Business & Economics

Archimedes principle explains helicopter flight.

The physics of how balloons float in the air can also be used to explain how helicopters hover. Results from experiments showed that in a stable hover, a 21g helicopter drone displaced at least 21g of air downward each second, to achieve buoyancy.
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  !"#$%&'(') +"%,#%+-' '.+-/%,) $'-%#0+1'" 2-%3$14 1 !"#$%&'(') +"%,#%+-' '.+-/%,) $'-%#0+1'" 2-%3$14 5"4 6%#$0-/) 7/,('--85%--) 5! 9:! !9!   56 !73 895:; <0,1/#1 '&/%-= ,%#>-&?3&.4#0& @'A)%1'= $11+)=BBA70C/,#CD'.+-/%,)D2-%3$14#0& E"'DE"%,1 FGH= 5945I569BJK484848L9:M4NN96OP << Q%#',)'= << RSDT! 649 U'CV0"()= !'"0(C,/&%#)P !"#$%&'(')P A70C/,#CP ("0,'P 2-%3$1P $'-%#0+1'"P $0W'"P -%21P +$C)%#)P !;)1"/#1 This paper demonstrates that the physics of how balloons float in the air can also be used to explain how helicopters hover. Results from experiments showed that in a stable hover, a 21g helicopter drone displaced at least 21g of air downward each second, to achieve buoyancy. Similar experiments have produced similar results. This is not a coincidence. Specifically, all helicopters achieve buoyancy by displacing a mass of air downward each second that is at least equal to its own mass. This new theory of flight that can be applied to any object that remains airborne by pushing air down, such as airplanes, insects, birds, frisbees, …... It is the first theory of flight to be supported by a scientific experiment on a real aircraft in realistic conditions. It provides a simple explanation for flight that is easily understandable. It is consistent with what is observed in reality, as well as the standard equations for force & kinetic energy. This is important as it provides new insight into the physics of flight that could improve aircraft design, pilot training, and aviation safety. Conventional physics has simply overlooked this critical aspect of flight. I.   I  NTRODUCTION    A.    Background. The physics explaining a helicopter hover is currently  provided by the ‘momentum theory’ of lift [1] [2]. The rotors exert a force that directly accelerates a small mass of air downwards, which then displaces more air below it. The ‘equal & opposite’ upward force pushes the helicopter up. But the momentum theory is incomplete. Not only are the upward & downward forces equal; the masses involved are also equal. The total mass of air (directly & indirectly) displaced at least equals the mass of the helicopter in a hover. See Fig. 1a. Fig. 1a Equal forces and masses in a hover. The physics for how a drone hovers is similar to how  balloons floats in the air. For example, a 21g floating balloon will displace 21g of air downward. Similarly, a 21g helicopter drone in a hover displaces at least 21g of air downwards each second. A key difference is that balloons passively displace air down (static buoyancy), while drones actively displace air down (dynamic buoyancy; which is a new concept). See Fig. 1b. Fig. 1b Balloon and helicopter drone. Calculations show that an 80 kg skydiver in free-fall, at terminal velocity displaced 80 kg/s of air downwards [15]. The skydiver accelerated until they displaced a mass of air each second that was equal to their own mass; and consequently achieved buoyancy. Not only are the upward & downward forces equal; the masses involved are also equal. Similarly, calculations showed that an 80 kg skydiver can be suspended mid-air by a large fan blowing 80 kg/s of air upwards. This is the same principle for how helicopters hover. See Fig 1c. Fig 1c. Skydivers achieve buoyancy.  !"#$%&'(') +"%,#%+-' '.+-/%,) $'-%#0+1'" 2-%3$14 2 This paper concludes that Archimedes 2,000 year-old  principle of buoyancy can be re-stated to include the dimension of time. Specifically: “To float, an object must displace at least a mass of fluid each second  that is at least equal to its own mass.” It could be relatively easy to validate or refute this theory of flight based on buoyancy. For example, fluid flow analysis (e.g. CFD) could be used to calculate the volume (and therefore the mass) of air displaced downward by an airplane in flight. If this data showed that all aircraft achieved buoyancy each second, then this would be significant.  B.   Two drone experiments. Two experiments were completed with the objective to measure the mass of air displaced downward by a helicopter drone in a stable hover. First, a simple experiment demonstrated that a drone in an open box, in a stable hover, exerted a downward force on a scale at least equal to the drone’s mass when at rest on the ground. See Fig. 1d. Fig. 1d Helicopter drones at rest and in a hover. Others have done similar experiments previously, obtained similar results, but didn’t make any conclusions related to  buoyancy. Specifically, this first experiment is similar to the textbook “bee in a jar,” [3] or the “birds in a truck” [4] physics experiments, as well as a similar drone experiment [ 5 ]. The results of all these experiments showed that the insect, bird or drone in a hover exerted a downward force that generated a weight (and mass) on the ground just below it, equal to their own weight (and mass) when at rest on the ground. For example, commenting on the “birds in a truck” experiment David Lentink, a mechanical engineer at Stanford University is quoted in 2015 in the New Scientist magazine: “the cargo [of birds in flight in a truck] will, on average, maintain a stable weight [compared to when the birds are at rest on the ground] [4].” To correctly analyze the physics of flight, it is the mass of the object that is significant, not its weight. The analysis in these other experiments incorrectly focused on weight (not mass). In addition, these other experiments ignored: the fact that the drone, insect, bird, etc… was circulating the air (and not just  pushing the air downwards); downwash velocity; ground effect and inefficiencies in generating lift. The second experiment of this paper, which has not been done elsewhere before, measured the drone’s rotor area (at 28.3 cm 2 ) and the vertical downwash velocity (at 6.8 m/s) in a stable hover. Calculations based on this data confirmed that the drone was displacing a mass of air downwards each second, that was at least equal to the drone’s mass. See Fig. 1d. C.   The theory of flight is unresolved & controversial. Many academics & engineers interviewed in the research for this paper wrongly claimed that the theory of lift for aircraft was known, proven and agreed. Engineers, academics, pilots and other pundits (NASA) continue to debate the theory of flight without an agreed resolution or proof with a scientific experiment on a real aircraft in realistic conditions [6]. Journals and the media occasionally comment on this ongoing dispute;  but it gains relatively little mainstream attention. There are at least twelve different explanations of lift and aerodynamic forces that are debated [6]. Experts still dispute whether an airplane is pushed or pulled upwards. NASA describes some theories commonly believed to be true as ‘incorrect’ [1]; including the theory taught to pilots at flying school (based on the Venturi effect and Bernoulli’s principles of fluid dynamics). Academics’ & engineers’ preference for  Navier-Stokes is not taken seriously by either NASA, pilots, aviation authorities or Burt Rutand (a famous aircraft designer). The book ‘Understanding Flight’ [6] came close to identifying buoyancy as an explanation of flight for airplanes. It  provided a back-of-the-envelope calculation to estimate that: “… a Cessna 172 at cruise is diverting about five times its own weight in air per second (downwards) to produce lift [6].” The  book claimed that: “The lift of a wing is proportional to the amount of air diverted per time times the vertical velocity of that air [6]. In addition, David F. Anderson a co-author of this book, stated in a lecture that he estimated: “..... a 250 ton airplane (ie. commercial airliner) ..... is diverting it’s own weight (in air), per second, to keep in the air [8].” However, this book did not explore buoyancy as an explanation of flight. It is also worth noting that both the co-authors of this book ‘Understanding Flight’, David F. Anderson & Scott Eberhardt, had private pilot’s licenses (as well as PhD’s from prestigious US universities). To understand the physics of flight, it appears essential to have practical experience actually flying a real airplane; which few academics & aeronautical engineers have.  D.   The philosophy of buoyancy. Some logical reasoning supports the application of the  physics of buoyancy to helicopters: • There is no reason why the physics of buoyancy cannot be applied to helicopters in a hover. Applying buoyancy to flight is consistent with all laws and principles of physics. No energy, momentum or mass is gained or lost in the process. • The principle of buoyancy is universal; it applies to both stationary and moving objects. For example, both a stationary and a moving boat must both still achieve buoyancy to float. • The principle of buoyancy is time-dependent. It matters how much water/air is displaced each second. A boat that ceases to displace enough water to float will sink immediately. Helicopters are seen to achieve buoyancy over a time period of one second in the experiments in this paper. • It is logical that in a stable hover both the upward & downward forces on a helicopter are equal; as well as the masses  !"#$%&'(') +"%,#%+-' '.+-/%,) $'-%#0+1'" 2-%3$14 3 involved (of the helicopter pushed up and the mass of air displaced down). • In physics, a standard definition of mass is that it is the quantity of matter of an object, which is independent of its volume or any forces acting on it [13]. Therefore, for an object to stay airborne, it is logical that a solid mass could be supported  by an equivalent mass of air underneath it, each second. • Applying the principle of buoyancy to explain flight fits Occam’s Razor, where the simplest explanation with the fewest assumptions is often correct.  E.   Why is this paper important? Applying buoyancy to moving objects provides new insight into the physics of flight that could be used to improve aircraft design, pilot training and aviation safety for all types of aircraft. In the past, aircraft design has been done primarily by trial and error [10]. Different designs were simply tested to see which worked the best with limited understanding of the correct reason how lift and flight were achieved. Pilots are trained to fly the aircraft within its known technical limits, established by trial & error (practical testing). The factors listed above would help improve aviation safety. Despite improvements in technology, general aviation (non-commercial) accident rates remain consistently high in the US according to the NTSB [11]. Most crashes result from a pilot’s loss of control of the aircraft and an uncontrolled descent into the ground. High accident rates are not surprising given that  pilots are taught the wrong theory of flight. In emergencies,  pilots are likely to make poor decisions where survival depends on an accurate understanding of the physics of flight.  F.    Additional information. Supplementary Materials : Explanatory videos are available on the website: . Author Contributions : This paper is entirely the work of the author, Mr. Nicholas Landell-Mills. This paper was completed after extensive research and discussions with academics, engineers, pilots & aviation authorities. Author background: The author was a pilot (PPL) and is British. He flew and maintained a ‘home-built’, single-engine airplane (Europa XS reg: G-OSJN). The raison d’être of this paper:  The author was motivated to write this article because the theory of flight that he was taught at flying school (as approved by the aviation authorities) was false, as NASA Glenn Research Centre [1] also claims. Past work: The author has previously published articles related to buoyancy and flight in open access journals: !   Buoyancy Explains How Planes Fly [14]. !   Buoyancy Explains Terminal Velocity Skydiving [15]. !   Calculation of the Air Displaced by a Wing Error! Reference source not found. . !   Buoyancy Explains Flight [14]. Pre-print papers: !  Lift Paradox (Lift < Weight) !  Newtons laws explain the equation for lift. !  The theory of flight remains unresolved. Author contact details:  75 Ch. Sous Mollards, Chamonix, 74400 France. T: 0033 638 77 39 40. Email: Affiliations : The author is a graduate of the Dept. of Humanities & Social Sciences, The University of Edinburgh, Edinburgh, UK. He was awarded a M.A. degree class 2:1. Disclaimer : The author confirms and states that all data in the manuscript are authentic, there are no conflicts of interest, and all sources of data used in the paper are acknowledged. Funding : This paper was self-funded by the author. Corresponding authors : None. Acknowledgments : None.  !"#$%&'(') +"%,#%+-' '.+-/%,) $'-%#0+1'" 2-%3$14 4 II.   B ACKGROUND -   T HEORETICAL FRAMEWORK     A.    Definition of Archimedes principle of buoyancy. “The principle of buoyancy holds that the buoyant or lifting force of an object submerged in a fluid is equal to the weight of the fluid it has displaced [18].” Or: “Archimedes Principle states that the buoyant force on a submerged object is equal to the weight of the fluid that is displaced by the object.” [1]. So for a balloon: Buoyancy Force Up  = Weight of Air Displaced Down => Force Up  = Mass AIR DISPLACED DOWN  x Gravity In equilibrium, the upward buoyancy force will equal the downward force from the weight of the balloon. So, the weight of the balloon equals the weight of the air pushed down. Gravity is a universal constant for the equation on weights, being on  both sides of this equation below, so cancels out. In turn, the mass of the balloon also equals the mass of the air pushed downward. As shown by the equations: Force Up  = Force DOWN  => Weight BALLOON PUSHED UP  = Weight AIR DISPLACED DOWN  => Mass BALLOON x Gravity = Mass AIR DOWN  x Gravity => Mass BALLOON UP  = Mass AIR DOWN  The balloon will float at a constant altitude when the mass (or weight) of the balloon equals the mass (or weight) of air that it displaces downwards. See Fig 2a. Fig. 2a Buoyancy in a balloon. For a balloon there is no difference in-between the mass and weights involved, they are both equal. Critically, the application of buoyancy to explain a helicopter hover focuses on the mass of air displaced down, not the weight of this air. This is explained further below in section III.B ‘Mass, not weight.’ For example: “Hot air balloons rise into the air because the density of the (warmer) air inside the balloon is less dense than the (cooler) air outside the balloon.” [1] “The balloon is in balance when the air density and pressure differences between it and the volume air displaced are equal. If density and volumes are equal between the balloon and the air, then the air mass displaced and the mass of the balloon must also be equal (Given that Mass = Density x Volume). “ [1] => Mass BALLOON UP  = Mass AIR DOWN    B.    Definition of downwash. For the avoidance of doubt, definitions of downwash according to two reputable aeronautical dictionaries: • “Downwash (aerodynamic): Air forced down by aerodynamic action below and behind the wing of an airplane or rotor of a helicopter. Aerodynamic lift is produced when air is deflected downwards.” [19]. Or: • “Air deflected perpendicular to the motion of the airfoil. Air that has been accelerated downwards by the action of the main rotor of a helicopter.” [20]. C.   The physics of a helicopter hover. In summary, for a helicopter drone (quadcopter) in a stable hover, the rotors accelerates a small mass of air downward, to creates a downward force. As the air pushed down causes air elsewhere to be pushed up, this circulates a larger mass of air around the drone. See Fig 2b. The forces and masses are both in  balance. These points are described by the equations: -   Downward Force = ma = m/dt x dv -   Upward Force (lift) = Downward Force -   Mass of drone pushed up = Mass of air displaced down Fig. 2b Drone circulating airflows. (1) In a stable hover, a small mass of air each second (m/dt) is directly accelerated downwards to a given velocity (dv) by the rotors to create a downward force. This air accelerated downwards then indirectly displaces the air below it. See Fig 2c. Fig. 2c Downward and upward forces. This is described by the equation: Downward Force = ma = m x dv/dt [1] = m/dt x dv => Mass of air displaced down. This equation is consistent with Newtons 2 nd  law of motion (F = ma) [1]. The total mass of air displaced includes air directly accelerated by the rotors, and air subsequently indirectly displaced.  !"#$%&'(') +"%,#%+-' '.+-/%,) $'-%#0+1'" 2-%3$14 5  Note that: -   m = mass of air directly accelerated down by the rotors. -   dt = change in time (per second). -   dv = change in velocity of the air. -   ‘m/dt’ is also called the ‘mass flow rate.’ [1] -   By definition, a downward force of 1 N will accelerate 1 kg of air mass by 1 m/s 2 . [1] [21] The physics of a helicopter hover is illustrated by taking the example where the rotors’ rotational speed is increased. This will increase the mass of air directly displaced by the rotors each second (i.e. ‘m/dt’ increases), as well as the velocity to which this air is accelerated downwards (i.e. ‘dv’ increases). In turn, this increases the downward force (as Force = m/dt x dv); as well as the air pressure under the helicopter (as Pressure = Force/Area). As a result, a greater mass of air is also indirectly displaced. Correspondingly, the ‘equal & opposite’ upward force (lift) increases. This is consistent with what is observed in reality; an increased rotor speed will generate more lift. Taking another example. If the helicopter increases its rotor’s angle of attack to the air, this will increase the mass of air directly displaced downwards by the rotors each second,  boosting lift. i.e. ‘m/dt’ in the equation above increases. This is consistent with what is observed by pilots in reality; a wider angle of attack on the rotor will generate more lift, under normal circumstances. This equation is also similar to that provided by the book “Understanding Flight.” “In the simplest form, lift is generated  by the wing diverting air down, creating the downwash.” [6]. “From Newton’s second law, one can state the relationship  between the lift on a wing and its downwash: The lift of a wing is proportional to the amount of air diverted (down) per time times the vertical velocity of that air.” [6]. ie. Lift = Force = ma = m/dt x dv. (2) An upward force (lift) that is equal and opposite to the downward force, pushes the drone upwards; consistent with  Newtons 3 rd  law of motion of action and reaction [1]. It is  possible to describe this by the equation: Upward Force (lift) = Downward Force (3) If the upward and downward forces are equal, it follows logically that the masses involved will also be equal. The mass of air displaced down will equal the mass of the drone pushed up. See Fig 1a. It is possible to describe this by the equation: Mass of drone pushed up = Mass of air displaced down (4) By definition, buoyancy is achieved when the mass (or weight) of air displaced downward equals to the mass (or weight) of the drone pushed up. See Fig 2d. Fig. 2d A drone achieves buoyancy. (5) Note that as Pressure = Force/Area [1]; High pressure arises below the drone (from the downward force). (6) Similar to a balloon floating in the air, the drone is able to  be suspended in the air due to sufficient air mass being displaced downwards. Balloons achieve buoyancy by displacing air downward to due to the volume of space that they passively occupy. Whereas, helicopters achieve buoyancy by actively displacing air downward. (7) The critical factor for buoyancy & flight being achieved is the drone’s ability to circulate enough air mass, which is not directly dependent on gravity. This means that gravity (as a universal constant) is not directly a critical consideration for whether a drone can hover.  D.    Illustrations of how helicopters generate lift. (1) Helicopter drones cannot fly in a vacuum. Other simple ‘classroom’ type of experiments demonstrate that small drones cannot fly in a vacuum (in an enclosed box with no air) [22]. To fly, something (air) needs to be pushed down [1]. See Fig 2e. In short, there is no air mass to displace downwards, and therefore no method to generate and equal & opposite upward lift force. So the drone cannot fly. This aspect of flight is consistent with the conclusions of this paper. Fig. 2e Helicopter drone in a vacuum in a box. (2) Helicopter (drone) performance in winter. It is well established that as air density increases, for example in the cold conditions of winter, helicopter performance (the lift generated) increases [23]. The same amount of power & energy from the engine produces greater lift in denser air. It is easier for a helicopter to hover in denser air, The explanation is that in cold conditions, a helicopter will generate lift more efficiently. The helicopter is displacing a greater mass of air downwards with each rotor turn, and with each 1 m 3  of air displaced down. This aspect of flight is consistent with the conclusions of this paper.
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