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According to Wikipedia, the Ingenuity helicopter is an astonishing piece of engineering having some attributes beyond the equivalent of the classical helicopters:

The lower gravity of Mars (..) only partially offsets the thinness of the 95% carbon dioxide atmosphere of Mars thus making it much harder for an aircraft to generate adequate lift. The atmospheric density of the Red Planet is about 1/100 as that of Earth at sea level, or approximately the same as 87,000 ft (27,000 m), an altitude never reached by existing helicopters. To keep Ingenuity aloft, its specially shaped blades of enlarged size must rotate at a speed at least 2400 and up to 2900 rpm, or about 10 times faster than what is needed on Earth.

I am wondering if the research done to make Ingenuity work can be used to improve helicopters flying in Earth's atmosphere or it is solely useful for very narrow cases like flying in very low pressure? Is the special design applicable to increase the maximum altitude for helicopters flying over Earth?

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Physics and aerodynamics are not magically different between Earth and Mars, so yes, everything we know about helicopters in general and about Ingenuity in particular is applicable on both planets.

You could apply the design of Ingenuity to a helicopter on Earth. You would need to make it three times lighter, of course, to counteract the triple gravity.

But the question is not so much: can we fly something like Ingenuity on Earth. The question is: what's the point of a helicopter that can fly at 27000m but runs out of battery after 180 seconds, is only the size of the palm of your hand, and can't carry any payload? It doesn't even have enough power to get up to 27000m on its own, so you need to deploy it from somewhere … but if you have an aircraft at 27000m anyway, then what do you need the helicopter for?

Helicopters are terrible aircraft. They are very complex, very inefficient, and very expensive. Their only advantage is their maneuverability in tight spaces and their ability to hover in place … but there aren't a lot of tight spaces at 27000m. And at 27000m altitude, an airplane flying in a circle is, for all intents and purposes, hovering in place compared to the ground. So there is simply no need to have a helicopter there.

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    $\begingroup$ I disagree until the end. There is one use for helicopters at such altitudes--high altitude rescue. Your circling plane can't pluck someone off Everest. I can't see Himalayan rescue being a big enough business to cause anyone to design a helicopter to operate up there, though. (I won't say search and rescue because there should be no search involved.) $\endgroup$ Dec 31, 2021 at 23:22
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    $\begingroup$ @LorenPechtel “can't see Himalayan rescue being a big enough business” – well, Mt Everest alone has become such a popular place to go (if not popular place to die...) that I could see this being financed all by the rich people who wish to brag about having climbed up the highest place on Earth but never mind the unglamorous-but-deadly climb down. $\endgroup$ Dec 31, 2021 at 23:51
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    $\begingroup$ @LorenPechtel: 27000m and Everest are two very different beasts, though. The altitude record for helicopters is already way beyond Everest, and it did not require any modifications to the helicopter other than stripping 200kg of mass (and adding oxygen for the pilot, obviously). The helicopter that was used for the record Everest landing was a stock model with only 120kg removed. You don't really need radically new engineering for Everest, just more powerful engines, maybe some partial oxidizer injection. $\endgroup$ Jan 1, 2022 at 0:52
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    $\begingroup$ For reference, Mt. Everest doesn't even go up to 9 km, let alone 27 km. $\endgroup$ Jan 1, 2022 at 4:15
  • $\begingroup$ @JörgWMittag Yeah, I was thinking of higher than current helicopters can go, not 27km. $\endgroup$ Jan 1, 2022 at 5:45
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Ingenuity is an impressive feat of engineering - its flight dynamics run on an open source Linux loop at 500 Hz, using a standard off-the-shelf processor which is way faster than other processors NASA is using.

But not so much because of its rotor speed. Rotor rpm simply scales up/down with blade length to bring blade tip speed to the optimal Mach number. At 2,900 rpm = 303.7 rad/s, the tip speed of the 1.2 m rotor is 182 m/s, well below critical Mach in Mars' atmosphere.

enter image description here

Photo above shows the blades of the test vehicle, which were optimised for blade stiffness and drag at low density. It is from this document, which also mentions that the rotor was designed at normal blade tip speed:

Actual Mars atmosphere conditions are significantly colder (e.g. approx -50 C), resulting in different blade aerodynamics conditions such as the Mach number. However, since the helicopter is operated with conservative tip Mach speeds (≤ 0.6), flight dynamics and performance results at ambient temperatures (e.g. approx 20 C) can be easily matched to the Mars conditions.

The altitude limit for helicopters is not so much because of the rotor design, supported by the familiar shape of the rotors in the photo. It is because of the air breathing engines powering the helicopter - even with super- and turbochargers, there is simply not enough power generated to sustain flight. Fixed wing aeroplanes can fly higher:

  • They can fly faster, and compress incoming air with the ram effect.
  • They do not superimpose rotor tip speed on the flight speed.

Ingenuity can fly on Mars' thin atmosphere because it is powered by an electromotor, the flight tests on Earth were done at a lower weight by offloading power and avionics. The electrical wires provided power - but we cannot power an electrical helicopter with a 27,000 m extension chord.

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    $\begingroup$ Maybe we could power it with a directed microwave beam from a fixed-wing aircraft circling the site, or something like that? $\endgroup$ Dec 31, 2021 at 23:55
  • $\begingroup$ amusing myself thinking about a miniature version you can switch on and put in your microwave oven on high and it hovers in place while the microwave is on $\endgroup$
    – briantist
    Jan 1, 2022 at 21:46

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