DEARET OF I-EG R EARCH UNTVERSITY OF. MICHlGAN ' UNIVERSITY OF' MICHIGAN Ann Arbor EXTEAAL MORANDU M NO,. 8 Project 'X.-794 (AAF Contract W33-038- ac-! 222:) Project. "tWiz.ard "The. Head-on Collision- of a Shock Wave And a Rarefaction Wave in- One Dimension" Prepared: By 9.< 6 H. 'E. Moses August 21, 1947

ACKNOWLEDGEMENT The writer wishes to thank Professor A. M. Kuethe and other members of the Department of Aeronautical Engineering of the University of Michigan for their helpful advice.

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DEPARTMENT OFI E"iERINGG RESEARCH Report No. *UM- A8- U NERSITY OF MICIGAN age v LIST OF SYMBOLS c velocity of sound, ft per sec p pressure, lb per sq ft t time, sec u absolute velocity of the fluid, ft per sec v velocity of the fluid relative to the velocity of the shock, ft per sec x position in the fluid (xl refers to the position of the shock) ft A increment. ratio of specific heats r defined by 2 2 defined bye2 1 1+/ 2 excess pressure ratio, PP Po lp density, slug per cu ft Subscripts o applied to p, p v, u, and c to indicate low density side of shock 1 applied to p, p, v, u, and c to indicate high density side of shock applied to F to indicate initial value i applied to p,,p, v, u, and c to indicate initial constant values in a simple wave A initial conditions on low density side of shock B 'initial conditions on high.density side of shock

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DEPARTMENT OF ENGINEERING -RESEARCH Report No. U-~,8 |UNIVERSITY OF MICHIGAN C Page 1 SUMMARY A procedure for investigating the strengthening of a shock which collides head-on with a rarefaction wave is suggested and is carried through for the case in which the entropy jump across the shock is small enough to be negligible. INTRODUCTION The problem which is considered in the present paper is the strengthening and acceleration of a shock which moves in such a way that it collides with a rarefaction which approaches the shock from the low pressure side. Such a collision may be thought to take place in a semiinfinite tube as shown in Figure 1. The x-axis iS taken to be the axis. of the tube. A piston is initially moving with a constant velocity less than that of sound toward the left and the gas in the tube moves with the same velocity. (The restriction on the initial velocity of the piston is not essential; it is merely convenient for the purpose of describing the phenomenon). Wfhen the piston reaches the position x 0 at time t O, it is assumed that the piston accelerates"in some manner toward the left and thus produces a simple rarefaction wave which moves toward the right. At a large positive value of x a shock wave whose low pressure side faces the origin is assumed to exist. The shock wave will move toward the origin with uniform speed and constant strength until it meets the oncoming rarefaction wave at a distance L from the origin. The shock will then accelerate and be strengthened as it meets the gas of decreasing density.. Furthermore, the entropy jump across the shock, which is constant as long as the shock moves through the,gas which has not yet been disturbed by the rarefaction, will increase. Thus an entropy wave which moves with the fluid will be formed on the high pressure side of the shock so that the'flow on the high pressure side of the shock is no longer isentropic. In Reference 1, Courant and Friedrichs consider the interaction discussed above and conclude that the final result of such an interaction will be a shock wave moving towards the left and a rarefaction wave moving toward the right separated by a zone of gas of varying entropy. The calculations in the present paper have been made with these results kept in mind and will describe the actual process of interaction between the shock and rarefaction waves in more detail. It might be pointed out that the collision between the shock wave and rarefaction wave as discussed in the present paper is an idealization of processes which occur in intermittent Jet engines and supersonic wind tunnels which operate by permitting air from the outside -atmosphere to pass through the tunnel into a low pressure reservoir. PROCEDURE Before discussing the interaction between the rarefaction and shock, the properties of the rarefaction and shock waves willbe reviewed briefly. The terminology of Reference 2 will be used throughout.

DEPARTMENT OF ENGINEERING RESEARCH Report No. UM Page 2 |. UNIVERSITY OF MICHIGAN A simple wave in non-steady, one-dimensional, flov refers to a special isentropic flow in which the fluid velocity, pressure, density, and speed of sound assume constant values along each straight line of a one-parameter family of straight lines in the x-t plane. These values in general differ from line to line in this family. For general, non-steady, onedimensional, flows, two families of curves in the x-t plane play particularly important roles. These curves are called the characteristic curves and are defined by the differential equations dt u c, where u is the velocity of the fluid and c is the local velocity of sound. WVhen the flow is a simple wave, one family of these characteristics is the family of straight lines discussed above. A fundamental theorem on simple waves says that flows adjacent to flows of steady state are simple waves. Hence the rarefaction wave described above, produced by acceleration of the piston, is a simple wave, since the fluid is initially in a steady state. In the present case the family of characteristics which are straight lines is that one which has the plus sign. In a simple wave the pressure p, the density p, and the velocity of sound c are related to the velocity of the fluid by the following formulas: r u-ui' 2 Ci P-Pi Jl- ~ 2 Ci7 (1) 2 6-1 u-uiI - Dl+ — -... (2) = i l 2 2 ci c i = P= Ci 1 2 (3) where subscript i refers to the initial constant state of the gas before, the rarefaction has affected the flow and V' is the-ratio of specific heats. The properties of shock waves will now be discussed. Let the sub- - scripts o and l refer respectively. to thelow and high pressure side of the. shock.' There are three. shock conditions which arise from the conditions of conservation.. of mass, momentum, and energy a.cross: the shock. -These relations can be written PjUoVo+ Po = JlUV1 + P (5)

Pi Po In the above equations Vo = U - Xl (7) Vl = U1 - Xi (8) where xl indicates the position of the shock and the dot indicates differentiation with respect to time so that x1 is the velocity of the shock. The constant 2 equals 1 It will be convenient to measure the shock strength in terms of the excess pressure ratio / -. In terms of fi, the shock conditions Po may be written Vo Colf / ( = 12) (9) (o / c- (uo u- ) (10) 0 and from (9) and (10) (Ui-uo) 2, (U-U)2 2 () 2 4 = +(U1UO) %'22(ul-uo)2 2 (a Ulm u1, (12) C1 (= CoI 3+) (!+~2f) The process of interaction will now be considered. The quantities on the low and high pressure side of the shock before the interaction will be denoted by the subscripts A and B respectively. Thus before the interaction

Page 4 UNIVERSITY OF MICHIGAN IRepor No. UMMS of the shock wave and rarefaction wave PB = PB, ul = uBs PI = ftB' C1 = CB, Po = PA, uO = UAY,o =,A, co = CA. Using this notation, Equations 1, 2, and 3 may be rewritten as follows: 2r o=PAL1 +t1 UA (la) L CPd 2 cA J 2 co = cA [1 u-1 UouA I (a) C0 CA K 2 (3a) L CA Since, in the wave, uo uA (uo and uA are negative in our coordinates), it is seen that Po< PA'0,OA, Co <CA. Furthermore, the head of the wave travels with velocity uA + CA. Furthermore if f denotes the excess pressure ratio of the shock before interaction, tien 1 --CA.l uB - uA - Y,11V1 (12a) The rarefaction wave is completely determined by the velocity with which the piston is withdrawn from the tube. It Will therefore be assumed that uo, hence also po, Op, co are known functions of x and t. It is the problem of the present paper to find xl, x1, l, Ul, 1 as a function of time as the shock wave moves along its path. If u1 is known as a function of uo then the differential equation (lla) can be solved to give x;(t) and xl(t). Then also f (t) can be found from (10). Therefore in addition to Equation 11 another relation between v0, UO, Ul, is needed so that vo, ul, can be solved in terms of uo alone. Having found u1 as a function of uo the procedures outlined above can be used to find the desired quantities. As explained previously, the fluid on the high pressure side of the shock wave is not isentropic. Consider the shock at a given time. The region on the high pressure side can b e divided into small regions in which the pressure, density, velocity, and entropy are considered constant. In particular consider the small region immediately adjacent to the shock. The fluid particles which have just passed through the shock will move toward the left with a velocity greater (i.e., more negative in the

DEPARTMENT OF ENGINEERING RESEARCH Report No. UMME-8 UNIVERSITY OF MICHIGAN Page 5 coordinate system chosen) than the fluid which occupies the small region being considered. Hence the particles which have Just passed through the shock may be considered the front of a rarefaction wave which will pass through the small region. Inasmuch as the fluid is isentropic in the region considered, Equationldescribes the relation between pressure and velocity. In Equation 1 we shall write P = P1' U = Ul, C = c1 ui = u1 + ul, ci c1 + L C1 (14) Pi = P1 + L Pl passing to the limit, the following differential equation is obtained dpl _ Pl du1 c1 (15) By means of Equations 12, 13, and la, Equation 15 can be converted into a differential equation with j and uo as variables. By integrating this differential equation, a new relation which gives 5 as a function of uO is obtained. By substituting for f in Equation 12, ul - uo is found as a function of uO alone, and the procedure outlined above may be used to find xl(t), kl(t), F (t) and finally ul(t). A particularly simple and interesting case is the one in which the shock is weak. As shown in Reference 2, t e entropy jump across a sufficiently weak shock is proportional to.Thus if the shock is so weak that all powers of f higher than the second can be neglected,the fluid on the high pressure side of the shock can be considered isentropic. When the value of cl corresponding to isentropic flow is substituted in (15), we obtain 2 - uu-1 l-uB Pl2 B (b This result was to be expected from the manner of derivation of Equation 15. It is also to be expected from the fact that we have a non-uniform isentropic state adjacent to the constant state given by pB, uB,,O B, therefore the wave on the high pressure side of the shock must be a simple wave and the simple wave relations must hold.

DEPARTMENT OF ENGINEERING RESEARCH Page 6 UNIVERSITY OF MICHIGAN Report No. UMM-8 From Equation lb 2_ I ~-l ul-u l d-I+ 2- I C t -1 (1 And ~A = (1 h 1) (17a) For the special case of weak shocks, the general procedure outlined above will be modified as follows: From the nature of the problem it is expected that ul can be expanded in powers of 1. U1(,1, uo) = fo(Uo) + flf(u) fl((uo) *... (18) The functions fo, f,' ' ' ' etc. will be sought. Using Equation 18, 17a and 12a in Equation 16, f is obtained as a power series infS which involves the fi's as coefficients. This series is not valid beyond the second* power offl, because of the assumption of isentropic flow. The expression for so obtained is substituted in Equation 9 to obtain vo in an expansion in ', which again will not be valid beyond the second power of 1. By substituting ul as given by Equation 18 into Equation 11, an alternative expansion of vo in powers of f 1 is obtained. Comparison of the coefficients of powers of fl of both expansions yields the functions fo, fl and f2. Having found these functions, ul is known as a function of uO; and X1, XJi,, u1 uo can be found as functions of time as the shock moves along its path as explained previously.

DEPARTMENT OF ENGINEERING RESEARCH. Report No. UlWkll-8 UNIVERSITY OF MICHIGAN Page 7.RESULTS A. Infinitesimal Shocks For initially- infinitesimal shocks Fl - O. It is easily verified that f uo.and that ft = 0. Therefore an infinitesimal shock remains an infinitesimal- shock when' passing through a rarefaction wave. Furthermore x uo - co, which is the equation of: a:backward characteristic of The simple wave. Hence infinitesimal shocks moye along the characteristics of the simple wave. This result was to have been expected from the role played by characteristics as propagators of Small disturbances. B. Weak Shocks tWeak shocks are defined as those shocks for which the powers of f higher than the first can be neglected..A It is found that fl -. Furthermore,to this approximation, - uBB =uo uA (19) which shows that for weak shocks the increase in the velocity of the fluid behind the shock is equal to the increase in velocity which occurs ahead of the shock. To put the result in different words, the velocity of the fluid behind the shock differs from the velocity ahead of the shock only by a constant. The differential equation for the position of the shock is x U + 2 uB UA) + (20) The path:' followed bya weak shock.is therefore no longer'a characteristic of the rarefaction.wae. Moreover, since UB~ -uA is negative, the shock wave iw'11 travel faster then the'velocity of an infinitesimal disturbance' (i.e., a sound wave) through the rarefaction region. The strength of the shock, as measured by its excess preast:sre ratio, increases as it passes through the rarefaction region. The expression for 'F is 8 = 1 Co (21)

DEPARTMENT OF ENGINEERING 'RESEARCH Page 8 UNIVERSITY OF M-ICHiGAN i 'Report No. UMM-8 From Equation 3a it is clear that decreases so that f increases. An A initially weak shock may therefore become strong when interacting with the rarefaction. In order t'o present a specific example of the method of finding the velocity of the shock and of the fluid before and behind the shock, a special rarefaction wave will: be considered, namely the rarefaction wave which results when the-piston undergoes infinite acceleration in changing its initial velocity uA to some final constant velocity. Such a rarefaction wave isicalled a centered rarefaction wave. The velocity u0 is given by uO uA = (1 /r2) (t A ' uA) (22) For simplicity cA will be taken equal to unity, uA will be taken equal to zero and the distance from the origin to L at which the shock and rarefaction interact will also be taken as unity. Likewise, the time interval required by the head of the rarefaction wave to move from the origin to the position x = L will be taken as unity. These simplifications correspond merely to a choice of units and a frame of reference from which the phenomenon is viewed. Then for t\) 1 = t r-no'Bj t-2,2 _ 1 (23) I _ (1-F 4(lY 2 ) 2(2 ) 12 _ 4 2)(1 4t(l#) / _2/ c?!-2/..'2 )...4 (.. -2 2 -ul(t) =.Uo(t). - uB (26;;) and the various quantities which are desired are completely solved for. C. Moderately Strong Shocks A moderately strong shock is one in which all powers of 5 higher than the second can be neglected. For such shocks it is found that

DEPARTMENT OF.-ENGINEERING RESEARCH. Report No. UMM-81 UNIVERSITY'OF MICHIGAN. Page 9 f2= 2 CA (27) Therefore ul - uB -uo UA as for weak shocks. This result is somewhat surprising, inasmuch as one mightexpect the relation between ul and uo to be non-linear for moderately strong shocks. It therefore appears from this point' of view that instead of using the excess pressure ratib as a criterion of shock strength it would be more satisfactory to use the non-dimensional velocity difference c}~o Thus weak disturbances might better be defined as those such that all powers of beyond the first may be neglected. Co However, we shall continue to use our original definition. For moderately strong shocks, the excess pressure ratio is given by c-l- [ 1c L 4 Sl(: ~1)] (28) CA 0c Since o1, the strength of the moderately strong shock increases at a faster rate than that of a weak shock in terms of CA Also the equation of motion of the shock is ~_= - - - (29) Uo + (us UA)-c(uB UA)2(29) which shows that the moderately strong shock travels faster through the rarefaction region than the weak shock. As before, when uo is given as a function of x and t, Equation 29 may be integrated to give the position and velocity of the shock as a function of the time and then xl, xl u0,o, ul, may be found as functions of the time or position of the shock.

DEPARTMENT OF ENGINEERING- RESEARCH. Page 10 UNIVRSIY OF MICHIGAN eport No. UREFERENCES 1. Interaction of Shock and Rarefaction Waves in One-Dimensional Motion. By R. Courant and K. Friedrichs. OSRD No. 1567. 2. Supersonic Flow and Shock WaveS (Shock Wave Manual). AMP Report "38.2R.

RESTRICTED DEPARTMENT OF ENGiNEERING RESEARCH Report- No. UMM-8g UNIVERSITY OF MICHIGAN Page 11 DISTRIBUTION Distribution of this report is made in accordance with AN-GM Mailing List No. 3, dated May 1947, as corrected, including Part A; Part C, ardd Part DA.

OF MICHIGAN U 9015 03483 0730 Printed and Lithoprinted in U. S. A. University Lithoprinters, Ypsilanti, Michigan 1947