At what angle to the ground track does wind become favorable to lower fuel consumption?

Question

We can all agree, flying from point A to B, a headwind will increase fuel consumption, and a tail wind will reduce it. The "jet stream" inspires these types of thoughts.

Quartering headwind, crabbing into, still more fuel, right?

Quartering tailwind, probably less, but the aircraft still must oppose the cross wind component with some force which is ultimately derived from burning fuel.

Perhaps, from someone with airliner experience, at what point off my tail does wind stop assisting in fuel economy?

With the tail as 0 degrees, nose as 180, and beam as 90, my guess would be around 80 degrees, a cross wind with a bit of tailwind component.

Is there evidence from practical experience to support or refute this guess?

Answer 1

To solve this problem, let's look at the vector diagram of TAS, GS and Wind Speed (WS):

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In this diagram, the angle between the desired course (ground track) and the wind is $\delta$ (where 0° is tailwind and 180° is headwind). The resulting wind correction angle $\theta$ can be calculated as follows:

$$ \theta = \arcsin \left( \frac{\text{WS}}{\text{TAS}} \sin\delta \right) $$

The resulting ground speed (length of vector labelled "Desired course") is then given by

$$ \text{GS} = \text{TAS} \cos\theta + \text{WS}\cos\delta $$

The following plot shows the wind correction angle and ground speed for an example of 400 KTAS and 50 kt wind speed at different angles:

We are now looking for the angle $\delta$, where the ground speed is equal to the TAS. This will typically be close to 90°, but not exactly. In the example above, this angle is at ~86.4°. When solving the equation above with GS = TAS for $\delta$, the result is:

$$ \boxed{\delta = \pm \arccos \left( \frac{\text{WS}}{2 \text{TAS}} \right)} $$

This result depends on the ratio of WS/TAS, meaning there is no single answer to the question. We can however plot this angle, where the wind has no impact on ground speed, as a function of this ratio (only positive solution plotted):

When the wind angle is less than this value, we benefit from the tailwind. When the wind angle is higher than this value, we suffer from the cross- and headwind.