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I have searched if any explanation that bind Bernoulli's principle in lifting airplane. There is one here which receive many appreciations, but still did not answer the question. So far, it was taught, explained, or mentioned in any aerodynamic or airplane principle, but never bound the Bernoulli's formula to the lift formula. As we know, there are pressure conservation as below: Bernoulli's Principle

Due to height is considered are same below and above the wing (the different are very small), then the third part of the equation will cancel one each other. Then left only part one and part two of each side of the equation. As the velocity above (considered as V1) is said will be different below the wing which wind is faster in above, then pressure above the wing will be lower, which will lift the wing. Thats is commonly taught in every explanation. Then, the V beneath the wing

Meanwhile, lifting formula is expressed as below: Lifting formula to lift airplane wing V here is the airplane velocity, which is the wind hit the airfoil. Thence, V=V2 of the Bernoulli's equation above. V1 that hit upper side of the airfoils/wing, which is said faster than below, is unknown.

So, where is the Bernoulli's principle contributes in this case? How actually we calculate the lift force of an airplane?

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The statement "Because of continuity, so the air that flows in from the left side must flow out at the right, the upper flow must get there at the same time. But because that line is curved, air has to go faster to get there at the same time." is not necessarily correct. See Chapter 3.2 from this e-book, excerpted here. https://www.av8n.com/how/ The author simulated a wind tunnel, and documented air flow patterns around a wing, with different colored chunks of air in different colors. He demonstrates that air above and below the wing do not arrive at the same time. The book also discusses quite well in my opinion how downwards airflow at the back of the wing really contributes to the lift created, more so than any low pressure area above the wing. Air being pushed down = weight of aircraft being pushed up. (The old "opposite and equal reaction".)enter image description here

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  • $\begingroup$ So, can I say that there is no Bernoulli's countribution in lifting process? I agree with you, the air is forced downward by the curved airfoil will act as agaist gravity (mg). $\endgroup$ – AirCraft Lover Dec 13 '18 at 16:18
  • $\begingroup$ I don't know about No contribution, but I would suggest perhaps not the main player anyway. See chapter 3.6 of the same e-book. av8n.com/how/htm/airfoils.html#sec-airplane-air $\endgroup$ – CrossRoads Dec 13 '18 at 16:35
  • $\begingroup$ Probably that the best way to conclude, that not the main part. $\endgroup$ – AirCraft Lover Dec 13 '18 at 18:45
  • $\begingroup$ downwards airflow at the back of the wing really contributes to the lift created Not that it doesn't contribute, but most of the lift is created at the front of the wing. The lion's share of the downward airflow at the back of the wing was turned downward by the front of the wing. $\endgroup$ – TomMcW Dec 13 '18 at 18:56
  • $\begingroup$ Then the question is, "why is the airflow turned downward?" It's turned downward because of the low pressure above the wing. Both are happening simultaneously $\endgroup$ – TomMcW Dec 13 '18 at 18:57
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If I understand correctly you want to know how to get from the first formula to the second one. If that is the case, the answer is simple: you don't :)

The lift formula is a simplified one, which considers the wing as a whole. It doesn't use velocities on different points of the wing, it uses the speed of the airplane. All the factors such as profile, wing shape, etc are included in the lift coefficient. This formula is useful when you already know the behavior of the wing in different situations (the behavior is determined in wind-tunnel tests or with numerical methods).

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  • $\begingroup$ Yes, you got my point. So the question is, where is the bernoulli's principle in lifting the airplane? Is bernoulli's contributing in that thing? $\endgroup$ – AirCraft Lover Jan 14 at 0:50
  • $\begingroup$ @AirCraftLover No, not really. Bernoulli's principle is not a correct explanation for the creation of lift (grc.nasa.gov/www/k-12/airplane/wrong3.html). $\endgroup$ – Emil Jan 15 at 13:32
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The Bernoulli equation is defined between 2 points in a flow field. It has limitations, mainly that it's valid only for incompressible flow, so only valid for low airspeeds. As you noted accurately, the height term is usually ignored, but it's more to do with the low density of air than the size, as for water or mercury it would make a large difference even with small lengths.

So the equation you derived (apart from rho being at the wrong place, it should be p/rho) is accurate between any 2 points in the flow field. Lift is caused by the pressure difference between the two sides of the airfoil, so if you wanted to calculate the lift, you would need to evaluate pressures at all points on the upper and lower side of the airfoil. As the Bernoulli equation states, where the flow speeds up, the pressure drops, and this is the upper side of the airfoil. See picture from wikipedia, the bottom streamline is almost straight, so its shorter than the curved one on the top. Because of continuity, so the air that flows in from the left side must flow out at the right, the upper flow must get there at the same time. But because that line is curved, air has to go faster to get there at the same time. Faster flow = lower pressure. enter image description here

So you can use the Bernoulli principle to estimate the pressure at every point, provided you know the speed at the given point. Once you do that, you get pressure distributions over the airfoil, something like this: enter image description here

Note that these distributions are not calculated by hand, but rather with some CFD code, or measured in wind tunnels. Once you have the distribution, it can be summed to get the total forces. To make life easier, engineers measured all these forces as a function of altitude, speed (Re number) and angle of attack, and made it available for you as a CL-alpha curve. Then the lift equation is a similarity law essentially, where you turn the non-dimensional pressure distribution, eg the CL into actual force that you can use to calculate.

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  • $\begingroup$ @D Sziroczak, sorry for the wrong equation. Hope now it is correct. $\endgroup$ – AirCraft Lover Dec 13 '18 at 13:44
  • $\begingroup$ Dear Sziroczak, I quote this: "the air that flows in from the left side must flow out at the right, the upper flow must get there at the same time. But because that line is curved, air has to go faster to get there at the same time". I heard that one very often. But there is no evident confirming that claim. Also NASA in their website still wrote that the Bernoulli's contribution in lift is still in debate. $\endgroup$ – AirCraft Lover Dec 13 '18 at 13:58
  • $\begingroup$ From this statement: To make life easier, engineers measured all these forces as a function of altitude, speed (Re number) and angle of attack, and made it available for you as a CL-alpha curve. Then the lift equation is a similarity law essentially, where you turn the non-dimensional pressure distribution, eg the CL into actual force that you can use to calculate. Can I say that by using coefficient of lift in that lift formula as I wrote above, I have include the Bernoulli's contribution in that lifting force? Please, I need your feedback, Sir. $\endgroup$ – AirCraft Lover Dec 13 '18 at 14:01
  • $\begingroup$ Because of continuity, so the air that flows in from the left side must flow out at the right, the upper flow must get there at the same time.  What says it has to get there at the same time? It doesn't. In fact the upper flow generally gets there before the lower flow. $\endgroup$ – TomMcW Dec 13 '18 at 18:49
  • $\begingroup$ AirCraft Lover: The Bernoulli equation is a way to describe properties at 2 points in the flow space. It does not "contribute" to anything. The lift is generated because of the pressure differences. Individual pressure differences can be calculated using the Bernoulli equation (keeping in mind the limitations). Lift coefficient is a simplification based on similarity law, so you don't need to calculate pressure distribution all the time, but assume that under similar conditions, pressure distributions are similar. $\endgroup$ – D Sziroczak Dec 14 '18 at 17:17
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Easy to illustrate. Hold a piece of paper loosely by the corners so it is curved down hanging away from your face. Blow gently across the top of it and see it lift. There is no "push from below", only a suction from above.

As a 700+ hour pilot I enjoy watching my wings get 'sucked' into the wild blue yonder by the lower pressure above the wings and the relatively higher pressure below - also known as 'Lift'

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  • $\begingroup$ As I have explained in the question body with the two formulas, there is no equation which bind the bernoulli to lifting force. $\endgroup$ – AirCraft Lover Jan 13 at 16:37

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