In the real world, the physical properties of air determine, in part, the aerodynamic forces (i.e., lift and drag) acting on an aircraft, and affect the measurement of aircraft state elements, such as airspeed and altitude. Our model of flight dynamics does not take into account the nature of the fluid through which the aircraft travels, and so, requires no knowledge of atmospheric properties. However, if flight simulation is the goal, then the mathematical model should account for the effects of atmospheric temperature, pressure, and density on measured airspeeds and the conversion to Mach number.
The variation of these properties (i.e., pressure, temperature, and density), are complex functions of altitude, time, geographic location, sun activity, and more. The atmospheric model employed here, the ISA (International Standard Atmosphere), ignores this complexity and describes the variation of temperature, pressure and density with respect to a single factor, altitude.
The ISA is a model of the earths atmosphere which describes the altitude dependence of air temperature, pressure, and density for altitudes up to 100 kilometers. Only that portion of the model which deals with the lower regions of the earths atmosphere (i.e., troposphere and stratosphere) extending to 20 kilometers above sea level will be considered.
The temperature of the troposphere, the region lying between the earths surface and an altitude of 11 kilometers, is assumed by the ISA model to decrease linearly from 288.150 at sea level to 216.650 at its upper boundary. The atmospheric region lying between 11 and 20 kilometer, known as the stratosphere, is assumed by the model to be an isothermal region with a temperature of 216.650 . The ISA assumed variation of temperature, , with altitude, , can then be expressed by the equations,
where ; is the standard sea-level value of temperature; and is the stratospheric isothermal temperature. The functions given by Equations 5-1 and 5-2 are graphically depicted in Figure 5.1.
The dependence of air pressure, , and air density, , on altitude, , in the ISA model, is derived using the ideal gas law,
and by applying definitions of pressure and density. In equation 5-3, represents the quantity of gas, in moles, contained within the volume, . is the universal gas constant for air, whose value is 8.314 Joules/mole-. The pressure function, derived from application of basic definitions and Equations 5-1 through 5-3, for the thermal gradient (i.e., troposphere) and isothermal regions (i.e., stratosphere), is given by the equations,
where is the standard pressure at sea-level; is the standard pressure at an altitude of 11 kilometers; kilometers, the lower altitude limit of the stratosphere; and represents the mean gravitational acceleration at sea level. The density, , as a function of altitude, is expressed as,
The speed of sound in air is useful in simulating Pitot-static device outputs and performing airspeed conversions. The speed of sound, , is, approximately, related to atmospheric pressure and density according to the equation,
where is the mean specific heat ratio for air. Equation 5-8 can be recast into a form which relates the speed of sound to temperature by replacing and with Equations 5-4 through 5-6, respectively. The equation which results,
is valid for all altitudes less than 20 kilometers.
Estimates of airspeed are derived indirectly using equations of the ISA model and from measurements of outside air pressure. Measurements of pressure in the E-6 aircraft are made using a Pitot-static system. A simplified diagram, illustrating the essential features of such a device appears in Figure 5.2. The mouth of the Pitot tube is aligned parallel to the aircrafts flight path, where a measurement of total pressure, , is made. A measurement of static pressure, , is made at an orifice whose axis is perpendicular to the direction of flight. Measurements of static pressure are assumed to be those predicted by the ISA model. Measurements of total pressure, , and static pressure, , are fed to a differential pressure gauge which measures .
Bernoulli's equation for compressible fluid flow relates , , , , and true airspeed, . This relationship,
together with Equation 5-8 and the equation,
relating air pressures, and , and air densities, and , are used to derive an equation describing the dependence of true airspeed on quantities obtained from the ISA model (assumes altitude is known) and the measurement of by the Pitot-static system. Namely, true airspeed, , is given by,
If only the pressure difference, , is known then true airspeed cannot be computed, directly, using Equation 5-12. Airspeed indicators are designed to provide estimates of airspeed based only on measurements of , but using Equation 5-12 with sea-level values for static pressure, , and the speed of sound, , substituted for and . In the absence of instrument errors, these devices provide values of calibrated airspeed. Calibrated airspeed, , is defined by
Indicated airspeeds, , are the actual estimates of airspeed provided by airspeed indicators (i.e., differential pressure gauges calibrated in accordance with Equation 5-13). The difference between values of indicated and calibrated airspeed arise from inevitable instrument errors.
True airspeed is related to Mach number, , according to,
The right side of equation 5-14, when substituted for in equation 5-12, yields the expression for Mach number,