Problems in Motion of Compressible Fluids and Reaction Propulsion Citation Tsien, Hsue Shen (1939) Problems in Motion of Compressible Fluids and Reaction Propulsion. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2698-R671. https://resolver.caltech.edu/CaltechETD:etd-01122004-105646 Abstract none Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Aeronautics Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Awards: Caltech Distinguished Alumni Award, 1979 Thesis Availability: Public (worldwide access) Research Advisor(s): von Kármán, Theodore Group: GALCIT, Caltech Distinguished Alumni Award Thesis Committee: Unknown, Unknown Defense Date: 1 January 1938 Additional Information: Hsue Shen Tsien was the co-founding father of the Caltech Jet Propulsion Laboratory Record Number: CaltechETD:etd-01122004-105646 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-01122004-105646 DOI: 10.7907/2698-R671 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 133 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 12 Jan 2004 Last Modified: 28 Feb 2023 22:43 Thesis Files Preview PDF (Tsien_hs_1938.pdf) - Final Version See Usage Policy. 4MB Repository Staff Only: item control page

PROBLEMS IN MOTION OF COMPRESSIBLE FLUIDS AND REACTION PROPULSION Thesis by Hsue-sheu Tsisnu In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1938 ACKNOWLEDGEMENTS The author wants first to express his deep gratitude to Dr. Theodore von Karman. Not only the subjects for PART (I), (II) and (III) were suggested by Dr. von Karman but his constant guiding and help were also essential for carrying out these studies, His inspiration and his warm personality have won the author's highest respeot and love. The author also wants to thank Dr. Harry Bateman for his help in summing the series in PART (IV) and general discussions. During the preparation of PART (IV) the author was very much benefited by frequent discussions with Mr. Frenk J. Malina who was also very kind in helping the author in mumerical computations and figures in PART (IV). TABLE OF CONTENTS PART (I) Boundary Layer in Compressible Fluids .... Appendix to PART (I) cevcvccsccccccvvevecce References on PART (I) scvccceccccessecece PART (II) Supersonic Flow over an Inclined Body of REVOLUCION essonvccnvecrcevcevseressvevers References on PART (II) cscccceccevvcvecce PART (III) Application of Tschapligin's Transforme- tion to Two Dimensional Subsonic Flow eos. Appondix to PART (III) sssscosccscccevvece References on PART (III) cccccccevvcceceee PART (IV) Flight Analysis of a Sounding Rocket with Special Reference to Propulsion by Successive Impulses seccvescevsesvescccece References on PART (Iv) COP eeeeOsEF EH ESOS iii P age 1 2h 2 26 & 9h PART (1) BOUNDARY LAYER IN COMPRESSIBLE FLUIDS The solution of flow problems in which the density is variable is in general very difficult; hence, every case in which an exact or even an approximate solution of the equations of the motion of compressible fluids can be obtained has considerable theoretical interest. Several authors noticed that the theory of the laminar boundary layer can be extended to the case of compressible fluids moving with arbitrarily high velocities without encountering insurmountable mathematical difficulties. Busemamn (Ref. 1) established the equations and calculated the velocity profile for one speed ratio. (By speed ratio is understood the ratio of the sirspeed to the velocity of sound.) Franki(Ref. 2) also made an analysis of the same problem, however, it is complicated end depends on several arbitrary approximations. Von Karmen (Ref. 3) obtained a first approximation by a simple but apparently not sufficiently exact calculation. Hence, in Section (I), a better method for the solution of the problem is developed. The boundary layer theory for very high velocities is not without practical interest. First, the statement can be found often in technical and semi-technical literature on rockets and similer high-speed devices that the skin friction becomes more and more significant at high speeds. Of course, -l- it is krown that with increasing Reynolds Number, the skin friction coefficient is decreasing, i.e., the skin friction becomes relatively small in comparison with the drag produced by wave formation or direct shock. Since high-speed flicht will be performed mostly at high altitude where the air is of very low density, so that the kinematic viscosity is large, the resulting Reynolds Eumber will be relatively smell in spite of the high speed. Another interesting point in the theory of the boundary leyer in eompressible fluids is the thernodynamic aspect of the problem. In the case of low speeds the in- fluence of the heat produced in the boundary layer can be neglected both in the calculation of the drag and of the heat transferred to the wall. In the case of high speeds, however, the heat produced in the boundary layer is not negligible, but determines the direclion of heat Plow. In Section (II) a few simple examples of heat flow through the boundary layer are discussed. it hes been found necessary in most parts of this analysis to make the assumption of laminar flow. This assump- tion was found necessary because of the lamentable state of knowledge concerning the laws of turbulent flow of compressible fluids at high speeds. ‘his assumption is somewhat justified by the fact that - as mentioned above - in many problems where =~ the results of this paper can be applied, the Reynolds lumber is relatively small, so that a considerable portion of the boundary layer is probably, de facto, laminar. Ackeret (Ref ly) called attention to the possibility that the stability conditions in supersonic flow mizht be quite different from those occuring in flow with low velocities. Recently Kichemann. (Ref.5) studied the stability of a linear profile near a wall under small sinuous disturbance, and shown thet as the velocity ratio increases, the flow is unstable at increasing wave length of disturbance. However, he assumed that the gas is non-viscous and the sound velocity is constent. Both assumptions tend to limit the usefulness of the theory, especially the later one. Because the Constancy of velocity of gas implies the constancy of gas temperature, which unfortunately is far from the truth for a perfect gas as will be shown in later calculations of this section. In spite of these uncerteinties, some calculations of this section, as will be pointed out, are also applicable to turbulent flow. In other cases, as in the calculation of drag, the assumption of leminar flow surely gives at least the lower limit of its value. Section (1) If the X¥ -axis is taken along the plate in the Girecbiou of bie free stream, Lhe G \euxls perpendicular to the -3- plate, and & and Y indicate the AX and Y components of the velocity at any point, then the simplified equation of . : ; : ' ob . be gan motion in the boundary layer is , f-. <iatime Mur pug + py Be t= 3g (Ley) (1.1) where both the density; and the viscosit are variables. y y The equation of continuity in this case is 1.2 ig (P) t+ Hy £(py)=o . (1.2) A third equation determines the energy balance between the heat produced by viscous dissipation and the heat transferred by conduction and convection. With the same simplification as used in Eqs-(1.1 and (1.2), one can write 2 ? use (GT) +9059 (GT) = 37d) we (1.3) where Cp is the specific heat at constant pressure, and } is the coefficient of heat conduction. If Prendtl's nunber, Gm Id is assumed to be equal to 1, then it can be easily shown that both Eqs. (1.1) and (1.3) can be satisfied by equating the temperature T to a certain parabolic function of the velocity U& only. Indeed, introducing GJ= Hu) -),- into equation (1.3) and replecing Wy by Gy fle , one obtains } Hence Eq. (1.3) is reduced to Eq. (1.1) if f (a) = -f or pl =O + where C, , C are constants, Venoting the wall temperature { = 0) by Tay and remembering that T= 7, for f= U where eh = free stream velocity, C, C; can be expressed in terms 7 of Tor T and ro 7 -/ oe 1 Z-B-(B- fe + LIM (t-G) (1.1) vifferentiating Eo. (1./4) om obteins _L-/£L) _ Lf f-1 2 ELA OL WME YG). 09 where the subacript wr refers to conditions existing at the L surface of the plate. Now (34/24), is elways positive; therefore, if [p-/ yep’ > (e/a) -J heat is trensferred from the fluid to the wall; if [U-V/2] Me = Te/ ~ 1 there is no heat transfer between the fluid and the well; and a 27 foa~ [W-WefLl? < Taf 1 heat is transferred from the wall to the fluid. If there is ro heat transfer, the enerry content per unit mass (72) 4 G T is constant in the whole rersion of the boundary layer. (Ref.6) The pressure being constant the relation between Ff and TT is, he -6< The expression for the viscosity based on the kinetic theory of gases is tL f= fs & (1.7) However, the following formula is in closer agreement with experimental data 0.76 ~ b=! (= (1.78) Busemenn (Ref. 1) calculated the limiting case for which [G-Ve JI? = ( L./ Ee) —~ J using Eq. (1.7) and found that for a high Mach's number, the velocity profile is approximately linear. Von Karmen (Ref. 3), using the lineer velocity profile, the integral relation between the friction and the momentum, and Eq. (1.7) found that A Frictional drag per unit width of plate f ( £ U “/2 ) Xx length of plate _o[& ft pg? fae [17 FS The dimensions quantity -) shown in Table (1.1) is a x (1.8) function of Mach's mmber only. However, if Eq. (17a) is used, then -0.12 ne ee ee ae b 0 dy [11 a (1.88) ie) 1 2 5 10 ad M ©E 1.16 1.20 1.25 1.39 1.50 1.57 It is evident that this linear approxmiation is not satisfactory for small values of Mach's mmber. For /7=0 , the case is the same as the Blasius solution (Ref. 7) for in- compressible fluids for which 6) is 1.326. To solve the problem more rigorously, one has to resort to Eqs. (1.1) and (1.2). By introducing the streem function V which is defined by e,. Oe _ 8,2. 2 5 Oy %, 7 0X the equation of contimity, Eq. (1.2), is satisfied automatically. Now if in Eq. (1-1) ye is introdured as the independent variable as was done by von Mises (Ref. 8) in his simplification of the boundery leyer equation for incompressible fluids then oY $! Hp ; OX On 5, “ ap where )l is a coordinate measured in normal direction to -B«= V4 coordinate. Using these new coordinates, the following relations exist pu He FIV G or = Suge D4 _ uw 2b. fe 5y fEZE WP therefor Eq. (1.1) oan be written as ah 2 / SL. tt a ~ Fy la Mop (1.98) Eq. (1.9a) can be put into non-dimensional form by introducing the following set of new quantities: u*= u/U = #/L (1.90). w= (P/UL Sul fe, = (YUL Wi i 3/8. WO = bi ft where L is a convenient length, say length of the plate, end /7 is the corresponding Reynold's Number, then Eq. (1.9a) becomes 24 a ( ust its) (1.9) -9- | Eq. (1.9) can be further simplified by introducing a new dependent variable : = yt v n® , then ae aan f (ut stut $F) This can be seeot by the method of successive (1.10) approximations. As gt and pw are functions of temperature only as shown in Eqs. (1.6 and (1.7) or (1.78) and the temper- ‘ature is oa function of u* then by starting with the known Blasius' solution (Ref. 7) the right-hand side of Eq. (1.10) can be expressed in terms of C « Therefore, one can write u #e “ut = HE) end Eq. (1. 10) booomes -$ - Lif] Consequently, the solution of Bge (1.10) is 4 '¢ wa _f £4 y* = CLF a where f= at 7 (1.11) and C is determined by the boundary condition that at ¢ = 0° ur / + = [4 (1.118) In the actual computation, two methods of evaluating the integrals in Eqs. (1.11) and (1.1la) are used. For small values of ‘4 ’ C¢<02 , the function | an and S16) ere expanded in a power series of C . Due to the uniform convergence of the power series for sufficiently -10- small values of CG , the integration is carried out term by term. For values of ¢ >0.2 , numerical integration is used. | A second approximation can be made based upon the value of ur obtained from Eq. (1.11). It has been found in the cases investigated that the third and fourth approxima- tion gives sufficient accuracy, if the velocity profile of next smaller Mach's mumber is used as the starting point of calculat- ing the velocity profile of next larger Mach's number. Having computed the final H # » the Y corresponding to u* cen be calculated in the following wey: It is known from Eq. (1.9b) that S = re a Then remembering the definition of Vg » one has 9 _ AR aye fa e* uX However due to the small slope of stream lines, a de -ll- Hence ne | Ay* = a oH (1.12a) e & 1 Pde [ta (1.12) ite Here the expansion of g ¥ in a power series for mall values of C is especially useful due to the singularity of integrand at C=0 . The skin friction scan be computed by using momentum Drag = /) = ff salU-eyp lew, LeOey Using Eqe (1212a), on has ee L dy* Wee ut thus D = [es UE / cde AL -12- But n* = aX therefore Ly Ly Hence Thus the skin friction coefficient can be computed as p D 2/ (1- Ww dg — ee “oO = ; = f Syb /R (1.13) The velocity profile, the temperature distribution, and the frictional drag coefficient are calenletad for different values of the Mach's number of the free streem, for the case 2 [Q-vp2 Ji? =(Te/h)- 7 using the approximate viscosity relation of Eq. (1.7a). The results are shown in Figs. 1.2 and 1.3. The velocity profiles for high speeds are very nearly linear, but it can be seen that the wall temperature for greater Mach's numbers is very high. If the free stream temperature is 0°F., then the wall tempera- ture will be 1600°F., 3620°F., 6540°F., and 10,170°F. for -13- Machts numbers of , 6, 8, and 10, roepectively. Therefore, there is no doubt that the law of viscosity as expressed by Eq. (1.78) will not hold. Also at such high temperatures, the heat transfer due to radiation cannot be neglected. The effect of radiation will be the equalization of gas temperature. In the extreme case of complete equalization, the temperature will be constant throughout the layer and due to the assumed constant pressure throughout the field, the density and viscosity of gas will be also constant throuhgout the field. Then the velocity profile will be again that calculated by Blasius for incompressible fluid. By this reasoning, the actual velocity profile for large Mach's number when radiation cannot be neglected is something between the Blasius profile end that shown in Fig. 3. The change in the constant , JA is appreciable, but not great. It decreases from 1.328 for Lf =O to 0.975 for JI = 70 » or about 30 percent. However, for Ok Ld < 3 the change of the constant is very small. Fig. 1.2 also shows that Eq. (1.8a) which was obtained by using the linear approximation is fairly accurate for very high Mach'ts numbers. As examples, consider first a projectile and second, a wingless sounding rocket. Taking the diameter of -1h- the projectile to be 6 in., the length 2 in., the velocity 1500 ft./sec. and the altitude 32,800 ft. (10 km.), then the Reynold's Number based on the total length is 7.86 x 10° and the speed ratio is 1.52. From Fig 1.2 the skin friction coefficient is G, = (1.286 x 107)/ 7.86=0,000),59 Changing the skin friction coefficient (based on the skin area) to the drag coefficient (based on the maximm cross- _ section), on: obtains Cy, = 040055 The drag coefficient due to wave formation taken from Kent's experiments (Ref. 9) is Ly = 0.190 Therefore the ratio of skin friction to wave resistance is 0.0055/0.190 = 0.029. However, the ratio is greatly changed in the case of the rocket. Taking the diameter of the rocket to be 9 in., the length 8 ft., and the altitude of flight 50 km. (see Appendix) (164,000 ft.), the velocity 3,00 ft./sec., then the Reynold's Number based on a density ratio at that altitude of 0.00067 and temperature 25°C. (deduced from data on meteors) is 6.1, x 10”, and the speed ratio is 3.00. From Fig. 1.2, the skin friction coefficient is ( = (1.213 x 10° )//rish = 0.00360 Then The drag coefficient due to wave formation from Kent's experiment (Ref. 9) is C, = ~~ 0.100 lyr Therefore, the ratio of skin friction and wave resistance is now 0.123/0.100 = 1.23. If the boundary layer ia partly turbulent, the ratio will be even greater. This shows clearly the importance of skin friction in the oase of a slender body moving with high speed in extremely rarified air. It also disproves the belief that wave resistance would always he the predominating part in the totel drag of a body moving with a velocity higher than that of sound. The reason under- lying this fact can be easily understood when one recalls that the wave resistance of a body is approximately directly pro- portional to the velocity, while the skin friction is proportional to the velocity raised to a power between 1.5 and 2. Therefore, the ratic of skin friction to wave resistance increases with the speed. With very high velocities and high kinematic viscosity, the wave resistance may even be a negligible portion of the total drag of the body. Section (II) In order to point out the thermodynamic aspect of the problem two cases will be considered; the flow of a hot fluid along a surface which is kept at a constant temperature inferior to that of the fluid, and the case of a hot wall -16- cooled by a fluid of lower temperature. The problems treated in this part have been discussed before in two very interesting papers by L. Crocco (Ref. 6). He especially gives an elegant treatment of the cooling problem in the case of very high velocities ("Hyperaviation"). The author feel sthat his treat- ment is somewhat more general and extended than Crocco's previous analysis. An interesting general relation between the heat transferred through the wall and the frictional drag can be obtained using the assumption that Prandti's number, i.e., the ratio Gh d » is equal to unity. The same assumption was used also in the previous calculations. It is renarkable thet the relation holds also as well for laminar as for turbulent flow. The heat flow g per unit time and unit area of the wall surface is $= dy, (AT/ AY), and the frictional drag ZT per unit area is P= My, (04/24), Using Eq. (1.4) the ratio 4 / Zz can be calculated from the relation G _ L[i7-#), $ °F Al (-F a Eu] (1.1) where 7, is the absolute temperature, and (/ the velocity of the fluid in the free stream, 7, the absolute -17- temperature at the wall, Ay and Ay, are the heat con- duction and viscosity coefficients of the fluid corresponding to the wall temperature and Ld denotes Machts number, - Substituting J] =0 one obtains from Eq. (1.1/4) tL tw he 6fh-%) T My U F(A) This is the relation known as Prandtl's or G. I. Taylor's formula, first discovered by 0. Reynolds. Hence Eq. (1.1):) gives the correction of this result for compressibility effects, In the case Ip 2 Tor i.@., when the wall is colder than the free etream, the effect of compressibility is to increase the heat transferred through the wall. However, it would be erroneous to interpret this result as an "improve- ment” in cooling because at high speed the heat produced in the boundary layer is of the same order as the heat transferred through the wall. In order to determine the efficiency of the cooling a complete heat balance mist be made. For this purpose Eq. (1.11) does not give sufficient information and the velocity and the temperature distribution in the boundary layer must be - computed. Such calculations were carried out for the particular assumption bw = To A i.e., for the particular case in which the absolute temperature of the wall is kept constant -18- at a value equal to one-fourth of the temperature of the hot fluid. With the same assumption for the variation of fe as in Section (I), the results shown in Pig. 1.2 and Pig. 1.), were obtained, The variation of C Je with /7 is similar to that obtained in the case without heat conduction through the wall. Also tho highost tomporature in the boundary layer is very high for extreme Mach numbers, However, the temperature maximum occurs some distance from the wall, The heat transferred from the boundary layer to the wall can be calculated as follows: By means of Eq. (1.12), one has “4 f C= = GL) gy Hence L (Vl e Sar May | y# det / [he Sil WL fol He Lott 2 - 4 fe fae gt fed -19= Therefore, combining the above two equations, (1.16) Using Eq. (le7a) and substituting Eq. (1.16) into Eq. (1.5), then AT) Kk aR Oth, ( 2L,/2* (1.17) — where K = (4° 075+ & WR » os given in Table 1.2. Therefore, the heat transferred to a strip of unit width of the wall of length LL per unit time is equal to 2 2D) oy x KAutd® pide y J, (37), dy 2 aE | Now the increase in heat content of the gas per unit time by flowing from X=27 to Y= / can be caloulated as ~20~ fon) LY ee [ / / — 7 a Af SG (Te Tf = BURTIEE (0-7 =A TAR Cr Tle = (GVO [Bf 6-7 (1.19) gl Tl fl 7 Ue = BA, TLE The viscous dissipation of gas in the boundary layer of unit width plate per unit time is thus Q = Q, t a, (1.20) Table 1,2 4 ial e 0 1.53 1 1.93 2 3.12 5 10.53 10 33.98 The total heat balance at different Mach's numbers is shown in Fig. 1.5. The “dissipation” curve represents in dimensionless form the heat produced by friction per unit time and unit width of the plate. The lower curve shows the increase (or decrease) of the heat content per unit time and unit width. The difference of the ordinates corresponds to the heat trans- ferred through the wall, It is seen that cooling takes place for I. < 2.6. Beyond this limit more heat is produced by friction than the amount which can be transferred to the wall end, as a matter of fact, the fluid is heated. In the case Ty 7 T, i.e., when the wall is hotter than the free stream, the ratio between the heat transfer and the drag decreases with increasing Mach's mamber, This is shown in Fig. 1.6 where the ordinate represents the ratio between 4/T with compressibility effect (according to Eq. (1.1))) to 4./T without compressibility | effect (according) to Eq. (1.15)). The calculation was carried out for a gas temperature of - 55° F. and a wall temperature of 180° F, end 300°F, It is seen thet there is no cooling in the former case for 47 > 1,69 and in the latter case for JY > 2.08. However, the decrease of cooling efficiency is appreciable even at mich lower speeds. This emphasizes the benefit of the reduction of the speed of cooling air and the relatively poor efficiency of cooling surfaces exposed directly Pee to a high-speed airstream. The curves in Fig. 1.6 being derived from Eq. (1.14) apply to laminar as well as to turbulent motion. “230 APPENDIX TO PART (I) ON THE VALIDITY OF THEORY IN VERY RAREFIED AIR The hydrodynamic equation holds so long as the mean free path of the molecules is small in comparison with the thickness of the boundary layer. For this case the thickness of the boundary layer is zero at the nose, however, at a distance ¢ of the length of the rocket it already amounts to 3.2 ome, while the calculated mean free path of the air molecules at the altitude considered is about 1.1 x 10 om, Hence it appears that even for this case the theory can be safely applied. This conclusion is substantiated by the experimental results of H. Ebert in "Darstellung der Stromungsvorgange von Gasen bei ndiari gen Drucken mittels Reynoldsscher Zehlen", Zeitschrift fur Physik, Bde 85, S. 561-564, 1933. le 2. Je he De ory Te 8. 9s REFERENCES ON PaRT (1) Busemann, A., Gas-stromung mit laminaren Grenzschicht entlang einer Platte, Z.AM.M., Vol. 15, S. 23, 1935. Frankl, Laminar Boundary Layer of Compressible Fluids, Trans. of the Joukowsky Central Aero-Hydrodynamical Institute, Moscow, 193), (Russian). Karmen, Th. von, The Problem of Resistance in Compressible Fluids, V. Convengo della Fondazione Alessandro Volta (Tema: Le Alte Velocita in Aviazione), Reale Accademia DiItealia, Rome, Ackeret, J., Uber Luftkreft bei sehr grossen Geschwindig- keiten inshesondere bei ebenen Stromungen, Helvetica Physica Acta, Vol. 1, S. 301-322, (1928). Kuchemann, De, Storungsbewegungen in einer Gasstrommg mit Grenzschicht, Z2.4.M.M., Vol. 18, S. 207-222, (1938). Crocco, L., Su di un valore massimo del coefficient di transmissione del calore da uma lamina piana a un fluido scorrente, Rendiconti R. Accademia dei Lincei, Vol. 1h, fase. 90-96, 1931. Creceo, L., Sulla Transmissione del calore da una lemina piana un fluido scorrente ad alta velocita, L'tAerotecnica, Vol. 12, fase. 181-197, 1932. Blasius, H., Grenzschichten in Flussigkeiten mit kleiner Reibung, Zeit. F. Math. u. Phys., Bd. 56, S. 1, 1908. von Mises, Bemerkung zur Hydrodynamik, Z.AM.M., Vol. 7, S. 25, 1927. Kent, R. H., The Role of Model Experiment in Pro jeotile Design, Mechanical Engineering, Vol. 54, pages 61-4)6, 1932. -258 PART (II) SUPERSONIC FLOW OVER AN INCLINED BODY OF REVOLUTION The aerodynamic forces acting on a projectile can be divided into three parts: the resistance or drag in the direction of the axis of the body, the lift in the direction perpendicular to the axis of the body, and the forces due to the rotation of the hody (Magmis affect). Tha first somponent, the resistance, is, of course, the most important one, because it is the predominating factor in determining the range of the projectile. However, in the oase of an actual projectile, inclination end rotation are always present, and therefore, accurate calculation of range is impossible without considering the second aud third components of aerodynamic forces, 1s, the lift and the forces due to rotation of the body. It is found that the solution of von Karman and Moore (Ref. 1) for the linearized hydrodynamical equation of axial flow over a slender body of revolution can easily be generalized to the casem in which the projectile is inclined to the flight path. Strictly speaking, the solution is applicable only to a very slender body inclined at a small angle to the flight path, because second order quantities of the disturbance due to the ‘presence of the body are neglected. However, for the case of ~26~ axial flow over a cone, von Karman-Moore's first approximation (Ref. 1) differs very little from the exact solution of Taylor and Macooll (Ref. 2) for vortex angles up to 1;0°. Tharefore, it is expected that the first approximation of the lift force eas obtained in this paper can be applied to a pointed projectile with fair accuracye If B is the potentiel of the small disturbance v velocity due to the presence of a body of revolution whose axis coincides with the x-axis, then the linearized equation of motion of compressible fluids in cylindrical co-ordinates x, r, and 6 is Lg, 1 1 #8 _, ~ /- LE 727 jae 72? OBO wy in this equation, V is the velocity of the undisturbed flow for which the velocity of sound is c. If the direction of the undisturbed flow coincides with the axis of the body, then @ is independent of 0, and Eq. (%.1) reduced to (/- Vi) 42 , Lab + 2 = Cc axe — 2 an =0 We (162) The sulution of this equation when the velocity of the undisturbed flow is greater than the velocity of -27- sound, is the same as that for a two-dimensional wave diverging from a center, It was obtained by Levi-Civita (Ref. 3) and by H. Lamb (Ref. 4). Von Karman and Moore (Ref. 1) applied it to the present case and showed that it can be expressed as a source distribution given by the potential Oo — L / cosh m7) 2 a where (= [Wi d-1 . Analogy with a similer case of flow of an incompressible fluid leads one to expect the solution of Eq. (11) to be a doublet distribution given by the potential po g = -a@ card LX = fh cosh U) cosh U At (Melt) cash This can be shown to be true, because, if the solution of Eq. (Uel) is of the form - / | B= cad Alta) then Eq. (1.1) reduces to LO) aE f+ IF F£ap aye? Ft aE Ae (4.18) -28- Differentiation of Eq. (1.2) with respect to r gives fr EB et BOD» BED hl Comparing Eq. (11a) with Eq. (1.2a), it is easily seen that Bq (Uel}) is a solution of Eq. (%e1). The function f has to be determined by the boundary condition Uy = _ oe if f': “Yn th Cosh u)cosk “Hd (1.5) ae wa 24) where Y is the normal component of the velocity V of the undisturbed flow, and R is the radius of the body. The complete solution of flow over an inclined body of revolution is then obtained by superimposing a cross-flow upon an axial flow, i.e. g= This solution was also suggested independently by C. Ferrari (Ref. 5). From the velocity potential g, one can caloulate the pressure distribution over the body and then the aero- dynamic forces. However, since the theory is based upon the linearized equation, the cross product terms of derivatives of g, and $x in the pressure calculation cen be neglected. -29- Therefore the following simplification results: the resistance or drag can be calculated from the axial flow alone and the lift can be separately computed from the cross flow. Since the resistance was calculated before, (Ref.1), the following treatment is concerned only with the lift force. The lift acting in a direction perpendicular to the axis of the body end the moment about the vortex are thus Tp oo = Japadbeabae <2py (8 2b andor M=- Jaf fipntirttaay fipeaea' 1.26) where // p is the difference between the pressure at the surface of the body and that of the undisturbed flow and £ is the density of the fluid in the undisturbed flow. Eq. (1.5) is a non-homogeneous linear integral equation in f which does not have a general solution of simple form. However, it is interesting to see how Eq. (4.5) simplified in the limiting case when the radius of body approaches zero, It is convenient here to use & = K- an cape as the independent varieble, thenEq. (%.5) becomes M4 tpie-pyle — FE aoe fle \e-2d. W/ lire)? 7h? eG ADE a y= Ba Integrating by parts, one has A-dke A-kh hx gs / [AO le ®) / + f MLE) AE Now, if the projectile is pointed nose, the doublet strength must be zero at the nose =0 , thus Ho) =O ,. Let Ke —>O , end writing Y instead of & in the integrand the above equation reduces to yy n~ Le 0 > Be J Alt) dd moe) o Since the cross-sectional area of the body of revolution is J = he » Eq. (15a) can be written as x y= Ff fo ke or a x [Aodt= 2S Differentiating, one arrives at #li)= = 44 (4.7) In order to calculete the lift, one has first to find the axial component of disturbance velocity. Thus 7h = wes f& af cosh it) cosh t da heh Cosh ae = Ne ia NEMA EME / 2 W lu-~)? — 0. R? Xx ~ Cog ‘ -_ L448 »,)_ Goatd dS a J Pg) Ag = TE fa)= 2B! ge Substituting into Eq. (1.6), the lift force is obtained as oe 2 _ Uy cos'O AS - L= Zev ff 2S" db Ax = SVGA, Where A; = ares. of the base section of the body. Hence the lift coefficient can be evaluated as L V; Gogh - 2 x27 mo tL £ VA V 2 b in which W = angle of attack of the body. The moment arm d, i.6., the distance between the point of application of the resultant lift force end the vertex can be obtained by dividing the moment computed from -32= Eq. (1.6) by the lift force, and thus d= (Azfe) 1 (19) whore An = area of the mean section of the body, i.e., the volume of the body divided by its length, L x The results of Eq. (%.8) and Eq. (1.9) are identical to those found in Munk's theory of airships (Ref. 6). At first sight, this might be surprising. However, if the radius of the body approaches zero as assumed, the cross-flow pattern is the same as that for an infinitely long circular cylinder moving with its axes perpendicular to the flow. Therefore, in every plane perpendicular to the axis of the body, the flow can be considered as two dimensional, i.e., it is independent of the variable { e Hence Eq. (1.1) reduces simply to IE h oh ji? ae (%.1b) This is immediately recognized as the equation of motion for two dimensional flow of incompressible fluids, which is the basis of Munk's theory. Due to this two dimensional character of the flow, the distribution of doublets is not affected by the change in Mech's number, which is only comected with the independent variable y , and, therefore, the lift coefficient and the moment arm are also independent of Mach's number as shown by Eqe (1.28) and Eq. (2.9). This can also be seon from the fact that when r approaches zero, the variable E=f- “hr Cp ut —» X and thus the effect of o& , which is a function of Mach! s number, is removed. To study the effect of Mach's number on the lift of the body, one has to go back to Eq. (2.5). To avoid the difficulty of solving this integral equation, the “indirect method” of solution can be employed, 1.6., take a func tion f and determine the necessary shape of the body to comply with this function f. Taking the simplest case flt- tn whuj= A (24- th cosh 4) where K = @ constant, Then e g, = - Atos J (X- (h COU) Cosh 4 Ha cosh (~ | = wos 6 | f (¥j2, _ #h, ty “ Zi Wig)—! ~ B ash re And the boundary condition reduces to _ Aad? {hy fy y2 x = “3 [le yGe) —1 + cet ye) Therefore, the solution, Eq. (2./ja), is evidently a solution for a cone with half vertex angle € , if tot € = _# ° A By putting co € = GS the boundary condition can a -3) \= be written in the form 2 v= Hh IC fET y coh" (a5) For any given value of vertex angle and Mach's number, the corresponding value of K can be obteined from Eq. (2.5b) In order to calculate the lift, one has first to find the axial component of disturbance velocity. Thus from Eqe (1.4e) x2 ag ak abl fpxtye (ze) fe he 2 LNA)! Cee lee! HK cot b,/{S—/ Substituting into Eq. (1.6), the lift force is found to be " =o L=2evu, [ [G8 acore ab de = &tkfo~/ gv | 3 / 2h dy 2 where S = lateral surface area of the cone, Therefore G24 fede ot ew o But from Eq. (1.5b), 4 is obtained as — — a Vo a” {6,fCEi + couh “Cf Hence “Ae gap (2.210) (,= AY where AC = 26 f5e/_ C/SH1 + coh S in the limiting case when € approached zero, A, —= 2 which agrees with Eq. (1.8). Similarly from Eq. (1.6) the moment coefficient is C = moment about vertex 2 i =£2AYP 4 72 A L 3 (%e11) which satisfies Eq. (1.9). Both Eq. (28) and Eq. (2.10) show that the lift at a given Mach's number is proportional to the angle of attack of the -36- body. This is a general characteristic of flow around a body without separation, If the fluid separates from the body and creates a “dead water" region on lee side of the body, then the lift will be proportional to the square of the angle of attack as was shown by W. Bolley (Ref. 7). The problem whether the fluid separates or not can only be answered by experiments. From the experimental data now available, (Ref. 8), it seems that the flow is continuous without separation, end, therefore, the lift is proportional to the angle of attack of the body. Fig. %e2 is the result of computation using Eq. (4-10). Calculations were carried out for values of hk, 2], because the value of fe, = f corresponds to €=B ’ where pp is the wave angle. For A < 7 it is found thet p< € - This means that the weve engle is maller than the vertex angle which is, of course, impossible. There- fore, A, = / werks the limit of validity of this solution. In fact, even when A is near to 1, the solution must be f considered as qualtitave only, since in this region the effect of the surface of the body on the shock wave cannot be neglected. To generalize the solution for a body of revolution with a sharp point at the origin and cylindrical shape at infinity, it is simplest to use a step-wise doublet distribu- tion. Consider the points x ae , RP. - £, of the -37- meridien line of the bedy, and designate their co-ordinates by A, | A, j x, KR, . nr Ma Ry poo i hy, Ky and the correspond- ing values of Y- «4 by E Bo Ee Ew oa 4 a Then the boundary condition of Eq. (1.5) can be written as Gr- 5 Sy-'. - -£. * yor —~ Ish (by Se g, ) t=/ w Hy «Kn (2.5¢) Ny - Ses jy &. 2 dyn 3 K- 2 /) 7 ( a hy, oan ) / a ky vB," This condition actually gives a set of IV equations to determine the N constents A, « This set of equations can be solved rather easily because each following equation in the set only contains one more which does not appear in the preceding one of the set. When KH; 5 are determined, the lift can be calculated by using Eq. (1.6). The pressure over each section of the conical surface is constant. The lift and moment co- efficients are thus obtained as we fet J) hue steal) FA EE EP) yes it, 0. ae ml Oh ny CU oi Boe 1 ha [Zn LE “ft fous Jif (2.12 -386 where // is the last point on the meridian line, //, is the base radius and /£ is length of the body. Pig. 2.3 is the result of the caleulation using Eq. (2.12) for a body of revolution with "6-cliber head" and total length of 4.8 calibers, when it is travelling with a velocity 2.69 times that of sound ( a = 2.5)- The 1ifs coefficient is considerably higher than that of a cone at the same angle of attack and at the same Mach's number, evidently due to the cylindrical part of the body. The position of the resulting lift force is also shown in the figure. Since, as mentioned before, lift and drag are independent in first order approximation, the calculated lift coefficient can be combined with the drag coefficient taken from experiments and thus give some information on the magnitude and direction of the result- ing force. Fig. 2.1 shows the method epplied to this projectile with the drag coefficient taken from Kent's experiment (Ref. 9). If the projectile has a length of 1.3, diameters in- stead of 1.8 diameters and has the same nose shape, and, if its center of gravity is located at a point 2.68 diameters back of the nose, then the calculated moment about the center of gravity can be expressed as A] = S Vk, “yp AL where 4/) = 935 for + = 269 ° This compares closely with the value HE) = [0.7 extrapoleted from R. H. Fowler's experiment (Ref. 8), for a -39- projectile of the same proportions. This shows that the theory developed in this paper can be applied to a projectile with fair accuracy. -l,0- 1. ee. 36 De 6. Te 8. Ge REFERENCES ON PART (11) Th. von Karman and N. B. Moore: “The Resistance of Slender Bodies Moving with Supersonic Velocities with Special Reference to Projectiles", Tans. ASME Vol. 54, pp. 303-310, 1932. G. I. Taylor end J. W. Maccoll: "The Air Pressure on a Cone Moving at High Speeds", Proc. Royal Society (A), Vol. 139, pp. 278-298, 1933. Je We. Maccoll: "The Conical Shock Wave formed by a Cone moving at a High Speed", Proc. Royal Society (A), Vol. 159, pp. 459-72, 1937. The comparison was mentioned in a paper by G. I. Taylor: "Well Established Problems in High Speed Flow", Atti dei V Convengo "Volta": "Le alte velocite in aviazione", pp. 198-21), 1936, Reale Accademia d'Italia, Rome, Levi-Civita: "Nuovo Cimento" (l), Vol. 6, 1897. He. Lamb: “On Wave-Propageation in Two Dimensions", Pros. London Math. Society (1), Vol. 35, pe Ul, 1902. See also H. Lemb: "Hydrodynamics", p. 298, 6th Ed., Cambridge 1932. C. Ferrari: "Campi di corrente ipersonora attorno a solidi di rivoluzione™, L'Aerotecnica, Vol. 17, fasco. 507- 518, 1937» M. M. Munk: "The Aerodynamic Forces on Airship Hulls", NACA Technical Report No. 18), 19%). W. Bollay: “A Theory for Rectangular Wings of Small Aspect Ratio", Jour. Aeronautical Sciences, Vol. , p. R. H. Fowler, E. G. Gallop, C. N. H. Lock and H. W. Richmond: "The Aerodynamics of a Spinning Shell", Philo- sophical Trans. of Royal Society of London (A), Vol. 221, E. M. Kent: "The Role of Model Experiments in Projectile Design", Mechanical Engineering, Vol Sl, pp. 41-646, 1932. PART (Lil) APPLICATION OF TECHAPLIGIN'S TRANSFORMATION TO TWO DIMENSIONAL SUBSONIC FLOW The equations of two dimensional irrotational motion of compressible fluids, assuming that the pressure is a single-valved function of density only, can be reduced to a single non-linear equation of the velocity potential. In the anpersonic case, the problem is solved by Prendtl, Meyer and Busemann by means of the powerful method of cheracteristics. The essential difficulty of this problem lice in the subsonic case especially when the velocity is near to the velocity of sound. The first logical step is to linearize the equation based on the argument that the disturb- ance super-imposed on the parallel rectilinear flow due to the presence of a solid body is sufficiently small compared with parallel flow. This makes the second and higher order terms of disturbance potential to be negligible. An exemple of this method is the well-known theory of thin airfoil due to Prandtl and Gleuert. But the presence of stagnation point at the nose of the airfoil makes the application of the linearized theory questionable, at least near this region; because thare the disturbance due to the presence of the body is no longer small, On the seme ground, the theory breaks down in case of -1\2- bodies whose dimension across the stream is not small compared with the dimension parallel to the stream. The next method is thet derived originally by Janzen and Lord Rayleigh. They solved the equation by successive approximations. However, the process is very tedious and the method convergent very slowly if the velocity approaches that of sound. Molenbroock (Ref. 1) and Tschapligin (Ref. 2) suggested the use of the magnitude of velocity and inclination of velocity to the x-axis as independent variables and were able thus to reduce the equation of velocity potential to a linear equation, This equation was solved by Tschapligin (Ref. 2) and recently put in a more convenient form by F. Cleuser and M. Cleuser (Ref. 3). The solution is essentially a series each term of which is a product of hypergeometric function and circular function. The chief difficulty in practical application of this solution is to obtain a proper set of boundary conditions in the transformed plane, or the hodograph plane. Tschapligin (Ref. 2) showed that a great simplifica~ tion of the equation in hodegraph plane results if the ratio of specific heats of the gas is equal to -1. Then the equation becomes the equation of minimal surface whose solution is well- kmown, However, at first, the hypothetical valuo of ratio of specific heats (all real gas has the value for this ratio renging from 1.00 to 2.00) makes the practical application of ole Tschapligin's theory questionable. It was Demtschenko (Ref. )) and Busemann (Ref. 5) who made the meaning of this special value of ratio clear, They found that this special value of ratio of specific heats really corresponds to take the tangent of p-v curve of gas instead p- VT curve itself. However, they limit themselves to use the tangent at the state of rest of the gas. Thus their theory can only apply to velocities up to 0.5 times sound velootiy. In this part , the theory is generalized to use the tangent at the state of gas corresponding to the undisturbed parallel flow. Therefore the range of usefulness of the theory is greatly extended. In the first section, the general theory will be developed. In the second section, the theory will be applied to the case of symmetrical Joukowsky airfoil at zero angle of attack. Section (1) If ‘4 is the pressure and thy is the density of gas, the adiabatic process is expressed as a curve in the bv plane as shown in Fig. 3.1. Now conditions naar to the point hi, V;, can be approximated by the tangent at this point. The equation of the tangent at this point oan be written as p,-p = C(%,- r)=0(¢- 8) (3.2) where S is the density of the gas. Now the slope C ohh must be equal to the slope of the curve at the point BY 2 Therefore C = - 4° @ Hence the approximate p- S relation near b, , Ss, can be written as 2,2/f / p-p= asi (f- =<) (3.2) From the generalized Bernoulli's theorem, the following relation is obtained Jy fug-fuj- | 4 2% 2“ 1 3 (3.3) where 4 is the velocity of gas, the subscripts 2, 3 indicate two conditions. But from Eq. (3.2), A can be expressed as a function of is » thus 2 Ap= 4 aa , (3.4) 2 4F S Substituting into the integrand in Eq. (3.3) and integrate, the following relation is obtained na ~ @ (3.5) where the subscbript O denotes the rest state of the gas. If the sound velocity @ is defined as the derivative of Fa with respect to § , then Eq. (3.4) gives a e = ae = 4,°§ "= constant (3.6) Therefore Eq. (3.5) can be written as f \2 wr? (B= /- ge or ? 2 7 /-4& (367) Furthermore, from Eq. (3.6), 974? = §,° 4,7 ’ thus Eq. (3.7) can also be written as S, — 2 _ We. (3.8) c /4 It is interesting at this stage, ta notice that from Eq. (3.8), the density decreases as velocity increases, as it is expected. Thus from Eq. (3.7), the . welooity of sound of the gas will increase as tho velocity is ~h6- increased. This is just opposite to real gas, because in the case of an adiabatic flow, it is well-known that the temperature of gas decreases as the velocity of gas is increased and thus the sound velocity also decreases, However from Eq. (3.7), the ratio a or Mach's number increases as the velocity «7 increases. But this ratio only reaches the value unity when WU = © » or from Eq. (3.8) when FP =a - Itis thus seen that the entire region of flow is subsonic and thus the equation of motion is always of elliptic type. This may be considered as the physical reason why the complex representation of velocity potential and stream function is possible in all cases, &8 will be shown in following pages. However one should bear in mind that the portion of tangent that could be used as an approximation to the true adiabatic curve, must lie within the first quadrant. Thus the upper limit of velocity for practical application of the theory is when a =O . By using Eqs. (3.2), (3-7) and (3.8), this upper limit is found to be Ona by wae tL) el max. (Z)N\e4 ' Or by putting a? = VA £ » the above equation / reduces to -h7- wy) ~ Lf 2 fay UW, loypy (4) (pi )=f1 (fy te The values of Cor hypay for different values of 4 . / are shown in Table 3.1. Table 3.1 z a Mak. ( 2, hen fe) © 2.186 0.2 10.91 2.195 0.4 5056 2.225 0.6 3.78 2.265 0,8 2.92 20335 1.0 2.05 21,05 It is thus seen that for most applications of this theory, A will remain positive. However due to large deviation from the true adiabatic process at high values of oa » one has probably to limit the ratio (Gr ) to about 2, -)\8- Now if the flow is irrotational, there exists a velocity potential such that VY ) OY (3.9) where “, V are the JV Y components of the velocity W . To satisfy the equation of continuity, the stream function yp is introduced. It is defined by Sys JP 8, __ ap 5, @ 34. B= OM (3-20) Now if the angle of inclination of the velocity “7 to the x-axis is Ve » then from Eqs. (3.9) and (3.10), one has Ag = uf ap ae tw “ ay Solving for dX ena i” Cah yg Imp S 4 a ay ae (3.12) dy. SnB Cath So yp t Ww de + Ww § So long as the correspondence between the physical plane and the hodoplane is one to one, or mathomatically Se FO one can express x , ¥ as functions of 4“ fp end so also g and WV as functions ‘4 and fp e hus =1)9- Ab = 4b dwt #, Ap A= Vo du + vs Ap (5.23) where primes indicate the derivative, and subscript indicate the variables with respect to which the functions are differentiated, Now substitute Eq. (3.13) into Eq. (3.12), the following relations are obtained / 7 dun (Ab, SP Ba Mut hy BEV My (3-44) Vi) / So ; in My =( TE + SE EY duct (Gat LEY My “WwW Since the left-hand sidesof Eq. (3.14) are exact differential, one can apply the reciprocity relation and obtains i (Ae - Sp & $. Vp (04 _ sn ab Sy 2 / Sinb 4 So hg SU / , aad We) Hy ake Carrying out the differentiation, and cancelling identical terms in left-hand and right-hand side, — Shh fb! _ OSB Spry! _ Cah f', Sib 87, Wy yp’ Ub Gu ~ ES Va = GEG +t EM 7 Jel COS / SNE & / y 7A 2 ; Af Su ~ EB yl = - ib 4! eh 81 p -50- Using Eq. (347), Eq. (3.16) becomes a7, ny! — . OSB fg’, WBS Af’ TB te SEB p= SEG ES aM Ww wo Ww / / / Jo / As in both equation. wy La and 7% s in bo equatio Ww ) qv: w ’ gb ; are connected with a proportional factor, one can solve for them, and (318) Now Eq. (3.18) can be further reduced if a new variable (” is introduced. Ww is defined as iv; W (3.19) Then Eq. (3.18) becomes g,= - We cD / / (3-20) 4° % \\ This is the fundamental set of equations for the present theory. It can be easily recognized as the Cauchy-Riemann differential equation, and thus gp Ft LP must be an analytic function of A 4+ 4W « However for the convenience of numerical calculation, a new variable WV is used instead of W =, such that Oo Or by integrating Eq. (3.19), 26, Ww W = (3.21) if A, +w* Ft tt, Hence by inverting, 4 By W (3.22) 4a?- W° | W = Thus by substituting into Eqe (3.8), & _ 4a, + W* 5.25) § 2 ° haze W If another set of new variables U- WV cas and / = W Sin B are used as independent variables, one has -52- : = oY BV 2 yy! in 2 pp 8 IW 220 OU + 7m ay = Weg Hopf a Using Eq. (3.24), Eq. (3.20) can be written as a . a 42 esp 2 4 snr = sing OF Cos 2 — $n ¢ S 2 _ av 17 snp BO coe 36 caspBl + sin BF These equations are aatisfied by _ 2b ab. op ay atv), a-V) ou (5.25) These are the Cauchy-Riemann differential equations, therefore the complex potential fF = O + iv is a function of U/ - LV = W . . Ore b+I1P = E(U-w- Hw) Hence p -; p _ El uri v)- F(W) (3.26) To transform from hodograph plane back to physical plane, the expression of Y and GY in terms of U/ and V must be found, By using Eqs. (3.22) and (3.23), Eq. (3.12) can be written as ee a LE S17 fe Zz 2 where ed = Uf t y/? » These equations can be combined into one equation by means of Eq. (3.26). Thus d= ditidy = a : oa (3.27) For practiesl application of the theory to tha flow over an obstacle, the computation proceeds as follows: (1) Find the complex potential for the flow of incompressible fluid over the obstacle, say w, GCE tg) ~ 0, GS) where 4, , is the velocity of parallel rectilinear undisturbed flow, and é ’ ” the space coordinate of the physical plane. (2) Now let f= W GUS) . Here We is the -5l- transformed undisturbed velocity, to be interpreted as Eq. (3.21). But the complex variable © has no direct physical meaning. (3) Using the above value of F- , Eqe (3.27) can be written as di= 45 - JVC Ey'& Integrating, fy [Ey u ° (3-28) Thus it is seen that the complex coordinate in the physical plane of compressible fluid is equal to the corresponding complex coordinate in the physical plane of incompressible fluid plus a correction term. The factor before the integral depends upon the Mach's number of the undisturbed flow only. By using Eqs. (3.7) and (3.8), and (3.21), one has ye? (4 | 7 MW d t (zx) = oD (3.29) 4 ay fie /- aap} The integration constant of the integral in Eq. (3.28) is not important, because it only means a translation of the whole 2 - pleneo. (4) The velocity ur corresponds to 2 can be computed by starting with -55- 7 AE _ | | We de = we ae = Ue By means of Eq. (3.29) _/ht an a We “Ml Thus by putting 4 = tu, , one obtains mn / / j- My Thus wh (330) mn - f1-£y 7 LaF q Z (a) Using Eqs. (3.30 and (3.29), the ratio & can be calculated easily. (5) To determine the pressure acting on the surface of the body one has to use Eq (3.2). With some manipuletion, the following relation is obtained: / 29% ey [Oe = Tae! | ] But - [te = wy? (GE) (3.31) /+ eye a aS /- (= Therefore (3.32) ft. a / 4 ig) ay f Fsue (By 1 -)) | + {pf Section (II) In this section, the general theory developed in Section (I) is applied to the simple case of flow over a sym- metrical Joukowsky airfoil at zero angle of attack, The complex potential in the circle-plane (see Fig. 3.2) is know to be (ube 2 on fip-8)+ | | | Wb (3.33) where @ = radius of the airfoil circle, ge eccentricity of the airfoil circle. The relation between the airfoil plane and circle-pjlane is the well-known Joukowsky transformation LL $= Ath (3.34) -57- if the radius of the transforming circle is unity. Now the starting point of the calculation is the function to find Vf JE wide ~ 42 FE de - (al 4, “a Therefore WapF = WI si f [fe +4035 -/ -Lf Thus the correction term in Eq. (3.28) is / . ig J Wee = Lh [rh -3f (3.35) where = Z lire ae + Ge | He _ a* Clacd "2 tI ER jeg? (a-bye J Ua-V) L=zg/ -~4 ,_ at) bi EI LGR t Tab a7 These integrals can be easily computed and simplified, noting thet a-£2= / . rf lui-b)=ne°? and _ 1 “2 -3, a j=afe 4#2eé a La tf{t- efor 8 Y, ylac’)} J =f [AI Log uh dP gue 2) -£ 2-20 “Hel AA Oe) Ee 4 eo #4 aaa (3.36) Separating the real and imaginary parts, and adding, lt (J ti,-i,)= “(3 cong -F cas 36) L/S U-A) yy / Ly 200d 3 + ¥ (1-A*) cos 26 + Ff Ltd) cos 30 f (3.37) In (14h-4) 2 a(- 9004+ s 30) + t { (424-29) sing - $( 1-22) Sin 26 -$ Lipd)sin 36 - (1-4) “to See ony These give the correction term to Xx and Y coordinates. The transformed velocity W over the surface of -59- the airfoil can be easily found by means of graphical method (Ref. 6). Then the actual velocity and pressure can be computed by using Eqs. (3.22) and (3.32) « Fig. 3.3 shows the result of calculation for the case “ = 1.20 and a = 0.550. The nose of the airfoil is somewhat rounded by transforming into the case of compressible fluid. However, the pressure gradient is steeper, as would be expected. The main defect of this type of cal- culation is that during the transformation from incompressible flow to compressible flow, the shape of the body is aiso changed. To isolate the effect of compressibility of the fluid, it is necessary to bring back the original shape of the body. This is done by first deforming the original Joukowsky eirfoil, such that the final profile after correction for compressibility is same as the original Joukowsky airfoil. The amount of deformation is obtained from the calculation assuming that the airfoil was a Joukowsky sirfoil at start. That is the effect of deformation on the correction term of Eqe (3428) is neglected. This is allowable because the quantity neglected is a second order quantity. This deformation of Joukowsky airfoil can be carried out by using the method developed by von Karman and Trefftz (Ref. 7). However, for some practical reasons, the Karman-Trefftz method is somewhat modified: aa Fig. 34a shows two airfoils, both having the same chord, one is Joukowsky airfoil desired and the other is the airfoil resulted from the first step calculation. Now apply the Joukowsky transformation to this figure, then the Joukowsky airfoil will become a circle C while the other airfoil a near-circular shape{, aishown in Fig. 3.ljb. The desired deformed Joukowsky airfoil will appear like C 2 in this figure. The difference between C, and C is just opposite and equal to that between C and C « Now let / C, be written as G,=h ef! » Obviously, A= [+t fle) (3.38) where 46) wil be small compared with 1, The function which established the conformal transformation of the outside of this boundary C, into outside of the circle C , may be denoted by 5, = 5, Lit AG)] (3.39) where C, ) CS 2 have. their origin at the center and the absolute value of fl Sn ) is again small compared with 1. Then it is shown (Ref. 7) that MO)t ke [ Mé.)]] =0 (3.0) In order to calculate 462) we develop the —-6 1» funetion He b ') in a Fourier series: /b WAN cos 1G HO) = * (3.11) Here only consine terms appear because the airfoil is symmetrical about the chord. One the other hand, the complex function f(. G+) has the form, for / $2 /7/ a“ C 7 7 mn 0 2 Now put C, 20 , then (3.40) is satisfied by | Cy = - ay Thus Lo) 2- 9 & (3.1) Ja 2s oe It can be easily seen that the velocity around the deformed Joukowsky airfoil can be calculated as “5 [Ze] aC (3 oly) where Wr = velocity around the Joukowsky airfoil. Now trom Eqs. (5-59 amd (4.45), ~62— 0 . 0 6,” = {4 Zila carn ¢ Pad tysn Neglecting small qumtities of second order, end noting Eq. (3-441), Pal = /+ Z) (2-1) dy os nb co co J 4 7! nay, 00s NO =P Oy 18 (3.45) 0 ra) \\ oo = /+4 GZ 4 in ng - fd) A trial calculation shows that the convergence of the coefficients ay is not very good. Therefore, one must avoid manipulation on the Fourier series as required by og Eq. (3.45). This is possible because — 7 4y cin #0 3S is known to mathematicians as the allied or conjugate series oo of 7! pn Cos 08 - It ie also lmown D os (Ref. 8) that if pi 4) = 7, Gy CoS 00 b TE <7, gg . Lf Mbtd)- Hos) _ Zi sn 78 = | — df =63= Therefore oo VA ys nnge-<L (8tE)-4/4 wD), fg 2 by 5in 19 = tft Zt & 2 Integrating by parts, “ages xe | --2 Lf " yore) ploy r [flaD) PY (/- cs F ) Hence Eq. (3.445) can be written as |Z 08) Map Hoo fp B= | zp [OO RNIN wy, fhe integral is evidently convergent for any continuous reguler function 4b 1) » because then the integrand is always finite. Its evaluation can be done numerically. Fig. 365 is the result of calculation for a Joukowsky airfoil with the thickness paremeter b =0.20 ’ at two speeds, — = 0.50 and 0.550. The suction a, peaks are considerably higher with higher speeds. Also the bye positions of pressure peaks tend to move backward with increas- ing speed. Both are in agreement with the experimental results obtained by J. Stack ( Ref. 9). The values of (, ppAYL ¢ ,* for a = 0.550 end AZ = 0.450 at which f ; real air will attain a velocity equal to the local sound velocity are equal to -1.654 and -2.755 respectively. It is thus seen that the effect of compressibility on pressure distribution is appreciable, even when nowhere the local sound velocity is reached. One should, however, bear in mind that the effect on the force coefficient of the airfoil will probably not be so marked as with the pressure distribution, because the resultant force on the airfoil is the algebraic difference of pressure force acting on two sides of the section. APPENDIX TO PART (I11) COMPARISON WITH OTHER METHODS Tn order to check the accuracy of the method developed in PART (III), the flow over a finite circular cylinder with its axis perpendicular to the direction of undisturbed flow is studied. The method exposed in Section (1I) of PART (III) for correction of shape of body is used. The following is the result of calculation for velocity at the top of tho ciroular section, compared with results by other methods [collected by E. Pistolesi (Ref. 10)/. We 0 Method at top of section Pert (III) 2.268 Rayleigh 2.206 Poggi 2.19) Taylor's Electric 2.188 Analogy a —i Ws 2.000 for incompressible fluid, ft Thus the present method gives a higher value. However, the flow over a cylinder is rather an extreme case. Because the difference between the velocities to be calculated and the ~66- and the undisturbed velocity is large, and thus this epproximate method involves lerger then usual error, - 67= l. ee Je he De Te 8. Ge 10. REFERENCES ON PART (III) P. Molenbroek; Hber einige Bewegungen eines Gases bei Annahme eines Geschwindigkeitspotentials, Arch d. Mathem. ue. Phys., Grunert Hoppe (1890), Reihe 2, Bd. 9, S. 157. A. Tschapligin: Scientific Memoirs of the Univ. Moscow, (In Russien) (1902). F. Clauser, M. Clauser: New Methods of Solving the Equations for the Flow of a Compressible Fluid, Unpublished Ph.D. B. Demtchenko: Sur les mouvements lents des fluides compressibles, Comptes Kendus, Vol. 19, p. 12168 (1952). Also, Variation de la résistence aux faibles vitesses sous ltinfluence de la compressibilité, Comptes Rendus, Vol. igh, pe 1720 (1952). A. Busemann: Die Expansionsberichtigung der Kontraktions- ziffer von Blenden, Forschung, Bd. h, 8S. 186-187 (1933). Also, Hodographenmmethode der Gasdynamik, Z.A.M.M. Bd. 12, S. 73-79 (1937). W. F. Durand: Aerodynamic Theory Vol. 2, p. 71-7, let Ed., Julius Springer, Berlin (1935). Th. von Karman, E. Trefftz: Potentialstromung dem gegebene Tragflachenqierschnitte, Z. F. M. Bd. 9, S- 111 (1918). Also, W. F. Durand: Aerodynamic Theory, Vol. 2, p.80-83 (1935) « G. H. Hardy, Je E. Littlewood: The Allied Series of a Fourier Series, Proc. of London Math. Soc. (2) Vol. 2h, pp.211-2)6 (1925). Je Stack: The Compressibility Burble, NACA Technical Note No. 543 (1935). E. Pistolesi: La portanza alle alte velocita inferiocri a qella del suono, Atti dei V. Convengo "Volta", fase 300, (1936) Reale Accademia d'Italia, Rome. . PART (IV) FLIGHT ANALYSIS OF A SOUNDING ROCKET WITH SPECIAL REFERENCE TO PROPULSION BY SUCCESSIVE IMPULSES. Introdution In 1919 R. H. Goddard (Ref. 1) published the historically importent paper which suggested the use of nitro- cellulose powder as a propellant for raising a sounding rocket to altitudes beyond the range of sounding balloons. To determine the feasibility of this propellant, oa series of ex- periments had been carried out and it was found that thermal efficiency of 50% could be expected if the powder was exploded in a properly designed chamber and the resulting gases were allowed to escape at high velocity through an expanding nozzle. In 1931 R. Tilling used a mixture of potassium chlorate and naphthalene as propellant and actually reached an altitude of 6,600 feet. More recently, L. Damblanc (Re. 2) made static tests with a slow burning black powder and from these estimated that a height of 10,000 feet could be reached using a two-step arrangement. The results so far reported offer an incentive to further analysis. The effect of decreasing gravitational acceleration on the maximum height reached by a rocket has been considered by A. Bartocci ( Ref. 53). However, he assumes that the -69- rocket itself has a constant acceleration during powered flight. L. Breguet and R. Devillers ( Ref. 4) also con- sidered the effect of the variation of g. To simplify the emealysis, they assumed that the acceleration of the rocket was equal to a constent multiple of g. Since the sounding rocket for practical reasons will be propelled by a nearly constant thrust or a uniform rate of successive impulses, in Section (II) the euthor has studied the problem anew according to this mode of propulsion. When the sounding rocket is ascending through the eir the maximum height reached is less than that reached for flight in vacuo. Recently, studies have been made of the problem by W. Ley and H. Schaefer (Ref. 5) and by F. J. Malina and A. M. O. Smith (Ref. 6). On the basis of the latter study a group of new performance parameters have been isolated from the generel performance equation, and these are discussed in Section (III). Notation Referring to Fig. (4.1), the following notation has been used throughout the paper: Ws weight of propellant and propellent container ejected per impulse, lbs. k «we ratio of container weight to sum of container and propellant weight ejected per impulse, Le (1 -k) We initial weight of rocket, lbs. J -70= L4, initial mass of the rocket, slugs. W,* Cc & i instantaneous weight of rocket, lbs. ratio of initiel weight of propellents to initial total weight of a rocket propelled by constant thrust. ratio of initial weight of propellents to initial total weight of a rocket propelled by successive impulses. cL number of impulses per second. total number of impulses occuring during powered flight. time, 8OCe interval between impulses, sece initial acceleration imparted to rocket, ft/sec”. acceleration of gravity at the starting point of flight, ft/sec ° acceleration of gravity above the starting point of flight, ft/sec“. | effective exhaust velocity of ejected propellant, ft/sec. instenteneous velocity, ft/sec. velocity imparted to rocket by the nt impulse, ft/sec. velocity at the end of the nth interval, ft/sec. velocity of sound corresponding to the atmospheric con- ditions at the starting point of the flight, ft/sec. velocity of sound corresponding to the atmospheric con- ditions at the height reached by the rocket at the time t; ft/sec. Mach's number hy, velocity of rocket at start of coasting flight, ft/sec. -71- ss h ys Fz \ y o = = i fl = velocity of rocket at start of coasting flight if ¢ is constant and equal to g, ft/sec. altitude above sea level, feet. height reached at the beginning of the 274 interval, feet. height reached at the end of the ath interval, feet, height traveled during powered flight, feet. height travelled during powered flight, if g is constant and equal to gy, feet. height travelled during coasting flight, feet, height travelled during coasting flight, if ¢ is constant and eqial to g,, feet. height travelled during powered flight and coasting flight, feet. height travelled during powered flight and coasting flight, if g is constant and equal to £5» feet. radius of earth, 2.088 x 108, feet. drag on rocket due to air resistance, lbs, drag coefficient of rocket shell, drag coefficient of rocket shell at the velocity of sound. drag-weight factor (discussed in the section on the effect of air resistance). mass density of air at the starting point of the flight, slugs per cu. ft. ratio of air densities at altitude and at the starting point of the flight. absolute temperature of the atmosphere at the height reached by the rocket at the time T, °F. absolute temperature of atmosphere at the starting point of flight, OF. -72~ = largest cross-sectional area of rocket shell, sq.ft. largest diameter of rocket shell, ft. \ ke a Wh \\ length of rocket shell, ft. Section (1) An approximate method of calculating the maximum height reached by a rocket propelled by powder was developed by R. H. Goddard (Ref. 1). To simplify the analysis a contin- uous loss of mass was assumed end the problem was so stated that a minimum mass of propellant necessary to lift one pound of mass at the end of the flight to any desired height was determinad. However, if highepowered powder is used, the rate of burning is so rapid that the propulsive action is insfanten- eous. The rocket is thus acted upon by an impulse rather than by a constent thrust, In the following analysis, it has, therefore, been assumed that the propulsive force is an impulsive force, ieds, the force acts for such a brief interval of time that the rocket does not change its position during the application of the force, although its velocity and its momentum receive a finite change. If the combustion process of the propulsive unit takes place at sonstant volume this assumption is justified. Further, a study of interior ballistics of small arms reveals thet the period between the ignition of the powder charge and the dullet's arrival at the end of a twoe- «73 foot barrel is of the order of 1 ten-thousandths of a second. If the gases are not restrained and their travel through the burning chamber end the nozzle is of much shorter length, as is the case for the rocket motor, even shorter pericds of duration of action can be expected. Assuming that the propulsive force acts as an im- pulse, then the motion of the rocket can be calculated by Newton's third law, which states that impulses between two bodies sre equal and opposite, Hence, eqating the momentum of the exhaust gases to the momentum imparted to the rocket, using the quantities defined in the list of notation and referring to Fig. 4.1, the following relation can be written for Plight in vaeno ——" AW o= Mn AY, 4, Is where A= [-k and W, = W-hw (4.2) or AY, = whe = Stel, / } (14.3) During the interval between impulses, At : the velocity is reduced by the action of gravity so that at the end of the 21/ interval, the velocity of the -7he rocket will be / / Uy = Ui- gat= vy, 4+ AY, - gat (rah) Therefore Sf. ue’ = AV. - At n= s 7 AF, (4.5) J=/ Substituting for b U; from Eq. (4.44) / S=A yla 2A Zz lng af Mi a Lo 7 G4 N fu bile) (1406) al. € Ale _ Or uy = we ng ae =f where J = / N The height gained during each interval will be represented by the area under the velocity curve in the interval, or (4.7) Therefore, at the end of the Nth interval which is the end of the powered flight, the height will be AN WV ; My = LZ, tat + 5G, (at) Substituting for uy, its value in Eq. (4.6) = at} Sr ~agat fr 4 G tat)” 0 a) N 7 / sz) ; <n Nr/- a t = Sifet 2 = oath Dio 4) A=! — _ Ode 2 = <0 At a) _ MY (st)? (4.8) where = 7) Mee Met A The maximum height reached will be the sum of the height at the end of powered flight and the height travelled during coasting or / 2 Frat ~ t, th, ~ Hp, , a me To calculate the maximum height one has first to evaluate the sums S, and Soe Noting that The o0 _/__ f ot 3 y /- 6 (&) 0 S| cen be written in the form fos) ven yous n) = Je File) ay 6 S=/ / tee (nH) 2 = -X é€ -€ J e re be 0 /- en Putting / - the above integral becomes _ a fet c Ny - - {y($)- fe (4.10) where YW (2) = Ay [og , the so-called psi- function (Ref. 7,8). Similarly, Sp can be summed as -T71- Substituting Eqs. (4.10 end (4.11) into Eqs. (4.6) and (4.8), and then into Eq. (4.9) finally We? ap? Ac { (hl - NE max, a4 v nm { (f° J (4-12) where w= ah Pp - VE. )- WE) \\ For convenience of calculation in Fig. }.2 the — quantity (Y is plotted against N for different values [a] Fy SN e It can easily be shown that when N = 1 so thet Eq. (4.12) reduces to Has, ~ in (4..12a) Also, as N —» oo Ws - bogli-Z/) thus Eq. (i.12) reduces to 2¢ aah 6) Lo (7- “ot Hn, = he [ A J fe 4/ | (yarer) 5 A where a, = ay Ae “ZG = a — JS, (4.12c) The quantity 2, can be considered as the initial acceleration of the rocket if N —> 0 » It is interesting to notice that Eq. (4.12b) is the equation ob- teined by Malina end Smith (Ref. 6) for calculating the meximim height of a constant thrust rocket, as expected. Fig. 4.3 shows the variation of Har, fp [e? with yic/¢ for different valves of CG,” and for four valves of N. These curves show that when the total number of impulses, N, becomes larger than 100, the maximum height reached is imperceptibly changed by increasing the number. At this point it is necessary to discuss the similarity existing between a rocket propelled by successive impulses and a rocket propelled by constant thrust. The former loses not only the mass of the propellant, but also the containers for the individual charges. The difference in | effect on the rocket between the propellant and its containers is that the propellant has an effective exhaust velocity, c, while the ejected containers leave the rocket without apprec~ iable velocity. The propulsive action, however, will remain the seme if the whole cartridge, that is, the propellant | charge end its container, is considered wholly as propellent but leaving the motor at a reduced effective exhaust velocity Al « The rocket propelled by constant thrust loses -79- only the mass equal to the propellant carried, therefore, it can be said to be equivalent to the "successive impulses" rocket if its effective oxhaust velocity and its total mass of propellant sare equal respectively to the reduced exhaust velocity and to the sum of the masses of all the containers of the “successive impulses" rocket. In other words, o is equal to AC and C isequlto G” . In Teble .1 the heights for four cases have been calculated to illustrate the effect of the exhuast gas velocity . and the total number of impulses given to a rocket whose weight ratio, Go is 0.70. It will be noticed that for flight in vacuo a greator height will be reached if a smaller number of impulses is employed. The lower portion of the Table shows the maximum height reached by an equivalent "constant thrust" rocket for the seme four cases with the initial acceleration given by Eq. (l.12c). The close agreement between . the maximum height reached by use of successive impulses, when the total numbor of impulses oxecods 100, and that roachod by the use of constant thrust simplifies the solution of the problem of decreasing acceleration of gravity with height, and enables prediction for flight with air resistance to be based on the results obtained for a rocket propelled by constant thrust (c.f. Ref. 6). These problems are considered in the following coctions. Section (II) It is well-known that the acceleration of gravity decreases with the height above the earth's surface according -O0— TABLE 4.1 22 | Hoan = 4el4p* . sueresse impulses Ls -P-N]f @ = DAC “hh Case N a Ft. /sec. Z 4 i Impulses Tax 6 Ac f per sec. Feet 1 10,000 0. 70 526 5 1,472,000 2 10,000 0.70 10 0. 092. 1,686,000 3 7,000 0.70 326 3 560,000 4 7,000 0.70 10 0.092 676,000 Constant Thrust / Har, = EIA [bey b- 2+ a [lagi 2)+e]f Jo Case Ft. /sec. Bo Umaxo C CS Feet 1 10,000 0. 70 32.2 1,468,000 2 10,000 0.70 32.8 1,468,000 3 7,000 0.70 12.9 555,000 4 7,000 0.70 12.9 555,000 to the following relation I= 4 (an? (98) At an altitude of 1000 miles the acceleration is only 0.64 times that at sea level. Therefore, for flights up to such altitudes the assumption that g is approximately constant is no longer valid. It was shown by Malina and Smith (Ref. 6) that a three-step rocket could theoretically reach such an altitude. Thus it is interesting to see how the decrease of g will increase the maximum height reached by the rocket, First the effect on powered flight in vacuo will be considered and then on coasting flight in vacuo, For powered flight the analysis is based on the assumption that the thrust is constant. However, the results can be applied to the case of propulsion by successive impulses if the total number of impulses, N, exceeds 100 as was justified in the previous section. The equivalent mass of gas flowing per seeond con- tinuously for the case of successive impulses is wn ceieonmm Se mM J (dye) Assuming that the rocket starts from rest at sea level the equation of motion in vacuo is -8l- dt hy? the =-g (147) + at? A l- gt (14015) This is a non-linear differential eqmation which can not be solved by usual means. However, for all practical pur- poses the ratio 4 during powered flight is much smaller than 1, therefore, only first order terms in R occurring in the expansions need to be retained. This approximation lineerizes the equation to the form Hh My 4 at L(A t) /- ft (44.16) The solution of this equation with the initial condition that h=0 and 4h. =0 when t = O is Sul 5a C gu ty ~ ku 14 = fed I~ sh [7h | e Gk | ets be Bl AE tf pal YS “4 4 (4.17) where f= 2h tte and a= /-4¢t At the end of the powered flight, the time is {-% = LAE P m (4.18) Therefore, the height at the end of the powered flight is of (1-6) £0) | £ * } (4019) If the hyperbolic cosine term and the integrals are expanded and only first order terms in w4 are retained in consistency with the linearization of Eq. (4.15), the equation becomes - = [3B ee fs cP felt ses] ¢, psf sty z ees Slog (1-3) 16 US sre) f (14.20) =H BE Ell) baglrs)+s (nc 1se r6]] 4 - Say Differentiating Eq. (4.17), and substituting the relation of Eq. (4.18), the maximum velocity at the end of powered flight is Jd. ir) /. g(r-s) ao, , WM Hy -5l) Ven 7 AE siahy| 3 Ue. fe i oe. “Je sf 21) -83- Again expending end retaining only first order terms in Ro : Eq. (4.21) becomes Mey LEB Mat. Un. - [Bef EB _ | c Log (1-5) 4 Layo lreftaglrsjees-36']} (4.22) It is seen that the second terms of Eq. (4.20) and (4.22) are the correction to be applied to Hp, and Vinay, to account for the variation of the acceleration of gravity. Since both corrections are first order approximations, they can be expected to epply approximately also to the case of successive impulses, even when the total number of impulses is less than 100, The coasting height reached by the rocket due to its velocity at the end of powered flight can be obtained by equating the increase of potential energy during coasting flight to the kinetic energy at the end of powered flight. Thus A, tH, ly, 2 dh 2 Voor WA (i+)? _ ale or a d= (pea) |— Var. ~f (4-23) MAX “4(L5 ols ia) (Hr4) 2 2 Putting 2/24 (iq) = M. which is coasting height obtained by assuming a constant gravitational acceleration of the value eqial to that at the height H, i.e., the height where coasting starts, then Eq. (4.23) oan ha written / d= (448) Wy HAR Upon expanding the second term this equation becomes, Me)’ Mm A + (Ge) Bue? + bg) rf a This eqaation shows that if the coasting flight starts from sea level, and if the maximum height reached is about 1000 miles, the increase due to the decrease in g is over 25%, which is considerable, -85- Section (III) | When the sounding rocket is ascending through the atmosphere instead of in vacuo, air resistance comes into pley, ceusing the acceleration of the rocket to be reduced, which decreases the maximum height reached. Since air resis- tence increases with the air density and with the square of the flight velocity, it is desirable to keep the rocket from ascending too rapidly through the lower layers of the atmos- phere where the air density is high. For this reason the op- timum initial acceleration will no longer be infinite as shown by Eq. (4.12b). For the case of constant thrust Malina and Smith (Ref. 6) have found thet the optimum acceleration is around 30ft./sec.“. For a total number of impulses greater then 100, the difference between propulsion by successive impulses and by constant thrust is very small, so one may expect the above optimum value of initial acceleration to hold for both cases of propulsion. The actual amount of reduction in maximum height due to air resistance can be calculated by the method of step-by-step integration, if fair accuracy is desired. This integration is carried out by using the fundamental equation for vertical rocket flight which, as given in the previous paper (Ref.6) is L uth <h&TV’ GA af, =4- G4 /- bla,t@) © 7}. Hat8)} Wr (425) * ~86- GA The significance of the ratio W, was (4 discussed in that paper (Ref, 6). Greater significance can, however, be attached to the various terms in the equation if it is transformed into the non-dimensional form | 2 ar a fy a t/ _ (oN ee BIA ee — (44026) where A= In Eq. (4.26) appear two types of significant "quantities. First, quantities, called “factors”, which are constent for any given family of rockets, and second, two quantities called "parameters", one of which is characteristic for a given family of rockets but ohanges in value along the flight path, and one which depends on the physical properties of the atmosphere. Thus there are the following factors; i) _—& wm ratio of initial acceleration to , “initial acceleration factor", a motor characteristic C = exheust velocity in ft./sec. ~ "exhaust velocity factor", a motot characteristic -87- A = "drageweight factor" ‘4 — Yratio of weight of combustibles to total initial weight of the rocket “v "loading factor" The first two factors, i.e., the “initial aecelera- tion factor" and the “exhuast velocity factor", determine the characteristics of the propelling unit for a given family of rockets while the "drag-weight factor" and the "loading factor” determine the physical dimensions of the rockets. The “drag- weight factor" is a ratio of the drag of the rocket at sea level when traveling with the velocity of sound to the initial weight of the rocket. Since for any given family of rocket shapes the only terms in the factor which can be varied are the maximum cross-sectional area A, and the initial weight W,, it is clear that if the initial weight is doubled then the cross- sectional area must also be doubled to keep the factor the same, The"loading factor needs to be discussed in some detail as it does not appear explicitly in Eq. (4.26). The Eq.(4.26) is a differential equation of the flight path which is satisfied at every point along the flight path. The loading factor C comes in only when this equation is integrated and the limits of integration are put in. For example, consider two rockets with identical performance factors and parameters, with the exception that one has a S of 0.90 and the other has a S of 0.50. The flight path of the two rockets will be identical up to the time that 0.50 times the initial weight of the rockets is used up as combustibles. At this point the rocket having a 7, of 0.50 will begin to decelerate while the one having a ¢ of 0.90 will continue to accelerate until the remaining combustibles are used up. It is thus seen that the value of S controls the maximum height reached. The two performance parameters are; og ~ physical properties of the atmosphere called Ty the “atmosphere parameter" G ~ serodynemic properties of the rocket shell a called the "form parameter" The “atmosphere parameter” for the earthb atmospheric » layer will, of course, be the same for all rockets if standard conditions are assumed end its value depends only on the height the rocket has reached above the starting point of the flight. The "form parameter" is determined by the shape of the curve of. Cy against B, This curve will be altered chiefly by the geometrical shape of the shell although it is also affected by the change in skin friction coefficient due to the change in Reynold's Number. As long as the rocket belongs to a family that has the same geometrical shape, which implies the same nose shape and the same Llha ratio, that is, the ratio of the length of the shell to the maximum diameter, the “form parameter" can be assumed to remain constant. ~89 It is thus seen that the performance curves calculated for a typical rocket will also hold for a whole family of rockets determined by the values of the "factors" and of the "parameters" of the typical rocket and the design of a rocket to meet certain prescribed requirements is greatly simplified. Furthermore, for a good rocket form design the variation of & iss é » the form parameter, at the same values of B is small. Also, the deviation from standard atmospheric characteristics cannot be very large, Then, in view of the fairly accurate but not exact basic ascumption of constant thrust, it is justified te use the same data for these two parameters for all cases. Thus, the performance problem is further simplified and depends only upon G. - the four performance factors 4 » ¢, LX ads , (1 908 CONCLUSION This study shows that a sounding rocket propelled by successive impulses can theoretically reach heights of much use to thoseinterested in obtaining data on the structure of the atmosphere and extra-terrestial phenomena if a propelling unit gives an exhmst velocity of 7000 ft. per second or more. The possibility of obtaining such exhaust velocities depends on two factors: first, the ability of the motor to transform efficiently the heat energy of the fuel into kinetic energy of the exhaust gases, and secondly, the amount of heat energy that een be liberated from the fuel. In an actual motor which burns its fuel at constant volume by igniting a powder charge in the combustion chamber the ratio of the chamber pressure to the cutlet pressure drops from a maximum at the beginning of the expansion to zero at the end of the process. {t is not possible to design a nozzle that will expand the products of combustion smoothly during the whole process. Therefore, the attainable efficiency must be less than that of a corresponding “constant pressure” motor which has a mixture of combustibles, e.g. gasoline and liquid oxygen, fed continu- ously into the combustion chamber at a constant pressure equal to the maximum pressure of the “constant volume” motor. However, very high maximum chamber pressures (up to 60,000 lbs. -91 per sqein.) can be developed in a motor using constant volume burning, while the chamber pressure of a motor using constant pressure burning is limited to mich lower pressures by the difficulty of feeding the combustibles, Therefore, the efficiency that can be obtained frommtors using sither of these processes should not be very different. As to the heat thet can be liberated per unit mass of fuel, the present fuel, such as nitro-cellulose powder for a constant volume motor, is much lower than the liquid combustibles such as gasoline and oxygen for a constant pressure motor. These considerations indicate that the sttainable exhaust velocity of a "constant volume" motor for propulsion by successive impulses will probably be lower than that of a “constant pressure” motor for supplying a continuous thrust. This is the reason why many experimenters abandoned the "constant volume" motor and turned to the "constant pressure" motor, the so-called liquid propellant motor. Theoretically, this defect of the “constant volume" motor can be compensated if a small total number of impulses (c.f. Fig. 4.3) is used. However, the use of few impulses is of doubtful practical value because the resulting extreme accelerations will be harmfal to instruments carried and will necessitate a heavier construction of the rocket, -92- However, even with the lower exhaust velocities of the “constant volume" motor it is shown by the analysis in this paper that with the exhaust velocity of 7000 ft./sec. obtained experimentally by R. H. Goddard (Ref. 1) it should be possible to build a poder rocket capable of rising above 100,000 feet, Thus it seems to the author that a rocket propelled by successive impulses has useful possibilities and further ex- perimental work is justified. l. Ze 36 Te 8. REFERENCES ON PART (IV) Goddard, R. He: “A Method of Reaching Extreme Altitudes", Smithsonian Miscellaneous Collections, Vol.71, No. 2, 1919. Damblanc, Le : "Les fusées autopropulsives a explosifs", L'Aerophile, Vol. 13 pp.205-209 and pp. 241,-2h7, 1935. Bartocci, As: "Le éscursioni in altezza col motore a reazione", L'Aerotecnica, Vol 13, pp. 1646-1666, 1933. Breguet, Le and Devillers, Re : "L'Aviation superatmos- phérique, les aérodynes propulsées par reaction directe”, La Science Aérienne, Vol. 5, pp. 183-222, 1936. Ley, W. amd Scheefer, He: "Les fusées volentes météoro- logiques", L'Aérophile, Vol. li, pp.228-232, 1936. Melina, F. J. and Smith, A.M.O. : "Analysis of the Sounding Rocket", Jour, of Aero. Sciences, Vol. 5, pp. 199-202, 1938. Whittaker and Watson: "Modern Analysis", pp. 26-247, lth Edition, 1927, Cambridge. Davis, H. T. :"fables of the Higher Mathematical Functions", Vol. 1, pp. 277-34), 1st Edition, 1933, Principia Press. 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