it^'l3^^ h. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1394 A FLAT WING WITH SHARP EDGES IN A SUPERSONIC STREAM By A. E. Donov Translation of Izvestiia-Akademia, NAUK, USSR, 1939. -.ITV OF FLORlDA_ ^ ■ARY V!LLE,''FL32611-70\tUSA Washington March 1956 lc:;-(,^^^'>^'\ -ytiHo^fi NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1594 A FLAT WING WITH SHARP EDGES IN A SUPERSONIC STREAM* By A. E. Donov In this work there is given aji approximate solution of the problem of a two-dimensional 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 immovable wing appears as a special case of the some- what more general problem of the investigation of the mechanical action of a moving gas on an immovable fixed wall constraining the motion of the gas. In our own explanation we begin with the formulation of this last problem in which we confine ourselves only to the consideration of the steady two-dimensional forces of ideal gases not subject to the action of gravitational forces. In the plane of motion of the gas we shall arrange an immovable rectangular coordinate system in such a manner 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 functions v will be determined by a pair of scalar functions of the independent variables. For these func- 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 x-axis, measured in the coiinterclockwise sense. In what follows we limit ourselves to the consideration only of flows for which the function p satisfies the condition 1L < B < i (1) *Izvestiia-Akademia, NAUK, USSR, 1959^ PP 603-626. NACA TM 139^ As Is well known, the study of the gas motion under consideration leads to the investigation of the following system of differential equations Svv Sv^ 'X ^''"x 1 3p V,, : + V,r + = ay Sv- y dy p bx y + V,; Svy 1 ^p + _ ri: = bx Sy P 3y S(PVX) S(pVy) bx AM axlpk Sy + V, ^\ = (2) Here k is the adiabatic exponent (for air k = l.iK)5). If the motion of the gas is constrained by an immovable frictionless fixed wall in the plane XOY, the gas will be adjacent to it along some curve. We shall call this curve the "contour K." — > Consider the unit vector t tangent to the contour K directed in such a manner that its projection on the x-axis is positive. Denote ^y Pk '^^^ angle which it makes with the x-axis. Clearly Pj^ 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 Pk(x) and assimie that it is continuous. If the function Pic(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 fonii tan |3i^(x)dx (5) NACA TM 139^ We can write this equation more briefly if we designate its right-hand side by y^{x) y = yk(x) i^) Since in the flows under consideration the direction of the velocity on the contoirr K must coincide with the vector t, the condition on the flow along an immovable fixed frictionless wall may be written in the following fashion P = Pk(x) (5) a-"t y = yk(^) • '^^^ condition (5) must be added to the system of equa- tions (2) as a qualifying boundary condition. Much work has been dedicated to the investigation of solutions of the system (2) subject to the con- dition (5) • Of these we are interested here only in those in which the flow is supersonic, condition i.e., flows at every point of which the following V > a (6) is satisfied, where a is the local speed of sound (7) 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 step-by-step calculation of a system of parti- cular values of the desired functions. (Works of Busemann, Kibelia, and Frankl.) The fundamental achievements of the methods represented by these works consist of the fact that by their use many actual practical problem.s may be solved quantitatively of which the solution by other methods would present great difficulties. In particular these methods solve thoroughly corner-nonpotential 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 NACA TM 139^ 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 axe based on the fact that in the case where vorticity is absent the character- istic system of differential equations (2) admit of integrable combina- tions. This theory leads to series of approximate results of any desired accuracy, giving a complete qualitative and quantitative pictiare of the flow. 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 theory. We introduce the stream function i|/ defined by the following relations Sx r^ = pv^ (8) = -P^x As is well known from equations (2), (7), and (8) the following relations follow without difficulty A= e (9) — + — 2 k (10) Where 9, tQ denote quantities which display the flow once and for all as a fimction only of ilf. With the help of eqimtions (2), (9), and (lO) it is easy to obtain the two equations SVy bvy^ dx by = a (11) NACA TM 139^ (a2 - V, ^2)^ , (,2 _ ,^2)^ _ ., J^ , ^ . (12) dx Sy ^dx Sy y where Q, denotes a quantity defined as a = p ^*0 a^ d In e dt k(k - 1) di (15) Equations (ll) and (l2) represent linear relationships between the first partial derivatives of the functions v^, Vy with respect to x and y. Since every flow iinder 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 (ll) and (l2) . For one family of characteristics, which we shall agree to call the first family, we obtain the equations dy = m-Lix (l^^) (a^ _ v2cos2p)in-i + v^sin p cos |3 f-.c\ d(v cos p) + m2d(v sin p) = fl dx ^■'-^^ v2cos2p - a2 and for the other, which we shall agree to call the second family, we have the equations dy = m2dx (l6) , , , , (a2 - v2cos2B)mo + v2sln p cos p , d(v cos p) + m-Ld(v sin p) = a — -^ dx (17) v2cos2p - a2 Here m]_, m2 denote the following expressions -v2sin p cos p + a\/v2 - a2 mi = ^! (18) a2 _ v2cos2p NACA TM 139^ -v2sln B cos R - a\/v2 - a2 mg = ^ (19) We now consider a supersonic stream with constant hydrodynamical elements (i.e., functions v, p, p, p, a). We shall call this flow the undisturbed flow. The values of the functions v, p, p, a in the undis- turbed stream will be denoted by w, pQ, Pq, oq respectively, and the ratio w/ao by M. Since the stream under consideration is super- sonic, M > 1. We shall choose the direction of the velocity of the undistixrbed stream to correspond, to the direction of the x-axis. We assume that the undisturbed stream strikes an immovable, fixed, frlctionless wall (contour K) , inclined in such a manner that in flowing around this wall the stream never detaches from it and remains super- sonic everywhere. We may distinguish two cases of flows of this type. Case I .- The contour K is situated in such a manner that the condition Pk(0) ^ (20) is fulfilled. . In this case, as is well known, there appears a curve of weak discontinuity OC (figs. 2(a), and 2(b)) proceeding from the origin and dividing the entire flow in two parts. On one side of the curve of weak discontinuity OC extends the region containing the undisturbed stream and on the other the region of flow around the wall. In the region of flow around the fixed frlctionless wall the hydrodynamical elements of the stream, generally speaking, are not constant but vary. In what fol- lows we shall call this part of the stream the dist\:irbed stream. In the entire region of the flow under 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 family since the hydrodynamical elements of the stream are con- stant. On this line the following relationships will hold in the entire region containing the stream to = ^ . J2l (21) 2 1: - 1 NACA TM 139^ 7 9 = Go (22) where Sg denotes a quantity defined as PO^ (23) Frorn equations (l3)j (21), and (22) we easily obtain ft - (24) i.e., the flow under consideration is irrotational. By virtue of rela- tion (2^+) the right-hand side of equations (15) and (I7) vanish and these equations can 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 of integrating equation (17) we have P - cp(v) = constant (26) satisfied along a characteristic of the second family. cp(v) denotes a function defined as cp(v) = /^^ arc tan,/^Il \[ZI^ _ arc tan VZ^ (27) - k + 1 a ^ Since on the curve of weak discontinuity OC the quantities v and p have the values w and 0, 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 TM I59I1 p +cp(v) = cp(w) (28) From equations 16, 26, and 28 it immediately follows that the char- acteristics of the second family (the curve OC being among these) are straight lines since along each of these characteristics the hydrodynam- 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? aJ^d. 5 "to construct expressions for the functions v, p, p, p, a in the region of the disturbed flow. However, the 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 velocity v as a function of the angle of inclination of this velocity with the x-axis 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 X-axis is a given fimction of x on the contour K there is the possibility of using equations 22, 21, 10, 3, 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 hydro- 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 a-Lpi^(x) + a.2_^^{^) + aj^^^U) + aj^Pj^^(x) + . . . (29) NACA TM 139^ where Pqw2 pQkM'^ ai= 2(m2 - l)-l/2 a2 = (m2 - l)-2^2 - m^ + ^^^^ M^) aj = (m2 - 1) ■7/s 5 5 6 6 ai, = (m2 - l)-5fl _ 2m2 + Li^M^ ^ ^2^^_^51lJiJ:M m^ + \5 5 6 12 1^ + 20k - 8k2 + 5k3 „8 ^ -21 - 20k + 5k2 + 2k5 ^iq ^ 5 + 2k - k2 ^ig M" + 12 ^8 48 M-i Case II .- The flow around a contour K is situated in such a manner that the following inequality is satisfied Pk(0) > (30) In this case, as is well known, a line of shock discontinuity OD appears (fig. 3) proceeding from the origin and dividing the entire flow under consideration in two parts. On one side of this line is the region of the undisturbed 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, p, p, p, a exhibit jump discontinuities on the shock-line OD. 10 NACA TM 159^ In the region of the distiirbed flow these fionctions must satisfy not only equations Z, 5> sxid^ 7 but also the dynamical conditions across the shock line. Considering the flow to be only slightly dist\;irbed, these conditions may be written in the following form 2k-12k-l V = w(l + biP + b2p2 + bjp? + bi^p^ + . . .) (52) where 1 bi = -(m2 - 1) 2 b2 = -(M2-l)-2^i.^M^) b. = -(m2 1)- i 1 + 1 m2 + 5(k _ D^i^ ^ 3k2 - 12k 4- ^ ^ ^ (k+l)^ m8 ^ 624 24 52 bi, = . (m2 - 1) -5 i + 2 m2 + liL±i2^ M^ + -1 - 27k -. ±2^- ^ ^ \2\ Q 2k 2k ^ + ^k - k2 + k? ^8 ^ -^ - k - 3k2 + 3k3 ^10] 16 1+8 y ^ = 1 + Zjp? + z^p^ + . . . (53) ^0 NACA TM 139^ 11 where Z3=^(^il_i).M6(M2- l)-t 12 I. = ^^^^ - ^^ m^(m2 _ D-^L + 2(k - 2)m2 - (k - 1)M^1 12 •- -^ dx where -^ = eo + e^p + egP^ + . . . (5^) 1 eo = (m2 - 1)" 2 e-i = M^(m2 - 1)' Condition 3I shows that, disregaarding the presence of jump discon- tinuities in the fimctions 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. However, condition 22 is not, generally speaking, fulfilled in the case now \inder consideration. However, there is the possibility of speaking of satisfying this condition approximately. In fact, consider equation 35- Its right-hand side does not contain terms in the first and second powers of p. 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 2h may be regarded as approximately satisfied, which means that 12 NACA TM 159^ equations 25 and 26 hold on characteristics. For values of p and v near and w, respectively, equation 28 may be written in the form of a series V - w(l + bi'p + b2'p2 + b.'p^ + b.'p^ + . . .) (35) where b-]_' = -(m2 - 1)" 2 = b-j_ b2 ' = -(m2-1)-2(|.^M^) b,' . -(Mg - l)-^f-J- ,. 2 M^ „ -IT -H 29k ^ ^ 3 - 19k + I6k2 ^ _^ ^ \24 8 24 24 3-2k-5k2 + 4k5 8 -3 + 8k - 7k2 + 2k5 ^A M + M I 32 96 / 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 CD. Since, on the other hand, along each characteristic of the first family equation 25 is approximately satisfied, equation 28 will be approximately satisfied throughout the entire region of disturbed flow. The approximate expressions for the functions v, p, 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 p in the right-hand side of equation 3^, we obtain NACA TM 139^ 15 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 pressiore 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 caiised to appear in the region of flow around the 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, but 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 the 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 circiiLation it is impossible to integrate equations I5 and I7, there is the possibility, however, of making up such combinations of differentials from eq\iations ik, 15^ 16, and I7, adding to these equations expressions for differentials of the stream 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 preceeding section. Assume that in this flow the hydrodynamical 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 ciorvelinear triangle made up of the curve OC2 (contour K), the shock line 00^, and the characteristic of the first family C]_C2 emerging from the lowest point of the contour K (fig- k) . Taking into consideration eqiiations 5, ik, 18, and 34, it is not difficult to conclude that with the assimiptions made j\:ist now relative to the hydrodynamical elements the ciurvelineajr triangle OC1C2 differs infinitely little from the isoceles straight-line triangle 0'Ci'C2' (fig- 5) where the equal sides O'Ci' and Ci'C2' are parallel to characteristics of the second and first families in the Ik NACA TN 1394 londlsturbed flow. As for the functions |3^(x), p, 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 insiire legitimacy of operations which are performed upon them. Moreover, we assume that in the flow under consideration the infinitesimal quantities Pj^(x), 3r^'(x), Pj^"(x), P, V - w, p - Pq, p - P(^, a - a„ have the same order of magnitude. Taking this last group of infinitesimals as fundamental (having ■'onit 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 sajne infinitesimal may be denoted by several different symbols. Thus, for example, if an infinitesimal a is denoted by ei^, the infinites- imal 2a may also be denoted by el^., and, moreover, the infinitesi- mals a and 2a may be denoted by e^, £2' ^l* On an arbitrary characteristic of the second or first family the equation d\^ = pv(sin pdx - cos pdy) (56) will be satisfied by virtue of equation 8 throughout the entire region filled by the flow. Eliminating dx and dy from equations ik, 1^, and 36 and taking into accoimt formulas 13 and 21, we arrive at the equation d(v cos p) + m2d(v sin p) - ^jd. In 9 (37) which is satisfied on any characteristic of the first family. Here $2. denotes the quantity a2 v2sin p cos p - (v^cosSp - a2)m-, 03_ = 1* (38) k(k - l)v(v2cos2p - a^)(m^ cos p - sin p) NACA TM 159^ 15 On the other hand, having the integral 25 of the equation d(v cos p) + m2d(v sin p) = (59) it is easy to find an integrating factor Lj^ of this equation, such that after multiplying by L^ it may be written in the form d[p + cp(v)] = (40) In order to determine In we have the obvious relationship L3_rd(v cos p) + m2d(v sin p) [ = dm + q)(v)| (4l) from which we obtain without difficulty Li(m2 cos p - sin p)vdp = d^ {k2) consequently Li = ± (43) v(m2 cos p - sin p) If now we multiply both sides of equation 57 by L^, this equation takes the form d[p + cp(v)] = H^d In 9 (hh) where H]_ denotes the quantity (v2cos2p _ a2)m-| - v2sin p cos p % = (45) k(k - l)v2 l6 ■ NACA TM 1394 We denote by Ejq the value of E2_ at v = w, p = 0. We have k(k - 1)m2 Equation kk may be rearranged In the following fashion dip + cp(v)] = HiQti In — + (H-L - Hio)d In — (1+7) ^0 ^0 Now 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 with the shock line by A (fig. 6). Integrating both sides of equation k'J along the above characteristic from point A to point S we obtain Ps + 9(vs) - Pa - ^(^a) = Hiolln -i - Iji -^ + / (Hi - Hio)d in — ®0 ^0/ ^AS °0 (hQ) where Pg, Vg, Q^ denote, respectively, the values of p, v, 9 at the point S and Pg^, Vg^, 69, denote the values of these quantities at the point A. Taking account of equation (32) we have Va = w(l + biPa + bgPa^ + bjpa^ + b^Pg^ + . . .) We introduce the quantity Vgi defined with the help of the expansion 35 in the following fashion Vai = w(l + biPa + b2pa^ + ^^'^a^ + H'Pa^ + • • •) (cf eq. (35) -Tr.) (50) NACA TM 139^ 17 By this definition of the q\aantity Yqj_ we have Pa + 9(vai) = cpCw) (51) With the help of formulas k^, ^, and 51 we rearrange the expression |3a. + Cp(va) in the following manner Pa + 9(va) = Pa + ^C^ai) + 9(va) - 9(vai) cp(w) + cp(va) - cp(val) cp(w) + cp'(w)|_(va - w) - (v^i - w)J icp"(v)[(va - v)2 - (v^-L - w)^] + . = 9(w) + wp' (w) (b5 - b3')|3a^ + (bi^ - b4')Pa^ w^"(w)bi(b3 - b5')Pa^ + €5 (52) Calcvilating cp ' (w) , aJ^d. ^k we easily find Pa + T(va) = 9(w) (bj - b^')^a^ + 2b2 1 -^(b3 - bj') - i(bi, - bi,') Pp^ + e. ^a (55) 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 (56) where 9p denotes the value of 6 at the point p. Taking logarithms of both sides of equation (33) we obtain In A= z^p3 + Zj^'p^ + 2^'p5 + (57) Since the values of the coefficients ll^\ ^'^' > • • • in what follows, we shall not calculate them. Using formulas (56) and (57) we easily see that will not be needed in ^ - nji ^ = Z3(Pp5 _ p^3) + zi,'(3p^ - Pa^) + £5 ^0 % (58) where p 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 (kS) , we easily find-L 4nstead, take a slightly more general assimiption admitting the part AS of the characteristic under consideration to be divided in the same finite n\:mber of parts in such manner that on each part the mean value theorem can be applied to the integral under investigation. NACA TM 139^ 19 / (Hi - Hio)d In — = l3(Hi - Hio)(Pp5 - 133^3) + e (59) AS ^0 ^ J (In consequence of this equation one must keep in mind that H-[_ - H^q = ^i) Here H]_ denotes the value of H]_ at some point on the characteristic under consideration betveen the points A and S. Using relations (55) > (58) j sJ^d (59) we write equation (48) in the following form Ps + 9(vs) = cp(w) - -^(bj - b3')Pg^5 + 2b2 (bj - bj') - -i-(bi^ - bi^') Pa^ + Hlob^Pp^ - Pa^) + %o^l|'(Pp^ - Pa^) ^3(Hl - Hio)(pp^ - 3a^) + £5 (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 + ^(^b) = ^(w) - —(1=5 - b5')Pa^ + 2b. ^:(b3 - bj') - _(bi, - bi,') Pa + Hlob^Po^ - Pa^) + HioH'(Po^ - Pa^) ^5(Hi - Hio)(Po5 - Pa^) + ^5 (61) 20 NACA TM 159^ where p^, v^ denote, respectively, the values of p and v at the point B and Po denotes the value of p at the point 0. We now proceed to the derivation of an expression for pg^. From fonavila (6o) we have From equation (62), using formula (35) we obtain Vs = w(l + biPs + b2ps^) + ej (65) and denoting by m2g the value of m2 at the point S we obtain, by using formulas (19) and (65) in23 = (m2 - l)-^/2 ^ k+^ ^(^2 _ i)-2 p^ ^ ^2 = eo + 2eips + eg (64) Analagous to the derivation of equation (k"]) , which holds on character- istics of the first family, we may derive equation P - cp(v) = H2oi In A + (H2 - H2o)d In |- (65) 90 90 which is valid on characteristics of the second family. Here Hg denotes the function defined as (v2cos2p - a2)m2 - v2sin p cos p H2 = {^^) k(k - l)v2 and H20 denotes the value of this function at p = 0, v = w. NACA TIA 1594 21 Now pass a characteristic of the second family throi;igh the point S and denote by Q its intersection with the contoirr K. Integrating both sides of equation (65) along; this characteristic from the point Q to the point S we obtain cp(vs) - c?(Vq) H20|ln^ -€- QS (H2 - H20)ci In — (6?) where p , v , 9 denote respective the values of p, v, M. M. SL point Q. Since the contour K is a stream line we have 9 at the ,(0) (68) where 9^^ denotes the value of 9 at 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), (57), and (68) Ps - 3q cp(vs) - cp(vq) (69) Applying formula (62) at the point Q we have Pq + ^(V " ^^^) + ^3 (70) Eliminating cp(w) from equations (62) and (70) we arrive at the fol- lowing equation Ps - 3q + 9(vs) - 9(Vn) = e; (71) 22 WACA TM 159^ Families (69) and (7I) give Ps = Pq + £5 (72) On the shock line we take an arbitrary point F (fig. 7) a^nd pass through it a characteristic of the second family in the region of the disturbed flow and we denote by p^., ni2f , respectively, the values p and m2 at the point F. Applying formula {6h) at the point F we obtain m2f = eg + 2e^3f + eg (75) We denote by — : the slope of the tangent to the shock line at the \ dx/j point F. From equations (3^) we have t ^ = eo + e^Pf + £2 (7^) Comparing formulas (75) and (7^) we see that the characteristic of the second family passing through F and the shock line at this intersection make an infinitesimal angle with each other moreover, if Pf > (75) the slope of the characteristic of the second family is greater than the slope of the shock line at the point F. o '-It is easy to show that if the shock line is unbroken and moreover condition (50) is satisfied the inequality p < is impossible on this line. As a matter of fact, in the opposite case the shock line is broken since with p < condition (3^) must be replaced by the following con- dition in virtue of Tsemplen's theorem g. -(eo - eiP + e2p2 - • • •) NACA Tt4 139^ 25 Denoting by L the intersection of the characteristic of the second family under consideration with the contour K and by x, abscissa of this point, we have Let <^i denote the value of p at the point L. Using equations (5) and (76) and MacLauren's formula we obtain pj = Pl,(0) + Pk'(0)_xj + €^ (77) Applying formula (72) at the point F we obtain Pf = Pi + £3 (78) As a consequence of equations (77) and (78) pf = pi^(O) + Pk'(0)xj + £3 (79) Since the point F was chosen arbitrarily on the shock line by use of equations (7^) j il^) } and (79) we can obtain the following differential equation for the shock line ^ = eo + eiP]^(0) + e2 (80) dx Consequently the equation of the shock line may be written in the fol- lowing form y = Bq + e^p^(O) X + eg (81) 2k NACA W. I59U Applying formulas {6k) and (72) to an arbitrary point situated on the characteristic of the second family FL we easily obtain the differential equation of this line from the following form dy — = eo + 2ejp^ + eg dx (82) Employing formulas (76) and (77) this equation may be written ^= Bq + 2e-Lpk(0) + €2 (85) Consequently the equation of the characteristic FL -may be written in the form r y ^ yi + [Sq + 2ei|3i5.(0) (x - xj) + eg (8i^) where y^ denotes the ordinate of the point L. On the other hand, taking account of formulas (5) and (76) we have yj - / tan Pk(x)dx = e^ ^ (85) Employing fonnulas (85) and (76) we may write equation (8^) in the form eo + 2eiPk(0) x - BqXi + eg (86) Applying foiTiulas (81) and (86) at the point F and denoting by Xf, y|. the coordinates of this point we obtain NACA TM 139h 25 yf r = Sq + ei|3k(0) Xf + £2 r 1 yf = I So + 2e^Pj^(0) |Xf - eQX^ + eg L J (87) From equation (87) we easily obtain 1 XX = — Xfpi^(o) + £2 ^0 (88) Replacing x^ in the right-hand side of equation (79) by the expression in formula (88) we obtain pf = Pl,(0) + ^Xfpj,(0)pi^'(0) + e, ^0 (89) We denote by Xg., yg, -"the coordinates of the point A and by x^, y^^ the coordinates of the point B. Applying formula (89) at the point A we arrive at the following result Pa = Pk(0) + — XaPk(0)pk'(0) + e. ^0 (90) We now express x^ in terms of x^. To this end, using formulas (ik) and (18), we write the differential equation for char-acteristics of the second family in the following fashion dy — = -eo + ei dx (91) 26 NACA TM 1594 Employing formula (9I) we write the equation for the characteristic AB of the first family in the form y = Yh - ®o(x - Xb) + e^ (92) Taking account of formula (?) we have y^ = / tan Pi^(x)±>c = e-^ consequently equation (92) may be written y = -eo(x - x^^) + e-^ (95) (94) On the other hand, equation (8I) for the shock line may be written in the form y = eo X + ei and applying formulas (9^) and (95) at "bhe point A we obtain ya = -eo(xa " ^b) + ^i ya = SQXa + e^ From equation (96) we easily find (95) (96) ^a = — + ^1 (97) NACA TO 139^^- 27 And consequently (98) Substituting this expression for Pg^ in the right-hand side of equa- tion (6l) and substituting pg f°^ Pk^*^) -^^ ^^^ fundamental formula (5) we obtain Pb + 9(vb) = cp(w) - — (b5 - b3')Pi^3(0) - 1 2b2 — (bl+ - bi^') J(b3 - bj') bl b,2 Pk'(o) 5e, 2er b^ - bj' + Hio^J XbPk^(0)pk'(0) + 65 (99) Employing relation (35) j we easily obtain from equation (99) r 1 + biPb + bgpb^ + bj'Pb^ + b^'p^^^ + (bj - b3')Pj^3(o) v^ = w 2b. b^ - bi,' ^(bj - b3') k 2bo :z: Pk (0) + -^(bj - b3')Pj^^(0)pb + 3e. 2eoL bj - b^' + Hio^3bi 1 >^bPk (o)Pk'(o) + e. (100) 28 NACA TK 159^ Substituting v^, p^-,, xt, for v, Pk(^) ^^'^ ^> respectively, in formula (lOO) we arrive at the following final expression for the velocity on the contour K: V = w 1 + bipj^(x) + b2Pi,^(x) + b3'pk^(x) + bi^'Pk^(x) + (bj - b3')Pj^5(o) + 2b2 bi^ - b^' - — (bj - bj') Pk (0) + ^^3 - b5')Pk5(0)Pi,(x) + i -^Ibj - b.' + Hio^jbi xpk5(0)Pi,'(0) + £c (101) We now proceed to the derivation of formulas from which the pressure on the contour K can be calculated. Clearly 2 v^O ^ PO (102) and moreover, on the contour K the following equation holds 1.= 9(0) (103) Employing formulas (7), (lO), (21), (23), (102), and (103) we easily obtain the following expression for the pressure on the contour K P = PO 1 k ~e(o)" k-l - ■n k-l 1 - 2 ^w2 1) / Qo (104) NACA TM 159^ 29 On the other hand, by virtue of eqimtions (5) and (33) the following equation holds j(0) = 1 + l^^^{0) + lk^^{0) + . . (105) e(o) , , Substituting the expressions for v and obtained in formulas (101) 90 and (105)^ respectively, in the right-hand side of equation (l04) we obtain after elementary transformations the desired formula for the calculation of the pressure on the contour K p = PO + q aipj^(x) + a2Pk^(x) + a^^i^^U) + aj^pj^^(x) + a-|_^Pj^5(o) a2dPk^(0) + a5(iPj^5(0)pj.(x) + ai^^Pj^3(o)p^'(o)xJ + €5 (106) where aid - -2(b3 - bj') - 21-. k(k - 1)m2 i±l mVm2 - 1,- i(- 1 . 1^ „2 . 2l^ „*) 4b c 21 ^2d = -2(bl^ - b4') + -^(b5 - bj') k '± k(k - 1)m' = M^(m2 - ±)-^- i±l + ^ -^ ^^ - ^' m2 + -10 - 3k 4- 6k2 - k3 ^^ _^ I 2 4 8 9 - 7k^ + 2k5 ^ _3 + k + 3k2 - k^ M^ 16 32 50 NACA TM 159^ ^5d - (^3 - >^3') (2M2bi - 2bi - -^1 + -— - 6 24 + -U + 3k + 6k2 - k3 K 5 - 7k - 7k2 + 3k3 ^ 2ii 96 3e ®0 i^ll^M8(M2-l)-5/_ 1 + 1^1 m2 + 5k^l M^^ For X = the fonnula (I06) takes the form P = PO + qklPk(O) + a2Pk^(0) + aj'(3j^3(o) + ai^'Pi^^(o) + 65 (IO7) where ax' = a^ + a-, ,, a]^' = a^ + a^. + a,.. Formula (107) may be used for the calculation of the pressure on a flat plate vhich is inclined at an angle Pv(0) to the undisturbed flow. In order to single out of the right-hand side of equation (106) those terms 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'O of finite length in such a manner that this arc is tangent to K at the point and is parallel to the x-axis at O' (fig. 8). Since the flow around such an additional contour is accomplished without the appearance of shocks (we suppose that the angle between the direction of flow and the X-axis and the derivative of this angle with respect to x are both infinitely small), formula (29) may be employed in the calculation of the pressixre on this contour. Comparing formulas (29) and (106) and denoting by '^stoss ^^^ pressure resulting from the presence of the shock front, we obtain i I NACA ™ 159ij- 51 ai+(0)x + €5 ' (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 at the front edge of the wing and arrange the coordinate system so that the positive x-axis corresponds 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 wings. The length of this curve will be denoted by T and the angle it makes with the x-axis by p. 52 NACA TM 15yi+ The form of the wings we are investigating is defined by a pair of ; contours like that investigated in the preceding section, possessing a i pair of common points 0, C2« Comparing ordinates of points on these contoiors having the same abscissa, we call the iipper contoior K^ that contoiir of which every point on the ordinate is greater than the corre- i sponding point on the ordinate of the other contoiur, which we call the lower contour K^. The function pjj.(x) for the upper contour we denote by Pku(^) ^^^ ^°^ '^^^ lower by Pj^-jCx). I 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 A^ emd Aj, respectively, the intersections of this straight line with the upper and lower contours. With the point A^ we associate a imit tangent vector t^ and at the point At a unit tangent vector t^. The vectors t^ and tj will be directed in such a manner that their projections on the direction OC2 are positive. We denote by p^ and p^, respectively, the angles these vectors make with the vector OC2. Clearly p^, p^ may be regarded as functions of t. ^M We denote by P^q and p,^ the values of p and Pj at the point and the values of the derivatives of P-^ and p^ with respect to t at the point by p^g' ^^^ ^Zo'' The abscissa x of A^ may be calculated from the formula t cos p - sin p / tan p^dt (ill) -' and that of the point A^ from the formula X = t cos p - sin p / tan p^dt (ll2) ^ The value of the functions Pjj^^(x) at the point A^ is detennined by the relation Pl,^(x) = P + P^ (113) I I NACA TM 139^ 35 and the value of the function Pij;j(x) at the point Aj by the relation Pkl(x) = P + Px (114) Moreover we have the relations pT / tan p^dt = (115) / tan p^dt = (ll6) We asstune that p and also P-^ and p^ and other derivatives with respect to t are infinitesimal quantities. From equations (ll5) aJ^d. (ll6) we easily obtain / p^dt = - I / Pu^it + £5 (117) /"T , pT 3 ^0 1 r. 3, / p^dt :. - i p^^dt + e (118) Proceeding now to the calculation of the lifting force and head resistance of the wing under consideration, we remark that on the top side of the wing a shock appears when and only when P + PuO > ^119) and at the bottom side when and only when P + PZO < (120) ^k NACA TM 159^ We introduce the quantities a3_^^ a2uj 3.-^-a> ^l+u defined as follows ^lu a2u ^Id a2d if p + 3uD > = a 3d (121) ^lu = a^ = aju = ^1+u = ° ^^ P + PlO - < (122) In an analogous way we define the quantities a-,,, Spij ^^jj a. 7 3-1.1 — ^Id ^21 = ^2d ^51 = ajd Hz = %d if P + P^Q < (123) ^11 = ^21 = ^31 = Hi = if p + p^o > (I2i+) Denoting by p^ the pressure on the upper contour K^ and by p^ the pressiire on the lower contour K^ we easily obtain, with the help of formulas (106), (121), (l22), (l23), (124) P^ = PO + q [aiPku(^) + a2Pku^(x) + a3Pi^^5(x) + a4Pj^^^(x) + aiuPku^(O) + ^2uPku^(0) + a3uPku^(0)Pku(^) + HuPku^(0)Pku'(°)^] + ^5 (125) I + ^5 NACA TO 139^ 55 aiiPkl^(O) + ^2i^^i^{0) + a5^Pj^^5(o)p^^(x) + aj^^p^j5(o)Pj^^ ' (0)x] (126) — * Let P denote the resultant vector of the hydrodynaralcal force acting on a unit length of the wing under consideration. We have then ^=0pnds (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 C-^. These coeffi- cients are defined by the formulas P. Cy = ^ (128) qT Cx = ^ (129) where Py and P^^ denote the projections of the vector P on the X- and y-axes, respectively. From formulas (127), (l28), (l29) we have Cy = - — / p^d^ + — f H^ (15°) ^ qT ^0 qT Jo 56 NACA TM 159^ I . nl Cx - + qT J'o '" ■^"" ' ^T Jo p^ tan Pj^^(x)iic / p^ tan Pj^^(x)dx (131) where I denotes the abscissa of the point C2. With the aid of formu- las (111), (112), (113), (114), (117), (118), (125), (126), (130), and (131) after a few elementary transformations we obtain Cy = Cyl + Cy2 + CyJ + Cy^ + C^ (l32) where Vl = -2aiP V = - T Jo {'^' - '''y^ '; 3 3 Cy3 = (ai - 2a3)p - ai^(p + P^q) " ail(P + Pzo) + -(ai - 3a3)p J (^Pu^ + Pz^)dt + y^ - a^j / (^Pu^ + p^^jdt - h I4. _ _ X Cyi^ = -a2u(P + Puo) + a2i(P + Pio) - a3up(p + Puo)^ + a3ip(p + Pio)^ - I ai^u(P + Pu0)^Pu0' + | a^zCP + Pzo)^Pzo' + y| ag - 6aiJp2 J (Pu^ - Pz2)cLt + l|| s.^ - h) P / (Pu^ " P^^)^^ ^/:(-^--> I NACA TM 159^ 57 where Cx2 = 2a^|3 + 2.-1 r^ Cx = Cx2 + Cx5 + Cx4 + 65 Pu^ + Pz^Ut -x3 5-2P r^ /„ 2 „ 2\.. . ^2 r^ /„ 5 „ 31 -l\-^ Pu - Pi P* Cxl^ = 2a3 - ^ p + a3_^p(p + P^q) + ^ll^(^ + Pio)^ + (153) 2 . „ 2 1 Ip - tY^ Jo ^^- " 'I p^ ■*■ Tp - y ,^ + p.^ dt + ± 4a. - -^ p Pu^ + PZ^V* + Then Let us consider a numerical example. Suppose k = 1.^5, M = 1.5, Pq = 1.033 kg/cm^ (15^) q = = 1«d33 kg/cm-^^ ai = 2(m2 - l)-^/2 = 1.789 ag = (m2 - 1)"^(2 - 3^2 + 1.205^^) = 2.296 a^ = (m2 - 1) -7/2(1.533 _ ai2 + 4.ooa^^ - l.SlJM^ + I /. n ■^ > U35) o.i+ooa4°) = 3.082 ai^ = (m2 - 1) "5(0.6667 - o.6667m2 + 5.6i6m^ - 3-82iM^ + 2.965^8 - 0.78i4OM^0 + 0.079934^^) = 8.290 (Equations continued on next page) . 58 NACA TM 139^ aid = 1-203M^(m2 - 1)"'''' ^(-0.5353 + 0. 265^42 - 0.0327B1^) = O.2766 a2d. = M^(m2 - 1)"5(-1.203 + 1.317m2 - 0.6J+31M^ + Q.Qk'^'^Gv^ + O.0J+855M8) = 0AMf8 a^cL = M^(m2 - l)"5(_o. 1+008 - 0.0025m2 + 0.3869M^ - O.I285M6) = 0.3318 a]^ = 0.56i5m8(m2 - l)"5(-i + o.7975m2 - 0.098131^) = 0.9035 Let us take as the functions p^, p^ p = -2p + M t T Moreover ve assume that /'155 1 I cone . (136) P < (137) The form and position of the profile of the wing, determined by equa- tions (136) and condition (l37), is shown in figure 11. It is easily- seen that the straight line S1S2 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 3 + PuO = -P > p + p^O = P< (138) ( NACA TM 159^ Consequently, 39 aiu = ^11 = an^ = 0.2766 a^u = ^21 = a.26. = 0-^^8 ^3u = ^31 = ^3d "^ 0.3318 ^kvL =" ai^l = ai^^ = 0.9035 (139) Using formulas (132), (l33), (l35), (136), and (139) we obtain Cy = -2aiP - - a2P^ + {-^ ^1 - ^a^j p^ + ''lO 56 \ -h ^2 - =j ai^ + 2a^^ + 2a^l(3 + e^ -3.578P - 3-06lp - 14.32P - 82.73P + = 5-963P^ + 9.l8i^P^ + 4o.8p + . . . (ito) Let 36 T = 100 cm (1^1) i^O Then NACA TM 159^ I Cv = 0.2936 Cx = 0.0i|-l68 (142) Knowing Cy, Cx, T, and q, we easily obtain Py = qTCy = k7.9 kg/cm Px = qTCx =6.81 kg/cm ^1 (lh3) SUMMARY 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 two-dimensional, stationary, supersonic and deprived of heat-conduction. In the initial part of the work, the problem is stated, and the well- known results obtained by Ackeret, Prandtl, and Busemann 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 characteristics 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 vortex-formation 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 integrable combinations of dif- ferential equations of the characteristics. The obtained combinations let us receive the approximate formula of pressure in any point of the contour of the wing investigated. From this foinula the term is easily segregated depending exclusively on the vortex formulation, caused by NACA TM 139-^ ^1 the change of entropy along the shock wave. The chajracteristic dis- tinction of this term of the obtained formiila of pressure from the other ones, is that it includes the curvatiore of the ving contoiir 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. Translated by R. Shaw Institute of Mathematical Sciences EEFERENCES 1. Meyer, Th. : tjber zweidimensionale Bewegungsvorgange in einem Gas, das mlt Uberschallgeschwindigkeit stromt. Forsch. -Arb. , Ing.-Wes., 62, 1908. 2. Prandtl, L., and Busemann, A.: Nahrungsverfahren siu: zeichnerischen Ermittlung von ebenen Stromungen mit ijberschallgeschwlndigkeiten. Stodola Festschrift, Zurich 1929- 5. Ackeret, I. : Gasdynamik. Handbuch der Physik, I9, B. VII. h. Busemann, A. : Aerodynamischer Auftrieb bei Uberschallgeschwindigkeit. Luf t f ahrtf or s Chung, 1955. k2 NACA TM 1594 Figure 1. NACA TM 13914- kl> Figure 2(a). y c / /^ A^ .-^ /^ .^-' _/S<^ N^ r^ ^\ XS; Figure 2(b). kk NACA TM 159^ y D /^^ A"""^^^^^^^^ -/<::^'::^ ^A<^^^^^'^ i>^ t Figure 3. y _^ c, Figure 4. NACA TM 159^ i^5 y c,' ^ /^ / \ / \ ^ / \ \pz ^ / 0' Figure 5, Figure 6, k6 NACA TM I39J+ Figure 7. y Ci Z^ /l-'-'Ov /---^"S'^ /---''^^^^^'^^^^-\ / -^-^-^^^^^^^^ /^ ^^^^^^^^^^^^'^ 0' Figure 8. NACA TM 139^ hi y C| /^ _AyJ;^^-^-\ J^^'j^^^^^^^Z-^\. i ^^^^°2 ^ ,^. 'Uf<^'^ 0' Figure 9. Figure 10, kd NACA TM 159^ NACA - Langlcy Field. Va. I » , . ^ ^ , 5 3 CO eg a en m M o CO a o 03 u en CO 3 < Z CO . U M H g CD z; •a T3 CO • > <: u <: (U CO O CD / o Li 2 << fab. Ix, Q Z < O. 1-* csi crs •<3' - a a D. CS 1 3 j3 o Li 3 CS c o t^ CO s < 6 o u CD CM CO O (0 a S '3 o Li c o 00 Li OJ §- (S 13 H O O bs .5 00 Li CO a- o en o s o p, g < CO o. Li n c s bD s a Li 3 m <1> c g i s s CO CO C35 c > ■3 o cd g 01 Li ■£ T3 0} < o •< OJ m >> o in > Z w < < — H to Hi .2 c s 13 Li o s 1 a bD c o 3 m Li C« bc Clj Li T3 i 3 s •o < Z K cd T3 > nl c n s H rt H UJ CQ o Li CO to 0) tl < <; c o ^5 hop CO C4 XI o c o a o J3 73 m a; "S. o z Z XI z < -t-t a Cll O > c^ w to c m CO 'S' o CJ CJ *^ OJ rt oJ o c c rj CQ •a T3 cs . ero ero ent low o << fa fc Ml, of 1/1 CO ; ra • M • £ ""J " H " z 2 ^ S .3 .CD - g s '" CD §^' Is Q Z H H M CO ^ §2 L, " 0) cn < W o Q — H S 0; g ffi g OT .«• o '^ =2 > z rt T3 „ <»■ = ** < 0! ^ Z Z < --H <; O g CD z: o CM Si'? S CO ■ S^ > S d Or c: H . o <3> Q .S WS . ^ cs <\ ynam ynam Com Supe

CD / << S (», u, QZ <£! a iH ci CO rji «a 3 . 00 -^ 1 3 ^2 ronautlcs. m A SUPER- irch 1956. 4 rom Akademl 626) H Li _i 0. d Ho . bO 11 CO ^ OS QJ ." mmittee for Ae SHARP EDGES E. Donov. M 1394. Trans, f a, 1939, p. 603- 0. c d 0) "3 x; c II ll bD,„ at small angles at the surface, e determined. 1 < M 1394 Advisory Co WING WITH TREAM. A. (NACA TM SSR, Izvestii to - g C CD |1 (D "^ t-. (U t a thin wing e distribution wave drag ar z 1 H-SH ra .m flow pas pressur and the ■§ NACA Nation A FLA SONIC diagrs NAUK 05 s < "^ m CO / ero ero ent low low < i^ > ~ / B£ U S bc U-. Q) m 01 B « ^5 !# 0) S XI ■a t, 4, g ,, 5 01 2 S .« IS 3 I" n o O. to £ I* r- a "> Z. < Zj Zi o.a < z o u ::i A- ^ " of und (1 ble 1.1. 1.2. S >^ -S-ii • M \ ics ics, pres rson (1 E. 1394 NAU 939, <\ S S S g. 2 2 ° 3 , A. TM miia iia, 1 -626. &&_-" " z? •0 -o ta . onov ACA kade vest 603 ero ero ent low low / <;< B P4 ta pz <:,r3 0. ^ « CO .^ji « a 3 .S;s i si ronautlcs. IN A SUPER- irch 1956. 4 rom Akademl 626) proximate so al supersonic of attack. T the lifting fo mmittee for Ae SHARP EDGES E. Donov. M 1394. Trans, f a, 1939, p.603- a B m ~ rt S I" S given for the f two-dimens at small ang at the surfa e determined i < < 94 Isory Co G WITH AM. A CATM Izvestli treatment is he problem t a thin wing e distribution wave drag ar i u M13 Adv WIN TRE (NA SSR, c ACAT ational FLAT 3NIC S lagrs. AUKS basic on of t ow pas ressur [id the aZ-. >.. T3 T3 o o Ur tH a E o ^ U £ o S E oi CO CQ CO M ^ P CO • S "'J " M 2Z2 ^ s .2 .to H c .3 CO Q ^ <; ^ d Tjl i-H HH ;-H S^a 1 V o: ojP. CJ P §S tl " V C/} <; w o Q ■ ■ W Q •0 iS 1-1 <; ■C S CD CNJ ^ u CD o Q > S e Pi K S < a ti m H IS OS m tH > Tl '& s < ^ H H < < B la < P>H ZZ< o Q . c» ^^ CO . T^ ta tq <; « r^ CO M . m »— I M ^ Z boP 0.25 m -o Z .3 J= t, o 5 E-i o Q) o .bo "'1 •=■! .3 ca t» ' ta : t, !**■'■ a ta o ■ ca o oj S 2 "si -^ a; a o :2 a "" ■a ca ■= A S ■ III M O ho rs| CO)? m ^^ £ 2S ea Q^ '*-' , 0) ca ■ U CD " tn o o- f ^ * rj B ;* 2 ° <« Ij C^ to g (1) f-t Jc •" *-^ 0) ■3-° S '^ .2 !« "a bo a 2 tl) S to 5 m h c -a ■< UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE UBRARY P.O. BOX 117011 GAINESVILLE, FL 32611-7011 USA