Aimed towards lighter piston planes including a powered glider, if a pilot wants to extend a plane's range unconventionally, could flying like a glider by climbing and then gliding down or would flying straight, lower or slower save fuel; even if it means extending flight time?
This approach of using an engine is called pulse and glide. It generally works because each engine has an optimal power setting at which it converts fuel into power most efficiently.
If the most efficient power is higher than is required, something should be done to accumulate and later use the excess energy. Raising the vehicle up looks like a solution, better than accelerating.
This method can only work if the power to efficiency curve of the machine is known. Otherwise, it is easy to do worse.
Here is one thing that WILL save fuel-- where possible-- such as when flying VFR. Note that this strategy only applies within the layer where thermal convection or orographic uplift are significant, not at higher altitudes where the air is generally not rising or sinking to any significant degree (apart from wave lift in which case the strategy will still apply!).
Don't try to hold a constant altitude on the second-to-second or minute-to-minute timescale. Instead, hold a constant airspeed, while leaving your power setting fixed. If your plane is climbing in an updraft, let it. If your plane is descending in a downdraft, let it. Then on a longer timescale-- maybe every few minutes or 5 minutes-- make a power adjustment (or airspeed adjustment) until you find the power / airspeed combination that gives an AVERAGE sink rate of zero over the long run.
The point being that if you make elevator inputs as needed to hold a constant altitude as you fly through updrafts and downdrafts, while operating on the front side of the power curve as we generally do, you end up speeding up in "lift" (updrafts) and slowing down in "sink" (downdrafts), which tends to maximize your time spend in downdrafts and minimize your time spent in updrafts -- the exact opposite of what a glider pilot would do to maximize efficiency. So for a given total fuel burn over a given range, your average airspeed will actually end up being lower when you fly the conventional way, i.e. when you maintain a constant altitude. Or for a given average airspeed, your fuel efficiency will be lower when you fly the conventional away.
You can take this idea one step further by slowing down a few knots in updrafts and speeding up a few knots in downdrafts-- providing that you don't take it to such an extent that it interferes with your ability to detect whether the air is rising or sinking, i.e. that the resulting changes in vertical speed don't dwarf the changes in vertical speed caused by the atmospheric movements.
Now, as for your idea of a powered climb followed by a (power-off) glide-- if you can feather the prop, there might be some benefit to this strategy, especially if you shut the engine down completely during the glides. As long as the engine is running, you might as well get the most out of it by producing enough power to climb.
Your question should emphasize "still air" conditions and not get into exploiting vertical air motion, which is a separate issue completely.
For normal airplanes, the basic issue is, you want to spend the enroute phase of the flight at the end of an optimal climb profile, with an optimal cruising altitude (which will depend on winds) for as long as possible, with a Top of Descent that starts at the perfect location to complete a power off glide to the runway. This is what airliners and pretty well everybody else does, or strives for.
Gliding and climbing takes you away from an extended time at optimal altitude and power settings and replaces them with intermittent climbs at non-optimal power settings and altitudes, with the glides in between. The total energy needed to complete the flight will be higher, for the same overall profile.
Bottom line is, if there was any efficiency benefit to doing that sort of thing, somebody somewhere would be doing it, and absolutely nobody does.
Keep in mind that no mechanical, power producing system can operate with 100% lossless efficiency. In heat engines, like those in aircraft, the Carnot model describes the theoretical maximum efficiency (power produced vs lost) in any given design. Another way of asking your question would be, "Can I save fuel by taking my foot off the accelerator in my car?" Perhaps, the answer could be "Yes" of you utilize the environment to introduce power back into the system - like allowing gravity to accelerate you down a hill. But ultimately, the net power lost would have to be less than in Vbe cruise flight.
The problem is, if you must constantly regain altitude lost from gliding , you will have a net loss in energy when you climb to regain it again. IF (that's a big 'if') you were in a light enough plane AND (that's a big 'and') you were able to introduce supplemental energy into the system from somewhere besides the engine, like rising in thermal updrafts, you might save fuel by doing this, but at that point, you're just flying a glider (and one that is poorly equipped to fly in that way).
Every meter of progress toward your destination is paid for by energy. At Vbe, you know you are getting the best price per meter out of your engine. Any meter flown above or below this speed means you are overpaying, unless it is offset by free energy from the environment - gravity, updrafts, tailwinds, etc. Rather than tallying over-expenditures vs situational energy gains - hoping for a net improvement, which would require a delicate balance of variables, many of which are out of your control and difficult to predict, the surest way to achieve maximum fuel efficiency for distance is to cruise at Vbe.
Remember when Major Gant (Clint Eastwood) stole one of the Firefoxes and was running out of fuel? There was a climb, a glide, and a dramatic landing on the ice floe (it was a movie). Was that the best way?
Perhaps so. One might imagine (for greatest distance), climbing at Vy, then gliding at Vbg. There would be some TAS benefit by getting as high as possible as well. This will also give you more options as to where to glide should gliding become necessary.
For lighter piston planes, with fixed pitch props, one can consult the POH to get fuel burn rate per hour and literally calculate best miles per gallon at a given power setting. So, one might try a Vy climb at optimal fuel burn RPM, if it is better than full open throttle (lower RPMs would allow a leaner fuel/air mixture).
But you can save fuel by getting best climb performance at Vy and best glide performance at Vbg.
Good trip planning should provide alternate landing sites to avoid this scenario, and, particularly when flying into a head wind, the field you passed may be your best choice to land.
To approach this with some napkin math / intuition... In general, you want to maximize time gliding and minimizing time climbing, as that'd minimize fuel burn. Lets say you had a O-320 that burned 12gph in climb at max power. And for your plane Vy was 75kts w/ 500fpm climb rate and a 100kt cruise was 8gph, and coincidentally best glide was 75kts w/ a 800fpm descent rate. These are ballpark of a Cherokee/Skyhawk...
If you climbed 12min at 500fpm you'd end up gliding 7.5min, because 500fpm/800fpm = 0.625 and 0.625*12min = 7.5min. Your total time for climb+glide would be 22.5min. Using a total fuel of 12gph * 0.2hr = 2.4gal fuel burn from the climb. You'd cover 28.125NM = 22.5min/60min/hr * 75kts. Giving a fuel economy of 28.125NM/2.4gal = 11.72NM/gal.
In comparison, in normal cruise you'd be doing 100kts/8gal/hr = 12.5NM/gal
If this planes best glide descent rate magically became 500fpm (same as the climb rate) then you'd get a 12.5NM/gal economy when using climb+glide. This convenience is only because best glide speed equals best climb speed, which are usually relatively close on many GA planes. Naturally, for powered gliders their glide performance is much better than their climb performance so the climb glide method would be more efficient + more range.
Although, gliding may not be the best choice for every plane? To really solve this you'd need to do a system of equations defining the physics of aircraft performance...
To solve this you'd need:
- the plane's drag polar, how much drag do I make?
- relationship of propeller efficiency to advance ratio, how much power do I need to put into the prop to make thrust = drag?
- relationship of engine's BSFC to RPM, how much fuel does it take to make the required power for the prop?
The tricky thing here is keeping assumptions reasonable, the classic "garbage in, garbage out" challenge.
Tangent story: Once when flying a tow-plane with a low-pitch fixed pitch prop, I did a similar climb/descend profile to get a faster average cruise speed. In level flight you could do only about 85kts before redlining the engine, yet you could still climb over 500fpm at full power at 85kts. So, I'd climb to 8000ft-ish then establish a partial-power 200fpm descent at +100kts to my destination. The end result wasn't much better, +10kts gain and certainly more fuel burn, but it made me feel better and was fun :)