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,
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.
| SI | English | |
|---|---|---|
| β | 1.458 × 10−6 kg/(s m K1/2) | 2.2697 × 10−8 slug/(s ft ° R1/2) |
| S visc | 110.4 K | 198.72 ° R |
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,
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
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
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
