tisdag 11 september 2012

AIAA Defends Old Incorrect Theory of Flight 1

Below is the the response from AIAA associate editor Greg Blaisdell on our rebuttal of the 
referee reports. An analysis of the position of AIAA is given in the next post.

Dear Professors Hoffman, Jansson and Johnson,

I have read your response to the reviewers’ comments and gone back over 
the reviews and your paper. I agree with the reviewers’ assessment of 
your paper. (Note: below I refer to Reviewers 1 and 2 as they are listed 
in the AIAA manuscript review system; on Professor Johnson’s website the 
two are reversed.) Reviewer 1, especially, has given a thorough, 
articulate and gentle explanation of the deficiencies of your paper. 
After reading your paper and your responses to the reviewers’ comments, 
I can see several misconceptions you have concerning basic concepts in 
fluid mechanics. My goal below is not simply to reiterate the comments 
of the reviewers, but to help you see some of the areas where you need 
to improve your understanding. I will try to be clear and direct in my 
comments to avoid any misunderstanding.

First, one of your major objections to the classical theory of the 
generation of lift is that the potential flow solution has a stagnation 
point at the (sharp) trailing edge and, therefore, produces a pressure 
distribution with a high value of pressure at the trailing edge. This 
high value of pressure is not observed in experimental measurements. 
Also, the pressure distribution from the potential flow solution does 
not give rise to any drag, which is incorrect from common experience 
(d’Alembert’s paradox). What you seem to be missing, which reviewer 1 
alluded to, is that this view of the classical theory is “truncated”; it 
is not a correct view.

High Reynolds number flow around an airfoil (or a wing in 3-D) is, as 
reviewer 1 said, a singular perturbation problem, where a small 
parameter (1/Re) multiplies the term in the momentum equation with the 
highest order derivative. The potential flow solution you object to is 
only the leading order inviscid (outer) solution. The complete theory 
treats the coupled viscous-inviscid interaction by determining the 
viscous boundary layer (inner) solution, finding the displacement 
thickness of the boundary layer, adding that to the starting geometry to 
find the effective shape of the airfoil (thus accounting for the viscous 
displacement of the streamlines in the outer inviscid part of the flow 
field), recomputing the potential flow solution, and then iterating this 
process until a converged solution is found. This results in a pressure 
distribution that does not have a high value of pressure at the trailing 
edge, in agreement with experimental measurements. This iteration 
process is what is done by the airfoil analysis program XFOIL, which 
reviewer 2 mentioned. XFOIL is widely used in teaching aerodynamics, and 
I use it in my classes.

The pressure near the trailing edge differs from that of the leading 
order inviscid potential flow solution because of the displacement 
effect of the viscous boundary layer. As a result the pressure does 
contribute to the drag; this is termed “form drag”. Both pressure and 
skin friction contribute to drag. Which one is dominant depends on the 
airfoil design, angle of attack and Reynolds number. At small angles of 
attack the skin friction contribution can be much larger than the 
contribution from the pressure; while pressure or form drag is more 
important at higher angles of attack where the airfoil acts more like a 
bluff body. D’Alembert’s paradox is no longer a paradox. His potential 
flow solution lacked two effects due to viscosity that create drag – 
skin friction and form drag due to the displacement effect of the 
boundary layer. Please note that, contrary to what is sometimes stated, 
form drag does not only occur due to boundary layer separation. Even if 
the boundary layer does not separate, the displacement effect of the 
boundary layer will alter the potential flow solution and result in drag 
due to pressure.

As reviewer 1 said, it is the truncated form of the classical theory 
that you object to. Unfortunately, the complete theory is not always 
taught, with the result that many students have misunderstandings 
concerning flow over airfoils and wings. In my department we do not 
require our students to take a course in perturbation methods; as 
reviewer 1 said, aerospace engineers today use computational fluid 
dynamics; and, as a result, they lose sight of some of the theoretical 
underpinnings of what we study.

Many of the concepts associated with viscous boundary layers, 
viscous-inviscid interactions, circulation, vorticity and the generation 
of lift, are not intuitively obvious. One of my main jobs as an educator 
in the area of fluid mechanics is to build up my students’ intuition by 
increasing their understanding of basic concepts.

Another set of basic concepts that you misunderstand deal with 
circulation and vorticity. They are related, but they are not the same, 
and both are important to understanding the fluid mechanics of airfoils 
and wings. I do not have the time or space to elaborate in detail; these 
are concepts that are taught over the course of a full semester. 
However, there are two points I want to clarify. First Kelvin’s theorem, 
that circulation does not change for a closed loop moving as a material 
curve (moving with the fluid), only depends on the assumptions of (i) 
inviscid flow, (ii) incompressible (low-speed) flow, and (iii) a 
conservative body force (e.g., gravity). The flow can be unsteady, 3-D 
and rotational (nonzero vorticity) and the theorem still holds. It is 
not invalidated by there being fluctuations in the freestream or 
instabilities in the flow (more on that below).

One important difference to see between circulation and vorticity is 
that when a vortex is stretched, as by a strain field discussed in your 
linear stability analysis, the vorticity is increased as the vortex 
radius decreases and vortex lines become concentrated, but the 
circulation remains fixed, in agreement with Kelvin’s theorem. The 
vorticity is twice the local rotation rate of fluid elements, and the 
increase in vorticity is similar to the increase in rotation rate of an 
ice skater when he spins and then brings his arms in tight. His rotation 
rate increases, but his angular momentum remains constant (not 
accounting for friction). Kelvin’s theorem deals with circulation, and 
it is in fact the circulation around the airfoil that is important to 
the lift.

The second point I want to make concerns the generation of vorticity. 
What reviewer 1 stated about vorticity not being generated in the 
interior of the flow under the assumptions of (i) incompressible flow 
and (ii) a conservative body force is correct. This can be proved 
mathematically from the basic governing equations. In the discussion 
above about a strained vortex, the vorticity increases locally because 
the vortex lines become more concentrated; however, no new vorticity is 
generated. New vorticity is generated on solid surfaces through the 
action of pressure gradients or unsteady motion of the solid surface. 
This point is discussed well in the text Incompressible Flow by Panton. 
The generation of vorticity on solid surfaces is one of the more 
difficult subjects included in my graduate introductory fluid mechanics 
course. The fact that some of the concepts are difficult to understand 
does not make them wrong.

The last technical point I want to make concerns the trailing edge 
instability and your linear analysis. It is well known that vorticity in
a strain field, such as near a stagnation point, results in vortex
stretching and an exponential increase in vorticity. The solution for
this is worked out in the book The Structure of Turbulent Shear Flow by
Townsend and is part of the rapid distortion theory of turbulence. As
discussed above, the increase in vorticity magnitude does not mean an
increase in circulation. The trailing edge vortices in your simulations
form in counter-rotating pairs. Their net circulation is zero, and their
presence does not alter the circulation or the lift on the airfoil. I
also want to point out a flaw in your analysis. Fluctuations in the flow
are not introduced by a non-conservative body force, as you state. For
flow over an airfoil or wing, there is no such non-conservative body
force. The only body force acting is gravity, and it is conservative
and, therefore, does not generate vorticity. Instead fluctuations come
from the freestream, as turbulence from upstream is convected toward the
airfoil; fluctuations near the trailing edge also come from turbulent
flow in the boundary layers or, in the case of a separated boundary
layer, turbulence produced in the free shear layer.

Lastly, both reviewer 1 and I had suggested that you contact colleagues
at KTH who could help explain some of these concepts to you. In response
to one of your previous comments, I want to say that doing so is not a
requirement for submitting papers to AIAA. That suggestion was made
because there are basic concepts in fluid mechanics that you do not
understand, and it is easier to explain those concepts face to face over
a period of time, rather than through the limited medium of email. I
know some of your colleagues in the Mechanics Department at KTH, and I
have a lot of respect for their knowledge of fundamental fluid mechanics
and applied mathematics. I strongly suggest you talk with them, or take
some of the courses they offer.


Greg Blaisdell
AIAA Journal Associate Editor

1 kommentar: