A four-question quiz: on fundamentals of fluid dynamics
By oversimplifying things, there is the risk of losing details that could affect the correct interpretation of a physical phenomenon. However, using common sense and logic based on premises derived from physical observations can lead to a better interpretation of veiled concepts that cannot be understood in more abstract ways.
This test consists of four questions, of which at least the first three seem to be so obvious that they could fall into the concept of silly questions; however, it is necessary to ask them to understand the last one and the whole meaning of this publication.
All pictures are retrieved from 'An Album of Fluid Motion' (van Dyke, 1982), in which a viscous flow (fluid: water) past a circular cylinder under different Reynolds number (Re) conditions.
1. Which of the following two images represents a fluid past at an extremely low Reynolds number?:
a)
b)
a) This option is a completely opposite condition, where inertial forces dominate over viscous ones, with fluid detachment despite the existence of viscosity.
b) According to the author's description, this is the correct answer since its Re corresponds to a near-zero value (Re<<1), thus viscous forces dominate over inertial ones. Notice that, naturally, viscosity inhibits the existence of vorticity in the surroundings (by viscous diffusion).
2. Coming from the previous viscous-dominated case, what should be the pattern for a slight increment in Re (by increasing the fluid velocity, which in the previous case was extremely low; 0.001 m/s), that is around Re=10?
a)
b)
a) As we know, the range of Re can go from around zero to millions in practical applications. Since most of the varying physical phenomena do not occur abruptly but require a gradual process, this option must be discarded, in fact, it represents a relatively high Re condition (Re=2000).
b) As has been observed experimentally by gradually increasing the Re of a fluid past a cylinder, both separation points tend to move upstream, towards the extreme top and the bottom points on the surface. Thus, such an image would represent the pattern observed for the low Re range (Re=9.6). Hence, this is the correct answer.
3. By following the previous sequence, which of the two next pictures shows a higher Re between them?
a)
b)
a) As said before, this photo was taken for a Re=2000, where the main disturbance behind the cylinder seems to be confined to a reduced region, around two diameters downstream.
b) It is clear that this picture shows a higher Re condition (Re=10000) since the main disturbance region extends far downstream, followed by a characteristic random pattern. Compared with the previous picture, this is the correct answer.
4. Until now, all the questions seemed easy to answer due to common sense, and the most simple logic has been applied. But, what happens if viscosity were 'almost' removed (e.g., leaving a millionth of a unit) from the last experiment? Which would be its new pattern?
a)
b)
a) The pattern will practically be the same because, at this Re, the inertial forces dominate over the viscous ones, no matter that fluid viscosity is practically removed, becoming a pure inviscid flow. In fact, compared with other fluids, the water (and air) viscosity value is relatively low, and its absence should not significantly affect its behavior for a specific Re condition (velocity and pressure fields remain practically unchanged).
b) As we see before, this pattern is obtained by a viscous-dominated condition (Re<<1), thus increasing instead of decreasing the fluid viscosity value (or reducing its velocity). This must be the INCORRECT answer.*
So, why does the current interpretation of potential flow theory (PFT) lead to a similar pattern to this last one for an ideal ("inviscid"; fully inertial-dominated) flow past a cylinder? Perhaps, such a solution is being modeled for low Re conditions? If this were true, how is it possible to obtain well-matched results (at least for lift coefficient) for the linear (viscous) region (CL vs alpha; without any boundary layer coupling!) for airfoils, wings, and airplanes, which operate in the high Re range?
The conclusion is that the current interpretation of the PFT is solving for an extremely high Re (Re→∞) condition with an infinitely thin boundary layer (enforced due to the imposed attached circulation to the surface). It should be remembered that, physically, such a thickness decreases as the fluid velocity increases; thus, at this point, it is at the limit due to the Re value! Of course, this cannot be properly considered an inviscid solution, and thus, drag 'force' is absent due to the lack of flow detachment. In potential methods (panel, vortex lattice, or lifting line), a true inviscid result can be obtained through a similar approach to that described in [1, 2], which is the base model for precisely obtaining lift, drag and moment coefficients on shell-bodies [3] (open-source codes are available), and could be extended to thick ones.
*Anyone who still thinks that this is the correct answer is inherently assuming that fluid flow separation is an abrupt (discontinuous; going from a turbulent to a laminar flow regime in an instant!) process triggered by an extremely small change in viscosity value, a non-physical behavior, at least from the observed macroscale scenario.
I invite you to read more related posts on this blog and share your thoughts! Meanwhile, check this short video:
Animation: A short presentation about my research.
"The pattern will practically be the same because, at this Re, the inertial forces dominate over the viscous ones, no matter that fluid viscosity is practically removed, becoming a pure inviscid flow."
ReplyDeleteno, the viscosity isn't removed, otherwise you wouldn't have any separation. That's the similarity between the "current interpretation of PFT", where you have no separation, and the "low Re" flow, in which the viscosity effect is so strong that it inhibits separation (especially with "easy" shapes, like a cylinder section).
You "obtain well-matched results (at least for lift coefficient) for the linear (viscous) region" with PFT because in that range the skin friction contribution is low and there is no separation of the flow, so the Kutta condition/imposing a wake detachment at the trailing edge allows you to get good results with a potential method.
In fact, it is not clear from you papers (and even less from the posts here) if your approach is intended only for fully separated flow, as per the paper's title, or in the whole range of incidences (you show results also for 5°). I definitely wouldn't mind a wake releasing from the whole surface of a fully separated body, but I don't see it working for a partially- or non-separated one; and I wouldn't call it "a new interpretation" of PFT, but more an extension of PFT+wake, which is one of the classical solutions of the PFT limitations (viz. Euler-D'Alembert) paradox.
Sorry, I cannot understand your point. Could you give me your option (a or b), please? As I understand it so far, are you saying that the existence of viscosity leads to fluid separation? If that is the case, then I cannot discuss more on this point because this constitutes the foundation of my development.
DeleteI have heard this argument before. So, where should the maximum viscous effects (including skin friction) be if not in the linear region? I mean, if fluid is 'completely' attached, there exists a maximum interaction between viscous layers, which causes more shear stresses that affect aerodynamic forces, mainly the lift one.
Originally, this method was developed to solve massively separated flows, however, as the abstract of the first paper (The Full Multi-wake VLM) states, the obtained solutions correspond to inviscid ones, which are supposed to exist independently of the AoA, thus a solution for 5 deg. is showing only the inviscid contribution to the lift, which must be shown in order to have a better comprehension of the phenomenon that is intended to be modeled (lift exists despite the lack of viscosity).
It's not viscosity per se, or its higher or lower value, it's the adherence condition: if the flow velocity must be zero at a stationary solid boundary, then an adverse pressure gradient will cause the boundary layer to separate. And I'm definitely not the one saying it.
DeleteWhen you say "the foundation of my development", would that be "viscosity doesn't lead to flow separation"? Note that separation effects aren't always important: for airfoils at low incidence you can ignore separation and still get good results, but information on boundary layer/wake must be provided in some other way (eg. Kutta condition).
Yes, it is often said that skin friction drag is much more important than pressure drag for attached flow on airfoils, but that's because pressure drag is minimal, not because there is "maximum interaction between viscous layers". Also notice that I'm talking about drag, whereas lift is practically due only to the pressure contribution: how could shear stress produce lift if it is a tangential stress, ie. directed along the surface? It would produce negative lift, if any, for an airfoil at incidence.
As for the application range, I suppose it is related to my point about viscosity and separation. However, even assuming that separation can also happen without viscosity (because of instability), how can you model the same wake for AoA 5 as that for AoA 45 (ie. detached all over the body, not the actual convected shape)?
It is not clear to me here the logic behind "the obtained solutions correspond to inviscid ones, which are supposed to exist independently of the AoA", I would appreciate if you could elaborate on that.
—As you know, in the N-S velocity-vorticity formulation, the pressure field is decoupled from the flow solution (which can be reconstructed from the streamline function); thus, from this particular approach, the pressure cannot be considered a cause of separation but the induced velocity field on elements of vorticity. Hence, I prefer to avoid discussing in terms of pressure as a cause of more complex phenomena such as fluid separation, since from this viewpoint it should be treated only as a consequence, not as a cause.
Delete—Yes, viscosity means attachment and no viscosity means detachment in the scope of my proposal; such simple concepts lead to obtaining all the numerical results shown so far.
—Yes, I should be more specific. When I say “viscous effects”, I am referring not only to skin friction but also to the curvilinear streamlines along an airfoil as a cause of viscosity and curvature, which IMO are enforced through the (current) potential flow and invariably have some effect on lift (a detached BL leads to clearly diminishing the lift force, not only affecting the drag one).
Related to this last, the most widespread belief is that in order to generate lift force, it is necessary to have fluid viscosity. Thus, why has the PFT not definitively solved such a fundamental question if we can obtain lift in a “totally inviscid” flow where “all viscous effects are negligible” along its entire application AoA range?
—In the unsteady solution, both convected shapes share the same characteristic pair of wingtip vortices, independently of the AoA. For the steady (straight wakes) case, the detached circulation over the body is only a conceptualization that is valid until around α=20 deg. (experimental inflection point) for the quadrangular flat plate configuration. When I say ‘conceptualization", I mean that is not the actual shape, but fortunately, as occurs between the VLM and the UVLM, the wake roll-up does not affect the circulation/vorticity field, and in consequence, the aerodynamic coefficients remain the same. That is the conclusion of the second paper (The Unsteady…).
—In order to answer the last request, I copy a paragraph from my first paper (The FMVLM: a detached…):
From the numerical (inviscid potential) point of view, the main hypothesis behind the previous proposal comes from the fact that, ideally, the detached flow must exist at the entire plate, independently of its AoA (due to their sharp edges and zero-viscosity assumption). It means that even for a flat plate at extremely low AoA (e.g., α=1 deg.), the detached flow is present on the entire plate, including their external edges (even along its leading edge; LE). Then, the intensity of such detached wakes must be determined by the instantaneous circulation strength calculated on the surface. In this particular case, the flow detachment could be negligible but nevertheless exists. On the other hand, from the physical (viscous) point of view, such a detached flow assumption should only be valid for the AoA range where viscous effects could be considered negligible (after the viscous-inviscid transition range; moderate-high AoA).
Thanks for maintaining the interest.
Thank you for your answer.
DeleteI'll reply first to the last points, about the application of your method and code. I have to admit that while discussing here I somewhat lost sight of the fact that you are dealing with flat plates and very thin wings with very low AR and/or triangular planform and sharp leading edge. With this in mind, I can understand why you would say that the flow is detached also at low incidences. Also, if I understand correctly, the inaccuracy in predicting the behaviour after stall (here, too, I'm too used to associate stall with flow separation) can be taken care of with some modifications concerning the LEV.
This being said, going back to the discussion on the more fundamental topics: I do agree that placing cause and effect labels isn't desirable (although not on the grounds of any formulation, in fact vorticity wouldn't be a cause either), but in the case of separation it should be intended in the sense that the boundary layer can't be attached and yet decelerate/go against the pressure gradient.
In this, the role of the adherence condition (introduced by viscosity) is essential. Without this condition, there would be no reason for the flow to separate, as it happens in the classical description of PFT. The condition is also essential to the generation of vorticity, which is ultimately related to the wake and the generation of lift and drag, overcoming the Euler-D'Alembert paradox.
In this sense viscosity is related to separation. The fact that at very low Reynolds the flow looks more and more attached is only a matter of magnitude of effects: keep in mind also that the Reynolds number is not a function of viscosity only. If flows at low Re look attached is because the conditions (both velocity and viscosity magnitudes) don't lead to separation. The difference for inviscid fluids is that, no matter the value of the velocity, there can't be any separation.
I'm not sure I've understood the role you attribute to viscosity in producing the curvature of the streamlines.
It is true that potential methods are able to describe the lift of airfoils pretty well in the linear regime, but that's only when they include some hypothesis coming from the theory of viscous fluids, such as the boundary layer/wake, the Kutta condition, etc. This doesn't make PFT any more complete by itself, nor should it. It just shows that the theory isn't enough to describe viscous fluids, but that was known from the beginning (as it includes no viscosity).
Batchelor gives an overview of this issues at the very beginning of chapter 6 of its treatise, including remarks on the limit as Re goes to infinity.
Yes, in fact, for the post-stall model (detachment model 2 in VortoNeX code), the circulation along the plate's LE is inverted, becoming a typical Kutta separation line. This leads to the characteristic abrupt loss in lift (through Kutta-Zhukovski calculation) according to experimental data, and it continues decreasing as the AoA increases (turbulence is not implemented yet), being zero for the perpendicular flow (AoA=90 deg.) case.
DeleteIMO, the cause of all effects, including separation, is velocity, which causes vorticity and pressure fields (velocity-vorticity and velocity-pressure formulations).
For the remaining aspects, it is clear that we both have completely opposite points of view, among them on the role of fluid viscosity.
I really appreciate your time and interest in discussing such complex concepts.
Best regards!
Not sure how velocity can be the cause of everything, if not in the sense that, if the velocity field is prescribed, then the other quantities can be computed (which wouldn't be true in the general case).
ReplyDeleteI see that our views on viscosity are opposite, and, instead of appealing to the authority (after all, I didn't come up with those concepts by myself), I'd like to ask you if you could please provide a description of the process of fluid separation given your views on viscosity (i.e. with no need for the adherence condition). In other words, could you try and convince me with detailed reasoning? (also related to the comment on your latest post ) I promise I'll try to be as open-minded as possible.
"Lift is generated by the difference in velocity between the solid object and the fluid. There must be motion between the object and the fluid: no motion, no lift."
Delete"For lift to be generated, the solid body must be in CONTACT with the fluid: no fluid, no lift." (a free-slip BC, as supposed in the PFT, cannot cause lift!).
These two ideas can be found here: https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/what-is-lift/#how-is-lift-generated
I understand that such definitions are for the public without any knowledge of aerodynamics, but precisely because of that, the most simple explanation should be interpreted as the compendium of all concepts behind such a phenomenon, which in the end should be considered totally valid, at least so far.
It is not so easy to try to explain in this format, among all, because I should explain viscosity since its microstructure (gaps between molecules, interaction between layers, etc.), and to be true, I am not interested in doing so far since it is not as easy as it appears (probably it will need a paper...); I prefer to go for the way of practical results and let each one take what serves them.
As you know, my proposal has only been applied to shell-bodies so far (for me, it is OK because I try to solve parachute aerodynamics), but what happens when someone applies it to almost-zero-thick ones? Imagine a stalled flat plate (e.g., AoA=60 deg.), where on the upper side the inviscid flow is massively detached from all its edges, while on the lower side it is 'completely' attached. In such a case, the lift force is being generated by both the detached and attached flow contributions to each side of the plate, thus creating a mix of viscous and inviscid effects where both viewpoints can coexist! Of course, this is only speculation; let's see what happens.
The most simple explanation should be preferred, but only if the explanation is sufficient: saying that there should be relative motion and the body should be in contact with the fluid (note that it doesn't say anything about the type of contact, it only excludes action-at-a-distance) are definitely necessary conditions, but just barely enough to start defining aerodynamic lift. Totally valid, yes, but I don't see how they are relevant here.
DeleteYou don't have to explain anything about the molecular aspect of viscosity, as the microscopic aspects are not important to continuum mechanics.
I'd say it's more important to specify the general concepts before: for example, I see that you use "viscous" and "inviscid" in a different way than their usual meaning. I understand that for you "viscous" means "attached" and "inviscid" means "detached", but if you switch from the one to the other it makes it really difficult to follow you.
I don't know what you mean with "shell-body", but the same reasoning you use for the 60 deg plate one can be applied for every kind of body, as you will have at least some portion over which the flow is attached. What viewpoint should be followed then? And, in fact, how could they even coexist?
Then, the issue remains about the mechanism of separation and how viscosity would keep the flow attached.
And finally, the issue of how all that would fit in the formulation of PFT, differently than the usual "PFT + wake + kutta condition".
IMO, they are relevant here because I am trying to answer you why "velocity is the cause of everything".
DeleteContact means physical interaction between a solid wall and the fluid at the interface. A free-slip (free-stream velocity) BC cannot allow any interaction between them, but only the no-slip (zero velocity at the surface) one.
Viscosity MUST BE understood from a microscale viewpoint, since gaps (void spaces) between molecules cause the 'viscous degree' of a fluid. Otherwise, its adherence to a curved surface cannot be physically justified. On the other side, an inviscid fluid (flow) where hypothetically there is no space to fill it flows without any resistance between its layers, and even the first one cannot be attached to a (perfectly smoothed) surface; in both cases, I am referring after a first infinitesimal time step but not to the fully developed state, thus justification as "adverse pressure gradient" cannot be used to try to justify fluid separation. Sorry, but I cannot explain it in a more detailed way, at least not in this format.
By the way, at this point, I noticed that you are arguing things that have already been exposed or can be inferred from my papers...For me, it could be almost impossible to try to explain a thing if we are not 'talking in the same language'. If you want to continue with the discussion about the implementation and its results, please write me via e-mail.
Thanks for your time.