Rudder and ailerons have different purposes and control rotation about two different axis, but their use is coordinated since a rotation about one axis induces a secondary rotation on the other one.
The rudder use is mostly to prevent unwanted yaw movement for safety and comfort, or to force yaw against the aircraft tendency to keep the aircraft facing the relative wind (e.g. for a cross-wind landing). See Understanding the use of rudder for a more comprehensive explanation.
A turn is performed using the ailerons, but for a non-small turn an adverse yaw opposite to the turn develops, the rudder is used to maintain the desired attitude and what is known as a coordinated turn.
When a yaw damper is available, coordinated turns can be conducted without pilot action on the rudder, as the yaw damper controls the rudder automatically.
When speed and altitude increase, using the rudder to turn around the yaw axis with the wings horizontal, tends to make the aircraft flies in crab in the same direction rather than to turn it. Turning this way would require a larger rudder area and would generate additional drag. So this is not the way an aircraft is steered in flight.
Banking turns the aircraft
An aircraft actually turns by changing the bank (or roll) angle. When the aircraft spins about this roll axis, the direction of the lift is changed, and can be seen as having two components:
The horizontal component of the lift moves the aircraft away from its original path and a turn starts.
Yawing aligns the aircraft with the wind
Lowering the aileron on a wing makes it more efficient, it generates more lift and raises. But generating more lift also induces more drag.
This drag in excess tends to slow the aircraft on the upper wing side. This creates an adverse yaw movement in the direction opposite to the one desired.
The pilot uses the rudder to counter the adverse yaw and keep the fuselage axis facing the airflow, countering slip and skid effects. This is known as performing a coordinated turn.
The amount of rudder to apply can be determined using (human senses and) different instruments, including the turn coordinator which is a sort of level usually associated with a bank or rate of turn indicator.
When executing a coordinated level turn without wind, at sea level, etc, the radius of the turn and the turn rate depend only on the airspeed and the bank angle:
- TAS 150kt, bank 20°: Radius 1,676m, 360° in 2:16.
- TAS 300kt, bank 30°: Radius 4,227m, 360° in 3:51.
See this question for additional information.
Ailerons deflection reduced in the turn
Note that when the desired bank angle is reached, the pilot reduces the ailerons deflection. The excess of lift and drag on the upper wing decreases too, as does the adverse yaw. The rudder deflection can be reduced too.
Normally the aircraft should maintain its bank and continue to turn with ailerons in neutral position, due to the upper wing flying faster and generating more lift.
However the aircraft in a rolled attitude actually slips into the turn and the dihedral effect tends to decrease the roll angle of the slipping aircraft (roll stability). So some degree of aileron deflection will be maintained.
The ailerons (and the rudder) are used to return the wings to their horizontal position and stop the turn more accurately.
In commercial airliners, the yaw control is assisted by one or two yaw damper(s) which function is to increase the yaw stability and prevent the development of the dangerous Dutch roll. Thanks to the yaw dampers, the pilot does not need to use the rudder pedals for coordinated turns.
Other means of controlling roll/lift
A mean of banking wings directly is required, including when Dutch roll and spin start developing and when flying or rolling with crosswind. But bank control can be obtained by other means than ailerons. While ailerons add lift by changing the camber of the wing,
spoilerons combine spoilers and ailerons, and break lift by disturbing the laminar airflow (B-52),
rudderons combine rudder and ailerons on flying wings (B-2A),
and now bleed air and Coandă effect can bank the aircraft without moving surface (Flaviir project).