This is leading away from the original question, but it's interesting.

Let's not forget that there is actually no such thing as viscosity. It's just a model to describe the effects of molecular interaction to support the model of a 'continuum flow'. As all models, it has it's limits. Likewise, the no-slip condition that is often used, turns out to be not exactly true if you look at a small enough scales, e.g. with the Boltzmann equations (or at high Knudsen numbers).

Adrin, on the question of vorticity generation: What are the competing ideas (viscous vs. inviscid)? How do I have to imagine inviscid generation of vorticity. On which scale is vorticity generated, anyway? If you consider the molecular scale, the definition of viscosity becomes questionable, so it wouldn't even be part of the discussion?

Do molecules exist or are they just models of atomic interaction? Do atoms really exist or are they just a way of modelling the probabalistic energy levels of electrons? Do electrons exist ... we cannot measure their position and momentum simultaneously, so they are strange beasts?

Should we explain the generation of vorticity at the sub-atomic level? If we cannot do this, do we not really understand it!!!

You can see molecules under a microscope, they are observable by sight and therefore it is easy to say, "Yes they exist". In my experience, I can see the different pouring characteristics of various liquids, this is explained well by a property that we call viscosity, so to me, viscosity is real

Your philosphical point (reality is in the eye of the beholder) is well taken, although I don't see that it adds much substance to the discussion. In fact, your argument annihilates all substance from any discussion we might have. It's a nice point, though.

... and I guess I missed one thing: Even vorticity itself may have to be re-defined, when looking at small scales. So, does it make sense to talk about vorticity generation on a scale where the 'continuum' assumption breaks down? What I was trying to suggest is that the distinction between 'viscous' or 'inviscid' effects may seem arbitrary at some point (especially as we enter Hmmm's realm of uncertain reality).

>>Let's not forget that there is actually no such thing as viscosity

It's funny I was thinking of the same thing this morning ... but this is more of a diversion from the original question, so let's not create new branches of questions

>>Adrin, on the question of vorticity generation: What are the competing ideas (viscous vs. inviscid)? How do I have to imagine inviscid generation of vorticity.

First, we are only looking at the continuum model (the navier stokes so-called equations). The question is precisely whether generation itself (not the evolution afterwards) of vorticity is viscous or inviscid. From an inviscid perspective, for example, you can see from the Poincare formulation (which is essentially the boundary integral equation solving the combined poisson equation for vorticity and the continuity equation, including velocity divergence) that the application of no-slip boundary condition leads to a distribution of _zero_ thickness vortex sheet right at the wall irrespective of whether or not there is viscosity in the fluid. Now this zero thickness vortex sheet, which sits at the wall, actually is a _fluid_ element not a _wall_ element (very important physical concept to appreciate), although from an "implementation" perspective it doesn't look any different from a potential velocity distribution at the wall (of course subsequent to accounting for directional effects). Then, one can argue that the effect of viscosity is to diffuse this zero thickness vorticity into the thicker layer of the boundary layer, _after_ vorticity is generated.

Now, clearly, a zero thickness vortex sheet is a mathematical construct (just as any singularity is), but so is the navier stokes "equations". I mean, to the extent that we're dealing with the continuum and not the micro, nano motion a zero thickness vortex sheet is as real as the (lumped parameter) viscosity.

The discussion is deeper than this, of course, and can be found in the literature. I just brought one example (my own perspective)

adrin

>>The fact that there is a velocity potential that defines the flow everywhere means that it is potential - a potential flow is one whose velocity can be determined from a "velocity potential"

OK, I'm not very clear what you mean above. I'm assuming "velocity potential" is a _scalar_ field, not vectorial. Are we in agreement on this or am I missing something in your statement above? IF it is only a scalar field then it cannot be related (directly) to the "vector potential" which yields vorticity. Again, going back to Hodge/Helmholtz decomposition we see that vorticity is linked to the vortical component of the velocity field and that the potential velocity is a "corrective" term that accounts for the boundary effects. The zero thickness shear layers behind an object can be viewed as a "boundary" above and below which we have potential flow. This does _not_ make the entire flow potential. _every_ flow field has a potential flow component (again, according to the abovementioned decomposition) ... I'm sure I'm missing something about what you're implying ...

>>You also have to be careful in talking about vorticity the way you are above - the problem with vorticity boundary conditions arises because vorticity is a "derived" rather than a physical quantity (it was originally introduced to exploit the relationship of the solenoidal condition to the Biot-Savart law). The physical quantities (the velocities) have boundary conditions but the derived quantities require derived boundary conditions which have to be determined by consistency with the physical ones!

Not exactly true on two counts. (1) Vorticity is as physical as pressure and/or velocity. What makes vorticity a derived quantity? because it is obtained by taking the curl of velocity or something else? If the former, then so is velocity because it is derived by time differentiating a length quantity (time and length are basic physical quantities that you measure "directly"; you need these two to "evaluate" velocity). I can argue that you can discuss and understand turbulence (and many physical phenomena) better using vorticity instead of velocity (alone).

As for the boundary conditions, in the case of the vorticity transport form of the navier stokes equations, one does not need a vorticity boundary condition to satisfy no-slip and no-flux boundary conditions. The velocity boundary conditions, themselves, provide the necessary consistent BCs. The Poincare formulation clearly shows this. Of course, the latter "creates" zero thickness vortex sheets at the wall which satisfy the necessary boundary conditions, but this is different from the more classical view of having to specify a vorticity boundary condition (note that a vortex sheet strength has units of velocity not vorticity)

> For Prandtl's boundary layer theory and vorticity generation just differentiate the boundary layer equations with respect to the wall normal to get the vorticity equation; i.e. vorticity generation is nothing more than the reduction in streamwise velocity to satisfy the no-slip condition. More precisely if y is the normal direction from the wall then the vorticity is approximately (u tangential to the wall) u_y=du/dy which is just the shear. Now as the freestream is approached u_y->0 (but u is nonzero) while at y=0 u=0. This implies that u_y must be nonzero in the boundary layer => vorticity generation mechanism.

Now this may be viewed as pedantry, but Prandtl theory does not explain the link between no-slip and vorticity at the wall, or more specifically vorticity generation (even in the vorticity formulation, which I've used in my earlier works), but the dynamics of boundary layer flow. I agree with your qualititative description of vorticity generation mechanism, assuming you're talking about a zero thickness boundary layer, which implies an inviscid process, even in a laminar flow case!

Anyway, it seems that we've diverted quite a bit

adrin

Adrin,

You seem to have thought about this quite a lot, and seem to know your math.

I am not quite sure about this statement though,

" ... that the application of no-slip boundary condition leads to a distribution of _zero_ thickness vortex sheet right at the wall ... "

To my mind, if you are talking about applying a no-slip condition at the wall, you are bringing the concept of viscosity into the argument and it therefore doesn't make sense to talk about inviscid generation of vorticity.

Enlighten me

>>To my mind, if you are talking about applying a no-slip condition at the wall, you are bringing the concept of viscosity into the argument and it therefore doesn't make sense to talk about inviscid generation of vorticity.

Let me start by saying that I don't have a strict position on whether or not vorticity generation is inviscid. As you pointed out, I think quite properly, the issue of scale is important to remember. That is, no-slip is a matter of fluid particles/molecules "sticking" to the wall, and at that scale there is no such thing as viscosity but a variety of "gravitational" forces. But if we take the continuum flow view, we need to explain the role of viscosity ...

At the heart of this debate (if we agree there is one ) is the concept that an infinite Reynolds number flow is not the same as inviscid flow - and that viscosity does have a role no matter how small. I agree that viscosity has a significant role. But, viscosity, or equivalentaly the diffusion equation/process, neither creates nor destroys vorticity (or anything else for that matter); it only acts to spread out an existing field. Therefore, one can argue that viscosity cannot possibly be responsible for vorticity _generation_ but only its spreading!

it's just some snack for thought

adrin

>>>In a boundary layer, viscosity leads to a velocity gradient due to the no slip condition which in turn leads to some vorticity. The same thing will happen in a shear layer.

Ahhhh, very good point/example. See my other responses to others. Viscosity does _not_ lead to a velocity gradient due to the no-slip but the _thickening_ or _smoothing_ of an already existing velocity gradient. Take, for example, impulsively started flow over a cylinder. At t=0 you have free-stream flow around the cylinder with velocity U (for simplicity). This is "potential flow" about the cylinder, but physically you must have no-slip at the wall. So, you end up having (demanding) a zero-valued velocity right at the wall and a non-zero-valued velocity, U, right above it. That is, you have a velocity jump, or an infinite-valued gradient/vorticity condition at the wall. In other words, you have a zero-thickness boundary layer, and that is precisely the amount of vorticity (vortex sheet strength) you generate at t=0. At t=0+ viscosity acts to diffuse this zero-thickness vorticity (with strength equal to infinity, but bounded integral value) to a finite-thickness boundary layer.

When viewed from a traditional finite difference point of view this is hard to comprehend since everything has to be finite valued and there is a finite height between the wall and the first layer of grids away from it (which automatically forces one to think of the role of viscosity according to your description).

adrin

"This does _not_ make the entire flow potential. _every_ flow field has a potential flow component (again, according to the abovementioned decomposition)"

You're trying to make a big deal out of something that is nothing more than a standard definition. If you can write u=grad{phi) then the flow is potential - it is as simple as that! In the case of a vortex sheet behind a body you have to cut the solution into two or more pieces and join them together through the boundary conditions. Not calling this a potential flow is akin to saying the Riemann-Hilbert problem has nothing to do with the theory of analytic/harmonic functions.

"Not exactly true on two counts. (1) Vorticity is as physical as pressure and/or velocity. What makes vorticity a derived quantity? because it is obtained by taking the curl of velocity or something else? If the former, then so is velocity because it is derived by time differentiating a length quantity (time and length are basic physical quantities that you measure "directly"; you need these two to "evaluate" velocity). I can argue that you can discuss and understand turbulence (and many physical phenomena) better using vorticity instead of velocity (alone)."

Firstly, in a Eulerian frame, particle position has very little meaning (in Hamiltonian fluid dynamics this is called particle relabling symmetry) and so the system is closed without the need to work out positions (in solid mechanics knowing the velocities would not be enough). This means that the step of thinking of the velocity as a derived quantity is not all that useful (in a Lagrangian frame this would not be true).

The main point is that the velocity and pressure are physical quantities because they are measurable - the vorticity is not! Another interpretation is that the vorticity equation is derived from the momentum equation and not vice-versa (it's not an independant equation). Indeed if it were not for the Biot-Savart law the resulting equatation would be of little or no use!

"As for the boundary conditions, in the case of the vorticity transport form of the navier stokes equations, one does not need a vorticity boundary condition to satisfy no-slip and no-flux boundary conditions. The velocity boundary conditions, themselves, provide the necessary consistent BCs."

In that case how do you treat the viscous term in the vorticity equation? You need a boundary condition on the vorticity in order for the equation to be solvable. This condition in turn must be consistent with the no-slip condition on velocity.

"Now this may be viewed as pedantry, but Prandtl theory does not explain the link between no-slip and vorticity at the wall, or more specifically vorticity generation (even in the vorticity formulation, which I've used in my earlier works), but the dynamics of boundary layer flow. I agree with your qualititative description of vorticity generation mechanism, assuming you're talking about a zero thickness boundary layer, which implies an inviscid process, even in a laminar flow case!"

The boundary layer does not need to be of zero-thickness. Indeed, as I pointed out in an earlier post, the viscous term in the incompressible Navier-Stokes equation can be written as -nu.curl(w) where nu is the kinematic viscosity and w is the vorticity. Now in order to satisfy the no-slip boundary condition this must be non-zero (if the term was identically zero it would in general be impossible to satisfy no-slip because the equations would be of lower-order). It trivially follows then that vorticity generation by the wall arises from the slowing down of fluid by viscosity in order to satisfy no-slip.

Speaking as a Mathematician whose worked in fluid mechanics for close to 20 years I think you need to step back from the formal mathematics and think about the mathematical manipulations, why you are performing them and what it all actually means in terms of the fluid physics. I've read/seen papers by theoretical physicists who have moved into the field of fluid mechanics who use a lot of the formalism you are discussing to obtain physically incorrect results; some examples of this are (i) proving stability results to a flow that is so unstable that the linearized full problem is ill-posed and (ii) suggesting inconsistent boundary conditions (i.e. no-slip when they should be stress-free).

>Now, clearly, a zero thickness vortex sheet is a mathematical construct (just as any singularity is), but so is the navier stokes "equations"

I agree, and actually that's what's giving me trouble. The vortex sheet is a mathematical construct, necessary to satisfy the no-slip boundary condition in inviscid flow (I think of it as reducing the boundary layer to zero thickness, while insisting on no-slip). I understand that the role of viscosity in that case would be limited to diffusion as opposed to generation of vorticity. I also understand that this model can be regarded as completely inviscid. However, it seems to me that we have simply avoided the question of vorticity generation by using a vortex sheet at the boundary. Where does this sheet come from? Is it created by an inviscid or a viscous mechanism, and what are these mechanisms? I was just curious to find out what the ideas are on underlying physical mechanisms... which this mathematical concept of a vortex sheet may not be able to explain.

Is the argument of the no-slip condition essential to generation of vorticity, and if so, what physical mechanism leads to zero slip? Is that a viscous effect?

My feeling is, that the latter question is ill-posed, because to gain a more detailed view on the boundary you'll have to leave the continuum model aside, and that contradicts the idea of viscosity. To stick to continuum flow, I think we'll have to accept no-slip as a given condition (without being able to explain or prove it), and everything else (including generation of vorticity) results from that condition and the governing equations, regardless if viscous or inviscid.

I'd like to throw in a comment (or question). I don't have the 20+ years experience in mathematics, so I am looking at this from a more physics point of view. If you guys cannot agree on how vorticity is generated (or how to best model it mathematically), maybe you can agree on an answer to this question first:

>Where< is vorticity generated? In the boundary layer? Locally at the wall?

My understanding is that vorticity (unless entering through other boundaries) is generated right at the wall, as an interaction between fluid and structure. Can this local effect be explained by continuum mechanics, or is the assumption of a no-slip boundary condition simply imposed on continuum models to simulate physics at a microscopic scale? Could this have anything to do with viscosity?

This is what I think: I understand that the diffusion of vorticity away from the wall and the generation of a boundary layer can be described by the effect of viscosity. However, the ultimate cause of the no-slip condition, and hence, the generation of vorticity, is something that cannot be explained within the continuum model.

I am aware that this argument is somewhat displaced from your mathematical discussion... but as long as you stick with the continuum model, the question of vorticity 'generation' is simply a question of how you apply the no-slip condition: Either allowing for a finite thickness boundary layer (viscous transport of vorticity, once it's created miraculously at the wall) or using a singular vortex sheet (inviscid, vorticity sticks to the wall, where it's miraculously created). Neither model explains the physics of vorticity generation, it's simply imposing it.

"My understanding is that vorticity (unless entering through other boundaries) is generated right at the wall, as an interaction between fluid and structure. Can this local effect be explained by continuum mechanics, or is the assumption of a no-slip boundary condition simply imposed on continuum models to simulate physics at a microscopic scale? Could this have anything to do with viscosity?"

A better way to think of this is that the no-slip condition creates a shear in the flow through the action of viscosity. This shear then can be mathematically identified with vorticity generation -the magnitude of this vorticity is related to the viscosity as can be seen for an attached boundary layer where it scales like the square root of the Reynolds number. The gerneration of a shear is more fundamental than the generation of vorticity (basically the vorticity generation is a consequence of the viscosity + no-slip condition - any boundary condition other than one of zero vorticity is going generate vorticity). After all, and this relates to your other question (I'm sticking to the Continuum), what is vorticity other than a mathematical quantitiy derived from the velocity vector? Most vortex dynamicists would take the view that "it is the nearest we can get to defining an angular momentum for a fluid". It is not however the angular momentum!

Another important point, I may have misinterpreted your above point and you already know this, is that in a 3D flow the vorticity is not conserved and vorticity can be generated by the stretching of a vortex tube in an invisicid flow (it's the cross-sectional area of the tube time the magnitude of the vorticity that's conserved). A Hamiltonian fluid dynamicist would undoubtedly take the stance that circulation was more important than vorticity for this reason.

The book "Vortex dynamics" by Saffman is well worth reading since it contains some discussion on the subject vorticity generation.

>The gerneration of a shear is more fundamental than the generation of vorticity (basically the vorticity generation is a consequence of the viscosity + no-slip condition - any boundary condition other than one of zero vorticity is going generate vorticity).

Looking at shear is fine with me, but doesn't change my perspective. Shear is a boundary condition, not created through the physics described by the governing equations, but imposed from outside. The effect of viscosity is then to transport momentum (or momentum defect) into the flow by enabling the flow to withstand some internal shear. Viscosity describes the interaction of fluid particles with each other, allowing for shear within the fluid. Generation of vorticity (and shear) at the boundary, however, must be a matter of interaction between fluid particles and structural wall.

Fluid viscosity will determine an equilibrium between generation and transport of vorticity, so the magnitude of boundary shear depends on fluid viscosity. However, the fact that there is wall friction (and no slip) in the first place, qualitatively, has nothing to do with the fluid viscosity but with the molecular events at the structural surface. So I would have posed your statement a little differently: vortex generation qualitatively is a consequence of the no-slip condition (regardless if the flow is viscous or inviscid), but the magnitude of shear or vorticity depends on viscosity (keeping in mind that zero viscosity does not necessarily mean zero vorticity).

>3D flow the vorticity is not conserved and vorticity can be generated by the stretching of a vortex tube in an invisicid flow

Good. This is a much better example of inviscid generation of vorticity.

OK, one last attempt ...

>>You're trying to make a big deal out of something that is nothing more than a standard definition. If you can write u=grad{phi) then the flow is potential - it is as simple as that! In the case of a vortex sheet behind a body you have to cut the solution into two or more pieces and join them together through the boundary conditions.

Please read my prior posts. I'm saying _precisely_ the same thing with consistency: if you can write u=grad(phi) then the flow is potential; otherwise it is _not_! The velocity at the vortex sheet violates the latter and curl(u) recovers the vortex sheet strength, not zero. Therefore, the flow is _not_ potential. As I said before, the flow is potential _above_ and _below_ the vortex sheet (which is _exactly_ the same thing as saying cutting the solution into two or more pieces/regions as you are saying above)

The _only_ reason "I'm making a big deal" out of this is to clarify this simple problem to the person who asked the original question. if a novice does not recognize the difference between potential and euler flow, to say that vortex sheets behind objects and/or waves are potential (without the "disclaimer") confuses the person - probably the reason the question was asked in the first place.

Let me bring a counter example. Two fluids in parallel with two different densities make up a variable density flow, and not constant density just because you can break up the flow into two separate constant density problems. Putting a contour around the vortex sheet and the body and solving the potential field (correctly) _outside_ that contour does not make the (entire) flow potential. Period. If that were the case, I could do the same for a bluff body flow - put a contour around the relatively compact vorticity field and solve the potential flow problem outside it and say the whole flow is potential. Anyway, I'm stopping this segment here.

AG>>"As for the boundary conditions, in the case of the vorticity transport form of the navier stokes equations, one does not need a vorticity boundary condition to satisfy no-slip and no-flux boundary conditions. The velocity boundary conditions, themselves, provide the necessary consistent BCs."

Tom>>In that case how do you treat the viscous term in the vorticity equation? You need a boundary condition on the vorticity in order for the equation to be solvable. This condition in turn must be consistent with the no-slip condition on velocity.

In an earlier post I'd said one does not need a vorticity boundary condition in the classical sense ("classical" meaning in the sense of grid-based vorticity-stream-function methods).

The no-slip and no-flux boundary conditions lead to a distribution of vortex sheets at the wall, _irrespective_ of viscosity. The strength of the vortex sheet has units of velocity, but it is still a _singular_ vortex. That is, you can write it as omega(x)=gamma*Delta(x), where x is vectorial _on_ the body, gamma is the vortex sheet strength and Delta is the delta function that says that vorticity at the wall _prior_ to diffusion is singular. This is the boundary condition for the diffusion equation for vorticity, which gives a stokes layer solution with the peak value being related to gamma, viscosity and time.

Again, note that we do _not_ need to specify a finite valued omega at the wall (the main source of difficulty in finite-difference type approaches), but just the vortex sheet strengths, which are themselves a manifestation of the velocity boundary conditions.

>>Speaking as a Mathematician whose worked in fluid mechanics for close to 20 years I think you need to step back from the formal mathematics and think about the mathematical manipulations, why you are performing them and what it all actually means in terms of the fluid physics.

I only pointed out to readers that there are two schools of thought on vorticity generation (by serious people). Physically, it is clear that viscosity is the cause of flow retardation near the wall (thus my earlier comment that an infinite Reynolds number flow is not euler flow). I've brought out the less common perspective as food for thought. The latter can only be dismissed on physical grounds (but at what scale?)

adrin

"vortex generation qualitatively is a consequence of the no-slip condition (regardless if the flow is viscous or inviscid)"

However, if the flow is inviscid the no-slip condition is meaningless (the no-slip condition is required for the Navier-Stokes equations and not the Euler equations by virtue of the viscosity multiplying the highest order derivative in the equations). The no-slip condition goes hand-in-hand with the viscous terms. You could always add a different boundary condiiton in place of no-slip of course - but the fact remains that a viscous flow requires more boundary conditions than an inviscid one.

"Let me bring a counter example. Two fluids in parallel with two different densities make up a variable density flow, and not constant density just because you can break up the flow into two separate constant density problems. Putting a contour around the vortex sheet and the body and solving the potential field (correctly) _outside_ that contour does not make the (entire) flow potential. Period. If that were the case, I could do the same for a bluff body flow - put a contour around the relatively compact vorticity field and solve the potential flow problem outside it and say the whole flow is potential. Anyway, I'm stopping this segment here."

This is not really a counter example since you are talking about "two fluids" and not a single variable density one. In this terminology air flowing over water would be a single stratified flow which it is not - air does not miraculously convert itself into water as the sea surface is crossed.

"Again, note that we do _not_ need to specify a finite valued omega at the wall (the main source of difficulty in finite-difference type approaches), but just the vortex sheet strengths, which are themselves a manifestation of the velocity boundary conditions."

All you've done is move the problem of finding a consistent surface vorticity to one of finding the vortex sheet strengths. Same problem different formulation.

I agree, in terms of the N-S equations and any real flow with a finite size boundary layer. Nevertheless is the combination of Euler or potential equations + singular vortex sheet a mathematically valid example of enforcing the no-slip condition (by assigning/creating vorticity at the wall) for an inviscid flow.

You will disagree, but I see the mechanism of flow sticking to a wall (no-slip: ultimate cause of shear and vorticity) and flow sticking to itself (viscosity: propagating vorticity) as two different physical mechanisms. The latter depends on the former, but not vice versa.

"I agree, in terms of the N-S equations and any real flow with a finite size boundary layer. Nevertheless is the combination of Euler or potential equations + singular vortex sheet a mathematically valid example of enforcing the no-slip condition (by assigning/creating vorticity at the wall) for an inviscid flow."

The problem with assigning a wall vorticity on the wall for the Euler equations in order to mimic an inifinitely thin boundary layer is problematic. Is your choice of vorticity consistent with the true solution? is it unique?

The correct way to do this type of problem, and this was done in aerodynamics for a number of years, is to use the "strong viscous-inviscid interaction" theory which is based upon the hight Reynolds number triple-deck theory. That is to say you need to solve the boundary-layer equations with a surface slip velocity obtained from the inviscid solution. This boundary-layer solution displaces the flow near to the surface which then in turn modifies the location of the surface as seen by the inviscid solution (through the displacement thickness). This in turn modifies the pressure seen by the boundary-layer => feedback loop.

Note that I've said nothing vorticity generation here! All I've mentioned is the displacement of the surface; i.e. instead of the zero normal velocity being applied at y=f(x) it is applied at y=f(x)+A(x) with A determined by the interaction of the boundary-layer with the inviscid flow. In this formulation the inviscid flow does not satisfy noslip (as it shouldn't!).

"You will disagree, but I see the mechanism of flow sticking to a wall (no-slip: ultimate cause of shear and vorticity) and flow sticking to itself (viscosity: propagating vorticity) as two different physical mechanisms. The latter depends on the former, but not vice versa."

If the fluid cannot stick to itself how can it stick to anything else?