
#55
Mar2905, 04:10 AM

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Andrew Mason (and in earlier thread, Jeff Reid) has pointed out that if the wing moves forwards, then a region behind&above it becomes evacuated, and that this indubitably occurring process must have some relation to lift.
It turns out that the evacuation (or, rather evacuation rate) is not as such directly responsible for lift; rather, it is the completely different response to such evacuation a viscous fluid displays (compared to the response of an inviscid fluid) which generates the actual lift. We need therefore to study in detail the inviscid fluid's response to an evacuation rate in order to appreciate the role of viscosity. Let us work within the ground frame, with both the wing and the air initially at rest. Also, I will solely concern myself with the development at the trailing edge; let the underside be horizontal, and the upper side of wing curved. Now, give the wing an acceleration (or, if you like, a jump velocity). In order to illustrate the evacuation rate, let us draw a following picture: Draw the "previous" curve the upper side inhabited. At the bottom, that is the position where the trailing edge was situated, draw a small horizontal segment to the trailing edge's new position, and draw the upper side where it now is. Thus, we have drawn an evacuated region, which is bounded below by the horizontal segment, and whose sides are the curved outlines of the upper side of the wing. Since the region is evacuated, the fluid elements adjoining it, will experience a net pressure force from the ambient fluid so that they rush into the evacuated zone. Now, pay attention to the fluid element directly beneath the HORIZONTAL line segment. Clearly, this will be accelerated UPWARDS into the evacuated zone, and will, in fact, hug the upper side as it speeds onwards. That is, a BACKFLOW is created along the upper side, and a stagnation point will develop somewhere on the upper side, when the back&upflowing fluid collides with downrushing fluid. This can then be negotiated as follows: The uprushing fluid bounces through a 180 degrees turn, i.e, twisting its velocity to gain the same direction as the rest of the fluid. Note however, how this is contrasted with the image of the flow given when the Kutta condition holds: There, the stagnation point was firmly fixed at the trailing edge, but here, the stagnation point might well be situated somewhere on the top side (the strong pressure there should clearly reduce the lift). We can also, of course, regard the upflowing fluid to generate counteracting circulation, and hence, liftreduction Now, before I proceed further on the inviscid theory, we can see the role of viscosity clearer: Take a pencil and thicken the actual upper wing somewhat, illustrating a boundary layer which remains firmly attached to the wing. Whereas the inrush of fluid into the evacuated region through the curved side of the wing's previous position is unaffected by that strip, not so at all with the uprush of fluid through the horizontal segment! On that fluid piece, there will be a strong resistive force acting upon it from the boundary layer. Thus, a viscous fluid actually favours downrush into the evacuated region above uprush hugging the airfoil.. I'll proceed further sometime later.. 



#56
Mar3005, 01:43 AM

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#57
Mar3005, 04:51 AM

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As to potential theory:
For all FINITE times (i.e, in the timedependent problem), the velocity at the trailing edge is finite. But this does not mean at all that when time has gone to infinity, (and the stationary picture in the wing's rest frame has developed) that the velocity must be finite there. In fact, it isn't. This highlights yet another mathematical reformulation of Kutta's condition : namely, that the velocity at the trailing edge must be finite. Only a single, nonzero circulation value is able to achieve this. An unnecessary note perhaps: D'Alembert's paradox pertains to the stationary, steady motion case. That is, potential theory certainly predicts forces to act upon an object in the nonstationary case, and those are in essence, the forces needed to accelerate fluid volumes with mass. For the general, nonstationary case, these inertial forces tends to swamp the effect of the viscous forces, so that potential theory remains very useful in many timedependent problems (but not in the evolution of flight..) As for the viscous case: Just to clarify, I do not mean that right from the start, there will be no upflow at all. Rather, there will be some upflow**, but that upflow doesn't carry the amount of momentum which would have been present in the inviscid case, and thus, the fluid is enabled to gradually shed it off in the form of vortices (see russ' excellent link on this process). Gradually, therefore, the stagnation point will be pushed downwards to the trailing edge (i.e, the establishment of kutta's "condition"). Thus we see that it is precisely BECAUSE viscous forces are dissipative that flight occurs: It is essentially the stronger dissipation of backflow than down/inflow which tilts the balance in favour of liftgeneration. That is, flight is the effect of a necessarily skewed spatial distribution of dissipation. **: Not of course, AT the actual surface, but (arbitrarily) close to it, inside the inner part of the boundary layer). Note that, mathematically speaking, it is that boundary condition we have to discard in the inviscid theory (no tangential velocity) which saves us. Thus, flight generation is really a striking illustration of a singular perturbation theory: If we try to only use the "outer" solution of NavierStokes (i.e, the solution of the Euler equations), our problem collapses into the evolution of D'Alembert's paradox. However thin, the "inner" solution (i.e, NavierStokes in the boundary layer) cannot be neglected if we want a realistic solution of the problem. Note: Just a slight correction to what you said: It is NOT necessary to take into account the effect of the boundary layer in the MAINTENANCE of the stagnation point at the trailing edge. Once enough momentum has been imparted to the fluid, and the unequal pressure distribution has been developed, we have a totally different inviscid picture than when we started with everything at rest and uniform pressure: Now, if we regard the situation from the ground frame, the fluid is already rushing down. If we therefore look at the evacuation picture again, that downrush is just sufficient to prevent any net upflow, i.e, the stagnation point reestablishes itself at the actual trailing edge. The inviscid fluid is therefore able to maintain flight. Let us analyze this new inviscid situation further. Suppose that we have gained a stationary flight situation, and proceeds to tilt the wing, maintaining its march velocity. We therefore enter a new, nonstationary phase; what circulation level should we expect to occur once stationary conditions becomes reestablished? When regarding the fluid as inviscid, but with circulation, we ought to expect from Kelvin's theorem that the circulation will remain CONSTANT. This is certainly true for the circulation on closed, material curves in the fluid; it necessarily remains so for the circulation about the wing in so far as it is correct to assume that the wing itself constitues a closed, MATERIAL curve for the fluid. Let us suppose it is..(there remains a tiny doubt, however: If the initial material curve gets kinked or something, can't it happen that the wing might pierce it somehow? I'm not entirely sure on this..) Leaving tiny doubts aside then, we should expect that if you tilt a wing (changing its geometry from a dynamical perspective) in a liftsustaining inviscid fluid, then that tilting wouldn't have any effect whatsoever on the lift which would ensue once stationary conditions reestablishes itself. That is, the lift would be the same as it were initially. Hence, tilting this wing would typically involve the evolution of a VIOLATION of Kutta's condition, we will NOT be able to gain the actual liftchange you would experience in a real, viscous fluid (where the Kutta condition will reestablish itself for the new geometry). On further reflection, that tiny doubt can be reduced into something minuscule: Since we know that the initial velocity distribution can be described by potential theory (essentially, a translatory field plus a point vortex distribution with the singularities hidden away inside the wing), tilting the wing should mathematically induce a redistribution of the point vortices inside the wing so that the boundary conditions remain fulfilled, and the net circulation kept constant during the timedependent phase. Since the potential solution is evidently a solution of the Euler equations, whatever doubt remains, is whether or not the Euler equations specify a unique solution or not.. 



#58
Mar3005, 07:46 AM

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The pilot ! cheers, Patrick. 



#59
Mar3005, 07:50 AM

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#60
Mar3005, 12:32 PM

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I must say that I find Eberhardt's "explanation" rather worthless.
Air is not actively pulled down by some Coanda hand; once a pressure gradient forms, air is accelerated in the direction of lowest pressure, whether or not that means that a given fluid element's path merely becomes curved or if the path remains straightlined (with acceleration along that). Another weakness is their confusion about Newton's laws. Let us see in some detail how a GLOBAL analysis should be done (in the stationary case): 1. Assume that the wing's rest frame is an INERTIAL frame. So, if there is a net lift force from the wing, there exists an independent external force acting upon the wing so that the velocity remains constant. (gravity is a good example, we will assume this in the following) 2. Let us describe the problem in the wing's rest frame. Let us surround the wing "W" by an annular control volume "V" of fluid, let for example the outer boundary of the annulus be a simple square S. Then, Newton's 2.law expressed for the fluid momentarily enclosed in V: [tex]\vec{P}+\vec{R}+\vec{W}=\int_{S+W}\rho\vec{v}\vec{v}\cdot\vec{n}dS[/tex] Where: a) [tex]\vec{W}=\int_{V}\rho{\vec{g}}dV[/tex] is the weight of the air in the control volume b)[tex]\vec{P}[/tex] is the surface forces acting upon S from the ambient air (when neglecting viscous forces, that is the net pressure force) c) [tex]\vec{R}[/tex] the force from the wing onto the fluid contained in "V" d) [tex]\int_{S+W}\rho\vec{v}\vec{v}\cdot\vec{n}dS=\int_{S}\rho\vec{v}\vec{v}\c dot\vec{n}dS[/tex] the net momentum flux through the boundaries of V; since [tex]\vec{v}\cdot\vec{n}=0[/tex] on W, it isd only through S there is a momentum flux. Now, the lift L is, by Newton's 3.law equal to the negative vertical component of [tex]\vec{R}[/tex] that is, we have: [tex]L=\vec{P}\cdot\vec{j}M_{air}g\int_{S}\rho{v}\vec{v}\cdot\vec{n}dS[/tex] where "v" is the vertical velocity component. Thus, only if we can disregard the other force terms acting upon our "V", can we state: [tex]L\approx\int_{S}\rho{v}\vec{v}\cdot\vec{n}dS[/tex] This will usually be the case if we let "V" be big enough. Hence, my use of the word "GLOBAL". Note, however, that there is full use of Newton's 2.law here, but FOR THE FLUID! By invoking Newton's 3.law, we find the force on THE WING. Eberhardt's miserable use of Newton's 1.law is best left uncommented.. 



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Mar3105, 02:10 AM

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#62
Mar3105, 02:29 AM

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#63
Mar3105, 03:52 AM

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There is a good topological argument for why a material curve should not usually get broken up:
Consider the positions of a material curve at times "t" and "t+dt". Since the constituent particles have finite velocities, we should expect that we can map the "t" curve onto the "t+dt" curve through a CONTINUOUS transformation (that is, given "sufficient" closeness of points on the "t" curve, their images will be satisfactorily close on the "t+dt"curve.) But, can a continuous transformation effect the radical topological change from "closed" to "not closed" (think of the famous rubber band analogy of topology)? This seems very unlikely; I am in fact, quite convinced it is untrue. From what I can see, such a pathology might only occur at points where such a curve gets tangentially kinked, or other such effects which signal a form of breakdown. As you readily can see, topology is NOT a strong side of mine.. However, from what I can see, it boils down to the following issue: Given an arbitrary initial velocity distribution, is there always a unique solution to IBVproblem posed by the Euler equations? I haven't studied uniqueness conditions sufficiently to give a rigourous proof either way, but the lack of uniqueness for the Euler equations would astound me.. I very much suspect that my "doubt" is just yet another lamentable result of my general ignorance.. 



#64
Mar3105, 06:23 AM

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I find it important to do some more Anderson&Eberhardt bashing.
This is a very revealing quote: Since these persons are reputedly employed at Fermi's National Accelerator Laboratory and still suffer from deep misunderstandings, I find it in order to review a few basics on pressure: Pressure at a "point" is a measure of the typical amount and intensity of molecular collisions at that "point" The "point" must be understood to be a tiny spatial region which is, however, so large that to speak of averaged quantities within that region (typical examples: velocity, temperature, density, pressure) is useful. If the region is too small, the merest random drift of molecules into that region would provide wild oscillations in these averages over time; that is, these averaged quantities would essentially lose their usefulness. As long as our region is big enough to contain gazillions of molecules, statistical arguments leads us to expect that such wild fluctuations in measured quantities die out. The region is incredibly tiny still, if I remember correctly, the typical linear dimension of such an "element region" for a notso dilute gas is about [tex]10^{7}m[/tex] (For liquids, like water, I think you can squeeze the linear dimension down at least a couple of orders of magnitude). Now, the pressure is given as a scalar, and the pressure force onto a surface at our "point" is in the "colliding" direction, i.e, directed along the inwards normal of the surface. Furthermore, and this is very important: Since our "point" really contains gazillions of molecules, there should within it be NO PREFERRED DIRECTIONS for the momentum transfers involved in the collisions. That is, the pressure force at "point" is equally strong in any direction. Mathematically, this means that the pressure at a point is not a function of the direction of the contact surface normal. Let us now consider a plate which is originally in contact&rest with a fluid (on one side of the plate, for simplicity). We keep the fluid inviscid, so that the "pure" pressure dynamics comes clearer into focus. Now, give the plate a jump velocity V directed away from the fluid (it so happens that the argument is easier to visualize in this manner, it is, of course, equally valid when speaking of a finite acceleration and its effect over time). Now, the pressure force on the plate at a given instant is evidently the accumulated effect of gazillions of molecules striking it at that moment. The molecules have a random velocity distribution; this also holds for that subset of molecules who happen to have a "colliding" velocity, i.e, those which are actually going to hit the plate. Let us see what happens in the jump velocity case (with some time gone..): Can we really say that suddenly there has appeared a tiny strip of complete vacuum between the plate and the fluid? Not really. Consider that subset of particles close to the plate which initially had a "colliding" velocity (a lot) bigger than "V". Clearly, these must be regarded to still strike the plate, but instead of say with their original striking velocity [tex]V_{0}[/tex] they do so with a new striking velocity [tex]V_{0}V[/tex] Thus, the only molecules which can be said to have been removed from the plate (relative to the case where the plate where at rest) are those whose original collision velocities satisfied the inequality [tex]0<V_{0}<V[/tex] Thus, unless V is very large, we cannot really expect a measurable density reduction at the plate. Since, therefore, in the new position there are still gazillions of molecules who have "followed" the plate, we have in reality established the boundary condition for the macroscopic velocity field, i.e, that at all times, the normal velocity of the fluid equals the normal velocity of the plate. The only dynamical feature we have gained, is a (significant) pressure DROP at the plate, which clearly follows from the argument above. (Since the total kinetic energy of a striking molecule ought to be the same as a nonstriking one, it follows that the striking molecules have a correspondingly less "tangential" velocity to start with, i.e, the actual amount of momentum transfers in local collisions remains nondirectional) Alternatively, we may say that we will get that pressure drop which is sufficient to accelerate the fluid so that the boundary condition of equal normal velocities is fulfilled.. Thus, there is absolutely no mystery involved in why a fluid tends to remain in contact with a surface, which Eberhardt&Anderson seems to think. In particular, we don't need to pose the existence of some ghostly hand reaching up from the surface to grab air molecules. An inviscid fluid is equally capable to fill out voids as a viscous fluid is; the pertinent feature is how either fluid goes about doing just that.. As we have seen, a viscous fluid prefers downrush about the wing, the inviscid fluid is not so picky. 



#65
Mar3105, 09:12 AM

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The one and only reason why airplanes fly is the fact the the orientation and shape of the wing in combination with the velocity relative to the air leads to more molecules hitting the wing from below than from above.
Consider a simple plane surface: if it is resting relatively to the air, the molecules hit both sides of the surface with the same rate and speed according to their density and thermal speed, i.e. there is no resultant net force; now consider the same surface moving such that it is orientated at a certain angle to its velocity vector. If you consider the air as a strictly inviscid medium (i.e. the molecules interact only with the surface but not with each other), then the surface facing into the relative airstream experiences a higher rate and speed of molecules and the other surface a lower (simply by the virtue of the velocity of the surface adding to or subtracting from the average thermal speed of the air molecules). This results in a corresponding net force proportional to cos(alpha) (where alpha is the angle between the normal of the surface and the airstream), which can then be decomposed into the horizontal component i.e. the drag (~ cos^2(alpha)) and the vertical component i.e. the lift (~sin(alpha)*cos(alpha)). Everything else like the airflow pattern around the wing etc. is only a secondary consequence of this due to the actual viscosity of the air, i.e. hydrodynamics may explain what effect an object moving through air has on the latter, but it does not actually give the causal reason why an airplane flies. 



#66
Mar3105, 09:14 AM

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AM 



#67
Mar3105, 11:35 AM

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What they are really saying, is that viscous normal forces are comparable to pressure forces at high Reynolds numbers. Nothing of what they present suggests that this is the case. 



#68
Mar3105, 11:43 AM

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A flat plate initially at rest (no circulation to begin with) in an inviscid fluid, and which then started to move, would develop D'Alembert's paradox once conditions in the plate's rest frame could be called stationary. 



#69
Mar3105, 12:28 PM

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#70
Mar3105, 01:32 PM

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isnt it so that airplanes can fly upside down?????
then this stuff with the wing pushing air down doesnt work really... 



#71
Mar3105, 01:53 PM

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That this tilted winggeometry is liftsustaining, is encapsulated in the fact that although somewhat inverted, the wing's EFFECTIVE angle of attack remains positive.. 



#72
Mar3105, 01:58 PM

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