A.3 THE EFFECT OF WIND
Wind is the single most important external factor that affects aircraft trajectory. In order to accurately simulate aircraft trajectory under various wind conditions, special attention must be paid to the formulation of aircraft movement when there is wind. In the following analysis, only wind components in the horizontal plane of the NED frame are considered because the vertical wind component (top shear) has a much smaller geometric scale, and its effect on the overall aircraft trajectory is limited. In this section, the effect of wind on aircraft ground speed is examined analytically first, then the computation of aircraft angle of attack and sideslip angle under wind conditions is described, and lastly the wind model used in the fast-time simulator is introduced.
A.3.1 The Effect of Wind on Aircraft Movement
Assume that aircraft is flying at a true airspeed Vr, subject to horizontal wind vector W, with flight path angle J. Further assume the angle between true airspeed and the horizontal plane is Jr. This angle represents the flight path angle relative to the air mass. The relationship between the true airspeed vector, the vertical speed vector (Vh), the ground speed vector, and the total speed vector (V), as collapsed into
the vertical plane, is shown in Fig. A-3a. The relationship between speed vectors as projected in the horizontal plane is shown in Fig. A-3b, whereOr denotes the incidental angle between the wind vector and the ground speed vector, and Pr denotes the angle between the projection of true airspeed and the ground speed vector.
Figure A-3a Relationship between flight path angle, true airspeed, and ground speed as collapsed in the vertical plane.
V
rFigure A-3b Relationship between wind speed, true airspeed, and ground speed as projected to horizontal plane.
As shown in Fig. A-3a and A-3b, the general relationship can be expressed in vector form as
W
V V V
V
h g r (A-18)Decomposing the wind vector into a headwind component Wh=WcosOr and a crosswind component Wc=WsinOr, from Eq. (A-18), in scalar form, the following equations can be derived
J
Assume 0d|Pr|<S/2, i.e. ground speed is positive, the following equations can be derived from Eq.
(A-19), (A-20) and (A-21)
Equation (A-22) shows that for given vertical rate Vh and true airspeed Vr, both headwind component Wh and the crosswind component Wc affect ground speed Vg. A positive headwind component causes the ground speed to decrease, and a negative headwind component (or tailwind component) causes the ground speed to increase; while the crosswind component always causes the ground speed to decrease no matter the crosswind component comes from left or right. However, the reduction of ground speed due to the crosswind component is much smaller than that due to the headwind component, if the wind speed is relatively small. This is easily seen from the following partial derivatives
1
On the right hand side of Eq. (A-27), the second term is the contribution of the crosswind component, and the third term is the contribution of the headwind component. The relationship represented by Eq.
(A-27) is illustrated in Fig. A-4, for several different values of the ratio of wind speed to true airspeed (W/Vr). In Fig. A-4, the vertical axis represents the ratio of ground speed to true airspeed (Vg /Vr), the horizontal axis represents angle Or. As angle Or varies from 0 deg to 90 deg, the wind vector changes from a pure headwind to a pure crosswind; and a Orof 180 deg gives a pure tailwind. The effect of wind on ground speed is the highest under headwind and tailwind conditions. The effect of a pure crosswind component on ground speed is illustrated in Fig. A-5. As seen in Fig. A-4 and A-5, the effect of the crosswind component can be ignored when the crosswind component is relatively small comparing to true airspeed. However, when the crosswind component is not small, or when precise results are required, the effect of crosswind component should not be ignored. In this simulator, both headwind component and crosswind component were taken into account.
0 0.1 0.2 0.3 0.4 0.5
Figure A-5 The effect of a pure crosswind component on ground speed as crosswind increases relative to true airspeed.
Figure A-4 The effect of wind on ground speed as wind direction varies relative to the direction of ground speed vector.
Since aircraft control and flight operation procedures are mostly based on airspeed, the effects of wind on aircraft ground speed will affect time-to-fly, and also the evolution of spacing between consecutive
aircraft pairs. The decrease of ground speed, such as in headwind conditions, will increase time-to-fly, and cause earlier separation compression relative to zero wind condition. The increase of ground speed, such as in tailwind conditions, will reduce time-to-fly, and delay separation compression relative to zero wind condition. Uncertainties in wind forecast, and/or change of wind will transfer into uncertainties in time-to-fly and uncertainties in spacing between aircraft.
Equation (A-24) represents the effect of wind on flight path angle. Because flight path angle is small under normal flight operation conditions, cosJ| 1, Eq. (A-24) can be approximated as sinJr | (Vg/Vr) sinJ. For a flight segment with fixed flight path angleJ, such as the initial constant speed segment in a Three Degree Decelerating Approach (TDDA) or MTDDA5, the decrease in ground speed due to headwind or crosswind will result in a shallower flight path angle relative to the air mass. This in turn will require higher thrust or less speedbrake usage to maintain the same airspeed. The increase of ground speed due to tailwind will result in a steeper flight path angle relative to the air mass, which in turn will require lower thrust or more speedbrake usage to maintain the same airspeed. Both thrust and speedbrake usage have noise implications. If the engine throttle setting is fixed, such as in idle, headwind and crosswind will cause aircraft to decelerate faster, and tailwind will cause aircraft to decelerate slower.
This will affect aircraft’s capability to follow a specific profile.
On the other hand, for a flight segment with given flight path angle relative to the air mass Jr, the decrease of ground speed due to headwind or crosswind will result in a steeper flight path angle. The increase of ground speed due to tailwind will result in a shallower flight path angle. For such a flight segment, the effect of wind is more complicated. Different wind conditions will result in different vertical flight paths. Not only the altitude above ground will be different, for the same airspeed profile, the longitudinal acceleration will also be different. This is because the relationship between true airspeed and calibrated airspeed will be different for different vertical profiles.