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.

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

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

Informative Blog. i liked it very much..

SvaraRaderacool math