Introduction to Flight Testing. James W. Gregory

Introduction to Flight Testing

(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) StartFraction p Over p Subscript r e f Baseline EndFraction equals e Superscript minus left-bracket g 0 slash left-parenthesis italic upper R upper T Super Subscript r e f Superscript right-parenthesis right-bracket left-parenthesis h minus h Super Subscript r e f Superscript right-parenthesis Baseline comma

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) StartFraction rho Over rho Subscript r e f Baseline EndFraction equals e Superscript minus left-bracket g 0 slash left-parenthesis italic upper R upper T Super Subscript r e f Superscript right-parenthesis right-bracket left-parenthesis h minus h Super Subscript r e f Superscript right-parenthesis Baseline period

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) StartFraction italic d p Over p EndFraction equals minus StartFraction g 0 Over italic a upper R EndFraction StartFraction italic d upper T Over upper T EndFraction period

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

(2.19) StartFraction p Over p Subscript r e f Baseline EndFraction equals left-parenthesis StartFraction upper T Over upper T Subscript r e f Baseline EndFraction right-parenthesis Superscript minus g 0 slash italic a upper R Baseline comma

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) StartFraction rho Over rho Subscript r e f Baseline EndFraction equals left-parenthesis StartFraction upper T Over upper T Subscript r e f Baseline EndFraction right-parenthesis Superscript minus left-parenthesis 1 plus g 0 slash italic a upper R right-parenthesis Baseline period

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) theta equals StartFraction upper T Over upper T Subscript upper S upper L Baseline EndFraction comma delta equals StartFraction p Over p Subscript upper S upper L Baseline EndFraction comma and sigma equals StartFraction rho Over rho Subscript upper S upper L Baseline EndFraction period

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) delta equals sigma theta period

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).

2.2.6 Operationalizing the Standard Atmosphere

Applying the equations developed above, we can take one of several approaches to implementing the standard atmosphere for flight testing work. Most simply, these equations form the basis for tabulated values of the standard atmosphere, which are tabulated by NOAA et al. (1976) or ICAO (1993). In addition, a limited subset of the U.S. Standard Atmosphere (NOAA et al. 1976) is reproduced in Appendix A. Alternatively, pre‐written standard atmosphere computer codes may be downloaded and used in a straightforward manner. Popular examples include the MATLAB code by Sartorius (2018) or the Fortran code by Carmichael (2018). If these are not suitable for a particular purpose, then custom code can be written, as described below in a form that simplifies the coding.

In the troposphere where the temperature gradient is a = dT/dh = − 6.5 K/km, the temperature distribution in Eq. (2.10) can be expressed as a linear function

(2.23) theta equals 1 minus k h comma

where h is the geopotential altitude and k = 2.256 × 10⁻⁵ m⁻¹ = 6.876 × 10⁻⁶ ft⁻¹ is a decaying rate. According to Eqs. (2.19) and (2.20), the pressure ratio and density ratio in the troposphere (0 ≤ h ≤ 11 km) are given by

(2.24) delta equals theta Superscript n

and

(2.25) sigma equals theta Superscript n minus 1 Baseline comma

where n = − g₀/aR

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