### AIRSPEED MEASUREMENT

The speed of an aircraft relative to the surrounding air is a critical part of operation and performance analysis. No direct method is available for measurement of velocity relative to the air so instead the velocity is inferred from measurements of dynamic pressure.

Dynamic pressure is defined as $q=1/2 ρV^2$

Various forms of dynamic pressure can be obtained depending on the instruments used to measure the pressure and the flight speed range of the vehicle. A simple airspeed indicator is shown in the section on Aircraft Instruments. All instruments rely on knowledge of Standard Atmosphere properties.

The primary atmospheric properties can be expressed in the form of ratios compared to Sea Level conditions.

$\table δ, = ,P/P_{ssl},\text" - Pressure Ratio"; θ, = ,T/T_{ssl},\text" - Temperature Ratio"; σ, = ,ρ/ρ_{ssl},\text" - Density Ratio"$

The approximate dynamic pressure can be measured by a Pitot-Static system,

$$q≈q_{ci}=P_0 - P_s$$

This pressure difference is known as the indicated impact pressure. For subsonic aircraft operating in a normal angle of attack range, total pressure ( P0 ) can be measured accurately. Static pressure ( Ps ) is a less accurate measure of the surrounding atmosphere pressure ( P ) and calibration factors may need to be applied to correct for measurement errors due to the position of the device.

$$q=q_i=P_0-P$$

where qi is defined as the impact pressure.

Applying the compressible flow Bernoulli equation gives the following expression for velocity,

$$V=√{{2γ}/{γ-1}P/ρ(({P_0-P}/P+1)^{{γ-1}/γ}-1)}$$

where γ = 1.4 is the ratio of specific heats for air.

For incompressible flow this simplifies to

$$V=√{2/ρ(P_0-P)}$$

As air density cannot be directly measured and static pressure measurements are subject to error, airspeed indicators use a calibrated ISA sea level pressure condition instead. The airspeed displayed on cockpit indicators is thus,

$$V=√{{2γ}/{γ-1}P_{ssl}/ρ_{ssl}(({P_0-P}/P_{ssl}+1)^{{γ-1}/γ}-1)}$$

or for incompressible flow,

$$V=√{2/ρ_{ssl}(P_0-P)}$$

For transonic and supersonic vehicles additional compressibility corrections may need to be applied. If shock waves form in front of the pitot tube then total pressure ( P0 ) will become inaccurate due to shock losses.

There is thus a translation sequence between the observed speed ( VO ) on the aircraft's instruments and the actual airspeed of the vehicle ( V ). For subsonic vehicles,

VO , Observed airspeed on instrument + Δ VIC (instrument correction)
=> VI , Indicated airspeed + Δ Vpos(static port position error)
=> VC , Calibrated Airspeed + Δ VC (compressibility correction)
=> VE , Equivalent Airspeed $×1/√{ρ}$ (Altitude correction)
=> VT , True Airspeed = V

For incompressible flows a rough estimate can be made that

$$V_0≈V_I≈V_C≈V_E$$

At high speed where compressibility correction becomes a large component then a Mach meter rather than an airspeed indicator may be used.

$$M=√{{2γ}/{γ-1}(({P_0-P}/P+1)^{{γ-1}/γ}-1)$$

Indicated Mach Number would thus be,

$$M_I=√{5(({P_0-P}/P_{ssl}+1)^{1/3.5}-1)$$

or

$${P_0-P}/P_{ssl}=(1+0.2M_I^2)^{3.5} - 1$$

For supersonic flight when a shock forms in front of the pitot tube an additional compressibility correction is required.

When

$${P_0-P}/P_{ssl}>0.893$$

then

$${P_0-P}/P_{ssl}={166.92M_I^7}/{(7M_I^2-1)^{3.5}}-1$$

Based on these airspeed relationships, dynamic pressure can be formulated in several ways,

$$q = 1/2 ρ_{ssl}V_E^2=1/2ρV_T^2=1/2γPM_T^2$$