Chapter 2. Special and gas dynamic functions

The following special and gas dynamic functions have been used in the current software:

standard atmospheric conditions [6],

empirical pressure losses curves for subsonic and supersonic inlets [4],

empirical curves for enthalpies [1, 3],

gas dynamic functions  , ,  [2].

To define the conditions at the different altitudes  the standard atmospheric conditions have been used [6]. The geopotential altitude is defined as

where  is a radius of earth. The temperature of air is

where  is a temperature on the sea level,  is a temperature at altitude  and  is a vertical temperature gradient. The atmospheric pressure is

where  is an atmospheric pressure on the sea level,  is an atmospheric pressure at altitude ,   is a gas constant and  is an acceleration of gravity at altitude . The acceleration of gravity can be defined as

where  is an acceleration of gravity on the sea level.

Empirical pressure losses curves for subsonic and supersonic inlets are extracted from Fig. 2.1 [4] by using Lagrange polynomial interpolation [5] with 7/9 points to provide smooth accurate curve approximation.

Fig. 2.1. Pressure losses in the inlets as a function of flying speed [4].

The empirical curves for enthalpies are extracted from Fig. 2.2-2.4 by using Lagrange polynomial interpolation [5] with 9 points.

Fig. 2.2. Enthalpies as a function of combustion temperature (200K-1100K) [1].

Fig. 2.3. Enthalpies as a function of combustion temperature (1100K-2000K) [1].

Fig. 2.4. Enthalpies as a function of combustion temperature (2000K-2900K) [1].

The gas dynamic function  is calculated as

 as

and  as

where   is normalized velocity. The normalized velocity is defined as

where is a speed and   is a critical speed of sound:

The gas dynamic functions can be calculated by using Mach number as well

where  is a speed of sound:

The relation between Mach number and normalized velocity can be written as