Theory and Formulas Behind (Airplane) Crosswind CalculationsThere are several mechanical and electronic flight computers available nowadays. Most problems can be solved with very simple arithmetics and using a simple pocket calculator but some calculations to predict wind effects require more complex formulas. Those are solved here.
|Airplane in crosswind|
This theory applies to all moving objects since it uses actual true air speeds involved in the situation. That will already account for all kind of factors such as air resistances etc.
|Jeppesen TechStar electronic flight computer can calculate wind effects|
It is assumed that the air moves evenly. The following figure illustrates the variables involved in the calculation.
|Variables used in the calculations|
When doing pre flight planning we usually want to know how wind affects our route legs
- 1) what is the resulting ground speed (Vg) and to which direction (HDG) to steer to compensate for the crosswind or
- 2) what is the wind when we know the ground speed (Vg) and air speed (Vo) and the heading (HDG) and Course Over Ground (COG).
Notice that if in the second case both air and ground speeds are equal and HDG and COG are equal then there is NO wind at all.
Object (Airplane) knowns
- Vo - air speed (desired cruise speed or pitot measured)
- COG - direction (desired COG or GPS measured)
Wind knowns or solve
- Vw - speed
- alpha - direction ("where from" is given)
- Vg - resulting ground speed (to solve time and fuel required)
- HDG - heading to steer (to assist to fly the trip)
The basic formulas are obtained from the facts that if we want to follow some predefined COG and if we have crosswind we have to steer into the wind so much that the cross track speed component of the wind is nulled. When that equation is solved we obtain also wind and side slip generated track speed components. All speed components are simply added along COG x and y axes. Using these formulas different unknowns can then be solved as required.
|To stay on cours (COG) Voy must be equal to Vwy|
- delta - side slip angle (add this to COG to get HDG)
- beta - effective wind angle ( = alpha - COG)
- Vwx and Vwy - wind speed components along COG
- Vox amd Voy - air speed components along COG
To stay on the desired COG track Voy must be Vwy so we can write:
Voy = Vwy (1)
and when that is met (by adjusting side slip angle delta) we also have the ground speed available along the COG x axis and for that we can write:
Vg = Vox - Vwx (2)
These two main equations can futher be written as:
Vo*sin(delta) = Vw*sin(beta) (3)
Vg = Vo*cos(delta) - Vw*cos(beta) (4)
Since all directions are given relative to the north we can also write:
HDG = COG + delta (5)
beta = alpha - COG (6)
Using these basic equations we can now solve the 2 cases given and any 2 unknowns if required.
Case 1) Unknown Heading and Ground Speed
To solve heading (HDG) first using (3) solve delta.
delta = asin(Vw*sin(beta)/Vo) (7)
and from (5), (6) and (7) we get.
HDG = COG + asin(Vw*sin(alpha - COG)/Vo) (8)
To solve ground speed using (4), (6) and (7) we get.
Vg = Vo*cos(delta) - Vw*cos(alpha - COG) (9)
Now since all variables in (8) and (9) are know we can solve Vg and HDG. Let's consider the following example.
During flight planning you determine that the forecast winds aloft at your cruising altitude are 080 degrees at 20 knots. Your course and true airspeed will be 030 degrees and 170 knots, respectively. Using the above equations, you compute the true heading and groundspeed as follows.
Vo = 170 knots
COG = 30 degr
Vw = 20 knots
alpha = 80 degr
Vg = ground speed, knots = ?
HDG = airplane heading, degr = ?
Using above formulas we get as follows.
delta = asin(Vw*sin(alpha - COG)/Vo)
= 5.17 degr
Vg = Vo*cos(delta) - Vw*cos(alpha - COG)
= 170*cos(5.17) - 20*cos(80-30)
= 156.45 knots
HDG = COG + delta
= 30 + 5.17
= 35.17 degr
So our ground speed will be 156.45 knots and we should steer to heading 35.17 degrees. This amount of side wind from given direction requires 5.17 degrees crab angle to the wind.
Case 2) Unknown Wind
In case of unknown wind we know the crab angle or heading and COG and also airplane Vo and ground speed Vg. Using (3) and (9) we get.
Vo*sin(delta) = Vw*sin(alpha - COG) (10)
Vg = Vo*cos(delta) - Vw*cos(alpha - COG) (11)
To solve alpha we first solve from (10) Vw
Vw = Vo*sin(delta)/sin(alpha - COG) (12)
and then insert Vw into (11)
Vg = Vo*cos(delta) -
= Vo*cos(delta) - Vo*sin(delta)/tan(alpha-COG)
and finally solve alpha.
tan(alpha-COG) = -Vo*sin(delta)/(Vg-Vo*cos(delta))
alpha-COG = atan(-Vo*sin(delta)/(Vg-Vo*cos(delta)))
alpha = atan(-Vo*sin(delta)/(Vg-Vo*cos(delta)))+COG (14)
Now calculate wind angle alpha first and then insert it into (12) to get wind speed Vw.
For this example you want to calculate the actual winds aloft using a course of 175 degrees, actual heading of 160 degrees, true airspeed of 180 knots, and actual groundspeed of 144 knots. Now, using the above formulas solve the wind variables.
COG = 175 degr
HDG = 160 degr
Vo = 180 knots
Vg = 144 knots
alpha = wind angle, degr = ?
Vw = wind speed, knots = ?
Using the formula (14) and (5) we get.
delta = HDG - COG = 160 - 175 = -15
alpha = atan(-Vo*sin(delta)/(Vg-Vo*cos(delta)))+COG
= atan(-180*sin(-15)/(144-180*cos(-15))) + 175
= 117.66 degr
and from (12) we get.
Vw = Vo*sin(delta)/sin(alpha - COG)
= 180*sin(-15)/sin(117.66 - 175)
= 55.34 knots
So our wind is 117.66 degrees at 55.34 knots.
xCalc - Expression Calculator
You can enter the above formulas to your computer using some programming language and that should do the job. If you need a desktop calculator for your Windows PC that can solve those expresions you can also download a free expression calculator "xCalc" from the link below. That will accept directly the above expressions. xCalc acts almost like the programming language BASIC interpreter used in the direct calculation mode.
Link to xCalc
Flight Computer Videos
E6B Flight Computer from Sporty's Pilot Shop
E6B Flight Computer: Ground Speed and True Heading
E6B Flight Computer: Unknown Winds
/1/ Jeppesen - Private Pilot Manual
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