hcAlW-!31f'f 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1369 
 
 FLAT PLATE CASCADES AT SUPERSONIC SPEED 
 
 By Rashad M. El Badrawy 
 
 Translation of "Ebene Plattengitter bei Uberschallgeschwindigkeit. " 
 Mitteilungen aus dem Institut fur Aerodynamik 
 an der E.T.H., no. 19, 1952 
 
 1 1 
 
 r,n\/FRSV 
 
 .^Y 
 
 
 32611-7011 Ui;^^ 
 
 Washington 
 May 1956 
 
CONTENTS 
 
 Page 
 
 PREFACE iii 
 
 INTRODUCTION 1 
 
 CHAPTER I. THE FLAT PLATE 4 
 
 1. General Considerations - Stipulations 4 
 
 2. Conditions at Expansion Around a Corner 6 
 
 5- Conditions of Oblique Compression Shock 8 
 
 k-. Lift and Drag of an Infinitely Thin Plate (Exact Solution) . . 10 
 
 5- Lift and Drag at High Mach Numbers 15 
 
 6. Calculation of Lift and Drag by Linearized Theory 15 
 
 7. Comparison of the Results of the Linearized Theory 
 
 With Those of the Exact Method I8 
 
 CHAPTER II. INTERSECTION, OVERTAKING AND REFLECTION OF 
 
 COMPRESSION SHOCKS AND EXPANSION WAVES 19 
 
 1. Introduction 19 
 
 2. Small Variations I9 
 
 5- Overtaking of Compression Shock and Expansion Wave 22 
 
 \. Intersection of Compression Shock and Expansion Wave 26 
 
 5. Crossing of Expansion Waves 31 
 
 6. Reflection of Compression Shocks and Expansion Waves 53 
 
 CHAPTER III. THE CASCADE PROBLEM 3^ 
 
 1. Problem 34 
 
 2. Method of Calculation 5^4- 
 
 5- Example 55 
 
 h. Calculation of Thrust, Tangential Force and Efficiency .... 57 
 
 CHAPTER IV. LINEARIZED CASCADE THEORY hO 
 
 1. Assumptions 40 
 
 2. Linearization of Cascade Problem •• k-Q) 
 
 5. Calculation of Lift and Drag •'+1 
 
 4. Numerical Example \h 
 
 5- Comparison With Exact Method ^4-5 
 
 CHAPTER V. SCHLIEREN PHOTOGRAPHS OF CASCADE FLOW if6 
 
 1. Cascade Geometry 46 
 
 2. Experimental Setup Kj 
 
 3. Schlieren Photographs 4-7 
 
 CHAPTER VI. THE FLAT PLATE CASCADE AT SUDDEN ANGLE-OF-ATTACK 
 
 CHANGE 49 
 
 1. Problem 49 
 
 2. The Unsteady Source 50 
 
Page 
 
 5. Pressiire and Velocity of a Periodically Arising 
 
 Source Distribution 52 
 
 k. Single Flat Plate in a Vertical Gust (Biot 19^4-5) 59 
 
 5. The Straight Cascade 61 
 
 6. N\:mierical Example 67 
 
 CHAPTER VII. EFFICIENCY OF A SUPERSONIC PROPELLER 7I 
 
 1. Introduction 7I 
 
 2, Effect of Friction on Cascade Efficiency 7I 
 
 5. Effect of Thickness 73 
 
 k. Appraisal of the Efficiency of a Supersonic Propeller .... "jk 
 
 SUMMARY 77 
 
 REFERENCES 78 
 
 TABLES 79 
 
 FIGURES 91 
 
 11 
 
PREFACE 
 
 The work on the present report was carried out at the institute for 
 Aerodynamics of the E.T.H., Ziirich, under the direction of Prof. 
 Dr. Ackeret, during the time from December 19^9 "to June 1951' 
 
 I want to express here my deep gratitude to Prof. Ackeret for his 
 s\;iggestions and for the great interest he took in my work. 
 
 I am very grateful to Dipl.-Eng. B. Chaix, scientific assistant at 
 the Institute, and to Mr. E. Hurlimann, precision mechanic, for their 
 indispensable help in taking the schlieren pictures. 
 
 I should like to acknowledge that the "Faruk University", Alexandria 
 
 (Egypt) made my studies in Zurick possible. 
 
 ixx 
 
Digitized by tlie Internet Arcliive 
 
 in 2011 witli funding from 
 
 University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation 
 
 http://www.archive.org/details/flatplatecascadeOOunit 
 
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM I569 
 
 FLAT PLATE CASCADES AT SUPERSONIC SPEED* 
 By Rashad M. El Badrawy 
 
 INTRODUCTION 
 
 The cascade problem in the subsonic range can be analyzed under 
 certain assumptions either by mapping or substitution of the blades by 
 singularities - sources, sinks and bound vortices - where the separation 
 of flow from the blades can cause various departures from the obtained 
 results. 
 
 Raising the flow velocity to a given value is accompanied by sonic 
 velocity within the cascade, which usually renders the solution of the 
 problem even more difficult. The same complication exists on the cascade 
 in flow at supersonic speed, in which the velocity is retarded to sub- 
 sonic by shocks. 
 
 But when the cascade operates entirely in the supersonic range, the 
 conditions become clearer. All disturbances act downstream only from the 
 soiirces of disturbance, so that the pressures and velocities at the svir- 
 face of a sufficiently thin airfoil in the stream can be readily determined. 
 
 The present report deals exclusively with problems of cascade flow 
 in the supersonic range . As is known the flat infinitely thin plate is 
 the best airfoil with respect to wave resistance in supersonic flows; 
 hence it is logical to start with the cascade of flat plates. The last 
 chapter deals with the case of finite thickness. 
 
 Lift and wave resistance of an Isolated plate are computed first 
 since the cascade problem can often be reduced to this special case. 
 The well-known theories of two-dimensional supersonic flow are applied - 
 that is, the laws of oblique compression shocks and the expansion around ' 
 a corner. 
 
 The air forces are then calculated again and compared with the pre- 
 viously obtained exact values by means of Ackeret's formulas of linearized 
 theory. 
 
 *"Ebene Plattengitter bei Uberschallgeschwindigkeit. " Mitteilungen 
 aus dem Institut fiir Aerodynamlk an der E.T.H. , no. 19, 1952. 
 
NACA TM 1369 
 
 The cascade problem was to be solved in such a way as to be free 
 from the inevitable inaccuracies of the graphical method. For this reason 
 the cases of overtaking, crossing and reflection of compression shocks and 
 expansion waves frequently occurring on supersonic cascade flows, which 
 usually are solved by graphical method, are analyzed in chapter II. 
 
 In chapter III the cascade problem is discussed and its solution 
 described in the light of the results obtained in chapter II. A nijmerical 
 example is also given. The same chapter gives further a definition of 
 the efficiency of the simple supersonic cascade and an evaluation for 
 several angles of stagger and attack. 
 
 The small angles of attack involved justified the use of a linearized 
 cascade theory.-'- This is done in chapter IV. The numerical example of 
 chapter III is thus linearized and the results compared with those of the 
 exact solution. The supersonic cascade flow at various angles of attack 
 was recorded by schlieren photographs of the flow between two parallel 
 plates, in the high-speed wind tunnel of the Institute (chapter V). 
 
 Chapter VT deals with the specific case of unsteady flow through 
 the cascade, caused by abrupt angle -of- attack changes. 
 
 -^According to Ackert's linearized theory, the lift and drag of a 
 double-wedge profile of thickness d and chord I at angle \|f in super- 
 sonic flow M is, in the presence of friction (cf) 
 
 ca = 
 
 k^ 
 
 \/ 
 
 Cw = 
 
 M^ 
 
 \/m2" 
 
 ^ . f^f . 2c, \/^ 
 
 For the best drar;-lift ratio e 
 
 — , put ^ = 0. This means that 
 Ca' dt 
 
 the wave resistance should be equal to the sum of friction drag and thick- 
 ness effect. In that event 
 
 ^opt 
 
 "Wopt 
 
 'opt 
 
 = 2^ 
 
 opt 
 
 21/^ + 2c. 
 
 ^Mi 
 
 Ass\iming possible values for d/Z and Cf results in comparatively 
 small optimum angles topt* 
 
NACA TM 1569 
 
 In chapter VTI the effect of friction and thickness in a special 
 case on the cascade efficiency is analyzed. Since there might be a 
 possible application of the supersonic cascade to the supersonic pro- 
 peller, a simple evaluation of the efficiency of such a propeller is 
 made. A parallel steady two-dimensional flow is - with exception of 
 chapter VI - postulated. 
 
 The conventional notation is used unless specifically stated other 
 wise in the text. 
 
NACA TM 1569 
 
 CHAPTER I. THE FLAT PLATE 
 1. General Considerations - Stipiilations 
 
 The general equation of continuity of any compressible flow is 
 
 Sp _^ 5(pu) _^ S(pv) _^ 5(pv) ^ Q 
 dt ^x Sy hz 
 
 (1) 
 
 The rate of propagation of a small disturbance, that is, the sonic 
 velocity, is, as is known 
 
 Sp p 
 Bp P 
 
 (2) 
 
 where k = 
 
 In flows, in which a flow potential cp exists, the continuity equa- 
 tion can be written as 
 
 o p p 
 1 |'Sp_^acpSp_^3cpBp_^^Sp\ B_cp _^ a_9 _^ B_^ ^ Q 
 
 pasyat ax Sx ay ay az azy ^^2 ^^2 ^^2 
 
 (5) 
 
 The momentum theorem gives the following relations 
 
 1 
 p 
 
 ap 
 ax 
 
 1 
 
 ap 
 
 p 
 
 Sy 
 
 1 
 
 Sp 
 
 p 
 
 az 
 
 ^^9 ^ ac^ aj^ ^ ac^ ^% ^ ^ ^% 
 
 ax at 
 
 ax -\ 2 5y ax ay az ax az 
 
 a cp acp a cp acp a cp acp a cp i 
 ay at ax ay ax ay ■% 2 az ay az 
 
 2222 
 a_cp^ ap a cp ap a cp acp a cp 
 
 az at 
 
 ax az ax ay az ay az ^^2 | 
 
 W 
 
NACA TM 1369 
 
 In two-dimensional flow the potential must therefore satisfy the 
 equation 
 
 S cp 
 ^2 
 
 1 - — ^ 
 
 a 
 
 2 dx 
 
 ;,2 
 
 d cp 
 
 
 1 - — ^ 
 
 2 2 
 
 25cpScpdq) 2^Scp 
 
 2 Sx By dx dy 2 Bx 3x St 
 
 2 2 
 
 2Scp Sep lScp_ 
 
 a2 By By Bt a^ St^ 
 
 (5) 
 
 The velocity of sound is then 
 
 a2 = a^2 
 
 (^ - 1) 
 
 L 
 
 ,Sx Uy/ 
 
 Bt 
 
 where aQ = velocity of sound in state of rest. 
 
 For the steady case, the equation is reduced to 
 
 ^2 
 d cp 
 
 Bx^ 
 
 1 - 
 
 l/Bcp\' 
 
 :s2 
 
 _^ 6 cp 
 
 1 - 
 
 l/Bcpf 
 
 2 
 
 2 Bq) Sep B cp 
 ^2 Sx By Bx By 
 
 = 
 
 (6) 
 
 (7) 
 
 and fhe flow is completely identified, if the function cp(x,y), which 
 is to satisfy the boundary conditions, is determined. 
 
 This equation is either elliptic, parabolic, or hyperbolic, depending 
 
 upon 
 
 (1 - m2) ^ 
 
 where 
 
 M = — grad cp 
 a I I 
 
 (8) 
 
 is the local Mach number. 
 
NACA TM 15b9 
 
 The use of this equation is difficult if its type in the particulair 
 range, as in the transonic range, is changed. However, the flows einalyzed 
 here, are of identical character everywhere, that is, the flow is of the 
 hyperbolic type. 
 
 One of the known solutions is the expansion around a corner, developed 
 by Prandtl and Meyer (ref. l) . 
 
 2. Conditions at Expansion Around a Corner 
 
 The two-dimensional flow past the wall AE at a Mach n\;mber M2_ 
 
 (fig. 1) is deflected by a convex bend at E through an angle 9, throxigh 
 which an expansion is initiated. The disturbance proceeding from E 
 spreads out solely in the range lying downstream of the Mach line EB3_, 
 
 where 
 
 ^ BiEA' = Mach angle p-i = sin"l ^ (9) 
 
 and stops at the Mach line EB2, where 
 
 ^ B2ED = M.2 = sin-1 — 
 
 M2 
 
 In it M2 is the Mach number of the flow after the expansion. 
 
 The streamlines in range B1EB2 are curved similarly and r\m 
 parallel to the wall ED downstream of this range. 
 
 It can be proved that the Mach lines in this flow are the character- 
 istics of the differential equation which define the potential. 
 
 When the expansion proceeds from a Mach ntmiber Mi = 1, ( ^^1 - o)' 
 the following relations can be proved (ref. 2) : 
 
 tan |i2 = A cot Tvtn (lO) 
 
 '^'\^ 
 
NACA TM 1369 
 
 P2 
 
 ic + 1 
 
 K-1 
 
 Pq i 2 cos^'kiij 
 
 (11) 
 
 (Pq = stagnation pressure) 
 
 Obviously 
 
 M2 =' 
 
 (k + 1) - 2 cos^Ato 
 ( Kc - l)cos ?\;a) 
 
 "^ 2 
 
 (12) 
 
 (13) 
 
 As function of M2 (ref. 3) 
 
 e = 
 
 cos 
 
 •1 -3^ + i A cos-l/l - 
 
 Mo 2 
 
 K + 1 
 
 K-1 2 
 
 1 + M2 
 
 (liv) 
 
 This equation gives a maximum angle of expansion Qmax) which cor- 
 responds to a Mach number M2 = " after expansion (k = l.ifOO) 
 
 ©max = 130.45° 
 
 If the Mach number before the expansion M]_ is assimied other than 1, 
 the maximum angle of expansion becomes obviously 
 
 e 
 
 max M]_ ~ ©max 
 
 - V 
 
 where v is the angle of expansion from M = 1 to M^. The values for 
 various Mach nimibers of the inflow (Mi) are 
 
 Ml 
 
 1.00 
 
 1.50 
 
 2.00 
 
 2.50 
 
 5 
 
 8 
 
 10 
 
 CX3 
 
 ®max Mi° 
 
 130.45 
 
 118.55 
 
 104.07 
 
 91.32 
 
 53.55 
 
 34.53 
 
 28.14 
 
 
 
NACA TM 1569 
 
 5. Conditions of Oblique Compression Shock 
 
 The discontinuities that may appear in supersonic flows and across 
 which velocity, pressure, density, temperature, and entropy undergo a 
 discontinuity, while the total energy, thermic and mechanical, remains 
 constant, were predicted by Riemann (1860) and Rankine and Hugoniot (1887) 
 as normal compression shocks. 
 
 In oblique shocks (Prandtl-Meyer) only the velocity component normal 
 to the shock front is modified. 
 
 In figure 2 the supersonic flow past the wall AE is deflected at E 
 by an angle B. A compression shock is produced and the shock front ES 
 is inclined at an angle 7 - the shock angle - toward the air flow 
 direction. 
 
 With subscript 1 denoting the state before the shock and subscript 2 
 that after the shock it can be proved that (refs. 3 and. k) 
 
 + 1\ 1 2k J 
 
 £2_ ^ 2k L 2„.^2., _ ^ - 1 ] 
 
 (15) 
 
 PO 
 
 . - i/£iy , 
 
 kK 
 
 k; + 1\ 
 
 POJ 
 
 {k + 1)2 
 
 .9 [k - 1 
 
 U + 1 
 
 1 + 
 
 kK . p 
 
 L (-^ - D' 
 
 li-1 
 
 (16) 
 
 1 K - 1\ 
 
 + 
 
 P2' " + Hu^^sln^y 2 
 
 (IT) 
 
 cot 6 = 
 
 [k + 1 Mi^ 
 
 2 • 2 
 
 lltan 7 
 
 \ '^ M]_^sin'^7 - 1 / 
 
 (18) 
 
 ^2 _ cos 7 
 
 u-|_ cos (7 - 5) 
 
 (19) 
 
NACA TM 1569 
 
 tan{y - 6) Pi 
 
 tan 6 
 
 K + 1 
 
 M2^sin2(7 - 6) 
 
 (20,21) 
 
 A direct relation between the Mach numbers before and after the 
 shock can be established 
 
 M2' 
 
 M-, cos (7 - & 
 
 cos y 
 
 (22) 
 
 The relation for the change of the static pressiore by the shock is 
 the same as for the normal shock when it is applied to the velocity com- 
 ponent perpendicular to the shock front. Consequently 
 
 PO1 
 
 1^ + 1 ., 2 • 2 
 2 1 ' 
 
 K 
 
 1 + 
 
 M-i^sin^y/ 
 
 2k .,2.2 ^ - 
 U + 1 -^ K + 2; 
 
 ,1-K 
 
 (25) 
 
 From these equations it follows that the shock angle 7 is greater 
 than the Mach angle, that is, the speed of propagation of a finite dis- 
 turbance is greater than the sonic velocity. When the angle of deflec- 
 tion 8 approaches zero, 7 = n and the shock changes to a Mach wave. 
 
 Also of interest is the shock angle at which the Mach number after 
 the minimum shock becomes equal to unity. Denoting this angle by 7^ 
 
 it can be proved (the weak stable compression shock is always allowed 
 for) (ref. 3, P- ^7) that 
 
 sin27s)(2 sin27g - ^ 
 
 = I sin273 - 1 
 
 ^sln^y^ 
 
 K + 1 
 
 (21+) 
 
10 NACA TM 1569 
 
 This equation is used to determine the maximum shock angle which 
 corresponds to a Mach niimber before the shock M-[_ = «> and a Mach n\im- 
 
 ber M2 ' = 1 after the shock. The result is 
 
 sin273 = ^-t2 (25) 
 
 hence 
 
 7s = 67.8° 
 at K = 1.400 (air) . 
 
 By equation (18) the corresponding deflection angle Sg is 
 
 5s = ^5-58P 
 
 2 
 Table 1 and figure 3 represent the values of 73 and 63 at 
 
 various Mach numbers M^. 
 
 k. Lift and Drag of an Infinitely Thin Plate (Exact Solution) 
 
 An infinitely thin plate ab in parallel flow at supersonic veloc- 
 ity Uj_ is placed at the angle 1^ . It is assumed that the width of the 
 
 plate transverse to the flow direction is 00, so that the problem is two 
 dimensional. 
 
 The streamlines above oa (fig. h) experience a deflection which 
 is associated with an expansion. So the state at the upper side of the 
 plate can be defined by equations (lO) to (l4). But below the plate a 
 compression shock ad occurs. The state of the flow on the lower side 
 of the plate is accordingly determined from the formulas (16) to (21). 
 
 The force on the plate per unit area Is 
 
 K = (pg' - P2) ' (26) 
 
 2 
 The weak stable shock is always taken into account. See Richter, 
 
 ZAMM, 19^8 and Thomas, Proc. N.A. Sc, Nov. I9I+8. 
 
NACA TM 1569 11 
 
 where pg ' and P2 represent the pressure on the lower and upper side 
 of the plate . 
 
 Obviously, the lift A and the drag W per unit width axe 
 
 A = K cos \|(L (27) 
 
 W = K sin i|fL . (28) 
 
 To compute a lift coefficient, a reference dynamic press\are of the 
 inflow 
 
 11 = 2 "l""! 
 
 or 
 
 ^1 = I Pl< (29) 
 
 is utilized. 
 
 As fimction of the Mach number Mj^, the ratio of dynamic to airstream 
 pressure is 
 
 '^l 1^ M 2 
 
 — = — Ml 
 
 Pi 2 
 
 that of dynamic to stagnation pressure is 
 
 !l= i!1m2= ^m2/!L^^Mi2 + 1^ ' (30) 
 
 PO 2 po 1 2 ^ I 2 ^ ; 
 
 The results are represented in table 1 and figure 5- 
 
12 
 
 NACA TM 1369 
 
 Lift and drag coefficients are herewith 
 
 c„ = 
 
 cos -if 
 
 sin >(/ 
 
 or, if all pressures are referred to stagnation pressure Pq^ 
 
 P2' P2 
 ^a, - cos \|/ ^M ~ sm \|( 
 
 P2 P2 
 
 Pq 
 
 ^ 
 
 Pq 
 
 (31) 
 
 (32) 
 
 The drag/lift ratio is 
 
 -w 
 
 e = -^ = tan \|/ 
 
 (35) 
 
 Table 2 gives the values of Cg,, c-^, and e up to y[^ = 10 as 
 computed by the formulas (52) and (35)' 
 
 In the calculation of the Mach numbers up to Mj = h, the tables 
 by Keenan and Kaye (ref . 6) as well as those by Ferri (ref . 3) were used 
 to define P2/P0 ^"^^ P2'/Pl (*^ = 1.400). 
 
 For higher Mach n\ambers, the formulas of sections 2 and 5 were 
 employed. At each Mach number, the angle of attack was varied up to 
 1's(M2' = !)• 
 
 Figures 6 and 7 show the variation of c and Cg^ over the angle 
 of attack \|;; figure 8 shows the polars Cg, plotted against c^. 
 
 The boundary curves show the maximum lift and drag coefficients that 
 can be expected without getting in the transonic range. 
 
NACA TM 1569 15 
 
 Other values for the boundary curve are given in table 5. Since the 
 pressure distribution on the upper and lower side is constant, the result- 
 ant force is applied at plate center and is normal to the plate. There is 
 no suction force as in subsonic flow. 
 
 5. Lift and Drag at High Mach Numbers 
 
 At high Mach numbers the angle of attack of the plate can exceed 
 the maximum expansion angle Qmax (section 2) corresponding to the Mach 
 
 number of the airstream ©max = ■*l's ^'^ ^1 = 6.J|). Hence, when assuming 
 continuous flow, an empty wedge-shaped zone between plate and flow appears. 
 This zone is largest at constant angle of attack when M3_ = ». in that 
 
 event, no deflection of flow is possible. 
 
 Owing to this vacuum space, the pressure at the upper side is zero. 
 The resultant force K is obtained then from the pressure on the lower 
 side, behind the compression shock. Hence, per unit area 
 
 K=P2'' (5^) 
 
 or, when referred to the dynamic pressure of the airstream. 
 
 K P2' 2 P2' 
 
 (55) 
 
 "^1 ^1 KMi^ Pi 
 Introducing P2'/Pl from equation (15) gives 
 
 K k . 2 K - 1 2 
 
 qi ^ + 1 ^ + 1 ^Mi^ 
 
 where the term containing 1M2. '^^^ ^® disregarded without great error. 
 
 K ^ . 2 
 
 <l± K + 1 
 
 sin'^7 (56) 
 
Ik 
 
 NACA TM 1569 
 
 So the lift and drag coefficient are 
 
 k . ? 
 
 — - — sin'^7 cos \|/ 
 
 K + 1 
 
 c^ = — - — sin'^y sm \li 
 K + 1 
 
 (57) 
 
 Both formulas are dependent on 7 and \(f only. Between these there 
 exist the relation given in equation (18), which can be written as 
 follows (5 = ^) : 
 
 cot ^ 
 
 'k + 1 
 
 2.2 1 
 sin'^7 - — ;: 
 
 1 tan 7 
 
 where the term can be disregarded again. Then 
 
 M' 
 
 a 
 
 / K + 1 \ 
 
 cot \l/ = I li tan y 
 
 2 sln^y 
 
 (38) 
 
 The values and curves designated with M = °o in table 2 and figures 6, 
 7 and 8 were defined by equations (57) ^^'^ (58). For comparison the lift 
 and drag was also computed by Newton's formula (the normal component) 
 
 Ca = 2 sin2\|f cos ^ 
 
 0-^=2 sin5\| 
 
 (59) 
 
 The corresponding values and curves cariy the subscript N. 
 
NACA TM 1369 
 
 15 
 
 6. Calculation of Lift and Drag by Linearized Theory 
 According to Ackert's linearized theory, the members of higher order 
 in — can be disregarded without great error in the potential equation 
 
 .2 
 
 cp 
 
 1 - 
 
 asW/ 
 
 _^ d cp 
 
 1 - 
 
 l/acp\2 
 a2\Sy/ 
 
 2 
 2 Sep Sep S cp 
 
 a2 Sx Sy Sx dy 
 
 = 
 
 for slender bodies at small angles of attack, because the interference 
 flows are small compared to that of the airstream. 
 
 The equation reads accordingly 
 
 ;,2 
 o cp 
 
 Sx^ 
 
 1 - u^^ 
 
 «2 Sx. 
 
 :,2 
 
 = 
 
 Inserting 
 
 U^ = m2 
 a2\axy 
 
 and observing that M is greater than unity, the equation reads 
 
 ^(m2 . 1) _ ^ = 
 
 (^0) 
 
 Sx^ 
 
 V 
 
 The general solution of this equation is 
 
 cp = f 
 
 fx - y\/i 
 
 M^ - 1 
 
 {hi) 
 
 It indicates, as stated in section 2, that the lines of constant 
 
 potential are the Mach lines of flow, and their slope has the Mach angle \i.. 
 
l6 NACA TM 1369 
 
 This solution shove further that the flow velocities 
 
 u = — ^ and V = — ^ 
 satisfy the condition 
 
 u = 
 
 (1.2) 
 
 \/m2 - 1 
 
 At the surface of a body in the streaim 
 
 U dx 
 
 is applicable . 
 
 The pressure variation by the momentum equation reads 
 
 ^ = - U AU = - Uu (43) 
 
 P 
 
 where U is flow velocity and u is interference flow in stream direc- 
 tion. 
 
 Accordingly 
 
 Z^ _ k dy 
 
 2' 
 
 IfU^ \/m2 - 1 
 
 ihh) 
 
NACA TM 1369 
 
 17 
 
 The pressure difference between both sides of a flat plate is 
 
 £sp 
 
 h dy 
 
 f 
 
 1 I / m2 _ 1 dx 
 
 (^5) 
 
 The lift and drag coefficients at the angles in question are 
 
 Co = 
 
 kji_ 
 
 \f 
 
 m2 - 1 
 
 'W 
 
 4\1(^ 
 
 / 
 
 m2 - 1 
 
 ih6) 
 
 The drag/lift ratio according to this theory is 
 
 -w 
 
 6 = — = * 
 
 (i+T) 
 
 Instead of the expansion wave and the compression shock at the 
 leading edge, it has simple Mach lines as interference line-s (fig. 9) j 
 in contrast to the exact theory. 
 
 The values given in table k and plotted in figures 10 and 11 were 
 computed by these formulas. The calculations were carried out at each 
 Mach number up to angle of attack i|(g - from the exact theory. The cor- 
 responding Cg. and Cv values lie on the curve G' . 
 
 At sonic velocity on the lower side of the plate P2'/Po ~ O.5283. 
 This value, introduced in the following directly obtainable relation 
 
 P2 
 
 Pi = 2 ^ = 
 
 2^1/ql 
 
 \/^ 
 
 (hQ) 
 
 - 1 
 
18 NACA TM 1569 
 
 and 
 
 P2 7po - Pl/Pp _ 
 
 _2JL 
 
 'll/Po ^Mi2 - 1 
 
 gives the angle of attack (ig^) corresponding to M2' = 1, which in gen- 
 eral is greater than i|/g . 
 
 7. Comparison of the Results of the Linearized Theory With 
 
 Those of the Exact Method 
 
 In table 5j "the difference is (cq - c^), where Cq is the coef- 
 ficient of the exact method and c^ is that of the linearized theory 
 at Mach numbers M^ = 1.^0 and M^ = 5'CK)- 
 
 It follows that the linearized theory is a very good approximation 
 for small angles (up to about 10°) • For greater angles the values of Cg^ 
 
 and c-^ are too small. 
 
 In figures 6 and 7j "the Cg^ and c^ curves by linearized theory 
 marked A' and A are included for comparison. 
 
NACA TM 1369 • 19 
 
 CHAPTER II. INTERSECTION, OVERTAKING AND REFLECTION 
 OF COMPRESSION SHOCKS AND EXPANSION WAVES 
 1. Introduction 
 
 Overtaking of expansion waves and compression shocks in supersonic 
 flows occurs when the marginal streamlines - or boundary walls - change 
 their direction twice in the opposite sense (fig. 12(a)). 
 
 If expansion waves or compression shocks strike a fixed wall and 
 their slope toward the wall does not exceed a given angle, they are 
 reflected as expansion waves or compression shocks (fig. 12(c)). Crossings 
 occ\ir in flows through channels and free jets (fig. 12(b)). All these 
 events can occur in cascade flows (fig. 12(d)). 
 
 2. Small Variations 
 
 (a) Suppose that a small expansion occurs at B in the supersonic 
 flow Ml, P2_, ai, p-j^ past the wall AB (fig. 13). The angle of 
 
 expansion is Zfi). If Z0 is sufficiently small, differential considera- 
 tions are permissible. 
 
 Bernoulli's equation gives 
 
 ^1 = " Pl"l ^1 ^^9) 
 
 where U is the magnitude of the velocity and AU its variation; /^ is 
 the pressure variation. 
 
 Since the vectorial velocity variation is normal to the Interference 
 line, the variation of U is 
 
 U-, Ae 
 a|u| = Ui tan m Ae = ^ (50) 
 
 Mi2 _ 1 
 
20 NACA TM I569 
 
 hence 
 
 ^ = - — ==I3 (51) 
 
 1/ 
 
 Mi^ - 1 
 
 But, as the dynamic pressure q is given "by 
 
 q = I PiUl^ = i KpiMi2 
 
 the pressure variation can be written as 
 
 kp-iMt^AB 
 ^ = - ^ ^ (52) 
 
 2 1 
 
 The variation of the Mach niomber M follows at (ref. 3^ P- 26) 
 
 Ml / . - 1 
 
 m = 1 + M-,2 Ae (55) 
 
 The Mach line BE^ forms with flow direction AB the Mach angle 11-, , 
 
 the Mach line BE2 at the end of the expansion the Mach line 112 • Now 
 
 it may be assumed that this small expansion takes place on the inter- 
 ference line BE ' , whereby 
 
 ^A'BE' = -^ (5^) 
 
 (b) A simple differentiation gives the change of the shock angle 7 
 as well as the pressure change P2' after the shock, due to a small vari- 
 ation of the angle of deflection 5, for the compression shock AB 
 (fig. Ik). 
 
NACA ™ 1369 
 
 21 
 
 Between 7 and 6 the relation (eq. I8) 
 
 ^Mi2 
 
 cot 5 = tan y\ - 1 
 
 iM-i^sin^y - 1 
 
 exists, and therefore 
 
 /^ = - sin^S 
 
 /A .\ .,2-2 2A 
 sec 7 1 - Mt sm 7 — 
 
 ^ B 1 T.2 
 
 A7 = C A7 (55) 
 
 where 
 
 A = 
 
 •^ + 1 ^A 2 
 
 and 
 
 B = ( Mi^sin27 - l\ 
 
 C = - sin'^^S 
 
 2 /A ,\ „ 2 • 2 2A 
 sec'^7j — - 1 - M-]_ sin'^7 -^ 
 
 lB 
 
 B^ 
 
 The 
 
 pressure pp' after the shock is (eq. I5) 
 
 2k; .. 2 • ? K + 1 
 P2' = Pi I ., , n M-^''sin'^7 
 
 IK + 1 
 
 K - 1 
 
 (56) 
 
 and the result for a small variation of the shock intensity is 
 
 2^t 
 
 AP2 = PiM^ sin 27 A7 
 
 k; + 1 
 
 (57) 
 
22 NACA TM I369 
 
 5. Overtaking of Compression Shock and Expansion Wave 
 
 (a) The supersonic flow Uj.) Pi^ P-\ past the wall AE (fig- 15(a)) 
 
 undergoes a directional change & at E. The compression shock EF and 
 the flow direction form the shock angle 7-, in zone (l). At C an expan- 
 sion takes place about an angle 9, and the expansion wave FCG overtakes 
 the compression shock at F. To simplify the calculation, the continuous 
 expression is replaced by a given number of expansion waves of finite 
 intensity, whereby a successive expansion through these waves is assured. 
 If n is the number of waves, the expansion due to a wave is &/n = A8. 
 The n\m.ber n must be so chosen that A6 is sufficiently small. 
 
 Now consider the intersecting of wave CF-i (i^ and compression 
 shock EFi(6), figure 15(b). 
 
 From Fi the compression shock advances with weeiker intensity in 
 direction G^, that is, it deflects the flow less - say by 6'. FiG^ forms 
 
 with the flow direction the angle 7' at (l). Indicating the various zones 
 by (1), (2), (3)^ and. (h) , the streamline through F^ splits the zone (k) 
 
 into the portions (i+o) and (^). The flow in (2) and (3) is fully known, 
 because the angles 6 and A0 are known. 
 
 To define the conditions in (k) , the streamlines S^, S2, and Sx 
 
 are examined. The directional change of S2 amounts to (5 - A©). But 
 along S^ the flow experiences the directional change 6'. To maintain 
 
 equilibrium in (h) , the pressure as well as the velocity direction above 
 and below the streamline Sj_ must be equal. 
 
 In general, the pressure change from (l) toward (3) is not the sajne 
 as from (l) to (k) , so that a reflected expansion wave - possibly a small 
 compression too - must appear between (3) and {k) , say along a line F2_Hi. 
 
 Supposing that this reflected wave is an expansion wave of inten- 
 sity A9' . By intensity" of an expansion wave or a compression shock is 
 meant the deflection, which the flow experiences in the process. 
 
 The pressure in (if^) is, (according to section 2) 
 
 Pi,,= P3(i-^=^Ae'] (58) 
 
NACA TM 1369 23 
 
 The difference of the deflection angles eunounts to (S' - 6) = Z^. In 
 general, this intensity decrease is small, because the compression shock 
 is much stronger than the expansion wave. The corresponding change in 
 shock angle 7 is A7. 
 
 The pressure in (4o) follows from the change in shock intensity. 
 Hence we can say that 
 
 Piio = P2' - ^2' (59) 
 
 This is again the equation for small variations derived for shocks from 
 equation (57) ■ Accordingly 
 
 Pk = Pp' - PnM-,^ ^ sin 27 Ay (60) 
 
 Posting PL = Pi, ^ gives 
 ■^o '^■u 
 
 kMt2 
 
 P3I1 - ■ A3' = P2' - PiMi^ -^ — sin 27 A7 (61) 
 ^m/ . 1 / ^ -^ 1 
 
 For the velocity direction in zone {k) to be lanequivocal, it must 
 
 ^ = /fi + A9' (62) 
 
 The relation between shock intensity variation and angle of shock 
 (eq. (55)) together with the two previous equations gives 
 
 4g - ^ 
 
 A9' = ^^ ^ (63) 
 
 JZ^QFG F J 
 
2k NACA TM 1569 
 
 with the constants 
 
 A7=^=^-^' (61.) 
 
 c c 
 
 itMj^ kM2^ M2_^sin 2y 
 
 Q = 
 
 x/Mj^ _ 1 \/M22 - 1 (Mi^sin^y - - 
 
 2k 
 
 The condition for equality of static pressures is not identical with 
 that for equality of velocity magnitude above and below the streamline S3_. 
 
 As the shock losses on either side of the intersection point F axe 
 unlike, the stagnation pressures in the wake above and below streamline Si 
 
 are different, hence there is a small vortex layer along this streamline. 
 
 Figure 16 represents the graphical solution of the problem by means 
 of the chajracteristics and the shock polars . The condition for equality 
 through equality of velocity magnitude in the entire zone (k) is 
 approximated . 
 
 (b) The reflected wave is disregarded: 
 
 In general, the angle of deflection Z^' - intensity of the reflected 
 wave - is very small (compare numerical example). Thus the pressure in 
 (5) is not much unlike that in (4), so that this reflected wave F-|_H-|_ can 
 
 be discounted. 
 
 In this event the flow directions in zones (5) and {k) are identical, 
 or in other words 
 
 Ae = -^ (65) 
 
 With equation (55) ^ can be defined and from it the new direction 7' 
 of the compression shock. The velocities in (3) and (k) have then obvi- 
 ously the same direction but not the same magnitude by reason of the small 
 vortex layer developing between (5) and (k) . 
 
NACA TM 1369 25 
 
 Numerical Example 
 Free stream: 
 
 Pl/Po = 0-12780 Ml = 2.000 
 
 Before overtaking: 
 
 shock intensity S = 6° 
 
 shock angle 7 = 35-24° 
 
 hence the state in zone (2) 
 
 P2/Pq = 0.1773 Mg =1.780 
 
 intensity of the expansion wave A0 = 1 
 state in zone (3): 
 
 P3/P0 = 0.1693 M3 = 1.818 
 
 Determination of constants : 
 
 c = 1.078 
 
 F = 3.0J+ 
 G = 3-015 
 
 Q = 3-175 
 Inserted in equation (63) and {6k) gives: intensity of the reflected wave 
 
 A7 = . 89° 
 Ae' = 0.06° 
 
 By equation (62) 
 
 z^ = Ae - A8' = 0.91+° 
 
 Therefore the shock intensity after overtaking is 
 
 6 = 5-04° 
 The new shock angle is 
 
 7' = 7 - A7 = 3^-55° 
 The reflected wave disregarded, leaves 
 
 A7 = c Z!S = c Ae = 1.078° 
 
 shock intensity B ' = 5° 
 
 angle of shock 7' = 3^-16° 
 
 It is readily apparent that the reflected wave is very small, hence 
 scarcely affects the pressure in zone (3)- 
 
26 NACA TM 1569 
 
 k. Intersection of Compression Shock and Expansion Wave 
 
 Figure l8(a) represents an expansion wave AG of intensity 9 
 proceeding from the corner A. At F this wave crosses a compression 
 shock of intensity 6 emanating from the corner B. 
 
 As before, the continued expansion is replaced again by n expansion 
 waves between which the flow is straight. The deflection by each wave is 
 
 n 
 
 After crossing (fig. 18(b)) the compression shock has an intensity 5' 
 and a shock angle 7'. Now the expansion wave has the intensity £0' . 
 
 The zones produced this way are numbered (l), (2), (3)j and (4) . 
 The streamline F^S splits the zone (k) into (^o) ^-^d. (^u)- 
 
 Looked for now is the shock intensity S', shock angle y' , and 
 expansion angle /^' after crossing, and the state of flow In (4), when 
 the state of flow in (l) 6, 7 and ^ are known. 
 
 According to chapter I the state of flow in (2) and (j) can be 
 determined directly. 
 
 The pressure in (^Hi) follows through a small expansion ^' according 
 to the laws in chapter II at 
 
 PV = P2 1 - Ae' (66) 
 
 \ \J^i^ - 1 j 
 
 where Z^ is still \inknown. 
 
 The method of solution consists in first making an assumption for 
 the shock intensity after intersecting, which is 
 
 5i = (& - AB) (67) 
 
 The corresponding angle of shock would be 7^^. 
 
NACA TM 1569 27 
 
 The state after this shock, indicated by ^q % can again be defined 
 according to chapter II. The pressure in (^q') i^ 
 
 no' - ^.[Tfi^i'^'^'^i - Hi] <'^«' 
 
 Now the flows in (^o) ^^'^ (2) have identical directions, but the 
 pressures and the magnitudes of the velocity are different. 
 
 To assure equilibrium within (k) , the pressures and velocity direc- 
 tions in (^o) 9-^d. (k^) must be equal. And to satisfy this condition the 
 assumed shock must be intensified by Z^-i_. 
 
 Obviously it shall 
 
 z:^! = Ae' (69) 
 
 The new pressure in (4o) is (according to section 2b) 
 
 2 2k 
 
 Pl^o = Pl^o' + P3M3 777 ^^"^ 27i A7i 
 
 (70) 
 
 where Ay^ is the change in shock angle y^ and is computed by 
 equation (55) 
 
 AS. 
 A7i = -pr- (Tl) 
 
 (For the calculation of C see section 2b) 
 
 Posting pij^ ^ "Pk, ' equations (66), (68), (70), and (7I) give 
 
 2 
 
 •^^2 \ j 2k 2 . p k - 1 \ 
 Pp 1 ^ \ = vj M^^sm^y. ] + 
 
 - 1 
 
 P3"3' dr ^^" ^^ ^ (T^> 
 
 This equation is linear in A0' . 
 
28 NACA TM 1369 
 
 Now the quantities 6' and 7' can be defined 
 
 S' = Si + Z^i = 5i + ZB' (75) 
 
 7' = 7i + A7j^ = 7i + (7^) 
 
 I 
 
 I 
 
 The pressiore in (^q) ^^^ ^^ obtained directly from equation (70) . 
 The Mach number Mj^ itself can be determined according to chapter I, 
 
 if S' and 7' are known. That in (4^) is likewise directly obtainable 
 from M2 by the isentropic expansion Z>0' • 
 
 The slight discrepancy between the values Mi^. and M}^ is due, 
 
 as stated in section J, to the fact that the condition for pressure 
 equality, owing to the change in static pressure after both shocks, does 
 not require equal magnitude of velocity. So a small vortex layer along 
 streamline F^S is to be expected. 
 
 Before intersecting the expansion wave forms with the flow direction 
 in zone (l) the angle (section 2) 
 
 X ^ ^^l + ^^3 - A9 
 
 After intersecting the angle with the stream direction in zone (2) 
 is 
 
 [in + \J^k - as' 
 
 y = 
 
 But there is a difference 5 between the flow directions in (l) 
 and (2), so that the looked-for directional change is 
 
 ^b = X + 5-Y= ^ + S (75) 
 
NACA TM 1569 29 
 
 The directional change of the shock front is 
 
 ): c = 7 - (7' + A9) (76) 
 
 A negative angle b and a positive angle c indicates that the 
 expansion wave or shock front after crossing is in more downstream 
 direction. 
 
 For illustrative and comparative purposes, the graphical solution 
 in figure 19 was made with the aid of the characteristics and shock polax. 
 Here also the condition for pressiore equality was replaced by velocity 
 equality. 
 
 The described mode of calculation is used in the following numerical 
 example for illustration. 
 
 Numerical Example 
 
 The flow in zone (l) is: 
 
 P-l/Pq = 0.22905 Ml = l.J+35 |i-^ = 1+4.18° (See cascade 
 example of the following chapter III, zone (5)). 
 
 Data before crossing: 
 
 intensity of expansion wave Z© = 1° 
 
 intensity of compression shock 5 = 3.05° 
 
 shock angle of compression shock 7 = 47-91° 
 
 With it the states in zone (5) become: 
 
 Pj/Pq^ = 0.28478 Mj =1.469 n = 42.89° 
 
 Therefore 
 
 P2/P1 = 1.157 
 P2/P01 = 0.34560 M2 =1-332 ^2 = ^7-75° 
 
 Assumed shock intensity 
 
 Si = 6 - A9= 3-03 - 1 = 2-03° 
 
30 
 
 NACA TM 1369 
 
 corresponding angle of shock y. =. 14.5.29° 
 pressure after shock Pi^ '/Pn ~ 0-3153 
 
 Determination of the constants 
 
 A = ^JL^M^^ = 2.592 
 
 B = (Mj-^sinS^i - 1) = 0.092 
 
 C = - sin25^ 
 
 ,2^ /A 
 
 . ? w 2 2A 
 
 n n^'v .M_ 
 
 sec'^y-;/— - 1 - sin^y-M 
 
 V ] - B' 
 
 1 -2 
 
 . 75014- 
 
 Equality of pressiore in (4o) and i\xi gives 
 
 Ph, = P2 1 
 
 KCMc 
 
 ^u 
 
 Ae' 
 
 w 
 
 1 2 2k . „ A3' 
 
 = P4„ = Pi,, + P5M3 ;rTT '"^ 2^i — 
 
 which inserted gives 
 
 Ae' = Z^i = 0. 
 
 A7i = — i = 1.19° 
 c 
 
 After the crossing: 
 
 shock intensity b ' = 8.;^ + i£>^ = 2.03 + O.89 = 2-92° 
 shock angle 7' = 7^ + Cij^ = I+5.29 + I.I9 = \G.h'^ 
 
NACA TM 1569 51 
 
 Hence 
 
 pjpo = 0.3303 ^i^ = 1^7-08° 
 Mk = 1.36^^-3 
 
 Mij. = i.36ifO 
 
 The comparison of the two Mach numbers indicates that the difference is 
 quite small and lies within the calciilation accuracy. 
 
 Directional changes: 
 
 Expansion Wave <^ b = ^4-3.036 + 3-03^ - h^.k^ = - l.k-° 
 
 Shock front <^ c = U7.91 - ^4-6.1^-8 + 1 = + 0.^4-3° 
 
 5. Crossing of Expansion Waves 
 
 Each expansion wave is again replaced by n small waves. In fig- 
 ure 20(a), two waves of intensity ZB]_ and ZBo cross each other in F. 
 
 After crossing, the intensities are ABn' and ZSo'- ^^ this case, only 
 
 one stream direction is obtained in zone (h) , when 
 
 /©l + /^2 = ^1' + ^2' ^^^^ 
 
 Application of the relations of section 2 results in 
 
 Vl^ = V^ - APj = P2 - ^2 
 
 that is 
 
 ^ A3g' = Po 1 - , A9i' (78) 
 
 MjS -11 \ l/MgS - 1 
 
32 NACA TM 1369 
 
 Since all other quantities are known, ABg' and ^^02_' can be com- 
 puted from these two equations. 
 
 The directional change of the Mach lines is like that in the pre- 
 ceding section 
 
 i^=- i i+A^L-^ 1 ^ (79) 
 
 Hi + II3 - 
 
 ■ Ae, 
 
 2 
 
 
 Hi + U2 - 
 
 ■ ^2 
 
 ^2-^ ^k- 
 
 ■ A8^' 
 
 2 
 
 
 U^+ ^k ■ 
 
 - Z^2' 
 
 ■^ c = + Z02 — (80) 
 
 Since all changes follow the same adiabatic curve, the condition for 
 pressure equality yields equal velocity values at both sides of the 
 streamline FS. Hence, no vortex layer will appear. Figure 20(b) 
 represents the graphical solution. 
 
 Numerical Example 
 Airstream: 
 
 pJpO = 0.3295 Ml =1.366 ^1 =: 1^7.05° 
 
 The intensity of the first expansion wave is: A9-]_ = 0.99°- Therefore 
 
 P2/P0 " 0-513^ M2 =1.1+02 H2 = ^5.50° 
 The second expansion wave intensity is: A8p = 1.06° 
 The conditions in zone (3) are then 
 
 Pj/pq = 0.312I15 M^ = l.^Ol+l Hj = 1+5 -1+33° 
 
 The two equations defining /^0^* and AS^' are: 
 
 A01 + z:^2 = 0-05579 = ^1' + ^2 
 
 2 2 
 
 K^^^ K;P2M2 ^ , 
 
 Pl+ = P3 - ^ ^ ^1 = P2 - ^2 
 
 \/m52 . 1 ^ 
 
 M2 - , 1 
 
NACA TM 1369 35 
 
 hence 
 
 AQ^.' = 0.01&1+ = 1.05^4-° 
 
 AG^' = 0.0174 = 0.997° 
 after which the conditions in zone (h) become: 
 
 PI^/PO = 0.29722 Mi^ = l.H ix^= 41^.01° 
 By equation (79) and (80) the directional changes of the waves are 
 
 ■^ b = + 0.5° and <^ c = - 0.02° 
 
 6. Reflection of Compression Shocks and Expansion Waves 
 
 No difficulties occur in the determination of the conditions existing 
 behind the reflected compression shock FB (fig- 21). Those in zone (2) 
 can be defined according to chapter I, if the state of the airstream and 
 the intensity 5 of shock AF are known. Obviously the reflected shock 
 is of the same intensity as the impinging shock, so that the shock angle 7 
 of the reflected shock and the conditions in zone (3) can be defined. 
 
 The same holds true for the reflection of expansion waves, when the 
 intensity of the expansion waves and their slope with respect to the wall 
 are known. 
 
Jlf- NACA TM 1569 
 
 CHAPTER III. THE CASCADE PROBLEM 
 1 . Problem 
 
 Visualize a cascade of infinitely many and infinitely thin flat 
 plates, of which two adjacent plates AB and A*B* are represented in 
 figure 22. The angle of stagger is 90 - P, the spacing t and the 
 blade chord L. This cascade is exposed at angle of attack a(/ to a 
 supersonic flow M3_, pj_, p-, . 
 
 It is assumed that the flow is the same in all planes perpendicular 
 to the plates and determines the force, that is, lift and drag as well 
 as the pressure variation along the plate (blade) . 
 
 2. Method of Calculation 
 
 To each plate there correspond interference lines (chapter I), that 
 is, the expansion wave issuing from the leading edge and compression 
 shock (fig. 22). 
 
 At wide spacing, the separate blades of the cascade will not affect 
 each other and the problem reduces to the single plate. 
 
 Now if the spacing decreases for constant chord, the Interference 
 lines of one plate intersect those of the other, without, however, any 
 force being exerted on the plates themselves for the time being. In 
 this event, the force on each plate is the same as on the single plate, 
 except that the wake flow is slightly disturbed. 
 
 The values of t/l, below which the interference line of a plate 
 begins to exert an effect on the adjacent one, are called ("^/L) ^^pj^-f^ 
 
 "critical chord-spacing ratio." 
 
 At t/l < [t/L^pj^-f;] the interference lines are reflected on the 
 
 plates. After the crossings and reflections, new zones appear on both 
 sides of the plate where the pressure as well as the velocities are 
 unlike the uniform pressures and velocities to be found at either side 
 of the plate. As a result, there is a change in the total force as well 
 as the lift and drag on each plate . 
 
 The mode of calculation consists in defining each intersection and 
 reflection with the laws of chapter II and from it determining the con- 
 ditions in the several zones. Integration of the various pressures on 
 both sides of the plate gives then the total force, that is, the lift 
 and drag. 
 
NACA TM 1569 35 
 
 The resultant force still is perpendicular to the plate, but no 
 longer through the plate center, hence produces a moment with respect 
 to the center. The position of the force is defined by statistical 
 methods . 
 
 This method is illustrated in the following example. 
 
 5 . Example 
 
 The cascade ABA'B' (fig. 23) with 30° angle of stagger, that is, 
 P = 60°, and at angle of attack of i|f = 3° is placed in a stream with 
 Mj_ = l-^OOij- (corresponding to v = 9°) • 
 
 The' blade spacing was assumed at the beginning, while the plate 
 chord was so chosen after completion of the calculation that the expansion 
 wave was reflected exactly once on the bottom side of the upper plate. 
 It was found that t/l = 0-5^+7. 
 
 The flow experiences a compression shock starting at the leading 
 edge a'. The shock angle 7 = ^9-57° is read from the shock tables 
 and the shock front A'a can be plotted. 
 
 Proceeding from the leading edge A, an expansion wave spreads out 
 between the Mach lines Ax and Ay. The first forms with the air stream 
 direction the angle (i-, = k'^.^6 . The characteristics tables give 
 
 My = 1.503, that is, the Mach nuunber which is obtained at an expansion 
 
 by 3° from the Mach number l.ij-OO^J-. The corresponding Mach angle, that 
 
 is, the angle which direction Ay forms with the plate, would be 
 
 |j.Y = Ul.70°. Instead of the continuous expansion, assume an expansion 
 
 in three stages, each corresponding to a 1° deflection. The conditions 
 in zones 1, 2, 3j and k are obtained from the characteristics tables, 
 after which the directions Aa, Ab, and Ac can be defined. 
 
 By applying the methods of chapter II to the calculation of the 
 crossings a, b, c, e, f, g, I, m, n, p, q, s and the reflec- 
 tions d, h, i, k, o, r, u, the static pressures, the Mach numbers 
 (table 6), and the intensities of the expansion waves and compression 
 shocks, as well as their directional changes (see table 7 ^^'^ fig- 2k) 
 in the several zones, can be determined. 
 
 The static pressures were referred to the standard stagnation pres- 
 sure Pq^. 
 
 The stagnation pressure changes were disregarded in the determina- 
 tion of the Mach number. This change is rather small according to table 6, 
 
56 
 
 NACA TM 1369 
 
 so that no appreciable advantage was to be gained by including it. 
 calculation of Po/Pn ' compare eq. (23).) 
 
 (For 
 
 The pressure distribution past the plate is obtained immediately and 
 represented in figure 25. There the passage of compression shocks and 
 expansion waves is accompanied by a sudden pressure variation. Since the 
 actual expansion is continuous, the serrated line is replaced by a smooth 
 curve, such that the areas decisive for the force calculation are identica 
 
 Note that the pressures on both sides of the plate cancel out over a 
 large portion of the chord. The resioltant force can be determined by 
 integration of the various pressures; the various spacings I are read 
 directly from figure 23. 
 
 The plate width was assumed at b 
 
 The result is 
 
 = 0.2963 upper side 
 = 0.2896 lower side 
 
 Downward resultant force 
 
 K 
 
 POn^ 
 
 0.0067 
 
 lift coefficient 
 
 K ^0. 
 
 POi^ ^ 
 
 cos ^ = 0.01532 
 
 drag coefficient 
 
 K ^Ol 
 c^ = sin \lf = 0.00082 
 
NACA TN 1369 
 
 37 
 
 k. Calculation of Thrust, Tangential Force and Efficiency 
 
 (a) The resultant force on the blade is resolved into two components. 
 One - the thrust S - is normal to the plane of the cascade, the other - 
 the tangential force T - parallel to it (fig. 26). 
 
 If 
 
 K = resultant force per unit of area 
 
 (90° - p) = cascade stagger angle 
 
 then 
 
 S = K cos p 
 T = K sin p 
 
 (81) 
 
 As functions of lift and drag 
 
 S = A cos(p - 1);) - W sin(p - \|() 
 
 T = A sin(p - i|/) + W cos(p - a|;) 
 
 Referring the force to the dynamic pressure q-j_ of the airstream, 
 'ives the coefficients 
 
 c„ = — cos p 
 
 ^n.= ^ Sin P 
 ^1 
 
 (82) 
 
 similar to the lift and drag coefficients, which can be obtained directly 
 from c^ and c^. 
 
58 NACA TN 1569 
 
 At fixed blade chord and fixed angle of attack the resultant force 
 reaches its maximum value when adjacent blades do not affect each other, 
 that is, when t/l > (t/L) ^^^^- In this event 
 
 K = P2' - P2 
 
 where P2 ' is pressure at lower side (behind compression shock) and 
 P2 is pressure at upper side (behind expansion wave) . 
 
 For a given Mach number of flow and angle of attack the thrust and 
 tangential force is maximum at |3 = 0° and p = 90°, respectively. 
 
 At a given angle p and a given Mach number, T and S increase 
 with increasing \|r. Owing to our assumptions \|f may not exceed vj/g, in 
 
 order to prevent subsonic flows on the bottom side of the plate. 
 
 (b) Definition of efficiency (no friction): 
 
 It is supposed that the air enters normal to the plane of the cascade 
 at a speed v (fig- 26) . The cascade moves with the tangential velocity \ 
 and finds itself accordingly in a relative flow with an angle of attack ^, 
 whereby tan(p - ^1^) = v/u. As a result of this flow, the two forces S 
 and T normal and parallel to the plane of the cascade act on the plate; 
 S and T are defined according to previous considerations. An efficiency 
 is defined as on a propeller, by visualizing the blade being driven at 
 speed u with respect to force T and so producing a force S in axial 
 direction on the flowing air. Then the power input is T X u, the power 
 output S X V and the efficiency is 
 
 n = 1^ (83) 
 
 Tu 
 
 or 
 
 r _ tan \|/ \ 
 ^ ^ tan(p - t) ^ V " tan p/ 
 
 tan p 1 + tan \j; tan p 
 
 The efficiency is seen to be dependent on \lr and p only. At 
 constant p it decreases with increasing i|/. At \|f = constant, tj has 
 a maximum, if 
 
 ^=0 (8lf) 
 
 5P 
 
NACA TM 1369 59 
 
 that is, when 
 
 tan (3 = tan ^ + \/(tan \|/) + 1 
 which approximately gives 
 
 P = ^5° + i^f • (85) 
 
 2 
 
 The maximum efficiency is then 
 
 .2 
 
 1 - tan I 
 %e.=( -\ (86) 
 
 •ma^ , ^ 
 
 \|/ = Constant \1 + tan 
 
 At small values of -i/, tan i|( = t and -^ is negligibly small, hence 
 
 -o - 1 
 
 at (3 = i+5 + - ilf 
 2 
 
 1 + \lf 
 
 The efficiency for various p and \(/ is represented in table 8 and 
 figure 27. 
 
kO NACA TM 1369 
 
 CHAPTER IV. LINEARIZED CASCADE THEORY 
 1. Assumptions 
 
 The theory is based upon the following: 
 
 (a) All disturbances are small in the sense that all interference 
 lines may be regarded as Mach lines. The expansions axe simply concen- 
 trated in a Mach line and the compression shocks replaced by Mach com- 
 pression waves. 
 
 (b) Intensity and direction of waves are not chajiged by intersection 
 of expansion ajid compression waves. The justification of this assumption 
 is indicated in the preceding numerical example, where it was shown that 
 the directional changes of the wave fronts are small, as a rule. 
 
 On these premises, the interference lines AA' and AA" parallel 
 to BB' and BB" start from the small disturbances A and B 
 (fig. 28(a)). At the intersection in a the directions of the waves AA' 
 and BB' as well as their intensities remain iinchanged. The pressure 
 ajid the velocities in the zones (2), (3), and (4) are defined by the laws 
 of chapter II. In the hodograph these assiomptions imply that the char- 
 acteristics network in the applied zone is replaced by a parallelogram 
 (fig. 28(b)). 
 
 2. Linearization of Cascade Problem 
 
 The application of these simplifications to the solution of the 
 cascade problem produces parallel Mach lines within the cascade, which 
 remain parallel after crossings or reflections (fig. 29(a)). (L = plate 
 chord, t = spacing and \|f = angle of attack.) The Mach lines Aa 
 and A' a emanate from the leading edges A and A'; the angles aA'X' 
 and aiiX axe Mach angles and both equal to ij.-, . On passing through A 'a, 
 
 the flow experiences a compression and a directional change \|f, along Aa 
 an expansion with the same directional change. 
 
 The pressure in (2) and (3) can be defined by the laws of isentropic 
 expansion and compression (chapter I); that of zone (4) is computed the 
 same way from the pressure in (2) and is obvioxxsly equal to p-,, as seen 
 
 in the hodograph (fig. 29(b)). But the flow direction in (4) differs 
 from that in (l) by an angle 2\|f. 
 
 The Mach line aC' intersects the plate at C' and is reflected 
 along C'E, whereby C'E is parallel to A'a. The pressure in zone (5) 
 is again equal to that in (3) and the flow is obviously parallel again 
 to the plate. 
 
NACA TM 1369 In 
 
 On passing through DE' the flow from (4) and (6) is compressed - 
 the reflected wave DE' - so that in (7) the direction and the velocity 
 of flow are the same as in (l); the same applies to the flows in (6) 
 and (2) . 
 
 Thiis it is seen that the corresponding zones repeat themselves, hence 
 that the further conditions are completely known without new calculations. 
 
 5. Calculation of Lift and Drag 
 
 The pressure variation on either side of the plate can be plotted 
 (figs. 29(c) ajid 29(d)). The pressure remains constant over the lengths 
 AC, CD, DE, EF and FB and over A'C', C'D', D'E', E'F' and F'B' 
 where the interference lines strike the plate. 
 
 Along CD the pressijres on both sides are equal and cancel out, 
 whereas a downward pressiore difference p:z - P2, obviously perpendicular 
 
 to the plate, acts on AC and EF, and an identical upward pressure dif- 
 ference on DE. 
 
 The pressure pattern in figure 29(e) repeats itself in length direc- 
 tion of the plate over the period Lj^. If the plate chord is chosen 
 
 exactly like Lj^ or a multiple of it, there is no resultant force, that 
 
 is, a plate of this length has neither lift nor wave resistance. 
 
 For the values of L, which satisfy the inequality 
 Lq < (L - nLi) < (Li - Lo) 
 
 whereby n can be = 0, 1, 2, . . . , the resultant force reaches its 
 maximxjm value, and then 
 
 K = (P3 - P2)Lo (88) 
 
 Hence it serves no usefiil pijrpose to make the plate longer than I^, 
 because there is no more lift increase anyhow. On the other hand, a 
 moment occurs and, in the presence of friction, the drag would increase 
 unnecessarily. The boundary Lo(= AC) is the plate length not touched 
 by interference lines of the other plate and can be defined geometrically 
 in terms of cascade spacing t and angles p, y, and \i. 
 
42 
 
 NACA TM 1569 
 
 sin p - (n^ + ^)\ sinTp + {^^ + )|/)| 
 Lo = t ^- -^ Li = 2Lo + t ^- i ^ (89) 
 
 sinda^ + \lf) 
 
 sin(ia-L - ^) 
 
 Accordingly the best ratio of spacing/chord is 
 
 t t 
 
 sin(ii-L + i|f) 
 
 L lo sin[p - (^i^ + ^^ 
 
 (90) 
 
 Now Ca. and c^ can be determined when 
 
 1. L = nL-|_ then <^a = "^w = ^ 
 
 2. Lq < (L - nL±) < (Li - Lq)? "the boundary values are 
 
 K 
 
 Cg^ = COS \|f = 
 
 P3 - P2 
 
 F COS \(/ 
 
 K P3 - P2 
 c^ = sin i|f = ^ F sm \)/ 
 
 q-^L 
 
 ^1 
 
 (91) 
 
 where 
 
 F= t 
 
 sm 
 
 [p - (n-L + ^)] 
 
 L sin(|j.-]_ + ij/) 
 
 5. Lq > (L - nil) 
 
 P3-P0 
 
 Cn = (L 
 
 -w 
 
 q-LL 
 
 P3 - P2 
 q-l_L 
 
 (L - 
 
 nLi)cos 
 
 
 
 ' 
 
 nL2.)sin 
 
 ^ 
 
 
 J 
 
 (92) 
 
NACA TM 1369 
 
 45 
 
 k. (L - nil) > (Li - Lo) 
 
 Co = 
 
 P3 - P2 
 
 q-j^L L 
 
 (L - nLi) - (Li - Lo) 
 
 cos ij/ 
 
 Cw = 
 
 _ P3 " ^2 
 
 ^1 
 
 L L 
 
 (L - nLn) - (Li - Lo) 
 
 sin \|; 
 
 (93) 
 
 The linearization can be extended to the pressures P2 and p^; 
 
 admittedly then only when the angle of attack is sufficiently small. ^ 
 
 The pressures can be defined by the laws of small variations 
 (chapter II). Thus 
 
 P2(3) = Pi 
 
 KM-, >!/ 
 
 1 + -^ 
 
 M-|_^ - 1 
 
 (9^) 
 
 Inserting these values in the above formulas for Cg^ and c-^^, while 
 expressing the dynamic pressure with 
 
 qi = I Kp^Mi^ 
 
 and the values 1 and ■<]/ for cos \(f and sin ^, gives as for the 
 isolated plate, 
 
 ^In the following table the pressiires after expansion of Pq = 0-31^04 
 (corresponding to M^ = 1.400^4-) are represented in terms of the expansion 
 angle: 
 
 P2 = pressure according to isentropic law of expansion 
 
 Pp^ = pressure according to the laws of small variations (chapter II) 
 
 V 
 
 1 
 
 P2/P0 0.29906 0.281+78 0.27114 0.25809 0.23363 
 
 P2l/p0 0.29865 0.28335 0.26718 0.25266 0.22196 
 
 (P2 - P2l)/p2 Pei'cent 0.1 O.5 I.5 2.1 5.0 
 
kk 
 
 NACA ™ 
 
 1369 \ 
 
 k^ 
 
 
 (95) 
 
 y^ 
 
 The factor F approaches 1 when t/L = t/Lg. The theory is now illus- 
 trated on the following numerical example. 
 
 k. Numerical Example 
 
 I 
 
 The cascade of the numerical example in chapter III is applied again 
 with the same airstream as by linearized theory, figure 50- 
 
 It was 
 
 t/L =0.5^7 3 = 60° 
 Ml = 1.400^4- f = 3° 
 
 p^/pq = o.3i^oit ui = 1+5.56° 
 
 The Mach lines within the cascade can now be plotted. By equation {i 
 
 Geometrically defined are 
 
 so that 
 
 Lq = O.lMl-L 
 
 (Li - Lq) = 0.T78L 
 
 (L - Li) = O.O78L 
 
NACA TM 1369 k^ 
 
 The tables of characteristics give 
 
 P2/P0-, = 0.2711 (expansion by 5° starting from P-,/pq ) 
 
 px/p„ = 0.56^ (isentropic compression by 3°) 
 Assuming the plate width at one cm, gives: 
 
 resultant force = (L - Li) — = 0.3^36 
 
 POl Po^ 
 
 resultant force per unit length = O.OO67 
 lift coefficient Cg, = O.OI56 
 drag coefficient c^ = O.OOO8 
 
 The pressure distribution on both sides and the resultant pressiire 
 are shown in figure 31- 
 
 5. Comparison With Exact Method 
 
 Instead of the lengthy calculations of all crossings and reflections, 
 the linearized theory affords a quick and simple solution of the cascade 
 problem. At small angles the results are reliable and the errors small, 
 as seen from the comparison with the numerical examples in section 3j 
 chapter III and the preceding section. 
 
 ca( exact) - ca( linearized) _ , 
 — ^ ^ = -2 percent 
 
 "-^a( exact) 
 
 The interference lines of the linearized solution within the cascade 
 the Mach lines - are included in figure 23 for comparison. It is seen 
 that the zones governing the resultant pressure are smaller by linearized 
 theory. 
 
 The pressure distribution of the linearized example is also shown 
 in figure 25. 
 
k6 NACA TM 1369 
 
 CHAPTER V. SCHLIEREN PHOTOGRAPHS OF CASCADE FLOW 
 1. Cascade Geometry 
 
 A disturbance in supersonic flow is known to spread out only down- 
 stream of the source of disturbance. So the pressures and velocities 
 on one of the sides of a profile, stipulated by the form of the sijrface, 
 are not influenced by the other side. 
 
 This property is used to represent the flow through a cascade con- 
 sisting of a number of infinitely thin plates. Two profiles with a flat 
 surface on one side are so assembled that their flat sides face each 
 other and are parallel. The flow between the parallel sides is exactly 
 the same as that between two adjacent plates of the cascades. 
 
 The two profiles can be moved apart or shifted relative to one 
 another, so that any desired ratio t/L and any stagger angle caxi be 
 obtained. 
 
 The experimental cascade was patterned after the cascade in the 
 numerical example of chapter III, which had the same angle of stagger 
 of 50°. The Mach number of flow was - as in previous calculations - 
 M = l.i+O; the spacing ratio was t/L = 0.517' The angle of attack i|f 
 ranged from 0°, 1-5°, 3° to I+.50. 
 
 The maximum profile thickness was so chosen that no blocking of the 
 tunnel (section 2) was produced at the selected Mach number and that the 
 deflection of the profiles at maximum axigle of attack is small. 
 
 Now at M = l.UO the deflection due to compression shock, which 
 exactly leads to sonic velocity, is Ss = 9°- As there is to be no sub- 
 sonic flow in the test section and since the angle of attack was assumed 
 at 4.5°> "the leading edge of the profile may at most form an angle of 
 about k° , which corresponds to the constructed profile. 
 
 The compression shock is not separated at the leading edge of an 
 infinitely thin plate or an infinitely shajrp wedge of sufficiently small 
 included angle. Therefore the leading edge shall be as sharp as possible. 
 It succeeded in attaining a thickness of O.O5 — O.O7 mm so that the 
 
 distance of the separated shock from the edge is scarcely visible. 
 
 The profile chord L was II8 mm, so that the cascade lies within 
 the tunnel window. Since the tunnel itself was i+OO mm wide, the width 
 of the profile was limited to 398 mm, figiire 32. 
 
 ^Hurit Co., Affoltern, Zurich. 
 
NACA TM 1569 k^ 
 
 2. Experimental Setup 
 
 The previously described profiles were mounted in the test section 
 of the supersonic tunnel of the Institute5 on four supports (fig. 33) • 
 
 The compression shock issuing from the leading edge of the top 
 profile could not be reflected at the upper tunnel wall at maximum \[(, 
 because the deflection to be made retrogressive at the wall was too great 
 for the Mach number prevailing behind the shock. To avoid blocking in 
 this region, a bend had to be made in the upper nozzle wall (fig. 3*+) • 
 The position of the bend was so chosen that the fan of expansions emanating 
 from it hits the cascade downstream from the entering edge. This adjusts 
 the wall to the flow direction after the shock to some extent as well as 
 raises the Mach number between the upper plate and the nozzle wall. 
 
 The Mach number in the test section before the cascade was deter- 
 mined by pressure measurements at the upper, lateral, and lower walls. 
 The investigation was carried out at a moisture content of air of about 
 . 5 g water/kg air . 
 
 3. Schlieren Photographs 
 
 The schlieren photographs illustrating the flow thro\jgh the plate 
 cascade at \|; = 0°, 1.5° and 3° are represented in figures 36, 31, 
 and 38- Since a conical jet regime is involved, the photographs appear 
 as shadows of the profiles. Figure 35 shows the position of the optical 
 axis with respect to the cascade; it is seen that the shadows of the pro- 
 files are distorted on the mirror. At the top profile the perspective 
 effect is more obvious, because the optical axis is closer to the bottom 
 profile. 
 
 The equality of Mach's angle in figure 37 (i = 0°) is' indicative 
 of an unchanged Mach number in the cascade. The visible disturbances 
 within the cascade may be due to the fact that the plate surfaces do not 
 exactly agree with the flow direction, or to thickening of the leading 
 edges by a boundary layer. 
 
 In figure 38 (i|/ = 1.5°) "the interference lines inside the cascade 
 are almost parallel, as stipulated by the linearized theory. 
 
 5See Report No. 8 of the Institute for Aerodynamics, at the E.T.H, 
 ref. 1. 
 
 For description of schlieren apparatus see Report No. 8 of E.T.H. 
 Institute . 
 
1+8 NACA TM 1569 
 
 At ilf = 5° (fig- 39) the deflection of the shock front at crossing 
 
 of the expansion wave emanating from the top leading edge is plainly- 
 visible. Figure kO represents an enlargement of the crossing to illus- 
 trate the numerical example in chapter III. The interference lines inside 
 the cascade for this example are again shown in figure kl at smaller scale 
 (compare also fig. 23), whereby the perspective effect is indicated. 
 
 In the majority of photographs the retardation of the flow near the 
 tunnel wall leads to separation of the head waves. 
 
 The flow in all photographs is from left to right. 
 
NACA TM 1569 ^9 
 
 CHAPTER VI. THE FLAT PLATE CASCADE AT SUDDEN 
 
 ANGLE-OF-ATTACK CHANGE 
 
 1. Problem 
 
 Visualize a cascade of flat plates in a flow with relative velocity W 
 at an angle of attack i/ . A supersonic flow which may be regarded as two- 
 dimensional prevails throughout the cascade. At a given moment the angle 
 of attack of the airstream changes from i|/ to i|f' within an infinitely 
 short time interval. The transition to the new state, which is to last 
 for a period, is analyzed. 
 
 Such a change in the angle of attack takes place when the cascade 
 moves in an absolute flow which has not the same speed at every point, 
 or when one of the velocity components of the flow, normal or parallel 
 to the plane of the cascade, varies with respect to time. 
 
 Resolving the velocity W in two components V and U (fig. h-2) 
 normal and parallel to the plates, the change of the angle of attack, 
 small in itself, can be regarded as a change of component V. This change 
 in V is obtained by superposition of a velocity vg, which has the same 
 
 direction as V and is obviously small compared to V and consequently 
 smaller than sonic velocity. From the assumption of a small angle of 
 attack, it follows that velocity component U remains greater than sonic 
 velocity. Besides, an eventual variation of this component U is 
 disregarded. 
 
 The problem therefore reduces to the study of the new forces on the 
 cascade, resulting from a gust vg 7 which, together with the velocity U 
 
 enters perpendicular to the plates. 
 
 Biot (ref . 5) solved the problem of an isolated plate by means of 
 "unsteady sources." This method is applied to the cascade problem. But 
 first the unsteady source is described in more detail. Since the plates 
 are to be partly replaced by such sources, the pressures and velocities 
 originating from a source distribution are analyzed. Then Biot's results 
 for the isolated plate are correlated and extended to the cascade. The 
 special case of straight cascade (nonstaggered) is examined. 
 
 By "gust" is meant a continued, uniform vertical velocity distri- 
 bution Vq. 
 
50 
 
 NACA TM 1569 
 
 2. The Unsteady Soijrce 
 
 According to linearized theory, the general potential equation (5) 
 for two-dimensional unsteady flow can be simplified to 
 
 ^1 . m2) + ^ - 2 ^ ^ 9 _ i ^ = 
 5x2 ^^2 aSxSt a2^^2 
 
 (96) 
 
 cp = flow potential. 
 
 For a system of coordinates moving with velocity U(u/a = M), that 
 is, air at rest at infinity, this equation gives the acoustic wave eqiia- 
 tion for two-dimensional motion 
 
 :i2 :,2 , ^^2 
 
 dcp^dcp ldcp_Q 
 
 bx^ By2 a2 at^ 
 
 (97) 
 
 One solution for a linear sound source is 
 
 1 , _i at 
 cp = k cosh -^ — 
 ^ r 
 
 (98) 
 
 where r = ux^ + y^ and K = constant with dimensional length times 
 velocity. 
 
 This solution is rewritten in the form 
 
 cp = klogel^.J^ 
 
 = k logg 
 
 -k log. 
 
 iat 
 
 W 
 
 a2t2 
 
 {' 
 
 I at + \la.^t^ - r^ 
 
 (99) 
 
NACA TM 1369 
 
 51 
 
 It represents a cylindrical wave varying in time rate. At t = r/a, 
 
 9 = 0, that is, if such a sing\ilarity appears in the zero point of the 
 
 coordinate system, its effect is diffused inside a circle of radius 
 r = a-t . 
 
 If such a source appears at the point (x,0) - on the x-axis - at 
 
 period t^, the potential in a point P(x,y) of the surroundings of this 
 
 source at a given period (fig. ^3) is 
 
 cp = k loge 
 
 i(t - t-i) + \/a'^(t - t 
 
 ,2 ^2 
 
 (100) 
 
 In this case 
 
 (x - xi)2 + y2 
 
 and the following velocity components are obtained by simple differentiation 
 
 ^ -]_ a(t - t] 
 
 v-p = — ^ = k — 
 Sr ^ 
 
 y^a2(t - ti) 
 
 2 - r2 
 
 (101a) 
 
 Sep 1 (^ - ^1) 
 ^x r'^ 
 
 i(t - ti) 
 
 ^a2(t - ti)2 - r2 
 
 (lOlb) 
 
 y by r2 
 
 a(t - ti) 
 
 \/a2(t - ti 
 
 2 - r2 
 
 (lOlc) 
 
 When y is small compared to a(t - t3_) - near the source 
 mula (lOla) becomes 
 
 the for- 
 
 Vr = k/r 
 
52 NACA TM 1569- 
 
 the saine as that of a steady wave in incompressible flow, hence with Q 
 denoting the strength of the source (dimensional length x speed) 
 
 2rt 
 
 The pressure in the same point is computed by 
 
 ^ = -p ^ = P^ ^ (102) 
 
 St 2rt 
 
 ^a2(t - ti)2 - r2 
 
 It will be noted that Vy always equals zero for y = 0, except at r = 
 
 in the source itself. It means that such a source delivers at no other 
 place on the x-axis a velocity component parallel to the y-axis. 
 
 3. Pressure and Velocity of a Periodically 
 
 Arising Source Distribution 
 
 Consider a continuous distribution of infinitely small sources over 
 the length OA (fig- k-k) along the negative x-axis. The distance OA 
 increases linearly with the time: OA = Ut, where U is a constant 
 velocity and the sources on the x-axis appear momentarily at the point 
 where A arrives at the moment. The strength of this source distri- 
 bution per unit length of OA is assumed equal to q (dimension of a 
 velocity) and remains constant in time. 
 
 (a) Pressure 
 
 At point P(x,y) (fig. 44) the pressure p of the source distri- 
 bution at time T is, by equation (l02) 
 
 p . P2i r-° "^ — (105, 
 
 '^'■■^^1=-"* \/a2{t - ti)2 - (x - xi)2 - ,2 
 
 t-j_ the time of origin of the source in point X]_. 
 
NACA TM 1369 
 
 55 
 
 With the following variable transformation 
 
 ^ = ^ 
 
 at 
 
 ^1 = 
 
 atn 
 
 at 
 
 a 
 
 
 1 
 
 
 sm |j = 
 
 
 U 
 
 
 M 
 
 (lOi^) 
 
 we eet 
 
 P 
 paq 
 
 1 
 2^ 
 
 d^ 
 
 -1/sin M >/uT^7^I^r7^2T77T^ 
 
 (105) 
 
 The boundaries shoiold be defined before the integral is evaluated. 
 For the function in the denominator is real only in the zone affected 
 by the source distribution; this is bound by the Mach line AM and the 
 circle with center and radius a-t. Hence the integration must be 
 made between the zero places of the function where it is real. 
 
 Posting 
 
 (1 + ^1 sin n)2 - (^ - ^i)2 _ Ti2 = 
 
 (106) 
 
 the new boundaries are found at 
 
 C 
 
 (1) 
 (2) 
 
 ■(^ + sin |j.) 
 
 4 
 
 (1 + sin ^l) - T) cos |i 
 
 -COS^IJ 
 
 (107) 
 
 To get an idea of the integrating process as function of the posi- 
 tion of point P, ^1^^ and ^i^^) are plotted in terms of ^. It 
 
 results in two curves of the second degree, which cross in point 
 q(^,^i) (fig. ii5(a)) whereby 
 
5ii 
 
 NACA ™ 1369 
 
 ^ = - 
 
 r\ cos n 
 
 sm ^L 
 
 ^1 = 
 
 cos u 
 
 ,sin ^i cos |i. 
 
 (1 
 
 The shaded area represents the r^^nge in which the integration should 
 be made. At small (, values up to ^ = \/l - r]^, integrate between ^q_^ '^ 
 
 and t!^2_ ^^^ then between ^-j ^ ' and the (, axis. In figure h-^ih) 
 
 the integral limits are shown p .ted in the x,y-plane for explanation. 
 The reason for not integrating over positive ^2_ values is the absence 
 of sources in the right-hand half plane. 
 
 Two integration cases are differentiated 
 
 1 - -n cos [1 
 
 sm 
 
 that is 
 
 OS ^\ ^ r 
 
 and 
 
 {^ 
 
 T)'^ < ^ < + 
 
 and 
 
 a - ^ cos ^\ ^ ^ ^ _ r2^2 _ ^2 
 sin |i / V ■' 
 
 \/ 
 
 a2t2 - y2 < 
 
 x<+ \l 
 
 a2t2 - y2 
 
 {^ 
 
 >(ic 
 
 In the first instance the pressure integral is 
 
 paq 2n J i-^{l 
 
 (2) 
 
 d^ 
 
 ^ \/(l+ Ci sin n)2 _ (^ _ ^^)2 _ Ti 
 
 (lie 
 
NACA TM 1569 55 
 
 But as ^-1^^ and ^-,(2) are the solutions of the expression below 
 the root, it can be rewritten as 
 
 c (2) .. 
 
 paq ~ 2it J p_(l) 
 
 ' /(..'^' - c.)(^.'" - 
 
 With the substitution 
 
 this integral gives 
 
 Pc 
 
 paq 2 cos [i 
 
 a formula that is independent of tj = y/at. 
 
 In the second case, if point P is so situated that 
 
 ^1 - Ti2 < ^ < + J] 
 
 it results in 
 
 « 
 
 paq 2n J/- (l) 
 
 ^1 \/(l + ^1 sin ^,f - U - ^i)2 - Ti^ 
 
 (111) 
 
 ^1= ^1^'^ + (Ci^2) - ^i(l))sin2e (112) 
 
 (115) 
 
 (11^) 
 
 Q 
 
 As long as the function in the denominator can be brought, with 
 the aid of the integral limits, into the form of equation (ill), the 
 integral gives the same value . 
 
56 
 
 NACA TM 1569 
 
 which after evaluation gives 
 
 paq jt cos |j. 
 
 COS' 
 
 (^ + sin ]i) 
 
 2 2 2 
 
 (1 + ^ sin |j) - T] cos n 
 
 (115) 
 
 The ensuing pressure pattern along a line y = Constant is repre- 
 sented in figure k6. For each y the pattern consists of two pieces. 
 In the first piece the pressure is constant and equal to p^,, that is, 
 
 along the length EF between the points where the Mach line emanating 
 from A and the circle with center and radius a-t intersects the 
 line y = Constant. The second piece is composed of length 
 
 FO' = - \/a2t2 - y2 and O'G = ■\-\j a^t^ - y2, where the pressure is vari- 
 able; at G the pressure is zero. At y = y^ the constant portion 
 
 disappears and wherever y = at, the pressure becomes zero. 
 
 At y = it represents Blot's case with the integral limits 
 
 sin \i. 
 
 < ^ <1 
 
 and 
 
 that is 
 
 at 
 
 sm n 
 
 < X < - at and 
 
 -1 < ^ < + 1 
 
 -at < X < + at 
 
 (116) 
 
 The pressure Pp has the same value as before 
 
 paq 2 cos \i. 
 
 (117)- 
 
 but the second piece of the pressiore distribution becomes 
 
 %=0 
 
 paq 2rt cos n 
 
 cos" 
 
 — - + sin n 
 at 
 
 \l + — sin |j. J 
 at 
 
 (118) 
 
 9q corresponds to Blot's 2Vf- 
 
NACA TM 1369 
 
 57 
 
 It should be noted that a pressure effect appears also outside the 
 area in which the sources are distributed, because the source in 
 affects the area inside the circle a-t as mentioned before. 
 
 (b) Velocity Calculation 
 
 In general, the velocity component is defined by the integration 
 of the portions stemming from a single source (eq. (lOl)). In our problem 
 the velocity Vy is of particular interest. 
 
 It becomes 
 
 q. 
 
 ■>xn=0 
 
 'xi=-Ut 
 
 a(t - ti)y dxj 
 
 (x - xi)2 + y- 
 
 -(t - t^)^ - (x - xi)^ + y^ 
 
 If r = \/(x - X3_) + y^ is small compared to a(t - t3_), that is, 
 for the places close to the x-axis, this equation simplifies to 
 
 2« 
 
 ^X1=0 
 
 -Ut 
 
 y '^1 
 
 [(x - xi)2 + y2] 
 
 _q^ 
 
 2rt 
 
 tan -'- 1 - tan"-'- — 
 
 (119) 
 
 Letting y approach 0, positive y, the results for negative values 
 of X are 
 
 1 
 
 -y=2 
 
 (120) 
 
 which may be designated by Vq (as in Biot's report). 
 
 For positive x values, v^. = 0. 
 
 It indicates that such a source distribution gives a uniform vertical 
 velocity Vq over the distance of the x-axis where the sources are. (For 
 negative y, inverse velocities result.) 
 
58 
 
 NACA TM 1569 
 
 Biot mentioned this fact in his report and used it to calculate the 
 pressijre distribution over a plate in a vertical giist (compare next sec- 
 tion) . 
 
 The same variable change as in the pressure calciilation gives 
 
 ^y==- 
 
 ^1 
 
 (2) 
 
 (1) 
 
 Ti(l + t,-^ sin ^i)d^-L 
 
 (^ - ^i)2 + r\^ 1/(1 + ^1 sin ^) 
 
 (121) 
 
 The arguments for the integral limit are the same as for the pres- 
 sure integral and ^]_^ ' ; ^2. is given by equation (l07)- Integrating: 
 
 between ^-]_ and t^j^ , that is, when 
 
 '1 - T) cos \x 
 sin |i 
 
 < 
 
 ^<l/ 
 
 1 - T)' 
 
 it is seen that the integral gives the value — , so that v.^, has the 
 constant Vq for this range of ^ . 
 
 For the second case 
 
 between ^-1^^ and indicates that 
 
 (- l/l - rjS < t^ < +y 1 - Ti2 j the integration 
 
 
 jt - tan" 
 
 ^\/l - (^2 + Ti2) 
 
 sin |i(^2 + T^2) _ ^ 
 
 (122) 
 
 This identifies the velocity distribution on the lines y = Constant 
 (fig- ^7)- 
 
 10, 
 
 See note on p. 66. 
 
NACA TM 1369 59 
 
 From the calculation of the pressure and velocity distribution over 
 the lines y = Constant, it is apparent that the flow outside the circle 
 of radius a-t is steady. This is true from the physical standpoint too, 
 since the gust front does not affect this axea. 
 
 k. Single Flat Plate in a Vertical Gust 
 
 (Biot 1945) 
 
 The flat plate AB of length I at supersonic velocity U' enters 
 a gust vrith the transverse velocity Vq (fig- ^8(a)). Since the trans- 
 verse velocity on the plate must be zero (no flow throijgh plate) an equal 
 and opposite velocity is superposed on the gust velocity vq in place 
 
 of the plate. This velocity can be visualized as reflection of the gust 
 on the plate (fig. ij-8(b)). 
 
 Since velocity U is greater than the sonic velocity, the sides of 
 the plates are not affected by one another, so that one side of the plate 
 can be analyzed separately. The pressure acting on one side is exactly 
 the same as on the other, except with inverse prefix. As the interference 
 velocity Vq is much smaller than velocity U, the linearized potential 
 
 equation can be applied to the stream potential. Selecting a system of 
 coordinates that moves with the velocity U, (eq. (97)) according to 
 which the disturbances are diffused with sonic velocity, can be applied 
 to the flow potential. 
 
 Biot's method replaces the part AO of the plate struck by the 
 gust, by unsteady sources. This source distribution, which increases 
 in time, yields a uniform velocity Vq normal to the plate, hence sat- 
 isfies the boundary condition on the bottom side of the plate. 
 
 If the plate enters the gust at time t = 0, the distance at time t 
 is AO = -Ut, the origin of the coordinate system being located in the 
 gust front . 
 
 The results of section 5 can be applied directly, and the pressure 
 variation along the plate defined (fig. k9{a)) . As it is dependent 
 solely on x/at the patterns are like those for the different Mach num- 
 bers. The total force on the plate - the lift - is obtained by integra- 
 tion of the pressure pattern. Three phases are involved here (fig. ^9): 
 
 I (U + a)t = Z 
 
 that is, the trailing edge is outside the effective range of the gust 
 front; 
 
6o 
 
 NACA TM 1369 
 
 II (U + a)t > Z > (U - a)t 
 
 that is, the trailing edge is inside the effective range of the gust 
 front ^ 
 
 III (U - a)t ^ Z 
 
 that is, the entire plate is outside the effective range of the gust 
 front, hence is no longer exposed to any unsteady effect. 
 
 The integration gives the following lift values of the three phases; 
 
 At 
 
 Ut 
 
 2pavoZ I sin |j. 
 
 (123) 
 
 ^11 
 
 2pavQZ Jt cos \x 
 
 cos 
 
 1 
 
 sm |_L 
 
 Ut 2 I 
 
 COS'^U I 
 
 I / 
 
 . ut . _i 
 + — sm -'- 
 
 Til 
 
 sin M.\Ut 
 
 1' +^- 
 
 (121|) 
 
 -'~4]he integral 
 
 I = 
 
 ^ d^ 
 
 (C - A)\ 1 - ^ 
 
 appears in the calculation of Aj and Ajj. With no boundary, the solu- 
 tion is 
 
 I = sin-1^ 
 
 A . Jl + Ai 
 sm--!- =• 
 
 \/a2 - 1 \A+ ^ 
 
 which gives 
 
 I = « 
 
 between the limits -1 and +1. 
 
 A 
 
 \/ 
 
 a2 - 1, 
 
NACA TM 1369 61 
 
 The sin"-^ to be taken between - q '^ a-^d. + — jt . 
 
 Am 
 
 2pavo^ cos p. 
 
 (125) 
 
 In phase I and II the lift increases continuously with the time and 
 reaches a maximum in phase III, where it becomes independent of the time. 
 In the last phase the lift is the same as on a plate at angle vq/U in 
 steady flow. 
 
 5- The Straight Cascade 
 
 The cascade problem is unlike that of the plate to the extent that 
 the plates mutually interfere. The sources replacing the portion of the 
 plate struck by the gust create a pressure on the adjacent plates. They 
 also produce a velocity v^, which in order to satisfy the boundary con- 
 dition of no through flow of the plate, makes a change in that source 
 distribution necessary. 
 
 Since the disturbances are small the solution of the single plate 
 can be superposed in the sense of the linearized theory of the adjacent 
 plate effect. 
 
 As shown in sections 2 and 5, the unsteady source - and the source 
 distribution - which lies on the x-axis, produces no vertical velocity 
 component along this axis, outside the distance, where it is. 
 
 This characteristic enables the velocity component Vy to be 
 
 replaced by an additive so\irce distribution along the particular parts 
 of the plate, which gives the velocity at each point. The new sources 
 create a further pressure on the plate itself - and in general react on 
 the adjacent plates. 
 
 The total force - the lift - on each plate consists then of the 
 lift of the undisturbed plate (A^), the lift from the pressure Py of 
 
 the sources of the adjacent plates {A^) and the lift (Ay) of the new 
 soiorce distribution due to velocity Vy. 
 
 Suppose that h is the plate spacing and L the plate chord of 
 the straight cascade ff (fig. 50).. The lines AM . . . represent the 
 
62 
 
 NACA TM 1369 
 
 Mach lines emanating from the leading edge, where ajigle MAB is the 
 Mach angle \i = sin" u/a. 
 
 Then the following approximation is made: the relative flow and 
 
 the plate form in reality the angle ^ - tan"-'-( vq/u) . But as vq is 
 
 small compared to U, this \|f is negligibly small with respect to ^i, 
 and it can be assumed that the Mach line itself rather than the relative 
 flow direction forms the angle |_i. In this event, the Mach lines form 
 the same angle with both sides of the plate. The Mach lines emerging 
 from the leading edge strike both sides of the plate at the same distance 
 AE from the leading edge. At (h/Z) > tan |i the points are not located 
 on the plates, and the plates do not influence each other. Consequently 
 the cases where (h/Z) < tan |j. are examined. 
 
 At time t = the cascade is directly in front of the gust; the 
 origin of the coordinates is placed in the gust front. In the first 
 time intervals of the phenomenon the disturbances have not spread out 
 enough to be able to influence the adjacent plates. As in figure ^1, 
 the distance is a-t < h, so that the circles with center and 
 radius a-t do not touch the plates. Lift and pressure distribution 
 are the same as on the single plate. 
 
 As soon as t > h/a, the plate AB comes within the effective range 
 of its adjacent plates. On EFG (fig. 52) the source distributions A'O' 
 and a"o" create an additional pressure which can be computed according 
 to section 5- The points F and G are then the points of intersection 
 of both circles with center O' and O" and radius a-t with plate AB. 
 It is readily apparent that the additive pressure on EF is constant and, 
 according to equation ( II3) , has the value 
 
 Pc 
 
 pavQ cos |i 
 
 (126) 
 
 If T] = h/at is inserted (y = h) in equation (II5), the pressure 
 on FG follows at 
 
 Ph 
 
 cos 
 
 -1 
 
 pavQ Jt cos |i 
 
 — + sm |i 
 .at 
 
 X ^ h2 p 
 
 1 + — sin [i\ - cos |j. 
 
 at ^ ^ 
 
 a2t2 
 
 The ensuing additive pressure is represented in figure ^2. 
 
 (127) 
 
NACA TM 1369 63 
 
 In addition, the following condition must be satisfied: The normal 
 velocities Vy = h created by the source distribution A'o' and a"o" 
 
 are reflected on EO, so that at that point the gust is partly compen- 
 sated. The source distribution to be applied is to compensate the veloc- 
 ity (vq - V ) . The velocity Vy is computed as in section 3> su^cL the 
 
 pressure py - along the particular plate - is obtained by integration 
 of the pressure contribution of each source. Assuming the local veloc- 
 ity Vy on a small distance dx-, to be constant, the yield of the source 
 
 distribution per unit length on this small distance is then q = 2Vy. 
 
 Along this area of the plate the source distribution produces the pres- 
 sure (compare section 2) 
 
 pavy dx-| 
 
 ^,.0 - ^ , ^ (128) 
 
 " \/a2(t - tj.)2 - 1-2 
 
 hence 
 
 ._-. rt'^Sl , ^y^i ^ (^, 
 
 '''° ^xi(l) " ,/,2(t . ti)2 _ r2 
 
 with Vy periodically and locally variable. It is best to solve the 
 
 integral graphically for each particular case . The arguments for the 
 integral limit are the same as before. The lift contribution Ay at 
 
 any instant is obtained by integration of the ensuing pressure plot. 
 
 To obtain the resultant pressure, this pressure is superimposed on 
 the two previous pressure distributions. 
 
 In the following, the pressure contribution due to the additive 
 pressure p^ is calculated. Three phases, depending on time and 
 ratio h/Z, are involved: 
 
6k NACA TM 1569 
 
 I, When I ^ (ut + ya^^ - ^^ ) (fig- 52), the lift is 
 
 n -\/a2t2-h2 ^ + \J a^t^-h^ 
 
 ^^ -^Z .Ut + -^-] J_l/a2t2-h2 ^ 
 
 tan ^i / 
 
 with the previously employed variable change and with y = h 
 
 
 -\fl^ +\/l^ P + \/l-Ti2 
 2atp„(0 „,. /, + 2at(pO / - at / , ^f-^)d^ 
 
 '. »t. y/t^ , - ^"^^^'-vr^ - "J-'v^ i^ 
 
 at 
 
 (150) 
 
 By eqiiation (127) 
 
 dp^ pavQ (1 + ^ sin ij.) - t]^ 
 
 d^ 
 
 (1 + ^ sin n)^ - Ti2cos2^ K/l _ ^2 _ ti2 
 
 (131) 
 
 hence the integral 
 
 r+x/lT^ dp^ 
 I = / ^— ^ ^i dC 
 
 r+ \/l^ (1 + ^ sin ^i - Ti2)^ 
 
 - \/l^ \/l - ^2 _ ^2 [^1 + ^ 3in ^)2 _ Ti2cos2^] 
 
 d^ 
 
NACA TM 1369 
 
 65 
 
 Its evaluation gives 
 
 rt 
 
 sin [X cos |j. 
 
 (cos ^L - 1) 
 
 1 + 
 
 T](cOS ^ + 1) 
 T] + COS H 
 
 (132) 
 
 consequently 
 
 \ 
 
 2pa2uQ 
 
 sm |a cos \i 
 
 (1 - T] cos |i) + (cos p, - 1] 
 
 ^ _^ Ti(cos [1+1) 
 
 (cos |1 + T)) 
 
 II. But if (fig. 53) 
 
 (133) 
 
 Ut 
 
 + V /a2t2 +'y2 ) > I > [i 
 
 ya^t^ + y2j > Z > [ut - \| a2t2 - y2 
 
 the evaluation of the integral gives the formula 
 
 „ I - Ut\ ^ 2pavo , 2 
 
 Ayii = 2pel : 1 + \/l - T]^ 
 
 at / cos u 
 
 2pavo 
 
 rt sm |i cos ^i 
 
 sm 
 
 ■^ 
 
 4 
 
 A . -I 1 + AS , Jt ,, 
 rrrr:^rr- sm--!- + — 1 
 
 A 
 
 'A^ - 1 
 
 A + S 2 
 
 ^f^Tlj 
 
 (f] - cos (l) 
 
 sin-^S 
 
 B 
 
 v^ 
 
 . _i 1 + BS ^ rt/, 
 sm -i- + —II - 
 
 B + S 2| 
 
 b2 - 1, 
 
 (I3i^) 
 
66 
 
 NACA TM 1569 
 
 where 
 
 S = 
 
 i - Ut 
 
 V 
 
 a2t2 - y2 
 
 1 + Tl COS |1 ^ T, 1 - n COS U 
 
 A = ■ ^— and B = ■ — 
 
 sin [i \'l - T]'2. 
 
 sin n\/ 1 - T^S 
 
 The pressure Pg is obtained from equation (127), when x = (^ - Ut) 
 is inserted, at 
 
 Pe 
 
 cos 
 
 -1 
 
 pavQ rt cos \i 
 
 Z - U t 
 at 
 
 + sm 
 
 -J 
 
 , Z - Ut . \2 h2 
 1 + sm n\ - 
 
 at / a2t2 
 
 cos2|j. 
 
 (135) 
 
 III. If Z ^ (U - at), py = Pc along the entire distance EB, so 
 that the additive lift 
 
 ^^III ^ 1 A _ h 
 2pavQ cos [i\ tan [i 
 
 (156) 
 
 reaches a value that is independent of the time. 
 
 Note on the Velocity Integral 
 
 By a simple transformation the integral can be rewritten in the 
 following form: 
 
 '^ 
 
 (2) (A + B^i)d^i 
 
 1= j^ (1) (^i2^a)^aCi2 + B^i-H y 
 
NACA TM 1369 67 
 
 This integral caxi be solved by means of tables (integral table, 
 Part I, by Grobner and Hofreiter, Springer Co.^ Wien) . Althoxogh the 
 general solution is quite complicated, the result is found to be inde- 
 pendent of ^2_> once the limits have been inserted. 
 
 Bearing in mind that the integration limits are the solutions of 
 the function below the root, the integral is rewritten as function of 
 the limits 
 
 (n\ (1 - Ci sin n)d5-L 
 
 ^-J.^ii) 
 
 (1 - ^^ sin ^)2 . cos\k^ - ^^(1)) (^^(2) _ ^^ 
 
 k. - iSAilS^ 
 
 ^1 - ^1^^' ^1^^^ -Ucos, 
 
 The 
 
 The following substitutions are made consecutively: 
 
 1. X=l-^j_sin|i 
 
 limits are thus Xj and X2 (X2 > X^) 
 
 1 2XnXo 
 
 2. Y = i ^^- - 1 
 
 X Xi + X2 
 
 3. Z = Y(X2 + Xi)/(X2 - Xi) 
 If. t = Z2 
 
 -1 ifX-|Xp - cos2^(x-| + Xp)^ 
 5. s = — i A' = —±-^ ^ — 
 
 A' + t cos2n(Xi - X2)^ 
 
 6. Numerical Example 
 Dimensions of cascade ' h/l =0.55 
 
 Mach number of flow M = l.ij-li*- (= \J 2.) corresponding to a 
 
 Mach angle ^ = ^+5° 
 
68 NACA TM 1369 
 
 The period tg up to the end of the phenomenon is determined by 
 
 (U - a)te = I 
 whence 
 
 7 
 
 t^ = 
 
 . klka 
 
 All time intervals are referred to y/a in order to obtain a dimen- 
 sionless ratio (= i/ti) . Then 
 
 ^=k.38 (y=h) 
 
 y/a 
 
 The three pressure contributions p^, p^^, and p^ are defined by 
 
 the formulas of the preceding section at various time intervals indicated 
 by the digits 0, 1, 2, 5, . . . 10. ^^ The time intervals were chosen as 
 follows : The time interval denoted by 5 represents the end of the first 
 phase of the undisturbed plate (compare section k) . At il = 1 (period: k 
 the influence of the adjacent plates begins and ends at t] = 0.2^4-4 
 (period: 9)- At t\ = O.707 (period: 6), the Mach line emerging from the 
 leading edge of the plate strikes the adjacent plate. 
 
 Fig\ire 5^ represents the position of the gust front and the area 
 disturbed by it at the different time intervals. 
 
 Figure 55 illustrates the pressure of the undisturbed plate p^_^. 
 
 Figure 56 illustrates the pressure contribution p^. 
 
 The pressure contribution py is computed graphically, the velocity 
 distributions Vy/vQ required for it are obtained by equation (l22) for 
 the time intervals 5j 6, 7> 8 and reproduced in figure 57- 
 
 ^he corresponding curves in figures 55j 5^, 57? 60, and 61 are 
 denoted by the same digits . 
 
NACA TM 1369 69 
 
 Since the integrand 
 
 becomes infinite at the two limits ^j}- ' and ^2 which are, as 
 known, the solution of the function below the root, the graphical solu- 
 tion is continued to ( ^x^ ^ " ^j ^^^ i^l " ^'j where e, e' are small 
 real values in comparison to ^-, . 
 
 Figure 58 represents several of the functions f for different time 
 intervals . 
 
 The integration over € and e' is made analytically, by putting 
 Vt,/vq = constant mean value . 
 
 The relation for t^/t at t] < O.'JOT, that is, when the additional 
 source distribution is boiind by a Mach line, is 
 
 t^/t = ri cos |a + ^"i sin \i 
 
 If the source distribution is limited by the circle of radius R = at, 
 a similar relation 
 
 t-^^/t = Ti cos p + ^-|^ sin p 
 
 is applicable (fig. 59)- 
 
 Figure 60 represents the pressure contribution p^, figure 61 the 
 resultant pressure distributions. 
 
 The lift of the plate (fig- 62) is obtained by graphical integration 
 over the resultant pressures. At the appearance of the adjacent plate 
 effect the lift decreases with the time interval; A' represents the 
 steady lift of the undisturbed plate . 
 
70 NACA TM 1569 
 
 Figure 63 shows the moment distribution M plotted against plate 
 center; Mst represents its steady value. Here the moment increases with 
 
 the time because of the built-up negative pressure from t = h/a. 
 
NACA TM 1369 71 
 
 CHAPTER VII. EFFICIENCY OF A SUPERSONIC PROPELLER 
 1. Introduction 
 
 The cascade efficiency defined from thrust and tangential force is 
 suitable also for the propeller. But in the preceding arguments the flow 
 was assumed parallel and the blades as infinitely thin plates, which now 
 must be modified. The friction at the plates must be allowed for and 
 the infinitely thin plates replaced by profiles of finite thickness. 
 Then the results are used to calculate the efficiency of a real propeller 
 in order to obtain an approximate picture of the efficiency to be expected. 
 
 2. Effect of Friction on Cascade Efficiency 
 
 When the friction at the plate surfaces is taken into account, the 
 resultant force K without friction defined in chapters III and XV, is 
 supplemented by an additional resistance F, so that K' is the total 
 force acting on the plate (fig. 6k). 
 
 The frictional force is parallel to the plate. But since its com- 
 ponent normal to the airstream direction is small at the angles of attack 
 in question, the total frictional force can be assxamed to be in the flow 
 direction. 
 
 The drag coefficient is expressed by 
 
 4^,2 
 
 c ' 
 
 + 2c^ (157) 
 
 w = , -^ '^^f 
 \/m2 - 1 
 
 F 
 C-p = is coefficient of friction of one side of the plate and q-. 
 
 2^1 
 is dynamic pressure of inflow. 
 
 The lift coefficient remains 
 
 \f 
 
 ^^^- (158) 
 
 m2 
 
 as for parallel flow. 
 
72 NACA TI4 1369 
 
 According to the definition introduced in chapter III, the cascade 
 efficiency is 
 
 n = ^ (139) 
 
 T'u 
 
 where S' and T' are the thrust ajid tangential force corresponding 
 to the new force K'. 
 
 In teriTis of angles (3, v and a (fig. 6U) the efficiency is 
 
 T] = tan a tan(p - \|/) 
 where 
 
 a = tan-1 — (l40) 
 
 Introducing the drag/lift ratio 
 
 / 
 
 I 
 
 W + F Cw' ( , . cfX/M^ - 1 
 
 ^ ^ -^^" - - (11,1) 
 
 A Ca \ 2ilr 
 
 the efficiency t] becomes 
 
 , = " - ^' ^^"(P - -^^ (142) 
 
 1 + e'/tan(|3 - if) 
 
 Hence it is apparent that, contrary to the earlier results, the 
 efficiency is now dependent on the Mach nvimber. 
 
NACA TM 1369 
 
 73 
 
 With the assiimption of a turbulent boundary layer 
 
 C^:■ = 
 
 0.455 
 
 logio Re 
 
 2.58 
 
 (1^+5) 
 
 according to Schlichting, where Re is the Reynolds number based upon 
 chord I and relative velocity w. 
 
 The values plotted in table 9 and figure 65 as functions of the 
 angle of stagger were calculated with the Mach numbers M = lAO and 
 M = 2.50, then at an angle \|/ = 5° and the optimum anglesl5 i|/ = 2.65° 
 (M = 1.40) and \|( = i+.ll° (M = 2.50). The Reynolds nimber assumed at 
 Re = 106 corresponds to cf = 0.00i^-5• 
 
 The effect of friction is illustrated in figure 65, along with the 
 efficiency curve for \|( = 3° with friction discounted. 
 
 3. Effect of Thickness 
 
 To assure minimum wave resistance the contour of a supersonic profile 
 must consist of straight lines and its ma:ximum thickness lie in the center, 
 hence a double-wedge profile is recommended (fig. 66). 
 
 By linearized airfoil theory (ref. 1) the thickness causes a drag 
 which increases quadratically with the thickness ratio d./l, and which 
 can be directly superposed on the lift coefficient and the frictional 
 drag of the plate . 
 
 Hence the drag coefficient of a profile of finite thickness ratio 
 with friction is 
 
 'W 
 
 M^ 
 
 ,2 ^ /d- 
 
 + Cf 
 
 \/m2 
 
 ilkh) 
 
 But the lift coefficient remains unchanged 
 
 Co = 
 
 k^ 
 
 / 
 
 M^ - 1 
 
 13 
 
 As stated in the introduction, ^^opt 
 
 the plate, obviously d/l = 0. 
 
 (d/Z)2 + Cf Vm2^- 1 ^ f^^ 
 
7^ NACA TM 1369 
 
 and the drag/lift ratio to be inserted in equation (l4l) in place of e' 
 is 
 
 e" = £i^ = ^ + 1 
 Ca \1/ 
 
 f f * c, ^ftin 
 
 (1^5) 
 
 Table 10 shows the efficiencies of two cascades of double-wedge 
 profiles and the relative maximum thickness ratio 
 
 d/l =0.05 and 0.10 - with friction 
 at M = 1.40 
 
 ^ = 3° 
 Cj = 0.0045 
 
 These values are also shown in figure 67 together with those for 
 d/Z = (the plate) for comparison. 
 
 The angle of stagger p - with friction and finite thickness - 
 for maximum efficiency at fixed angle of attack and fixed Mach number is 
 found by simple differentiation at 
 
 ^max 
 
 = tan-1 - e + \Je^ + 1 ] (l46) 
 
 = 45° + [4' - i tan-^ej 
 
 4. Appraisal of the Efficiency of a Supersonic Propeller 
 
 On a supersonic propeller the blades are struck at a relative speed 
 which at every point of the blade is greater than the sonic velocity. 
 Two types of propellers axe differentiated. The one moves forward at 
 supersonic speed, so that supersonic speed occurs at every rpm and every 
 On the other the supersonic speed is reached without it having to move for 
 ward with supersonic speed. The efficiency of the first type propeller | 
 is calculated. ' 
 
NACA TM 1569 75 
 
 The propeller has a forward speed v of about 4o6 m/sec (M = 1.20 - 
 sonic velocity a = 558 m/sec); it has four blades of 2m outside diameter 
 and Im inside diameter and O.5 hub ratio. The cross section of the blade 
 is a double-wedge profile, with maximum thickness ratio of d/Z = O.O7 
 at the hub, and tapering to d/Z =0.05 at the tip. 
 
 The maximum efficiency of a profile is reached with (3 = ^5°> 
 according to chapter III. At the blade tip where the thrust is highest, 
 this condition gives a tip speed of 400 m/sec; that corresponds to 
 3,820 rpm. For reasons of strength the blade chord tapers from Ljnj = ^0 cm 
 
 at the hub to Ls = JO cm at the tip. 
 
 In each coaxial cylindrical section - with respect to the propeller 
 axis - the angle between the relative flow direction and the profile axis - 
 the angle of attack - was assumed at 1'orit (compare introduction). To 
 
 satisfy this condition, the angle of stagger in each section, that is, 
 the angle between profile axis and direction of peripheral speed U must 
 be varied. The relation 
 
 tan(p - t) = v/u 
 
 must be satisfied. 
 
 With reference to a system of coordinates fixed in space, each point 
 of the propeller moves on a helical line. Disturbances issue from each 
 point which at the assumed pressure conditions and angles of attack can 
 be regarded only as sound disturbances. The zone disturbed by each blade 
 is then limited by the enveloping curve of all spheres whose centers lie 
 on the various helical lines and whose radii at the same time are equal 
 to sonic velocity x time. Figure 68 represents the disturbed zone of 
 an edge OA, which, for example, moves at a forward speed of 1.2 X a 
 and whose maxim\:mi tip speed equals the sonic velocity; O'A' represents 
 the position of the same edge after a time interval At, which corresponds 
 to a fourth of a revolution. 
 
 Considering that the blades are twisted, that the disturbances of 
 different sections can influence one another and be reflected on propeller 
 hub, it is readily apparent that an exact calculation of the forces on 
 each blade represents a difficult problem. When each blade is outside 
 the zone of disturbance of the other blade, the blade can be examined 
 separately. Assuming homogeneous flow and coaxial cylindrical areas, 
 that is, radial equilibrium, the blade forces can be determined from a 
 two-dimensional consideration of the developed. blade (fig. 69), by com- 
 puting the lift and drag and from it the thrust and tangential force in 
 each cross section by linearized theory. 
 
76 NACA TM 1569 
 
 At the velocities selected the boiindary effect is confined to a 
 moderately large zone compared to blade area, so that its effect within 
 the framework of the intended appraisal on the total forces can be 
 disregarded. 
 
 The thrust of the whole blade is then 
 
 
 The integration is made by graphical method (fig. 71(a)) with 
 
 ^- H pw2^ cos[(p - t) ^7] ^^^Q^ 
 
 ^ fn^ - 1 2 COS 7 
 
 computed for five sections (fig. 69). 
 
 The corresponding Reynolds number for all sections was assumed at 
 Cf = 0.004 (turbulent boundary layer). 
 
 The torque D of a blade is defined the same way as the thrust 
 by integration (fig. 7l(^))- The following relation applies: 
 
 ^ hub i^ J .^/j^2 _ 1 2 cos 7 
 
 The characteristics for the five sections are correlated in table 2. 
 The integration gives 
 
 Sblade = ^^25 kg 
 I>blade = 600 kg/: 
 
 m 
 
NACA TM 1369 77 
 
 hence for the propeller 
 
 S = 1700 kg 
 
 D = 21+00 kgm 
 The efficiency of the propeller is given by 
 
 Sv , 
 
 Tl = — - (150) 
 
 2rtnD 
 
 with the values inserted gives 
 
 T] = 71-8 percent 
 
 A quick and close estimate of the efficiency is obtainable directly 
 
 from the calculation of [ — and ( — ) , where these values are appli- 
 
 \±c Idri 
 
 V /M \ /M 
 
 cable to the whole blade. 
 
 SUMMARY 
 
 1. Lift and drag coefficients for the flat plate at various Mach 
 numbers, ranging from 1.20 to 10, and for different angles of incidence 
 axe calculated, account being taken of the exact flow over both sides of 
 the plate . These values are tabulated and also given in the form of 
 charts. The same coefficients are also calculated under the assumptions 
 of linearized flow over the plate, according to the Ackeret theory. A 
 comparison of both methods shows reasonable agreement between the lin- 
 earized theory and the exact method within the usual range of angles of 
 incidence (max 10°) and for the usual Mach numbers. Special formulas 
 for calculating the lift and drag coefficients for very high Mach num- 
 bers are derived. 
 
 2. An analytical solution of the problem of the interaction between 
 shock waves and expansion waves has been established. 
 
 5. A method for calculating the lift and drag coefficients for a 
 cascade of flat plates is described and applied to an example, with the 
 aid of the formulas derived in the foregoing item. A definition for the 
 
78 NACA TM 1369 
 
 efficiency of the cascade - without friction - is introduced and the 
 efficiency is evaluated for two Mach numbers and different angles of 
 blading. 
 
 k. A linearized theory for supers cnic flow through a cascade of flat 
 plates is established and applied to the example already treated. Com- 
 parison of the lift coefficients shows reasonable agreement. 
 
 5. For demonstration purposes, schlieren photographs were made showing 
 the flow between two flat surfaces. They serve to confirm the established 
 linearized theory for small angles of incidence and show clearly the inter- 
 action between shock and expansion waves. 
 
 6. Under the assumption that the flow through the cascade of flat 
 plates undergoes a small sudden change of direction, that is, a small 
 change in the angle of incidence, the nonstationary flow in the cascade 
 is discussed to show the kind of forces which act on the plates during 
 
 the transition period. An example has been calculated in detail. J 
 
 7- The definition of the efficiency mentioned in 5^ is especially 
 suitable for application to a supersonic propeller. The effect of fric- 
 tion and blade thickness on that efficiency is shown. A rough estimation 
 of the efficiency of a supersonic propeller is then made. j 
 
 Translated by J. Vanier 
 National Advisory Committee 
 for Aeronautics 
 
 REFEEENCES I 
 
 1. Ackeret, J.: Gasdynamlk, Handbuch der Physik, Bd. 8, S: 289-3^4-2, I925. J 
 
 Gasdynamik, Vorlesungen an der E.T.H., u.a. Mitteilung Nr. 8 des 1 
 Institutes fiir Aerodynamik. 
 
 2. Sauer: Einfilhrung in die technische Gasdynamik. Springer-Verlag, 
 
 Berlin, 19^3- 
 
 3- Ferri: Elements of Aerodynamics of Supersonic Flows. The Macmillan 
 Company, New York, I9U9. 
 
 k. Schubert: Zur Theorie der stationaren Verdichtungsstobe. Z.A.M.M., 
 Heft 5, Juni, 19i^•3• 
 
 5. Biot: Loads on a Supersonic Wing Striking a Sharp-Edged Gust. Joixrnal 
 
 of Aeronautical Sciences, May, 19^9- 
 
 6. Keenan and Kaye: Gas Tables. John Wiley & Sons, New York, 19^5- 
 
NACA TM 1569 
 
 79 
 
 TABLE 1 
 
 Ml 
 
 5s 
 
 
 Pl/PO 
 
 q/Po 
 
 q/pl 
 
 1.00 
 
 
 
 90 
 
 0.52830 
 
 0.36981 
 
 0.700 
 
 1.10 
 
 1.1+0 
 
 73.68 
 
 .46835 
 
 .39701+ 
 
 .81+7 
 
 1.20 
 
 3.70 
 
 68.08 
 
 .1+1238 
 
 .1+1567 
 
 1.008 
 
 1.30 
 
 6.32 
 
 65.12 
 
 . 36092 
 
 .1+2689 
 
 1.185 
 
 l.i^O 
 
 9.03 
 
 63.33 
 
 . 31I+2I+ 
 
 . I+5III+ 
 
 1.572 
 
 1.50 
 
 11.67 
 
 62.25 
 
 .2721+0 
 
 .1+3050 
 
 1.575 
 
 1.60 
 
 11+.24 
 
 61.65 
 
 •23527 
 
 .1+2182 
 
 1.792 
 
 1.70 
 
 16.63 
 
 61.37 
 
 . 20259 
 
 .1+0995 
 
 2.023 
 
 1.80 
 
 18.81+ 
 
 61.28 
 
 .171+01+ 
 
 • 39^+72 
 
 2.268 
 
 1.90 
 
 20.87 
 
 61.35 
 
 .11+921+ 
 
 •3771^+ 
 
 2.527 
 
 2.00 
 
 22.71 
 
 61.1+8 
 
 .12780 
 
 .35750 
 
 2.800 
 
 2.20 
 
 25.90 
 
 61.90 
 
 .09352 
 
 .3168I+ 
 
 3.388 
 
 2.50 
 
 29.67 
 
 62.1+0 
 
 .05853 
 
 .25610 
 
 4.575 
 
 3.00 
 
 3I+.01 
 
 63.77 
 
 .02722 
 
 .1711+3 
 
 6.300 
 
 l+.OO 
 
 58.75 
 
 65.25 
 
 .00658 
 
 .07375 
 
 11.20 
 
 5.00 
 
 1+-1.11 
 
 66.20 
 
 .00189 
 
 .03306 
 
 17.50 
 
 6.00 
 
 1+2.1+1+ 
 
 66.75 
 
 .0006^ 
 
 .01588 
 
 25.20 
 
 8.00 
 
 ^3.79 
 
 67.00 
 
 . 00010 
 
 .001+1+8 
 
 1+4.80 
 
 10.00 
 
 1+1+.J+3 
 
 67.12 
 
 .00002 
 
 .00165 ■ 
 
 70.00 
 
 00 
 
 U5.58 
 
 67.70 
 
 
 
 
 
 00 
 
80 
 
 NACA TM 1369 
 
 TABLE 2 
 
 LIFT AND DRAG COEFFICIENTS 
 
 Ml 
 
 *° 
 
 P2/P0 
 
 ^0 
 
 Pg'/Pl 
 
 Ps'/Po 
 
 ca 
 
 "w 
 
 e 
 
 1.20 
 
 1 
 
 o.39ii^5 
 
 58.75 
 
 1.056 
 
 0.45527 
 
 0.1054 
 
 0.0018 
 
 0.0175 
 
 
 2 
 
 . 57210 
 
 61.10 
 
 1.120 
 
 .46187 
 
 .215B 
 
 .0075 
 
 •0549 
 
 
 5 
 
 • 55^^03 
 
 64.57 
 
 1^199 
 
 .49444 
 
 .5573 
 
 .0177 
 
 .0524 
 
 
 3-7 
 
 •35952 
 
 68.08 
 
 1.277 
 
 .52681 
 
 .4011 
 
 .0259 
 
 .0641 
 
 i.iwo 
 
 1 
 
 . 29910 
 
 46.87 
 
 1.051 
 
 .55027 
 
 .0725 
 
 .0015 
 
 •0175 
 
 
 2 
 
 .28i^8o 
 
 48.19 
 
 1.104 
 
 . 54692 
 
 .1440 
 
 .0050 
 
 • 0549 
 
 
 3 
 
 . 27114 
 
 ^9^57 
 
 1.159 
 
 . 56420 
 
 .2156 
 
 .0115 
 
 .0524 
 
 
 
 .25824 
 
 51.15 
 
 1.219 
 
 .58506 
 
 .2888 
 
 .0202 
 
 .0699 
 
 
 6 
 
 . 25576 
 
 54.62 
 
 1.555 
 
 .42517 
 
 .4415 
 
 .0464 
 
 .1052 
 
 
 8 
 
 . 21150 
 
 59.56 
 
 1.527 
 
 . .47984 
 
 .6168 
 
 .0867 
 
 .l4o6 
 
 
 9.05 
 
 . 20050 
 
 65.17 
 
 1.655 
 
 . 52007 
 
 -7521 
 
 -1159 
 
 .1584 
 
 1.60 
 
 1 
 
 . 2257^ 
 
 59.67 
 
 1.060 
 
 .24959 
 
 .0608 
 
 .0011 
 
 •0175 
 
 
 2 
 
 .21 269 
 
 40.75 
 
 1.104 
 
 .25978 
 
 .1116 
 
 .0059 
 
 .0549 
 
 
 3 
 
 .20207 
 
 41.82 
 
 1.161 
 
 .27500 
 
 -1679 
 
 .0088 
 
 .0524 
 
 
 4 
 
 •19191 
 
 42.95 
 
 1.219 
 
 .28679 
 
 .2244 
 
 •0157 
 
 .0699 
 
 
 6 
 
 .17280 
 
 45.56 
 
 1.545 
 
 .51657 
 
 -5385 
 
 •0556 
 
 .1051 
 
 
 8 
 
 •15525 
 
 48.04 
 
 1.484 
 
 .54958 
 
 .4557 
 
 .0641 
 
 .1406 
 
 
 10 
 
 .15914 
 
 51.14 
 
 1.644 
 
 .58685 
 
 .5785 
 
 .1019 
 
 .1762 
 
 
 12 
 
 .12458 
 
 54.89 
 
 1.852 
 
 .45101 
 
 .7110 
 
 .1511 
 
 .2125 
 
 
 14. 2U 
 
 . 11022 
 
 61.65 
 
 2.145 
 
 .50418 
 
 .9052 
 
 • 2552 
 
 • 2552 
 
 1.80 
 
 1 
 
 .16505 
 
 54.64 
 
 1.054 
 
 .18552 
 
 .0468 
 
 .0008 
 
 • 0175 
 
 
 2 
 
 .15640 
 
 55^55 
 
 1.110 
 
 .19518 
 
 • 0951 
 
 .0055 
 
 • 0549 
 
 
 3 
 
 . 14815 
 
 56.48 
 
 1.170 
 
 .20565 
 
 .1404 
 
 .0074 
 
 .0524 
 
 
 ii 
 
 . 14019 
 
 57.44 
 
 1.230 
 
 . 21407 
 
 .1867 
 
 .0151 
 
 .0699 
 
 
 6 
 
 .12554 
 
 59^49 
 
 1.562 
 
 .25704 
 
 .2814 
 
 .0296 
 
 .1051 
 
 
 8 
 
 .11175 
 
 41.69 
 
 1.505 
 
 .26195 
 
 .5768 
 
 • 0550 
 
 .1406 
 
 
 10 
 
 .09955 
 
 44.06 
 
 1.661 
 
 .28908 
 
 .4754 
 
 .0854 
 
 .1763 
 
 
 12 
 
 .08806 
 
 46.70 
 
 1.855 
 
 -51956 
 
 -5732 
 
 .1218 
 
 .2126 
 
 
 15 
 
 •07503 
 
 51-55 
 
 2.159 
 
 •57227 
 
 .7325 
 
 .1962 
 
 .2679 
 
 
 18 
 
 .06011 
 
 58.00 
 
 2.551 
 
 .44598 
 
 .9249 
 
 • 5005 
 
 -3249 
 
 
 18. a 
 
 .05788 
 
 61.28 
 
 2.740 
 
 .47687 
 
 1.0045 
 
 •5427 
 
 .5^12 
 
 2.00 
 
 1 
 
 .12076 
 
 50.82 
 
 1.058 
 
 •15521 
 
 .0404 
 
 .0007 
 
 .0175 
 
 
 2 
 
 . 11401 
 
 31-65 
 
 1.118 
 
 .14288 
 
 .0807 
 
 .0028 
 
 .0549 
 
 
 3 
 
 •10757 
 
 52.58 
 
 1.181 
 
 1.15095 
 
 .1211 
 
 .065 
 
 •0525 
 
 
 4 
 
 . 10141 
 
 55^40 
 
 1.247 
 
 .15937 
 
 .1618 
 
 .0115 
 
 .0699 
 
 
 6 
 
 .08994 
 
 55^24 
 
 1.577 
 
 . 17726 
 
 .2429 
 
 .0255 
 
 .1051 
 
 
 8 
 
 .07949 
 
 57^22 
 
 1.559 
 
 .19668 
 
 .3246 
 
 .0456 
 
 .l4o6 
 
 
 10 
 
 .07005 
 
 59^52 
 
 1.707 
 
 .21815 
 
 .408 
 
 • 0719 
 
 .1765 
 
 
 12 
 
 .06149 
 
 41.59 
 
 1.889 
 
 . 24141 
 
 .4925 
 
 .1046 
 
 .2125 
 
 
 15 
 
 .05024 
 
 45.54 
 
 2.195 
 
 .28052 
 
 .6222 
 
 .1667 
 
 .2679 
 
 
 18 
 
 .04070 
 
 49^78 
 
 2.555 
 
 .52655 
 
 .7604 
 
 .2471 
 
 • 5249 
 
 
 21 
 
 .03265 
 
 55^67 
 
 5.014 
 
 .58519 
 
 .9207 
 
 .5554 
 
 • 5638 
 
 
 22.71 
 
 .02886 
 
 61.48 
 
 5-460 
 
 .44219 
 
 1.0659 
 
 .4462 
 
 .41&7 
 
NACA TM 1369 
 
 81 
 
 TABLE 2.- Concluded 
 
 LIFT AND DRAG COEFFICIENTS 
 
 Ml 
 
 *° 
 
 P2/P0 
 
 ''o 
 
 Ps'/Pi 
 
 P2 7po 
 
 ca 
 
 
 
 e 
 
 2.50 
 
 1 
 
 0.05296 
 
 24.35 
 
 1.068 
 
 0.06251 
 
 0.0373 
 
 0.0006 
 
 0.0175 
 
 
 2 
 
 .05113 
 
 25.05 
 
 l.l4l 
 
 .06678 
 
 .0611 
 
 .0021 
 
 .0349 
 
 
 3 
 
 .04772 
 
 25.82 
 
 1.216 
 
 .07117 
 
 .0914 
 
 .0048 
 
 .0524 
 
 
 k 
 
 .04450 
 
 26.62 
 
 1.296 
 
 .07585 
 
 .1221 
 
 .0085 
 
 .0699 
 
 
 6 
 
 •03857 
 
 28.27 
 
 1.452 
 
 .08498 
 
 .1826 
 
 .0189 
 
 . 1051 
 
 
 8 
 
 •03455 
 
 30.00 
 
 1.658 
 
 .09704 
 
 .2416 
 
 .0340 
 
 .1406 
 
 
 10 
 
 •02859 
 
 31.86 
 
 1.865 
 
 . 10916 
 
 .3098 
 
 .0536 
 
 .1763 
 
 
 12 
 
 .02445 
 
 33 •Si 
 
 2.091 
 
 .12232 
 
 .3738 
 
 .0795 
 
 .2126 
 
 
 15 
 
 .01917 
 
 36.95 
 
 2.467 
 
 .14439 
 
 .4723 
 
 .1266 
 
 .2679 
 
 
 18 
 
 .01484 
 
 40. 40 
 
 2.895 
 
 . 16949 
 
 .5742 
 
 .1865 
 
 .3249 
 
 
 22 
 
 .01035 
 
 45.62 
 
 2^557 
 
 .20816 
 
 .7162 
 
 .2893 
 
 .4o4o 
 
 
 28 
 
 •00575 
 
 56.35 
 
 4.885 
 
 .28592 
 
 .9659 
 
 .5136 
 
 .5317 
 
 
 29^67 
 
 .00482 
 
 62.65 
 
 5.602 
 
 .44220 
 
 1.0960 
 
 .6240 
 
 .5695 
 
 5.00 
 
 3^60 
 
 .00118 
 
 14 
 
 1.541 
 
 .00291 
 
 .0523 
 
 .0033 
 
 .0627 
 
 
 6.17 
 
 .00084 
 
 16 
 
 2.051 
 
 .00388 
 
 .0914 
 
 .0098 
 
 .1076 
 
 
 10.68 
 
 .00043 
 
 20 
 
 3^247 
 
 .00614 
 
 .1698 
 
 .0319 
 
 .1879 
 
 
 16.60 
 
 .00018 
 
 26 
 
 5.436 
 
 .01027 
 
 .2926 
 
 .0869 
 
 .2972 
 
 
 20.21 
 
 .00009 
 
 30 
 
 7.129 
 
 .01347 
 
 .3800 
 
 .1393 
 
 .3666 
 
 
 26.78 
 
 .00002 
 
 38 
 
 10.893 
 
 .02059 
 
 .5557 
 
 .2792 
 
 .5024 
 
 
 31.21 
 
 .000009 
 
 44 
 
 13.913 
 
 .02629 
 
 .6805 
 
 .4088 
 
 .6000 
 
 
 36.29 
 
 .000003 
 
 52 
 
 17.953 
 
 .03392 
 
 .8284 
 
 .6048 
 
 .7301 
 
 
 41.11 
 
 .000001 
 
 66 
 
 24 . 206 
 
 .04575 
 
 1.0427 
 
 .9098 
 
 .8726 
 
 10.00 
 
 3^21 
 
 
 8 
 
 2.091 
 
 
 .0228 
 
 .0013 
 
 .0560 
 
 
 7.65 
 
 
 12 
 
 4.877 
 
 
 .0614 
 
 .0082 
 
 .1340 
 
 
 13^31 
 
 
 18 
 
 10.981 
 
 
 .1549 
 
 .0365 
 
 ■ 2356 
 
 
 18.55 
 
 
 24 
 
 19.135 
 
 
 .2613 
 
 .0877 
 
 .3551 
 
 
 23^47 
 
 
 30 
 
 29.018 
 
 
 .3828 
 
 .1662 
 
 .4341 
 
 
 28.17 
 
 
 36 
 
 4o . l64 
 
 
 .5079 
 
 .2726 
 
 .5366 
 
 
 32.59 
 
 
 42 
 
 52.080 
 
 
 .6288 
 
 .4017 
 
 .6393 
 
 
 36.64 
 
 
 48 
 
 64.263 
 
 
 .7488 
 
 .5565 
 
 •7437 
 
 
 40.18 
 
 
 54 
 
 76.213 
 
 
 .8641 
 
 • 7^2 
 
 .8441 
 
 
 42.90 
 
 
 60 
 
 87.361 
 
 
 • 9529 
 
 .8862 
 
 • 9293 
 
 
 44.43 
 
 
 67.12 
 
 98.713 
 
 
 1.0125 
 
 .9925 
 
 .9803 
 
 00 
 
 8.37 
 
 
 10 
 
 
 
 • 0497 
 
 • 0073 
 
 .1471 
 
 
 16.57 
 
 
 20 
 
 
 
 • 1873 
 
 • 0558 
 
 .2973 
 
 
 24.50 
 
 
 30 
 
 
 
 • 3795 
 
 • 1731 
 
 .4557 
 
 
 28.31 
 
 
 35 
 
 
 
 .4741 
 
 • 2559 
 
 .5386 
 
 
 32.05 
 
 
 40 
 
 
 
 • 5843 
 
 .3662 
 
 .6261 
 
 
 35-55 
 
 
 1^5 
 
 
 
 .6812 
 
 .4846 
 
 .7146 
 
 
 38.81 
 
 
 50 
 
 
 
 • 7840 
 
 .6121 
 
 .8042 
 
 
 41.6 
 
 
 55 
 
 
 
 • 8735 
 
 .7751^ 
 
 .8876 
 
 
 43-9 
 
 
 60 
 
 
 
 .9452 
 
 .9123 
 
 .9623 
 
 
 45.58 
 
 
 67.8 
 
 
 
 1.0001 
 
 1.0204 
 
 1.0200 
 
 N 
 
 5 
 10 
 15 
 20 
 25 
 30 
 55 
 ko 
 ^5 
 50 
 
 
 
 
 
 .0152 
 .0594 
 .1294 
 .2198 
 .3238 
 .4330 
 .5388 
 
 .6531 
 .7068 
 
 ■7547 
 
 .0013 
 .0105 
 .0347 
 .0801 
 .1509 
 .2500 
 .3772 
 
 .5311 
 .7068 
 .8992 
 
 .0875 
 .1763 
 .2679 
 .3640 
 .4663 
 .5774 
 .7002 
 
 .8391 
 
 1.0000 
 
 1.1918 
 
82 
 
 KACA TM 1569 
 
 TABLE 3 
 
 LIFT AND DRAG COEFFICIENTS OF THE BOUNDARY CURVE 
 
 Ml 
 
 ♦s° 
 
 P2/P0 
 
 7s 
 ' 
 
 Ps'/Pi 
 
 ^2 7^0 
 
 Ca 
 
 Cw 
 
 6 
 
 1.10 
 
 1.1^ 
 
 O.i+3110 
 
 73 
 
 11+ 
 
 1.130 
 
 O.5292I+ 
 
 0. 21+71 
 
 . 0101+ 
 
 O.O2I+8 
 
 1.20 
 
 5.70 
 
 •35952 
 
 68 
 
 5 
 
 1.277 
 
 .52681 
 
 .1+011 
 
 .0259 
 
 .061+1 
 
 1.30 
 
 6.32 
 
 .26668 
 
 65 
 
 7 
 
 1.457 
 
 .52586 
 
 .6035 
 
 .0668 
 
 .1107 
 
 l.i+O 
 
 9-03 
 
 . 20050 
 
 63 
 
 10 
 
 1.656 
 
 . 52009 
 
 .7321 
 
 .1159 
 
 .I58I+ 
 
 1.50 
 
 11.67 
 
 . 11+392 
 
 62 
 
 15 
 
 1.891+ 
 
 .51593 
 
 .8339 
 
 .1721+ 
 
 .2068 
 
 1.60 
 
 lij-.2J+ 
 
 . 11022 
 
 61 
 
 39 
 
 2.11+3 
 
 . 50I+I8 
 
 .9052 
 
 .2297 
 
 .2532 
 
 1.70 
 
 16.63 
 
 .07918 
 
 61 
 
 22 
 
 2.1+39 
 
 .1+91+12 
 
 .9697 
 
 .2897 
 
 .2988 
 
 1.80 
 
 18.81+ 
 
 .05788 
 
 61 
 
 17 
 
 2.7I+I 
 
 .1+7687 
 
 l.OOi+5 
 
 .31+27 
 
 .31+12 
 
 1.90 
 
 20.87 
 
 .01+0 1+7 
 
 61 
 
 21 
 
 3.097 
 
 . 1+6220 
 
 I.OI+I+9 
 
 .3983 
 
 .3812 
 
 2.00 
 
 22.71 
 
 .02886 
 
 61 
 
 29 
 
 3.1+60 
 
 .1+1+219 
 
 1.0665 
 
 .1+1+65 
 
 .1+187 
 
 2.20 
 
 25.90 
 
 .011+17 
 
 61 
 
 54 
 
 I+.250 
 
 .3871+6 
 
 1.0883 
 
 .528I+ 
 
 .1+855 
 
 2.50 
 
 29.67 
 
 .001+82 
 
 62 
 
 39 
 
 5.602 
 
 .1+1+221 
 
 1.0960 
 
 .621+2 
 
 .5695 
 
 5.00 
 
 3I+.01 
 
 .00073 
 
 63 
 
 1+6 
 
 8.385 
 
 . 2282I+ 
 
 1.0735 
 
 .721+3 
 
 .671+7 
 
 l+.OO 
 
 38.75 
 
 .000011+ 
 
 65 
 
 15 
 
 15.25 
 
 . 10060 
 
 1.0650 
 
 .8551 
 
 .8025 
 
 5.00 
 
 1+1.11 
 
 .000001 
 
 66 
 
 12 
 
 2I+.I+0 
 
 .01+612 
 
 1.0515 
 
 .9107 
 
 .8652 
 
 6.00 
 
 42.41+ 
 
 
 
 6G 
 
 45 
 
 55.25 
 
 .02231 
 
 1.0362 
 
 .91+32 
 
 .9136 
 
 8.00 
 
 1+3.79 
 
 
 
 67 
 
 
 
 63.11 
 
 .00630 
 
 1.0171 
 
 .9692 
 
 .9601 
 
 10.00 
 
 1+1+.I+3 
 
 
 
 67 
 
 7 
 
 98.90 
 
 .00233 
 
 1.0121 
 
 .9895 
 
 .9803 
 
 00 
 
 1+5. 58 
 
 
 
 67 
 
 1+1 
 
 1 
 
 
 1.0050 
 
 1.0185 
 
 1.0200 
 
 
NACA TM 1569 
 
 85 
 
 TABLE k 
 
 LIFT AMD DRAG COEFFICIENTS ACCORDING TO THE LINEARIZED THEORY 
 
 Ml 
 
 ♦° 
 
 <^a 
 
 ■^w 
 
 e 
 
 1.20 
 
 ^ 1 
 
 0.10553 
 
 0.00018 
 
 0.0175 
 
 
 2 
 
 .2101*6 
 
 .00731* 
 
 ■ 051*9 
 
 
 3 
 
 .31600 
 
 .01655 
 
 ■0521* 
 
 
 5-7 
 
 •38957 
 
 .02516 
 
 .061*6 
 
 i.to 
 
 1 
 
 .071'*1( 
 
 .00125 
 
 •0175 
 
 
 2 
 
 .11*21*8 
 
 .001*97 
 
 .031*9 
 
 
 3 
 
 .21392 
 
 .01121 
 
 .0521* 
 
 
 k 
 
 .281*96 
 
 .01989 
 
 .0698 
 
 
 6 
 
 .1*271*1* 
 
 .OM*80 
 
 .101*7 
 
 
 8 
 
 ■56992 
 
 .07950 
 
 ■1596 
 
 
 9.03 
 
 .61*259 
 
 .10112 
 
 .1571* 
 
 1.60 
 
 1 
 
 .05601* 
 
 .00089 
 
 ■0175 
 
 
 2 
 
 .11177 
 
 .00390 
 
 ■031*9 
 
 
 5 
 
 . 16781 
 
 .00879 
 
 ■0521* 
 
 
 h 
 
 •22553 
 
 .01560 
 
 .0698 
 
 
 6 
 
 .33530 
 
 .031*58 
 
 .101*7 
 
 
 8 
 
 .1*1*707 
 
 .0621*0 
 
 •1396 
 
 
 10 
 
 • 55881* 
 
 •09752 
 
 .171*5 
 
 
 12 
 
 .67060 
 
 • ll*Ol*0 
 
 .2091* 
 
 
 ik.zk 
 
 .79582 
 
 .19772 
 
 .21*85 
 
 1.80 
 
 1 
 
 .01*676 
 
 .00082 
 
 •0175 
 
 
 2 
 
 ■ 09325 
 
 .00326 
 
 .031*9 
 
 
 5 
 
 .11*001 
 
 .00751* 
 
 .0521* 
 
 
 1( 
 
 .18650 
 
 .01502 
 
 .0698 
 
 
 6 
 
 .27976 
 
 .02932 
 
 .101*7 
 
 
 8 
 
 •57301 
 
 .05205 
 
 • 1596 
 
 
 10 
 
 .1*6626 
 
 .08135 
 
 • 171*5 
 
 
 12 
 
 .55952 
 
 .11711* 
 
 .2091* 
 
 
 15 
 
 •69953 
 
 .18311+ 
 
 .2618 
 
 
 18.81* 
 
 •87855 
 
 .28881* 
 
 .3288 
 
 2.00 
 
 1 
 
 .01(01*2 
 
 .00070 
 
 •0175 
 
 
 2 
 
 .08060 
 
 .00281 
 
 .031*9 
 
 
 3 
 
 .12102 
 
 .00651* 
 
 .0521* 
 
 
 4 
 
 .16120 
 
 .01125 
 
 .0698 
 
 
 6 
 
 .21*180 
 
 .02552 
 
 .101*7 
 
 
 8 
 
 • 3221*0 
 
 .01*995 
 
 .1396 
 
 
 10 
 
 .1*0301 
 
 ■07050 
 
 ■ 171*5 
 
 
 12 
 
 .1*8361 
 
 .10125 
 
 .2091* 
 
 
 15 
 
 .6o!*$3 
 
 .15829 
 
 .2618 
 
 
 18 
 
 .72561* 
 
 .22802 
 
 .511*2 
 
 
 21 
 
 .81*61*5 
 
 .3101*0 
 
 ■5665 
 
 
 22. Tl 
 
 .91502 
 
 •36259 
 
 . .3962 
 
 2.50 
 
 1 
 
 .03051* 
 
 .00055 
 
 .0175 
 
 
 2 
 
 .06091 
 
 .00215 
 
 .051*9 
 
 
 3 
 
 .0911*5 
 
 .001*79 
 
 .0521* 
 
 
 1* 
 
 .12181 
 
 .00851 
 
 .0698 
 
 
 6 
 
 .18272 
 
 .01915 
 
 .101*7 
 
 
 8 
 
 .21*565 
 
 .031*00 
 
 .1596 
 
 
 10 
 
 .501*51* 
 
 .05318 
 
 .171*5 
 
 
 12 
 
 .3651*1* 
 
 .07650 
 
 .2091* 
 
 
 15 
 
 .1*5689 
 
 .11962 
 
 .2618 
 
 
 18 
 
 .51*851^ 
 
 .17230 
 
 .511*2 
 
 
 22 
 
 .67016 
 
 .2571*2 
 
 .381*0 
 
 
 28 
 
 .85288 
 
 .U1675 
 
 .1*887 
 
 
 29.67 
 
 .90366 
 
 .1*6789 
 
 .5178 
 
 5.00 
 
 k 
 
 .05699 
 
 .00398 
 
 .0698 
 
 
 6 
 
 .0851*9 
 
 .00896 
 
 .101*7 
 
 
 12 
 
 .17097 
 
 •03579 
 
 .2091* 
 
 
 18 
 
 .25651* 
 
 .08061 
 
 .311*2 
 
 
 2U 
 
 .51*205 
 
 . 11*321 
 
 .1*189 
 
 
 50 
 
 .1*2752 
 
 •2?585 
 
 .1*712 
 
 
 36 
 
 .1*7050 
 
 .52232 
 
 .5760 
 
 
 Ifl.ll 
 
 .58581* 
 
 .1*2055 
 
 •7175 
 
 10.00 
 
 It 
 
 .02806 
 
 .00196 
 
 .0698 
 
 
 6 
 
 .01*209 
 
 .001*1*1 
 
 .101*7 
 
 
 12 
 
 .081*18 
 
 .01762 
 
 .209"* 
 
 
 18 
 
 .12631 
 
 .05969 
 
 .511*2 
 
 
 2lt 
 
 .1681*0 
 
 .07051 
 
 .1*189 
 
 
 30 
 
 .2101*9 
 
 .11022 
 
 .1*712 
 
 
 36 
 
 .25258 
 
 .15800 
 
 • 5760 
 
 
 U2 
 
 .291*67 
 
 .21600 
 
 .7330 
 
 
 Ult.U5 
 
 .51172 
 
 .21*177 
 
 •7755 
 
8h 
 
 NACA TM 1369 
 
 TABLE 5 
 
 Ml 
 
 ^0 
 
 (■^aG - ^al) 
 
 ^aG 
 
 (cwG - ^wl) 
 
 ^wG 
 
 1.1+0 
 
 1 
 
 0.000 81+ 
 
 0.07228 
 
 
 
 0.0013 
 
 
 2 
 
 .00151 
 
 .14399 
 
 
 
 .0050 
 
 
 3 
 
 .00163 
 
 .21555 
 
 .0001 
 
 .0113 
 
 
 k 
 
 .00385 
 
 .28881 
 
 .0002 
 
 .0202 
 
 
 6 
 
 .011+08 
 
 .1+1+152 
 
 .0010 
 
 .01+61+ 
 
 
 8 
 
 .01+690 
 
 .61682 
 
 .0072 
 
 .0867 
 
 
 9.03 
 
 .08950 
 
 •73209 
 
 .011+8 
 
 •1159 
 
 5.00 
 
 k 
 
 .0005 
 
 • 0523 
 
 .0009 
 
 .0033 
 
 
 6 
 
 .0030 
 
 .0911+ 
 
 .0010 
 
 .0098 
 
 
 12 
 
 .0255 
 
 .1698 
 
 .0063 
 
 .0319 
 
 
 18 
 
 .0696 
 
 .2996 
 
 .021+6 
 
 .0869 
 
 
 2k 
 
 .1391 
 
 .3800 
 
 .031+1 
 
 •1395 
 
 
 30 
 
 .181+6 
 
 • 5557 
 
 .11+53 
 
 •2792 
 
 
 36 
 
 • 3972 
 
 .6805 
 
 .2698 
 
 .1+088 
 
 
 
 1+1.11 
 
 .1+669 
 
 .8281+ 
 
 .1+895 
 
 .6oi+8 
 
NACA TM 1569 
 
 85 
 
 TABLE 6 
 
 Region 
 
 P/POl 
 
 M 
 
 vP 
 
 Po/POi 
 
 1 
 
 . 51^0l^■ 
 
 I.I+OOI+ 
 
 ^5.5 
 
 1 
 
 2 
 
 .36I+5 
 
 1.293 
 
 50.6 
 
 .9996 
 
 5 
 
 .2991 
 
 l.i+35 
 
 44.2 
 
 1 
 
 h 
 
 • 5^+55 
 
 1.332 
 
 48.8 
 
 .9997 
 
 5 
 
 .2848 
 
 1.1+69 
 
 42.9 
 
 1 
 
 6 
 
 .5301+ 
 
 1.36I+ 
 
 47.1 
 
 .9997 
 
 7 
 
 .2711 
 
 1.503 
 
 41.7 
 
 1 
 
 8 
 
 .3139 
 
 1.1+01 
 
 45.7 
 
 .9998 
 
 9 
 
 • 3285 
 
 1.368 
 
 47.0 
 
 .9997 
 
 10 
 
 .3139 
 
 1.1+01 
 
 45.6 
 
 .9997 
 
 11 
 
 .2985 
 
 1.436 
 
 44.1 
 
 .9997 
 
 12 
 
 .2999 
 
 1.433 
 
 44.2 
 
 .9997 
 
 15 
 
 .2850 
 
 1.1+68 
 
 42.9 
 
 .9997 
 
 Ik 
 
 .2708 
 
 I.50I+ 
 
 41.7 
 
 .9997 
 
 15 
 
 .3621+ 
 
 1.15I+ 
 
 50.1 
 
 .9982 
 
 16 
 
 .3UI+5 
 
 1.334 
 
 48.6 
 
 .9997 
 
 17 
 
 .3295 
 
 1.366 
 
 47.0 
 
 .9997 
 
 18 
 
 .3121+ 
 
 l.4o4 
 
 45.4 
 
 .9997 
 
 19 
 
 .3276 
 
 1.370 
 
 46.9 
 
 .9998 
 
 20 
 
 • 313^^ 
 
 1.402 
 
 45.5 
 
 .9997 
 
 21 
 
 .2972 
 
 1.439 
 
 44.0 
 
 .9997 
 
 22 
 
 .2996 
 
 1.434 
 
 44.8 
 
 .9997 
 
 23 
 
 .281+2 
 
 1.483 
 
 42.8 
 
 .9997 
 
 21+ 
 
 .2696 
 
 1.507 
 
 41.6 
 
 .9997 
 
 25 
 
 .3610 
 
 
 
 
86 
 
 NACA TM 1569 
 
 TABLE 7 
 
 (a) Angle of deflection and shock angle of the 
 shocks between the zones 
 
 
 Angle of deflection 
 
 Shock angle 
 
 Regions 
 
 6° 
 
 7° 
 
 1-2 
 
 5.00 
 
 49.57 
 
 ^-h 
 
 5-05 
 
 47.91 
 
 5-6 
 
 2.93 
 
 46.48 
 
 7-8 
 
 2.95 
 
 45.13 
 
 8-15 
 
 2.93 
 
 49.43 
 
 11-16 
 
 2.92 
 
 47.78 
 
 15-17 
 
 2.90 
 
 46.40 
 
 14-18 
 
 2.93 
 
 45.03 
 
 18-25 
 
 2.93 
 
 49.30 
 
 (b) Intensity of expansion waves 
 between the zones 
 
 
 Intensity 
 
 
 Intensity 
 
 Regions 
 
 Z^o 
 
 Regions 
 
 /^° 
 
 1-3 
 
 1.00 
 
 13-14 
 
 1.02 
 
 2-4 
 
 1.03 
 
 15-16 
 
 1.00 
 
 5-5 
 
 1.00 
 
 16-17 
 
 .88 
 
 4-6 
 
 .89 
 
 16-19 
 
 1.00 
 
 4-9 
 
 1.03 
 
 17-18 
 
 1.06 
 
 5-7 
 
 1.00 
 
 • 17-20 
 
 ■ 99 
 
 6-8 
 
 1.00 
 
 18-21 
 
 1.00 
 
 8-11 
 
 1.01 
 
 19-20 
 
 .89 
 
 9-10 
 
 .92 
 
 20-21 
 
 1.05 
 
 10-11 
 
 1.00 
 
 21-23 
 
 .89 
 
 10-12 
 
 .92 
 
 22-23 
 
 1.05 
 
 11-13 
 
 .91 
 
 25-24 
 
 1.05 
 
 12-13 
 
 1.02 
 
 
 
NACA TM 1369 
 
 87 
 
 TABLE 8 
 
 CASCADE EFFICIENCY t\ PERCENT = f(p,4/) (NO FRICTION) 
 
 ^^^^° 
 
 1 
 
 3 
 
 5 
 
 7 
 
 10 
 
 3° ^^^-^ 
 
 
 
 
 
 
 10 
 
 89.8 
 
 69.6 
 
 1+9-6 
 
 29.72 
 
 
 20 
 
 94.7 
 
 83.9 
 
 73.6 
 
 63. i+ 
 
 48.4 
 
 30 
 
 96.0 
 
 88.3 
 
 80.8 
 
 73-5 
 
 63.1 
 
 40 
 
 96.5 
 
 89.8 
 
 83.1+ 
 
 77-3 
 
 68.8 
 
 50 
 
 96.6 
 
 90.00 
 
 83.8 
 
 78.2 
 
 70.4 
 
 60 
 
 96.0 
 
 88.8 
 
 82.4 
 
 76.5 
 
 68.8 
 
 70 
 
 9i|.8 
 
 85.7 
 
 78.2 
 
 71.4 
 
 63.1 
 
 80 
 
 90.6 
 
 76.4 
 
 65.8 
 
 57.65 
 
 48.4 
 
 po for max. i] 
 
 45.5 
 
 i+6.5 
 
 i+T.5 
 
 48.5 
 
 50.0 
 
 nmax^ percent 
 
 96.5 
 
 90.2 
 
 83.7 
 
 78.2 
 
 70.4 
 
88 
 
 NACA TM 1369 
 
 TABLE 9 
 
 CASCADE EFFICIENCY WITH FRICTION ALLOWED FOR 
 
 po 
 
 M = 
 
 l.4o 
 
 M = 
 
 2.50 
 
 ^ = 5° 
 
 ^ = 2.65 
 
 ^1, = 50 
 
 1|( = 4.11 
 
 10 
 
 55-8 
 
 57.0 
 
 1^4.1 
 
 Ifl.l 
 
 20 
 
 Ih 
 
 2 
 
 7i^.6 
 
 63.9 
 
 63.7 
 
 50 
 
 80 
 
 3 
 
 80A 
 
 71.2 
 
 71.8 
 
 IK) 
 
 82 
 
 5 
 
 82.8 
 
 73.9 
 
 74.8 
 
 50 
 
 82 
 
 6 
 
 82.7 
 
 73.5 
 
 1^-7 
 
 60 
 
 80 
 
 5 
 
 80.6 
 
 69.9 
 
 71.8 
 
 TO 
 
 71^ 
 
 7 
 
 71^.6 
 
 60.6 
 
 63.8 
 
 80 
 
 57 
 
 8 
 
 57.2 
 
 33.5 
 
 41.4 
 
NACA TM 1369 
 
 89 
 
 TABLE 10 
 
 CASCADE EFFICIENCY WITH DOUBLE -WEDGE PROFILE AND FRICTION 
 
 M = l.i^-O \1( = 30 
 
 po 
 
 d/Z = 0.05 
 
 d/Z =0.10 
 
 10 
 
 ^5-55 
 
 29.01+ 
 
 20 
 
 65.28 
 
 U7.22 
 
 50 
 
 72.52 
 
 7i+.78 
 
 i^O 
 
 75.11 
 
 56.93 
 
 50 
 
 7^^.83 
 
 5if.82 
 
 60 
 
 71.50 
 
 ^7-31 
 
 70 
 
 62.71 
 
 29.25 
 
 80 
 
 37-18 
 
 
90 
 
 NACA TM 1369 
 
 TABLE 11 
 
 Section 
 
 1 
 
 hub 
 
 N 
 
 3 
 
 center 
 M 
 
 5 
 
 tip 
 
 S 
 
 Radius, rm 
 
 Profile chord, m 
 
 Thickness ratio, d/Z . . . 
 Max. thickness, dm ... . 
 Tip speed, um/sec . . . . 
 Relative velocity, wm/sec 
 Relative Mach nimiber, M 
 
 Angle of attack, \|/opt- \ ^ 
 
 1° 
 Lift coefficient, Cg^ . . . 
 
 Drag/lift ratio, e . . . . 
 Gliding angle, 7° . . . . 
 
 Stagger, (3° 
 
 dS/dr, kg/m 
 
 dO/dr, kg 
 
 0.500 
 0,ii50 
 0.070 
 
 0.0315 
 200 
 
 1+52.7 
 1-339 
 
 0.0818 
 4.67 
 
 0.3672 
 
 0.1637 
 
 9.30 
 
 68.8 
 
 612 
 
 1026 
 
 0.625 
 O.I125 
 0.065 
 0.0282 
 250 
 i+76.8 
 l.iill 
 
 0.0787 
 If. 52 
 
 0.3164 
 
 0.1572 
 
 8.95 
 
 62.9 
 
 748 
 
 1118 
 
 0.750 
 0.400 
 
 0.06 
 
 0.024 
 
 300 
 
 504.6 
 
 1.^93 
 
 0.0761 
 4.38 
 
 0.2745 
 
 0.1523 
 8.66 
 
 57.9 
 825 
 
 1173 
 
 0.875 
 0.375 
 0.055 
 0.0206 
 350 
 536.0 
 1.586 
 
 0.0742 
 4.25 
 
 0.2407 
 
 0.1484 
 
 8.43 
 
 53.5 
 
 878 
 
 1212 
 
 1.000 
 0.350 
 0.050 
 0.0175 
 400 
 570.0 
 1.687 
 
 0.0721 
 
 4.13 
 
 0.2120 
 
 . 1442 
 
 8.21 
 
 49-58 
 
 904 
 
 1224 
 
NACA TM 1369 
 
 91 
 
 F^M,U, 
 
 Pg M2 Ug 
 
 Figure 1.- Expansion around a corner. 
 
 E S 
 
 Figure 2.- Oblique compression shock. 
 
92 
 
 NACA IM 1569 
 
 80 
 
 
 
 
 
 
 
 
 
 
 
 
 
 70 
 
 
 \ 
 
 
 
 Ys- 
 
 \ 
 
 
 
 
 
 
 
 
 \^ 
 
 ^^ 
 
 
 
 \ 
 
 
 
 
 
 
 
 60 
 50 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 b^ 
 
 
 
 
 
 i — 
 
 00 
 
 40 
 30 
 2n 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 4 
 
 8 9 10 M 
 
 Figure 3,- Shock and flow deflection angles for M2 = 1. 
 
NACA TM 1569 
 
 95 
 
 M^ P^ U, 
 
 Figure 4,- Flat plate in supersonic flow. 
 
 0.7 
 0.6 
 0.5 
 0.4 
 
 0.3 
 0.2 
 Ql 
 
 
 
 a 
 
 23 456789 
 
 (a) Dynamic pressure/stagnation pressure 
 
 (b) Dynamic pressure/ inflow pressure 
 
 10 M 
 
 1 
 
 Figure 5. 
 
9h 
 
 ^kCk TM 1369 
 
 Figure 6.- Lift of the flat plate. 
 
MCA TM 1369 
 
 95 
 
 Fieure 7.- Drag of flat plate. 
 
96 
 
 NACA TM 1369 
 
 5 
 
 O 
 
 d 
 
 00 
 
 
 
 
 
 h- 
 
 
 
 
 
 
 • 
 
 
 0) 
 
 
 -4-> 
 
 
 0) 
 
 
 1 — 1 
 
 10 
 
 D4 
 
 
 
 -s 
 
 
 .— 1 
 
 
 •M 
 
 
 «M 
 
 
 
 
 
 w 
 
 in 
 
 ^1 
 
 O" 
 
 .—1 
 
 
 
 P^ 
 
 
 • 
 
 ^ 
 
 CD 
 
 
 
 S 
 
 
 hD 
 
 
 •r-H 
 
 
 fo 
 
 ro 
 
 
 O' 
 
 
 C7^. 
 
 00 
 
 N- 
 
 U3 
 
 in ^. 
 
 ro 
 
 OJ 
 
 0-0 
 
 
 
 d 
 
 
 
 d 
 
 d 
 
 
 
 
 
 
 
 
 
 
 
 d 
 
 o 
 
 o 
 
NACA TM 1369 
 
 97 
 
 U M2P2U2 
 
 II r->" I ■" 
 
 d^ M2P2U2 
 
 Figure 9.- Flat plate in linearized flow. 
 
 Figiire 10.- Lift by linearized theory. 
 
98 
 
 KACA TM 1569 
 
 10 15 20 25 30 3 
 
 5 40 ,0 40 
 
 Figure 11.- Drag by linearized theory. 
 
NACA TM 1569 
 
 99 
 
 w 
 
 
 
 
 
 '.^_ 
 
 L. 
 
 tf) 
 
 o 
 
 
 
 "l_ 
 
 
 
 0) 
 
 CL 
 
 
 
 
 
 
 
 
 JC 
 
 1_ 
 
 
 
 
 
 
 
 ^ 
 
 jC 
 
 
 
 C/5 
 
 0) 
 
 0) 
 0) 
 
 01 
 
 c 
 
 0) 
 
 > 
 
 o 
 
 5 
 o 
 
 c 
 
 .9 Q, 
 
 _ <u o 
 
 o cr o 
 
 01 
 
 c 
 
 w 
 to 
 o 
 
 O XI o "'^ 
 
 o 
 ■^ 
 
 0) 
 
 u 
 
 ft 
 
 o 
 o 
 
 I 
 
 o 
 
 •53 
 
 a . 
 
 !=! O 
 
 
 .^ 
 
 O 
 
 CD 
 
 I 
 
 ho 
 
100 
 
 NACA TM 1569 
 
 Figure 13.- Small expansion. 
 
 U4 
 
 VTTT? 
 
 A E S 
 
 Figure 14.- Small variation for one compression shock. 
 
 (0) 
 
 (b) 
 
 Figure 15.- Compression shock overtaken by expansion wave. 
 
NACA TM 1369 
 
 101 
 
 Characteristics 
 Shock polar 
 
 Figure 16.- Schematic representation of the graphical solution. 
 
 Figure 17.- Reflected wave disregarded. 
 
 B ^S 
 (a) 
 
 ■^^Cs\ A® ^ 
 
 (b) 
 
 Figure 18.- Expansion v;ave crosses compression shock. 
 
102 
 
 NACA TM 1369 
 
 AX- 
 
 Figure 19.- Schematic representation of graphical solution. 
 
 Figure 20. - Crossing of expansion waves. 
 
NACA TM 1369 
 
 103 
 
 //// 
 
 ZZkiT 
 
 A/ B 
 
 Figure 21.- Reflection of expansion waves and compression shocks. 
 
 Figure 22.- The cascade problem. 
 
lOil 
 
 NACA TiA 1569 
 
 X 
 
 CD 
 
 (D 
 
 a 
 a 
 ui 
 
 O 
 
 CO 
 
 • 1—1 
 
NACA TM 1369 
 
 IC? 
 
 
 
 Q. 
 
 
 
 . . 
 
 
 
 
 
 ^ 
 
 
 CM 
 
 ^ ^ 
 
 
 X ~ 
 
 s <>J 
 
 
 
 \ ^ 
 
 ^ 
 
 \ /^ 
 
 
 \/ 
 
 
 -«- 
 
 
 V^ 
 
 /. 
 
 00 ^ y^ ^ 
 
 
 
 o> wVv 5i°- t^ 
 
 10 ^.i 
 
 
 OD \ 
 
 f 
 
 
 
 ' 0/ \0 ' 
 
 - /^ 
 
 
 - / 
 
 <' 
 
 / s 
 
 
 .' \ 
 
 / 
 
 
 
 / 
 
 
 '' ID X 
 
 
 52 
 
 / 
 
 / 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 CJ 
 
 
 
 
 
 
 *~ 
 
 
 
 
 a> 
 
 
 cvj 
 
 .c 
 
 C\J 
 
 
 
 
 s 
 
 *" ID 
 
 
 
 1^ 
 
 ID toVf^ 
 
 
 S , y 
 
 
 
 
 
 CM / 
 
 0) 
 
 !:: !3S<^- 3: 
 
 g : 
 
 ^ 
 
 
 ^ 
 
 / to 
 
 9' \o 
 
 
 
 • 
 
 ^ 
 
 
 1\ "^J 1 
 
 y s 
 
 
 
 • 
 
 
 
 
 \ 
 
 / s 
 
 
 
 / 
 
 
 
 
 \ 
 
 •^ 
 
 
 |0 V, 
 
 / 
 
 
 
 
 \ 
 \ 
 
 
 
 
 <j> 
 
 
 CM 
 
 
 
 
 E 
 
 a> 
 
 T3 
 
 CM i- 
 _ CM 
 
 00 
 
 
 ^- 
 
 00 
 
 
 
 
 <£) 
 
 
 
 \. / 
 
 s cr> 
 
 •- 
 
 
 ♦ 
 
 in ^ 
 
 \ // 
 
 
 
 ^v^ 
 
 
 
 
 ^^^ 
 
 9"' \ 
 
 VO 
 
 
 • 
 
 / 
 \ 
 
 
 1 
 
 \ CM 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 \ 
 
 
 
 
 
 \ 
 
 in 
 
 
 ""■ 
 
 
 
 
 \ 
 
 
 
 
 CvJ 
 
 
 2? 
 
 
 jn 
 
 — 
 
 CVJ 
 
 M 
 
 en 
 
 U) 
 
 
 to 
 
 
 
 
 CVJ 
 
 \v ,'' 
 
 
 Sv • 
 
 \ 
 
 
 -»■ 
 
 00 / 
 
 
 
 i< y^ 
 
 «. l\v X 
 
 
 
 ^/> 
 
 in ^/VV 
 
 "t 
 
 si^-X"" = 
 
 
 CO 
 
 \ 
 
 / CD 
 
 / X^ 
 
 
 / -^ 
 
 ,-' \ 
 
 
 
 \ 
 
 
 
 
 
 
 X 
 
 • ^^ 
 
 
 CO " 
 
 ' ^ 
 
 
 
 V 
 
 ro 
 
 
 
 cvJ 
 
 
 in 
 
 
 
 
 CT 
 
 
 cr 
 
 in ^ 
 
 
 
 to 
 
 W 
 
 
 
 ^ 
 
 -%4 
 
 eg 
 
 - -^ Vrt ^ 
 
 / N 
 
 
 90 
 
 
 K 
 
 V 
 
 • ^ \ 
 
 
 • \ 
 
 • \ 
 
 
 
 \ 
 
 
 
 
 f^ N 
 
 
 
 \ 
 
 
 
 
 
 
 r^\ 
 
 0) o 
 
 i^ 
 OJ .,-1 
 
 (D Q) 
 
 Si -*^ 
 
 t4_, M 
 
 |. 
 
 ^ .§ 
 CD 'O 
 O 
 
 
 
 Kl -5 
 o 0) 
 
 
 u •- 
 
 CD W 
 
 -t; OJ 
 
 ^ <A 
 
 ■^ o 
 
 •g N 
 
 O) o 
 ho C, 
 
 d (D 
 
 oj -a 
 
 o 
 
 (/: 
 
 u ^ 
 
 
 
 fafl 
 
 '^ m 
 d O 
 
 '^ d 
 w o 
 
 fe <^ 
 
 d^ 
 
 .2 (D 
 
 w .d 
 
 (D o :g 
 
 <fH X3 hD 
 
 ° g-C 
 
 P! >>2 
 
 w i; •+-! 
 
 O 0^ 
 
 " g a 
 
 WO) 
 (D r^ W 
 
 g ^ g 
 
 -^ ti -5 
 
106 
 
 NACA m 1369 
 
 P/Po 
 0.6 
 
 04 
 
 Q2 
 
 
 
 02 
 
 
 
 
 
 
 
 
 
 
 
 
 ~^-Xi 
 
 H^ 
 
 
 
 
 
 
 r"'^'^ 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 rfimn 
 
 
 1 Inn 
 
 
 
 
 
 
 
 
 
 LI 
 
 1 
 
 
 
 
 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t.O 
 
 l/L 
 Pressure at upper side 
 
 Pressure at lower side 
 
 /////// Resultant pressure 
 
 Resultont pressure by linearized theory 
 
 Figure 25.- Pressure variation along the plate (exact method). 
 
 Figure 26.- Definition of thrust and tangential force. 
 
NACA TM 1569 
 
 107 
 
 too 
 
 90 
 80 
 70 
 60 
 
 50 
 
 40 
 30 
 
 20 
 
 
 
 -— 
 
 
 
 
 
 
 
 
 
 
 »//7[°: 
 
 5"^^ 
 
 • 
 
 
 ^ 
 
 ^^ 
 
 'i'--i^ 
 
 
 N, 
 
 \ 
 
 1/ 
 
 ' = 5*^ 
 
 ?~-^^ 
 
 
 
 ^. 
 
 ^ 
 
 ^ 
 
 1 ^ 
 
 "N 
 
 K 
 
 \^ \ 
 
 \ 
 
 / 
 
 
 A 
 
 ^ 
 
 1 
 
 \^ 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 V 
 
 \ 
 
 
 // 
 
 
 
 
 
 
 
 \ 
 
 
 / / 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 10 20 30 40 50 60 70 80 90 
 
 Figure 27.- Cascade efficiency (no friction). 
 
 Figure 28. - Linearization of crossing. 
 
108 
 
 NACA ™ 1369 
 
 ^ |_ 
 
 -oHfAb' 
 
 1 r 
 
 p p 
 J I. 
 
 (c) 
 
 r 
 
 C^J 
 
 ± 
 
 P-P Jf 
 
 Tp -p 
 
 T 
 
 r~ (^) 
 
 2,(6) (b) 
 
 1, (7) 
 
 Figure 29, - Linearized cascade theory. 
 
NACA TM 1569 
 
 109 
 
 Figure 30.- Numerical example of linearized cascade theory. 
 
 Q6r 
 
 04 
 
 0.2 
 
 
 
 -0.2 
 
 1 ( 1 
 
 I 
 
 
 
 0.1 0.2 0.3 0.4 0.5 0.6 0.7 08 0.9 t.O 
 
 l/P 
 
 Pressure on upper side 
 
 Pressure on lower side 
 
 ////// Resultant pressure 
 
 Figure 31.- Pressure distribution over plate by linearized cascade 
 
 theory. 
 
110 
 
 NACA ™ 1569 
 
 Figure 32,- The profile. 
 
NACA TM 1569 
 
 111 
 
 Figure 33.- Profile in test section of tunnel. 
 
 A Tunnel axis 
 
 W Tunnel wall 
 
 F Window 
 
 P Profiles 
 
 S Supports 
 V Compression shock 
 K Bend 
 
 E Expansion branch released 
 from bend 
 
 Figure 34.- Schematic representation of the bend in the upper tunnel 
 
 wall. 
 
112 
 
 NACA TM 1569 
 
 O Light source 
 K Tunnel 
 
 S Mirror 
 P Profiles 
 
 Figure 35.- Position of optical axis and shadow formation, x = 8 cm; 
 
 y = 4.5 cm; Z = 300 cm. 
 
 Figure 36.- Profile at starting. 
 
 Figure 37.- Schlieren diaphragm vertical, t = 0°. 
 
NACA TM 1569 
 
 115 
 
 Figure 38.- Schlieren diaphragm vertical. + = 1.5°. 
 
 Figure 39.- Schlieren diaphragm horizontal, i = S.O*- 
 
114 
 
 NACA TM 1569 
 
 Figure 40. - Photograph of crossing. 
 
 Figure 41.- Perspective distortion of figure 23. For comparison with 
 schlieren photograph in figure 39. (O represents the position of 
 the light source; the finer lines represent the shadow boundaries of 
 the plate and interference lines.) 
 
NACA TO 1569 
 
 115 
 
 {) u/Sv '''-^w '/'-/w 
 
 Figure 42.- Sudden change in angle of attack. 
 
 T^ 
 
 1 
 
 
 \ 
 
 [hr 
 
 
 / 
 
 ^ 
 
 
 P 
 
 V\ 
 
 
 / 
 
 at 
 
 \ 
 
 J 
 
 1 \ 
 
 y \ 
 
 
 
 
 ^'^i 
 
 X 
 
 t- 
 
 1 ; 
 
 
 \ 
 
 
 
 Figure 43.- The unsteady source. 
 
 Figure 44.- The periodically created source distribution. 
 
116 
 
 NACA TM 1369 
 
 (a) 
 
 (b) 
 
 Figure 45.- The integration limit. 
 
 Figure 46.- Pressure distribution at y = constant lines. 
 
NACA TM 1569 
 
 117 
 
 Figure 47.- Velocity distribution at y = constant lines. 
 
 y^ 
 
 'oTTTTTTT 
 
 B A 
 
 1 
 
 -^ — u 
 
 v„i U i . 
 
 
 
 
 (a) 
 
 (b) 
 
 Figure 48. - The flat plate in a gust. 
 
118 
 
 NACA TM 1569 
 
 2Pc 
 
 
 u t 
 
 2P, 
 
 y=0 
 
 
 
 -at- 
 
 (0) 
 
 B 
 
 .-X 
 
 -^v 
 
 ^A 
 
 ,^ 
 
 ,^ 
 
 2P 
 
 y=o 
 
 I— at 
 ut - 
 
 - I 
 
 (b) 
 
 
 
 B 
 
 
 _J 
 
 
 1 1 
 
 1 1 
 
 
 
 
 
 
 
 B 
 
 VnT 
 
 "t ^'^^^ ^*^\^ ^ *^\^ 
 
 I 
 
 7.' 
 
 (c) 
 
 Figure 49, - Pressure distribution along the plate. 
 
NACA IM 1569 
 
 119 
 
 ^0 m 
 
 ^° M M t t ^ 
 
 ^oTTZTZT 
 
 Figure 50, - The straight cascade in vertical gust. 
 
 Figure 51.- Start of process. 
 
120 
 
 NACA TM 1569 
 
 ^o3II 
 
 Figure 52.- t > h/a. 
 
 Figure 53,- Second phase of additional pressure. 
 
 I 
 
NACA TM 1369 
 
 121 
 
 
 /OD 
 
 m\ 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 / 
 
 
 \ 
 
 \ 
 
 v' 
 
 A 
 
 1 
 
 
 \ 
 
 '} 
 
 K 
 
 / 
 
 Ll_ 
 
 
 
 
 \CQ 
 
 Q a 
 1. 6 
 
 OD 
 
 
 ^ 
 
 X 
 
 / 
 
 * 6 
 
 II 
 
 
 10 
 
 
 OJ 
 
 
 ^ 
 
 
 
 0^ 
 
 " * 
 
 ^|o „ 
 
 
 ^ P~ 
 
 ^ p- 
 
 
 
 
 m CD 
 
 DO -Q 
 
 
 
 in 
 
 rM 
 
 in 
 
 
 
 s- 
 
 10 
 
 
 
 CvJ 
 
 
 
 
 Mo 
 
 p; 
 
 tl° 
 
 ^ 
 
 >,|0 " 
 2i F^ 
 
 / 
 
 - _^_^ \ 
 
 ^1 
 
 
 
 
 \ 
 
 
 N 
 
 - ■y'\-- 
 
 j 
 
 \ 
 
 -/ 
 
 m 
 :=! 
 o 
 
 u 
 
 > 
 
 ■a 
 
 w 
 cti 
 
 CD 
 
 u 
 
 ni 
 CD 
 
 73 
 
 +-> 
 
 > 
 
 c 
 
 fH 
 
 
 
 0) 
 
 u 
 
 +-> 
 
 ■M 
 
 (-1 
 
 
 •rH 
 
 -t-J 
 
 
 t/J 
 
 <l> 
 
 ^ 
 
 M 
 
 
 ^M 
 
 -)~> 
 
 u 
 
 
 Pi 
 
 
 C) 
 
 
 •r-^ 
 
 
 -4-J 
 
 
 oj 
 
 
 n 
 
 
 
 
 
 J 
 
 
 r 
 • 
 
 
 Hi 
 
 
 0) 
 
 
 ^ 
 
 
 ho 
 
 
 •r— i 
 
 
 P^ 
 
 
122 
 
 NACA TM 1369 
 
 P/A 
 
 0.9 
 o.e 
 0.7- 
 0.6- 
 
 0.5 
 0.4 
 
 0.3 H 
 
 0.2 
 
 0.1 
 
 
 
 Figure 55.- Pressure of undisturbed plate d 
 
 Figure 56. - Pressure contribution 
 
 Ph- 
 
NACA TM 1369 
 
 125 
 
 Figure 57.- Velocity distributions Vy/vg at times 5, 6, 7, and 8. 
 
 7?= 0.85 
 
 0.5 0.3 0-\ 
 
 f 
 
 9 
 
 8 
 
 ■7 
 6 
 
 ■5 
 4 
 
 ■3 
 2 
 ■1 
 
 
 0.99 0.95 
 1.0 0.9 0.8 0.7 
 
 0.6 0.5 0.4 0^ 
 
 Figure 58. - Functions f for various time intervals. 
 
124 
 
 NACA TM 1369 
 
 Figure 59. 
 
 O5 iB'Og 
 
 Figure 60.- Pressure contribution p. 
 
 V 
 
 t/. 
 
 0-B 
 06 
 OA 
 0.2 
 O 
 0.2 
 
 0.4i 
 0.6 
 
 o.e 
 1-J 
 
 TT'icrii-rp R1 - Rp.cmltpnt nrpp;.<=;nrp di Ktributions. 
 
NACA TO 1569 
 
 125 
 
 A/A' 
 
 0.8 
 0.6 
 04 
 02 
 
 
 
 
 t/il 
 a 
 
 Figure 62.- Lift of plate. 
 
 M/M, 
 
 1 
 
 t. 
 
 08 
 0.6 
 04 
 02 
 
 
 
 
 
 
 ^ 
 
 
 
 --t 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 ■ 
 
 / 
 
 ^ 
 
 ' 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 t/h. 
 a 
 
 Figure 63.- Moment distribution M referred to plate center. 
 
126 
 
 NACA TM 1569 
 
 V°/o 
 
 90 
 80 
 70 
 60 
 
 50 
 40 
 30 
 
 Figiire 64.- Allowance for friction. 
 
 
 
 
 a 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 b' 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 / / 
 
 
 ^j-.^ 
 
 
 b 
 
 
 \ 
 
 
 ^ 
 
 // 
 f / 
 
 /^ 
 
 
 
 c 
 
 
 
 
 ' / 
 
 
 
 
 
 
 
 // 
 
 
 
 
 
 
 
 \\ 
 
 
 1 
 
 
 
 
 
 
 \\ 
 
 7 
 
 f 
 
 
 
 
 
 
 \\ 
 
 '/ 
 
 '1 
 
 
 
 
 
 
 
 II 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 10 20 30 40 50 60 ^o 80 
 
 P 
 
 (a) No friction, 
 
 (b) With friction. M = 1.40, (c) With friction. M = 2.50, 
 
 * = 30. t = 30. 
 
 (b') With friction. M = 1.40, (C) With friction. M = 2.50, 
 
 i = 2.65°. ^ = 4.11°. 
 
 Figure 65.- Cascade efficiency. 
 
NACA TM 1569 
 
 127 
 
 Figure 66.- Allowance of finite thickness ratio. 
 
 77% 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 20 30 40 50 60 ^° 80 
 
 (a) (S/1--0 (b) d/i=0.05 (c)d/Z =0.10 M=1.40,^=3' 
 
 
 a 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 \ 
 
 \ 
 
 
 /J 
 
 
 
 
 \ 
 
 V 
 
 
 
 c 
 
 ^ 
 
 
 
 \ 
 
 
 ' 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 Figure 67.- Efficiency of cascade of double -wedge profiles. 
 
128 
 
 NACA TM 1569 
 
 3 
 
 A' P' 0' 
 
 Figure 68, - Area disturbed by leading edge of a blade assumed as 
 
 flat plate. 
 
NACA TM 1369 
 
 129 
 
 r m 
 1.00 
 
 0.75 
 
 0.50 
 
 T 1 1 1 1 r 
 
 2 4 6% 
 
 6/1 
 
 Figure 69.- Blade chord, maximum thickness, stagger, and location 
 
 of cross sections. 
 
130 
 
 NACA TM 1569 
 
 Figure 70.- Blade form, blade form without angular rotation. 
 
 1.25 
 
 1.0 
 
 0.75 
 
 0.5 
 
 b 
 
 0.5 0.75 1 m 
 
 (N) (M) (S) 
 
 (a) dS/drxiO"^kg/m. (b) dD/drxiO"^ kg 
 
 Figure 71. 
 
 NACA - Langley Field, Va. 
 

 Oi 
 
 a. 
 
 3 
 
 (0 
 
 01 
 
 UJ 
 
 ■o 
 
 ',' ' 
 
 ^ 
 
 rt 
 
 o n 
 
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