JUNE,1930 73

THE VERTICAL DESCENT

The significance alld importance of the drag coefficientrests on the physical fact, borne out by experience andsupported by theory, that this combination of the quantitiesinvolv d is nearer to constant than any other simple com­bination. It does not rest on its more or less clever andingenious definition. Over a very lArge range, the drag co­efficient can be considered as absolutely constant for mostpractical applications. It is therefore possible and prac­tical to discuss the drag coefficient as an absolute physicalconstant, the value of which is independent of and has no

reference to any particular density ofthe air, velocity of the fall and diame­ter of the propeller.

In order to apply the drag coefficientCD to actual design problems, we have to reverse the aboveequation of definition of CD. We obtain then

D = Cn V2 S Q/ 2and

v == I w--V CD S Q / 2

The supported weight is thus demonstrated to be propor­tional to the drag coefficient, to the area, to the density,and to the square of the velocity. The velocity of fall isproportional to the square root of the weight, and inverseto the quare root of the drag coefficient of the area andof the density of the air.

From all equations follows the desirability of a large dragcoefficient for vertical descent, quite contrary to the dragcoefficient requirements of structural parts of aircraft forordinary flight. A drag coefficient four times as largeallows half the velocity of fall or one-fourth of the para­chute area.

After these explanations, the problem of the verticaldescent is reduced to the question: How large is thelargest drag coefficient obtainable by a parachute, or, inthis case, by a freely spinning propeller? Thi questionasks for an experimental investigation, and subordinatedto it is the question: Does a theoretical maximum value ofthis drag coefficient exist, and if so, how large is it?

Some persons have advanced the opinion that the maxi­nut11 drag coefficient theoretically possible is equal to ONE.They assume the propeller to move through resting air,and to transfer the velocity of fall to all air passed withinthe cylinder with the propeller diameter. It is true thatthis gives formally the maximum drag coefficient ONE,but the argument and this drag coeffiCient are not convinc­ing. The argument should never have been advanced, be­cause it has long been known that the drag coefficient ofdiscs and of strips is larger than ONE. The autogiropaper of 1922 cited above gives 1.7, not for the maximumdrag coefficient thinkable, but for one actually obtained ina test. Flying tests with autogiros are reported to indicatedrag coefficients as large as FOUR. There is indeed noreason why the air should not be accellerated beyond thevelocity of fall, and the argument leading to the value ofONE seems to be poorly founded.

De la Cierva, in a recent paper, comes back to this anomaly,as it seems to him, still under the influence of the doubtfulargument just discussed. The vortices are conjured up inorder to show that a propeller slipstream with diminishingvelocity diverges, a fact which is more readily -seen,without vortices. There are also smoke tests cited in sup­port of that obvious fact. De la Cierva's explanation isfinally that the factor ONE is correct, but the referencearea should be taken larger than is ordinarily done-notequal to the propeller disc area but equal to the final cross­section of the propeller jet.

The matter appears in a different light to me, and I shallproceed now to discuss my explanation and to deduce anumerical value of the maximum drag coefficient consistentwith theory larger than ONE. I shall avoid carefully all,"ortices and similar difficult conceptions. excusable andeven useful for mathemati- (Contin/oted on page 182)

Consulting EngineerDr. Max M. Munk

V 2 Q/ 2

D

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I N this article I shall discuss a sub­ject that has recently receivedmuch attention from progressive

aviation engineers-the magnitude ofthe drag on a propeller that is spinning and descendingunder the action of gravity.

My first autogiro paper of 1922 describes tests only andgives the results of these tests. The reception it foundwithin the organization which ought to have protected it,discouraged me at that time from giving further thoughtto the matter. After reviewing the paper briefly, I shalladd another interesting theoretical contribution suggestedby one of the several facts first brought out in that paper.

There were three principal new facts introduced in myautogiro paper of 1922; others may become prominent later.In the first place, this was the first paper in which the at­tention of engineers was invited to the fact that a freelyspinning propeller may be used as a substitute for the or­dinary wing.

In the second place, the tests show the direction of rota­tion of the propellers to be opposite to that which was at

• time commonly supposed; the autogiro propeller ro­o"'; in the same direction as a helicopter propeller of the

same shape.The thir Il-esult of these tests was the experimental proof

of th large-parachute effect of autogiros: the drag ofautogiros de cending vertically may be much greater thanthe drag of plane discs or of parachutes of the same diame­ter. Becau. e of this fact, the vertical descent of heavier-than­air craft is brought into the region of practical possibility.It is this third feature of the paper which I wish to discusson theoretical grounds in this article.

vVe consider an autogiro propeller descending verticallywith the velocity V under the action of gra\-ity. The di­ameter of the propeller may be D. giving the disc areaS = D2rt/ 4 • The propeller thrust may be denoted by T,and during a steady. descent it is equal to the weight of theautogiro plane. The problem arising is :r Is there a theoreticalmaximum of the thrust, if the diameter and the density oft .I' are given; and if so, how large is the theoreticalt II ,,:>t, and the velocity of descent?

The problem is rendered simpler and clearer by the in­troduction of the so-called thrust coefficient, or drag co­efficient CD. This is the thrust, or weight, or drag, dividedby the area of the propeller di·sc, S, and by the dynamicpressure of the velocity of fall. This latter dynamic pres­sure is equal to the product of the square of the velocityand of half the mass denisty Q of the air. Accordingly. thedrag coefficient is equal to

74JUNE, 1

1930.T UN E,

TlJ"C

t there are two phases of the General's contempla­which overshadow the rest of the book. The firstI do with comparative conditions in aeronautics, con­19 what Europe has done with what America is try­I do.not nice to be compelled to admit that this nation ismght flat-footed. But it is difficult to dodge the con­I that the country.which saw the first ah-plane fly must:ro s the sea to discover what came of it. I am awarerery now and then someone publishes a lot of figuresre that America leads the world in aeronautics, that

-Oldil LIle \Jenerarllln1se".-.-----:<n'irr-,....,-"y""o;;u~ca;;-:l;;-l-;r::::e::::a-::r::-al=-ld-::;--c::-:a:-:-r:-:-e--:-a 7ir-p:11a-'-les fly higher and farther here than anywhere, thatto do so, you should borrow three dollars from the near- our airways are longer and our air mail system more com­est grease monkey and buy yourself a copy. prehensive than anything on ( Continued on page 204),

THE VERTICAL DESCENT(Continued from page 73)

cal investigations but unsuitable for clarifying a problem. . (Contil/ifed fro·11/, preceding page)Let us assume the air to be perfect; that is, free of fric- to the aX.ls, IS equal to the velocity of fall, V, and the final

tion and compressibility. Let us also assume the propeller ~:eloclty IS ec!ual to twice this, to-2V. The air flowsblades to be free of any friction drag, commonly called mally mto a Jet moving ahead of the falling o-iro.profile drag. We neglect, of course, the mechanical fric- . The volume of the air taking part in this ~lotioJ1 can notion of the mechanism. long~r b.e cOJ11puted from the inflow velocity, becaus the

The argument leading to a drag coefficient ONE is as relative mflow velocity is zero. We have to make an esti-follows: The air is originally at rest and passes the pro- J11.ate for the cross-section of the initial and of the finalpeller disc with half the velocity of fall V, that is, with slipstream. It seems a good guess to assume that both;~ V. The air assumes finally twice this value, V, in keep- together ~CCU?y the entire propeller disc area, half of thising with a general relation between the initial velocity, in- cross-.sectlOn 1~1 front of t.he giro; that is, half of it beingRow velocity and final velocity, requiring that the inflow occu~led?y all" approachmg the propeller and half of it"elocity is the arithmetic mean of the other two velocities. ~)y an' be1l1g r~pelle?, ~y t!le propeller and falling downUnder these circumstances the mass of air passing the ~head of the giro. I hIS gwes the mass l? V S/2 and thepropeller disc per unit time is l? S V / 2 where l? denotes tp.e crag equal to the change of the momentumair density and S the area of the propeller disc, D 2rr./4. T = l? V 2 V S /2The chan!!e of momentum per unit mass of air is T V, so that .the drag coefficient becomes now TWO rather th

~ ONE 111 til' I ,anso that the change per unit time of momentum becomes le c asslca case. The power absorbed is now;~ l? S V2. This is the thrust, or drag, and divided by equal to the entire gravitational energy available.o l? S V2 gives the drag coefficient ONE. . The actu~l flow created by a descending giro will be

It should be noted that the energy absorbed by the ai r ~~me\Vhat different from the described one, but it willas kinetic energy is 0 T V per unit time. This is only l~,.obably possess a. dra~ coefficient governed, to a certailhalf of the gravitational energy becoming free, the other ~x~ent, by the r~latlOn tound. The air is unlikely to flowhalf has to be dissipated. This is the weak point of the I ~tl ~nYrgreat c!lsta.nce ahead of the falling giro; th~ re-argument. It must be shown that there exists no other flow pc e s Ipstream Will probably soon be dissipated. 1'1absorbing more than half the energy available for the crea- !:llngs ~hou~cl be investigated experimentally. The pre;:~~tion of drag. If it does, its drag coefficient will be larger t leor~ IS SUItable as a basis for new experiments, and the'than ONE, and ONE will not be the theoretical maximum. III tUll1 Will serve as a new basis for further improvemen/ I

In order to arrive at such solution, we have to conceive I) [ the theory.in our mind a picture of a flow different from the one J'ust [ll the mean time, in the light of the preceding d' .r . h ISCUSSlOndiscussed. The old computation is based on two assump- .. remall1 .somew at skeptic.al about the reported value of"tions: (a) that the transfer of pressure takes place strictly J. OUR ot the drag: coefficient. Until further evidence isin the propeller plane, and (b) that the mass of the air collected, my mmd IS open fLlr values near Tv'/O II 111 S If d . I on y, astaking part in the transfer is equal to the product of the y e measure. elg lt years ago. Exact measurementspropeller disc area and of the inflow velocity. Both as- o~ t~e do~~~ ~eloclty at~e very difficult to make, and ap-sumptions are well fitted for propellers under ordinary con- jJIecJable eliOIS can easily occur. It is also possible thatditions, but they seem to be unfit for propellers under ~.n the cases reported a steady state of motion was not yetextreme loading, as in the present case. leached, and the propel!er was supplying energy as a fI ,_

wheel. b )vVe proceed by considering a flow where the air ap- Tl b"

proaches the propel!er so that it is entirely stopped relative f' ~e SL1 stltutlOn of the maximum drag coefficient TWto it when it has reached it. The inflow velocity relative 101 t

fle °lld value ONE, until such time when our knowledge

h 11 ' b 1] h laS urt ler advanced is 1 t . dto t e prope er IS zero, ut nevert le ess t ere passes an t tl' . ,wla we mten ed to contributeappreciable volume of air. This is possible because the ~ 11.S time t.o the discussion of the complex problems con­effects of the propeller blades reach beyond the propeller _1:ontmg u~ 1I1 the autogiro. The arguments brought outplane, to points wel! in front and rear of it. The air is <11 e not o~~{ useful for the deduction of the mathematicalnot stopped in its motion in all directions, only the com- ~t:stant'J .71/0, .but beyond this immediate purpose, theyponent of its motion parallel to the original motion and to r~~Ot ~ ,)r;ght ~lgh.t on the mechanism of the creation ofthe propeller axis is zero. In the vicinity of the propeller If )y 1ee 'I Spll111lng propellers and on the aerodynamicsthe air flows substantially parallel to the propeller disc. 0 an autogIro descendmg vertically or moving- otherwise.The air finally returns in the direction from which it came,­the final velocity becoming equal and opposite to the in­itial velocity.

Relative to the undisturbed air, the initial velocity is thenzero. The inflow velocity, or rather its component parallel

(CM/tinned on 1'le.'Vt page)

than the wave velocity, the drag or air re­sistance of the waves tends to swell them.The transverse fabric tension is only a sec­ondary factor for the longitudinal wavevelocity; so, too, are the length of the paneland the stiffness of the fabric. The mainfactor for the longitudinal wave velocity isthe logitudinal tension. The outcome of thepreliminary tests encourages us, therefore,to follow this line of argument further.

The theoretical value of the wave velocity(string formula) is:

Va=y'TXgjWwhere Va denotes the wave velocity, T thetension per unit length, W the weight perarea, and 9 the acceleration of gravity. Thewave velocity computed from this formulais inserted into the plot.

It becomes apparent that the critical veloc­ity follows this theoretical wave velocity,as does the stiffness of the fabric.

The tests therefore support the theory thatthe critical velocity of flutter is practicallythe same as the natural wave velocity. Ahelpful thought, showing the limitation ofwhat can be reached and suggesting the wayfor improvement. Dampening increases thegap between wave velocity and critical veloc­ity, aAd such dampening could be increasedby the choice of special fabrics and by ar­tificial steps. The main improvement willcome through increasing the theoretical wavevelocity, by putting more importance on thelongitudinal fabric stress and less on thecircular fabric stress than is done now.

TEMPERATUREINDICATOR

AUGUST, 1930

THE General Electric type DO-18 ther-mocouple engine-temperature indicator

accurately measures the temperature of cyl­inder heads, cylinder walls, or any hot spotsthat may be of particular significance. It isinstantaneous in operation and, therefore,gives immediate indications of heating' orcooling. A complete indicator consists of athermocouple, twin-conductor leads, and awid j unction, temperature compensated, re­mote-indicating instrument. Two differentforms of thermocouples are optional withthe purchaser. One is a straight threadeddesign arranged to be screwed into a quar­ter-inch tapped hole in the engine. Theother is in the shape of a washer and canbe mounted between the spark plug and cyl­inder head in the same manner as a gasket.

Thermocouples have been used in the lab­oratory for some time. However, their ap­plication has been limited, because the oper­ator had to correct for varying air tempera­tures. In the General Electric DO-18 allcorrections are made automatically by theinstrument.

The type DO-18 engine-temperature indi­cator is accurate to within three degreesFahrenheit for ordinary air temperaturechanges. 'VIThere extreme variations areencountered, such as between sea level andhigh altitudes, the maximum error is notmore than ten degrees Fahrenheit.

40aoplotted against the

to be interesting.The tests were conducted by E. D. Per­

kins upon my suggestion. A wood frame,four feet by eight feet was constructed,which had three loose corner connections,covered by an envelope of linen fabricweighing 0.135 kilograms per square meter.The tension of the fabric could be variedlengthwise and crosswise by loading it witlidifferent weights, pulleys being interposedwhere necessary. For each combination oftensions, the air flow of the tunnel wasstarted and gradually increased until flappingbegan. The velocity that started the flap­ping was noted; it was always sharply de­fined and could easily be observed. Thesame tests were made with a frame sub­divided in an up-stream and low-stream por­tion of unequal size, giving two combinationsby simply turning the frame around. Theresult with the undivided frame is repre­sented in the illustration. The critical airspeed is plotted to the right, and against itthe longitudinal loading is plotted upwards.The di fferent curves represent di fferenttransverse loadings.

It is apparent from this plot (and is con­firmed by the two other test series with thesubdivided frame, not reproduced here) thatthe transverse fabric tension is the dominat­ing factor in determining the critical velocityof flapping. The transverse tension has onlya secondary effect and so, too, has the longi­tudinal extension of the frame.

That much about the tests actually per­formed.. The tests intended but not per­formed were of the same character, withdifferent kind of fabrics, varying in weight,stiffness and internal dampening.

In the scientific analysis of the problem, itappears logical to compare the critical veloc­ity with the natural velocity of the fabricwaves, the fabric waving like a rope orviolin string. If the natural waves have avelocity greater than the velocity of thepassing air, they must overcome drag andconsequently are dampened out. On theother hand, if the air velocity is greater

m s.

~JIDn@m~

FLAPPING OF AIRSHIP COVERSBy Dr. Max M. Munk

10 20Critical flapping velocity and theoretical wave velocity,

longitudinal tension

o

THE

90

T HE lighter-than-air field. has I~t~ly

strengthened its strategic pOSItionrelative to the heavier-than-air. It

is again looked on with favor and expecta­tion by many who thought it doomed asunpractical and out-of-date. This type ofaircraft still possesses the proverbial "in­fancy" complex; we expect the airships toget better every day in every way. Thethree principal ways in which progress isbeing made are (1) improving of the struc­ture (and of structural material); (2) less­ening of drag by means of the application ofmodern aerodynamic inventions, resulting ina saving of horsepower; and (3) increasingthe speed. This last point is particularlyimportant, for the relatively low speed ofthe airship at present is its outstandinghandicap; this low speed is partly the resultof limitati~ns of the size of the power plant.The power required increases as the cube ofthe speed, and each reasonable increase ofspeed without aerodynamic improvements re­quires an unreasonable increase of the out­put of the power plants. It also necessitatesstronger and heavier structures.

There is an additional obstacle to the in­crease of the speed of Zeppelin type dirig­ibles. They are covered with a light fabric,and at a certain critical speed this coverbegins to flap, considerably increasing theair resistance of the ship and thereby pre­venting further incr~ase in speed.

Some of the thoughts on the flappingproblem expressed here may prove helpfulin overcoming this difficulty. The argu­ments are supported by the results of sometests which were made at the N.A.C.A.laboratory in 1924, and which have neverbeen published. These simple tests wereeminently successful and showed promise offurnishing further valuable information forthe solution of the problem by merely re­peating in systematic variations the testmethods then developed. For administrativereasons it became necessary at that time, togive up this line of research, just when itsvalue was established and the results began

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