This post covers some basic aspects about optimizing turn performance. While we perform turns on a daily basis, they do not necessary get the attention they deserve, as overestimating your aircraft’s turn capability can be as embarrassing as lethal. If you are faced with a VFR approach in mountainous terrain or a demanding circling approach, it is certainly wise to do the math before take-off. In particular, being able to calculate turn radius for other than standard-rate turns is vital. See below some handy formulas:
Figure 1: Formulas for level turns from my notes based on [1] and [3]
A short literature review reveals, that turn performance can be limited by several factors, such as aerodynamic, structural or performance constraints [1]-[4].
In military aviation, good turn performance is critical and military pilots are thus familiar with the term “corner-speed”, the speed where maximum turn capability is achieved [2]. The idea is simple: Fly the limit load factor at the lowest possible speed. This makes sense when looking at the equations above. If you are familiar with an aircraft’s maneuvering envelope, it is obvious that this speed is equal to the "maneuvering speed". There is a catch however: The power/thrust required increases dramatically, in fact it increases so much, that almost no aircraft is capable of maintaining “corner-speed” at the limit load-factor [3]. The secret lies to a large extent in the composition of the drag coefficient:
Figure 2: Drag coefficient composition (my notes based on [2])
The induced drag coefficient (red) is proportional to the square of the lift-coefficient. This equation holds a very important message: When you are flying a relatively “low” airspeed (or high lift-coefficient) and you try to turn, the increase in power/thrust-required is much greater than when flying “fast” (or low lift-coefficient) and then doing the same. You might have noticed this when doing slow-turns. This is also relevant when an airplane with normally-aspirated engines performs turns at high altitude, as your typical cruise-EAS decreases with altitude [4]. Sustained turn performance is therefore often limited by thrust/power available.
Saarlas [4] demonstrates, that maximum sustained turn performance is achieved, when the aircraft is operated at the maximum lift/drag ratio and simultaneously at the maximum thrust/weight ratio.
The following graph from my notes is based on information from [1] and aims to summarize the contraints:
Figure 3: Typical turn capability (Radius vs. EAS) constraints at a given altitude (my notes based on [1])
Point A shows the minimum sustainable turn radius, whereas point B depicts the minimum instantaneous turn radius. Usually, the aerodynamic and structural limit come into play for the instantaneous case, whereas the power/thrust and aerodynamic limit dictate the sustained case.
In this context, it is worth thinking about what happens to the flight path angle (FPA), if you perform a turn from straight and level flight without adding power/thrust (maybe because there simply is no excess power/thrust available). This consideration is based on a derivation from Russel [3], assuming you maintain the cruise-lift-coefficient (i.e. AOA) and is valid for shallow descents only.
Figure 4: FPA during turn with level-cruise thrust (my notes based on [3])
The message here: When there is no power/thrust to spare and we want to optimize FPA for shallowest descent, we should maintain the highest lift/drag ratio.
I tried to avoid posting absolute numbers because there may be significant differences amongst different aircraft types. The idea is to have a conceptual understanding of what the limits might be and then make yourself familiar with a particular airframe's performance.
Rev/20181022
References
[1] Roskam & Lan, Airplane Aerodynamics and Performance, DARcorp. 1997
[2] Anderson, Aircraft Performance and Design, Tata McGraw-Hill, 1999
[3] Russel, Performance and Stability of Aircraft, 1st edition, Butterworth-Heinemann, 1996
[4] Saarlas, Aircraft Performance, Wiley, 2007