/\//lrAl'/^-lwl C^JtiLEADON NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1270 THE GAS KINETICS OF VERY HIGH FLIGHT SPEEDS By Eugen Sanger Translation of ZWB Forschungsbericht Nr. 972, May 1938 Washington May 1950 DOCUMENTS DEPARTMENT NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1270 THE GAS KINETICS OF VERT HIGH FLIGHT SPEEDS* By Eugen Sanger ABSTRACT In ordinary gas dynamics we use assumptions which also agree with kinetic theory of gases for small mean free paths of the air molecules. The air forces thus calculated have to be re— examined, if the mean free path is comparable with the dimensions of the moving body or even with its boundary layer. This case is very difficult to calculate. The con- ditions, however, are more simple, if the mean free path is large com- pared to the body length, so that the collisions of molecules with each other can be neglected compared to the collisions with the body surface. In order to study the influence of the large mean free path, calcu- lations are first carried out for the case of e3ctreme rarefaction. Furthermore, the calculations on this "completely ideal" gas will be carried out under consideration of Maxwell's velocity distribution and under the assumption of certain experimentally established reflection laws for the translational and nontranslational molecular degrees of freedom. The results thus obtained allow us to find, besides the air pressure forces perpendicular to the surface, also the friction stresses parallel to the surface. Their general results are calculated out for two practically important cases: for the thin smooth plate and for a projectile— shaped body moving axially. The mathematical part of the investigation was carried out primarily by Dr. Irene Bredt. *"Ga3kinetik sehr hoher Fluggeschwindigkeiten. " Forschungsbericht Nr. 972, May 31, 1938. MCA TM 1270 OUTLINE I. INTRODUCTION II. AZR FOECES ON THE FRONT SIDE OF A FLAT PLATE OBLIQUE TO THE AIR STREAM III. AIR FORCES ON THE FRONT SIDE OF A FLAT PLATE VERTICAL TO THE MR STREAM IV. AIR FORCES ON THE BACK SIDE OF A FLAT PLATE V. APPLICATION EXAMPLES VI. SUMMARY I. INTRODUCTION With the motion of todies at very great atmospheric heights, the air can no longer he considered a continuous medium, in the sense of flow theory. At over 50 kilometers altitude, the mean free path of the air molecules will he of the magnitude of houndary layer thickness and, at over 100 kilometers altitude, of the magnitude of the moving body itself. The mean free path at greater heights will be definitely greater than the body dimensions of the moving body, and the especially simple conditions of very rarefied "completely ideal" gases are valid where the effect of the collisions of the molecules with each other disappears compared to the effect of the collisions with the moving body. The air molecules collide then against the moving body as individual particles, independent of each other, and are reflected with a mechanism which deviates more or less from the known Newtonian principle of air drag, as shown by the results of the kinetics of very rarefied gases. The velocity of the body will be denoted by v (meters per second) in the following discussion. If we Imagine as ususlL that for the consideration of the flow process the body stands still and the medium moves, then v equals the uniform undisturbed flow velocity of all the air molecules. NACA TM 1270 The air molecules have also their ideal random thermal motion. The individual molecular thermal velocities are distributed com- pletely at random in all directions and are of completely arhitrary magnitudes, where the various absolute velocity magnitudes group around a most probable value, c (meters per second), according to Maxwell's distribution, which has the ratio yf2: J3" to the most applied gas kinetic value of "average molecular speed c"" (square root of the average of the squares of all the velocities present). According to the known Maxwell speed distribution law, letting p(kg s^/m^) equal the total molecular mass per unit volume, the mass dp of those molecules having velocities between c-j;- and c^ + dc^ is: dp _ h Cx P V7 c3 2 2 -72 Cx dc^ (1) If we consider only this mass dp of an otherwise motion- less (v = 0) gas, the molecules of which are moving with the particular speed c-^ in random directions, then the quantity dp 01 molecules striking per second on a unit surface of a motionless plane wall can be calculated, imaging that all velocities c-^- are plotted from the center of a sphere with radius Cx. The conditions of figure 1 are the result of an inclination angle ^ between the wall normal and the velocity direction under consideration, and of the molecular quan- tity dp — ^^ which passes through the striped area dT of the sphere surface . Then : ^'^ = rV c^ cos dP = ^ C^ NACA TM 1270 If, with the aid of Maxwell's equation, we include in our calcu- lation all molecules with the various possihle speeds c-^- between and 00, the total mass p of the molecules colliding per second will "be: ■ P = dp Cx=0 fr 3 — ^3 dc^ = Cx=0 PC 2\rn (2) as one can find in any textbook of gas kinetics. The pressure of the motionless gas against the stationary wall can be calculated similarly. The impulse di^ perpendicular to the wall with which molecules of particular speed c^ are striking the wall at an angle will be: di. dp ij-c^^rt 'n/2 c-^ cos^ AF dp 2 6 '^ 0=0 and the impulse of all molecules striking the wall: ^P = dp = ^ 2 P cx=0 3 \n? h "X ^=. = ,^ o2 Cv=0 (3) If we double this impulse value because of the elastic rebound always assumed for motionless gases, then we obtain the gas rest pressure £c2 P -2 or — c The value of the total impulse is also interesting, i.e., the sum of all single molecular impulses which strike per second on the unit area. NACA TM 1270 The totaJ. impulse of the molecular mass dp corresponding to a particiilar c^ will "be: J- J- iP 2 di = dpcx = Y ^^ It is thus 1.5 times greater than the effective impulse dip against the plate. The total impulse i is also greater in the same ratio. II. ATR FORCES ON TEE FRONT SIDE OF A FLAT PLATE OBLIQUE TO THE AIR STREAM If we consider again the mass dp of molecules with speeds between cx and Cx + dcx (almost equal) and if we examine its action on a flat plate in an air stream with an angle of attack a, then this process can be illustrated by figure 2, if we further assume that V sin a < Cx- The uniform velocity v of the individual molecule combines with the ideal random velocity (which can have any space direction) to give a resultant, the components of which are: perpendicular to the plate: v sin a + Cx cos parallel to the plate and to v cos a: v cos a + Cx sin cos "if parallel to the plate and perpendicular to V cos a: c^ sin sin \^ and for which the absolute value is therefore: w = y v^ + 2vcx (sin a cos + cos a sin cos \|f + c 2) From the sphere of all possible directions of c , a spherical sector with the half opening angle cos^f = v sin a/cx is excluded, in which the speed component v sin a + Cx cos is directed away from the plate, i.e., the molecviles of this range do not strike the plate. The integration over the velocity directions of all colliding molecules is not from 0=0 to ^ as with the motionless gas, but from 0=0 to 3t = ^. NACA TO 1270 The moleciilar mass colliding per second against the unit plate area with the selected speed c^r is therefore : dp = dp Uc/n 'lT->t (v sin a + c-^ cos i|')2cx it sin d0 (to dp / ^ . V sin^a -p-lcx + 2v sm a + ^ V c„ For V sin a > c-j;- the integration extends over the whole sphere from 0=0 to jt and the molecular mass colliding against the plate with selected speed c-j^. is dp = dp Ucx^rr rw 0=0 (v sin a + Cx cos 0)20^ n sin d0 = dpv sin a NACA TM 1270 Both, equations naturally give the same value for v sin a = c^. With the aid of Maxwell's distribution equation the total molecular mass colliding on the unit area per unit time within the speed range c^j- = to 00 will be : P = c.j^=v sm a K- ^x + 2v sin a + v^sin^a dp + 'V 3in a V sin a dp J^x=0 = _e_ "> 00 Cy=v sin a (; y^ 2vc 2 v^c ^ + 1- sin a + -^ -a)< - sin^aJe ^^ dc. ^^) »v sin a 2 VCy sin a e dc. cx=0 ? ? V sin a v sin a •V sm a _ V sm a c^ + ^ '- — - + 2 _ cx e c dc. Thus the number of the colliding molecules is known, and for cal- culation of the forces acting on the plate, we must now determine what impulse the molecular mass under consideration produces in the direc- tions in question. The impulse perpendicular to the plate, ip and the impulse parallel to it ix will be examined separately. 8 NACA TM 1270 We find for the impulse per second perpendicular to the plate J pit- Pi? (v sin CL + c^ cos 0)2dF 0=0 ^c 2 6 ^ /l + ^^ sin aV dp 2. If V sin a > c^: di = ( (v sin a + c^ cos (^)^dF P ), 2 ^^x « J0=O dp 2 6 ""^ (2 + 6 ^52^ — sin^a) This summation of the impulse components over all possible direc- tions yields the total impulse^ perpendicular to the plate^ of the air molecules striking the unit area per second with a speed c^ deter- mined by dp. NACA TM 1270 The further summation of the impulses over all speeds c^ with the aid of Maxwell's distribution equation results in the total impulse i acting per second perpendicular to the plate : ^P = 'V sin a c^=0 dp ^ 2(r 2+6 sin^al + cx2 ■^oo Cv=v sm a dp 2/-, V . \' -^ Cv (1+ sin ar 6 \ <=3C / .|P Pv sin a ), — ^i^ 1 "^ e c2 Cy=0 fn c3 (a . 6 iL ,i„2,) dc. nco Cy=v sm a 1 ^x k -2?5 ^ c3 II + — sm a)- 'dc. _P_ J v sin a /1 v2gin2a - i cv sin y/lFl ^ V3 3 a + |c2), 2 • ? 2 Ilpsin^a + I c2\ + ^v2sin2a + | cs') pv sin a _ ^ c2 dcx Cx=0 ] (5) If we set a = then equation (5) gives the impulse of the motion- less gas against the motionless wall, equation (3)- Also a = gives the impulse of the motionless gas, which will not "be influenced through uniform motion of the gas mass parallel to the flat plate. We find the impulse parallel to the plate in the direc- tion of V cos a in the following manner: The molecular "beam with a particular speed c^ and with a particu- lar direction (the latter determined by the inclination angle between the velocity c^ and the perpendicular to the plate, and by the angle T|f 10 WACA TM 1270 "between the projection c^ coa and the direction v cos a) gives the molecular mass colliding per second according to figure 2. dp = dp c„ sin d0 di^ kc/n rv sin a + c^ cos 0j The velocity component of this beam parallel to the plate is: V cos a + Cx sin cos >|r and the impulse of the beam with a particular c^, 0j and "^ will be ^ sin d0 di)f/ dp ^ — ^ (v sin a + Cx cos 4jf \ Vv cos a + Cx sin cos i^ j If one integrates over all ^ and 0, one obtains the impulse of the beam with a particular c^. For V Bin a < c-u-: '2jt Mr=Oj 'it-P€- 0=0 rv sin a+ Cx cos ) ("^ cos a+Cx sin cos ikjsin d0 d^ dp 2M v3 . 2 ^ v^ . , , -7- c-jr sm'^a cos a + 3 sm a coa a + — — cos a 6 ^ \2 cx3 cx2 2 c 3 I. and for v sin cl > c^ri ,. dp diT= 5— '2Tt \|fcO 0-0 rv sin a + Cx coa 0j(v cos a+Cx sin cos \|/jsln d0 d\(f = dp v^sin a cos a NACA TM 1270 11 The integration over all c-j;. with the aid of Maxwell's distribution equation gives finally the total impulse in the required direction parallel to the plate: 'V sm a Cx=0 poo dpv'^ sin a cos a + Cj(^v sm a dp 2/3 v-^ . 2 -7- c-v- ^ — =r sm'^a cos a 6 V2 ^^3 + 3 ■^— o sin a cos a + -^ -^ cos a 0^2 2 Ox = P 2 • ? 2 ^ vc cos a e (~ v^sin a cos a) 1 + — =■ 2 ^\ c\\ ^ ov sin a - liais£a c2 e dc. Cx=0 (6) We can start out from the impulse of the gas stream, given by equations (5) and (6), (perpendicular and parallel to the plate) in order to calculate the forces produced by the air on the front side of the plate oblique to the air stream, including the force perpendicular to the plate (normal pressure p) and the force parallel (friction stress r ) . For this calculation, we have to maJce some assumptions on the trans- fer of this impulse to the plate and on the change of the kinetic trans— lational energy of the molecules into other ener^ forms for which pre- sent gas kinetics furnish only a partial basis. If we first assume monatomic gases, so that inner degrees of free- dom for ener®^ absorbance do not exist, and further assume that the struck molecules of the wall are in such a temperature condition that they also cannot taJce over any energy from the colliding molecules. 12 NACA TM 1270 then the molecules have to leave the wall again with the same speed with which they arrived. The collision is thus completely elastic and we have only to derive the direction of reflection. Gas kinetics distinguishes two different possibilities for this: Mirror Reflection where the assumption is made (following Newton) that the angle of incidence and the angle of reflection are equal and "both beams are in the same perpendiciilar plane. Diffuse Reflection where it is assumed (following Knudsenl) that the reflection direction is not at all dependent on the direction of the impinging beam and is completely diffuse, i.e., that the colliding molecules first submerge in the wall surfaces, then after a finite time of "adherence" leave again, in a completely arbitrary direction. This last hypothesis is generally accepted in flow theory, where the adherence of the frictional boundary layer on the surface is explained by diffuse recoil of the molecules. In the case of a motionless gas (v = O) both assumptions lead to the same distribution of rebound molecules and thus to the same forces on the wall, as the striking molecules are in completely random direc- tions and this then is also true for the rebound molecules under both assumptions. In the case of a gas in motion, the two assumptions lead to very different air forces. With elastic "mirror" reflections, the impulse ij of the gas flow parallel to the wall stays unchanged. Shear stresses on the front side of the plate t^ are not transferred to the wall. The friction forces are zero. T V = (7) The impulse i of the impinging molecules normal to the wall is destroyed completely and an equal but opposite impulse is produced by -'-Knudsen, M. : Annalen der Physik, Vol. 28, p. 75, (1909); Vol. 28, p. Hk, (1909); Vol. 28, p. 999, (1909); Vol. 31. pp. 205, 633, (1910); Vol. 35, p. 389, (1911); Vol. 3^+, p. 593, (1911); Vol. k&, p. III3, (1915); Vol. 50, p. k72, (1916); and Vol. 83, p. 797, (1927). NACA TM 1270 13 the completely elastic recoil. The pressure on the wall caused by this process thus equals twice the impulse i.,: Py = 2ip (8) With diffuse-elastic reflection, the impulse i^ of the gas stream parallel to the wall will be given up entirely to the wall and the fric- tion stress equals i^ : \ = V (9) The wall normal impulse ip of the arriving molecules is destroyed again, whereby is created a partial pressure Pj^ = ip. The second part of the pressure, due to the diffuse— elastic recoil, has to be investigated more closely. To imitate the real process, an impulse value of the magnitude of the complete impulse i of the beam striking per unit plate area is distributed evenly on a hemisphere as if all gas particles started from the center of this hemisphere, and finally the resultant of this impulse distribution perpendicular to the plate is ascertained. The totaJ. impulse i of the arriving beam is derived, according to the proceeding impulse calculations, in the following manner: The molecular mass striking per unit time on the known area section df = c^2 sin d0 d\|; selected for a certain Cx, 0, and ^ is: dp = dpfcj; sin d0 di^ Ikc^ ^)\^ ^^^ a + c^ cos 0j The effective speed of this ray is w = i/v + 2vcx(3in a cos + cos a sin cos ^) + c^^ lU NACA TM 1270 and the impiilBe per aecond thus : ^- ^ sin d0 d\^/ dpw = dp f^ — ^ — -V V sin a + c-jj-cos 0j uv + 2Tc^fsin a cos + cos a sin cos ^j+ c^ This Impulse integrated over all Cy_, 0, and ^ will give finally the total impulse of the "beam. The actual carrying out of this integration is so difficult that the impulse will evaluate "by successive approxi- mation. It is started with the vector sum of the impulse resultants i and Ij , perpendicular and parallel to the plate. T2 .2 .2 ^ = ip + ^ This impulse resultant is smaller than the total impulse. It is found in connection with equation (3) that the total impulse of a gas at rest (v = O) is 1.5 times the impulse resultant. Total impulse and resultant are equal for uniformity flowing gas without heat motion (c^ = O). For conditions lying in hetween, we assume a constant relationship for the factor with which the impulse resultant has to be multiplied to obtain the total impulse i. For instance:^ 1 = 1.5 + ^ sin a + 1.5(5-)^ - 1 . ^ sin a . 1.5(J)2 This total impulse, according to our assumption, is now considered as the completely uniform impulse radiation per area unit in all directions outward from the surface. 2 Interpolation by Prof. Busemann, Braunschweig. NACA TM 1270 15 The "beam pressure pp perpendicular to the surface is then equal to i/2, as shown "by a simple integration over all normal components. Thus, the pressure vertical to the wall in the case of diffuse- elastic reflection is: p^ = i + i/2 (10) In order to get a first view on the numerous conditions of the air forces Just found, figure 3 shows the relationship between the pres- sure p or the shear force t and the dynamic pressiire q = £ v for a hypothetical atmosphere of monatomic hydrogen with t = 0° C tempera- ture (c = 212J+ m/s) and for flight speeds up to v = 8OOO m/s, for either mirrorlike or diffuse recoil. It is seen how different the air forces can be according to the assumptions made: mirrorlike or diffuse. In gas kinetics an attempt is made to approach the real conditions "by assuming that the reflection for a fraction f of all striking molecules is diffuse, while the remainder (l — f ) will be repelled mirrorlike. The fraction of diffuse reflections depends on the kind of striking molecules and particularly on the material, surface conditions, and temperature of the struck wall. According to numerous measurements^, the plate can be considered as completely rough under conditions usually prevailing in flight technique, i.e., the mirror reflected part (l — f ) is negligibly small, so the reflection will be almost completely diffuse. An experiment ELlly obtained dependence of f on the angle of attack, such that the reflection will be more mirrorlike with smaller angle of attack, is according to previous measiirements of flight relations, too insignificant to be considered. Knauer and Stern^ assume from optical analogies that the angle of attack at which mirror reflection begins is such that the surface roughness ^For example, Karl Jellinek, Lehrbuch der physikalischen Chemie, Vol. 1, p. 270, 1928. \hauer, F,, and Stem, 0.: Zs. f. Phys. "Vol. 50, pp. 766, 799 (1929). l6 NACA TM 1270 height, projected on the 'beanij must he smaller than De Broglie's wavelength X, of the molecular heam. With a wave length of 10~° cm and a roughness height of 10~5 to 10""", one ohtains for the angle of attack which is of interest sin a = X/h = 10-3^ i.e., an angle range of a few minutes, which is insignificant in flight technique. With regard to the reflection direction, we assume In the following analysis that f = 1, I.e., completely diffuse recoil. So far, for the reflection speed, perfect elasticity of the recoil was assumed, which means Individual recoil speed Is equal to the colliding speed. The struck wall will, in fact, be much colder than the gas molecule temperature after its submergence in the wall surface (and so after Its complete braking on the plate to the velocity v), so that we have to assume heat transmission to the wall molecules from the colliding molecule which remains a finite time in the plate surface. Figures k and 5 show the internal energies U for molecular nitrogen or hydrogen and their combination from the Individual degrees of freedom of the molecular motions as a function of the temperature, starting from an internal energy Uq of the gas at rest corresponding to a temperature of 0°C, with the other internal energy values equal to the kinetic energy corresponding to v. For this, the relation between U and v is stated as U = Uq + Av^/2g. The graph goes up to a U = 80OO kcal/kg, corres- ponding to a flight speed range up to v = 8OOO m/s. The specific heat at constant volume c^ was calculated under the usual assumptions on energy absorption by translation, rotation, and oscillation of the molecules (the latter according to Planck's formula) after the collision. We see from both figures that, especially for the N2, very high temperatures correspond to the high flight speeds. A complete tempera- ture equalization of the colliding molecule to the wall temperature would be equal to a total annihilation of the recoil speed, or an almost completely inelastic collision. It should be observed that the wall accommodates itself in a short time to the temperature of the colliding molecules, because of the very small heat capacity of the thin metal walls of the moving body. The molecule mass, colliding on the oblique ixnlt surface per second at very high flight speeds is, for example, about equal to pv sin a. NACA TM 1270 17 and thus the arriving energy E = Uppv ain a. With pp = 7 = .10~" kg/m^^ a = 7°j and v = 8OOO m/s the value of the energy hrought in is E = 7.8 kcal/m^ sec, almost independent of the composition of the atmos- phere. If the accommodation coefficient of the arriving gas molecules is one, then the wall would obtain this energy in the form of heat and this heat quantity should he given away by radiation, where a temperature increase AT of approximately 580° is necessary for "black body radia- tion, i.e., the plate stays in fact pretty cold compar-ed to the colliding molecules, and a very intensive, lasting energy delivery by the colliding molecules is out of the question. According to existing measurement s5, this temperature equalization is not 100 percent, however, an accommodation coefficient of 30 percent was found under certain conditions, i.e., the reflected gas mass contains still 70 percent of its internal energy U which it possessed at the moment of collision. The reflection velocity for a monatcmic atmosphere is established this way. The remaining internal energy of a molecular atmosphere will dis- tribute Itself quite differently over the existing degrees of freedom of the reflected molecule than assumed for the colliding energy, which con- sisted primarily of kinetic energy l/2 Av^/g, and only in small measiire of the internal energy U of the gas at rest, which latter was distri- buted evenly over all degrees of freedom. For a diatomic molecule with three translational and two rotational degrees of freedom, the individual shares of Uq, for an "average" velocity c", axe for each kilogram of gas 3/6 Ac^/g for the three translational degrees and 2/6 Ac^/g for the two rotational degrees of freedom of the molecules . On collision, all degrees of freedom will take part in the energy distribution change, and it can be assumed for further estimation of the diatomic molecule between the perfectly elastic and the perfectly inelastic collision conditions, for instance, that the total energy A/g[- v2 + 3. 52 ^ 2 jj2j ^ A/g[i "^^ "^ I ^^) d-istributes itself on the average evenly over all these five degrees of freedom. It can be taken from figures k and 5 that very high temperatures are associated with the high colliding molecular speeds at which another motional degree of freedom is excited, that of mutual molecular oscillation. Wien-BarmB, Handbuch der Experimentalphyslk, Vol. VTIl/2, p. 638, 1929. l8 NACA TM 1270 The known Boltzmarm^e equilization rule on the energies is not of value for these oscillational degrees of freedom. While the three translational and the two rotational degrees of freedom of a diatomic molecule have each the same energy admission Ux = - AET = - Ac^/g (kcal/kg) 2 6 that is together U, N = ^ AET = 5 AC^/g (trans. + rot,j 2 6 the energy admission of the oscillational degree of freedom Ug at low temperatures is practically zero and Increases at higher temperatxires according to Planck's equation: TT AE* 2 A _2 U„ = ; = — — C'^ e-' - 1 T(e*/^ - 1) approaching the limiting value ^ valid for high temperatures, of full exitation of the oscillational degree of freedom, Uf, = - AET = ^ :^ c^ ^2 b g In the last equation: A = the mechanical equivalent of heat, I/U27 kcal/kg E = the individual gas constant m/° T = the absolute temperature ^ $ = a temperature characteristic for each material, which is, for example, for nitrogen Ng = 3350°K, for hydrogen Hg = 6lOO^. The temperatures of the colliding molecules at high flight speeds are so high (according to figs, h and 5) that the molecular gas here NACA TM 1270 19 already diasociates strongly into Its atoms under normal equillbritun conditions . The transformation of the gases hydrogen and nitrogen, which are of importance in the higher atmospheric layers, into their monatomic, active modification belongs to the most energetic endothermic chemical processes which are known (H2 = 2H - 513OO kcal/kgj Ng = 2N - 605O kcal/kg), and the dissociation (if it actually occurs) would ahaorb extraordinary amoiints of energy and wo\ild make the collision almost completely Inelastic. So far it has not yet been proven by experiments that these molecules really dissociate on collision against a fixed wall at the speeds here considered. However the results of known tests with electrons colliding against the molecules of very rarefied Hg or Ng gases let us guess that the energy of the collision with a molecular speed up to V = 8000 m/a ia not sufficient to disturb the molecular bond. With the electrons colliding against Ng or H2 molecules dis- sociations are observed" only when the energy of the colliding electron was several times the dissociation energy of the struck molecule. This transferred on our case would yield colliding speeds of over V = 10,000 m/s for a nitrogen atmosphere or over v = 35,000 m/e for a hydrogen atmosphere, which lies outside of the range of our investigations . For the calculation of the forces on a plate oblique to the air- stream, we shall therefore not assume the dissociation of the colliding molecules. The degree of elasticity of the recoil will only be derived from the energy distribution of the wall molecules and of the proper rotational and oscillational degrees of freedom of the colliding molecule. This degree of elasticity, i.e., the ratio of the molecuiar reflection velocity, when energy division occurs, to the reflection velocity when rio energy division occurs, is estimated as follows: 6 Wien-Harms; Handbuch der Experimentalphysik, Vol. Till /I, pp. 10k, 706 (1929). NACA TM 1270 First, correepoiiding to the measured accomodation coefficient, )0 percent of the collision energy coiresponding to t he communicated .e wall molecules . The remaining collision energy. *M¥^^f^^^r^)-M¥'^^f^^) -d he dlstrihuted over the three traaslational degrees, the two ional degrees and the oscillational degree of freedom evenly and •ding to the exitation degree so that each translational degree of om has the following energy: Er H¥ ^ ^ I -=1 5 + 20 T(e*^ - 1) ondition of motion of the diffusely reflected molecules is the same sumed with equation (3), only there the energy content of a trans— nal degree of freedom was A/g 7- c^. Here the internal energy ging to the three translational degrees of freedom is given = 3Er and the translational speed is now: Jo.7 2 5 A \ T(e^/^ - 1) ad of: c. :fr FT ^ + c" when no energy distribution takes place en the wall and the translational degrees of freedom. (The latter comes from the energy balance: i.%2 = 1^2 2 ^i^') MCA TM 1270 21 The recoil Impulses should fall off approximately like speeds olitained from energetic considerations, so that: e = <¥ ^' * I =') if^^h") 5 +■ (e*/" - 1) i^'^h') (11) This degree of elasticity is drawn in figure 6 for nitrogen and hydrogen as a function of the flight speed v, using figure ii- or 5 for the relation "between v and T. The pressure vertical to the wall on the plate oblique to the air stream, in the case of diffuse— inelastic reflection, can he obtained from equation 11: Pv = ip+^2 Corresponding to figure 3, figure 7 shows the relation between the pressure p or the shear t and the dynamic pressure q for hydrogen (c = 1508 m/s) and nitrogen (c = ^4-06 m/s) for air angle of attack a = 4° and for flight speeds to v = 80OO m/s. In figure 7 the two impulse contributions due to collision (ip) and recoil (ei/2) are separated. 22 NACA TM 1270 The collision impulse could be given vithout objection from purely mechanical relationships. ■ The reflection impulse is estimated on the basis of a series of rather axbitraiy assumptions on direction and speed of the reflection. Further information, proceeding out of the conjectures presented here^ on the retention of single air molecules after collision with the wall with high speed should be found by experiment. Such tests may be Joined with the well— known molecular beam Investigations, where the usual beam speeds are to be increased greatly by correspondingly greater energy delivery to the molecules under investigation. One can thus obtain a type of wind tunnel investigation in which single molecules fly with extremely high velocities, and the effects of their collision with a solid body can be observed. With the help of very rapid molecular beams, a number of questions should be cleared up experimentally: how often a reflection of the molecules from the struck wall actually occurs, what factors change the completely elastic collision into a more or less inelastic collision through the transformation of translational energy into other forms of energy (i.e., rotational, osclllatlonal or dissociatlonal energy of the gas molecules or the wall molecules), and what direction law the final reflection follows, whether mirrorlike reflection, or predominantly diffuse reflection, or following a different law. These investigations can use to advantage De Broglie*s analogy between molecular beams and x— rays . III. AIR FOECES ON THE FRONT SIDE OF A FLAT PLATE VERTICAL TO THE AIR STREAM Vertical flow against the flat plate [a = — ) represents the limiting of oblique flow. The relations obtained in section 2 are preferably discussed here for this special case. NACA TM 1270 23 The total molecule masSj colliding per unit area per unit time ia given ty equation k: 1 k ijc-x- + 2v + — |dp + vdp |Cx=V lcx=0 _P_ ■> 00 Jc^^^ \c3 ^ c3 ""03/ Pt ^2(3 c , ^.-.2- e c dc. Cv=0 /, 1 c2 1x3 c^ V e c v^ 2 + 2 JfT ihe.) The complete integration of equation k{a.) (in contrast to equation k) is possihle "by series development, because one can put v/c»l for the high flight speeds under consideration, which was not possible for V sin a/c in equation k. 2k NACA TM 1270 The total tmpiilBe of this gas mass striking Tertically against the plate per unit time and imit area is given "by equation 5; t A^ ^ kJ Cx=0 2 = pY^ + — pC 1 2 P v-^ VC Y^ 2 "^ ^\3c 2 3 v2 If J as assumed at the start of the integraticn. v/c » 1, the terms _ II of the equation multiplied "by e c^ can he neglected compared with the first two tenns of the equation. For instance, the influence of the variety of ahsolute molecular velocities (following Maxwell's distribution), which is presented hy these higher terms, is less than O.3 percent of the value of the first two terms, when v/c is 2 or more. G?heref ore, for v/c > 2, the impulse can he calculated as if nil the molecules had the same velocity c; thus, considering only the first two terms of equation ^{a.) , i^ = pv2 + i pc2 (5h) Under the assumption of a completely inelastic collision, this Impulse is equal to the required pressure on the plate: p = i_ = pv^ + i pc2 (10a) NACA TM 1270 25 If it Is assumed that the air molecules have not lost their velocity after the collision vith the plate, but that they rebound perfectly elastically and "mlrror^like, " then the impulse given to the plate is simply doubled, and equation 10(a) for elastic mirrorlike collision is written: p = 2ip = 2/pv2 + I c^\ (lOb) The Influence of the random molecular motion, represented by the second term, is at 21 = 2 approximately 12.5 percent of the pure Newtonian air force, which is represented by the first term. However, the influence decreases, from ^ = 5 on^ to under 2 percent of the Newtonian value, contrary to the oblique conditions with small angles of attack, where the influence of the molecular velocity is still great even with high flight speeds. If it is assumed that the molecule reflection is completely elastic but diffuse, then the pressure on the plate decreases to a value: 2 V 1 + TT — sin 3 c »^Q^ p = ip + 2 = sp -^ °-'^5 1 + — sin c -lar 1 + TT — sin a + ( — ) = p(t + ^ c^;) 1 + 0.75 (10c) 2 2 V 3 / V \ 1 + — sin a + — { — ) c 2 \ c/ 26 NACA TM 1270 If finally the molecule reflection 'becoines not only diffuse "but also partly inelastic in the sense of equation 12, then the pressure on the plate reduces again to: p = i + e i = p[y^ + 1 c2J 1 + 0.75 5 + i^^^h'') 20 T(e*^ - J / 2 3 2\ 1 + sin 3 c -i^J 1 + — sin c » ^ Hzf (lOd) The graphs in figures 8 and 9 shov the pressure conditions according to equations 10(a) to 10(d) versus the dynamic presstire of nitrogen or hydrogen. For the calculation of the pressure of a motionless gas on the walls of the gas container which has the same temperature as the gas kinetics assumes completely elastic collisions for the normal ran^e of molecular speeds J i.e., the molecule keeps the same translational energy, on an average, as it had "before the shock. It is not important to Imow if the reflection is mirrorlike or diffuse, because both assumptions lead to the same molecular picture for the gas at rest. If the molecules are polyatomic then the distribution of the total energy to each possible degree of freedom is already uniform before the shock; and this uniform distribution need not change after the shock. In equation 10(a) (for completely inelastic collisions) for the air pressure against a plate vertical to the air stream, p = pv^ + £. c^ the first term pv^ corresponds to the dynamic pressure of molecules having no random motion against the plate (Newton), while the second term ^ pc^ corresponds exactly to the pressure of the static atmospheric air. MCA TM 1270 27 Howerer, this explaxiatlon of the Individual terms is cmly formally correct, since the resting air pressure is calculated assuming elastic molecular collisions. With the Inelastic collision, the decrease in the "stopping" pressure due to loss of the recoil Impulse will he off- set "by the mixed term in the square of the sum of the tvo speeds (t and c), in the velocity range under consideration. This condition can he recognized more clearly from equation 10(b) for the completely elastic collision p = 2pv2 + pc^ where after sub— 1 P traction of the static air pressure — pc there remains a pressure of 2pv + — pc^, which contains besides the Newtonian term, 2pv , 12 also an additional term of — pc , which reflects the effect of the mixed term. Wo resulting impulse is given parallel to the vertical, plate because of the symmetry of the total system, i.e., friction forces are transferred in the plate plane, but the sum of these forces outward is zero. IV". AIR FOECES ON THE BACK SIDE OF A FLAT PLATE If the mean free path of a molecule is small compared to the dimensions of an entity space into which the gas is flowing, then the flow— in speed can be greater than the most probable molecule speed. In the flow of diatomic gases into a complete vacuum, the directed velocity, Cj^g^^, of the total flowing mass can sturpass the probable molecule speed c by a factor of about I.87, according to the laws of gas dynamics. If, however, the molecular mean free path is comparable to the empty space dimensions or even greater than these as assumed here, then the number of molecule collisions behind the rapidly moving plate during the flow— in is not sufficient to produce the mentioned acceleration, and the molecuJ.es move with their usual speed c into the empty space behind the plate. According to figure 10, collisions between the air molecxiles and the back side of the plate cannot take place, i.e., the pressure against the "suction side" of the plate must have become zero, as soon as > c V = sm a 28 HACA TM 1270 This "border line is, however, strongly 'blurred because of Maxwell's distribution. The effectiTe forces against the hack side of the plate can he derived by the same process which led to equations h to 12, where now the uniform velocity v is directed away from the plate at the angle a, while before it was directed toward the plate (fig. ll). It is assumed first that v sin a % the resulting molecular velocity is directed away from the plate. Therefore no collision with the plate takes place. The integration over all colliding directions is extended from 0=0 to c-jj- there are no collisions with the plate, so that this case need not be treated. NACA TM 1270 29 The total molecule mass colliding on the unit area per luiit time is ohtalned with the aid of Maxwell's distritutlon equation for the velocity range c^ = v sin a to <» : Jcx=v sin a ■j-[cx — 2v sin a + v'^sln'2a/c.^\dp P fCj=T sin a 2 y^c — 2 — ^^ sin a + c3 c3 c3 ^X^ ^ TCx sin^ale dc. T^sin^a poo P c r- — Y sin a/c e dc. tfc^=T sin a P c ^ 2 v2sln2a r2 , 2/ ■" + 2v sin a/c ' V sin a i— e dc. |c^=0 (13) 30 NACA TM 1270 Similarly for the lnipulae verticai to the plate; ox dlT dp 4cx^rt ^ 9^=0 \Cjr COB (f — Y sin a] dJ" iP „ 2/ = -^ c-j;- 11 — 3 ::— sin a +• 3 — — a in a -^ sin 2 v3 ^) h- \ ^/l^ 3 ^sin Jcx= V sin a a + 3 — rs sin^a 3in3aJ dp cx cx3 y P T^sin^a V sin a Y^sin a + — cv sin a c ;e 3 3 2 ' h- 2 . 2 \l V^ 12 / c \ 2 't sin a — 0=0 dci (li^) WACA TM 1270 31 Finally, for the Impulse parallel to the plate; nM" iiy dp 4cx^n ^(=0 (cy_ COS (^ — y sin ajv cos a diF dp T f ^ v3 P \ Icy V COS a — 2Tn8iii a cos a + — sin a cos a; ■J 00 ix if Cy t COS a — 2v sin a cos a + — sin cx a cos a I dp U3x=v sin a _ V sin^a P i c c^ Y sin a ■—r Y COS aiTT e VJt \2 c e dc. Jc35-=T sin a nTT V cos a T^sin a — ::^~ t» T sin a J ■^T sin a 2 ""^Ldc -HI 2 °-^x 2 Cx=0 e — ■ (15) The Impulse resultant vertical to the wall as a consequence of the elastic-diffuse molecular recoil is from the total impulse i of the colliding molectiles: l.|Z3ina.(^f i . 1.5 £ ^^ ^i^2 ^ i^2 1 + ^ sin c -l(f)^ 32 NACA TM 1270 Equation (lO) again holds for the total pressure against the suction side of the plate -with elastic-diffuse recoil. (Translator's note: Formula missing in original German report.) While for the total shear stress on the suction side of the plate, equation 9 is used: T = i • r T The influence of a certain inelasticity of recoil can be estimated, particularly for small angles of attack hy the same procedure which led to equation (ll), according to which the degree of inelasticity can also he specified. For the total pressure against the suction side, equation (12) is valid: Pr = ip + ^1 Corresponding to figure 7, "the graphs in figure 12 show the relation "between the pressure p or the shear t and the dynamic pressure q for hydJTOgen and nitrogen at an angle of attack a = 4° and flight speeds up to v = 8000 m/s. V. APPUCATION EXAMPLES With the help of the previously mentioned relations, it is possihle to estimate all the air forces acting on the surfaces of a flying hody of any shape, which is moving at flight altitudes of over 100 km with speeds between about 2000 m/s and 8000 m/s, if definite assumptions are made on the composition of the air at this height. The air forces were differentiated into those which act perpendicular to the surface under observation (pressures), and those which act parallel to the surface (friction) . The pressure stresses as well as the shear stresses were found to be a function only of the angle of attack and the flight speed, for a particular gas. In figures 13 and ik is shown this dependence of the air forces on all possible angles of attack and on flight speeds between v = 2000 m/s and V = 8000 m/s for an atmosphere of molecular hydrogen. NACA TM 1270 33 It is to be noted in figure 13 that the air pressure vertical to the plate st2Pongly increases with increasing velocity, even vith an angle of attack a = 0, if the molecular recoil is diffuse. This representation can he used as a hasis for the calculation of air force coefficients for certain flight bodies in hydrogen, treating each flat surface section separately, with its ovn angle of attack, or, if the body surface is curved, analyzing it into a great number of small areas with individual angles of attack (flat areas or symmetrical cone areas), and then Investigating these. As the simplest example, the flat. Infinitely thin plate will be treated first . The Usual symbols are A lift W drag F wing surface and the air force coefficients are: ^ ;Pt Pr\ /Tv -"A ^ c„ = — = I Icos a — I — + — jsin a --(■ W /Pv Pr\ ^ f\ ^r\ c^ = — = I Isin a + I — + — Icos a gF \4 <1/ \tl 1/ and the glide ratio: Cw sin a ^k NACA TM 1270 In figure 15 are ira-wn the lift coefficients and in figure I6 the glide ratios of the flat, thin plate according to the above relations. On account of the extraordinarily great shear„ forces very had glide ratios result, which are appro xinately e = — =1.9 at 2000 m/s ■^a with the most favorahle angle of attack and which got worse at higher speeds, for example, at 80OO m/s, e = ahout 2.7. The most favorahle angles of attack are comparatively great at small speeds, i.e., at v = 2000 m/s, a = 25° approximately, and decrease with increasing speed to ahout 7° at v = 80OO m/s. Similarly to the infinitely thin plate, high speed profiles of finite thickness can also he calculated, i.e., wedge— shaped and lenticular airfoils. Their air force coefficients hardly deviate from those of the smooth plate, if they are of moderate thickness.. In general, the wings investigated here in the gas kinetics flow range "behave worse than in the gas dynamics range, where already the glide ratios are worse than in the usual aerodynamical flow region. The full effect of this unfortunate behavior will he corrected to some extent hy a flight technique such that at the high flight velocities under consideration, Inertial forces are developed hy the concave down- ward flight path, which support the wing. Figure 17 treats the question of how great the air drag is in the gas kinetics flow raxige for a hody of rotation (projectile form) moving axially, with an ogival nose of three calihers radius and cylindrical hody, and how far the air drag can he improved hy a truncated cone hevel at the end of the missile. These questions can he easily answered with the aid of figures 13 and lU if the ogive is divided into a large number of truncated cones, each of which represents a surface with a definite angle of attack. The extraordinary value of the drag coefficient is again striking; it can he traced to the very great friction forces in the extremely rarefied air. A noticeable Improvement of the drag coefficient could be obtained by beveling the end of the projectile; the improvement is about 7 per^ cent of the original value . Somewhat more tediously but in basically the same manner, the air forces on a projectile, airship, etc., at an oblique angle of attack can be determined, using figures I3 and 1^^-. NACA TM 1270 35 VT. SUMMART The air forces on "bodies of artitrEury shape are investigated when the hodles move with speeds of 2000 to 3000 m/s in such thin air that the mean free path of the air molecules is greater than the dimensions of the moving "body. The air pressure acting perpendicular to the tody surface, as well as the friction forces acting parallel to the surface, are derived with the aid of the calculation procedure of gas kinetics for surfaces facing hoth toward and away from the air stream at any angle. The air forces for an atmosphere of definite composition (molecular hydrogen) are calculated as a function of the flight velocity at all posaihle angles of attack of a surface and shown in graphs. Therehy the friction stresses "between air and "body surface prove to he of the same magnitude as the dynamic pressure and as the air pressures vertical to the hody surface, i.e., 300 times greater than in the aerodynamic flow range . The application of the general calculation results to particular technically Important cases, like thin airfoils and projectile shapes, results in extraordinarily high air drag coefficients and poor glide ratios even for the theoretically test wing sections Translated hy Bureau of Aeronautics Technical Information Navy Department 36 NACA TM 1270 dF=2Cv 7rsin d<^ Figure 1.- Velocity vectors of the thermal motion of molecules of a motionless gas and their position relative to a fixed boundary wall. NACA TM 1270 37 0) >— I —I cd •J3 a' o bo o ■1-1 ■<-> o a o o 13 o o •*-J o 0) > o o O) > I CD hJD .1-1 ft. 38 NACA TM 1270 2000 4000 6000 8000 Figure 3.- Air pressures p and shear stresses t on the front side of a flat plate at 4° angle of attack in an atmosphere of atomic hydrogen under the assumption of elastic diffuse or mirror-like recoil of the atoms from the vail. NACA TM 1270 39 4000 2000- 10000 20000 30000 Figure 4.- Colliding speed and associated internal energy of molecular nitrogen in relation to the colliding temperature of the gas. ko NACA TM 1270 8000 4000 2000 1000 2000 3000 Figure 5.- Colliding speed and associated internal energy of molecular hydrogen in relation to the colliding temperature of the gas. NACA TM 1270 41 1.0 0.8 0.6 0.4 0.2 ~~~~~ ^ 1 HYDROGEN NITROGEN 1 vCm/s] 2000 4000 6000 8000 Figure 6.- Degree of elasticity of recoil for nitrogen or hydrogen molecules from the struck wall. 1.0 0.9+^ 0.8 0.7 6 0.5- 4 0.3 0.2 0.1 p/q r/q ip/q 1 \ V\ 1 \\ \ \ \ ^ WipO sHgTiq 1 \\^ \ V xHaPq ^_ ' ^ ^ N2P:q V .^^ N2T:q ■ N"i-n . t^p^j. vtm/s] - ^ ■ 2000 4000 6000 8000 Figure 7.- Air pressure p and shear stress t on the front side of a flat, plate at 4° angle of attack in an atmosphere of molecular hydrogen or nitrogen under the assumption of diffiise and semielastic molecular recoil from the wall. k2 NACA TM 1270 9- 1 7 _ 1 \ i\ 1 \ V ELASTIC ; MIRROR LIKE COLLISION 4- \ ' -^ ELAST C DIFFU SE COLL ISION ■I _ A /^ tUAL CO LLISION - ^ INELAS AO TIC COL _ISION 2 1 _ 1 • vCm/s] >~ 2000 4000 6000 8000 Figure 8.- Air pressure p on the wall vertical to a stream of molecular hydrogen, under various assumptions on the collision process. NACA TM 1270 h3 a- '^ 4 ELASTIC MIRRQR-LIKE COLLISION E:LASTIC- DIFFUSE COLLISION ACTUAL COLLISION INELASTIC COLLISION 2000 4000 v[m/s] 6000 8000 Figure 9.- Air pressiire p on the wall vertical to a stream of molecular nitrogen, under various assumptions on the collision process. Figure 10.- Velocity vectors of the molecular motion for collision on the back side of the flat plate. kk NACA TM 1270 cj d •V Cl) -t-i rt • — 1 iX -t-i ol •♦H 0) 4:^ ■*-i MH (1) T1 •rH W ^ r> rrl J^ fli 4:; (D C rt G, a 01 u n -M (/I . — 1 -♦-> ■*-> +-> f1 •r-t s ^ r! Jh n Ol ^ 15 f) C) tl) (1 i — 1 n fi tl) Cl) 4U x^ .+-> -l-> MH rn u -t-J C) tl) > >> -t-> •(— ) i — 1 0) > ORIENTATION OF THEl PLATE IN THE AIRSTREAM •rH NACA TM 1270 i^5 2000 4000 6000 8000 Figure 12.- Air pressures p and shear stress t on the back side of a flat plate at 4° angle of attack in an atmosphere of molecular hydrogen or nitrogen, under the assumption of diffuse and semielastic molecular recoil from the wall. 46 NACA TM 1270 t.vj- i I 90° 80° ■^ *i 70° R ^q 60° Y 6.U~ K 50° Y ^= ■ — — __ d.O 40° \ ^ ---_ O- \, ^^v^ t .IJ 30° \ ^-- 1 •=;- \ 20° X ■"""—- — 1 n- 10° '~~~~~~~ ■ n «^- 0° \ *^ -~— __ . -50°\ -60° A -70°\\ -80°\\ -90°.\V -10° -20° \ -30° 1 vTn fl/s] ^ / T Figure 13.- Coefficient p/q of the air pressure vertical to the plate for all angles of attack and for flight speeds between 2000 m/s and 8000 m/s in atmosphere of molecular hydrogen. NACA TM 1270 1^7 1.5 1.0- 0.5. r/q ♦ 40" -+50' + 30' + 60' + 20' + 70' tlO" -70"-, 190°:^\ 0' + 80 -10' 40° -20° 2000 4000 6000 8000 Figure 14.- Coefficients r/q of the shear stress between air and plate for all angles of attack and for flight speeds between 2000 and 8000 m/s in an atmosphere of molecular hydrogen. 2000 4000 6000 8000 Figure 15.- Lift coefficients for the flat infinitely thin plate. k3 NACA TtA 1270 2000 4000 6000 8000 Figure 16.- Reciprocals of the glide ratios for the flat infinitely tMn plate, and best values of the glide ratio (dotted line) with corresponding angles of attack. NACA TM 1270 h9 2000 4000 6000 8000 Figure 17.- Coefficients of the pressure drag, friction drag, and total drag for a projectile -shaped body of rotation, with different missile bottoms. 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