Speed ​​to fly theory

The speed -to- fly or MacCready theory is used to maximize the cruising speed of gliders , hang-gliders and paragliders . Since these aircraft have no drive, they are dependent on using updrafts for flights over long distances . You linger in the updraft for some time - mostly in circles - to allow yourself to be carried up by it. Then they use the altitude to cover the next part of the flight route in gliding flight .

The fly-to-fly theory helps the pilot to decide how long he should use the updraft in order to make progress as quickly as possible. It also gives tips on how fast you should fly in updraft and glide. The distance to the next thermal , its strength and the characteristics of the aircraft are taken into account.

Basic assumptions

Model of a cross-country flight

The speed-to-fly theory refers to a model that is shown on the right. The aircraft glides from one updraft to the next, covers a distance and loses altitude in the process. When the next updraft arrives, it circles back up to its original altitude. If the pilot flies too fast, he will arrive lower in the updraft and must climb for a very long time. If it flies too slowly, it arrives high but is very slow. Between these two extremes lies the optimal pre-flight speed, at which the cruising speed is maximum.

The speed-to-fly theory is based on the following assumptions:

1. The aircraft only climbs when circling in the updraft , while gliding it sinks.
2. The next updraft is always reached.
3. The pilot can estimate the strength of the next updraft.

The last two assumptions in particular are not always given in practice. If the rise of the next updraft does not correspond to the expectation, then the flight will also be carried out at a non-optimal speed. If the next updraft cannot be reached safely, it may make sense to take another trip.

Mathematical derivation

The cruising speed

${\displaystyle v_{\mathrm {gl} }=}$ Airspeed (gliding)

${\displaystyle v_{\mathrm {si} }=}$ Rate of descent (positive number!)

${\displaystyle v_{\mathrm {st} }=}$ Rate of climb in the updraft

${\displaystyle v_{\mathrm {Reise} }=}$ Cruising speed (average speed)

${\displaystyle E=}$ Distance to the next updraft

${\displaystyle H=}$ Height that is slipped and then climbed again

${\displaystyle t_{\mathrm {gl} }=}$ Time of sliding

${\displaystyle t_{\mathrm {st} }=}$ Time to climb

${\displaystyle t=}$ total time

(1)	${\displaystyle v_{\mathrm {Reise} }={\frac {E}{t}}}$	Cruising speed = distance / time
(2)	${\displaystyle t=t_{\mathrm {gl} }+t_{\mathrm {st} }}$	Total time = pre-flight time + climb time
(3)	${\displaystyle H=t_{\mathrm {gl} }\cdot v_{\mathrm {si} }}$	Altitude = pre-flight time * sink
(4)	${\displaystyle H=t_{\mathrm {st} }\cdot v_{\mathrm {st} }}$	Height = climb time * climb speed
	${\displaystyle t_{\mathrm {gl} }\cdot v_{\mathrm {si} }=t_{\mathrm {st} }\cdot v_{\mathrm {st} }}$	Equate (3) with (4)
${\displaystyle \Rightarrow }$ (5)	${\displaystyle t_{\mathrm {st} }={\frac {t_{\mathrm {gl} }\cdot v_{\mathrm {si} }}{v_{\mathrm {st} }}}}$	Dissolve after rising time
(6)	${\displaystyle t_{\mathrm {gl} }={\frac {E}{v_{\mathrm {gl} }}}}$	Flextime = distance / airspeed
(7)	${\displaystyle t_{\mathrm {st} }={\frac {E\cdot v_{\mathrm {si} }}{v_{\mathrm {gl} }\cdot v_{\mathrm {st} }}}}$	Insert (6) into (5)
	${\displaystyle t={\frac {E}{v_{\mathrm {gl} }}}+{\frac {E\cdot v_{\mathrm {si} }}{v_{\mathrm {gl} }\cdot v_{\mathrm {st} }}}}$	Insert (6) and (7) in (2)
${\displaystyle \Rightarrow }$ (8th)	${\displaystyle t=E\cdot ({\frac {v_{\mathrm {st} }+v_{\mathrm {si} }}{v_{\mathrm {gl} }\cdot v_{\mathrm {st} }}})}$	total time
(9)	${\displaystyle v_{\mathrm {Reise} }={\frac {v_{\mathrm {gl} }\cdot v_{\mathrm {st} }}{v_{\mathrm {st} }+v_{\mathrm {si} }}}}$	Insert (8) into (1)

Equation (9) applies to every speed flown and every aircraft descent.

The optimal pre-flight speed

${\displaystyle v_{\mathrm {gl} }=}$ Airspeed (gliding)

${\displaystyle v_{\mathrm {si} }=}$ Descent speed of the aircraft compared to the air mass (positive)

${\displaystyle v_{\mathrm {st} }=}$ (Expected) rate of climb in the updraft

${\displaystyle v_{\mathrm {met} }=}$ meteorological decrease in air mass when gliding (positive)

${\displaystyle t_{\mathrm {gl} }=}$ Time of sliding

${\displaystyle t_{\mathrm {si} }=}$ Time to climb

${\displaystyle t=}$ total time

(1)	${\displaystyle t=t_{\mathrm {gl} }+t_{\mathrm {st} }}$	Total time = pre-flight time + climb time
(2)	${\displaystyle t={\frac {E}{v_{\mathrm {gl} }}}+{\frac {H}{v_{\mathrm {st} }}}}$	Replace times with the quotient of distance and speed
(3)	${\displaystyle H={\frac {v_{\mathrm {si} }+v_{\mathrm {met} }}{v_{\mathrm {gl} }}}\cdot E}$	The loss of altitude results from the sum of sinking (meteorological + airplane) by speed * distance.
	${\displaystyle t={\frac {E}{v_{\mathrm {gl} }}}+{\frac {(v_{\mathrm {si} }+v_{\mathrm {met} })\cdot E}{v_{\mathrm {gl} }\cdot v_{\mathrm {st} }}}}$	Insert (3) in (2).
${\displaystyle \Rightarrow }$ (4)	${\displaystyle t=E\cdot ({\frac {1}{v_{\mathrm {gl} }}}+{\frac {v_{\mathrm {si} }+v_{\mathrm {met} }}{v_{\mathrm {gl} }\cdot v_{\mathrm {st} }}})}$	This is the total time as a function of the (pre-flight) speed. In order to determine the optimum now, one differentiates this equation according to the speed and sets this to zero.
(5)	${\displaystyle {\frac {\mathrm {d} t}{\mathrm {d} v_{\mathrm {gl} }}}=E\cdot (-{\frac {1}{v_{\mathrm {gl} }^{2}}}+{\frac {{\frac {\mathrm {d} v_{\mathrm {si} }}{\mathrm {d} v_{\mathrm {gl} }}}\cdot v_{\mathrm {gl} }\cdot v_{\mathrm {st} }-(v_{\mathrm {si} }+v_{\mathrm {met} })\cdot v_{\mathrm {st} }}{(v_{\mathrm {gl} }\cdot v_{\mathrm {st} })^{2}}})}$	The distance is always not equal to zero, so the expression in brackets must be zero.
${\displaystyle \Rightarrow }$	${\displaystyle -1+{\frac {{\frac {\mathrm {d} v_{\mathrm {si} }}{\mathrm {d} v_{\mathrm {gl} }}}\cdot v_{\mathrm {gl} }-v_{\mathrm {si} }-v_{\mathrm {met} }}{v_{\mathrm {st} }}}=0}$	This result can also be written down as a ratio equation.
(6)	${\displaystyle {\frac {\mathrm {d} v_{\mathrm {si} }}{\mathrm {d} v_{\mathrm {gl} }}}\cdot v_{\mathrm {gl} }=v_{\mathrm {si} }+v_{\mathrm {met} }+v_{\mathrm {st} }}$	The so-called speed-to-fly equation.

Determination of the optimal pre-flight speed

The term can be plotted as a point on the Y-axis. The optimal speed is obtained by applying a tangent to the aircraft polar. This is expressed by the term on the left side of equation (6). The determined, optimal travel can be read on the Y-axis. ${\displaystyle v_{\mathrm {si} }+v_{\mathrm {met} }+v_{\mathrm {st} }}$

The practice goes something like this: The pilot first estimates the strength of the next updraft and sets this value on his speed command (mostly integrated in the variometer ). During the gliding flight, it varies the speed depending on the meteorological up and down winds ( ) according to the specifications of the speed command sensor. When the air mass drops, the pilot flies faster; when it rises, slower. With this approach, he optimizes his speed and reaches his goal faster overall. For this purpose, the glider pilot, physicist and engineer Paul MacCready invented a rotatable ring that is attached to the variometer. You can read off the optimal speed directly. Paul MacCready won the World Gliding Championship in 1956. ${\displaystyle v_{\mathrm {met} }}$

Approximate calculation of the optimal pre-flight speed

The speed polar of the aircraft represents the relationship between the airspeed and the aircraft's own rate of descent. The polar can be approximated by a quadratic function . By inserting the approximation equation into the speed-to-fly equation, the optimal trip can easily be calculated.

${\displaystyle v_{\mathrm {gl} }=}$ Airspeed (gliding)

${\displaystyle v_{\mathrm {si} }=}$ Rate of descent (positive number!)

${\displaystyle v_{\mathrm {st} }=}$ (Expected) rate of climb in the updraft

${\displaystyle v_{\mathrm {met} }=}$ Meteorological movement of the air mass (rise / fall) when gliding

${\displaystyle a,b,c=}$ Quadratic function coefficients, they depend on the type of aircraft and the wing loading.

(1)	${\displaystyle v_{\mathrm {si} }=a\cdot v_{\mathrm {gl} }^{2}+b\cdot v_{\mathrm {gl} }+c}$	Airplane polar, approximated
(2)	${\displaystyle v_{\mathrm {si} }'=2\cdot a\cdot v_{\mathrm {gl} }+b}$	The approximation equation differentiates according to the speed.
	${\displaystyle (2\cdot a\cdot v_{\mathrm {gl} }+b)\cdot v_{\mathrm {gl} }=a\cdot v_{\mathrm {gl} }^{2}+b\cdot v_{\mathrm {gl} }+c+v_{\mathrm {met} }+v_{\mathrm {st} }}$	(1) and (2) have been inserted into equation (6) of the previous chapter (The optimal pre-flight speed).
${\displaystyle \Rightarrow }$	${\displaystyle 2\cdot a\cdot v_{\mathrm {gl} }^{2}+b\cdot v_{\mathrm {gl} }=a\cdot v_{\mathrm {gl} }^{2}+b\cdot v_{\mathrm {gl} }+c+v_{\mathrm {met} }+v_{\mathrm {st} }}$
${\displaystyle \Rightarrow }$	${\displaystyle a\cdot v_{\mathrm {gl} }^{2}=c+v_{\mathrm {met} }+v_{\mathrm {st} }}$
${\displaystyle \Rightarrow }$ (3)	${\displaystyle v_{\mathrm {gl} }={\sqrt {\frac {c+v_{\mathrm {met} }+v_{\mathrm {st} }}{a}}}}$	The optimal speed (speed to fly) as a function of the coefficients a and c as well as the meteorological air movement during gliding and the expected strength of the next updraft.

Scale of the target travel ring

Speed-to-fly curves according to the MacCready theory

An example should explain how the above speed-to-fly equation (6) is mapped on a scale and is used in practice in the speed-to-fly ring according to MacCready.

The present diagram is based on the speed polar of an LS4 : The value of the total sinking is plotted on the x-axis, i.e. practically the display of the variometer, and the sliding speed is plotted on the y-axis. The dashed curve is the result of the formula (6) for an expected increase (the differential quotient was determined numerically from the polar). If you have an expected climb , the curve shifts to the left by the corresponding value, which is done in practice by turning the MacCready ring around the estimated climb value. The strongly highlighted red curve corresponds to the speed to fly for an expected increase in and is used in the following example. ${\displaystyle v_{\mathrm {si} }+v_{\mathrm {met} }}$ ${\displaystyle v_{\mathrm {st} }=0}$ ${\displaystyle v_{\mathrm {st} }>0}$ ${\displaystyle v_{\mathrm {st} }=2m/s}$

The diagram also shows the curves of total sinking according to the polar for various values ​​of meteorological air sinking for . ${\displaystyle v_{\mathrm {met} }=0\ldots 5m/s}$

As an example, we take an expected increase of 2 m / s and a meteorological air mass decrease of 4 m / s as a basis; So the speed-to-fly curve highlighted in red and the polar in green apply. Assume we are now flying in the gliding phase at a speed of 100 km / h (red point); then the variometer shows a sink of 4.7 m / s according to the green curve. The red MacCready curve gives an optimal speed of 185 km / h for this descent. If we press the stick to reach this speed, the new sink rate is about 6.7 m / s. The optimal value for the gliding speed here is 200 km / h, and so on ... As you can see from the gray arrows, we iteratively approach the optimal value through this specification, which is the intersection of the polar with the MacCready curve is defined (blue point); we always move along the curve for 4 m / s (air mass decrease here assumed to be constant).

MacCready ring with setting 2 m / s for expected climb; schematic

The MacCready Ring (see picture on the right) has a circular scale that is labeled non-linearly according to the dashed curve and is attached to the variometer so that it can be rotated. If you set the zero point of the ring (marked by an arrow) to the variometer value of the expected climb (i.e. to a positive climb value), the corresponding optimal gliding speed can be read off the ring for each sink value in gliding flight . Of course, you can only approximate the optimal guideline value, in practice it makes no sense to want to achieve exactly this value.

Since the polar changes with increasing flight altitude due to the decreasing air density , the use of the speed control ring is basically only a useful approximation. Modern electronic speed control sensors combine all existing measured values ​​automatically and also make the corresponding corrections, which means that the speed control ring is not in the form described here has more of its original meaning.

swell

• Helmut Reichmann - distance gliding , Motorbuchverlag, Stuttgart 2005, ISBN 3-613-02479-9
• John Cochrane - A little faster please , German translation , original (English) ( MS Word ; 144 kB)

Individual evidence

1. ^ List of world gliding champions ( memento from February 6, 2010 in the Internet Archive ) of the FAI

Web links

• The optimal speed (speed to fly)