ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR PROGRESS REPORT SIMPLE WAVES IN THE STEADY, SUPERSONIC, PLANE, ROTATIONAL FLOW OF A COMPRESSIBLE POLYTROPIC GAS.....~ ~. N; CO(BUM-RN Project 2201 ORDNANCE CORPS, U. S. ARMY, DETROIT ORDNANCE DISTRICT CONTRACT NO. DA-20-018-ORD-13282 DA PROJECT NO. 599-01-004, ORD PROJECT NO. TB2-001(892) September, 1954

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ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN TABLE OF CONTENTS Page 1. Introduction 1 2. The Basic Relation in Terms of Characteristic 2 Variables 3. The Intrinsic Conditions for Rotational (or Irrota- 6 tional) Motion in Terms of Characteristic Variables 4. Canonical Form of the System (2.25), (3.10), and (3.12) 9 5. The Case of Plane Flows 11 6. Simple Waves of Type I in Plane Isentropic 13 Rotational Flows and the Metric Coefficients 7. The Case of Rotational Plane Isentropic Flows at 15 Mach Number One 8. Properties of Plane Rotational Isentropic Flows with 17 Simple Waves of Type I (q> c and y l 1): A Class of Simple Waves 9. The General Theory of Simple Waves of Type I in Plane, 20 Rotational, Isentropic Flow of a Polytropic Gas, (7 = 5/3) 10. Simple Waves of Type II 22 References 29 ii

ENGINEERING RESEARCH:INSTITUTE ~ UNIVERSITY OF MICHIGAN SIMPLE WAVES IN THE STEADY, SUPERSONIC, PLANE, ROTATIONAL FLOW OF A COMPRESSIBLE POLYTROPIC GAS 1. Introduction The purpose of this paper is to express the characteristic relations for the steady, three-dimensional, supersonic motion of a polytropic gas in' intrinsic form and to apply these relations to the study of simple waves. This means that the characteristic relations shall be written in such a form that they express relations between curvatures associated with the characteristic manifolds, the rate of change of the magnitude of the velocity, qu, and the sound speed, c, with respect to displacements along the normal to the characteristic manifolds and along two independent directions in these manifolds. Then, by specifying the curvatures or the rate of change of q and c, degenerate characteristic manifolds such as generalized simple waves may be studied. First, intrinsic forms of the characteristic relations for general three-dimensional nonisentropic rotational flows are obtained and canonical forms of these relations are determined. Then, application is made to the case of plane isentropic rotational flows. It is shown that for the limiting case when the Mach number of the flow is one, the bicharacteristics are always a single family of radial straight lines. For this type of rotational flow, the following conditions are satisfied: (1) the stream lines are orthogonal to the straight line bicharacteristics; and (2) the magnitude of the velocity and the sound. speed are constant along a given Stream line, Further, the sound speed and the magnitude of the velocity vary (for y = 5/3) as the onethird power of the distance from the intersection of the radial lines, The flow is a vortex flow. This leads to a study of simnple waves of type I, These are defined by the condition that the bicharacteristics form a family of straight lines, It is shown that at least one such family exists for all y, except y = 1. The flows are vortex flows with the following properties: (1) the bicharacteristics are tangent lines to a circle, (2) this circle is the limiting line of tieflow, (3) the sound speed varies (for y = 5/3) as the one-third power of l the distance measured along a bicharacteristic from the limiting line. Finally, the necessary and sufficient condition for the existence of simple waves of

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN of type I is found. Present computations indicate that the above class of vortex flows are the only two-dimensional rotational flows with simple waves of type I. Simple waves of type II are those for which the Mach number is constant along a bicharacteristic. It is shown that the above vortex flows, with q = c, are the only flows with simple waves of type II and straight line bicharacteristics. A detailed analysis of simple waves of type II is made. As a result, it is shown that a four-parameter family of simple waves of type II exists. The explicit determination of these simple waves depends on the solutions of four ordinary differential equations. One special case is studied where the bicharacteristics are spirals. 2. The Basic Relation in Terms of Characteristic Variables Let xJ, j = 1,2,3, denote a Cartesian orthogonal coordinate system in Euclidean three-space and let us denote derivatives by the symbolism ij -J (2.1) 6xj In a Cartesian orthogonal coordinate system, covariant and contravariant quantities are equivalent. However, in order to use the Einstein summation convention of summing on repeated lower and upper indices, these two types of equivalent quantities shall be used. Now, the equations of continuity, motion, and energy will be considered If p denotes the specific density and vJ is the velocity vector, then the continuity relation is vJajp + p jvj = 0. (2.2) For nonisentropic flows of polytropic gases, the equations of motion may be expressed in the form - YPvjSjvk + ak(pc) = 0, (2.3) where c is the local sound speed which may be defined in terms of the pressure, p, and the specific entropy S by c = and 7 is the constant of the gas law p = pTf(S).

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN In addition, an energy relation in-the foriA-'of'the Bernoulli equation is assumed jvi C.+ 0=, (2,4) where q2 = Vjvj is the magnitude squared of the velocity vector., By definition, the -characteristic manifolds of the system (2.2), (2.3), and (2.4) are those surfaces along which discontinuities in the derivatives, ajp, ajc, bjvk - can occur. Using the methods of a previous papera2 these surfaces can be easily determined. Howevery these methods are not pertinent to the present problem; hence, the procedure which follows will be by analogy. By'eliminating p between (2.2') and~ (2.13) and then by eliminating' ajc2 in the resulting equation through use.of (2.4), the following relation is obtained (ivJk - c2gJk) ajvk = 0o (2.5) where gjk represents the metric tensor, Thusf the samnie basic equation (2.5) is valid in the nonisentropic as well as in the isentropic case3s If the symmetric tensor aJk is defined by aJk vjvk - c2gjk j (2.6) then for isentropic flows the characteristic manifolds of Equation (2.5) are determined by aikn = (27) j~'k 0, (27) where nj is a unit vector orthogonal to one family of characteristic surfaces* From the known theory of characteristic manifqldS4, it follows that v cnj + tj, (2.8) where t1 is the unit vector tangent to the bicharacteristics. By use of (2.6) and(2, ) it is found that aJknk c tq2c t (2.*9) aJktk c qq2c2 nj + (qa-2c2) tj (2.10) It can be shown that (2.7) through (2,10) are valid in the nlonisentropic ca-se2..

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN NowJ a few basic relations from differential geometry are needed5, Since nj is a unit vector orthogonal to ol surfacesr the following may be written; ank Sjk + njuk, sjknk = uknk = O, (2,11) where sjk is the symmetric second fundamental tensor of the mc surfaces orthogonal to nj aLd u.k is the curvature vector of the nj congruence of curveE Further, let lj denote a unit vector field which is orthogonal to both tj and nj so that ljs nj, and tj form an orthogonal triple at each point. In future work, the term, gjikjtk, will have to be evaluated in terms of the congruences,- nj and lj and their curvature vectors, u.j and.tj, respectivelyj where 1jjlk U k * (2,12) From the basic decomposition of the metric tensor gjk tjtk + nJnk + lJlk (2.13) it can be found that gJk0jtk = titk ajtk + nJnkaj tk + lJlkljtkJ (2.14) In view of the relations t'k 1, lktk 3 0, nktk = 0, (2.15) it follows by differentiation that tk'jtk = 0, lkjtk = - tkjjlk, nk!jtk = -tkjjnk. (2.16) The first relation of (2.16) shows that titk ajtk = 0 Further, from (2*11), (2.12), and (2,16)s it is seen that nJnk ajtk = -tk(njink) = -tkuk, and lJlk ajtk = tk(ljlk ) =.tkThus, Equation (2*14) may be expressed in the form gjk ajtk =,tk(uk +?pk:) (2,17) 4

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN In this paragraph the previously derived relations will be used to express the basic relation (2.5) in terms of rates of change of q and c with respect to displacements along tj and nj, respectively, and in terms of the curvatures sjks uj, and U.j. The following notation for directional derivatives will be used. at- m t6J- nJi (2.18) Thus, 6/6t represents rate of change with respect to displacement along tj and 6/6n represents rate of change with respect to displacement along nj, By differentiation of (2.8), it is found that ajvk = c ajnk + j2-c2 jtk + nkajC + tkj 2-. (2.19) Multiplying (2,19) by aJk and using the relations (2.9) and (2.10), it can be shown thataJkajvk = c aJktjnk + lq-cm ajkjtk+C q 2 —2 + -(q2-2c2) \Ot n t (2.20) Next, the curvature terms, aJk6jnk, of the above equation must be evaluated. From (2.6) and (2.8) the following result is obtained ajk = c2(nJnk-gJk-titk) + q2tjtk + c Jq:-c(n Jtk + njtk) (221) Forming the scalar product of (2,21) with (2.11) it is seen that ajlkjnk = c ~ tkuk + (q2-c2) sjktJtk - c2M, (2.22) where M is the mean curvature of the characteristic surfaces; M gksjk By use of the relations (2.11) and (2.6), it can be shown that ninkajtk = -tknJijnk = tkuk, nJtkj tjtk = tknk = 0 tInkI jtk = -tJtkajnk = -Sjk titk

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN These last relations and (2.22) show that ajkljtk = -c2tkuk -c /q2-c2 sjktat-k -c2gjkijtk. (2.23) By use of (2o17), the above equation reduces to aik8jtk = -c 2 2 sjktJtk + c2tkuk. (2.24) If (2.22) and (2.24) are substituted into (2.20) and the left hand side of the latter equation is equated to zero, an intrinsic form of the basic relation (2,5) is obtained in terms of characteristic variables 0 = c q2J-~~-c~ ~ - ~. ~ 0 = ____2 an ) + (q2-2c2) a+ c24 q2c2 t(uk+) _ C3M, (2.25) 3- The Intrinsic Conditions for Rotational (or Irrotational) Motion in Terms of Characteristic Variables If eiik denotes the permutation tensor, then the vorticity vector, oiJ, is defined by oJ = eJlkalv~ * (3.1) Assuming that the ordered triad, tj, nj$ and ljforms a right hand system, then the following cross-product relations are valid ij - e jpktPnk tj = ejpknPlk nj = ej Ptk If each of the above equations is multiplied by eimq wad the fact used that eimqejpk _ ep J s q where ~ is the Kronecker delta tensor, it can be shown that ejmqlj = tmnq-tnm,, eiJmqtj = nmlq-qlm I (.32) eimqnj = lmtq-lqtm.m

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Forming the scalar product of (3.1) with the vectors tj, nfj, and lj, through use of (3.2) the following formulae are obtained citj = (nPlk - nklP) apvk mJnj = (ptk - lktp) pVk (33) lj = (tPnk - tknP)avk j pk A lengthy but direct computation using (2.11), (2.16), (2.19) shows that (353) may be written as ~J~tj= ku -ac +Iq l(nPlk - nklP) ptk, (3.4),n. =.a +q7 t Jtkaj 1 (3'5) Cmli = _ac q-c _2_c2sjktjtk- c tkuk. (3.6) 6t an In order to obtain an intrinsic formulation in terms of a characters istic variable of the left hand sides of (3.14) through (3.6), it should be noted thate lk 6jho - Taj S = ejlkv X. (3.7) Here, T is the absolute temperature, S is the specific entropy, and ho is the stagnation enthalpy h _c + -- a (3.8) 7-l 2 where y is the ratio of the specific heats of the polytropic gas. Since the stagnation enthalpy is constant along a stream line (see 2.4), with the aid of (2.8) it is found that ah q ] ~ - ah0 - ~ (3.9) 6n 6t and S satisfies a similar relation. To express (3.7) in intrinsic form, the. scalar product of this relation with the vectors tj, nj,. and lj is found.. Using (2.8), (3.2), and (3.4) through (3.6),, the, following formulae are obtained

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN -T aS = c(c 2 sJk t-tk - c t'%k), (3.O10) ah-T aS = q2.c2 ( -C-q S jktt - tN2kud n nat (3.11) ___ T = q + (q2-c2) tJtkaJlk - c21ulk - cq 2nJlk (ajtk-ktj). (3.12) Evidently, (3.11) is a consequence of (3.9) and (3.10). The above equations can be described by saying that they form a system consisting of three equations (3.10), (3.12), and (2,25) in the unknowns at an al where ho and S are prescribed functions which are constant along a stream line. The sound speeds c, is determined as a function of the magnitude of the velocity, q, by the Bernoulli relation (3.8). Thus, the roles of c and q may be interchanged. In the applications, the stream lines are unknown. Hence, the problem is to determine the functions ho, Sj and q so that two relations of the type (3.9) are satisfied (one equation in the derivatives of ho and the other in the derivatives of S) and also equations (3.10), (3.12) and (2.25) are valid. The- curvature term, K = nPlk(%tk - bt), (3.13) in the right hand side of (3,12) will be briefly considered. Through use of (2.11) and.(2.16), this term may be written as K = -(nPtk lk- tPlking) = -nPtklk + s ptlk. (3.14) If the unit vector field. lj, is orthogonal to vl surfaces (as is the case in plane and axial-symmetric flows) then bilk = rjk + ljk, lkJk = lk = 0, 8 ~ ~ l ik O

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN where rJk is the symmetric second fundamental tensor of these surfaces and Uk is the curvature vector of 1k congruence. In this case, (3.14) reduces to K = -rjknjtk + sjktJlk (3.15) However, if the vector field. of the bicharacteristics, t-j, is orthogonal to ool surfaces (as in the case of plane flows) then K = 0, (3,16) 4. Canonical Form of the System (2,25): (3:10), and (3,12) In this section the equations (2,25), (3,10), and (3,12) shall be written so that one equation contains only the directional derivative, 6/6t, a second equation contains only the directional derivative, 6/6n, and the final equation contains only the directional derivative, 6/6l. If (3*10) is multiplied by 4qc — and the resulting equation is added to (2.25), the following relation is obtained 2c Iq2 j- c2 a c + (qZ2c2)... c. +?q2-c2< tkU c(q2-c2) sjktJtk - 3M = at at =q2 (- ast (4 s)1)| at at The first two terms on the left hand. side of (4,1) may be replaced by 2c q1-ecac + (q2-2C2) ( q2c (q2c). a q 1 (4,2) Hence, (4.1) may be written as ~ q 2 _ c2 a - tk, + jktJtk + c3 M + 1 -T at;I-~-~ J;;~~IS"t,4as this is the first equation of the desired canonical system for q; ce Now, the second equation of the desired system will be determined. First, (3.10) will be multiplied byv qJ'-c2 and the resulting equation will be subtracted from (2.25). It is found that...~~~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 2c q- + (q2-2c) a + c2 tk(2uk+'-) an 6t -C3 M + c(q2-c2) sjk tjtk = -q2c2 T ) (4 4) \t 6t 44 Both S and ho are constant along stream lines, and hence, by (3.9) the right hand side of (4.4) may be written as Nrq a )E ( 6 -T a (4.5) To evaluate the second terms on the left hand side of (4.4), note by use of (3.8) that Y CC2 ho:" a r~ r c -?] ~ (4.6) t aC [t 2(7-1) Through use of (3,10), (4.6) reduces to equation Jbj-c2 ___1 6 b +r aE [T bt - 7 q -_-1 bCt y-1 at 3n -c J9 sjktJtk c2tkuk]}. (4.7) Converting aho/6t, aS/at into aho/6n, aS/&n through use of (3.8)J it is found. that (4.7) reduces to ___ 1_ r 2 c + 6lS! ~at s~L(y-l) qn tl n 3n + c c2 sjkttk + c2tkukj]. (4.8) Substituting (4.5) and (4.8) into (4.4), the latter equation becomes (7-3)qe= + 4cr2;7::'c 2q (+3 sjktJtk + c2 M+ { (-J[(3-) 4] tyk - c C-.o~ tk. + (_3)q2 _ (-5)c2 + 2q2(+)c T as (y-1) (q2.o~). an (r-z) (q2-2 ) a(4.9) 10

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The equation (4.9) is the second of the desired canonical relations for q f c. Evidently, Equation (3.12) is the third equation of the desired canonical system7. This equation can be written in the form q -q = h -T aS + c21kuk. (q2-c2) tJtk6jlk + c 4q2-c2 K, ai ai ai (4.10) where K is the curvature defined in (3.13). 5. The Case of Plane Flows The congruence determined by the unit vector, lj, consists of parallel straight lines perpendicular to the plane of the flow, and hence, Eh = 0. These lines lie in one principal direction of the cylindrical characteristic surfaces. The principal normals of the bicharacteristics coincide with nj, the unit normal to the characteristic surfaces, and hence the bicharacteristics are the curves in the second principal direction. Thus, M = Sjkttk, (51) where K is the curvature of the bicharacteristic curves. Further, if; denotes the curvature of the normal congruence, nk, then uk = Ktk. (5*2) Finally, in the present case, the congruence determined by tk is orthogonal to wz surfaces and thus the curvature K of (35.13) must vanish. Now, the canonical equations (4.3) and (4.9) may be expressed in terms of an orthogonal Cartesian coordinate system, x, y in the plane, It should be noted that the third canonical equation (4.10) is identically satisfied for plane flows. Consider two families of parameterized curves, a _ constant denoting the family of bicharacteristic curves and = constant denoting the orthogonal trajectories of these curves (and hence with tangent vector, nj). With respect to these curves, the arc length element becomes (ds)2 = (Ada)2 + (B)2 (53) where A and B are metric coefficients. In addition, let @(a,,) denote the angle between the ox-axis and the tangents to the bicharacteristic curves, a = constant. Then8s, 1L

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN = Bcos @, = Bsin@, (5.4) x =- Asin@ -A cose, (5.5) 1 =..... aB - - -- (5.6) B; AB a ' 1 ar 1 AB c = - - = - -- (~.5) K11 1 (5.7) A ba AB a6 a 8 s+ a 0 a O (5.8) If (5.1) and (5.6) are substituted into (4.3), the following relation is obtained q2 e2 a 2a, a s *(. cq2 2 cq2 a This last relation may be.written in the more symmetric form cqLc 4 [a.n - 1 - -NT s)= -. (5.10) c DP Nle _,2 q~ ( This is the first canonical relation expressed in terms of net when q f c, c ~ O. If the appropriate relations of the set (5,1) through (5.7) are substituted into (4.9), the latter equation becomes (Z-3)q2 + 4c2 q..2... a c[(r-3)q2+4c2] 6@ 2 a=(q2-2c2)- ln B - Jq~, aa (7_3)__(__ 5)_2 2q2-(7+3)c2 as +a q 2(5.11) 12

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The relation (5.11) expresses the second canonical equation when q / c in terms of quantities associated with the a, $ net. In addition, the conditions that ho and S are constant along a stream line (see 3.9) lead to the equations c h0 0_ __ + 0 (5.12) A a B aB and c as +;s =0 (513) c~ + =0~ (5.13) T ac~ a B 7 Since (5.12) and (5.13) each possess one independent integral, it follows that S = S(ho). (5.14) Finally, the equation (5.8) which expresses the vanishing of the Riemann tensor of the plane is valid. Thus, the basic system consists of Equations (5.8), (5.10), (5.11), (5.12), and (5.14). Note that the previous equations can be used to determine a quasicharacteristic system for steady, nonisentropic, rotational, plane, supersonic flow of a polytropic gas (T = c2/7R). In this system of seven equations for the seven dependent variables, q, c, ho, S, Q, A, B, and two independent variables a and P (the characteristic variable), two of the equations are S = S(ho) ho = q2 + C2l 2 of71 ' and of the remaining five first order partial differential equations, two equations are such that each equation contains derivatives with respect to only one variable (see(5.10), (5.11)), and three equations -C ah + O aQ 1 aB i = OB P contain derivatives with respect to both a and B. By use of similar methods, two characteristic variables can be introduced and a full characteristic system obtained.9 6. Simple Waves of Type I in Plane Isentropic Rotati-onal Flovws and the Metric Coefficients Simple waves of t_ are defined by requiring that the family of bicharacteristics are Straight lines or 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN o = @(a). (6.1) Substituting (6.1) into (5.6) it is seen that B = B(P). (6.2) On integrating (57), it is found that A -= - (a) B () +g(c), (6.3) where 9 (a) denotes the derivative of Q(a) with respect to a, g(a) is an arbitrary function of a, and B(g) = ( ) d. (6.4) By proper choice of scale factor along the bicharacteristics, a = constant. B(P) = 1 1 (6 5) may be chosen. Then P is the distance along these straight lines and the metric coefficient~ A, of (6.4) becomes A = - 0 (a) + g(a). (6.6) In section 8, the solution of the basic equations (5.10) through (5.12) is studied for the case when "B 1, A = aa - P (6.7) where a is a constant. These relations imply that the function, g(c), of (6*3) is linear in a and that = a. (6.8) The net of a, P curves corresponding to these metric coeffirients may be determined by integrating (5.4), (5*5). The result is x = (P-aa) cos cz + a sin ca + xo j (6.9) y = (P-aa) sin - a cos C + yo (6.10) where the constants of integration have been denoted by x0 and y0, The geometric meaning of the mapping relations (6.9) and (6*10) is as follows. Consider a circle of radius a with center at the origin (x; = y. = 0). 14

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The tangent lines to this circle are the bicharacteristics, a = constant; the circle is the limiting line of the flow. This can be seen from the following figure. y Q' Q Fig. 1 The coordinates of the point P are xp = a sin a, yp = a cos a. If the distance PQ is denoted by r, where r = a ca -, (6.11) then the coordinates of the point Q are given by (6.9) and (6.10). The point Q1 can be obtained by letting PQ1 = rT, where r' = f - a C. (6.12) However, only one of these two permissible mappings may be used in any given study of a flow. In particular, if a = 0, then the bicharacteristics are radial lines (a fan). 7. The Case of Rotational Plane Isentropic Flows at Mach Number One Since the canonical relations (5.11) and (5.12) are valid only if the Mach number is greater than one (q > c), the basic relations (2o25), (3011), and (5.12) must be used. The basic relation (2.25) in the case, q = c, leads to the condition M = - 0 (7-1) Thus, the bicharacteristics must be straight lines or simple waves of type I. From (3.8), it can be seen that both c and q are constant along a stream line. The formula (2.8) shows that the stream lines are orthogonal to the bicharacteristics. Further, from (5.12) it follows that ho is a function of 15

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN P only. The basic relation (3.11) is an identity and the relation (3.10) furnishes the equation (by use of (5.2) and (5.7)) X =cdc -c2 B dQ (7.2) dPfi d A dc By choosing a scale factor so that B: = 1, and using the relation (6.3) for A, it is found that the second term of the right hand side of (7.2) becomes 2 B dO c2=g A dC -P0' + g(a) Since c and ho are functions of P, the relation (7.2) implies that g(a) kt (7.3) where k is some constant. Thus, (7.2) becomes de c2 da ad (k-n) and the metric coeffcients are B = 1, A = t( (7.5) By integration of (5.4) and (5.5) for the case of the metric coefficients (7- 5) the following is obtained x = r cos @, y = r sin (7.6) where r = 1-k.(77) For the present case of Mach number one, the relation (3.8) shows that c2 = q2 = 2_(7-1) 2( 1 ho.(7.8) With the aid of (7-7) and (7.8), the relations (7.4) may be integrated and furnishes the equations ho = ho r(71), c2 - 2(7-1) ho r(7-1).....16..

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN where ho is a constant. Thus, it is seen that for the case of Mach number one, the bicharacteristics form a family of radial straight lines along which ho and c2 vary according to the (y-1) power of the distance from the origin. The flow is a vortex flow in which the stream lines are the family of circles orthogonal to the radial linesl~. It should be noted that for polytropic gases with 7 = 5/5, c varies as the one-third power of the dis' tance from the origin. 8. Properties of Plane Rotational Isentropic Flows with Simple Waves of Type (q> c and 8 q 1)> A Class of Simple Waves Now some general properties of the solutions of Equations (5.10) through (5.12) will be considered before studying a particular solution of these equations for the cases when the metric coefficients are (6.5) and (b.7). Substituting (6.1) and (3.8) into (5.10), the following relation is obtained (l-y) c2 a + [(7-3) q2 + 2c2] f = O. (8.11) Since (8.1) is homogeneous in c2, q2, a simple computation shows that q2 = f2(Ca) c2 (O-)/('-3)+ c2, 7 1, (8.2) where f(a) is an unknown function of a. If (8.2) is substituted into (3.8) a relation between c and ho is found, namely, 2(y-1) ho = (7+1) c2 + (7-l)[f(a) (7-3)/(7-1)2, 7 1. (8.3) It should be noted that (8.2) and (8.3) are valid for all isentropic rotational flows with simple waves of type I. Also, in the remainder of the work, it will be assumed that f(a) o.0 This is due to the fact that when, f(a) = O, Equation (8.2) reduces to, q = c; the case discussed in Section 7. It should be noted that for the irrotational isentropic case ho = constant, S = constant. By use of (6.2) the second canonical relation (5.11) reduces to?= e (8.4) 17

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The first canonical relation (5.10) and (3.8) lead to the well-known condition that c and q are constant along a given simple wave. Thus, the relation (8.4) may be written in the well-known formll dO The conditions (3.8) and (8.5) completely specify the family of simple waves. It has been noted in the irrotational isentropic case that q and c are constant along a given bicharacteristic. This means that both of these quantities are functions of a only. For rotational isntroic flnow, neither of these quantities, c and a, cn =e constant along a ivyen bicharacteristic. This can be shown by noting that if one of the quantities q or c is a function of a, then use of (8.2) shows that the other of these quantities (and also ho and by (8.3)) is a function of a. From (5.12), it follows that in thi case, c vanishes and cavitation occurs. Thus, no continuous rotational flows consisting of simple waves can exist adjacent to a flow region in which the velocity and density are constant,z2 or adjacent to a simple wave region in the irrotational flow. Another interesting question is, "What is the relation of simple waves as defined by (6.1) to degenerate mappingsl3 of the hodograph plane?" In the irrotational isentropic plane case, the hodograph plane is determined by the two components of the velocity vector, vJ, j = 1, 2. For a degenerate mapping, each component of vj is a function of only one variable, say *i. Since ho of (3.8) is a constant, c and q are functions of A. Thus, the angle between'the velocity vector and the x-axis, and the Mach angle depend only on i. So that angle 9 (of Section 5) is a function only of pI. In short, the parameters a and l' can be identified. For the case of the rotational isentropic plane flow of a gas, the flow must be mapped in a hodograph space determined by vJ, J = 1,2 and c, the sound speed. In this case, the theory of degenerate mappings of the hodograph space into a curve is similar to that outlined above. Thus, in the preceding paragraph, it was shown that no such mapping with -straight line bicharacteristics s-rDssible. l Now, it shall be shown that a family of simple waves of type I exists for any isentropic flow of a polytropic gas, when r7 1. This class of flows is characterized by the following: (1) magnitude of the velocity q, is constant along a stream line; hence, (2) the stream lines consist of concentric circlesC1 (with center at the origin, see Figs 1); and (3) the circle of Fig. 1 is a limiting line and the flow is a vortex flow. To verify these results, a class of simple waves of type I is considered for a particular nonisentropic flow. It is assumed that the metric coefficients are given by (6.7). Further solutions of Equations (5.10) through (5.13) are sought such that c, q, and hence ho, and also S are functions of the one variable 18

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN r = A = aa -. (8.6) Equations (5.12) and (5.13) lead to the relation q2 = ( C (8.7) *-(87) By use of the Bernoulli relation (3.8), Equation (8.7), and the polytropic gas relation T = c2/7R, the Equations (5.10) and (5.11) reduce to two ordinary first order differential equations in the dependent variables, c2, S. In general, such a system possesses a unique solution (to within constants of integration) for c2(r) and S(r). By use of the substitution principle due to M. Munk and R. C. Prim-4, an isentropic flow with the same stream lines and Mach number as the above nonisentropic flow can be found. Hence, the Mach angle remains unaltered and the bicharacteristics are still straight lines (simple waves of type I). Further, the relation (8.7) is still valid. From this relation and (8.2), with f(a) replaced by a constant (say f), q and c can be determined as functions of the one variable, r. Then, the Bernoulli relation (3.8) determines ho as a function of r. Further, a direct but lengthy computation (see Section 9) shows that for arbitrary values of the constants, a, f, Equations (8.2), (8.7), and (3.8) imply that c2 = Co (r-1) (8.8) q2 =r c (r- ) + (r (8.9) 2(7+1) ho = c L+) ()r () + (7-l) r (8.10) where co2 = f(7~'), r0 = a. (8.11) The bicharacteristics, a = constant, are determined by (6.9) and (6.10) (see Fig. 1). For the important case, 7 = 5/3, the formulas (8.8) through (8.10) reduce to c= c (r/ro)/, (8.12)............ ~19

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN cZ = Cco [(r/ro)2/3 (+rr. (8.13) 2ho = [4(r/r) 2/ + (r/ro)-4/3 (8.14) Thus, on the limiting line, q and ho become infinite. From the formulas (8.8) through (8.10), it is seen that q is constant when ho is constant. Thus, q is constant along the stream lines, r = constant. These curves are circles (see Fig. 1); the flow is a vortex flow. 9. The General Theory of Simple Waves oflteI in Plane, Rotational, Isentropic Flow of a Polytropic Gas, y = 5/3 In this section, it shall be proved that -the vortex flows of Section 8 are the only plane, rotational, supersonic flows with simple waves of type I. To verify this result, consider (8.2), the integral of (5.1G), (5.11) and (5.12). For 7 = 5/3, (8.2) becomes f2 cqi = c2 + C. (9.1) If a function p(a) and functions c(a, ), q(a,B), ho(a,B) are defined by p3(a) = f(a), (9.2') C = p, q = pq, ho = p ho (93) then, (9.1) leads to the relation qc2 = 4+ C2. (9.4) Similarily, (8.3) leads to 2 ho = C4+ 47J2 If the new dependent variables are introduced, =,3 s,= p'/p and the metric coefficients (6.5) and (6.6) associated with simple waves of type I are used B = 1, A = - off + g, 20

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN then (5.11) and (5.12) reduce, respectively, to =- (9.6) cxa 2(1-2y2) _ _. + 6.+ (9.7) ~a A A(1-2y2) The theory for solving the overdetermined system, (9.6) and (9.7), is well known. The integrability condition, aa2 a is formed. 'This leads to a functional relation of the type f yY,s,,A,s',, - - ) 0 When this last relation is differentiated and ompared with (9.6) and (9.7), then proper choice of s, e,' A, must lead to identities. This is due to.the result shown in Section 8 that no relation of the type k(y,st, ~@") = 0 can exist for simple waves (that is, c cannot be a function of ca only). The details of this procedure are carried o.t for the case when s = o, p(c) = consta nt Then (9.6) and (9.7) reduace to ',,, _ g t-' * (9.8) Oa a6 A The integrability condition of (9.8) leads to A a3 a A 21

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Substituting from (9.8) into this last equation, the following result is obtained y = A a).(9.9) This relation determines y (or c) as a function of c, >. It must now be determined whether Qtg can be chosen so that (9.8) is satisfied. From (9.9) it follows that the second relation of (9.8) is identically satisfied. The first relation is satisfied if and only if g = a @ 1 (9.10) where a is a constant. From (9.10) it follows that (9.9) can be written as y 1 (- P + a G). (9.11) From (9.3) and the fact that p(c) is constant, it is seen that C3 =, r a where r = - B + a 9. Further, from (5.4) and (5.5), it follows that the metric coefficients B 1, A - - A = t + a 0 0' determine the mapping discussed in Section 6 (see (6.9), (6.10)) with @ replacing the variable QC. The flows are the vortex flows of Section 8. The case when p(a) is not constant can be treated in the same manner. But in this general case where s f 0 (see (9.6), (9.7)), the computations are rather difficult. Present calculations indicated that no simple waves of type I exist in this case. 10. Simple Waves of Tne II This class of simple waves will be defined by the condition that the Mach number is constant along a bicharacteristic or M = q/c = M(ce). (10.1) First, note that for irrotational flows of a polytropic gas, M is always constant along a simple wave. Again, for rotational flows of polytropic gases (7 = 5/3), it follows from (3.8) that 2h - (M+ 3) c2. (10.2) _ _ _ _ _ _ _ _ _ _ _ _ _,,.. 22

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN If simple waves are defined by the condition that the bicharacteristics are straight lines (see (6.1)), then (5.10) shows that when (10.1) is valid and q> c, c = c (c) Since ho is a function of only a, by (5.12) the flows are cavitation flows (c = 0). Thus, the vortex flows of Section 7 for q = c are the only flows in which both the bicharacteristics are straight lines and the Mach number is constant along a bicharacteristic. Consider the properties of the class of a simple wave of type II defined by (10.1). Expressing (5.10) in terms of the Mach number, M, and using the defining relation (10.1), the relation 3 ln c + X M2 In + + f(a) = 0 (10.3) is obtained, where f(a) is an arbitrary function of c and = JM2TiM2. (10.4) Now, (5.12) may be written as 2aa a8a~~~~c2 B - [(M2+3) c2] +AD1Mi ( + 3) = 0. (10.5) Simplifying this last relation, the following equation is obtained B In [(M2+ 3) c2] + 2A JM2-i 0. (l1.6 By use of (5.6) and (10.6), it is seen that lnB - A a l n (M2 + 3))c2] a7 B 2 ~n__ __~c.~ (10.7) as Use of (5.10) or (10.3) leads to the result a B = 2M2 -C2 [in (M2 + 3) c2]. (10.8) The equation (5.11) can now be replaced by (see (10.8) or (10.3)) ( 9(M2-2). (cAM' -c -42) [1n(M2 + 3) c2] + c- [3 In c +XM in X+ ba 4(M -3)2 ac a2 23

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN + d + 2M2-5 c a [(M 1+3) C2]. (10.9) da 4 bX M2(M2-3) a The last relation may be written as U- -3 (ln c) + W = 0, (10.10) 2a da where: = x M2+ 9(-2) _) 3X - (2M2-5) (M2+3) 2( 0-3) 2% M2(M -3) (10.11) W = d (d) + 9(M-2)%M df _ d( nx(2-M ) da 2 (M2-3) (Me+3) ac do 2%M(M2-3) Since (10.10) is linear in (ln c), the solution of this equation is in c = -e35 fUe-3 dc + e3i g(p), (10.12) where g(p) is an arbitrary function of P and The relation (10.8) can now be integrated to find the metric coefficient, B. It is found that in B 35 dM + 3 1al n c dM +h(), JM(M+3) M2 aM where h(p) is an arbitrary function of B. By use of (10.12), this last formula becomes n1B =7 3 dM + 3 d e-3K X W e3X dcl dM M(Mp2+3) J aM2 ac u j(o) + 3g(f) [1 d (e3X) dc + h(f). (10.13) jM dcz

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN By use of (10.6) and (10.12), the metric coefficient, A, may be determined. Since the expression for this coefficient is complicated and will not be used in the remainder of the paper, it will not be discussed any further. There remains one further problem: to determine the conditions on the functions M(a), f(a), g(p), and h(p) in order that the Riemann equation (5.8) may be satisfied. In view of the fact that Q(a,c) is determined by (10.4) in terms of M(a), f(a), and c(ca,), it is necessary only to show that (5.6) and (5-7) are satisfied. Now (10.7) implies that (5.6) is satisfied. Hence, only (5-.7) remains to be satisfied; this relation determines the conditions on the functions M(C), f(cZ), g(p), and h(P). By use of (10.3) and (10.12), (5.7) leads to - M.= H(a) + K(cz) g(p), (10.14) B 6P where H(ca) =M_ l- _ (x. M2 in ) - e e d dac d:X (10.15) K(a) = - 3 e3. Through use of (10.6),it is found that 1 aA _ 1 B ln(M+3) c -.; I(+) (10.16) B as B a6 L2 A n an a' Expanding the right-hand side of (10.16), through the use of (10.13) and (10.12) the following relation is obtained 1 6A = Lh dd i + R F + d (g j\ + 3Q rr1 ( e3 da _BL _ + 3R f1 /d e3AX da] g (10.17) where dh h'= d g = dg (a) e dCX u e-3f da + in (M+3 (10.18) R(aC) = 1 d.e x e3M# dac 25

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN From (10.14) and (10.17), find that g(p), h(p), M(a), and f(a) must be such that H+ Kg = h + d 1 + (10.19) L I dp g ' go d, g' This is the basic condition on M(G), f(a), g, and h(P), where H = H-3QfL (d e5)( da2, I = K-beR s n t1 d sien) cd (10.20) It shall be shown that a and sufficient condition thatd (10.19) possess solutions for a is that the following -system of differential equations may be satisfied Q = cl R + c3 K, H =-C2 R + c4 K, (10.21) g = C3 1 - 4 p = -cl 1 + C2 where cl1 c2, c3, and c4 are arbitrary constants and (for g' f 0), 1 _ h'g + d> g '/ d~ kgt J ( (10. 22) p - hog + d g' dp gJ First, it will be shown that (10.21) is necessary. In terms of 1(W), p(P), (10.19) may be written H + Kg = Q1 + Rp. (10.23) By differentiation of (10.23) with respect to P, the following equation is obtained, 0 = - Kg' + Q1' + Rp', (10.24) where primes denote differentiation with respect to B. Solving (10.24) for g' and differentiating the resulting equation with respect to a, it follows that d-(Q\i d d K. (10.25) 26

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Separating variables in (10.25), the equations d = R - -, (10 d 26 dp dl da K da K dp dp are obtained, where cl is an arbitrary constant. Integrating (10.26), it is found that Q = cl R + c3 K, p -c1l + c2, (10.27) where c2 and c3 are arbitrary constants. Thus, two relations of the set (10.21) have been found. Substituting (10.27) into (10.23) results in H R - c2 - g + c 1. (10.28) K K But (10.28) implies that H = c2 R + c4 K, g = C3 1 -c4 (10.29) where c4 is an arbitrary constant. Relations (10.29) are the last two relations of the set (10.21). By substituting (10.21) into (10.23), it is seer.n that the.I.atter are satisfied. Thus, the conditions (10.21) are sufficient o In addition to (10.21), other special solutions of (10.19) may be found. For instance, such a solution is h = aS, g = (10.30) K = aR, H = aQ + R, where a is a constant. In particular, if a = 0, h = 0, g =, (10.31) K = 0 H = R is a special solution. Thus, (10.13) leads to In B = m(c) + P n(a) (10.32) and (10.3) and (10.12), with g(f) = P, leads to 9 = r(C) + B s(c), (lo.33) 27

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN where the particular functions m, n, r, and s are of no significance in the future work. In fact, from (10.32), (10.33), and (5.4), the geometric structure of the bicharacteristic, cl = constant can immediately be determined. Integrating (5.4), it is found that x - xo = B cos(4 - Y), y - yo = B sin (@ - 9), (10.34) where tank = s/sS2 +n2 B B/ Js2 + n2 and xo, and y are functions of ca. From (10o34), it follows that the bicharacteristics, a = constant, are spirals. 28

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN REFERENCES 1. The relation (2.3) is easily derived. See "Intrinsic Relations Satisfied by the Velocity and Vorticity Vectors", N. Coburn, Michigan Math J, I, No. 2 (1952). 2. ":Discontinuities in Compressible Fluid Flows"', N. Coburn, Mathematics Magazine (May-June, 1954). The theory of discontinuity manifolds for nonisentropic flows is similar to that of isentropic flows arid was worked out by Mr. J. McCully (not published yet). 3. "Supersonic Flow and Shock Waves', R. Courant and K. 0. Friedrichs, Interscience Publishers, see formula (102A.6) p.247 for the isentropic two-dimensional case. 4. "The Method of Characteristics in the Three-Dimensional Stationary Supersonic Flow of a Compressible Gas", N. Coburn and C. L. Dolph; Proc. of the First Symposium of Applied Math., 1947, Am. Math. Soc., 1947. 5. 'EinfUhrung in die Neuren Methoden der Differentialgeometriee,: J. A. Schouten and. D. J. Striuk, P. Noordhoff, Groningen, Batavia., Vol. II, 1938, p.38. 6. See Reference 3, p 22, formula (14.03). The relations (3.7) are equivalent to the equations of motion. If ho is eliminated from these equations through the use of (3.8), then the resulting equations, and (2.5), and the energy relation (2.4) form a system of five eqgatiions in. t:he. five unknowns, c, S, vS. 7. "Superso;.zic Flow and Shock Waves", R. Courant and K. 0, Friedrichss, Interscience Press, N. Y., (1948), pr 40; "Characteristlic Directions in Three-Dimensional Supersonic Flow", N. Coburn, Proc. Am. at,,th.., 1 So. 2, 241-245 (April, 1950). It should be noted that the ca-o.Tical system has one very important property; each equation possesses directional derivatives inz only one direction. This is a highly desired property. Actually this property can also be extended to characteristic systems by replacing (3.11) by the equation corresponding to (3.10) for the second character,istic direction. But a price must be paid for this advantage; the righthand sides of (4.3), (4,9), and (4.10) contain curvatures of the various manifolds. Finally, it should be noted that when the unit vectors tJ, nJ, and lj do not fovrmia triply orthogonal bcoordinate system, then it may be advantageous to use a full characteristic system, since the unit vectors tJ, nj, and 1J (where lJ is a unit vector along the intersection of the characteristic manifolds) always determine such a coordinate system.....- 29

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN REFERENCES (cont. ) 8. Relations (5.4) through (5.8) are well known in differnetial geometry. Equation (5.8) implies that the Riemann tensor of the plane vanishes; this equation is the integrability condition for Equations (5.6) and (5.7) 9. For another formulation see, "Characteristic Conditions for Three-Dimensional Flows with Vorticity", R. J. Clippinger and J. H. Giese, Ballistic Research Labs, No. 615. 10. "Steady Rotational Flow of an Ideal Gas", R. C. Prim, J. Rational Mech. Anal. I., No. 3, 425-497 (1952). 11. See Reference 3, p.265, formula (108.08). 12. In the irrotational case, discussed in Reference 3, p. 60, the diametrically opposite theorem is shown to be valid. 13. See Reference 3, p 59-62. Also see "Compressible Flow with Degenerate Hodograph", J H Giese, Quart. Appl. Math., 9, 237-246, (1951). 14. See Reference 10, p 428. Also see "Vorticity and the Thermodynamic State in a Gas Flow", C. A. Truesdell, Memorial des Sci. Math., No. 119, 36-39, Gauthier-Villars, Paris (1952). 15. The Absolute Differential Calculus, T. Levi-Civita, Blackie and Son, London, 1929, pe 15. 30

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