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UNIVERSITY OF ILLINOIS 
 GRADUATE COLLEGE 
 DIGITAL COMPUTER LABORATORY 
 
 
 REPORT NO. 70 
 SINGULAR SHOCKS WITH STRAIGHT BRANCHES 
 
 by Y. S. Chow 
 September 15, 1955 
 
 This work has been supported by 
 the National Science Foundation 
 under Grant G-1221 
 
I. Introduction 
 
 The purpose of this report is to discuss the situation which arises in 
 a two-dimensional stationary or pseudo- stationary flow when a shock front consists 
 of one straight shock and one analytic curved shock. All the notations used below 
 have the same meaning as those given in (l), unless they are defined explicitly. 
 We shall suppose the flow is polytropic and the flow ahead of the shock is uniform, 
 i.e. all the flow quantities u, , u p , m, p, $ are constants ahead of the shock. 
 
 Let the shock configuration in a two-dimensional pseudo- stationary 
 (stationary) flow at time t he represented by the equations 
 
 x. = ta. (s ) 
 1 1 
 
 i = 1,2. 
 
 [x. = a.(s)] 
 
 where x, and x are fixed Cartesian coordinates in the plane in which the shock 
 is moving, and s is a parameter along the shock which may be chosen so that 
 
 a! (s)a!(s) = 1, where 
 li 7 
 
 da (s) 
 (s) = 
 
 ds 
 
 at regular points . 
 
 We shall assume that at time t each point of the region behind the 
 
 shock with a singularity is occupied by a single fluid particle which has crossed 
 
 the shock at some time t earlier than t. Let s^ be the value of s corresponding 
 
 o * " 
 
 to the singular point of the shock and call the singularity as the point § , and 
 
 ■jfr Jg. y y y 
 
 denote all the values of flow variables at s by u, , u , f , c . ro • Behind the 
 shock, the streak line S of s^ and one of the characteristics at s , C, divide 
 the whole region into three parts I, II and III, say, see Figure 1. In part I, 
 
 the flow is uniform, for the shock front 
 is straight there and the flow ahead of 
 
 
 UNIF. 
 I 
 
 *S UNIF, 
 
 
 
 -+► CL. 
 
 the shock is uniform. In part III, the 
 flow is non-isentropic, rotational, for 
 the shock front is curved there and the 
 flow ahead of the shock is uniform. In 
 part II, the flow is irrotational and 
 isentropic, since the flow is continuous 
 
 in crossing C. The boundary S is a 
 
 (2) 
 
 slip-stream , and in stationary case, 
 
 C is a straight line. 
 
 -1- 
 
II. The Coordinate System in Part I 
 
 Let the shock front be x. = ta.(s), a< (s )a / . (s ) = 1 , 
 
 1 1 ' 11 ' 
 
 where a. ; (s) = 0, when s > s^ + 0. n = 2, 3, k, . . . . 
 
 a 
 
 ) n *(s) = when s < s - 0, n ■ 2, 3, k, . . ., N-l, 
 
 but a! N ^(s-) ^ for i = 1 or 2, 
 
 3b.(s,r) 
 and let — — = u. - b.(s.r), b.(s,0) = a.(s). 
 
 then in Part I, u. = constants, b.(fe, 1 &') = u. + (a.(s) - u. )e 
 
 ' i ' i ' 11 i 
 
 Let T be the arc length of a.(s,Z") for fixed s, and T = when <T"= 0, then 
 T = - ^(a.Cs) - u.y^a.(s) - u.^e"^- l) = - \[F (e'^ - l), 
 
 where A = fa.(s) - u.Va.(s) - u.\ = "v(,s,0) jA = V^= const „~| , 
 
 Then a.(s,T) = u ± + (a.(s) - u.)(l - ~=) = a.(s) - (a.(s) - u.) -|- , (2.1) 
 
 where a.(s,T) = b.(s,t) \\(s,T) = a.(s) + u. y= I 
 
 U.(d,T) = -(a.(s) - u.j(l - ^) , V 2 = A(l - ^) 2 , V -JT- T. (2.2) 
 
 3a, T (a (s) - u .)ra'(s)(a.(s)- u.)lT 
 
 \ S ^T ■ ^'H 1 " gT> + ~ ^ J ^ |.Va:(s) = const.] ( 2. 3 ) 
 
 v -^ 
 
 2 
 
 2(a ; (s)(a (s) - u)) 2 [ a! (s ) (a (s ) - u )] 
 
 = (1 -J.) 2 + [a!(s)(a (s) - u )| 2 h§| - -^ ) 
 
 v/A ii x \/A 3 A 2 
 
 3a 
 
 bt^ = - ( a i (s) - \)^r 
 
 -2- 
 
a^ a:(sj [a.(s) - u. ]fo(s)(a.(s) - u.)j 
 
 3T 
 
 \TST 
 
 \/~A : 
 (s,T) - u_ a (s) - u, 
 
 a 2 (s,T) - a 2 (s) u p 
 
 From (2.1) we have r — =r = 7— r I / r fa \ 7 — r = — 
 
 a 1 (fe,T) - u x a 1 (s) - 1^ I a^e/r) - a 1 (s) 1^ 
 
 Therefore the streak lines in part I are straight and lie on the straight lines 
 passing through the points (s,0) and (u,,u ) . Put y - a + p, — . 
 
 Then 
 
 U 2 a g (s) - u 2 
 
 if 2 2 2 /° 
 
 tan J = 77- = — 7 — v = independent of T. [r = indep. of s and T 
 
 if U-, a, \s ) — U-, ~* c. 
 
 V n, 
 
 3T 11 
 
 = 0, 
 
 n = 1,2, .... 
 
 "»% 9n / 2 
 
 From (2.2) / U . = \.U. = -(l 
 t 11 
 
 w- 
 
 L Ss n 3T" 
 
 [a.(s)(a.(s) - u.y 
 
 = 0, n = 1,2, 
 
 (2.5) 
 
 U, = const 
 
 •] 
 
 (2.6) 
 
 U n = Wl - [-iW(«teW " U 2 ] " »2 (s)(a l< 8 > - V] (1 - jff f' 
 
 [U = 
 
 L. n 
 
 const, 
 
 3 
 
 as 
 
 T=0 
 
 ■If 
 
 5 
 
 8s 
 
 n 
 
 T=0 
 
 = 
 
 3\ 
 
 = 0, 
 
 2^ 5^ 
 
 aT = a 2j dT + c 2 
 
 g ig c \ COS |1 
 
 3s : 2 mc 
 
 U 
 
 n 
 
 aV 
 
 ^U a'(s)(a (s) - u ) 
 
 3T y/X ' 8T n 
 
 = 0. n = 2, 3, ^, ... 
 
 n = 3, 4, 
 
 (2.7) 
 
 (2.8) 
 
 (2.9) 
 
 -3- 
 
9U. Q _ 9a. U.U. 
 U i pT V ^T U i 3T V 
 
 il=- i 
 
 £>T 
 
 g - O] (2.10) 
 
 L3T 
 
 By rotating the coordinate axes of a-.,a p plane we can always suppose that 
 the straight shock is parallel to the a p axis, i.e.: 
 
 a, (s) = a = const, a (s) = s, when s > s^. 
 
 a,' (s) = 0, a'(s) = 1, when s > s*. 
 
 Then t \t \ i <\2 
 
 (a - u , )(s - u ) T (s - Up) 
 
 H = -7Ti T ' ^ = 1 -^r + ^r- T 
 
 U t - - (■ - u 2 )(l - $-), n = - (a - Ul ){l - ^) 2 
 
 s - u . fk D - AT + (s - u ) T 
 
 Tan |i = — , tan a = —? r? c= , 
 
 a - u, , _T_ (a - u n )(s - u Q )T ' 
 
 1 ■*■ " u 
 
 (s - Ug) - (a - u x ) 
 
 T 
 
 sin »* = 7T\ > cos ^ = ~uk (1 • W ] 
 
 A 
 
 i 2T / x2 T 2 / x2 
 = 1 - - (a - u x ) + - (a - u x ) 
 
 III. The Boundary C Between Part I and Part II. 
 
 /~~2 
 r^ n r ds _ , _ sin u i vm - 1 cos u. 
 On C, s = ^ = - a + , where a + = — *• - —* 
 
 — — A.(l - m cos \i) 
 
 In (s,T) plane, the image of Figure 1 is as the following Figure 2. 
 
 -h- 
 
From Figure 1, we know s* >u , s* > u 
 
 _ 2 : 
 
 - "1 
 
 (V*<°' ^nC< 
 
 cos n* < 0, sin u* < 
 
 
 , -r For the shock front is continuous at s , hence 
 cos |t* < 0, sin |i* <0 
 
 Figure 2 
 
 2 2 
 1 - m cos n > 
 
 -Of* < - Q!* 
 
 — + 
 
 Therefore C is a C -characteristic, i.e. along C, we have s= -a . 
 
 For psuedo- stationary flow, suppose on a curve F: x. = tb. (fe,T) 
 
 which lies in I and on which m E 1> then V a c = const. Also by (2.2) we 
 
 have, along F, 
 
 £Y_ ' __J__ _ S " U 2 
 dT ~ " e/A~ " /3T 
 
 s - 1 = - X sin i4«s -1 = 
 
 3 = 
 
 X sin (-L 
 
 On the other hand 
 
 a. = 
 
 + K sin p. 
 
 s = - a 
 
 Therefore F is a characteristic. Now suppose m* = 1, then V = c, i.e. 
 
 V =v/A"- T = /(a - U;L ) 2 + (s - Ul )' 
 
 - T = c 
 
 -5- 
 
gives a relation between s and T, therefore it determines a curve in a, ,a p plane, 
 on which 
 
 V =/( a;L (s,T) - u x ) 2 + (a 2 (s,T) - u 2 )' 
 
 = c 
 
 i.e. it determines a sonic circle; this circle should he the boundary C, because 
 the existance of C through each point is unique. For stationary case, if m* = 1 
 then m = 1 in part I, so m = 1 does not give an equation of a curve. But in this 
 case, on the boundary C we still have m = 1, and C is a straight line. 
 
 For both stationary and pseudo-stationary cases, if m* = 1, then we have 
 proved that, along C m = 1, therefore along C, we have: 
 
 ij 
 
 I id 
 
 \ sin n 
 
 , n . n 
 a. sin \i 
 
 1 cot |i 
 
 1 
 
 1 n cot [i 
 1 
 
 (3-D 
 
 The only shock configurations with singularity which can exist are those such that 
 [i, (11.20)] 
 
 B N+1 A N+1 = 
 
 ij J 
 
 (3-2) 
 
 where 
 
 A 11 A 11 
 
 A. = A. .a ., 
 
 „ 2 .2 
 am sin n 
 
 a i " r+ 1 ' 
 
 2 /-, 1 sin u.\ /^„\ 
 
 -^n (i - — s + ■ —n) ( >o ) 
 
 2 " f-\ 
 
 m 
 
 <r c 
 
 and 
 
 B n . 
 
 A dS / 
 
 A. . + -r= . . 
 
 ij dT ij 
 
 n t ^n 
 
 At s* we have 
 
 N+l 
 
 1 . N+l ' K Y 
 sin |i 
 
 1 / 2 sin 2u 2 sin u s 
 
 — ( r+ 1 + n cot ^'771 ' ~2 } = 
 
 sin 2|-l 
 
 n 
 
 ■v ,v . N+l V 2 
 ^+1) sm m- 
 
 Cf5 + 2) 
 
 -6- 
 
,N+1 
 
 . 2 
 sin [i 
 
 . N+l y+ 1 
 sin m. ' 
 
 f ^.^ 2 . N-l 
 [f+1) a- sin n 
 
 B 1 . 
 
 X 
 
 sin |i sin |i 
 
 0-0 
 
 COS [i 
 
 1 
 sin [i 
 
 sin |i 
 
 T = 
 
 sec [i 
 
 
 
 B n . 
 
 T* = 
 
 If N = 0, then (B! . A'.) 
 
 = 
 
 n > 2 
 
 implies 
 
 * 
 sec (i • 
 
 r+ i 2 . -l * 
 
 cr sm |i 
 
 = 
 
 i.e. tan \i* = 
 
 2 2 
 
 1 - (m* cos u*) = sin |i* > 0, 
 
 tan n* ^ 0. Therefore 
 
 we get a contradiction. Hence, for N = we have m* f 1. For N > 0, if m* =1 
 the conditions (3-2) are always satisfied, therefore we can possibly have the case 
 m* = 1. 
 
 • 
 
 IV. Stationary Case , (m > l) 
 
 In part I, j, and y p are constants; so are they on the boundary C. By the 
 result on j_III, p. 60^ , in a neighborhood of C in part II, the flow is a simple 
 wave, i.e. 5, = f(j p ) for some function f. Suppose the flow in part II is analytic 
 then the flow in the whole region II is simple. Because C is a C , the flow in 
 Part II is a /_ wave (i.e., the image of part II in J,, J_ plane lies on a certain 
 characteristic /_ in J,, 7 p plane) and along each C in part II J , and j are 
 constants [[ill, p. 60-6l~] . Therefore by Bernolli's law, V is constant along 
 every C in part II, for the flow there is isentropic, and u, and u p are constants 
 along every characteristic C in part II. Applying the result proved on [ill, p. 60J , 
 we know every C in II is a straight line. 
 
 •7- 
 
In II every C cannot intersect C. (otherwise passing through some point 
 on C there exist two C 's), so it must intersect the boundary S. For p, j, u, , 
 u p , c, m are all constants along C in II, therefore, if we know all these values 
 on S, we know the flow in II. On S we have: 
 
 a.(s*,T) = a i (s^) + a^(s^,0) T + a J (s^,0) ^7 + 
 
 where 
 
 'hv'i ■ 
 
 Si 
 
 a^s.T) 
 
 _) s 
 T 
 
 = 8, 
 
 = o : 
 
 a:( s ^,o) -^ 
 
 T = = cos 3 2 
 
 s = s 
 
 = sin T 
 
 if i = 1, 
 
 if i = 2 
 
 * ..-^ 
 
 aJ(s^O) = -sin J 2 -K =0 (i = l) 
 
 = cos fZ'*** = ° (i = 2) 
 
 a N (s^ ,0) = - sinf* •K^ 1 =0 (i = l) 
 = cos f* •K^ 1 =0 (i = 2) 
 
 a ( s„, 0; = - sm 
 
 = cos 
 
 fa'** 
 
 (i = 1) 
 
 (i = 2) 
 
 In a small neighborhood of s^, we can omit all the terms whose power 
 are greater than N+2, then we have 
 
 i x ( s^T) = a + cos f* T - sin f* • K^~ ^gyi 
 
 _* * * ,pN+2 
 
 a 2 ( s^T) = s* + sin f£ T + cos ^ • ^" J^l 
 
 (*.l) 
 
 -8- 
 
where K. = Ap k„ which can he calculated. 
 
 / „ . <° *- TT la* *N *- T 
 
 (f 2 (8„T) -f a+ K„ fr=?2 +A 2 S. IT 
 
 f ^i) - £ ♦ *« * I (u - 2) 
 
 For the flow in part II is isentropic and irrotational, so we can deter- 
 mine c(s ,T), and hy Bernolli's relation we can find the value of V(s ,T). There- 
 fore we can determine u,(g^,T) = V cos jp, u ? (s^,T) = V sinjl. Thus we know the 
 values of the flow variahles on the stream line S. By the result proved in 
 LlII, p. ^2 J , we know the direction of the characteristic C at each point on S. 
 For C is straight, so we know the C itself. Along every C the flow variahles 
 are constants, so we can determine the flow in part II if we know the singular 
 shock and the uniform flow ahead of the shock. 
 
 V. The determination of the Curved Shock . (Stationary Case) (m > l) 
 
 Now we try to discuss another question: If we are given a uniform flow 
 ahead of a singular shock and a simple flow *r = f (j, ) in part II, then what can 
 we say ahout the curvature and the derivative of curvatures at the singularity 
 B # ? Tauh [i, (8.2) and (8.8)] proved that 
 
 ^(s,0) 
 
 = ka, (5.1) 
 
 9s i 
 
 2 2 
 
 2m sin 2u 2 /n 1 sinuu 
 
 where ^ = r+1 . * 2 = yTT (l - ^ + ~^) 
 
 — = = A. . — nT = kA. .a. = kA! on the shock. (5.2) 
 
 ^T ij 5>S ij J 1 KS i 
 
 In part II, f 2 = t(f^) 
 
 1 
 
 ■h ...tSh 
 
 = f 
 
 <?T W 1'5T 
 
 ifj_)^T (5 * 3) 
 
 -9- 
 
For the simple wave should be a '_ wave, hence 
 
 rtf^- ^^ (5» 
 
 By (5-2) and (5-3) we have, at § . 
 
 li- f'A^^ 1 - "^-=-i a} = (5.5) 
 
 m 
 
 if k" > 0. Hence 
 
 where 
 
 a p sinh© + d coshO = d sinh© + a p cosh©, .'. a = d. (5«6) 
 
 /m* - 1 2/m 2 - 1 . _ , , Q /~2 I" 
 
 = r— a, = — : — Bin 2u, tanh© =v/ m - 1 cot u.. 
 
 21)^+1 ' 
 
 m 
 If k~ = k~ = = k" x = 0, k^ ^ 0, then 
 
 9 "5l * f 2 „ f/<? » t 
 
 = n = 1,2, N-l, N. and — ^p = f'(j^) — ^77- at 
 
 (5-7) 
 
 5>t"/ s>r* T " * p 
 
 ^ + l "I'o^l 
 
 By Tauh [i, (9.4) and (9. 5) J, we have, at s , 
 
 a N+1 5. 
 
 1 - A? +1 k7„, (5.8) 
 
 jH+1 - "i "(H) 
 
 (Ml./SlM) =Q ( > 
 
 A 2 
 
 m 
 
 a sinh (N + 1) + d cosh (N + l) = d sinh (N + l) + a p cosh (N + l) 
 
 a 2 = d. (5.IO) 
 
 -10- 
 
On the shock: 
 
 1 2 f 9 2 . „ 2 ■ ,r, \ 
 
 - — = -s — 7 ( ;r~ m .sin 2 M- + m — - sin 2pJ 
 
 gs <>r+ i v ps CK ps 
 
 2 _ 2 / 1 ^m 3 / glnjx x 
 
 ^s r+ 1 I" m 3 ps ^s k ^2 ; 
 
 By Taub [i, (8.1), (8.5)] we have 
 
 t:-^i > S-* (v!-^- 1 -^)-'^-) 
 
 2 2 2+Jjr- i)^- 2 
 
 m cos jj. = * — 
 
 {r+ 1) + 2 (<r d - 1) 
 
 5m •, / \ 
 
 — = k« (function of ^ m. and \i) 
 
 = k*(function of ^-, m, and p.) 
 
 — = kb., then -J. 
 
 i 2 
 
 Suppose -^ = kb.. , then r = k.,a. + k b. (5-13.) 
 
 <P s 
 
 5T 2 *Ij 5s 2 ^T ps A ij Ps 3 s 
 
 = A. . (k, a . + k b . ) + -—?■ ka . + A. . — ^ ka f . 
 
 On the shock, A. . are functions of m, \x only, therefore we can suppose 
 3A. . <3A . 
 
 -11- 
 
and 
 
 Jl 
 
 ^ 
 
 J = pS 2 ^ \a. + k 2 B? 2 ^ where B^ = A? 2 A . + C . .a . + A. .D ., a. (5-15) 
 
 2 i 1 1 ' i ij j ij J ij jlck 
 
 From (5«3) we have 
 
 £!i = f(f 1 )(^) 2 + f'(/ 1 ) tk. hence 
 
 <? 
 
 T 2 
 
 9 
 
 # ' 
 
 B 2 - fC^KA^) 2 - f(j^)B 2 
 
 = 
 
 (5.H») 
 
 _-* 
 
 Therefore if k" 4 °> "then 
 
 B 2 - ftfjU^f - fC-f^B 2 = 
 
 (5-1^) 
 
 -•* 
 
 and we cannot determine k 
 
 By the same reasoning as before, we can suppose 
 
 ^U 
 
 JT 
 
 ^T = A?k + 3kk n B? + k 3 C? 
 
 3 i 2 1 i i 
 
 Differentiating (5«3) twice and using (5.15), we have 
 3k kl (B3 _ fn{ fj A l A 2 . fl{ Jj B 3) + k 3 (c 3 _ f m ( f i)(A l)3 . 3f ,. ( jP ) A 1 B 2 
 
 (5.16.) 
 
 - f (f 1 ).c^) = o 
 
 (5.17) 
 
 If k~* 4 0, and B^ - f"(^)ATA^ - f'CJ'' )B^ ^ 0, we can determine k" for given k" 
 
 _•* _ * 
 Similarly we can get relations for k , k , 
 
 -12- 
 
Bibliography 
 
 1. Taub, A.H., Determination of Flows Behind Stationary and Pseudo-Stationary 
 Shocks, Annals of Mathematics, 62:300-325 (1955) 
 
 2. Taub, A. H. Singularities on Shocks, American Math. Monthly, 61:11-12 (195*0 
 
 3. Courant and Friedrichs, Supersonic Flow and Shock Waves, Intersciences, 
 p. 60 (19^8) 
 
 -13-