Introduction to Flight Testing. James W. Gregory
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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 × dhG, and the forces acting upon it (here, hG 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
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 dhG), 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
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), hA is the distance from the center of the earth (defined here as the absolute altitude2),