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NATIONAL ADVISORY COMMITTEE FOR AEr.ONAUTICS TECHNICAL MEMORANDUM 1241 TWODDIEESIOIAL POTENTIAL FLOWS By Manfred Schilfer and W. Tollmien Outline: I. CHARACTERISTIC DIFFERENTIAL EQUATIONS INITIAL AND BOUNDARY CONDITIONS II. TITEGRATION OF THE SECOND CHARACTERISTIC DIFFERENTIAL EQUATIONS III. DIRECT APPLICATION OF MEYER'S CHARACTERISTIC HODOGRAPH TABLE FOR CONSTRUCTION OF TWODIMENSIONAL POTENTIAL FLOWS IV. PRANDTLBUSEMANN METHOD V. DEVELOPMENT OF THE PRESSURE VARIATION FOR SMALL DEFLEC TION ANGLES VI. NUMERICAL TABLE: RELATION BETWEEN DEFLECTION, PRESSURE, VELOCITY, MACH NUMBER AND MACH ANGLE FOR ISENTROPIC CHANGES OF STATE ACCORDING TO PRANDTLMEYER FOR AIR (K = 1.405) VII. REFERENCES I. CHARACTERISTIC DIFFERENTIAL EQUATIONS  INITIAL AND BOUNDARY CONDITIONS. For setting up the characteristic differential equations one starts from the differential equation for the velocity potential since the velocity components can be expressed more simply by the velocity potential than by the stream function. This differential equation reads, according to I(15): S2 %x '2 av ,,y 2 + 0 (1) with u = q v = p (2) *IEbene Potentialstromungen. Technische Hochschule Dresden, Archiv Nr. 44/3, Kapitel ITT, March 22, 1941. NACA TM 1241 and the sonic velocity a determined by a2 = ( 1 q2 (?) in the selected nondimensional representation. The characteristic condition (cf. chapter 11(8), NACA TM 1242) is a y2 + 2xvy + 2 (4) If one puts u = q cos 3 v = q sin ( (5) the characteristic condition is written a2 q2cos2, 2 + 2q sin' cosa x (a2 q2sln2 2 = 0 (6) Hence result two roots X' and X" for k, the slope of the charac dx' teristic base curves toward the xaxis: qsin 3 cos 3 + a. q2 a2 (a) q2cos23 a2 q2sin cos 6 a lq2 a2 () q2cos2S a2 As differential equation for the first family of the characteristic base curves results dy X' dx = 0 (8a) and for the second dy dx = 0 (8b) IJACA TM 1243 3 For explanation of these relations (somewhat difficult to survey due to the complicated form of X' and X") one uses an artifice which is permissible in twodimensional flows. Since in the twodimensional flow, for instance, in contrast to the rotatlonallysymmetrical flow, no direction is preferred, one may place the xaxis of a Cartesian xy system in the direction of the flow at the location under investigation; thus there becomes S= 0 a a: (9) X" k Sa  2 a2 The Mach angle a is defined by sin a =  q so that tan a = a (10) \q a2 This signifies according to (8) that the characteristic base curves form with the stream lines the Mach angle a. The first family (8a) of characteristic base curves forms the Mach angle toward the left (looking in the flow direction), the second family (8b) forms the same angle toward the right. The first family of characteristicbase curves were thereupon denoted as lefthand, the second as righthand Mach waves. The second characteristic equation of the first q family of characteristics is, according to II(28b), (NACA TM 1242): q2sin cos 9 a q2 2 du + dv = 0 (lla) q2 cos 2 a2 NACA TM 1243 and for the second family of characteristics, according to II(29b) (TM 1242): 2 2 2 q sin 1: cos i + a /q2 a du + sin cos o + aq dv = 0 (11b) q2cos 2 a2 with du = cos 9 dq q sin 3 d8 (12) dv = sin i dq + q cos do If one applies the same artifice as in selecting the special coordinate system at the considered location, one obtains, since a there becomes 0, du = dq, dv = q dO and one has on the first family of characteristics dq a q de = 0 (13a) \/q2 a2 and on the second dq +  q d = (13b) The two relations (13) contain only quantities independent of the special selection of a coordinate system. Together with the remark made initially on the admissibility of the last used coordinate system there results, accordingly, the general validity of the relation (13a) for the first, of the relation (13b) for the second family of charac teristics. The relations (13) are often, in an elementary manner, deduced from the fact that the infinitesimal velocity variation is perpendicular to the Mach wave: NACA TM 1243 sin a q + dq _in_ sin ( + a dd) cos a S cos a + sin m da 9 = 1 tan a d3 dq q tan a d1 a = q d1 \q2 a2 Thus the velocity variation in crossing a lefthand Mach wave is regulated according to this equation (of. (13b)); the crossing has to take place along a righthand Mach wave. Regarding this elementary derivation the fundamental remark has to be made that here tacitly the existence of a relation between u and v alone (and between q and i alone, respectively) is assumed. In the rotationallysymmetrical case where this presupposition no longer holds, one obtains accordingly another characteristic equation although the velocity increment occurs as before perpendicularly to the Mach wave. A few remarks concerning the secondary conditions which supervene the differential equation of the twodimensional potential flow are to be inserted at this point. The secondary conditions may for instance be given by initial conditions, that is, on an initial curve (for example, in a certain cross section of a channel) a few or all flow values are prescribed. The front of a compression shock also may serve as initial curve. The initial distributions for the approximation method discussed here are approximated by distributions constant over small distances. Therein it was often used as approximation principle that the jumps in i are to be of a certain magnitude, for instance +10 or +20. Correspondingly, the boundary distributions which are given by boundary conditions, the main types of which will now be discussed, are also approximated. NACA TM 1243 J is prescribed at a solid wall. If one denotes as compression wave a Mach wave behind which a pressure increase and a velocity decrease takes place, compression waves are reflected at a solid wall as compres sion waves. This can readily be seen in the figure. The crossing of the righthand Mach compression wave occurs along a lefthand Mach wave; thus the pre supposed decrease of q according to (13a) is connected with a decrease of 3. The crossing of the adjoining left hand wave along a righthand wave must cause due to the boundary condition at the wall which is assumed to be unbroken an increase of 3 which according to (13b) produces a decrease of q, therefore a compression. At a free let boundary the pressure p is prescribed as constant; thus the velocity q also is a known constant there. The free jet boundary must because of the cinematic boundary condition of vanishing normal velocity coincide with a stream line which will be determined in the course of the solution of the flow problem. A compression wave is reflected at a free jet boundary as rarefaction wave since the drop in velocity which occurred first must be made good again by an increase, in order to satisfy the condition of constant velocity at the free jet boundary. 2. INTEGRATION OF THE SECOND CHARACTERISTIC DIFFERENTIAL EQUATIONS. Of the characteristic equations (8a), (8b), (13a), (l?b) the last pair can be very easily integrated; this was already done by Th. Meyer (cf. Th. Meyer, particularly p. 38). Following, a derivation is given which fits into our general theory. On the lefthand family of characteristics S q (13a) vq2 a2 with a2_ 1 ( q2). Hence 3 may be determined as function 2 of q by quadrature. The execution of this elementary integration results in NACA TM 1243 K + 1  are tan KiI  arc tan arc tan a* 2 +1 K I + V( )(1 q2) K r1 1 1 1 are tan + 4 a2 a*2 a2 *2 1 + 1 or =  V 1 rt 1 3 v2 are tan 1 arc tan 1 K + 1\a2 a2 1 1) is the critical velocity Ii and 81 represents J(K + 1) + a7, where a* an Integration constant. It has to be noted that all velocities have been made non / 2K Po dimensional by Vm = 1 po K P (Po tank pressure, po tank density). For this relation a table of data particularly convenient for the practical calculation has been given by 0. Walchner for air (K = 1.405) (cf. pp. 2223). Aside from the velocities q referred to the sonic velocity a and the critical sonic velocity a*, the pressure p, referred to the tank pressure p, is given, for which the equation L = (1 q2)K/(K1) Po is valid. In the last column the Mach angle a is indicated. The integration constant 01 is selected as O, v is used instead of b for this special integration constant, the reason for this will be shown in the next paragraph. The application of the MeyerWalchner table for the approximated construction of twodimensional potential flows will be discussed in the next paragraph. / 1 V 2 K +1 +1 V( 1)(1 q2) K 1 1 a* (14) NACA TM 1243 For the righthand characteristics one obtains correspondingly by integration of (13b) 1 1 arc tan a arc tan I (15) a* \ a*2 Va2 a*2 r or . = +i are tan i 1 arc tan 1  arc tan V 2 a2 with br being an integration constant. On the lefthand character istics increasing q is, according to (13a), connected with increasing 0, on the righthand characteristics increasing q, accord ing to (13b), with decreasing 0. Hence the designation lefthand and righthand, at first introduced through the characteristic base curves of the Mach waves, is immediately comprehensive also for consideration of the "characteristic hodographs" the equation of which is given in polar coordinates (q radius vector, angular coordinate) in differential form by (13a) and (13b), in the integrated form by (14) and (15). Cl. Thiessen (1926) first drew attention to an interesting geometrical interpretation of these characteristic hodographs. A geometrical proof which, however, requires longer preparation, has been given by A. Busemann. Following, we give our own proof, relin quishing the nondimensional representation, in order to conform to customary representation. According to Thiessen the characteristic hodograph curves are epicycloids originating from the rolling of a circle of diameter vm a* on a circle of radius a* NACA TM 1243 Since the rolling point A, the point Q the fixed circle 0  increment dq lies on circle rolls in the denoted position about the  which has the distance q from the center of traces a small circular arc about A. Thus the the line QB, since QB is perpendicular to QA. On the other hand, dq is, according to the end of the previous paragraph, for twodimensional potential flows unequivocally determined by the fact that dq is perpendicular to the Mach wave which with q (here represented by OQ forms the Mach angle a = arc sin a/q. Thus the proof will be given if the angle 4 OQA is found to be equal to the Mach angle. If one puts preliminarily 4OQA = and an QBO = one has 4. OB = n/2 + e 4 QAO = n/2 + 0 According to the sine one obtains theorem applied to the triangles OQB and OQA sin( + sin sin E sin( + + cos C sin P sin e cos B Vm q a* Q From these two equations the angle c which is of interest to us may be calculated by elimination of 0. This brief calculation may be performed about as follows. One has directly sin 0 = ' cos Vm cos = sin F a* NACA TM 1243 Hence follows by squaring and adding or q2 cosB2 E + \ m S2 sin2, + v2 m a n 1 an2) K + sln2 E 1 K  Next, there results sin2 = K  2 2 2 Vm q2 q2 a2 q2 This, however, is just the equation for the Mach angle; thus E equals the Mach angle. Therewith the fundamental statement dealing with the behavior of dq with regard to the characteristics has been obtained again. The directional field produced by rolling of the circle coin cides, therefore, with the directional field of the characteristic hodo graph curves. The characteristic hodograph curves are epicycloids. I. DIRECT APPLICATION OF MEYER'S CHARACTERISTIC HODOGRAPH TABLE FOR CONSTRUCTION OF TWODIMENSIONAL POTENTIAL FLOWS. Meyer's solution for the characteristic hodographs indicated in the previous paragraph has been used directly for the construction of twodimensional potential flows by J. Ackeret and, recently, by 0. Walchner. We shall explain the method in one of the examples cal culated by 0. Walchner which for the first time showed the general validity of the method. First it is to be noted that the one table given actually is sufficient for all characteristic hodographs, since, according to (14), for the lefthand characteristic hodographs .a = v + (a6 (16) NACA TM 1243 according to (15) for the righthand characteristic hodographs b = V+ + r r (17) It has to be pointed out that the crossing of a righthand Mach wave takes place along a lefthand characteristic, so that (16) is valid. The crossing of a lefthand Mach wave occurs along a righthand charac teristic with (17) being valid. The transition from one field with the index the index i + 1 is regulated at the crossing of wave, thus according to the equations: i to the next with a righthand Mach whence follows, with S= Vi +b =+ vi+1 + 19 8 eliminated: v+1 = v + ) For crossing of a lefthand Mach wave di = Vi + 1+1" i +1 + r (18) NACA TM 1243 must be valid or, after elimination of Or: 1 v+l 1 (i+1 (19) According to the two equations (18) and (19) Meyer's table is used for the approximated calculation of twodimensional potential flows. The selected example of Walchner (fig. 3 of Walchner's report Lufo Bd. 14 (Aviation Research, vol. 14) p. 55, 1937) deals with the flow about a biplane at an angle of attack, with relatively rough approxi mation since large Jumps in direction from one field to the next are admitted. Furthermore compression shocks are approximated by Mach waves which in view of the weakness of the occurring shocks is still justified (cf. chapter V, par. 5). The field numbers are put as indices to the pertaining flow ql pi values. In the initial field = 1.64, = 0.221, 91 = 0 is al Po prescribed. According to the table the value V1 = 160 pertains to this value of q/a or p/po. In field 2 02 = 10, due to the geometrical boundary condition. Since in the transition from field 1 to field 2 a righthand Mach wave is crossed, v2 = 60 according to (18); ql according to the table, the pertinent values are = 1.293, aP Pi S= 0.363. For field 3 one has, for geometrical reasons, 33 = 40; the Po transition from field 1 to field 2 leads, according to (19), to v3 = 120 3 P3 with = 1.504, = 0.270. The calculation of the q and bvalues a3 p0 in the next field 4 represents the general case of the method. From field 3 one arrives at field 4 by the crossing of a righthand Mach wave, thus according to (18): vq = v3 + 4 3 = 80 + V4. From field 2 one arrives at field 4 by the crossing of a lefthand Mach wave, thus according to (19): V4 = V2 4 2) = 40 04. From the two equations for 4 and V4 set up just now follows 4 = 60, v4 = 2o q4 P4 with = 1.132, = 0.449. In all remaining fields 0 is prescribed a4 Po NACA TM 1243 by geometrical boundary conditions so that the calculation is easy and takes place very similarly to that for field 2 and field 3. One has 5 = 4, V = 120 = 1.504, = 0.270; q6 P6 6 =100, v = 60, = 1.293, 6 = 0.363; 6 O 7 = "30, 7 = 190, = 1.743, = 0.190; a7 Po 08 P8 g = 30, vg = 130 = 1.538, = 0.257 8a8 p0 A drawing machine is desirable for plotting the Mach waves which close any newly calculated field. With a finer subdivision of the angular variations, a greater accuracy seems attainable by means of this method than with the aid of the PrandtlBusemann method described below. 4. PRANDTLBUSEMANN METHOD: For approximated construction of twodimensional potential flows mostly the PrandtlBusemann method is used, the main expedient of which is a diagram with the characteristic hodograph curves (14) and (15), respectively. The PrandtlBusemann method is, according to our terminology introduced in chapter II, (NACA TM 1242), a field method, that is, a pair of values q, d is coordinated to each field formed by the characteristic base curves or Mach waves. For the sake of a simple representation of the method we assume this pair of values q, j to be valid precisely for the field center, the definition of which was given in chapter II, paragraph 7, NACA TM 1242.1 We now visualize the field centers as connected with each other; these connecting curves 1 The field centers are very useful for explanation of the method; they are, however, in case of twodimensional potential flows, in contrast to rotationallysymmetrical ones, not required for the construction so that the exact definition of the field centers would here not yet be necessary. NACA TM 1243 give, as was shown, again characteristic base curves, thus here the Mach waves. To these Mach waves, not perhaps to the Mach waves of the field boundaries, the characteristic hodograph curves were coordinated. The net of the characteristic hodographs may be drawn once and for all according to the expositions in section 2 for a given K. According to former representations the crossing of a lefthand Mach wave in the flow plane is connected with a progressing along a righthand charac teristic hodograph in the velocity plane. Correspondingly, crossing of a righthand Mach wave is coupled with progressing along a lefthand characteristic hodograph. Busemann and Preiswerk gave a net of charac teristic hodographs for K = 1.405. This diagram is customarily denoted simply as "characteristics diagram". In order to calculate from the known / pairs of values q, B in field I and II m the unknown pair of values q, B in the field III adjoining downstream, one pro iT gresses from the point in the character istics diagram corresponding to field I along a righthand epicycloid, since one has crossed a lefthand Mach wave in the transition from I to III; correspondingly one progresses from the point of the characteristics diagram corres ponding to field II along a lefthand epicycloid. The point of intersection gives the pair of values q, 6 for the field III. The field that had been open so far is then closed by two Mach waves the direction of which is determined from the values of q and 9 found just now. The modifications of the method for fields at the boundary of the region are obvious. In order to facilitate the reading of the q and Svalues from the characteristics diagram, one may take the net of the characteristic hodographs as net of coordinates. According to (14) and (16), respec tively, one has for the lefthand epicycloids: q)J =' (20) according to (15) and (17), respectively, for the righthand epicycloids v(q = r (21) Thus 3 and through the tabulated function v(q) q as well may be very easily expressed by the parameters O1 and 9r of the epicycloids. NACA TM 1243 Instead of the parameters 3 and 3, which probably first seemed obvious, Busemann selected others with only the starting points of the count shifted from and V. The angles 5 and v are measured in degrees. The degree sign (0) is omitted below. One may then express Busemann's epicycloid numbering so that as equation of the lefthand epicycloids v(q) = 2(X 400) (22) as equation of the righthand epicycloids 9 v(q) = 2(i 600) (23) is written with the new parameters X and p. Hence there results 200 8 = p X (24) 1000 V(q) = p + X (25) Thus the difference of the new parameters' X and p gives the angle b except for an insignificant shifting of the initial point and reversal of the sense in which one is counting. The center line of Busemann's characteristics diagram (9 = 0) obtains the "direction number" p X equal to 200, whereas the sum of X and p yields the function v(q) and therewith also q and the pressure p. The numbering of the epicycloids according to (22) and (23) is carried out very easily if one considers additionally that v(q) just vanishes for q = a* (critical velocity). Eusemann writes the parameters X and p as field numbers" into the fields of the flow plane; a table for the connection of the "pressure number" p + X with q and p must be given as supplement. If one approximates the initial and boundary conditions in such a manner that one replaces the prescribed angles by sectionally constant distributions with jumps of +10 or +20, one may assure by a suitably finemeshed characteristics diagram that one gets by without interpola tion. The customary characteristics diagrams are in their main part arranged for angular jumps of 10. NACA TM 1243 Additionally developed graphical expedients for facilitated plotting of the Mach waves will not be discussed, since one can dispense with them when a drawing machine is used. We will be content with these observations regarding the Prandtl Busemann method and will cmit the carrying out of a standard example since the method has been represented in detail by Heybey (HVP Archiv Nr. 66/31 and 66/32). 5. DEVELOPMENT OF THE PRESSURE VARIATION FOR SMALL DEFLECTION ANGLES. In a flow unilaterally bounded by a wall the flow variations en forced by the boundary conditions are propagated from the wall along one family of Mach waves; in the figure it is the lefthand family. 60> This property of twodimensional potential flows follows immediately from the characteristic differential equations (13) which connect velocity and directional variation. This property is by no means transferable to other than twodimensional potential flows, for instance potential flows with rotational symmetry. Since for the conditions assumed in the figure the flow variations occur at the crossing of lefthand Mach waves, they may be calculated by progressing along a righthand characteristic, thus according to (13b) from aq dq = da (13b) 2 a2 A development of the pressure difference for deflection of the flow from the angle 3= 0 to the angle 3 = 5 is to be given; the deflection angle 5 is to be small and the third powers of 5 are still to be included in the development. Busemann has for the first time set up such a development, using a method totally different from NACA TM 1243 17 ours. In order to develop the pressure, first the velocity q2 which is to pertain to = 8 is developed according to equation (13b) with respect to 6. For b = 0, q is to equal ql. We set up: 2 = q1 + c + 02 +2 +3 3 (26) and determine the unknown coefficients c, c2, c3 fran (13b) by com parison of the coefficients. One has only to equate = Cl+ 2c2 +3c3 aq with the expression originating from when q is replaced q2 a2 by q2 according to (26). Therein is a2 = 1 (Vm2 q2). It is most convenient to square the two sides of the relation mentioned before comparing the coefficients. There results 1 __ = 2 q 1)ql + 2aI a12 4 (q2 a 12)2 L 3 = a1q, 1)(2K 3)q6 (27) 12 2 a12 7/2 U + 9( )a12q14 + 6a1 q2 + 2a16 In order to obtain frcm the development for q2 that of the pressure P2, one starts from the isentropic pressure equation 21 i 2 2 A P2 m .2 2 21 q1 2 = =m K (28) p1 2 q i2 2 q m m NACA TM 1243 This expression is binomially expanded, making use of (26) and (27). For transformation of same of the terms one applies the formula 2K 91 2 Pl K 1 2 q12 01 which follows from the energy theorem (1(6)), and al2 1 m2 12) with the Mach number M = ql/al one then finally obtains S P 2 52(M2 2)2 + KM 2 1 1 4(M2 1)2 &3 (29) + (a + 1)M8 12(M2 1)7/ (/2 + (292 7' 5)M6 + I0(K 1)M 12M2+8 j The first term of this development may be found already in Th. Meyer's report; Busemann gives the development up to 53 (Volta congress, p. ?37). It is true that the coefficient of 53 as calculated by us, completely differs from Busemann's. The comparison of the pressure difference occurring for isentropic deflection with the one in case of a compres sion shock is made in chapter V, paragraph 5. In transferring formula (29) to the case of the flow variation taking place at the crossing of righthand Mach waves, one has to establish the connection with the characteristic differential equation (13a). The simple rule results that one has to select the sign of 5 in (29) as it corresponds to a reflection on the freestream direction of the conditions assumed above (cf. the figures on p. 16). NACA TM 1243 1i As application of this formula (29) we use the calculation of ca and ow for circular segment profiles in second approximation which was already performed by A. Busemann. yr According to the figure one has the profile 8ob = 0 Bob and for If one replaces cos 6 by 1, sin 6 c t Ca 2 q12f t0o 2 a 2_ cw 0 t2 for the by the upper side (index ob) of lower side (index u) 8u = 0. 8, one has Pu Pob) dl (Pup Pob PObPob) dl with the integration to be extended over the entire wing chord. If one uses formula (29) to the second term in 6 for the expression of pu and pob: NACA TM 1243 S1 2 2 M2 ( 2 2)2 M4 P Pl = I +  2 1 (M2 )2 J for which one writes abbreviatedly: 2 51 q 1 + C252) P P1 (18 + C2 2 (M2 2)2+ +M4 S M 1M s C2 4(M2 )2 one obtains Ca = 2Ci Pob2 d t Therein it is already taken into account that f Bob dl vanishes (for the expression for yo we shall also use the vanishing of O ob dl for circular segment profiles). In the foregoing deriva tion ca is then determined to include terms which are quadratic in the angles P and. ob. For ow one obtains analogously the expression including terms of the third order in the angles 0 and Pob: cw = 2Cji2 tot+ f d P T o2 dZ 0 ob d  ob ft 0 NACA TM 1243 Since one has for approximation of the circular arc by a parabolic arc 81d 4d Pob = t one obtains ob2 16 t 0 ob 3 t Therewith becomes ca 2C1  c = 2C1,2 d2 \t/ C1 4d\2 4d2 + 3M C2 t) It is noteworthy that for the angle of attack P = 0 a negative lift of the circular segment profiles exists which was also determined experi mentally. NACA TM 1243 TABLE RELATION BETWEEN DEFLECTION, PRESSURE, VELOCITY, MACH NUMBER AND MACH ANGLE FOR ISENTROPIC CHANGES OF STATE ACCORDING TO PRANDTLMEYER FOR AIR (K = 1.405). p q m a a P ' a, = arc sin a Po a* a q 00 0.527 1.000 1.000 90000' 1 .478 1.067 1.081 67 45 2 .449 1.106 1.132 62 05 3 .424 1.140 1.177 58 10 4 .402 1.170 1.217 55 20 5 .382 1.199 1.256 52 50 6 .363 1.225 1.293 50 40 7 .346 1.250 1.330 48 45 8 .329 1.275 1366 47 05 9 .314 1.299 1.401 45 35 10 .299 1.322 1.436 44 10 11 .284 1.345 1.470 42 50 12 .270 1.367 1.504 41 40 13 257 1.388 1.538 4o 30 14 .244 1.408 1.572 3930 15 .232 1.428 1.606 38 30 16 .221 1.447 1.640 37 35 17 .210 1.466 1.674 36 40 18 .200 1.485 1.708 35 50 19 .190 1.504 1.743 35 00 20 .18o 1.522 1.778 34 15 21 .170 1.540 1.812 33 30 22 .161 1.557 1.848 32 45 23 .153 1.575 1.883 32 05 24 .145 1.592 1.918 31 25 25 .137 1.609 1.954 30 50 26 .130 1.625 1.990 30 10 27 .123 1.642 2.027 29 35 28 .116 1.658 2.083 29 00 29 .109 1.674 2.100 28 25 30 .103 1.689 2.138 27 55 NACA TM 1243 RELATION BETWEEN DEFLECTION, PRESSURE, VELOCITY, MACH NUMBER AND MACH ANGLE FOR ISENTROPIC CHANGES OF STATE ACCORDING TO PRANDTLMEYER FOR AIR (K = 1.405) Continued. S q ~ arc sin Po a* a q 310 0.097 1704 2.177 27020' 32 .091 1.720 2.215 26 50 33 .086 1.735 2.255 26 20 34 .080 1.750 2.295 25 50 35 .076 1.765 2.335 25 20 36 .071 1.780 2.376 24 55 37 .067 1.794 2.418 24 25 38 .062 1.808 2.460 24 00 39 .058 1.822 2.502 23 35 40 .055 .836 2.545 23 10 41 .051 1.850 2.590 22 45 42 .047 1.864 2.635 22 20 43 .044 1.877 2.661 21 55 44 .041 1.890 2.728 21 30 45 .038 1.903 2.778 21 10 46 .035 1.915 2.823 20 45 47 .033 1.928 2.872 20 20 48 .030 1.940 2.922 20 00 49 .028 1.952 2.974 19 40 50 .026 1.984 3.027 19 20 51 .024 .976 .081 18 55 52 .023 1.988 3.136 18.35 53 .021 1.999 3.191 18.15 54 .019 2.011 3.247 1753 55 .018 2.022 3304 17 35 56 .016 2.033 3363 17 20 57 .015 2.044 3.424 17 00 58 .013 2.055 3.487 16 40 SNACA NACA TM 1243 REFERENCES 1. Meyer, Th.: Uber zweidimensionale Bevegungsvorgange in einem Gas, das mit Uberschallgeschwindigkeit stromt. Forsch.Arb. Ing. Wesen Nr. 622, 1908. 2. Ackeret, J.: Luftkrafte auf Fligel, die mit grSsserer Geschwindlg keit als Schallgeschwindigkeit bewegt werden. ZFM Bd. 16, pp. 7274, 1925. 3. Walchner, O.: Zur Frage der Viderstandsverringerung von Tragflu geln bei Uberschallgeschwindigkeit durch Doppeldeckeranordnung. Lufo Bd. 14, pp. 5562, 1937. 4. Prandtl, L., and Busemann, A.: Naherungsverfahren zur zeichneri schen Ermittelung von ebenen Stramungen mit Uberschallgeschwindig keit. Stodolafeetschrift, Zurich 1929. 5. Busemann, A.: Article from. "Gasdynamik", Handbuch der Experimental physik Bd. 4, Teilband 1, pp. 423459, 1931. 6. Preiswerk, E.: Anwendung gasdynamischer Methoden auf WaBserstrb' mungen mit freler Oberflache (Mittellungen aus dem Institut fir Aerodynamik, Eidgenossische Technische Hochschule, herausgegeben von Prof. Dr. J. Ackeret, Nr. 7, Zirich 1938.). 7. Heybey, W., Das PrandtlBusemannsche Niherungsverfahren zur zeich nerischen Verfolgung ebener Uberschalletr&mungen. HVP Archiv Nr. 66/31 and Nr. 66/32, 1940. 8. Reale Accademia d' Italia, Fondazione Allessandro Volta Atti del Convegni 5, Convegno di Science Fieiche, Matematiche e Naturali 30 Sett. 6 Ott. 1935. Tema: Le alte velocity in aviazione Rcma 1936: Busemann, A.: Aerodynamischer Auftrieb bel Uberschallgeschwindigkeiten pp. 328368 (abgedruckt in Lufo Bd. 12, pp. 210220, 1935.) 9. Busemann, A., and Walchner, 0.: Profileigenschaften bel Uberschall geschwindigkeit. Forschg. Ing.Wes. Bd. 4, pp. 8792, 1933 Translation by Mary L. Mahler National Advisory Committee for Aeronautics. 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