— . -. .!, + ..? .- - . .%- . .-. .. NATIONALADVISORY COMMITTEE FOR AERONAUTICS - TECHNICAL MEMORANDUM 1270 -J m # m OF VERY HIGH FLIGHT SPEEIX By Eugen Sanger 0 -4- Translation of ZWB Forschungsbericht Nr. 972, , May 19-38 i -+ Washington May 1950 . i. .. ----- . .. .. .... .-. , -L .- ● . NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS IIKENICAL WORANDUM 1270 TKE (MS KCNETIC!SOF WRY HI(2EFLIG3T SPEEDS* By Eugen Sanger ABSTRACT Ih ordinary gas dynamics we use assumptions which also agree with kbetic theoq of gases for smell mean free paths of the air molecules. The air forces thus calculated have to be r=xsmined, if the mean free path is comp=able with the dimensions of the moving body or even with its boundsry layer. This case is very clifficult to celculate. The con- ditions, however, are more simple, if the meen free path is large c- pared to the body length, so that the collisias of molecules with each other can be neglected compared to the collisions with We body surface. Ih order to study the influence of the lsrge mean free path, calcu- lations ere first carried out for the case of extreme rarefaction. Furthermore, the calculatims on this “completely ideal” gas will.be carried out under consideration of -Maxwelltsveloci@ distribution and under the assumption of certain experimental.lyestablished reflection laws for the trsnsl.ational end nontrenslational molecular de~ees of freedao The results thus obtained allow us to find, besides the air pressure forces perpendicular to the surface, also the friction stresses parsJ2el to the surface. — Their general results are calculated out for two practically important cases: for the thin smooth plate end for a projectile-shaped body moving axisUy. The mathematical part of the investigation was carried out prinmrily by Dr. Irene Bredt. *“Gaskinetik sehr hoher Fluggeschwindigketten.” Forschungsbericht Nr. 972, May 31, 1938. 2 - NACA TM 1270 . — —- . I. INTRODUCTION 11. AIR FORCXSON TEE FRONT SIDEOF AI?LATPIA’ZEOELIQUETOT53 AIR STREWM .- 111. AIR FORCES ON TEE FRONT SIDE 03’A FIAT PIATE VERTICAL TO 53 AIR s- . .. ... m. AIR FORCZS ON TH33BACK SIDE OF A FIAT PIATE — v. APPLmm’IoN 13xmmEs — ----- VI. 1. INTRODUCTION With the motion of lmdies at very great atmospheric heights, the air can no longer be considered a centinuous medium, in the sense of .- flow theory. At over ~0 kilmeters altitude, the mean free path of the.air ● molecules till be of the ma@tude of boun- lsyer thickness agd, at over 100 kilometers altitude, of the magnitude Of the moving body itself. ✎ The mean free path at greater heights wiIl”be defjn~tely greater than the body dimensions of the moving body, and ,theespecially simple ‘ conditions of very rmef ied “completely ideal” gases are valid where the effect of the collisions of the molecules with each other disappears ✎ ✎ compared to the effect of the collisions with the moving body. ✎ The air molecules colllde then against the moving b@y as individual -particles,in.dep@@ of each.”other, and are I%flected with a mechsniam ✚ whi~h deviates more or less from the lamwn Newt-onianprinciple of air. .. ✍✎ drag, as shown by the results of the kinetics of very rarefied gases. The velocity of the body will be denoted by v (meters per second) in the following discussion. .— H we imagine a6 usual.that for the consideration of=.theflow process the body stands still and the medium moves, then v equals the uniform undisturbed flow velocity of dl the air molecules. ✎ . ✎ ,. NACA TM 1270 3 The air molecules have also their ideal random thermal motion. The individual moleculsr thermel velocities sre distributed ccJ& pletely at randcm in all directions and are of completely srbitrary magnitudes, where.the vsrious absolute velocity magnitudes group around a most pro%able value, c (meters per second), according to Maxwell’s distribution, which has the ratio @ ~ to the most applied gas kinetic value of “average molaculer speed E“ (sqyare root of the average of the squqres of all the velocities present). According to the known hk.xwellspeed distribution law, letting P(kg 52/!) equal the total moleculsr mass per unit volume, the mass dp of those molecules having velocities between q and Cx + dcx iS: 2 .% M 4 <e C2 dc (1) T=ij7?c3 x E we consider cmly this mass dp of an otherwise motion– . less (v = O) ~S, the molecules of whioh are moving with the particular speed c= in random directions, &en the quentj.w Q– of molecules striking per seccmd on a unit surface of a motionless plane wall can be calculated, imaging that ell.1velocities Cx are plotted from the ‘ center of a sphere with rsdius Cx. The conditions of figure 1 are the result of sn inclination angle @ between the well.normal and the veloci@ directicm under consideration, and of the moleculer qusn– tity @ ~ which passes through the striped srea cll of the spliere 4CX% surface. Then: — --- — ...— - 4 NACA TM i270 . — .- If, with,the aid .ofM&weld.’s equatlmn, we include in our cd.cu- . Mkion aUI.molecules with the various pos~ible speeds Cx between.–O snd .w, the total mass F of the molecules”colliding per second will”. be: as one can find in any tetibook of gas kinetics. me pressure of the — motionless gas against the stationary wall can be calculated similsrly.— The impulse dip ,pe~endfcular to the,.- @th [email protected] of _ .._. particulti speed Cx are striking the wall at an LU@e $ will be: . -z n/2 dp 2 . dip = — Cx COS2 g al?= y CX2-” 4CX% . I$&O “ “- and the impulse of all molecules striking the wall: . .2 . (3) --- JCx=o If we double this impulse value because of the elastic rebound alwqys assumed for motionless gases, then we obtain the gas rest pressure ...... , ..* .“”” —.— .— — .— The value of the total impulse is also interesting, i.e., the sum . ..— -. of all single molecular iqulses which strike per second on the unit- area. —. NACA TM 2.!270 The totel impulse of the molecular mass d~ corresponding to a ??@iCti~ Cx will be: It iS thus 1.5 times greater than the effective impulse %? agaimt the plate. The total impulse 1 is also greater in the same ratio. II. AIR FORCES ON THE FRONT SIDE OF A FIAT PIATE OBIZQ~ TO THE AIR STREAM H we consider again the mass dp of molecules with speeds between Cx and Cx + dcx (sJnost equal)’ snd if we exam3ne its action on a flat plate in an air stream with en angle of attack a, then this process can be illustrated by figure 2, if we further as6um& that v sin a < Cx. The uniform velocity v of the individual molecule ccmibineswith the ideal random velocity (which csn have any space direction) to give a resultant, the components of which are: perpendicular to the -plate: vsina+cxcos@ pa&eJJel to the plate and to v Cos a: vcosa+cYsinticos.- $ ~arallel to,the plate end perpendicular to v cos a: Cxzsin @ sin $ snd for which the absolute value is therefore: w= v2 + 2vcx (sin a cos @ + cos a sin @ cos ~ + cx2) From the sphere of all.possible directions of Gx, a spherical sector with the half opening sngle COSZ= v sin cL/cx is excluded, in which the speed component v sin a + Cx cos @ is directed a- from the plate, i.e., the molecules of this @ range do not strike the’plate. The integration over the velocity directions of all col.lidtagmolecules is not from @ = O to ~ as with the motionless gas, but from @ . 0 to n =X* , —. 6 mcA m 1270” . ..- —- The molecular mms. co~iding per second against tbe unit plate ‘“-- .,: area with the selected speed Cx is therefore: .— — — m— % .. —. .+ .dp d~ = (v sin a + Cx Cos * )2C=% -- 4CX2’“J@o .— — . For v sin a > cx the integration extends over t~e whole sphere _. .7 from @ = O to YC &d the “moleculsrmass coiliding against the ylate _ -. ..—= with selected.speed Cx is m d~=~ (v sin a + Cx cos @)2cx2fi‘sin @ d~ = “dpv gin a“ ““ .- — . “2” J$&O . NACA ‘IW11270 . Both equations naturally give the same value for v sin a = Cx. With the aid of Maxwell’s distritition equation the total moleculsr mass colliding on the unit area per unit tfme within the speed range Cx =0 to mwil.l be: m v sins 1 F= +2. Tsinadp E Cx f3iIla++sm2”dp+c~ J(c=- sin a MCx=0 2 (4) 2 vsina 2 _—Cx Vcx ~’ ~c ● 4P sinae ~ x v f Cx=o Thus the number of the colliding molecules is known, and for cal- culation of the forces acting on the plate, we must now detemnine what hrpulse the molectiar mass under consideration produces in the direct- ions in question. The impulse perpendicular to the plate, ip and the impulse psrallel to it iT will be examined separately. ● . 8 NACATM;270 . We find for the impulse per second perpendicular to tke plate --- 3-C-2 dp ..—. 1. If T sln a < Cx: dip = (v sin a,+ Cx Cos @)%? “X2” J+0 .- &fl+Lst ia 3 ““ Cx Y( ) .. 1-r dp —, 2.~ VSill CL>Cx: dip- (v 19inal+ c~ Cos @)% =“+0 J — .- dpc22+~ — 6X ( This sumation of the impulse components over allpos~ible direc- tions yields the total impulse, perpendicular to.the plate~of the air —— : molecules striking t~e unit area per second with.a speed Cx dete~ mined by” dp. NACA TM 3.270 9 . The further summaticm of the impulses over KLl speeds c= with the aid of Mexwellts distribution equation results in the total impulse ‘P acting per second perpendicular to the plate: , Vsina g 1 c~ k- ~2 +’ =$P 2+6— Si112adcx ~Te ( CX2 ) [JCx=o 1 If we set a= O then equation (5) gives the impulse of the motion- ~ess gas against the motionless wall, equaticm (3).