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TECHNICAL MOTES

“’’”““”’’ylm!m@—._.....

No. 599.—. ——— —

w’-””— -— _-

.-

CHARTS EXPRESSING THE TIME, V“ELOCITY, AND ALTITUDE ._ _ “:

RELATIONS FOR AN &(RPLANE DIVING

IN A STANDARD ATMOSPHERE

33y H. A. PearsonLangley Memorial Aeronautical Laboratory

—.- --- .-.

-....:: .-. :l.

. ---

... t-.-—- - ----.-.:.

Washington —

April 1937.-

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https://ntrs.nasa.gov/search.jsp?R=19930081344 2018-08-19T18:15:23+00:00Z

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.._—. —. ..=

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

——— -_ —

TECHNICAL NOTE NO. 5S9

—-— —

CHART’S EXPRZiSSING THE TIME, VELOCITY, AND ALTITUDE

RXLATIONS FOR AN AIRPLAITE DIVING

IN A STANDARD ATMOSPHERE > -

By H. A. Pearson

SUMMARY

In this report charts are given showing the_relationbetween time, velocity, end altitude for airplanes havingvarious terminal velocities diving in a standard atmos-phere. The range of starting altitudes is from 8,000 to32,000 feet, and the terminal velocities vary from 150 to ‘“-550 mi3.es per hour. A comparison is made between an ex-perimental case and the results obtained from the charts;Examples pointing out the use of the charts are included.

Il~TRODUCTION

The velocity-altitude relations for airplanes in adive have been treated, in the past by several writers.Diehl (reference 1) assumed a constant density atmosphere.~ilson (reference 2) and Becker (reference 3), who havetaken the variation of density into account, though attack-ing the pro%lem differently, have given no expression forthe time to dive. Regardle~s of the manner In which thedensity is taken into account, ‘the velocity-altitude equa-tions become too lengthy and complicated for general use.

Because of this fact, and also because the time varia-%le has previously been omitted, it is the purpose of thispaper to present in a readily usable fo”rm charts express-ing all three relations with density variation taken intoaccount. #

N.A.C.A. Technical Note No; 599

CHARTS

The charts shown in figures 1 to 9 cover a range ofterminal velocities from 150 to 550 miles per hour by 50-mile-per-hour increments. The starting altitudes varyfrom 8,000 to 16,000 feet by 2,000-Fcot intervals, andfrom 16,000 to 32,000 feet by 4,000-foo.t increments. Thislatitude in both speed and altitude will, it is believed,cover all cases. of interest from the extremely slow air-plane UT to the fastest airplane available. ..

The terminal velocity U by which each chart is des-ignated, is that which would be obtained fn an atmosphereof constant sea--level density. The abscissas of the curvesare the true, not” the indicated, velocities.

In the establishment of the velocity-altitude curves,the equation develhped in reference 2 was used with slightmodifications to make it agree with the now recognizedIrstandard atmosphere!’ of reference 4~ The modified equa-tion was similar in form to that given in reference 2 ex-cept that-the factors 3 and 1,200 occtirring in the origi-nal were replaced by the factors 2.7 and 1,254, respec-tively. The new equation was:

f)125A a 1254 2———~

(

u H ()---w-Va = 2g 1 + –2*7h

; (1 + 6i;;;)dh (1)

64000

where 7 is the true air speed in f~p.s.

.-.

●

I

g, gravity constant, 32.2 .m

h, altitude i.n feet above sea level

H, start.in~ altitude in feet above sea level

U, terminal velocity of airplane in air &t stand-ard sea-level density, f.p.s.

..

The placing of the time network on the charts was accom-plished by using three methods: (1) by the use of” the vac-uum i?ormula, (2) by use of an equation developed in the .’appendix which takes variation of density into account ~and (3) by use of a step-by-step process of integration. &Xach of the foregoing methods had its particular rOgiOn in

9

I

N.A.C.A. Techhical Note No. 599 3

which it was more easil~ apylied than the others f-or thesame degree of accuracy.

The acceleration of an airplane in a vertical dive,at any instant, is given by

a= 3’*2 (’ -$3

where a is the acceleration in ft./sec.a

~! dynamic pressure

qu, dynamic pressure at terminal velocity19.h.iu

It is obvious from this equation that during the early ‘:”.::.jJ””[J

. .. ..-

part of the dive the acceleration differs only slig-ht-ly “( 1from g; hence the vacuum formula is applicable, wi:h_iy_ ._ - __the plotting accu~acy, for this range of the charts. Thisrange increases with increase in starting altitude and_.._terminal velocity. All timing lines below 6 seconds” ar”ecomputed by using the vacuum formula and in some cases the -range has been extended to 8 seconds. Above. these time-j .

● values, use was made of a step-by-step integration of thevelocity-altitude curves when the terminal velocities were

. 300 miles pe_r hour, or more. Below a terminal velocity of● 300 miles mer hour, the time-altitude formula developed in

the appegd~x was used, Application of this formula showed .that it was necessary to carry a large nu”m%er of termswhen the terminal velocity was greater.than 300 mile~ per”hour, and also when the altitude loss was ‘small, regard-less of the value of the terminal velocity. consequently,its practical use iS limited to altitude losses of morethan 2,000 feet and limiting speeds of less than 250 mile_sper hour. The foregoing limits are, of c_oyrs~, perfectlyflexible depending both upon the combination of U andd-h and upon the degree of accuracy desir-ed.

The charts, &Qthough derived for a vertical diyestarting from rest, may be used .to include various divingangle~ snd sts.rting velocities. If U is the terminalvelocity of an airplane in a vertical dive, its terminalvelocity in a dive where the flight path makes an ~gle/.Y with the ground is

.. —.— -.——. UJsinY

-.

-,

.-

4 N,A.C,A. Technical Note No. 599

The vertical component ofi this velocity, which is the on+used in choosing the appropriate chart, is

u? = u (sin Y)3 /2

These factors will be found tabulated on each of thecharts, except figure 1. The effe.ct..oflan “initial divingspeed is taken into accdmst in the char% simply by con-sidering that the airplane started .to dim from a higheraltitude.

PRll!CISION●

As far as the charts are concerned, the only errorsare those due to plotting! to neglecting terms in the timeequation, and to the discrepancies that will occur in any

-—

step-by-step integration. The error in the time lines du-eto the last two sources is believed to be within “2 ~ercent;the plotting error in the velocity-altitude curves is neg-ligible.

In the application of—the charts to diving airplanes,however, several. uncontrollable sources of error will ex- .ist; namely , those due to:

1. Vaiiation of.‘~~rn=~jher-e”fr”om-“s-t~~.dqrd.”—

2. Scale. and compressibility effect on the air-pl-e drag, .- -.

I

3. Manner of entry into the dive. 1

4. Dive-angle variation from that assumed.

5, Propeller effect.

The error due to the first source is negligible and neednot be considered. Present knowledge indicates that theerror due to source 2 is small up to 500 m~l-es per hourwith small angles of atitack (reference 5). The manner of––entry into the, dive will be relatively unimportant in thelonger dives but may be important in the ehorter ones;however, in practice, its effect may be obviatti if de-si~able. The effect Of errors in the dive angle may becomeconsiderable in the shallower dives. .

.

N.A.C.A. Technical Note No. 599 5

.

.’

Probably the largest individual source of error isthat due to the propeller effect. In the computation ofthe values of U with”which to enter th-e c“h-~rts$liK 6f-fect should, if possible, be taken into account. .-

—.

In order to determine the cumulative effect of someof the a’hove-listed errors a comparison is made in-figure10 between an experimental case and the results given bythe charts. The initial air speed is taketi as 120 milesper hour in order to eliminate the entry into-the divefrom level flight. It can be seen that a satisfactory”- ”---agreement exists %etween experiment and the results ‘given- ‘

.—

by the charts. If the dive.h.ad been ,longer and the start-ing altitude greater, the agreement would have been muchbetter.

EXAMPLES

1. Given :

a. Airplane with U of 496 m.p.h.

-b. Altitude at start H, 16,000 ft.

c. Velocity at start, 100 m.p.h.

d. Dive angle 60° to the ground.

Required:

a! Velocity at 6,000 ft. in m.p.h.

l), Time to dive, sec.

Solution: \

Ur = U (sin p)3’a = 496 (0.866)3’a = 400 m.p.h. -

Using figure 6, point A is located at 100 miles per hourand 16,000 feet. An interpolated curve is drawn %etwe-enthe existing curves down to B at 6,000 feet. ~“roj8ct”ingfrom 3 to c, the velocity is found to be 406 miles perhour (an ind..icated velocity of 3?’2 m.p.h.). The time is28.3 - 4.6 = 23.7 seconds, found by subtracting the” t“imecorresponding to A from that at B.

,-

6 l?.A.C,A. Technical Note No. !399

2. Given :

a, Airplane with U of 500 m.p.h.

b, Altitude at start H, 14,000 ft.

c. Velocity at start, O m.fi.h.

d. Starts gull-out at 3,000 ft.

Required:

a, The velocity at pull-out.

1

.

—

Solution:

Using figure 8 and f-ollowing 14=000-foot curvedown to 3,000-foot altitude, the true velocity is

—

449 miles per hour or the indicated velocity is 430miles, per hour.

Langley Memorial Aeronautical Laboratory,National Advisory Committee for Aeronautics,

Langley Field, Vs., May 2, 1934...,

7.

. ..APPWJDIX ‘

Derivation of Time-Altitude Pormula

As stated in the introduction, ~revious investigatorshave not included an equation connecting time and alti-tude taking density variation into account. Consequently,the following derivation is made and is based Upori Wilson~svelocity-altitude equation (reference 2), modified to con-form to the standard atmosphere of reference.4.

.,

In equation (1) of the text, let && = a, and

(%) =C*dh

Then dt = .——___

f

————-—____

c’ H2g (1 + ah] J (1 + ah)-c’ dh

hfrom whence

t =J-—dh -

—— —————. + cl. ————————— ———— -

I~a

2g (1 + ah] ~H (1 + ah)-c2.’dh

If t = O when h = H, Cl disappears. After integratingthat portion under the. radical, the foregoing expression ,is -.=.. ___-—

t dh.JH —-.—— — —

/,

—.— ____ _Ca ~’.”l

2g (l+ah)[(l+aH)l-c” - (l+ah)l-c9

a T:P 1

NOT,2g

letting —--—._. = -ka and (l+aH)l-c2 = Ka (l-c=) ,.,

(Note:2g

––—-- is negative for U < 1254)a (l-ca)

N.A.C.A. Technical Note No. 5998

then t 1 f l__________q._______—=—‘h

J( l+ ah). K(l + ah)ca

Letting “=~ ‘he” ‘h= :~du

Subst-ituting, there results

This expression cannct be integrat~ directly but may beput- into a f~rm that can, lIy expanding bin.omiall.y the ex-pression

[

2(W)]-*1-Ku.

J!The resulting series converges when K u

2(C2-1)~ 1,

which is the case” here.

When the foregoing expansion is made the expressionfor t becomes

/li-aH

t’=— f[1 + ~ K~2(c2-’) + ~ ~~ U4(C2-1)

a’ m-8

1

& ~3”u6(Ca-1) + .,.: au+ 16

which becemes upon integrating . .

[

~2c2-1t2=— u+*K _ + 3 2 ~4c= -3

ak~K~~

2C2 - 1 -3

5 ~ ~6c2-5

1

~l+aHK+-— ––~— +

16 6c-5 ““” Gl-ca

Remembering that u = J l+ah and K = (1 + “&E) ,the upper limit is

.-

—.-:

,.-

i—.:

,

.

■

.H.A. C.A, Technical Note No. 59”9 9

.

.

+ . . ● l“~”s”””(an-1) 1

2.4.6 ...2n 2nca - (2n - 1) 1

In ‘the same manner the lower limit 3ecomes -

*. , .1.3.5 ...2n - 1..——2,~,~Q..2n

Combination of the two

1-——211C2 - (2n - 1) (Hs(’’’a)]

.

-.

limits gives an equation for the ,time in terms of the altitude,

t[

:k x===np-~ 1— ““--” “=— = [211] 2nca - .(2n - 1)

1.—-—(;T:)n(& -l)]2nc2 - (2n - 1)

.

.

10 N.A.C.A. Technical Note No. 599.

REFERENCES ‘..

,-,. , .-= -.=

1. Diehl , Walter S.: Engineering Aerodynamics. TheRonald Press Company, New York; 192B, p. 222. .—.

2. Wilson, .~.dwin Bid’ivell: The Limiting Velociby in,

I?alling from a Great Height. T.R. ~0. 78, N.A.C,A. ,1919. -.. .-

3. Becker, Fritz: Der Sturzflug in yer&nderlicher. Luft- . ._’_ _.dicht-e. Z.I?OM., November 28, 1932; pp. 659-663,

4. Diehl, Walter S.: Standard Atmosphere - Tables andData. T.R. NC). 218, N.A.C.A. , 1925.

. .-:’

!5. Stack, John: ‘~~ N.~.C,A. High-Speed Wind Tun~Q and ‘——

Tests of Six Propeller Sections. T.R. NO. 463,N.A.C.A. , ,19Z3.

.—

●

.

.

..

..-

N.A.C.A. T.N. 599 Fig. 1

32,000

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28.000

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24.000ft.,o [ 40 80 [120 I 160 f28.000ft.,O \ 40 I 80 I 120! l~u IKIU IZ4U I II II

._—-

.-..—..-.:

!_v!_J-!w- 1- I “ “ - .

“32,000ft.,6 40 80 120 160 200 240 . 1!

Figure 1.- Time-altitude-velocity relations for airplanes diving in astandard atmosphere. u= 150 m.p.h.

N.A.C.A. T.N. 599 Fig. 2

32,000 0 w v

30,000

28,000

26,000

24,000

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~ 20,000d:QJ18,000

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m I6,000g

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.91OI182

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32,000ft.,0 40 80 120 160 200 240 ““,, .

—

..—

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.—

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—

Figure 2.- Time-altitude-velocity relations for airplanes diving in astandard atmosphere. U = 200 m.p.h.

.

.

N.A.C.A. T.N. 599 Fig. 3

““ I I i I ) r I I \ I

JI

~ 20,000I I

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! 1 1 I I 1 1 t t ! 1 , , r

2.(300I rl I I I 11 I I I I I I I I I.-,

“l-+ + + + + + 1 1 1 + -t

Figure 3.- Time-altitude-velocity relations for airplanes diving in astandard atmosphere. u= 250 m.p.h.

.-

. .

.

. ---

.

.

1“

N.A.C.A. T. N. 599 Fig. 4

32,000

30,000

28,000

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1 .

6,000 \vI

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4.000

2,

8,00012,0

1 —

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.-

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—..- —

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Figure 4.- Time–altitude-velocity relations for airplanes diving in astandard atmosphere. u= 300 m.p.h.

● ✎ .● ●

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N.A .C.A. T.N. 599 Fig. 6

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pdmh p -1 p16,000ft.,O I 80 j ]60 I2 0 I320 \4r’i3I48020.000ft..O I 80 \160 [240 [320 ~ 400 !480 “1,!, ‘1 M

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32,000ft.,O 80 240 320 400 480 ,, - .

Figure 6.- Time-altitude-velocity relations for airplanes diving in astandard atmosphere. U = 400 m.p.h.

-. .

*- ,-.- .,

N.A .C.A. T.N. 599 Fig. 7b

b

32,000

30,000

28,000

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32,000ft.,O 80 160 240 320 400 4 0 560 M

Figure 7.- Time-altitude-velocity relations for airplanes diving in astandard atmosphere, U = 450 m.p.h.

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N.A..C.A.T. N. 599 g. a

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24,000ft.,O \ 80 I160 [240 ~320 ]400 I480 I5q0 ~ ,, ~~128,000ft.,O I 80 I 160 [240 320 400 \480

_—560 ; r, .,

>

32,000ft.,O 80 160 2 0 320 400 4i0 560 ;,””.— -:.:— .

Figure 8.– Time-altitude-velocity relatlons for airplanes diving in astandard atmosphere. u= 500 m.p.h. —-

●

N.A.C.A. T.N. 595 Fig. 9—

E!.-=. —.—.—

u’4195:

,.-—--

.-

.-.

—..—

___

28.000ft.,O ! 8’0 !160 ! 240 I 320 I4t!0I“460!560 !640 !. tI -1 ‘“32,000ft.,0 8’0

Figure 9.- Time-altitude-velocityQ standard atmosphere.,C

160 240 320 460 4d0 560 640 ~

relations for airplanes diving in aU = 550 m.p.h. -.—— ,-

.

N.A.C.A. T.N. !599 Pig. 10 —

40

0

Experimental, for vertical dive on 1?6C-4.Initial velocity 120 m.p.h., initial altitude12,000 ft.Calculated, assuming a standard atmosphere.

4

Figure 10.-

10,000

9,000— .—

8,000

3,000

2,000

1,000

.—

..—

.-.

-——

-—

..

.

_—--

— -—

8 12 16 20 24 28 0Time, seconds

Comparison between experimentaland calculated results.