There are two changes in the form of presentation beginning in this part, following suggestions made by members. (Thanks to Rolland Everitt) henceforth V squared, which is V times V, or V raised to the power 2, will be shown as  V^2. (The single star * that I had been using is normally used for multiplication in computer text and might be distracting to programmers. I had avoided a multiplication mark to conform to the old convention of adjacent symbols being multiplied, as readers who dig deeper will find in standard aerodynamic references.) Second, John Rawson suggested that it might be easier for both of us if I drew the figures (in ASCII) rather than trying to tell you how to do it. I hope that the imperfect results are not more misleading than helpful.

A SHORT REVIEW

So far in this series: In Part 1 we saw how sail forces can be resolved into Thrust and Side Force. To sail in any direction except downwind, the latter must be opposed by underwater forces developed by the hull, or its appendages. The cost of supplying that side force to resist leeway is an incremental drag that is added to the resistance of the hull's motion through the water, so becoming an additional charge against available Thrust. In Parts 2 and 3 we examined the ideas that Lift is a natural result of the addition of a fluid stream and a circulation. In discussing SECTIONS in Part 4, we saw how viscosity initiates circulation around a foil. It does so with a STRENGTH [depending on shape (mostly camber) and angle of attack] just sufficient to ensure that the fluid flows smoothly off the trailing edge with a minimum disruption of the universe. When this condition can't be met, the section "stalls" and Lift is lost. In Part 4 we looked at the dynamic "two dimensional" meaning of flow around a section. After definining its shape parameters, we also conducted a series of imaginary experiments in a closed wind tunnel on a rectangular "sample section." From the "observations" of those experiments, we deduced the standard force equation, relating Force to the factors of: mass density (which we examined); Force Coefficient; Area; and Velocity. We discussed boundary layers and separation, and related them to common experience. We saw the part they play in causing the slight deterioration of real performance compared with what might have been "ideal" performance (if something could start a circulation in an "ideal" fluid. (Just for the record, nothing could.) We have perhaps begun to feel how important it is to take into account Reynold's Number as a factor in "flow shape."

Now we're ready to go ahead.

Suppose that we widen the "test and observation" portion of our wind tunnel. This moves its sidewalls beyond the influence of our SECTION test sample. At the same time, we take steps to ensure that density and stream Velocity remain constant.

Now, if we put smoke in the widened tunnel, we will see trailing vortices extending downstream from near each wing tip. If we look upstream from behind the wing, we will see that the vortices are opposed. Assuming positive Lift, the port-side vortex is clockwise, the starboard counter-clockwise. A new flow-shape has been introduced to the system, implying different forces. Indeed, when we look, the balances indicate a (possibly-only-slight) reduction of Lift, and an increase of Drag. Our SECTION sample (which, during our examination in Part 4, had been acting as though it were indefinitely long, and hence two-dimensional) has become a finite, three-dimensional WING.

The THEORY OF WINGS (actually, there is more than one theory) is a record of a series of attempts by various investigators to give better explanations and definitions of the three-dimensional "flow shape" about a finite wing. From that, the aim is to predict the performance of the wing.

For wings of ordinary shapes (meaning PLANFORMS, the outline shape of the wing as you look down on it) the simplest useful theory is based on the idea of the "Lifting Line." In this approach, the wing is replaced by a line of equal span. The theory predicts the pattern of three dimensional flow about the line in terms of the parameters of the wing it replaces. Then it permits the application of SECTION data to that "transformed" three dimensional flow. That enables prediction of WING Lift and Drag (and pitching Moment for those who are interested.)

The Limiting Premises of the Lifting Line Theory

The lifting line theory supposes that:

1. the stream Velocity (direction and speed) is uniform across the span.
2. there are no "discontinuities," meaning SUDDEN changes of section (shape, chord, or angle of attack) over a short change of span.
3. any spanwise component of velocity on the wing is small compared to the free stream velocity, V.

Symmetry suggests that whatever is happening on one side of the wing is balanced by its mirror image on the other side. Clearly, no fluid flows across the central plane of the wing. (Otherwise the flow could not be symmetrical.) Each half-wing may be considered independent of the other, with no change in its flow shape, so long as that central two-dimensional flow is maintained. Suppose we insert a central plate in the wing, to prevent cross-over flow. Then each half wing no longer sees its real counterpart, but a VIRTUAL image of itself in the central plate. We may then remove one half wing. The remaining half will have an unchanged flow shape, nearly identical with what it had while it was part of the whole wing. (The variation may be in some turbulence effects at the plate.)

For many discussions of wing performance, and shape, it is more convenient to work with the half wing (even without thoughts of separating the halves.) For dealing with half wings, the parameter "half span," symbol s (where s = b/2) is useful.

For much of the dynamic analysis of a wing, the pattern of the distribution of Lift (called the LOAD DISTRIBUTION) along its half span is more significant than the plan-form of the wing itself. Indeed, there may be only a marginal relationship between shape and loading. As we will see, the one exception (in which wing loading matches the planform) gives us a clear insight into wing performance.

The Meaning of Wing Load Distribution

Many members could probably do with a shorter development than I offer here. But the idea of wing Load Distribution helps us to account for the trailing vortices that reduce Lift and cause an increment of Drag. And it helps us to understand the relative merits of different planforms, and the effects of twist. Hence, in keeping with my hope of making this series understandable to every intelligent sailor who wants to know, regardless of background, I'll review the ideas of wing load distribution in some detail.

In one way of looking at it, Lift is produced by a difference of pressures between upper and lower surfaces. Below the wing, pressure is higher than free stream pressure. Above the wing, it is lower. At the wing tips, there is no surface to separate the air flow, hence there can be no pressure difference.

It might be interesting to see how small these pressure differences are, for sailboat elements, compared to the absolute pressure of the air. This "side-bar" might help to develop the idea of wing loading too:

• Imagine a horizontal panel of sail one foot high, at a point where the chord is 10 feet. Operating at a Lift Coefficient 1.0, in a breeze of 8.9 knots, that ten square feet of sail develops a lift force of 1 pound. Assume that the pressure differential is distributed uniformly along the chord. (If it isn't we'll average it out.) Then the pressure difference between opposing surfaces of the panel is 0.1 pound per square foot (psf). That may be developed by a pressure increase on the lower (low velocity) or windward side, of 0.05 psf, and a pressure deficit on the upper surface of .05 psf. Meanwhile, the absolute pressure of the free stream air is one atmosphere, or approximately 14.7 pounds per square inch (psi). If we assume that the air pressure at the moment of our observation is exactly 14.7000 psi, then the pressure of the free stream is 2116.80 pounds per square foot. So the sail is producing its force by generating a lee-side pressure of 2116.85 psf, and a weather side pressure of 2116.75 psf. In relative terms, this is an overall difference of 0.0047 percent of normal air pressure.  Even at 40 knots and Lift Coefficient 1.0* the force per square foot is 2 pounds. This is also the pressure differential. Here it is about 0.045 percent increment and decrement of standard air pressure. *(In these conditions you'd probably be trying to reduce the Lift Coefficient to a lower value, using flat sails and low angles of attack, to achieve lower pressures and forces.)
• Perhaps this is a good place to dispose of the common notion of a "suction" on top of the wing. With a positive air pressure of more than 2000 psf above the wing, it requires a leap of faith to agree that Lift is caused by the wing's urgent desire to fill some kind of magic "vacuum" in the space above it. At times, at some places along the chord, the increase of velocity, and hence the pressure DECREMENT above the wing may be greater than the reduction of Velocity, and hence the pressure INCREMENT below the wing. But the Lift is provided by the combined effect of these pressure differences taken over the area.
• What we have just done also justifies our earlier assumption that gases are essentially "incompressible" in sub-sonic aerodynamics. The volume of a given mass of gas varies inversely as its absolute pressure. With such slight changes of absolute pressure, and hence volume, there is little error in taking the change of volume as being zero. (Meaning the gas suffers no change of density, and hence may be regarded as incompressible.)

FIGURE 5.1 (An Attempt to show the Pressure Distribution along the Span (sadly limited by ASCII symbols))
===================================================================
Now you can draw a curve of the total presure differential. (Fig 5.2) Work with just the left half of the wing, and use the straight line as base. Move the line PQ vertically upwards until Q is on the wing (or base line.) The new position of P may be marked R. Divide your half span into sections one foot wide. (I can't do that and show a reasonably-shaped curve because of the limitations of ASCII text.) At the first mark from center you have the points P' (P prime) and Q' (Q prime). Raise this line until Q' is on the base line and mark the new position of P' as R'. Continue this process on to the wingtip, where R with s primes lies on the base line.
===================================================================
                       .R'  .(P)R
           .R"               |        (Here the points R are spaced
       .R"'                  |        to give a reasonable shape to
     .R""                    |        the curve)
    .R""'                    |
    :-------------------Q
               s
FIGURE 5.2
===================================================================
Now, the Lift Force at each of your stations is the pressure differential times the area of the panel [Note F=P*S or P=F/S.] Assuming your span-wise panels are one foot wide, the area is equal to the chord c. So we can multiply our chord (area) of c times the pressure differential R (with whatever prime) at each station and get the Force F = Rc (with the same prime) at each station. If we join all the points F from wing centre to tip, we have drawn a Load Distribution Curve for the wing. For other plan forms, we would multiply R by the actual chord at each section to get F.

All wings of likely interest, without twist or other complications, with half plans ranging from triangular to rectangular (and all shapes in between) tend to have Lift Distribution Curves (LDCs) that lie BETWEEN the triangular and rectangular forms. In other words, The LDC of a rectangular plan form doesn't quite fill the rectangle, and the LDC of a triangular half plan has what would be called a "roach" if it were a sail.
===================================================================
           rectangular planform
     |- - - - - - - - - - -.- - - - - - - - - - - - - -|
     |       .                                            -    |
     |    .\    range of LDCs         -          |
     |  .    \<======     .        -                 |
     | .       \ .    - triangular planform    |
     |.     .   -        (dashes)                     |
     |    -                                                   |
     :-----------------------------------------O
     s                                            (midspan)

FIGURE 5.3 (An Attempt to Show the Range of Load Curves for Untwisted Planforms ranging from triangular to rectangular shape.)
===================================================================

Factors beyond planform affect the LDC of a given wing. The Lift at each station depends on section lift Coefficient (C-l) as well as chord. The Coefficient may be altered by progressive changes of sectional shape or angle of attack along the span. Such changes may be deliberate (in hard wings), and more likely accidental in sails. Additional factors in the loading of SAILS are spanwise changes in direction and strength of apparent wind resulting from wind shear.

A twist which REDUCES the angle of attack as the wingtip is approached is called "washout." (This is the direction of twist that we think of as the "head falling off" in sails.) It is sometimes used in aircraft wings to reduce tip loading and so delay "tip stalling." The opposite twist, called "washin" is rarely if ever used in aircraft wings. For a given *sail*, it would require detailed study to determine if if it were washed in or out, considering its twist, and the speed and direction of local apparent wind caused by wind shear. Even if it appeared to be neutral, one is then left with the problem that wind shear also increases the apparent wind speed with height.* With pressures and forces varying as the square of the increasing speeds, this adds its own complication to the LCD for sails, tending to increase tip loading if there is no washout.

• Change of velocity (direction or speed) along the span violates a premise of the lifting line theory, which assumes that the wing, in a uniform stream, is neither yawing, rolling nor pitching. (Note that yawing of a wing corresponds to pitching of a boat, and pitching of a wing is yawing of a boat. Rolling is the same action for both wing and sail.
• Despite the breaks with lifting line theory, we can still learn a lot from it, even for sails.

• * means :
  Here, as in earlier Parts of this Series, the hyphen is an ASCII-enforced attempt to indicate that the second letter is a subscript on the first, which is how you will see these parameters symbolized in books. (I now wish that I had used an underline.) The only reason I can think of for the original selection of subscript "s" for what we commonly think of as the "root chord," would be to indicate the chord at semi-span.

The remaining significant wing parameter might be thought of as a "slenderness quotient." !!( Trumpets sound a fanfare off stage )!! Called ASPECT RATIO, symbol A, it is defined as the square of the span divided by the area, or A = (b^2)/S.

• For a rectangular plan, A = (b^2/(bc) = b/c or the ratio of span to chord.
• For a triangular half plan, A = (b^2)/[(c-s)b/2] = 2b/(c-s). So, for a wing with span 6 (half span 3), root chord 1, and zero taper ratio, the aerodynamic aspect ratio, A = 12 (not 3 and not 6).
• The area of an elliptical wing of span b and root chord (c-s) is S = (Pi)b(c-s). So for the ellipse,
                    A = b^2/(Pi)b(c-s) = b/(Pi)(c-s).

No other ratio representing slenderness has any AHdynamic meaning. One commonly hears (and reads) talk of luff-to-foot ratios of sails, and depth-to-width ratios of keels as "Aspect Ratio." But, if the intent is to convey AHdynamic significance, that is wrong and misleading. Attempting to apply such notions to analysis of performance would lead to erroneous results. [In part 6 we'll look at ways of estimating (guesstimating?) the effective Aspect Ratio of sailboat elements.]

Now let's try to account for the trailing vortices caused by the "physical" wing. Look again at the pressure distribution sketch (FIG 5.1). We see, at the wing tip, free-stream air facing an invitation to move into the space above the wing to equalize the pressure decrement there. At the same time, higher pressure air below the wing is inclined to move laterally towards and beyond the tips in a similar bid to equalize pressure. At mid span, there is no tendency to lateral motion. But at all other points along the half span, a small component of lateral velocity appears in the flow. Under these conditions, the chordwise speed component of velocity is effectively unchanged. But at every point of the trailing edge (except at midspan) air flows now meet with a slightly opposing crossflow, outboard below, and inboard above the wing, in an effect increasing towards the tips. At every non-central point of the wing, this "shearing effect" causes a vortex, so tending to produce a "vortex sheet" behind the wing. (Not to be confused with the Karman Vortex Street of alternating vortices shed by bluff bodies.) In a moment, we'll look into the idea that a vortex sheet is unstable, and must find something else to do with itself.

Now the time has come to replace the physical wing with the lifting line. It has no surface to support differences of air pressure. But we can use the equally valid idea of circulation and free stream to account for Lift. From the L = (rho)V(gamma) description, we can account for changes of Lift along the span as changes in the strength of circulation alone [because (rho) and V are constant.] To create the image, we imagine the root section circulation as a bundle of so-called "bound vortices") laid along the lifting line. At every point along the span where there is a reduction of Lift compared with the next-inboard section, we consider that enough vortex filaments leave the bundle and turn downstream to account for the reduction of Lift. At the tip, where Lift is zero, all bound vortices have left the wing. This gives us an equally valid way of accounting for that INCIPIENT VORTEX SHEET behind the wing (lifting line.)

• Here we see that earlier statements-questions(?) about trailing vortices being circulation somehow "peeling itself off the wing" was not such a bad notion after all. Having myself planted the idea in Part 2, I was objecting more to readers suggestions that it was necessarily "waste" than anything else.  Put another way, in an aircraft, it is always waste. Later in this part 5 and more in Part 6,  for sailboats, we'll see why it may provide at least as much off-wind benefit as upwind cost.

Another of the premises of the Lifting line theory is that the lateral distance between the centres of the trailing vortices is near enough the span of the wing to yield useful results. [Except for wings of very-low aspect ratio, requiring a different theory, the premise is close enough to being true to have successfully designed thousands of airplanes.]

Now look at our lifting line from behind. We see two contra-rotating circulations. They tend to generate an upward component of flow outside the wingtips. Inboard, their effect is to impose a downward velocity component, called the "induced velocity," given symbol w, at every point along the span. In general, w varies from section to section along the half span.

If we look from the side, and add w (vectorially of course) to the free stream velocity V, we will see that the resultant local flow, (still of magnitude V when w is small compared to V), is redirected somewhat downwards at an "induced angle" with the "natural measure" w/V. *
===================================================================
                        V (free stream)
            <---------------------------------------------O
            |                                   -         Lifting
        w |                    -                         line
            |<   -
                 \  Flow Velocity at the lifting line redirected
                   \  downwards at the "induced angle" w/V

FIGURE 5.4
===================================================================
This "induced" angle at any station along the span takes the symbol (alpha-i) because it has an effect on the angle of attack. In speech and writing, it is commonly called "downwash."

• Perhaps the idea of "natural angle" deserves a few words.  The division of a circle into quarters, points, degrees, or grads, is arbitrary. Angles expressed in those terms have no direct physical meaning because of their artificial origin. But there is a natural, or physical, way of expressing angles, called "radian measure." In this system, the angle is expressed as the length of arc swung out by the turning radius of a circle, divided by the length of the radius. In other words, as angular measure properly should be, it is Length over Length, and hence dimensionless. The circumference of a circle is 2(Pi)r, where r is the radius. So the circle contains 2(Pi)r/r = 2(Pi) radians.  It also contains 360 degrees. So one radian contains 360/2(Pi) = 57.3 degrees. *For small angles,* the length of the arc of a circle is effectively equal to the length of a segment, so we are justified in saying that the natural angle of the redirection of the flow is w/V (radians.)

           G -
              \   geometrical angle (alpha-g)
                   {or sometimes just (alpha) }

• In this figure, the chord of the section (if it were shown, which    it is not, because we have replaced the wing with the lifting    line) would lie on the line OG. The angle (alpha-i) is induced by    the downwash, or we might say, "given away" by the wing. This    leaves only (alpha-e) = (alpha-g) - (alpha-i) as the angle of    attack of the redirected airflow upon the section at any    span-wise location. This is what the section sees as "section    angle of attack" (alpha-0) that we studied in Part 4.

   For the record:
    (alpha-g) = (alpha-i) + (alpha-e)
    or
    (alpha-e) = (alpha-g) - (alpha-i) = (alpha-0)

For wings of ordinary aspect ratio (to which the lifting line theory applies) the cross-flow velocity (i.e. spanwise along the wing) is slight compared to V. So we take it (without significant error) that each section along the span performs in the locally redirected stream as though it were in two-dimensional flow at section angle of attack (alpha-0). In other words, from the section's point of view, the wing has "given away" the downwash and it must live and work in what's left.

Please note the difference between the downward component of velocity GENERATED by the section (discussed in part 4, which is a measure of Lift) and the downwash. The latter is not something PRODUCED by the section, but something it ENCOUNTERS and lives in by virtue of its place in a finite wing.

The general solution of the equations for downwash across the span in terms of arbitrary load distributions, planforms, span-wise changes of section, twist, and sweepback (rake of aerodynamic centres) is extremely difficult. But a useful theory of wing performance has developed from a fairly simple beginning for a special case (an elliptical load distribution.) From that easy beginning, by way of some high-powered mathematics whose results have been shown in tables and graphs, a complete, workable, and fairly simple theory for the performance, and practical design of arbitrary wings has been developed. Let's follow that trail a little.

First, Prandtl (in "Applications of Modern Hydrodynamics to Aeronautics," NACA Report No. 116, 1921) showed that for an ELLIPTICAL LOAD DISTRIBUTION (notice please that I said Load, not planform) the angle of downwash is UNIFORM across the span, with the value

         (alpha-i) = (C-L)/A(Pi) radians.                        (5-1)

• (alpha-i) is the downwash,
• (C-L) stands for WING Lift Coefficient (normal to the free stream, not the rotated flow),
• A is Aspect Ratio,
• (Pi) has its usual mathematical meaning (ratio of circumference to diameter of a circle.)

Now, at any section, the SECTION lift vector is normal to the redirected stream (the effect of downwash.) But we are interested in a WING Lift at right angles to the FREE stream. At small angles of downwash, the wing Lift, and the section Lift are sensibly equal in magnitude. We can resolve the SECTION Lift vector into a vertical (cross main stream) Wing LIFT, and a downstream Wing DRAG vector. (Figure 5.6) This downstream component of section Lift is called the Induced Drag. Note that, *unlike* Profile Drag, it is not an effect of viscosity on the section, but a result of the rotation of the section Lift vector by the downwash.
===================================================================
                                     Component of rotated Lift
                                     of section in the downstream
                                     direction, therefor a DRAG.
                                              /
               rotated SECTION Lift    <========|  WING Lift
                                       \        |  in direction of
                                         \      |  interest, i.e
                                           \    |  across the free
                                             \  |  stream
            direction of the free stream       \|
                     - - - - - - - - - - - - - -O
                                             -
                                        -
                                   -
                              -     rotated stream at the section
                       _         (angles exaggerated by ASCII)
                    <
FIGURE 5.6
===================================================================

As before, it is convenient to reduce forces to Coefficients.  In Figure 5.6, let's replace the WING Lift with its Coefficient (C-L), and induced Drag with the induced Drag Coefficient (C-D-i).

If we compare the downwash angle w/V, and the induced Drag and Lift coefficients, we see (by similar triangles) that induced Drag coefficient is to Lift Coefficient as w is to V, or:

   (C-D-i)/(C-L) = w/V.

Transposing, this leads to

   (C-D-i) = (C-L)w/V

But w/V is the downwash, which equation 5.1 shows to be (C-L)/A(Pi). Hence

   (C-D-i) = (C-L)^2/A(Pi)                                  (5-2)

• or induced Drag Coefficient of the wing (applicable to our standard force equation) equals the square of the Wing Lift Coefficient divided by the product of Aspect Ratio and (Pi).
• Remember that this relationship is only strictly true for elliptical wing loadings (which produce a constant downwash along the span. Refinements for other shapes come later.)

• A runner on an outdoor track will provide an example. Assume she is running at 10 fps. With no true wind, at every point on her loop, her (apparent wind) resistance is proportional to 10^2 = 100. If a 5 fps true wind springs up, her downstream relative wind is 5 fps, with a drag proportional to 25. But her upstream drag in the 15 fps wind is 225. So her average up-and-down wind resistance is      (225+25)/2 = 250/2=125,
  compared to the 100 when there was no variation in apparent wind. In other words, any true wind hurts the runner more than it helps help her around a closed circuit. Similarly, variations of downwash hurt the wing by increasing overall induced Drag. Thinking of this will help us to understand the relative merits of different wing shapes.