Expert: Paul Soderman - 6/17/2011

Question
QUESTION: 1. The free stream becomes faster at the venturi of the duct.
A free stream becomes faster over both the leading slopes of an airfoil at zero angle of attack.
By Newton's first law, an external force, by a body, should get applied on the free stream - in the direction of the stream (a force in the opposite direction will slow the free stream)
Q:Which body applies such a force? By which law?
Q: Has this question been answered directly?

2. From the above cases one fact: A free stream becomes faster over a smooth slope.
Q: Which body applies the force? The slope? Can the stationary slope apply? If so, how and by which law?

3. Lesser is the degree of sloping faster is stream.
Q: Why?

ANSWER: This sounds like a homework problem, which I don't generally do anymore.  But I can give you some ideas.
1.  A stream accelerates over the leading edge of an airfoil because of the Coanda effect, which forces it to follow the surface.  Centrifugal forces then come into play.
2.  Question unclear.
3.  A flow slows in an expanding flow because the effective flow area becomes larger and the conservation of mass flux leads to a velocity reduction.


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QUESTION: Sincere thanks for the reply. Again I request your reply.

The questions are not home work ones but are for thought because direct answers are not available in literature. Your reply also doesn't tell the body that applies the speeding force on the free stream.

= Due to Coanda effect a fluid bends. It does not get speeded up.

In an airfoil it alone cannot be the cause of bending because, above the stall angle, the upper stream flows separated from the upper surface.

= In a venturi, one has to imply the area is the body that applies the speeding or slowing force on the free stream. This cross sectional area is not a physical body to apply a force.

Another question for thought.

Q: Which body applies the lifting force on the airfoil at zero angle of attack?

By Bernoulli's theorem, one has to imply it is the lower stream. At zero AOA it flows parallel to the lower surface or even away. It is a neutral body, so cannot apply the lifting force perpendicular to its direction of flow. There is no law that states that neutral bodies apply such force and gives the magnitude of the force.

Try to identify the bodies and the law/s by which they apply the forces. One lands in difficulties.   

The questions are based on the fundamental law, Newton's first law. So, any explanation that doesn't or cannot give direct answers is fundamentally flawed.

Thanks again

Vidhyasagar

ANSWER: OK Vidhyasagar - I am starting to understand your questions.  And you are correct, all fluid motions are ultimately explained by Newton's laws.  Perhaps the best explanation of the aerodynamics of an airfoil can be found in the paper by Weltner and Ingleman-Sundberg:  http://user.uni-frankfurt.de/~weltner/Flight/PHYSIC4.htm.  They show that a tiny volume element of air passing up the leading edge upper surface of an airfoil will be forced to follow the surface via the Coanda effect.  That will cause centrifugal forces on the element that reduce the pressure on the element compared to volumes above and upstream of the element.  Those pressure gradients create the Newtonian forces that cause the element to accelerate. We could say that the body is the initial source of these forces, but it is the pressure gradients that more clearly cause fluid accelerations.  Weltner et al give similar arguments explaining how the fluid decelerates on the downhill section of the upper surface.  Note that it is the pressure gradient that causes the velocity change and not the velocity change that causes the pressure gradient.  Bernoulli's law relates pressure and velocity in a flow but does not tell us which is the driver - pressure or velocity.  From Weltner et al we can conclude that the pressure drives the velocity field using Newton's law as you suggest.

These same pressures develop on a symmetrical airfoil at zero angle of attack, but the integrated forces of lift cancel out from the upper and lower surfaces.  I guess I don't understand your question for that case.  Perhaps your are talking about a cambered airfoil at zero angle of attack. Again we analyze the surface pressures and will find that the upward force on the upper surface is greater than the downward force on the lower surface, so we get lift.  I don't see any mystery there.

For a venturi, the wall pressure will cause the elemental volume to accelerate in such way to stay tangent to the wall.  Similarly a pressure gradient will be established causing the elemental volume to accelerate toward the throat.  The conservation of mass flux will be maintained. All these forces satisfy Newton's law.

Take at look at Weltner et al's paper and get back to me if you still have some confusion.
Paul

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QUESTION: Many thanks for the reply and for the willingness for further reply.

I have read articles very similar to that of Weltner et all. I strongly feel that it is deficient in the following fundamental aspects.

1. Explanation of the lift of the airfoil:

A. Because of the experiment and the use of the word 'air' - not air stream - I assume that it is the explanation of lift of an airfoil moving in static air.

   Why does the airfoil always send the air only downwards - not upwards or backwards or any other direction? It means the airfoil obeys a law. So, the following question has to be answered.

Q: By which force and by which law the airfoil accelerates the static air downwards?       Kindly try to answer.

Then, by Newton's third law, the air lifts the airfoil by a reaction force.

B. Try this explanation for the lift of the same airfoil while stationary in static air and an air stream flows across.
The explanation will be (given in many text books etc): " The airfoil accelerates the stream downwards; in reaction the stream lifts the airfoil"

 The explanation specifies the airfoil applies the action force on the stream. The airfoil is an electrically neutral body. So, this repulsive force can only be a mechanical force.

A fact:
A neutral body can apply, by Newton's third law, a reaction mechanical force to a collision force on it by another body. It cannot apply an action force. There is no law that states that it applies and gives its direction and magnitude. Keep two stationary bodies touching each other. Neither will ever apply a force on the other and push it out.

So, the explanation is based on a wrong statement.

2. Pressure gradient causes velocity change of the stream:

 Before the air stream flows normal atmosphere/static air surrounds the airfoil. There is no pressure gradient in it. There is also no pressure gradient in the free stream up to the leading slope of the airfoil. Then the free stream speeds up. If it is due to pressure gradient:
Q: Where is the pressure gradient - in the air or in the (free) air stream?
Q: How did this pressure gradient develop?

To develop a pressure gradient requires a force of a body and energy.

Q: Which body, by which law, by which force, by which energy produces the pressure gradient.

So, you end up in my questions to be answered.

       You may accept the deficiencies if you have simple alternative explanations e.g. for speeding up of the stream and development of pressure gradient:

One will agree static air cannot apply the speeding force. The slope, being stationary, cannot apply an action force to speed up. Its reaction force to the collision force of the stream on it will only deflect/reflect the stream. Reaction forces do not speed up. There is no law that states that it does and gives the degree of speeding up. There is no other body there.

So, the only possibility is part of the stream speeds up the rest by collision and in turn is speeded up by the following part just like what happens when a crowd passes through a gate; one man pushes the man in the front and in turn gets pushed by the man following behind.
Euler found that part of the stream bends before reaching the slope (Chapter 1.1. "Fundamentals of aerodynamics" by John D Anderson Jr,4th edition). Draw the leading upper slope of the airfoil. Draw the free stream reaching it. You will find that the lower part of the stream reaches the slope first, collides, gets reflected, will collide on the upper part. This collision in the direction of motion of the upper part will speed up the upper part. Then, it will be speeded up by the following reflected lower part. Thus the whole stream is speeded up.      

 This stream will deflect off static air molecules over the upper surface; it can deflect because its (average) kinetic energy is greater. Due to this work, its kinetic energy, so its velocity, so its collision force, so its pressure goes on decreasing downstream. Thus a pressure gradient develops.

Thanks again,
Vidhyasagar

Answer
Vidhyasagar - I am not sure I can clear up your confusion without going through the fundamentals of aerodynamics step by step, which is beyond the scope of this forum.  But here are some key ideas.

A wing moving through still air can be visualized by placing a coordinate system fixed in the air and letting the wing fly by or by placing a coordinate system on the wing and letting the air flow by. The aerodynamic forces are the same in both cases, but we usually fix the coordinate system to the wing to get a better steady state picture of the flow. Thus, in that coordinate system the entire field of air is moving even far ahead and far behind the airfoil. In that case it is clear that a wing at angle of attack deflects a large mass of air downward.  Locally, some air has to flow up and over the wing, but the area integrated change in momentum of the air results in a predominantly downward flow.  From Newton's law the wing experiences a reaction force upward because of the momentum of air deflected downward.  This is the momentum theory of aerodynamics.  Exactly the same aerodynamic forces can be found by studying the surface pressures around the airfoil such as Weltner does in his analysis.  This is the more common type of analysis.

I am not sure what you mean by 'a neutral body cannot apply an action force'.  That does not make sense to me.

As for pressure gradients, it is clear that when a moving stream hits the stagnation point on the nose of an airfoil a high pressure develops in that region. Then as the air flows along the surface, the pressure drops.  I think Weltner gives a good explanation of why the pressure drops, but let us just accept the fact for now.  So now we have a high pressure region next to a lower pressure region; i.e., we have a pressure gradient.  From thermodynamics we know that pressure is a force per unit area and air will flow from high pressure regions to low pressure regions.  That causes the air to accelerate over the wing upper surface for example.  By integrating all the surface pressures or by computing all the surface velocities and getting pressures from Bernoulli's law, we arrive at a total force or lift on the wing.  So there you have it - the interaction of the air with the wing causes pressure gradients, which in turn cause velocity changes - all of which causes forces on the wing.  Newton's law is satisfied.
Paul