KoAaalstKiat 10 - DELFT REPORT No. 57
1 9 JULI 1952
THE COLLEGE OF A E R O N A U T I C S
CRANFIELD
OPTIMUM CLIMB TECHNIQUE FOR A JET PROPELLED
AIRCRAFT
by
WING COMMANDER L. KELLY. B.Sc, D.C.Ae.E., A.M.I.Mech.E., A.F.R.Ae.S.
This Report must not be reproduced without the permissior) of the Principal of the CoHege of Aeronautics.
1 9 JULI 1952
REPORT NO. 57APRIL. 1952
T H E C O L L E G E O F A E R O N A U T I C S
C R A N F I E L D
Optimun Clitib Technique f o r a J e t P r o p e l l e d A i r c r a f t *
b y
-Wing Corxiander L. K e l l y ,
B . S c , D.C.Ae., A.M.I.Mech.E., A . F . R . A e . S .
S U M M A R Y
The present nethods for obtaining a climb technique for jet propelled aircraft do not include the effect of kinetic energy variation v/ith height. By introducing the concept of
'energy height' to include the geometric height and the height equivalent of the kinetic energy, a nore exact treatnent of the optinim technique has been possible. A new nethod has been suggested for obtaining the energy height clinb function.
The energy height optinun clinb has been compared 7/ith the existing techniques to assess the advantages vrhen specified end conditions are included, the conparison being illustrated by considering a modern fighter project.
Although no great advantage caaes fror.^ using the energy height optitiun clinb betvreen given energy heights, a suggestion for zoom clinbing at the end of the steady clinb has been shown to give a saving of 1 nin. 43 sees, in the nininun
time to a height of 40,000 ft. for the aircraft considered.
An extension of the energy height nethod shows how a saving of up to 10 sees, in the tine to Liaxinim speed at sea level is possible.
Part of Thesis presented for Diploma, Jvixie 195'1.
COrWENTf Page 1, Introduction 4 2, Notation 5 3, P a r t I , Optimum Climb 6 3 . 1 . I n t r o d u c t i o n 6 3.2. The fundamental equation 7
3 . 3 . Approximations used i n the s o l u t i o n of the 7 fundamental equation
3»3,1 The v a r i a b l e s involved 7 2
• 3 . 3 , 2 The terra K D ^ ( ^ . ^ ) 8
3.3,3 The term in dV/dH 8
3.4. The concept of 'energy height' 9
3.5. Application of the 'energy height' concept 10 to the derivation of optim\an climb technique
3.5.1 Method I 11 3.5.2 Method II . 1 2 3.5.3 Method III 15 3.6. Results 15 3.6.1 Method I 15 3.6.2 Method II 19 3.6.3 Method III 20
3.7. Appreciation of the separately deduced 20 optimum climb techniques
3.8. The end conditions 22
3.9. Qiiantitative assessment of the advantage of 23 using the energy height technique
3.10. Optimum climb to height when the end conditions 25 are included
3.10.1 Optimum climb to geometric height H 25 and the minimum control speed V . (EAS)
^ mm '
«
3.10.2 Optimum climb t o geometric h e i g h t H 27 and the maximian speed V
max
3.11. Method of obtaining the optimum climb r e - 28 l a t i o n s h i p for any p a r t i c u l a r a i r c r a f t
3.12. Conclusions 29
4. Part II. Distance Climb 30
4.1. Introduction 30
4.2. Minimtra tine to height at a particular 30 radius when fuel econony is not essential
4.3. Best time to height at a particular radius 31 when fuel econony is a prime consideration
4.4. Conclusions 32
5. Part III. Initial Climb for Long Range Flight 33
6. Part IV, Initial Clinb for Tine Endurance Plight 34
References 34
Appendices
I. Specification of 1950 Fighter Project and 35 assimptions -ased in the analysis
II. Proof that the nininun tine to energy height 37 leads to the nininun tir.ie to ge cos trie height
for the same end conditions
The full tactical operation of a military aircraft requires the knowledge of a clinb technique which enables the aircraft to reach the greatest possible height and distance in the shortest possible tine, using the nininun amount of fuel, and at the sane tine reaching the highest possible end speed. A study of the paraneters involved shows that there is no gen-eral solution to this problen. It vra.s decided, therefore, to divide the study into four ncin parts, each part covering a specific operational requirenent, and capable of further
sub-division to neet the detailed tactical enplqynent of the aircraft.
Part I deals with the first operational case 'optinun clinb'. This is defined as the case meeting the requirenent to reach a certain height and end speed as quickly as possible, It is, in essence, the technique required of a fighter to inter-cept a bonber which is in visual or radar sight of the operational unit.
Part II covers the 'distance clir.ib' case. It rep-resents the requirements of a fighter to intercept the eneny at a particular radi\as f ran a nilitary objective, and so prevent the jettisoning of bonbs dtiring the interception phase causing unplanned, but nevertheless serious, damage.
Part III detemines the initial clinb technique in a long range flight. This is the heavy bonber, reconnaissance, sea patrol, or heavy bonber fighter-escort case.
Part IV deals with the initial climb requirenent in a tine endurance flight. This case covers all 'patrol' tactical reqtiirenents, whether it is the fighter awaiting vectoring
instructions, or the close support aircraft awaiting information on targets from the ground troops, or the search aircraft in s. coastal or reconnaissance role.
2. Notation
T Thrust
V True a i r speed
V. Equivalent a i r speed =: Y y c
V Rate of clLnb
V ,V. Optinijn speeds for clii.ib
C xO
V Q , V . „
'Quasi-optinim' speeds
V ,,V. -, Speed for minimijm di-ag
md' ind ^ '^
H Geometric height of a i r c r a f t above sea l e v e l
H Energy height of a i r c r a f t = H + vV2g
f(Hg,V) d H / d t
0(H^,V) dt/dH^
':X(H , v ) d i i ^ d t
D Drag of aircraft
D, Drag of aircraft in straight level flight
KD, Induced drag of aircraft in straight level flight
W Weight of aircraft
\
V\d
n K /^. v öT^mm
cr Relative a i r density
v.A. .
r m d
I n c l i n a t i o n of f l i g h t path above h o r i z o n t a l
_ K _ ^W)= __ ,_,,,, , g j
Ind-uced drag f a c t o r i n equation Gj, = C^. + EC.
/(TCA)z
b Wing-span
3, PART I. OPTDCTJ CLU'/IB
3.1. Introduction
The short duration of radar warning, and the high speed and altitude of bomber aircraft has made the ninimum time Inquired to climb to any given altitude from ground level the most important characteristic of fighter perfomance. A few
seconds lost can spoil the chances of interception,
In general the optinim technique will be markedly influenced by the end conditions. Ideally, an acadenic survey shotild determine the shortest possible time to pass from any one combination of height and speed to any other, and the flight technique to be adopted dxjring the transition between the two states. Such a generalisation would almost certainly prove too unwieldy in practice. The pilot under the high stress and emotion of an operational duty cannot be expected to have the nicety of mathematical appreciation of the tranquil scientist. Consequently the end conditions studied in this report have been restricted to those most likely to occ\jr in nilitary operations, and the transits on stage between end conditions reduced to the simplest possible terns.
The high rate of fuel consumption of turbo-jet air-craft makes it probable that fighter airair-craft will start the interception direct fraa take-off without a prior period of patrol. This consideration sets the initial conditions for the climb. When the required interception altitude is reached there are two tactical possibilities:
(i) for a head-on or bean attack the speed of the inter-ceptor is tininportant, and the nain consideration is that height should be reached in the least possible tine so as to cover the case of least warning
(ii) for a stem attack, or any attack in which 'closing-in' or pursuit of the enemy is essential then the desired height should be reached at the greatest possible speed.
These two reqtiirenents determine the final end con-ditions for the most likely tactical enplqynent of the aircraft,
3, 2. The Fundamental E q m t i o n
The fundamental equation for the longitudinal notion
of the aircraft climbing in g at angle Y ^^ still air can be
written approximately
W dV / \
T - D - W sin Y = -
I T• (l)
g dt
If we further denote the total drag of the aircraft
in straight level flight by D, and the induced drag in this
condition by K D , then for rectilinear flight at the sane
speed along a path inclined to the horizontal at angle Y the
induced drag is K D, cos y and equation (1) may be written»
T - |(1_K)
\ + ^ \
cos^
r\ - ^^
sin Y = I ^
2 ¥/• dV i . e . T - D, + K D, s i n y - ''^ s i n Y = - T T • h h g d t ^ . . 1 dH But s m y = :^ ^ ,therefore the above may be written:
T-D, . K D J I § f - I § = ^ § (2)
h h \V d t / V d t g d t ' W dH f, V
°^ V dt f + i
i ] -~'-\^^\(j §ƒ• -(3)
This is the fundamental equation for the climb from
which the reqioisite technique must be deduced.
3.3» Approximations used in the solution of the Fundamental
Equation
3.3,1 The variables involved
The variables T, D, and K in the fundanental
eqijation are functions of air pressure, air temperatvire, W
and V, In a given atmosphere, therefore, they are functions
of Iff, V, and H,
The weight of the aircraft W is a variable during
the clinb, but as a first approximation vre may take W as
constant, and use the standard methods of perfomance reduction
t o d e t e m i n e t h e e f f e c t on t h e c l i n b t e c h n i q u e of w e i g h t v a r i a -t i o n w i -t h h e i g h -t . \7e nay t h e n s a y : , , ^ f (H, V, j - : - ) . 3 . 3 , 2 The t e r n K
c D, r i
^f
/l '-"'^T V 2 The tern K D^ (" ^•.':j i.e. K D, sin y represents the effect of the reduction, in induced drag due to inclination of the flight path to the horizontal. This tern is small except in zoom climbs, and will be ignored in the detemination of the first approximation of the clinb technique between specified end conditions. Its effect on a continuous clinb for a fighter is shown in a later stage of this report to be less than 0.2 per cent.The fiindamental e q u a t i o n now becomes:
[•
w an K ^ v dv = ^ -
D ,.
V d t I g dH ( h
3. 3, 3 The term in d^l/dE
For propellered aircraft the term in dV/dH, which is a measure of the acceleration along the flight path, may be neglected, because the rate of clinb is sirfficiently small for
the acceleration to be neglected.
Thus for propellered aircraft the fundamental equation becomes:
i - o = * - V f f - w
In the case of jet propelled aircraft, however, thespeed in climb is so high that speed variations mean large changes in the energy transfomed, and the term can no longer be omitted. Calculations in a later part of this report show that for a modern project fighter with 40 svreepback the effect of this t e m on estimates of the rate of a continuous climb is
2
proportional to V , and nay be as high as 12 per cent. Simpli-fication of the eqtiation without a.iitting the acceleration tern was achieved by the G e m a n s who introduced the concept of 'energy height', (reference l ) .
3.4. The concept of 'enerjTy h e i g h t '
The enei'gy height H i s defined as the sum of the geometric, or standerd a l t i m e t e r h e i g h t , and tlie speed height, which i s the height t o ?^iich the a i r c r a f t coiold c l i n b by v i r t u e
of i t s K.E.
Thus H^ = H 4- -^
Fran tlii.'s definition, novenent along a curve of con-stant energy height proceeds ideally withoiit consunption of energy. This neans that the speed along the path with which a definite energy hjeight is attained is immaterial because any other speed nay be reached at the sraae energy height without consijuptlon of enex'gy. Calculations done, therefore, with the energy height, instead of the height of flig?it, need no correc-tion for acceleracorrec-tion in the flight path. This can readily be demonstrated from the equation of energy for the clinb along a small element of the flight path between heights H. and H»
(so that T and D may be assumed constant).
We have from equation (l):
TV 5t = D^ V 5t + Wdl^-H^) + W (V^ - Y^) /(2g) H^-H^ + (Y^ _ ^,2^ /(2g) ,^ i.e. 6t = T eg e^ T - D ^ - \ W • V • V
But V ( T - D , ) /¥/• from eqv^ation (4) i s the r a t e of c l i n b v for fliglrb uncorrected for a c c e l e r a t i o n .
Therefore 5 t = (H - H^ ) / V „ . (6)
Sp ®>i "
The f\mdamental equation approximated as i n d i c a t e d i n t h i s t e x t , v i z :
T - D, h
W dJI r V dV]
V dt [_ g ^-i J "
may be written in terms of this new concept;
d Hg/'db = (T - D^) V/W .
Since T and D, are functions of H and V, they
KanaolsUaot 10 - DELFT
-10- 1 9 J Ü Ü 1 9 5 2
are a l s o fimctions of H and V and so -we may w r i t e :
d H ^ d t = (T - D^)V^ = f(Hg,V) (7)
Th\is for a degree of approximation in which sin'" y is neglected, the climb performance of an aircraft of given weight depends on two variables.
3.5. Application of the 'energy height' concept to the derivation of optimum climb technique
Consideration of the energy height concept gives a new approach to the solution of the fundamental equation when the climbing speed is so great that the acceleration term must be included. Previous practice has been to base the best climbing speed on 'partial climbs' made at constant equivalent air speed V. = V v ' ^ and mean height H. At each height on the climb the air speed chosen is such that;
3v /SV. = 0 at H = constant, c 1
V is the rate of climb in a partial climb at con-c
stant equivalent air speed through a small ijange of altitude about the height in question. The partial differentiation is made with H constant and the speed chosen is independent of the acceleration along the climb path. The rate of climb by
this method is, therefore, overestimated in the ratio 11
- e ^ m
+ — ^Y. ^ >On introducing the new concept, 'energy height', the climb equation, including the acceleration term is (equation 7 ) :
dH „
and t h e r e f o r e , i f t^ i s the time required t o change fron
to H
1 ^2
H to H according to a particular technique;
Gj, e
n\
1*2 l/dH /dt dH ' ^ \ 0 (H^,V) dH^ , say. H a E 0 _. "1 ^1The requirement i s f o r the v a r i a t i o n of v e l o c i t y with energy h e i g h t , v i z . V = 'y-(H ) , t o be so chosen t h a t t h i s i n t e g r a l i s a minimtira.
E u l e r ' s condition for a s t a t i o n a r y value of t h i s i n
-t e g r a l reduces -t o
dH = 0 H c o n s t , eThus there is a stationary value of the 'time to climb' integral whenever the partial derivative with respect to air-speed of the rate of change of energy height, ?d.th the energy height kept constant, is zero,
It will be noted that the conditions have been deter-mined far a minimum time to energy height, but it is by no means
self evident that this is equivalent to minimum time to geometric height. The equivalence of the conditions is proved in Appendix II,
The quantity dH /dt is eq-ual to the rate of climb which would be obtained in a partial clinb made at constant true airspeed. Th\as the optimum climb teclinique night be derived from partial climbs made at constant true airspeed, with the mean energy height kept constant.
From the considerations of the previous paragraphs three methods have been developed for assessing the optimum climb technique in the transition stage between specified end conditions. In each case the method has been applied to the 1950 Fighter Project designed in the Department of Design of the College of Aeronautics. The specification of the project is detailed in Appendix I.
3.5,1 Method I
This nethod is attributable to Lush although it was not explicitly used in reports A. and A.E.E./Res/237 and A. andA,E,E./Res/243.^'^
In this method numerical valvies for thrust and drag obtained either by experiment or analytical assessment are used to determine dH /dt and thus 0 ( H ,V) at constant H .
0 ( H , V ) is then plotted against true air speed V as in
Figures 1 and 2, The values of V and 0 ( H , V ) corresponding
*° ÖV
Vl|/
=r Ü are then read from the curves and theIH const,
curve of ^(Hg,V)^^^^^^ against H^ plotted as in Figure 5.
The tine taken to clinb between two energy heights H and H
°1 ®2 is the area tinder this curve betvreen the respective ordinates.
The values obtained for H , V and dH /dt are then used to obtain a solution of the nore general performance equation
(including the sin y term) by successive approximation.
For comparison, the values of optimum clinbing speed and time to height as deduced from the R,Ae,S. data sheet EG 3/l have been obtained.
3.5,2 Method II
Lush has used an approximate nethod based on the theory o u t l i n e d below.
The fvindamental equation, n e g l e c t i n g the s i n " y term, has been shown t o be of the form
d H ^ d t = f(H^,V) = '?^(H,V),
and the Euler condition states that the air speed required for optimim climb is such that
df/av = 0.
Since èf/dV depends on derivatives of thrust and drag measured on an energy height scale, it is not readily expressible in terms of familiar quantities. The value d /Vav is more easily deduced, as the thrust and drag terns are now expressed in geometric height units,
For the relationship between these two we nay proceed as follows:
since H and V are independent variables,
and f (H^,V) = f (H + vV2g,V) = 9^(H,V) it follows that and V
i
af af a A an ^ sv ~ d Y e af d7s an ~ a H • /Therefore ..,Therefore - - r ^ + -rrr = -r^ ,
g a H av a VThus the condition af/av = 0 i s equivalent t o
d9{ Y d7\
a V ~ g a H
Since (v/g) (d'/^/an) is small and negative (see Figure 11) it follows that the true optimum speed is a little higher than the
quasi optinim speed at which d9^/èY is zero. Lush claims that the difference is usually about 5 per cent if no compressibility effects a3re present. Method I gave a value of 6 per cent for the exanple chosen, at heights up to 20,000 ft., but this
in-creased to 8|- per cent at 28,000 ft. at which height compressibility effects became apparent: This method in its essence is an
approximate estimate of the optimum speed made by estimating the quasi optimum speed and adding 5 per cent to it.
An analytical expression for the quasi optimim speed may be deduced as follows:
9( = (T - D ) -J by equation (7) and the quasi
optimimi speed is given by the condition -r^ = 0. 2
If we neglect the sin y term, we may write:
^ = ^100 ^
{TOO)
^ ~
^'00\^ooJ
2 d.
(
V \ 1 c 00/JLf
VI00/This occurs when
i . e . when
md 100^ V imd 100 Then 2 tr = / D . mm •^100 ^100 '^100 "^. ^ i o o ;
= 2 » 1/4 •-^/^loo
"^100/and we
and we may write say.
P i
D~r" = 2
rain T h e r e f o r e s u b s t i t u t i o n i n t h e f u n d a n e n t a l e q i i a t i o n y i e l d s : For j e t a i r c r a f t c l i n b i n g a t a i r speeds n e a r t h e o p t i n i m , y i s n o t l a r g e and W/D . i s n e a r l y e q u a l t o t h e maximum r a t i o( L / D ) which i s a c o n s t a n t f o r t h e a i r c r a f t . Hence t h e above e q u a t i o n may be w r i t t e n :
Oi^c- / L \
TT1 / 2 1
N imd ^ max | mm \ m / s i n c e V = m V, , / 7 ^ . I t f o l l o w s t l i a t tlie c o n d i t i o n è %/dY = 0 i s e q u i v a l e n t t o : 0 = — . ^ èY \D . - 2 (^^ + ~2 ] ' + "^
( NX. . uv M ^ " ^ 3 / ÖV f • JD. av " M " " '
1 fon - M ^ 1
~ ~^ ) ÖV
f-mm V m / f-mm \ "^ ' ' „ . am Vg^ m S i n c e — = : ^ = V ' imd we o b t a i n n m T 1 / 2° =v
ID
. -^ n
mm V ^ 1 \ n aT m / 1 \• " T + D . d v - v " p ^ - i : )
n / ' mm \ ^ m / , 2 1 2 r / , V ^ T ^I . e . 3m - - ^ = J I '^ "^ T av / ~ ^^^'
m min V, / Hence i f /N i s t h e q u a s i optimim v a l u e of m,^Q=•/! (^ *h * y
since the real and positive solution is the relevant one.
Thus TSg is expressible as a fxmction of x only. This fxmction which is plotted in Pi gure 12 was used to detemine
the quasi optimun speeds for the 1950 Fighter Project, An esti-mate of the true optimum speed was then deduced and the resvilts are contained in Table VI, p. 19.
3,5,3 Method III
It is considered that this nethod has sane advantages over the two previously discussed.
Gmrves of equal rate of climb v , v/ithout correction c
for acceleration (either measured and reduced to unaccelerated flight, or calculated) are plotted in a V, H, H network in Figure 9. These curves were obtained from the v , V curves drawn in Figure 8,
It has been shown that the time 5t to clinb between two energy heights H , and H + 6H is 6t = &H /v . More-over in Appendix II it has been shown that the conditions for optimum climb between two energy heights are equivalent to the conditions for optimum climb betvreen t\TO geometric heights. Hence the optimim climb path V = " ^ ( H ) must occia: at the speeds where the curves of constant energy height are tangential to the v
c curves, i.e. where
a / N a
— ^^c\
=a-v
^e-
- H dH e dt= 0
e
In Figure 9 the optimum climb path is clearly A, B, C, D, - - - - and this is identical with the one obtained by the more laborious process of Method I.
The climbing times are obtained by graphicsal or numerical integration from the relation
t =
H
where v is the rate of climb uncorrected for acceleration. c
3.6. RESULTS
3.6,1 Method I
The results obtained for the 1950 Fighter Project using Method I for the determination of the optimum climb path
are summarised in Table I. Table I H
ft.
1,000 5,000 10,000 15,000 20,000 ' 5 , 0 0 0 50,000 50,000 55,000 •JO,000 dV dH f . p . s , / f .0016 . 0 0 4 .0072 .0108 . 0 1 2 0 .0138 .0157 - . 0 0 4 - . 0 0 4 0 V f . p . s . 674 6 9 0 . 8 720.1 7 6 2 . 7 814 870 927 Allow 9 0 5 . 5 8 8 5 . 4 881 V dVg an
.0325 .0847 .1583 .2536 .3030 . 3 7 6 3 .4632 ring f o r C - , 1125 - , 1 1 0 0 00
.00592 .00636 .00715 .00781 ,00875 .00994 .01182 1 168.91 157.23 139.86 1 2 8 . 0 4 114.28 100.60 84.60 V c 1 6 3 . 4 144.95 120.75 102.14 87.71 73.09 57.82 ! o m p r e s s i b i l i t y E f f e c t s .01149 .013^S .01750 8 7 . 0 3 73.21 5 7 . 1 4 9 8 . 0 6 82,26 5 7 . 1 4 mph 460 471 491 520 555 593 632 6 1 7 . 4 604 601 He 8 , 0 5 4 12,410 18,051 24,032 30,289 36,753 4 3 , 3 4 5 42,725 47,172 5 2 , 0 5 3The following relevant curves have been drawn:
Pigs. 1 and 2 - Curves of 0 ( H ,V) against V at a series of constant energy heights
Fig. 3
Fig. 4
Fig. 5
The optinum climb path starts at 46O m.p.h. at sea level and this corresponds to an energy height value of 7,069 ft. From this datum the time to various energy heights and the
corresponding geometric heights is as set out in Table II (neglecting compressibility).
- Curve of 0 ( H ,V) against V at 45,000 ft. energy height with and without allowance for conpressibility drag
- Optimum climb curve V against H in a network of constant energy heights
- Curve of 0 ^ ^ ^^e'"^^ against H^ giving minimum time to energy height.
Table I I
TIME TO HEIGHT UNDER OPTPMvI CLDffi CONDITIONS
Energy
Height
" ^ . . .7,069
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
Geometric
Height
H
f t .
S.L.
2,778
7,300
11,600
15,863
19,784
23,650
27,449
31,200
Time
Taken
sees.
-17.4
49.4
84.2
122.4
164.0
209.9
259.9
309.5
The r e s u l t s quoted i n T a b l e s I and I I v/ere deduced from t h e fundamental e q u a t i o n W "^^e WA m T^ tr T^ - 2
V
-ït = WTn = T - \ + K Dj^ sm Y
G 2 n e g l e c t i n g t h e t e m i n s i n y . A f i r s t a p p r o x i m a t i o n t o the e f f e c t of t h i s term on 0 ( H , V ) i s g i v e n i n Table I I I . T a b l e I I IVARIATION IN 0 ( H ,V) Y/HEN THE s i n ^ Y TERM G ^ .
IS INCLUDED PT THE FUNDAI.IENTAL EQUATION
Energy
Height
«e
10,000
20,000
30,000
40,000
Geometric
Height
H
2,778
11,600
19,784
27,449
Value of
KD^ S±n\
9.747
5.315
2.765
1.276
% V a r i a t i o n
i n
0
0.15
0.11
0.07
0,04
Since the variation in 0 is less than 0.2 per cent,
it is considered that the emission of the sin y term is justi-fied.
Calculations for the 1950 Fighter Project based on the Royal Aeronautical Society Data Sheet EG 3/l were done to enable
a comparison to be made between the existing technique and the one suggested by the energy height concept. These calculations are summarised in Table IV.
Table IV
OPTBIUM CLIMB PATH FOR 1950 FIGHTER PROJECT
AS' DEDUCED FROM R.Ae.S. DA.TA SHEET EG 3/l
Height H f t . S.L. 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 V , mph 430.4 439.4 451 463.8 476.7 491.3 504.6 518 535 % s 631.3 644.2 661.5 680.2 699 720.6 740 760 784.7 dV
df
.001 .0028 .0033 .0037 .0041 .0041 .0041 .0045 .0050 V dV- 3.4
- 8.8- 9.7
-10.1 -11.6- 9.4
- 6.4
- 7.7
- 5.29 V °1 (Not allowing for a c c -e l -e r a t i o n ) 174.5 157.5 143.3 129.4 115.5 102.1 88.7 72.7 . 4 3 . 4 V°2
(Allovdng for a c c e l -e r a t i o n ) 171.1 148.7 133.6 119.3 103.9 92.7 82.3 65.0 38.1 1_ V°2
.006725 .007485 .008382 .009624 .010787 .01215 .01538 .02624 Equiv-a l e n t Energy Height f t . 114W. 16794 22184 27587 33063 38503 43969 49562 Value of0
.006306 .006979 .007716 .008619 .009771 .011386 .01265 ,01546The Rcyal Aeronautical Society Data Sheet states that it is unnecessary in practice to correct the optimum climbing speed for acceleration along the flight path. A correction for acceleration is, however, given for the rate of climb, and this has been incorporated in the results of Table IV. The optimum climb speed obtained in this way has been plotted on Figure 4 for comparison with that deduced by Method I. The comparative rates of climb have been plotted in Figure 6 and the reciprocal of the rate of climb plotted against height in Figure 7 to give the times
to altitude. These times are summarised in Table V.
Table V
TIMES TO GEOMETRIC HEIGHT FOR 1950 FIGHTER PP.O.TECT AS DEDUCED FROM R.Ae.S. DATA SHEET EG 3/l
Ge came t r i e H e i g h t H Sea L e v e l 5,000 10,000 15,000 20,000 25,000 30,000 Time Taken s e e s .
-31.9
67.5
107.1
156.6
202.7 2 6 0 . 3 3.6,2 Method IIThe method of calculation for the optimum climb tech-nique using Lush's Approximate Method are detailed in Reference 1.
The results for the present example are summarised in Table VI.
Table V I
OPTnriBl CLBiBIMG SPEEDS AS DEDUCED BY LUSH'S
AEPROXIi,iATE METHOD Geometric H e i g h t H f t . Sea L e v e l 10,000 20,000 30,000 (vdthout s o m p r e s s i b i l i t y ) 30,000 (with c o m p r e s s i b i l i t y ) 40,000 (without o c a n p r e s s i b i l i t y ) 40,000 (with 3 c m p r e s s i b i l i t y ) V a s g i v e n by e n e r g y h e i g h t method. m. p . h .
459
491
555
•^632
1617
r730
J1.601
V a s g i v e n by R . A e . S . DateSheet m. p , h,430
451
477
505
-535
— V a s g i v e n by L u s h ' s a p p r o x i m a t i on m, p . h.457
483
545
599
-658
-Alethod III .,,
Kanaalstiaat 10 - DELPT
2 0
-3.6,3 Method I I I
The results obtained for the optimum climb path by Method III were, within the accuracy of the curve drawing,
ident-ical with those obtained by the more lengthy process of Method I.
The following relevant curves have been drawn:
Fig. 8 - Uncorrected rate of climb against speed at a series of gecmetric heights
Pig. 9 - Constant uncorrected rates of climb against speed in a network of constant energy heights.
3.7. Appreciation of the Separately Deduced Optimum Climb Techniques
The optimum climb techniques discïussed in § 5 3.5,1 -3.5,3 fall into two distinct classes. One may be termed the
'energy height'' technique, and the other is the one normally used viz. that given by the Royal Aeronautical Society Data Sheet EG 3/1. Lush's approximate method is an approximation to the
energy height technique.
The technique detailed in EG 3/l gives a lower forward speed and claims a higher rate of climb than does the energy height technique. In effect it states that if V = "^(H) is a
2 (dt/dH)dH is a «1
solution of dH/dt = Ï S ^ ( T - D , ) when
minimum, then V = ^ (H) is, in practice, a sufficiently close solution, under the same conditions, to the more exact equation»
f k
- ^ A + I ^ V - t^(T - D
)
dt - dt l^^ * g dny -
fm-^ \^
'
This is clearly true when dV/dH is small, i.e. when the acceleration along the flight path is negligible. In the example taken, which is a typical modern fighter project, the acceleration during the climb is too great for this approximation to be warranted, and there is a wide disparity in the two optinum climb functions. This is shown in the curves of Figures 4 and 9.
The rate of climb has been corrected for acceleration by subtracting ( — ^ j ^ from -üie rate of clinb deduced from the simplified equation, and gives results very close to
the true values. The true values have been calculated by reducing the optimim climb function V = '^^.(H) to the energy
height function 0 ( H ,V) = dt/dH and subsequent evaluation
by a step by step nvmerical process. The estimated and true values for rate of climb are shown in Table VII ajid are illus-trated in Figures 6 and 7.
Table VII
Gecmetric
Height
5,000
10,000
15,000 '
20,000
25,000
30,000
Rate of Climb
as estimated
by EG 3/1
f . s .
148.7
133.6
119.3
103.9
92.7
82.3
True r a t e
of
clinb
f . s .
150.2
134.2
120.2
105.4
93.7
81.9
The Royal Aeronautical Society technique, therefore, gives a higher rate cf climb and a lower forward speed than the energy height technique. The energy height technique relies on relatively high kinetic energy being acquired at low altitudes where the maximum acceleration is attainable, for conversion into potential energy (i.e. height) at higher altitudes. Longer time is required for the acceleration up to the higher initial climb speed at sea level, and a rather greater time is spent at the lower altitudes, but the excess speed acquired can quickly be converted into height, and the advantages or otherwise of this new technique can only be decided on the overall assessment of the steady climb ccxibined vdth the end conditions.
To enable a direct ccmparison to be made, the optimum clinb path given by the method of EG 3/l has been expressed in the terms of the energy height concept and plotted on Figure 5. The values of 0 obtained are progressively higher as the energy height increases, showing that this technique gives an incjrease in the time required to energy height. Therefore, when the end conditions are considered, and kinetic energy can be translated into potential energy, the energy height optimum climb technique must be of greater advantage. The extent of this advantage can only be assessed after a detailed study of the end conditions.
3.8, The End Conditions
The transition period of steady climb between the end conditions has been resolved into two techniques. In one a particular gecxnetric height is reached in a shorter time, but at a lower flight speed. It has been shown separately that if the kinetic energy of the aircraft can be rapidly converted into its height equivalent, then the shortest time to a particular height and speed is obtained by using the technique deduced fron the energy height concept. In this technique, during a steady climb, a particular geometric height is reached in a longer time, but at a greater flight speed, and the kinetic energy in hand more than compensates for the longer time taken.
Kinetic and potential energy are convertible into
height and speed equivalents by 'zoaii' climbing and diving. The rate of conversion, however, is dependent on the permissible
manoeuvrability of the aircraft. For jet propelled aircraft
travelling at high speed, manoeuvrability factors are low. Even at sea level the maximum permissible acceleration is likely to be of the order of 4 g, and at 40,000 ft. geometric height this figure would be of the order of 1.5 g. Near the ceiling, zoon climbing or diving could not be effected. The advantage of using the energy height optimum will depend, therefore, on the margin of kinetic energy gained, and the rapidity with which this can be converted into its geometric height equivalent with the prac-tical limitations on manoeuvrability imposed,
The shortestpossible time to height is of vital import-ance only to the interceptor fighter. An enemy is unlikely to limit his attack to those heights at which the interceptor jet fighter is most efficient, and consequently the end conditions must be studied at all heights. The tactical employment of the fighter, however, demands, in general, only t'ïvo possible speeds at the interception height. In a head on attack the minimum control speed is sufficient. For an attack which involves pursuit of the enemy the maximum speed will be required. A head on attack is an unlikely tactical operation with conventional armament, but may be possible with air to air missiles having radar, acoustic, or infra red honing devices. Moreover it may be the only form of attack available to the defence. This will
occur when the closing speed of fighter and bomber is small, and insufficient warning of attack is given.
In the work which follows an attempt has been made to assess quantitatively the advantages of using the energy height
optimum climb technique. The extent to which this advantage can be used in meeting the tactical requirements has been studied; and finally, the optimun climb path from the initial conditions to the actual interception has been determined.
3.9. Quantitative Assessment of the Advantage of using the Energy Height Technique
It will be evident at this stage that no spectaciiLar advantage can be gained by using the energy height technique.
In Figure 5 the function 0(^ ,V) has been plotted against H for the optimum climb defined by the Royal Aero-nautical Society, and for the climb defined by the energy height concept. The time taken to pass between two energy heights, H and H is,-^1 ^2 (iH ^2 i2f(H V) dH .
JH
®1Hence the time saved is represented by the area between the two curves. The total time which can be gained in passing fron an energy height of 7069 ft, (which represents the initial conditions for the energy height clinb and is equivalent to 46O n.p.h. at sea level) up to an energy height of 45,000 ft., is of the order of 15 sees. This advantage cannot be completely attained in practice, as it assun^s that kinetic and potential energies can be converted into one another without time loss, and without any restrictions on manoeuvrability being imposed,
A number of examples have been evaluated for two air-craft, one using the R.Ae.S. climb and the other the energy height climb, and then both reaching the sane end conditions. To make direct comparison possible, the end conditions at each height have been taken, firstly as the optimun climb speed for the R.Ae.S, climb, and secondly a speed of 0.94 M, which is near the maximum attainable for the aircraft considered,
The optinum climb curves plotted in a network of con-stant energy heights as in Figures 4 and 9 nay be interpreted as giving the theoretical means of attaining the same end conditions, They are therefore liseful for indicating the best manoeuvre for attaining any particular speed and height relation.
In the first case both aircraft start at 430 m.p.h. at sea level. One climbs in accordance with the curve C Q D
(Figure 4) reaching 25,000 ft. and 491 m.p.h. (Q). The other accelerates at sea level from 430 m.p.h. to 46O m.p.h., and then climbs at the greater speeds represented by A P R to P, where the constant energy height line through Q intersects the energy height optimum climb curve. If the extra kinetic energy at P could be converted immediately into potential energy, it would be equivalent to the geometric height difference between P and Q, and would represent a zoom at infinite speed from P to Q. The time difference between the paths CQ and CAPQ represents therefore the maximum theoretical advantage attainable. This can be shown to be of the order of 52 sees,
In practice the aircraft would clinb to a point about 2,000 ft. bel<3W P and then zoom the height difference to Q, In the time taken for this zoom, energy would be put into the aircraft proportional to the energy height difference between the start of the zoom, and the energy height of P and Q. The advantage gained is likely to be about half the theoretical obtainable.
In the second case the first aircraft climbs in accord-ance with the curve CQD to Q, and then accelerates to S in level flight at 25,000 ft. to 0,94 M flight speed. The second aircraft accelerates at sea level frcxi 430 to 46O m.p.h. and then climbs in accordance with the curve APR to R i/rfiich is the point of intersection of the constant energy height line through S and the energy height optimum climb curve. From reasoning identical with that in the first case the time difference between these paths represents the maximum theoretical advantage attainable. This can be shown to be of the order of 7^ sees.
In practice a dive would be commenced at about 26,000 ft. geometric height corresponding to the point Y on the clinb
curve. In the time taken for this dive, energy would be put into the aircraft proportional to the energy height difference between Y and R. The advantage gained is likely to be about 5 sees.
Two further examples have been evaluated, namely, frcci initial conditions to 30,000 ft. and 505 m.p.h,, and initial conditions to 30,000 ft. and 648 m.p.h. The theoretical advan-tages attainable have been shown to be 8g- and 13 sees, respectively,
It would appear that some advantage is to be gained by building up kinetic energy at a low level and then zoon climbing to height. It was necessary to investigate, therefore, whether it would be advantageous to increase speed at sea level up to near the maximum, and then zoom climb to a height and speed on the optimum climb curve, A steady climb would be continued from this position.
In the analysis the aircraft took 65 sees, to acceler-ate from 460 m.p.h. to 680 m.p.h. at sea level. The higher speed ccrresponds to an energy height of 15,400 ft. If the in-creased kinetic energy gained could be converted immediately into its height equivalent, the aircraft would zoan to the point on the optimum climb curve corresponding to 473 m.p.h. and 7,900 ft. geometric height. In practice we should have to allow for the energy input from the engine vdth a result that a vertical zoon would take the aircraft to 8,750 ft. and a speed of 487 m.p.h. The tine taken^ allowing 2 sees, loss of advantage at each end for transition from zoom to steady climb conditions, would be 14 sees., giving a total time of 79 sees,
The time taken to reach the sane end conditions by proceeding along the optimum clinb curve v>rould be 60 sees., rep-resenting a saving of 19 sees, for the operation. Thus tine is lost in increasing the speed to near the maximum at sea level and then zoom clinbing. This result is to be expected fron the theory in support of the energy height optinun clinb technique. An increase of speed at sea level represents an increase of energy
height, and deductions fron Euler's condition for the nininun tine between two energy heights shov;s that this is best achieved under the conditions set by the optinun climb curve,
3.10. Optinum Climb to Height when the End Conditions are Included
The end conditions in this investigation have been defined as the minimum control speed and the maximum speed at a stated gecmetric height.
3.10,1 Optimum climb to geometric height H and the minimum control speed V . (EAS)
^' n m — — ^
It is evident from the foregoing study that the theoretical optimum clinb is to accelerate at sea level to the
Kanaalstraat 10 — DELFT 2 6
-evaluated initial conditions (46O n.p.h. for the 1950 Fighter Project) and then clinb steadily following the optimum climb
p
curve to an energy height of H + V . /(2crg) finishing with a zoom clinb to the gecmetric height H. In practice the steady
P p
climb w i l l end a t an energy height H + V . /(2crg) - 6 where
2 ^'^^
8 depends on the manoeuvrability factor of the a i r c r a f t a t
2
t h a t height. The difference i n energy heights 6 between the
t h e o r e t i c a l optimum and p r a c t i c a l w i l l be a measure of the energy
put i n t o the a i r c r a f t during the zoon,
To i l l u s t r a t e these deductions l e t the nininun control
speed a t 40,000 f t . be 180 m.p.h. EAS or 363 m.p.h. TAS. The
energy height eqidvalent of the end conditions i s 44,394 ^''t.
This energy height i s reached on the optinuni climb curve a t a
geometric height of 31,000 f t . and a t a f l i g h t speed of 635 m.p.h.
TAS. Theoretically a zoon could be done frcan t h i s gecxaetric
height and speed converting the excess k i n e t i c energy i n t o the
gecmetric height difference between 31,000 f t . and 40,000 f t .
The t o t a l time taken f o r the climb fron the i n i t i a l conditions
to the required energy height i s obtained fron the curve i n
Figure 5 as P ^ ' ^ ^ 4 ^(jj ^y) g^ ^^^ ^^ ^^ ^^^^^ jj^g .^ime
07069 ^ ^
taken t o do a steady climb to 40,000 f t , using the R.Ae.S.
reecmnendations would be (from Figure 7) 415 s e e s . Thus the
t h e o r e t i c a l advantage a t t a i n a b l e i s of the order of 1 n i n . 51
sees.
In p r a c t i c e the zoon w i l l connience a t a lower geonetrie
height and take a f i n i t e t i n e , but the g r e a t e r portion of the
t h e o r e t i c a l advantage can be a t t a i n e d . The l o s s in t h e o r e t i c a l
advantage i s equal t o the t i n e talcen for the zoon l e s s the
difference i n t i n e s taken to reach an energy height of 44, 394 f t .
and the energy height a t which the zoon comnences. If the zoon
connences a t 30,000 f t . geonetrie height i n s t e a d of the t h e o r e t i c a l
31,000 f t . , the l o s s i n advantage would be the t i n e for the zocci
l e s s 11 s e e s . Thus even allowing a zocm t i n e of 20 s e e s . , an
overall advantage i n the t i n e to clinb 40,000 f t . of 1 min. 42
sees, would be a t t a i n e d .
By way of i l l u s t r a t i o n , a Meteor w i l l zoom from
35,000 f t . t o 40,000 f t . i n 9 s e e s . , the speed f a l l i n g frcsn
520 m.p.h. to 392 m.p.h,
Thus the absolute minimum tine to height for the 1950
Fighter P r o j e c t i s achieved i n three s t a g e s ,
-(a) An a c c e l e r a t i o n a t sea l e v e l to 46O m.p.h.
H f t ,
0
5,000
10,000
15,000
20,000
25,000
30,000*
(30,000"^
<35, OOO"*"J40,000'*'
V mph, C460
471
491
520
555
593
632
617
604
601
V. mph.
e
460
437
k22
413
405
397
387
377
336
298
* Not allowing for compressibility
+ Allowing for compressibility
(c) A zoom climb to height. The zoom w i l l commence a t an
energy height 500 t o 1,500 f t . belcjw the energy height value
given by the required geometric height, and the minimum flying
c o n t r o l speed of the a i r c r a f t a t t h a t height. The lower figure
quoted w i l l apply t o lower a l t i t u d e s , and the g r e a t e r figure for
a l t i t u d e s i n the neighbourhood of 40,000 f t . The figure cannot
be determined accurately by a n a l y s i s , but i s r e a d i l y obtainable
from a f l i g h t t e s t .
At heights very near the c e i l i n g , the nininun control
speed, and the optinun c l i n b speed, are nearly equal and a zocaa
c l i n b i s not p o s s i b l e .
3.10, 2 Optir-iun clinb t o geonetrie height H and the maximum
I t i s evident t h a t the t h e o r e t i c a l optimum c l i n b i s
t o a c c e l e r a t e a t sea l e v e l t o the evaluated i n i t i a l conditions,
and then clinb s t e a d i l y following the optinun clinb to an energy
height of H + V / ( 2 g ) f i n i s h i n g vdth a zoan dive t o the
nax
geonetrie height H.
The optinum climb speed approaches the maximum speed
a t height and consequently the geometric teight of overshoot
decreases. At 25,000 f t , geometric height the overshoot i s
of the order of 1,800 f t . and the theoreticjal advantage a t t a i n
-able i s 7^ s e e s . , w h i l s t a t sea l e v e l the maximum speed i s b e s t
attainable theoretically by a clinb to 7,750 ft. with a possible time saving of 22 sees.
In practice, lack of manoeuvrability of a high speed jet aircraft sets a severe limit on the steepness of the zocn dive. Even with 4 g manoeuvrability, ï/hich is only possible at low levels, the following 'pull out' radii are requiredi
V m.p.h. Radius of Pull-out ft.
684 10,300 497 5,480 311 2,150 186 768
At 30,000 ft. the factor is likely to be down to 2 g and at 40,000 ft., 1.5 g.
In practice the theoretical advantage is not fully attainable. The dive must be canmenced at a lower energy height than the theoretical and approximately half the theoret-ical advantage can be achieved.
Calculations show that a saving of 10 sees, in the time to a speed of M = 0.9 at sea level can be gained by a climb to 5,000 ft. geometric height and a zoom dive of 22 50' frcan the horizontal. This dive is illustrated in Figure 10. At 25,000 ft, the overshoot in practice ydll be of the order
of 1,000 ft. and the time saved 4 sees. The magnitude of the overshoot will best be detemined by flight testing when the maximum diving angle is known.
3.11 Method of Obtaining the Optimun Climb Relationship for any Particular Mrcraft
If the thrust and drag figures for the aircraft are known, the optinum climb curve can be obtained fron the methods of analysis described.
In flight it has been shown that the requisite inform-ation is given by partial climbs at constant energy height. Owing to the small time intervals involved this is not likely to prove accurate, A better method would be to determine the accelerations in straight and level flight at a series of
gecmetric heights. This would give values for (T - D, ) from which 0 ( H , V ) could be determined by the methods of analysis previously quoted,
3.12, Conclusions
1, The optimun climb curve obtained by using the energy height concept gives the most rapid clinb between specified end conditions,
2, The gain in tine of this clinb over that deduced fron the R.Ae.S. Data Sheet EG 3/l betvreen two energy heights is small, being less than 15 sees, between energy heights of 7,000 and
40,000 ft. for the particular aircraft considered.
3, The minimum tine to height is achieved by finishing the steady climb vdth a zoom climb until the aircraft speed reaches the minimum value for effective control. In this climb a saving of 1 min. 41 sees., for the exanple taken, as ccmpared with a clinb following the R.Ae.S, Data Sheet nethod is achieved
in going from 46O m,p. h. at sea level, to ninimum effective con-trol speed at 40,000 ft.
4, The mininun tine to maximum speed at a teight below the service ceiling is achieved by overshooting the geometric height by an anount in theory varying fron 7,400 ft. at sea level to 1,800 ft. at 25,000 ft. resulting in an overall time saving of 22 sees, and 7^ sees, respectively as ccmpared for the aircraft considered with the results given by the R,Ae,S, Data Sheet method. In practice, allowing for manoeuvrability limit-ations, these figures become 5,000 ft. and 1,000 ft., and the time savings 10 sees, and 4 sees, for the exanple chosen.
5, The optimum clinb is best obtained in flight testing by carrying out a series of acceleration measurements in straight and level flight at vailous geometric heights.
6, The approximate method for determining the optimum 2
climb speed given by Lush is satisfactory at low altitudes but is in error by approximately 7 per cent at heights in the
4. PART II, Distance Climb
4.1. Introduction
In this section of the report the best climb to inter-cept the eneny at a particular radius from a military objective is investigated.
There are two nain aspects to be considered. Firstly, the tactical requirenent may be for interception in the shortest possible time regardless of fuel.expenditure. This vdll occur when the range of the fighter is not unduly limited by the anount
of fuel carried, or the interception is so critical that the tactical commander is prepared to accept a shorter time of con-tact with the enenry. In this case success in the operation may depend on the number of fighters engaged. Secondly, the
requirement nay be for interception in the least time, vdth fuel economy an over-riding consideration, to permit as long a time of contact with the eneny as possible,
Between these two eases there is an infinite number of variations. It is felt, however, that a detailed study of these particular end conditions will give a guide to the best technique to be adopted in any practical case.
4. 2. Mininun tine to heipjit at a particular radius v;hen fuel econoTiiy is not essential
The horizontal distance covered in the optinum climb
from the initial conditions of 46O m.p.h. at sea level to 40,000 ft, is of the order of 25 miles. This distance is proportionately less for lower heights of interception. It is assumed that the radius at which contact is to be made is appreciably greater than these distances.
Maximum speed (M = 0,96) at sea level is 731 m.p.h. and this figure decreases vdth height, becoming 634 m.p.h, at 40,000 ft. Thus a 15 per cent saving in tine from this con-sideration alone is achieved if the interception radius can be covered at sea level. The fuel consumption at maximum speed at sea level, hciwever, is more than 3 times greater than that at 40,000 ft.
Calculations have shown that the maximum speed at sea level can best be attained by an optimum clinb from the initial conditions (46O m.p.h.) until an energy height is reached which
is slightly less than that corresponding to sea level and maxi-mum speed, namely 0 + (l072) /(2g) ft. The aircraft is then
dived at an angle not exceeding about 23 to sea level,
In the exanple chosen a theoretical advantage of 32 sees, is attainable in reaching a speed of 0.9 M, and a 10 see, practical advantage is probable. The theoretical energy heiglit is 15,4D0 ft. corresponding to a geometric height of 7,750 ft. and 4B3 m.p.h. In the practical ease the 10 sec. advantage is gained by a climb to a geometric height of 5,000 ft. followed by a 23° (measured from the horizontal) dive. For a final speed of 0.96 M the theoretical advantage would be approximately 45 sees, and the bulk of this could be gained in an interception at considerable radius. The aircraft would be climbed initially to a geonetrie height of 10,000 ft. and a long dive would follc3W.
The clinb to height vrould start at about 20 niles from the required radius of interception. The excess kinetic energy would be converted into its height equivalent by a zoom clinb to a point on the optimum clinb curve. This zoon would take the aircraft to about 12,000 ft. A steady climb would follav in acccrdance with the relation previously established. The steady climb would then be followed by a zoom to the nininun control speed for the minimum overall time to be achieved. If, however, it is required to reach the interception height at the naxinun speed, the steady clinb will be continued beyond the interception height, and the aircraft dived to the naxinun speed. These two procedures have been detailed in Part I of this report.
Two further advantages of ecïvering part of the distance to the interception radius at sea level are:
(a) the manoeuvrability factor being higher, alterations of direction due to changes of vectoring instructions can more readily be carried out
(b) the final clinb to height vdll be achieved in less time owing to the decrease in aircraft v/eight corresponding to the consumption of fuel.
4. 3. Best time to hei.n:ht at a particular radius v/hen fuel economy is a prime consideration
If fuel economy is a requirenent up to the time of contact vdth the enemy, then attainment of maximum speed at sea
l e v e l i s precluded. I t i s necessary t o a t t a i n height as quickly
as p o s s i b l e . The a i r c r a f t should be climbed on the optimum climb
to an energy height H + V / ( 2 g ) , and then dived to reach
max
maximum speed at geometric height H. Flight is maintained at this speed up to the required radius.
A further saving in fuel could be effected, at the expense of tine, by zoom climbing from the optimum climb to the required height, and then accelerating from the mimnum control speed to the maximum speed in straight level flight.
4.4. Conclusions
The minimum time to height at a particular radius is, therefore, achieved in the follovdng stages:
(a) The maximum speed at sea level (0.96 M) is reached by follovdng the optimum climb to approximately 10,000 ft. and then diving at a small angle to sea level.
(b) Maximum speed is maintained at sea level until vdthin twenty miles of the required interception radius, and then a zoom climb of approximately 10,000 ft. to a point on the optimum steady climb curve.
(e) The optimum clinb is then followed until vdthin zooming reach of the required height, the height being attained at the minimum control speed. For interception at 40,000 ft. the aircraft is vdthin zooming reach at 30,000 ft.
The minimum time to height at a particular radius and maximum speed is obtained by carrying out procedures (a) and (b) above, and then continuing the steady climb on the optimum clinb
p
curve to an energy height slightly less than H + V' /(2g) . nax
The aircraft is then dived as steeply as possible to reach maximum speed at geometric height H.
When fuel economy is to be considered, the best procedure is as follows:
(a) A steady climb following the optimum is made to an p
energy height H + V /(2g) followed by a dive to reach maximum speed at geometric height H.
(b) Maximum speed at geometric height H is maintained until the required radius is reached.
An even greater fuel economy, at the expense of tine, can be effected by zoom clinbing from the optinun climb curve to reach the geometric height H at the mininun control speed,
and then accelerating in straight and level flight to the
re-quired radius,
5. PART III. Initial Climb for Long Range Flight
The Specific Air Range (S.A.R.) is given by:
o A r, 60 V 8.1 ,. T
S.A.R. = -gg- X c X Drag nautical m.p.g.,
vdiere C = Specific fuel consumption, Ib./hr./lb. thrust, and assuming a fuel density of 8.1 lb./gal.
The maximum specific air range is thus obtained at the
1 -a
value of V. for which
Drag _
V " ^
B.„„/S.f. '-
100 i^ioo_; (v./ioo)2J \
is a minimum if C is assumed independent of speed. This occurs when m + l/m is a minimum, where n = V. /V. , ,
' ' 1 ' imd *
i.e. when m = 3 "^ = 1.31.
7-17
Hence maximum S.A.R. = • -^i, -^ [• i. nautical m.p.g. 100 ^100 ^^'^
C, i n f a c t , decreases vdth increase i n a l t i t u d e .
Thus i n a t t a i n i n g the b e s t s p e c i f i c a i r range the
over-r i d i n g v a over-r i a b l e i s height, and n e i t h e over-r V. noover-r W aover-re c over-r i t i c a l
2N
^Soo'^'^n
1
For the example considered, the s p e c i f i c fuel
consump-t i o n i s ^ ^ 11,600 r.p.m.
The maximum range, t h e r e f o r e , i s a t t a i n e d by climbing
t o the c e i l i n g following an energy height optinun climb obtained
by using t h r u s t f i g u r e s corresponding to 11,600 r.p.m. Near
the c e i l i n g manoeuvrability f a c t o r s are too lov/ for zooms to be
effected. The a i r c r a f t v d l l continue t o climb slowly vdth
Y^ = 1.31 V ^ ^ as W, and therefore d^^^ decreases due t o the
consumption of fuel.
KanaalsUaat 10 - DELFT
-34-6, Part IV. Initial Clinb for Tine Endurance Flight
'The speed for maxinum endurance is clearly the nininun drag speed V. ,. The specific fuel consumption decreases with height. Consequently increase of height neans an increase in the time endurance of an aircraft.
For the example chosen the specific fuel consunption is a minimum at 11, 600 r. p. m.
The best clinb to height for a tine endurance flight is, therefore, to clinb at an energy height optinuni clinb using the thrust giving minimum specific fuel consunption. In Ihe example chosen this is achieved by using 11,600 engine r.p.m. The energy height clinb path can be obtained by one of the nethods detailed in Part I.
REFERENCES
1, Lush K.J. A reviev/ of the problen of choosing a clinb technique vdth proposals for a new clinb technique for high perfon:iance aircraft. A,R,C, R, and M. No. 2557, 1951.
2, Lush, K.J. The loss in clinb performance, relative to the optinun, arising from the use of a prac-tical clinb technique.
Report No. A. and A.E.E./fees/243, Aug. 1949.
3, Royal Aeronautical Society Performance Data Sheet EG 3/l, October, 1950.
APPENDIX I
Specification of 1950 Fighter Project Assumptions used in the Analysis
and S p e c i f i c a t i o n Wina F u s e l a g e T a i l - p l a n e Gross a r e a Net a r e a t / o r o o t t / c t i p SMC T r a n s i t i o n p o i n t P o s i t i o n of max. t h i c k n e s s Span Angle of sweepback V/etted a r e a Maximum d i a n e t e r Length T r a n s i t i o n p o i n t Net a r e a STUD P o s i t i o n of max. t h i c k n e s s t / c T r a n s i t i o n p o i n t P i n and Rudder Net a r e a SI^ P o s i t i o n of max. t h i c k n e s s t / c T r a n s i t i o n p o i n t Drag Summary Wing T a i l - p l a n e F3.n and r u d d e r F u s e l a g e Roughness- vdng f u s e l a g e t a i l I n t e r f e r e n c e - vdng/body t a i l / b o d y C o n t r o l gaps - t a i l vdng canopy 602 s q . f t . 536 s q . f t . 0 . 0 8 0 . 0 7 11.92 f t . 10 p e r c e n t 35 p e r c e n t 5 0 . 5 f t . 40° lOlfO s q . f t . 5.75 f t . approx. 58 f t . Nose 1 4 3 . 5 s q . f t . 7 . 6 f t . 40 p e r c e n t 0 . 0 7 L,E. ( s i n c e p r o b a b l y i n wing walce) 56.7 s q . f t . 7 . 2 f t . 35 p e r c e n t 0.07 10 p e r c e n t 3 2 . 4 l b . 8.2 l b . 3 . 6 l b . 2 2 . 2 l b . 7.1 l b . 1.8 l b , 2 , 8 l b . 1.6 l b . 1.0 l b . 3.0 l b . 1.6 l b . 1.5 l b .
86.8 l b .
+ 12 p e r c e n t l e a k s , excsrescencea e t c ,97 l b .
and C-. = 0.0135
/P^or
For drag increase id-th Mach Number, the following
assumptions have been
nades-M
^ D
< 0 . 9 0
0.0135
0.91
0,0142
0.92
0,0154
0.93
0.0175
0.94
0.0207
0,95
0.0250
0.96
0.0305
W i s assigned constant a t 95 per cent of the take off
weight, namely 26,600 l b s ,
Engines
Two t u r b o j e t s of s t a t i c t h r u s t 30 per cent g r e a t e r
than the R o l l s Royee Nene,
Engine s e t t i n g taken as 12,300 r,p.m.
General
K, the induced drag factor is assigned to be 1.1,
Atmospheric conditions are taken as standard I.C.A.N.
APPBNDIX II
Proof that the minimum time to Energy Hei.^ht leads to the minimum time to Geometric Height for
the same end conditions
The time to height may be vvritten in either of the alternative forms:
\H
t = ) 0(Hg,V) dHg iH (A,1)or
t =pH.
H,
F(H,V^_,dV/dHjctH (A. 2) where H^ = H + V /(2g) (A. 3)It follows from §3.5 that Euler's condition for a minimum value of ( A . I ) is equivalent to
dV
= 0H const. e
(A. 4)
and the condition for (A.2) t o be a minimum i s
SF
av
v^ = d^r/dH.
d d H f OF 0ll,Yj
From equation (A. 3) ffl = (l + v, V/g) dH therefore
0(Hg,V) (1 + Vj^ V/g) = F(H,V,Vj^) (A. 5)
Treating V, H and v, . as independent v a r i a b l e s vre have,
from equation (A.5):
ÖP öV " ~ J H,v^
a^
av
^H
V i ^
V
hi^h] ^ ^ ••••'^•«
H,V
= l0
g '^
(A.7)
/Therefore
Therefore ^ f §^
But M = M
^^^ dH an
I & ^ "^ g dH H,V/ V'
-^i^l -Iv
(A.8) (A.9) Hand hence e q u a t i o n (A.8) n a y be w r i t t e n d_ dH
ap
av.
' h '1IH,V g0
1 È^
g ÖH 1 + V I V / a i ^ + g ^h> "^ g av H / e , (A.10)S u b t r a c t i n g (A. 10) frcan (A. 6 ) :
3F
av
d_ / a p dH ( av, 'H,v,av
H,V, H Hence t h e c o n d i t i o nav
= 0 leads to the
H
equivalent condition
ap
av
d^
dH
H,v,
H,V
= 0
Thus t h e mininun t i n e t o energy h e i g h t l e a d s t o t h e minimum t i n e t o g e o m e t r i c h e i g h t a l l o w i n g f o r t h e s a n e end c o n -d i t i o n s .
o
O o O (0 in O o tn O l/s «o O o8
O in oo
o
^ ^ m8
o
m o Oo
d. ECURVES OF ^ CH«, V) AT A SERIES OF CONSTANT
ENERGY
HEIGHTS.
RFPORT No. S7. AiniaissaadwoD H U M ,000^ ^
5
^ « ^ 0 a, ECURVES OF Sf(^«.V) AT A SERES OF CONSTANT
ENERGY HEIGHTS
0'023 0 0 2 I 0-0I9
iJ
(H«,
V)
0'0I7 0-0I5 0-0I3 O'OII 6 0 0 «- ]
• >•i
1
U I t » ^ ^WITHOUT COMPRESH« =
4 S , 0 0 0 f f SIBILITY ^ ' - ^ 620 V m.p.h. 6 4 0 6 6 0VARIATION OF ^ (Hc,V) WITH V AT 45 0 0 0 t l ENERGY HEIGHT
WITH AND WITHOUT ALLOWANCES FOR COMPRESSIBILITY
V ^ < ' W > . 25,000 2 0 , 0 0 0 IS.OOO 10.000 5 , 0 0 0 0 ^ " /
*-'ƒ
SI
vi j ^\ oc ƒ4
/ /^i^
/ / / /T/
rsL
1 ( ^
1 1
/'N
^ S w / 5» / / \/ "V
/ >s25,C \>»^o,ooo'
> > ^ 5 . 0 0 0 ' \ vlO.OOo'N.
?|" FIG. 4 .
T I
'T
V i >v^40*000' \ N . \ 3 5 , 0 0 0 ' 1 SJ'O.OOO ^ ^ \ \ \i V f^.»
300' ; 2 >k 5'
< \ c l 0 1 • 1 5 \ 1 "O \ 0 \ C 1 \\\ 5 » 4 0 0 5 0 0y
m.p.h. 6 0 0 7 0 0•Olio
•doo
--• 0 0 9 0^ ( H . , V )
•0080 •0070 • 0 0 6 0 •0052 40,000COLLEGE OF AERONAUTICS REPORT No. S7.
S^OOO 10,000 15,000 20,000 25^000
H. ft.
VARIATION OF RATE OF CLIMB WITH HEIGHT, R.Ac. S. AND ENERGY HEIGHT OPTIMUM CLIMBS
O-Oll VALUES OF ^ ESTIMATED BY dH METHOD OF REF 3. 0 ' 0 0 6
lopoo
20,000 3 0 , 0 0 0 H ft.VARIATION OF '^Vc WITH HEIGHT FOR TIME TO
HEIGHT ESTIMATION. R.Ae.S. CLIMB.
REPORT No. 57. 180 I 6 0 I 4 0 lOO OOO OOO OOO
