The distance an airplane floats with respect to the ground during deceleration at constant altitude is analyzed taking into account the effects of a constant wind. By use of suitable nondimensionalizing parameters, data applicable to all airplanes are presented by means of a single family of curves

Phillips, W. H.

English

NASA Technical Reports Server

NASA Technical Paper 1157
Ground Distance Covered
During Airborne Horizontal
Deceleration of an Airplane
APRIL 1978
'
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NASA
https://ntrs.nasa.gov/search.jsp?R=19780012172 2020-03-22T04:31:04+00:00Z
NASA Technical Paper 1157
Ground Distance Covered
During Airborne Horizontal
Deceleration of an Airplane
William H. Phillips
'Langley Research Center
Hampton, Virginia
NASA
National Aeronautics
and Space Administration
Scientific and Technical
Information Office
1978
SUMMARY
An analysis is presented of the distance an airplane floats with respect
to the ground during deceleration at constant altitude, taking into account the
effects of a constant wind. By use of suitable nondimensionalizing parameters
suggested previously in the literature, data applicable to all airplanes are
presented by means of a single family of curves. The application of these
curves in a typical example is included.
INTRODUCTION
A study of the phase of flight in which an airplane decelerates at constant
altitude has several practical applications. If an airplane completes the land-
ing flare with excessive speed, the pilot may elect to hold the airplane off the
ground until a suitable touchdown speed or attitude is reached. General-aviation
airplanes operating at large airports frequently approach at speeds higher than
normal to maintain their spacing in the traffic pattern. Measurements have
shown that, even in routine operations, many pilots land with excessive speed
(ref. 1). The flight path of an airplane after an aborted take-off is another
situation which may involve deceleration at constant altitude. Some situations
exist involving an excessive tendency to float during a landing. For example,
landing with a tail wind increases the distance covered. The effect of encoun-
tering a head wind at low altitude is an increased initial airspeed which may
also lengthen the floating distance. Sailplanes, because of their low drag,
may float excessively in landing, particularly if a deceleration device such as
a drag parachute fails to work. In all these applications, the presence of wind
has an important effect on the ground distance covered.
A rational analysis of the landing of an airplane is given in reference 2.
This analysis includes the phases of glide, flare, deceleration at constant alti-
tude (called the float phase), and the ground run. Constant deceleration (corre-
sponding to constant lift-drag ratio) was assumed during the float phase, an
approximation that was adequate for the purpose under consideration. Approxi-
mate methods for the calculation of landing distance sometimes neglect the float
phase entirely (ref. 3). Some consideration of the role of the float phase in
determining certification standards for the landing distance of transport air-
planes is given in reference 4 but again the assumption of constant lift-drag
ratio was made. In cases in which the flare is completed with considerable
excess speed, however, a more accurate method for calculating the distance
covered during the float phase would be desirable.
A solution of the equations governing deceleration in level flight is pre-
sented in reference 5. The analysis of reference 5 shows that by use of suit-
able nondimensionalizing parameters, the solutions for air distance and airspeed
may be expressed by means of a pair of curves applicable to .all types of air-
planes. The present report extends these results by including the effects of a
constant wind and solving for the ground distance covered during deceleration at
constant altitude. By use of the same nondimensionalizing parameters as those
suggested in reference 5, the results for a series of values of wind velocity
are given by a single family of curves applicable to all types of airplanes.
SYMBOLS
C constant of integration
CD drag coefficient, p
P U2S
2
CD,o profile drag coefficient
CL lift coefficient, L
P U2S
2
CL,max maximum lift coefficient
D drag
g acceleration due to gravity (Ig = 9.807 m/sec2)
K variation of CD with C^2
L lift
lp aerodynamic penetration, 2m
PSCD,0
m mass
S wing area
s float distance with respect to ground
s1 nondimensional float distance with respect to ground, s
O true airspeed
Ug ground speed
Ur reference velocity,
Os true stall speed
Uw wind speed
U1 nondimensional airspeed, u
Or
Ua' nondimensional ground speed, _£
Ur
Us' nondimensional stall speed, s
"r
Uw' nondimensional wind speed, w
Ur
W weight
x = U'2
Y flight-path angle
p air density
A dot over a symbol denotes differentiation with respect to time.
ANALYSIS
The vertical and longitudinal forces acting on an airplane in gliding
flight with wings level are shown in figure 1. The longitudinal equations
in flight-path axes are
2
U + g sin Y + (CD,0 + KCL2)? = 0 (1)
m
P
 02S
Ut + g cos Y - CL i _ = 0 (2)
m
In level flight, Y = Y = 0- Equation (2) then becomes
Substituting this value in equation (1) gives
•§ U2s 2U + CDf0 2 _ + K_Eil_=0 (4)
' . § "2s
If the velocity over the ground is defined as Ug = U + Uw, where Uw is a
steady wind directed along the flight path, the differential equation govern-
ing the distance an airplane will float over the ground at constant altitude is
U = u £% =
2m pS(Ug .
or
^g
 = CD>oPS (Ug - Uw)2 _ 2Kmg2 (5)
ds 2m ug pSUg(Ug - Uw)2
As shown in reference 5, the solution of this equation may be simplified by
expressing distance in terms of a reference length called the aerodynamic pene-
tration which is defined by the following equation:
= 2m
c
 PSCD,0
and by expressing velocity in terms of a reference velocity given by
This reference velocity is the speed at which CDfO = KC^2 and is therefore
the speed for the maximum ratio of lift to drag. Let s1 = s/Zp, Ug' = Ug/Ur,
and Uw' = Uw/Ur. Equation (5) then becomes
d
 < - u w ' ) 2 1
ds' Uo' n 'fn ' — n M 29 ug vug uw '
Separating the variables yields
dUg'
(V - "w')2 +
ug '
 ug' (V ~ u w') 2
If equation (7) is integrated,
•!'• ' (Ug ' - U w ' ) 2 dUg'(U' - U w ' ) 4 + 1 (8)Jg ~ W
Let U1 = Ug' - Uw', or Ug' = U' + Uw', and dUg' = dU'. Then equation (8)
becomes
_s, = A"' ^ U W ' ) U ' 2 du, = /"U'3 ^ U w ' U ' 2 du,
] U'4 + 1 J U'4 + 1
(9)
Considering the terms separately gives
(10)-s' = f"'3 dU' + V fu'2 du'
J U'4 + 1 J U'4 + 1
Let x = U'2 and dU1 = dx/2U'. Equation (10) can then be written
_st _ 1 C x dx + uw' f\yx dx (11)
2
 J x2 + 1 2 J x2 + 1
These integrals may be found in a table of integrals such as that given in ref-
erence 6. The solution is
-s' + C I ln(x2 + 1) - HSL I lnML± ,_
4 2\f2 2 \x - \/2x + 1
+ 1 _ tan-l
1 - x
(12)
The numerical solution of equation (12) may lead to difficulties if x = 1,
because of the infinity occurring in the1 argument of the tan~l term. The term
itself, however, approaches rr/2 under these conditions. The correct quadrant
for the tan"-'- term must be used. If the value of the term \/2x/(l - x) is posi-
tive (x < 1), the angle should be between 0 and ii/2, whereas if the value is
negative (x > 1), the angle should be between ir/2 and IT.
RESULTS
The curves of figure 2 show in nondimensional form the float distance s1
required to decelerate as a function of airspeed U1, for various values of tail
wind or head wind. The results, based on equation (12), are presented in terms
of the nondimensional parameters
= s U uwJw
where lp is the aerodynamic penetration defined as 2m/pSCDfO. As was
mentioned previously, these nondimensionalizing parameters are suggested in
reference 5.
The values of s1 are plotted as positive quantities in figure 2. The
negative sign of s1 shown in equation (12) is ignored. This negative sign
in the formula results from the fact that the distance is increasing in a posi-
tive direction as the airplane decelerates. If the point s1 = 0 corresponds
to the point where the ground speed is zero, the airplane is, strictly speaking,
in the region of negative values of s1 during the period of deceleration. .
The constant of integration C in equation (12) may be used to place the
zero of the scale of s1 at some arbitrary location. In the data presented,
the value of C is determined such that, for tail winds, the value of s1 = 0
when U1 =0, and for head winds, the value of s1 = 0 when U' = 0 or
u- = -uw'
q' 
In practice, the float distance required is the distance covered between
some initial and final values of airspeed. This distance may be obtained as
the difference between the values of s1 at these values of airspeed. The
maximum float distance is usually obtained if the final airspeed is the stall
speed. The nondimensional stall speed is
A vertical line may be drawn in figure 2 showing the airspeed at touchdown to
facilitate reading the values of float distance for various values of wind
speed. In the unlikely event that a head wind exists with velocity greater
than the touchdown speed, the forward progress of the airplane stops when
U1 = -Uw'. In this case, the maximum forward penetration of the airplane may
be obtained from the value of s1 at the initial airspeed, inasmuch as the
final value is plotted as zero when U1 = -Uw'.
DISCUSSION
The results of this analysis are based on the assumption that the drag is
given by an expression of the form CD = CDfO + KCj^ . If the airplane is float-
ing at low altitude, the values of CDfO and K may be influenced by ground
effect. Values appropriate to the altitude under consideration should therefore
be used. The primary effect of proximity to the ground is a reduction in the
induced drag as expressed by the value of K. At high values of lift coeffi-
cient, the ground effect may also reduce the effective dynamic pressure. If
the airspeed is considered to be the speed with respect to the distant air mass,
as is required for calculation of the distance covered, the effect of this
reduction in dynamic pressure may be approximated by appropriate modifications
in C£)fO and K. The effect of an idling propeller or jet engine may intro-
duce drag or thrust terms that differ in form from the assumed expression
Cjj = CDfO + KC^ . A more detailed analysis would be required to account for
these effects.
The value of float distance for a particular case may be read from figure 2
and converted back to dimensional form. If more accuracy is required, the value
may be calculated from equation (12). Alternatively, the entire plot presented
in figure 2 may be converted back to dimensional form for application to a par-
ticular airplane. An example is shown in figure 3, which presents the float dis-
tances for a general-aviation airplane with the characteristics shown in the
figure. The drag characteristics assumed correspond to a lift-drag ratio of
6.98 at the stall (CL = 1.2). This value of lift-drag ratio is representative
of the condition of flaps and landing gear down.
In reference 5, the float distance and time are each given as functions of
airspeed by expressions similar to the first and second terms, respectively, of
equation (12). Both time and distance are involved in the present analysis
because, with a head wind, the distance the airplane moves with respect to the
air must be corrected for the movement of the air with respect to the ground
during the time of flight. In the derivation presented herein, however, the
ground distance is determined as a function of airspeed directly, without the
intermediate step of determining time of flight.
CONCLUDING REMARKS
An analysis is presented of the distance an airplane floats with respect
to the ground during deceleration at constant altitude, taking into account the
effects of a constant wind. By use of suitable nondimensionalizing parameters,
data applicable to all airplanes are presented by means of a single family of
curves. These results provide a rational approach for estimating the contribu-
tion of this floating regime of flight to the distance needed for landings or
aborted take-offs.
Langley Research Center
National Aeronautics and Space Administration
Hampton, VA 23665
March 13, 1978
REFERENCES
1. Goode, Maxwell W.; O'Bryan, Thomas C.; Yenni, Kenneth R.; Cannaday, Robert L.;
and Mayo, Marna H.: Landing Practices of General Aviation Pilots in Single-
Engine Light Airplanes. NASA TN D-8283, 1976.
2. Glauert, H.: The Landing of Aeroplanes (Part I). Technical Report of
the Advisory Committee for Aeronautics for the Year 1919-20, Vol. II,
pp. 513-525. (Also available as R. & M. No. 666.)
3. Perkins, Courtland D.; and Hage, Robert E.: Airplane Performance Stability
and Control. John Wiley & Sons, Inc., c.1949, pp. 198-199.
4. White, Maurice D.: Proposed Analytical Model for the Final Stages of Landing
a Transport Airplane. NASA TN D-4438, 1968.
5. Larrabee, E. Eugene: Aerodynamic Penetration and Radius as Unifying Concepts
in Flight Mechanics. J. Aircr., vol. 4, no. 1, Jan.-Feb. 1967, pp. 28-35.
6. Korn, Granino A.; and Korn, Theresa M.: Mathematical Handbook for Scientists
and Engineers. Second "ed. McGraw-Hill Book'Co., Inc., c.1968, pp. 934, 938.
Forces acting °
n an
2.0
c
SS
12
c
I -8o
.4
Nondimensional
wind velocity, Uw
> Tail wind
> Head wind
0 1 2 3 4
Nondimensional true airspeed, U'
Figure 2.- Nondimensional float distance required to decelerate as a function of
initial airspeed for various values of wind, s1 = £_; u1 = U
V »
Jw 2m
PSCD,O
2000
1600
1200
o>
o
5 800
400
Stall speed, U
CL = 1.2
Wind velocity, U . m/sec
w
20.91,
>Tail wind
Headwind
True airspeed. U, m/sec
Figure 3.- Float distance for a typical general-aviation airplane as a
function of initial airspeed. Cp = 0.1 + 0.050^ ; ra/S = 68.72
p = 1.226 kg/m3; lp = 1121 m.
10
1. Report No.
NASA TP-1157
2. Government Accession No. 3. Recipient's Catalog No.
4. Title and Subtitle
GROUND DISTANCE COVERED DURING AIRBORNE HORIZONTAL
DECELERATION OF AN AIRPLANE
5. Report Date
April 1978
6. Performing Organization Code
7. Author(s)
William H. Phillips
8. Performing Organization Report No.
L-12008
9. Performing Organization Name and Address
NASA Langley Research Center
Hampton, VA 23665
10. Work Unit No.
505-06-63-02
11. Contract or Grant No.
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, DC 20546
13. Type of Report and Period Covered
Technical Paper
14. Sponsoring Agency Code
15. Supplementary Notes
16. Abstract
An analysis is presented of the distance an airplane floats with respect to the
ground during deceleration at constant altitude, taking into account the effects of
a constant wind. By use of suitable nondimensionalizing parameters, data applicable
to all airplanes are presented by means of a single family of curves.
17. Key Words (Suggested by Author(s))
Float distance
Landing distance
18. Distribution Statement
Unclassified - Unlimited
Subject Category 05
19. Security Qassif. (of this report)
Unclassified
20. Security Classif, (of this page)
Unclassified
21. No. of Pages
10
22. Price*
$4.00
* For sale by the National Technical Information Service, Springfield, Virginia 22161
NASA-Langley, 1978
National Aeronautics and
Space Administration
Washington, D.C.
20546
Official Business
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Ground distance covered during airborne horizontal deceleration of an airplane

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