the dividing line between the troposphere andthe stratosphere is called the tropopause. Because of the assumed temperature profile, the equations defining temperature, pressure, and density can be written as dτ = β dh dp/p = −(g/R)dh/τ dρ/ρ = −(g/R + β)dh/τ (3.3)

3.1. Standard Atmosphere

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120 104,990 ft β = 5.4864Ε−4 65,617 ft β=0 40 β = −3.5662Ε−3 0 36,089 ft

154,200 ft

80 hx (ft) 10-3

Stratosphere

Tropopause Troposphere300

400 τ (deg R)

500

600

Figure 3.1: Temperature Distribution - 1962 U.S. Standard Atmosphere where β is a constant for each layer of the atmosphere. For the troposphere (β = −3.5662E−3 ◦ R/ft), these equations can be integrated to obtain τ = 518.69 − 3.5662E−3 h p = 1.1376E−11 τ 5.2560 ρ = 6.6277E−15 τ 4.2560 where the standard sea level conditions τs = 518.69 ◦ R, ps = 2116.2...