If you let go of the steering wheel of a (properly maintained) car, it will typically go in a straight or nearly straight line. Would a powered airplane in no-wind conditions do the same thing?
If the airplane is properly trimmed, the airmass is smooth, and the aircraft is inherently stable in its design, then yes - many airplanes are capable of flying straight with only very light and occasional inputs. In fact pilots are trained to trim so that very little correction is needed to maintain straight and level flight.
P.S. "Wind" is simply the relative movement of a mass of air across the surface of the earth. This concept is lost on the airplane, that knows only its own movement through this airmass. The airplane responds only to instability of the airmass in the form of turbulence, gusts, or windshear, all of which have the potential to upset its equilibrium. The conditional statement "without wind" is actually irrelevant to the question.
Yes. Most aircraft are designed to be inherently stable. That is, if you just let go of the controls, the aircraft will return to whatever it is trimmed to - typically stable and level flight.
Trims are redundant controls (rudder, elevator, ailerons) which are controlled by a "set-and-stay" knob rather than a dynamically movable yoke or pedals. Turning the elevator trim knob has the same effect as pushing or pulling the yoke, for instance. So if the pilot wants to hold a rate of climb or descent (often 0), they set the trim knob rather than use muscle to hold the yoke, which would be fatiguing.
If the trims are used conventionally, the pilot doesn't need to hold the yoke at all, and the airplane will keep doing what the trims are set to do, typically straight and level flight.
Generally, aircraft which are not inherently stable will inherently fall off-center, which will cause a vicious cycle and will rapidly exceed flight maximums. They must be kept at center by constant correction. The workload involved is generally too intense for pilots, so they must have some sort of automation to do it for them.
The real answer is that it would fly fairly straight. This was realised by the Wright brothers who were bicycle makers.
Both for bicycles and aircraft you have a choice between stability and manoeuverability. Some very early aircraft were too stable. Set on a straight course into a hillside they would follow it. The ideal is to be slightly unstable with a long natural period. That way a straight course can be imposed, by pilot or autopilot, with very little exertion, but if a turn is needed it can be done quickly
Just to add something to the already given answers.
Hands-off, an airplane is stable but it moves a bit around this stability.
As already explained, if the airplane is stable (inherently or artificially) and trimmed then it is going to fly just dead straight. Anyway even within this boring flight path there exist "stable-instabilities" which are not necessarily triggered by wind/gust and which are going to give a
Let's consider a jetliner and let's say that, due to whatever reason (except obviously wind/gust: pilot nudges on the pedals, turbolence over airplane's surface is not perfectly symmetrical among left and right side, ...), it yaws a bit nose-right. The following picture shows this yaw in an exaggerated manner:
What happen now? Well, as said the airplane is stable so it somehow reacts to this initial yaw trying to return to its straight position... but not that simply. Qualitatively:
- Drag and lift generated on the left wing possess now a bigger leverarm in respect to the CG and therefore the left wing tends to go up (bigger lift's leverarm) and back (bigger drag's leverarm) creating a left-up roll and a yaw opposite to the initial yaw.
- The left wing rotates/moves against the airflow and therefore its local airspeed increases, while on the right wing the local airspeed decreases. Bigger speed means bigger lift and drag and therefore the left wing tends to go up (bigger lift) and back (bigger drag) creating a left-up roll and a yaw opposite to the initial yaw (just like in 1.)
- The vertical stabiliser possesses now an AoA different from zero and produces a lateral force. This lateral force is also acting against the initial yaw but!
- It is also located above the CG (seen from the back) and it creates therefore a left-up roll as well.
- Due to this roll, the local AoA on the lowering right wing increases while on the rising left wing, it decrease. Lift becomes therefore bigger on the right wing and this bigger lift opposes to the initial roll and levels the airplane off.
That's a lot of events for only a light nudge on the pedals. And it is not yet the end of the story: these restoring yaw and roll don't make only the airplane go back to a straight flight but they actually make the airplane overshoot it's initial straight condition. The airplane continues the yaw and the roll movement on the other side! And then, since it is stable, backward again! This nice picture depicts this "stable instability":
It is called Dutch roll and it is one of the few typical and well known "dynamic modes" of airplanes.
So, hands-off an airplane is stable but it moves a bit around this stability.
If the airplane is configured to fly straight and level, then it'll remain that way until the pilot decides to touch the controls. This is especially true if the aircraft has an autopilot system. In a no wind scenario, the airplane will settle in straight and level flight at a particular angle of attack at a given airspeed. As long as the aircraft elevator can be trimmed to maintain this angle of attack, then the pilot will not have to give pressure to the yoke. If the air was not 100% calm, then adjustments may have to be made, but depending on the positive stability of the aircraft, it'll stabilize and regain straight and level flight. So the answer to your question is, generally yes.
There are many airplanes which will eventually tend to slowly roll into a banked turn if the pilot lets go of the controls (and takes his or her feet off the rudder pedals) in smooth air. If the aircraft is truly perfectly trimmed in roll, and the air is free of gusts (not wind-- a steady wind is irrelevant), then the direction of the roll and turn (left vs right) will be random.
An airplane's tendency to stay wings-level with no control inputs is called "roll stability" or "lateral stability".
Design features that tend to enhance roll stability include dihedral, sweep, and a high-mounted wing. Design features that tend to make an aircraft unstable in roll ("spirally unstable") include a low-mounted wing, a long wingspan (because if the airplane starts to turn at all, the outboard wingtip describes a larger radius and thus experiences more airspeed and generates more lift than the inboard wingtip), and a very large vertical fin (because it minimizes sideslip in turning flight, and sideslip is fundamental to how dihedral, sweep, and high-wing placement all contribute to roll stability.)
Examples of aircraft that will eventually roll into a banked turn in the absence of control inputs from the pilot (or autopilot), even when flying in smooth air and trimmed to be perfectly balanced in roll, include nearly all sailplanes (and motorgliders), and all purpose-designed aerobatic airplanes with a mid-wing configuration with zero dihedral. All modern hang gliders also exhibit this behavior -- earlier "Rogallo" designs did not, and would stay wings-level when flown for extended periods without the pilot touching the control bar.
In an aircraft that tends to roll into a turn when flown hands-off, as the bank angle increases, the airspeed (and therefore the wing loading, which is essentially lift force per unit weight) tends to increase.1 The flight path tends to descend. The result is often called a "diving spiral". In some cases-- but not in all cases-- in the absence of further control inputs from the pilot, the bank angle will tend to reach a maximum value and stop increasing before the aircraft is damaged. In other cases, in the absence of pilot intervention, the aircraft is likely to be damaged or destroyed either due to overloading, or due to aerodynamic flutter from overspeeding.
I suppose that in theory, if the air were absolutely smooth (free of gusts), and the test were initiated with the wings perfectly level, an airplane that was trimmed to be perfectly balanced in roll would not ever enter a turn, even if it had a configuration such that it was inherently unstable in roll. But those conditions are never present in reality. To expect this to happen in the real world, would be like expecting a nail balanced on its point on a hard marble slab to stay vertically upright, because there is no reason for it to fall in any one particular direction. In the real world, to stay wings-level indefinitely in the absence of corrective inputs from the pilot or autopilot, and not eventually roll into a turn, an airplane must have positive lateral stability (roll stability), and not all airplanes meet this criterion.
There's a more fundamental point at play here though-- up to this point, this answer has addressed an airplane's tendency, or lack thereof, to roll into a banked turn. But even in an aircraft with strong "lateral stability" or "roll stability", that will roll back to wings-level after any temporary disturbance, there is no tendency for the heading to stay constant over the long run. Each time the bank angle is slightly disturbed, the heading will change a little bit as the airplane turns slightly. Expecting the heading to stay perfectly constant over the long run, is like flipping a coin a hundred times and expecting to come up with an exactly 50/50 split between "heads" and "tails". Just as if you had a car that was correctly aligned so that it had no tendency to turn-- if you drove out on a flat dry lakebed where you could let the car run for miles on end without touching the steering wheel, you wouldn't expect the car to to end up heading in exactly the same direction it started out on, unless perhaps it was following a rutted track left by another vehicle. In essence, an airplane with positive "roll stability" or "lateral stability" "knows" which way is up and actively returns to wings-level after a disturbance, but it has no way to "know" which direction it was travelling before that disturbance.2
The angle-of-attack actually tends to decrease somewhat as the bank angle increases and the flight path curves, but not nearly enough to stop the airspeed from increasing.
An anecdote-- I once watched a novice radio-control flyer fly his model airplane-- he was constantly trimming the rudder, and I asked him why. He said "well, I want it to keep flying into the wind when I take my hands of the sticks, but it won't. It wants to turn out of the wind." He thought that he could trim the plane to hold a desired heading, and he was mis-interpreting the plane's random deviation from the desired heading as a specific tendency to turn downwind. It doesn't work that way-- the best the pilot can do is to trim out any tendency to enter a banked turn, so that it will tend to return to wings-level after any slight disturbance. He can't trim the plane to hold a specific heading. By changing the rudder trim whenever the airplane randomly wandered to a slightly different heading, the pilot was actually interfering with, not enhancing, the airplane's inherent "roll stability" or tendency to return to wings-level after a disturbance. A pilot should trim to remove any consistent rolling or turning tendency, not in response to any random deviation from the desired heading.