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3 LI"d I ) J0 UL I I L NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORY ,NDUM NO. 1196 ONSSTATIONIARY GAS FIOW IN THIN PIPES OF VARIABIE CROSS SECTION* By G. Guderley iM 1siLCT Characteristic methods for nonstationary flows have been published only for the special case of the isentropic flow up until the present, although they are applicable in various places to more difficult questions, too. The present report derives the characteristic method for the flows which depend only on the position coordinates and the time. At the same time the .treatment of compression shocks is shown. To simplify the application numerous examples are worked out. *'"ichtstationrre GasstrEmungen in d6nnen Rohren verfnderlichen Quersehnitts." Zentrale fi~r wissenschaftliches Berichtsvesen der Luftfahrtforschung des Generalluftzeugmeisters (ZWB) BerlinAdlershof, Forschungsberioht Nr. 1744, Braunschweig, Oct. 22, 1942 NACA TM No. 1196 1. INTRODUCTION In papers by F. SchultzGrunow and R. Sauer methods have been developed recently for completely solving the problem of nonstationary isentropic gas flows in a pipe of constant cross section. An expanded view of the problem is the basis for the present report. Flows are considered, which likewise depend only on the position coordinate; however, the cross section of the tube need no longer be constant and the entropy may vary from particle to particle. The method of solution applied here has been discovered almost simultaneously in several places, by Adam Schmidt, W. During, and F. Pfelffer, among others. The application of the characteristic method is possible without a previous substantial knowledge of mathematics. Corre spondingly, if a derivation was desired too, one could be had which did not make any special mathematical demands on the reader. As a model, the Busemann derivation of the characteristic method 2 for twodimensional stationary gas flows might possibly do It is actually possible to apply this derivation immediately to the 1 SchultzGrunow, F.: Nichtstationhre eindimensionale Gas bewegung. Forschung auf dem Geblet des Ingenieurwesen; Bd. 13' (1942) pp. 125 to 134. Sauer, R.: Charakteristikenverfahren f'Mr die eindimensionale instationare Gasstr6mung, IngenieurArchiv, XIII Vol. (1942) pp. 79 to 89. Vorbereitende Untersuchungen sowie Anwendungen finden s1ch in den Arbeiten von H. Pfriem. Zur Theorie ebener Druckwellen mit steller Front Akustische Zeitschrift Jahrg. 6 (1941) part 4. Die ebene ungediapfte Druckwelle grosser Schwingungswelte, Forschung Vol. 12 (194) pp. 51 to 64  Reflexionsgesetze fUr ebene Druckwellen grosser Schwingungswelte, Forschung Vol. 12 (1941) pp. 244 to 256 Zur gegenseitigen Uberlagerung ungedampfter ebener Gasswellen grosser Schwingungsweite, Akustische Zeitschrift Jahrg. 7 (1942) part 2 Zur Frage der oberen Grenze von Geschossgeschwindigkeiten Zeitschrift f. techn. Physik 22 (1941) pp. 255 to 260. Eine weitere Anwendung findet sich bel G. Damk'dhler und A. Schmidt, Gasdynamische Beltrage zur Auswertung von Flammenversuchen in Rohrstrecken. Zeitschrift fir Elektrotechnik Vol. 47 (1941) pp. 547 to 567. 2Busemann, A.: Beitraa Gaestnamik in Handbuch der Experimental physik (WienHarms) BE. 4, Tell 1, p. 421 and adjoining pages. NACA TM No. 1196 isentropic nonstationary flow In a pipe of constant cross section and from this by means of some supplementary physical concepts succeed in getting a treatment of flows in a tube of variable cross section; this is the course which had been taken, originally. In comparison to the mathematical theory of characteristics, however, these considerations operate with a Jack of clarity sufficient so that the mathematical theory for ths engineer too can be represented as the best approach to the characteristics method. Considerations necessary for the present problem are now 3 brought forward from the characteristics theory As a result, equations for the directions of the characteristics as well as conditions which must be satisfied along the characteristics are obtained. Proceeding from these relations, the next sections develop the actual method of computation. Next, the characteristics method for the case which is familiar by now, that of isentropic flows in a pipe of constant cross section,is deduced again and the transformations appearing there are used. to simplify the computation in complicated ctses, too. Since this is not always possible, the most general form of the characteristics method is shown in a later section. A'ter this, the formulas obtained for the special case of an ideal gas with constant specific heat are simplified and the consideration of boundary conditions explained. The remaining sections deal with celculatior of compression shocks; the known relations which connect the phase befor and behind a compression shock with one another are set forth in a convenient form for the present problems and with that the calculation of a compression shock in a flow is carried out. The theory is illustrated with suitable examples treated in detail. In that regard, it semed advantageous to avoid definite problems of technical interest, in doing so gaining the possibility of working out examples under very general assumptions without excessive effort. It Is hoped that, nevertheless, the application of the method to physical problems offers no additional difficulties worth mentioning inasmuch as thr earlier publications contain such applications. The author expresses his thanks to Dr. Hans Lehmann for working out the examples. 3Ccmpare CourantHilbert. "Methoden der mathematischen Physik II", p 291. Guderley follows the representation given by H. Seifert at the same institute for Gas Dynamics in lectures. NACA TM No. 1196 2. BASIC EQUATICKS Consider nonstationary, perfect gas flows in a pipe with a crosssection that varies in space and time4 in the neighborhood of the flow tube; that is, it is assumed that the velocity and the phase over a cross section of the pipe may be considered as sufficiently constant. In general, this assumption is Justifiable only if the thichIess of the tube, relative to its length, changes slowly enough. Only for flows which have as surfaces of constant phase parallel planes, coaxial cylinders or concentric spheres need this limitation be ignored. 'Thse flows with plane, cylindrical, or spherical wave propagation are included as special cases In the present problem. To stress the relationship to stationary twodimensional flows, let the axis of the pipe be vertical, the position coordinate be y, the time be t and plotted horizontally. In this ytdiagram the flows pre investigated. (Compare fig. 1.) Let p pressure a entropy per unit mass 0 density v velocity F = F(yt) the cross section of the pipe, let F be given In a region free of compression shocks, the flow is described by the dependency of the density on pressure and entropy, the Newtonian principle, the equation of continuity and the statement that the entropy of a particle is preserved, as follows: p = p(s,p) (1) 1 pE + v L + 8 = 0 (2) S+ (4) Problems with time variations in the crosssectional area are rare; they were included, since they can be handled without additional difficulty. NACA TM No. 116 5 In this the derivatives of p should be replaced by the derivatives of p and s, for this purpose 0o 1 (9) ~p 2 a is introduced. Therefore, instead of equation (3) V op y be v 1 Nr o s 1nF + znF SB.+ v + p + +  pv + p = 0 (3a) a2 y sB 6y )y a2 ot sB at Y at is obtained. 3. FROM THE THEORY O' CHARACTERISTICS In regard to the system of equations (2), (3a), and (4), the familiar question is raised from the theory of haracteristics. In a region of the ytplane let the solution of this system of equations and its derivations be finite throughout. On a curve C placed in this range let the values p, v, and s which correspond to this solution and, therefore, the appropriate derivatives taken in the direction of C be known. The question is asked whether the derivatives in other directions may be cmnnuted with the aid of the system of differential equations, and under what conditions. To answer this, a curvilinear coordinate system Ss is introduced in which a curve = constant coincides with C (fig. 1). All the derivatives with respect to along this curve = constant are given, the derivatives with respect to are sought. This trans formation is carried out and terms are arranged so that the unknown derivatives with respect to are on the loft and only known quantities are on the right. That is = ,(y,t) n = r(y,t) a..=a +. + dt 0 dt q o t NACA TM No. 11q6 With this form (2), (3(a)), and (4) are obtained a]2 .Iv V + 4 _n 1vb 1( 3' 3 0 y+ ',+  3+)v LS / ( L v&2 )y at/ a ) 3y 3qn y 6t as P y aat v o + v L9 +  and the unknowns, themselves, are obtained by Cramer's rule as the quotient of two determinants. The determinant in the denominator is the same for all unknowns. It always ives a singlevalued solution for the system of equations, if the determinant in the denominator is different from 0. In the other case with a determinant in the denominator that vanishes, it is a necessary condition for the existence of solutions that remain finite that the determinants in the numerator also vanish. In this case, however, the solution of the system of equations is only defined over any portion of the solution of the homogenous system. In application to the system of equations (6) signifies the following: The determinant in the denominator is formed from the coefficients of the unknowns. Considering a fixed point on C, at which p, v and s are known by assumption, the coefficients depend on andi n fo, that is the direction of C. If C is so directed that the determinant in the denominator does not vanish anywhere, the P soetc. are computable as single valued. Of greater interest for our considerations ps the other case, namely, that the determinant in the denominator is zero at every point of C. Such a curve is termed a characteristic. Because of the assumption of finite derivatives the determinants in the the r.ssulmption of finite derivatives the determinants in the NACA TM No. 1196 numerator also vanish. Relations are obtained thereby, in which the right side of (6) and, therefore, the derivatives of p, v, and s along C appear as essential ingredients. These relations represent the starting point of the graphical numerical method of solution. Since the solutions of a linear system of equations are no longer single valued for vanishing numerator and denominator determinants, the pursuit of a given solution of a characteristic is possible in various ways. These different possibilities actually appear on changing the initial and boundary conditions. 1. THE DIRECTIONS OF THE CHARACTERISTICS To find the directions for which the curve = constant is a characteristic, the determinant in the denominator must be set equal to zero in the solutions of the system of equations (6). v 0 Ia +. WI/ This gives v ay at o __ .Ly 1 a2 ( y + / 0 From this are obtained the conditions S + = or v + a) + = 0 %. Fy^ 5t (Ba) oyr + ta 3,7 at) oy ot NACA TM No. 1196 (v a) 3 = 0 3y cit (8b) The slope of any curve = constant is given by dt aS ). From (7) and (8), together with this, the slopes of the charac teristics are = v dt Iz = V + a At _ v a dt (ioa) (l1b) The characteristics defined by (9) are pathtime curves for the individual gas particle) they might be termed life lines of the particles. According to (10), velocities are determined from the slope of the other characteristics, which differ from the velocity of the particles by ta. For stationary flows the Mach waves correspond to these last characteristics; this designation will be adopted. Therefore, let Mach waves of the first family be those which spread out with the velocity v + a and Mach waves of the second family be associated with the velocity v a. 5. THE CCOSISTECY CONDITIONS CN THE CHARACTERISTICS As shown in section 3, along the characteristics, certain conditions must be complied with by the derivatives which result NACA TM No. 1196 from the vanishing of the determinant in the numerator. These conditions are called consistency conditions, for p, v, and are subject to them, if the derivatives with respect to are to remain finite. If the right side of (6) is designated Ri, and R in sequence, then the following is obtained for the determinant in the numerator of the quotient for op: 67 v + F 3y Ft V \ I R 0 This determinant must vanish to give teristics. Substituting (7) gives R1 0 R2 o ~.y 0 R O0 0 + 0 +St at J (11) the directions of the charac 0 0 =0 0 According to this the determinant (11) vanishes (8a), that is, for a Mach wave 1 Ra a 1 0' R2 0 a 3 0 a 3 .Y by itself. =0 With is obtained, or NACA TM No. 1196 [ oaRl a2RP + O aR 0 In this, is certainly different from zero, as long as v and a are finite and grad E 0. As the condition for Ahe Mach wave 1 is t'btained (12a) RI aR2 + a 3 R3 = 0 The consistency conditions for the Mach waves 2 is, if a is replaced by a oR1 + aR2 a R = 0 (12b) A condition for the life line is obtained if determinant in the numerator in the quotient from (6) requires S)y LAs a2 )y a v1 y + ot 0 + } the vanishing of the for 8 to be got = 0 p2 R3 3 For the Mach vave this equation for the life line is is satisfied by itself, the condition R3 = 0 3 (13) The determinant in the nu''Lerator of could be investigated, too; however, this vould not give any new consistency conditions. NACA TM No. 1196 The values of Rl, From (12a), R2, and R3 are still to be put in. + 0 i i (v 1W 1;V + a) + 1 = apv ~nF a .InF =p y Ft. is obtained. The direction d' along a Mach wave 1 dt (10a); on that account is given by d. + = (v + a) + 'l is valid for it. With that the consistency condition for the Mach wave 1 can be written in the form A+ dv a tv +F + )71n\1 ao dt dt n T + (14a) The consistency condition substituting a for a for a Mach wave 2 is obtained, by _dp dv a v 2nF ap at dt T + Jr. 6t (14b) From (13) for a life line is obtained 5s = This may be integrated immediately a = constant (15) Naturally this constant will differ from particle to particle, in general. H (v+ a) + a d][d t NACA TM No. 1196 6. FLOWS WITH CONSTANT ENTROPY AND UNIFORM PIPE CROSS SECTION Equations (14), (1), (9), end (10) just obtained are certainly useful, fundamentally, as a starting point of a characteristics method in fact, there are examples, where it is necessary to revert ro them comparee section 11); in most cases, however, there are still other transformations suitable. The direction in which to proceed for these is obtained if an attempt is made to derive the characteristic method for isentropic flows in a pipe of uniform cross section from equations (14) and (15) possibly in the form applied by SchultzGrunow. To emphasize the fundamental ideas, no assumptions of any kind are made therein of the characteristics of the flow medium. On account of the hypothesis of constant entropy, equation (15) satisfies itself. In equations (14) the right sides are omitted since it concerns a tube of uniformm cross section. Further, on account of the hypothesis of constant entropy the state of the gas is still dependent as only one variable, perhaps the pressure, or the temperature; the quantities appearing on the left side and a are accordingly functions of this variable. It is possible, therefore, to consider the expression LP as a differential. Let pa T temperature i heat content (enthalpy) s entropy By the second main theorem T ds = di i dp From this, on account of the hypothesis of constant entropy, o uldT .dTm With that, it follows that a 1 (di' dT pa a dTl NACA TM No. 1196 W(T) = T d dT aoT a 'dT o (17) is introduced in which ao is the sonic velocity of a comparison phase which was added to make W dimensionless. The phase of the gas may be characterized by W from W = W(T). It follows that T = T(W) (18a) Further, it is valid that With the use of a ao p = p(T) = p(W) a = a(T) = a(W) etc. W equations (14) appear in the form1 dW + dv = 0 for a Mach wave 1 dW dv = 0 for a Mach vave 2 (18b) (lic) Bringing in (19a) S= W+ ' a0 So S= W. .L. Uo (19b) these last relations change to a form which may be integrated. This gives X = constant for Mach wave 1 (20a) S= constant for Mach wave 2 (20b) If the magnitudes of X and j are known for a point of the ytdiagram, the velocity is thereby completely defined as well as the thermodynamic phase. It is, to be exact, WX+I 2 ao 2 (21a) (21b) NACA TM No. 1196 and on account of equations (iR) p = p(X + u) (22a) a = a(X + ,) (22b) The next sections explain this transformation and the application of equations (20) to an example of an ideal gas whose specific heat is a function of tmriperature. 7. TFERMODYNAMIC RELATIONS FOR PN IDE.L GAS WHOSE SPECIFIC EEAT IS A FICTION OF TEMPERATURE; COMPUTATION OF W Let c specific heat at constant pressure cv specific heat at constant volume R gas constant For an ideal gas E PT (23) 0 According to the second main theorem, if p end T are considered as independent variables ds = 1 i dT + 1 d~ dpp (24) T 'T T ,p pT Since ds is a perfect differential, S11 1 i T i 1 Bp ',T T/ BT T p T Accordingly, si.ibstituting p fron (23), the following known fact is obtained p 3p NACA TM No. 1196 that is i = i(T) Fp = Cp p = DC (T) From (24) as a result =p ds =  T dT R dL P and front this by integration dT n .L T Po the index o Introducing characterizes a comparison phase. I/ SR P = e P  Considering c.(T) as known, p by (28) and p by (23), are given as functions of T and s; the thermodynamic properties of the medium can be calculated in principal, therefore. The now therefore (25) (26) (2ha) s so /T C, R T/ R (27a) (27b) (28) NACA TM No. 11Q6 quantity is by all According a is also defined by p and p. The computation of means simpler if carried out in the following way. to (5) for which the entropy is to be kept constant. For constant entropy from (24o) dT R= dR T c. p by differentiation from (23) d do dT P p T From the last two relations together with the familiar relation Cp C = R is obtained .c a(T) =/P2 RT SCv (29) The relations discovered up until now describe the properties of the gas and must always be knownT it makes no difference which variation of the characteristic method is chosen for the calculation of the flow. In contrast, the introduction of the functions W, X, and L serve only as prenarai.lon for carrying out of the charac teristics method in the form presented in the preceding section. Next, to compute W. Fron (25) it follows (\, .'5 01 * '6F/ with (26) , 6T3 p 8~\ c 7a NACA TM No. 1196 17 Setting the last equation as well as (29) into (17) .gives iTV = Ti (30) 2 o To ET For the present case, 9 = constant = so (28) becomes (3 'T R T L (T) = e (2a) ~0 Po Now the following can be formed T = T(% + p) a = a(X + P) P P(X + u) These calculaEti.?ns were crr1?d. out, numerically for carbon dioxide. The relation between the specific heat and temperature was taken from Hi.,tte with the aid of these values (i ,,)a a/ao, .P, and W can be co1i !.:ed from 3qi.:ations (26), (29), (27a), and (30) as functions of the temperature. (See figs. 2(.) nd s(b).) Figure 3 shows a.a P, and T plotted as functions of X L = 2LW. 8. THE COIISTRUCTTOi OF TEE FLOtJ FIELD The following problem. should be dealt with: Along a curve K of the ytriagra., which has at the most one point in corrimon with each characteristic, let p,p and v/ be giv;n (fig. ). The flow should be constructed f.or the following times as far as it is defined b,' the portion of K iion. Ther~f'.re, it is concerned here with the coTiputation o.f thl part of th flow defined by the initial conditions vhich by th same rga,.mcnts appear everywhere in the interior, too. Before t]h construction of th: flow can be started fitte, 27th edition, Vol. 1, p. 48, table 5, Berlin 1941, Wilhelm Ernst und Sohn, publishers. NACA TM No. 1196 the initial values p/p0 and v/a0 must be expressed in terms of the variables X and p.. Since the entropy was assumed constant, p/p0 and P coincide. With the aid of the relation presented in figure 3, between P and X + p and equation (21b), X and p may be ascertained without difficulties. Figure 4 shows en the right, the ytdiagramon the left, the diagram of the assumed values P/Po and v/ao as well as those of the computed quantities X and i as functions of y. Proceeding from the individual points of K if the network of Mach waves had been brought in. the ohase at each lattice point would be determined thereby; according to (20) X is constant along Mach vave 1, p along Mach wave 2, and on that account, equal to the values at those omins of K from which the Mach waves spread out. By (21b) end (22) the phase is given by X rid p. To be able to dray the nei'ork of Mach vaves, only their directions are still needed. These are given at the lattice points by (10); a/!a is a functimn of X + p in figure 3, v/ao is computed as X . 2 The direction for the portion of a Mach wave between two lattice Dnin+s is approximated as the average value of the corresponding directions at the lattice points. The construction becomes especially Lnple if the Mach waves are drawn for equidistant values of X and p. The directions of the Mach waves appearing can be co.nputed beforehand and possibly prepared in the form of table I. The interval between adjacent values of x or u was selected as 0.1, the size of the interval depends on the accuracy desired. In the table the upper column headings and signs refer to Mach wave 1, the lower to Mach wave 2. The numbers entered in the table represent the average values for (v + a)/a, and (v a)/ao. For Mach wave 1 for which X = 0.3 and which leads from a point with p = 0.2 to a point with 4 = 0.1, in the column with the heading X = 0.3 the value is to be taken from the row p = 1.5, that is, (v + a)/ao = 1.103. In the flow diagram the values of X valid there are entered to the lift of the lattice point and the values of p to the right. To determine, for example, the position of C from the points A and B since the phass of C is given beforehand by X = 1.1 and = 0.5 the average directions of the Mach waves (v + a)/a0 = 1.422, (v a)/as = 0.778 can be taken from table I and dra'n in the ytdiagram. The auxiliary diagram on the left in NACA TM No. 1196 19 figure 4 can be used for this. There the direction of a Mach wave for which (v + e.)/ao 0.8 is drawn in. Similar diagrams can be used as aids for the following exa:,ples, too. The portions of the Mach ,vae goin0 out from K really require a special co~ioutation since the average values of X or for them do not agree, in general, with the valL.es of ta3be I. The small deviation was tolo'rable, however. 9. FLOWS WITH CONSTANT E!TTROPY IN A PIPE OF VARIABLE CROSS SECTION If the cross section of the pipe is net constant, the right side of equations (14) from which it is necessary to start out, here too, are reserved. With that, there is the possibility of undertaking that integration along the Mach waves which led to equations (20). Nevertheless, the introduction of X and L still re.nains useful. Setting + = M (31) then dxk 84 (32a) dt is obtained as the consistency cndllon fir Mch wave 1 and .a a 1. (32b) dt o fr Mach 'ave 2. The consistency conditions in 'he fori. of (0i) contain at any given time the differen.ial of only one of the Lnknoun quantities X or p while the differentials of both p and v appear in (14) already. This implies an appreciable improvement in the numerical calculation. The construction of the flow rests on the fact that equations (32) are considered different equations. LAt GA be the value which a quantity G assumes at the ooint A, CGA the difference GB GA and GmBA an average value of G takun between A and B. NACA TM No. 1106 Applying equations (32) to commute from two known points A and B the phase at a third, C, which is on the same Mach wave, in the form XC XA = 'XC,A = MACao tA C( PC LB = = .CB MBCao tCB For the determination of the flow XC and PC have to be computed by memns of these last aquatins and, at the same time, the position ascertained of the points sought in the ytdiagram by the use of equations (10). The calculation process might be explained by an example. The flow is cnsidered as givcn along a curve of the ytdiagram and, admittedly by X and 4 (fig. 5, table TI). In addition, the pine cross secti'n must be r known fimncion of y and t. For that it is only necessary to require that F can be differentiaLed 'ith respect to Tnsltion and tine, a premise which is always fulfilled In Practice. For this exa sle F is taken in the form F = F t From ('1) for M M = ( 2 +  a\ ao0 y a0 The positions y = 0 and t = 0 for which M goes to infinity do not belong to this region of flow where such singularities appear (for example at the center of spherical waves); it is necessary to make special investigations which cannot be entereA into in the present report . The best way to follow the calculation is by means of the systematic calculation in table II. To facilitate comparison with the description the columns are numbered. The first column contains the designation of the point which is to be computed, the second column gives the known point which, in common with the point to be computed, is on Mach wave 1. Column 3 contains the corresponding 7Compare G. Guderley. "Starke kugelige older zylindrische Verdichtungsstisse in der IHhe des Kugealittelpunktes oder der Zylinderachse." Luftfahrtforschung, Bd. 19 (1942).pp.302312. The concerns itself with a complicated special case of such a singularity. NACA TM No. 1196 point for Mach wave 2. The first five rows reproduce the initial values as vell as some further values that hold at the given points which are necessary for a later calculation. The calculation of a new point is carried out in the form of an iteration method; as an example the point 4 will be explained. Next, the values for X4 and u14 are estimated (columns 4 and 5). In order not to use too favorable an estimate, it is assumed that )4 = 1 and P4 = p2 The quantity (X + 0)4 is determined for these magnitudes and fro'i that, with the aid of figure 3 and, \ ao.4 / farther on ( (columns 6 and 8). With these values v + a .o4 ao ,4 anda are computed columnss 9 and 10) Now the average \o J4 directions for Mach waves 1 and 2 v + a and v a \o ml,4 a4 are formed (columns 14 and 19) and the Mach waves are plotted on the ytdiagram. From this Y4 and aot4 (columns 12 and 13) are obtained. With these values M4 (column 11) and the average values Mml,4 and Mm2,. (columns 15 and 20) are computed. To continue for Mach waves 1 and 2 Aaot4,1 = aot4 aot1 and Saot4,2 = aot4 aot2 have to be computed (columns 16 and 21) and can be substituted in equations (33). The quantities AX4,1 and Ap J. as well as X4 and g4 (columns 17, 18, 22, 23) are obtaineds If the values X and p calculated in this manner do not agree well enough with tho. original estimate, the calculation must be repeated in which X and p just calculated appear in place of the earlier estimates. Naturally, the Mach wevee must be plotted over again, too, in the ytdiagram for this. These figures only show the final form at any instant. For that reason all the steps in the iteration method are put in the tables. A good view of the results of the calculation as well as insight into the estimates to be carried out by the iteration method is obtained, if the flow is followed simultaneously in a Xpdiagram, as well as the ytdiagran (fig. 5, right). There the Xaxis was selected slanting up to the right at 450 and the paxis downward at 45. With a suitable vertical scale X t, and therefore v/aos is obtained in ediately on a horizontal scale X + i or W and with the use of unequal distributions a/a and P and, for isentropic flows D/p, too. The X and 4aics were inclined 450 to obtain the quantities of physical interest v/a a/ao, etc. in a coordinate eystein with the conventional arrangement. 22 NACA TH No. 1196 10. FLOW OF AN IDEAL GAS WITH ENTROPY DIFFERENCES The introduction of X and g with the object of obtaining equations in only one unknown, at any time, with the iteration method for the determination of the flow was possible up until now because the expression dL with constant entropy might have been Da considered as the differential of a function W independently of the characteristics of the incident gas. Naturally, that is no longer possible with variable entropy. The computation of the flow must, in general, therefore, return to (14). The ideal gases constitute an excOption. Here, as recognized in (30), the function W which essentially agrees with f IT for constant entropy, Uca depends on the temperatilre alone, and no longer on the entropy. If the expression 1d is considered, therefore, in the, case of oa variable entropy as dependent on the variables T and s the effect of change in entropy is separated, thtn the rnst can be written here as a differential and X and ii ccn be introduced as previously. The change in the entropy along the Mach waves must naturally be regarded separat:.ly. This is oossibl3 without especial difficultiio since the entrony is constant along the life lines. The transformation re carried through in the following mfnner. From the second law T ds di 1 do 0 kingg into acco nt (26 and (29) 2 d T as /Cpc dT /v d' a a a RT / Inrroducinc t, X, and p as before, the consistency conditions are nbained in the form __ = j lF + a + /"v T J de (34a) for Mach wave 1 dt ao t Vc R CI dt du a (vtnF + IrF + /c T 1 _s (34b) for Mach wave 2 dt .o )y At / R a, dt NACA TM No. 1106 23 The differential quotients ds/dt formed along the Mach waves interfere: The following transformations are possible. Analogous to the flow function of twodimensional stationary flows, a function is introduced, VI is constant along the life lines. This can be achieved by requiring that l4 F _2_ (35a) E Fo PO F .P (35b) ay Fo o0 Along any curve of the ytdiagram + dt (36) Along a life line d = v therefore dt SF o F a v y" Fo o, F Po v that la is actually, constant along the life line. At each point of the ytdiagram \ itself can be defined by a line Irjtngrn] that leads from a fixed point A at which might be zero to B. BB Sdy + T (37) B The physical significance of can be recognized as follows: Let C (fig. 6) be the intersection point of the life line through A with the line t z constant through B. To begin with, the path of integration is along the life line fromn A to C and, from there, out along the line t = constant to B. Along the life line AC, is constant S= A = 0 24 NACA TM Nc. 11Q6 along the section CB, dt is equal to zero, accordingly .B IB = / LFL d B C F., 00 From this, it is evident that I represents the mass which is enclosed between the par+icles at an instant in time for which J is zer. The fact that s is constant along a life line can be written with the use of I1 in the forn (38) S = s(4) For ds/dt then ds ds d dt W a dt for which dX. is to be taken, just as dt the Mach wave considered. From (35) and (10) Wt 2 P a dt F0 pO d Fo 0O d' F, P ds/dt previously, along for Mach wave 1 for Mach wave 2 Substituting these in equations (34), allowing for (23), (28), and (29) replacing is according dd to (27b) by 1d and po/o 7 d 0 C V by o a0 yields the f'lloving consistency; conditions: Po NACA TM No. 11q6 For Mach .ave 1 X_ = a a v nlnF dt n a, \ y For Mach wave 2 at a v dt o \ ^ Here P is a function of X + j (fig. 3), a function of y and t. From (38) and ( F/IF is known to be 27b) it follows that n = n() and from this Id li (*) = d, d^' constant for a life line For the sake of corntactness, introducing C C F d N the for Then (39) goee over into the forn dX a (M + N) dt o S=ao( N) dt (41a) for Mach wave 1 (41b) for Mach wave 2 Equations (40) and (41) supplant the previous consistency conditions (15) and (14). Cv F p Cpo F p0 o and dEj + ao 1 + so (39a) 3?nF, + t ) C0 Cpo ^_ I P dL (39b) (40) bInF dt NACA TM No. 1196 Before starting the characteristic construction, the problem arises here, too, of computing X and p, and now da besides, from the initial values. Along a curve K of the ytdiagram let the velocity be given by v/ao, the phase of the gas by P/Po and T. From T with the aid of figure 3 X + p is obtained, from v/ao, X i; with this X and p are known. Since p/Po are given, and P as a function of T is to be gathered from figure 2, n is obtained iLmmediately from (28). As a result of plotting it against the values of y from the curve K and differentiating is obtained. From (36) and (35) together with dy (23) d_ for the curve K may be computed for the curve K and, dy finally, with that V dy dy is deterlined. In many cases 1hese computations are superfluous; if entrnoy differences arise fr'm compression shocks, the determination of dT X and p includes their calculation. The dV way the computation of f3ow has to be carried out is shown in figure 7 and table III with points 4, 5, 6 as exunples. The related Xpdiagram is right center. (The points included, in addition, in the table and the figures relate to a later section.) Along the curve K (points 13) X, p, and dA are assumed as dt known, in the auxiliary diagram dn has been reproduced as a function of y. The computation of a new point take point 4 as an example begins, here too, with an estimate of X and i (table III, columns 4 and 5). After that, as before, the following ev/ v + a\ are computed (X + p)4; (a/ao)4; (v/ao)4 ; ( Lv4Y a( o /4 a /4 ( ac ; va i 1 ; (columns 610, 19 and 24), the position a ml,4 a0 /m2,4 of 4 is indicated in the ytdlagram and y4 and at4 in the table (columns 11 and 12) assumed. The determination of d6 with the aid of the life lines enters in as something new. It should be sufficient for this to draw in a multitude of life lines, simultaneous NACA TM No. 1106 with the construction of the Mach waves and going back over these to learn the desired value 1 from the auxiliary diagram. The di kXdiagram is useful for a quick determination of the direction of the life lines. The position of the intersection points of the life lines with the Mach waves may be estimated there without difficulty, and then the average velocity learned. (Compare points 14 and 15 in the yt and in the Xpdiagrarn.) After dL has beenn found and, in addition, P has been learned from diagram 3 (columns 13 and 14), MI ani N, as well as (M N)4 and (M + N)4 may be co'Dured (c.lumns l'18), the average values (M lN)i 4 and (M + N)2, for the Mach waves be forced (colurrns 20 and 25) and vith 'he aid of A a t (columns 21 and 26) from equations (4l) comute AX and A& and, ulitimately with that X and i. (Coluns 22, 23, 27, and 29.) Where the original estimates were too bad, ihe computation 'as reTpeated. 11. THE GENEPALIZED FORM OF THE CHARACTERISTICS METHOD An outline shall be giv:n of how to proceed if the simplifications given above are no longer possible or if the flow is so small that the prepared computations as given at the end of section 7 do not pay. As an example, let the caipuitcation of the point 4 be carried through from the points 1 and 2 of figure 7.(See fi.. 8.) The quantities pl/p = 1.44; nl = 1.2; Vl/a = 0.425 p2/Po = 1.866; r2 = 1.332; v2/ao = 0.400 correspond to the initial values assumed there. For the medium to be investigated p and a must be giv=n %s functions of p and it. In this case P is obtained, first of all, from (28) and from that and fiGure 2(b), T. Then c/o and a/a are obtained with (23) and (29). H.:nce al/ao = 1.021; 01/00 = 1.375; = 1.46 0 a ao '71 a2/ao = 1.037; o02 / = 1.710; = 0.637 \ o .2 28 NACA TM No. 1196 besides M1 = 1.747; M2 = 1.442 can be computed. Here, too, an estimate is made in computing a new point. For example I1 + '2 4/po = p/p = 1.44; v4/ao = v/ao = 0.425; 14 2 1.266 With this P4 = (p4/Pol/4 = 1.137 is obtained, whence T = 282.5 Continuing further a4a = 1.014; 04/0 = 1.390; ( + a4ao 1.439 74 a) o = 0.589; (v + a)l, a/o = 1.443; (v a)m2, / = 0.613 With that the position of point 4 in the ytdiagram may be found, giving Y4 = 1.446; a0t4 = 1.258; (aot)4,1 = 0.048; A(aot)4,2 = 0.092 and, after further calculation M4 = 1.403 The average values are found to be /)l, = 1.33; a/aon14 = 1.0175; l, = 1.439 (/o) = 1.550; a/a 1.026; M,4 = 1.423 O ,/mn2,4 0 m2 ,4 NACA 2M No. 1196 Considering equations (14) as difference equations then7 00 1, 4 mlnJ,4N p0 1 o3 0, ,n 4 d m2,k a 0 4 \ a 2,4Po 0 1Po  a 0 aot r 0\ = a tk "ot)ml,4 = (ao t aot)m2,4 Replacing C by 2 Cpo from (29) and (23) gives C 0 A 7 )p 4 Po0' l,4 o C 0 CO m2, a /m2,k ao V+ a, v4 ao ao C \o \ a p1 CPO\ /mlp\ a )ml,4po  ,4(aot Aot) C So 0 a P2 S m2, ma 2,4 T0 0 M a t a 2 ,4 0 4 o 2 For ideal gases the first tern of the left. side of (14) may be written 1 d"np a, then o/o does not have to be computed k dt o separately. To permit the procedure to be applicable in more general cases, this simplification is not used here. + ao  NACA TM No. 1196 Putting in numerical values gives as a result 0.547 P4/po + v4/ao = 1.1430 0.o84 p/p v4/a0 = 0.3715 P4/Po = 1.470 v,/a = 0.342 4 O From the velocity computed above v4/ao and the velocity at a point 4', estimated for the present, of the connecting line 1.2, the average direction of the lif line passing through 4 is obtained by an approximation method. If this is proceeding from 4 backwards, the more accurate position of 4' is obtained. By interpolation between 1 and 2 V = n = 1.243 is obtained. Since the values Pi v4 P, , PT4 do not agree sufficiently well yet with the originally Po a' estinated values, the computation must be repeated with the magnitudes just obtained as starting values. This gives S/p = 1.478; v/a = 0.3383; 4 = 1.243 3.SIMPLIFICATIONS FOR IDEAL GASES WITH CONSTANT SPECIFIC HEATS Generally the flowing medium is an ideal gas with constant specific heat or at least can be considered as such, as an approxd .nation. In such a case appreciable simplifications are possible. Let k = C /C then k 1 C ; C R P ki k i NACA TM No. 1106 From equations (27s), (29), and (30) k Sk1 (T 5 aL P=\)r. iT 0T a = */. kPT, 0 / o0 W += + 2 / a 2 k 1 \ , 2 k1\0 With this, it follonE that and from that a = 1 + I ( + i) & 4 (42c) consequently, P= 1 + ( + )I k1 (42d) The directions of the characteristics are obtained from (9) and (10) in the for!n d= ao   _ a( + Ltd x dt 4 dy ao  k + 1 d for the life lines   ) for 4pech waves 1 + .. Xk for Mcch waves 2 4 / 2 . k lao k I a 0 1' r \ (42a) (42b) NACA TM No. 1196 The consistency conditions for the Mach waves remain unchanged in the form (40) and (41). M and N are expressed as follows, now, M I= i + + (X + )' I z F 1 + inF 4 a2 y at 2k 1 F =+ h: 1( + k1 T4 4 Ur, The directions of the characteristics may now be found very conveniently graphically. A construction which is suitable if the simultaneous treatment of the flow in a Xpdiagram is avoided is the contribution of Adam Schmidt.(Seo fig.9.)For the determination of the direction dy/dt for a life line, two vertical scales at a distance of 1 apart are used with plotted on the right one 2 and t5 on the left one as above. A life line for a phase which 2 is given by X and LI has the direction of the connecting line of the points concerned on the function scales. Similarly, there are scales to use for a Mach wave 1, which give 3 k on the k 1 on the left and 1 + k 1 on the right. For Mach wave 2 k + 1 3 ]4 has been plotted on the left and 1 + X on the right. In figure 9, the direction of Mach waves 1 and the life line is given for X = 1.1 and 0.6. If the chases in the course of the construction of a Xpdiagram are followed up, the following "ethodis suitable (fig. 10, right). A vertical line is sent through the 0point of the Xjsystem and the poles Pl' Po2 and PL are determined, where PL is on a level with the origin of the Xpsystem and *2 avay from it. P1 and P2 are directly below and above PL, respectively, and likewise the distance /2 froa it. To find the direction of the characteristics for a given phase, a horizontal ray and two rays slanting upward and downward at an angle arc tan k 1 are drawn. These intersect the vertical line through the origin of the Xpsystem at the points Q1, Qr, and QL. The connecting lines PiQ , P2Q2, and PLQL are the directions of Mach waves 1 and 2 and the life lino. In figure 10 the construction for point 4 is carried out. NACA TM No. 1106 This construction is especially convenient with a triangle having an angle arc tan k .. Figure 10 and table 4 give an example of 2 an application for the same initial values as in figure 7 and with Op/Cv = constant = 1.4. 13. BOUNDARY CONDITIONS If the flowing gas column is not infinite, the variation of the flow is determined by the phase at the start, in addition, also by conditions at its bour.darles. For example, a gas can be closed off by a Diston or rigid wall, flow out into a space with a given pressure, or be sucked out of th same. Generally, the boundary conditions may be formulated so that relations between the phase magnitudes of thu gas and its velocity along a curve of the ytdiagream are prescribed. The number of conditions which are needed for the boundary curve corresponds to thi number of charac teristics which run out from there into the interior of the flow. For example, the gas flows out of the end of the pipe into a snace with constant prssure, with v< n, thn tbe line y = constant is the curve for the nipe for vhich the boundary conditions are given. A family of Mich waves spreads out fro.r it inward, while the other family and the life lines reach this curve, approaching it from within. In this case the condition can be prescribed that the pressure in the exit section be equal to the outside pressure. If the gas is sucked in from outside, Mach waves of the one family proceed front the curve of the boundary conditions as well as the life lines. Accordingly, two conditions must be given. The one states that the entropy of the entering particle is the same as the entropy in the outer space, as a second it would be required perhaps that the phase. of the gas in the entrance section be related to the phase in the outer space through Ecrnoulli's equation8. (An exact formulation is difficult, since the flow at this location is no longer onedimensional.) If the characteristics of all three families cf a given curv. lead out into the iterior of the region to le computed, there ar. three, conditions to jrsecribo; this is the initial value problem already treated. The other extreme, that at the boundary of the region of Interest, generally, no condition can b. fulfilled,is physically conceivable too. For example, if a gas with v> a flows in a space at constant pressure, generally no characteristic goes inward from the outflow section. compare SchultzGrunow, loc. cit. NACA TM No. 1196 Actually here disregarding boundary conditions which force compression shocks no effect on the flow variation in the interior is possible frao outside. The treatment of boundary conditions is explained with two examples which are connected with the flow in figure 7. The oaf putation is entered in table III, as far as possible. The first example includes points 7 to 9 and, admittedly, it has been assumed that the gas column is bounded by a piston whose life line is represented in the ytdiagram as the curve 3, 7, 9. (Whether it is practicable to realize such a piston in a tube of variable cross section is unimportant for carrying out the computation.) The XpdiaCram referred to is in figure 7, upper riSit. To begin with, an estimate is made of the phase at 7 which has been chosen 7 = X = 0.00C, 7 = 13 = 0.050. Since the line 3.7 is the life of a particle, (dT is alread; known ard is uqual to ( . With this the values in columns 610 and 19 are calculated. As a result of drawing in the Mach wave 5.7, Y7 and aot7 (columns 11 and 12) are obtained and besides v7/ao from the direction of the life line at point 7 which has been reached. (This quantity is found in column C under the value computed from the initial estimates.) Now the quantities in columns 14 to 18 and 20 to 23 may be computed, the value v7/ao obtained from the boundary conditions will be used. With that X7 is already known. The quantity 17 is obtained from the relation v Xp ao 2 Inserting numbers 0.323 = 1/2. (0.47 n7); n = 0.229. Since the first estimate was too poor, the computation must be repeated. Point L is computed from 6 and 7 by the method explained in section 9. From 6, point 9 is obtained in the way Just described. This method of calculation is useful for any laws of motion of the pipe; a special argument is necessary only if a discontinuity appears. The discontinuity in the velocity is to be considered attained on transition of the boundary from a continuous velocity variation at very large acceleration. In the ytdiagram that means NACA TM Na, 1196 that the life line of the piston which has a bend at the instant of the velocity discontinuity is rounded off immediately. Then the flow may be drawn accurately just as previously. To obtain sufficient accuracy, enough points must be taken on the rounding off so that the velocity of the pistcn does not change excessively from point to point, and at each point a Mach wave of the first family may converge and a Mach wave of the second family may divergt from there. First of all X must be computed for the converging Mach wave and than from X and the velocity at the incident point I determined for each Mach wave. If the rounding off becomes smaller and smaller, these points on the rounding off draw closer and closer. With that the values of ) approach a single value, which may be computed from the field before the bend. The Much waves 2 spread out in the share of a fan from the bjnd end the fan includes all values of p which lie between the values of p for the velocity before and after the velocity discontinuity. For the second example, there is at the position y = yl an op en pipe end, through which gs is sucked in from outside and for which two conditions 'ust. bo specified along the boandarycondition curve. The curve is the curve 1, 30, 13 in figure 7. In the outer space let i = il; for the entering particle therefore di = 0. This is one boundary condition. As tha second boundary condition there is tha require nent th'nt the phase in the inflow section be related to the phase in the outer space by th? Bernoulli equation. This condition may be satisfied, already, :t point 1, accordingly i + v2/2 = 1i + 2 or also i 1 vi2 11 i 1( ll 2 S 10 v\2 ao (l = constant To determine these constants, from figure 3 the temperature T1 is taken for (X + 1), from figure 2(a) for T1, (i1 io)/ao. Then il 10 V'12 +o  + 0.292 a2 a NACA TM No. 1196 Since (i io)/ao2 is a function of T and, therefore, pf i + n, L = L this boundary condition can be plotted as the curve K o 2 in the Xcdiagram (fig. 7, lower right). At best, the computation of point 10 begins anew with an estimate for X and V so that the boundary conditioi e are already satisfied (columns 4 and 5). With this, the qurantities in columns 6, 7, 8, 10, and 24 are computed and Mach wave 2 drawn in with that. The quantity a tl1 is obtained in column 12, the values yl0 = yl and = 0. (Columns 12 dx and 11 are given beforehand.) Now the quantities in columns 14 to 18 can be obtained. To determine, with this, the quantity (M + N) in column 25 it is to be noted that (M + N) for the particle originally in the pipe has the value, perhaps, at point 4 and changes dis continuously for the particle recently sucked into the quantity (M + )10. On that account the life line is drawn, which separates the particles in the interior originally fron those particles flowing in from outside. This intersects Mch wave 4, 10 at point 11. Then the following is obtained (column 25) (M + N)m4 10 = Lrt(aot)4,11(M + N) + L(aot)11,10(M + N)10] The quantities in columns 26, 27, and 28 may be computed now. As a result of inspecting the curve of the boundary condition in the Xudiagram with the value of i found, X is obtained (column 23). The computation is repeated with the values found in this way. From points 6 and 10, point 12 is obtained in the manner described in section 9. In connection with that the difficulty just described appears again in finding the average value for (M + N). From 12 and the boundary condition, point 13 may be computed by the method just presented. The XiLdiagrams of the two last exaples were k3pt separate from thr Xpdiagram draTn for points 16 for the sake of clarity. NACA TM No. 11q6 If the various figures are visualized as being joined the upper diagram connected to the middle one ut the line 6, 5, 3, the middle one with the lower one at the lne 1., 4, 6 it is recognized that the plane is covered with several sheets which are connected long the figures of the characteristics. There is such a superposition, already, in the low1r Xpdiasram; tncre arc to be imagined inclosed the quadrilateral 10. 4, 6, 12 along 10, 4, the tri.ngle 1, 4, 10, along 10, 12 the triangle 10, 3, 12. In addition to the boundary conditions, transitional conditions can also appear in the interior of Lhe flow. In the example just discussed just that woula have been the caee. if in the outer space it vere different front at. At the location of such a discontinuity for n arpenent of pr3ssuro and velocity must be required. To go into such questions with greater detail lies beyond the scope of this report. 14. TRANSITIlOAL CONDITIONS AT COMPRESSION SHOCKS The flow in a given part of the ytplane is defined by the initial and boundary ccnditicn3 end tc calculabl. ty the methods derived up until. now. It 13 possibl that it might hcppon during th3 construction that regions of the ytpl'rz are covered with phase quantities several times. This is the sign for the appear ance of compression shocks. The entropy is no longer.constant after tho passage of a compression shock. On that account the computation of compression shorks simultan nously includes the determination of the function es() or d.(), too, for the di region of the ytplane behind the compression shock. For the mathematical treatmient, a compra essin shock is to be considered a curve along which two flows collide, which are related to one another and to the direction of this curve by transition conditions. It will be th. problem of this section to deriv, these (known of themselves) transition conditions in a convenint form for the present purpose. Proceeding front a setaionary compression shock, that is from a conrression shock whose front is at rest rlative to the coordinate system selected, let the index I lcsignae the phese before th'. Conpare Ackeret for instance Beitrag Gasiriamik in Handbuch der Physik, Md. VII, p. 324 and following pages, Berlin 1927. 38 NACA TM No. 1196 shock, the index II the phase after the shock. The additional index a might point out that this concerns the calculation of a stationary shock. Then the momentum and the energy theorems as well as the equation of continuity are written in the form DIs + IsVI = PIIs + PIIVI 2 (43a) S+1 2 = + v 2 (43b) Is 2 Is 'Is 2 IIu o v = v (43c) Is Is =IT ITs Furthermore, the characteristics of the ga. concerned must be known, possibly in the form p = p(I, o) (43d) If the quantities in advance of the shorck i., olT, and VIs are known then +he co,'pressi'on shock is therevith calculable. Actually all threE quantities enter into the general gs laws, too RB parameters. TIn order to carry out the computation practically, in such a case, z from (43) an from 443b) have to be expressed as functions of vIis and the known uantitles and then substituted in (43a). With that, an account of (43d), p too, is a IIs function of vIT and the known quzntitics in advance of the shock. ITo In this manner an equation for vlls alone is obtained which must be solved numerically in a suitable manner. For an ideal gas for which c is not constant, equations (43) transform with the aid P of (23) as follows: I1 RTIs + v v (4a) IlIs UIs (Ts) s= (Tis) v 2 (4b) Is (44c) VIs VII OITs NACA TM No. 11Q6 Since os appears here only in the combination p /pII only TIs and v still remain as par eters upon which thc phases behind the shock depend. To calculate the shock curves nmnericslly, it is useful, first to regard Tjs and TIIs as parameters and determine v., from this subsequently. The computation process is the following: From (44a) ard (44c) RTI RTII  + I I + s Ie = I s VIIs Vs '"!Is (45a) As a result of squaring this R2TI 2  + RTI + VIs2 Is  TTI + 2PT 2 v  2I + 2IIs IIs VIn Introducing gives Ai = iTis is I2 2 2\I Is I VT from (44b). Putting this in obtained as VIs 1 A 2 (TIIs L% (h5b), the desired equation for * TI + IBS2 412 + 4FAi TIIs l i 2 ( T 22 l l s 1 2\1 2R290 2L1 =  TIs 0 If vIs is determined, then vIIe and PIs /PIs are computed in turn with the aid of (46) and (44c); flrially IIs/Ps = ITs/ Is Ts Is/T (45b) (46)  TIS) NACA TM No. 1196 For an ideal gas with constant specific heats, the following trans formations may be undertaken. According to the familiar relations k 1 ki and a? = kRT Equations (45a) end (44b) are v.ritten in the form aTs2a  is Is Is Viis 2 2 2 2 2 Tk +1 v I k 1 IIs+ i13 k 'a I+ 2 'Is kl Sfv~o'r als 1 si 9 ) :. ^ 1 +s '11s '~is  k I + IVTs k 1 By this, als,/als andl V ls/ale and, with that, the other quantities, too, depend on the parameter v /'als alone. To compute Vls./als, aIIs/aIs)2is eliminated: k + VIIs I Ials ) v / s k als V aIs r v \2 1 + l = 0 . 2 a ) is obtained as a result. VTIs aIF (47a) (47b) a Is\ + + VIs NACA TM No. 1196 The solution of this equation is fund, immediately, if it is borne in mind that on account of the form of (47) a solution is represented by "IIs/ais = VIs/aIs then 2 als k + 1 vI SIs + k 1 Vs 2 a Is, Using this, the following is obtained from (47b) a' '*2 ( alls " be ; "^isy 1k 1 2 oIIs/ Is Is /a. I IIS) IIs In = = s'als als /VIs Is' Is Is' U~s alS/al B \ The change of entropy is of interest, as well; with the aid of (27) and (28), these expressions result SI 1s S s k  R k 1 PIs SIIs sI Rie I B = ?k __1_ k 1 a s 2 k2 i I1 k 1 ais  2n I1 aIs V n I als + 7n VIIs 27n aII Sis al + In IIs aIs VIs al3 olls/ fTlis /T I NACA TM No. 1196 From this 2 (ITs 8 k1 v IskI ^ VIIs Is Is TIe als als VIIs Arbitrary compression shocks result from the stationary compression shocks Just calculated because a velocity is superimposed. In doing so, the thermodjnamic phase quantities before and after the shock for which accordingly LhE index s can be omitted are retained and moreover the velocity differences. Since the phase in advance of the shock is already given in the construction of flows, before the shock is computed, th.e relative velocities with respect to the phase in advanc. of the shock are formed. Lut u absolute velocity of shock front ,u relative velocity of shock front with respect to particles in advance of shock Then v v a Au Is. V V 2 Is I n1 a II II I k + 3 a The signs opp':aring in this are not astonishing. A stationary compression shock in a gas which :noves in the positive direction propagates itself in a negative direction relative to the material ahead of ths shock, and in so doing, produces a change in velocity in the direction of its propagation velocity, that is, in the negative direction, toe. Naturally, compression shocks, which travel in the positive direction in the material at rest are also possible, the signs of the velocities hare to be changed for these. The thermodynamic phase quantities of this ar3 not touched upon. Corre sponding to the distinction which had been met in Msch waves, these last compression shocks are designated compression shocks of the first tyDe, those which propagate in th3 negative direction as compression shocks of the second type. In figure 11 the pressure ratio, for an ideal gas with k = 1.405 the propagation velocity of the compression shock and the change in entropy (.xpreseod by it /(l) has boon presented as a function of the velocity change Lv IT. For compression shocks of the first type Au and Avi are to be taken with positive sign, for compression shocks of the NACA TM No. 1196 second type with negative sign. The fundamental numerical values appear in table V. Such a diagram would have to be used to apply the characteristics method in the form given in section 11 in the computation of compression shccks. How are these relations for the compression shock expressed in terms of X and p? If two compression shocks which only arise separately from superposition of a velocity they are distinguished by the inlexss a and P  are represented in a Xpdiagram, that is, if the phases in advance of the shock X1 ; ila X 10; I IB, and the phases behind Che shock ars plotted, then here, too, the expression must be iarived at that the thermoilynamic phases in advance of and behind the shock, as well as 'he velocii.y differences for both compression shocks are the samTe. Accordingly, Ia I. + u, IIa+ PTI, = I I I, 11 SII,a  ( x P = (x II,a/ !,a I,a) ( IIP, By subtraction of the fi:'st two equations  .) II,a I,a) =,,  IP) + / ("IIP I) Rearranging terrm in the third equation gives ,a \ I I, I Ia) IIcL Ia (lip From the last two equations it follows that IIa I C Ii,az I,0 II,' I,p 1143 143 xII ,a Ia = II,P I, IP ) IIP 1) " II,p)  %TO I) NACA TM.No. 1196 that is, the changes in X and tainted in the superposition of a are designated by I in a compression shock are main velocity. Accordingly, the shocks X11I,I = %II " x 11,1 = CII LI The following relations hold for heats, according to (42) 11,1 II I II I k  ideal gases with constant specific v V II _ II,I II I 2 =I k 1 a 0 a  2 ( \ a k 1 a1 0 1I by (42.z). For the V V +II I T _ a /ao is to be computed from ,I and L expressions in curved brackets AX = 2 II k 1 aI a II V a a a o/ o o I a  4 0 r NACA TM No. 1196 are introduced. These quantities, as well as Au/aT, Ey/fi, and p.l/n1 denend only on vs /a according to relations previously developed. They are plotted in figures 12(a) and 12(b), and. admittedly, the .upper designations refer to the compression shocks of the first type, and he lo:er designations to compression shocks of the second type. Figure 12(b) represents an increased section of figure 12(a), with the appropriate numerical values in table V. The following example shows a first application of this diagram. In a .Ie of constant cross section there is a quiescent gas of constant entropy and constant pressure, the sonic velocity is taken to be aI = a Suddenly, a piston is drivn into the pipe at a uniform speed of 0.5ao. What is the ensuing flow like? Figure 13 shows the ytdiagram. The starting point of the piston motion lies at the origin of the coordinate system. The life line of the piston is shown with hatching. A compression shock forms in front of the piston, which imparts the velocity of the piston to th. particles, so that the particles behind the compression shock move with constant velocity. Corresponding to the phase in front of the compression shock is 1 = 0; pI = 0 The velocity behind the compression shock is II = 0.5a therefore, X1 xIT ):= 0.5 II II = 1 From this, on account of X = 0 and 4 = 0 II,I II,I Since ei/ao = 1 this gives A Ai = 1 NACA TM 1o. 1196 As a result of causing this straight line in the Xts4diagram (fig. 12(b)) to intersect the shock curve, the following is obtained: AX = 1.022; A~ = 0.022; 4" = 1.346; 'ii/i = 0.970 aI II = 1.022; Ii = 0.022; u = 1.346 From II and rp, P is computed by (42d), from this by (28) pII/pi = 1.970 The goal would be reached somewhat quicker in this by application of diagram 11. 15. PRELIMINARY ARGU=METS IN THE DETERM4IATION OF A COMPRESSION SHOCK IN THE FLOW FIELD It is the object of this section to show first of all by what data a compression shock in a flow is determined, and, secondly, to give a method by which the computation of ouch a compression shock is possible. AP can be readily show, the velocity of a compression shock is larger than 1he velocity of a Mach wave in the material. This means, that the flow field in advance of the compression shock remains unaffected by this and can be computed independent of it. It will be assumed to be Imown vhat follows. For the field behind the shock, a compression shock of the first typo represents on the one hand the start of life lines and Mach waves 2, on the other hand the terminal of Mach waves 1. It follows, from this, that the flow behind the shock and the shock itself are mutually related and can only be computed together. This is the reason, therefore, that the computation of the compression shocks becomes, essentially, more complicated than the computation of other parts of the flow. Next will be shown how examples can be conceived of flow fields with compression shocks. If in the ytdiagram (figs. 14(a) and 14(b), the flow field in front of the compression shocks and the portion CD of the life line of the compression shock is given, then the phases behind the shock are also determined. From the slope of the life NACA TM No. 1196 line the propagation velocity of the compression shock is given, namely for each point of CD. Beside, the phases in front of the shock can be learned for the points of CD; with this the phases behind the shock are calculable. From the phases behind the shock, a portion of the flow field behind the shock, namely the region CED (fig. 14(a)) may be computed, or if the entropy is known for the life lines at the lower end of C. The region CFD (fig. 14(b)) as well. It is necessary to go forward along the life lines and Msch waves 2, backwards along Mach waves 1. Imagine in figure 14(a) that'the computed life line CE is realized through the motion of a piston, then there is a flow in which a compression shock appears and which satisfies a boundary condition (if not prescribed, too). In figure 14(b) it is necessary to imagine another flow field ad.oined continuously at the lower end of CF; here the compression shock and the flov determined by it satisfy the condition that it is compatible along the Mach wave CF with another flow. Froi these flo fields the following is recognized; tho compressIon shock through the portion CE of he life line of the piston or CF of the Mach wave is defined as far as it is reached by Mach vaves of its tyno (here the first, therefore). A change of the life line of the piston outside of CE or the Mach waves outside of CF propagates along Mach wava 1 in the ytdiagram, to be exact, and neglecting cases in which a second compression shock arises, attains the compression shock at the upper end of D, certainly. On the other hand a change brought about between C and E or between C and F in the boundry or junction conditions takes effect at that position on the compression shock where tre Mach wave 1 concerned reaches iL, that is, the portion CD is certainly changed. If the life line of the piston is known beyond E to G or the Mach wave beyond F to H, then a further portion of the flow field is thereby determined, without the necessity for knowing the continuation of the compression shock beyond D; it concerns the regions CEGJD or CFHFD. It will now be shown how to procede funaamentilly to compute a compression shock for specified boundary or junction conditions. As a concrete example assume the compression shock to be produced by a piston which experiences a sudden jump in velocity. (Soc fig. 15.) The starting point of the compression shock is that point of the life line of the piston at which the velocity jump appears. The phase immediately behind M can be ascertained immediately by the method applied to the example of the last section. The compression shock as in previous examples of Mach waves i computed in individual sections, which are so ,sall that the phase quantities 43 NACA IM No. 1196 for 'hen may be regarded as varying linearly. As just carried out, the phases behind the compression shock are calculable, if the velocity of the shock is known. The velocity at M is known. Along the portib.n of the compression shock to be computed, M, N, the nhase change and, with i*, the change in propagation velocity of rhe conrreesion shock, too, are regarded as linear. Accordingly, for all possible shocks which satisfy the transition conditions, the nortinn M, N, of the compression shock depends only on a single paraneter, the velocity change between M and N, to be exact. As a result of coipu*ing the field behind the compression shock for various values nf this Drraneter, by interpolation, that shock iey be ascertained which is consisl.ent with the specified piston iLovelent. At best, for this N is permitted to travel on a fixed life line in the field in advance of the shock. Let C be the point on the life line for which the Mach wave 1 passing through N proceeds. Now the region OPQN may be computed in a familiar manner. For the determination of the extension of the compression shock NR the phase behind the' compression shock at the point N may be regarded as given everywhere along the entire Mach wave NQ. On the other hand, that value of velocity changes between N and R has to be determined by interpolation, which relates to a flow field that continuously joins the known field along NQ. With these two typEs, ntmely the computation of a compression shock going out from a piston or wall and the computation of a compression shock continuing into or arising in the interior of the flow, the most important problems have been mastered that can appear here. The interpolation methods described become pretty tedious; instead of them, iteration methods will be used, which actually lead to the goal more quickly. The interpolation method was mentioned previously, however, since it affords better insight into the basic relations. 16. EXAMPLES OF THE COMPUTATION OF COMPRESSION SHOCKS IN THE FLOW FIELD Exaples rill be given of how the problems formulated in the mnrceding section can bj solved by aeanp of iteration methods. Let the flo'" be thai computed in figure 10 and table IV. As the start of the nev portion of the conpression shock to be computed, point 1 is chosen in every case, accordingly it is identified with the point M (fig. 15) once and with the point N a second time. The nev portion of the compression shock to be computed that corresponds to M or NR, accordingly, is assumed to end an the life line 8, 9 NACA TM No. 1196 of figure 10. The phases in front of the shock for N or R are obtained as a result of interpolation along this line. For these calculations it is necessary, on that account, to have the knov ledge of the flow field in front of the shock at the points 1 (M or I) and 8 and 9. In table VI which has the same arrangement as table IV these values hare been recorded. While it sufficed to know !d fnr the construction of the flow field, here i itself d .!r must be known. These quantities for points 1, 8, and 9 are located in column 26. In the designations, in these examples, the only deviation from figure 15 is that only points on the compression shock are characterized by letters. Numbers are used for points of the flow field, corresponding to previous use. We begin with the more elementary problem of continuing a compression shock in the interior of the flow. For this the phase behind the bhock at the point N and the phases along the Mach wave N 11,10 (fig. 16(a)) may be considered known. The phases at NI and at point 10 appear in table VI, phases in between are found by linear interpolation; moreover, for NII the velocity of the compression shock and n have been given (columns 25 and 26). Besides 2d for the life lines lying below may be viewed as daS computed. It was entered for point 10 in the corresponding column. If the distances between points on the compression shock are not chosen too large, it is sufficient to regard d. between them as as constant. In the following this has happened throughout. Since NI and 10 lie on a Mach wave, the consistency condition must naturally be satisfied. Tn connection with .h flow calculation rhe existing data are to be taken from the preceding calculation steps. The real ccnoutaiion begins with the fact that the different n from its value at the starting point of the portion of the compression shock to be computed (N here) is ascertained for the life line up to vhich the compression shock is to be. comnjut'ed (8, 9 here). This computation is carried through along the curve of the initial values in figure 10, the life line 8, 9 used here passes through point 7 there. By (37) NACA TM No. 1196 S v dt) F o 00 \ 0 v a t7, o 7,N By (23) and (42a) = r1 + k (1 + l + + Just as for figure 10, F has the Dorm F = Fo t For point 7 y = 1.450; at = 1.180; , = 0.66; 4 = 0.16; 7 = n = 0.849 7 8 For N the corresponding values appear in table VI. With this the following is obtained: (I F o 0 P = 2.690; p 07 N\ 2.170; o a a4 ,7 " F (F\ 0 J ,7 = 1.110 = 0.930 = 1.020 7 % so) i F 0 a = 2.42r x 0.07T + 1.020 X 0.03 = 0.2122 7,F o 'IH GIN g OP oAN. S .7 7,N It K : 7,N n kl II) NACA TM No. 1106 In figure 10 kd had already been given, it must be the same as dla that found from the quantities just computed. In fact 7 = .Oh.2 = 0.230 ( 7,N A7 0.212 This is the average value of 1T. as can be gathered for the d,', stretch 1.7 from the auxiliary diagram in figure 10. After these preparations, the actual iteration method i2 reached. To begin with, the phases at the points Ri and 11 are estimated, in that 11 is the intersection point of the Mach wave 1 leading backwards from R with the giren Mach Wvve 1,]0. Since no better reference point exists for the estimate, these phases are equated to the phase at 1N .* Moreover, otill another estimate is needed for S behind the shock; for this, the same value that prevails dr at the lower end of I is. chosen. With these assumptions, the figure ay beigurer, 1 n be .w 16(a). Starting with the life line of the cormrssEion D:ock I:P, whose, direction here is the same as the direction of the compression shock at Tl (t9ble VI), P is obtained as the in,.erseetion point with the life line 8, 9. Then the Mach rqave E,11 is irawn in proceeding frorr, E ba',7kwards. The direction of this Mach wave ,,as taken in the familiar manner fron a Xjdisgrai (nor given here). Fron this figure I.he position of P in advance of the shr.ck is learned ly interpolation along N,10 the ohase ai 11. (See table VI.) Prom this may be obtained the values entered further on in the r.soective lines which ara necessary; for later conpu 'stion. YProcecding from XI1 by neans of the consistency conditions, the quantity PTRIT is comiputed for the Mach vsve (l1,RI). For this the initial estimates for the phase in R1T are taken as a basis and then colurn s 6 to 13, 17, 15, 16 and 18 Lo 20 compu'ed. For XRII so obtained the properties of the compression shock ar.; taken from the shock diagram 12(b). The following computations are essntiel to this AX XRI R = 0.95 = a 1II,/(a/l) = 0.9/1.040 0.918 NACA TM No. 1196 From the shock diagram A = 0.0130; ]RII/RI = 0.978; From this it is computed that = 0.0135; R = 0.090; RII r = 0.830; EII = 1.307 &u R = 1.360 a o u o = VR /a + u,/ao = 1.661 R I A portion of these results are given in table VI (columns 24 to 26). Moreover dr PII NTI = 0.830 0.781 = 0.30 d ~ B N 0.2122 To improve these values, IeL a second iteration step be carried out. First, the figure N,R,11 has to be drawn again for the values just obtained. The average direction of the compreselon shock is U.R (uN + R) = 1.733 Then RI and 11 are crbttined b. incrpolation, XpRI consistency condition for the Mach wave 11,RI . from the To find the characteristics of the shock, it is necessary to carry out the following computation RII,I 1.46 0.493 = 0.963; L = 0.927 P1,IIII From the shock diagram S= 0.0130; RII/T" R = 0.980; Au " = 1.310 ap From this is obtained R,II = 0.087; it I= 0.828; u a0 = 1.657; RIT R I3 o ' 1II,I du = 0.220 Ap NACA TM No. io6 An additional iteration step is not necessary any more. In the second example (fig. 16(b)) the compression shock is produced ty the sudden velocity change of a piston. The point of the ytdiagran at which this velocity jump takes place let it be designated M in agreement with figure 15 is to coincide with point 1 of figure 10. From the point M the piston has the velocity corre stoinding to the life line in the field in front of the shock, in particular the velocity at M in front of the velocity jump is 0.425ao. At M the velocity changes, suddenly, to the value VM = 0.95ao and rises until the instant aot = 1.3 to the magnitude 0.97",9. This and the flow field as determined by the initial conditions and the niston Tnotion uo to the point M is given. Next the phase behind the shock at the point M is computed. ao aa!7 4,aII = 0.483 al1 S X = 0.483 As a result of this line in the shock diagram 12(b) intersecting the shock curve, the following is obtained = 0.996; T1 = 0.020 $1,II/I' = 0.977; AuM = 1.333 agI From this M,II = 1.620; ,II= 0.229 = 0.781; 1.80o 0 '1L NACA TM No. 1196 The chase at M i is known ":ith that. (table VI.) Now the difference must be cormputed., over again, from the life lire of the piston for the life line up Lo which it is desired to compute the coapr.ssion shock. It is desired to allow the compression shock to end at the life line 8, 9, here too and take the phases in 8 and 9 (teble VI) froi the preceding example and nM = = 0.2122 8,M N,M The computation of the compression shock makes use of figure M,S, II, 11. (See fig. 16(b)). M, S, N Is the life line of the compression chock; N, 11 is the Mach wave 1 returning from N; 11, S is the Mach wave 2 retuining from 11. To begin, an estimate of the phase at the points N I, 11 and SII is mad d and this is chosen equal everywhere to the phase at M I. In addition, an estimate for is necessary. Let k = 0.230 as a start. Figure M, N, 11, S d'4 may be drawn with th.se assumed values. The' ord.r in which the points were named correBnonds to the order in which they came up in the drawing. For th positions of T and 11. obtained thereby the phase in front of th,; shock (se.e tabl.. VT) or thc velocity of the life line is oltaind! by; interpolation. The iteration method begins at point 11 and it can be shown that 11 can be only slightly different from STI because the line ele'nant S ,l11 is 'TI I smsll relative to the other dimensions. The quantity pS can TIT differ from "MII only slightly, since it originates in linear interpolation betweeMn M1 and T, and N li,s v:cr close to M. Therefore pl1 PMII is chosen as a starting point. If the 11 M,II velority of the Distor at 11 that is known from the boundary conditions is use(P for :his kXl1 'na. be comnuted. From the consistency condi tion for Ihe Mach vave 11,N XTIIy is obtained. NTov the following co.ipuui:ation 11NII = 1.017 1,1 = 0.988 and from the shock dia.ar Ai = 0.020; I/i = 0.973; 334 3, NACA TM No. 1196 from this ,11 = 0.077; II 0827; = 1678 Further it is calculated that dn = 0.217 The phase at S is obtained by interpolation between M and N. With the aid of the consistency condition for the Mach wave S,11ll,1 is.finally obtained, and )ll from the boundary condition for point 11. The first iteration step nds with that. It is necessary to check whether the quantities Xll, 1l1 Ni 4NIIJ) tII, and computed agree sufficiently with the original estimates. To increase the accuracy a second iteration stev might be carried out. On ths basis of the values just computed, tn figure is redesigned and the computation is carried out in the manner .jur.t described. The vailu fir P11 just computed is taktn as a beginning. The following calculation is obtained for the determination of the characteristics of the shock lX = 1.003; LX = 0.967 NIT,I N From the shock diagran t u S= 0.0189; 'i I =0.97; I = 1.327 from this rI = 0.0o3; I = 0.828; 1.678; : 0.221 The computation is continued in the mariner givcn until the phase at point 11 is obtained, again. An additional iterhtion step is not neccsstry. NACA TM No. 1196 17. SUMMARY The differential equation system for norstationary, one diqersional flo"s possesses three families of characteristics; the thermodynamic and the flow phase are described by three variables. As a result of setJing up consistency conditions for the charac teristics passing through the pcin+ for which the conditions have been set up, there? equations are obtained from which the phase may be obtained. In that a possibility for the computation of the flow has been given fuindn.mentally. The report carries out these idees, in generl, and brings the simplifications which are possible under special assumptions, as well as detailed examples. Compression shocks appear, in this, as transitional conditions in the interior of the flow and are likewise investigated in detail. Translated by Dave Feingold Nitional Advisory Committee for Aeronautics w s w A & t I otna 0 < I il il il il fl il r rr < l rI1 rt w oIn ol < ** *. *. * AS 0 R 4 M S g M M a A 4 4 r l H H1 H4 H 9 M o 4 a l r: l l r4 l 4l l 4 4l 4l l r 4 a* S A 4 U r4 4 ? 4 s SH H H H H H 1 H H H H H H H N a g .? OR r a I4 : .I r4  M. gio tS A : .4 A r: .4 H 4 M .4 0 CO CU C U 0 i p $ *t & a. a w * r4 ra r4<__r ___r4 C WN a en AS H S S W r4 In Ch If * o 0 0 C 0 0 0 0 0 %d'&C* I H 1 I R O 2 Ct 1 i r; CR t a 1 a 1 a 0 aU at A NACA TM No. 1196 5 B ! g 19 5 ; Q4 4 S 4 q B0 UN o 0 u u 0 O4 0 WN HC m4 00 00 *0  i* jS r4 . I.I I I *.1 1 II I" Cd .i ." .4 00 S A4 .4.4 .4. mm P1 n ( n a l 4 ll l 4 14 1114 .41 .41 A A 1.4 .4 .4 m 4.9 1;;l rla 4 4 Mr. r4 M r. 4 .r4  d rt( 0D u W% en 01 on  0 1 00 m 414 r4 Mo 4 1 r4 H oI I I I a 0 Ha U Cd d i in .4 400 .4 CM ( .t Ifi '0 NACA TM No. 1196 I H I 'B g4 Cii ) ( t m m e m f mt m Q .0 .. ,. . N" S 1s ss Rse ..,. ., * A Qp.44!7 M0 9 99 u I I I I I II q 0 0__ __ __ W _ t .. . i.s a A .. "T^' iiitt iT 'T m ;  tt .4 g S S 8 9 a In 8 . 0 I c o N 4. .4 .4.4 .4 4 4, 4 .4,, a r 3 SI s gg VBA a 4 J .44 M 14 I 4 A 1 4 .4 A .4 = ^0 & S & R & % 9 g 01 ri .4i CC? .. mm N C. o u    I p" u 8 1  l  o, C 2,0  cu .4 71odt aO C C E 0 NACA TM No. 1196 I ft N S p Ii I U ia 0 0 I' I c R 8 I If tl a a i 0 I I 0 I I S3 A S .. 4 4 4 4 4 4 0 l S ml aP Ir r a l 0 ocu c c 0 n in 00 W.4 .4 4 I .4 t I4 .R 61 .4 .4 .4 " t0 I I o I. " r4 0 04 r4 C8 . oc o ,n r I m rH r 0 a 0 0 0 0n .4 .4 4 4 4 4 4 .1 0 08 8 n 9 AO wr 4 t C * 4S 0. 1, 11 1  gal 1 h ~  "~  '  oj ^ro ac Ti c VIa NACA TM No. 1196 r1 I 8 Sii 03 a I C II NACA TM No. 1196 TANE V NBcSTATICmAr CCGoFRSSII sHo (K 1.405) /Au A/a A ,I, /a1 I/z ,/AI T. A 1.00 1.00 0.0 1.00 1.00 0 0 0.98 1.02041 0.03360 1.04817 .0.99998 0.06721 0.00001 .96 1.0167 .06791 1.09938 .99991 .13588 .00006 .94 1.06383 .10298 1.15391 .99970 .20615 .00019 .92 1.08696 .13884 1.21203 .99927 .27816 .00047 .90 1.11111 .17556 1.27406 .99862 .35206 .00094 .88 1.13636 .21319 1.34037 .99748 .42806 .00167 .86 1.16279 .25180 1.41137 .99587 .50633 .00273 .84 1.19048 .29146 1.48748 .99369 .58711 .00419 .82 1.21951 .33224 1.56925 .99080 .67062 .00615 .80 1.25 .37422 1.65722 .98708 .75713 .00869 .78 1.28205 .41751 1.75201 .98240 .84694 .01193 .76 1.31579 .4620 1.8 545 .97659 .94038 .01599 .74 1.35135 .50840 1.96527 .96951 1.03781 .02101 .72 1.38889 .55625 2.08544 .96101 1.13965 .02716 .70 1.42857 .60588 2.21608 .95091 1.24634 .03458 .68 1.47059 .65745 2.35841 .93899 1.35843 .04345 .66 1.51515 .71114 2.51387 .92506 1.47647 .05418 .64 1.5625 .76715 2.68413 .90903 1.6011 .06684 .62 1.61290 .82570 2.87113 .99067 1.73320 .08180 .60 1.66667 .88704 3.07714 .86974 1.87350 .09941 .58 1.72414 .95147 3.3"1484 .84602 2.02302 .12009 .56 1.78571 1.01931 3.55735 .81954 2.18291 .14430 .54 1.85185 1.09094 3.83845 .79006 2.35447 .17260 .52 1.92308 1.16680 4.15261 .75746 2.53923 .20563 .50 2.0 1.24740 4.50520 .72182 2.73895 .24414 .48 2.08333 1.33333 4.90276 .68305 2.95572 .28905 .46 2.17391 1.42529 5.35333 .64131 3.19120 .34141 .44 2 .2773 1.52410 5.86637 .59679 3.45070 .40251 .42 2.38095 1.63073 6.45517 .54980 3.73531 .47385 .40 2.5 1.74636 7.13401 .50078 4.05005 .55732 .38 2.63158 1.87242 7.92319 .45026 4.40174 .65517 .36 2.77778 2.01063 8.84701 .39889 4.79144 .77018 .34 2.94118 2.16314 9.93888 .34751 5.23210 .90582 .32 3.125 2.332&6 11.2418 .29698 5.73172 1.06644 .30 3.33333 2.52252 12.8138 .24825 6.30261 1.25756 .28 3.57143 2.73716 14.7347 .20234 6.96o61 1.48630 .26 3.84615 2.98225 17.1156 .16016 7.72656 1.76206 .24 4.16667 3.26542 20.1167 .12251 8.62809 2.09725 .22 4.54545 3.59705 23.9721 .090026 9.70301 2.50891 .20 5.0 3.99169 29.0416 .063089 11.0043 3.0a088 .18 5.55556 4.47032 35.8933 .041760 12.6082 3.66757 .16 6.25 5.o6445 45.4722 .025786 14.6295 4.50062 .14 7.14286 5.82359 61.6909 .014596 17.2482 5.60104 .12 8.33333 6.83019 80.9708 .007388 20.7642 7.10387 .10 10.0 8.23285 116.672 .003219 25.7176 9.25191 .08 12.5 10.3285 182.394 .001133 33.1878 12.53078 .06 16.6667" 13.8101 324.386 .000288 45.6935 18.07325 .04 25 20,7568 730.082 .000040 70.7902 29.27672 .02 50 41.5634 2920.83 .000001 146.254 63.12755 .00 o oo o 0 o oo NACA TM No. 1196 co i uln I doinst muy pmmo DIoIUI r oj ld a U1r winCs ,C. 9 7 .4 7 E ; ** *" . ,' c. a ... .. i . & 8 88 8 1 ft 9 C p p a i f 3 41t C!4: 0 . f . ."" " ?0 r jI ccj B 8 B 0 C C f f .. J. f It p f 0 C p o 33 ft C. ft f t i1 ft 1Cd ft a C1 0% S f ^j &UJ ?0 ? CC ? C Cl a  0 .5z  C      NACA TM No. 1196 Figure 1. Curvilinear coordinate system t . CFO, 64 NACA TM No. 1196 TrC Figure 2a. Relation between i and T for CO2. NACA TM No. 1196 3 115 a is  i Z5 0 055 2,5 0 0 0 50 100 150 200 Figure 2b. ; P; W as functions of the temperature a for CO2. 'C 00 1105 / 5B ea 5 *LUL~~~4' ) ^ r .^ \ I Figure 3. a; P; T as functions of W, ao NACA TM No. 1196 2 1,J 1 or A+,, for CO2. NACA TM No. 1196 j I I I I I I I I I I I Seo _ _ ~ I ft 68 NACA TM No. 1196 00 C% / / />a 4 VX V. rI a a o Ca 1 ou ,4 o, bD a) Oo 0 '4 . '*K  5  S  '!  "Is a) ^ ,4 4 TNACA TM No. 1196 The physical significance of . 69 pethline of a particle Figure 6. NACA TM No. 1196 .I Figure 7. Any flow of an ideal gas with variable specific heat. Treatment nf hninrinr rnnrfitinn.. NACA TM No. 1196 1,1 1,2 13 Figure 8. NACA TM No. 1196 0,5 0 0   1, 1,5 0,5 1,' 0.5 0 qs 1,0 1,5 Figure 9. I qs NACA TM No. 1196 LA I  * UI U 4 0 O 0 4, LO 4I 0 o cv Cd 41 U (U 0 c0 a 0 4 mb :~ : ; ; NACA TM No. 1196 Figure 11. Characteristics of compression shocks K = 1.405. NACA TM No. 1196 >I' \ I t.. In !' _ s S  \   S ______ __ __ __ . v14 NACA TM No. 1196 NACA TM No. 1196 77 path line of a particle Figure 13. 78 NACA TM No. 1196 y y D D K 6 C CE tt Figure 14b. Figure 14a. NACA TM No. 1196 79 y R N 0 P P4I Figure 15. NACA TM No. 1196 Figure 16a. NACA TM No. 1196 Figure 16b. Sr ' 1I f < V7 O%, 0  0 r < o ) I z Il de 0 ou 0 4L 0 o i ( OH o0 (4 0' PA r11 * t 8 SO O 4 0 4+ 0 1) 0 4 U 0 1 3 0 U OOO C p u  co m co 0r 0 4 t ro W to 0 u *D 03 MHC B 0 m M O H 0 0 al v a 00 >H r o 4. C m aL) 4 A U 4 Ca 0 l ' o S .4) o + A o o o< m o a) 0 a a 0 0m 4 0 ad P  0 4 0 C) 0A 1 c (D040 I C0d .0 0 0D.0 me Cd OOa e Cd OO 0 4D H 4) +d H H 0000 0 00 Sa(D 1 4 U *"  oi d H a C We H 0C 0 P4 0 (D 000 0 000 S4r 0 0 rX A: PO t 0 0 OriE) 0 AM 4 4 oH ~ I~ C f Qml I i 04 Cd Ok Cd C.Cd;4 40 H P 0 0 D 4100 4'm 3 a D 0 od oOP4O O 4 C0 0 Cd P (D0) a.l 00 i4 A ' r4 d p d rm 41 C1 0 a 0 0 i 0 4) r d 00) 0 0 0)f ScIl PT 0 000D c 4 d l D UnIVcniiT ur rLuMIu 
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