James W. Gregory Introduction to Flight Testing

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 TSL = 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 Svisc 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 μSL = 1.7894 × 10−5 kg/(s m) = 3.7372 × 10−7 slug/(s ft) at a temperature of TSL = 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 (href) 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 pref and Tref 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 TSL = 288.15 K, pSL = 101.325 kPa, and ρSL = 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 hG 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)

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)

and

(2.25)

where n =  − g0/aR = 5.2559.

Figure 2.4 The normalized temperature, pressure, and density distributions in the standard atmosphere.

In the lower stratosphere (11 km < h ≤ 20 km), the atmospheric temperature is constant at 216.65 K. If we define the critical altitude at the tropopause to be htrop = 11 km, then the temperature and pressure ratios at the tropopause are θtrop = 0.7518 and δtrop = 0.2233, respectively. Recasting (2.16) in terms of these ratios, we obtain

(2.26)

for the pressure ratio in the lower stratosphere. Finally, the density ratio in the lower stratosphere is simply found by the ideal gas law,

(2.27)

Figure 2.4 shows the pressure, density, and temperature distributions normalized by the sea level conditions in the standard atmosphere.

2.2.7 Comparison with Experimental Data

The above equations describe the idealized atmosphere where the parameters are considered as the mean values of the measured quantities. However, as indicated in The U.S. Standard Atmosphere (NOAA et al. 1976), measurement data show considerable variations of the atmospheric parameters depending on time (day and season) and geographic location, which should be considered in flight testing.

Experimental measurements may be compared with the theoretical variation of pressure and temperature derived from the standard atmosphere. Atmospheric data can be collected by a weather balloon (Figure 2.5), which ascends through the atmosphere and measures pressure and temperature throughout the flight. For the case presented here, the balloon ascended to an altitude of 30.161 km (98,953 ft) before bursting and descending via parachute back to Earth. Data throughout the ascent and descent were collected and are presented in Figures 2.6 and 2.7. The temperature data shown in Figure 2.6 show similar trends to the standard temperature profile, but the agreement is not very good. This is not surprising, since the details of the temperature profile are strongly dependent on local weather, time of year, latitude, etc. However, some of the similarities are noteworthy: the experimental temperature lapse rate is approximately the same as the standard lapse rate, particularly at low altitudes. Also, the location of the tropopause, corresponding to a change to an isothermal temperature profile, is in good agreement. Finally, the slope of the high‐altitude lapse rate is also in fairly good agreement with the standard temperature profile. Pressure data, in Figure 2.7, show excellent agreement with the standard atmosphere. This is also expected since the hydrostatic equation is a good descriptor of the physics of pressure variation with altitude. The good agreement shown here underscores the utility of using pressure measurement for measuring altitude on aircraft (see Chapter 3 for further details on how altimeters operate).

Figure 2.5 Launch of a high‐altitude weather balloon from the oval of The Ohio State University.

Figure 2.6 Comparison of the standard atmosphere with temperature data measured by a weather balloon.

Figure 2.7 Comparison of the standard atmosphere with pressure data measured by a weather balloon.

2.3 Altitudes Used in Aviation

We will now conclude this chapter with a discussion of different altitude definitions used in aviation. We have already introduced several definitions of altitude for the preceding discussion on the standard atmosphere. To recap, these include absolute altitude, geometric altitude, and geopotential altitude. Absolute altitude, hA, is measured from the center of the Earth and is only relevant when determining the value of gravitational acceleration at a particular altitude. Geometric altitude, hG, is the height of an aircraft above mean sea level. And, geopotential altitude, h, is the height above sea level with the assumption of constant gravitational acceleration. Geopotential altitude is only relevant in the context of deriving the standard atmosphere, so should not be used elsewhere. For the remainder of this book, we will presume that the differences between geometric altitude and geopotential altitude are small and will simply refer to the geometric altitude as h.

However, these altitude definitions are limited to an engineering context. To make things interesting, we also have a set of altitudes that are defined for the aviation community. And, to make things more interesting, some of the aviation altitudes use the same terms but different definitions! The aviation set of altitudes include true altitude, indicated altitude, pressure altitude, density altitude, and absolute altitude. We will discuss each of these as follows.

True altitude is the height above mean sea level. In the aviation community, this altitude is often abbreviated as MSL. When an aircraft altimeter is referenced to the local barometric pressure reading, it indicates true altitude. (Details on altimeter operation are provided in Chapter 3.) Pilots around the world often refer to this setting of the altimeter as QNH. Note that the aviation definition of true altitude is identical to the engineering definition of geometric altitude.

Similarly, indicated altitude is a direct reading from the altimeter, no matter how the altimeter is set. This may or may not be the same as true altitude, depending on the reference pressure used on the altimeter. (The reference pressure essentially shifts the calibration of the altimeter to match local barometric pressure, instead of standard sea level pressure.)

Pressure altitude, in the aviation realm, is defined as the altitude read from the altimeter when it is set to a reference pressure of 29.92 inHg, which is the standard sea level pressure. In many locations around the world, barometric pressure readings are reported in millibars or hPa, where 1013 mbar (=1013 hPa) is equal to 29.92 inHg (both being standard sea level pressure). Pilots refer to this setting of the altimeter – to provide pressure altitude – as QNE. In engineering terms, pressure altitude has essentially the same meaning. An engineer would express pressure altitude as the altitude corresponding to a given pressure in the standard atmosphere. Both the engineering and aviation definitions for pressure altitude are equivalent, since the altimeter referenced to 29.92 inHg (1013 mbar) is calibrated based on the standard atmosphere.

Density altitude is defined, in aviation terms, as the pressure altitude corrected for non‐standard temperature. If the temperature on a given day at a particular altitude is hotter than the standard temperature, then the density altitude will be higher. In engineering terms, density altitude is defined as the altitude corresponding to a given density in the standard atmosphere. Aircraft performance depends significantly on local air density, so density altitude is a direct indication of aircraft performance. Higher density altitude (corresponding to lower density) will lead to longer takeoff ground roll, slower rates of climb, higher true airspeed for stall, etc.

Finally, pilots are also concerned with the height of the aircraft above the local terrain, which is termed absolute altitude. In the aviation realm, absolute altitude is often also termed height above ground level, so the acronym AGL is often used. On aviation charts, both true altitude (MSL) and absolute altitude (AGL) are reported for various obstacles. For example, the top of a 500‐ft tall radio tower mounted on ground that is 1500 ft above sea level will have a maximum height of 2000 ft MSL or 500 ft AGL. Thus, pilots pay close attention to the absolute altitude (also referred to as QFE) as well as the true altitude (QNH). Note that absolute altitude (AGL) in an aviation context is not the same as absolute altitude (hA) in an engineering context. The engineering definition of absolute altitude is seldom used in aerospace or aviation, outside of discussions of the standard atmosphere.

A selection of the most significant of these altitudes is illustrated in Figure 2.8. The aircraft depicted in this figure is cruising at a true altitude of 5000 ft (MSL), but because its flight is over mountains rising 2000 ft above sea level, the aircraft is at an absolute altitude of only 3000 ft AGL. On this given day, the local barometric pressure reading is lower than standard, causing the pressure altitude to be higher than the true altitude. And, if the temperature on this day is higher than standard, then the density altitude will be even higher than pressure altitude or true altitude. Thus, we could easily have a situation where absolute altitude, true altitude, pressure altitude, and density altitude are all different. In Chapter 3, as we move into instrumentation used for flight testing, we will discuss the operation of the altimeter in greater detail.

Figure 2.8 Illustration of different altitudes used in aviation.

Nomenclature

a

temperature lapse rate,

dT

/

dh

c

chord

g

gravitational acceleration

g0

gravitational acceleration at sea level

h

geopotential altitude

hA

absolute altitude (height relative to the center of the Earth)

hG

geometric altitude (height above mean sea level)

k

constant,

a

/

T

SL

m

mass of air in the control volume

n

constant, −

g

0

/

aR

p

pressure

R

gas constant for air

rEarth

Earth's mean radius

Rec

Reynolds number based on chord

Svisc

Sutherland's constant

T

temperature

U∞

freestream velocity

W

weight of air in the control volume

x

length of control volume element

y

width of control volume element

β

constant used in Sutherland's Law

δ

pressure ratio,

p

/