The important thing to realize that flaps itself does not in any way increase the turn rate. It's essentially a secondary mechanism to use in trying to maintain a very delicate balance between the turn speed and turn radius.
In high speed fights, they assist by dragging the speed down. In low speed fights, they react in a very different manner as the effect of lowering the stall speed comes in more important than the 'speed brake' aspect of flap use.
Usually, or rather, almost instinctively people understand the importance of maintaining a tighter turn radius - that's where the flaps come in. The US planes are usually equipped with combat flaps which is applicable up to about 400mph.
Deploying the flaps will drag your speed down, which then enables you to pull a tighter turn radius - imagine that your car is travelling a corner. It is especially useful when fighting planes like the La-7 or the Bf109G-10 - both planes are extremely fast, and yet their ability to utilize the speed is limited.
In the case of the Bf109 the elevator authority is greatly effected by speed, which in this case the US plane utilizing combat flaps will very easily turn inside the target plane. In the case of the La-7, this bird is just itching to accelerate, and the vast majority of mediocre pilots that populate the La-7's cockpit usually means that they turn with full throttle engaged - which will lead them to a critical black-out. To avoid this the pilot commonly pulls only a limited amount of stick deflection(instead of thinking to reduce his own speed) - which will allow the the US plane (which is both deccelerating faster and pulling more stick deflection) to turn inside its target.
The problem is, as said, engaging combat flaps to turn achieves a better turn radius by killing one's own speed - it's a double-edged sword in that by doing so, the loss of speed may lead him to a extremely disadvantaged position where he is unable to escape, should he ever be overshot by the enemy. So, generally, using flaps to chase an enemy plane in a turn usually means you are prepared to commit to that fight, and leave the field as the winner - since clearly, if you fails, you will be the loser.
Thus, many pilots may choose to just momentarily engage the combat flaps to assist in a short, pre-determined length of chase during a turn. Meaning: if the use of flaps will enable your plane to turn inside the target enough to get a shot in, do it. If it looks like you're not going to be able to get a shot in, don't use the flap, save your speed, and prepare/maneuver for a second pass.
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Now, the more complicated use of flaps is related usually with slow-speed fights, referred commonly to as "stall fights" or "knife fights". Essentially the flaps work in the same manner as in high speed fights, but there is also another effect of lowering stall speed which comes to consideration.
Many planes in AH(almost all of them) can start deploying flaps when is typically under 200mph TAS - as you may realize, this is an EXTREMELY low speed to maneuver at, which holds of potential dangers of stalling your plane at a critical moment. In this case, a plane may be able to pull a tight radius, and then immediately stall out because the speed is too low to stay in the air.
Let's imagine a fight between a Bf109G-2 and a Spit9 - both planes are simular in top speed. The 109 holds an advantage in acceleration and climb, the Spit9 holds an advantage in armament and maneuverability. At a low speed the Spit9 starts to turn, and the 109 tries to stay behind. Essentially, as the speed goes under 200mph the 109 starts to lose ground in the turn fight because he can't maintain a high angle of attack as the Spit9. Also, the 109 is nearing its stall speed.
In this instance, one can try engaging flaps - even more than 2~3 notches if necessary. As explained with the use of flaps during high speed fights, the flaps will kill the speed. What's more however, is at the same time the speed is dropping, the flaps will also lower your stall speed. For the sake of reasoning, if we assume that the 109 stalls out and cannot maintain the turn anymore at about 100mph, the engagement of flaps will enable the 109 to stay afloat for extra 10mph and allow it to maneuver at 90mph minimum.
So, when the Spit9 starts turning, the 109 pilot, as he begins the chase, will momentarily engage 1~2 notches of flaps. His speed drops down, but the stall speed is lowered. So, through the sacrifice of the turn speed, the turn radius can be tightened a bit - since the lowering of stall speed will enable the pilot to pull the stick a bit more and turn a little bit harder.
But the problem is, the flaps keep dragging down the speed, so the positive effect of pulling a harder turn cannot remain indefinate (since, despite the short increase in the ability to pull a tighter turn, the speed will drop down again).
So what happens, is basically as the Spit9 turns at a certain speed and certain radius, the 109, seeing that he cannot match that turning speed and radius, gives up on the speed part and while moving at a very slow rate, pulls harder on the turn - the ideas to get a shot in and disable the enemy plane before the moment passes.
After the moment passes, the 109 pilot has to make a critical decision if he fails to bring the enemy plane down - the use of flaps has enabled him to pull a tight radius, but the loss of speed takes its toll if he keeps that way. Even if he may be able to pull a tighter turn radius than the Spit9, the speed is so low, that the Spit9, while pulling a larger turn radius, still might end up crawling towards the six of the 109.
What typically happens when the 109 pilot keeps his flaps down and still tries to follow the Spit, is that he is turning a tighter radius than the Spit, but not enough to get a second chance at shooting. In the 109 pilot's shoes, it would seem like the Spit9 is suddenly moving faster and faster. He is thinking "if I can pull a bit more..", but the odds are that's never gonna happen. So the Spit9 seems to accelerate, just out of reach from the gunsights, and then slowly but firmly closes in behind the 109.
So, the critical decision is this:
First choice is to keep the chase. It may be possible to bring the Spit9 down if its pilot is not very skilled. If the Spitfire is using full throttle to turn, there's a good chance that by keeping the chase with flaps dropped down, the 109 would get a second chance to get a shot in.
The second, and more logical choice is to raise flaps and slowly pull out of the turn. Keep a keen eye upon Spit9 - if he continues the tight turn which you cannot follow, raise flaps, engage WEP and slowly increase your turn radius until you reach level flight. The Spit has to turn a large circle to get behind you, and by the time he does, your acceleration and speed will push the 109 away from the Spit9 to a safe distance - where you may start climbing, utilizing the climb advantage.
If the Spit9, seeing that you do that, decides to scissor into you(this, is more dangerous to you if the Spit pilot is skilled), engage flaps and lower throttle (to dump speed once more). While the end-result of scissoring is more dangerous to you, the beginning phase of the scissors usually gives you another opportunity to shoot at the Spit9, as it will most likely cross in front of your sights.
Another possible move, seeing that the Spitfire starts to scissor into you, is to try your luck, momentarily turn into him, and accelerate away as fast as possible. If you can leave outside the gunnery range unhurt, the end result would be safer than the above choice to engage flaps again and try to shoot the Spit9 down while scissoring: this is because your 109 began accelerating before the Spit started to scissor into you. And as the flight path crosses once, the Spitfire will have to change flightpath again(which takes away some speed) to aim at you. (This move is more practical in IL2/FB than in AH, due to the difference of fatal gunnery ranges)
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So, as with the example of the Spit9 and the G-2 described, using flaps to do anything during low speed fighting, is usually associated with globs of confidence - absolute confidence that your opponent is less skilled than you in everyway, so it is worth the risk to engage him in such a manner.
That is why usually, in normal conditions, getting into a fight which may require the use of flaps in the first place, is not a very wise thing to do - as you never know the enemy pilot may be more skilled than you. The general rule of thumb is, don't risk a flap-using fight unless your plane can maneuver better at low speeds than your opponents.
