Sunday, 19 June 2011

When the wind blows, you always lose on the roundtrip

You sometimes hear pilots claiming "I don't mind the headwind on the way out, because the tailwind on the way back will 'blow me all the way home' and make up for it". You probably know this to be untrue, but I thought it would be useful to prove that this is wholly untrue. With some very simple maths, we can quickly see that whenever the wind blows, you always lose out in round-trip time and fuel consumed. Here's why:

First, some simplifying assumptions to make the points clear:
• TAS (true airspeed) is constant along the route, denoted V
• Windspeed is in constant direction with constant speed, denoted W
• Rate of fuel consumption, denoted R, is constant along the route
• Outbound leg is directly into wind. Groundspeed outbound is therefore V-W
• Return leg is directly downwind. Groundspeed inbound is therefore V+W
With these simple assumptions, and for a leg distance denoted D (each way), we arrive at the following results for time en-route, denoted T, and fuel consumed, denoted F, for each leg:

Outbound
D/(V-W)D*R/(V-W)As expected, the effect of the headwind is to reduce the magnitude of the denominator, thereby increasing the time and fuel consumed for the leg
Inbound
(tailwind)
D/(V+W)D*R/(V+W)By contrast, the effect of the tailwind is to increase the magnitude of the denominator, thereby decreasing the time and fuel consumed for the leg

It is instructive to define these results with respect to the difference with the corresponding values in zero-wind, which we'll denote by T'=D/V and F'=D*R/V respectively. The differences are denoted ΔT=T-T' and ΔF=F-F'. We thus obtain the following expressions for the relative (or fractional) differences compared with the zero-wind baseline, where we have used for convenience the symbol r=W/V (ratio of windspeed to TAS):

ΔT/T' (or equivalently ΔF/F') Comments
Outbound
r/(1-r)The difference is positive, as expected for the headwind
Inbound
(tailwind)
-r/(1+r)By contrast, the difference is negative for the tailwind. However, and moreover, the denominator in the tailwind expression (1+r) is always numerically larger than in the headwind expression (1-r), for any r, which means that the advantage offered by the tailwind is always numerically smaller than the penalty from the headwind. The two effects never cancel-out: the headwind always wins, you always lose.

It is instructive to now consider the total round-trip by adding the differences for the outbound and return legs, and dividing by the zero-wind round-trip values (2*T' and 2*F' for time and fuel, respectively) to give the relative (fractional) difference in round-trip time and fuel as:

Penalty= (ΔT/2T') = (ΔF/2F') = r2/(1-r2)

...from which we can make the following observations:
• The expression is always positive for any value of windspeed and TAS, which means you always lose out on the round-trip. This is the central result of this article.
• As windspeed increases, the round-trip penalty increases by greater-and-greater amounts (due primarily to (1-r2) in the denominator. This means that the higher the windspeed, the greater the penalty.
• The expression is only valid for values of r from 0 to 1. At r=0, this represents the zero-wind condition, and the expressions evaluates to 0, as expected.  At r=1, the windspeed equals the airspeed and the penalty is infinite since the outbound groundspeed is zero and the leg is never completed.
Let's now look at some typical realistic values:

r=W/VPenalty
r2/(1-r2)
expressed as %

V=100 kts; W=10 kts

0.1

1%

Negligible penalty since 1% ought to be well within your planned fuel margin

V=100 kts; W=20 kts

0.2

4.2%

Penalty more significant at 4%, but still ought to be within your planned fuel margin

V=100 kts; W=30 kts

0.3

9.9%

Penalty significant at 10%. Starts to impact on fuel margin. Moreover, a 30 kt wind is not uncommon at GA altitudes.

V=100 kts; W=40 kts

0.4

19%

Penalty significant at 19%. Serious impact on fuel margin unless distance is short. Probably should not fly into a 40 kt headwind in a light aircraft for any significant distance unless there is adequate fuel margin.

V=100 kts; W=50 kts

0.5

33%

Penalty huge at 33%. However, it is highly unlikely you would fly into a 50 kt headwind in a light aircraft for any significant distance. That said, for an ultralight with a TAS of, say 70 kts, the ratio of r=0.5 is quite easily encountered i.e., with a headwind of 35 kts. Beware: you will require 44% more fuel for a round-trip under such conditions !

V=100 kts; W=70.7 kts

0.707

100%

Out of fun/curiosity, you can work-back the expression to demonstrate that the penalty becomes 100% (i.e., you double your en-route time and fuel for r=(1/√2) or 0.707). Unlikely you would ever encounter such extreme headwind (unless you were in an airship, or such!).

We can now explore further variations on the theme using iNavCalc which performs all such "velocity triangle" calculations effortlessly. Normally, you would use iNavCalc to automatically calculate for the given actual/forecast weather along the planned route (we call this "AutoMETic"). However, you can optionally switch-off the automated weather functionality and specify the weather (winds aloft etc) manually. This is particularly useful for demonstration/educational/training purposes, and so we will make use of that capability now.

To get started, let's consider the idealized scenario presented earlier. Namely, a round-trip flight conducted with the outbound leg directly into wind, and the inbound leg directly downwind. Let's assume the following parameters for the calculations:

ParameterValueNotes
IAS (cruise) 96 kts This gives a TAS of 100 kts at 3000 ft altitude in ISA conditions
Altitude 3000 ft Typical altitude for GA VFR flight
Start fuel 150 litres Typical available fuel (full tanks) for light aircraft
Fuel consumption (cruise) 40 litres-per-hour Typical cruise fuel consumption for light aircraft
IAS (climb) 80 kts Typical climb IAS for light aircraft. Note: iNavCalc accounts for climb as well as cruise in the time and fuel calculations, so we need to specify the climb IAS.
Fuel consumption (climb) 40 litres-per-hour iNavCalc accounts for enhanced fuel consumption in climb, so we can specify accordingly.
IAS (descent) 96 kts Typical descent IAS for light aircraft. Note: iNavCalc accounts for descent as well as cruise in the time and fuel calculations, so we need to specify the descent IAS. In this example, we will specify same as for cruise.
Fuel consumption (descent) 40 litres-per-hour iNavCalc accounts for enhanced fuel consumption in descent, so we can specify accordingly.
In this example, we will specify same as for cruise.
Climb rate 1100 feet per minute Typical climb-rate for light aircraft. Required for time and distance calculations in climb.
Descent rate 600 feet per minute Typical descent-rate. Required for time and distance calculations in descent.
Sea level outside air temperature 15 C ISA conditions (note: iNavCalc would normally determine this automatically. However, we are switching-off AutoMETic for now, so we need to specify).
Sea level atmospheric pressure 1013.25 mb ISA conditions (note: iNavCalc would normally determine this automatically. However, we are switching-off AutoMETic for now, so we need to specify).

To specify the route, let's choose an arbitrary starting point given by the following coordinates: 540500N 0043724W (this happens to be Ronaldsway Airport, Isle of Man ICAO: EGNS). We will fly due (true-)north for 100 nm, then due (true-)south for 100 nm, completing the round-trip.

Let's start with the baseline (zero-wind) case. In which case, we will set the windspeed to zero, and the complete iNavCalc parameter-string, encapsulating all of the foregoing, is given by:

route={540500N 0043724W MyStart}, >360/100, >180/100; bearingtype=true; culture=UK; startfuel=150; ias=96; alt=3000; fuelflow=40; iasclimb=80; iasdescent=96; fuelflowclimb=65; fuelflowdescent=40; climbrate=1100; descentrate=600; met=manual; oat=15; qnh=1013.25; wind=360/0

(Click here for full documentation describing all  iNavCalc parameters and their usage.)

All we need to do now is send an email to plogs@flylogical.com with the above parameter-string as the subject-line. The resulting response email  contains a comprehensive navigation planning log (PLOG) for the flight. From this, we can pick out the desired values for present purposes. Namely:

• Total distance: 200 nm
• Total flight time: 2hrs 0 mins
• Total fuel consumed: 81.2 litres

These represent the baseline (zero-wind) values for the round-trip.

We can now add non-zero wind in the northerly direction (direct headwind). For example, for the windspeed of 10 kts, we simply specify the following parameter-string:

route={540500N 0043724W MyStart}, >360/100, >180/100;  ias=96; alt=3000; met=manual; oat=15; qnh=1013.25; wind=360/10

(Note: we are taking advantage of the "sticky" parameter functionality whereby we do not need to specify all of them again if they haven't changed. Click here for full documentation describing all  iNavCalc parameters and their usage.)

The resulting PLOG reveals:

• Total flight time: 2hrs 1 mins (0.83% penalty, compared with 1% from earlier analytical expression for r=0.1)
• Total fuel consumed: 82 litres (0.98%  penalty, compared with 1% from earlier analytical expression for r=0.1)
...which are in close agreement with the results of the earlier analytical calculation (i.e., 1% penalty for r=0.1). The differences are due to the fact that  iNavCalc takes careful account of climb and descent performance (on time and fuel calculations) as well as cruise, whereas the analytical calculations pertain to idealized cruise conditions.

We can continue with the exercise by putting successively higher windspeed values into  iNavCalc, and recording the results obtained. They are summarised below alongside the corresponding analtyical result.

r=W/VAnalytically
derived
penalty
r2/(1-r2)
expressed as %
Penalty from
iNavCalc
results
(time, fuel)
0.11%0.8%, 1%
0.24.2%4.2%, 4.2%
0.39.9%10%, 9.9%
0.4 19%19.2%, 18.8%
0.5 33%33.3%, 32.9%
0.707100%99.2%, 97.8%

Clearly there is very good agreement between the simple analytical result and the more detailed calculations from  iNavCalc, confirming that when the wind blows, we lose out on the round-trip.

But what about other scenarios that are not directly into-/down- wind ? We could quite easily derive analytical expressions for these scenarios, but they would not be as simple because of the need to include angular offsets and trigonometry. Instead, we can simply use  iNavCalc to explore these scenarios.

Let's start with the intermediate crosswind situation where the wind is off the nose by 45 degrees on the outbound leg (and likewise 45 degrees off the tail inbound). This is simple to evaluate using  iNavCalc: all we have to do is re-specify the wind direction. For example, the  iNavCalc parameter string for a wind of 045 degrees at 10 kts would be:

route={540500N 0043724W MyStart}, >360/100, >180/100;  ias=96; alt=3000; met=manual; oat=15; qnh=1013.25; wind=045/10

The table below shows the results computed by  iNavCalc for the range of windspeeds at 045 degrees, compared with the corresponding penalty for the into-/down-wind scenario.

r=W/VAnalytically
derived
penalty
for into-/down-wind
scenario
Penalty from
iNavCalc
results for wind
at 045 degrees
(time, fuel)
0.11%0.8%, 0.7%
0.24.2%3.3%, 3.1%
0.39.9%7.5%, 7.4%
0.4 19%14.2%, 14%
0.5 33%25%, 24.4%
0.707100%73.3%, 71.8%

The penalties are still inevitable, though are not as severe as for the direct into-/down-wind scenario.  Finally, let's explore the pure crosswind scenario, wherebey the easterly wind is specified in the  iNavCalc parameter-string as:

route={540500N 0043724W MyStart}, >360/100, >180/100;  ias=96; alt=3000; met=manual; oat=15; qnh=1013.25; wind=090/10

r=W/VAnalytically
derived
penalty
for into-/down-wind
scenario
Penalty from
iNavCalc
results for wind
at 090 degrees
(time, fuel)
0.11%0.8%, 0.5%
0.24.2%2.5%, 2.1%
0.39.9%5%, 4.8%
0.4 19%9.2%, 9%
0.5 33%15.8%, 15.3%
0.707100%41.7%, 40.8%

Again, the penalties are inevitable (some fuel must be burned to counteract the crosswind), but are the most benign compared with the direct into-.down-wind scenario.

Gathering all these results together, the following conclusions can be drawn:

• Whenever the wind blows, there will inevitably be a round-trip cost in terms of time and fuel compared with the zero-wind scenario
• The worst-case penalty occurs when the wind is a direct headwind/tailwind. In which case, the round-trip percentage penalty is approximated by the analytical expression 100*r2/(1-r2) where r=W/V
• The least penalty occurs when the wind is directly across the track. In which case, the round-trip percentage penalty is approximately half the worst-case, i.e.,  50*r2/(1-r2where r=W/V
• The intermediate case when the wind is 45 degrees off-track gives a round-trip percentage penalty of approximately three-quarters the worst-case, i.e.,  75*r2/(1-r2where r=W/V. This is a useful rule-of-thumb when initially planning a trip.
•  iNavCalc is a powerful and convenient tool for performing such calculations for training/educational purposes (as in this article), or for actual flight planning using its built-in automated weather feeds. Sign up now -- it is *free*.

Remember -- it is wholly incorrect that the tailwind on the way back will 'blow you all the way home' and make up for the headwind on the way out. In fact, this is the worst-case scenario in terms of time and fuel penalty on a roundtrip. Pilots beware!