1. K. LEADON NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1196 NONSTATIONARY GAS FLOW IN THIN PIPES OF VARIABLE CROSS SECTION By G. Guderley Translation of ZWB, Forschiingsbericht No. 1744, October 1942 30 Washington December 1948 1X3^^" ^" J) SW-f I f Lf MTIOML ADVISORY C0MMII1EE FOE AEROlLUlTICS TECMICL MSMCE'^WriUM WO. 11 96 KONS'mTIOIJAEY GAS FLOW Hi THIH PIPES OF 7ARIABIE CROSS SECTION* By G. Guderley I3S'£Pj\C'j} Characteristic methods for nonstationary flows have "been putllshed only for the special caso of the isentropic flow up •until the present, altliough they are applicahle in various places to more difficult questiors^ 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 mimerous o:'-aiaples are irorlced out. *'!l!ifichtstationare Gasstromttngen in di3Tmen Rohren veranderlichen Querechnitts. " Zentrale fur wissenschaftliches Berichtswesen der Luftfahrtf ore Chung des GeneralltTftzeugmeisters (ZTO) Berlin -Adlershof, Forechungsterioht Nr. Ijhhf Braimschweig, Oct. 22, lc/k2 WACA TM No. Iin6 1. INTRODUCTION In -papers "by F. Schultz-Griuiov and E. Sauer methods have teen developed recently for completely solving the problem of nonstationary isentropic gas f.lows in a pipe of constant cross section. An expanded view of the problem ia the haGis for the present report. Flovs are considered, which likewise depend only on the position coordinate^ however, the cross section of the tube need no longer be constant aril 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. D6'ring, and F. Pfeiffer, 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 ma.ke any special mathematical demands on the reader. As a model, the Busemann derivation of the characteristic method for two-dimensional stationary gas flows might possibly do . It is actually possible to apply this derivaojon immediately to the SchultiZ-Grunow, F.: Nichtatationare eindlmensionale Gas- bewegung. Forschung auf dem Gebiet des Ingenieurwesen^ Bd. 13' (l9l)-2) pp. 125 to 134. Sauer, R.: Charakt-eristlkenverfahren fur die elndimensionale instationare Gasstromung, Ingenieur--Archiv, XIII Vol. (19^2) pp. 79 to 89. Vorbereitende Untersuchungen sowle Anwendungen finden sich in den Arbeiten von H. Pfriem, Zur Theorie ebener Druckwellen mit steiler Front Akustische Zeitschrift Jahrg. 6 (l9^l) part U. — Die ebene ungedampfte Druckwelle grosser Schwingungsweite, Forschung Vol. 12 (l9^) pp. 51 "to 6k - Eeflexionsgesetze fiir ebene Druckwellen grosser Schwingimgsweite, Forschung Vol. 12 (19^1) pp. '2hk to 256 - Zur gegenseitigen Uberlagerung ungedampf ter ebener Gasswellen grosser Schwingungsweite, Akustische Zeitschrift Jahrg. 7 (l9^^2) part 2 - Zur Frage der oberen Grenze von Geschossgescbwindigkeiten Zeitschrift f . techn. Physik 22 (19^1) pp. 255 to 260. Eine weitere Anwendung findet sich bei G. Damkohler und A. Schmidt, Gasdynamische Beltrage zur Auswertung von Flammenversuchen in Rohrstrecken. Zeitschrift fur Elektrotechnik Vol. ^7 (19U) pp. 5^7 to 567. Busemann, A.: Beltrag GaedjTiamlk in Handbuch der Experlmental- physik (Vien-fiarms) Bd. U, Teil 1,.P« ^21 and adjoining pages. NACA TM No. II96 isentronic nonstatlonary ficw jn l- Dipe of constant cross section and from this ty means of some supplementaiy ph.ysical concepts succeed in getting a treatment of flows in a tube of variable cross section; tin's is the corrse which had been taken, originally. Ib compariscn to the mathematical theory of chai-acteristics, however, . these considerations ot)erate with a .lack of clarity sufficient so that the msthematicnl theory - for the engineer too - can be represented as the best approach to the characteristics method. Considerations necessary for the present problem are now broiight forward from the characteristics theory . As a resu-lt, "equations for the directions of the characteristics as well as conditions which must be satisfied along the characteristics are obtained. Proceeding from these reD-ations, tha 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 const^int cross section, is deduced again and the trcnsf ormations appearing there are used to simplify the computation in complicated c.?ses, too. Since this is not always possible, the most general form of the characteristics method is shown in a later section. Aft'^r this, the formulas obtained for the special case of an ideal gas with constant specific heat are simplified and the consideration of boundai^y conditions explained. The remaining sections deal with calculation of compression shocks; the known relations which connect the phase befor- and behind a compression shock with one another are set forth In e convenient' form for the present problems arid with that the calculation of a compression shock :n a flow is carried out. The theory is illustrated with suitable examples treated in detail . In that regard, it seemed advantageous to avoid dwf inite problems of technical interest, in doing so gaining the possibility of woi'king out examples under very general ascimaV)tions without excessive effort. It is hoped that, nevertheless, the application of the method to physical problems offers no additional difficulties worth mentioning inasmuch as the earlier publications contain such aprilications . The author expresses his thanks to Dr. Hans Lehmann for working out the examples . "Compare Courant-Hilbert. "Methoden der mathematischen Physik II", p 291. Guderley follova the representation given by H. Soifert at the same Institute for Gaa Dynamic e in lectures. NACA TM No. 1106 2 . BASIC EQUATIONS Ccoislder nonetationary, porfect gas flovs in a pipe with a cross— section that varies in space and tiaie* in the neighborhood of the flow tuho; bhat is. it in aQSumed that the velocity and the phase over a cross section of the pipe may be considered as sufficiently constant. In general, this assxnnption is Jtistiflable only if the thickness of the tube, relative to its length, changes slowly enough. Only for flows which have as svirfaces of constant phase parallel planes, coaxiel cylinders or concentric spheres need this liniitation be ignored. S.ese flows with plane, cylindrical, or spherical wave propagation are Included as special cases in the present problem. To stress the relationship to stationary two-diuensionsl flows, let the axis of the pipe be vertical, the position coordinate be y, the time be t and plotted horizontally. In this yt-diagram the flows are investigated. (Compare fig. 1.) Let p pressure s entropy per unit mass 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 e particle is preserved, as folloi'^: P = p(s,p) (1) (2) (3) (M •4 '. ProDlems with time variations in the cross-aectionel area are rare^ they were included, since they can be handled without additional difficulty. 1 ^ + V 1^ + ^ = p dy dy dt V ^P + p?'' + 5^ + pv ^l^ + p ^T-'-^ = dy dy dt dy dt '|^3t = ° NACA m No. 1106 5 In this the derivatives of p should ho replaced hy the derivatives of p and s, for this purpose bp 1 dv - ^2 (5) is introdviced. Therefore, instead of equation (3) |P + v|ll^+o^+-^l^-^^^^pv^+p^ = (3a) X _^ , ^ , a,2 By " ' Ss Sy ' '^ Sy ' a^ ot ' ds St is obtained. 3. FEOM THE THEORY Ot CHARACTERISTICS In regard to the systsm of equations (?), (3a), and {k) , the familiar question is raised from the theory of 'haracteristics . In a region of the yt-plane 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, tho appropriate derivatives taken in the direction of C ho knovn. The question is asked whether the derivatives in other directions may. be com-iutod with the aid of the system of differential equations, and under what conditions. To answar this, a curvilinear coordinate system S^t] Is lntrcduo«i in which a curve g = constant coincides with C (fig. l) • All the derivatives with respect to ^ along this curve S = 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 left and only known quantities are on the right. That is g = S(y.t) 11 = n(y,t) Sy _ _-^ ai + 5 at] 'ii ay M ay §t = .-^. ^ + a_ an a^ at an at 6 ^ NACA m Wo. iiq6 with this form (2), (3(a)), ^d (h) are obtained 5t) 1 BS, ^V / BS , 5£ \ Sp 1 CtTl Sv ( M , M _ _=i -\. t V — ^ + — =- = ^ — _ 1 V -— + — - ^g ^y ag'\ Sy Sty St ay an', Sy St ss a2 V ay at/ a^ ^ ay ag as\ ay -N at _ ap i_[ ail _!_ aTi_'\ _av an _ as f '^n _^ an\ a_p am ? M q2\^ ay at/ a^ ay aq\^ ay at j as ay ^t\ ay ^ at/ " "an \J ay ^ at^ A linear system of equations is obtained for the unknovnis -"-St- 5p Sv ac, ac, end r-nj the unknowns, themselves, are obtained by Cramer* 's rule as the quotient of tvo determinants. The determinant in the denominator is the same for all unknowns . It always gives a single-valued 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 a] so 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 equationo (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 £9 and ^, that ay at is the direction of C If C ia so directed that the determinant in the denominator does not vanish anywhere, the 2E etc. are as computable as single va3aied. Of greater interest for our considerations .'s the other case, namely, that the determinant in the denominator is zero at every point of C. Such a cui'va is termed a characteristic. Because of the assumption of finite derivatives the determinants in the NACA TM No , II96 numerator also vanish. Relations are obtained thereby, in vrhich the right side of (6) and, therefore, the derivatives of p, v, and s along C apuear as essential ingredients. These relations represent the starting point of the graphical numerical method of solution. Since the solutions of a linear aystem of eq.uations are no longer single valued for vanishing numei-ator and denominator deteimlnantSj the pursuit of a given solution of a characteristic is possible in various vays. These different possibilities actually appear on changing the initial and boundary conditions. h. TEE DIRECTIONS OF THE CHARACTERISTICS To find the directions for vhich the curve b = constant is a characteristic, the determinant in the denominator must be set equal to zero in the solutions of the system of equations (6) . 1 5& p 5y 1. fv ^ + ^^ 7 This gives bd, V ^ + s| Ss \ 3y Sty = at, = From this are obtained the conditions i^f :^.'^ ^i^i=° (7) or fv + a^ ^ + 51 = > 5y 5t (Ba) 8 or WACA IM No. 1196 (v.a)|.||.0 (8b) The slopo of any avrve ^ = constant is given "by dt ^ Sy Frcm (7) and (8), together with this, the slopes of the charac- teristics are or 4Z = V (9) at dZ = V + a (10a) dt or ^ = V - a - (10b) dt The characteristics defined hy (9) are path-time 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 v;ill 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 CONSISTMCY CCNDITIONS (M THE CHAEACTERTSTICS As shown in section 3, along the characteristics, certain conditions must be complied with by the derivatives which result NACA 1M No. 1196 from the vanishing of the detenninant in the niimerator. These conditions are called consistency conditions, for p, v, and s are suhject to them, if the derivatives with respect to i, are to remain finite. If the right side of (6) is designated E-j.^ ^2^ and R-^ in sequence, then the following is obtained for the determinant in the numerator of the quotient for 2£: E R. R, Sy at p ss ^y Sp : V S^ 4- a^ ss \ ^Y §t;/ I ■■. Sy St ; (11) This determinant must vanish to give the dii-ections of the charac- teristics. Substituting (7) gives R. Rr E 3 Sy = According to this the determinant (ll) vanishes by itself. With (8a) , that is, for a Mach wave 1 R. -a Sy ay R. i -^P a^I =0 5s 5y as ; 57 1 is obtained, or 10 NACA m No. 1196 ( ^\' -^ da 5 - In this, ^ If ^ 8 certalnljf different frora zero, as long as v and a are finite and grad ^ ^ 0. Aa the condition for *ihe Mach wave 1 is ■obtained ■oR , -aE2 + a|£E3 = (12a) The consistency conditions for the Mach waves 2 is, if a is replaced by -a ■o\ + a^a - a |2. R^ , (I2h) A condition for the lifs line is obtained if the vanishing of the determinant in the n^,iinerator in the .'luotient for 2£ to be sot as from (6) requires 1 b| p f^y i-fv^H-l^ ,2 y 3y a^ f aS sS^ ^ \ ay ot y ^i R-, E = For the Mach wave this equation is satisfied by itself, the condition for the life line is E3 = (13) The determinant in the nu'nera-'or of 2^ could be investigated, too; however, this would not give any new consistency conditions. MCA TM No . 1196 11 The values of IR-,, 1^2 » ^^^ -^^ ^^® still to be put in. From (l2a) , a dil "sn (v + a) + |a" oy at Sv ^a (v + a) + ^ j py ot_ -apV i: SliiF . = 5.:iiF' Sy at is obtained. The direction il along a Mach vave 1 is given by dt (lOa); on that account dt §7 dt 5t Sy ^^ ^^ at is valid for it. Wi'h that the consistency condition for the Mach wave 1 can be vritten in the form i i£ + ^ = -a ( v ^M. + ^^riFJ ap dt dt \ Sy at / The consistency condition for a Mach wave 2 is obtained, by substituting -a for a (ll^a) i ^ . ^ :. -a ap dt dt y aZnF + airiF ] ay at / (li+b) Froifl (13) for a life line is obtained I- This may be integrated Immediately s = constant (15) Natiirally this constant will differ from particle to particle, in general . 12 WACA m No. 1196 6 . FLOWS WTH CONSTANT MTROPY AND UNIFORM PIPE CPOSS SECTION EquationB {ih) , (15), (9), end (lO) just obtained are certainly ;\8eful, fundamentally, as a starting point of a characteristics method - in fact, there are "eianples, where it is necessary to I'evert t"0 them (cooroare section 11); in most cases, however, there ar<5 still other transfoi-ma Lions suitable. The direction in which to ■proceed for these is obtainod if an attempt is made to derive the characteristic method for Isen tropic flows in a pipe of uniform cross seci.ion from equations (ik) and (if^) possibly in the form applied by Schultz-Grunow. To emphasize the fundamental ideas, no assuTiptions of any kind are made therein of the characteristics of the flow medJuia. On accoiait of the hypothesis of constant entropy, equation (15) satisfies itself. In equations (lU) the right sides are omitted since it concerns a tube of -miforni 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 ^ as a differential. Let pa T temperature ' • i heat content (enthalpy) s entropy By the second main theorem T ds = di - i dD P From this, on account of the hypothesis of constant entropy. Uclv\ _ fdi}. pWi'>^ .dTi With that, it follows that ^ = i f§l\ dT pa a \dT^ WACA TM Wo. 1196 13 o is introduced in which a^ ns the sonic velocity of a comparison phase which was added to make W dimenslonless . The pha.se of the gas may he characterized hy VJ from VJ = ¥(t). It follows that T - T(V) (18a) Further, it is valid that p = p(T) - p(W) (iBt) a = a(T) = a(W) etc. (l8c) With the use of W equations (lU) appear in the fomi a dW + dv = for a Mach wave 1 a^^ dW - dv = for a Mach wave 2 Bringing in \ = W + -21 (19a) H - W . -L (19b) these last relations change to a form which may he integrated- This gives X = constant for Mach wave 1 (20a) |i = constant for Mach wave 2 (20b) If the magnitudes of \ and \jl are known for a point of the yt-diagram_, the velocity is thereby completely defined as well as the thermodynamic phase. It is, to be exact, W = ^-li (21a) X = ^^~Z-^ (21b) ^o 2 ' ll; NACA TM Wo. 11q6 and on account of equations (l^) p = p(\ + u) (22a) a = a(X. + n) , (22b) The next sectione explain this transformation and the application of equations (2o) to an example of an ideal gas whose specific heat is a fionction of t^^mperature . 7 . THKFMODyNAMIC RELATIONS FOR M IDML GAS WHOSE SPECIFIC HEAT IS A FIKCTION OF TEMPERATURE; COMPUTATION OF W Let c specific heat at constant nressnTe ? c^ specific heat at constant volume E gas constant For an ideal gas £ - RT (23) P According to the second main ^heorem, if p end T are considered as independent variables ds = i ^ dT + i ^ dp - -L. dTD (2U) T St T Sp pT ' Since ds is a perfect differential, dp \T bl/ ST \ T 9p oT Accordingly, substituting p fron (23), the following knoi-m fact is obtained NACA TM No. II96 I5 that is now therefore From (2U) as a result i = i(T) (25) (Dp ^ ^p ^''^ 'v = %^^) ds = _E dT - R io (21+a) T P and fro'U this hy integration s - s^ /■^'^ c £ = / !2 ^ -In X the Index o characterizes a comparison phase. Introducing T / s dT p = e '% E T n e ^ - gives (27a) (27h) -E- = gn (28) Considering o^{T) as known, p ty (28) and p by (23), are given as functions of T end s; the thermodynamic properties of the medium can be calculated In principal, therefore. The l6 NACA TM No. in6 quantity a is also defined by p and p. The computation of a is "by all means simpler if carried out in the following way. According to ([?) 3 _ ^ -V. for which the entropy is to be kept constant. For constant entropy from (2i4.a) dT ^ _R_ d2_ t' c^ p by differentiation from (23) dp _ do _ dT P " P " T From the last two relations together with the familiar relation Cp - c^ = E is obtained a(T) =,/!2^RT (29) The relations discovered up until now describe the properties of the gas and must always be knovn; it makes no difference which variation of the characteristic ^lethod is chosen for the calculation of the flovr. In conira.Tt, the introduction of the functions W, X, and \i serve only as preparation for carrying out of the charac- teristics method in tJie fonn presented in the preceding section. Next, to compute W. From (2^) it follows with (26) i^±\ =c TIACA m No. 1196 17 Setting the last eq lation as well as (29) into (17) .gives T ¥ = kUi = i„ / /!i!i a.T (30) For the present case, s - constant = s (?8) "becorass o ■.T / /■ ■./ rp E T la ^ -i^ (T) = e (28a) ^o Wow the following can "be formed T = T(X + m) a = a{X + \i) P = P(X + u) These calculations were carried out numerically for carbon dioxide. The relation hetwesn the specific heat and temperature was 5 o taken from Hiitte with the aid of these valuos (i - i^)/ap'"^ a/a^. P, and W can he co'iputed from equations (26), (29), (^Ta), and (30) as fijnctions of the temperature. (See figs. 2(m) and 2(h).) Figure 3 shows a/a , P, and T plotted as functions of X, + H = 2W. 8 . THE CONSTRUCTION OF THE FLOW FIELD The following problem should be dealt with: Along a curve K of the yt-cliagra'T., which has at the most one point in common with each characterisric, let v/Vo' ^^^ ^/^o ^® given (fig- h) . The flow should be constructed for the following times as far as it is defined by the portion of K given. Therefore, it is concerned here with the computation of the part of the flow defined by the initial conditions which by the sajne arguments appear everywhere in the interior, too. Before the construct ion of the flow can be start ed. H-utte, 27th edition, Vol. 1, p. kS , table 5, Berlin 19kl, Wilhelm Ernst und Sohn^ publishers. 18 NACA TM Nr* . 1196 the initial values p/p and v/a must be expressed in terms of the variables X and \i . Since the entropy was assumed constant, p/p and P coincide. With the aid of the relation presented in figure 3, hetweon P and \ + n and equation (Slb;^ X and \i nay be ascertained without difficulties. Figure h shows en the ri^t, the yt-dlagram, on the left, tha diagram of the assumed values p/Pq and "I'/aQ ^3 well as thos-^ of the computed quantities X and n fes - functions of y. Proceeding from the individual poin-^s of K if the network of Mach waves had been broufjit in. the nhase at each lattice point would be deter lined thereby; according to (£0) X is constant along Mach wave 1, n along Mach wave P., and on that account, equal to the values at those uoinis of K fro-n which the Mach waves spread out. By (Sib) and (22) the -nhase is given by X raid n- To be able -^o drav the nett^'ork of Mach waves, only their directions are still needod. These are given at ihe lattice points by (lO) • a/a^^ is a function of X + |i in figure 3, v/a-o is computed X - u as 2 The direction for the portion of a Mach wave between two lattice points is approxi'Tiated as the average value of the corresponding direciions at the lattice points. The construction beco'nes especially aiinple if the Mach waves are di-awn for equidistant values of X and n- The directions of the Ifech waves appearii^ can be computed beforehand and possibly prepared in the form of table I. The 3'.nterval 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)/aQ and (v - a.) /e.^. For Mach wave 1 for which X = O.3 and which leads from a point with n = .2 to point with n = 0.1, in the column with the heading X = O.3 the value is to be taken from the row ^ = I.5, that is, (v + &) /a^ = I.103. In the flow diagram the values of X valid there are entered to the left of the lattice point and the values of p to the right. To determine, for examiole, the position of C from the points A and B since the phase of C is given beforehand by X = 1.1 and (1 = 0.5 the average directions of the Mach wavoe (v + a)/a^ = 1.U22, (v - a) /a^ = -0.778 can be taken from table. I end dra-^n-in the yt-diagraTi. The auxiliary diagram on the left in NACA TM No. II96 19 figure k can be used for thif.. ITiere the direction of a Mach wave for vhich (v + ?)/&,-, - 0.3 I3 drawn in. Simllai- diagrams can be ueed as aids lor the following exaaiples, too. The portions of the Mach wave goinr; out from K really require a special coiiroutatlon since the averajje values of X or \i for them do not agree, in genera]., with the values of table I. The small deviation was tolerable, however. 9. FLOWS WITH CONSTANT 2MTE0PY IS A PIPE OF VAKI.ABLE CEOSS SECTION If the c^oss section of the pipe ir net constant, the right side of equations (l^i-) from which it is necessary to start out, here too, are Tjreserved. 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 |.i still re:aalns useful. Setting (j^^m.^ i.o^E.) = u (31) \a^ dy a^ dt / then ^4 = -^^^ (32a) at ' is obtained as the cons^stcncv condHlon for Mach wave 1 and ^.i =^-a M (32b) dt o f ^r Mach wave 2 The consistency conditions in '•he foKn of (32) contain at any given time the differencial of only one of the uDknown quantities X or n Willie the differentials of both p and v ap-pear In (l^) already. Tills implies an appreciable Improvement in the numerical calculation. The construction of the flow rests on the fact that equations (32) are considered different equations. Lot G;^ be the value which a quantity G assumes at the point A, AGji^ the difference G^ - G^ and G^. an average value of G taken between A and B. 20 NACA TM Wo . 11q6 Applying equations (32) to com-aute from two known points A and B the phase at a third, C, which is on the same Mach wave, in the form ^A - '-^C,A -^nL^C^o ^*CA > [1^ - ii-Q = .AX^^3 = -W^C^o A^CB (33) For the detemiination of the flow Xq and u-q ^^'^s "to be computed "by mefins of these last equations and, at the same time, the position ascertained of the points sought In the yt-diagraai "by the use of equations (lO) . The calculation process might he explained hy an exaratjle . The flow is considered as given along a curve of the yt-diagrara and, admittedly hy X and u (fig. "?, tahle II). In addiiion, the pirie cross section must he r known function of 3' and t. For that it is only necessary ■'o require that F can he differentiated with respect to nosition and tl-ne, a premise which is always fulfilled in practice. F^r ^hia exa aple F is taken in the form F = F,^y^t From (31) for M M = -^ a„ \ a, o J' / The positions y = and t = for which M goes to infinity do not belong to this region of flow where such singularities appear (for ezample at the center of spherical waves); it is n«cepf3ary to make special investigations which oannot "be entere'' into in. the present report . The best way to follow the calculation is by means of the sycteuiatic calculation in table II. To facilitate comparison with the description the columns are numbered. The first column containa the designation of the point which is to be computed, the second colurin gives the known point which, in coDHiion with the point to be computed, is on Ifech wave 1. Column 3 contains the corresponding "^Compare G. Guderley. "Starke kugelige oder zylindrische Verdichtungsstosse in der Mhe des KugeJmittelpunlctes oder der Sylinderachse." Luftfahrtforachung, Bd. I9 (I9I+2) . pp. 302-312 . Th^e concerns itself with a complicated special case of such a singularity. MCA 1M No. 1196 21 point for Mach wsve 2 . The first five rows reproduce the initial values as veil as some firrther values that hold at the given points vhlch are necessary for a later calculation. The calculation of a new point is carried out in the form, of an iteration method,' as exi example the point k will be explained. Next, the values for Xl). and nij. are estii.aated (colu^^ins h and 5) • In order not to use too favorable an estimate, it is assumed that Xj^ = X-^ and M,r = i-ip • The quantity (X + |i)r is determined for these magnitudes and fro'-i that, with the aid of figure 3 [ —\ farther on I JL\ (ooluTins 6 and 8) . With these values and [ J- ■"- ?■■ \ are computed (coluiiins 9 and 10 directions for Mach waves 1 and 2 ! /_ !^.. ^. j and are formed (columns l'+ and 19) and the Mach waves are plotted on the yt-diagram. From this y^ and ao^ (columns 12 and 13) a^re obtained. With these values Mj^ (column 11) and the average values Mjjj^ j^ and M^ ^ (columns 15 and 20) are computed. To continue for Mach waves 1 and 2 Aa t. = a^t)^ - a^t-j^ and Aa^tw g = a^tj^ - ^q^^. have to be computed (columns 16 and 2l) and can be substituted in equations (33) • Tlie quantities Mj, -j^ and A|j.r p as well as Xl and jil (columns 17, I8, 22, 23) are obtained'. If the values X and \x. calculated in this manner do not agree well enough wi.th the original esti^nate, the calculation must be repeated in which X and n just calculated appear in place of the earlier estimates. Natiurally, the Mach waves must be plotted over again, too, in the yt-diagraia for this. These figures only show the final form at any instant. For that reason all the steps in the iteration method are p\it 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 X|j.-dlagram, as well as the yt-diagran (fig. 5, right) . There the X-axis was selected slanting up to the right at U5'^ and the la-axis downward at ^5°- With a suitable vertical scale X - m-, and therefore ^/^q, is obtained immediately on a horizontal scale X + n or W and with the use of -unequal distributions a/a and P and, for isentropic flows p/p too. The X- and ij-axes were inclined ^5° to obtain the quantities of physical interest "v/a , a/a^, etc. in a coordinate system with the conventional arrangement. 22 NACA TM No. II96 10. FLOW OF AN IDEAi GAS WITH ENTROPY DIFFERENCES The introduction of X and \i with the object of ottaining equations in only one unknovTi, at any time, with the iteration method for the determination of the flow was possihle up until now because the expression i£. with constant entropy might have been oa considered as the differential of a function W independently of the characteristics of the incident gas. Naturally, that is no longer possihle with variable en ti' opy . The computation of the flow must, in general, therefore, return to (ik) . The ideal gases constitute an exception. Here, as recognised in (30), the function W which essentially agrees with / ^2. for constant entropy, J pa depends on the temperature alone, and no longer on the entropy. If the expression §^ is considered, therefore, in the case of pa variable entropy as dependent on the variables T and s the effect, of change in entropy is se'parated, then the rest can be written here as a differential and X and ii can be introduced as previously. The change in the entropy along the Mach waves must natui'aily be regarded separately . This is possible without especial difficulties since the entropy is constant along the life lines. The transformations are cari'ied through in the following m&nner. From the second law T ds - di - i dT3 "aking in'-o account (26) and (29) i_ QT1 = ^ - 0^ ^s = , / ^P^^ flT - / '^^ oa a - T i^- = ./ :KJ- dT - tX T.. ds a / RT \/ c.^ E Inrrodacing W, X, and |i as before, the consistency conditions are ob^'ained in the form ^ = - £_(v ^toF + 3TnF I + /S_ T J_ ds. it a.n V 5y 9t J V c^, R a^ dt (3^a) for Mach wave 1 du = - ^ [ V ^T^ + MrF_] + /^v^ i_ ds (3i^t) for Mach wave 2 'i't ^o \ ^J St/ ^ S R a^ dt NACA TM No. II96 23 The differential quotients ds/dt formed along the Mach waves interfere: The following transformations are possible. Analogous to the flow function of two-dimensional stationary flows, a function \1/ is introduced, ij; is constant along the life lines . This can he achieved hy requiring that ^ = -Z.SLr (35a) St " ^o Po |1^ = ^ -£■ (35h) 5y Fq Oo Along any curve of the yt -diagram ^ = ^ + ^ it (36) dy ■ rij hi dy Along a life line ^ = v therefore dt dy" " F^ p^ ■ F^ p^ 7 = that l3 "^ is actually, constant along the life line. At each point of the yt-diagram \if itself can "bo d^^flnod l^y a line integral that leads from a fixed point A at which 'I' might be zero to B . \ = Jm ^ - ^ ^ = / f JL ^ dy - F. ^.at") (37) B <./A\c)y 5t J Ja^v^Fo P^ F^ p^ J The physical significance of "if can he recognized as follows: Let C (fig. 6) he the Intersection point of the life line through A with the line t ^ constant through B. To begin with, the path of integration is along the life line from A to C and, from there, out along the line t = constant to B. Along the life line AC . '^ is constant ^ M/c = ^ = 2k MCA TM No. 1196 along the section CB, dt is equal to zero, accordingly • B dy From this, it is evident that V represents the raass vhich is enclosed heit-^een "he particles at an instant in time for which ^ is zer^^. The fact -i-hat s is cons rant along a life line can "be written with the use of '^\^ in the foi^n = b(^) (38) For ds /dt then ds ^ ds ^ dt d^ dt for which =3L is to be taken. Just as ds/dt previcusly, along dt the Mach wave considered. From (35) and (lO) o '^o for Mach wave 1 d^i/ _ F D dt F p o *^o for Mach wave 2 Substituting these in equations (~h) , allowing for (23), (2fi), and (29) replacing 1^ according to (2Tb) by -1 ^ and x>^/o^ dlj; TT rill/ ° oy Q — a^ yields the f-illowing consistency conditions : WACA 1M No. 11Q6 25 For Mach ■'■-ave 1 ^= a dt _v i^l nF ^ _1_ SZnF] _ ^o F p dn % \a^ ~§y a^ dt So^- dij/ (39a) For Mach wave 2 ^ = a dt o Pn O P F. q\i (39t) Here P is a function of \ + n (fig. 3), ^/^o ^^ knovn to "be a function of y and t. From (38) and (27b) it follows that n = 3t(^J/) and from this dJt^ = [^ (ii/) = constant for a life line d\}/ dd/ (^0) For the sake of co'npactness, introducing ^o F An Then (39) ff;'^es over into the forii d>^ = a (-M + K) dt ° (ij-la) for Mach wave 1 ^ = aJ-U - N) dt ° (i)-lb) for Mach wave 2 Equations (kO) and (Ul) suxiplant the previous consistency conditions (15) and {Ik) . 26 NACA TM no. II96 Before starting the characteristic construction, the problem arises here, too, of computing X and [i, and now ^ iDesides, dV from the initial values. Along a curve K of the yt-diagram let the velocity be given by "V/'a , the phase of the gas by v/Vq and T . From T vith the aid of figure 3 X + |i is obtained, from v/a^, X, - u; with this \ and p. are knoi-Tn. Since p/Pq are given, and P as a function of T is to be gathered, from figure 2, It is obtained immediately from (28). As a result of plotting Jt against the values of y frora the curve K and differentiating ^ is obtained. From (36) and (3'5) together with (23) £;_ for the curve K may be computed for the curve K and, finally, with that dit ^ dn di]/ (?V dy dy is determined. In many cases these computations are superfluous^ if entroTDy differences arise fr'^m com-oression shocks, the determination of ^, X and (i includes their calculation. The d^' way the computation of f].ov has to be carried out is shown in figure 7 and table III with points h, 5, 6 as examples. The related X^i-diagram is right center. (The points Included, in addition, in the table and the figures relate to a later section.) Along the curve K (points 1-3) X, \i, and ^Jt, are assumed as known, in the auxiliary diagram ^ has been reproduced as a Soli function of y. The computation of a new point - take point k as an example - begins, here too, with an estimate of X and |.i (table III, colmims k and 5) • After that, as before, the following are com"DUted \ o /l^. \ o /'l^ (columns 6-10, 19 and 2U), the position of h is indicated in the yt-diagram and y. and ^ri\ ^ 'the table (columns 11 and 12) assumed. The determination of ~ 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. Iiq6 27 with the construction of the Mach waves and going back over these to learn the desired value — from the auxiliary diagram. The d\}/ Xn -diagram 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 he estimated there without difficulty, and then the average velocity learned. (Compare points Ik and 15 in the yt- and in the \p.-dlagrain .) After ^ has been found d\|/ and, in addition, P has been learned from diagram 3 (columns I3 and Ik) , Mj and Wj, as veil as ( -M - K)l^ and (-M + l Pit lU \/V^ /V^ P2 p ^ P''ra2,lJ^ ^^ni2,l| T^o ' ^o " % V ^ 4?, l}^ ^42,1+ ^o -S;-^n2A(%\ - V2 '''For ideal gases the first term of the left side of (l^i-) may he T-rritten i _ i£!ia, then 0/0 does not have to be computed k dt ' ' ' o separately. To per^iit the procediire to be applicable in -iiore general cases, this simplification is not used here. 30 NACA TM No. II96 Putting in numerical values gives as a resialt 0.5i+7 Pi^./Pq + \/a^ = 1.11^30 OMk p^/p^ - v^^/a^ = 0.3715 ' . ' V, /a = O-Sl+S 4 o From the velocity computed atove v^/a and the velocity at a point k\ estimated for the present, of the connecting line 1.2, the average direction of the life line passing through h is ottained^ hy an approximation method. If this is proceeding from k backwards, the more accurate position of U' is ohtained. By interpolation between 1 and 2 jr. ' = nj^ - 1.2l^■3 is ob tamed. Since the values PIl vk _Lj _-l^ jt, do not agree sufficiently well yet with the originally ^o 'o estimated values, the computation m\ist be repeated with the magnitudes Just obtained as starting values . This gives v./v = l>hlQ; V. /a = O.33B3; n. = I.2I13 4- O 40 M- 3£. SIMPLIFICATIONS FOB IDE/i GA5ES T-TITH CONSTANT SPECIFIC HEATS Generally the flowing 'nedium is an ideal gas with constant specific heat or at least can be considered as such, as an approxi- '■latlon. In such a case appreciable simplifications are possible. Let k = /C V V then "" — ■'• ■0 • p _ 1 ^p " rrj ^-^ ^v - i^TT NACA 7M. Wo. 11q6 3I From equations (27a), (29), and (30) With this, it follows that ^ - iV^o y a^ 2 / a , V T (i^2l)) f^ = k and from that J- = 1 + k.^(^ + ,i) (l;Oc) ^O consequently^ i_ ^ _! The directions of the chajracterl sties are oTDta:ne(^ from (9) an.d (lO) in the form If = a^ ^_I_ii for the life lines dt o 2 |i = a fl + kj_l X - i^ (a) for Mach waves 1 dt *- V k k- / ^ = a^f-l- i?-_L- u. + 2-:-i£ x) for Me.ch waves 2, 32 NACA TM No. II96 The consistency conditions for the Mach waves remain unchanged in the fern (■'^0) and (^1). M and N are expressed as follows, now, '^ ^'-A 2 ay a^ ^r) o 21c . N ^ ^o L. k J d\!/ The directions of the characteristics may now he found very conveniently graphically. A construction which is suitatle if the simult.aneous treatment of the flow in a X.(j.-dlagrazn is avoided is the contribution of Adam Schmidt . (See 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 and '<- on the left one as ahove . A life line for a phase which is given by X and \i has the direction of the connecting line of the points concerned on the fimction scales. Similarly, there are scales to use for a Mach wave 1, which give -^. ." . . \i on the left and 1 + — r — X on the right. For Mach wave 2 f "*" , - (i has "been plotted on the left and -1 + ■■ 1 ■ ' X c«n the right. In figure 9, the direction of Mach waves 1 and the life line is given for X = ]. .1 and n - 0.6. If the phases in the course of the constru.ction of a X|i-diagram are followed up, the following nethod-ls suitable (fig. 10, right)- A vertical line is sent through the 0-point of the XM.-syBtora and the poles P-,, Pr,, and ?-r are deter.'^ined, where P-j^ is on a level with the origin of the X|j,-system and ''2 away from it. P-[_ and Pp are directly below and above Tj^, respectively, and likewise the distance '-^2 from it. To f-lr^d the direction of the characteristics for a given phase, a horizontal ray and two rays slanting upward and down'VTard at an angle arc tan ^ " 1 are drawn. 2 These intersect the vertical line through the origin of the X(i-system at the points Q-]_, Qg, and Q^ . The connecting lines P]_Qi, P2Q2, and PtQ-j. are the directions of Mach waves 1 and 2 and the life lino. In figure 10 the construction for point h is carried out. WACA TM Wo. 1106 33 This ccnstruction is especially convenient with a triangle having an angle pre tan ^— ^ — • Figure 10 and table k give an example o-f an application for the same initial values as in figu.r* 7 and with o-dAv ~ constant = 1,1|. 13. BOUNDAEY CCWDITICNS If the flowing gas column is not Infinite, the variation of' the flow is deteiTiined hy the phase at the start, in addition, also "by conditions at its boiJiidaries, For example^ a gas can he closed off hy a piston or rigid wall, flow out into a space with a given pressure, or he sucked out of the same. Generally, the houn«''ary conditions may be formu3.ated so that relations between the phase magnitudes of thu gas and fts ve].oc3ty along a curve of the 'yt-diagra'Ti are prescribed. The nuraber of conditions which are needed for the boundary curve corresponds to the numbur of charac- teristics which run out fron there into the interior of the flow. For example, the gas flows out of the end of the pipe into a space with constant pressi;re, with v< a, then the line y = constant is the curve for the "Pipe for which the boundary conditions are given . A f a'nily of Mach vraves spreads out from it inward, while the other family and the life lines reach this c^^rvo, approaching it from within. In this case the condition can be prescribed that the pressujr-e in the exit section be equal to the outsid.i pressure. If the. gas is sucked in from outside, Mach waves of the one family proceed from the cm^ve 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 woitld be required perhaps that the phase of the gas in the entrance section be related to the phase in the outer space through Bernoulli's equation^. (An exact formulation is difficult, since tlie flow at this location is no longer one -dimensional.) I? the characteristics of all three families of a given curve lead out into the interior of the region to >e computed, thure are three conditions to prescribe; this is the initial value problem already treated. The other extreme, that at th(3 boundary of the region of interest, generally, no condition can b'j fulfilled, is physicaD.ly conceivable, too. For example, if a gas with V > a flows in a space at constcnt XJ^'-ssure, generally no characteristic goes inward from the outflow section. Tompare Schultz-Gnmow, loc . cit. 31; NACA :M No. II96 Actually here — disregarding boundary conditions vhich force compression ehocks — no effect on the flow variation in the interior is possible frciii outside. The treatmeiit of boundaxy conditions is explained with tvo examples vrhich are connected with the flovr in figure ?• The com- putation is entered in table III, as far as possible. The first example includes poin'cs 7 to 9 s-nd, adMttedly, it has been assumed that the gas coli-'iiui is bounded by a piston whose life line is represented in the yt>-dia£iraiii as the curve 3^ If 9» (Whether it is practicable to realize such a piston in a tube of variable cross section is unimportant for carrying out the comp'atation. ) The X|i-diagram referred to is in figure 7* upper riglit. To begin \riLth, an estimate is made of the phase at 7 which has been chosen X = X = O.oOO, \j.rj = II, = 0.050. Since the line 3.7 is the life of a particle, { — ) is already- Iciown and is uqiial to f — ) . V^M 7 \6.-^J 3 With this the values in coliimns 6-10 and I9 are calcvilated. As a result of di'awing in the i'lach wave y.'J, yv aJ^d apty (colirans 11 and 12) are obtained and besides vy/aQ from the direction of the life line at point 7 which has been reached. (Tliis quantity is found in column G under the value computed from the initial estimates.) Now the guantities in colurjis l-^ to I8 and 20 to 23 may be computed, the value Vy/ap obtained from the boundar;^r concUtions will be used. With that X„ is already knoim. The quantity [in is obtained from b, he relation V a„ Inserting numbers 0.323 = 1/2. (0.U7 - ix^); Uy = -0.229. Since the first estimate was too poor, the computation must be repeated. Point C is computed from 6 and 7 by the method explained in section 9. From o, point 9 is obtained in the way just described. This method of calculation is usefu.l for an;'- laws of motion of the pipe; a special argument la necessaiy onl;^ if a dlscontlmiity 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 y1>-dlagram that means NACA 1M No* 1196 35 that the life line of the piston which has a 'bend at the instant of the velocity discontinuity is rounded off iimnediately . Then the flow may "be drawn accui'ately Just as previously. To ohtain sufficient accuracy, enough points must he taken on the rounding off so that the velocity of the piston does not change excessively from point to point, and at each point a Mach vave of the first family may converge and a Mach wave of the second family may diverge from there. First of all X must be computed for the converging Mach wave and then from X and the velocity at the incident point n detonnined for each Mach wave. If the roimding off "becomes smaller and smaller, these points on the rounding off draw closer ond closer. With that the values of X approach a single value, which may be computed from the field before the bend. The Mach waves 2 spread out in the shape of a fan from the bend and the fan includes all values of \i vhich lie between the values of i-i for the velocity before and after the velocity discontinuity . For the second example, there is at the position y = y;j_ an open pipe end, through which gas is sucked in from outside and for which two conditions i.iust bo specified along the boundary-condition curve. The curve is the curve 1, 10, 13 in figure J. In the outer space let jt = it-, : for the entering particle therefore ^ = 0. ^ ' d^ This is one boundary condition. As the second boimdary condition there is the require. iient that the phase in the inflow section be related to the phase in the outer spsce by the Bernoulli equation. This condition may be satisfied, already, at point 1, accordingly i + v2/2 = ii + v-l7: or also ^ - ^o lfr\^ h-^o l^lf constant To determine these constants, from figure 3 the temperature T-j_ is taken for (X + |i), frcm figure 2(a) for T]_, (li - iQ)/a.Q^. Then i. - i. --Lf 2\ / Ti 0.292 36 IIACA TM No . 1196 2 Since (i - iQ)/ag- is a function of T and, therefore, pf X + n, = ^^-^-^ this "boundary condition can te plotted as fcbe curve K V -0 2 in the Xn-diagram (fig- 1, lower right) . At "best, the computatiai of point 10 "begins anew with an estimate for X and li so that the houndary conditions are already satisfied (colvjmis k and 5) • With thin, the quantities in colutnns 6, 7, 8, 10, and 2U are computed and Mach wave 2 drawn in with that. The quantity s. t-, q is ohtained in coluinn 12, the values y-^Q - ^i ^"^"^ Si = 0. (Columns 12 di|/ and 13 are given beforehand.) Now the quantities in columns Ik to 18 can be obtained. To detcmine, wi'th this, the quantity (-M + N) in column 25 it is to be noted that (-M + K) for the particle originally in the pipe has the value, perhaps, at point k and changes dis- continuously for the particle recently sucked into the quantity (-M+N)^Q. On that account the life line is drawn, which separates the particles in the interior originally from those particles flowing in from outside. This intersects Mnch wave k, 10 at point 11. Then the following is obtained (column 25) -HA(a,t)^3_^3_Q(-M-^N)3_oJ 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 X^-dia3I'am with the value of p- found, X is obtained (column 23) The computation is repeated with the values found in bhis 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 ,1ust presented. The Xn" diagrams of the two last examples were kept separate from the X|i-diagram drawn for points 1-6 foi- the sake of clarity. NACA ™. Ro . 11q6 37 If the various figures are visualized as "being joined - the upper diagram connected to the middle one at the line 6, 5, 3, the middle one with the lower one at the line 1, '+, 6- it is recognized that the p].anft is covered with several sheets which arc- connected along the figures of the characteristics. There is such a snperposition, already, in the lower Xia-diegram; there arc to ho iiiiagined inclosed the quadrilateral 10, \, 6, 12 along 10, ■'-(•, the trifJigle 1, h, 10, along 10, 12 the triangle 10, 3, 12. In addition to the boundary conditions, transitional conditions can also appear in the interior of the flow. In the example just discussed just that would have "been the case, if in the outer space rt were different fron n^ • At the location of such a discontinuity for n agreenent of pressure and velocity must "be required. To go into such questions with greater detail lies "beyond the scope of this report. li+. TRANSITIOn.AL CGNDITIdlS AT COMPRESSION SHOCKS The flow in a given part of the yt-plane is defined "by the initial and "boundary conditions and it; calcula'ble "by the methods derived up unti.l now. It is possTola that it might happen during the construction that regions of the yt-plane are covered with phase quatxtities several times. This is the sign for the appear- ance of compression shocks. The entropy is no longer .constant after the passage of a compression shock. On that account the computation of compression shocks simuJ-taneously inclvides the determination of the fimction E(\lf) or ^(-{r), too, for the region of the yt-plane "behind the compression shock. For the mathematical treatcient, a compression shock is to he considered a curve along which two flows collid'i, which are related to one another and to the direction of this curve by transition conditions. It will be the^^problem of this section to derive these (kiiown of themselves)^' transition conditions in a convenient form for the present purpose . Proceeding fron a stationary compression shock, that is from a co-apression shock whose front Is at rest relative to the coordinate system selected, let the index I designate the phase before the 9 ' " Coi'.ipare Ackeret for instance. Beitrag Gasdynamik in Handbiich der Physik, Bd . "TTI, p. 32k and following pages^ Berlin 192?. 38 . NACA TM Wo . 1196 shock, the index II the phase after the shock. The additional index s might po:ixit out that this concernR the calculation of a stationary shock. Then the momsntijia .and the enerry theorems as well as the equation of continuity are -written in the form ^Is ^ ^Is^ls'^ = ^Ils ^ Plls^iis- (^3a) i + Jv^ 2 = ^ +1^ 2 (i^3t) Is ? Is Us 2 Il3 o V = V {^30) Is Is lis lis Furtherniore, the characteristics of the gas concerned must be knotm, posslDly in the form P = p(i, 0) (U3d) If the quantities in advance of the shock ixc » Ox-,. 9-nd Vt„ are i^'-/ ' ^-iz>' xs knovr^ then +he co'ipression shock is therewith calculable. Actually all three quantities enter into the general gas lavs, too,- as parameters. In order to carry out the computation practically, in such a case, Ptt- fi'om ('l-3c) and 1 from (^3h) have to be expressed lis lis as func'tlons of "Vjjg and the known quantities and then substituted in (H3a) • With that, an account of C+Bd), n , too, is a lis function of v and the known quantities in advance of the shock, lis In this manner an equation for v^^. alone is obtained which miast ^ lis be solved numerically in a suitable manne-^. For an ideal gas for whjch c is not constexit, equations (l+B) transform with the aid of (23) as follows: .^S- HT^ ^ ^- v^ £ = BT^^ + v^^ 2 (Ivl^a) o Is o Is lis Us ^ ' '^IIs ^Ils ^•(%s)-^^is' = i(^iiB)-j-ii;' ('^^^) -IS ^= -IIS ^^^^) ■-lis WACA TM No . 11q6 39 Since o antieRrs here only in the combination o^^ /p^ only Is ' " Ila Is T^ and ■"■-. still re^iain as parameters upon which thj phases behind ths shock depend. To calculate the shock curves mTaerically^ it is useful, first to regard Tjg and Tjjg as parameters and determine v-j- from this suhsequently . The computation process is the following: From (kha) and {kkc) ^I3 ^'^ ""lis ^^^ As a result of squaring this O O ■ O o E"T ^ + 2ET^ + v^J = ^- + 2ET-rT„ + V-, ^ ' ""'^lE " "Is - ,, 2 ^ "^"'^IIs ■ "lis (i^5b) ^Is ^11.-^ Introducing gives from (i^^h) ^■^ " ^IIs " "^lE ^Ils^ = ^is' - 2Ai ■ (^6) Putting this in {h'i'h) , the desired equation for v^^'^ is obtained as V j^l 2A1 - 2B,fTj^^ - Tj \j + vj^^ j -k^^^ + lmi(T^^^ - Tj^ L_ 7 ■^^('^lls^ - Tis^) -SE^Tjg^^i =0 If Vt^ is determined, then v^^ and Pt-x„/Pto ^^® computed in J.O ' xlS XX 3 J-o turn with the aid of (hb) an.d {hkc) ; firilly lis Is lis Is Il£> -^s i^O NACA TM. Wo. II96 For an ideal gas vlth constant specific heats, the following trans- formations ':iay be undertalren . According to the familiar- relations i = —^ — RT k - 1 and a? = kET Equations (U^a) &nd (^i-l+'b) are isTitten in the form a^ 2 2 2§-+ r^- ...-i-Ig-, -!- V. or Ir-r Is ~ ■.,„ ■ ^IIS ^- Is ^^IlB 2 2 . 2 2 p , 2 r^ ^Is ^ ^Is = r^ "-IIS ■" ^IIB + lis ^ if^iiv 1- + Ins ^ ^Is/^Is ^'Is ' ^^^""1^/ ""lls/^ls ^I (i^Ta) By this, ^TTs/'^T'=i ^"^^ "^TTq/^T° ^^^> ^irith that, the other qiiantities^ too, depend on the parameter "Vib/^Ts slone . To compute ■^TTa/^is' .f^ll'^/^ls) -^^ eliminated: is obtained as a result. NACA TM No . II96 la The solution of this equation is found, iamediately, if it is tome in mind that on account of the form of (^7) a solution is represented by '^IIs/^Is = ^Is/^ls then Us /^ 2 / !ls + k - 1 ""is ^ ^ 1 \ ^IB 2 a. Using this, the following is obtained from (kT^) and / \2 lis ', ^ -^ + k - 1 /t^ '-,2 /t ^ \2 f _Is \ _ J lis \^ib; \ i^ y ^Ils/^IB =Cib/^sX"ib/\is) ^113''^I3 = (pIIs/Pi.3^)(tiis/Ti8) = v^ /a a^ /v^ [a^^ /a-r \ ■ Is' Is Is' lis I lls'^ Is I \ / The change of entropy is of interest, as well; with the aid of (27) and (28) , these expressions result s - s , J?3 Ts ^ E k - 1 'vrj^ /T \ p k— irl -lis \ _ in^IIs ^IIs ' ^la _ 2k in. k - 1 a. IL§. - In -X?. + 2n _Ss - 22n Us a. Is ^Is ^Is 2 7n -^Sa - In II^ + in ila k - 1 ^Is 'Ts Ts 1^2 NACA TM Wo. II96 From this it-TT- / B \ V a ^'IIS _ «IIs \ ^IB Is >IbV «IB \ St^ ; "j^ Vjj^ Arbitrary compression shocks result from the station-'=jry compression shocks ,1uGt calculated tecauso a velocity is superimposed. In doing so, the thermodynamic phase quantities before and. after the shock for vhich accordingly the index s can he omitted are retained and moreover the velocity differences. Since the phase in advance of the shock Is elrea'ly given in the construction of flows, before the shock is computed, the relative velocities with respect to the phase in advanc-j of the shock are formed. Let \\ absolute velocitjr of shock front Au relative velocity of shock ifront with respect to particles in advance of shock Then -V /v Au __^ _. Is . ^^ ^ -^ - ^ ^ _ _L_. _I3 . a^ R^ ' II ,1 II I k + 1 \ a^ The signs opnoaring in this are not astonishing. A stationary compression shock in a gas which moves in the positive direction propagates itself in a negative direction relative to the material ahead of the shock, and in so doing, produces a change in velocity in the direction of its propagation velocity, that is, in the negative direction, too. Naturally, compression shocks, which travel in the positive direction in the material at rest are also possible, the signs of the velocities have to be ch.anged for these. The thermodynamic phase quantities of this are not touched upon. Corre- sponding to the distinction which had been met in Mach waves, these last compression shocks are designated compression shocks of the first type, those which propagate in the negative direction as compression shocks of the second type. In figur-e 11 the pressure ratio, for an ideal gas with k = 1.1+05 the propagation velocity of the compression shock and the change in entropy (expressed by n /-Hj) has been presented as a fimction of the velocity change Av . For compression shocks of the first type Au and Av J- J. yX X J- ^X are to be taken with positive sign, for compression shocks of the riACA TM No. 1196 J+3 second type with negatlTe sign. The fundamsntal nrumerlcal valties appear in table- Y. Such a diagram, would have to be used to apply the characteriBtics method in the form given in section 11 in the computation of compression shocks . How are these i-elations for the compression shock expressed in terms of X and ii ? If two compression shocks which only arise separately from superposition of a velocity - they are distinguished by the indexes a and p - are represented in a X|i-dia.gram^ that is, if the phases :"n advance of the shock \^ : ii-r ; X-rp ; \i-rr, ^'^^^ the phases behind ohe shock are plotted, then here, too, the expression must be arrived at that the thermxodynamic phases in advance of and behind the shock, as well as the velocity differences for both compression shocks are the same. Accordingly, I, a I, a IP Ip 4l.a " ''ll,a ^ ^Iip "^ ^11, p (\ - \i \ - ( X -|i \ = 1 \ - \.i \ - ( \ - \i ) \ II, a II, a/ y I^a I,a/ ^ ^.P ^^A' \ ^P ^P/ By subtraction of the first two equations (x ~ X ] + ( \i - \i ] = l\ - X, A + AiT-rn - ^-^rt^ \II,a I, ay y 11,0. I, ay >, II, p IP, I Iip ipy Eearranging teire in the third equation gives {X - X \ - [ \J- - i-i \ = (x - X \ - i\s - |i \ \;il,a I,a; , V Il.a l,al y Up l,pj y Up iPy' From the last two equations it foD.lows that A. "A, — A. " X Il,a I, a II, p I,p II, a I, a II, p I,P hh NACA TM.No. 11q6 that is, the changes in X and p. in a compressicn shock are main- tained in the superposition of a velocity. Accordingly, the shocks are designated by ^11, 1 ~ 4l ' ^I and The follovlng relations hold for Meal gases with constant specific heats, according to {h2) A ^ = X - X = , II, I II I ^ - -L \ ^o a 2 r^ii '^A^IlI.Il \ o o / o o !ii . A .^11 k - 1\ a^ An. II, I II = ^^^ - ^^ = ]£ h(-- \ II a II a V-r °/' a. 'II _ ^11 - ^I a I k - 1 V ar a /a is to be computed from X-j. and \i-£ by (^2c) . For the I' o expressions in curved brackets AX = a -! . V - V _. _II . l) ^^ 1 An _2 k - 1 • In \ a \. I - 1 V - V II _i NACA TM No. II96 1^5 are introduced. These quantities, as veil as Au/a-r, '^uAt? and p. T-/"nj de-nend only on v-j. /a^ according to relations previously developed. They are plotted in figures l£(a) and 12(1)), and. admittedly, the .upper designations refer to the coLnpression shocks of the first type, and the lover designations to compression shocks of the second tyT)e . Figure 12(1)) represents an increased section of figuj^e 12(a), vith the apiDropriate numerical values in table V. The folloving exatiDle sho-'-s a first application of this diagram. In a pnne of constant cross section there is a quiescent gas of constant entropy and constant pressure, the sonic velocity is taken to be a^ = a . Suddenly, a piston is driven into the pipe at a imif orm speed of .5a • '^s'hat is the ensuing flow like? Figure 13 shovs the yt-diagram. The starting point of the piston motion lies at the origin of the coordinate system. The life line of the piston is shovn vith hatching. A compression shock forms in front of the piston, vhich imparts the velocity of the piston to the particles, so that the particles behind the compression shock move vith constant velocity. Corresponding to the phase in front of the compression shock ia Xj =.0; ^l-J- = The velocity behind the compression shock is ^11 = •5ao . • ■ therefore. i'ihi - '-^11] = <^-^ \i " ^11 = ^ From this, on account of X = and n = AX - A^ =1 II, I II,,I Since s^/a = 1 this gives AX - Ai = 1 1^6 NACA TM So* 1196 As a result of causing this straight line in the ZvX/jH-diagram (fig. lP(b)) to Intersect the shock curve, the following is ohtained: AX - 1.022^ All = 0.022; ^ = 1.3li6; Tij-r/-n-r = 0.9J0 \ = 1.022; u = 0.022; u = I.3U6 From Xjj and \Xjj, P is computed hy {h2d.) , from this hy (28) Vjj/^1 = 1-970 The goal would be reached somewhat quicker in this by application of diagram 11. 15. PEELIMIKARY ABGIMENTS Hi THE DETEEMINATIOW OF A COMPEESSICN SHOCK IN THE FLOW FIELD It is the objject of this section to show first of all by what data a co^nprespion shock in a flow is determined, and, secondly, to give a method by which +he computation of such a compression shock is possible . Ap can be readily sho^vn, the velocity of a compression shock is larger than ihe velocity of a Mach wave in the material. This means, that the flov^ field in advance of the conpression shock remains unaffected by this and can be computed independent of it. It will be assumed to be Icnovn what follows . For the field behind the shock, a cofnpression shock of the first typo repi-esents 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 shovn how examples can be conceived of flow fields with compression shocks. If In the yt-diagram (figs, lil-(a) and lU(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 1M No. 1196 I+7 line the pz'opagation velocity of the compreseion shock is given, namely for each point of CD. Beside, the phases in front of the shock can he learned for the points of CD; with this the phases "behind the shock are calciilahle . Fro:a the phases tehind the shock, a portion of the flow field "behind the shock, namely the region CED (fig. lU(a)) may he computed, or if the entropy is known for the life lines at the lower end of C. The region CFD (fig. lU(h)) as well. It is necessary to go f oi-T^ard along the life lines ar.d Mach waves 2^ "backwards along Mach waves 1. Imagine in figure l^(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 pre3crl"bed, too). In figure li|(h) it is necessary to imagine another flow field ad.ioined continuously at the lowe,'r end of CF; here the compression shock and the flow deteriuined "by it satisfy the condition tJiat it is comDa+ible along the M.^ch wave CF wf th another flow. From these flow fields the following is recognized; the coipression shock through the portion CE of the life line of the piston or CF of the Mach wave is defined as far as it is reached by Mach waves of its Lyne (here the first, therefore) . A change of the Ijfe line of the piston outside of CE or the Mach waves outside of CF propagates along Mach wave 1 in the yt-diagram, to he exact, and neglecting cases in which a second compression shock arises, attains the comprsseion 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 boimdeiry or junction conditions takes effect at that position on the compression shock where the Mach wave 1 concerned reaches it, that is, the portion CD is certainly changed. If the life line of the piston is Imown 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 necusslty for knowing the continuation of the compression shock beyond Dj it concerns the regions CEGJD or CFHKD. It will now be shown how to procerie fimdamentally to r:ompute a compression shock for specified boimdary or junction conditions. As a concrete example assiime the compression shock to be produced by a piston which experiences a sudden jump in velocity. (See 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 iriLmediately behind M can be ascertained inimediately by the method applied to the example of the last section. The compression shock - as in previous examples of Mach waves - is computed 3n individual sections, which are so small that the phase quantities h^ MCA m No. 1196 for -^.hem, mp-y be regarded as varying linearly. As ,iust carried out, the phases behind the compression shock are calculGblo, if the velocity of the shock is knovn. The velocity at M is known. Along the portion of the co.Tipression shock to bo computed, M, N, the nhase change and, vith i-^,, the change in propagation velocity of ihe co'-'irrepsion shock, too, are regarded as linear. Accordingly, for all possible shocks which sa+isfy thd transition conditions, the Tjoriion M, N, of the conprossion shock depends only on a single paraneter, the velocity change between M and N, to be exact. As a result of co'ipu'-ing the field behind the compression shock for various values of this psra^eter, by interpolation, that shock lay be ascertained which is consistent with the specified piston '■iiovenont. At best, for this K is penaitted 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 K proceeds. Now the region OPQW may be computed in a familiar manner. For the determination of the extension of the compression shock NR the phase behind th'-o compresrjjon shock at the point N may be regarded as given everywhere along the entire Mach wavj NQ. On the other hand, that value of velocity changes betvreen N and R has to be determined by interpolation, -which relates to a flow field that continuously ,1cins the known field along KQ. With these two types, namely 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 COMPUTATICK OF COMPRESSICN SEOCKS IN THE FLOW FIELD Exa-'ples will be given of how the problems formulated in the Tirecedlng section can bo solved by neans of iteration methods. Let • the flow be that computed in figure 10 and table IV. As the start of the new 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 H a second time. The new portion of the compression shock to be computed that corresponds to MtJ or NR, accordingly, is assvmied to end on the life line 8, 9 NACA m No. iiq6 {4.9 of figure 10. The phages, in front of the shoclc for N or E are obtained ae a result of interpolation along this line. For these calculations it is necessary, on that account, to have the Imow- lelf^e of the flow field in front of the shock at the points 1 (M or W) and 8 and 9. In table VI which has the same arrangement as table IV these values hav-e been recorded. While it sufficed to know SJL for the construction of the flow field, here -n itself dVr must be knoT>?n. These quantities for points 1, 8, and 9 are located in column 26. In the designations, in these examples, the only deviation from figure I5 is that only points on the compression shock are characterized by letters. Kurnbers 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 shock at the pojjit N and the phases along the Mach wave R-j.-j-,10 (fig. 16(a)) may be considered knovn. The phases at N and at point 10 appear in table VI, phases in between are foimd by linear interpolation; moreover, for N^^ the velocity of the compression shock 3n.d n have been given (columns 25 and 26) • Besides 25. for the life lines lying below N may be viewed as di{; 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 -5. between them as dj/ as constant. In the following this has happened throughout. Since N and 10 lie on a Mach wave, the consistency condition must naturally be satisfied. Tn connection with the flow calculation the existing data are to be taken from the preceding calculation steps. The real coiit)utation begins with the fact that the difference in \|; from its value at the starting point of the portion of the corapression shock to be computed (K here) is ascertained for the life line up to vhich the coTipression shock is to be computed (8, 9 here) . This co'^putation is carried through along the curve of the initial values in figure 10, the life line 8, 9 used here passes through point T there. By (37) 50 NACA m Wo. 1196 f^ / \ /' \ By (23) and (V2d) Po Just as for figure 10, F has the form F = F^y-t For point 7 y = 1A5O; at = I.18O; X = 0.66; [i = -O.I6; jt^ = n = 0.81+9 For N the corresponding values appear in tatle VI. With this the following is obtained: JL L.\ . 2.680; fl- ^ X-\ = 1.110 I. £_\ . 2. 170 J fJL £- jlN . 0.930 F F^ P. a. V. o oA \ o ^0 o \ H £-\ = ?.1|2-.: /"l- iL Jt}.. = 1.020 " - ] F D a 7tN,T \^o o oy,^ ^ ^o ^ A\I/ = 2.U25 X 0.075 + 1.020 X 0.03 = 0.2122 7,N WACA TM Wo. 11Q6 In figure 10 SS. had already been given, It must "be the same as tha+ found from the quantities ,1ust computed. In fact /S£.\ = "t " \ = 0'0h9 = 0.230 llhis is the average value of M. as can "be gathered for the d^' stretch I.7 from the auxiD.iary diagram in figure 10. .After these preparations, the actual iteration method ie reached. To begin with, the phases at the points R_^ snd 11 are estimated, in that 11 is the intersection point of the Mach wave 1 leading hackwords from E with the given Mach wave K,10. Since no better reference point exjsts for the estimate, these phases are equated to the phase at W^^ . Moreover, still another estimate is needed for ~ behind the shock: for this, the same value that prevails dif; ' at the lower end of W is chosen. With these assimiptions, the figure K, E, 11 may be drr-wn in fi^airo l6(a) . Starting with the life line of the compression shock KE, whose direction here is the same as the direction of the compression shock at N (table VI), E is ob+ained as the intersection point with the life line 8, 9. Then the Mach vave E,l]- is dra-sm in proceeding frora E backwards. The direction of this Mach wave v/as taken in the familiar manner froi a X(i-diagram (not given here) . From this figure the position of E in advance of the shock is learned by interpolation along W,10 the ohase at 11. (See table VI.) From i:his may be obtained the velues entereafl fur-t-her on in the resDective lines which are necessary for later computation . Proceeding from Xjj by means of the consistency conditions, the quantity '^-djt ^^' computed for the Mach wave (ll,E^^) . For this the initial estimates for the phase in Ejj are taken as a basis and then colisnns 6 to 13, 17, 15, 16 and 18 to 20 compuied. For X ^ so obtained the Exl properties of the compression shock are taken from the shock diagram 12(b) . The following computations are essential to this AX = X - X = .95^ EII,I EII EI E . '•"EII,Ijl 52 WACA m No . 1196 From the shock dingram / Ail ^ = 0.0130; «j^jj/«Kj = 0.978; — £ = 1.307 From this it is computed that Au A^ - 0.0135; n = 0.830; —^ = 1.360 EII.I RII - ' a ' o ■ H = -0.090; u /a = v^^/a + Au-^/a = I.661 EII ' R o RI' o P/ o A portion of these resu.lts are given in tahle ""/l (colunms 2k to 26). Moreover drt - "eI I " ^11 ^ .330 - 0.781 ^ Q poQ dV " A^U ' 0.2ia2 To inprove these values, let a second iteration step he carried out. First, the figure N,R,11 has to he draxm again for the values JuBt obtained. The average direction of the compression shock is Then E^ and 11 are ohtained hy interpolation, ^j-r from the consistency condition for the Mach wave 3.1, E-j.^. To find the characteristics of the shock, it is necessary to carry out the following computation AX = l.U?6 - 0.'493 = 0.963; aL = 0.927 EII, I ^ From the shock diagram _ Au Ap. = O.OI3O; n k^ = 0.980; — ^ - 1.310 ' EII/ EI ^I From this is ohtained \i = -0.087; It = 0.328; u /a = 1.657; — = 0.220 E,ii EII EIl/ o dj; NACA IM No. 1196 53 An additional iteratj.on step is not necessary any more. In the second example (fig. 16(1))) the coiiipression shock is produced ly the sudden velocity change of a piston. The point of the yt-diagrem at which this velocity Jump takes place - let it he 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- s-poinding 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 0.1<-25aQ. At M the velocity changes, suddenly, to the value V = 0.925a^ and rises until the instant a^t =1.3 to the •nagnitude 0.97'=a,^. This and the flow field as determined hy the initial conditions and the -oiston r.otion xm to the point M is given. Next the phase hehlnd the shock at the point M is computed. Av, MET, I % V"o/M,i V "''M,II VWM,I \^o/w 0.'?00 ^f^hl = .I.R3 1(AX - An) = 0M3 2 As a result of this line in the shock diagram 12(1)) intersecting the shock curve, the following is obtained /U = O.9B6; All = 0.020 jt. m,iiAmi = 0-977; ^ = 1.333 From this Vii = ''^''^* %ii = -°*''^ V ^^ = 0.781; ^ = 1.80^ M,II a o ^h NACA TM No. II96 The -Dhase at M^-j- is 1010^11 \rlth that, (tntle VI.) Now the dif-ference must be conputed, over again, froru tlie 13 fe lire of the -oiston for the life line up to which it is desired to compute the coiiipression shook. 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 (t&hle VI) froju the preceding exiiraple and A\L- = /^ =0 .2122 *B,M W,M The computation of the compression shock makes use of figure M,S, K, 11. (See fig. l6(t)).). M, S, N is the life line of the compression shock; W, 11 is the Mach wave 1 returning from Wj 11, S is the Mach wave 2 returning from 11. To begin, an estiiaate of the phase at the points N^.-, 11 and S-^ is mado and this is chosen equal evei'where to the phase at Mtt • In addition, an estimate for — II d\[r is necossarv. Let ~ = O.230 as a start. Figure M, N, 11, S dv may he dravm with these assumed values . The ord(=r in which the points were named corresponds to the order in which they came up in the drawing. For tho positions of N and- 11 obtained thereby the phase in front of the shock (see table VI) or the velocity of the life line is obtained by interpolation. The iteration method begins at point 11 and it can be shown that [x^.. can be only slightly different from (i„ because the line element S^-j-,11 is small relative to the other'dimensions . The quantity |io can ^11 differ from n,, only slightly, since it originates in linear interpolation between M and N, and K lies very close to M. Therefore |i = M-w ^-p is chosen as a starting point. If the velocity of the niston at 11 that is known from the boundary conditions is used for "his X^-, nay be computed. From the consistency condition for the Mach wave 11 ,N %^y is obtained. Now the following co-aputatlon ^11 I " -^'^-^-^ ^ " 0,^8 and from the shock diagram AM^= 0.020; «j,,i/«i,i = 0.973; 3^'*^^'' ^I KACA TM No. II96 55 fro'Ti thiP .,,-^^^ = -0.077; 'Vii- 0.827; ^=1.678 Further it is ca3cu-\atcd that ^ = 0.217 The phase at S is ohtalned bjr interpolation hetween M and W . With the aid of the consistericj'- condition for the Mach wave S,ll,|j.-[2 is. finally obtained^ and X-^-^ from the houndary condition for point 11. The first iteration step e-nds with that. It is necessary to check whether the quantities X-,-,, ^M^ \ttt' ^^KII' %II' and i3i computed agree sufficiently with the original estimates.' To Increase the accuracy a second iteration stop might "be carried out. On the hasis of the ■■ralues just cojaputed, the figure is redesigned and the coiaputaticn is carried oi^t in the manner ,jur.t deecrihed. The value for i-i-,^ just computed is taken as a heginniig. The following calculation is ohtalned for the determination of the characteristics of the shock AX = ] .003: AX = 0.967 NIX, I K From the shock diagram ^l from this u ^i„.,_ = -0.083; Tt„_ = 0.828; Ji = 1.678; ^ = 0.221 The computation is continued in the manner given until the phase at point 11 is ohtained, again. An additiona.1 iteration step is not necessary. 56 NACA TM No. ir76 1?. SUIMAEY The differential equation sysLe-Ti for nor stationary, one- di.nensional flovB pos'seeses three faTiilies of characteristicsj the thernioHynamic and the flow phase are described "by three variables . As a result of setcing up consistency conditions for the charac- teristics passing throiigh the point for vhich the conditions have teen set up, three equations are obtained from which the phase may be obtained. In that a possibility for the computation of the flow has been given fundc?Jiien tally . The report carries out these idee.s, in general, and brings the siuiplif ications which are possible under special assunptions, as well as detaiJ.ed examples. Compression shocks appear, in this, as transitional conditions in the interior of th3 flow and are likewise investigated in detail. Translated by Dave Feingcld National Advisory Committee for Aeronautics NACA TM No. II96 57 • • ^ CO V£> S m r- ^ in rH t- CO 3 i X a. rH H H H H rH H rH i-t t-i H rH i-A r^ r-^ CI I .4 :1 r-l • H • rH A 1 iH 0\ tPv m • iH 1 rH • rH • CJ S « CVJ ? ^ ^ S ^ ° H OJ S S ,< a. rH iH H rH rH rH H iH rH rH r^ rH f-t rH rH • ■ 1 no g sf ^ CO OJ » 1^ m « ^ a l~i H H rH rH H H H H H rH r-{ rH fH rH I ^ SO 2 Ox « §^ ^ 01 1 5 s IfN ? ,< 3. H H H H rH rH iH H rH H H H H • I • 1 • ft « • 5 rH • H • rH rH I-* 1 • t 00 • ■ 00 S m ff On S § S & -* g t- 1 S IfN H CO ^ a H rH <-H i-H iH rH rH rH rH rH ■ • CO • H • r-t on rH • •H • • • rH • • f ? 1 ^ • • ^ a. • H H H 1 « H 1 • «H PO • • rH f 1 ^ S 1 1 if\ I X a r-l • • H J*- H • • 1 • ^ ". i 1 1 • 5 • ^ ■ ■ m g 0\ & 00 ^ s s m CO ^ ^ ?■ S ^ ,< a. l-t H rH rH H on H § ^ S J- s s "S ^ ° S CJ 1^ lA X a rH H •H rH CJ § S 8 CO 1 ^ S ^ pn t IfN 8i IfN 1^ X a H H H H % m § On rH ^ 1 ^ ^ • IfN • « ■^ 1 ,< a H ft i B kt ^ 8 ^ CVJ ITN 2.2. Q .-t i-t :if->e 01 8-^ V 1 1 1 1 1 t 1 fr« rH rH Si a • • "^ ""i 1 ■ 1 ■ 1 1 ■p H «° ^R 0\ Q gg CM < • • 00 »? ^S a\ CT\CD in uS Ji r^ p-i W CJ r^ H «H r-l r^ rA a ■) ty\a5 m ir\ vovo fT. 1 10 m vY?S< ®S H ► » 1 1 1 1 1 {.•g 1 • ^ CO l^^ (y\ H rH S -^^5 s 4'^ • • h- ^ s»^ 5?|i cSvS rH ? ' • • I 1 \o •p i-i -° ^^ rH r^ ifMTN S^R < m J* »S ^* ss w mjf m m W r^ .-1 ir\ Jt «g t-t^ s* l>> J^ i< ^ 55 RR H H H r-lrH I-I H M r-\ J1 X 1 5 n CD IfN m rH rH CU CM H 'H rH r^H rH H •-{ rH d ^ s;^ ^ Iv^ f^S^ a\ep c^a^ 1 IfN VO VD t-^o vovo 3 ►■ « • • t • • 1 1 1 1 I 1 •1 ^ p; § ;?!8 57 if! ?^% o\ + -" ^ J* m ro m ro _fc H tH H rH rH H W rH rH ^ P ^^ ^5 on m m Hit • • • rH C- m 5^^ OS CO 1 lA 'i'S t- m C^0| H H 1 r 1 1 1 !! •H 1 1 1 p m m i 5 ^ cy .H H 1 1 1 r-t r^ •H rH 1 1 3 % 1 1 ^3 rH H 1 1 tn *B m WW rH rH 1 1 1 1 H B S 1 58 S^ § §1 ^ m ro ^ OJ 01 S3 lAir. H X 5 1 f-1 3 •H r-t CO ir. OJ 0\ 00 rH rH >C m rH H H rH (^ Vh r\ H rH H OJPJ 1 1 1 §1 H rH i 1 83 cy CJ rH rH 8g CJ W rH rH. Sy q1 >H rH ^ s § «^ f §§ f ^ S) a 1-1 r-l H iH iH )H mm po H rH r^ H H 1 1 H H • • rH H 3s H rH d K 1 SIS. II s 1 1 jS§ M rH rH j H H l(NlA H rH rH rH m m • • s 1 .° 1 1 1 1 1 I 1 1 t ^1 1 1 1 1 1 1 mrH CTv t] 4- ! t-t (H r-l iH H •H 00 if\ m CJ m r*~> OJ OJ in m OJ CJ rH rH 1 rH rH 00 1 ^ S^ ^ 0\0 OJ w ir\rr mHJr-t rH CJ OJ i^ gl 3? PS 8S t- CO «H rH rH r-( K a Si 00 rH rH H I-i H 3. + 1 1 1^ 1 m m 1 § 8 If ^ ^ m m ^J ^1 if\ 3. s 1 a a I 1 \ 1 ^ mo. 1 1 CU 01 I 1 OJ cy 1 1 S 5 f? 1 ( ovo cy S ■ 1 J* X If 1 m. Q t— m 5 lA t- ^ VO S! CJ ^ OJ -* ir\ VD CD ^ •H ^moi 1-4 OJ m J ir\ VO e^ CO o\ s a 60 NACA TM No. II96 ^ calcu- lated 1 8 • • • r-< 1 I • 1 1 1 1 m i IfN t^ s ^ s g 1 1 1 1 1 8i 5° 1 • • • • • • \^° ^ 9^ J- m ^ »H H r^ i-t rH rH H rH H t- H«° in 8 ^ CO H m ^ CO ST vo a. + m 1 CO 8 in rH J- iTv • • • ir\ •0 ■P D 8 rH 1 0^ rH 1 • 1 • 1 1 ^ 1 rH in § S m f»-» 1 O (M m ITS cu iH OJ J- rH iUTOd f-i CJ r^ J» ITN VO NACA TM No. II96 61 TABLE V HCHSTATICIIART COMPSSSSKS SHOCKS (K - 1.1»05) a J /An Au/*j ^^II,l/-I ^J^l "ll/"l AX 1.00 1.00 0.0 1.00 1.00 0.96 1.0201*1 0.03360 i.oi»8i7 .0.99998 0.06721 .00001 .96 1.01*167 .06791 1.09938 .99991 .13588 .00006 .9lt 1.06383 .10298 1.15391 .99970 .20615 .00019 .92 1.08696 .13881* 1.21203 .99927 .27816 .00047 .90 1.11111 .17556 1.271*06 .99862 .35206 .00094 .88 1.13636 .21319 1.3'»037 .99748 .42806 .00167 .86 1.16279 .25180 1.'H137 .99587 .50633 .00273 .8U 1.1901*8 .2911*6 l,A87ii6 .99369 •58711 .00419 .82 1.21951 .33221* 1.56925 .99080 .67062 .00615 .80 1.25 .37U22 1.65722 .96708 •75713 .00869 .78 1.28205 .1*1751 1.75201* .96240 .84694 .01193 .76 1.31579 .1*6220 1.851*1*5 .97659 .94038 .01599 .71* 1.35135 .5081*0 1.96527 .96951 1.03781 •02101 .72 1.38889 .55625 2.0851*1* .96101 1.13965 •02716 .70 1.1(2857 .60588 2.21606 .95091 1.24634 .03458 .68 1.1*7059 .6571*5 2.35841 .93899 1.35843 .04345 .66 1.51515 •71111* 2.51387 .92506 1.47647 .05418 .6k 1.5625 .76715 2.681*13 .90903 1.60114 .06684 .62 1.61290 .82570 2 .87113 .89067 1^73320 .08180 .60 1.66667 .88701* 3.07714 .86974 1.87350 .09941 .58 I.72IHU .951'*7 3. 301.81* .84602 2.02302 .120Q9 .56 1.78571 1 .01931 3.55735 .81954 2.18291 .14430 .51* 1.85185 1 .09091* 3.8381*5 .79006 2.35447 .17260 .52 1.92308 1.16680 1* .15261 .75746 2.53923 .20563 .50 2.0 1 .21*71*0 4.50520 .72182 2.73895 .24414 .Ii8 2.06333 1.33333 4.90276 .68305 2.95572 .28905 .U6 2.17391 1.1*2529 5.35333 .64131 3.19120 .34141 ,W 2.27273 1.521*10 5.86637 .59673 3^45070 .40251 .i*a 2.38095 1.63073 6.45517 .54960 3.73531 .47385 .Uo 2.5 1.71*636 7.1340? .50078 4.05005 .55732 .38 2.63158 1.8721*2 7.92315 .45026 4.40174 .65517 .36 2.77778 2 .01063 8.84701 •39689 4.79144 .77018 .B"* 2.91*118 2 .16311* 9.93888 .34751 5.23210 •90582 .32 3.125 2. 332 61* 11.2418 .29698 5.73172 1.06644 .30 3.33333 2.52252 12.8138 .24825 6.30261 1.25756 .28 3.5711*3 2.73716 U.7347 .20234 6.96061 1.48630 .26 3.81*615 2.98225 17.U56 .16016 7.72656 1.76206 .21* 1* .16667 3.2651*2 20.1167 .12251 8.62809 2.09725 .22 1* .5^*51*5 3.59705 23.9721 .090026 9.70301 2.50891 .20 5.0 3.99169 29.0416 .063089 11.0043 3.02088 .18 5.55556 1* .1*7032 35.8933 .041760 12.6082 3.66757 .16 6.25 5.061*1*5 45.4722 .025786 14.6295 4.50062 .11* 7.11*286 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 1*1.5631* 2920.83 .000001 146.254 63.12755 .00 CO CO 0=. <30 00 62 NACA TM No. II96 ( ^./s. —J < — _/^ ) C '^~" — " — ) « """" -" ; s « r4 1 ! 1 & <= 00 (CfC ty 1 «2 1 1 «! 2|^ § ^ Oj 3. ? ■" * ' • s 8 ! ? 5S~ r^ OCi 9 f ? 7 ? 8 Sj ^ tvi 8( p n i g g 8 g n g ^ g s 8 f ^ s i { e s ? « S! i 9 s 39 g fi •, a fl 8 1 = S 8 _" g § s<^s -ll *> 5 "^ Qi i:-;i.^si^t a~5 US a-* <^ S S S* '^ " a M W a fl 0] fl S S 1' A £"35 S V g r-teD § l~* n-> ^o vn -t <« ^ V VD s^3a s SR s « a 5s 3 s H ^ rH r-t r-l rH r^ r4 ^ H ^ ^ ^ ^ ^ CTs 3 1^ p 8^^ 8 8 g 8 I? '^ g J) ^ g ■ g 3? g 2 e S 2 8 2 rH $ ( f[ ^ t t i? i S c 8 ;; 8 8 £ 8 < 8 H ^g 8 •J - |g I""! 8 V ^ f^-"^^ a B SaTS a" 5'^-a l^-s° 5I i 5<^S ° 59 8 t^ 1 & ^ = t U Q , Q " Q 0. s ^ 5 ■7 s Trl^l S 1 ?ri " a M T a fl lAO s (T s ^ 3S 3 5 5( c A ^ rH rH rH rt « H .-t r\ 1 ? f 1 Ol 1 ^ 1 u- g ii '^ ^ rH M rH .-< r-> j "' u i' ^ ^ ^ " ■ ■-* «H ■~i ^ p ■- OS "° M ^ >f __, b tin :^^S a H b e ^:h '^ Wfd K a OS B "-1 ^ ■^ K z «s R r^ (C tn rH tc NACA TM No. II96 63 Figure 1.- Curvilinear coordinate system 5 , tj , Gk ]^kCk TM No. 1196 /-/, 50 100 150 200 Figure 2a, - Relation between i and T for CO2 T'^C NACA TM No. II96 65 w i; - P ■% ft 1 .? 15 ¥ 1 f ^.s 1 ?5 125 rW 2.0 1 ? 10 a y ay 1 ',5 1 15 7.5 1 / /P in 1 1 5 M ^r 0.5 1 I 05 2,5 A V 1 J^ y^ \ 7 ''c 50 100 150 200 Figure 2b.- — ; P ; W as functions of the temperature for CO9. oO /i G6 NACA TM No. II96 -?? ^A^M Figures.- — ; P ; T as functions of W, or > +^ , for COg. NACA TM Wo. 1196 67 if •1-1 > m oJ fao CD O "T-j .rH 3 m o M o 5 •r-l TO o w O) ^ p. o rt =>-ie **-tap =1. y O XI I ■stH - \l ^ — ""^ 1 »s • i. s 5. S I. : J- 2 B •iH ^ , ro ■i-> n1 >. bD 1— 1 fTi d 0> T) :4:3 •i-H ^ (1) CO ^M w w ^ c; t— t «*H 0) -g M-H oj -M 0) H crt ^1 >< • rH -t-> fl) ^H c ft rl < d 1 i—i -1 1 f) lO oi •1— t /< ; 1 ^1 T) rt C ^ > Cti hJD NACA TM No. 1196 69 pathline of a particle / Figure 6,- The physical significance of f 70 NACA TM No. II96 o

5>/ ♦N _3^ >« »i X b> s- i 3- iJ- C 3. ii =:« fN ~" • C3- "S a •ii CTJ ,p_^ X! g 1 0) a. -i-> r< a) P a; :S 10 n • T-H «4-l II w (]) M f1 • f-H r-H -i-> •iH -*-j .— ( :a 0) •pH > 1=: rt w ^ rt M f3 1— 1 crt 0) TJ •i-H j5 cd «fH ^ 1-4 \^ I • tH 0) j3 hfl p^ Ih NACA TM No. II96 -I- N, • \ / \ \ \ \ / \ \ \ \ / N ^ \ s \ / \ s / \ \ V /I ' \ > / \ \ / / \ V \ •^^ / \ s. \ \ / \ \ \ \ N \ \ \ \ H*"--- s 4 s c ■ 3 c 5 \ 5Ji-iC » fi i. a t : '. '.U i & 9 ; i i. : %. i «. i > 3 b * > ^ K " I 40 ^ » 1* r ' pl^^^^j. 5^ •■ ' D sf *"- CL|M.rN *= 5q Figure 11.- Characteristics of compression shocks K = 1.405, NACA TM No. II96 75 |3 >♦ \ «*> \ fit «•> t*t \ \, 00 \ k •M \ \ !P > \, \ «N \, \ ?♦ s s. \ .4 < \ \ •N \ \ \i 9> \ \, \ \ \ >» ^ \ / ■N \ / \ \_ 3 g «> = " K, i^ ^ ^ A 10 s f Ol u^ r c if a — ^ — ^ --^ ■ — s N i < ^ — e > I — ^^ — s 1) 00 r ' s > — « 3 ii — » > ' -» (^ o II M w O o Xi w o w CO Q) U I- o o w o • r-1 w •f-( Q) o O P4 76 NACA TM No. II96 10 O II M m M o o m o w w 0) & o o m o •1-1 -t-> w •r-< ?H 0) -M O n) (h Xi W 0) fo "t3 6> w «< «0 BT rs," »^- •«.■> »»■" •»- •».» rf) «i ^^ ft NACA TM No. II96 77 '^a.t Figure 13. 78 NACA TM No. II96 Figure 14a. Figure 14b. NACA TM No. II96 79 - I Figure 15. 80 NACA TM No. II96 Figure 16a. NACA TM No. II96 81 Oo' Figure 16b. u^ rc:ps(TY OF FLORiP^^j^-^ IF^FL 32611-7011 USA ^ O m ft •H P4 ^ ^'•'k r. U\ <^ [<) -^ z/ > r o a / H o / Ph •H ■P CD Rj =2 vo a CU o o • t>3 r-\ -p (D ^ B (D rHCO -4- P cd •H U • ON f^ P j < (D •^ ^ u m fe P o o (D •H CQ O m (D W g o (d U -P m ci H » H r?g H H CO ffj -H ^ • ON C P 0) O H o o •H (D ■S ^ u P CO CD ei EH rO p CQ • Q in CO Cj < s G O Q o o u ^»j < © So PQ &; R •H CQ (D (0 U > w P CQ CVJ ■d •H CQ t» g CD u en o f^ CCJ o •H p p CO a o o •H p o CD CD M o 5h cl:> rH c5 vo o rH CO • c> O H S < < ^ p p> m to •H >H -d -d ! T) P ,C| O O -d P ;G 8 © O H O -H H -H P Cm H O P Vi H CD CO) • 43 -H G > ID ■d -H CO 00 tH iH m "d T-l CD CD tH -H p G -H G bo fi O p q; -H c! W3 .5 < ^£0 -H <; ^G CD -H P -H O 05 p -H g cd © d) fn ^ p P ^ -d G 3 fH o CD H CD o cd a ,^ ^ o cd G rCi q ^ O 53 > H G >H cd > H (D (fl O O ft s5 ft ^ ^ >j^ H g U ^ f>j-H H i •d P ,G cd E^ CD H -d ^ «3 P G CD •d '^ > S H -H P Ch H CD G © Cm CO © ^ CD © CD P OJ Jh © M -H ft •p U U ft ft H Id ,0 'r:i -r-i ta co -h -H < P G -H G «) ^ CD -H P" nH cd © © ^ -H P P U -d G (n P 1 © ference charac ary one plicabl flow c xamples Ch Cm G ft © tH cd +J "d -H CO p p >5'd :i < ft tr) rH 4-> ^ ^ ^ tn rd H G G cd ft I ft p C Jh cd S UNIVERSITY OF FLORIDA 3 1262 08106 457 7 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY PO. BOX 117011 lESVILLE, FL 32611-7011 USA