Deterministic disturbances

The velocity and direction of the mean wind with respect to the ground usually is not constant along the flight-path. This variation of the mean wind along the flight-path is called windshear.¹ The influence of windshear upon the motions of the aircraft is of particular importance during the final approach and landing, and during take-off and initial climb.

An idealized profile of the mean wind as a function of height above the ground has been shown in figure 4.1. Much more extreme wind profiles in lower atmosphere have been recorded; some data from actual measurements in lower atmosphere can be found in ref.[1]. A very dangerous type of windshear is encountered in so-called microbursts: large pockets of air moving rapidly downwards to the ground in thun-derstorms. An aircraft flying through a microburst can be faced with a sudden in-crease in headwind, followed by a severe downdraft that is immediately followed by a sudden increase in tailwind, all within a matter of seconds. Such conditions

1In some textbooks the term ‘windshear’ is used to denote the local variations of the wind and turbulence with respect to the ground. In this report turbulence will be considered separately.

44 Chapter 4. External atmospheric disturbances

0 0.5 1 1.5 2 2.5 3

0 50 100 150 200 250 300 350 400 450 500

V_w[ ]m/s H [ ]m

Figure 4.1:Wind profile for λ= −0.0065 Km⁻¹and Vw_9.15 =1 ms⁻¹

may exceed the aerodynamic and propulsive performance of airplanes, causing the aircraft to loose lift and altitude, which can be very hazardous at low altitudes.

One of the best researched accidents involving a microburst was the June 24, 1975 crash of Eastern Air Lines Flight 66 during final approach to John F. Kennedy airport in New York. Figure 4.2 shows the reconstructed wind profile and the recorded flight-path of this airplane. Based on this data, numerical models of two-dimensional flow-patterns for thunderstorms have been developed [11]; such models are often used for flight crew training in flightsimulators. Although this subject will not be explored in further detail in this report, it is useful to acknowledge the hazards of such extreme weather conditions, and to realize that the idealized wind model that will be pre-sented next will not always be sufficient; for some purposes, it may be necessary to apply additional models, or even measurements of actual wind velocities during extreme weather conditions. However, that goes beyond the current scope of this report.

The atmosphere model used in section 3.5 was based on the ICAO standard atmos-phere model, which is characterized by a standard temperature lapse-rate λ=dT/dH=

−0.0065 Km⁻¹; see for instance ref.[30] for more details. The following equations rep-resent a typical idealized windspeed profile that is valid for this particular value of the temperature lapse-rate [1]:

Vw = Vw_9.15

H^0.2545−0.4097

1.3470 (0< h<300m) Vw = 2.86585 Vw_9.15 (h≥300m)

(4.1) V_w9.15 is the windspeed at an altitude of 9.15 m (approximately 30 ft, which is a com-monly used reference height for meteorological experiments). The wind profile in

4.1. Deterministic disturbances 45

Distance from threshold [ ]

m

m

Figure 4.2:Microburst wind pattern that caused the crash of Eastern flight 66

figure 4.1 is based upon a value V_w9.15 = 1 ms⁻¹. Ref.[1] presents some alternative wind profiles, which are typical for other values of the temperature lapse rate.

In sections 2.3 and 3.3.4, we described how the influence of the wind on the motions of the airplane can be expressed in terms of correction terms for the external force components in the body-fixed reference frame. If we know the wind velocity relative to the Earth, we can obtain the components of the wind along the aircraft’s body-axes using the following axes transformation:

V^B_w= T_E→B·_V^E_w (4.2)

where V_wrepresents the wind vector, the superscript B represents the aircraft’s body axes, and the superscript E represents the Earth axes. T_E→B is the transformation matrix from Earth to body axes, which involves the three consecutive Euler rotations shown in figure A.2 in appendix A:

TE→B ≡ TV→B = T_Φ·T_Θ·T_Ψ (4.3)

This transformation has been written out in more detail in equation (A.4) of appen-dix A.

In practice, wind is usually not given in terms of velocity component along the Earth axes, but rather in terms of windspeed and wind direction. The first represents the magnitude of the windspeed vector, and the latter represents the direction on the compass rose from where the wind emanates; for instance, ‘northerly wind’ means that the wind is blowing from the north. This notation does not take into account vertical wind components, because such vertical windspeeds are often short-lived, whereas the horizontal wind pattern is usually much more steady.

46 Chapter 4. External atmospheric disturbances

If V_whor is the horizontal wind velocity and ψ_w is the wind direction, the horizontal wind velocity components in the Earth axes are equal to:

u_w^E = V_whor·cos(ψ_w−π)

v_w^E = V_whor·sin(ψ_w−π) (4.4)

For sake of completeness, we will also introduce the ‘vertical wind direction’ angle γw, for cases where the wind also has a vertical component; this angle is positive when the wind is blowing upwards. Using that angle, the horizontal wind velocity is found to be:

V_whor =V_w·cos γ_w (4.5)

In this generalized situation, the vertical wind velocity component can be non-zero:

w^E_w= −V_w·sin γ_w (4.6)

where the minus sign reflects the fact that the Z_Epoints downwards.

Equations (4.3) to (4.6) can be used to model a steady wind pattern, while equa-tion (4.1) can be used to describe the changing wind velocity in the Earth’s boundary layer. Notice however, that the horizontal wind direction ψ_wwill normally also vary with height. The above given representation of the vertical wind component may be practicable for modelling microburst wind patterns like the one shown in figure 4.2, but for most other purposes the vertical windspeed can simply be neglected (i.e. γw

can be kept identical zero).