LI B HAHY OF THE UNIVERSITY OF ILLI NOIS 510,84 I^6T no. 156-163 c o p o 2 Digitized by the Internet Archive in 2013 http://archive.org/details/singularsolution156gear No. 84 ItGr no. \5Q f DIGITAL COMPUTER LABORATORY UNIVERSITY OF ILLINOIS URBANA, ILLINOIS REPORT NO. 156 SINGULAR SOLUTIONS AT BOUNDARY INTERSECTIONS IN TWO DIMENSIONAL QUASI-LINEAR HOMOGENEOUS PARTIAL DIFFERENTIAL EQUATIONS by C. W. Gear December 9 } 1963 1. Introduction This paper is concerned with the nonexistence of analytic solutions to the equations of two dimensional stationary nonviscous flow when the boundary conditions are prescribed by a wedge and an attached shock. This nonanalyticity is shown to occur for a general set of two dimensional quasi- linear elliptic differential equations whose coefficiants are independent of the independent variables when some boundary conditions are prescribed analytically in terms of the dependent variables on one line and the remainder are prescribed similarly on another intersecting line. SHOCK (Boundary l) Oncoming Flow A WEDGE (Boundary 2) Figure 1. Shock Attached To a. Wedge Previous authors A3A A. A ) ] a&v - e discussed the impossibility of satisfying the local boundary conditions analytically at A unless the wedge angle x satisfies conditions which can be met only for a countable number of angles when the flow in front of the shock is uniform and the wedge is straight . This is the case which will be considered in detail. A transformation, singular at the intersection of the two boundaries, will be applied to the independent variables and the solution expressed as a power series in one of the new independent variables with coefficients which are functions of the other variable. This transformation will depend on one parameter t which will be determined by the boundary conditions at the inter- section of the two boundary lines, which is the apex of the wedge A in the -1- case of the flow equations. The case of the flow equations will be examined in greater detail because in this case it is possible to find a closed expression for this parameter t. The transformation used and the value obtained for t determine the nature of the singularity of the solution. (h) We start with the equations of flow as expressed by Taub v , equations (5.7) to (5.15), namely &t dm d~T d5 o . . ii 3r~' x ' 3 ~ > ' 1 fi 7-1 2s = - - (1 + ^ — m ) m dT da sin u. d"T " \ *t, dc dr d| - 1 5 1 d~T~ (1.1) where T is the arc length along a stream line from a.n arbitrary curve crossing all stream lines, taken here as the shock front, s is a parameter along the shock front, and constant along a stream line, \ is the scale factor between the parameter s and actual length in the physical plane, that is, \ds is a length, |, = — log p where p is the pressure and y is the ratio of specific heats, n |p = p + the inclination of the stream lines to the horizontal, m = v/c is the local Mach number of the flow where v is velocity and c is the local speed of sound, l_i is the angle between the normal to the curve T constant and the tangent to the curve s constant (positive when counter clockwise), a = lp "M- ^ s ^ ie inclination of the stream line T constant to the horizontal, -9- a... = X sin [i m -m cos |i (— -l) cos |i sin n and the summation convention is used, Writing m = | [X X 3 h h mc v = mc = | equations (l.l) can be written as ^t ij c^ir 1,J = 1,2, ...6 (1.2) where A = \(l-m cos |j.) sin |i (1 2 ) cos n m -[2+(7-l)m ] sin [i (l £■) cos u m — sin u m m cos |i sin p. 2 . 2 m sin u cos \x 2 2 -m [2+(7-l)m ] cos u 2 2 \ cos \x (l-m cos |j,) -mc cos u ■3- The boundary conditions for the wedge problem are that I , the inclination of the flow, is prescribed along s = (the wedge) a.nd that the £ . (i = 1, . . .6) are functions of a. single parameter a (the shock angle) on T = (the shock). The latter conditions are the Rankine-Hugoniot equations or R-H equations (see Ta,ub equations (h.k) to (k.12)), and can be expressed parametrically as i ± (a,0) = f. (a (a)) i = 1,...6. (1.3) where the f . are functions determined by the R-H equa.tions and a is an arbitrary function of s. By eliminating a, the boundary conditions ca.n be expressed as C.U (s,0)) =0 J = 1,...5 J (l.k) D g(l) = lo " X = ° (x is the wedge angle) where the C. are a set of scalar functions of the vector J i. - (I]_j ig' ^y %i±> ^5^ ^5) It is convenient to view the problem in this way so that the problem of the set of two dimensional quasi -linear equations Ji _ . JA J = l,...n (1.5) where A. . depends only on |_, subject to the boundary conditions ■'■J C,(l (s,0)) - i = 1,2,.,, r (1.6a) 1 — D (|_ (0,T)) = j=r+l,...n (l.6b) J where 1 < r < n -1, can be discussed simultaneously. (if r is or n, then the boundary conditions are expressed along a. single line only. ) -k- Section 2 of this paper will examine the nonexi.u. doc of analytic solutions in the general case. Section 3 introduces a. singular transformation of the independent varia.bles s and T to attempt to over- come the difficulty, and discusses a series solution in the new variables. 2. The Nonexistence of Analytic Solutions If an analytic solution exists, it can be expressed as a. power series in s and T about the point s = T = inside the region under consideration. At the point (0,0), equations (1.6) are a set of n simultaneous equations for the n unknowns £ (0,0) which will be assumed to admit at least one solution. (The R-H equations have 2 solutions for wedge angles sufficiently acute.) Differentiating equations (l.6a) and (l.6b) w.r.t. s and T respectively, we get dc. dt, = i = 1, ...r (2.1a) sqar and Substituting (1.5) in (2.1) we get dD. d| SfV^T- ' =r + l,...n (2.1c) q Equations (2.1a) and (2.1c) form a. set of n homogeneous simultaneous linear equations for ^ — evaluated at s = T = 0. They only admit the zero solution unless det [M] = (2.2) -5- where M ik ■ 3d. i < r sq A # i > r+1 v J In general, this condition is not satisfied. If it is not, then &t 57"=°' and therefore "by (1.5) 5t~ = o, Differentiating (2.1a) and (2.1b) w.r.t. s and T respectively and setting s = T = 0, we get l / k ^k" W = (2.3a) *>j ^\ SI q \Bt" (2.3b) Differentiating (1.5) w.r.t. s and T separately, we get dsdT d £ . dA. . d| . = A. 1J a J , iJ 'J 2 (Ts 3s s\ d | . dA. . d| ij "J d^ A i j cHSt + c^T** (SIT ' -6- At (0,0) the last two terms vanish, therefore = A §T 2 y(0,0) iJ V s V(0,0) where A . . is the ij element of the square of the matrix A. Substituting -L J this in (2.3b) we get SD 1 2 9 \ s q n os (2.3a) and (2.3c) are a. set of linear simultaneous equations for which only admit a nonzero solution if *\ (2.3c) ctet [Mg] = (2A) Ife ik ] 5T qk ^ i < r i > r+1 Again, (2.U) is not satisfied in general. Continuing this process it can then be proved that the only analytic solution is the constant solution unless one of the matrices M is singular. This will not be the case in P (5) general. For the flow problem it has been shown by Thomas that those angles of incidence a of the shock for which det [Ml =0 for some p > 1 p — are dense over the set of all a for which the flow behind the shock is subsonic . 3. A Solution Singular at the Origin We start by transforming the s-T plane into a "distorted" polar coordinate plane R-(3 by the transformation -7- s = R 1 ' 1 sin p T = R 1 / 1 cos p (3-D where t is a constant to be determined. (For the equations of flow it will he convenient to apply a linear transformation to the s-T plane first, say to the s'-T" plane, in order to simplify the solution. This step does not affect the nature of the solution obtained.) Expressing g as a. power series in R with coefficients which are functions of P by: 6 (s,T) = E i vi (f»* v p=o y we ha.ve and d | . 00 , /. ^ E R P_1/ [pt| pi (p) sin p + | pi (p) cos p] p=o > gii = E RP" 1 /* [ p t| (p) cos p p=o | p .(p) sin p] (3.2) where £ .(p) is the derivative of £ .(p) w.r.t. p. Substituting these into pi pi (1.5) we get -1 / b E | .R P [S.. sin p+A. . cos p] = tR -1 ^ E | . R P [-A.. sin p+5. . cos p] p=o PO 10 ij p=o PO ij 10 (3.3) where 5. . is the Kronecker delta, 10 -8- A is a function of |_ only. Assuming that it is an analytic function of £ in the region of interest, it can be expressed by 00 a = Z r p a (^(p), ^(p),..., r(p)) p=o where f (p) is the n vector U pl (P), I Jp),..., 6 (p)). Substitute this into (3 '3) and equate terms in R ' to get [I sin p+A^ cos P]i p (p) = pt [I cos p-A^ sin p] ipCP)*^^,,^,^,^, . . .'l^) (3.M where I is the nxn identity matrix, A_ is the matrix whose elements are A„. . - ' -0 ( Oij and e is a catchall for terms not involving |_ or |_ . Equation (3-*0 expresses |_ as a function of |~, i.-!*'*-! -it by means of linear ordinary differential equations if p > 1 and by quasi-linear ordinary differential equations if p = 0. Consider equation (3«M for p = 0. e^ = 0, so the equations are homogeneous of the form (I sin p + A cos p) j =0 These only admit the solution j* = provided that det (I sin p + A_. cos p) / 0, as is certainly true if the original equations are elliptic, since then the matrix A has complex eigenvalues. In the case of the flow equations the matrix A has four eigenvalues of zero, which correspond to the matrix (I sin p + A^ cos p) becoming singular at p = 0. This only means that certain of the flow variables may have discontinuities across the stream line P = 0, as is well known. For all other p, the equations are nonsingular if the flow is subsonic. Parenthetically we note that if the condition det (I sin p + Aq cos p) = (3-5) -9- is satisfied, then equation (3-M may have zero or ma.ny solutions. Specifically, let us suppose that the rank of the matrix [I sin B + A~ cos Bl is v < n. For equation (3-M to have solutions, the right hand side must satisfy the same n-w relations of linear dependence that the left hand sides do. The leads to a set of new algebraic relations RjCl^..., i p _ P io*---* ip> P) = o 3 = w + l,...n ■which express E in terms of 8, t «... | -, and | , . . . | .. . These n-w ^ ^p K > ^o 7 — p-1 ■ 3 -o' p-1 relations, together with w linearly independent equations from (3.*0 give a set of n equations which may serve to determine |_ . For p = 0, the right hand side of (3.*0 is null, so that the n-w algebraic relations are missing. However, for a matrix A corresponding to a. general quasi-linear problem, the requirement that the matrix [ I sin B + A cos 6] has rank w implies n-w algebraic conditions on A , and hence on the dependent variables . If these n-w algebraic conditions plus w differential equations have a solution, the solution has the property that the dependent variables are many valued at the origin. A solution of such a type is know to exist in supersonic flow, namely the Prandtl -Meyer fan. (e.g. see '). To return to subsonic flow, or elliptic equations, we assume that (3-5) is never satisfied for any B to be considered, except possible on the boundary, and only then if the boundary conditions are such that the variables in which discontinuties can occur do not explicitly occur in the conditions on these boundaries. Under such conditions, these singularities can be neglected. The general solution of (3»M is of the form | . = K .F. . (6,pt) + F .(B) (3-6) b pi pj ij ^^> c pi v ^' where F . (p) is a particular solution of (3»^)> F. ,(p,pt) j = 1,2, ...,n are the n independent solutions of the homogeneous equations I = ptN(p)| (3.7) -10- where N(p) ■ (i sin p + A cos p)~ (i cos p - A sin p), and the K are arbitrary constants. In the case under study, j> = and due to the p on the right hand side of (3-3), e in (3.*0 does not depend on |^, therefore e, is zero so that a, particular solution is F 1 .(p) =0 i = 1, ...n . Therefore in - Vij (p ' t5 (3 - 8) where the n coefficient K, . are to be determined by the boundary conditions (1.6). t is as yet undetermined. Differentiating (l.6a) and (l.6b) w.r.t. R and noting that s = R on p = _- and T = R on p = 0, = i = l,...r (3.9a) = j = r+l, ...n (3.9b) Evaluating these at R = and using the fact that we get mP-JjM'^M-Wi&'V be. i < r srv^ ^) = o j > r+i ■li- These equations admit a nonzero solution for the K, , and hence for f, if Ik ~1 and only if det [M 1 (t)] = (3-10) where dc. 3T F q.i { 2> t} > \ V Vi (t) - ( dD. 3-i F . (0, t), '{ 13 i < r i > r+1 (3.10) is an equation for t. When this is satisfied, the remainder of the boundary conditions can be satisfied if the equations t- + terms in ' v < p R=0 ft- 1 P- 2 i < r and = l s q L q p + terms in % v < p ^R v J R=0 P=0 i > r+1 can be solved. Substituting from (3.6), ve get M, \ . .K „ = terms in £ . (q < p) plus boundary condition terms (3-ll) (p)ij PJ 1J - (no summation on p) •12- where 5T F a.i<§' *> *q qj pij Now we require that 3d. s-r^ F .(0,pt) d P q.i ' *q qj i < r i > r+1 J (3.12) det [M . .] ± pij r (3-13) of if it is zero for some p, that the right hand side of (3.11) satisfies a. consistency condition. Without further knowledge of the equations little can be said, but it would appear that the condition (3-13) is met in general. In the particular case of the equations of flow behind a, shock attached to a, straight wedge, t can be determined explicitly so that equations (3-7) and hence (3. 5) can be numerically integrated if so desired. To do this, it is convenient to first transform the s-T plane into the s'-T' plane by the transformation ft 2 \ l/2 s' = s(l - m ) cos (i T" = s sin |a + T where m and \x are the values of m and u. defined in (l.l) at the apex A of the wedge immediately behind the shock. Under this transformation the shock front boundary becomes the line a a 4. "I s ' 4. "I = P = "tan jjp- = tan (l-m ) ' cot |i 1 -13- while the wedge boundary remains 8 = 0- Therefore the equations developed in this section must be satisfied at 8 = 6 instead of at 8 = — . The o 2 reason for making this change is that the matrix N(p) defined in equation (3.7) becomes NO) = < \l '6l N 12 ■V 1 N )n (p) N,,_(S) k2' N 5] _(P) N 52 (p) N 2 and impossible for odd n. However for n = 2, there ■ik- always exists a. transformation of the independent and dependent variables which will put A Q in this form provided that A has complex eigenvalues, i.e., provided that it represents an elliptic equation. These transformations for the matrix A o = I- C a b d are: l) Transform to new dependent variables £ and t by: — a-d — t 2 = i 2 2) Transform to new independent variables s'and T' by: s* = s y-bc (a-d)< (ad-bc) T , =T + a + d s 2(ad-bc) Under these transformations the differential equations become d| di dT' A cTs 7 where at s = T = 0, A has the form: ~bc-(^) 2 » .i^-bo-(^) 2 ■15- The transformations are defined and real since c / 0, ad - be / 0, and Ubc + (a-d) < in order that the original matrix ha.s complex eigenvalues . The flow equations are essentially of degree 2 with side conditions. This is evidenced in the matrix N of (3-1*0 • The variables £ and i are interdependent only on themselves. Using (3.1*0 in (3-7) we get a set of solutions of (3-6) for p = 1 which are: I 1X (P) = m i (K ll Sln tP + K 12 Sln tf3) 6 12 (p) = -(l-m^) l/2 (K n cos tp - K 12 sin t0) (3-15) e ij (p) = W M) + K i2 I W p ' t) + K ijj (sin p)t a - 3,... 6 The F and F ca.nnot be explicitly integrated, but fortunately they do -LJ-J- ■*-o < — not play a part in determining t. The boundary conditions to be applied to £_ are that along the wedge the flow is straight, i.e., that l 12 (0) = (3.16a) and that on the shock f3 = 3 at the apex of the wedge A where R is zero, df . «* <"»„> - sr <°> § (3 - l6b) The latter equation arises by differentiating (1.3) w.r.t. R. — is the dn "curvature" of the shock w.r.t. the parameter R along the shock. Substituting from (3-15) into (3°l6a), we get K 1X = (3.17a) and substituting into (3»l6b) for i - 1 and 2 K i/i ■"" *p = & I (3 - 17b) •16- K 12 (l ' m i )l/2 Sin tP o da dR (3.1 Dividing (3.17b) by (3.17c) we get cot t = df x (l-m^) 1 / 2 L f da m. da tan -MI-iilT) ' cot u. (3.18) df x df 2 - — and ■= — can be evaluated from the R-H equations, thus determining t da da ' & explicitly, leaving K to he determined by nonlocal boundary conditions. This expression gives a, countably infinite number of values for t. The denominator of (3»l8) is 3 and by geometrical considerations is the -1 2 l/2 principal value of tan [(l-m ) ' cot [i, ], however the presumably be any positive value of cot -1 rdf da 1 f . 2x1/2 2 2 — ( l-m. ) ' m n -z — ' x 1 J 1 da numerator can As a changes from the value for which m = 1 to the value for which I is extreme (maximum deflection of flow), the smallest value of t changes from + 00 to 0. The value of a for which the smallest t is 1 is (2) known as the Crocoo angle a . At this angle the flow is analytic; for t larger than 1 (a smaller than a ) the shock has zero curvature at the c apex of the wedge; for t less than 1 (a larger than a ) the shock can have infinite curvature at the apex of the wedge, although, by taking other values of the inverse cotangent in (3.18), t can be greater than 1, so that the shock can have zero curvature at A. h. Inhomogeneous Equations The treatment has been restricted to homogeneous equations of the type in (1.5) where A does not depend on s or T and the boundary conditions do not depend on s or T. If either of these conditions are violated, as for example when the oncoming flow in front of the shock is nonuniform or the -17- (fc.l) wedge is not straight, then two new "dependent" va.ria.bles can be introduced which happen to be identical to s and T. The matrix A and the boundary conditions no longer depend on s and T explicitly, but the price is that the equations are now of the form where d depends on |_. This type of equation also arises directly in the case of pseudo-stationary flow. (See Taub ' equations ( 5.10) -( 5.15) • ) The effect of this is that equations (2.1c) will now include some inhomogeneous terms, so that a solution exists if equation (2.2) is not satisfied. This is the unique analytic solution under these conditions. Uniqueness is not a satisfactory state of a.ffairs because, both in this case and more general cases, some degree of freedom is desirable in order to satisfy nonlocal boundary conditions, which in the flow problem might be expected to be some conditions on where the sonic line meets the wedge down stream from the shock. If an expansion of the form used in (3-2) is assumed for £_ in equations (4.1), then the first two terms in (4.1) contain powers of R of the form R ' , whereas the last term contains powers of the form R . Unless t takes the particularly simple form l/p for some integer p, it is not possible to equate terms in powers of R to zero. To overcome this problem, a double sum for g of the form SZ| (p) H 1 ""/* n m — nm can be tried. Terms in R' ' can then be equated. This technique introduces a degree of freedom to the solution. In fact, since equation (3.18) defined an infinite number of values for t, a multiple power series i n+m /t ,+m /t p +. . . of the form R could be used. However, when equation (2.2) is satisfied, a, solution can only be found if the inhomogeneous terms in (2.1c) satisfy a consistency condition. •18- In stationary flow, this amounts to a condition on the oncoming flow and the shape of the wedge behind the shock. This condition has been discussed by Tsien for irrotational isenthalpic flow ahead of a shock attached to a wedge with zero angle, when it reduces to a condition on the ratio of the curvatures of the stream lines ahead and behind the shock. (3) Lin and Rubinov discuss a solution involving a singularity of the 1/2 shock angle of the form s ' so that the curvature of the shock has an infinite curvature at the origin. This solution relaxes Tsien' s condition to an inequality. A singularity of this form would arise if t = 1/2 in the preceeding equations. This value of t does not satisfy (3 .18) which leads only to the solutions t = 1, 3, 5* • • • • Therefore the class of solutions presented in this paper do not include the Lin and Rubinov solution. 5. Conclusion A class of solutions to the general homogeneous problem where boundary conditions are specified on two intersecting lines has been given. In general these are singular solutions since there does not exist a nonsingular solution. However, the inhomogeneous equations can cause problems in precisely those situations when there is an analytic solution of the homogeneous equations. No solution has been found in those cases; it certainly is not one of the class of singularities discussed in this paper. The solutions found have parameters which are presumably determined by nonlocal boundary conditions not discussed here. Since power series were used on the two boundaries, additional conditions on these boundaries must involve singularities away from the intersection. Such conditions could be taken into account by demanding that the power series solution have a suitable radius of convergence, or that the nature of the equations change, as happens, for example, at the sonic line for the equations of flow. ■19- BIBLIOGRAPHY 1. Bargmann, V., Montgomery D. : Prandtl -Meyer Zones in Mach Reflection. U. S. Office of Scientific Research and Development. Report #5011. 2. Crocco, L. : Singolaritia Delia Corrente Gassosa Iperacustica Nell' intorno di una Prora a Diedro. Aerotechnica 17 (1937)- 3. Lin, C, Rubinov, S. E.: On the Flow Behind Curved Shocks. Journ. Math and Physics 27 (19^8). pp. 105-129. k. Taub, A. H.: Determination of Flows Behind Stationary and Pseudo- Stationary Flow. Ann. Math. 62 No. 2 (1955). pp. 300-325. 5. Thomas, T. Y. : The Distribution of Singular Shock Directions. Journ. Math and Physics 28 No. 3 (19^9). pp. 153-171- 6. Tsien, H. S.: Flow Conditions Near the Intersection of Shock Waves with Solid Boundaries. Journ. Math and Physics 26 (19^7). pp. 69-75. -20- ju» 2 Sim