Intuition for why flow follows convex surfaces

lopp · February 22, 2021, 17:59

I'm trying to understand how airplanes fly. The crux of this mystery to me is why the air should follow the upper airfoil surface instead of always creating a bubble behind it.

I have some understanding about an adverse pressure gradient, and how it could cause flow reversal close to the surface, which forces at least a local separation. However, I haven't found anything that explains why the flow should bother staying attached in the first place, even before it encounters a significant pressure rise. It seems like the "centrifugal forces" around a convex surface might cause separation first.

Doug McLean (arguing against the coanda effect as an explanation for air following the airfoil) says in his book "... ideal inviscid flows represented by solutions to the potential equation have no trouble following curved surfaces ... the natural tendency of the boundary layer is to remain attached unless it is provoked to separate by an adverse pressure gradient that is too strong...". I'm looking for an explanation to this natural tendency that actually to me seems kind of unnatural.

Does anyone have any intuition about how/why this happens? If you initialized a flow with a stagnant or circulating bubble behind an object (say a wing or a cylinder) how/why would the bubble clear and the flow return to hug the surface? In potential flow and/or real air?

Thank you.

Edit: Problem Statement Update and Thread Summary (as of 2021 March)

After much discussion in this thread, I have included below some clarification and redirection about what I'm currently looking for and a summary of the progress so far. Since there doesn't seem to be much interest in getting to the bottom of this, I might give this thread some time off and see if someone like-minded comes along and wants to contribute some ideas or re-initiate the conversation.

Clarifications

Not specific to flight. This thread is currently focused about how fluids follow surfaces, particularly on the back side where it seems less natural. Understanding how fluids follow surfaces was initially motivated by seeking an understanding for lift, but understanding lift is not the focus at present. The current focus of the thread is the curve-following mechanism.

Separation, not cavitation or vacuum. When I say "follow surfaces" I'm not suggesting that we need to understand why the fluid doesn't fly off the back side leaving a complete vacancy of primary fluid. We can imagine that when the velocity is reasonably low that the environmental pressure would prevent this. Rather, we need to understand why we don't see a stagnant or recirculating bubble of fluid behind the object.

"Deterministic-cause-effect explanation" vs "after-the-fact proof". I believe that an after-the-fact proof would be relatively easy. For example, since invsicid fluid cannot introduce any vorticity, we know that there cannot be a stagnant bubble (with a shear layer between the bubble and the moving fluid) since that would be vorticity by definition. Similarly, if there was a recirculating bubble I'm guessing that would require vorticity as well. However, while such proofs may be simple and convincing, they don't provide much intuition into the mechanism that prevents separation.

Intuitive-model-based vs Mathematical. Related to the above point, mathematics is welcome but not satisfactory by itself. Like an "after-the-fact-proof", referring to a mathematical property does not help us understand the mechanism. If math is used, the math must be intimately tied into the model so we feel that we can "see" the mechanism in the math. Consider the idealized model to be more fundamental than the idealized mathematics.

Properties of the curve-following effect in the presence of perturbations and/or vorticity. The primary mechanism of curve-following is found in incompressible inviscid fluid, and such ideal fluids have no way of introducing vorticity when initialized at rest. Despite this fact, we are still interested in understanding the how the curve-effect behaves in the presence of both pertubations and vorticity. These pertubations and vorticities may originate from initialization, or from upstream, or be arbitrarily produced from external forces. Some may wonder why we should be interested in vorticity if an ideal mathematical representation of fluids has no vorticity at all. One reason to be interested is simply that pertubations and vorticity exist in real life, and that while we understand that the primary mechanism for curve-following is present in ideal fluids, that does not prove that the curve-following mechanism is effective in real life without the aid of a secondary mechanism enabled by viscosity. For example, consider the case of a divergent channel (see attachment, lower channel with stagnant fluid) with careful initialization. Here the curve-effect completely fails. Are there similar equilibrium for inviscid external flow with a bubble (of finite length) behind a cylinder or airfoil (see attachment with bubble behind cylinder)? If so is it stable? What is the mechanism of clearing bubbles in both ideal (if it exists) and viscous (we know it exists since planes can recover from stall) fluid?

Now for some explanations-in-progress for the curve-following mechanism:

Relation to the Heat Equation as a Hint about the Applied Pressure Gradient

We know that incompressible irrotational flow must adhere to laplace's equation $\nabla^2\phi=0$ where $\phi$ is a "potential field" whose gradient is the flow velocity field.

Mathematically, this is the same equation as the steady state heat equation, where the potential field would be temperature and the velocity field would be heat flux. While this enables us to see that fluid flow could go around the back of an object just like heat flux could, it doesn't tell us how this happens. The potential field/function is not typically given any intuitive or physical meaning (only its gradient, which is velocity). However, I have found a special case where we can give the potential function some meaning. During the time when the fluid field is unsteady (such as when an object accelerates through the flow or the flow is put in uniform accelerating motion), the field "accelerations" can be considered to be caused by an applied pressure gradient (superimposed on the momentum-induced pressure field) proportional to the potential function. This applied pressure gradient is explained further in the next two sections.

Existence of an Applied Quasi-Static Pressure Gradient

For any fluid we know from the mathematics of fields that (

is velocity, not volume)

$\frac{Dv}{Dt} \equiv \frac{\partial v}{\partial t} + v \cdot \nabla v$

Now considering incompressible irrotational (there may be more restrictions such as constant density) fluid initialized at rest, we find for a quasi-static fluid, (which we define as $v \approx 0$ ),

$\rho \frac{\partial v}{\partial t} = -\nabla p \equiv -\nabla p_{applied}$

We know that incompressible irrotational fluid velocities must occur in what we will call a "potential flow pattern" at all times and so we conclude that for a quasi-static fluid the rate of change of field velocities and applied pressure gradient must also follow this pattern.

I hypothesize that an intuitive explanation for the form of the applied quasi-static pressure gradient can be found by strong analogy to the way temperatures equilibrate, and that this equilibrium occurs instantaneously due to the incompressibility of the fluid. See the next section for more details.

Now consider that the flow is steady but no longer quasi-static. We have

$\rho (v \cdot \nabla v) =- \nabla p \equiv -\nabla p_{steady}$

Now consider that the flow may be both unsteady and non-quasi-static.

$\rho \frac{\partial v}{\partial t} + \rho (v \cdot \nabla v) = -\nabla p_{applied} + -\nabla p_{steady} = -\nabla p$

So for general flow of incompressible invsicid fluid initialized at rest we can think of the pressure field as a combination of the steady-state independent pressure field (supported by the momentum of the fluid itself) and an applied quasi-static pressure field (externally applied somehow, or due to the movement of the frame of reference from the surface itself moving).

Origin of the Applied Quasi-Static Pressure Gradient

I propose that the applied quasi-static pressure gradient is developed analogously to how steady-state temperature gradients develop. Specifically, the pressure or mass is analogous to temperature or energy content, and the massflow is analogous to heat flux. The development towards steady-state occurs instantaneously in the limit of our model becoming fully incompressible, just as the development towards steady-state would occur instantaneously in the limit of our thermal model losing all heat capacity.

Note that, in the limit of incompressibility, there is infintesimally small mass flux due to the applied pressure gradient, so there is no momentum flux to complicate the flow pattern.

Summary of Heat Equation Analogy and Applied Quasi-Static Pressure Gradient Explanation

So the intuitive explanation so far is:

Fluid fields (at least the particular ideal fluid fields we considered above) can only be acted on in the form of developed "applied quasi-static" pressure gradients which are analogous to temperature gradients. It is therefore intuitive how the pressure field would push fluid around the back of an object without separation.
For an unknown reason the fluid is happy to keep this same flow pattern
Conclusion: it stays attached.

A few things are incomplete about this explanation:

Steady flow continuation (no separation) after applied quasi-static pressure gradient removed. This is what I was trying to get at with the streamline/train track analogy (see post #35). If fluid parcels did follow streamlines like trains follow tracks (even if we image the track to shift), then we might be able to imagine how the streamline that ran across the surface couldn't detach since it would leave a vacuum. However, if the streamline/train analogy is flawed (see response) maybe we need a different explanation.
Extension to rotational but still inviscid fluid. The above mechanism breaks down in the example of the divergent channel with stationary fluid. Is a steady-state bubble possible behind a cylinder or airfoil? Are bubble-attached-states stable or unstable equilibrium in invsicid fluids and why? Do we have to bring viscosity back into the picture to explain their removal?

Shear impossibility mechanism (a "half explanation")

The "shear impossibility mechanism" is what I'm calling a "half explanation" because it's an idea of an explanation but it isn't very substantial. The "shear impossibility mechanism" may explain how the fluid flow pattern continues after the applied pressure gradient is removed, or may even be sufficient without even referencing the quasi-static pressure gradient at all.

Consisder a parcel of fluid located at rest on the edge of the divergent channel (see attachment with parcel sections labeled "top" and "bottom"). Now we know that there cannot be shear layer developed between the top and bottom sections because that would be vorticity which is impossible to introduce in our ideal fluid.

Now we need to know the mechanism or to be able to "see" why this should be the case. I'm not sure if this is possible on a local level without appealing to the entire flowfield. If you have an idea please share.

Separation "Push-of-War" (another "half explanation")

YouTube user or channel "learnfluidmechanics" briefly expressed his belief that close to the separation point, fluid leaves the surface vertically/normal before it bends downstream. (see attachment with separation "wall".) If this is the case (and if we can explain why it is the case) then maybe we can imagine separation (or lack thereof) as a "push-of-war" between the boundary layer momentum and favorable viscous stresses upstream and the adverse pressure gradient downstream, where each is pushing against the vertical wall where separation is currently occurring. If the upstream momentum "wins" and pushes the separation all the way off the object, then there is no separation. This push-of-war may explain how airfoils can recover from stall.

I suspect that the local-normal-wall of separation (if it even exists) requires viscosity to exist. If viscosity is in fact required, then this mechanism does not explain the primary non-viscous curve-following effect.

Note that if the flow at the separation point "wall" could be oblique (as I suspect to be the case for at least inviscid flow) then the wall can simply be bent over instead of pushed downstream, and this argument falls apart.

sbaffini · February 23, 2021, 07:09

I think it is useful if you first understand that potential flows cannot actually separate. They won't even give you lift, it's the insight from latest century pioneers that allowed us to trick potential flows into computing lifting bodies.

So, when speaking about physics of lift, we necessarily need to include viscous effects.

Still, honestly, I think you (and most people reasoning about lift) are largely overthinking this. Most airfoils will start to produce the small separation you mention at an angle of attack of just 10°. I mean, try to visualize that for a moment.

Also, lift is just the result of the pressure integral around the body and perpendicular to the free stream velocity. It just turns out that, for airfolis, most of it comes from the flow acceleration on the upper surface rather than compression on the lower one.

So, the trick an aerodynamicist tries to achieve is to accelerate the upper flow as most as he can while still being able to decelerate it without separation toward the trailing edge. Note that separation can only happen in the deceleration part, but at that point the flow has also benefited from the acceleration (it has more energy to deploy before an adverse pressure gradient can hurt it) and even more if turbulent.

In the end, it is just a matter of how large a curvature angle can be for a given state of the flow before it separates. There are correlations and, honestly, expecting separation for anything above 0, independently from the state of the flow, is just not realistic.

If you wonder what keeps a flow attached in a high curvature acceleration, well, it just turns out that there is a pressure gradient normal to the streamlines and pointing away from the center of curvature. But don't look at this as a cause-effect thing. The acceleration and the pressure field are aspects of the same flow that mutually interact.

However, no Coanda effect is involved in the production of lift in its traditional meaning (i.e., airplane lift). Of course, you can use the Coanda effect for generating forces that actually resemble lift, but they have a different nature (as long as we agree on what the Coanda effect actually is).

lopp · February 23, 2021, 13:48

Thanks for your reply

Quote:

I think it is useful if you first understand that potential flows cannot actually separate... I think you are largely overthinking this ... it just turns out that there is a pressure gradient normal to the streamlines and pointing away from the center of curvature.

But why - why, even with extreme corners, no stagnant or recirculating bubble? How do the pressure gradients develop to always be sufficiently strong? Anything besides "it just turns out"?

If there was a (stagnant) bubble in a potential flow (see attachment), would the pressure in the back be close to atmospheric or maybe even higher, while the pressure near the object would be lower? And so the bubble would get pushed (at least partway) from the back out the front sides?

Is it mathematically possible that there could be a bubble in potential flow? Do we just restrict ourselves to the bubble-less mathematical solutions that match the real world?

Quote:

However, no Coanda effect is involved in the production of lift... (as long as we agree on what the Coanda effect actually is).

That wasn't the main question but since you mentioned it I am interested. If we think of the Coanda effect as a low-pressure created at a surface because the fluid that was occupying the space is being carried away (not just accelerated, in the case of resisting an adverse pressure gradient) by an adjacent fluid (not strictly a jet), does it play a role in lift? Maybe near stall?

sbaffini · February 24, 2021, 07:57

Ok, let's make some order.

First note that, when stating "you are largely overthinking this", I was actually referring to the presence/lack of separation in most practical cases, which is quite opposite to extreme corners. Most aerofoils will indeed start to have those small separations at just 10°. If you take a goniometer and try to visualize that, there is no extreme corner at all.

But, besides very specific cases, those separations don't usually start at the point where, now I see, you seem to have more concerns, which is near the leading edge where the flow accelerates and reaches its peak velocity. They, instead, typically start from the rear part of the aerofoil, once the flow has lost all its energy and failed to reach the trailing edge.

As I said, specific cases exist, where indeed such separations occur near the leading edge, forming a bubble. But note that, not only these are more common than you could expect (indeed, it's not obvious at all how to avoid them in certain cases, and that's why we rely on CFD in the first place), but they still are viscous effects, and potential flows (which, however, do not exist) don't have bubbles.

But let me digress a little bit on potential flows here (steady and incompressible). These flows obey an equation of the form:

$\nabla^2 \phi = 0$

with $\phi$ assigned infinitely away from the body and with a null normal gradient on the body. Mathematically, this $\phi$ has the same solution of the temperature equation in the same domain with the same boundary conditions. The velocity field is just the gradient of such $\phi$ (which means that away from the body it has just a linear variation along the flow direction, as implied by the boundry conditions). If you are able to visualize/imagine an equilibrium temperature solution for a given domain, well, you see that it has no way to form any sort of bubbles. This is not only physical, but it is part of the mathematical nature of the temperature/potential equation.

What is the effect, then, of a null normal gradient body inserted in an otherwise linear potential field? You can have a grasp here in 2D (please deactivate the kutta condition for now):

http://dimanov.com/airfoil/af_prj.html

Basically, every potential line must be perpendicular to the body, so original vertical lines away from the body are split by it according to its superficial curvature. Note that, as long as you have the kutta condition off, it is largely ininfluential, at this stage, where exactly they split because it is purely mathematical and only depends from the geometry (said otherwise, it is just the result of a mathematical transform between the cylinder solution and the given geometry).

It is interesting now to activate the kutta condition to see what happens. Basically, it is just a consideration of the fact that, such a potential solution is not relevant because the flow would need an infinite velocity to actually turn around the trailing edge. Such very high velocity, in reality would force viscous effects to release a vortex and establish a contrarian circulation on the profile in order to move the rear stagnation point at the trailing edge. This actually happens in reality, and it does so for every change in the angle of attack (some vorticity is spit out and a new circulation sets in on the profile in order to take the rear stagnation point where viscosity claims it to be). Also note that, the same effect exists if you actually forget the potential theory and just solve the full euler equations with some convective scheme with artificial viscosity.

So, what we do in potential flow computations for airfoils (and is automatically present in Euler ones) is to add the kutta condition to take into account the viscous effects at the trailing edge. This is equivalent to actually add a circulation to the original potential solution so that nothing unreasonable happens at the trailing edge. Turns out that the only lift we can have in such a potential solution is indeed the one dictated by such additional kutta circulation. So:

potential flow -> No lift
potential flow + Kutta condition -> Lift

It also turns out that what was a clever mathematical machinery actually produces very accurate lift predictions for small angles of attack. So, while it is not the full story, one can certainly claim it has some merit in also explaining one of the main mechanisms behind lift (note that the release of vorticity when an airfoil starts or changes angle of attack is an experimental evidence). So with just these two pieces, in my opinion, we can already explain a great deal of the lift and how airfoils work. And note that in this picture, from the physical point of view, we have:

1) Potential solution determined by the geometry and boundary conditions = continuity equation

2) Kutta condition = clever account of some major viscous effect

2) Pressure distribution = just a result of momentum conservation

Or, if you want, the potential solution is determined first by the mass conservation, then altered by viscous effects. Finally pressure is the result of the bernoulli equation. At least thisis how we do it in computations, sequentially (i.e., pressure doesn't affect the potential, it's a result of it).

But this machinery still fails when we increment the angle of attack beyond some small value. Kutta condition would force the leading edge flow around larger and larger curvatures, going beyond physically sound values.

So, if I am allowed to translate your concerns, the real point here is understanding when reality will deviate again from the model (it always does, we here mean in a relevant way). When the flow acceleration (and attachment) at the leading edge is not anymore realistic?

Before proceeding, let me restate again what we have now. The sharp trailing edge, which is an essential feature of an airfoil, trough viscous effects, limits the flow on the lower side of the airfoil. It kind of blocks it by straightening the streamline leaving the trailing edge. So more of it must go on the upper side, with resulting additional velocity/energy. This energy is in addition to the one that would be originally available in the underlying potential flow without kutta condition.

At this point, however, we haven't yet considered, at all, the camber and thickness of the airfoil. Indeed, what I wrote is the same also for a purely flat plate with no thickness (you can check that in the app above as well, using $\nu = 0$ ). In this case, as you can imagine, a separation bubble at the leading edge would appear, for the same exact viscous reasons of the trailing edge, even for very little angles of attack. It is trough proper camber and thickness distributions, which also shape the airfoil leading edge, that the aerodynamicist is actually able to avoid this early leading edge separation for low angles of attack. But again, there is no magic here, we can't go to 30° without separation, we are talking about 10° in most cases, sometimes 15°.

So, as long as you would accept an explanation (with some solid experimental evidence) which follows from a potential flow theory enriched by viscous effects, the velocity around the airfoil is determined by the continuity equation and a strong viscous effect at the trailing edge. Pressure just adapts to it, as long as the whole potential picture largely holds. When considering what happens at the leading edge, viscous effects (and the specific airfoil geometry) must be taken into account, but details on the boundary layer here become more relevant. As you predict, there will be indeed separation bubbles at the leading edge beyond certain angles, nothing very magical at all. But at that point the potential picture also starts to be far from the reality.

This, hopefully, also explains how, in fact, lift has nothing to do with the Coanda effect.

FMDenaro · February 24, 2021, 12:18

First of all, no, potential flows cannot be used as a model for separation, this is for energetic consideration (total pressure is conserved and the process along a body is reversible).
In a real (viscous) flow you could immagine that the particles of the fluid (assume steady conditions) have some kinetic energy when they encouter the airfoil. The path on the upper surface is different form the path on the lower surface and the particles differently transform their kinetic energy in the process. Somehow, the particle of fluid along some point of the body will become "exhausted", without enough kinetic energy to follow the body shape. In the upper part (due to the curvature) its path simply departs from the line following the body geometry (like if the body it can actually follow is formed by the original body plus the bubble). In this region, where the streamline detaches from the body, other particles are "sucted" owing to the change in the pressure and creates the bubble.

Of course, there are more rigorous explanations.

And of course, an airplane moves in a fluid at the rest, so particles do not move along the airfoil

niranjan863 · February 24, 2021, 12:31

In a pure potential flow there cannot be any recirculating bubble.
Let me clarify what I mean be a pure potential flow here. In a pure potential flow
$\nabla^2 \phi = 0$
everywhere. This is because if there was a recirculating bubble then in the area enclosed by this bubble there should be net vorticity. But in a pure potential flow vorticity is zero everywhere. So a recirculating bubble cannot exist.

lopp · February 25, 2021, 12:46

Thank you for "digressing" about potential flow in a mathematical sense and in particular the relation to a steady state temperature distribution. The temperature connection you gave may be sufficient proof that a steady state inviscid incompressible irrotational flow must follow a curved surface. However, I'm interested in gaining more intuition about exactly why/how it comes about.

The magnitude of the angle of attack doesn't matter to me, and the kutta condition or other ideas about lift is not my focus at the moment (they would be goals down the road, and thank you for the work it took to write them out). Right now I'm trying to understand why an inviscid fluid follows curved surfaces so religiously no matter how small or large the curvature.

I think I may have been unaware of the full meaning of potential flow when I wrote my previous posts. In addition to being inviscid and incompressible, a potential flow must also be steady state with no vorticity?

I think I found a better example than a cylinder or wing to get at the crux of my problem. The example is a divergent channel shown in the attachments. The first attachment shows my expected solution for a potential flow. The second attachment shows an alternative solution that I believe to be perfectly valid for a steady state incompressible inviscid fluid (not potential flow since the lower fluid is stationary and hence mathematically that means that there is vorticity).

This is interesting because it means that fluid can flow without following curved surfaces - could it be actually possible (in theory) for there to be a steady state inviscid solution where there is a bubble behind a wing at even a low angle of attack?

I have one idea for how exactly fluid could develop to start following the curved surface of the channel instead of just going straight along the upper surface. When the fluid is first accelerated from rest relative to the channel, there must of course be a pressure differential to cause this acceleration. (Side question - if the channel was accelerated instead of the fluid could something different happen?) The pressure in fluid is by nature omni-directional. Thus whatever pressure is at the bend that would push the fluid straight forward, must also push it down along the curve. There is no centrifugal force around the curve since the fluid was not moving. After the pressure gradient is gone, fluid flow along the curve is maintained by the vacuum that would otherwise develop behind the fluid already in the curve.

Not very rigorous and kind of fuzzy but do you think I'm on to something?

blackjack · March 5, 2021, 19:09

Potential flow can model separation. It doesn't normally model separation because a potential exists only in irrotational flow, i.e. no vorticity. No vorticity means no shear layer. No shear layer means no separation.

The second image of the duct flow from a previous post shows a region marked stationary. So a shear layer exists between the stagnant region and the jet above.

The "trick" to getting lift from an airfoil in potential flow is to add a shear layer (wake) at the trailing edge. The shear layer is not in the potential flow, it is a boundary to the flow. Even an inviscid flow must separate at the trailing edge when the pressure is less than atmospheric, i.e. the flow cavitates.

Take a sphere in uniform flow. Calculate the boundary layer on the sphere assuming attached flow. Nash & Hicks' method will predict separation around 135 degree from the upstream stagnation point. (I have not found a boundary layer method that predicts the laminar separation on a sphere.) Now add a shear layer at the separation point. The strength of the shear layer is the surface velocity upstream of the separation. Inside the separation the flow is stagnant. That is, the flow inside the separation has a different total energy from the outside flow.

The calculated and experimental pressures are compared in the figure. Maybe it's a trick.

Separation regions with large variations in total energy cannot be modeled with a single potential-flow bubble.

sbaffini · March 6, 2021, 05:50

Quote:

Originally Posted by lopp

Thank you for "digressing" about potential flow in a mathematical sense and in particular the relation to a steady state temperature distribution. The temperature connection you gave may be sufficient proof that a steady state inviscid incompressible irrotational flow must follow a curved surface. However, I'm interested in gaining more intuition about exactly why/how it comes about.

The magnitude of the angle of attack doesn't matter to me, and the kutta condition or other ideas about lift is not my focus at the moment (they would be goals down the road, and thank you for the work it took to write them out). Right now I'm trying to understand why an inviscid fluid follows curved surfaces so religiously no matter how small or large the curvature.

I think I may have been unaware of the full meaning of potential flow when I wrote my previous posts. In addition to being inviscid and incompressible, a potential flow must also be steady state with no vorticity?

I think I found a better example than a cylinder or wing to get at the crux of my problem. The example is a divergent channel shown in the attachments. The first attachment shows my expected solution for a potential flow. The second attachment shows an alternative solution that I believe to be perfectly valid for a steady state incompressible inviscid fluid (not potential flow since the lower fluid is stationary and hence mathematically that means that there is vorticity).

This is interesting because it means that fluid can flow without following curved surfaces - could it be actually possible (in theory) for there to be a steady state inviscid solution where there is a bubble behind a wing at even a low angle of attack?

I have one idea for how exactly fluid could develop to start following the curved surface of the channel instead of just going straight along the upper surface. When the fluid is first accelerated from rest relative to the channel, there must of course be a pressure differential to cause this acceleration. (Side question - if the channel was accelerated instead of the fluid could something different happen?) The pressure in fluid is by nature omni-directional. Thus whatever pressure is at the bend that would push the fluid straight forward, must also push it down along the curve. There is no centrifugal force around the curve since the fluid was not moving. After the pressure gradient is gone, fluid flow along the curve is maintained by the vacuum that would otherwise develop behind the fluid already in the curve.

Not very rigorous and kind of fuzzy but do you think I'm on to something?

A potential flow can be unsteady, but must still be irrotational because, by definition, the velocity is the gradient of a scalar function.

Now, as the lift example hopefully clarifies, you can actually build potential flows that do the most strange things, by just superposing fundamental solutions of the equations that satisfy all your constraints. But the additional constraints like the Kutta condition are not exactly appropriate, they use effects outside the scope of the original potential equation and its formally required boundary conditions. It is in this sense that potential flows do not admit separations.

For example if you know a purely potential criteria that in a viscous flow would make it separate at the leading edge, you could embed it in the potential solution just like the kutta condition.

FMDenaro · March 6, 2021, 05:57

The discussion about the potential and the modelling of the separation remind me old topics I studied as a student.

For example the conformal map of the flow around a plate with separation.
However, one should be aware that the potential function is actually a function in the complex plane. The imaginary part is a potential function we can still introduce for viscous flows since its curl provides the velocity that can produce vorticity.
A further aspect if simple potential functions satisfy the NSE (but not the whole BCs)

sbaffini · March 6, 2021, 06:04

Quote:

Originally Posted by FMDenaro

The discussion about the potential and the modelling of the separation remind me old topics I studied as a student.

For example the conformal map of the flow around a plate with separation.
However, one should be aware that the potential function is actually a function in the complex plane. The imaginary part is a potential function we can still introduce for viscous flows since its curl provides the velocity that can produce vorticity.
A further aspect if simple potential functions satisfy the NSE (but not the whole BCs)

Do you mean vorticity introduced by actual cuts in the complex plane and the relative potential jump?

But, from a formal point of view, is there any information in the problem and pysics at hand (i.e., potential) that allows you to introduce such cuts? I mean, beyond the fact that you simply can

FMDenaro · March 6, 2021, 07:29

Quote:

Originally Posted by sbaffini

Do you mean vorticity introduced by actual cuts in the complex plane and the relative potential jump?

But, from a formal point of view, is there any information in the problem and pysics at hand (i.e., potential) that allows you to introduce such cuts? I mean, beyond the fact that you simply can

Well, there are some specific methods, for example using a conformal map like this https://www.researchgate.net/publica...n/figures?lo=1

lopp · March 8, 2021, 14:58

OK I think I finally have the right definition for potential flow: incompressible and irrotational. Potential flow can be time varying, but the velocity field can only scale uniformly in magnitude and cannot change form at all (as long as the object/boundary does not change form). It is more restrictive than inviscid flow, since while inviscid implies constant vorticity/rotation, potential flow must have this constant vorticity equal to zero. Do I have this all right?

Assuming I'm now on the right page with vocabulary/definitions, let me try to re-ask the original question while summarizing the thread so far:

I'm trying to get a thoughough, intuitive, and causal understanding of how fluids follow convex curved surfaces. This is the crux of my understanding about how airplanes fly, but the focus of the question isn't about how airplanes fly. The question is not focused on how to modify potential flow to account for separation or explain lift.

One explanation for how the fluid could follow a surface is the Coanda effect - the flow over the object entrains (by viscous effects) the air that would be forming a bubble on the downstream surface of the object, and by removing the bubble, we are left with a low pressure region that pulls the air down. However, we don't feel comfortable with this viscous explaination because we find that in our mathematical solutions of inviscid fluids, fluid still follows convex surfaces.

Setting asside viscosity as the explaination for how fluid follows surfaces, we consider two types of flow. The first type ("Type I") is potential flow. The second type ("Type II") is similar to potential flow in that it is incompressible and inviscid, but we relax the requirement that it be initialized with zero rotation.

For the first type ("Type I", potential flow), we can immediately see that by definition there can be no separation/bubble. If there was a bubble we would have rotation at the bubble/non-bubble interface. So this an interesting proof after-the-fact that we could not have separation. However, it doesn't explain causally how this came about. It's just a proof, not an intuitive causal explanation.

For the second type ("Type II", incompressible, inviscid flow), we get another interesting result - bubbles are actually completely possible (at least in the divergent channel in my previous post). We conclude that air following convex surfaces is not a strict requirement of inviscid flow but rather a developmental phenomena - it tends to develop that way if initialized with no rotation, but if there is already a bubble on the surface "all bets are off".

1. What is the causal, preferably locally based, intuition (not after-the-fact proof) for how fluids develop to follow convex surfaces in potential flow ("Type I")?
2. Are bubbles in "Type II" fluids possible besides the divergent channel example that I gave? Are these bubbles stable equilibrium or do the bubbles tend to clear and get blown downstream? Do we need viscous effects to explain the removal of bubbles?

Thanks again

sbaffini · March 10, 2021, 03:28

I won't give you an answer yet, but let me first try to elucidate better what are the differences between the two types.

You start from the NS equations and get the Euler inviscid equations by just setting to 0 the molecular coefficients (viscosity and conductivity).

Now, if we can focus on incompressible, isothermal flows, what you have is that initial 0 vorticity in the inviscid flow will just stay so, unless you explicitly introduce it at the inlet.

So, while Euler inviscid equations allow a vorticity, I think none of the mechanisms that produce it are of interest in what we are discussing here. That is, the effect you want to understand is not related to compressibility and/or density changes. Not even unsteadiness.

So why a flow follows a convex surface, I think, has the same explanation for type I steady and the more general Euler equations.

Along the same lines, as bubbles are not possible for potential flows, if they are possible for Euler equations is only trough vorticity specified at inlet, compressibility or other density change effects. Do you think they are relevant for such bubbles you want to investigate?

LuckyTran · March 10, 2021, 05:23

I just wanted to point out that the question doesn't really concern separation bubbles. You can always remove the separation phenomena by virtually inserting a (slip) wall along the streamlines. So the question is, why/how do Type 1 flows spontaneously expand to fill their volumes? Doesn't have to be a sudden expansion, could be a gradual wavy duct.

I know ergodic theorem can explain how this takes place, but a nice simple intuitive answer, I don't have at the moment.

sbaffini · March 10, 2021, 05:47

I would have said that it has to do with the properties of Harmonic functions, but that's an interesting perspective as well

lopp · March 10, 2021, 22:48

Quote:

So why a flow follows a convex surface, I think, has the same explanation for type I steady and the more general Euler equations.

I could believe that. But what is the explanation?

Quote:

...if [bubbles] are possible for Euler equations is only trough vorticity specified at inlet, compressibility or other density change effects. Do you think they are relevant for such bubbles you want to investigate?

I am interested on whether the mechanism for following curved surfaces breaks down for Type II systems when there is vorticity or a bubble present at the object surface (as the mechanism seems to break down in the divergent channel example). I am also interested in how a Type II system could clear bubbles and develop towards following the surface (if it even can), in spite of vorticity or bubbles present at initialization, introduced at the inlet, or created spontaneously by God.

sbaffini · March 11, 2021, 05:18

Quote:

Originally Posted by lopp

I could believe that. But what is the explanation?

As we are talking about a purely mathematical model, I have difficulties explaining its behavior with anything except mathematics. As I wrote above:

Quote:

Originally Posted by sbaffini

I would have said that it has to do with the properties of Harmonic functions

The incompressible, potential, steady equation is a mass conservation statement. That is, an inviscid, incompressible, constant density fluid (within this idealized model we can surely assume it is the fluid to have such properties) under steady irrotational conditions at the boundary, will obey very restrictive conditions of Harmonic functions due to mass conservation.

Roughly speaking, mass will simply diffuse, thus it will get all the space you can give to it.

Actually, there is a well known physical scenario where a potential like flow develops, which is the Hele-Shaw flow.

For what concern your Type II flow question, I have no idea. But, generally speaking, once you have vorticity in the flow, it will mostly behave like a real turbulent flow, so anything of what you suggest can happen.

lopp · March 11, 2021, 12:47

Quote:

As we are talking about a purely mathematical model, I have difficulties explaining its behavior with anything except mathematics...Roughly speaking, mass will simply diffuse, thus it will get all the space you can give to it.

Can you tie the math into a physical deterministic cause-effect phenomena at the surface of the object? I don't have a problem understanding math. The analogy you gave to the heat equation has convinced me that the math produces the curve-following behavior. The relationship between temperature and heat flow is intuitive. It's just I feel unsatisfied about my understanding about how it happens in fluids since it is fluid flow and not heat flow. Temperature is a physical thing that produces heat flow, potential function is ... dunno ... somehow causes velocity ... wait I thought that was pressure?

Can you build off (or discredit) these two approaches (or some combination). Both assume the fluid starts at rest and analyze acceleration at infinitesimally small velocities:
1. All fluid acceleration in type I or II fluids is caused by a pressure field over time. I wonder if we can mathematically consider "virtual" pressures at single points isolated from the other pressures that develop in the flow. If it could be shown that such a "virtual pressure point" at any point in the field could not cause fluid to accelerate off the surface (see attachment where dotted lines show path of detachment) without also accelerating the fluid along and down the surface. If this could be shown to hold for all individual points, then it must be true for the total field?
2. Starting at rest imagine we apply a small impulse of a differential pressure between the left and right boundary conditions (at infinity or wherever is typically done) across the entire incompressible flow in order to accelerate it. The resulting pressure gradient during this impulse must be locally proportional to the potential function itself if the accelerated fluid is to follow the pattern of potential flow (which we know it does). Can we see an intuitive reason why the pressure field in a quasi-static fluid must develop in the same manner as temperature in a heat conduction problem?

Both approaches would then have to account for fluid motion once it had begun to accelerate.

Quote:

For what concern your Type II flow question, I have no idea

If vorticity and bubbles can cripple the ability of fluids to follow surfaces, this is interesting. It may strengthen insight into how fluids follow surfaces without bubbles. Real air has vorticity and bubbles - why should the equilibrium be that bubbles and vorticity clear objects instead of collecting on and behind them? Do we have to rely on viscosity to explain how bubbles clear? If so, then viscosity may be needed to explain how fluids follow surfaces after all. In any case, it seems important to me to have a thorough understanding of what's going on.

FMDenaro · March 11, 2021, 12:58

I suggest to forget about the potential function phi for a physical intuition and going directly to the equation for the other potential function psi. In a 2D case you get a physical intuition about the behavior of the stream function as it obeys the equation

Lap psi = vorticity

so that if the source term vanishes you have the psi being an harmonic function. That justifies the flow pattern in the potential case.

February 23, 2021, 07:09		#2
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	I think it is useful if you first understand that potential flows cannot actually separate. They won't even give you lift, it's the insight from latest century pioneers that allowed us to trick potential flows into computing lifting bodies. So, when speaking about physics of lift, we necessarily need to include viscous effects. Still, honestly, I think you (and most people reasoning about lift) are largely overthinking this. Most airfoils will start to produce the small separation you mention at an angle of attack of just 10°. I mean, try to visualize that for a moment. Also, lift is just the result of the pressure integral around the body and perpendicular to the free stream velocity. It just turns out that, for airfolis, most of it comes from the flow acceleration on the upper surface rather than compression on the lower one. So, the trick an aerodynamicist tries to achieve is to accelerate the upper flow as most as he can while still being able to decelerate it without separation toward the trailing edge. Note that separation can only happen in the deceleration part, but at that point the flow has also benefited from the acceleration (it has more energy to deploy before an adverse pressure gradient can hurt it) and even more if turbulent. In the end, it is just a matter of how large a curvature angle can be for a given state of the flow before it separates. There are correlations and, honestly, expecting separation for anything above 0, independently from the state of the flow, is just not realistic. If you wonder what keeps a flow attached in a high curvature acceleration, well, it just turns out that there is a pressure gradient normal to the streamlines and pointing away from the center of curvature. But don't look at this as a cause-effect thing. The acceleration and the pressure field are aspects of the same flow that mutually interact. However, no Coanda effect is involved in the production of lift in its traditional meaning (i.e., airplane lift). Of course, you can use the Coanda effect for generating forces that actually resemble lift, but they have a different nature (as long as we agree on what the Coanda effect actually is). aero_head likes this.

February 24, 2021, 07:57		#4
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Ok, let's make some order. First note that, when stating "you are largely overthinking this", I was actually referring to the presence/lack of separation in most practical cases, which is quite opposite to extreme corners. Most aerofoils will indeed start to have those small separations at just 10°. If you take a goniometer and try to visualize that, there is no extreme corner at all. But, besides very specific cases, those separations don't usually start at the point where, now I see, you seem to have more concerns, which is near the leading edge where the flow accelerates and reaches its peak velocity. They, instead, typically start from the rear part of the aerofoil, once the flow has lost all its energy and failed to reach the trailing edge. As I said, specific cases exist, where indeed such separations occur near the leading edge, forming a bubble. But note that, not only these are more common than you could expect (indeed, it's not obvious at all how to avoid them in certain cases, and that's why we rely on CFD in the first place), but they still are viscous effects, and potential flows (which, however, do not exist) don't have bubbles. But let me digress a little bit on potential flows here (steady and incompressible). These flows obey an equation of the form: $\nabla^2 \phi = 0$ with $\phi$ assigned infinitely away from the body and with a null normal gradient on the body. Mathematically, this $\phi$ has the same solution of the temperature equation in the same domain with the same boundary conditions. The velocity field is just the gradient of such $\phi$ (which means that away from the body it has just a linear variation along the flow direction, as implied by the boundry conditions). If you are able to visualize/imagine an equilibrium temperature solution for a given domain, well, you see that it has no way to form any sort of bubbles. This is not only physical, but it is part of the mathematical nature of the temperature/potential equation. What is the effect, then, of a null normal gradient body inserted in an otherwise linear potential field? You can have a grasp here in 2D (please deactivate the kutta condition for now): http://dimanov.com/airfoil/af_prj.html Basically, every potential line must be perpendicular to the body, so original vertical lines away from the body are split by it according to its superficial curvature. Note that, as long as you have the kutta condition off, it is largely ininfluential, at this stage, where exactly they split because it is purely mathematical and only depends from the geometry (said otherwise, it is just the result of a mathematical transform between the cylinder solution and the given geometry). It is interesting now to activate the kutta condition to see what happens. Basically, it is just a consideration of the fact that, such a potential solution is not relevant because the flow would need an infinite velocity to actually turn around the trailing edge. Such very high velocity, in reality would force viscous effects to release a vortex and establish a contrarian circulation on the profile in order to move the rear stagnation point at the trailing edge. This actually happens in reality, and it does so for every change in the angle of attack (some vorticity is spit out and a new circulation sets in on the profile in order to take the rear stagnation point where viscosity claims it to be). Also note that, the same effect exists if you actually forget the potential theory and just solve the full euler equations with some convective scheme with artificial viscosity. So, what we do in potential flow computations for airfoils (and is automatically present in Euler ones) is to add the kutta condition to take into account the viscous effects at the trailing edge. This is equivalent to actually add a circulation to the original potential solution so that nothing unreasonable happens at the trailing edge. Turns out that the only lift we can have in such a potential solution is indeed the one dictated by such additional kutta circulation. So: potential flow -> No lift potential flow + Kutta condition -> Lift It also turns out that what was a clever mathematical machinery actually produces very accurate lift predictions for small angles of attack. So, while it is not the full story, one can certainly claim it has some merit in also explaining one of the main mechanisms behind lift (note that the release of vorticity when an airfoil starts or changes angle of attack is an experimental evidence). So with just these two pieces, in my opinion, we can already explain a great deal of the lift and how airfoils work. And note that in this picture, from the physical point of view, we have: 1) Potential solution determined by the geometry and boundary conditions = continuity equation 2) Kutta condition = clever account of some major viscous effect 2) Pressure distribution = just a result of momentum conservation Or, if you want, the potential solution is determined first by the mass conservation, then altered by viscous effects. Finally pressure is the result of the bernoulli equation. At least thisis how we do it in computations, sequentially (i.e., pressure doesn't affect the potential, it's a result of it). But this machinery still fails when we increment the angle of attack beyond some small value. Kutta condition would force the leading edge flow around larger and larger curvatures, going beyond physically sound values. So, if I am allowed to translate your concerns, the real point here is understanding when reality will deviate again from the model (it always does, we here mean in a relevant way). When the flow acceleration (and attachment) at the leading edge is not anymore realistic? Before proceeding, let me restate again what we have now. The sharp trailing edge, which is an essential feature of an airfoil, trough viscous effects, limits the flow on the lower side of the airfoil. It kind of blocks it by straightening the streamline leaving the trailing edge. So more of it must go on the upper side, with resulting additional velocity/energy. This energy is in addition to the one that would be originally available in the underlying potential flow without kutta condition. At this point, however, we haven't yet considered, at all, the camber and thickness of the airfoil. Indeed, what I wrote is the same also for a purely flat plate with no thickness (you can check that in the app above as well, using $\nu = 0$ ). In this case, as you can imagine, a separation bubble at the leading edge would appear, for the same exact viscous reasons of the trailing edge, even for very little angles of attack. It is trough proper camber and thickness distributions, which also shape the airfoil leading edge, that the aerodynamicist is actually able to avoid this early leading edge separation for low angles of attack. But again, there is no magic here, we can't go to 30° without separation, we are talking about 10° in most cases, sometimes 15°. So, as long as you would accept an explanation (with some solid experimental evidence) which follows from a potential flow theory enriched by viscous effects, the velocity around the airfoil is determined by the continuity equation and a strong viscous effect at the trailing edge. Pressure just adapts to it, as long as the whole potential picture largely holds. When considering what happens at the leading edge, viscous effects (and the specific airfoil geometry) must be taken into account, but details on the boundary layer here become more relevant. As you predict, there will be indeed separation bubbles at the leading edge beyond certain angles, nothing very magical at all. But at that point the potential picture also starts to be far from the reality. This, hopefully, also explains how, in fact, lift has nothing to do with the Coanda effect. aero_head likes this.

February 24, 2021, 12:18		#5
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,877 Rep Power: 73	First of all, no, potential flows cannot be used as a model for separation, this is for energetic consideration (total pressure is conserved and the process along a body is reversible). In a real (viscous) flow you could immagine that the particles of the fluid (assume steady conditions) have some kinetic energy when they encouter the airfoil. The path on the upper surface is different form the path on the lower surface and the particles differently transform their kinetic energy in the process. Somehow, the particle of fluid along some point of the body will become "exhausted", without enough kinetic energy to follow the body shape. In the upper part (due to the curvature) its path simply departs from the line following the body geometry (like if the body it can actually follow is formed by the original body plus the bubble). In this region, where the streamline detaches from the body, other particles are "sucted" owing to the change in the pressure and creates the bubble. Of course, there are more rigorous explanations. And of course, an airplane moves in a fluid at the rest, so particles do not move along the airfoil aero_head likes this.

March 6, 2021, 05:57		#10
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,877 Rep Power: 73	The discussion about the potential and the modelling of the separation remind me old topics I studied as a student. For example the conformal map of the flow around a plate with separation. However, one should be aware that the potential function is actually a function in the complex plane. The imaginary part is a potential function we can still introduce for viscous flows since its curl provides the velocity that can produce vorticity. A further aspect if simple potential functions satisfy the NSE (but not the whole BCs) aero_head likes this.

March 11, 2021, 12:58		#20
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,877 Rep Power: 73	I suggest to forget about the potential function phi for a physical intuition and going directly to the equation for the other potential function psi. In a 2D case you get a physical intuition about the behavior of the stream function as it obeys the equation Lap psi = vorticity so that if the source term vanishes you have the psi being an harmonic function. That justifies the flow pattern in the potential case. sbaffini and aero_head like this. Last edited by FMDenaro; March 11, 2021 at 15:25.

February 24, 2021, 12:31		#6
niranjan863 New Member Niranjan Join Date: Feb 2021 Posts: 3 Rep Power: 5	In a pure potential flow there cannot be any recirculating bubble. Let me clarify what I mean be a pure potential flow here. In a pure potential flow $\nabla^2 \phi = 0$ everywhere. This is because if there was a recirculating bubble then in the area enclosed by this bubble there should be net vorticity. But in a pure potential flow vorticity is zero everywhere. So a recirculating bubble cannot exist.

March 8, 2021, 14:58		#13
lopp New Member Join Date: Feb 2021 Posts: 16 Rep Power: 5	OK I think I finally have the right definition for potential flow: incompressible and irrotational. Potential flow can be time varying, but the velocity field can only scale uniformly in magnitude and cannot change form at all (as long as the object/boundary does not change form). It is more restrictive than inviscid flow, since while inviscid implies constant vorticity/rotation, potential flow must have this constant vorticity equal to zero. Do I have this all right? Assuming I'm now on the right page with vocabulary/definitions, let me try to re-ask the original question while summarizing the thread so far: I'm trying to get a thoughough, intuitive, and causal understanding of how fluids follow convex curved surfaces. This is the crux of my understanding about how airplanes fly, but the focus of the question isn't about how airplanes fly. The question is not focused on how to modify potential flow to account for separation or explain lift. One explanation for how the fluid could follow a surface is the Coanda effect - the flow over the object entrains (by viscous effects) the air that would be forming a bubble on the downstream surface of the object, and by removing the bubble, we are left with a low pressure region that pulls the air down. However, we don't feel comfortable with this viscous explaination because we find that in our mathematical solutions of inviscid fluids, fluid still follows convex surfaces. Setting asside viscosity as the explaination for how fluid follows surfaces, we consider two types of flow. The first type ("Type I") is potential flow. The second type ("Type II") is similar to potential flow in that it is incompressible and inviscid, but we relax the requirement that it be initialized with zero rotation. For the first type ("Type I", potential flow), we can immediately see that by definition there can be no separation/bubble. If there was a bubble we would have rotation at the bubble/non-bubble interface. So this an interesting proof after-the-fact that we could not have separation. However, it doesn't explain causally how this came about. It's just a proof, not an intuitive causal explanation. For the second type ("Type II", incompressible, inviscid flow), we get another interesting result - bubbles are actually completely possible (at least in the divergent channel in my previous post). We conclude that air following convex surfaces is not a strict requirement of inviscid flow but rather a developmental phenomena - it tends to develop that way if initialized with no rotation, but if there is already a bubble on the surface "all bets are off". 1. What is the causal, preferably locally based, intuition (not after-the-fact proof) for how fluids develop to follow convex surfaces in potential flow ("Type I")? 2. Are bubbles in "Type II" fluids possible besides the divergent channel example that I gave? Are these bubbles stable equilibrium or do the bubbles tend to clear and get blown downstream? Do we need viscous effects to explain the removal of bubbles? Thanks again

March 10, 2021, 03:28		#14
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	I won't give you an answer yet, but let me first try to elucidate better what are the differences between the two types. You start from the NS equations and get the Euler inviscid equations by just setting to 0 the molecular coefficients (viscosity and conductivity). Now, if we can focus on incompressible, isothermal flows, what you have is that initial 0 vorticity in the inviscid flow will just stay so, unless you explicitly introduce it at the inlet. So, while Euler inviscid equations allow a vorticity, I think none of the mechanisms that produce it are of interest in what we are discussing here. That is, the effect you want to understand is not related to compressibility and/or density changes. Not even unsteadiness. So why a flow follows a convex surface, I think, has the same explanation for type I steady and the more general Euler equations. Along the same lines, as bubbles are not possible for potential flows, if they are possible for Euler equations is only trough vorticity specified at inlet, compressibility or other density change effects. Do you think they are relevant for such bubbles you want to investigate?

March 10, 2021, 05:23		#15
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,751 Rep Power: 66	I just wanted to point out that the question doesn't really concern separation bubbles. You can always remove the separation phenomena by virtually inserting a (slip) wall along the streamlines. So the question is, why/how do Type 1 flows spontaneously expand to fill their volumes? Doesn't have to be a sudden expansion, could be a gradual wavy duct. I know ergodic theorem can explain how this takes place, but a nice simple intuitive answer, I don't have at the moment.

March 10, 2021, 05:47		#16
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	I would have said that it has to do with the properties of Harmonic functions, but that's an interesting perspective as well

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Will the results of steady state solver and transient solver be same?	carye	OpenFOAM Running, Solving & CFD	9	December 28, 2019 06:21
mass flow through surfaces in CFD-Post	rfavalli	CFX	2	August 21, 2015 08:33
Flow meter Design	CD adapco Group Marketing	Siemens	3	June 21, 2011 09:33
Question on 3D potential flow	Adrin Gharakhani	Main CFD Forum	13	June 21, 1999 06:18
computation about flow around a yawed cone	Tylor Xie	Main CFD Forum	0	June 9, 1999 08:33