By "Energy conserved" I meant, energy transfer between Pressure and kinetic energy, no energy is lost by "Work done".

Hello Adnan Shahriar ,

I am no expert on this subject, but I am a student who also has been befuddled by poorly and/or incompletely explained phenomena.

There are many reasons why you are right to question the use of Bernoulli's equation as the sole explanation of lift with respect airfoils. First, some books do mention that in the case of real airfoils, viscosity and compressibility of the fluid (air) are always present, and Bernoulli's equation assumes that the fluid is inviscid and incompressible. Second, form drag and viscous drag are always present for real airfoils, i.e., there is no free lunch, lift comes at a price. Sure, gliders seem to partake of a free lunch, but this is only an appearance. Gliders are, in reality always falling, the only reason they appear to soar is due to them riding thermals (thermal air columns), wherein the air is rising faster than they (the gliders) are falling. They circle around a thermal for as long as they can or it lasts, and then they glide over to the next thermal and resume their circling riding up the thermal. The glider pilots usually watch for vultures or other soaring birds for clues as to where the next thermal is located.

Third, how does Bernoulli's equation explain how an airplane can fly upside down? Some books mention this question,

Jearl Walker; The Flying Circus of Physics with Answers; John Wiley & Sons, Inc.; 1977; pp. 84 (section 4.31), 254-255 (answer to section 4.31).

Since I am not a pilot, I cannot elaborate on whether or not the wing control surfaces (flaps, which move in unison, and ailerons, which move in opposition) are in the neutral position or not, and what the angle of attack has to be.

Some books (including Walker's book), mention that Bernoulli's equation is not the sole explanation for lift and that circulation of the fluid around the airfoil enhances lift, see the diagram in the following book,

Walter Boyne, Dr. Tom D. Crouch, et al.; How Things Work, Flight; Time-Life Books; 1990; p. 39 (Figure labelled "THE SECRET OF LIFT).

This circulating flow, which adds to the fluid velocity at the top of the airfoil and subtracts from the fluid velocity at the bottom of the airfoil, is referred to as Kutta- Joukowsky theory by Walker in his answer to section 4.31.

Note, you are not the first student (or teacher for that matter) to have had doubts about the facile explanation offered for the phenomenon of lift given in physics textbooks. Walker's book cites four different articles in journals/magazines such as Physics Teacher, American Journal of Physics, and Scientific American, which discuss the same question as you posed. In any event, you are in good company.

Regards,

Tom Cuff

Bernoulli is a very simple way to address the lift produced by the pressure differences over an airfoil but is a model that approximates the reality (as always in engineering problem). There are different theories to approach the genesis of lift.

The total pressure is conserved but the integral allows for conversion of kinetic energy in a mechanical reversible work. The irreversibile work is disregarded but is then considered when the BL theory is taken into account. Note that for fluid having low viscosity, in the BL the pressure normal to the wall is almost equal to the pressure in The Bernoulli equation.

Hello Adnan Shahriar ,

One last thing, which I forgot to mention: the gulf between theory and practice (analysis versus synthesis). If Bernoulli's equation was both necessary and sufficient for the analysis of an airfoil, then why does it not uniquely specify the design of an airfoil cross section (profile)? No matter whether the airfoil is for a fixed wing aircraft or a rotating wing aircraft (propeller or helicopter blade), designing the profile, length, and taper involves a lot more than Bernoulli's equation. For a good history of the design of fixed wings and rotating airfoils, see the following book,

Walter G. Vincenti; What Engineers Know and How They Know It, Analytical Studies from Aeronautical History; Johns Hopkins University Press; 1990; pp. 16-50 (Chapter 3 - Design and Growth of Knowledge: The Davis Wing and the Problem of Airfoil Design) & 137-169 (Chapter 5 - Data for Design: The Air-Propeller Tests of W. F. Durand and E. P. Lesley).

Regards,

Tom Cuff

Yes, I agree with Professor Filippo Maria Denaro .

It is very easy to understand the process of energy conversion. But please mention that in the pure Bernoulli equation the condition is assumed that the fluid is inviscid, adiabatic, and no energy loss. So the process of energy conversion is isentropic. So, the process is ideal.

Thank you.

Best regards.

Bernoulli's equation is the simplest form in this issue while minimizing the components to the least for fast and practical applications. However, many approaches exist describing this phenomenon and not necessarily deals with Bernoulli's equation. Nonetheless, considering the effect of the boundary layer on the surface of the airfoil, the presence of energy easily could be seen.

Thanks Dr Filippo Maria Denaro .

I was not sure, if Bernoulli's equation really help understand or approximate Lift force.

Not only energy, but also I didn't understand where the Force is coming from in momentum perspective. When there is Force, there must be change in Momentum or Transfer of Momentum from one object to another. But, when Bernoulli's principle has been used, I can't see any Momentum transfer from air to Aerofoil which cause Lift force on the Aerofoil. Before the Aerofoil and After it the Air leaves with the same Kinetic and Pressure Energy. So, in terms of both Energy or Momentum, this makes me confused.

I guess, for this ideal case, Lift force shouldn't be generated. Its probably more equivalent to D'Alembert's paradox where no drag is generated for ideal case.

Thanks a lot Thomas Cuff . The circulation theory I have been through and as far I understood, without viscosity, you can't have any lift! Boundary layer separation may makes thing complicated but I am really not deep into it, but hopefully learn more in near future. Specially your references would be very helpful.

This explanation with BL makes sense, but explaining Lift without viscosity makes me confused as I can't see any momentum or energy going in or out.

Thanks Nazaruddin Sinaga . Yes, that's my point is. In the process the Energy in conserved into Kinetic and Pressure for the Air. But, I am getting Lift force, while I can't see any momentum or energy transfer from air to the object when Bernoulli's principal is used. This makes me confused. I guess, in ideal case there shouldn't be any lift at all!

Adnan Shahriar

the transfer in the momentum is governed by the change between kinetic energy and the reversible part of mechanical work (pressure). The constant is the total pressure. That is an adiabatic and isoentropic transfer of a form of energy into another.

Just to add a different view, consider the general form of the conservation of total energy on a material volume

dE/dt= W-Q

assuming a steady, adiabatic, reversible process. That is Bernoulli.

You have NS equations, then Euler equations and finally Bernoulli equations. Each has its own application. The simplest approximate equations to guess pressure distribution around an object is Bernoulli's equation with assumptions like: incompressible, low Reynolds no., Inviscid flow.

In fact a theory known as d'Aembert's paradox explains how by approximate solution you will end up with zero drag.

D'Alembert proved that – for incompressible and inviscid potential flow – the drag force is zero on a body moving with constant velocity relative to the fluid.

Bernoulli's equation don't explain the lift phenomena, it's used only to relate velocity and pressure over the surface of the object with values of the free stream. If the flow is rotational and viscous, then the constant in the Bernoulli equation must be different between streamlines, this is why we don't apply Bernoulli in that case. Bernoulli is only a simplification of the problem where the constant is the same in the domain (invicid and irrotational). Now, there are difference pressure and difference velocity up and down of the object related by Bernoulli, why?, which cause the incremente of velocity ? Bernoulli don't explain the nature or origin of this question, only explian the consequence. The origin of lift is the curvature of the surface or the angle of attack, both cause the curvature of the streamlines around the object and this cause the different of pressure, and then, by Bernoulli, the difference of velocities.

Thanks a lot Christian Muñoz . I have been clarified.

Thanks Dr Filippo Maria Denaro , Mohsen Jahanmiri and Amin Jaberi

I agree with Christian Muñoz, really thanks for your explain.