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by Karel Termaat** **

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Some years ago, I found that when landing our new sailplane it usually made
a couple of bumps onto the ground before it decided to stay there; very
interesting to friends observing your flying skills with the new toy of
course. Another surprising thing
happened to me with this sailplane when I made a long cross-country flight,
came back a little late and low and decided to divert from my shortest route to
the airfield to an area where small flocks of cumulus clouds were still
forming. To my surprise I could not climb there, even though the air was quite
turbulent and going up on average. I tried another identical spot with the same
disappointing result. I flew out of this promising area and found a weak smooth
thermal which brought me home. Quite a frustrating experience.

After a while, my son
Ronald and I began to suspect the lift-curve of the wing of having an unusual
part at the lower speeds, which could explain our findings that pitch control
just prior to touching down had hardly any effect on sink rate and that the
good climbing performance of our new sailplane degraded substantially when
circling in turbulent thermals.

Because of our observations
we contacted prof. ir. Loek Boermans at Delft Technical University. "We
improved the not so good climbing performance of a standard class sailplane in
turbulent conditions already some years ago”, he said. “A new wing profile and winglets were
applied. Apart from a recent phone call of Ronald flying at the EC in Leszno, I
heard no complaints about the climbing performance of the new breed of
sailplanes with flaps though I was indeed worried about this. I advised Ronald
to fly somewhat faster in turbulent thermals with more banking angle. I think
there are ways to solve this problem also for sailplanes with flaps as we did
for the standard class glider”.

Shortly after the meeting,
prof. Boermans came up with a slightly redesigned profile promising a better
climbing rate in turbulent conditions while maintaining optimum glide rates. I
myself started to develop some software of my own to better understand the
effects of the flatter part in the lift-curves of modern sailplanes with flaps
and came up with a couple of practical ideas when flying in turbulent air or
when landing at low speed. Slightly modifying the wing profile is of course no
option to pilots having one of the current breed of high performance
sailplanes.

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The lift-curves:

The lift of a sailplane is
controlled by the well known lift formula:
L = ½.r.V^{2}.S.C_{L }. With this formula, together with the C_{L}-a graph we can study the performance of a sailplane in
smooth and turbulent air. Modern wings are quite thin to minimise profile drag
and are normally operated at small values of a within the so called ‘laminar drag bucket’ where
maximum length’s of laminar boundary layers on both the upper and under
surfaces of the wing are realized. The lift formula shows that at low
thermalling speeds or when landing, Cl must be as large as possible to properly
carry the weight of the sailplane. Maximising C_{L} at low speeds and
small values of a is
controlled by a balancing process, where with increasing a the lift in the forward section of the wing grows as
expected, while the lift on the aft part of the wing breaks down at about the
same rate because of earlier laminar boundary layer transitions and flow
seperations at the flap hinge area. The net effect of this is that the mean
lift coefficient remains about constant over quite some interval of a, especially for the larger flap settings, as shown in
figure 1.

So, a linear relationship
between** **C_{L} and a, as common in earlier years of wing design, does not
apply to modern wing profiles with flaps. Almost all have a more or less
horizontal ** step** in their lift curves. Beyond the step, C

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Figure 1: Measured C_{L}- a curves (typical for any modern sailplane)

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Now consider the case of a
pilot on finals to the airfield close to touching down. No water in the
sailplane (mass=450kg) and FL=20°, ie in the landing position. Assume an
approach speed V = 76km/h. Then from formula (2) it folows that C_{L} = 1.46. The FL=20° curve
of figure 1 indicates that for this value of C_{L}, a = 2.5°
which is just in front of the step as indicated. In rounding off prior to
touching the ground, the pilot lifts the nose of the sailplane to increase a with the idea of slowing down the descent rate of the
sailplane. a increases alright, but C_{L} doesn’t do that unless the nose of the sailplane is
tilted so high that a has
increased from 2.5° to more than 7.0°. Only then C_{L} will increase to above 1.46 and the sailplane will stop
its downward motion. Usually a pilot’s action in rotating the sailplane over
this large angle is too cautious. The sailplane will contact the ground
prematurely and will usually bounce a couple of times. Bumpy landings are quite
familiar to pilots flying modern competition sailplanes and can frequently be
observed. The solution to this problem is to stay more in front of the step
region by flying a little faster than the minimum approach speed and use
airbrakes rather than pitch to control descent rate. Once on the ground, good
wheelbrakes should be applied to bring the sailplane to a halt.

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In turbulent air, serious
changes in a occur
because of vertical movements of air hitting the wing surface. When gliding straight
on at small values of a close
to 0°, which is usually the case, one can see from the C_{L}- a graphs
of figure 1, that upgusts will increase C_{L}
and downgusts will decrease C_{L}
in the same manner. On the average C_{L}
will stay constant with time and no altitude gain or loss appears in this
situation where gusts have a random distribution in strength and direction. But
especially at low wing loading one may expect a rough ride because of
significant positive and negative accelerations due to the more or less strong
variations in C_{L} with
time.

However, now consider a
sailplane flying nose up with a = 4.0°
and FL=15°, where C_{L}=1.40
as indicated in figure1. So right at the beginning of the step where upgusts
have no effect on C_{L},
since an increase of some degrees in a falls right into the step where C_{L} is constant. However,
downgusts will reduce a with
the same some degrees, thereby reducing C_{L.}
So C_{L} swaps between
1.40 for positive gusts, and some lower value, say 1.20 for negative gusts. On
average C_{L }= 1.30,
i.e. 7 % less than the 1.40 required to carry the weight of the sailplane.
Therefore the sailplane starts a dynamic downward motion as long as the
turbulences are present. Not what one really wants of course.

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To get a better insight
into the climb rate of the sailplane under turbulent conditions I wrote a
small, but effective, computer routine. The first part of the routine is an
outer loop, which is rather straightforward and describes vertical speed as the
difference between thermal climb rate and polar descent rate in smooth thermals
for 300 values of circling speeds in a practical range.

I used a theoretical
thermal model with a parabolic shape and accurate speed polars as measured by
Idaflieg (the German academic flight test group). I used cubic spline functions
to make the ploar curves accessible to the computer routine as a function of
flying speed. The straight flight sink rate data of the polar were corrected
for mass and bank angle. With these formulae for the thermal model and polar
descent rate, stable climbing rate as a function of flying speed can be
calculated as: Vs_(th+pol) =
Vs_thermal + Vs_polar ......(3)
(purple curve in figure 3)

To calculate the additional
effect of turbulence, some sensible dynamics equations had to be defined. In
the computer routine, these equations are enclosed in an iterative inner loop
having small time steps dt = 0.01s.
Turbulence is described as sine waves with a time constant of 2 seconds and
random amplitude. In this way the effect of turbulence, Vs_gusts, is calculated
during 30 seconds for each of the 300 flying speeds considered and plotted in
graphs (blue curve in figure 3). Then total climbing rate is: Vs_total = Vs_(th+pol) + Vs_gusts ......(4)
(black curve in figure 3)

A key issue in the routine
is the application of Newton’s second law to find the vertical speed, Vs(t), of
the sailplane from the forces acting on the wing due to the airgusts present.

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Figure
2 shows the development of vertical
speed Vs(t) of the sailplane for the first couple of turbulence gusts, Vg(t),
encountered. One can say that the results are quite dramatic when looking at
the blue Vs(t) curve which averages out at -0.59m/s. The sine wave shapes of
the turbulence can be recognised, as well as their randomised amplitudes. More
erratic forms are likely of course, but these are not expected to make a
significant difference to the results of this analysis.

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Figure 2: Development of sink
rate Vs(t) due to sine wave shaped turbulences.

More
practical results with the computer routine are given in figure 3 which shows one of the many
graphs studied. The a line
shows the step as expected. Considering a smooth thermal, a best climbing rate
of Vs_total = Vs_(th+pol) = 1.80m/s is obtained. Circling with a speed just
above something like 100km/h is OK and comfortable, but of course this is
dependent on the shape of the thermal.

Considering also the
turbulence in figure 2, total climbing rate is now Vs_total = Vs_(th-pol) +
Vs_gusts as indicated. For speeds above, say, 120km/h, the defined turbulence
has no effect because a is
small (even negative) and quite less than 4.0° where the step in the C_{L}-a curve for FL=15° begins. Then positive and negative gusts cancel out as is shown by the
Vs_gusts curve. However, for speeds less than 120km/h, positive gusts raise a to into the flat part of the C_{L}-a curve of figure 1 and give a reduced contribution and
at 99km/h no contribution at all to the lift of the wing. Negative gusts
however still fully reduce C_{L}
as usual. Therefore with turbulences
present, the nice total climbing rate of 1.80m/s at 100km/h for the smooth
thermal is dramatically reduced to 1.20m/s only, a loss of 33 per cent. The
only way out of this is to circle some 10km/h faster. In this case a climbing
rate of about 1.40m/s is attained at a
speed of 110km/h. So, still a loss of 22 per cent due to the turbulence.

To get a beneficial effect out
of random air gusts, a pilot may try to fly with an a just behind the step in the C_{L}-a curve
where positive gusts push the sailplane up and negative gusts have no effect.
Figure 3 promises a nice climbing rate when circling with a speed of say
96km/h. Some pilots seem succesful in doing so, however the sailplane may be
rather difficult to control and high drag may deminish this special effect
substantially.

Other calculations with the
routine show that having a dip in the lift curve is still worse than a flat
step, but some earlier high-performance sailplanes have that. A small positive
gradient in the step area, as suggested by Loek Boermans, improves climbing rate with turbulences
present quite considerably, so this looks the way to go in the design process
of new wing profiles. Just now, some new sailplanes having this idea are coming
on the market.

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Figure 3: Reduced climbing
rate in a turbulent thermal

It should be mentioned that
the above calculations were performed assuming that the effects of the flat
part in the liftcurve occur at the same moment over the total span of the wing.
This not the case however due to the random distribution of turbulences in size
and in space and because of different wing profiles and Re-numbers in the
spanwise direction of the wing. Therefore the results of the above calculations
maybe somewhat exaggerated for the practical situation.

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Recent optimisations of
wing profiles with flaps have resulted in C_{L}-a curves with a more or less ‘flat part’ in the lower
speed range. Because of this, landing behaviour and climbing performance in
turbulent thermals are somewhat disappointing.

This article shows that
pilots can partly take care of these drawbacks by flying with a (angle of attack) well in front of the ‘step’ in the
C_{L}- a curve.

The main actions to achieve
that are:

a. When trying to make
perfect landings, approach speeds should be a little higher than usual and speed
brakes gently applied rather than pitch to finely control the descent rate in
rounding off just before touching the ground.

b. When optimising total
climbing rate in turbulent thermals, flying speed and bank angle should be
higher than a pilot would usually prefer.

c. Thermals should be
entered with redundant speed to avoid that the sudden upflow encountered pushes
a into the step area.

This study shows that
slightly modifying the C_{L}-a curve with a positive gradient in the step area will
improve sailplane performance substantially. A better control of the descent
rate during landing will then be possible and an increase in average climbing
rate of some 20 per cent may be expected when circling in turbulent thermals. I
understand that only minor modifications of the wing profile are necessary to
obtain these improvements without a penalty in the high speed range of the
sailplane (as suggested by Prof. Ir. Loek Boermans).

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I would like to thank my
friends for the inspiring discussions about the step in the C_{L}-a curve of modern sailplanes – especially my son Ronald
and Loek Boermans, both of whom gave in their own specific way of practical
experience and theoretical knowledge, a substantial support in the realisation
of this work. Additionally the books and articles of John Anderson, Helmut
Reichmann, Fred Thomas, Loek Boermans and others and many articles found on the
internet were also quite inspiring.

ir. K.P. Termaat

Arnhem, NL

20 oct. 2011** **

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