Lift as a consequence of streamline arguments

[arildno]

I'll now move onto a "global" analysis, by which I mean the following:

Consider a fixed, geometric control volume surrounding the wing and a portion of the fluid.

Furthermore, let us make that control volume so big that the external pressure forces acting upon the contained fluid from the fluid outside the control volume cancel out (that is, the pressure forces)

Then, since we have a steady state situation in the wing's rest frame, it follows from the appropriate version of Newton's 2.law that the net momentum flux out of the control volume equals the force acted upon the fluid from the wing.
(If our control volume were smaller, we would have needed to subtract from the net momentum flux the contributions from the external pressure forces to find the fforce from the wing).

By the use of Newton's 3.law, then, we may calculate the force acted upon the wing from the fluid.

When we say that air flow has TURNED, or has experienced a net downwards deflection, this means that the net momentum flux out of the control volume has a net DOWNWARDS component.
That is, an initial, horizontal free-stream has gain vertical momentum downwards as a result of passing by the wing; by Newton's 3. law, therefore, the wing experience a LIFT.

Why then, have I focused so strongly on "centripetal acceleration" rather than pointing to net, downwards deflection of the air?
For the simplest reason possible:
Since the fluid has TURNED away from straight-line motion, it must necessarily have experienced CENTRIPETAL ACCELERATION (by passing the wing), that is, centripetal acceleration is the (absolutely necessary) local acceleration concept which negotiates the turning of the flow!

This is why I have focused on the c.a. concept in my earlier posts; I'll post a few more thoughts later on.

 

[rcgldr]

I'm not an expert on aeroplanes, but I think Arildno analysis is emphatized on non symmetric wings with zero angle of attack. He wants to discover the lift phenomena without involving angles of attack, and without having to consider a stream deflection at the rear.

If there's lift, there's stream deflection, and an effective angle of attack. The typical usage of the term angle of attack can zero or less than zero even when a wing is producing lift, because it's just a measure of the leading and trailing edges of the wing. I prefer to use the term effective angle of attack, which is zero when lift is zero.

When air "follows a curve" due to friction and viscosity, it slows down (relative to the wing). Otherwise air follows a curve, because if it didn't, a void would be created. How it follows this curve affects the lift. If the flow is more vortice like than laminar, there's less lift produced. However delta wings take advantage of the vortices created at the leating edge of the wing when at high angle of attacks, about doubling the maximum angle of attack for peak lift over conventional wings.

This is my point, a wing introduces a "void", even in the case of a curved airfoil. I'm using "void" to refer to the low pressure region created above a wing prodcing lift. Air will accelerate towards this low pressure region from all directions (not just across the top of a wing), except that the wing itself prevents the air from flowing upwards through the wing (some air will flow around the wing, reducing the amount of lift). Since the acceleratin is in all directions but upwards through the wing, a net downwards acceleration of air occurs.

When a wing produces lift, there is a dowwards defelection of stream flow.

The example of deflection is usual in turbomachinery. I would like to discuss about the lift phenomena at blades. I would like to know if such pressure differential is the responsible of lift forces on the blades instead of the proper stream deflection as Euler equation of Turbomachinery states.

Some of the lift of a typical wing is due to simple deflection of air. However, a significant portion of this lift is lost because the low pressure area above the wing draws some of the air stream away at the leading edge, separating more of the flow to above the wing, leaving less air to be deflected below. In the case of a flat board type wing, or symmetrical air foil, the angle of attack determines the "lift", and the calculations for total air flow are probably not that senstive to whether or not the flow is generated from above or below the wing (in front of in back of the blade).

If a turbine is at rest (with the air), but spinning there's a signifcant low pressure region in front of the turbine, and a high pressure region behind the intake blades.

 

[arildno]

Since Jeff Reid is rather insistent upon the "void effect", I think it is proper to comment on this.
I would have liked first to map out those features seen in the stationary, lift-sustaining situation, and then moved onto initial, transient features which could be said to CAUSE lift in the first place.

However, I will present the ideas here now, because Jeff Reid's "void effect", although correctly identifying the mechanism in the production of a low pressure zone/point above the wing, is insufficient in explaining the lift, since an important detail has been left out:
The crucial role played by VISCOSITY in lift-production!

In order to justify this, we must go into some detail in why an inviscid fluid can be modeled as lift-sustaining, but not really lift-generating.
(Note that I have emphasized that my earlier post should not be seen as detailing the cause of lift, but as a mapping of the elements in an interconnected, lift-sustaining whole).

Now, consider a stationary 2-D stream with a circle/cylinder embedded within it.
In potential theory, we would typically find that no forces act upon it, in accordance to D'Alembert's paradox.

However, there exists a lot of OTHER potential solutions of this problem which DO predict a net force on the cylinder acting normal to the flow direction!

These can be found by placing a POINT VORTEX (which can be derived from a velocity potential) at the origin (of the circle), and which has a given magnitude.
Note that this point vortex readily satisfy the boundary conditions of zero normal velocity on the cylinder surface, and vanishing in infinity.

That is, the superposition of the non-lift generating "fundamental" solution, and a point vortex of arbitrary magnitude (or circulation, if you,like) produces an arbitrary force on the cylinder, directed normal to the free-stream.

This is called the Magnus effect.

Now, it can be shown that in the case for a body of ARBITRARY shape immersed in an infinite fliud, D'Alembert's paradox still holds.

NO HYDRODYNAMIC FORCE (neither lift or drag) CAN ACT UPON AN IMMERSED BODY FROM UNLESS IT HAS AN ASSOCIATED CIRCULATION ASSIGNED TO IT!
(in which case a lift is given).
(That is, in the stationary case where the body is immersed in an infinite, ambient, inviscid fluid)
First, let us see what this means for solving Laplace's equation about a typical wing:
The (analytical) procedure is as follows:
1. Let a vortex distribution fill the wing (or wing surface) (The singularities are therefore, themselves not (in the interior) part of the fluid).
2) Derive equations consistent with no normal velocity on the wing surface
3) Make use of the additional Kutta condition to find a unique, lift-generating solution.

The Kutta condition is a very interesting condition; it is the requirement that the fluid should leave the trailing edge in a SMOOTH, TANGENTIAL MANNER!

This is, of course, the only realistic inviscid flow about a wing profile; we cannot expect to predict the wake region properly, and neither can we handle separation phenomena properly either.
(I'll leave a more detailed discussion of the relation to Kelvin's theorem later)


Important for our purposes, remember that there does exist (an unrealistic) solution which does not provide a lift (has zero circulation):
The velocity profile generated by this is of high importance:
Several streamlines from the bottom has twisted themselves around the sharp trailing edge, then follows the upper edge a bit, and then twists again, leaving the wing.

Streamlines starting further up on the upper edge, separates from the wing when meeting this bundle of streamlines having sneaked themselves around the trailing edge.

Through this mechanism, then, there has been no net turning of the flow, and we get NO lift.


We are now in the position to evaluate Jeff Reid's void effect, and resolve the sketched dilemma by pointing on the role played by viscosity.

1) First, if a fluid were really inviscid, and at rest at start (i.e seen from the ground frame), the no-lift stationary situation (stationary, that is from the wing's rest frame) is the one which would be achieved.

The tiny flaw in the "void effect" argument is that it underestimates the net UPWARDS flow into the exposed cavity/vacuum from the lower half of the fluid. As we have seen, this will stabilize itself for a truly inviscid fluid in the rather weird streamline pattern as sketched.
2) So then, what is the role of viscosity?
At the actual, upper foil, the air STICKS to the surface.
When the wing suddenly accelerates forwards, there will FOLLOW WITH it a tiny band of air (with a width typically proportional to the viscosity).
(Note: It is NOT to be confused with the boundary layer, it should be much thinner than an ordinary boundary layer)
That is, the region which will momentarily be sparsely populated (i.e, the void), and hence associated with low pressure, is not AT the actual wing surface, but some small distance away from it.
Imagine this as a small cavity, or point of low pressure.
3)
Now, the subtle effect of the sticky layer kicks in:
Its presence provides a RESISTANCE against the fluid from moving into the low pressure zone from underneath!

That is, fluid situated initially where the upper foil was, gets a HEAD START in filling the void, compared to the fluid to get in there from the exposed lower half.

4) The filling in of the cavity, will typically be in the formation of a VORTEX about the low pressure point.
If the ambient fluid pressure is strong enough, that vortex will dwindle in radius; the low pressure point will be squeezed onto the airfoil (which is crucial for lift.)
5) From the viewpoint of the inviscid region above the wing then, there is an initial, transitory SEPARATION phase, but if the fluid pressure is strong enough, the inviscid streamlines will become latched onto the airfoil (i.e, produce a situation analyzable in terms of streamline arguments).
But this is seen to be a physical "production" of the Kutta condition..

To sum up the two most important, initial, upwing features of getting the plane off, then:
Jeff Reid's "void effect" produce a low pressure zone.
The presence of viscosity hinders the lift-reducing upflow so that lift is physically realizable.

As I have thought about this stuff when developing this thread (and having received extremely valuable ideas&support from most posters), I have come to that I need to say more about the circulation issue at a later stage.

 

[rcgldr]

in accordance to D'Alembert's paradox.

The limit of drag as viscosity approaches 0 isn't 0. Other models for the flow of an invisicid fluid, where separation occurs downstream of an object seem to be better candidates. This stuff is beyond me, so here's a link to a pdf file that discusses this:

http://locus.siam.org/fulltext/SIREV/volume-23/1023063.pdf

This link can be found on this page:

http://locus.siam.org/SIREV/volume-23/art_1023063.html

 

[arildno]

Please don't lecture me on this.
I am perfectly well familiar with both the regions where potential theory are excellent, and where it cannot be used.

In essence, Navier-Stokes is a SINGULAR PERTURBATION equation as the viscosity parameter goes to zero; hence, it is to be EXPECTED that the asymptotic (and correct) limit solution occasionally will differ significantly from the solution where the viscosity parameter is simply set equal to zero.

As I've shown quite clearly, lift CANNOT be understood without non-zero viscosity.

If you want to continue, please come with intelligent remarks instead.

 

[rcgldr]

I read up a bit more on this, and have found a few references to "triple deck", where the 1st deck is the boundary layer created by viscosity. Apparently at higher Reynolds numbers, the boundary layer is so relatively small compared to the other layers in the "triple deck", that it can be ignored for doing lift calculations or lift models.

As previously mentioned, the limit as viscosity, and the thickness of the boundary later approach zero, isn't zero, and there's a separation of flow.

 

[rcgldr]

lift CANNOT be understood without non-zero viscosity.

Ok, but apparently, the vicosity component is so small that it can be ignored when calculating lift.

Also, can you explain to me about the articles like the one I posted that seem to contradict zero drag in an incompressable, invicid fluid? From what I read, the alternative theory is that separation (non-laminar?) occurs even in these therotical cases.

As posted I don't understand all of this myself, so I have to rely on what information I can find about this.

The reason I use the "void" theory, is that it's easy to understand, like in the case of a bus traveling down a highway. It's the "void" at the back of the bus that contributes to most of the drag.

 

[arildno]

Jeff Reid:
First I'll apologize for barking at you; I'm really glad you chose to continue the discussion!

Now, I haven't had time to look over in detail the articles you posted, but I'll mention two things of some relevance here:
1) Reynolds number
It is, in the non-turbulent cases generally true, that at large Reynolds numbers, viscous effects can be ignored.
However, we must be VERY careful in what is to be a proper measure of the Reynolds number!
Let's take the case where the plane has a big HORIZONTAL velocity U.
Wherever is it stated that we can utilize the implied Reynolds number (i.e, with U as the velocity) when we are considering effects in the VERTICAL direction?
After all, lift is a "vertical" phenomenon, so we need to tread with a lot of care here..

(I haven't checked what your articles say here, possibly the say the same thing in a much better way than myself)

2) Void theory
I ABSOLUTELY AGREE WITH YOU!
This is a fundamental insight, but as I have shown, unless we tread with extreme caution (that is, remembering viscosity) we'll run straight into D'Alembert's paradox
(And we won't end up there, or what..)

 

[rcgldr]

D'Alembert's paradox

A few articles I've read seem to disagree with this. One issue pointed out is that limit for drag as viscosity approaches zero isn't zero. Another issue is the assumption that the flow will not separate and create vortice like flow patterns behind the cylindrical or spherical shape. Now there are some that believe the flow patterns would be similar to the flow patterns of fluids with very low viscosities, at least for objects with a large radius.

Since there aren't gases or fluids with zero vicosity, it's just a theory and an instresting math exercise.

Lastly what if the object is a flat plate with relatively sharp edges? Will this create turbulent flow even with an invicid fluid? If the flat plate has some "angle of attack", can lift be generated, even if all off the lift is the result of turbulent flow? Or better yet, a delta type wing which takes advantage of leading edge vortices flowing along the front edge of the wing and then across the top of the wing?

 

[arildno]

"A few articles I've read seem to disagree with this. One issue pointed out is that limit for drag as viscosity approaches zero isn't zero."
It is PRECISELY this which can happen in so-called singular perturbation problems.

To be specific, the limit solution of Navier-Stokes as the friction parameter goes to zero, would satisfy a demand that ALL Navier-Stokes solutions satisfy:
Namely, NO slip at the boundary.

But when you use the Euler equations (N-S with friction coefficient set to zero), you CANNOT find a solution which satisfy this demand along with all the other demands.
That is, all solutions of Euler equations violate the no-slip condition at boundaries.

Now, for many problems, the zone close to a boundary is so thin (where the no-slip condition must hold) , and the macroscopic effects of it so tiny, that the Euler equations yields excellent approximations.

But, as stated, for other problems, the Euler equations cannot be used to get appreciable results (D'Alembert's paradox is one of many examples).

So, we should appreciate the simplification we gain by using Euler equations, but we should use them with care and caution, and know when we cannot use them.

I'll look over your other comments in a while.

 

[Clausius2]

One more question

I have never simulated numerically the potential flows equations. Also I know little about potential flow. Let's suppose the flow is incompressible and steady. There is a flow over a two-dimensional airfoil with some angle of attack (in general distinct of zero). If I use the potential flow equation for incompressible flow $$\nabla ^2 \phi=0$$:

i) would I obtain a value of lift different of zero?. If so, we have concluded that lift has to do with circulation, and in last term with vorticity. Potential flow is based on irrotational behavior, so we shouldn't have any vorticity field and so any lift force.

What is the error in the above reasonement?

ii) would I obtain a value of drag different of zero?

 

[arildno]

ii) There's no drag in 2-D theory, however I seem to remember (I'll verify it later) that in 3-D flow theory, there is an induced drag.

i) We have at our disposal the point vortex potential, in polar coordinates:
$$\Phi(r,\theta)=\frac{\Gamma}{2\pi}\theta$$
This induces the velocity field:
$$\vec{v}=\frac{\Gamma}{2\pi{r}}\vec{i}_{\theta}$$
For any simple, closed curve containing zero, the circulation is $$\Gamma$$
(wherefore $$\Gamma$$ is called the circulation constant)

However, the vorticity at any point where the velocity field is defined is zero.
This means that the LOCAL angular velocity (about a point where the velocity field is defined) is zero everywhere, and that's what irrotationality is about.
(Since the velocity field isn't definded at the origin, the point vortex field doesn't contradict the irrotationality property)


The standard uniqueness theorem for a solution of Laplace's equation (given the usual boundary conditions) presumes that the region we're looking at has the topological property that every simple closed curve in our domain is reducible.
A reducible curve means basically that you can "shrink" it to point, and still remain within your region.
This is easiest to understand by looking at the opposite case, where there exists irreducible curves:
Suppose the region you wish to look at is the whole plane MINUS some figure lying within it (for example, the cross-section of a wing).
Now, if you draw a closed curve NOT bounding your figure, you can "shrink" this curve, and still be within your region.
However, this shrinking can't be done for a closed curve enclosing the figure within it!
You can shrink your curve down to the boundary of your figure, but no further (that would take you outside your region).

In order to gain a unique solution of Laplace's equation in such cases, you may do so by specifying the CIRCULATION about the figure, in addition to the usual boundary conditions.
It may be noted that zero circulation corresponds to d'Alembert's paradox, non-zero circulation values will generate a net force (lift) perpendicular to the direction of the free-stream velocity (Kutta-Jakowski's theorem).

The Kutta condition in flow theory specifies "smooth,tangential flow at the rear edge",
and this ensures a unique (and, the only realistic one) solution, by basically picking out that unique circulation value which yields smooth flow.
(In reality, it is the effect of viscosity to ensure smooth flow, and, for large Reynolds numbers in the non-stalling case, that is the only effect of viscosity upon the lift (that is, the actual value of the lift may be found by inviscid theory). At low Reynolds numbers, say for insects, the importance of viscosity in determining the calculated lift cannot be neglected)


Lastly, the basic way of computing the flow field (as far as I know), is to make a point vortex distribution along the boundary of the object, and determine this to satisfy Kutta's condition along with the other conditions.
(This, gives you some integral equations to solve, I believe..)

Another somewhat related issue is to figure out:
What (wing) geometry is consistent with a given vortex distribution?
That is a simpler problem, from what I know of it.

 

[Clausius2]

Another somewhat related issue is to figure out:
What (wing) geometry is consistent with a given vortex distribution?
That is a simpler problem, from what I know of it.

Ok, I think I've got it. It seems to me that the time for starting to learn deeper potential flow is coming. It is my weakest point.

I have heard a lot of times about the "inverse" problem when calculating an airfoil. I mean, aerodynamicists usually begin assuming an external flow and then solve for the airfoil shape precised to enhance such flow field. Is it what you're referring to?

Regards.
Javier.

 

[arildno]

Yep, that's what I referred to.
All you need to solve for when you've got the vortex distribution, is the mean-camber line of the wing (i.e, the fundamentally relevant geometrical info for the lift problem).

Arild

 

[Clausius2]

Another question

Currently I am searching for the sources of Vorticity within a flow field. I have found the way to derive the equation for vorticity in inviscid compressible flow:

$$\frac{D\overline{\omega}}{Dt}=-\overline{\omega}\cdot\nabla\overline{v}+\rho^{-2}(\nabla P x \nabla \rho)+\nabla x \overline{f}$$

where I don't understand what is the meaning of the first term on the right. Also I have understood the baroclinic term (second on the right) and the curl of the mass force (third on the right).

Two questions:
-do you understand what is the meaning of the first term on the right?. I have heard it has to do with Cauchy Integral.

-do you know an analogous expression for a general flow including viscous terms?.

Thanks.

 

[arildno]

Hmm..your equation is wrong (there shouldn't be a minus sign in front of the first term on the right-hand side).
Restricting myself to the case of constant density&dynamic viscosity and conservative volume forces, taking the curl of Navier-Stokes yields:
$$\frac{D\vec{c}}{dt}=(\vec{c}\cdot\nabla)\vec{v}+\nu\nabla^{2}\vec{c},\vec{c}\equiv\nabla\times\vec{v},\frac{D}{dt}\equiv\frac{\partial}{\partial{t}}+\vec{v}\cdot\nabla$$
In the inviscid case (with volume forces being a gradient field), therefore, we have:
$$\frac{D\vec{c}}{dt}=(\vec{c}\cdot\nabla)\vec{v}$$
(Note that this is also exact in the inviscid, barotropic case, $$\rho=\rho(p)$$)
As for the interpretation of the right-hand side, we note that if two points A,B in the fluid is separated by the vector $$\vec{AB}=dq\vec{s}$$ (where dq is an infinitesemal quantity), we have:
$$\vec{v}_{B}-\vec{v}_{A}=((\vec{AB}\cdot\nabla)\vec{v})\mid_{A}=dq(\vec{s}\cdot\nabla\vec{v})\mid_{A}$$
This is proven as follows:
$$\vec{v}_{B}=\vec{v}(x_{A}+dx,y_{A}+dy,z_{A}+dz,t)=\vec{v}_{A}+dq(\vec{s}\cdot\nabla)\vec{v}\mid_{A},(dx,dy,dz)=dq\vec{s}$$
If therefore A, B is fluid-particles at the same instantaneous vortex line (i.e, defined with $$\vec{c}$$ as the tangent vector at time t), we have, by the vorticity equation:
$$\vec{v}_{B}-\vec{v}_{A}=dq\frac{D\vec{c}}{dt}\mid_{A}$$
(at time t)
That is, the vorticity change, as experienced by particle A is only affected by the perceived velocity difference along the vortex line to which A instantaneously belongs.

To proceed further, let $$\vec{AB}_{t+\bigtriangleup{t}}$$ be the vector joining A, B an instant later.
We have therefore the equality:
$$\vec{AB}_{t+\bigtriangleup{t}}=\vec{AB}+(\vec{v}_{B}-\vec{v}_{A})\bigtriangleup{t}=dq(\vec{c}+\frac{D\vec{c}}{dt}\bigtriangleup{t})\mid_{A}=dq\vec{c}(t+\bigtriangleup{t})\mid_{A}$$
That is, particles A and B remain joined to the SAME vortex line at time $$t+\bigtriangleup{t}$$ as they were on time t.
(At time $$t+\bigtriangleup{t}$$ the vorticity experienced by particle A is $$\vec{c}(t+\bigtriangleup{t})$$

Hence, we may conclude that in the inviscid case, vortex lines are MATERIAL curves, in that they consist of the same particles throughout time.

 

[arildno]

The Clebsch decomposition

That vortex lines are material curves, is neatly illustrated in the special case where a Clebsch decomposition of the velocity field is possible, that is:
$$\vec{v}=\nabla\Phi+\alpha\nabla\beta\to\vec{c}=\nabla\alpha\times\nabla\beta$$
In this case, the inertia term can be transformed as:
$$\frac{D\vec{v}}{dt}=\nabla\frac{\partial\Phi}{\partial{t}}+\nabla\beta\frac{\partial\alpha}{\partial{t}}+\nabla(\alpha\frac{\partial\beta}{\partial{t}})-\nabla\alpha\frac{\partial\beta}{\partial{t}}+\nabla(\frac{1}{2}\vec{v}^{2})+(\vec{v}\cdot\nabla\alpha)\nabla\beta-(\vec{v}\cdot\nabla\beta)\nabla\alpha=\nabla(\frac{\partial\Phi}{\partial{t}}+\alpha\frac{\partial\beta}{\partial{t}}+\frac{1}{2}\vec{v}^{2})+\nabla\beta\frac{D\alpha}{dt}-\nabla\alpha\frac{D\beta}{dt}$$
Hence, assuming constant density (and volume forces derivable from a potential V), we may eliminate the pressure as an independent variable, through setting:
$$\frac{D\alpha}{dt}=0,\frac{D\beta}{dt}=0, \nabla\cdot\vec{v}=0$$
(the last equation being the equation of continuity),
and the pressure fullfills:
$$\frac{p}{\rho}+\frac{\partial\Phi}{\partial{t}}+\alpha\frac{\partial\beta}{\partial{t}}+\frac{1}{2}\vec{v}^{2}+V=K$$
(where K is some constant).

That is, $$\alpha,\beta$$ are conserved quantities for the fluid particle, and the vortex lines are the intersections between $$\alpha,\beta$$ surfaces, since $$\vec{c}=\nabla\alpha\times\nabla\beta$$

 

[Clausius2]

Some clarification?

I will use the remainder of this post to focus on the separation phenomenon; in particular, I would like to post some thought on how this is related to STALLING, i.e., the dramatic lift reduction commonly associated with separation.

Now, separation will typically occur at points on the surface where the wall shear is zero, i.e, close to the separation point, there will be some backflow, so that a SEPARATION BUBBLE/vortex forms between the foil surface, and the streamline roughly denoting the limit betweeen the viscous and inviscid layers of the fluid.

There, is however, a subtlety connected to this which is of crucial importance for our purposes in getting some grip on the stalling phenomenon:

At first, we can say that the centers of curvature for the motion (trajectories)of the fluid in the viscous layer is no longer coincident with with the centres of curvature as determined by the foil surface (that is the case when the Prandtl equations are valid).

Rather, the centres of curvature in the viscous layer has now a complicated spatial distribution, in the form of vortex centres situated in the viscous layer itself.

Let us fix our attention to a single vortex (in the stationary case):
Locally, therefore, the particle trajectories has the approximate form of concentric "circles" about the vortex centre (that is, after all, the archetypal vortex representation), and let us furthermore assume that the local, associated velocity field is dominantly dependent on the radial variable (measured from the vortex centre).

It then follows, from Navier-Stokes, that only the pressure gradient can provide the centripetal acceleration associated with the vortex motion (viscous forces affects the tangential accelerations).
That is, the pressure must increase radially outward from the vortex centre.

Yes, my equation was wrong. Thanks for the clarification. I will post the doubts arised.

Now I want to return at the discussion of stalling. A free vortex has a velocity distribution:

$$v_{\theta}=\frac{A}{r}$$ where $$A=\frac{\Gamma}{2\pi}$$

The N-S equations yields:

$$\frac{v_{\theta}^2}{r}=\frac{\partial P}{\partial r}$$

that is,

$$\frac{A^2}{r^3}=\frac{\partial P}{\partial r}$$

so that,

$$P(r)=-\frac{1}{2}\frac{A^2}{r^2}+C$$

where C is a constant solved when imposing an external pressure $$r\rightarrow\infty / P\rightarrow P_{\infty}$$

Therefore:

$$P(r)=-\frac{1}{2}\frac{A^2}{r^2}+P_{\infty}$$

It was that what you were referring to, when you said the vortex provokes points of despressurizing?. I have assumed $$v_r=0$$ which I don't if that's true.

Regards.

 

[arildno]

I'll get back to this, but I'll give what I roughly had in mind:
1) We know that viscosity tends to oppose velocity gradients; in particular, it is unrealistic to assume that infinite velocity gradients can occur in a real fluid.
It therefore seems permissible to say that viscosity imposes a BOUND upon the magnitude of the velocity gradient.
2) In an inviscid flow which is assumed to follow a curved object smoothly (no separation) ,say, a sphere, consider the following case:
Let the free-stream velocity be U, and the object a sphere with radius R.
Now, we can by potential theory solve this problem easily for an arbitrary R.
But:
Letting U be a constant, and regarding R as tiny, this flow has a huge pressure gradient associated with it!

Now let's look at the real case:
We know there will be some separation, and this will be stronger as the radius is smaller (free-stream velocity kept constant). That is, a REAL fluid is unable to set up the pressure gradient which must provide the centripetal acceleration of the fluid predicted by inviscid theory.

3)But cannot we then say that viscosity sets a BOUND upon the magnitude of the pressure gradient (indirectly by putting a bound on the magnitude of the velocity gradient)?
This idea is rather natural to assume from 1), and approaches certainty when considering 2)

Regarding this idea as our "primary", cannot we say that separation occurs whenever the required centripetal acceleration to follow the object becomes so large as to exceed the bound upon the pressure gradient dictated by viscosity?

4) This was basically my rough idea as to why separation will occur.
It is of interest to note, that in the d'Alembert's paradox case with a wing, the presssure gradient at the trailing edge is, indeed infinite (and so is the velocity there).
Hence, in a real fluid, we may say that it is the same mechanism which prevents the evolution of d'Alembert's paradox which also dictates the onset of separation (the mechanism being that the magnitude of the pressure gradient is always subject to some bound due to viscosity).

EDIT:
The argument in the cited paragraph is connected with the effort of finding a rough estimate of the pressure value at the wing. As I read your response again, was this what you were asking about?
In addition, note that the archetypal (only radially dependent) vortex may be of the form:
$$\vec{v}=(\frac{A}{r}+Br)\vec{i}_{\theta}$$
(The last vortex contribution, is of course, the velocity field associated with rigid-body motion, whereas the first is the potential point vortex field)

 

[Clausius2]

A break

I know I am storing threads of yours that must be answered. But before that, do you mind answering me this question, which appears in one past exam that I have?. It rushes me, because I have an exam on Monday.

"Under what conditions is the solution of potential flow compatible with the Euler equations?."

I know potential flow is for irrotational flow, and Euler equations can deal with both irrotational and rotational behaviors. So maybe the condition would be the flow must be irrotational?. But the logics of the question doesn't appear to be satisfied with that answer.

What do you think?

Thanks for bearing me.

 

[arildno]

Kelvin's theorem states that for an inviscid, baratropic fluid, the circulation around any material curve is constant through time.
Hence, if at say t=0 the circulation about EVERY material curve is zero, it follows they will remain so throughout time.
But, now, for example by invoking Stokes' theorem, it follows that the flow is irrotational
EDIT: The above argument, assumes, of course, that the Euler equations are at all times the "correct" equations.

Kelvin's theorem&lift:
It is repeatedly stated that lift is generated by the production of a vortex pair in the fluid, one remaining whizzing about the wing, the other (having opposite circulation) being displaced into infinity, (or dissipated by viscous forces).
This explanation is apparently consistent with:
a) An initial position of rest (prior to the plane starting to accelerate)
In this case, the circulation is certainly zero everywhere about any material curve.
b) In the time-dependent period, since the vortices are counter-spinning, the net circulation about any material curve containing both vortices remain zero.
c) The stationary case (when time has gone to infinity), where we can regard the counterspinning vortex to have gone off to infinity, so that for any material curve of finite extent bounding the wing, there remains a net circulation.

However, the trouble with this scheme, is that it introduces a singularitiy in the fluid domain IN A FINITE PERIOD OF TIME, namely that counter-spinning vortex.
This can simply not be accepted!
(I confess I've accepted this myself at some earlier point as well, but, now, I don't think the counter-vortex argument holds water, due to the finite-time singularity it implies)
The only time-dependent inviscid solution which does not introduce any singularities in the fluid domain during a finite period of time, is the solution generating d'Alembert's paradox (which, in an INfinite period of time establishes a singularity).
Hence, lift PRODUCTION cannot be explained solely within inviscid theory; there is an actual generation of NET circulation going on (in the time-dependent phase); i.e, the effect of viscosity is essential.
Inviscid theory can be shown to be lift-SUSTAINING, not lift-producing.
(It can be regarded as lift-sustaining, since Kelvin's theorem states that a given material curve will maintains whatever circulation it has through time, whenever we may approximate the flow as inviscid).

EDIT: This means that, IMO, the Euler equations cannot be regarded as the "correct" equations to view the lift-situation from (throughout time); rather, it is the Navier-Stokes equations which undergoes a surprising simplification when stationary conditions has been reached (i.e, potential theory may be used to determine the lift)