- - AEDC-TR-70-32 ARCHIVE COPY DO NOT LOAN SPHERE DRAG IN THE FREE-MOLECULAR AND TRANSITIONAL FLOW REGIMES David L. Whitfield and William B. Stephensn ARO, Inc. April 1970 This dcument has been apprved fr public release and sale; its distributin is unlimited. TECHEIT^L REPORTS ~ ^ = 1 b3 ^^m : =j =r~ ; 3 =siu mi 3 mi DCTE * < =i^3 S= Q ' -in ^==ru 1 a r^ jcj!.._. ; s_-ul = ^^, AEROSPACE ENVIRONMENTAL FACILITY ARNOLD ENGINEERING DEVELOPMENT CENTER AIR FORCE SYSTEMS COMMAND ARNOLD AIR FORCE STATION, TENNESSEE p?ertyo?u.s.aia^ce E0 a-.tc L3E?.4EI F40Ö00-S9-C-OOO1 - - - a
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SPHERE DRAG IN THE FREE-MOLECULAR AND TRANSITIONAL FLOW REGIMES David L. Whrtfield and William B. Stephensn ARO, Inc. This dcument has been apprved fr public release and sale; its distributin is unlimited.
FOREWORD This wrk was spnsred by the Air Frce Cambridge Research Labratries (AFCRL) (CRMP), Bedfrd, Massachusetts, under Prgram Element 621 OIF, Prject 6690, Task 669002. The results presented were btained by ARO, Inc. (a subsidiary f Sverdrup & Parcel and Assciates, Inc.), cntract peratr f the Arnld Engineering Develpment Center (AEDC), Air Frce Systems Cmmand (AFSC), Arnld Air Frce Statin, Tennessee, under Cntract F40600-69-C-0001. This wrk was cnducted frm April 14 t September 26, 1969, under ARO Prject N. SB0913. The manuscript was submitted fr publicatin n January 2, 1970. The authrs wish t acknwledge the assistance f tw cperative engineering students in perfrming this wrk, Jack L. Wmack (University f Tennessee) and Nrman O. Speakman (Auburn University). This technical reprt has been reviewed and is apprved. Rbert T. Ott Ry R. Cry, Jr. Majr, USAF Clnel, USAF AF Representative, AEF Directr f Test Directrate f Test n
ABSTRACT Results are presented f sphere drag measurements made in the free-mlecular and transitinal flw regimes. The drag data were btained using a drag balance and the free-flight technique. Cnditins fr which measurements were made are within the ranges 1 < Kn» < 32, 2.9 < M. < 11.2, 0.8 < T w /T. < 16.0, and 2.5 < % < 3.2. An analysis f the drag n a sphere in rarefied flw is als presented, and the results are cmpared with experimental data and ther theries. The analytical results well predict the trend f the experimental data in the transitin regime and remain valid at Knudsen numbers fr which previus theries are nt applicable. in
CONTENTS ABSTRACT üi NOMENCLATURE vi I. INTRODUCTION 1 II. EXPERIMENTAL PROCEDURE 1 III. TRANSITION DRAG ANALYSIS 4 IV. RESULTS AND DISCUSSION 7 V. CONCLUSIONS 10 REFERENCES 11 I. ILLUSTRATIONS Figure APPENDIXES 1. Flw Cnditins in the M3 and M6 Nzzles IS 2. Schematic f Lw Density Tunnel, ARC (8V) 16 3. Effect f Viscus Crrectin n Mach Number and Dynamic Pressure at the M6 Nzzle Exit 17 4. Pitt Prbe Viscus Crrectin Data 18 5. Increase in C/CD fm and Decrease in HQ^/H 0 fr the M6 Nzzle (T 0 = 800 K) 19 6. Tare Frce and Ttal Drag Measurements Using 0.25-in.-diam Sphere n Drag Balance in M6 Nzzle Flw 20 7. Falling Spheres Using the Free-Flight Technique 21 8. Curve Fit f the Axial Displacement f a Sphere versus Time 22 9. Assumed Mdel f the Rarefied Flw abut a Sphere 23 10. Incident Flux f Mlecules n a Sphere in Free-Mlecular Flw 24 11. Free-Mlecular Sphere Drag Cefficients fr Diffuse Reflectin and Cmplete Accmmdatin 25 12. Sphere Drag Cefficients in the Transitinal Flw Regime Accrding t Eqs. (19) and (20) 26 13. Effect f 0 n Kn w /Kn 27 14. Sphere Drag Cefficients fr Cnstant and Variable X w with Sw = 1 28 15. Effect f Nrmalizing C D by C Dfm 29 16. Balance and Free-Flight Sphere Drag Data with Analytical Results 30 17. Trend f Analytical Results and Experimental Data in the Transitinal Flw Regime 31 18. Analytical Result with S w = 1 and Sme Experimental Data with Little Scatter 32 19. Cmparisn f Eq. (19) with Other Theries and Available Experimental Data fr S» near 1.6 33 Page
Page II. TABLE I. Sphere Drag Cefficients and Flw Cnditins 34 NOMENCLATURE A Crss-sectinal area f sphere, m 2 C D Drag cefficient, defined by Eq. (1) Cp Cv Specific heat f gas at cnstant pressure Specific heat f gas at cnstant vlume Cl, C2, C3, Cnstants, Eq. (2) D d Drag Sphere diameter erf(x) Errr functin, 2A/F/ x e -t dt H Kn M m N N n P P' Pp Lcal ttal enthalpy Knudsen number Mach number Mass f sphere Number flux f mlecules Number f mlecules per unit time Number density f mlecules Pressure Ttal pressure immediately dwnstream f a nrmal shck Measured pitt pressure q Dynamic pressure, pu 2 /2 R Gas cnstant VI
Re Reynlds number based n sphere diameter, als pitt prbe diameter in Fig. 4 Re 0,r» Re* Re» r Nzzle reservir Reynlds number, p 0^/2H 0 r*//* P.U..d/u w P-U d//i Radius f sphere r* Nzzle thrat radius S Speed rati, U/(2RT)** Sw UJ(2RT W )* s. Uj(2RTja T t U V X Temperature Time Velcity Velcity f mlecules emitted frm the sphere surface Axial crdinate 7 Rati f specific heats, C p /C v X Mean free path, (JT/2RT)W n/p t* Viscsity P a Gas mass density Effective mlecular diameter SUBSCRIPTS 2 Cnditin immediately dwnstream f a nrmal shck <L Nzzle centerline C Cllisin mlecules vu
E fr i s w Nzzle exit Free-mlecular value Incident mlecules Reservir (ttal cnditin) Surviving mlecules Sphere wall cnditin 00 Free-stream cnditin VUl
SECTION I INTRODUCTION Investigatins f atmspheric winds, density, pressure, and temperature using spherical satellites and falling ballns (Ref. 1) have stimulated interest in sphere drag frm subsnic t hypersnic flw and frm the cntinuum t the free-mlecular regime. Besides the dependency f drag n the Mach number, r speed rati, drag in the free-mlecular and transitinal flw regimes depends significantly n the sphere wall and free-stream gas temperatures. Als, there is the pssibility that varius materials will have different accmmdatin cefficients at satellite velcities and thereby influence the drag. An example f a bdy which may have all f these factrs influencing the drag simultaneusly wuld be a satellite designed with surface materials f different emissivities t prvide thermal cntrl. An accurate measurement f the drag f such a bdy is very difficult because (1) in rder t simulate high Knudsen number (Kn_ = \Jd) flw in present lw density test facilities, the mdel must be relatively small and this makes actual simulatin f the mdel difficult, and (2) the actual surface cnditin f the materials is difficult t reprduce even if the same materials and paints are used, because f cndensed gases, etc., n the surface f the mdel which may nt be present in space. The effect f wall temperature n drag can be measured in wind tunnel experiments, and the effect f surface cnditin n drag is usually nt as large and can be investigated better using a mlecular beam. Therefre, the effect f surface cnditin n drag will nt be discussed ther than the presentatin f data taken using a sphere with a black and then a gld surface. The purpse f this reprt is t present the results f sphere drag measurements made in the free-mlecular and transitinal flw regimes. A theretical analysis f the drag n a sphere in the transitin regime is als presented, and the theretical and experimental results are cmpared. This analysis predicts the free-mlecular drag value as Kn_ -*b and the Newtnian drag value as Kn_ -* 0. The present results are cmpared with ther analytical and experimental investigatins (Refs. 2 thrugh 9). SECTION II EXPERIMENTAL PROCEDURE The experimental data were taken in the Aerspace Research Chamber (ARC) (8V) f the Aerspace Envirnmental Facility (AEF) at AEDC. This chamber is 10 ft in diameter, 20 ft in length, and uses crygenic surfaces t pump the gas frm the wind tunnel nzzle. The principal pumping capacity f the ARC (8V) chamber was prvided by 620 ft 2 f 77 K liquid-nitrgen crysurfaces and 240 ft 2 f 15 t 20 K gaseus-helium crysurfaces. A ttal f 4-kw gaseus-helium refrigeratin capacity was used. This arrangement permitted the cntinuus pumping f nitrgen r argn at a mass flw rate f 10 gm/sec at a chamber pressure f abut 10 5 mm Hg. The pumping requirements fr this test were less than 10 gm/sec.
Drag measurements were made using a balance and a free-flight technique. The design, calibratin, and peratin f the equipment and instrumentatin necessary fr these tw methds f measuring drag are discussed in detail in Ref. 10. The data reductin techniques are discussed here. 2.1 WIND TUNNEL NOZZLES Tw lw density nzzles, designated as the Mach number 3 (M3) and Mach number 6 (M6) nzzles, were used t prvide the flw cnditins. The wall cling n the M6 nzzle was recently mdified by replacing the aluminum tubing thrugh which liquid nitrgen was frced, by a cmplete liquid-nitrgen jacket similar t the M3 nzzle. The result was t prduce a cnstant 89 K wall temperature which was a 100 K decrease in wall temperature at the nzzle exit. This reductin in wall temperature reduced the bundary-layer grwth n the nzzle wall and als imprved the repeatability f the flw in the nzzle since the wall temperature remained cnstant with time. Sme results n the flw cnditins in the M3 nzzle using nitrgen r argn as the test gas are given in Ref. 11. Bundary-layer prfiles and axial-flw cnditins fr the M3 nzzle, and mre results n the flw cnditins in the M3 and M6 nzzles, fr nitrgen nly, are given in Refs. 12 and 13. A summary f the perating ranges f these nzzles is given in Fig. 1 (Appendix I). 2.2 NOZZLE CALIBRATION The ttal pressure, P 0, was btained using an MKS Baratrn (variable capacitance transducer), which measured the average f the pressure at tw lcatins in the reservir as shwn in Fig. 2. The ttal temperature, T 0, was measured using a ttal temperature prbe lcated n the centerline f the reservir (Fig. 2). The flw was calibrated using a l-in.-diam, 10-deg internally chamfered pitt prbe. A Baratrn was used t measure the pitt pressure. The pitt prbe was munted n a scanner mechanism as shwn in Fig. 2. The mvement f the scanner, which was in the plane f the figure, permitted the measurement f pitt pressures frm 5 in. inside t 20 in. dwnstream f the nzzle exit planes and frm 5 in. belw t 20 in. abve the nzzle centerlines. The vertical mvement was sufficient t traverse the exit radii f the tw nzzles. Fr the rarefied flw cnditins f interest, the viscus crrectins t the pitt pressure measurements were significant. The magnitude f this effect n Mach number and dynamic pressure, q.., is illustrated in Fig. 3 fr the M6 nzzle. Since the drag cefficient is inversely prprtinal t q_, it is bvius frm Fig. 3 that serius errr is intrduced if the viscus crrectin is nt knwn accurately. The viscus crrectin data used t crrect the pitt pressure measurements were btained in a previus test prgram, and these data are given in Fig. 4 with data frm ther surces (Refs. 14 thrugh 17). A prblem assciated with lw density wind tunnel testing is that f bundary-layer merging, i.e., the bundary layer cmpletely filling the nzzle. It has been previusly assumed that merging in the M6 nzzle ccurred at the pint f minimum
Mach number (Fig. 3).This assumptin has since been fund t be invalid fr the fllwing reasn. It was bserved that an abrupt increase in the drag cefficient ccurred at values f Kn. where the drag cefficient shuld be free mlecular r at least be asympttically appraching the free-mlecular limit. In rder t explain the unexpected increase in CD, an investigatin was made t determine whether r nt the flw was merged, because if it was, the pitt pressure measurements wuld be incrrect and the free-stream prperties, q_, T, etc., calculated using P 0, T 0, and P 0 ', wuld als be incrrect. The prblem f merged flw was investigated using the fully viscus slender-channel prgram develped by Rae (Ref. 18). It was fund frm the numerical slutins that a significant decrease in the lcal ttal enthalpy, H, ccurred at the M6 nzzle exit centerline at abut the same pint, i.e., value f P 0 fr T 0 = cnstant, at which the increase in CD ccurred (Fig. 5). The decrease in Hq, E indicated the flw was nt isentrpic, and hence the q used t determine CD fr P < 1 mm Hg was incrrect. The value f nzzle reservir Reynlds number, Re 0ir», at which merging ccurred fr this value f the nzzle wall t reservir temperature rati (which was 0.11), was Re 0(I * = 900. The thrat radius fr this nzzle was r* = 1.54 in. It is interesting t nte that merging actually ccurred fr the cnditins in Fig. 3 at P 0» 1 mm Hg, which was a significantly larger value f P 0 than the value at which the minimum Mach number ccurred. Numerical slutins were als btained fr the flw in the M3 nzzle; hwever, data are presented nly fr isentrpic flw cnditins. 2.3 DATA REDUCTION 2.3.1 Drag Balance Data reductin f the frces measured using the drag balance was straightfrward. After frce measurements were made n the sphere munted n the sting at varius flw cnditins, the sphere was remved, and a secnd sphere, independently supprted, was psitined slightly upstream f the sting (Fig. 6). The tare frces n the sting were measured using this arrangement fr the same range f flw cnditins fr which ttal frce measurements were made. An example f the tare frce and ttal drag frce measurements is given in Fig. 6. Fr this 0.25-in.-diam sphere, the tare frce was frm 15 t 20 percent f the ttal drag frce. If the sphere size was reduced, the percent f ttal frce which was attributable t tare increased. The difference between tare and ttal frce measurements is the sphere drag, D. The drag cefficient, CD, was btained frm the definitin C D = D 2 = -2_ (1) A secnd sphere, identical t the ne used n the balance sting t measure the drag, was used t determine the sphere wall temperature. This secnd sphere was supprted by the leads t a thermcuple which was used t measure the sphere temperature. It was attached t the same supprt as the balance and was simultaneusly subjected t the same flw cnditins as the sphere munted n the balance sting. It was assumed that the temperature f the sphere was unifrm, and therefre the measured temperature was taken as the sphere wall temperature.
2.3.2 Free Flight Free-flight drag data were btained frm phtgraphs f the trajectries f spheres (Fig. 7) which were drpped thrugh the nzzle test sectin. The phtgraphs were multiple expsures f spheres illuminated at cnstant time intervals by a strbe light perating at the rate f 120 flashes/sec. Frm ne t three aluminum and/r magnesium spheres were drpped simultaneusly. The spheres were initially lcated in the drp mechanism a minimum f ten diameters apart in a plane perpendicular t the plane f the phtgraph in Fig. 7. Immediately after the spheres were drpped, the minimum separatin distance increased, and it was assumed that n interactin amng the spheres ccurred. Sphere diameters f 1/32, 1/16, and 1/8 in. were used. The sphere wall temperatures were determined by measuring the temperature f the drp mechanism. The change in sphere temperature fr the perid f time required fr the spheres t traverse the test sectin was negligible. The drag frce n a sphere was determined by applying Newtn's secnd law f mtin t the sphere in the directin f the free-stream velcity. The distance, x, which a cnstant mass bdy wuld mve in a cnstant frce field is a quadratic equatin in time, t, i.e., x = cit 2 + c 2 t + c 3 (2) Using Newtn's law ' i-fä (3) and substituting Eq. (2) int Eq. (3) and differentiating gives D = 2mci (4) where m is the mass f the sphere. Therefre, if ci can be determined, the drag can be calculated. The cnstant ci was determined by fitting a secnd-rder plynmial thrugh the data f a plt f x versus t (Fig. 8). The value f x fr each t was determined by measuring the axial displacement f the sphere frm pint t pint in Fig. 7 at cnstant (knwn) intervals f time. These measurements were made frm the phtgraphic negatives using a film reader. SECTION III TRANSITION DRAG ANALYSIS A simple analysis is presented fr the drag f spheres in the transitinal flw regime. The mathematical mdel is based n the fllwing fur assumptins: (1) The flw Mach number is sufficiently high that the number flux f mlecules which has a chance f clliding with the sphere is given by N = n»u, (2) all mlecules are emitted nrmal t the sphere surface, (3) each emitted mlecule experiences nly ne cllisin with an ncming free-stream mlecule, after which neither mlecule is cnsidered, and (4) all cllisins ccur at a distance X w frm the sphere surface (Fig. 9). 4
There are certain restrictins and cnditins which need t be cnsidered because f the basic assumptins. The first assumptin requires that the randm mlecular velcity be less than the mean flw velcity. Physically this means that the mlecules tend t mve in straight lines parallel t the mean flw directin, and therefre, the net number which crss the bundary f the imaginary cylinder swept ut by the sphere per unit time, i.e., irr 2 U m, is negligible. This is nt a particularly stringent assumptin, as indicated by the results given in Fig. 10 fr the incident flux in free-mlecular flw (see Ref. 19 fr a derivatin f the incident flux). Frm Fig. 10 the apprximatin Nj * N = njj m is gd fr S_ > 1 r (2RTJ W < U. The secnd assumptin permits ne t neglect the distributin functin f the mlecules emitted frm the sphere surface, and it is justified nly by the final results. The third and furth assumptins, tgether, imply that the sphere is in free-mlecular flw with respect t mlecules which survive the cllisin surface (Fig. 9) since there are n cllisins between free-stream and emitted mlecules in the regin between the cllisin surface and the sphere. Therefre, by determining the number flux f surviving mlecules, n s U, the drag n a sphere can be determined frm D CD n s ~^~ = ch7 ra = ~^T (5) since the drag n the sphere is the same as the free-mlecular drag with a free-stream number density f n s. That is, fr the same free-stream and sphere wall cnditins, the drag is prprtinal t the free-stream number density, r, fr the case f the surviving mlecules, t the effective free-stream number density, r^. Befre slving fr n s U, it is interesting t nte an errr intrduced by cnsidering the rati CD/CD fm in Eq. (5). It was assumed in writing Eq. (5) that the free-stream cnditins f the surviving mlecules, ther than number density, did nt change, and als that the sphere wall cnditins did nt change. The assumptin f cnstant free-stream cnditins is valid because f the fur basic assumptins, but the ne cncerning the wall cnditins is nt. This is because fewer mlecules are available t transfer heat between the gas and the sphere because f the "shielding" effect prduced by the cllisins, and therefre, T w is nt the same fr the shielded and unshielded sphere. Hwever, fr a diffusely reflecting wall, which was the mdel assumed in calculating CD fin fr the experimental data, the effect f T w decreases as S» increases (Fig. 11), and it is assumed that the effect f the difference between wall temperatures n CQ is negligible. Of curse fr a specularly reflecting wall, CD is independent f T w, and n errr is intrduced. In the slutin fr n s U w the mean distance between cllisins f emitted mlecules with ncming mlecules, X w, will be taken as a functin f ^ as given by (Refs. 2 and 20).. 7r 0 n [(U + v w cs ifi) + (v w sin ^) 2 P
In terms f the mean free path f the mlecules in the free stream. \ = \((y/2ira 2 n c.). Eq. (6) can be written A w \J2 v w 1 = (7) A«, tuj f 2U«,v w cs^ + v 2 w ]^ Since the number flux which has a chance f clliding with the sphere is n.u«, then nly thse cllisins which ccur at the "effective cllisin surface" (which is that part f the cllisin surface within the cylindrical vlume swept ut by the sphere, Fig. 9) will affect the surviving number fjux which reaches the sphere. The number f mlecules per unit time which survive, N s, can be expressed as where N s = PL -N c (8) N. = njw 2 (9) is the number f free-stream mlecules per unit time which wuld cllide with the sphere if n cllisins between emitted and free-stream mlecules ccurred, and N c is the number f mlecules per unit time which are emitted tward the effective cllisin surface and hence suffer cllisins with free-stream mlecules. Therefre, N c is the number per unit time which are remved frm N... Since the mlecules were assumed t be emitted nrmal t the sphere surface, the number f mlecules emitted frm the sphere surface within the angle 9 is equal t N c as expressed by e N c = nguap = n s U / 2m 2 sin 0 cs ^ d^ (10) where n s is the number density behind the cllisin surface, and the integral, Ap, is the prjected surface area f the sphere within 6 (Fig. 9). Integrating Eq. (10) gives Expressing N s as N c = n s UOT 2 sin 2 6 (in N s = n s U«77r 2 (12) which is the ttal number per unit time which survive t cllide with the sphere, and then substituting Eqs. (9), (11), and (12) int Eq. (8) gives n s U = nu«,-n s Ü sin 2 Ö (13) Slving fr n s /n«,ne has Ji 1 + sin 6 (14) Substituting Eq. (14) int Eq. (5) and using Ap/Afr sin 2 0 (Eqs. (10) and (11)) gives CD CD fm 1 + sin 2 6 A (15)
The angle 0 depends n X w since 0 is the angle at which the distance between the cllisin surface and sphere surface in the 0 directin is equal t X w (Fig. 9). Frm the gemetry in Fig. 9 ne can write Substituting Eq. (16) int Eq. (15) gives sin0, 1 (16) C D i C Dfm 1 + (17) Using Eq. (7) fr X w. the term X* /r becmes BS)'-G9--* where Kn_ = XJd. S w - UJ(2RT W )*, and v w = (9JTRT W /8) % (Ref. 5). Substituting Eq. (18) int Eq. (17) gives fr the drag cefficient C, l - ^ F P } 2 V 2 K» M f f \ ty/n I \ 3vV J ; An expressin fr 0 used in Eq. (19) can be btained by substituting Eq. (18) int Eq. (16) t get (18) sin 0-2\''2 Kna (20) Equatin (20) is a transcendental equatin which must be slved fr 0 fr each value f Kn, and S w. The slutin fr 0 which satisfies 0 < 0 < JT/2 (Fig. 9) was used in Eq. (19) t calculate CQ/CD,,,, fr each value f Kn«, and S w. 4.1 ANALYTICAL SECTION IV RESULTS AND DISCUSSION It is f interest t examine the limits f Eq. (19). In the limit Kru^^the result fr the drag cefficient is C/CD fm = 1. which is, f curse, expected. In the limit Kri»-* 0 the result fr the drag cefficient is C D /C Dfm = 0.5. This result can be reasned physically by recalling that ne f the assumptins was that nly ne cllisin between
emitted and free-stream mlecules was cnsidered. Therefre, when the mean free path X. = 0 (and hence X,, - 0, see Eq. (18)), nly ne-half the mlecules which have a chance f clliding with the sphere culd actually cllide, since each mlecule at the sphere surface prevented anther mlecule frm clliding with the surface. This result is the same as the result btained frm the Newtnian thery if CD fm is based n specular reflectin. Newtnian thery assumes that all the mlecules directly upstream f the sphere hit the sphere but that they impart mmentum because f the cllisin nly, since they leave parallel t the surface. Whereas, in the present analysis, nly ne-half the mlecules directly upstream f the sphere hit the sphere, but they impart mmentum frm bth the cllisin and the reflectin. This is the reasn fr a sensible result in the cntinuum limit. Sme results frm Eq. (19) are given in Fig. 12 where 0 was btained frm the slutin t Eq. (20) fr each value f Kn» and S w. Ntice in Fig. 12 that CD/CD fm decreases as S w increases. That is, fr cnstant free-stream cnditins, the cllisin surface appraches the sphere surface as S w increases, because f decreasing T w and hence decreasing X w, and cnsequently mre shielding ccurs, and the drag is reduced. A simplificatin in the calculatin f the drag cefficient is btained if X w in Eq. (17) is taken as a functin f Kn«, and S w nly, i.e., if 0 in Eq. (18) is taken as cnstant. The effect f the 0 term in Eq. (18) is shwn in Fig. 13 t have relatively little effect n the value f X w, r Kn w, fr cnstant S w. The assumptin f cnstant 6 in Eq. (18) eliminates the necessity f slving Eq. (20). The resulting drag cefficients as calculated using cnstant and variable X w are illustrated in Fig. 14 fr Sw = 1. Fr small values f Kn_ the result with X w (0 = TT/2) gives slightly better agreement with variable Xw, whereas fr large values f Kn_ the result with X w (0 = 0) gives better agreement. This result is expected since fr small Kn», 6 «* ir/2, and fr large Kn, 0 «0 (Fig. 9). In general, fr all Kn«, the result with X w (0 = 0) gives better agreement with the result fr variable X w (Fig. 14). It shuld be pinted ut that the assumptin f cnstant 0 in Eq. (18) is justified nly because X w is a weak functin f 0 and a strng functin f Kn. and S w. 4.2 EXPERIMENTAL 4.2.1 Balance Data Drag measurements were made using the balance in the M3 and M6 nzzle flws. The reservir (ttal) temperature was 800 K, and the ttal pressure was varied between 0.1 and 5.0 mm Hg, depending n the nzzle used, t prduce free-stream Knudsen numbers f 1 < Kn. < S, based n a 0.2S-in.-diam sphere. The balance prved capable f measuring the change in sphere drag which was caused nly by a changing wall temperature. The wall temperature f the 0.25-in.-diam sphere used t take the data in Fig. IS varied between 253 and 314 K, and the free-stream gas static temperature, T, varied between 74 and 62 K. Taking int accunt the apprpriate S and T w /T rati m the calculatin f Cn fm, the drag data nrmalized by C fm have less deviatin than the CD data as shwn in Fig. 15. The assumptin used fr the calculatin f Cp fm was that the mlecules were cmpletely accmmdated and reflected diffusely frm the sphere surface (Fig. 11). 8
The balance drag data are shwn in Fig. 16 by the clsed symbls. The data taken in the M6 nzzle, i.e., 7.0 < M. < 7.7, have less than 2-percent scatter, and the data taken in the M3 nzzle have slightly mre scatter. The difference, if any, in the drag between the tw spheres with painted surfaces, ne cated with a flatblack lacquer paint f high emissivity, which was used fr thermal cntrl n satellites, and the ther cated with gld, culd nt be detected. The drag cefficients and flw cnditins fr the balance data are tabulated in Table I (Appendix II). 4.2.2 Free-Flight Data The free-flight drag data using nitrgen and argn as the test gas are als given in Fig. 16. The scatter in the free-flight data is bviusly much wrse than the scatter in the balance data. There are several pssible explanatins fr the scatter in the free-flight data. Amng them are: (1) The spheres were drpped thrugh the test sectin with initial directin nrmal t the free stream rather than directly upstream as is usually dne in free-flight measurements, and therefre, small axial displacements f the spheres were btained because f the lw free-stream dynamic pressure and the requirement that the spheres be at least 1/32 in. in diameter in rder t be phtgraphed, (2) the strbe light used was limited t 120 flashes/sec, and this restricted the number f expsures f the spheres which culd be made during the free fall, (3) errr in reading the sphere trajectries frm the phtgraph negatives, (4) large distance, 9 ft, frm the camera and strbe light t the spheres, and (5) the prtin f the test sectin thrugh which the spheres fell had an axial Mach number gradient f abut 0.0S/in., and the flw cnditins used t reduce the sphere data crrespnded t that pint in the flw where the spheres crssed the nzzle axis (the maximum axial displacement f the spheres was abut 6 in.). Suggestins t eliminate r reduce the magnitude f these effects are given in Ref. 10. The drag cefficients and flw cnditins fr the free-flight data are tabulated in Table I. 4.3 ANALYTICAL AND EXPERIMENTAL COMPARISONS Cmpared with the balance and free-flight drag data in Fig. 16 is the analytical result given by Eq. (19) fr S w = 2.7, which is a mean value f the wall speed rati fr these data. Frm this cmparisn the analysis appears t prvide a reasnable predictin f the drag in the near free-mlecular regime. Certainly the agreement between thery and experiment is gd, cnsidering the simple mathematical mdel used t analyze the drag in the transitin regime. In rder t determine the qualitative trend f the analysis thrughut the transitinal flw regime, Eq. (19) is cmpared with drag data frm varius surces in Fig. 17 fr tw values f S w which essentially bund the experimental data. As can be seen in Fig. 17, the trend f the experimental data in the transitin regime is relatively well predicted. Hwever, the experimental data f Fig. 17 d nt always vary with Sw as the
analytical results predict. Therefre, the slutin fr S w = 1 was taken t cmpare with sme data in Fig. 18 which have little scatter. It was fund that the slutin fr Sw = 1 prvided the best agreement with the bulk f the experimental data. Furthermre, if X w is taken as cnstant with respect t 8, then the curve in Fig. 18 can be rather accurately apprximated by Cp c Dfm = p! i _ (21) i + r L 1 + 1.615 KnJ IF which, in terms f M., Re«, and 7» can be written as C D 1 C Df m 1 + 1 1 + 2.02\/y M» Ree (22) A cmparisn is made in Fig. 19 f the present result with the theries f Baker and Charwat (Ref. 2), Rse (Ref. 3), and Willis (Ref. 4) and sme experimental data fr Sw near 1.6. The theries f Rse and Willis depend n S n and are calculated nly fr the cnditins f Sims' data (Ref. 7). They shuld nt be expected t cmpare with the ther data, althugh the cnditins f the data f Ref. 6 are clse t thse f Sims' data (the different test gases used shuld be nted, hwever). The present result and the thery f Baker and Charwat d nt depend n S and therefre are applicable t all the data in Fig. 19. As can be seen in Fig. 19, the thery" f Willis and the present result well predict these drag data in the near free-mlecular regime. Als, the present result cntinues t predict these drag data at Knudsen numbers belw which the existing theries shwn in Fig. 19 are nt applicable. SECTION V CONCLUSIONS Sphere drag measurements have been made using the balance and free-fught techniques within the ranges 1 < Kn» < 32, 2.9 < M> < 11.2, 0.8 < T w /T. < 16.0, and 2.5 < Sw < 3.2-The balance data had less than ± 1-percent scatter, but the free-flight data had as much as ± 10-percent scatter. Drag measurements were made n tw spheres which had different surface catings, ne painted black and the ther gld, and any difference in the drag cefficients f the tw spheres was less than the experimental errr f the balance data. It was fund that the balance used was sensitive enugh t measure the increase in drag n a 0.25-in.-diam sphere which was caused nly by an increasing sphere wall temperature. Prper accunting f T w in calculating C fm fr nrmalizing CD smthed the drag data. The sphere drag analysis prvided a reasnable predictin f the drag cefficient thrughut the transitinal flw regime. It was in gd agreement with the thery f 10
Willis, which appeared t be the mst accurate f the three theries with which it was cmpared in the near free-mlecular regime fr the cnditins cnsidered. Furthermre, the present analysis was in gd agreement with the experimental data fr Knudsen numbers belw which the previus theries were nt valid. REFERENCES 1. Engler, N. A. "Develpment f Methds t Determine Winds, Density, Pressure, and Temperature frm the Rbin Falling Balln." University f Daytn Research Institute Reprt N. AF19(604)-7450, AFCRL-65-448, May 1965. 2. Baker, R.M.L., Jr. and Charwat, A. F. "Transitinal Crrectin t the Drag f a Sphere in Free Mlecule Flw." The Physics f Fluids, Vl. 1, N. 2, March-April, 1958, pp. 73-81. 3. Rse, M. H. "Drag n an Object in Nearly-Free Mlecular Flw." The Physics f Fluids, Vl. 7, N. 8, August 1964, pp. 1262-1269. 4. Willis, D. R. "Methds f Analysis f Nearly Mlecular Flw fr a Satellite r Other Space Vehicle." (AD241900), General Electric C., Space Sciences Labs, August 1960. 5. Kinslw, M. and Ptter, J. L. "The Drag f Spheres in Rarefied Hypervelcity Flw." AEDC-TDR-62-205 (AD290519), December 1962. 6. Ptter, J. L. and Miller,J. T. "Cnsideratin f Simulatin Parameters fr Blunt Thick Bdies in Rarefied High-Speed Flws." AEDC-TR-68-242 (AD678159), Nvember 1968. 7. Sims, W. H. "Experimental Sphere Drag Results in the Near-Free Mlecule Regime." Rarefied Gas Dynamics, Vl. 1, Academic Press, New Yrk, 1969, pp. 751-756. 8. Smlderen, J. J., Wendt, J. F., Naveau, J., and Bramlette, T. T. "Sphere and Cne Drag Cefficients in Hypersnic Transitinal Flw." Rarefied Gas Dynamics, Vl. 1, Academic Press, New Yrk, 1969, pp. 903-907. 9. Maslach, G. J., Willis, R. D., Tang, S., and K, D. "Recent Experimental and Theretical Extensins f Nearly Free Mlecular Flw." Rarefied Gas Dynamics, Vl. 1, Academic Press, New Yrk, 1965, pp. 433-443. 10. Stephensn, W. B. and Whitfield, D. L. "Drag Measurements in a Lw Density Gas Stream." AEDC-TR-70-25. (t be published). 11. Stephensn, W. B. "The Cllectin f a Nrmally Incident Lw Density Supersnic Stream by a Crygenic Surface." AEDC-TR-67-201 (AD663755), January 1968. 11
12. Whitfleld, D. L. "Theretical and Experimental Investigatin f Bundary Layers in Lw Density Hypersnic Axisymmetric Nzzles." AEDC-TR-68-193 (AD674597), September 1968. 13. Whitfield, D. L. and Lewis, C. H. "Analysis f Bundary Layers in Lw Density Hypersnic Axisymmetric Nzzles, Including the Effects f Displacement, First-Order Transverse Curvature, Velcity Slip, and Temperature Jump." AIAA Paper 69-653, June 1969. 14. Ptter, J. L. and Bailey, A. B. "Pressures in the Stagnatin Regins f Blunt Bdies in the Viscus-Layer t Merged-Layer Regimes f Rarefied Flw." AEDC-TDR-63-168 (AD416004), September 1963. 15. Enkenhus, K. R. "Pressure Prbes at Very Lw Density." UTIA-R-43 (AD 126534), January 1957. 16. White, R. B. "Hypersnic Viscus Interactin and Rarefactin Effects n Impact Prbes." AIAA Student Jurnal, Vl. 5, N. 2, April 1967, pp. 46-49. 17. Maslach, G. J. "Sme Prblems Assciated with the Measurement f Very Lw Pressures." AGARD Reprt 175, March 1958. 18. Rae, W. J. "Sme Numerical Results n Viscus Lw-Density Nzzle Flws in the Slender-Channel Apprximatin." AIAA Paper 69-654, June 1969. 19. Shidlvskii, V. P. Intrductin t Dynamics f Rarefied Gases. American Elsevier Publishing Cmpany, Inc., New Yrk, 1967, pp. 32-33. 20. Perepukhv, V. A. "Aerdynamic Characteristics f a Sphere and Blunt-Nsed Cne in a Highly Rarefied Gas Flw." (Translated frm Russian) FTD-MT-24-135-68 (AD681686), 1967. 12
APPENDIXES I. ILLUSTRATIONS II. TABLE 13
1000 M6 Nzzle (Nitrgen) M3 Nzzle (Nitrgen) ^ 100 Js 0.^^ 2 E c >» Of :# M6 Nzzle (Argn) 1.0 10 I 6 8 Mach Number l 10 12 14 Fig. 1 Flw Cnditins in the M3 and M6 Nzzles 15
O ARC (BV)Chamber Plenum Chamber- Radiatin Shield - Resistive Heating El-ements- Preheater (Resistive Heating Elements)- -Pitch Mechanism fr Balance Alignment -Turnbuckle1 0\ -Pitch Drive Mtr T Baratrn " /// Nitrgen and Ttal Pressure/// Argn Inbleed Measurements^/ Fig. 2 Schematic f Lw Density Tunnel, ARC (8V)
0.020 0.018 & Uncrrected fr Viscus Effects Crrected fr Viscus Effects 0.016?5 0.014 0.012 0.010 0.008 0.006 7.2 r- Mm 6.4-2 3 Ttal Pressure, mm Hg Fig. 3 Effect f Viscus Crrectin n Mach Number and Dynamic Pressure at the M6 Nzzle Exit 17
2.0 M S 25,5, T 1400 K (Ref. 16) 0.1 < 11 < 0.67 300 K (Ref. 17) M = 0 (Incmpressible, Ref. 15) > m O -J Ü M 1.8 P 1.6 900 K 300 K 870 K 300 K 500 K 900 K 300 K 00 1.2 1.0-0.8 0.1 J L J ' I I I I I I 0.2 0.4 0.6 0.8 1 4 6 R e (Ufc/Hjj) J I I I I I I I 8 10 20 40 60 80 100 Fig. 4 Pitt Prbe Viscus Crrectin Data
' AEDC-TR-70-32 fra 1.1 i- 1.0-0.9-0.8 0 1 l cx> 1 1 1 0 Merged Flw -*-l 1 0.7 " 1 1 1 1 1 1 1 _ J 1.1 1.0.r 900 Frm Fully Viscus Channel Flw Slutin Using Numerical Methd f Rae (Ref. 18). H Merged Flw _J 0.8 0.7 2 Ttal Pressure, min Hg Fig. 5 Increase in C D /C Dfm and Decrease in H«/H 0 fr the M6 Nzzle (T 0 = 800 K) 19
Ttal Drag Sphere Balance Sting Ttal Drag Tare Frce 10 20 Drag, dynes 30 Fig. 6 Tare Frce and Ttal Drag Measurements Using 0.25-in.-diam Sphere n Drag Balance in M6 Nzzle Flw 20
AEDC TR-70-32 Fig. 7 Falling Spheres Using the Free-Flight Technique 21
1730 i- 1630 H 50 u 1530 1430 Measured Sphere Lcatin Secnd-Order Curve Fit <* 1330 E t tsj 1230 = 1130 4 1030 930 830 1.00 1.80 2.60 3.40 4.20 5.00 5.80 6.60 7.40 Time, Number f Strbes 8.20 9.00 9.80 10.60 11.40 12.20 Fig. 8 Curve Fit f the Axial Displacement f a Sphere versus Time
Cllisin Surface Effective Cllisin Surface (that Part f Cllisin Surface Bunded by the Effective Cylinder) Fig. 9 Assumed Mdel f the Rarefied Flw abut e Sphere 23
2.5 + VT? S [1 + erf < S J]} > m N - n U 00 00 00 Cd M 2.0 2.0 (S = H ) 00 09 t 1.5 1.0 Q.5 1.0 1.5 2.0 2.4 Fig. 10 Incident Flux f Mlecules n a Sphere in Free-Mlecular Flw
6 5 'D fffl 3 in T A 0 (Same as fr Specular Reflectin) 1 1 I 123456789 Speed Rati, S Fig. 11 Free-Mlecular Sphere Drag Cefficients fr Diffuse Reflectin and Cmplete Accmmdatin 10 > m O H 3) U
> m D 9 ü t 'fm Fig. 12 Sphere Drag Cefficients in the Transitinal Flw Regime Accrding t Eqs. (19) end (20)
1.6 I- Kn Kn Fig. 13 Effect f 0 n Kn w /Kn. 27
1.0 0 - T/2 in Equatin fr \/\ > m a 3) I? u.g 0.8 r* [&# &*) :] 1/2 t 00 'fm 0.7 6-0 in Equatin fr \,/\ 0.6 6 - Variable in Equatin fr \/\ 0.5 0.4 10 >-2 10" 10" Kn 10 J 10' Fig. 14 Sphere Drag Cefficients fr Cnstant and Variable X* with S* - 1
AEDC TR-70-32 2.5 O CK 0 93 2.4 2.3 2.2 7.0 < M < 7.7 00 3.4 < T A < 5.1 W 2.8 < S w < 3.2 2.1 1.0 fm 0.9 ~ 0.8 J I I I I L-L-J 10 Kn Fig. 15 Effect f Nrmalizing C D by C Dfr 29
1.11 w- O D D O > m a H a SI u 'fm i. 0 I- - 0.99 I- " V 4 1 m O O <* O O O Q <9 O 8 < ^ Eqs. (19) and ( 20) with S w - 2.7 I Sym M 09 V T - fw Test Gas Methd 7.0 t 7.7 3.4 t 5.1 2.8 t 3.2 Nitrgen Balance (Sphere Surface Painted Black) 7 " 7.2 t 7.7 4.0 t 5.4 2.8 t 3.0 Nitrgen Balance (Sphere Surface Painted Gld) 4 2.9 t 3.3 0.8 t 1.2 2.5 t 2.7 Nitrgen Balance 6.7 t 7.8 4.4 t 4.8 2.7 t 3.0 Nitrgen Free Flight 8.4 t 11.2 9.0 t 10.0 2.6 Argn _L 1 1 1 _l_ I II.. Free Flight _L l_ 1 l_l 1 1 1 10 100 Kn Fig. 16 Balance and Free-Flight Sphere Drag Data with Analytical Results
1.1 r- 1.0-0.9-0.8 - W fm 0.7-0.6-0.5-0.4 8.3 13.0 10.7 3.7 4.1 9.96 0 t 21.5 t 21.0 10.5 t 10.8 7.0 t 7.7 9 t 3.3 6.7 t 7.8 8.4 t 11.2 15 57 39 3.6 4.54 1.92 1.06 t 1.69 4.84 t 11.09 1.98 t t t t 3.4 0.8 4.4 9.0 t 16. 4.00 5.4 1.2 4.8 1.78 1.58 1.57 1.65 1.61 6.0 8.8 t 16.5 5.3 4.5 2.8 t 2.5 t 2.7 t 2.6 t 6.7 t 6.3 3.2 2.7 3.0 Ref. 8 '8 8 7 6 6 6 6 5 Present Present Present Present ' I I I I IN I I I I I I I I J I I I I I III I I I I I III 10 10" 10 100 Fig. 17 Trend f Analytical Results and Experimental Data in the Transitinal Flw Regime Kn > m O 3) i «J O Ü N
l.lr- > m O O? u M 'fm 100 Fig. 18 Analytical Result with S* = 1 and Sme Experimental Data with Little Scatter
1.0 r- 0.9 - Eqs. (19) and (20) with S.- 1.6 0.8 - 'fm 0.7 - Baker and Charwat (Ref. 2) 0.6-0.5-0.4 10 Willis (Ref. 4) 10-1 10* Kn Syn» Test Gas Ref. & 3.7 3.6 1.65 Carbn Dixide 7 O 8.3 15 1.78 Air 8 A 13.0 57 1.58 Argn 8 O 10.7 39 1.57 Argn 8 V 4.1 4.54 1.61 Nitrgen 6 I I I I I I I I I 1 I I I I I HI 10' 10' Fig. 19 Cmparisn f Eq. (19) with Other Theries and Available Experimental Data fr S* near 1.6 > m D O H 3)»J u
TABLE I SPHERE DRAG COEFFICIENTS AND FLOW CONDITIONS a. Balance Data M. Re, Kn. Tw/T. C D CD /CD fm 6.98 2.12 4.92 3.40 3.16 2.460 1.012 7.02 2.30 4.55 3.49 3.14 2.437 1.001 7.06 2.47 4.26 3.58 3.12 2.462 1.011 7.10 2.68 3.95 3.68 3.09 2.445 1.003 7.13 2.82 3.77 3.77 3.07 2.450 1.003 7.19 3.10 3.45 3.89 3.05 2.460 1.006 7.25 3.37 3.21 4.01 3.03 2.465 1.008 7.31 3.59 3.03 4.15 3.00 2.455 1.003 7.41 4.01 2.75 4.34 2.97 2.425 0.990 7.49 4.75 2.35 4.51 2.95 2.380 0.972 7.57 5.72 1.97 4.67 2.93 2.310 0.942 7.65 7.40 1.54-4.87 2.90 2.210 0.901 7.70 9.25 1.24 * 5.05 2.86 2.152 0.875 7.22 3.20 3.36 4.08 2.99 2.480 1.012 7.40 3.95 2.79 4.43 2.94 2.437 0.993 7.50 4.81 2.32 4.70 2.89 2.418 0.984 7.57 5.70 1.98 4.86 2.87 2.315 0.942 7.61 6.51 1.74 5.00 2.85 2.265 0.919 7.65 7.36 1.55 5.13 2.82 2.220 0.900 7.68 8.30 1.38 5.25 2.80 2.172 0.880 7.70 9.03 1.27 5.37 2.78 2.156 0.871 2.90 1.19 3.65 0.80 2.71 2.615 0.951 2.92 1.23 3.55 0.85 2.65 2.630 0.955 3.07 2.16 2.12 0.90 2.71 2.405 0.890 3.19 2.64 1.80 1.00 2.67 2.420 0.890 3.28 3.23 1.51 1.10 2.61 2.250 0.820 3.34 3.89 1.28 1.20 2.55 2.140 0.778 Test Gas Nitrgen 34
TABLE 1 (Cncluded) b. Free-Flight Data M- Re. Tw/T. S w C D CD/C Dfm Test Gas 7.65 4.63 2.46 4.69 2.95 2.010 0.828 Nitrgen 7.69 2.28 5.02 4.73 2.96 2.340 0.966 7.79 1.11 10.5 4.85 2.96 2.352 0.976 7.59 2.10 5.37 4.62 2.95 2.379 0.979 7.69 1.04 11.00 4.73 2.96 2.447 1.011 7.50 3.82 2.92 4.52. - 2.95 2.162 0.878 7.53 1.88 5.99 4.55 2.95 2.300 0.934 7.45 3.40 3.27 4.47 2.95 2.153 0.875 ;, 7.47 1.67 6.66 4.49 2.95 2.237 0.908 ' 7.51 0.83 13.40 4.53 2.95 2.440 1.000 * 7.51 0.85 13.20 4.53 2.95 2.750 1.118 7.37 3.00 3.66 4.38 2.94 2.412 0.986 7.45 1.45 7.64 4.47 2.95 >. 2.507 0.913 7.29 2.55 4.26 4.29 2.94 2.228 0.911 7.23 1.30 8.31 4.23 2.94 2.341 0.956 7.15 2.14 4.97 4.14 2.94 2.203 0.899 7.11 1.08 9.82 4.10 2.94 2.480 1.010 7.02 0.83 12.65 4.01 2.93 2.390 0.969 6.71 0.63 15.80 3.69 2.92 2.337 0.946 6.69 0.65 15.43 3.67 2.92 2.310 0.935 6.78 0.32 31.50 3.76 2.92 2.532 1.029 7.32 1.14 9.54 4.47 2.89 2.542 1.040 7.48 1.45 7.68 4.65 2.90 2.385 0.971 7.58 1.63 6.95 4.77 2.90 2.500 1.020 7.65 1.84 6.22,4.85 2.90 2.397 0.990 7.72 2.00 5.74 4.93 2.91 2.440 1.009 7.76 2.22 5.22 4.97 2.91 2.623 1.070 8.41 1.67 8.23 8.52 2.63 2.664 1.071 Argn 8.42 0.84 16.3 8.52 2.63 2.700 1.083 9.42 2.20 6.96 11.17 2.61 2.257 0.905 9.52 2.24 7.26 11.29 2.59 2.482 0.998 10.04 2.73 6.00 12.60 2.59 2.772 1.117 10.06 2.74 5.98 12.68 2.58 2.427 0.977 11.12 4.67 3.88 15.38 2.59 2.395 0.967 11.14 4.87 3.73 15.50 2.59 2.380 0.962 35
UNCLASSIFIED SecurityClassMicatin DOCUMENT CONTROL DATA -R&D (Security elassllicatln l title, bdy f abstract and Indexing anntatin must be an lend when the verall reprt la elaeeltled) I. ORIGINATING ACTIVITY (Crprate authr) Arnld Engineering Develpment Center, ARO, Inc., Operating Cntractr, Arnld Air Frce Statin, Tennessee 37389 9. REPORT TITLE 2«. REPORT SECURITY CLASSIFICATION UNCLASSIFIED 2b. GROUP SPHERE DRAG IN THE FREE-MOLECULAR AND TRANSITIONAL FLOW REGIMES 4. DESCRIPTIVE NOTES (Typ» l reprt and Inctualve dates) Final Reprt - April 14 t September 26, 1969 S. AUTHOR(S) (First name, middle Initial, laat name) David L. Whitfield and William B. Stephensn, ARO, Inc. N/A 6. REPORT DATE April 1970 8a. CONTRACT OR GRANT NO. b. PROJECT NO. 6690 F40600-69-C-0001 7a. TOTAL NO. OF PACES 7b. NO. OF REFS 43 20 9a. ORIGINATOR'S REPORT NUMBERISJ AEDC-TR-70-32 e. Prgram Element 62101F d. Task 669002 10. DISTRIBUTION STATEMENT Sb. OTHER REPORT NO(S) (Any ther numbers that may be aeelgned thle reprt) N/A This dcument has been apprved fr public release and sale; its distributin is unlimited. II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Frce Cambridge Research Available in DDC Labratry (CRMP), L.G. Hanscm Field, Bedfrd, Mass. 01730 13. ABSTRACT Results are presented f sphere drag measurements made in the freemlecular and transitinal flw regimes. The drag data were btained using a drag balance and the free-flight technique. Cnditins fr which measurements were made are within the ranges 1 < Kn«, < 32, 2.9 < Ma, < 11.2, 0.8 < T w /T < 16.0, and 2.5 < S w < 3.2. An analysis f the drag n a sphere in rarefied flw is als presented, and the results are cmpared with experimental data and ther theries. The analytical results well predict the trend f the experimental data in the transitin regime and remain valid at Knudsen numbers fr which previus theries are nt applicable. DD, F N O R V\.1473 UNCLASSIFIED Security Classificatin
UNCLASSIFIED Security CUasificatin 14. KKV WORDS LINK A ROH WT LINK LINK C ROL8 WT ROLK WT aerdynamic drag weight indicatrs free mlecule flw transitin flw 2-Kxrudsen flw free flight trajectries rarefied gas dynamics /, MIUIIH UNCLASSIFIED Security Classificatin