K)AC/1-TM/^Y3^F . & i«i - .^Aocw 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1243 
 
 TWO-DIMENSIONAL POTENTLAlL FLOWS 
 By Manfred Schafer and W. Tollmien 
 
 Translation of "Ebene Potentialstromungen." Technische 
 Hochschule Dresden, Archiv Nr. 44/3, Kapitel in, March 22, 1941 
 
 Washington 
 November 1949 
 
 DOCUMENTS DEPARTMENT 
 
1^? S^3HI 
 
 NATIONAL ADVISOEY COMMITTEE FOR AEt>ONAUTICS 
 
 TECHNICAL MEMORANDUM 12^3 
 
 TWO-DIMENSIONAL POTENTIAL FLOWS* 
 
 By Manfred Schaf er and W . Toll mi en 
 
 Outline: I- CHARACTERISTIC DIFFERENTIAL EQUATIONS - INITIAL AND 
 BOUNDARY CONDITIONS 
 II. INTEGRATION OF THE SECOND CHARACTERISTIC DIFFERENTIAL 
 EQUATIONS 
 III. DIRECT APPLICATION OF MEYER'S CHARACTERISTIC HODOGSAPH 
 TABLE FOR CONSTRUCTION OF TWO-DIMENSIONAL POTENTIAL 
 FLOWS 
 IV. PRANDTL-BUSEMANN METHOD 
 
 V. DEVELOPMENT OF THE PRESSURE VARIATION FOR SMALL DEFLEC- 
 TION ANGELES 
 VI. NUMERICAL TABLE: RELATION BETWEEN DEFLECTION, PRESSURE, 
 VELOCITY, MACH NUMBER AND MACH ANGLE FOR ISENTROPIC 
 CHANGES OF STATE ACCORDING TO PRANDTL -MEYER FOR AIR 
 (k = l.i^05) 
 VII. REFERENCIB 
 
 I. CHARACTERISTIC DIFFEREOTIAL 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 l(l5) • 
 
 with 
 
 a / a \ a 
 
 ^, V = qy (2) 
 
 Ebene Potentialstromungen. " Technische Hochschule Dresden, 
 Archiv Nr. hh/3, Kapitel III, March 22, 19I+I. 
 
KACA TM 121*-^ 
 
 and the sonic velocity a determined ty 
 
 2 ^ LJ_1 A . ,2\ (3) 
 
 a^ = 1 - q' 
 
 2 V 
 
 In the selected non-dimensional representation. 
 
 The characteristic condition (cf. chapter Il(8) , KACA TM 121|2) Is 
 
 (, - p ^ ^-^ ^ (i - p - (« 
 
 If one puts 
 
 u = q cos i3 V = q sin •fl (5) 
 
 the characteristic condition is written 
 
 (a^ - q2co8^^jy2 + 2q23in-3 cobt3 xj + (a^ - q^sin^iSji = 
 
 (6) 
 
 Hence result two roots X.' and X." for ~, the slope of the charac- 
 teristic "base curves toward the x-axis: 
 
 , _ q^sln t3 cos t3 + a\|q^ - a^ , , 
 
 A, — (iB.) 
 
 2 2 2 
 q^^cos i3 - a*^ 
 
 . ti q^sln -a cos i3 - a /q^ - a^ ,_ , 
 
 A. = (Tb) 
 
 q^cos^iS - a^ 
 
 As differential equation for the first family of the characteristic 
 "base curves resiolts 
 
 dy - X,' dx = (8a) 
 
 and for the second 
 
 dy - X," djc = (Bh) 
 
NACA TM 1243 
 
 For explanation of these relations (aomewhat dlfficiilt to survey 
 due to the complicated form of X, ' and x") one uses an artifice which 
 Is permissihle In two-dimensional flows. Since In the two-dimensional 
 flow, for Instance, In contrast to the rotatlonally-symmetrlcal flow, 
 no direction Is preferred, one may place the x-axls of a Cartesian xy 
 system In the direction of the flow at the location under Investigation; 
 thus there becomes 
 
 ■a = 
 
 X' = 
 
 \" = 
 
 ? - a2 
 
 \/^2V-,2 
 
 y 
 
 (9) 
 
 The Mach angle a Is defined by 
 
 sin a = — 
 
 q. 
 
 so that 
 
 tan a = 
 
 ^ 
 
 ,^ - «2 
 
 (10) 
 
 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 characteristic base curves were 
 
 thereupon denoted as left-hand , the second as 
 
 right-hand Mach waves . 
 
 The second characteristic equation of the first 
 family of characteristics is, according to Il(28b), 
 (RACA TM 121^-2) : 
 
 du + 
 
 q^sln ^ cos -a - a yq^ - a^ 
 
 dv = 
 
 q cos t3 
 
 (lla) 
 
KACA TM 1214.3 
 
 and for the second family of characteristics, according to 11(2911) 
 (TM I2I+2) : 
 
 q sin <3 cos t3 + a Vq^ - a^ , , 
 
 du + — dv = (lib) 
 
 ? ? 2 
 
 q'^cos -a - a 
 
 with 
 
 du = cos ■^ dq - q sin '^ d.'S 
 dv = aln i3 dq + q cos -d di3 
 
 (12) 
 
 If one applies the same artifice as In selecting the special coordinate 
 system at the considered location, one obtains, since t3 there becomes 
 
 du = dq, dv = q di3 
 and one has on the first family of characteristics 
 
 dq - ————— q diS = (l3a) 
 
 i 
 
 q2 - a2 
 
 and on the second 
 
 dq + ^ q d^ = (13b) 
 
 \^ 
 
 fq2 - a2 
 
 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 resiilts, accordingly, the general validity of the relation (l3a) 
 for the first, of the relation (l3h) for the second family of charac- 
 teristics. 
 
 The relations (13) a^e often, in an elementary manner, deduced 
 from the fact that the infinitesimal velocity variation ±3 perpendicular 
 to the Mach wave: 
 
WACA TM 121|3 
 
 q + dd Ml - ^ 
 
 <1 
 
 ^1 + a - d^ j 
 
 sin 
 
 COB a 
 
 cos a + sin a diS 
 
 = 1 - tan a di3 
 dq. = -q tan a di3 
 
 V- 
 
 a 
 :=: 1 di3 
 
 q2 - a2 
 
 Thus the velocity variation in crossing a left-hand Mach wave is 
 regulated according to this equation (cf. (1312)); the crossing has to 
 take place along a right-hand Mach wave. Regarding this elementary 
 derivation the fundamental remark has to he made that here tacitly the 
 existence of a relation hetween u and v alone (and hetween q 
 and i3 alone, respectively) is assixmed- In the rotationally -symmetrical 
 case where this presupposition no longer holds, one obtains accordingly 
 another characteristic equation althoijgh 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 two-dimensional 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 i3 are 
 to be of a certain magnitude, for instance +1° or +2°. Correspondingly, 
 the boundary distributions which are given by boundary conditions, the 
 main types of which will now be discussed, are also approximated. 
 
MCA TM l2i^-3 
 
 i3 ia prescri"bed at a solid wall. If one denotes as ccmpression 
 wave a Mach wave tehind which a pressiire 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 right-hand Mach compression wave 
 occurs along a left-hand Mach wave^ thus the pre- ////////y/ ////)//7 
 supposed decrease of q. according to (13^) is 
 connected with a decrease of t3 • The crossing of the adjoining left- 
 hand wave along a right-hand wave must cause - due to the boundary 
 condition at the wall which is assumed to "be unbroken - an increase 
 of i3 which according to (13^) 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 kinematic 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. INTEGflAriOW OF THE SECOND CHARACTERISTIC 
 DIFFERENTIAL EQUATIONS. 
 
 Of the characteristic equations (8a), (8b), (l3a) , (l3b) 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 left-hand family of characteristics 
 
 r= 1 (13a) 
 
 ^ _ 
 
 di3 
 
 \/42 - a2 
 
 2 K_2_l ,^ _ o 
 
 with er = (1 - q2) . Hence t3 may be determined as function 
 
 2 
 
 of q by quadrature. The execution of this elementary integration 
 results in 
 
NACA TM 1214.3 
 
 /K + 1 , I/k - 1 / 2 K + 1 
 
 ^ = 1/ arc tan 
 
 IC - 1 W K + 1 
 
 (K - 1)(1 - <i2) 
 
 , 2 K + 1 
 
 arc tan , / - r + " 
 
 ,<). 
 
 (•c - 1)(1 - q2) 
 
 Tn. ^ 1- ilk) 
 
 r-rl 
 
 — arc tan "S a* , I— - — ( - arc tan J— - — - + ^, 
 
 a* \/a2 a*2| \/a2 a*2 Z 
 
 or 13 =1/ arc tan ,/ /*- - 1 - arc tan 1'^ - 1 + ^7, where a* 
 
 \h - 1 Vk + 1\ a2 >'-2 i' 
 
 is the critical velocity v / ; — f and Oi represents an Integrati 
 
 V(k + 1) ' 
 
 constant. It has to be noted that all velocities have been made non- 
 
 on 
 
 / 2k Po , > 
 
 dimensional by Vjj^ = / — ^-r — (pq tank pressure, Pq tank density). 
 
 y^ - 1 pQ 
 
 For this relation a table of data particvilarly convenient for the 
 practical calculation has been given by 0. Walchner for air (k = l.lj-05) 
 (cf. pp. 22-23). 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 Po Is given, for which the equation 
 
 -^ = (1 - (i2)'^/(^-l) 
 
 is valid. In the last column the Mach angle a Is indicated. The 
 integration constant ^i is selected as 0, v is used Instead of ^ 
 for this special Integration constant; the reason for this will be 
 shown In the next parag3:'aph. The application of the Meyer -Walchner 
 table for the approximated construction of two-dimensional potential 
 flows will be discussed In the next paragraph. 
 
WACA TM 12I4.3 
 
 For the right-hand characteristics one obtains correspondingly "by 
 integration of (l3b) 
 
 "■ - arc tan. -x 
 
 1 
 a*2 
 
 (15) 
 
 or 
 
 + 1 
 
 arc tan\ 
 
 f K - 1 
 K + 1 
 
 arc tan, 
 
 a! 
 
 with iSr "being an integration constant- On the left-hand character- 
 istics increasing q is, according to (l3a) , connected with 
 
 increasing 
 
 i3. 
 
 on the right-hand characteristics increasing q, accord- 
 
 ing to (13^), with decreasing ^. Hence the designation left-hand and 
 right-hand, 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, t3 angular coordinate) in 
 differential form "by (13a) and (13b) , in the integrated form by {ik) 
 and (15)' CI. 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 non-dimensional 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 m^ 
 on a circle of radius a*. 
 
NACA TM 12U3 
 
 Since the rolling circle rolls In the denoted position about the 
 point A, the point Q - which has the distance q. from the center of 
 the fixed circle - traces a small circular arc about A. Thus the 
 increment dq lies on the line QB, since Q3 is perpendlciilar to QA. 
 
 On the other hand, dq. is, according to the end of the previous 
 paragraph, for tvo -dimena 1 onal 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 
 
 4- OQA = € 
 and . 
 
 4-QBO = p 
 one has 
 
 4- 0Q3 = Jt/2 + e 
 
 4- QAO = n/2 + p 
 
 According to the sine theorem applied to the triangles 0Q3 and OQA 
 one obtains 
 
 <l ^ ^ 
 
 \2 / ^ cos € _ V 
 sin P sin 3 q 
 
 sin e 
 
 sin € 
 cos 3 
 
 a* 
 
 sin ft + 3) 
 
 a 
 
 From these two equations the angle e which is of interest to us 
 may be calculated by elimination of p. This brief calculation may be 
 performed about as follows. One has directly 
 
 q 
 
 sin 3 = — cos e 
 
 Vm 
 
 COS 3 = — sin e 
 a* 
 
10 , NACA TM 121+3 
 
 Hence follows "by squaring and adding 
 
 q^2 / cos^ e _^ sin^ 6 \ ^ ^ 
 
 V 2 a*2 
 
 'm 
 
 or 
 
 Next, there results 
 
 .2 K - 1 m 
 sin e = — 
 
 V„^ - d^ a2 
 
 2 q2 ^2 
 
 This, however, is Just the equation for the Mach angles thus e equals 
 the Mach angle. Therewith the fundamental statement dealing with the 
 "behavior of dq with regard to the characteristics has "been o"btained 
 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. 
 
 3. DIEIECT APPLICATION OF MEYER'S CHARACTERISTIC HODOGKAPH 
 
 TABLE FOR CONSTRUCTION OF TWO-DIMENSIONAL 
 
 POTENTIAL FLOWS. 
 
 Meyer's solution for the characteristic hodographs indicated 
 in the previous paragraph has "been used directly for the construction 
 of two-dimensional potential flows by J- Ackeret and, recently, "by 
 0. Walchner. We shall explain the method in one of the examples cal- 
 ciilated "by 0. Walchner which for the first time showed the general 
 validity of the method. 
 
 First it is to "be noted that the one ta"ble given actually is 
 sufficient for all characteristic hodographs, since, according to (ih) , 
 for the left-hand characteristic hodographs 
 
 ■a = V + -di (16) 
 
NACA TM 121+3 
 
 11 
 
 according to (15) for the right-hand characteristic hodogrepha 
 
 t3 = -V + -a 
 
 (IT) 
 
 It has to he pointed out that the crossing of a right-hand Mach wave 
 takes place along a left-hand characteristic, so that (l6) is valid. 
 The crossing of a left-hand Mach wave occurs along a right-hand charac- 
 teristic with (17) "being valid. 
 
 The transition from one field with the index i to the next with 
 the index i + 1 is regulated at the crossing of a right-hand Mach 
 wave , thus according to the equations: 
 
 ^i = ^i-'H 
 
 h+i = ^i+i + h 
 
 whence follows, with iS^ eliminated: 
 
 i+1 i \ i+1 ±j 
 
 (18) 
 
 For crossing of a left-hand Mach wave 
 
 % = -Vi + ^^ 
 
 %+l = -^i+1 + ^r 
 
12 NACA TM 1243 
 
 most "be valid or, after elimination of ^r: 
 
 Vi = ^i ■ f^i+i - ^iH ^^9) 
 
 According to the two equations (18) and (19) Meyer's tatle is used for 
 the approximated calculation of tvo-dimensional potential flows. 
 
 The selected example of Walchner (fig. 3 of Walchner's report Lufo 
 Bd. Ik (Aviation Eesearch, vol. Ik) p. 55, 1937) deals with the flow 
 atout a hlplane at an angle of attack, with relatively rough approxi- 
 mation since large Jumps in direction frcaa one field to the next are 
 admitted. Furthermore compression shocks are approximated "by Mach 
 waves which - in view of the weakness of the occurrix^g shocks - is 
 still Justified (of. chapter V, par. 5) • 
 
 The field numbers are put as indices to the pertaining flow 
 
 ^1 ^ Pi 
 
 values. In the initial field — = l.&v, — = 0.221, d-i = is 
 
 ^1 ^o • 
 
 prescrlted. According to the tatle the value v^ = l6° pertains to 
 
 this value of q./a or p/Pq- In field 2 iSg = -10°, due to the 
 
 geometrical "boundary condition. Since in the transition from field 1 
 
 to field 2 a right-hand Mach wave is crossed, V2 = 6° according to (18) j 
 
 ^1 
 according to the table, the pertinent values are — = 1.293> 
 
 Pi 
 
 — = 0.363. For field 3 one has, for geometrical reasons, -So = k'-'; the 
 
 Po 
 
 transition from field 1 to field 2 leads, according to (19), to v^, = 12° 
 
 "^3 ^3 
 
 with — = 1.50^4-, — = 0.270. The calculation of the q.- and i3-values 
 
 3 ^o 
 
 in the next field k represents the general case of the method. From 
 
 field 3 one arrives at field k hy the crossing of a right-hand Mach 
 
 wave, thus according to (18) : \ = '^•a + \ ~ t^o = 80 + i3|^. From 
 
 field 2 one arrives at field k "by the crossing of a left-hand Mach wave, 
 thus according to (19): v^^ = ^^ ~(^]^. " ^2) ^ "^° ~ '^ k' ^°^ ^^^ *^° 
 
 equations for i3|^ and ^ i^ set up Just now follows '^1^ = -6°, ^^^ = 20 
 
 "i-k Pk 
 
 with -~ = 1.132, — = 0.ivl4-9. In all remaining fields i3 ia preBcribed 
 
NACA TM 121(.3 13 
 
 by geometrical ■boundary conditions so that the calciilatlon is easy and 
 takes place very similarly to that for field 2 and field 3- One has 
 
 ^5 = ko, v^ = 12°, ^ = i.<^ok, ^ = 0.270; 
 
 ^g = -10°, v^ = 6°, ^ = 1.293, ^ = 0.363; 
 
 6 o 
 
 1 P 
 ^7 = -3°, v„ = 19°, £ = 1.71+3, ^ = 0.190; 
 ' 7 ^o 
 
 ^8 = "3^ ^8 = 13°, - = 1-538, - = 0.257 
 o o ag Pq 
 
 A drawing machine is desirable for plotting the Mach waves which 
 close any newly calculated field. 
 
 With a finer subdivision of the singular variations, a greater 
 accuracy seems attainable by means of this method than with the aid 
 of the Prandtl-Busemann method described below. 
 
 k. PRANDTL-BUSEMAM METHOD: 
 
 For approximated construction of two-dimensional potential flows 
 mostly the Prandtl-Busemann method is used, the main expedient of which 
 is a diagram with the characteristic hodograph curves (ik) and (15) , 
 respectively. 
 
 The Prandtl-Busemann method is, according to our terminology 
 Introduced in chapter II, (NACA TM 12l|2) , a field method, that is, a 
 pair of values q, ^ 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, it to 
 be valid precisely for the field center, the definition of which was 
 given in chapter II, paragraph 7, NACA TM 12I4.2.I We now visualize the 
 field centers as connected with each other; these connecting curves 
 
 The field centers are very useful for explanation of the method; 
 they are, however, in case of two -dimensional potential flows, in contrast 
 to rotationally-symmetrical ones, not required for the construction so 
 that the exact definition of the field centers would here not yet be 
 necessary. 
 
Ik WACA TM 12J+3 
 
 give, as was shewn, 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 left-hand Mach wave in the 
 flow plane is connected with a progressing along a right-hand charac- 
 teristic hodograph In the velocity plane. Correspondingly, crossing of 
 a right-hand Mach wave is coupled with progressing along a left-hand 
 characteristic hodograph. Busemann and Prelswerk gave a net of charac- ■ 
 terlstlc hodographs for *^ = l.i^-OJ* This diagram is customarily denoted 
 simply as 'characteristics diagram.'. 
 
 In order to calculate from the known 
 pairs of values q, i3 in field I and II 
 the unknown pair of values q., -3 in the 
 field III adjoining downstream, one pro- 
 gresses from the point in the character- 
 istics diagram corresponding to field I 
 along a right-hand epicycloid, since one 
 has crossed a left-hand 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 left-hand epicycloid. The point of 
 Intersection gives the pair of values q_, S 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 i3 found 
 Just now- The modifications of the method for fields at the "boundary 
 of the region are o"bvlous. 
 
 In order to facilitate the reading of the q- and -3 -values from 
 the characteristics diagram, one may take the net of the characteristic 
 hodographs as net of coordinates. According to (ik) and (l6) , respec- 
 tively, one has for the left-hand epicycloids: 
 
 ^- Hq) ='\ (20) 
 
 according to (15) and (17) , respectively, for the right-hand epicycloids 
 
 - -d - v(q) = -^ r (21) 
 
 Thus t3 and - through the ta"bulated function v(q) - q as well may be 
 very easily expressed by the parameters i3^ and d of the epicycloids. 
 
NACA TM 121+3 15 
 
 Instead of the parameters ^i and ^^ which probably first seemed 
 
 obvious, Busemann selected others with only the starting points of the 
 count shifted from "3 and v. The angles -9 and v are measured 
 in degrees. The degree sign (°) is omitted below- One may then express 
 Busemann' 3 epicycloid numbering so that as equation of the left-hand 
 epicycloids 
 
 d - V(q) = 2(X - UOO) (22) 
 
 as equation of the right-hand epicycloids 
 
 -S - v(q) = 2(n - 600) (23) 
 
 is written with the new parameters A, and m-- Hence there results 
 
 200 - ^ = V. - \ (24) 
 
 1000 - V (q) = ^l + X (25) 
 
 Thus the difference of the new parameters' X and ^ gives the angle i3 
 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 (^ = O) obtains the direction number 
 [1 - X equal to 200, whereas the sum of X and ^l 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) . Busemann writes the parameters X and [i 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 sectlonally constant 
 distributions with Jumps of +1° or +2°, one may assure by a suitably 
 fine-meshed characteristics diagram that one gets by without interpola- 
 tion. The customary characteristics diagrams are in their main part 
 arranged for angular Jumps of 1°. 
 
I6 ■ NACA TM l2l^■3 
 
 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 omit the carrying out of a standard example 
 since the method has been represented in detail by Heybey (HVT - Archiv 
 Wr. 66/31 and 66/32). 
 
 5. DEVELOPMENT OF THE PRESStKE VARIATION 
 FOE 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 left-hand family. 
 
 S>0 
 
 6 <0 
 
 This property of two-dimensional potential flows follows Immediately 
 from the characteristic differential eq.uations (13) which connect 
 velocity and directional variation. This property is by no means 
 transferable to other than two-dimensional 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 left-hand Mach waves, they may be calciilated 
 by progressing along a right-hand characteristic, thus according to 
 (13b) from 
 
 aci 
 
 dq = , ^ d.^ (13b) 
 
 Vc2 - °2 
 
 'q.^ - a*- 
 
 A development of the pressure difference for deflection of the 
 flow from the angle '^ = to the angle -8 = 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 1214-3 
 
 IT 
 
 ours. In order to develop the pressure, first the velocity ^2 which 
 
 is to pertain to i3 = 6 is developed according to equation (l3h) with 
 respect to 5. For i3 = 0, q is to equal q-, • We set up: 
 
 qg = q-L + c^ 6 + Cg 5 + c 6 
 
 3 
 
 (26) 
 
 and determine the unknown coefficients c-j^, C2, Co from (l3h) by com- 
 
 dq 2 
 
 parison of the coefficients- One has only to equate — = c-]^+ 2c2 5+3co 5 
 
 with the expression originating from 
 
 aq 
 
 ^ 
 
 2 2 
 q - a 
 
 when q is replaced 
 
 by q2 according to (26) . Therein is a^ = 
 
 k; - 1 
 
 (v/ - .^). 
 
 It is 
 
 most convenient to eqixare the two sides of the relation mentioned before 
 comparing the coefficients. There results 
 
 ci = 
 
 ^lll 
 
 Co = - 
 
 ll 
 
 tl^ 
 
 ai^ 
 
 C3 = 
 
 ) ^2 
 
 iltll 
 
 U(qi2 - ai2) = 
 
 [(. - Dq^^ + 2ai^ 
 
 I2(q,2 . ,^2)7/2 
 
 (k - 1)(2k - 3)q^6 
 
 > 
 
 + 9(^ - ^)^<i.^ + ^^^^ + 2ai6 ^ 
 
 (27) 
 
 In order to obtain from the development for q2 that of the pressure 
 P2, one starts from the isentropic pressure equation 
 
 Pi 
 
 Vm^ - 12^1^ 
 
 V 2 - n 2 
 m ^1 
 
 = <1 - 
 
 v^ 
 
 12^ " 1l^ 
 
 in 
 
 2 2 
 
 m - I1 
 
 (28) 
 
18 
 
 NACA TM 1243 
 
 This expression is tilncamlally expanded, making use of (26) and (27) 
 For trans fonnat ion of some of the terms one applies the formula 
 
 2k 
 
 ^"' yJ-^1^ 
 
 = Pi 
 
 which follows from the energy theorem (l(6)), and 
 
 K - 1/ 
 
 2 . n 2 
 
 2 V 
 
 V - ^1 
 
 with the Mach numher M = <l]_/a2_ one then finally ohtains 
 
 !__&__ ^ .p(m2 - 2)2 + kM^ 
 ^m2 - 1 
 
 oJ " c2 * 
 
 Pg - Pi = Pi<li2<-:= + 5 - 
 
 i+(M2 - D' 
 
 63 
 
 12(m2 - l)T/2 
 
 (k + 1)m8 
 
 (29) 
 
 (2^2 
 
 7*^ - 5)m6 + 1o(k - l)M^ - 12M2 + 8 
 
 The first term of this development may "be found already in Th. Meyer's 
 report j Busemann gives the development up to &3 (Volta congress, p. 337)- 
 It is true that the coefficient of 63 as calculated by us, completely 
 differs frcm 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 right-hand Mach waves, one has to 
 estahlish the connection with the characteristic differential equation 
 (13a) . The simple rule results that one has to select the sign of 6 in 
 (29) as it corresponds to a reflection on the freestream. direction of the 
 conditions assumed ahove (cf. the figures on p. 16) . 
 
MCA TM 1243 
 
 19 
 
 As application of this formula (29) we tise the calculation of Ca 
 and Ov for circular segment profiles in second approximation which 
 was already performed by A. Busemann. 
 
 According to the figure one has for the upper side (index oh) of 
 the profile b^-^ = -p - 3^^ and for the lower side (index u) 5-u = p. 
 
 If one replaces cos 5 "by 1, sin 6 hy 
 
 8, one has 
 
 Pob) 
 
 dl 
 
 <V 
 
 PuP - PohP - PohPoh) ^^ 
 
 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 p,j 
 and Vob- 
 
20 
 
 NACA TM I2k3 
 
 Pill 
 P - Pi = —J- 
 
 26 
 
 4^^ 
 
 + 6^ 
 
 m2 - 2)2 + kM^ 
 
 ,(m2 
 
 for which one writes ahtrevlatedly: 
 
 P - Pi = 
 
 ^ -'■ ^Ct5 + CgS^ 
 
 one ottalns 
 
 2 (m2 - 2)2 + kM^ 
 \/m2 - 1' ^ ' 4(m2 - l)2 
 
 ca = 2CiP - ^ 
 
 Poh ^^ 
 
 Therein It Is already taken into account that 
 
 P -u dZ vanishes 
 
 (for the expression for c^ we shall also use the vanishing of 
 / P - dl for circular segment profiles) . In the foregoing derlva- 
 
 
 
 oh 
 
 tion Cq Is then determined to Include terms which are quadratic In the 
 angles P and ^qi,- For c^^ one obtains analogously the expression 
 Including terms of the third order in the angles P and Pq-j^: 
 
 o2 ^ "1 
 
 Cn 1'^ . 3C2 _ 
 
 Cw = 2CiP^ + - P^^^ dZ 
 '0 
 
 t / ■ oh t I o'b 
 
 P 1 3.V dZ 
 
NACA TM 121+3 21 
 
 Since one has for approiimatlon of the circular arc by a parabolic arc 
 
 one obtains 
 
 /. 
 
 3ob' ^^ = - - 
 °^ 3 t 
 
 Therewith becomes 
 
 Ca = 2C 
 
 .^- - ^f (^) 
 
 . .0,3= . 1(^5 - 0.f^ 
 
 Kri-^^y 
 
 It is noteworthy that for the angle of attack P = a negative lift of 
 the circvilar segment profiles exists which waa also determined experi- 
 mentally. 
 
22 
 
 MCA TM 12J+3 
 
 TABLE 
 RELATION BETWEEN DEFLECTION, PRESSURE, VELOCITY, MACH NUMBER AND 
 MACH ANGLE FOR ISENTROPIC CHANGES OF STATE ACCORDING TO 
 PRANDTL -MEYER FOR AIR (k = l-J+O?) • 
 
 
 V 
 
 p 
 
 q. 
 
 1 
 
 , a 
 a = arc aln — 
 
 
 
 Po 
 
 a* 
 
 a 
 
 a 
 
 
 0° 
 
 0.527 
 
 1.000 
 
 1.000 
 
 90°00 ' 
 
 
 1 
 
 .1^78 
 
 1.067 
 
 1.081 
 
 67 45 
 
 
 2 
 
 .kk9 
 
 1.106 
 
 1.132 
 
 62 05 
 
 
 3 
 
 .k2k 
 
 l.l4o 
 
 1.177 
 
 58 10 
 
 
 h 
 
 .k02 
 
 1.170 
 
 1.217 
 
 55 20 
 
 
 5 
 
 .382 
 
 1.199 
 
 1.256 
 
 52 50 
 
 
 6 
 
 .363 
 
 1.225 
 
 1.293 
 
 50 4o 
 
 
 7 
 
 .3k6 
 
 1.250 
 
 1.330 
 
 48 45 
 
 
 8 
 
 .329 
 
 1.275 
 
 1.366 
 
 47 05 
 
 
 9 
 
 .3U 
 
 1.299 
 
 1.401 
 
 45 35 
 
 
 10 
 
 .299 
 
 1.322 
 
 1.4^6 
 
 44 10 
 
 
 11 
 
 .28U 
 
 1.3^5 
 
 1.470 
 
 42 50 
 
 
 12 
 
 .270 
 
 1.367 
 
 1.504 
 
 41 40 
 
 
 13 
 
 .257 
 
 1.388 
 
 1.538 
 
 40 30 
 
 
 Ik 
 
 .2kk 
 
 1.1+08 
 
 1.572 
 
 39-30 
 
 
 15 
 
 .232 
 
 1.428 
 
 1.606 
 
 38 30 
 
 
 16 
 
 .221 
 
 l.khl 
 
 1.640 
 
 37 35 
 
 
 17 
 
 .210 
 
 l.k66 
 
 1.674 
 
 36 40 
 
 
 18 
 
 .200 
 
 l.ii85 
 
 1.708 
 
 35 50 
 
 
 19 
 
 .190 
 
 1.504 
 
 1.743 
 
 35 00 
 
 
 20 
 
 .180 
 
 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 
 
 
 2k 
 
 .U5 
 
 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 121+3 
 
 23 
 
 RELATION BETWEEN DEFLECTION, PRESSURE, VELOCITY, MACH NUMBER AND 
 MACS ANGELE FOR ISENTROPIC CHANGES OF STATE ACCORDING TO 
 PRANDTL -MEYER FOR AIR (k = 1.405)- " Continued. 
 
 V 
 
 P. 
 
 0. 
 
 1 
 
 a 
 a = arc sin — 
 
 
 Po 
 
 a* 
 
 a 
 
 1 
 
 31° 
 
 0.097 
 
 I.70I+ 
 
 2.177 
 
 27020 ' 
 
 32 
 
 .091 
 
 1.720 
 
 2.215 
 
 26 50 
 
 33 
 
 .086 
 
 1.735 
 
 2.255 
 
 26 20 
 
 3^^ 
 
 .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 
 
 .0& 
 
 1.808 
 
 2.460 
 
 24 00 
 
 39 
 
 .058 
 
 1.822 
 
 2.502 
 
 23 35 
 
 ko 
 
 • 055 
 
 .836 
 
 2.5^5 
 
 23 10 
 
 1+1 
 
 .051 
 
 1.850 
 
 2.590 
 
 22 45 
 
 k2 
 
 .oi+7 
 
 1.864 
 
 2.6^5 
 
 22 20 
 
 h3 
 
 .okk 
 
 1.877 
 
 2.661 
 
 21 55 
 
 kk 
 
 • oi+i 
 
 1.890 
 
 2.728 
 
 21 30 
 
 ^5 
 
 .038 
 
 1.903 
 
 2.778 
 
 21 10 
 
 k6 
 
 .035 
 
 1.915 
 
 2.823 
 
 20 45 
 
 kl 
 
 .033 
 
 1.928 
 
 2.872 
 
 20 20 
 
 k8 
 
 .030 
 
 1.9^^0 
 
 2.922 
 
 20 00 
 
 h9 
 
 .028 
 
 1.952 
 
 2.974 
 
 19 40 
 
 50 
 
 .026 
 
 1.984 
 
 3.027 
 
 19 20 
 
 51 
 
 .021+ 
 
 .976 
 
 .081 
 
 18 55 
 
 52 
 
 .023 
 
 1.988 
 
 3.136 
 
 18.35 
 
 53 
 
 .021 
 
 1.999 
 
 3-191 
 
 18.15 
 
 ^ 
 
 .019 
 
 2.011 
 
 3.247 
 
 17.53 
 
 55 
 
 .018 
 
 2.022 
 
 3.304 
 
 17 35 
 
 56 
 
 .016 
 
 2.033 
 
 3-363 
 
 17 20 
 
 57 
 
 .015 
 
 2.044 
 
 3.424 
 
 17 00 
 
 58 
 
 .013 
 
 2.055 
 
 3.i^87 
 
 16 40 
 
2k NACA TM 1243 
 
 EEFEREWCES 
 
 1. Meyer, Th. : Uber zweldlmenslonale Bewegimgsvorgange In einem (Jas, 
 
 das mlt Uberschallgeschwindigkeit strarnt. Forsch. -Art- Tng. - 
 Wesen Nr. 622, 1908. 
 
 2. Ackeret, J.: Luftkrafte auf Flugel, die mlt grosserer Geschvindig- 
 
 kelt als Sciiallgeschwlndigkelt bewegt werden. ZEM Bd. 16, 
 pp. 72-^k, 1925. 
 
 3- Walcliner, 0-;^ Zirr Frage der Wider standsverrlngerung von Tragflii- 
 geln tel Uberschallgeschwindigkeit durch Doppel decker anordnung. 
 Lufo Bd. Ik, pp. 55-62, 1937. 
 
 k' Prandtl, L-, and Busemann, A.: Naherungaverfahren zur zelchnerl- 
 
 schen Ennlttelung von ehenen Stramiingen mlt Uberschallgeschwindig- 
 keit. Stodolafestschrlft, Zurich 1929- 
 
 5- Busemann, A.: Article from "Gasdynamlk", Handbuch der Experimental - 
 physlk Bd. k, Tellband 1, pp. U23-i^59, 1931. 
 
 6. Prelswerk, E.: Anwendung gasdynamlscher Methoden auf Wasserstro- 
 mungen mlt freler Oberfiache (Mlttellungen aus dem Instltut fiir 
 Aerodynamlk, Eldgenoaslsche Technlsche Hochschule, herausgegeben 
 von Prof. rtr. J. Ackeret, Nr. 7, Zurich 19380- 
 
 7' Heybey, W- , Das Prandtl -Busemannsche Naherungsverfahren zur zelch- 
 nerlschen Verfolgung ebener UberBchallstromungen. HVT Archlv Nr. 
 66/31 and Nr. 66/32, 19I+O. 
 
 8. Reale Accademla d' Italia, Fondazlone Allessandro Volta Attl del 
 
 Convegni 5> Convegno dl Science Flalche, Matematlche e Naturali 
 30 Sett. - 6 Ott. 1935. Tema: Le alte veloclta In avlazlone 
 Eoma 1936: Busemann, A. : Aerodynamlscher Auftrleb bel 
 Uberschallgeschwindlgkelten pp. 328-368 (abgedruckt in Lufo Bd- 
 12, pp. 210-220, 1935- )• 
 
 9' Busemann, A., and Walchner, 0.: Profllelgenschaften bel Uberschall- 
 geachwlndlgkeit. Eorschg. Ing.-Wes. Bd. k, pp. 87-92, 1933- 
 
 Translation by Mary L- Mahler 
 National Advisory Committee 
 for Aeronautics. 
 
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