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Richard Miller's Home Page

WING LIFT & FLIGHT

~~WING~~FLIGHT~~AIRFOIL~~WINGTIP~~VORTECES~~

An Apology of Sorts

~~~ This page has been abandoned, sort of. It was meant, in any event, as prologue to the more immediate subject of conserving the losses due to the up-flow around the ends of the wing, and I had arrived at the threshold of that subject on this page before events - the flu, inclement weather, computer down time - intervened, followed by notice from dcregistry, the sponsor by whose courtesy this page exists, that a new version, 2.0, much improved, was availablle and that page editors, such as I, or me, should begin to migrate from the one to the other.

~~~ Meanwhile events increasingly turned by attention to the wing tip, its configuration, actual and possible, and the process I call emarginated dispersion, the most familiar expression of which are the splayed primary feathers of the land soaring birds. The case is this: For every real wing, which is to say wing of finite span, there occurs, in flight, simultaneously, the co-processes of loading and unloading. The first is the generation of dynamic pressures of opposite polarities, quantified as the coefficient of lift, by which sustentation is achieved; the second the equalization, at the tip, of those pressures by a flow from the lower (+) to the upper (-) surface. Thus vorteces and contrails and more than likely a substantial energy loss to the system.

~~~ The loading and the unloading of the wing in flight are both 3-dimensional processes. The wing operates on a mass of air that has, in addition to the two dimensions corresponding to its surface, a third, vertical, proportional to the chord (and the lift coefficient) that varies with aircraft size, weight and speed. It would seem superfluous, even gratuitous to mention something so seemingly obvious were it not that Simon Newcomb, the great nay-sayer of powered man flight, failed to grasp this fact, and said many foolish things in consequence, and that the assumption of two-dimensional wing loading is perpetuated in contemporary articles on loading and scale effect.

~~~The unloading of the wing occurs at the extremities of the block of air processed by the wing. The upwash around the wing tip occurs in what can be visualized as a 3-dimensional window at, and slightly overlapping the end of the wing. The dimensions of this window are proportional to the chord (squared) and to the lift coefficient, the latter which determines the magnitude, in depth and width, thru which the flow takes place.

~~~What one brings away from this that may be profitably used is the fact that, as wing loading varies with the third power of the span, so the process of unloading varies with the third power of the (effective) tip chord. It also makes clear the rationalization for emarginated dispersion. Consider that the ratio of the sums representing the flow potential around a truncacted tip with a chord of, say, 4, and that of 4 discrete elements, each with a chord of 1, is 16:1. The fraction is 6.25, and one to conjured with.

~~~Between the real, the loss equal to 64 incurred by the solid wing tip, and that of 4 with the brachiated tip, lies the shadow -the fragmentation of the wing, the incorporation of those slots by which one gets from a unitary surface to one that is segmented. Some loss must be anticipated and the question is how to minimize it.

~~~In all previous attempts to emulate Nature's model of the splayed wing feathers, and it dates back to almost the beginings of powered flight, the experimentors have retaied (at least so far as I have been able to determine, the basic configuration common to the birds, a cascade in which the leading element is in the highest position, those following in descending order; for purposes of definition, a descending or negative echelon .

~~~An analysis of the basic structural problem, that of segmenting the wing tip, and with it the specific characteristics of feathers, leads to the conclusion that there appears to be no practical alternative, insofar as the bird's wing is concerned, to the negative echelon. Whether this is the superior configuration, however, is open to question, as the alignment of the slots does little to impede the flow upward and rearward as it moves around the end of the wing. A positive echelon, on the other hand, directs the freestream flow through the slots between the vanes in such a mannner as to inhibit or suppress the tendency of the disparate pressures to mix, or to keep them from doing so until they are able to do so with the minimum possible fuss.

A Personal Note In the very long while of my involvement with classical mechanical theory, and with the losses due to flow at the wing tip in particular, I have been struck, and with increasing force, with the propensity for abstraction on the part of those who have written the surviving texts. Their realm is mathematics and their preoccupation with the manipulation of numbers representing supposedly real entities. Excursions into the world from which these entities are derived are brief, only long enough to abstract yet another number to be added to the calculations.

~~~This has been a little difficult for me to come to terms with due to my own propensity, which is to endeavor to follow, say, a molecule, attempting to determine the forces to which it is subject and thereby its likely path as it approaches and passes the wing or the wing tip. Invariably I have been mystified by the results of using this approach on existing tip configurations. These I would tend to term formulaic, the abstractions derived from the real world, dealt with mathematically, then fed back to it in what seems like a good idea at the time. The ones I have examined simply dont make sense to me in terms of, for example, the molecule posed just ahead and slightly below the tip, what I consider the hot point in terms of impressed forces and the inducements of change direction and velocity with extremem rapidity. If any molecule must be met head on it is this one, but typically it is let pass with at most a nudge or some minor redirection that has no more than a minimal effect on the total tip flow. End of Jeremiad.

~~~And beginning of the cri de coeur. Why, then, has the aviation community slept through all this? Why this continued ignorance of the proven potential of wingtip vanes? Why the use of a vastly inferior alternative, and sailplane aspect ratios of 30 and 40:1 for the sole purpose of reducing tip losses? Hmm? It is scandalous, and a wake-up call is needed, which is what this is all about.

Original Page Text

Prologue

~~~ This page, in examining the subject of wings and flight, will a address a number of curiossities, some physical, but mainly those that are the product of human ingenuity and imagination, a category Euler called fictive hypotheses. First among these is the momentum theory of lift, theheart of which is an exchange of that momentum generated by the convergence of the wing and the relative airstream. This derives ultimately from Newton. It dates back to the Principia and bears the master's imprimatur in the invocation of his Third Law, that of equal and opposite reaction.

~~~ If the reader is so inclined he, or she, may wish to draw a diagram to the following specifications. First, a simple airfoil at a positive angle of attack. Around it, a circle/sphere representing the are in which the redirection of forces takes place: The Field of Transaction. From left to right, horizontally, a broad arrow , its tip or head penetrating that circle; and from below to above, vertically, coincident tothe gravity vector, another broad arrow, its tip contingent to the upper perimeter of that confine.

~~~The horizontal arrow represents the foce inherent in the accelerated mass of the airfoil vis-a-vis the atmospheric freestream; it is labelled F/s-1, system force one. The vertical arrow represents lift; it is labelled F/s-2, system force two.

(MORE TO COME BEFORE THIS SEGMENT IS FINISHED)

~~~The phenomenon of flight is defined as the sustentation of a plane, or wing, by the difference in potential between a greater dynamic pressure bearing on the bottom surface and a lesser dynamic pressure on the upper surface. Near the extremities of the wing the course of least resistance for these pressures is toward the tip on the under side of the wing and away from it on the upper side, which results in a flow at the tip in which energy is dissipated and lost to the system.

~The fact of this loss has been recognized for about a century, and for the better part of that time, possibly as a concomitant to the confusion attendant on the preposterous theory advanced to account for it, substantially less attention has been paid to the issue of tip flow losses than to other comparable problems in aircraft design. This page is in part an effort to redress that oversight.

~~The issue can be summed up in two related imperatives, each with its negative corollary. These are -

~~~1/ ~The total force represented by the convergence of the airfoil and the airstream - the masses of the components x their accelerations, save only for that minor portion lost in the process of conversion- must be transferred to the wing, half as momentum extracted, half as most momentum transmitted, from/to the upper and lower surfaces respectively; and there is no residual force nor momentum to manifest in any other form, and certainly not beyond the trailing edge of the wing.

~~~2/ ~The sustaining forces, which is to say the momentum involved in such exchanges, must operate at right angles, thus vertically, upward, to the free-stream flow; and as a consequence trigonometric functions have no place in in the determination of these phenomena beyond representing the slight rearward inclination of the lift resultant due to drag.

~Neither the reputed intellectual prowess of the brain surgeon, nor that of a rocket scientist is required to acknowledge the validity of these statements - they being nothing more than a representations of the equivalence of the input and output of precisely defined quantities - nor that, together, they completely vitiate the allied concepts of a direct-contact transmission of momentum bewteen the wing and the airstream, of a planing force, and of an induced downwash. Thus it follows, with the same provisions, that an alternative explanation is required.

I~The Received Wisdom

~The basic elements that comprise the theory of lift in its contemporary form were present in the principles
of classical mechanics as formulated by Sir Isaac Newton. These have been summed up by
Theodore vonKarman in the initial chapter of his work Aerodynamics as follows:

~~~The formula or law known generally as Newton's sine-square law of air resistance refers to a force acting on an inclined flat plate exposed to a uniform airstream... In the 34th Proposition of his Principia he calculated the total force acting on the surface of bodies by computing and adding the forces caused by the impact of air particles, which supposedly move in a straight line until they hit the surface... It is supposed that after the impact the particles follow the direction of the plate. Then one obtains the change in momentum of the fluid mass hitting the plate in unit time by multiplying this mass by the velocity component created by the impact."

~The system of fluid mechanics to which this description applies was developed within the constraints of a strict determinism, or more exactly of a mentality confined by such constraints. Its givens, ponderable bodies and the transmission of momentum between such bodies by direct contact, are unspecified inferences that are attested to by a pattern of resultant forces that do not differ substantially from what one would anticipate in the movement of billiard balls.

~It is a further inference that there is, or was no alternative given to these hypotheses, nor would there be the likelihood of one presenting itself until the advent of field theory in the latter half of the 19th century. When the theory of lift as we know it today was formulated in the early years of the present century it was under the abiding influence of Newton and the concepts of classical mechanics, and the results were basically an elaboration of the model derived from the Principia. ...

O+O+O+O+O+O+O

~The wing does not function as a turning vane, nor does it derive any of its lifting force as a consequence of air being decelerated doowneard, nor does the downwash contribute in any direct way to the process of sustentation.

~The total force represented by the mass of the wing accelerated in relation to the kinetic (free-stream) flow, less that fraction attributable to drag of one sort or another, must be transmitted to the airfoil as lift. The idea that it is possible to achieve sustentation with some fraction of that force, as, in climbing, having one's foot slip backward half the length of each stemp forward, so defies common sense as to be innane, and the invocation of the 3rd law of motion to justify it is fatuous.

~ In almost every significant particular there is a failure of the circulation-downwash-induced-drag theory to accord with observation and measured values. To the initial objection that the sine-square formulation yields too small a force, and by a significant margin, there is besides no reasonable correspondence between the concept of the wing sustained by a planing force and the actual chordwise distribution of pressure across the lower surface and the consequent location of the aerodynamic center. To this may be added the fact that the role of the upper surface of the wing, to which the major burden of the work is shifted as a result, is treated in a superficial manner.

II~Lift as a Field Phenomenon

~In the relative motion between any fluid and the surface of a solid body there is not, nor
can there be any direct transmission of momentum. In all cases a field mediates. The
forces within the free-stream flow, subjected to any change in velocity, give rise to a
dynamic field within which forces are intensified by the accretion of momentum.

~At all times and in all aspects these fields mediate the flow and the extent and nature of
impressed forces. All instances of the transfer of momentum with a fluid are compound:
By a surface through the mediation of a field to a flow, as when the life field, entrained
by the wing, induces the lift wave; By a flow through the mediation of a field to a
surface, as when the dynamic forces within the lift wave are transmitted to the airfoil.

The Field Defined

~Field is defined for the present circumstances as the continuous distribution of some quantity or quality, or, negatively, as the absence of such specified quantities or qualities. As they relate to fluid dynamics they are of two sorts: Kinetic, that of the undisturbed free stream, in which all components are identical and have vectors of equal magnitude and constant direction, in which all defining streamlines are equidistant, and in which there are no gradients; and dynamic fields derived from changes of velocity within the kinetic in which streamlines diverge from uniform spacing and across which streamlines there are field gradients.

~The formation of a dynamic field within a kinetic flow is a practical representation of Bernoulli's insight into conservation theory, that energy is constant along streamlines. In geometricla terms it consists of two vectors representing reciprocal forces on transverse axes. For every element of deceleration, >V, within the kinetic flow there is a perpendicular correlate of an internal pressure, ~This is precisely

III~The Enigma of Lift

THE DIAGRAM

~~~1/ Draw a 4-in. datum line. Label the left end (a), the mid-point (b) and the right end (c).
~~~2/ Draw a sine curve with an amplitude of one inch beginning at (a) and ending at (c). Place a vertical line through (b) and label the upper intersection (d). Place a point (representing the provisional center of pressure) 1/4-in. below point (d).
~~~3/ Devise an airfoil with a chord of one inch and a positive angle of attack. On the center line of the airfoil, 1/4-in. from the leading edge, place a point representing the aerodynamic center.
~~~4/ Superpose the aerodynamic center of the airfoil on that of the wave.

~~~5/ On vertical lines thru (a) and (c) place points 1/2-in. above and 1/2-in. below the datum. Label the upper points (e) and (e'), the lower ones (f) and (f")

~~~6/ With a colored pen draw these three lines: a sine curve from (e) thru (d) to (e'); a second sine curve from (a) to and thru the airfoil, following the mid-line, and from the trailing edge to (c), and a straight line from (f) to (f').

~~~7/ The upper area bounded by colored lines diminishes in depth from the left margin of the diagram to the mid-point, then diminishes to its normal vertical dimension. The inverse is true for the lower area.

~~~8/ The center line

IV~Emarginated Dispersion

~The losses consequent to the production of lift accounted for in classical aerodynamic theory are in fact the dissipation of system energy as a result of airflow at the ends of the wing. The locus of this flow is a three-dimensional window at, and overlapping the the wing tip. This window has one relatively fixed dimension, the longitudinal, that of the wing chord in the area of maximum flow, Its lateral dimension, the distance in from the tip the spanwise flow effectively begins, and to the extent it overlaps the end of the wing, as well as its depth, are direct concomitants of the potential across the wing, which is to say the coefficient of lift. The loss from tip flow is thus proportional to the square of the chord multiplied by the lift coefficient, which, as chance would have it, is identical to the classical equation for induced drag once its redundancies have been excised.

SUMMATION

~The wing is sustained by the potential between the positive and negative dynamic pressures on its under and upper surfaces.
~This potential resolves its as a flow around the ends of the wing when that is the path of least resistance.
~This flow occurs thru a window the volume of which varies as the 3rd power of the tip chord.
~The calculations - losses reduced to the third power of the number of emarginated segments - are encouraging in the highest degree, and provide a clear rational for the principles embodied in the tip feathers of the land soaring birds.


DIAGRAM No.2

~~~Within a box 1/2" high and 4" long draw a simple flat-bottomed airfoil. With three equidistant vertical and three equidistant horizontal lines divide the box into 16 identical segments. Label the horizontal divisions 1,2,3,& 4, the vertical a,b,c,& d.
~~~In boxes 1a, 2b, and 3c draw the same airfoil, 1/4 its original size. Ignore 4d. Add a line of plus signs - + + + + + + + - just below the airfoil in the box and a matching row of minus signs - - - - - - - - above it.
~~~With a colored pen draw vectors beginning ahead of the airfoil, passing over the airfoils in areas 1a, 2b, and 3c, and ending in the row of + + + + +'s below the box.