Expert: Paul Soderman - 7/1/2011

Question
QUESTION: Thanks again and thanks for your repeated responses.

1. Yes, NSSL is inaccurate but only quantitatively i.e the force the fluid applies is not as given by its equation. But it is accurate in qualitative aspects (page 9 and 15 of the article by Von Karman):
  (i) The force is always normal to the surface
  (ii) It increases as density of the stream increases
  (iii) It increases as area increases
  (iv) It increases as the angle of collision increases
  (v) The converse of the above are true.
You can see these aspects operating in kite flights.   

To explain you need only qualitative aspects of an equation. To estimate only you need its quantitative aspect. So, any explanation by NSSL will be correct.

Note:
(i) NSSL is quantitatively inaccurate only for sub-sonic air flows and for liquid flows. It is found to be reasonably accurate for super-sonic air flows (Sec. 1.1 and 14.3, Fundamentals of aerodynamics by Prof. John D Anderson Jr. 4th edition, 2007)  
(ii) Sec.1.1 gives the cause for the inaccuracy proposed by Euler: Only part of stream collides directly, the rest bends before reaching the surface.
  Q: Why doesn't part of the stream collide?

2. See whether my understanding of Weltner is correct.

  Before the free stream flows the airfoil is surrounded by normal atmosphere/static air whose molecules are traveling randomly in all directions, hence colliding on the airfoil surfaces, so applying force/pressure on them.
Consider the free stream first reaching the airfoil surface. It goes on interacting with the surface all along(by Coanda effect, centrifugal force creating Newtonian forces - lot of forces!),till the trailing edge, resulting in pressure gradients in itself (- surely not in the static air ahead of it), resulting in its velocity changes throughout downstream. --- Correct?

If so, the following need consideration:
(i) Q: Why does the degree of interaction go on changing downstream?
(ii) Change the curvature of the surface. The pressure distribution/gradients change, resulting in change of lift.
Q: Why?
(iii) Change the curvature to flatness. The velocity of the free stream will not practically change.
Q: Why doesn't the stream interact now?
(iv) Above the stall angle the stream flows separated from the rear part of the upper surface only.
Q: Why is there no interaction in this part of the surface only?
(v)Q: Is this interaction between neutral bodies defined in science? I think not.
Q: Is it supported by laws or is it only an assumption?
(vi) Q:Is this interaction accepted by all?          
Prof. Anderson gives a different explanation (his book "Introduction to flight":
  " The stream is 'squeezed (!)' between the static air and the airfoil surface just like in a venturi. In a cambered airfoil, due to the camber, the upper stream is squeezed more. So, to obey the conservation law/s and its equation pAV = Constant, its velocity goes on changing"  

So, there are more than one explanation. I have read that if there are more than one explanation for a phenomenon by experts then none is correct and the correct explanation has not been found. So, I have thought of a new explanation by collision laws.

3. Consider the following collisions:
(i) When a propeller is rotated its blades collide on the static air molecules resulting in its propulsion.
(ii) When a (combustion) gas stream flows on to a turbine rotor, it collides on the blades and rotate the turbine.
(iii) When a (combustion) gas collides on the piston top in a piston engine in power stroke the piston is pushed down resulting in propulsion.

Note that the efficiencies of these devices are very low around 30% even after more than 150 years of research. Still, surprisingly, we have been studying them in terms of temperature, pressure, velocity, volume change but not in terms of collision, its laws, logic and measurements. God knows what such a study will reveal.

Thanks
Vidhyasagar

ANSWER: Vidhyasagar - It does not matter in science if one person or a million people believe in a law or theorem.  What matters is if the law or theorem predict or explains a physical phenomenon with reasonable accuracy.  If one verifiable experiment is devised that disproves the law or theorem it must be abandoned.  Thus is the case with Newton's sine-square law (NSSL) for lift on a thin wing, which curiously was not proposed by Newton but was proposed by others who read Newton's manuscripts. NSSL gives the normal force F on the wing as:

F = rho S U^2 sin^2(alpha), where rho = density, S = wing area, U = airspeed, alpha = angle of attack.  The lift normal to the airstream would then be L = F cos(alpha).

This law was devised early in the age of flight, before wind tunnels were developed.  Since then, countless wind tunnel studies have shown that the lift on a wing is given by:

L = CL rho U^2 S/2  where CL is the lift coefficient found by integrating the section properties along the span and is often found experimentally. It can be approximated for simple elliptical wings below stall as CL = 2 pi alpha.  Alpha is in radians.  This equation for wing lift has little resemblance to NSSL and NSSL must be discarded.  NSSL does not correspond to actual data.  To say it is qualitatively accurate is nonsensical, especially when a much better relationship has been found.  I don't believe NSSL works for supersonic flows either.

You must study Weltner's derivation of the centrifugal forces on a fluid element following a curved surface to see how pressure gradients on an airfoil are related to geometry (http://user.uni-frankfurt.de/~weltner/Flight/PHYSIC4.htm).  The centrifugal forces depend on the radius of curvature of the surface.  Anderson's idea that the flow over the top of the airfoil is squeezed like a venturi and must accelerate is correct and another way of looking at the field using conservation of mass.  But that idea cannot quantify the surface velocities because the area between the airfoil and infinity (or equivalent flow region) is unknown.

Your efficiencies are low.  A variable pitch propeller can easily operate from 80% to 90% efficiency.
Paul



---------- FOLLOW-UP ----------

QUESTION: Thanks again

1. (i) Yes, a variable pitch propeller has high efficiency.

Q1: Do fixed pitch propellers have such high efficiencies?

  (ii) Combustion gas piston and turbines (the jet engines in air crafts) do have low, less than 45% efficiencies. Even the non-combustion gas turbine - the wind turbines- have such low efficiencies. A gas turbine's maximum theoretical efficiency, called Bet's limit, is only 59%. Please verify.
If so,

Q2: Is it not high time to study these devices and the low efficiency aerodynamic lifts in terms of their actual work, collision, collision laws, logic etc,?

2. NSSL accuracy: Please verify your views.
  
  (i) 2007 edition of Prof.Anderson's book "Fundamentals of aerodynamics" gives the following:
     (a) Section 14.3, first para: "Newtonian theory (- based on NSSL and his ideas on fluid motions)is used frequently to estimate pressure distribution over the surface of a supersonic body"
     (b) Second para: "... collisions with the surface(fluid) particles lose their components of momentum normal to the surface but tangential momentum is preserved.
     (c)Page 774, 4th/5th para: "The quantitative and qualitative results ( of Newtonian theory)presented here are reasonable representations of hypersonic aerodynamic characteristics of a number of practical hypersonic vehicles."

  (ii) Please go through one page in the two books in the Books section of Google search typing 'Newtonian theory hypersonic flow"
     (a) In page 1 of this site, the book, "Fundamentals of modern unsteady aerodynamics" 2010 edition by Ulgen Gulat: In page 14: (NSSL) is valid only at hypersonic speeds and at high angles (more than 50 degree) of attack." This is also given in page 6, sec.1.1. of Anderson's book.
     (b) In page 3 of this site the book," Dynamics of atmospheric re-entry" 1999 edition by Reagen et all: Page 358: "To understand why newonial theory does so well qualitatively .."

So, to repeat: NSSL is accurate qualitatively under all conditions. Qualitative aspects (how/why?) explain. Quantitative aspects (how much) estimate. My doubts are only in explanations.

It is in accurate for lower speeds and lower angles of attack. Till recently, such high speeds and angles were not encountered. In fact at angles higher than 15 degree the lift is lower. So, scientist had to device means to accurately estimate/predict lifts. They found e.g. Weltner's article, the famous Circulation theory, thin airfoil theory.

3. Weltner et all article:
  
  (i) A doubt already asked but not addressed by you:
" The free stream becomes faster at the leading slopes of airfoil. This requires work with force and energy transfer by an external body. By the article, the external body is the airfoil using cenrtifugal and Newtonian forces. But

Q3: Which energy the airfoil has to go on transferring to the stream?

  (ii) Another doubt already raised but not addressed.

There are many pressures here.
     (a) Steady/lateral pressure of the stream - Bernoulli's pressure perpendicular to the direction of flow.
     (b) Dynamic pressure of the stream in the direction of the stream's flow due it can apply on a surface by NSSL.
     (c) Pressure inside the volume of the stream.
     (d) Atmosphere's/still air's pressure in it.
     (e) Atmosphere's pressure on the airfoil surface due to the collision forces of air's molecules on the surface.

Q4: Which of these pressure is changed into gradients by the airfoil?

Q5: To induce pressure gradients also requires energy. Which energy the airfoil uses.

  (iii) Another doubt (June 23) raised but I am unable to follow and have been brooding over it:

My doubt: " There two opposite pressure gradients over the airfoil surface. If the gradients induced by airfoil are in the stream then the stream must flow both downstream and upstream. Instead it flows only downstream. Why?"

Your reply: ... the pressure gradient is not strong enough to overcome velocities induced by the free stream and drive the flow upstream"

If the pressure gradient is in the stream with higher pressure at the trailing edge and lower pressure at the point x at the crest,

Q6: Does it mean that the stream flows against thermodynamic gas diffusion laws?

If the pressure gradient is in still air then,

Q6: Does it mean that it is not strong enough to slow down the on-coming stream, stop it and reverse its direction?

Q7: If so, which body induces pressure gradients using which energy in still air?.    

Q8: What is meant by the phrase,' velocities induced by the free stream' in your reply? In which body does the free stream induce velocities?

Before proceeding further please confirm the following very important observations:
(a) At + AOA, the free stream collides on the whole under surface of airfoil (symmetric or cambered) from the leading edge to the trailing edge; some free stream collides only on the leading slope of the upper surface, not on the whole upper surface.
(b)But at zero AOA, the free stream collides only on the leading (upper and lower) slopes of airfoil (symmetric or cambered), not on the rest of the surfaces, where the upper and lower stream (formed from the free stream) flow in stream lines with the inner most (boundary) layers colliding/interacting) with the irregularities of the surface resulting in friction, slowing down, vortices formation (? Weltner's centrifugal forces).
(vortices formation is given in page 311 and 794 in Anderson's book)

From Weltner's article one infers/has to that all lifts occur due to NTL. Let us verify.

  (i) At +AOA, the airfoil (symmetric or asymmetric) is lifted by NTL (to be exact by my NSSL). But at zero AOA, the symmetric airfoil is not lifted.

Q9: Why doesn't NTL lift?

Note: The upper and lower streams are both faster than the free stream but equally faster.

  (ii) But the cambered airfoil is lifted.
Note: (a) Since the lift must be due to the shape/asymmetry/cambering some call this lift "shape lift" (b) Here also, both the upper and lower streams are faster than the free stream but the upper stream is faster than the lower stream. Why is debated by experts till date (Anderson accepts this debate.

Q10: How does NTL lift here?

This question made me think a long time. Let me apply this law here. Tell me if I am wrong.
" The leading lower slope deflects the (colliding) free stream downward; in reaction the stream lifts the airfoil up" But, upper slope will deflect the colliding free stream upward; in reaction the stream will push down the airfoil. So the opposing forces, if they are equal, will either cancel/balance when there will be no lift or result in a net force. If the net force is directed up (i.e force of the free stream on the lower slope is greater) then will it lift? No, due to its point of application on the slope it will cause rotation/ torque/moment of the airfoil around the axis at the centre of pressure or at trailing edge (this torque/moment is given in sec.1.5 of Anderson's book)."     

So, NTL (not even NSSL) can cause lift. Then,

Q11: Which body, by which force, by which law, using which energy lifts?

This is another question that made me think. The only answer I got from some experts is,' lower stream by Bernoulli theorem.' But I couldn't accept this answer. Reasons:

Q12: Will the lateral pressure's force be strong enough to lift the heavy airfoil or aircraft?

I think no. One logical reason is given by the lift of the hand held paper when an airstream is blown across its upper surface (which resembles the upper surface of an airfoil). If the lateral pressure is strong to lift airfoils then the lateral pressure of this upper stream must push down the paper; instead the paper is lifted up.

 (iii) The same airfoil is lifted when it moves in still air at zero angle of attack.

Q13: How can NTL explain this lift?

Note that there is no free stream sent here towards the airfoil.

Q14: Are upper and lower streams formed here? Is the upper stream faster than the lower stream? If so,

Q15: How are they formed? Why is the upper faster than the lower stream?  Which body by which force by which law using which energy lifts?

Note: This lift in terms of motion of airfoils is not explained in literature.

Finally, I request you to answer the doubts/questions, as far as possible so that my doubts get fully cleared. Kindly bear the pain. It may be worthwhile.

Thanks

Vidhyasagr.

Answer
Vidhyasagr - Your questions are getting longer and longer.  I wish you would take a course in aerodynamics - some of your questions require understanding of basic concepts that I don't have time to get into here in depth.  Anyway, here are some answers to your numbered questions:

1 - Fixed pitch propellers can have high efficiency but only at one flight condition.  See Dommasch: Elements of Propeller and Helicopter Aerodynamics.  Betz's Law applies only to a wind turbine extracting energy from wind.  The maximum possible energy derived is 0.593 of the kinetic energy in the wind.
2 - Yes, it is always good to study aerodynamics of propulsion devices.
Yes, NSSL seems to be valid for hypersonic flow only, but with some modifications proposed by various authors such as Van Dyke.  It is not valid for subsonic flow.
3 - I don't quite understand your question.  A wing requires energy from a propulsion system; that energy goes into work done on the stream.  A force times velocity (i.e., drag times airspeed) equals power which is the time rate at which work is done.  Eventually, the energy goes into heating the air and the airplane, but that is usually outside our scope of interest.
4 - Bernoulli's equation describes three pressures on a streamline:  total or ram pressure (pressure developed when flow is brought to rest), static pressure (pressure felt while moving with a stream or pressure on a surface parallel to the stream), and dynamic pressure (kinetic energy per unit volume).  To get the forces on a wing we need to find the surface pressures which are the static pressures in Bernoulli's equation.  However, we usually know total pressure and since all three pressures are related, we can work with static pressures or velocities to deduce forces.  The static pressures and velocities change with pressure gradients on the wing.  This is clearest for a coordinate system attached to the wing.
5 - See answer 3.
6 - Question unclear.  The air flows over the wing at the nominal airspeed of the airplane.  The pressure gradients slow or accelerate the air relative to the airplane airspeed.  I don't know how to make it clearer.
7 - See answer 3.
8 - See answer 6.
9 - I've answered this before.  A symmetrical airfoil at zero angle of attack has pressures on the upper and lower surface that in the vertical direction are equal and opposite.  The total lift is zero.
10 - A cambered airfoil has lower pressures on the upper surface than on the lower surface, so the forces integrated all around the airfoil are upward (lift) at angles of attack equal to or greater than zero.  Lift forces are obviously strong enough to lift an airplane. I think you are caught up in this collision idea and not understanding pressures.  Pressures are key to understanding aerodynamics.  Integration is key to understanding aerodynamics.
11 - See answer 10.
12 - See answer 10.
13 - See answer 10.
14 - Yes
15 - See Weltner.... again.  And Euler.  And Newton.  And Bernoulli.
Paul

p.s.  When I reread your questions I see you are caught up in the idea that the airfoil deflects the air away.  But from the Coanda effect we know that the air follows the airfoil surface.  Even the upper side air follows the surface and flows downward at the trailing edge if the airfoil has angle of attack or is cambered.  The net effect is lift.  This is also true for a sheet of paper, but the flow field is more complex because the upper surface flow creates a separation bubble that effectively forms a pseudo thick cambered airfoil.  Our professors used to try to trick us on that one.