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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1594 A FLAT WINGC WIER SKARP EDGES IN A SUPERSONIC STREAM*~ By A. E. Donov In this work there is given an approximate solution of the problem of a twodimensional steady supersonic stream of ideal gas, neglecting heat conduction, flowing around a thin wing with sharp edges at small angles of attack. (Determination of the law of distribution of pressure along the wing, lifting force and head resistance of the wing.) PART I The problem of the investigation of the mechanical action of a moving gas on an immiovable wing appears as a special case of the ;ome what more general problem of the Investigation of the mechanical action of a moving gas on a11In imovable fixed wall constraining the motion of' the gas. In our own explanastion we begin with the formulation of this last problem in which we confine ourselves only to the consideration of the steady twodimensional forces of ideal gases not subject to the action of gravitational forces. In the plane of motion of thle gas we shall arrange an immovable rectan~gular coordinate system in such a mann~er that it is situated as in figure 1. We introduce three functions v, p, and p of the independent variables x and y defined, respectively, as the veloc ity, density, and pressure. The vector frictions v will be determined by a pair of scalar functions of the independent variables. For these fune tions we shall agree to take either the functions vx, vy defined as the projections of the velocity of the axis x and y, respectively, or the functions v and p, defined, respectively, as the absolute value of the velocity and its angle with respect to the positive direction of the xaxis, measured in the counterclockwise sense. In what follows we limit yourselves to the consideration only of flows for which thle function P satisfies the condition 2 2 *IzvestiiaAkad~emia, NIAUK, USSR, 1959, pp 603626. 2 N~ACA TM 1334 As is well known, the study of the gas motion under consideration leads to the investigation of the following system of differential equations ax avx 1_ ap v, +v + =o ax bx y p ax avy a7y 1 ap vx + Vy + =0O ax ay P ay (2) a(pyx) apvy)=O ax ay vx axl + "y = oI Here k is the adiabatic exponent (for air k = 1_.k05). If the motion of the gas is constrained by an immvable frictionless fixed wall in the plane XGY, the gas will be adjacent to it along some curve. We shall call this curve~ the "contour K." Consider the unit vtector t tangent to the contour K directed in such a manner that its projection on the xaxis is positive. Denote by Pk the angle which it makes with the xaxis. Clearly Bk may be regarded as a function of the abscissa x of that point of the contour K associated with the vector t. We denote this function by Bk(x) and assume that it is continuous. If the function Pk(x) is prescribed and, moreover, the coordinates of any point of the contour K are given, the form and position of the contour is completely determined. We agree to take as origin the left edge of the contour K. Then the: equation, of this contour will have the form y = tan Pk(x)dx (5) NACA TM 1594 We can write this equation more briefly if we designate its righthand side by, yk(x) y =yk(x) (4) Since ini the' flows under consideration the direction of the velocity on the contour K must coincide with the vector t, the condition on the flow along an immorovable fixed frictionless wall may be written in the following fashion at y = ;/k(x). The condition (5) must be added to the system of equa tion3 (2) as a qualifying boundary condition. Much work has been dedicated tor the investigation of solutions of the system (2) subject to the con dition~ (9). Of these we are interested here only in those in which the~ f'low is ;upersonic, i.e., flows at every point of which the following condition v >a (6) is ~stified, where a is the local speed of sound The investigations contained in these works divide in two fundamental directions. The first direction is represented in works in which solu tions of the problem are achieved with the help of numerical or graphical processes permitting the stepbystep calculation of a system of parti cuilar values of the desired functions. (Works of Busanann, Kibelia, anid Frankll.) The fundamental achievements of the methods represented by these worKs consist of the fact that by their use many actual practical probllem ray be solved quantitatively of which the solution by other methods would present great difficulties. In particular these methods solve thoroughly cornernonpotential problems. The chief defect of these methods is that the solutions obtained are numerical so that it is impossible to obtain a general qualitative estimate of the phenomena 4 NACA TM 1594 under investigation. The second direction is represented by the works of Meyer, Ackeret, Prandtl, and Busemann, which are confined to a culti vation of an exact theory of irrotational flows. The results are based on the fact that in the case where vorticity is absent the character istic system of differential equations (2) adn~it of integrable combina tions. This theory leads to series of approximate results of any desired accuracy, giving a complete qualitative and quanitative picture of' the flo~w. Since our investigation is mostly connected with the theory of irrotational flows we give below a brief introduction to the fundamental methods and results of this the~ory. We introduce the stream function Jrdefined by the following relations 9 p ax P" (8) ay As is well known from equations (2), ("1), and (8) the following relations follow without difficulty e (9) pk v2 a2  + tO (10) 2 kr 1 Where 6, tO denote quantities which display the~ flow once and for all as a function only of J. With the help, of equations (2), (9), and (10) it is easy to obtain the two equations by, aVx 0 (ll) ax ay Here ml* m2 denote the following expressions v2sin B cos B + a\ 2 3 ml = a2 2cos2B NACA ?TMl 194 (a2 vx2) Yx+ (a2 q2)bi _ ax ay ax ay j (12) where R denotes a quantity defined as (13) Equations (11) and (12) represent linear relationships between the first partial derivatives of the functions vx, vy With respect to x and y. Since every flow under consideration is supersonic, the entire region of the flow may be covered by a pair of families of characteristics. 'The differential equations of these characteristics are obtained easily by the use of equations (11) and (12). For one family of characteristics, which we shall agree to call the first family, we obtain the equations (14) (15) dy = mldx S(a2 v2cos2p)ml + v~sla P cos B x d(v cos 0) + m2d(v sin P) = and for the other, which we have the equations v2cos 2P a shall agree to call the second family, we (16) dy = m~dx sinp) 0(a2 v2cos2B)m2 + v2sin B cos p x v2cos2B a2 d(v cos p) + mld(v (17) (18) d~ k(k 1)a~ &n NACA TMr 1394 vfsin p cos B a\/v2 a2~ n2: = (1.9) a2 v2cos2p We now consider a supersonic stream with constant hiyird;redynamical elements (i.e., functions v, p, p, p, a). We shall call th~is flow th~e undisturbed flow. The values of the functions v, p, p, s in the unidis turbed stream will be denoted by v, pO' p0, aO respectively,, aInd the ratio w/a0 by M. Since the stream under consideration is super sonic, M > 1. We shall choose the direction of the velocity of the: undisturbed stream to correspond, to the direction, of the xaxiS. We assume that thLe undisturbed stream strikes anl Lmmovable, fixe~d, frictionless wall (contour K), inclined in such a marner that in flowing around this wall the stream never detaches from it adi~ remains suiper sonic evreryvbere. We~ may distinglluish two cases of flo1ws of thijS tyrpe. Case I. The contour K is situated in such a manner1 that the condit ion Pk(0) (O (20) is fulfilled. .In this case, as is well known, there appea~rS a ciirve of weak discontinuity OC (figs. 2(a), and 2(b)) procee~dingl from thie origin and dividing the entire flow in two parts. On one slide ofr the cui~rve~ of weak discontinuity OC extends the reg~ionl containing thie uCdisJturbe~d stream and on the other the region of flow around the wall. In the region of flow around the fixed frictionless wall the hydrodynamical elements of the stream, generally zpestingv, are not constant but vary. In what fol lows we shall call this part of th~e stream the disturbed stream. In the entire region of the flow undr consideration the functions v, P, p, p, a are continuous but their partial, derivatives with respect to x and y (all or only some) exhibit jump discontinuities, at least on the curve of weak discontinuity OC. The same curve OC appears as a characteristic of the second fam;ilyr since the hydrodynamical, elements of the stream. are con stant. On this line the following relationships will hold in th~e entire region containing the stream ,2 aO2 tO f + (21 2 k 1 NdACA 1TM 1394 (22) 9 = 80 where 903 delotes a quantity defined as 80 pk Froim equadtions (13), (21), and (22) we easily obtain 0 = O (25) (24) i.e., thel flo)W under consideration is irrotational. By virtue of rela tion (24) the righthand side of equations (15) and (1'1) vanish and these equations canl be integrated. As a result of integration of equation (15) we obtain the relationship p + cp(v) = constant (25) satisfied along any characteristic of the first family, and as a result ojf integra~ting equation (171) we have p rp(v) = constant (26) satisfiedj along a characteristic of the second family. cP(v) function defined as cp(v)  are tan \va.2 are tan \r denotes a 2a a (27) Since on! the curve of weak discontinuity OC the quantities v and P have the values w and O, respectively, the following relation is satisfied along every characteristic of the first family intersecting this line and consequently in the entire region of the disturbed stream: NACA TM1 1394 s +t r(v) = rp(w) (28) From equations 16, 26, and 28 it immediately follows that the char acteristics of the second family (the curve OC being am~ong these) are straight lines since along each of these characteristics the hydrojdynamu ical elements are constant. Making use of these circumstances it is not difficult with the aid of equations 28, 26, 22, 21, 16, 10, 7, and 5 to construct expressions for the functions v, p, p, p?, a in the region of the disturbed flow. HoweJver, thne construction of these expressions is not of great interest since our chief interest is centered on the construction of an expres sion for the pressure on the contour K: which may be accomplished without the use of these expressions for the hydrodynamical elements of the flow. Actually equation 28 allows us to determine the velccity v as a function of the angle of inclination of this velocity with the xaxis at every point of the region filled by the disturbed flow. Since by virtue of equation 5 the angle of inclination of the velocity with respect to the xaxis is a given function of x on the contour K thr'e is thle possibility of using equations 22, 21_, 10, 9, and 7 to determine the pressure p as a function of x on the flow around a contour. If we limit ourselves to the consideration of slightly disturbed flows, i.e., flows whose hyjdro dynamical elements differ but little from the~ hydrodynamical elements of the undisturbed flow, the expression for the pressure on the flow around a contour K may be written in the form of a series. This series has the form p = pO + q ali(x) + a28k2(x) + a pk5(x) + a pk(x) + .. .] (29) NACA TM 1394 where pokM2 2 p ~w2 2 = al=2(2 a2 = (MB 1)~2 2 ~2+ 4 2 4 4 +1 3 36 6 ak = (M?2 1)5 1 2 M2 + + 19k M4 + 21 43k + 18k2 M6 + \3 3 6 12 12 48 48 Case II. The flow around a contour K is situated in such a manner that the following inequality is satisfied Pk(0) > O (30) In this case, as is well known, a line of shock discontinuity OD appears (fig. 3) proceeding from the origin 0 and dividing the entire flow under consideration in two parts. On one side of this line is the region of the Lundisturbed stream and on, the other the region in which the fluid flows around the fixed frictionless wall. Just as in case I we call the flow in the region in which the stream around the fixed frictionless wall is accomplished the disturbed flow. In the present case, in contrast to case I, the functions v, B, p, p, a exhibit jump discontinuities on the shockline OD. 10 HIACA TM 1396 In, the region of the disturbed flow these functions must, satisfy, not only equations 2, 5, and 7 but also the dynamical conditions across the shock line. Considering the flow to be only slightly distuirbed, these conditions may be written in the following form v2 a2 w2 &0O  + (31) 2 k 2 k 1 v = w(1+ bly+ b202 b pi + b48 + .) (2 where b1 = (M2 1) 2 +l M" b2 = (M2  1) 21 + 6 5 1 1) +  1)M4 + Sk2 12k + 5 Mo 24 1M2 2 5 2 8 Sj(k 1 27k + 12k2 1 24 17 + 29k e 24 b4 (M2  +5 k Sk2t +h MkpD) 5 + 5k k2 + k3 M8 16 O +zg +2p (' 3~) (1 + 1)2 M 32 NACA TM 1594 where k(k2 ) 11 = ~M (M 1)2 12 34 = 2(k2 1) N (M2  12 = c + elB + esp2 + ... (34) where 1 eO = (M2 1) 2 el k+ IM (M2 1)2 Condition 31 shows that, disregarding the presence of jump discon tinuities in the functions v, p, p, p, a, equation 21, just as in case I, is valid throughout the entire region filled by the flow under considera tion. BHowever, condition 22 is not, generally speaking, fulfilled in the case now under consideration. However, there is the possibility of speaking of satisfying this condition approximately. In fact, consider equation 33. Its righthand side does not contain terms in the first and second powers of B. Therefore, for slightly disturbed flows, equa tion 22 may be regarded as approximately satisfied on the line OD and con sequently throughout the entire region filled by the flow under considera tion. From this it follows that in the region of disturbed flow equa tion 24 may be regarded as approximately satisfied, which means that 1) 4 + 2(k 2)M2 (k 1)M4 12 NACA TM 1394 equations 25 and 26 hold on characteristics. For values of B and v near 0 and v, respectively, equation 28 may be written in the form of a series v = v(1 + bl'p + b2 2~ + b 'p) + b 'p4 + .) (Sp) where 1. bl' = (M2 1) = bl b2' = (M2 1) 2 k 1 b)= b2 2 b = (M2 1) +12 M2 + 2k 1)M + 19k + 16k2 M6 + 24 24 8 +17 + 29k p l 24 b =(M2  +3 + 8k 7k2 + 2k59 M10 j 2kr 5k2 + 4kk M8 52 Comparing equations 32 and 35 we see that for slightly disturbed streams the first may be substituted for the second with good approxi mation. Consequently, for slightly disturbed flows, equation 28 will be approximately satisfied along the line OD. Since, on the other hand, along each characteristic of th~e first family equation 2) is approximately satisfied, equation 28 will be approximately satisfied throughout the entire region of disturbed flow. The approximate expressions for the functions v, B, p, p, a are constituted exactly like the accurate expres sions for these functions in case I. Substituting the approximate expres sion for the function B in the righthand side of equation 34, we obtain +2k2 5k + 3 "] 12 NACA TMI 1394 13 a differential equation of the first degree for the approximate deter mination of the form of the shock line. Summing up our considerations we can deduce that the accurate results contained in case I can serve as approximate results for case II, and further that expression 29 can serve as an approximate expression for the pressure on the flow around a contour in case II. These same considerations show that there is no sense in cal culating all terms in this expression. It is sufficient to limit ourselves to the first two or three terms. From all that has been said about cases I and II one may conclude that the form of the contour K may be made up in such a manner that art fully constructed shocks may be caused to appear in the region of flow around th~e fixed frictionless wall. In such cases when we pay attention to this phenomenon, the results we have obtained are valid, not for the entire region of flow around the fixed frictionless wall, bu~t only for that part in the neighborhood of the front side of the flow around a contour. The fundamental problem of the present work is the3 construction of approximate expressions for the pressure on the flow around a contour in case II, with the calculation of the circulation of the flow occasioned by the presence of the shock discontinuity OD. In spite of the fact that in the case of the presence of circulation it is impossible to integrate equations 15 and 17l, there is the possibility, however, of making up such combinations of differentials from equations 14, 15, 16, and 17, adding to these equations expressions for differentials of the streak function, that with the aid of these combinations it is possible to construct expres sions which we shall integrate. Investigations concerning the preceding construction constitute the contents of the following section. PART II Suppose we have a flow corresponding to case II of the proceeding section. Assume that in this flow the hydrodyna~nical elements in the region of the disturbed stream differ infinitely little from the hydro dynamical elements in the region of the undisturbed flow. We revamp somewhat our notion of the region of disturbed flow. Shortly before we agreed to apply this name to the region bounded by the curvelinear triangle made up of the curve OC2 (contour K), the shock line OC1, and the characteristic of the first family CIC2 emerging from the lowest point of the contour K (fig. 4). Taking into consideration equations 5, 14, 18, and Sk, it is not difficult to conclude that with the assumptions made just now relative to the hydrodynamical elements the curvelinear triangle OCIC2 differs infinitely little from the isoceles straightline triangle O'Cl'C2' (fig. 5) where the equal sides O'Cl' and Cl'C2' are parallel to characteristics of the second and first families in the NACA TN 1394 undisturbed flow. As for the functions Bk(x), a, v, P, p, a we assume that they all have the properties of differentiability and continuity to as many degrees as may be necessary to insure le~gitim~acy of operations which are performed upon them. Moreover, we assume that in the flow under consideration the infinitesimal quantities Bk(x), Pk'(x), Bkrl(x), B, v w, p pO' P 0 a O aO have the same3 order of magnitude. Taking this last group of infinitesimals as fundamental (having Lunit order of magnitude) we shall agree in, what follows to adhere to the fol lowing system of notations appearing in investigations involving infinitely small quantities. By Em (m being any positive integer) let us denote an infinitesimal whose order of magnitude is not less than m. Clearly such a mode of notation does not exclude the possibility of several dif ferent infinitesimals being denoted by the same symbol_, and, vice versa. The same infinitesimal may be denoted by several different symbols. Thus, for example, if an infinitesimal a is denoted by E4, the infinites imal 2a may also be denoted by E4, and, moreover, the infinitesi mals a and 2a may be denoted by E E E . On an arbitrary characteristic of the second or first family the equation d~ = py(sin pdx cos Bdy) (36) will be satisfied by virtue of equation 8 throughout the entire region filled by the flow. Eliminating dx and dy from equations 14, 15, and 36 and taking into account formulas 13 and 21, we arrive at the equation d(v cos p) + m2d(v sin. 8) = (9ld In 9 (ST) which is satisfied on any characteristic of the first family. Here 61 denotes the quantity a2 viesin B cos p (v~cos2p a2)ml k(k 1)v(v2cos2p 2)(ml cos B sin P) NACA TM~ 15394 15 On the other hand, having the integral 25 of the equation d(v cos B) + m2d(v sin P) = 0 (39) it is easy to find an integrating factor L1 of this equation, such that after multiplying by L1 it may be written in the form ap + ,(v) = o (4o) In order to determine L1 we have the obvious relationship L1 d(v cos p) + m~d(v sin B) = d[B + q(v) (41) from which we obtain without difficulty L1 m2 cos P sin P)vap = dB (42) consequently v(m2 cos p sin p) If now we multiply both sides of equation 37 by L1, this equation takes the form a~ +[p +(v)] = Hia In e (44) where H1 denotes the quantity H (v~coseg a2)ml v2sin p cos P (~ k(k 1)v2 HTACA TM 1394 We! denote by R10 the value of B:1 at v = w, P = O. We have 10 1 (M2 1)1/2 k(k 1):M2 (46) Equation 44 may be rearranged in the following fashion di a+ q(v)= KlOdIn + (H1 E10)d (47) Nov choose an arbitrary point S in the region of disturbed flow and lead a characteristic of the first family through it. We denote the point of' intersection of this characteristic vith the shock line by A (fig. 6). Integrating both sides of equation 47 along the above characteristic from point A to point S we obtain  Ba q~va)= 8100S In + S3(q1 810)d in Ps + cp(vs) (48) where ps, vs> es denote, respectively, the values of B, v, 9 at the point S and p,, v,, 63 denote the values of these quantities at the point A. Taking account of equation (32) we have va =w( + blia + b2Pa + b pg$ + b4Ba4 ) We introduce the quantity val defined with the help of the expansion 35 in the following fashion val = v(1 + b1 a + b2 a2 + b '8a3 + b4'Pa4 + .) (cf eq. (35) Tr.) (50) NACA TK 1594 17T By this definition of thet quantity val we~ have Ps + rp Yal) = cP(w) (51) With the help, of formulas 49, 50, and 51 we rearrange~the expression a, + rpYa) in the following manner Pa + cP(va) = Pa + CP(val) + 9(PVa) cplval) = cpr) + cp(va) cp(val) = 9(v) + 9'(w) (va V) (val ') tp"'(w) (va u Y# (Val ')2] i = q(w) + vy'(v) (b J b )paJ + (b4 b4')sdI + wip"(v)bl(by b ')Ba + E5 (e) Calculating cp'(w), cp"(w) we obtain cp'(w) =1(55) vb1 9 (() = 2b 2 v2b13 where Bp denotes the value of p at the point p. Assuming that the mean value theorem is applicable to the integral arising from the right hand side of equation (48), we easily find IInstead, take a slightly more general assumption admitting the part AS of the characteristic under consideration to be divided in the same finite number of parts in such manner that on each part the mean value~ theorem can be applied to the integral under investigation. NACA TM 1594 Using formulas 52, 53, and 54 we easily find 1 Ba + cp(va) = cP(w) (bj b 1  b~')Ba3 + Now pass a stream line through the point S and denote by P the point of intersection of this line with the shock line. Since the stream function C is constant along this line we have 9, = Bp (56] where 8p denotes the value of 6 at the point p. Taking logatrithms of both sides of equation (33) we obtain In 0 3jP3 + 1 'p+ 3 'Pi + . Since the values of the coefficients 14', 19', ... in what follows, we shall not calculate the~m. Using formulas (56) and (57) we easily see that will not be needed ines O in ea 3(BP, ba3)+ 24 '(Pp B ) + E O 1 ~(bs bj')b4 'b4 8a4+E5 bl. IJACA TM~ 13'94 AS (Hi 81)d In e= Z(Hy 10)(p p a,) E O0 (9) (In consequence of this equation one must keep in mind that Kl' 10 = E ). He re HI de~notes the~ value of HI1 at some point on the characteristic uinder consideration between the points A and S. Using relations (BB), (58), and (59) we write equation (48) in. the following~ form Ps + qp(vc) = cp(w) lb3 (y b ')Pa3 + 1023g(Pp3 p, ) + H1034'(pp4 a 4) + 35(H1~ R10) ip paJ) + Sy (60) We denote by B the intersection of the characteristic of the first family under consideration with the contour K. Applying formula (60) to the point B (which is possible, since the point S was chosen arbitrarily) we obtain Pb +' Vb) = q(w) (b3 b ')Pa$ + I(b3 b ') 1 (b4 b 1 3 884+ H10 5 0o3 a 3) + lo34'(Po~ ak4) + 23(H1 Hio) 803 Ba3) + E (61) (b b ') 1 (b4 b4' ) pa+ (v2cos28 a2)m2 2sin p cos B k(k 1)v2 NACA TM 1594 where Bb, vb denote, respectively, the vaus of a and v ~point B and SO denotes the3 value of B at the point 0. We now proceed to the3 derivation of an expression for pa. formula (60) WE? have Ps + cP 's) = (P') + From equation (62), using formula (35) we obtain vs = (1 + blBs + b2Bs2) +E at the From (62) (63) and denoting by m2s the value of m2 using formulas (19) and (63) at the point S we obtain, by mes = (M2 1)1/2 k = eO + 2eliS + E2 b (M2 1)2 s, + E2 (64) Analagous to the derivation of equation (47), which holds on character istics of the first family, we may deri~ve equation dp q(v) = H20d I 0 + (112 B[20)d InO (65) which is valid on characteristics of the second family. Bcere B2 denotes the function defined as (66) Hf2 = and H20 denotes the value of this function at a = 0, v = v. HACA TMr 1594 Ilow pass a characteristic of the second family through the point S and denote by Q its intersection with the contour K. Integrating both sides of equation (65) along this characteristic from the point Q to the point S we obtain QS (H2 H20)d In (67) where P vq, 6q denote respective the values of P, v, 9 at the point Q. Since the contour K is a stream line we have Gg = (0) (6g where e( ) denotes the value of 9 a~t the point 0. Assuming that the mean value theorem can be applied to the integral arising from the right hand side of equation (67) we easily find, with the aid of formulas (56), (69) Applying~ formula (62) at the point Q we have Pg + 9(vq) = cp(w) + E (70) E~liminating cq(w) from equations (62) and (7O) lowing equation we arrive at the fol (71) P, Pq (vs) I; 979 = 2 I In + PsPg`C'p(v,) m(vg~l= Ej Ps P, +[rp(v,)cp(vq~l=~j the slope of the characteristic of the second family is greater than the slope of the shock line at the point F.2 2It is easy to show that if the shock line is unbroken andJ moreover condition (50) is satisfied the inequality P < O is impossible on this linze. As a matter of fact, in the opposite case the shock line is broken since with F < 0 condition (34) must be replaced by the~ following con dition in virtue of Tsemplen's theorem HIACA TM 1394 Families (69) and (71) give (72) a, = Pq + 63 On the shock line we take an arbitrary point F (fig. 7) and pass through it a characteristic of the second family in the region of the disturbed flow and we denote by Pf, m2f, :respectively1, the values P and m2 at the point F. Applying formua (64) at the point F we obtain (73) m2f = eO + 2el f j 2 We denote by '~ the slope of the tangent to the shock line at the point F. From equations (74) we have =r eO + el f + Eg (7 ) Comparing formulas (73) and (74) second family passing~ through F make an infinitesimal angle with we see that the characteristic of' thte and the shock line a~t this intersection each other moreover, if (75) Br > o dy e l 2p . d = (O ep+er"""* HIACA TM4 1394 Denioting by L the intersection of the characteristic of the second family under consideration with the contour K and by x2 abscissa of this point, we have (6) xZ = E L~et PZ denote 'the value of p at the point L. and (76) and MacLauren's formula we obtain Using equations (5) Pt = Pk(0) + Bk'(0)xZ + f (77) Applying7 formula (72) at the point F ve obtain (78) Bf = BI + E As a consequence of equations (77) and (78) Pc = Pk(0) + Pk'(0)xZ + E (79) Since the point F was chosen arbitrarily on the shock line by use of equitions (74), (76), and (79) we can obtain the following differential Iequ~at~ion for the shock line eO + el k(0) + Eg dx (80) Conse~~qucntly the equation of the shock line may be written in the fol (81) y = eO+ el k(0 x + E2 NACA TM 1394 Applying formlulats (64) and (72) to acn arbitrary point situated on the characteristic of the second family FL weF easily obtain. the differential equation of this line from the following form dy eO + 2el 3 + E ix (82) Employing formulas (76) and (77) this equation may be writer. eO + 2elpk(0) + E2 dx (85) Consequentlyr the equation of the characteristic FL;may be written in the form (84) where y denotes the ordinate of the point L. On the other hand, taking account of formulas (3) and (76) we have yZ = O (E0) tan Pk(x)dx = E Emnploying: formulas (85) and (76) we may write equation (84) in. the form (86) Applying formulas (81) and (86) at the point F and denoting by xf, yf the coordinates of this point we obtain y =yZ R) + 2e~l k(0 (x xZ) + Ep y = e(O +2li()x egx1 + E2 N~ACA ITMl 1594 Yf. =eO t elik(0)]* xy + (87) yf = eO + 2el k(0) xr egxy + E2 From equation (giT) we easily obtain eg Replacing x2 in the righthand side of equation (79) by the expression in formua (88) we obtain eg We denote by xa, y, 121e coordinates of the point A and by xb, yb the coordinates of the point B. Applying formula (89) at the point A we ar2rive at the following result pa, =k(0) + elxaik(0)Bk'(0) + E, (90) eO We now express xat in terms of xb. To this end, using formnulas (1 ) an~d (18), we write the differential equation for characteristics of the seco~ndi family in the following fashion dy eO 1 E (91) dx 26 NACA TM~Y 1394 Emplo)ying formula (91) we~ write the~ equation for the characteristic AB of the~ first family in the form Y = ;Yb eO(x xb) + E1 (92) Taking account of formula (3) we have ybJ x tan Pk(x)dx = El (95r) consequently equation (92) may be written. y = eO(x xb) 1E (9 ) On the other hand, equation (81) for the shock line may be written in the form y = eO x + el Si and applying formulas (94) and (95) at the point A we obtain ya = eO(xa xb) + EI(6 ya = egxa + El From equation (96~) we easily final xa + El 97) HAlrCA TM4 1594 And consequently el Pa = Pk(0) + 1e xbik(0)Pk'(0) + q (98) Substiturting this expression for Pa in the righthand side of equa tion (61) and substituting P0 for Pk(0) in the fundamental formula (5) we obtain Pb + cPvb) = cPY) 1 (b3 bj')Pki(0)  bl L(b4 b ') 2tb12(bJ b3')j pk (0) Sel b b 'v + 310 3 x k()Pk'( 0)+ E ~ 99) 2e00 bl Alpi;loyig relation (35), we easily obtain from equation (99) b b4' ~2(b1 b.') pk (0) + 'u(byb b ')Pk5(0)Pb + be rb'+ 1 bxk(0k() + 1) 28 N~ACA TM4 1594 Substituting vb, Pb, xb for v, Bk(x) and x, respectively~, in formua (100) we arrive at the following final expression for the velocity on the contour K: v 1 + bl k(x) + b28k2(x) + by' kJ(x) +b' k (x + b )k50 b by' ( b ') pk4 (0) +2(b3 b ')Pk3(0)8k~x) + 2 b3 b + HIfO }bl x k (0)Pk(0)0) + E5 (101) We now proceed to the derivation of formulas from which the pressure on the contour K can be calculated. Clearly a2 = k pO(102) PO and maoreover, on the contour K the following equation holds AL= e(0) (1os) Employing formulas ('1), (10), (21), (25), (102), and (105) we easily obtain the following expression for the pressure on the contour K 1 k P k1 hl k k1 lIACA TM4 1394 On the other hand, by virtue of equations (5) and (33) the following equation holds e(0)1+op50)+ 4pk (0) + .. (105) e(o) Substituting the expressions for v and obtained in formulas (101) 80 and (105), respectively, in the righthand side of eggsation (104) we obtain after elementary transformations the desired formula for the calculation ,of the pressure on the contour K P = 0 +2 alk~x + gg2() +a30k(x)+ a4k (x) + ald k3(0) + asd k () + a 4pk5(0)Sk(x) + aggPk5)(0)Pk'(0)x + Eg (106) whe re 213 ald = 2(b3 b ') k(k 1)M2 = M (M2 1) + M2 S24 M 4b2 214 Sad =2(b4 b') + (by b ') bl k(k 1)M2 Sk + 1 + 3k 2k;2 2 10 Sk + 6k2 kj M4 2 4 8 16 32 SO NACA TM 1394 agd = (b3 b ') 2421 21  S6M2 15 jk + 1 7 + 2k 5ik2M 6 24 4 + Sk+ 6r~ k2 kS 5 Tk 7k2 + k5 24 96 ag = e ~(bj b + .5103 bl) (k + 1)2 kk = M8(M2 1) + M2k + M 16 2 8 For x = 0 the formua (106) takes the form P = Po + q Ial k(0) + app (0) + a 'DkJ(0) + ag'pk i(01 + Ej (107) where a = a3 + ald, ag = a4 + a~d + a . Formula (107) may be used for the calculation of the pressure on a flat plate which is inclined at an angle Pk(0) to the undisturbed flow. In order to single out of the righthand side of equation (106) those terns which depend exclusively on the presence of the shock in front of the contour K, we add to the contour K under consideration an arc O'0 of finite length in such a manner that this are is tangent to K at the point 0 and is parallel to the xaxis at O' (fig. 8). Since the flow around such an additional contour is accomplished without the appe~iarace of shocks (we suppose that the angle between the direction of flow and the xaxis and the derivative of this angle with respect to x aire both infinitely small), formua (29) may be employed in the calculation of the pressure on this contour. Comparing formls (29) and (106~) and denoting by Cpstoss the pressure resulting fran the presence of the shock front, we obtain NACA TM4 1394 CPstoss = q aldki(O) +a2d k l(0) + a~di5(0)8k(x) + aggppi(0)p,'(o)x + eg (108) We mnay, in turn single out of the expression for aCpstoss the term depending solely on the vorticity caused by the presence of the shock. In order to do this we add to the contour K(OC2I) a straightline segment tangent at the point 0 (segment O'O in fig. 9). With the contour O'OC2 a shock is formed, but the shock line O'Cl is straight so that vortex formation is absent. Calculati~ng the pressure on the portion OC2 of the contour O'OC2, we obtain P. = 9 + q algg(x) + appk2(x) + a pk5(x) + aqpk (x) + a.1dPk~(0) + a2d k(0) + a 4pk5(0)Pk(x ) + 6 (109) Comparing formulas (106) and (109) and denoting by Cprot the pressure due to vortex formation caused by the shock, we obtain rot = g~aggk (0)Pk'(0)x + E5 (110) PART III We now. apply the results obtained to the calculation of the lifting force and head resistance of a flat wing with sharp front and rear edges placed in a supersonic stream having constant hydrodynamical elements. We place the origin 0 at the front edge of the wing and arrange the coordinate system so that the positive xaxis corresponlds to the direction of the velocity of the undisturbed flow and measure angles in the manner used heretofore. Segment OC2 connecting the front and rear edges (fig. 10) will be called the chord of the wing as in the theory of wihgs. The length of this curve will be denoted by T and the angle it makes with the xaxis by P. NACA TMu 13596 The form of the~ wings we are investigating is dlefined by a pair of contours like that investigated in the preceding section, possessing a pair of common points O, C2. Comparing ordinates of points on these contours having the same abscissa, we call the upper contour Ku that contour of which every point on the ordinate is greater t'han the corre sponding point on the ordinate of th other contour, Adhell we call the lower contour K2. The function Pk(x) for the upper contour we denote by Pku(x) and for the lower by 13kl(x). We choose an arbitrary point A on the chord of the wing and denote the distance OA by t. Through A we pass a straight line perpendicular to the chord of the wing and denote by An and At, respectively, the intersections of this straight line wit the~ upper and lower contours. With the point Ag we associate a unit tangenrt vtector ti, and at the point At a unit tangenrt vector t'l Thne vectors tu and t2 will be directed in such a manner that their projections on the direction OC2 are positive. We denote by Bu and P3, respectively, thne angles these vectors makre with the vctor OC2. Clearly pu, Pt may be regarded as functions of t. We denote by BuO and B20 the vaue of Pu and Pl at the point 0 and the values of the derivatives of pu and St with respect to t at the point 0 by Bu0' and PZOr The abscissa x of Au may be calculated from the formula x = t cos p sin 5 tan pudt (111) and that of the point At from the formula x = t cos p ,sin tan Pgdt (112) The value of the functions Pku(x) at the point Aul is determined by the relation eku(x) = + pu NACA 1TM 1394 and the value of the function PkZ(x.) at the point A2 by the relation PkZ(x) = + pl P (114) Moreover we have the relations ST 0, tan Pudt = 0 (116) tan P~dt = 0 We alssulme that and also pU and pZ and respect to t are infinitesimal quantities. (116:) we easily obtain other derivatives with From equations (115) and +~ E~ 117) + E5 (118) SgTudt = dT~ u at Ot pgVdT I  Pzt =  Procreeding now to the calculation of the lifting force and head of the wing under consideration, we remark that on the top side wing6 a shock appears when and only when resistance of the S+ Puo > o (119) anld at the bottom side when and only when S+ P10 < O (120) Sk NACA TMl 1594 We introduce the quantities alu, a2us a5u, a4a defined as follows alu = &ld a2u = a2d if aju = a54 abu = agg alu = a2u = aju = abu = 0 (121) (122) In an analogous way we define' the quantities all, a223, aj~ &4, all = ald a;23 = a2d a32 = a3d a42 = aba if S+ BzO < 0 (125) if + B30 0 O (124) az2 = a23 = )3t = a42 = 0 Denoting by pu the pressure on the~ upper contour Ku and by p2 the pressure on the lower contour K2 we easily obtain, with the~ hel~p of formulas (106), (121), (122), (123), (124) Pu P 9 + 9 aliku(x) + a2 ku2(x) + a Pku(x) + aggku (x) + aluPku(0) + a2uiku ()+auk50pux b u()k() (12$) B + Bu0 > O if B + Pz0 < 0 HApCA, Tpr 1396 P2 PO + 4 ~ 1kl(x) + a2 32(x) a Pk;#(x) + a Pk1 (x) ~alhIS(0) + a2l3 k/(0) + a 7Pk35(0)P k3(x) + a 3pk3(0)p3 '(0)xll(~x + E (126) LetP denote the resultant vector of the hydrodynanical force acting on a unit length of the wing under consideration. We have then P =pads (127) where n denotes a unit vector normal to the contour of the wing and directed inwards. We introduce the dimensionless coefficient of the lifting force Cy and the dimensionless coefficient of head resistance Cx. These coeffi cients are defined by the formulas SPY cy 7 P, xx qT where Py and Px denote the projections of the vector P on the (128) (129) xr and yaxes, respectively. From formulas (127), (128), (129) we have CY = 1T s udx + r sop x (150) NACA TM! 1394 Pu tan Pk~u(x)dx S Cx pZ tan Pk3(x)dx (131) where 2 denotes the abscissa of the point C2. With the aid of formu and (131) after a few elementary transformations we obtain Cy = Cyl + Cy2 + Cy3 + Cy4 + 9 (1J2) where =   032 d up2 Cy2  altl@ + Pz30) + Cy5 = (a 2a7)8 aluf@+ puo) So (u2 + +1 al 9 ai r Bsu3 q5 8dt l(al Sa3) Cyk = a2ulf + Buo)4 + a22($+ p2o)4 a~u ( + Buo)3 + a uP(P + Bzo)5 _t~ ak u + Buo)p + a1(P + BZo)3P Zo' + as 6ay)( (B~u2 pl )dt + P as la4)~ ju 7 (BuZ BIS)dt  &T4JOT  1 adt u , NACA TM 1594 (133) Cx = Cx2 + ox} + Cx4 + 65 where SOT pu 312 at Cx2 = 2alp +  pl2 dt +OgT C,~ = cxb = 2a3 )T (u2 (u " 8 dt ~ p + aluB i~ uo)+ al P( + to)J + + a3 S~ 0 6a3 5  'I u + p at u3 Let us consider a numerical example. Suppose (134) k =1.405, Po = 1.o53~ kg/cm2 M = .5> Thein po'kM2 q 1.6~3) kg/cm2 al =2(M2 1)1/2 = 1789 a, = (M2 1) (2 ME + 1.20544) = 2.2196 5 a = (M2 1)7/2(1.533 BR + 4.oo8R~ 1.8194 + 0.4008M~8) = 5.082 ag = (M2 1)5(o.6667 o.6667M2 + 5.616M4 3.824M6 2.96948 0.786aglo + 0.079924I12) = 8.290 S(155) (Equations continued on next ~page). . 3d t + NACA UK 1594 ala = 1.203M4(42 17/2(0.5553 + 0.26584 0.o3271M4) = 0.2766 ad = M (M2 1)(5~1.203 + 1.51'7M2 0.645mbI + ook556M6 + 0.0485=J18) = 0.448 a~d = (g6~2 1)5(0.4oo8 0.oo25M2 + 0.386944C 0.1285M6) = 0.5318 akd = 0.3615M8(M2 1)5(1 + 0.7975M2 0.09813M4) = 0.9035 Let us take as the functions Pus St Bu = 28 +  llt T Bz = o (136) Moreover we assume that p < (137) The form and position of the profile~ of thre wing, determined by equa tions (136) and condition (157), is shown in figure 11. It is easily seen that the straight line S182 drawn perpendicular to the wing through its mid point is the axis of symmetry of the profile under consideration. From equations (136) and condition (137) we have S+ Bu0 = P > 0 S+ Bzo = B< 0 (138) 135~ NJACA TM~ 15394 LConsequently, alu = all =: ald = 0.2766 a2u = "23 = a2d = 0.4448 a~u = ap2 = azd = 0.5318 aku = an1 = age = 0.9035 (155), (135), (156), and (139) we obtain 42 2 i 1 6aj)8 + (139) Using formulas (152), Cy = 2a~l (lo 56  as2 s+25d+2 + E5 = 3.5788 3.061p2 14.327 82.7381 + > 10~91~ 26 3 al 6 4 = 5.965P + 9.1845p3 + 40.8p~ + .. 36 T = 100 em (14c) (141) NACA TM~ 1394 Then c, = 0.2936 Ox = 0.04168 12 Knowing Cy, Cx, T, and q, we easily obtain Py = qTCy = 47.9 kg/cm 13 Pg = qTCx = 6;.81 kg/cn In the~ present work the problem of a flow of stream of ideal gas around a thin wing at small angles of attack is investigated, this stream being supposed to be twodimensionatl, stationary, supersonic and deprived of heatcon~duction. In the initial part of the work, the problem is stated, and the well known results obtained by Ackeret, Prandt1, and Busernann are cited. These results, as known, are obtained on the basis of the potential supersonic streams theory, which is founded on the existence of integrable combina tions of charracteristics of differential equations concerning this problem, and in which some peculiarities of the dynamical conditions on the line of the shock wave are utilized. In, the second part the approximate solution of the problem is given with an allowance for vortexformation caused by the change of entropy along the shock wave, when receding from the leading edge of the wing, near which this shock wave is formed. For this purpose differential equations of characteristics non admitting integrable combinations are to be dealt with. The solution is obtained by means of' a special method, which enables us to find the approximate int;grable co~mbination~s of dif ferential equations of the characteristics. Thie ob~tained combinations let us receive the approximate formula of pressure in any point of' the contour of the~ wing investigated. From this formla the term is easily segregated depending exclusively on the vortex formuilation, caused by NACA TM 1J94 the change of entropy along the shock wave. The characteristic dis tinction of this term of the obtained formula of pressure from. the other ones, is that it includes the curvature of the wing contour at the leading edge and the distance from this edge up to the element of the wing for which the pressure is calculated. In the third part of the work the expressions for lift and drag coefficients of the wing are given, on the base of the formula of pres sure obtained above. In conclusion a numerical example is studied. TraLnslated by R. Shaw Institute of Mathematical Sciences REFERENCES 1. Meyer, Th.:l Uber zweidimensionale Bevegungsvorg~inge in einem Gas, das mit Uberschallge schwindigkeit strijmt. Forsch. Arb., Ing.Wes., 62, 1908. 2. Prandt1, L., and Busemann, A.: Nabrungsverfahren sur zeichnerischen Ermittlung von ebenen Strijmungen mit ijbers challge schwindigkeiten. Stodola Festschrift, Ziirich 1929. 3. Ackeret, I.: Gasdynamik. B~and~bu~ch der Ph~ysik, 19, B3. VII. 4. Busemann, A.: Aerodynamischer Auftrieb bei tjberschallge schwindigkeit. Luftfabrtforschung, 1355. Ph NJACA TM 1394 Figure? 1. C MI I m T Ol~" nn1 NIACA TM 15394 Figure 2(a). Figure 2(b). NACA ITM 1394 Figure 3. Figure 4. / \ c' NJACA TM 1594 Figure 5. Figure 6. 46 NACA TM 1394 V CI F C2 L. x 0 Figure 7. y Cl / C2 Fig are 8. NrACA TM 139 4 CI O x O'O Figure 9. W It AuU LL 1 I LX Figure 10. NACA TM 1394 X w SI /S2L Figure 11. 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