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.
Sincerely,
Greg Blaisdell
AIAA Journal Associate Editor
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.
Sincerely,
Greg Blaisdell
AIAA Journal Associate Editor
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