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_RM A55I15
NACA RESEARCH MEMORANDUM A METHOD FOR SIMULATING THE ATMOSPHERIC ENTRY OF LONGRANGE BALLISTIC MISSILES By A. J. Eggers, Jr. Ames Aeronautical Laboratory Moffett Field, Calif. UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT GAINESVILLE, FL 326117011 USA NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WASHINGTON December 28, 1955 Declassified June 20, 1957 NACA RM A55H5 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS RESEARCH MEMORANDUM A METHOD FOR SIMULATING THE ATMOSPHERIC ENTRY OF LONGRANGE BALLISTIC MISSILES By A. J. Eggers, Jr. SUMMARY It is demonstrated with the aid of similitude arguments that a model launched from a hypervelocity gun upstream through a special supersonic nozzle should experience aerodynamic heating and resulting thermal stresses like those encountered by a longrange ballistic missile entering the earth's atmosphere. This demonstration hinges on the requirements that model and missile be geometrically similar and made of the same material, and that they have the same flight speed and Reynolds number (based on conditions just outside the boundary layer) at corresponding points in their trajectories. The hypervelocity gun provides the model with the required initial speed, while the nozzle scales the atmosphere, in terms of density variation, to provide the model with required speeds and Reynolds numbers over its entire trajectory. Since both the motion and aerodynamic heating of a missile tend to be simulated in the model tests, this combination of hypervelocity gun and supersonic nozzle is termed an atmospheric entry simulator. INTRODUCTION The aerodynamic heating of a longrange ballistic missile entering the earth's atmosphere poses problems of so serious a magnitude that the success or failure of the missile may well hinge on their solution. The extent to which these problems are solved can be fully determined only by flight tests, but such tests are, per se, very difficult, time consuming, and costly. It is appropriate to inquire, therefore, if a method can be devised for simulating with relatively simple equipment on the ground, the aerodynamic heating and resulting thermal stresses in a ballistic missile. This question forms the starting point of the present paper which undertakes first to establish conditions of simulation and then to determine a practical method of simulation. In the course of events it will be discovered that simulation of heating goes hand in hand with simulation of motion of a missile; accordingly, the subject of this paper is, more generally, an atmospheric entry simulator. NACA RM A55H5 SYMBOLS reference area for drag evaluation, sq ft rate constant (see eq. (Al)), sec" rate constant (see eq. (A2)), cu ft/, sec drag coefficient, dimensionless local skinfriction coefficient based on conditions just outside of boundary layer, dimensionless thermal conductivity, ftlb/sec ft2(R/ft) length of nozzle, ft flight Mach number, dimensionless mass of missile or model, slugs concentration of particles (i.e., number of particles per unit volume), ft"3 distance normal to boundary, ft reduced normal distance, n/D, dimensionless nozzle reservoir pressure, lb/sq ft total heat transferred, ftlb radius of curvature of body surface at stagnation point, ft equivalent skinfriction coefficient dimensionless typical model or missile dimension, ft Young's modulus, lb/sq ft heat transferred per unit area, ftlb/ft2 diffusivity, sq ft/sec NACA RM A55H5 3 5 surface area, sq ft T temperature, R T0 nozzle reservoir temperature, F t time, sec t reduced time, g, sec/sq ft D V velocity, ft/sec y' altitude, ft x,y,z rectangular coordinates, ft (except when appearing as subscripts on stresses) x,y,z reduced coordinates (x = x/D, y = y/D, z = z/D), dimensionless a coefficient of thermal expansion, ft/ftR (3 constant in densityaltitude relation, ft_1(see eq. (l)) 6 typical flow deflection angle. deg 0 angle of flight path with respect to horizontal, deg v Poisson's ratio, dimensionless p air density, slugs/cu ft pQ reference density, slugs/cu ft (reservoir density in nozzle) ax>ay>az tensile or compressive stresses, lb/sq ft (see sketch, p. 9) TXy,Txz,TyZ shear stresses, lb/sq ft (see sketch, p. 9, and note Txy = Tyx> Txz = Tzx> an(^ Tyz = Tzy) ,T), direction cosines of normal to a boundary, dimensionless Subscripts av average conditions e conditions at "entrance" to earth's atmosphere or simulator 1 local conditions just outside of boundary layer It NACA RM A55I15 mi missile mo model s stagnation conditions ANALYSIS In order to clarify the basic requirements of a simulator, this study is initiated with a review of some of the more important causes and effects of aerodynamic heating of longrange ballistic missiles. Following this review, the conditions of simulation will be set forth in a mathematical form designed to facilitate the choice of a practical simulator. It is a fundamental characteristic of ballistic missiles (see ref. l) that speed begets range, with the result that hypervelocities in excess of 10,000 feet per second are required in order to obtain long ranges in excess of 1,000 miles. A longrange ballistic missile first attains hypervelocities near the end of powered flight. In this phase of flight and throughout the large majority of unpowered flight the vehicle should pass more or less unimpeded through the rarefied upper atmosphere of the earth, corresponding to altitudes in excess of several hundred thousand feet.1 Its trajectory terminates, however, with a very rapid descent through all or part of the earth's relatively dense lower atmosphere. In this phase of flight, termed the atmospheric entry, retardation and severe aerodynamic heating of the missile can almost certainly be expected to occur (see ref. 2). Retardation during atmospheric entry is caused by the combined action of pressure and viscous forces, while aerodynamic heating stems in the main from work done by viscous forces. In both cases it is aerodynamic rather than gravity forces which play the predominant role, with the result that motion and heating of the missile emerge as closely related phenomena. Thus changes in missile shape which affect motion will also affect heating. This fact can, as discussed in reference 2, be exploited in the design of missiles with reduced aerodynamic heating. The potential for excessive heating remains, however, as an unavoidable property of the longrange ballistic missile which enters the earth's atmosphere at hypervelocity. 1It is presumed that the missile is, unlike the usual meteor, so large (say of the order of feet in typical dimension) that free molecule phenomena play a minor role in its motion and heating._ Motion and Heating of LongRange Ballistic Missiles NACA RM A55H5 5 Excessive heating can have several effects. First and perhaps most serious of these effects is the development of high thermal stresses in the structure of the missile. These stresses tend, for example, to far overshadow the pressure induced stresses. In addition, there is the natural weakening of the missile material at high temperatures. so that structural failure may occur during atmospheric entry. There is, of course, the further possibility that intense aerodynamic heating will, as with meteors, cause burning and the ultimate destruction of the missile. These, then, are some of the important causes and effects of aerodynamic heating of a longrange ballistic missile entering the atmosphere. They suggest that an atmospheric entry simulator might logically include means of simulating missile velocity, missile configuration (e.g., shape), and the lower portion of the earth's atmosphere. We will proceed from this suggestion to formulate our conditions of simulation. Conditions of Simulation The analysis to follow tacitly presumes the validity of many timetested assumptions of aerodynamics, thermodynamics, and solid mechanics. In addition, however, it is predicated on the assumptions that during atmospheric entry (l) radiation has a secondary effect on missile heating, (2) gravity has a secondary effect on missile motion, (3) the flowfield freeze principle of Oswatitsch (ref. 3) holds for the missile, and (h) the thermal properties (e.g., specific heats) and transport properties (i.e., thermal conductivity and viscosity) of air are functions of temperature only. The first two assumptions are suggested by the calculations of references 1 and 2,2 while the third assumption hinges essentially on the requirement that the square of the hypersonic similarity parameter for a missile be large compared to 1 during entry (i.e., M2sin25 > > l). For missiles of normal slenderness, the hyperspeed of entry tends to insure the satisfying of this requirement with the result that flight Mach number loses its significance as an important similarity parameter.3 The last assumption is restrictive only in the event air tends to dissociate 2The first assumption is, of course, best suited to "relatively light" missiles which are designed on the socalled "heat sink" principle, or, more generally, which are designed to maintain relatively cool surfaces. 3It is demonstrated in reference 3 that Mach number and stream angle in the disturbed flow are independent of flight Mach number, provided M2sin2& > > 1, and provided the air behaves ideally. In the event nonideal behavior, like changes in the specific heats, occurs as a result of high temperatures in the disturbed flow, then flight velocity tends to replace flight Mach number as the important index of motion since it is through this velocity (via kinetic energy) that high disturbed air temperatures are created. o NACA RM A55H5 (or possibly ionize) and it will therefore be treated in a discussion of these phenomena later in the paper (see section on "Performance Limitations"). Now it is convenient in discussing similitude to imagine a model counterpart to the missile and a test chamber counterpart to the atmosphere. Furthermore it is permissible for our purposes to proceed from the simplified equations of reference 2 for the convective heating of ballistic missiles. Thus for an isothermal atmosphere (which closely approximates the earth's lower atmosphere, see ref. 2) f = e'Py' (1) po and there follows: (a) Heat absorbed per unit mass at altitude y' Q l/Cf'S m k VCrA where ve2 v2) CDPoA Py' 2 P Bm sin 0 V2 = Ve2e 90 e (3) (b) Average rate of heat transfer per unit area 3 3 CpPpA _3y dHav Cf'P0Ve Py' 2 (3m sin 8P 6 { ' dt h e e (c) Rate of heat transfer to stagnation point Py' 3 CDppA _Py. ^ = 6.8xl0"6 /S Ve3e" 2 e" 2 Pm sin 0e 6 (5) dt v r e KJ> According to equation (l) the density of the air in the test chamber must vary exponentially with the distance corresponding to altitude in the atmosphere. The more general implication is, of course, that the test chamber must duplicate variations of p/pn in the atmosphere, whatever NACA RM A55H5 7 they may be, although the absolute magnitudes of p and p0 may be quite different from those in the atmosphere. The static temperature of the air in the test chamber is, as in the case of the atmosphere, presumed to be small by comparison to missile recovery temperature (see ref. 2 in connection with this point as it relates to the derivation of eqs. (2) through (5)) It will be stipulated now that model and missile be geometrically similar in structure and configuration, and made of the same material. Furthermore, the condition is imposed that the model enter the test chamber at the same speed and temperature as the missile enters the atmosphere. Finally, it is required that model and missile have the same Reynolds numbers (based on local conditions outside the boundary layer) at corresponding points By1 in their trajectories. By corresponding points it is meant where the product By1 is the same for model and missile. It should be recognized, of course, that, in general, B and y' will individually be grossly different for model and missile. It follows from these requirements and equation (2) that the heat transfer per unit mass Q/m to model and missile will be the same at corresponding points By' provided V is the same, since Cf' and Cf'S/CjA are the same. But from equation (3) the velocity V will be the same at corresponding By1 only if Cj)P0A/Bm sin 0e is the same. If the subscript mo refers to model and mi to missile, then the last provision may be written (pQD)mi AgA2 (y'/sin 0e) (poD)mo (y'/sin Ge) (6) But model pnd missile Reynolds numbers, velocities, and disturbed air temperatures4 are the same, hence (PoD)mi = (P0D)mo (7) 4 This observation with regard to disturbed air temperatures is easily verified by considering flow near the surface of a missile with a stagnation point. Assuming for simplicity that air in the disturbed flow behaves ideally, we have at the stagnation point Ts = V /2Cp since M2 > > 1. Then the temperature of the air just outside the boundary layer is given by the expression T^ = ( 1 + *^= Mj J Hence if V and M^ 2Cp \ 2 / are the same for model and missile, then T^ and are the same, independent of ambient air temperature. We are assured of equal M^'s by the freeze principle (see footnote 3) 8 NACA RM A55I15 and equation (6) may be written ?e\ = L = ( ye' ) (Anc.) (8) 2 sin 0e/mo Vsin 0e/mi \Ani, This expression fixes the length L of the test chamber in terms of the portion of the atmosphere to be simulated therein and the ratio of model to missile size.5 If equation (8) and the previously set forth requirements are satisfied, then model and missile should experience equal heat transfer per unit mass, and hence equal average temperature rise at corresponding points in their trajectories. These quantities are significant, of course, because they tend to determine whether a missile will melt or perhaps burn during flight. The next question is how do the heattransfer rates compare in the case of model and missile? It is easily deduced from equations (h) and (5) and the conditions for equal heat transfer per unit mass that cLHaA Ani ^ /mo Alio V dt /mi IV (9) and '^s\ Ani dt /mo Ano ^dt ' (10) at corresponding points Py'. That is to say, the average and stagnation point heattransfer rates are higher for the model in proportion to the ratio of missile to model size. But perhaps the foremost importance of heattransfer rates is, as discussed earlier, in how they influence thermal stresses in the missile structure and, for example, lead to ablation of surface material. Evidently, then, it would be most desirable if equations (9) and (10) implied equal thermal stresses in model and 5Since model and missile velocities are the same at corresponding points in their trajectories, it follows from equation (8) that the time of flight in the test chamber is reduced below that of atmospheric entry by the ratio (Dmo/Ani)2 NACA RM A55H5 9 missile. This possibility is easily checked using modified equilibrium thermalstress equations for an unrestrained isotropic elastic body (see ref. (k) and sketch) Z Y Stress notation Thus Sax + Sr^y + )txz aE St Sic dy dz l 2v dx dtty + ^TXy + ^TyZ ^T dy a\x clz 1 2v dy Saz Stxz dryz aE St ^ + + i= 0 dz dx dy 1 2v Sz 7 (ID where the stresses are not true thermal stresses, but rather they are the stresses produced by the "body forces, aE ST 1 2v bx , etc. 10 The corresponding compatibility conditions are 2 x w y aa / v pm V v + "TT =  V2T + 2 where 1 s2t aE ( V 1+ V 3x2 = 1 2v vlv 1 aE ( V 1+ V Sy2 1 2v [lv 1 b2i aE (v 1 + v 3z2 = 1 2V Vlv V2^ + Tzk = ( H^tVT + 2 p 1 S2^ aE S2T V2Tyz += 2 1+v dydz 1 2v dyc3z 2 1 c!2t aE c32T V2Txy + 1+v bxdz 1 2v Sxdj 1 S2^ ^ aE S2T 1+v bxby 1 2v dxdy b2 S2 dS V2 = o + p + dx2 dy2 Sz2 * = x + ay + az and the boundary conditions are u aET ^vT + Tyz^ + Txvi xyi oET 1 2v azt + Txzl + V11 aET 1 2v NACA RM A55H5 11 The heatflow equation is (see ref. (5)) v2t a2T a2T a2T Sx2 Sy2 dz2 i St k at (13) with boundary conditions and (probably) (Twal1) i = known \ waii' mitial k ar an k ^ k an outer surface inner surface r  dt = 0 > (Hi) Now the thermal stresses are given by superposing the "hydrostatic" pressure aET/l2v on solutions to equations (ll) consistent with the compatibility and boundary conditions (eqs. (12)) and the solution to the heatflow equation (13) along with its boundary conditions (eqs. 1*0 But if equations (9) and (10) along with the requirements necessary for their development are satisfied, then equations (ll) through (lM (and the "hydrostatic" pressure) are mathematically identical for model and missile. Hence, the thermal stresses must be the same in model and missile at corresponding By' in their trajectories. Necessary conditions for simulating aerodynamic heating and resulting thermal stresses in a ballistic missile are then, according to this analysis, as follows. First, model and missile must be geometrically similar and made of the same material. In addition, they must have the same flight speeds and Reynolds numbers (based on local conditions outside the boundary layer) at corresponding points in their trajectories. Finally, in order to meet these conditions and to insure equal heating of model and missile, the test chamber must contain air at relatively low temperature and with variations in p/pQ along the model flight path equal to those in the atmosphere along the missile flight path. With this knowledge we are in a position to consider the practical problems cf simulating atmospheric entry. 12 NACA RM A55I15 A PRACTICAL ATMOSPHERIC ENTRY SIMULATOR It is appropriate to determine first how to provide a model with the correct initial hypervelocity required for simulation. For this purpose it is suggested that a hypervelocity gun can be employed. A. CCharters and his group at Ames Laboratory (see ref. 6) have, for example, launched a 22caliber model from a hypervelocity (lightgas) gun at speeds up to 15j^00 feet per second. This speed is 60 percent of satellite speed and corresponds to that of a ballistic missile (see ref. l) with up to 2,000 miles range. Substantially higher speeds (corresponding to longer ranges) appear to be realizable with lightgas guns. Both speed and range fall, then, into the categories of interest in this paper. The next question is how to provide a model test chamber which simulates the lower portion of the earth's atmosphere. For this purpose it is suggested that a special supersonic nozzle can be used to advantage. To illustrate, it was found in reference 2 (see fig. h therein) that the major part of the aerodynamic heating of a ballistic missile entering the atmosphere occurs over a range of about 100,000 feet in altitude. In this altitude range p/pQ varies by a factor of about 10"2. A corresponding variation in density can be obtained between the settling chamber and exit section of a Mach number 5 supersonic nozzle (see ref. 7) Imagine, then, a hypervelocity gun positioned to launch a model upstream along the axis of such a nozzle. The nozzle is designed to provide an essentially exponential variation in density along its axis, the density decreasing from a very high value in the settling chamber to a very low value at the exit. Accordingly, a model proceeding upstream through the nozzle encounters an increasingly dense atmosphere like that presented by the earth to a descending ballistic missile. Now, to be sure, unlike atmospheric air, the air in the nozzle is in motion. However, the air velocity is small by comparison to the hypervelocity of the model, and therefore this difference between nozzle air and atmospheric air should mar only slightly the function of the nozzle as a test chamber.6 The combination of hypervelocity gun and supersonic nozzle merits attention, then, as an atmospheric entry simulator. An example simulator of this type will therefore be considered next. 6It is interesting to note that the supersonic nozzle when used in the proposed manner has compensating features; namely, as model velocity decreases, the air velocity decreases and, as model recovery temperature decreases, the air static temperature rises up toward atmospheric air temperature. These features are, of course, favorable to simulating atmospheric entry in that they tend to preserve the required similarity between atmospheric air and nozzle air In this manner, too, the possibility of correcting for small differences between these media is enhanced. NACA RM A55H5 13 Example Simulator Consistent with the previous discussion, a Mach number 5 nozzle is chosen to simulate the earth's atmosphere over 100,000 feet of altitude. Now let us assume that ballistic missiles with up to U,000 miles range are to be studied with this simulator. The corresponding range of atmospheric entrance angles 9e is from ^5 down to about 30 (see ref. l). The length of the nozzle is fixed according to equation (8) by the maximum values of (ye'/sin 0e)mi and D^/D^. This value of (ye'/sin Qe)m is, from the above specifications, '= 200,000 feet. The maximum value sin 30u of DmoAmi will be dictated by the size of the largest model which can be launched by the special gun available and by the size of the smallest missile to be simulated. For the purpose of this discussion it suffices to observe that the model size will probably be of the order of a fraction of an inch, while the missile size will probably be of the order of several feet. It follows that the maximum value of DmoAmi should be of the order of 10"2. In this event we have from equation (8) that the length of the nozzle is of the order of 20 feet, and from equation (7) the nozzle stagnation density is of the order of 100 times sea level air density. A missile 3 feet in diameter and weighing 5,000 pounds would thus be simulated by a model O.36 inch in diameter and weighing 0.005 pound. On the basis of these considerations our example simulator might appear something like the one shown in figure 1. The required nozzle stagnation densities are obtained for running times of the order of a second by using a settling chamber charged with high pressure air. A wide range of settlingchamber pressures and, hence, densities makes it possible to vary the range of altitudes simulated. The storage heater in the settling chamber maintains the air temperatures above those for which liquefaction can occur in the nozzle. The nozzle contour will tend to have very small slopes in the streamwise direction. Hence it is anticipated that flow in the nozzle will not depart radically from the onedimensional type. Air from the nozzle passes into a vacuum tank which is of sufficient size to maintain the nozzle compression ratios required for supersonic flow during the course of a test. Observation windows are located at short intervals along the nozzle side walls to permit photographing models in flight and determining effects of aerodynamic heating. In this regard it would quite likely prove desirable to employ a spectrograph (or spectrographs) to identify the sources of radiation energy emitted in the vicinity of models. The simulator would be instrumented for measuring settlingchamber pressure and temperature and, of course, the timedistance history of models. Models would be launched from a hypervelocity gun located at or near the end of the nozzle, and they would be arrested in a catcher located in the settling chamber. NACA RM A55I15 We have, then, some idea of how the proposed simulator might be employed in practice. It is important, however, to be aware of the points of difficulty which may limit the performance of the device as a s imulator. Performance Limitations First, it should be noted that pressureinduced stresses in the model will be higher than those in the missile by the ratio Ani/Ano" This point may prove troublesome, although not unduly so if these stresses are very low (as they tend to be) in the missile. It should be remembered too that by proper model design (e.g., the use of internal pressurization or other bracing) this problem can be minimized. Careful attention must, of course, be given to both model and sabot design from the standpoint of minimizing stresses in the model during launching. Another point of difficulty may be encountered if material properties (eg'> yield point) are a significant function of time under rapid heating conditions. There is some indication (see ref. 8 and papers cited therein) that at the very high heating rates of longrange ballistic missiles (corresponding to temperaturerise rates of the order of 100 F/sec and more), the importance of time is small. It is indicated too that this remains true, and, more important, that material properties remain essentially the same at the much higher heating rates produced in the simulator.7 Far more information is needed, however, before the significance of time in the sense of this discussion can be fully assessed. A further question which should be raised is whether or not a body can distort fast enough with changes in temperature to remain in stressstrain equilibrium (i.e., acceleration terms negligible) as was assumed in deriving equations (ll) and (12). This situation would be more serious in the case of the model than in the case of the missile. A crude check on the matter is easily obtained for a body which behaves elastically. To illustrate, the time required for the model to make small adjustments in stressstrain equilibrium should be of the same order as the time required for an elastic wave to travel the length of the model. For a typical steel model (D = l/3 in.) in the simulator, this latter time would be about 106 seconds. If we assume that the model experiences a total temperature rise of 1500 F while traveling through the simulator, then 10"6 seconds is also the time required for only about 1 temperature change in the model material. Accordingly, only small adjustments in stressstrain equilibrium are evidently required in this time, and hence This discussion should not be construed to mean that material properties are the same under conditions of high rates of heating as they are under steady state conditions (see ref. 8). NACA RM A55H5 15 equilibrium should tend to be realized in model as well as missile. If timedependent plastic deformation should become significant, then this statement obviously no longer holds.8 Thus, for example, the simulator may not (in view of its foreshortened time scale) duplicate more than qualitatively a fracture process (see ref. 9) although it should tend to duplicate thermal deformation up to and including the beginning of fracture. There is, too, the possibility of a missile being aerodynamically heated to temperatures where it will burn during descent through the earth's atmosphere. The simulator should tend to duplicate conditions leading up to this phenomenon; however, there is a question as to how well burning would be duplicated. The complexity and lack of complete understanding of metal burning (see, e.g., ref. 10) preclude the possibility of obtaining a quantitative answer to this question at the present time. From the qualitative viewpoint it is reasonable to expect that the effects of increased partial pressure of oxygen (acting to increase burning rate) and the reduced model size (acting to decrease the amount of material to be burned) should combine in the simulator to compensate for its foreshortened time scale, thereby more nearly providing simulation of missile burning. Burning should be accurately simulated for cases where burning rate is directly proportional to the partial pressure of oxygen. As a final point, it is appropriate to consider the matter of dissociation and association of the oxygen and nitrogen in air. The simulator produces essentially the same disturbed air temperatures as the missile entering the atmosphere. Accordingly, the potential for dissociation and association is duplicated by the simulator. At the present time, however, it is felt that these two phenomena obey different rate laws (see Appendix). In this event, neither the phenomena nor their effects on heating of a ballistic missile can be duplicated except by 1 to 1 simulation (i.e., the equivalent of flight tests with the fullscale missile). An indication of the possible error that this situation may introduce in tests with the proposed simulator can be obtained from the calculations of reference 11 which suggest that the net effect of equilibrium dissociation and association in free flight may be to increase the rate of heat transfer to a stagnation point by only about 50 percent.9 This increase should be well within the design safety factor of a missile. 8Thus with a slow process, like creep, model deformation would not simulate missile deformation (see, e.g., ref. 9), but rather, it would be less by the ratio of (Dmo/Ani)2 due "to the foreshortened time scale in the simulator. Even in the case of the missile, however, the entry time is so short (of the order of seconds) it seems unlikely that creep could play a major role in deforming the vehicle. g Dissociation and association may have especially marked effects on stagnation point heat transfer since these phenomena can strongly affect both the inviscid and viscous flows in this region._ 16 NACA RM A55H5 CONCLUDING REMARKS It has been found that an atmospheric entry simulator consisting of a hypervelocity gun combined with a special supersonic nozzle may be used to study the conditions of extreme heat transfer and thermal stress which introduce such serious problems in the flight of a longrange ballistic missile. The effects of aerodynamic heating on the model can be observed with relative ease, and further, the tests can be conducted at a cost which is negligible by comparison to that of flight tests. Indeed, in the simplest test, the simulator could provide with one photograph of a model rather substantial evidence as to whether or not the corresponding missile would remain essentially intact while traversing the atmosphere. Ames Aeronautical Laboratory National Advisory Committee for Aeronautics Moffett Field, Calif., Sept. 15, 1955 NACA RM A55I15 17 APPENDIX A DISSOCIATION AND ASSOCIATION RATES The dissociation of O2 and N2 is thought (see, e.g., refs. 12 and 13) to obey the linear law dN dt d(T)N (Al) where N is the concentration (or density in particles per unit volume) and Ci(T) is the rate constant which usually depends only upon temperature. On the other hand, the corresponding association process (e.g., 0 + 0> O2) is thought to obey the secondorder law1 ^ = C2(T)N2 (A2) dt Now disturbed air velocities are the same in the case of model and missile, while disturbed air densities are higher in the case of the model by the ratio of Dmi/jDmo* It follows then that the simulator will tend to duplicate a rate process in which the percentage rate of change of concentration of a given type of particle is proportional to the concentration of that particle, namely, = C(T)N (A3) N at But the simulator tends to duplicate temperature and type of particle; hence, it should duplicate the "association" rate process given by equation (A2) since C(t) = C2(t). It will apparently not, however, duplicate the "dissociation" rate process given by equation (Al). Actually there is some question as to the correctness of either equation (Al) or (A2) (see e.g., ref. (l^)) for pure media and there is the further complication of impurities which could lead, for example, to a thirdorder rate process. 18 NACA RM A55H5 REFERENCES 1. Eggers, A. J., Jr., Allen, H. Julian, and Neice, Stanford E.: A Comparative Analysis of the Performance of LongRange Hypervelocity Vehicles. NACA RM A5^L10, 1955 2. Allen, H. Julian, and Eggers, A. J., Jr.: A Study of the Motion and Aerodynamic Heating of Missiles Entering the Earth's Atmosphere at High Supersonic Speeds. NACA RM A53D28, 1953 3 Oswatitsch, Klaus: Similarity Laws for Hypersonic Flow. KTH Aero. TN 16, Royal Inst, of Tech., Division of Aeronautics, Stockholm, Sweden, 1950. h. Timoshenko, S., and Goodier, J. N.: Theory of Elasticity. McGrawHill Book Co., 1951, pp. h23k2k. 5. Carslaw, H. S., and Jaeger, J. C: Conduction of Heat in Solids. Oxford Univ. Press, 19^+8, p. 8. 6. Charters, A. C, Denardo, B. Pat, and Rossow, Vernon J.: Develop ment of HighVelocity FreeFlight Launcher The Ames LightGas Gun. NACA RM A55G11, 1955 7. Ames Research Staff: Equations, Tables, and Charts for Compressible Flow. NACA Rep. 1135, 1953 8. Heimerl, George J., and Inge, John E.: Tensile Properties of Some Sheet Materials Under RapidHeating Conditions. NACA RM L55E12b, 1955 9. Nadai, A.: Theory of Flow and Fracture of Solids. McGrawHill Book Co., 1950, pp. 395397 10. Coffin, Kenneth P.: Burning Times of Magnesium Ribbons in Various Atmospheres, NACA TN 3332, 195k. 11. Eggers, A. J., Jr., Hansen, C. Frederick, and Cunningham, Bernard E.: Theoretical and Experimental Investigation of the Effect of Yaw on Heat Transfer to Cylinders in Hypersonic Flow. NACA RM A55E02, 1955 12. Bethe, H. E., and Teller, E.: Deviations from Thermal Equilibrium in Shock Waves. BRL Rep. X117, Aberdeen Proving Grounds, 19^5 13. Krieger, F. J., and White, W. B.: The Composition and Thermodynamic Properties of Air at Temperatures from 500 to 8000 K and Pressures from 0.00001 to 100 Atmospheres. RAND Rep. R1^9, 19^9 1^. Prutton, Carl F., and Maron, Samuel H.: Fundamental Principles of Physical Chemistry. The MacMillan Co., 191+1+, pp. 6lO6ll. Figure I Example atmospheric entry simulator NACA RM A55I15 National Advisory Committee for Aeronautics. A METHOD FOR SIMULATING THE ATMOSPHERIC ENTRY OF LONGRANGE BALLISTIC MISSILES. A. J. Eggers, Jr. December 1955. 19p. diagr. (NACA RM A55I15) It is demonstrated with the aid of similitude arguments that a model fired from a hypervelocity gun upstream through a special supersonic nozzle should undergo motion and experience aerodynamic heating and resulting thermal stresses like those encountered by a longrange ballistic missile entering the earth's atmosphere. This combination of hypervelocity gun and supersonic nozzle is therefore termed an "Atmospheric Entry Simulator. Copies obtainable from NACA, Washington 1. Aerodynamics With Heat (1. 1.4) 2. Research Equipment and Techniques (9) I. Eggers, Alfred J., Jr. II. NACA RM A55I15 NACA RM A55I15 National Advisory Committee for Aeronautics. A METHOD FOR SIMULATING THE ATMOSPHERIC ENTRY OF LONGRANGE BALLISTIC MISSILES. A. J. Eggers, Jr. December 1955. 19p. diagr. (NACA RM A55I15) It is demonstrated with the aid of similitude arguments that a model fired from a hypervelocity gun upstream through a special supersonic nozzle should undergo motion and experience aerodynamic heating and resulting thermal stresses like those encountered by a longrange ballistic missile entering the earth's atmosphere. This combination of hypervelocity gun and supersonic nozzle is therefore termed an "Atmospheric Entry Simulator." Aerodynamics With Heat (1.1.4) Research Equipment and Techniques (9) Eggers, Alfred J., Jr. NACA RM A55I15 Copies obtainable from NACA, Washington NACA RM A55I15 National Advisory Committee for Aeronautics. A METHOD FOR SIMULATING THE ATMOSPHERIC ENTRY OF LONGRANGE BALLISTIC MISSILES. A. J. Eggers, Jr. December 1955. 19p. diagr. (NACA RM A55I15) It is demonstrated with the aid of similitude arguments that a model fired from a hypervelocity gun upstream through a special supersonic nozzle should undergo motion and experience aerodynamic heating and resulting thermal stresses like those encountered by a longrange ballistic missile entering the earth's atmosphere. This combination of hypervelocity gun and supersonic nozzle is therefore termed an "Atmospheric Entry Simulator. " Copies obtainable from NACA, Washington I. II. Aerodynamics With Heat (11.4) Research Equipment and Techniques (9) Eggers, Alfred J., Jr. NACA RM A55I15 NACA RM A55I15 National Advisory Committee for Aeronautics. A METHOD FOR SIMULATING THE ATMOSPHERIC ENTRY OF LONGRANGE BALLISTIC MISSILES. A. J. Eggers, Jr. December 1955. 19p. diagr. (NACA RM A55I15) It is demonstrated with the aid of similitude arguments that a model fired from a hypervelocity gun upstream through a special supersonic nozzle should undergo motion and experience aerodynamic heating and resulting thermal stresses like those encountered by a longrange ballistic missile entering the earth's atmosphere. This combination of hypervelocity gun and supersonic nozzle is therefore termed an "Atmospheric Entry Simulator. " Copies obtainable from NACA, Washington Aerodynamics With Heat (1.14) Research Equipment and Techniques (9) Eggers, Alfred J., Jr. NACA RM A55I15 naca NACA RM A55I15 National Advisory Committee for Aeronautics. A METHOD FOR SIMULATING THE ATMOSPHERIC ENTRY OF LONGRANGE BALLISTIC MISSILES. A. J. Eggers, Jr. December 1955. 19p. diagr. (NACA RM A55U5) It is demonstrated with the aid of similitude arguments that a model fired from a hypervelocity gun upstream through a special supersonic nozzle should undergo motion and experience aerodynamic heating and resulting thermal stresses like those encountered by a longrange ballistic missile entering the earth's atmosphere. This combination of hypervelocity gun and supersonic nozzle is therefore termed an "Atmospheric Entry Simulator. I. II. Aerodynamics With Heat (1.1.4) Research Equipment and Techniques (9) Eggers, Alfred J., Jr. NACA RM A55I15 Copies obtainable from NACA, Washington NACA RM A55I15 National Advisory Committee for Aeronautics. A METHOD FOR SIMULATING THE ATMOSPHERIC ENTRY OF LONGRANGE BALLISTIC MISSILES. A. J. Eggers, Jr. December 1955. 19p. diagr. (NACA RM A55I15) It is demonstrated with the aid of similitude arguments that a model fired from a hypervelocity gun upstream through a special supersonic nozzle should undergo motion and experience aerodynamic heating and resulting thermal stresses like those encountered by a longrange ballistic missile entering the earth's atmosphere. This combination of hypervelocity gun and supersonic nozzle is therefore termed an "Atmospheric Entry Simulator. " Copies obtainable from NACA, Washington Aerodynamics With Heat (114) Research Equipment and Techniques (9) Eggers, Alfred J., Jr. NACA RM A55I15 NACA RM A55I15 National Advisory Committee for Aeronautics. A METHOD FOR SIMULATING THE ATMOSPHERIC ENTRY OF LONGRANGE BALLISTIC MISSILES. A. J. Eggers, Jr. December 1955. 19p. diagr. (NACA RM A55I15) It is demonstrated with the aid of similitude arguments that a model fired from a hypervelocity gun upstream through a special supersonic nozzle should undergo motion and experience aerodynamic heating and resulting thermal stresses like those encountered by a longrange ballistic missile entering the earth's atmosphere. This combination of hypervelocity gun and supersonic nozzle is therefore termed an "Atmospheric Entry Simulator. " Copie obtainable from NACA, Washington 1. Aerodynamics With Heat (1.14) 2. Research Equipment and Techniques (9) I. Eggers, Alfred J., Jr. II. NACA RM A55I15 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY RO. BOX 117011 GAINESVILLE, FL 326117011 USA t I I A31262081066002A 