UNCLASSIFIED AD RepAoduced. by, the ARMED SERVICES TECHNICAL INFORMATION AGENCY ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA UNCLASSIFIED

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1 UNCLASSIFIED AD RepAduced by, the ARMED SERVICES TECHNICAL INFORMATION AGENCY ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA UNCLASSIFIED

2 NOTICE: When gvernment r ther drawings, specificatins r ther data are used fr any purpse ther than in cnnectin with a definitely related gvernment prcurement peratin, the U. S. Gvernment thereby incurs n respnslhillty, nr any bligatin whatsever; and the fact that the Gvernment may have frmulated, furnished, r in any way supplied the said drawings, specificatins, r ther data is nt t be regarded by implicatin r therwise as in any manner licensing the hlder r any ther persn r crpratin, r cnveying any rights r pennissln t manufacture, use r sell any patented inventin that may in any way be related theret.

3 Best Available Cpy

4 . i mmn OF MIUIIMIVSICS UNIVERSITY OF TORONTO C t % Lfijl-]- ^ O C DRAG MEASUREMENTS ON CIRCULAR CYLINDERS AND SPHERES IN THE TRANSITION REGIME AT A MACH NUMBER OF 2 by %~* $*/ A. K. Sreekanth :,- ' ) 1 fffsst* : r ') : - - L \ \ ' : " '. ^H^ ' 1! ^.r-v-r-lxilj I APRIL, 1961 UTIA REPORT NO. 74 ARL - 53

5 ^ DRAG MEASUREMENTS ON CIRCULAR CYLINDERS AND SPHERES IN THE TRANSITION REGIME AT A MACH NUMBER OF 2 by A. K. Sreekanth APRIL, 1961 UTIA REPORT NO. 74 ARL - 53 A

6 ACKNOWLEDGEMENTS The authr wishes t express his sincere gratitude t Dr. G. N. Pattersn, fr having given him the pprtunity t d graduate wrk at the Institute f Aerphysics under his guidance and fr his steady interest and encuragement during the curse f this wrk. The authr is indebted t Dr. J. H. deleeuw fr his friendly interest in the reprted wrk and his valuable advice. Thanks als g t Mr. B. G. Dawsn f the Institute's Shp Staff fr making the prbes and mdels used in this investigatin. The c-peratin f the lw density grup at the Institute f Engineering Research, University f Califrnia (Berkeley) wh furnished the design fr the drag balance used is gratefully acknwledged. This investigatin was made pssible thrugh the financial assistance rendered by the Wright Air Develpment Divisin, United States Air Frce (Cntract AF 33(616)-6990) and the Defence Research Bard f Canada.

7 SUMMARY Measurements f the drag f circular cylinders placed transverse tb the flw and spheres at a Mach number f 2 in air were btained in the UTIA lw density wind tunnel. The mean free path f the air in the test flw was " and the mdel sizes were such that Knudsen numbers in the range 0. 2 t 6 fr the cylinders and 0. 1 t 0. 8 fr the spheres were cvered. The drag cefficient f circular cylinders calculated frm the measured frces was fund t increase with increasing Knudsen number and reach a value f at Kn = 3. There was n apparent increase as the inudsen number was further increased. In cntrast, the theretical value fr free mlecule flw cnditins is 3.7 if cmpletely diffuse reflectin and cmplete temperature accmmdatin are assumed and if cmplete sipecular reflectin ccurs. This shws that at a Knudsen number f apprximately 5 the drag cefficient is still significantly lwer than the free mlecule flw value. On the ther hand the experimental results n sphere drag in the same flw indicate that the thery and experiment are essentially in agreement. It is suggested that the discrepancy between the theretical and measured values fr the case f circular cylinders is sciated with the fact that in this case nt all dimensins are smaller than the mean free path. This cntentin was supprted by additinal experiments cnducted in subsnic flw; pressure readings taken by m eans f an rifice n the side f a cylinder nrmal t the flw prved t e dependent n the cylinder's length. Frm these findings it was ncluded that the validity f the cnventinal assumptin that the free m nlecule flw cnditins shuld be applicable at a Knudsen number f apprxmately 5 is in dubt fr the case f cylinders transverse t the flw.

8 NOTATION (i) TABLE OF NTENTS Page I. INTRODUCTION II. EXPERIMENTAL APPARATUS 1 Lw Density Wind Tunnel 2 Frce Balance 3 Mdels III. DESCRIPTION OF EXPERIMENTS IV. V. VI. VII VIII Flw Calibratin Alignment f Balance and Mdel Static Calibratin f the Balance Determinatin f the Methd f Mdel Supprt and the Best Pstin f the Shield with Respect t the Mdel Frce Measurements REDUCTION OF THE DATA DISCUSSION OF THE RESULTS PRESSURE MEASUREMENTS TO DETERMINE THE EFFECT OF THE LENGTH OF THE CIRCULAR CYLINDER TRANS- VERSE TO THE FLOW 16 EXPERIMENTAL ERRORS NCLUSIONS REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E TABLES I t VI 5 7 7a

9 1 (ii) NOTATION A prjected area f cylinder r sphere I C2 ( cnstants in the Sutherland's frmula (see Appendix D). n ^ «. drag frce Cn drag cefficient = r s 6 7: If V 2 A D drag frce K a factr which accunts fr interference and end effects Kn Knudsen number 1 length f cylinder mdel M mach number P static pressure P stagnatin pressure P'OK^^O measured impact pressure P' 0 ideal impact pressure in the absence f viscus effects R Gas cnstant P e Reynlds number T 0 stagnatin temperature Tj free-stream temperature V free-stream velcity w frce per unit length n the cylinder mdel. W frce n the supprting sting x l' yi' y2 lever arm lengths Y rati f specific heats A mean free path

10 (iü) j density IX viscsity

11 (1) I. INTRODUCTION Interest in the field f high-speed, high-altitude aerdynamics has increased in recent years with the advent f missiles and satellites. In rder t fully understand the flw phenmena ccuring at the high altitudes ne has t resrt t either the kinetic thery f gases r cntinuum fluid mechanics, depending upn the degree f rarefactin f the gas medium in which the vehicle is mving. The basic parameter that indicates the degree f rarefactin f a gas is the Knudsen number, Kn, defined as, Kn= ± where A is the mlecular mean free path (i. e., the average distance traversed by mlecules between cllisins) and L is sme significant dimensin in the flw field. The Knudsen number can als be expressed in terms f the Mach number, M and the Reynlds number, Re (the tw basic parameters used in cntinuum mechanics) by the relatin Kn = 1. 26J If { / = rati f specific heats) ' Re where bth Kn and Re are based n the same characteristic length. Gasdynamics can be divided rughly in t the fllwing regimes accrding t the degree f rarefactin measured by the Knudsen number based n the free stream value f \ and a characteristic bdy length J Cntinuum flw Kn < Slip flw. OK Kn < 0. 1 Transitin flw 0. 1 < Kn < 5 Free mlecule flw Kn 7 5 Hwever, this rugh divisin is n lnger cnsidered adequate at hypersnic speeds r in cases Where there exists a large temperature difference between the adiabatic bdy temperature and actual bdy temperature. It has been fund that at such cnditins the lcal value f the mean free path, rather than the free-stream value, must be used fr determining the Knudsen number and classifying the flw. At speeds with a Mach number smaller than 5, n the ther hand, the classificatin given prves' quite useful. The analyses f transitin and free mlecule flws are based n the kinetic thery f gases, whereas the cntinuum and slip flws are characterized by the Navier-Stkes equatins f mtin with the apprpriate

12 (2) LrhtT.r 111 "" 0 ' 18 - de, ' VlVed - s «"-"y n the basi. f cntinuu m fluid particles and the bdy. In such a raqp th«fi,, * " uexween gas ^+ ^, ^ öut-n a case tne iluxes f incident and rpfl^n+ri Z thexr ^T ^ m0le - 1 rf-e interaclns Zt be cnsidered" <Ref. 1 and 2 ^ "-"earch grup at the University f Califrnia high speed free^ll^lvrjltdnrrstald 0^6 "^J^^^ '" «(Ref. 3, wh.neasured the d, fctr^x^^rrle^the gas stream In free mlecule flw Apart frm tm* L,,u near iree mlecule flws have been reprted t this date. investigated.he^s"lrrd;:r 0 f C a h s a ;h W e a r t e ( fn ef - ^ 'T ^^ ^nzzr ^ neglact thf -rms m e t rn a fth r ;m e!cr e s C t flow - frmed in a Mach 2 flw f at ' experiments we -e per- il. EXPERIMENTAL APPARATUS 1- Lw Density Wind Tunnel The experiments were perfrmpd in the TTTTA I~ I a vacuum pump drtve Z^ T ^^^J^tTT*

13 (3) " static pressure range frm 1 t 70 micrns Hg. is shwn in Plate 1. A side view f the tunnel An axially symmetric pen jet nzzle designed t give a Mach number f 2 at a static, pressure f abut 20 micrns Hg was used in the present experiments. Details f the perfrmance and the characteristics f the Mach 2 nzzle can be fund in Ref. 5. The nzzle has an exit diameter f 5. 82" but the unifrm cre f Mach 2 flw was nly 1" in diameter at the center f the nzzle, the remaining prtin f the jet being filled with bundary layer. Atmspheric air passes thrugh a dryer in t a needle valve which is used t regulate the mass flw rate by thrttling the air dwn t lw pressure befre it enters the stagnatin chamber f the tunnel. In this chamber any desired value f stagnatin temperature up t 150 F culd be set by means f a heated liner. A series f six bster pumps n the dwnstream side f the nzzle maintain a cntinuus flw f air thrugh the test sectin. A butterfly -type flap n ne f the bster pumps allws fine cntrl f the test chamber static pressureby changing the pumping speed slightly. The desired flw cnditins are set by prper manipulatin f bth the needle valve and the pumping speed. 2. Frce Balance A single cmpnent micrbalance similar t ne designed by Latz (Ref. 6) with slight mdificatins was used. Schematic diagrams f the balance are given in Figs. 1 and 2, and a phtgraph is shwn in Plate 2. It is a remte cntrl, beam-type, null balance with crssed flexural pivts. The flexure pivts cnsist f tw pairs f crssed wires rigidly attached t tw jaws. The upper jaw is fastened t a base plate and the lwer is free t rtate abut a flexural pint. A j" diameter brass shaft is attached t the lwer jaw and is passed thrugh hles cut in the base plate and the upper jaw, A small platfrm f abut 1" x 3/4" size with tw pins fr psitining the base that supprts the mdel is attached t the tp f the brass rd. A beam is attached t the lwer jaw. The cre f a LVDT (linear variable differential transfrmer) is attached t ne end f the beam and serves t detect deviatins frm the null psitin; a helical quartz spring is attached t the ther end. The ther attachment pint f this spring can be mved by means f a mtr-driven lead screw. The extensin f the spring is indicated by a Veeder-Rt cunter. Referring t the Fig. 2 applicatin f a frce n the mdel in the flw directin will prduce a cunter-clckwise mment n the flexure pivts. This mment will cause the lwer jaw t rtate. The resulting displacement f the cre f the LVDT frm its initial psitin will change the circuit current which is indicated by a galvanmeter, see the circuit diagram in Fig. 3. Null balancing is achieved by extending the quartz spring until the galvanmeter again indicates the null psitin.

14 (4) The spring extensin is a measure f the spring frce, and the actual frce n the mdel is determined by the spring cnstant and lever arm lengths. Fr the purpse f determining the cnstant f the spring, a weight pan is attached t the beam.- Fr damping the scillatins f the balance suspensin, an aluminum vane is attached t the lwer jaw and is made t mve in a ' magnetic field prduced by tw hrseshe shaped Alinc permanent magnets. The damping factr is varied by simply changing the distance f the magnets t the vane. The balance is designed t measure a frce accurate t 0. 1 mg. and has an angular null sensitivity f degrees rtatin. 3. Mdels a) Cylinder Mdels A sketch f the mdels used t determine the drag f the cylinder is shwn in Fig. 4. Stainless-steel hypdermic tubing was used fr making these mdels. The diameters f the cylinders tested varied between " and ". Since the unifrm cre f the Mach 2 flw in the jet was nly 1" in diameter, the maximum length f the cylinders were restricted t abut 0. 8". The cre f the cylinder was filled with slf slder. b) Sphere Mdels Brass and steel bearing balls were used as mdels in the sphere drag experiments. Tw types f mdels were tested. They differ m the manner in which they are supprted, (see Fig. 5). A small hle was drilled in the sphere and it was attached t the mdel supprt by a push fit. The sphere supprt piece cnsisted f a fine, tapered sewing needle. The diameters f the spheres tested varied between 1/16" and 7/16. The sizes f the spheres and supprt rds are given in Table I. Fr the case f the 1/16" diameter mdels, the spheres used were steel bearing balls and these were welded n t the supprting rd. The spheres were separated frm their supprting rds fr the purpse f determining the tare drag by just breaking the weld. A phtgraph f typical cylinder and sphere mdels is shwn in Plate. 3. c) Shields As mentined previusly the diameter f the unifrm Mach 2 cre was nly abut an inch, the remaining prtin f the nzzle being

15 (5) filled with bundary layer. The mdel which was abut 0. 8" in length has t be supprted by a rd frm the balance. Since the exit diameter f the nzzle is 5. 82'] the mdel requires a supprting rd with a minimum length f 2. 91" r 2. 51" depending whether it is expsed t the flw hrizntally and supprted frm behind r vertically and supprted at ne end. Under these cnditins the frce n the supprting sting will be many times that n the mdel itself. One methd t minimize the supprt frce, by making it extremely thin, culd nt be used as there were sme vibratins present in the tunnel intrduced by the pumping units. In rder t knw the frce n the mdel, the frce n the supprting sting has t be subtracted frm the ttal frce. Even a slight experimental scatter in the ttal measured frces n the mdel and its supprt may then appear as a large errr in the frce n the mdel itself. T imprve this situatin,use f a shield fr the supprting sting as anther methd f minimizing the supprt frce has been made. Care then has t be taken t ensure that the shield has n interference effect n the mdel. Extensive experiments were cnducted t determine the best psitin f the shield with respect t the mdel t minimize interference effects. The effect f the shield n the flw is discussed in Sectin 111, 4, and als in Appendix A. The dimensins f the shield used are shwn in Fig. 6. d) Pressure Prbe The dimensins f the pressure prbe that was used in subsnic flw t determine the effect f the length f the prbe n the pressure readings is given in Fig. 7. It was munted transverse t the flw n a specially designed rtating mechanism by means f which the angular psitin f the rifice with respect t the mass flw directin culd be varied. The pressures were recrded by a thermistr gauge, which is described in detail in Ref DESCRIPTION OF EXPERIMENTS 1. Flw Calibratin All the cylinder and sphere drag measurements were dne in a Mach 2 air flw. The nzzle that prvided this flw is designed t perate at a stagnatin pressure f micrns Hg and a test chamber static pressure f 20 micrns Hg. An impact prbe f " dia. with 10 external chamfer was used t calibrate the flw. The calibratins were perfrmed with the drag mdels remved frm the balance but leaving the shield in the flw. First the stagnatin pressure was set at the designed value by means f the air inlet valve. The test chamber pressure was set at the design value f 20 micrns Hg by manipulating the bster pump valves. The impact prbe readings were taken n the nzzle center line at the nzzle exit and 3/4" dwnstream. The pressure in the test

16 (6) chamber was varied slightly by changing the pumping speed f ne f the bster pumps by means f a flap until the difference in impact prbe readings at these tw psitins was less than percent. The absence f fluctuatins in these impact pressure readings with time and the cnstancy f pressure reading at all pint alng the nzzle centerline fr a distance f 3/4" frm the nzzle exit indicated that the jet was smth and well balanced and that there were n axial gradients in the flw field at the center f the jet. A mercury McLed gauge was used t measure the impact and stagnatin pressures. T get cntinuus readings, a thermistr gauge was cnnected t the impact prbe while balancing the jet and checking fr fluctuatins in the flw. Pressure prbe traverses were made alng the nzzle centerline and acrss the jet at distances f 4" and 2" frm nzzle exit. At the very lw Reynlds numbers f the flw at which these experiments were dne the measured impact pressures (P'Q meas ) depart radically frm the ideal values wing t viscus effects. The viscus crrectin t the measured pressure can be expressed in the fllwing frm, Supersnic flw; pv- ideal P' meas. f (Mach N., Reynlds N. based n prbe diameter, and prbe shape). Subsnic flw; P' meas "P' ideal % 5V f {Mach N., Reynlds N., and prbe shape) where $ and V are free stream density and velcity respectively. The relatin between the impact pressure and stagnatin pressure (p 0 ) in a supersnic isentrpic flw is given by P' ideal P (Y+ DM* 2 + ( ^ - 1) M 2 T-l T+l ^YM^O-O r-i 0 was calculated frm measured impact pressure by ideal using the viscus crrectin chart given in Ref. 7. The centerline Mach number was calculated by the rati f p' 0.,. t p assuming isentrpic flw, which is apprpriate because the presence f a unifrm cre is indicated by the measurements. As the true Mach number and the Reynlds number are bth unknwn, an iterative prcedure was applied, starting by assuming the viscus crrectin t be zer and using successively mre accurate values f M and Re untircnvergence was btained. Frm

17 P (7) the static P ressure the calculated Mach number and the value f P'ideal (p) at the pint n the centerline was determined by the Rayleigh supersnic pitt frmula Y r 1 y-i Y-l P'ideal y +1 2 ivr Y-H ZTM^'p-O This calculated static pressure was used t prvide a check n the calculated values f M by cmparing it with the wall tap pressure near the nzzle exit. These tw pressures agreed within a fractin f a micrn f Hg. Frm this it was inferred that the static pressure acrss the jet was essentially cnstant and the evaluatin f the Mach number fr pints away frm the centerline, fr which the isentrpic relatin t the stagnatin cnditins is n lnger valid wing t the viscus effects, is based n the rati f impact pressure and the cnstant static pressure. The viscus effect n the prbe readings was again eliminated by an iterative prcedure. The Mach number prfile acrss the jet at a statin i" frm the nzzle exit plane is shwn in Fig. 8. In all the drag frce tests the center f the mdel was placed 5" dwnstream frm the nzzle exit. As there were n axial gradients in the flw fr a distance f 3/4" frm nzzle exit, the whle mdel was subjected t the same flw cnditins. A cmplete calibratin f the flw field was dne peridically but the centerline Mach number at \" frm nzzle exit was checked befre the start and at the end f each experiment. During the drag frce measurements the maintainence f the calibrated flw was checked by measuring the stagnatin, wall tap static, and test chamber pressures. 2. Alignment f Balance and Mdel It was necessary t determine the plane f mtin allwed by the balance suspensin. T d this, a lng pinter was attached t the suspensin which was then allwed t scillate. The tip f the pinter thus described a line in the plane f scillatin. The rientatin f this line was then marked n the balance base plate. This line was further checked by ptical means. When the ptical axis f a telescpe was exactly aligned with the plane f mtin there was n apparent lateral mvement f the image f the pinter with respect t the telescpe crss-hair. Having lcated the plane f mtin it was then pssible t check the crrespndence with the line previusly drawn n the base plate. It was cncluded frm bservatins that the deviatin f the line marked n the balance frm being parallel t the plane f scillatin f the mdel suspensin was within The balance was placed n a circular turntable inside the tunnel test sectin. The mdel t be tested was put n the balance and was rigidly attached t it by means f a special spring clip. The psitin f the mdel and the balance with respect t the nzzle was set apprximately t the desired psitin by eye. A circular steel plate 62" diameter and

18 (7a) 3/16" thick having a 4" lng, 5" square steel bar attached t it at its center and perpendicular t it was held tightly against the nzzle exit. The steel bar was fund t be perpendicular t its base within 1/10 f a degree. Fr the final adjustment, the balance was then rtated by means f the turntable until the line marked n the balance was set parallel t the steel bar. It was estimated that the frce measuring directin f the balance was parallel within I degree t the nzzle centerline (i. e., the directin f mass flw) thereby ensuring that the balance was measuring the drag frce. This drag frce psitin was duble-checked by rtating the turntable and measuring the frce n a mdel set vertical t the flw directin. The m easured frce wa.s quite symmetrical abut this psitin, decreasing in either directin f rtatin. These measurements indicated that the balance was aligned t the flw directin within 0. 5 degrees. A mirrr having tw thin crssed lines marked n it was held tight against the nzzle exit s that ne f the lines was hrizntal and passed thrugh the center f the nzzle. The cylinder mdel was aligned with the help f this mirrr s that it was hrizntal and nrmal t the flw directin. Fr the case f spheres, the center f the sphere was set at the center f the jet. The lever arm lengths f the suspensin system were measured by means f an ptical cmparatr and a cathetmeter befre the balance was put in the tunnel. After having aligned the balance and the mdel, the sensitivity f the balance was adjusted. This was dne by mving the vertical cunter weight (see Fig. 2) until placing a weight f 4^2 mg. in the weight pan prduced a galvanmeter deflectin f 6 divisins frm its null psitin. This crrespnded t a frce f less than 0. 1 mg n the mdel fr ne galvanmeter divisin deflectin frm its null psitin, i. e. the balance was able t sense a frce difference f 0. 1 mg n the mdel. 3. Static Calibratin f the Balance Having adjusted the sensitivity f the balance, the galvanmeter needle was set at zer by means f the helipt cnnected in the LVDT circuit. The reading f the Veeder-rt indicatr which was cnnected t the quartz spring actuating mechanism was recrded. A knwn weight was placed in the calibrating pan. The quartz spring was elngated until the galvanmeter indicated the zer psitin. The extensin f the spring as indicated by the Veeder-rt cunter was nted. The weight was then remved and the tensin n the quartz spring was released until the galvanmeter came back t the riginal psitin. The reading f the cunter was checked against the initially recrded reading. In mst cases the initial and final readings were the same but in few cases they differed slightly. This might have been due t the hysteresis effect in flexures and pssible backlash in the gear mechanism. The calibratin was repeated fr these

19 (8) cases until the initial and final cunter readings were the same. The prcedure was repeated fr different weights in the pan. A typical calibratin curve is shwn in Fig. 9. experiment. The static calibratin was perfrmed befre and after each The variatins in these calibratins were negligible. 4. Determinatin f the Methd f Mdel Supprt and the Best Pstin f the Shield with Respect t the Mdel a) Cylinder Mdels There are varius ways f supprting a cylinder mdel t measure its drag in a wind tunnel. The fllwing set f figures indicate the mst cmmnly used methds. (a) Shield Cylinder Mdel -Flw Shield Nzzle T Balance Mdel and shields can be either hrizntal r vertical. If the shields are nq,t disturbing the flw arund the mdel, then the measured frce cefficient is equal t that f a tw dimensinal mdel as the end effects are eliminated by the presence f shields.

20 (9) (b) Mdel Hrizntal -? / - Shield- Nzzle T Balance GROSS FORCE ARRANGEMENT SSSSSS/ rfl Ö NOZZLE T Balance TARE FORCE ARRANGEMENT The measured net frce n the mdel in the abve arrangement is influenced by end effects and therefre is a functin f the mdel's aspect-rati (c) (i) Dummy Sting Shield /U Mdel Vertical Nzzle T Balance GROSSFORCEARRANGEMENT

21 (ü) (10) CU "_"' Nzzle TARE FORCE ARRANGEMENT In this arrangement the frce n the mdel is again a functin f its aspect-rati. The abve arrangements f the mdels shw sme f the methds t measure the drag n cylinders in cnventinal wind tunnels. In the curse f sme preliminary wrk dne in the UTIA lw density wind tunnel it was nticed that the shield used t cver the supprting rd was disturbing the flw field slightly. The shield extended frm the center f the jet t the uter edge and the disturbance caused by it was prpagated upstream thrugh the subsnic prtin f the bundary layer t affect the flw in the supersnic cre f the jet. It was bserved frm the impact prbe survey that the shield was causing a disturbance in the supersnic regin f the flw fr a distance f abut M upstream frm its leading edge. Cnsequently, it was decided t place the mdel at a distance f at least 1" upstream frm the leading edge f the shield s that the mdel was free frm any interference frm the shield. Sme experiments were dne in which the cylinder mdels were placed just dwnstream f the shield as shwn in Fig. (a) abve, where the shields cvered parts f the mdel itself, s that the cylinders were actually in the disturbed flw. A cmplete descriptin f these tests and their results are given in Appendix A. The mean free path in the flw in which the experiments were perfrmed was apprximately ". The diameter f the cylinder mdels varied frm " t ". As mentined previusly in Sectin II, 3, the length f the mdels was restricted t 0. 8" because the Mach 2 flw was unifrm nly ver a diameter f 1". Since the mdel had t be upstream f the shield it was placed hrizntal and transverse t the flw and supprted frm behind, with the shield cvering the vertical prtin f the sting. A schematic diagram f this type f supprt and the pstin f the mdel with respect t the shield is shwn in Fig. 10. A phtgraph f the actual arrangement is shwn in Plate 4.

22 (11) The mdel supprt and the shield used in the sphere drag experiments were similar and are referred t - in the text as tajlsting supprts. Plate 5 shws a phtgraph f this set up and Fig. 11 gives a schematic diagram f the arrangement used t measure the sphere drag. 5. Frce Measurements (a) Cylinder Mdels Having determined the methd f mdel supprt and the psitin f the shield with respect t the mdel, the prblem left was t islate the tare r the supprting rd frce frm grss frce. The cnventinal methd f separating the mdel frm the supprt rd and suspending it by a dummy rd and then measuring the tare frce will nt wrk in the present case because the sizes f the supprting rd and the mdels happen t be f the same rder f magnitude. The net frce n the cylinder btained by taking the difference between the measured grss frce and the tare frce will nt be the same as that n an islated cylinder (i. e.. a cylinder nt supprted by any stings) as n accunt is taken f the interference effects between the supprting rd and the mdel. Mrever, it was desired t crrect fr the aspect rati effects n the frce readings s that the final result culd be extraplated t a tw dimensinal cylinder It was therefre decided t measure the drag f the cylinder by a new technique in which the length f the cylinder was successively diminished and the grss frce was measured in each case. After placing the mdel in psitin and making a static calibratin f the balance, the tunnel was evacuated and kept under vacuum fr sme hurs fr ut-gassing purpses. The galvanmeter needle was brught t the zer psitin and the reading f the cunter was nted. As the recvery temperature f a mdel is always higher than the stagnatin temperature in free mlecule flw r near-free mlecule flw and as the drag frce is a functin f this temperature, a perid f 10 t 15 minutes was allwed after starting the flw fr the mdel t reach an equilibrium temperature, befre the drag frce was measured. After this, the flw was shut ff and the null psitin f the balance was checked. A minimum f three frce readings were recrded fr each experiment t make sure that the measured frces were crrect. The variatins between these measured frces were within t %. First the mdel having abut 0. 8" length was put in the flw and the frce n it and n the supprting rd was measured. The tunnel was then let t atmsphere and the balance suspensin was rigidly held by means f clamps prvided n the balance base plate t prtect the fragile suspensin. The psitin f the mdel was measured with a telescpe placed utside the tunnel. The mdel and its supprting base was very carefully remved frm the balance and it was placed n a specially designed jig in which the length f it was reduced t abut 0. 6" by shrtening its ends by 0. 1". The mdel was then put back n the balance, the whle

23 0 (12) peratin being carried ut withut bending r distrting the mdel supprt. The psitin f the mdel was checked t see whether it had gne back t its riginal psitin. (In cases in which thelelescpe indicated that the mdel did nt return t its riginal psitin, because f sme distrtins caused while shrtening its length, the experiment was abandned and a new ne was started). The frce n the shrtened mdel was then measured. The experiment was repeated fr varius different lengths f the mdel ranging frm 0. 8" t 0. 15". Frces n a minimum f fur different lengths f the mdel were measured fr all the mdels except the " diameter ne. Fr the latter, frces n nly 3 different lengths were measured as it was desired t have an appreciable difference between their readings. (b) Sphere Mdels Experiments perfrmed by Sherman and Kane, n the sphere drag in a lw density flw, Ref. 8, indicated that the mdel supprt had an influence n the measured drag. The interference caused by the crss-stream supprt was much greater than that due t the tailsting arrangement. The same tw types f supprt were used in the present experiments, see Fig. 5. First the grss frce was measured. During this run, a dummy sting (f a size equal t that f the mdel supprt) which later became the mdel supprt during the tare frce measurements, was attached t the traversing mechanism and was placed vertically n tp f the mdel center s that it was almst tuching the surface. The effect n the measured frce f the presence r absence f this sting near the mdel was bserved. It served t indicate the magnitude f the interference between the sting and the mdel and its supprt. The mdel was then separated frm the supprt and subsequently suspended by the dummy sting, and the tare drag was measured. Schematic diagrams f this arrangement fr bth types f mdel supprt are shwn in Figs. 11 and 12. Phtgraphs f the arrangement during a run are shwn in Plates 5 and 6. It shuld be nted that fr crss-stream supprted mdels n shield was used fr cvering part f the supprt. The difference between the grss and tare frces gave the frce n the sphere. In the absence f any interference effects due t the supprts this frce becmes the true net drag n the sphere. The prcedure fr measuring the frces was the same as that fr the cylinders, and a minimum number f three frce readings were taken in each experiment. IV. REDUCTION OF THE DATA a) Cylinder Mdels

24 (13) cw FLOW F = Spring frce Let w 1 W F K frce per unit length n the cylinder mdel length f the cylinder net frce n the mdel supprt acting at a center f pressure situated at a distance f Y frm the flexural pint spring frce a factr which accunts fr bth the interference f the supprting sting n the mdel and the cylinder end effects. Taking mments abut the flexural pint, fr equilibrium wlyj + Wy 2 + K = F^, In the experiments nly 1 was varied S JL- ^wlyj + Wy 2 + K - FX, "^ =0 df X, r w - if it is assumed that the factr W y and K are dl T " z independent f 1. K will be independent f 1 if the supprting sting influences nly a very shrt length f the cylinder at its center and if the end effects are cnfined t a shrt length adjacent t the ends. This is because any disturbances prduced in the free mlecule r near-free mlecule flws are felt mstly within a radius f ne mean free path. Since the mean free path in the test flw was " and the first mdel length was abut 0. 8", there is an appreciable part f this length (abut 0. 6") which is

25 (14) unaffected by supprt interferences r end effects. The assumptin that the disturbance effects are cnfined t a very small prtin f the length f the cylinder and that it is cnstant as the mdel length is varied was justified by the fllwing fact. The drag cefficient values derived frm the slpe f the free-vs-length curve agree clsely with the drag measured in anther experiment in which the cylinder mdel was placed dwnstream f tw shields. This is further discussed in Appendix A. The derived expressin fr w, the frce acting n unit length f the cylinder is als based n the assumptin that Wy2(mment abut flexural pint due t frces acting n mdel supprt) is independent f the length f the cylinder mdel. This was justified by cnducting an additinal experiment in which the supprt rd sizes were varied. Appendix B gives details f this experiment and the results. The expressin w = (df/dl) ix 1 /y 1 ) indicates that the graph f F vs 1 shuld be a straight line. Knwing the slpe df/dl ne can determine w, since the lever arm lengths xi and yj are knwn (measured). In all the experiments the graph f F vs 1 was pltted and the value f w determined frm the slpe. In every case the graphs f F vs 1 were straight lines in that prtin f the curve where 1 was greater than 0. 2". In sme experiments in which the length f the mdel was reduced t less than 0. 2" the curve deviated frm this straight line belw the pint indicating that the supprting sting interference cntributin t K became dependent n the length f the mdel r that the end effects started t merge fr this length. Hence the slpe f the curve between 0. 2" and 0. 8" was expected t give the frce per unit length. A typical graph f F vs 1 is shwn in Fig. 13. The effect f shrtening the mdel t a length less than 0. 2" is shwn in Fig. 14. b) Sphere Mdels The frce n the sphere was btained directly by calculating the difference between the grss drag frce and the tare frce. The experiments described in Sectin III, 5b, in which a dummy rd was brught clse t the sphere mdel during a grss frce run, gave the fllwing results. Except fr the case f the 1/16" diameter sphere mdel there was n nticeable change in the frce when the dummy rd was in psitin as cmpared with that when the rd was absent, indicating the negligible influence f the dummy rd n the mdel. Hwever, there was quite an appreciable change in the frce n the 1/16" dia. sphere mdel when this dummy rd was brught near the sphere. The measured drag frces n cylinders and spheres are listed in Tables II and III respectively.

26 V. DISCUSSION OF THE RESULTS (a) Cylinder Mdels (15) cefficient «is defined Let D be as the dra S frc e n the cylinder.y-^ue-i i f xengtn length L. T The Th drag ^ c, D cylinder). cylinder and equal t L. d (d = dia. t This was reduces e t V th U e e drl W^ '".T *** ""* length t th % ^Mer. The nlt f n t th e drag cefficient by dividing it by i y v 2 d ie shtn 1^ F? g as 1 5. fuucuon 0f KnildSen n b " ^ n cylinder diatneter cmplete W^n'ct^nJlT, TlT 1 ' V&1 ' leb f CD fr the ca - ' ^ihlii^hr^nf^---^^^^^^^^ further increase in the Knudsen number frm 3 06 ln P ^ " (Ref. 3) n the drag f a cytind r at ZKT-ZT ^^^ ^ Stald " " ** gave lhe same apprximately 3 tr <- TK tt!, " value f free mlecale y flw r af a^ma^vumrer T Iff? f ^ f0r, a ^^^ ^ reflectin, n net heat transfer frm the mdel and's'st^' 6 """ff specular T-pfi+n-^v, - inuaei ana d. d4 fr cmplete be'amstavf rrt Jth P e e rfw\r CylinderS ^ m1^ most SUrfaCeS f engineering interest L disuse i e the^ ^T ^ cefficient is equal t ne r^ ' ^ mmentum accmmdatin 20% between ihtelxzen^ ZT^ ^^ is " disc^^ f abut reflectin t be valtd ThT. ^ theretical values assuming diffuse f the cylinder ircnsidethr 1 ; ' explained ^ the ^^t that the length free m^cule cndmns " ^ ^T" ^ the mean free P ath s ndltl0ns are nt ^at experiments attain^ at the Knudsen numbers f the the the effect Jt^S^^^ ^^ mdel ^ -uld see

27 (16) Let AA and BE be tw infinitesimal prtins f the cylinder situated a little less than ne mean free path apart.. Mlecules rebunding frm the prtin AA will nt significantly influence the flux f particles striking AA because the cllisins take place many diameters ahead f AA but the rebunding mlecules will cllide after travelling ne mean free ' path with sme f the mlecules directed t BB just in frnt f it and hence will be prevented frm reaching it. Similarly sme mlecules that wuld nt have struck BB will strike it due t these cllisins. The case f reflected mlecules frm BB is similar. The basic pstulate f free mlecule flw (n cllisins between reflected and incident mlecules clse t the bdy) is thereby vilated and prblem becmes ne similar t that f transitin r near-free mlecule flw in which cllisins between reflected and incident mlecules have t be taken int accunt. /b) Sphere Mdels The drag cefficient f the spheres as a functin f Knudsen number based n sphere diameter is pltted in Fig. 16. A cmparisn is made in Fig. 17 f the present sphere drag results with thse measured by Kane and Sherman (Ref. 8) and Jensen (Ref. 9). Except fr the case f 1/16" dia. sphere the drag cefficients f the spheres were the same fr bth types f supprts thereby indicating that the mdel supprt rd had a negligible effect n the drag cefficient. Since the dimensins f a sphere are the same in all the directins there shuld be a better crrelatin between experiment and thery fr the case f spheres in free mlecule flw. Unfrtunately it was nt pssible t measure the drag f a sphere at a Knudsen number larger than ne because f size limitatins. Hwever, the present tests indicate a trend that shws fr the case f spheres the drag cefficient at larger Knudsen numbers will nt be seriusly lwer than the theretical tree mlecule value. VI. PRESSURE MEASUREMENTS TO DETERMINE THE EFFECT OF THE LENGTH OF A CIRCULAR CYLINDER TRANSVERSE TO THE FLOW The. lar r ge difference between the experimental drag cefficient f a tw dimensinal cylinder and its theretical free mlecule fzuftjt^ i 1^36 reflecti^ called Kr father investigatins t fmd ut the effect f the length. Hence it was decided t make pressure measurements arund a cylinder transverse t the flw. Since the diameter f the pressure prbe had t be cnsiderably larger than the diameter f nhtinh m0 +t ^ ^ desired ValUeS 0f the Knudsen ambers culd nly be btained in the lw density wind tunnel at lwer pressures and lw speed ratis, a subsnic nzzle was used in these pressure prbe experiments. T-,, K /.,. ^n rifice P rbe similar t the ne tested and used by Enkenhus (Ref. 7) was used t calibrate the flw. An " diameter rifice was drilled thrugh a thin sheet ( " thick) f aluminum fil

28 (17) and cemented ver a 0, 030" dia, hle drilled in the side f a " dia stainless steel tubing, with the rifice carefully psitined at the exact center f the hle. The pressure readings at three specific angular psitins f the rifice (0, 90 and 180 t the flw directins) will give the speed rati (Ref. 11). In particular the 90 pressure reading will give the stream static pressure when thermal transpiratin is taken int accunt. After having calibrated the flw, the rifice prbe was remved and anther cylindrical prbe with a diameter f " was put in the flw. This prbe was almst identical t the rifice prbe except that instead f having an rifice thrugh an aluminum fil, an " dia. hle was grilled in the tubing itself (see Fig, 7). This hle had a length f " s that the rati f length t diameter was ne and hence a shrt tube prbe resulted rather than an rifice prbe. Effrts were made t have this hle drilled as clse t the free end f the tube as pssible, the end being plugged by sft slder. Pressure readings were taken at varius angular psitins f the rifice. A secnd set f measurements was made with anther cylinder f the same diameter as that f the prbe attached t the traversing mechanism and psitined vertically abve the rifice prbe and almst tuching it. This cmbinatin gave the effect f a pressure hle essentially in the middle f a lng tube. A phtgraph f this arrangement is shwn in Plate 7. Pressure readings were taken with this cnfiguratin at the same angular psitins as befre. The results btained are shwn in Fig. 18 in which the pressure ratis with and withut the added length is shwn at varius angular psitins f the hle t the flw directin (0 crrespnds t the psitin f the hle at the stagnatin pint). Frm this it can be seen that atknudsen number f abut 5 and a speed rati f the difference in the tw readings is apprxi* mately 4, 3%. Since this 4. 3% represents the cntributin frmnly ne half f an infinite cylinder and since an equal cntributin may be expected frm the prbe (because its cnstructin makes it essentially the ther half f the infinite cylinder) the ttal errr intrduced by an infinite cylinder wuld prbably be 8, 6%. It can als be seen in the figure that at Knudsen number as high as 14 there is still a significant effect due t cylinder length and free mlecule cnditins have nt been reached yet. As these experiments were dne nly t prve that there is an effect f the length (r tw dimensinality f the bjects) n frce and pressure readings, n effrt was made t investigate any functinal relatinship between the length effect and speed rati. I ' t i i 111 Is It shuld be nted that since the pressure prbe experiments indicated that the length f the prbe had an effect n pressure readings the speed ratis f the flw as calibrated by the rifice prbe may be in errr.

29 (18) VII. EXPERIMENTAL ERRORS It is difficult t give an accurate estimate abut the errrs f the present experiments but the prbable magnitude f the errrs are as fllws. The flw parameters. Mach Number and static pressure were accurate t - 1%. The lever arm lengths were measured with an ptical cmparatr and a cathetmeter and the errr in this is less than 0. 1%. The quartz spring calibratin varied by + 1% due t hysteresis f the flexures and the backlash in the gear mechanism. The errr in the alignment f the balance with respect t the flw directin was estimated t be less than In the sphere drag results significant errrs may ccur as a result f taking the difference f tw large numbers t get the drag. In tw sets f experiments carried ut in which the spheres were supprted by tw different methds viz the crss-stream supprt and the tailsting supprt the frce n the mdels were the same in bth cases but the frces n the supprting sting in ne case was many times larger than that in the ther. The clseness f the final results indicate that the errr were within reasnable limits except fr the 1/16" dia. mdel. The drag frce n the cylinders was calculated by the slpe f the ttal frce vs the mdel length which the discussin in Sectin IVa. and experiment shwed t be a straight line. Hence the errr in these results shuld cme mainly frm the spring calibratin errr which wuld be n mre than abut t 1%. Since the slpe had an errr f nly T 1% the ttal errr n the frce measurements is +2%. The drag cefficient was btained by dividing the measured frce by j.fv A. The variatin in the flw Mach number will vary the static pressure and at a Mach number f 2 at which these experiments were perfrmed, a 1% errr in Mach number intrduces abut 2% maximum errr in the value f f V. The ttal maximum errr that can ccur will be t 4% but the absence f scatter in the final results shw that actual errr was well belw this value. Repeatability f the Experiments Tw experiments n the drag f a " dia. cylinder were perfmred at an interval f abut 10 weeks using tw different quartz springs. The results were as fllws Dia. Kn CD " This shws that the experiments were quite repeatable.

30 VIII. NCLUSIONS (19) Drag frces f spheres and cylinders were measured in air at a Mach number M = 2 in the Knudsen number range f 0. 1 < Kn < 1 fr spheres and 0. 2 < Kn < 7 fr cylinders. There is n thery available at present that will predict the aerdynamic frces n bdies in transitin r near-free mlecule flws at mderate Mach numbers where thermal mtin f the incident stream is nt negligible. Cnsequently it was nt pssible t cmpare the experimental data with relevant theries. The drag results fr the cylinders shw that free mlecule flw is nt reached at a Knudsen number f 5 based n cylinder diameter in cntrast t statements by previus wrkers (Ref. 15). The measured drag cefficient at this value f the Knudsen number was abut 20% lwer than the theretical free mlecule flw values fr the case f cmplete diffuse reflectin and 10% lwer fr specular reflectin. On the ther hand the available experimental results n the sphere drag indicate that the free mlecule flw thery and experiment will mst likely agree at Knudsen numbers nly slightly larger than unity since at a Knudsen number f abut 0. 6 the measured drag was already higher than the theretical free mlecule flw value based n specular reflectin. It is suggested that the discrepancy between the theretical and measured values fr the case f circular cylinders is assciated with the fact that in this case nt all dimensins are smaller than the mean free path. This cntentin was supprted by additinal experiments cnducted in subsnic flw. Pressure readings taken by an rifice prbe indicated that there is an appreciable effect f the length f the cylinder n pressure readings at Knudsen numbers as high as 9. On the basis f these results, it is suggested that fr flws ver cylindrical bdies nrmal t the stream the value f the Knudsen number based n the cylinder diameter is inadequate t classify the type f flw ver the bdy. Perhaps this cnventinal Knudsen number based n cylinder diameter culd be suitablly mdified by a mdel aspect-rati term t shw its apprpriate free mlecule flw limit. The theretical wrk dne by Lunc and Lubnski (Ref. 16) n the aerdynamic frce n an infinite strip in a high speed flw shws that at a Knudsen number f 5 based n the width f the strip, the theretical value f the drag is abut 7. 5% lwer than the crrespnding free mlecule flw value. Their calculatins als shw that as the Knudsen number is further increased, this difference decreases. This is cnsistent with the physical reasning because, fr an infinitely lng mdel nrmal t the flw, the free mlecule frce values shuld be asympttically reached as the Knudsen number based n mdel's width is increased. Cntrary t this, the present results shw that a drag cefficient value f is reached at a Knudsen number f 3 and that there was n apparent further increase as the Knudsen number was further increased. A pssible explanatin fr

31 (20) this, is that the range f Knudsen numbers (viz. 3. t 6) cvered in the present experiments may nt be large enugh t psitively indicate that the drag cefficient values becme independent f Knudsen number. In additin, the experimental accuracy might have been lwer in these regins due t a lw magnitude f the frces measured as cmpared t the neighburing pints. Finally it is suggested that further wrk shuld be dne with a different nzzle t permit the use f a wider range f high Knudsen numbers than was pssible in the present wrk in rder t reslve sme f these uncertanties.

32 (21) REFERENCES 1. Schaaf, S. A. Chambre, P. L. 2. Schaaf, S. A. Talbt, L. 3. Stalder, J. R. Gdwin, G. Creager, M.O. 4. Baker, R.M.L. Charwat, A. F. 5. Enkenhus, K. R. 6. Latz, R. N. 7. Enkenhus, K.R. 8. Sherman, F. S. Kane, E.D. 9. Jensen, N. A. 10. Harris, E. L. 11. Pattersn, G. N. Flw f Rarefied Gases, Sectin H, Fundamentals f Gas Dynamics, Vl. Ill, High Speed Aerdynamics and Jet Prpulsin. Princetn University Press, Mechanics f Rarefied Gases, Sectin 16, Handbk f Supersnic Aerdynamics, NAVORD Reprt 1488 (Vl. 5), A cmparisn f Thery and Experiment fr High Speed Free Mlecule Flw. NACA Tech. Nte. 2244, Transitinal Crrectin t the Drag f a Sphere in Free Mlecule Flw. Physics f Fluids, Vl. 1, March-April The Design, Instrumentatin and Operatin f the UTIA Lw Density Wind Tunnel. UTIA Rep. N. 44, June Design f a Tw Cmpnent Micrbalance fr Lw Density Wind Tunnels. University f Califrnia Engineering Prject. Reprt HE , Aug., Pressure Prbes at Very Lw Density. UTIA Rep. N. 43, Jan Supplementary Data n Sphere Drag Tests. University f Califrnia Engineering Prject Reprt HE , August Supplementary Data n Sphere Drag Tests - Part 2 - University f Califrnia Engineering Prject Reprt HE , Sept., Investigatin f Free Mlecule and Transitin Flws Near the Leading Edge f a Flat Plate. UTIA Rep. N. 53, Nv Thery f Free Mlecule, Orifice Type Pressure Prbes in Isentrpic and Nn-Isentrpic Flws. UTIA Reprt N. 41 (Revised) May, 1959.

33 (22) 12. Gwen, F. E. Perkins, E. W. 13. Pattersn, G.N. 14. Stalder, J. R. Zurick, V.J. 15. StaXder,J. R. Gdwin, G Creager, M. O. 16. Lunc, M. Lubnski, J. Drag f Circular Cylinders fr a Wide Range f Reynlds numbers and Mach Number NACA TN 2960, July, Mlecular Flw f Gases. Jhn Wiley and Sns Inc., Theretical Aerdynamic Characteristics f Bdies in a Free-Mlecule Flw Field NACA TN 2423, July, Heat Transfer t Bdies in a High Speed Rarefied Gas Stream. NACA Rep. 1093, Sur Une Slutin Apprchee Du Prbleme De L'e culement D'un Gaz Rarefie Autur D'un Obstacle. Nadbitka Z Archiwum Mechaniki Stswanej Tm VIII Zeszyt 4, Warsaw, 1956.

34 (23) APPENDIX A Drag Measurements Using Mvable Shields In rder t knw the Mach number at any pint in the flw ne has t knw the impact and static pressures at that pint. Due t the very lw magnitude f the Reynlds number f the flw, prbe readings are subject t high viscus crrectins. At the Reynlds and Mach numbers at which the UTIA lw density wind tunnel perates nly viscus crrectins t the impact prbe readings are knwn. Hence the flw has t be calibrated using impact prbes alne. The usual assumptin which is experimentally verified is made that the flw is isentrpic at the centre f the jet and the static pressure is cnstant acrss it. When a shield is placed in the flw the accurate determinatin f the Mach number dwnstream f the leading edge f the shield is difficult and there is an uncertainty f the flw Mach number clse t the shield as the flw may nt be isentrpic in these regins. At the beginning f this research prject it was decided t make drag measurements by munting a mdel 5" in length vertically n the balance and expsing nly 0. 8" f the cylinder at the nzzle centre t the flw by cvering the remaining length by tw shields. These shields were munted n a specially designed traversing mechanism by means f which they culd be mved in r ut relative t the nzzle centreline. This permitted the length f the cylinder expsed t the Mach 2 flw t be varied remtely while the flw was n (Plate 8). An impact prbe survey made n the nzzle centreline with the mdel remved but shields left n is shwn in Fig, (19). Referring t this figure it can be seen that shields disturb the flw quite appreciably even upstream. There is a significant change in the impact prbe readings at the nzzle exit plane with and withut the shields when they are placed s that their leading edges are " frm the exit plane. There is als a sudden change in the flw immediately dwnstream f the shields. (It is wrthwhile t mentin it here that when nly ne shield was placed in the flw abut 1-1/8" frm the nzzle exit t cver the supprting sting fr the experiments described in the main part f this reprt, there was n change in the flw Mach number at the nzzle exit). The impact prbe reading alne will nt give the Mach number in this regin as the flw may nt be isentrpic. Because f the uncertainty f the flw velcity immediately dwnstream f the shields the idea f keeping the mdel dwnstream f them had t be given up. Hwever, fr cmparisn purpses sme experiments were perfrmed with the mdel placed dwnstream f the shield. Care was taken t place the mdel as clse t the leading edge as practicable withut tuching it. First abut 0. 8" f the mdel was expsed t the flw and the frce n it measured. The length was then reduced in steps f 0. 1" and frce measured in each case. Frm the slpe f the frce vs length graph (a typical curve is shwn in Fig. 20) the drag was evaluated assuming that any influence f the shields n the cylinder wuld be cnfined t a very shrt length adjacent t the shields. The curve wuld be a straight line whse slpe gives the drag per unit

35 (24) length. Results thus btained n cylinders f different diameters are shwn m Fig. (21). The Mach number was calculated by measuring the impact prbe pressure n the nzzle centreline at a distance f 0 034" the ^^ ^^ 0f ^ Shield assumin g -entrpic flw. ^TM/ s'tr (This is the apprximate psitin at which the mdels were placed). On the same figure the tw sets f drag cefficients measured by placing cylxnders dwnstream f the shield and far upstream f it (as reprted in the main sectin f this reprt) are shwn. reprted m Sme significant cnclusins can be drawn frm the cmparisn f these tw experimental data. Fr the case in which the cylinder mdel was placed dwnstream f the shields, the drag cefficient ^innlztt^ e «lly that 0f a Cylinder f infinite -Pect rati Any influence f the shields n the mdel was assumed t be cnfined t a ^therl^re T ^ T^ ^ ^ m0del le^th^ ^ J.e ther hand the experiments with the mdel far upstream f the shield end ef^rr th^s T^ f ^ length " ^ Again ü^ assumes ^^ ^e end ellects (r the aspect-rati effects) were cnstant as the mdel length z-rr^r extr t the frce - -^ * -" i - w* + In these tw ex P er iments the "end-effects" were due t tw smelds "rth 111 T 6 CaS. e ' the effect WaS due t0 the P-sence"f th shields, in the ther it was due t the finite aspect rati. In either case hwever, these effects were eliminated mathematically by taking mea^e- ZTJ: tz e z\i:t 8 ' Th : clse agreement f the^ relts indicates that the technique used was quite satisfactry and crrect thereby justifying assumptins riginally made n phys reasn^. e

36 1 (25) APPENDIX B Effect f the Supprting Sting n the Cylinder Mdel All the cylinder mdels were supprted by a stainless steel tubular sting " diameter and 1. 1" lng (Fig. 4). In cmputing the drag frce frm the data it was assumed that the interference effect f the supprting sting was cnfined t a very small prtin f the length f the mdel at the center where the mdel was sldered t the sting and that the frce n the supprt remained cnstant as the mdel length varied. T check these assumptins an experiment was cnducted in which the diameter and the length f the supprt rd were increased t " and 2. 1" respectively, see Figure belw. 0.8 * L " dia. "< 2. 1"- rr " dia. i) The results were as fllws. Diameter f the mdel Length f the Dia. f the CQ Supprting Sting Supprting Sting Kn 0.049" 0.049" 1.1" 0.020" " " The clse agreement f the values fund fr the final drag cefficient justifies the abve assumptins.

37 (26) APPENDIX C Temperature Measurements CDi = drag cefficient due t incident mlecules = c D r = drag cefficient due t reflected mlecules = T = stagnatin temperature = 5450 R Tj = free stream static temp = Rat M = 2 its supprt t^x t duu andletce nence thl^w^f tnere was sme ^ t0 heat, thermauy transfer insulate A n nnq" the m H,^ ^ ^m W "Tw'tt^ ^r-^16 WaS USed t0 -ersur^the mde a temperant ZZTl a^r Xr^r th 0ne^ ^ ^ piete^ hllw mdels Rth 1 i.? thermcuple inserted inside the tabul^fn Tablf IV ' OSt ^^^ readings The - resu^ are (abut 1250 F f J^^i:^ ^^ de -ed value f temperature value f the rir^ ^ inserted m Equatin (A2 9) the theretical value i the drag cefficient is which is IP^C thr, iw i ^ 1% lwer than that f the adiabatic mdel valup m ^ TJ u frm 1 18 t! 15? Thi^ I P^tlcular ^se this value is changed

38 1 (27) APPENDIX D Calculatin f the Mean Free Path In rder t calculate the Knudsen number, it is necessary t knw the value f the mlecular mean free path in the flw. is given by In Ref. 13, it is shwn that the mlecular mean free path k" u JL - where ß- = cefficient f viscsity J TT = density = temperature Substituting fr f in terms f p(pressure) and RTj, the abve expressin reduces t The viscsity is a functin f temperature nly and may be represented adequately by Sutherland's relatin where (T, and Cg are cnstants fr a particular gas. The frmula fr air is _ i. - 3 J \ 5 Zlt X\0 J f Ti + ZlO-6 where A is i n inches, p is in micrns Hg., and T^s in degrees Rankine. The value f TjWas fund frm the flw Mach number and the stagnatin temperature assuming adiabatic flw.

39 (28) APPENDIX E Theretical Aerdynamic Characteristics f Bdies in a Free Mlecule Flw Field NOTATION Cj) drag cefficient, Drag frce/ i S V 2 A C L lift cefficient. Lift frce/i? V 2 A CJJJ mst prbable mlecular speed E internal energy flux, energy/unit time x unit area erf(s) errr functin - 4r / «. dx \/n- y f mlecular distributin functin IQ [ I ^f ) mdified Bessel functins f the first kind j number f mlecular degrees f freedm m mlecular mass. N number flux p nrmal mmentum R gas cnstant s speed rati V//J2RT T temperature, abslute T w wall temperature ^wequi adiabatic wall temp. Uj^ cmpnents f inlecular velcity V mass velcity 9 lcal angle f attack i thermal accmmdatin cefficient if rati f specific heats

40 (30) If the gas thrugh which a bdy is mving is sufficiently rarefied, the mtin f the mlecules impinging n the bdy will be essentially unaltered by cllisins with reflected mlecules. Under these cnditins the ttal frce and energy imparted t the bdy by the mlecules can be brken dwn int tw cmpnents; ne arising frm the impingement f incident mlecules and the ther frm the re-emissin f the mlecules frm the surface. The velcity distributin f the gas mlecules until they strike the bdy will therefre be that f a gas at rest, namely the Maxwellian velcity distributin. In rder t cmpute the frces imparted t the surface by the reflected mlecules, it is necessary t make sme assumptins regarding the nature f the mlecular interactins with the surface. The cncepts f specular and diffuse reflectin have been recgnized since the early studies f Knudsen and thers. If the walls are perfectly smth, specular reflectin will ccur in which the cmpnent f the mlecular velcity tangent t the surface remains unchanged while the cmpnent nrmal t the surface, n cntact with the wall reverses its directin with n change in magnitude. Hwever a real surface is mre r less rugh and the mlecules are reflected quite randmly s that all traces f their past histry becme entirely r almst entirely lst. This type f reflectin is called diffuse. If the reflectin is cmpletely diffuse, all directins f emissin abut the nrmal t the surface are equally prbable; they then bey a csine law similar t that f a surface emitting radiant energy. In the case f cmpletely diffuse reflectin the velcity distributin f the re-emitted particles is Maxwellian and is cnsistent with the surface temperature. Fr mst surfaces f interest in engineering the re-emissin prcess deviates slightly frm cmpletely diffuse reflectin. It is then pssible t characterize the reflectin prcess frm a given surface material by defining a quantity representing the average "diffuseness" f the re-emissin, r, what amunts t the same, the degree f accmmdatin t the wall cnditins f the re-emitted mlecules. This quantity takes the frm f a cefficient ranging frm 0 t 1 as the re-emissin changes frm cmpletely specular t cmpletely diffuse. Such "accmmdatin cefficients" are defined separately fr energy (thermal accmmdatin) and fr the tw cmpnents f mmentum (tangential and nrmal t the surface). Thus, the thermal accmmdatin cefficient is defined as d " t ' - c/s'iv where de i = incident energy flux (per unit time and area) de r de w = re-emitted energy flux frm the surface = the re-emitted energy flux fr cmplete diffuse reflectin

41 (31) cefficient is ZZeäVs^^ m0mentum exchan g e ^he accmmdatin 5- = ^r^- (A - 2 > where T. = injidenl tangential mmentum flux C r = re-emitted tangential mmentum flux Similarly fr the nrmal mmentum exchange we have ^ C P: ~ Pw (A " 3) rt^c^a^dt 6 ^mit^ ^ 7^^ ^ 0f n0rmal m0ment - reflectin " al mmentum f r cmplete diffuse It can be seen that fr cmplete diffuse reflectin ck = (S^ =<rj. ^ 1 whereas fr specular reflectin ^ _ g- -5- _ ^ Analysis mlecular vel^tl (? 1,C2 :K 3) ^ the Cm P nents f the randm (thermal) where c m = ^ 2RT = the mst prbable mlecular speed. y (A-4) (-u,. \ «j, -uu2 9, - u^awth". U) alng the ^ c-rdinate " 'H^ With axpc; velcit (v y-v' ^ having \ cmpnents i_ ^?, = "1 +C!, =. u 2 + c 2, y s ^ u3 + c 3 in the bdy then^crm^ ^^ ^ -ferred t axes fixed

42 / = (frr C m y (32) f&[c^^c^c^] We shall first derive expressins fr the frces and the heat transfer arising frm cllisins f the mlecules with a surface element f the bdy. The aerdynamic quantities fr the whle bdy are then fund by integratin ver all surface elements. Let us cnsider a surface element f area da and chse a c-rdmate system such that the element lies in the (x, XQ) plane s that X! is nrmal t da (see Fig. A-l). 1*3 FIG. A-l an angle ö If the incident velcity V lies in the (x^ x 9 )plane and is at with the plane f the element, then t^s M'Sir^Q and J^ d^ The number f mlecules with velcities in the range If striking an element f area da n the frnt side f the surface in unit time will lie in a cylinder f base da and length )-, with its axis ' in the directin f J- ; the vlume f this cylinder is W da. If m is the number density f the incident stream, then the number'f mlecules striking da per unit time is 0 +»0, ^ ',0 M; H ^.UH^A ^ t- - Y) t 'cm 4- -U, _! L 0-+ er f cz r" e. c We nw define the speed rati s = v/c rrii and recall that u, = VsinQ Then ui =c mi sin9s.

43 (33) Thus Vt C m ; K JA (A-5) where 7 = -^itrssin^ Cl + m i S. St'rt*) The number f mlecules striking the rear side f the element may be fund by the same methd, except that the limits f integratin fr ^ becme - «^ ^ ^ 0 -The result is r da (A-6) where 4 Si.r\ 6 vffr SSind ^/ - er/ s.scnö} Nrmal Mmentum Each mlecule striking a surface element carries a mmentum cmpnent nrmal t the surface f magnitude ni where m is the mass f the gas mlecule The number f mlecules with ve cities in the range ^ t f+jt which strike the frnt surface will impart t the surface element a mmentum equal t The ttal mmentum imparted t the frnt surface by incident mlecules is btained by integrating the abve expressin ver all pssible velcities. The result is f c ^srfe da 3. I -5 Sin0 yfrr S SVn* 1 yl Zs's^JJiA-l) where f. = m ^ = density f the incident mlecules. Similarly, the incident nrmal mmentum due t mlecules striking the rear surface can be calculated as (^ - ftvw* JA (A-8) Tangential Mmentum Each mlecule carries a tangential mmentum f Yr\\ The ttal tangential mmentum imparted t the element by the incident mlecules striking the frnt side is

44 (34) (^) F _- 6 J- (? ^^ mr>^ I j I UfAS^SjS - 5 S ir/e (A-9) Similarly fr the rear side the tangential mmentum is -S Sir,'' (--0 yftt 5 Sing (^i~eff jr. S,- n e) (A-10) Mmentum due t Mlecular Emissin frm a Surface It is assumed that the emitted stream has a Maxwellian velcity distributin crrespnding t a gas in equilibrium at an unspecified temperature T w. Since the reflectin is diffuse the mlecules can be cnsidered as thugh cming frm a ficticius gas n the rear side f the surface at a mst prbable mlecular velcity c rnw crrespnding t the temperature T w. Let flw be the number density f the reflected mlecules. The nrmal mmentum imparted t the frnt surface by the mlecules rebunding frm it is +» + " 0 i, ^ u a\ (^F YnYi v l{n ^wf^ 0 _ - 0 C "wfr / dcyae^dc^ c/a % Similarly the nrmal mmentum imparted t the element due t mlecules reflected frm the rear side f the element is (K 2. (A-12) If the frnt surface is thermally insulated frm the rear, then (T W ) F and (T W ) R will differ. f w and ^ als differ. In rder t btain values fr JW F and 5 W in terms f knwn quantities the cnservatin laws are applied. The number f mlecules reflected frm the surface N v is equated t the number f mlecules striking the surface (Ni). (A-13) N (T w ) p

45 (35) (^V- ^#-^0^ -(NO, iy/tt have Substituting this value in equatins (A-ll) and (A-12) we Wc = T^J-~X^A (A _ 14) 4 s- and (K\^ -lill J 1 -^- 7'dA 45 Z (A-15) By symmetry cnsideratins (r w ) and (r w ) are ::w i 1' the net tangential m0 ^m diffx renectl/rie- cules is zer. The ValueS h f < T w)f r nt and ( T w ) rear depend n the energy exchange. By equating the ttal incident energy t the energy the'fr^f? H 6 ' 1 :^^ m0lecules P1US the heat lss^ tte bdy'fn the frm f cnductxn and radiatin, ne culd determine (T w ) fr^ and surface. We are nw in a psitin t calculate the actual frces n th* bdy by the use f accmmdatin cefficients defined previusly Nrmal mmentum: P = Pi + Pr = (2 - G- N ) pi + (5- N p w Tangential mmentum: IT- C i -TV = <r T XT; in the ^ve relatins the ntf^8 ^ Z* 1 * ^ Pi ' ^ and ^ imparted't surf expr * SS10 *. fr nrmal *** tangential mmentum imparted t a surface element in free mlecule flw are thus

46 (36) ^--^r-d*)^* (* ** ) v ^ Zs^sifJ Jlrs.s, - S^Sirfc ZS X 4 Ti -5 Sin e -V fit S.S.'ö ^/ + ^r/. S.Sirö) {A-16a) 1 rea\ r 2.. Z _S S.n Ö e. ^ <JA/ XsW T^. i. -s si e y/v SS.nfi^l-er/SS.ne) (A-16b) r. ffcti^- ^v J A j (5" cse Sim S 5in 6 y/i S,Sine ^l + ey/s.s.n^ (A-17a)?cV Tcai d/4 ) (51 «rse Scnö S^Sin^ ytst S.S.nö - Qi-ex^sSinö) (A-17b) If the frnt and rear surfaces are in perfect thermal cntact, then (T w )f rn.( ; = (T w ) rear. The ttal nrmal mmentum imparted t the surface element as a whle is

47 (37) P fjfw ** 'rear,.- hi** % [z- 6 7.) v/lf s S Sin Ö Sine -t 2 (Z-G'H) er/- s.s.v^ (s^e + J^^ vtir Sm^ \J ^-J ( (A-18) Similarly ^- ^i^ Tei' c* _:^ JAN ^s^csesm«-s SV ^ v/it S SinB -+ er^ ssiö (A-19) The nrmal and tangential mmentum n the surface element can nw be reslved int cmpnents alng the flw directin and nrmal t it t btain drag and lift frces respectively. In nn-dimensinal frm (thrugh divisin by i^v'im ) the results becme as fllws, Lift cefficient L ii<ny 1. i < cse (3.-<sr N -<irr)sin6 v/lf s 2S 1 ^W f + ^l-t-e.'r/s.sinej cs Sin e (2'5- N -<s^) i ^r- ^ flüf 5,n6 Ä-2Üa)

48 (38) C I-rear ~ / * (i-erfs.s(e) cab ß^6 (2-f N -<r T ) t tok. - ^ji-fr F^r** ^ Hj ^L* ^Li^\ -V ^L Ye/ (A-20:b1 2- a - 5 S.> Q C, - ) SinÖCöse e fr 5 2^'öV<r T J ^ 5jLca&Sif>& fir jlk Cs&fzsirfeCz-si-fl)* ~ er/ssc* 6 Drag Cefficient (A-21) c fr^v e ^F -S Sip fx-^s^fi-v -^ Sen 0/7 2^ H- ST ^> 25 V T 7 {lterfs^r&)si,e ^'^(S^+J-^ + fk^sl^l^m «j- 6VCß3, v 6 (A-22a) 'Drec'i T"^ ^g" T^ (A-22b)

49 : ^"p - ^Dft*,* (39) C, C n =.5 sin e AHT 5 ' X' -[ei/(s. Sen*)"] Sir«[2^-^)(^n^ + ~.) +X(r T f«a el (A-23) Drag f a Sphere ^A u. The ex P ressin fr th e drag frce exerted n an element f area da when bth f its sides are expsed t the flw is (frm Eq. A-23) D = isvda - -ss^e r L z (Z'<r N ) scse + 2^«s x e t ^LJ^ Sir, 1 «J Tw ATTTS L -1 Tt + [er/^s^s^e^*'^)^* + ^v) + 2cr r^vj( (A-24) This expressin can be integrated ver the surface f the sphere t btain, the drag f a sphere. In general, if the surface f the bdy is a nn-cnducting material the temperature T w will vary frm ne surface element t anther. In the special case where the bdy is a perfect cnductr, s that T w is cnstant ver the surface. Equatin (A-24) may be integrated ver all surface elements. Rcie Element f area = 2 TR 2 cs G d V ^=9- R = radius f the sphere Frnt Elemental Ring Rear Elemental ring FIG. A-2

50 (40) Thus fl^ -^- = 2 n^cse de ^- / 2 (s-c H ) ^0 + 2<r r c^e 7 + S.^ ^ f ^ J I ^5 i i J 5 ' h ferfa. Sle) Sifi6pl{2-<r N ) (sij-6 + JJlj H- i^c^e 1 ( ^T &r f S -^ 77= J f I +S J e ^ i -r- The prjected area f sphere = TTR'.4.A. ^ g'^^t 4^45-/ ^,AU) (A-25) Fr diffuse reflectin 3 s 1 T; C D sphere diff use = Fr specular reflectin (j- g. (j- - C D sphere S P ecular = ^ [ d^i-^5 +^ ( A xj ( (A-27) Drag f a Cylinder Transverse t the Flw Assuming again that the cylinder is a perfect cnductr and hence T w is cnstant ver the whle surface the expressin fr the drag n an element da given by the Equatin (A. 24) can be integrated ver the surface f the cylinder.

51 (41) Frnt side f elemental area Area f the element LRdG y FIG. A-3 Rear side f elemental area D cylinder _ 4V Vx The values f the integrals encuntered are given in Ref. 14. The result is v =tf ( vv^/ ^+ ipifa.^^^^^ + h^^a^') i^j] i J c n '/zsxtlr (A-28) 2A? = prjected area f cylinder where i(0 mdified Bessel functin f first kind and rder. zer mdified Bessel functin f first kind and first rder Fr diffuse reflectin <5^s(S^a

52 (42) C 0 diffuse = ^ t 5 ^ jy^^;^^ Zj J lci)ia~29) Fr specular reflectin ^f^ r (Tl ' 0 Energy Exhange 2. 4 g~t. /F /^. i. 2 - (i+f ji.^tft^:,^' (A-30) The aerdynamic frces n an element (in free mlecule flw) due t the reflected stream f mlecules depend n the wall temperature T w. This value f T w depends upn the efficiency f the energy transfer prcess that ccurs between the slid surface and impinging stream and is described by the intrductin f the thermal accmmdatin cefficient ^ (Eq. A-l) dej - der then If d^ is the net cnvective heat transfer t the surface dq = de,; -d r «de^ = incident energy, bth translatin and internal. CK. is assumed t be cnstant fr bth kinds f energy transfer dq - <* C de t ' d ^ Incident Energy n an Element f Area da The translatinal energy incident n the frnt side f an element in unit time is ^-Vv-W hn^ttow^ 2.. If the gas is mnatmic then the ttal incident energy is given by the abve expressins. If the gas is cmpsed f diatmic mlecules then each mlecule carries an additinal amunt f energy called the internal energy. By the principle f equipartitin f energy the amunt f internal energy carried by each mlecules is (j/2)m RT ; ;

53 (43) where i is the number f degrees f freedm f mtin. Fr air at nrmal temperatures l^-z. 1 is related t the rati f specific heats y by the relatin 0 -~ TT-I Therefre the interal energy f mlecules striking the frnt surface is /'if v T % V v ' ^XM mlecules striking da in unit time = T^T" T ^Tl Nl^A where N ' fr h v is number f ^ " rip " \ +\?t5.scnechfe<f S."S0n ) Thus the ttal incident energy n frnt surface is Similarly fr the rear surface (A-31)) fm } (A-32) Energy carried by the Reflected Stream The energy de w carried by the reflected stream f mlecules when in Maxwellian equilibrium with the surface, can be calculated in a similar manner. Fr the frnt surface

54 (44) Thus the ttal energy - de Wfr gr f n) + d Ewfr gn ) - de^^ - I W : ^ +JirSSifle Ci+eY/s.s^e) (A-33) vtilr - y J Similarly, fr the rear surface ^r transfer transler ^ L (A^4) If <Tö 117^1 reflectin ^ = 0 and hen - there is n energy If 0< ^ 0 and if there is n heat transfer (dfl = 0) then the ele- Tu" "This"? ' ^"rt Ure ^^ as ^ r'equüibrium tempera- Srecte^e^:^ Can ^^ ^ ^^ ^ ^^ ^ ^^ The fllwing expressins thus derived give the rati f equilibrium temperature t incident (free-stream) temperature Frnt surface f the element 7-7 "Iz+i/*.) ] --s^ctfe _., ~ x / Rear Surface f the element ^ (T^e.) rpaf. i_ c ^^j^ ^ - ^ ^f ^f^j^i! j) 1 ^ ' / ^ «s ^ firs-s^&ci' ef f ssin > J Frnt and rear surfaces in perfect thermal cntact 7; IVg^uil ' Equilibrium Temperature f a Sphere If T w is cnstant ver the entire surface area then the betnte^r^dt integrated t ^ btain ^^ the equilibrium ^ ^^ temperature ^^ f 0n a sphere. an element can Incident energy n an element da in unit time is 2? <* T O 3/ -<" c-^*(^4) ^ 5, Si e (, Ws. s,. n,)c 5 vf + n J The elemental area chsen fr integratin is the same as that used in the drag frce calculatins (see Fig. A-2)

55 (45) da = 2 TT r 2 cs 9 de (r = radius f the sphere) Nt Vrrr'' 3/ V2- \l7fr I *" *" J - ^ fr n net heat transfer (adiabatic mde]) T;. ' ^JA L - fr-^ L - ^ Z e +f!terf5(5-f ^j\ 5C^ Fr a mnatmic gas j - 0. fr diatmic gas i. i reflected mdules, fr a sphere^^^we Lw '^ ^ diffusely Equili^^^j^^ra.^e^a^ Fig AS) T The eie - men i f area ch^n fr integratin is shwn in Fig. A 3). T W is assumed cnstant ver the entire surface f the cylinder.

56 (46) da = L. r. de where r = rad. f the cylinder, L = length. : * % ~~t?i( * T^k(L.r)\2le^*'* <i zw _ zsictji)^ a-ir Sir 6 cr 0 ctr) ^-^[i, ( )( s \ s'a+i) -miyi ii)iau\ u)i w =1^4}^%r E i ~ E w fr n net heat l ss. T k?0ij _ J ^ ^^ i^^/'^vf^j.^^^^o^^t^q, ^c 5 ix 5%^^x'^r;^ HW, + Table V gives the th eretically calculated values f equilibr um temperature and drag cefficient (due t incident and diffusely reflected mlecules) fr a cylinder in free mlecule flw dl " USely

57 TABLE I SPHERE AND SUPPORT ROD SIZES Sphere Diameter (in.) Supprt Rd Dia. (in.) Rd. Dia. Sphere Dia Nte: Supprts used fr crss-stream and tailsting supprted mdels were f the same size.

58 TABLE II CYLINDER DRAG DATA Mach N. = 2 00 Mean Free Path s Cylinder Dia Knudsen (in. ) N ! ! i 40 0, , 08 c U D ,

59 Q U ^ (N T I rh C- * OS O T-H m t> CM CV] CM CM CM CM CM 0) (l,i S I rn I u (11 3 rj u ^ a tud +-> 0) rl a rh B OS CM 00 lo CT) CM cd in in CM 03 t CM CM m in c m CM CM c cd c Jn cd PQ < 4-> u Q a Ü 3 < «id Q rf (U W fn «cq l K dn O IH U OJ 0) u d CM O CM" II cd 5 p4 a Ü U =3 ID a 13 W Xi G 3 ^ X 0 cd % M 13 0 H C n 13 crj Cfl EQ Ü m CM ^ 03 c- 03 I> 00 CM t- in 00 CM O O c- m in CM CM M CM i 1 CM CD If I>- 03 ^ t- 03 CM 1. 1 CM CM in c- T-H CM O 03 T-H rh r-h I> in cd <U -in a c in in in m in c~ IT- t- m CM *, CM 03 ^ CM r-h T-k TH O

60 ü 00 ^ CM t- un r i O 00 in r-t Cvl CM CM CM p «S ; S ^ 3 cfl r 0 ^ Si -M w ;3 _ ^ g - - ^H r-l CD vh C-' > 1 CM Oi T ( CSI CD 4-: <1 Q a ^ sw hh Ü XI -H cd hh 0) Qi CJ (i)?-l fn 0 fe d rt k q CÜ <u PH l;-< s IH b IH 1 I CD LO U-J 00 CD * LO in * u HH CH bß W Q.J W f-4 ^ w c<3 en O 00 t- 00 CM "* 03 t- LTJ CM r-h CM ^ CD a Ü CÜ ""* CD vi< t- UJ 03 c~- GO CM 00 r ( CM IO C- c- CQ a) w Xi i-h. ß > > ; 00 CM ^ «tp c^ t*" l> en a> OS 03 OS LO Q _ 0) XI a CZ3 LO O CM d LO LO CM CD O LO CM O LO CM CD O

61 TABLE 4 REVERY TEMPERATURE OF CYLINDER MODELS. Stagnatin Temperature = 85 0 F. Mach N. = Cylinder Dia. Knudsen (in. ) N. Temp, f the Mdel 0 F

62 TABLE V EQUILIBRIUM TEMPERATURE AND DRAG OF A TRANSVERSE CYLINDER IN FREE MOLECULE FLOW S speed rati T w temp, f the cylinder Ti ^ 'mn free stream temp. mnatmic gas diatmic gas CDi ^Dj.^mn ^Di^dia C D ~ C D C D, drag cefficient due t incident mlecules drag cefficient due t reflected mlecules (mnatmic gas) drag cefficient due t reflected mlecules (diatmic gas) ttal drag cefficient Values were cmputed assuming that ^]\j = 6"rp = 1 (cmplete diffuse reflectin) and that there be n heat transfer frm the cylinder, (i. e, the cylinder is at equilibrium temperature) T 1 w "" T WequiL s KJ Tw equi - Ti. mn T w equi Ti c Di dia (c D ) ^r mn (C >. r dia

63 TABLE V cnt'd s rrt T w equi i w equ i C D- (C D r > m n T i - mn_ Ti i dia (C D L^J* >, dia Ill :

64 TABLE VI EQUILIBRIUM TEMPERATURE AND DRAG OF A SPHERE IN FREE MOLECULE FLOW ' S T w speed rati temp, f the sphere Ti mn ( >dia CDi free stream temp. mnatmic gas diatmic gas drag cefficient due t incident mlecules ( c D r ) mn drag cefficient due t reflected mlecules (mnatmic gas) C D " CDi + C D r (C DrW dra S cefficient due t reflected mlecules (diatmic gas) ttal drag cefficient Values were cmputed assuming that (5~ N = cr T = 1 (cmplete diffuse reflectin) and that there be n heat transfer frm the sphere (i. e., the sphere is at equilibrium temperature). T w T w equilibrium w equi T i J mn w equi dia D,- (C D ) (C n ) U r mn D r'dia

65 TABLE VI cnt'd Ti 'mn T w equi / dia c Dl ^Dr'mn '^r'dia , '

66 y Cylinder Mdel Mdel Supprt Base Platfrm Linear Variable Differential Transfrmer ioi ] Damping Vane FIG. 1 SCHEMATIC DIAGRAM OF BALANCE

67 EH g OH w > i I ( s H U PQ O u I ( ^ CM d

68 V ß CD +-> r-l C 0) A 01 ^H XI 0) - in rt Q > ^-1 F n3 B OQ >> u U IM T3 O nj W IH Ü ^HVWWVSr-^- a? H O H H I? W K W i t Q W Ö OJ <U S q > - i Ü > w g 1 ( UiUULfiJL^ SH 0) 0» u S 0 «H Ö cd H n r väää/n (H r-i 0 O O 1 1 ^ r-h O Oi 03 n p i > r-h UJ th O OJ XJ -1 Ü << l>> rh AjlÄÄflÄJ «K < an % % 9 ^.

69 0.1" h z 1.1" 0.020" O. D. L- 0. 8" J 3.6" 0.065" dia. T^.^TTi "h- FIG. 4 DETAILS OF CYLINDER MODEL. MODEL AND SUPPORTING MOUNT MADE OUT OF STAINLESS STEEL HYPODERMIC TUBING. MODEL SILVER boldered TO THE STING. DIAMETER OF THE MODEL VARIED FROM 0.008" t 0 180"

70 a T3 0) (D ä ä H 0) M T) (Ü a ai H O U O & a =3 I 0) i CQ O u U in T3 OJ (D Ö (s c? 0) U3 T3 (1).2 T3 In _L u a a q -H i l TH -if CD LO CM H J Q W W 121 Q W w H pcj < a w H «Q W S S OH «PH fe M O fc CQ O «w H W Q W Q I Ü

71 I 0.35^1*- 3.2" Material: " thick shim stck. 01 '0.05"R FIG. 6. DETAILS OF THE SHIELD.

72 Plugged end-v 0.008"dia. Orifice H J 0.049"0.D 0033" I.D 0109 OD. 0085"l.D. ^_ "dia ^ FIG. 7 GEOMETRY OF ORIFICE PROBE

73 CM II O H i < w <A Q> XL O C (O X < u. u. UJ z < a, H H 9 W g ^ n ACRO RATIN w w d E & 2 a < «C^J 0) Q W Q 3^ H j W S) u si CD I I ft,

74 z m OH N <; D Ö I h H O U Ü S Ü g 125 «W H W Q «O fa W > u H < m M j u <J u ft 00 6 Ö Ö saqdm '3- ö i NOISN31X3 r CVJ Ö Ö Ö 9NlddS a"

75 -^a, ^ Flw Shield NOZZLE ^1 Shield supprt balance FIG. 10 POSITION OF THE SHIELD WITH RESPECT TO THE CYLINDER MODEL DURING RUN. SHIELD WAS USED TO VER ONLY THE TRANSVEksE STING ( SIMILAR ARRANGEMENT USED FOR TAILSTTNr SUPPORTED SPHERE MODELS ). TAILSTI NG

76 J N SI 2.a GO O G w «w E OH W CQ H 2 H PH W D «H 00 W <1 03 w H g U g H» H fc S Q S W N O T3 u G n3 r i H W W u «O c^ Ü H i i U n H Q t i-i 0 H Ü w HI Ü 1 1

77 O 0) ü a 03 a H fc <J h D H «h w g «OT 0 «H «H ^ w H < Q M, < X! 02 U 9 ^ s H M Z. Q 5 3 P w ^ a ^w S «a. P 3 w OH P W OH a s H y U H ^ H^ p H «h W 5 w O B; Ü rt pa -j 0 s W

78 FIG. 13 MODEL LENGTH, inches irä^^^^^^^^ MOO EL AND SUPPORT

79 DIA- CYLINDER MODEL MODEL LENGTH, inches FIG N^Sr G p F M 0 0 R D C E E I. 0 L F E C N Y c L T, NI>RICAL MODEL AND SÜPrRT AS *

80 O r i ' I.. ln3ioijd30d O cvi OVdQ IO Ü

81 i E <U fe D e fl Vita 0>tf>.es 5. QL O? <u "> Q. Q. (75 X w P IM ^ g w LÜ en Q ^ s <: w M w H S K 0 OH Q Op a«p w U M üg p 5 in i i 3 l ln3ididd303 OVdQ

82 0 l ln3ioidd303 OVdd

83 f; = Pressure reading f the rifice prbe alne (rifice clse t ne end) P 2 = Pressure reading f the rifice prbe with added length a.- QCM FIG ANGLE OF ROTATION, Ö (degrees) A A r3s OF PRESSURE READINGS' WITH AND WITHOUT ^v ^Y CYLINDER ON TOP OF THE PROBE OBTAINED s^ltjzvz^t^ ORIFICE PROBEIN

84 N g CM K U H j I w u i h ü % J <: w «> < K H W pq O «H u '~, 0 ii. 1 6H suiiui '3MnsS3Ud iovdwi

85 IA E UJ < DC O OK* 1 70 V S 00647" DIA. CYLINDER MODEL y K^- 60 y 50 v^ i ^ y i* X s* >^ 40 X ^ r! _ 30. > ^ SHIELDy / MODEL?0 y ^ ' 10 M y i SHIELD ; 1 p 1111 NOZZLE 1 T Drag Balance 0. 0.i.; «J MODEL LENGTH, inches FIG. 20 DRAG FORCE OF A CYLINDER MODEL PLACED DOWNSTREAM OF THE SHIELDS AS A FUNCTION OF THE CYLINDER LENGTH.

86 inhiiddi vya

87 w iz; h Q H W W Q O I I H

88

89 PLATE 3 TYPICAL CYLINDER AND SPHERE MODELS 3

90 - A PLATE 4 TEST SECTION OF UTIA LOW-DENSITY WIND TUNNEL - WITH MACH-2 NOZZLE INSTALLED. A CYLINDRICAL MODEL MOUNTED ON THE DRAG BALANCE IS PLACED IN THE CENTER OF THE FLOW. THE VERTICAL POR- TION OF THE MOUNT IS SHIELDED. THE FLAT PLATE AT THE LOWER NOZZLE EDGE IS A BAFFLE, USED TO CUT DOWN THE CROSS FLOW IN THE BALANCE REGION

91 I I. PLATE 5 ARRANGEMENT OF PLATE 4 WITH THE CYLINDRICAL MODEL REPLACED BY A SPHERICAL MODEL.

92 % ~.' ( PLATE 6 I A SPHERICAL MODEL MOUNTED ON A CROSS-STKEAM SUPPORT FOR MEASURING THE TARE FORCE THE SUPPORTS ARE NOT SHIELDED

93 f t- 1 u H W Q i-h H w W ^ H W Q O E w Q PC? J < U g W n > P u <; I I W PQ S w u i-h M W Q iz; i i U D Q Q ^5 W H N g W O O fe OH m W W S W K w H H aw:.l^**- w < J

94 i PLATE 8 A CYLINDRICAL MODEL. TWO MOVEABLE SHIELDS ARE MOUNTED UPSTREAM OF THE MODEL.

95 n, m ^ ^ T3 "O C _ c rt.-3 a) rt s s? «d) O > H 0 ^ C- ^ O 0 3 Q, J= S IM DO a < z «0 ft w S P D " % MS s ^ -^ 2 a c u ; a OJ > H 0 5 t ^ C 3 O CD P St Iß 0 i J= a ^ a;.5 i 0 i S g 1 n «i (0 a ^ t? Ü <u u 1 I < 3 Q a: nj cs c = i 0 r CD tn L- 01 Tl 1H a (D 5 fc rd OJ -c 2 S S s«^ cd H ^ C " 3 c 0 d i» a t i 2 " a b!h OJ a QS < ^ ^ "" c 3 XJ U ^.3 'S 3 * U 0).5 C ä 0 5 M ^ M r: j T3 M u B -- Ifl a QJ -c ss E ä: C rt Q; JI O B «f 3 ^ < n 3 «c S a; -c c N a 3'" ^ -H * j & N 5 q K S ; G - z '5») a -C H S J t^ < ^ :- J. 0 4 a - in 1 < < a.n.s K ^ t fi ' t- c a «< ~" - ^ ^ -D _ T3 S C (U c «-g.g a ä * M >, E ^ «5.9 ^ -a -a >, - * «a, c ^ td z ii; s2 u 0 n, ^ u a? ä OJ OS as- i» < H Ü d cylinde were b in the 5 in the ed. ^ L, h u ^ I 'a«ü h ;3 5 a 2 «^ ß ^ -. 0 c M ^ a T: 3i S E C rt (u w tt) 0) 3 fe s g-i _ m Ci 03,- S,2 i' E S ^ * c - K «i -t -55 fe > n 5 0 2; J^ Iran he I as 0 t 6 TJ ; *" g<s OJ -ir? <C 2 ~ O 0 ^ a; cylin were in th 3 in t cd. lar air air t>er ver g -s 112 U N w = _ - 0 C S 0 0 J= S» «S 3^»> ^ -9.. B K ^ s :- ^ 0 ^ < r a a J 2 s ^ r S the 1 nu 1 fn hat sph Z^vup a (U f-< t, c T a S < s t