Guide to calculating turn rates of aircraft

[Laurelix97]

You can calculate the turn rates

For example:
Ki-84
Loaded Weight: 3600kg
Wing Area: 21m2
Air density at sea level: 1.225kg/m3
Flap Area: 2.436m2
Wing Lift Coefficient: (according to its manual)
Flaps at 0 degrees: 1.46 CL_Max
Flaps at 15 degrees: 1.70 CL_Max
Flaps at 30 degrees 1.92 CL_Max

Step 1:
Find out the Lift Force:
Weight (kg) X Gravity
3600 X 9.81 = 35,316 Newtons

Step 2:
Look at the formula and fill in the numbers

V= Stall Speed
2L = 2x Liftforce (2x 35316)
CL = wing lift coefficient
p = Air density (kg/m3)
a = Wing Area (m2)

Note:
If you are looking for stall speed without using flaps (0 degrees) you do not add the flap area to the wing area in the calculation. It's only when you start calculating the stall speed with flaps deployed, that's when you add the 2.436m2 to the 21.0m2
So the wing area that will be used for calculating flaps 15 and 30 degrees is (23.436m2)
Also if you want to know the stall speed at different altitudes, change the air density. So if you want to find out stall speed at 3000m altitude,find out the air density at 3000m and use that in your calculation.
If you want to find out the stall speed on Jupiter, change the gravity, on Earth it's 9.81 and change the air density
Also finding out the stall speed at different weight is also an option.
If I wanted to find out Ki-84 stall speed at 3400kg, I'd just do 3400 X 9.81 and use that as lift force in the equation.

Step 3:
Do the working out, use scientific calculator which lets you write the entire format. Top is just 2x L, bottom is CL X p X a
The top is divided by the bottom and everything is square rooted.
The answer you will get is the stall speed in m/s, you need to multiply it by 3.6 to convert it to Km/h.

If you're struggling with the math, here's the easiest way.
Airplane Aircraft Wing Lift Design Equations Formulas Calculator - Velocity

Ki-84
At 3600kg weight , Sea level
Stall speed is:
Flaps at 0 degrees: 156km/h
Flaps at 15 degrees (Combat): 137km/h
Flaps at 30 degrees (Landing): 129km/h

Step 4:
Work out the Power to weight ratio
Weight (Kg) / Power (HP)
3600 / 1990 = 1.81kg/HP

Step 5:
Now you can compare the turn rate against other planes.
Yak-3
2692kg loaded weight
1300hp engine
That's 2.07kg/HP power to weight ratio

Yak-3 stall speed at 2692kg, sea level, No flaps (0 degrees)
163km/h

Which ever plane has lower stall speed, that plane has smaller turn radius and sharper turns. This is the initial turn.
Ki-84 not only has lower stall speed (better initial turn) than Yak-3, it has better power to weight ratio than Yak-3 (carries less weight per horsepower). This means the Ki-84 sustains it's speed better than Yak-3.

There are instances where planes have very good initial turn (they turn very well with energy) but their engine is underpowered and as soon as they get into prolonged turns, they turn into whales that do small turn radius but they take long time to complete one because the plane is fighting stall thus the plane isn't turning efficiently because it doesn't have enough energy to perform manuevers. One example would be the F6F-3. Also note that use of flaps in combat decreases stall speed (improved initial turn) but the drag makes the plane lose speed which has negative effects on the sustained turn. It's important that the pilot knows when to use flaps and when not to.

Yak-3 sustained turn (no flaps, 1000m altitude)
18 seconds to complete 360 horizontal turn
-
Ki-84 is rated at 17 seconds
La-7 is rated at 19 seconds
Ki-44-II Hei rated at 18 seconds
J2M3 rated at 19 seconds
F4U-1a rated at 21 seconds
N1K2-J rated about 17.5 seconds, not as good as Ki-84 but not as bad as Yak-3
F4U-4 rated at 20 seconds
BF-109G-10 rated at 19 seconds


I tend not to compare turn rates when both planes use flaps because Ki-84 is the only plane Where I know it's flap area and I only know the wing lift coefficient when flaps are used for Ki-84 and BF-109E.
I compare sustained turn rates of planes when both are not using flaps.
Laminar Flow wings have worse wing lift coefficient, usually in the 1.3-1.33 CL_Max area
Normal Wings can have CL_Max of 1.36 and as high as 1.5 without using flaps.
Ki-84 wings give very high amount of lift (1.46 CL_Max)
Planes like BF-109E/F/G/K have 1.4 CL_Max (no flaps used)
Ki-61/100 have 1.44 CL_Max
P-63's have 1.42 CL_Max
Yak-3 has 1.41 CL_Max
J2M's and N1K's actually used laminar flow.
P-51D will definitely have about 1.32 CL_Max with its laminar wings (less lift, less drag)

 

[drgondog]

The equations presented for Stall speed are correct for symmetrical (level) flight conditions.

For asymmetrical flight (i.e. turn/level, turn descending) several factors come into play:
1.) unequal Lift between higher CL wing (up), and lower CL wing --------> also leading to trim drag due to rudder and elevator deflections to maintain steady - higher G - constant altitude turn. Simply stated CLmax as frequently used - 'Isn't" - and in fact appreciably lower.
2.) Power Available for High RPM/Lower speed prop/engine system has a lower efficiency factor at lower velocity, than at top speed. Therefore, Thp is lower than expected for a given velocity
3.) Power Required soars due to higher drag conditions in high AoA/asymmetrical flight when such factors as cooling drag become much more important than in level/high speed fight

Not much difference between Instantaneous turn rate and sustained turn rate derivative initially, but the difference becomes pronounced as the turn dynamics cited above come into play.

 

[GregP]

I like to think of the turn rate calculated above as the best turn rate you can get. It deteriorates in the real world as Bill stated above, but you can get, instantaneously, very close to the calculated rate at relatively low altitudes. If you're at 25,000 feet, you won't get anywhere close to that, but at 10,000 - 14,000 feet, you can come close until you are some 30° - 60° into the turn, after which all the factors above will have made them selves apparent.

In something like a Cessna 172, you are taught to roll into about a 30° bank and pull harder on the wheel to get to 45° - 60°. It is different in a fighter that is not inherently stable, and you can roll directly into a 60°+ bank and pull. Most fighters will not tend to steepen or shallow up, but will maintain the bank rather easily. It is a fighter design goal achieved by most, but not all. WWII fighters don't have enough specific power to maintain high-g for a complete circle, much less a number of circles, and steep turns degenerate into a contest of who makes the first mistake near stall within a short time if a turning dogfight is the order of the day, unless you ALSO descend to maintain energy. You can only descend until the ground interferes with your descent, and many Soviet Front 1943 and on fights were turning battles just over the Russian steppes, where the La-5/7 and Yak-3/9 had some advantages over the Bf 109 / Fw 190 series. The Soviet fighters, were VERY good low-altitude turners.

 

[wlewisiii]

Ok, I get the math. It's a fairly straight forward equation.

However, where do we find out the CL on various aircraft? And would a biplane require modification of the equation as well?

Thanks!

 

[Laurelix97]

All planes turn their best at sea level because as you get higher the air gets thinner and stall speed increases which also makes wider turn. La's are known to turn well at low altitude because above 3000 metres they lost vast amount of power which meant worse power to weight ratio and that has very negative effect in sustained turns.

 

[Laurelix97]

Ok, I get the math. It's a fairly straight forward equation.

However, where do we find out the CL on various aircraft? And would a biplane require modification of the equation as well?

Thanks!

I keep away from biplanes because I don't know.
If I don't know the CL_Max of a plane I look up wether it had laminar flow wings or not. If it has laminar wings I automatically assign 1.32 because I know it's roughly going to be in that area and the rough estimate I get will be 1-3km/h from the real value, it might even be bang on correct.

Non laminar wings (standard wing flow) I usually use 1.4 CL max to have an estimate but I take it with a pinch of salt because I could be 4km/h off the real value but it's still a very good estimate. If I see a Lavochkin or BF-109 with wing slats and normal wings, I know for sure that it won't be lower than 1.4

 

[pbehn]

All planes turn their best at sea level because as you get higher the air gets thinner and stall speed increases which also makes wider turn. La's are known to turn well at low altitude because above 3000 metres they lost vast amount of power which meant worse power to weight ratio and that has very negative effect in sustained turns.

That is only true of the absolute value, there was no merit in being the best in a theoretical place of research. The Bf 109 was better than the P 47 in turning performance at most altitudes but the higher the encounter the less marked it was. At very high altitudes above 25,000 ft the P47 possibly had the advantage, but as a bomber escort high altitude was where they frequently met.

The Spitfire always had great turning performance not only in absolute values shown in calculations but also in the "feel" the pilot had when approaching the limit. Attempts to improve the Spitfire with the Spiteful and its laminar flow wings were not a success, there may have been a theoretical or even actual improvement but the stall characteristics were vicious.

 

[Laurelix97]

That is only true of the absolute value, there was no merit in being the best in a theoretical place of research. The Bf 109 was better than the P 47 in turning performance at most altitudes but the higher the encounter the less marked it was. At very high altitudes above 25,000 ft the P47 possibly had the advantage, but as a bomber escort high altitude was where they frequently met.

The Spitfire always had great turning performance not only in absolute values shown in calculations but also in the "feel" the pilot had when approaching the limit. Attempts to improve the Spitfire with the Spiteful and its laminar flow wings were not a success, there may have been a theoretical or even actual improvement but the stall characteristics were vicious.

Also the reason for Ta-152H wing shape

 

[pbehn]

Also the reason for Ta-152H wing shape

Why does the Ta-152 come into almost every discussion? There were many more Spiteful, Seafang and Attacker wings flown than Ta-152s.

 

[Shortround6]

It also takes a while to get down to the speed at which the infamous slats work in a turn.
Say for example our 109 pilot is moving along at 350mph and then racks his plane over into a near vertical bank (80 degrees+) and pulls a 6 'G' turn. Speed starts bleeding off (it started bleeding off during the bank) and as speed drops if he can continue to hold 6 'G's the turn will tighten up in radius by several hundred feet Perhaps 400ft ?) however the slats won't open until near stall which may be between 250-300mph depending on altitude? Plane is dropping like a rock if he holds 6 'g's for very long. Nose is pointed down at over 25 degrees?
At fewer 'G's the radius is larger for a given speed but then the stall speed is lower so the slats don't come out until closer to 200mph?

 

[Shortround6]

Why does the Ta-152 come into almost every discussion? There were many more Spiteful, Seafang and Attacker wings flown than Ta-152s.

simply painting them on an aircraft makes it fly faster, higher and turn tighter

 

[Ivan1GFP]

It seems to me that this rather basic calculation misses a lot of real world data.
Drgondog has already commented that Thrust is the important factor rather than Horsepower.
Along with that, the propeller efficiency and pitch limits may come into play to limit thrust especially at low advance ratios.
There are also quite a few more factors affecting energy bleed as GregP pointed out. Oswald Efficiency value should come into play somewhere in this calculation because it will affect the maximum rate of turn VERY quickly.
Some aircraft such as the FW 190 and Me 109 had a maximum CL quite a bit above the average 1.4 that you are using for non Laminar Flow.
Also, as mentioned in another thread, while the J2M Raiden technically had a "Laminar Flow" airfoil, the maximum camber was IIRC at about 35% which isn't very different from the typical non-Laminar Flow airfoil used for many aircraft at this time.

- Ivan.

 

[pbehn]

View attachment 485671

simply painting them on an aircraft makes it fly faster, higher and turn tighter

In the case of the Ta 152 as far as I can see much is not known so much is just made up into mythical performance.

 

[Ivan1GFP]

It also takes a while to get down to the speed at which the infamous slats work in a turn.
Say for example our 109 pilot is moving along at 350mph and then racks his plane over into a near vertical bank (80 degrees+) and pulls a 6 'G' turn. Speed starts bleeding off (it started bleeding off during the bank) and as speed drops if he can continue to hold 6 'G's the turn will tighten up in radius by several hundred feet Perhaps 400ft ?) however the slats won't open until near stall which may be between 250-300mph depending on altitude? Plane is dropping like a rock if he holds 6 'g's for very long. Nose is pointed down at over 25 degrees?
At fewer 'G's the radius is larger for a given speed but then the stall speed is lower so the slats don't come out until closer to 200mph?

Hello Shortround6,

Perhaps you are already stating this. I am not sure, but I thought I would try to describe the slats as I understand them.
There is nothing really magical about them. They are normally held closed by dynamic (air) pressure in regular flight attitudes but as the Angle of Attack increases to a certain value, they drop from their own weight. Thus what they really do is allow a higher AoA than one would get from a wing without slats and thus generate a higher Coefficient of Lift at the higher AoA. Of course energy bleed would also increase, so nothing is free.

- Ivan.

P.S. Those crosses are amazing. I feel as if I were flying faster just by looking at them!

 

[Shortround6]

I would note that you really have to be familiar with an aircraft to start trying to calculate wing areas or coefficient of lifts with planes that used "combat flaps".

Not everybody's Fowler flaps actually moved the same. That is some moved further back than others before tilting down. Or they tilted down slightly as the they moved aft and then tilted with greater speed for only a little more aft movement.

I believe the Japanese planes used a setting of 8 degrees for combat but I don't know how far aft they traveled at that setting. A P-38 could use up to "15 degrees" for combat. However it was limited to not lowering the flaps at speeds greater than 250 mph ASI, full flap was prohibited over 150 mph ASI.

I don't know if this was the combat setting or not.
however this was full flap.

Now even if you know the area of the flaps, how much of that area do you ADD to the wing in the top picture?

 

[BiffF15]

It also takes a while to get down to the speed at which the infamous slats work in a turn.
Say for example our 109 pilot is moving along at 350mph and then racks his plane over into a near vertical bank (80 degrees+) and pulls a 6 'G' turn. Speed starts bleeding off (it started bleeding off during the bank) and as speed drops if he can continue to hold 6 'G's the turn will tighten up in radius by several hundred feet Perhaps 400ft ?) however the slats won't open until near stall which may be between 250-300mph depending on altitude? Plane is dropping like a rock if he holds 6 'g's for very long. Nose is pointed down at over 25 degrees?
At fewer 'G's the radius is larger for a given speed but then the stall speed is lower so the slats don't come out until closer to 200mph?

SR6,

Excellent post! However I would like to add that in your example the Bf-109 is flying what looks like downward spiral in a fairly steep angle. While putting the nose down to maintain energy (airspeed) helps keep the "G" on or circle small it also causes a large altitude loss. This altitude lost is turning room in the vertical for an adversary. Or in today's vernacular the 109 is giving up turning room. In the big scheme of things that's not good as it allows a less maneuverable aircraft an entry (or opening) to bring his nose to bear and employ weapons.

Cheers,
Biff

 

[Shortround6]

I don't know about later aircraft but the 109E and Spit I were lucky if they could maintain a 2 1/2-3 G turn without losing altitude once they bled off the initial speed they may have entered a turn with. Later planes with more powerful engines could do somewhat better. This is what makes climb rate related to turning.
The climb rate is an indication of surplus power near the stall speed and is the power/thrust the plane can put into maintaining altitude in a turn. What jets can do with their 25-40,000fpm climb rates is a whole new world

However the point I was trying to make was that in some cases the leading edge slots/slats won't do a darn thing for the turn rate in the initial few seconds of a turn, or longer depending on how long it takes to bleed of the speed.
Plus the often over looked fact that they only affect the co-efficient of lift on the outer part of the wing often around 1/3 of the wing area even when they do deploy.

 

[Zipper730]

The equations presented for Stall speed are correct for symmetrical (level) flight conditions.

Are there any rules of thumb to guesstimate this?

 

[drgondog]

Are there any rules of thumb to guesstimate this?

Guesstimate = level flight equations w/o consideration for the factors I presented for clarity

 

[Ivan1GFP]

I don't know about later aircraft but the 109E and Spit I were lucky if they could maintain a 2 1/2-3 G turn without losing altitude once they bled off the initial speed they may have entered a turn with. Later planes with more powerful engines could do somewhat better. This is what makes climb rate related to turning.
The climb rate is an indication of surplus power near the stall speed and is the power/thrust the plane can put into maintaining altitude in a turn. What jets can do with their 25-40,000fpm climb rates is a whole new world

However the point I was trying to make was that in some cases the leading edge slots/slats won't do a darn thing for the turn rate in the initial few seconds of a turn, or longer depending on how long it takes to bleed of the speed.
Plus the often over looked fact that they only affect the co-efficient of lift on the outer part of the wing often around 1/3 of the wing area even when they do deploy.

Hello Shorround6,
I believe that the method described by Laurelix97 wasn't really about steady state turn performance. It was more about instantaneous turn rate, thus engine power / thrust really has no effect in the instantaneous case other than to provide an additional vector (because of AoA) to supplement lift.

The other point I was trying to make regarding the leading edge slats on the Me 109 is that it really doesn't depend on bleeding off speed at all. It is simply a matter of balanced forces which at a certain AoA allow the slats to deploy and raise the angle for stall a few more degrees. If the control forces allow enough control to pull to that AoA ....AND.... if the aircraft structural limits allow it, then there is no reason that the slats could not deploy at any speed,

- Ivan.