Yup. As **GusNSpot **so eloquently said: curved, bent, short, narrow, bumpy, & twisty runways are all found all over the rural part of the world. And work OK-ish for bush planes and the like. He lived and worked in that world for decades and lived to tell his tales. So it can be done.
For bigger or faster airplanes the curve would have to be so slight that no real benefit would result. There also aren’t that many places that have both a need for bigger or faster airplanes and also don’t have the economic resources to build an adequate straight-runway airport almost regardless of the local topography.
Having said that, there are plenty of bizjet or airliner airports in the world where the runways are straight but due to terrain in the immediate area, the aerial path to or from the runway includes pretty aggressive turning pretty much immediately after lift-off or immediately before landing.
And there are some one-way airports where terrain is so close to one end that you can’t takeoff towards it or land over it. Since landings & takeoffs are in opposite directions that poses some interesting air traffic management problems to avoid head-on flight. Plus no matter which way the wind is blowing at any time, either takeoffs or landings will be downwind.
One-way airports also have the fun feature that when you are landing over the (relative) flats and aimed at the mountain, you may not be able to successfully go around from a screwed up landing or other failure in close to the runway. After a certain point on the approach you become committed to landing on the runway or doing the best crash you can as close to the runway as you can. Flying away for a second try is no longer an option.
Okay, I want to play, too! Suppose instead of just a curve or dogleg in the runway, it’s an actual banked track, like a motor speedway. Banking the plane isn’t necessary since it’s already at an angle, and the runway is banked at a proper angle to keep the wings from scraping the ground.
For any radius of curve and amount of bankedness, there is exactly one speed which balances gravity & centrifugal/centripetal force such that no “turning” is required and the vehicle naturally follows the curve.
You can experience this on any cloverleaf freeway ramp. Crawl around it in traffic and the car wants to turn inwards & fall into the center. Conversely, hit it going too fast and the car wants to high-side. But if you hit it exactly at the design speed you don’t need to make much steering input at all.
Cloverleaf ramps are typically designed so at the posted speed you still need to steer inwards; the bank only offsets part of the centripetal/centrifugal force. But there is a speed, usually about 15-20 mph below posted where you can drive up to the start of the banked part, release the steering wheel and the car will follow the curve on its own. Speed up to go higher, and slow down to go lower; you can steer with the gas pedal. It’s fun to play with as long as you’re not obstructing somebody close behind you.
All this physics poses a problem for a curved runway. On takeoff no two airplanes accelerate exactly the same. So no matter what curvature & bankedness is chosen, it won’t be right for most takeoffs by most aircraft. Landings are similar; everybody touches down at a different speed and slows a little differently than everybody else.
To be sure, all this means is we could use bankedness on our curved runway to offset some, but not all, of the problems with a curved runway. Careful engineering towards a happy medium could in principle make this a manageable variable, just as we now manage crosswinds.
BUT …
By definition, a takeoff is a continuously accelerating maneuver. IOW, we start slow and end fast. So as we roll down the runway the ideal shape would have increasing radius and/or increasing bankedness to offset the increasing speed.
And landing is, by definition a continuously decelerating maneuver. IOW, we start fast and end slow. So as we roll down the runway the idea shape would have decreasing radius and/or decreasing bankedness to offset the decreasing speed.
So you can’t use the same curved banked runway for takeoffs and for landings. Oops. And that’s the straw which IMO breaks this silly camel’s back.
Doesn’t this also increase the G-forces the passengers and crew experience? Those forces would be the vector sum of the plane accelerating and the inherent one to traveling in a circle. And they wouldn’t be at the normal angle of straight back into the seats, it would be a vector sum that would point back and to the side relative to a passenger or crewmember facing forwards. You could rotate the seats but that would create even more problems.
If the curvature and bankedness of the runway was exactly correct for the speed at every point along the takeoff roll you’d only feel slightly increased G straight down from head towards feet.
That extra force being straight down is exactly the result of the correct fit of bank & curve versus speed. Which is also what makes the vehicle turn appropriately without the need for steering inputs.
If the bank was too steep, or the curve too gradual for the speed of the moment the aircraft would start to fall into the center of the curve. If the pilots let that happen the people would still feel only vertical G in their reference frame. If the pilots instead steered to maintain track, they’d be steering to the high side and then the people would be feeling themselves wanting to slide sideways out of their seat towards the low side.
The opposite results & sensations would occur if the bank was too shallow or the curve too tight for the speed of the moment.
In either case people will feel the same directly rearward acceleration they do now simply due to increasing speed. And again you can feel this yourself on any freeway cloverleaf. Heavily bolstered car seats make the sensation a bit more subtle than it was back in the old bench seat days.
As an aside, humans have two reinforcing defects in their fore-aft accelerometers.
One is that under acceleration their neck is not fully rigid. So the head tends to tilt backwards and most folks coordinate transform and stabilization system isn’t perfectly calibrated. Nor is it instantaneous. So they perceive some amount of body rotation that isn’t real.
The other defect is crosstalk coupling between sensors that perceive rotational motion in the head-nodding plane with separate sensors that perceive direct nose-to-back-of-head acceleration.
Normally the visual system provides a damping input that drives the overall percept closer to the true motion & acceleration of the whole person. But in situations with limited or no external visual cues, such as flight in clouds or sitting in the center seats of a widebody aircraft, these defects produce illusions where pitch upwards or downwards in the aircraft’s pitch axis is misinterpreted as acceleration fore and aft, i.e. change in speed. Or vice versa. I wrote a bit more about this here http://boards.straightdope.com/sdmb/showpost.php?p=18502169&postcount=103