How to describe the wind sprial in computer system?

How to describe the wind sprial in computer system?

2017-02-21 刘崇军 风螺旋线

If we want get the approval of ICAO, we must separate the theory to different parts, which is easy to be adopted and which is not. Let’s talk about the easy part of wind spiral theory first.

Easy part 1: Where is the E_θ in aeronautical speed triangle?

Chart 1. Aeronautical speed triangle

What ever the acft flying a straight line or a circle line, it will be affected by the wind accumulated by time. E_θ will always equal to wind speed multiple with time we counted. As the time pass by, GS will be pushed to the position of GS’ followed the direction of wind speed. GS’ is the combination of two movements, so it’s certainly to be the outer boundary of flying, and also be the tangent line when flying a circle.

Easy part 2: How to connect a straight movement and a circle movement?

If we continuous counted the time from a point, it will be soomthly connected with the outer boundary with the wind spiral as showed in chart 2.

Chart 2. The connection of two kinds of movement

In chart 2, the ws1 (wind spiral) expanded to ws2 follow the direction of E_θ, that means the direction of E_θ represents the most efficient direction to expand. With the foumla E_θ = w/R*θ, we can see E_θ is directly proportional with θ. So when we rotate a spiral such as ws1 in chart 2 anticlockwise, we can get the new spiral ws2 which is smoothly connected with the outer boundary of straight flight.

Easy part 3: How to describe a wind spiral?

There are two kinds of method to describe wind spiral in polar coordinates system. The first one is (α, ρ), the second is (r, θ) show in the chart 3.

Chart 3. Symbol used in polar coordinates system

The formula of wind spiral which expands in clockwise:

The (r, θ) is the accurate method to describe the wind spiral, (α,ρ)
is not. For example, we can get the tangent line’s direction with
θ-DA+90 (clock wise wind spiral), while it’s hard to get tangent line
with angle α.

Except the turn parameters we usually used, the basic parameters of a spiral should include:

a.position of center point

b.draft angle (DA)

c.initial angle of radius

d.initial offset (E_θ at the initial position)

The
wind spiral show in chart 3 can be described as follow: initial angle 0
degree with initial offset 0m, or initial angle 90 degree with initial
offset E₉₀. In fact, they are the same spiral at point O.

Easypart 4: Calculation of common tangent line.

If two spirals have common tangent line, it means there will have an E_θ in each sprial that tangent to the same common tangent line.

Chart 4. Common tangent line of two wind spiral

The different length of two E_θ show
in chart 4 is caused by the deference in initial angle and initial
offset.The distance of center point also play import role in calcuate
angle β.

β= arcsin( (E_θ2-E_θ1)/ Distance_o1o2 )

When β is gotted, the common tanget line’s direction is gotted too.

Supplement 1.  Formula is NOT important

If
you are working with mathematical stuff, formula is the easy way for
calculation. But if you only need to draw the wind spiral, there’s
another simple way to get the same result.

Chart5. Geometrical method for drawing wind spiral

First
step is to draw the normal track of circle. Second step is to devide
the circle to equal parts with radius in same interval of angle. Third
step is to extend the radius line, rotate the part beyond the circle
with angle of DA, and use E_θ value to determine the end of E_θ. Fourth step is to connect the ending point of each E_θ, a wind spiral will be gotted.

Supplement 2. Anticlockwise spiral is quite different with clokwise one

There
are different axis system in use, so the formula maybe various. The
relationship in the triangle is the fundamental element for wind spiral
calculation.

Chart6. Different axis used.

The
point in the wind spiral always delayed compare with the related point
in the circle. For this reason, the left spiral in chart 6 will expands
clockwise, and the right one will expands anticlockwise.

Supplement 3. The ρ in spiral MUST bigger than zero

When
the wind spiral extends into the normal circle, it will lose the
characteristic of expanding with equal distance. That means you couldn’t
get the anticlockwise spiral by extend the clockwise spiral or give the
negative parameter to the formula.

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想要撩起 ICAO 的注意，英语是必须的，所以从今天开始，争取把英语好好的练一下。欢迎各路大神批评指正，说不清楚的地方一定是本人英语太烂的缘故，敬请谅解。

 

 

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