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Why does viscosity help air bend?

  1. Dec 17, 2009 #1
    Hello all,
    I'm trying to nail down why the viscosity of air helps it bend around a curve. I have a specific, simplified example in mind to cut away other phenomena.

    Imagine the region immediately behind the camber of the top of a typical airfoil. That is, the place where the slope of the airfoil first becomes negative. Why do the air streams bend down to follow the surface. I know that it is due to a pressure differential between the streams in the boundary layer, but what causes this pressure differential?

    It cannot simply be the wake effect (as the foil moves forward, the region immediately behind the curve is deprived of air since it was where the win "used to be.") Clearly, such an effect occurs regardless of viscosity. The instantaneous absence of air in the wake cannot be the sole reason as evidenced by the turbulence and separation of boundary layer found in the wake of a sphere.

    So, I'm trying to understand what it is about the viscosity of air that causes a pressure differential that (in term) allows this bending to occur.

    A separate, but related question, I can see how the pressure differential allows for "bending" of air streams around a surface, but the literature I have read suggests this pressure differential is required even after a bend to keep air streams parallel to a surface, or else the layers separate. Why is that necessary? If two air streams are parallel to one another, why is any pressure gradient needed to keep them from separating? I suspect it has something to do with one stream having a different velocity than the other but would appreciate a clear description.
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  3. Dec 17, 2009 #2


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    I call it 'void effect', or more appropriately, 'void abhorrent effect'. The wiki article on wing also explains this:

    In that case a low pressure region is generated on the upper surface of the wing which draws the air above the wing downwards towards what would otherwise be a void after the wing had passed.


    As an extreme example, think of a moving bus. At the back of a the bus, a moving void is introduced into the air, which fills in that void (otherwise you'd have an absolute vacuum), resluting in a low pressure zone aft of the bus. In the case of a typical bus, some of the affected air's acceleration is inwards, but most of it is forwards, corresponding to drag. Even without visocity, the void introduced at the back of the moving bus results in a low pressure region that the air has to fill in somehow.

    Now imagine a streamlined bus with a tapered tail, that gradually introduces the void. This allow the air to fill that void by mostly accelerating inwards instead of forwards, reducing the drag.

    At the nose of the bus you have a similar situation, but the effects are less as most of the affected air will tend to flow around the bus instead of being accelerated forwards.

    Getting back to the wing, once past the peak of the cambered surface, you have a situation similar to the bus with a tapered tail (except the tapered tail is offset at an angle to produce lift).

    It helps, up to a point. If you have two adjacent flows of air, one moving slower than the other, then friction between the flows causes them to interact. If the inner flow is being accelerated downwards (or inwards) over a cambered surface, then friction between adjacent flows accelerates them similarly but to a lower amount. Rather than being a series of streamlines, what you have is a continous gradient of air flow. At the surface the air isn't moving, and just a few mm or cm away, the air is moving at speed, called the shear boundary. Once beyond the shear boundary, the relative speed generally decreases with distance from the surface of a wing.

    Where viscosity becomes an issue is if the speed and/or acceleration between adjacent flows becomes excessive, resulting in turbulence (vortices, like mini-tornadoes).

    The pressure differential is the result of an effective angle of attack and forward speed. You can take a flat board and produce lift, which is how those small balsa gliders work. You can stick you hand out the window of a moving car and angle it up and down to produce lift.

    The bottom surface simply deflects the air downwards which increases pressure. The upper surface draws air downwards because of what I called 'void effect', which decreases pressure. Because of the lower pressure zone above a wing and/or higher pressure below, a volume air senses this pressure differential (at the speed of sound) before the wing actually arrives, intially flowing upwards. At the leading edge of the wing, this upwards flow is then diverted back downwards to approximately follow the effective angle of attack of that wing.

    If you seperate the moving air into idealized streamlines, then since mass flow is constant, and compression and expansion factors are small, those streamlines get thinner as they move faster. However those flows can't be parallel if all them are simply getting thinner. The streamlines further away have to have a larger deflection angle so that the streamlines somewhat converge at the fastest and thinnest flow zone.

    It's a dynamic situation. The air isn't just accelerating and flowing parallel to the wing surface, it's also accelerating and flowing downwards towards the upper wing surface, which is moving away from the air as the wing moves fowards (with the surfaces moving downwards because of the effective angle of attack).

    It's probably easier to understand this from the air's perpective. The wing is a solid object with an effective angle of attack, passing through the air, accelerating the air downwards (lift) and a bit forwards (drag). Most of this acceleration for a normal wing is due to reduction of pressure above the wing. Initially, there's an upwards acceleration just before the leading edge of the wing arrives, followed by a downwards acceleration that creates the lift. After the wing passes by, the air affected by the wing ends up flowing mostly downwards and somewhat forwards before it dissapates. The forces don't dissapate, the downwards force from gravity on the aircraft is transferred through the air via momentum, eventually reaching the surface of the earth.

    At all but the lowest of speeds and angle of attacks, the air flow will separate and become turbulent, generally about 1/3rd behind the leading edge. There are special laminar aifoils that increase this to 40% or more behind the leading edge. As long as the turbulent flow mostly consists of tiny vortices, it's not an issue, and in the case of delta wing, these lowere pressure vortices flow along the leading surface of the wing, allowing those wings to operate at high angle of attack (20 degrees or more). If the angle of attack and or speed is excessive, then the size of the vortices increase to where it may just be one gigantic vortice, with a big decrease in lift and large increase in drag. The air is now filling in that void more by accelerating forward than downwards than it did at a lower angle of attack or speed. This called the stall region, where increasing angle of attack results in decreased lift.
    Last edited: Dec 17, 2009
  4. Dec 17, 2009 #3
    I think I came up with a reasonably simple solution... please let me know what you think.

    First, an observation: The change in velocity as you come away from the wing is less and less until, at some point, the air streams are flowing at the "ambient" velocity.

    For example, if the ambient velocity is 100 units, you might see adjacent air streams with the following velocities:

    Airstream A: 91 units
    Airstream B: 91 units
    Airstream C: 90 units
    Airstream D: 86 units
    Airstream E: 77 units
    Airstream F: 61 units
    Airstream G: 36 units
    Airstream H: 0 units

    [I'm not claiming the above are accurate, just a depiction of the idea that the difference in velocity is less and less as you move away until it stops changing at the end of the boundary layer.]

    Now, what causes this change in the change of velocity? We know that viscosity is caused by diffusion of momentum between airstreams. In other words, particles in one airstream slip into another. If those air streams have the same velocity, this doesn't matter, but when two adjacent air streams have different velocities, the faster one is slowed down and the slower one is sped up, allowing entrainment.

    This leads to the observation that particles in stream C are not (in this simplified model) passing into stream B at all. Since stream B is not getting any help from Stream A but is the same velocity as stream A, we must have that stream B is not being slowed down by stream C at all...but stream C is being sightly sped up by stream B, meaning stream B is losing particles to stream C but not accepting any. In other words the pressure in stream B is greater than the pressure in stream C. Similarly, for stream C to maintain a speed so much closer to stream B than stream D, while Stream D is being kept an even greater speed above stream E, we must adduce that there is a net migration from stream C to stream D, meaning a pressure differential there as well.

    This continues down the line. At each point more particles come down than go up, tending to push the streams toward the surface. If this pressure gradient is not great enough to maintain this effect, the layers separate, leading to a stall.

    [I realize that speaking of "adjacent" air streams in this way is a bit dodgy...I could instead speak of "representative" air streams at different distances away from the wing...and speak of net ingress or egress between regions, etc.]
  5. Dec 17, 2009 #4

    Andy Resnick

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    Are you thinking of the Coanda effect?

  6. Dec 17, 2009 #5


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    Along the upper surface of the wing the airstream velocity will be higher than the relative ambient stream, Airstream A - 110 units, ...

    Also these idealized streamlines get thinner as speeds increase, so their relative angles with respect to the wing, have to increase with distance from the surface of a wing so that they essentially converge at the point of fastest flow. The futher away from the wing, the larger the angle of the flow (a larger component of perpendicular flow).

    Although not mentioned yet, Bernoulli can lead to confusion by causing the reader to focus on the relative velocities. It's easier to understand how lift and drag are created by noting that a wing deflects the air by producing pressure differentials (with repect to ambient) which in turn cause the air to accelerate. The greatest amount of forces (lift and drag) correspond with the greatest amounts of acceleration, not the greatest amount of speed, which are the end result of those accelerations. Note that the accelerartion near the leading upper surface of a wing has a significant component perpendicular to the flow, which doesn't involve any local change in speed, although the corresponding lower pressure zone will accelerate the adjacent air into that flow.
  7. Dec 17, 2009 #6
    Thanks so much for taking the time to write this, but I'm afraid much of it dances around what I'm really getting at. I tried to limit my question to something very specific: why does viscosity cause streamlines to bend? What causes the the change in pressure?

    You state that the change in pressure is due to the angle of attack. While it is true that the angle of attack determines the net pressure differential, it is not responsible for the effect I am speaking of. That was why I picked a very specific point to look at it, the region immediately behind the camber. Streams will tend to deflect downward, trying to follow the contour regardless of the angle of attack. There may be no net lift induced if the angle of attack is poor, but regardless the air will tend to bend around the curve due to viscosity, and I"m trying to understand why it bends as laminar flow in this case but would not bend if air were inviscid.

    Thus, the only tool we have is the viscosity of air....and the only understanding I have of viscosity of air is based on "diffusion of momentum" across streams (the tendency of particles in one stream to move to another, so that faster streams can speed up slower ones and vice versa...i.e. entrainment.)

    Thus, I'm not assuming the air is necessarily going faster there (though I realize it is) because this tendency for air to follow a contour prevails even if the air nearest the wing is the same speed as the air further away (when it first encounters the slope).

    I have put a version of a "very dumb wing" online to help illustrate precisely what I am trying to understand. Why does viscosity account for the difference here, where the understanding of viscosity is "the tendency of one airstream to affect another by diffusion of momentum?"


  8. Dec 17, 2009 #7
    Yes. And I'm trying to figure out why it occurs in a gas.

    In a liquid it is easy to understand due to Van der Waal forces.

    In a gas, where the only interaction is diffusion of momentum between adjacent streams, it is harder to see. The wikipedia article does not indicate the cause, even in its section on "cause."
  9. Dec 17, 2009 #8
    Hmm...air stream P was meant to be the 0-velocity layer "stuck" to the wing...I'm in the rest-frame of the wing here.

    By "ambient" here I don't mean the air FAR above the wing, but rather the faster layer of air immediately above the boundary layer [which is moving faster due to the semi-venturi affect of the camber.

    I need to see a picture of this.

    It's dodgy to apply Bernoulli at all within the boundary layer...or possibly within any discussion of flight at all since the plane is doing work on the air. That is why I have omitted the B-word. You may notice that I am essentially trying to work in an explanation based on a conceptualization of the Navier-Stokes equations instead.

    But that's just the thing! Where are these pressure differentials coming from? As you mention, what is necessary is the deflection of air downward. This entire thread is aimed at understanding why viscosity allows that deflection in the first place. Lift is impossible in an inviscid fluid.

    Since this deflection or bending to follow a contour occurs regardless of the things that occur prior to reaching the camber [where streamlines are accelerated due to the fore-part of the camber], I want to see an explanation of this bending that does not require a velocity differential generated before reaching the apex of the camber (hence my "dumb wing" picture.) I know this bending, at its core, is due to viscosity...and I'm trying to understand where the specific pressure differentials that allow this bending come from (which are not the same as the net pressure differential created by the angle of attack, since the air will bend around the top even when there is no net pressure differential...it will just not create net downwash when added to the airstreams coming along the bottom.)
  10. Dec 17, 2009 #9


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    Youtube video, just press pause. This is a real world case, so the stream lines converge at the peak of the cambered surface. Note that the streamlines are deflected downwards, but by much less angle than the wing surface, an example of the detached flow (usually turbulent at higher Reynolds numbers). The upper and lower surfaces of the wing tunnel interfere with the flow, preventing any significant net downwash of air, so what you see isn't quite reflective of what happens in the real world, where there's significant downwash up to 1/3rd wing span above the center of a wing.

    The effect is more appropriately described as a Conada like effect that includes the void effect I mentioned above.

    From the movement of the wings surfaces with respect to the air, combined with momentum of the air. The air accelerates in order to approximately follow the surfaces of the wing, and there's a resultant force corresponding to the affected air's mass times acceleration. This force per unit area corresponds to the pressure gradient in that affected volume of air.

    Or forwards as in the case of the bus.

    You're thinking of D'Alembert's paradox which requires incompressable fluid as well as inviscid fluid, in order for no drag to occur.


    If the fluid is just inviscid (zero viscosity), but is compressable, the same as any real world object, where everything responds to a contact force with some type of deformation, and has mass (momentum) then lift is possible in an inviscid fluid, it's just less. As an example, a very high reynolds number can be achieved by using a low viscosity gas, which is commonly done in wind tunnels (to simulate high reynolds numbers). In this somewhat clever experiment, a flame from such a gas was directed at a glowing airfoil at varying angles in this series of pictures. There's virtually no "attached" flow, but the flow over the wing is deflected just fine due to "void effect", it's just less deflection than a higher viscosity gas. Pause the video at about 29 seconds for a good view of this.

    Some of the bending is due to friction at the surface and viscosity of the air, but in spite of all of that, what you have is a surface that attempts to introduce a void into air the with a surface that recedes from the air. The air is going to follow that receding surface in order to prevent the creation of a void. The air has momentum, so it can't exactly follow that surface, so it expands a bit and its pressure is reduced. The resulting lower pressure zone results in the surrounding air also being accelerated towards that lower pressure zone.

    What's important is how that air follows that surface. The goal of an efficient wing is to get the air to follow that surface with a high ratio of downwards acceleration versus forwards acceleration to produce a high lift to drag ratio. In the case of a severe stall, you end up with a large vortice which still has lower than ambient pressure, but not as much of a reduction in pressure as when the air mostly follows the surface of the wing with a minimal amount of detachment and turbulence.

    Note that turbulence will almost always occur when the pressure gradient transitions from negative (pressure decreases over distance) to positive (the pressure increases over distance). However turbulent flow can follow surfaces easier than laminar flow, so generally you have laminar flow, a detachment zone, then turbulent flow and reattachment. For powered aircraft, this isn't a big deal, but gliders try to optimized this effect with roughed up wing surfaces and/or turbulator strips. Oil flow tests are used to confirm the setups on gliders. In this article, the poor guy initially put the strips too far back, with them ending up in the detachment zone, instead of having the desired effect of triggering the detachment zone at a controlled location.

    http://www.standardcirrus.org/OilFlows.html [Broken]

    Very small and light hand launch (now called discus launch) model gliders have a more severe issue with laminar flow, so most of these use turbulator strips.
    Last edited by a moderator: May 4, 2017
  11. Dec 17, 2009 #10
    Based on the you-tube video, it appears you are referring to the lines prior to the camber. My discussion involved the lines immediately after the camber.

    You are referring to an off-hand remark I made concerning the affect on the streams before reaching the top of the camber. At that point there is no "void" in affect. I believe you would see this phenomenon even in a totally inviscid liquid.

    But, again, my question is why? And, in particular, how does viscosity allow a laminar flow to follow the surface of a wing far more than would be found in an invisicid fluid?

    Yes, I'm referring to subsonic speeds low enough where air can be incompressible, ~< Mach 0.3


    As I keep on asking...my question is why does the viscosity of air allow this affect? That has been my question all along: There is clearly a distinct, qualitative difference between the kind of contour-following brought about by viscosity and that simply due to wake (where air can come in from anywhere and the streamlines are just responding to a pressure difference anterior to the surface rather than viscosity effects of the boundary layer.)

    For air, viscosity is due to one streamline entraining another, and curvature in the boundary layer is due to a pressure differential between air at the surface and air at the top of boundary layer. This is what I am asking about...not the more general pressure differential between air below the wing and air above the boundary layer. That is why I displayed the ludicrous "dumb wing" drawings to get at this effect without any regard to angle of attack or anything that happens before the camber.

    My question has been, and still is, how does viscosity (i.e. diffusion of momentum between streams of differing velocity) allow a far more distinct, laminar Coanda effect compared to what would be expected by the simple wake of a moving wing in the absence of viscosity (which is less pronounced and leads to turbulence where the layers separate).

    That is the whole point of the Coanda effect, the fluid is attracted to a curved surface far more than is expected in the inviscid case.

    By itself, this explanation would not suffice because it would lead to separation, partially because in the inviscid case chaos can ensue since ANY air will rush into the depleted region in the wake of the wing (not just the streamlines you are looking at), as is shown in the picture...but it does lead me to hypothesize that the reason viscosity allows air to follow a surface so strongly is that the air streams immediately next to the surface have less momentum (since they are moving slower)...and hence their ability to react to the low pressure of the depleted region is much greater than other air (for example, the air that could get sucked back into the region as shown in my diagram).

    So, if that is a valid explanation of the importance of viscosity, I'm left with my other question: why is a pressure differential required to maintain the boundary layer over a flat surface. I'm no longer talking about making the bend around the camber here, but now I'm referring to the streamlines going straight (and slightly downward) along the rest of the top part of the wing.

    In other words, in as simple language as possible, what causes boundary layer separation? What causes flow reversal?

  12. Dec 18, 2009 #11


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    As you can see in that video, those streamlines are expanding as the pressure gradient has transitioned to positive (increasing), and the flow has somewhat detatched.

    Most of the web site articles I've read only mention viscosity having a signficant effect within the thin boundary layer near the wing where the air transitions from zero relative speed at the surface to high speed just a few mm or cm away from the surface. Once beyond the boundary layer, the rate of change in speed versus distance from the surface of a wing is relatively small, so there's little interaction between those adjacent flows outside of the boundary layer.

    The compression and expansion factors for a typical wing at mach 0.3 are about 5%, not insignificant, but small enough that changes in density can be ignored if doing a crude approximation of lift and drag for an airfoil.

    A true incompressable fluid model is problematic, for example the speed of sound or pressure information is infinite in an incompressable fluid, so how could a pressure differential exist within an incompressable fluid, which leads to D'Alembert's paradox where there's no drag on an object because there's no pressure differential within an incompressable fluid inspite of any object moving within that fluid.

    The boundary layer is relatively thin. Mose of the contour following occurs outside this boundary layer. Outside the boundary layer, the main effect of viscosity is related to the Reynolds number (the lower the viscosity, the higher the Reynolds Number), which in turn is related to laminar versus turbulent flows:


    To increase Reynolds Numbers in a wind tunnel on a scale model, the pressure (and therefore density) can be increased, or for very high Reynolds numbers a low viscosity gas like helium can be used.

    If it's a submerged object, like a rudder or prop in water, I'm not sure how signifcant the boundary layer effect is. If it's a thin stream of water flowing down the back side of a spoon, then almost all of that flow is within the boundary layer, which is a different case.

    Wiki link about boundary layers:


    Which is essentially what happens. From the air's perspective, the surrounding air acclerates towards the lower pressure zone above a wing from all directions except upwards through the wing itself because the wing itself blocks that flow. Since the lower pressure zone is moving with respect to the air, the path of the affected air is curved.

    The flow below a wing is a bit more complex. The wing increases pressure and imparts momentum into the flow, giving it direction. The increased pressure means that the air continues to accelerate as it's pressure decreases, and the flow should constrict as it's speed increases since mass flow is constant and compression factor is small within the flow, but the surrounding air is going to interact with this flow. This Nasa article shows what I'm getting at with the constricting flow aft of a prop, but doesn't explain how the surrounding air would interact with that flow:


    The static pressure of the air within the boundary layer is essentially the same as the pressure of the air just outside the boundary layer. This is why static ports mounted on the sides of aircraft fuselages can sense the pressure of the ambient air, by "hiding" within the boundary layer, regardless of the relative speed of the ambient pressure air flowing just outside the boundary layer.
    Last edited: Dec 18, 2009
  13. Dec 18, 2009 #12

    But, Jeff, it is precisely the bending within this layer that I'm interested in...because it is the bending within this layer that is due to viscosity...which is what I keep asking about. I'm not asking "why does air deflect due to a wake" because that is obvious. What I"ve been asking is "how does viscosity allow air to follow a contour far more closely than one be expected in an inviscid fluid?"

    Can you please, please, just answer that? (Based on what viscosity in air means: entrainment of streams, velocity distribution in boundary layer, etc.)

    And, while the size of this layer is small compared to the entire volume deflected, it is evidently critical to that deflection since low-speed flight would be essentially impossible without viscosity, and viscosity has its manifestation in the boundary layer.

    I'm not sure what you are saying about my possible explanation of how viscosity allows the boundary layer to hug the surface far better than inviscid air does. Can you please comment on that?

    You said that the reason air (in general) cannot perfectly hug a contour is that it has momentum, which makes perfect sense. My hypothesis was that the air in the boundary layer has less forward momentum than other air near the near-future wake/void/depleted zone and hence is better able to claim it for its own, meaning that laminar, contour-hugging flow persists in the boundary layer because (unlike in the inviscid case shown in my picture) air from foreign streams are not rushing into the void to separate layers.
  14. Dec 18, 2009 #13

    Andy Resnick

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    Ok, so the question is equivalent to "can the Coanda effect occur in inviscid fluids?". Amazingly, I can't find *any* quantitative information about the Coanda effect. Tritton's "Physical Fluid Dynamics" has a section on flow attachment, but there's no quantitation.

    Perhaps this means that viscosity is an essential component of this effect; are you then asking for the molecular mechanism of viscosity? As an aside, intermolecular forces are not the 'cause' of viscous dissipation; the forces are conservative.
  15. Dec 18, 2009 #14
    My understanding is that the Coanda Effect is completely due to viscosity because the "Coanda Effect" can be defined as "Why fluids follow their surfaces more than would be expected without viscosity."

    I'm not so much looking for a molecular understanding of viscosity itself, but rather the actual relationship between the understanding of viscosity I have and the Coanda effect. I've given two of these in this thread.

    My understanding of viscosity in gases is:
    1. The particles immediately touching the surface are considered to have zero velocity relative to the surface

    2. particles can (quite understandably) slip from one air stream to another (or, similarly, particles in one air stream influence another), and this means that if one air stream is adjacent to another with a different speed, then each will mitigate the other's speed: the faster one will tend to speed up the slower one and vice versa.

    [Note, I don't mean that velocities change along a streamline due to this interaction...what I mean is that the velocity in one stream line is "different from what we would expect without any interactions." I hope that is clear enough...]

    I can understand how this sets up the velocity profile in a boundary layer.

    Now, what I"m trying to do is get from that understanding of viscosity to an understanding of why streamlines of a gas in the boundary layer can hug a contour much better than they would were there no viscosity.

    In the liquid case, Van der Waal forces would make this a piece of cake...
  16. Dec 18, 2009 #15
    Wikipedia claims that Frank White's 5th edition of Fluid Mechanics (2002, McGraw Hill) describes why viscosity allows the Coanda effect. They source this text for the second half of the quote below.

  17. Dec 18, 2009 #16


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    I don't know that it does except at very low speeds were viscous effects are dominant over inertia effects. As mentioned in the wiki boundary layer link, within the boundary layer viscous forces are dominant, and outside the boundary layer, inertia forces are dominant, but I don't know how much viscosity contributes to bending of the boundary layer, as Andy mentioned, I haven't seen any quantitative information stating that x% of the bending of the airflow within the boundary layer is due to viscosity, versus void effect and pressure differential drawing the boundary layer (and the adjacent air) around the cambered surface of an airfoil.
    Last edited: Dec 18, 2009
  18. Dec 19, 2009 #17


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    A re-think on this.

    Imagine a wing with a cambered surface initially at rest within an inviscid gas or fluid, that then starts to accelerate. During the acceleration of the solid object, 'void effect' would be a factor. The surrounding gas or fluid, would be accelerating to fill in what would otherwise be a void left by the wing. The issue is if the fluid accelerates downwards or forwards in order to fill in that void. If the cambered surface is gentle enough, then the gas or fluid accelerates primarily downwards since that is the way in which the void is being introduced into the gas or fluid. In this case, even when the wing stops accelerating and moves at constant velocity, you end up with the 'wake' effect you mentioned before, and as the wing flows through the fluid, a downwards and somewhat forwards and somewhat rotating moving wake moves along with the wing.

    What viscosity adds to the boundary layer is a shearing torque, you have a forwards skin friction force, and a backwards viscosity related forces until you get far enough way that the speed gradient versus distance from wing surface diminishes to some small amount. It would seem that if the flow remains laminar, than the shearing torque would "bend" the flow. If the flow becomes turbulent, then the shearing torque produces vortices in addition to "bending" the flow. It turns out that depending on the speed, it's easier for somewhat turbulent flow to remain attached (or to reattach) than laminar flow, so in some cases, the turbulent flow "bends" better than the laminar flow.

    Last edited: Dec 19, 2009
  19. Dec 20, 2009 #18
    I came to similar conclusions this morning when I decided to go back to the basic governing equation for flow lines: The Navier-Stokes equation. And when I did that, the answer became evident.

    The solution combines the two ideas I gave earlier:



    And my "combination" here I don't mean "a little bit of both."

    Rather, if you look at the plight of a stream-line that has just had "the bottom fall out from under it" (that is, the surface is curving away), you can speak of its momentum carrying it forward and pressure pushing it down.

    I earlier mentioned that the fact that these particles are going more slowly should help them "turn the corner" but when I looked at the Navier-Stokes equations I realized they also are helped by the velocity gradient I discussed in my first option (post number 3 in this thread).

    If you look at the velocity profile in the boundary layer, you see that its second derivative (wrt the distance from the wing) is decidedly negative. (This was what I was getting at in post 3), but the Navier-Stokes equation (for incompressible liquid) indicates that this is exactly what accomplishes the viscous side of acceleration in a viscous liquid:

    density * Acceleration = (negative of pressure gradient) + viscosity * (vector laplacian of Velocity).

    In the case of the velocity profile near a wing, this "vector laplacian of velocity" goes against the flow in the airstream.

    Thus, when bending the corner there is a "downward" force due to the void effect and a "backward" force due to viscosity (the "shear stress" you mention.)

    Upon thinking a bit more I realized what the Navier-Stokes equation is really saying: because the boundary layer is concave up, a given stream is being slowed (by the streams below it) more than it is being sped up (by the streams above it). Thus, there is a slight negative acceleration along a streamline. This is why the boundary layer widens as you go along the back of a an airfoil.
    This negative acceleration is, of course, exactly what is needed to help a streamline catch the curve and crowd out other streamlines that are vying for the void.
  20. Dec 20, 2009 #19


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    The effect of this is limited by the flow just outside the boundary layer, which is dominated by 'void effect'. If that external flow doesn't bend as much, it ends up causing the boundary layer to become thicker and with more turbulence, reducing it's bend, perhaps detaching, so it would seem that there is a competing balance between shear torque and void effect, depending on airfoil, AOA, speed, ....
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