Myth of Bernoulli's Principle

[mtb]

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(The following is an excerpt from an article on wind tunnel testing of America’s Cup sailboat keel designs from the Seattle Post Intelligencer.)

Paul Robertson, who created the scale model of the keel bulb for the test, said the wind-tunnel test is great for testing flow, but it’s not quite accurate to say that the behavior in air and water is always going to be identical.

“Air is pretty much always just air,” said Robertson, president of Aeronautical Testing Service in Arlington. “But water is not always just water.”

That is, a wind tunnel is great for airplanes or testing behavior in air because the nature of air remains fairly consistent. But water, as a substance to move through, can vary by quite a bit, he said. (Emphasis added.)

“A lot of garbage in the water can cause greater turbulence,” Robertson said. Sailing closer to land, where the water contains a lot of silt and dead organisms, requires a different configuration for maximum efficiency than sailing in deep water, he said.

(The last paragraph is of particular interest to surfboard fin design and dynamics.)

Surfboards are ridden in a high energy zone, with lots of residual turbulence present at various length-scales and with varying degrees of intensity as the result of the shoaling, transformation, and breaking of waves. Therefore the most representative and available lift curves for foils operating in this environment (short of measurements/calculations in the presence of similar turbulence) are probably those measured/computed for foils with “standard roughness”.

The differences in foil characteristics between a foil with roughness, versus a smooth foil, are that the presence of roughness normally results a higher drag coefficient coefficient, and a lower angle of attack at stall (and hence a reduced maximum lift force). However, the change in lift per unit change in angle of attack (up to the point of stalling) is virtually unchanged by the presence of roughness. The presence of turbulence in the water should also not affect the lift slope (but may either adversely or beneficially affect the stall angle and drag coefficient, depending on the characteristics of the foil section, and the properties of the turbulence).

Obviously one of the prime sources of “garbage” at surf breaks is the presence of sea weed(s)–fortunately the swept-back nature of most surfboard fins (especially keel fins) makes this effect, while potentially very significant, of only a transient nature.

While it is true that in general “air is pretty much air”, that is also not always true. For example, airfoil sections that are designed to maintain laminar flow as long as possible (to minimize drag) are sensitive to small levels of roughness on the foil surface. Practical experience has shown that flying through rain with some foil designs can result in an associated increase in effective roughness that results in a reduced lift. In fact, it was discovered that with one canard configured aircraft (small wing forward, main wing aft), flying in heavy rain resulted in such a reduction in maximum lift on the canard that it was impossible to keep the nose up enough to maintain level flight until exiting the area of most intense rain.

[mtb]

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……in future when refering to whats happening over the foil i could use the term bernoulli/coanda effect…

i wont give away the bernoulli effect , look closely at the coanda effect and its still intertwined with the bernoulli principal……

……function before fashion…

regards

BERT

Here’s a couple of different foil sections (from: Aerodynamics, Aeronautics, and Flight Mechanics. Barnes W. McCormick. 1979. John Wiley and Sons Pub. NYC), and associated lift curves (lift coefficient as a function of the angle of attack) that may assist in conceptually exploring the Bournelli and Coanda effects. There are two sets of curves. One for a Reynolds number of 420,000, the other for 128,000. The larger corresponds to a fin with an average chord length of 3.5" (e.g. ~ 4 inch base) moving through water at about 11 mph. Measured max board speeds (shortboards, bodyboards, with riders of varying skill levels) in head-high (+/-) waves using the GPS system described by “solo” in the thread “Surfing speed challenge” seem to run about 16 to 19 mph, so a Reynolds number of 700,000 would probably be more representative value when trimming across the face of a wave. But curves for that Renolds number weren’t available, and 420,000 is probably reasonably representative when executing maneuvers (especially toward the end of the maneuver). In any case, you can get a feel for the effect of increasing Reynolds number by comparing the curves for the 420K and 128K Reynolds numbers.

There are 5 foils with drastically different section profiles, ranging from a very thick (25%) section with a very rounded leading edge (625), to a curved thin sheet(417A), and a thin flat plate. One might expect that lift associated with the Bernoulli effect would be maximal with the thick section (625), and minimal with the flat or curved thin sheet. However, lift curves in the second figure show the change in lift with changing angle of attack for each of the foils (for the two Reynolds numbers mentioned above). [In this case, the angle of attack has been chosen such that zero angle of attack corresponds to zero lift (rather than based on a line from the leading edge to the trailing edge) of the foil.]

One of the more obvious features of the lift plots is that both thin sections (flat plate, 417A) generate more lift per unit angle of attack up to the point of stalling than do the thicker, and more rounded foil sections. In fact, the thickest and most rounded section generates the least lift per unit angle of attack. The major advantage of the more rounded sections is that the onset of stalling of the section is delayed to a larger angle of attack–and hence to a larger maximum lift.

Unfortunately, the curves for a Reynolds number of 420,000 only extend up to an angle of attack of about 12-15 degrees. However, some indication of the effects on lift after reaching the stall angle angle might be estimated from the curve for a Reynolds number of 128,000. Here we see that although the rounded sections have a higher maximum lift coefficient, the consequences of stalling on the magnitude of the lift are more abrupt and result in a greater reduction (as a percent of the maximum) with further increased angle of attack. Hence both the flat plate and the thin curved foil (417A) sections should be easier to control as the foil angle of attack moves through the stall angle–and it may even be that at angles of attack in excess of 20 degrees, the curved thin section may generate more lift than the more rounded sections. [Note, however, that with the small aspect ratios, and high degree of sweep present in typical surfboard fins, this difference will be much less evident than on these plots. The latter correspond to zero sweep and an infinite aspect ratio.]

The two sections N60 and N60R are essentially the same foil section, but with the latter incorporating a small upsweep (reflex) in the trailing edge. Reflex is commonly used in “flying wing” designs to offset the “pitch down” characteristic of sections without reflex. On conventional airplanes, the horizontal stabilizer on the tail of the aircraft generates a down force that compensates for this pitching moment. I would expect that the lift created with the reflexed foil (N60R) would be less than for the same foil without the reflex (N60) since the upswept trailing edge would be expected to reduce the downward velocity (momentum) imparted to the fluid flowing over and under it. However, the apparent equality shown in the figure may simply be a consequence of setting the angle-of-attack as used in the figures to be zero at zero lift (and in reality, the N60R section requires a more positive angle of attack, relative to the horizon, than does the N60 section when both are producing zero lift). It is clear, however, that the presence of reflex reduces the maximum lift that can be generated (consistent with the momentum conceptual approach to generating lift).

[blakestah]

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Unfortunately, the curves for a Reynolds number of 420,000 only extend up to an angle of attack of about 12-15 degrees. ...

This is very close to the relevant dynamic range on surfboards…

[mtb]

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Unfortunately, the curves for a Reynolds number of 420,000 only extend up to an angle of attack of about 12-15 degrees. …

This is very close to the relevant dynamic range on surfboards…

Agreed (most of the time). However there are some maneuvers that involve “skidding” the rear of the board, during which time the foil is either ventilated, or stalled, or both-- in which case the post stall condition becomes relevant with regard to recovery from the maneuver (with a stalled foil having some lift/drag properties roughly in common with a ventilated foil). Hence it would have been nice to see the lift curves for a Reynolds number of 428,000 extended out to a larger angle of attack so that one could examine the post stall characteristics at these speeds/Reynolds numbers.

Just to clarify the application of these foil section lift curves to the typical surfboard fin (and in a very approximate manner)…

While it is correct that the relevant dyanmic range of foils is approximately zero to 12-15 degrees, it should also be noted that the lift coefficients for the foil sections shown in the graphics can be expected to be about one-half the values shown when incorporated into typical surfboard (side) fins.

By way of example, let’s examine a Futures Eric Arakawa 450 foil planform and assume that the foil section (of infinite aspect ratio, as in the figure) stalls at an angle of attack of 15 degrees. The aspect ratio for the FEA 450 foil is about 1.41. The end plate effect of the bottom of the board will raise the effective aspect ratio to something approximating 2.42. [Actually, it will be a bit less, but as I don’t know the exact value–which depends in part on the placement of the foil relative to the rail–I’ll use a factor of 2 (as for a perfect end-plate) to get the approximate value of 2.42.]

For an unswept foil, the slope of the of the lift coefficient plotted vs the angle of attack for an aspect ratio of 2.82 will be reduced by a factor of about 0.71 from that for the (infinite aspect ratio) section slope, while the max lift coefficient remains essentially the same. Hence the angle of attack at the stall increases to about 21 degrees.

Assuming a leading edge sweep of 45 degrees, the slope of the lift coefficient vs the angle of attack will be reduced by an additional factor of about 0.90, resulting in a stall angle of approximately 23.5 degrees–provided that the foil stalls at the same maximum lift coefficient value as for the same unswept foil section. However, the maximum lift coefficient for a swept wing decreases roughly as the square of the cosine of the sweep angle. Hence for a 45 degree sweep, the maximum lift coefficient would be roughly one-half of the maximum lift coefficient for the unswept wing (in practice, it will be a bit more).

Thus the maximum lift coefficient will be reached, and the stall will occur, at approximately one-half the 23.5 degrees that we computed above assuming that there was no reduction in the maximum lift coefficient. Hence the combination of the finite aspect ratio and swept foil results in a stall angle of ~12 degrees – i.e. comparable to the stall angle of an unswept foil of infinite aspect ratio and the same foil section, but the magnitude of the lift coefficient is only about one-half that of the unswept infinite aspect ratio foil at the point of stall.

[blakestah]

I put fins on boards and allowed them to rotate to get my answer. When I couldn’t make a useful turn because the free rotation was too much, I assumed I had reached the relevant angle of attack for surfboards with 30-35 degree sweep angles.

10 degrees was pushing it. Barely able to turn.

[SOUP]

are you thinking that “lift” can either be upward or downward depending on the hull and fins, and cause the surfboard to rise up or squat down? Are you talking about both horizontal and vertical lift, maybe with horozontal lift that creates pressure between the fins to reduce “chatter” or to speed up flow to the center fin or something? The lift is a force that can act in any direction depending on the design, and is usually centered near the fins… creating stability and a pivot point?

[mtb]

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are you thinking that “lift” can either be upward or downward depending on the hull and fins, and cause the surfboard to rise up or squat down? Are you talking about both horizontal and vertical lift, maybe with horozontal lift that creates pressure between the fins to reduce “chatter” or to speed up flow to the center fin or something? The lift is a force that can act in any direction depending on the design, and is usually centered near the fins… creating stability and a pivot point?

“Lift” is normally defined as the component of the force on a foil that is perpendicular to the direction of motion of the foil through the fluid; “drag” is the component of the force parallel to the direction of motion. Hence on a foil of nearly constant thickness along its span, the lift force lies in a plane approximately perpendicular to the plane of the foil, and perpendicular to the free stream motion (i.e. the direction of the flow well away from the influence of the foil) within that plane. Hence a vertical fin generates lateral lift forces; a horizontal fin creates vertical lift forces, a canted foil generates both horizontal and vertical forces whose the magnitudes (in the absence of the presence/influence of the bottom of the board or the sea surface) are proportional to the cosine and sine of the cant angle, respectively.

The magnitude and direction (+/-) of the lift force depends on the angle of attack relative to the angle of attack for zero lift (the latter is zero for symmetrical foils, around -4 degrees for typical cambered foils). Hence horizontal or canted foils can produce a lift force that is directed upward or downward (relative to the cambered side of the foil, if asymmetrical), depending on the angle of attack relative to the bottom of the board, and the angle of attack of the bottom of the board relative to the free stream motion of the water.

This effect was used back in the 60’s to assist in nose riding by adding a pair of small horizontal fins to the primary skeg/fin to produce a downward force on the tail of the board (accomplishing the same thing as “kick in the tail” combined with the Coanda effect)–and the star fin can contribute a similar function (depending on the angle of attack of the tip foils), but with the increased benefit of an “end plate” effect to increases the effective aspect ratio of the primary fin/foil.

The horizontal component of lift for a vertical fin is a bit more complicated as the wetted area of the bottom of a surfboard (and most boat hulls) are typically of low aspect ratio (e.g. 0.5 to 2), and hence there can be a significant cross flow generated as water moving toward the hull “chooses” to veer off to either side, rather than getting deflected downward and passing under the board/hull (end result: the board must have a greater angle of attack to provide the lift required to support the weight of the board and rider, leading to increased induced drag, which varies approximately as the square of the angle of attack). Because of this cross flow (and the non-zero lift at at an angle of attack of zero for an asymmetrical foil section, if present), it is possible that in some circumstances both side fins may be generating lift forces directed away from the centerline of the board (and creating induced drag), even though they may have some toe-in.

As to whether toed-in side fins increase the flow over a following center fin, I can’t say. Although one might argue that because the trailing edges of the two side fins have less separation than the leading edges, and hence the speed of the flow must increase, examination of the flow speeds (as illustrated in one of the links contained in the first post in this thread) shows that the speed of the flow below a single foil at an angle of attack that produces an outward lift force is reduced below the free stream speed on the inward side of the foil (due to the stagnation point for the flow moving down the inside foil face and redirecting part of the flow over the outward face–as mentioned in a previous post). With a pair of fins, the situation becomes less certain. Whether the downwash from the two fins merging, and just the interference between the two fins is sufficient to offset, or speed up the flow between them, and over the trailing center fin, will depend on a variety of factors including the toe-in angles, the cross-flow generated by the planing bottom of the board, the separation between the two side foils, the fore/aft separation between the side foils and the center foil, the relative areas and foil sections (and aspect ratios?) of the three fins, etc… A further complication is that in a turn, the outboard side foil may not even be in the water. This complexity is beyond my knowledge and my ability to reliably estimate, so I can’t comment on the resultant effect.

[SOUP]

thanks. for what its worth, i could be flat wrong but it seems that whirlpools like to be upright… i have watched a whirlpool on its side struggle to get upright. fish like halibut and manta rays are not fast swimmers, but fish that are not dorso-ventrally flattened can be. Maybe eddys (of the size we are talking about) naturally orient themselves uprght, and surfboard fins therefore are better aligned than the hull to tap into this power?

[mtb]

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thanks. for what its worth, i could be flat wrong but it seems that whirlpools like to be upright… i have watched a whirlpool on its side struggle to get upright. fish like halibut and manta rays are not fast swimmers, but fish that are not dorso-ventrally flattened can be. Maybe eddys (of the size we are talking about) naturally orient themselves uprght, and surfboard fins therefore are better aligned than the hull to tap into this power?

This is definitely an area with which I’m not familiar. But keeping in mind that I’m speaking out of ignorance, my guess would be that (assuming that you really mean “whirlpool” vs “eddy”) gravity and the presence of a free surface (i.e. air-water) interface tends to drive whirlpools to a vertical orientation. However, “submerged” eddies would not be subject to this alignment tendencey–as supported by the photo of the persistent (horizontally aligned) tip vortices generated by the jet plane in one of the two links in the initial post in this thread.

As regards flat fish versus vertical fish, my guess would be that flat fish are primarily bottom feeders (giant Manta probably being an exception)–as supported by the location of the mouth–and hence don’t need to swim as fast as pelagic predators. I also seem to remember reading in the Scientific American some time ago that the tail and body structures of pelagic fish differ between those that have a high degree of maneuverability (low aspect ratio tail fins and taller bodies) and those capable of high speeds (high aspect ratio tail fins and more streamlined bodies).

[mtb]

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I put fins on boards and allowed them to rotate to get my answer. When I couldn’t make a useful turn because the free rotation was too much, I assumed I had reached the relevant angle of attack for surfboards with 30-35 degree sweep angles.

10 degrees was pushing it. Barely able to turn.

Certainly controlled experimentation is one of the best (if laborious) ways to optimize a complex system.

When you say that the free rotation was too much at 30-35 degree sweep, I interpret that to mean a 30-35 degree rotation about the yaw axis of the foil in it’s box (rather than the sweep angle of the leading edge or midchord of the foil)? I also interpret “too much” to mean that one has exceeded the stall angle (or at least reached a point on the lift curve where the change in lift per unit change in angle of attack has noticably decreased)?

What seems to be missing to determine the angle-of-attack of the foil with respect to the free stream flow is the alignment of the longitudinal axis of the board relative to the free stream velocity–the sum (or difference depending on how the rotations are measured) of this angle, and the 30-35 degrees of rotation for the foil in the box yielding the angle-of-attack of the foil relative to the free stream motion. Although possible, I do not believe that is fair, nor necessarily a good approximation, to assume that the mean stream flow is aligned with the longitudinal axis of the board. Have you measured this?

Another problem in computing lift curves for surfboard fins from presently available data and relatively simple models (versus using more complex–but more accurate-- CFD numerical simulations) is that typical surfboard fins tend to fall in-between the most commonly published configurations with swept leading edges: (1) a swept wing and (2) a delta wing.

In my example calculation, approximating the surfboard foil as a swept wing, I estimated that the maximum lift coefficient for a surfboard foil with a leading edge sweep of 45 deg. would be about one half the maximum lift coefficient value for the same, but unswept foil–and that the corresponding angle-of-attack would be about 12 degrees. However, as I commented paranthetically, the equation used to get the maximum lift from the sweep angle typically underestimates the maximum lift coefficient. Assuming that the actual lift swept is about 70 percent of that when unswept (vs 50 percent in the prior example) would raise the stall angle to about 17 degrees.

However, a typical swept wing does not have as great a difference between the root chord and the tip chord as does the surfboard fin. In that respect, the surfboard fin is more like a delta wing. On the other hand, the typical delta wing has a trailing edge that is more or less perpendicular to the fore-and-aft axis of the foil, while the trailing edge of the surfboard fin is relieved forward towards the root of the foil.

A plot of maximum lift coefficient vs aspect ratio for a “plain delta wing” (McCormick, p.303) indicates the maximum lift coefficient for an aspect ratio of 2.8 is about 1.00. Assuming (as mentioned above) that the maximum lift coefficient for a swept wing with 45 degrees of sweep is 0.7, and the maximum lift coefficient for an unswept wing is 1.5, the maximum lift coefficient would be about 1.05 (for the presumably underestimated ratio of 0.5, the max lift coeff. would be about 0.75).

Thus it would seem that the maximum lift coefficient for the surfboard foil in this illustration would be about 1.0–independent of whether or not it is treated as a swept wing, or a delta wing. However, the angle-of-attack corresponding to this lift coefficient differs between these two wing forms. Although the lift curve is not illustrated in McCormick for an aspect ratio of 2.82 (the illustrations only go up to 2.0), the maximum lift coefficients for a delta wing with an aspect ratio of 1.0, and 2.0, both appear to occur at an angle-of-attack of about 25 degrees. There are a couple of reasons why one would expect the maximum lift coefficient to decrease as the aspect ratio increases beyond 2.0, so it would seem that the angle-of-attack for the surfboard foil, based on the data for a plain delta wing, will be a bit less than 25 degrees.

Therefore I would estimate that for the foil I used in my example, the maximum lift coefficient would be attained with an angle-of-attack somewhere between 17 and 25 degrees (the estimates for the swept wing, and the delta wing). Hence if I assume that my example foil is a reasonable approximation to your foil, then it would seem to imply that the angle of attack (about the yaw axis of the surfboard) between the longitudinal axis of the board and free stream flow might range between something like 5 and 15 degrees (and probably biased more towards the smaller angle).

It would be interesting to determine the actual value for a comparison.

I’m guessing that perhaps your incremental changes were on the order of 5-degrees? …and hence the foil stalls, or it’s response diminishes, somwhere between fin rotation angle of 25-30 deg, and/or 30-35 degrees?

[blakestah]

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I put fins on boards and allowed them to rotate to get my answer. When I couldn’t make a useful turn because the free rotation was too much, I assumed I had reached the relevant angle of attack for surfboards with 30-35 degree sweep angles.

10 degrees was pushing it. Barely able to turn.

Certainly controlled experimentation is one of the best (if laborious) ways to optimize a complex system.

When you say that the free rotation was too much at 30-35 degree sweep, I interpret that to mean a 30-35 degree rotation about the yaw axis of the foil in it’s box (rather than the sweep angle of the leading edge or midchord of the foil)? I also interpret “too much” to mean that one has exceeded the stall angle (or at least reached a point on the lift curve where the change in lift per unit change in angle of attack has noticably decreased)?

The sweep angle is the same as the sweep angle on generic fins. Thruster fin leading edges have a sweep close to 35 degrees, and most singlefins have sweep from 25 to 30 degrees.

I meant only to imply that was the sort of fin in the box. I made pie-shaped cutouts to limit rotational angle, and initially used very little side forces. So, the fin just flopped side too side, and could be engaged, firmly, at the limits of rotation.

At 10 degrees of rotation relative to the board centerline, only very short radius turns were possible, and spins were common. This was too much rotation, if you tried to toe-in a thruster rail fin this much it would be difficult to get any drive out of it.

With 10 degrees, however, it is possible to create some drive at angles of attack of the board that are entirely within the normal range for me. As a system optimization problem, I made the next boxes with a lower limit on rotation. I am pretty sure that on this shortboard if I had used 12 degrees, I would not be able too turn at all.

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Although possible, I do not believe that is fair, nor necessarily a good approximation, to assume that the mean stream flow is aligned with the longitudinal axis of the board. Have you measured this?

No, my only measure has been the impact of the end-of-rotation on the usability of the fin. With no side forces, and because the water flow is very powerful in re-aligning the fin, you can get ballpark estimates from this.

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Assuming that the actual lift swept is about 70 percent of that when unswept (vs 50 percent in the prior example) would raise the stall angle to about 17 degrees.

However, a typical swept wing does not have as great a difference between the root chord and the tip chord as does the surfboard fin. In that respect, the surfboard fin is more like a delta wing. On the other hand, the typical delta wing has a trailing edge that is more or less perpendicular to the fore-and-aft axis of the foil, while the trailing edge of the surfboard fin is relieved forward towards the root of the foil.

A plot of maximum lift coefficient vs aspect ratio for a “plain delta wing” (McCormick, p.303) indicates the maximum lift coefficient for an aspect ratio of 2.8 is about 1.00. Assuming (as mentioned above) that the maximum lift coefficient for a swept wing with 45 degrees of sweep is 0.7, and the maximum lift coefficient for an unswept wing is 1.5, the maximum lift coefficient would be about 1.05 (for the presumably underestimated ratio of 0.5, the max lift coeff. would be about 0.75).

Thus it would seem that the maximum lift coefficient for the surfboard foil in this illustration would be about 1.0–independent of whether or not it is treated as a swept wing, or a delta wing. However, the angle-of-attack corresponding to this lift coefficient differs between these two wing forms. Although the lift curve is not illustrated in McCormick for an aspect ratio of 2.82 (the illustrations only go up to 2.0), the maximum lift coefficients for a delta wing with an aspect ratio of 1.0, and 2.0, both appear to occur at an angle-of-attack of about 25 degrees. There are a couple of reasons why one would expect the maximum lift coefficient to decrease as the aspect ratio increases beyond 2.0, so it would seem that the angle-of-attack for the surfboard foil, based on the data for a plain delta wing, will be a bit less than 25 degrees.

Therefore I would estimate that for the foil I used in my example, the maximum lift coefficient would be attained with an angle-of-attack somewhere between 17 and 25 degrees (the estimates for the swept wing, and the delta wing). Hence if I assume that my example foil is a reasonable approximation to your foil, then it would seem to imply that the angle of attack (about the yaw axis of the surfboard) between the longitudinal axis of the board and free stream flow might range between something like 5 and 15 degrees (and probably biased more towards the smaller angle).

It would be interesting to determine the actual value for a comparison.

I’m guessing that perhaps your incremental changes were on the order of 5-degrees? …and hence the foil stalls, or it’s response diminishes, somwhere between fin rotation angle of 25-30 deg, and/or 30-35 degrees?

I did 5 degree fin series in fin sweep angle. But the “limit of rotation” angles were not as systematic. If you use an end-limit that is slightly too high, you can use appropriate side force elements to completely control fin rotation, and make the end-limit only a safety net for when the side-forces lessen with use.

But the question I addressed with the 10 degree box is

“If you have a fin that is only engaged at an angle of attack of the stringerline of 10 degrees or more, how much can the fin be engaged?”

The answer is not very much. However, you get a different answer at angles not very much smaller. My prediction is that a fin that only engaged at 12 degrees or greater would be pretty useless. This doesn’t necessarily related to the stall angle - although it would be nice if fins were optimized so that the stall angle was close to the end of the normal working range.

[wouter]

is alive

http://en.wikipedia.org/wiki/Coand%C4%83_effect_movies

just so that everyone finds coanda easily now

[riffraff]

Thanks for posting this - cant quite beleive what Im reading!!!

Dont get me wrong - Im not an expert in aero dynamics other than I have an eye for a nice line and I think I understand airflow quite well from working on some of the projects I have and seeing the results but this statement (following) must be one of the stupidest comments I have ever heard…

"In order to explain why the air goes faster over the top of the wing, many have resorted to the geometric argument that the distance the air must travel is directly related to its speed. The usual claim is that when the air separates at the leading edge, the part that goes over the top must converge at the trailing edge with the part that goes under the bottom. This is the so-called “principle of equal transit times”.

If I had known that this is what was being taught to pilots I would have never flown in a plane. Its like saying if you put your hand in a bucket of water then pull it out again all the bits of water will join up exactly as they were before you put your hand in…

[riffraff]

Im in vic.

did you get to cats on sunday??

I was too… um… recovering…

s.