Introduction to Flight Testing. James W. Gregory

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