JfakTH-Mj 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1343 
 
 THE EXCITATION OF UNSTABLE PERTURBATIONS 
 
 IN A LAMINAR FRICTION LAYER 
 
 By J. Pretsch 
 
 Translation of "Die Anfachung instabiler Stbrungen in einer laminaren 
 
 Reibungsschicht." Bericht der Aerodynamischen Versuchsanstalt 
 
 Gbttingen E. V., Institut fur Forschungsflugbetrieb und 
 
 Flugwesen, Jahrbuch der deutschen Luftfahrt- 
 
 forschung, August 1942. 
 
 Washington 
 September 1952 
 
 "Y OF FLORIDA 
 
 
 
 ENT 
 
T«0V"» VV'* 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 13^3 
 
 THE EXCITATION OF UNSTABLE PERTURBATIONS 
 
 IN A LAMINAR FRICTION LAYER* 
 
 By J. Pretsch 
 
 With the aid of the method of small oscillations which was used 
 successfully in the investigation of the stability of laminar velocity 
 distributions in the presence of two-dimensional perturbations, the 
 excitation of the unstable perturbations for the Hartree velocity 
 distributions occurring in plane boundary -layer flow for decreasing and 
 increasing pressure is calculated as a supplement to a former report. 
 The results of this investigation are to make a contribution toward 
 calculation of the transition point on cylindrical bodies. 
 
 OUTLINE 
 
 I. STATEMENT OF THE PROBLEM 
 II. THE GENERAL DIFFERENTIAL EQUATION DESCRIBING THE PERTURBATION 
 III. THE SOLUTIONS cp-j*, cp 2 * OF THE DIFFERENTIAL EQUATION DESCRIBING 
 
 THE FRICTIONLESS PERTURBATION FOR FINITE EXCITATION 
 
 (a) Binomial Velocity Distribution (Pressure Decrease) 
 
 (b) Sinusoidal Velocity Distribution (Pressure Increase) 
 IV. THE FRICTION SOLUTION cp^* FOR FINITE EXCITATION 
 
 V. STATEMENT AND SOLUTION OF THE EIGENVALUE PROBLEM 
 VI. RESULTS OF THE CALCULATION 
 VII. DISCUSSION OF THE RESULTS 
 VIII. SUMMARY 
 IX. REFERENCES 
 
 *"Die Anfachung instabiler Storungen in einer laminaren Reibungs- 
 schicht." Bericht der Aerodynamischen Versuchsanstalt Gottingen E. V., 
 Institut fur Forschungsflugbetrieb und Flugwesen, Jahrbuch der deutschen 
 Luftfahrtforschung, August 19^2, p. 15I+-I7I. 
 
KACA TM 13^3 
 
 I. STATEMENT OF THE PROBLEM 
 
 If one wants to make a theoretical calculation of the profile drag 
 of bodies in a flow for a certain direction of air flow, one must know - 
 in addition to the pressure distribution - the position of the transition 
 zone where the laminar boundary layer becomes turbulent. The separation 
 point of the laminar layer forms a rearward limit for the transition 
 point on a section of this body surrounded by the flow. It lies in the 
 region of the pressure increase at the point where the velocity distri- 
 bution in the boundary layer has the wall shearing stress zero. This 
 separation point is a fixed point of the profile in the flow, the 
 position of which does not shift due to a variation of the Reynolds 
 
 number Re = -^— (JJ m = velocity of air flow, t = chord of the body) . 
 
 One may calculate it according to the well-known approximation method 
 of Pohlhausen which, for prescribed pressure distribution, provides for 
 every profile point a velocity distribution of the boundary layer with 
 a certain form parameter X (X = -12 separation). As forward limit for 
 the transition point, one may take the stability limit of the laminar 
 layer with respect to small two-dimensional perturbations which were 
 calculated in references 2 and 3 according to the method developed by 
 W. Tollmien (ref. 1). According to this method, there exists for every 
 form of velocity distribution in the boundary layer a so-called critical 
 
 /u S*\ / 
 Reynolds number Re* ... = -§ — (U a = local potential velocity, 
 
 V v '^ 
 \ • cr 
 
 5* = local displacement thickness) below which all perturbations are 
 
 damped; its value increases as the form of the velocity profile becomes 
 
 TT FvJf- 
 
 fuller (with increasing X.) . Where the Reynolds number Re* = — - — 
 
 formed with the actual potential velocity U a and the actual displace- 
 ment thickness 5* exceeds this critical Reynolds number, there begins 
 the instability of the boundary layer. In contrast to the laminar 
 separation point, this stability limit is therefore not fixed on the 
 profile for a normal pressure distribution but travels forward toward 
 the stagnation point on bodies in flows with increasing Reynolds 
 number Re (see fig. l). 
 
 The actual transition point which lies between these two limits - 
 stability limit and separation point - is known to likewise shift forward 
 with increasing Reynolds number Re. That it lies only a certain 
 distance behind the stability limit (as shown by a comparison of experi- 
 mental transition points and theoretical stability limits) is plausible, 
 because the excitation of the unstable perturbations starts only at this 
 limit point of the stability and must obviously have attained a certain 
 degree before the instability further downstream leads to the breakdown 
 of the laminar flow configuration. 
 
NACA TM 13^3 
 
 For this reason it seemed necessary to calculate, or at least to 
 estimate, for the velocity distributions in the laminar boundary layer 
 in the entire instability range of the latter, the excitation of the 
 unstable perturbations as well, with the objective of making an improved 
 calculation of the transition point possible. 
 
 For the velocity distribution on the flat plate in longitudinal 
 flow (Blasius profile), H. Schlichting (ref. k) has already determined 
 the excitation quantity as a function of perturbation frequency and 
 Reynolds number in a part of the instability range; and for the velocity 
 distributions in the region of pressure increase, W. Tollmien (ref. 5) 
 has explained the behavior of the excitation (in first asymptotic 
 approximation for very large Reynolds numbers) in a very general manner, 
 neglecting the effects of internal friction. 
 
 II. THE GENERAL DIFFERENTIAL EQUATION 
 DESCRIBING THE PERTURBATION 
 
 Since the bases for the method of stability investigation have been 
 discussed in detail in an earlier report (ref. 3)> w e can refer to the 
 results attained there. 
 
 Let U(x,y) and V(x,y) be the tangential and normal components 
 of a plane steady boundary-layer flow; let x denote the length of the 
 arc and y the normal to the profile contour. The stream function of 
 the two-dimensional perturbation motion which we superpose on this 
 basic flow is assumed to be 
 
 ■ / .\ / \ ia(x-ct) / n Bit i(ax-3r"t) 
 i(x,y,t) = cp(x,y)e = rp(x,y)e x e 
 
 c = c r + iCj_ Bj_ = c^a B r = C r a 
 
 J 
 
 (1) 
 
 here t denotes the time, a the spatial circular frequency of the 
 perturbation, the real part c r of c its phase velocity, and the 
 
 imaginary part Cj_ a measure for its excitation (cj_> 0) or damping 
 (c-j_ < 0); besides Bi is the logarithmic increment of the excitation 
 of the perturbation amplitude, and Br the circular frequency in time 
 of the perturbation. 
 
NACA TM 13 1+3 
 
 If we substitute the motion originating by superposition of the 
 boundary -layer flow U, V, and the perturbation motion 
 
 u = ** = q,'e ia ( x - ct ) 
 
 v = • 
 
 Sy 
 
 (2) 
 
 5x 
 
 iacp + &P\,ia(x-ct) 
 dx ; 
 
 into Navier -Stokes' differential equations, we obtain - as was proved 
 in reference 3 in detail - the differential equation describing the 
 perturbation in the form 
 
 (U - c)(V' - a 2 cp) - U"q> = — — (q^V - 2a 2 cp" + cAp) 
 \ / aRe* v ' 
 
 U a S* 
 
 Re* 
 
 J 
 
 (3) 
 
 Here the prime ( ' ) denotes differentiation with respect to the wall 
 distance yj the velocities are referred to the local velocity Ua at 
 
 the boundary- layer limit, and the wall distance as well as the wave 
 
 length A = 
 
 2* 
 
 a 
 
 are referred to the local displacement thickness 5*, 
 
 In order to avoid misunderstandings it should again be emphasized 
 that - in spite of the assumption that U, V, and cp be functions of 
 the arc length x - only the form of the local velocity distribution is 
 decisive for the stability investigation as one recognizes from 
 equation (3). The immediate effect of the pressure gradient, however, 
 is negligibly small as is also the influence of the x-dependence of the 
 perturbation amplitude cp. 
 
 The boundary conditions of the differential equation (3) result 
 from the condition that the perturbation velocities u, v vanish at 
 the wall and that the friction effect at the outer boundary -layer limit 
 (U' ' = 0) has disappeared. With their aid, the calculation of the 
 excitation of the unstable perturbations may be reduced to an eigenvalue 
 problem, the solution of which is discussed in section V. 
 
 We deal first with establishing the particular solutions of the 
 differential equation (3), limiting ourselves to small values of the 
 
NACA TM 13V3 
 
 excitation quantity c-jj thus, the general solution of the differential 
 equation describing the perturbation (3) may be represented in the form 
 
 v=l v=l V U / 
 
 There cp v denotes the particular solutions for cj_ = which are 
 
 obtained in the calculation of the limiting curve of the instability 
 range (neutral curve) in the a, Re*-plane (ref. 3)> and the o> v 
 signifies additional functions for c^ > 0. 
 
 We turn first to calculating the integrals cp-.*, qv>* and the 
 additional functions od-, , CDp. 
 
 III. THE SOLUTIONS cpj*, y * OF THE DIFFERENTIAL 
 
 EQUATION DESCRIBING THE FRICTIONLESS 
 PERTURBATION FOR FINITE EXCITATION 
 
 If aRe* is assumed to be very large, the differential equation (3) 
 is simplified to the so-called frictionless-perturbation equation 
 
 (U - c)(cp" - a 2 rp) - U"cp = (5) 
 
 This differential equation has a pole of the first order at the location 
 U = c = c r + ici to which we coordinate the point y c * of the complex 
 
 y-plane. In the neighborhood of this singularity, a fundamental system 
 may be easily indicated by series development. In order to establish 
 the connection with the case of the purely real c treated before 
 (see ref. 3)> we first give the relation between the complex y c * and 
 the wall distance y c of the "critical" layer U = c r . 
 
 From 
 
 u(y c ) = c r ( 6) 
 
NACA TM 13^3 
 
 and 
 
 u (y C *) = c r + ici 
 
 there follows 
 
 u(y c *) - U(y c ) = ici = (y c * - y c )u ' + . . . (8) 
 and, with limitation to the terms linear in c-^, therefore 
 
 y* = y c + ^L (9) 
 
 u o 
 
 We shall now indicate the construction of the solutions cp-i*, r Po* 
 
 of equation (5) for those special velocity distributions U(y) by which 
 ve shall approximate the laminar boundary profiles of Hartree (ref . 6) 
 for the calculation of the excitation of the unstable perturbations 
 in the same manner as we did before in reference 3 for the calcula- 
 tion of their critical Reynolds numbers. 
 
 (a) Binomial Velocity Distribution 
 (Pressure Decrease) 
 
 In the region of the pressure decrease, we used the approximation 
 function 
 
 U = 1 - (a - y) n n = 2,3A . . . (10) 
 
 where a denotes the coordinate of the point of junction to the potential 
 velocity. 
 
 With the new variable 
 
 y - y c 
 
 v l 
 
 a - y c 
 
 11 
 
NACA TM 13^+3 
 
 and with 
 
 cx-l = a(a - y c . 
 
 (12) 
 
 the perturbation equation (5) for indifferent perturbations then reads 
 
 £ 
 
 1 - (1 - yi )n 
 
 d^P 
 
 n-2 r 
 
 a^cp + n(n - l)(l - y^"^) = 
 
 13 
 
 Using the relations (9) and (11), we now introduce the complex variable 
 
 yi* = : — — = yi + - — — = yi -* (ik) 
 
 a - y. 
 
 a - y c 
 
 u '(a - y c ) 
 
 With the abbreviation 
 
 c i 
 
 u o'( a - yc) n(c r - i; 
 
 (15: 
 
 the differential equation describing the frictionless perturbation for 
 nondisappearing excitation then is transformed into the form 
 
 1 - y,*) n - 
 
 nif^l - (1 - y) 
 
 n-1 
 
 ■ay 
 
 2 *\ 
 
 n(n - 1, 
 
 1 - yi ** n - 2 
 
 + (n - 2) if -A - y{ 
 
 *\n-3 
 
 cp* = (16) 
 
 in which, according to our presupposition, only the terms linear in 
 f^ have been taken into consideration. 
 
8 NACA TM 13^3 
 
 If one writes equation (l6) after multiplication by 
 
 n* 2 
 
 1.(1 - 3 r 1 *) I ' + nlf 1 [l-(l- 3 r 1 *) 1 '- 1 ] 
 
 and after division in the form 
 
 d^tp 
 
 y *2 J2L + q>* J_ PiV 1 = (IT) 
 
 dy*2 !Zi 
 
 the first solution qp-j* is given by 
 
 yi* 52 ^i*" = — — K L <°l) (l8) 
 
 a - y c V=0 a - y c \ u Q ' 
 
 with 
 
 «>i - 2_ §vyi v (19) 
 
 v=o 
 
 The series coefficients e * are obtained from the 3 V according 
 to the recursion formula 
 
 v=l 
 
 v(v + l)e v * + ^ *e * =0 e * = 1 (20) 
 
 u=o 
 
 The solution cp 2 for all y,* is found to be 
 
 ■Pi 
 
 %* - V/^ * - 1 CA W * -1* r^ >» '1* <-' 
 
NACA TM 13^3 
 
 In this equation, the logarithms are, however, ambiguous; one must cut 
 the complex yy*-plane in such a manner that 
 
 3 * JT 
 
 - i « < arg yi < _ 
 
 If we put 
 
 *v*yi* v = C Wn v + ifi YZ *wi v 
 
 V=2 V=2 V=l 
 
 (22) 
 
 cp 2 may he represented for y-^ = real part of y-^* > at first in the 
 form 
 
 ^2* = 1 + XZ Vi v + if i5Z *wi v 
 
 V=2 
 
 V=l 
 
 I 1 " "^ * lf i £ wj 
 
 In y-L + 
 
 ifi 
 
 lCj 
 
 — = <P2 r ^ 
 
 7l ) u 
 
 (23) 
 
 with 
 
 av 
 
 a - yc 
 
 fci hyi + i-^yl- lny i) + 
 
 2e i in y i y~^y/ 
 
 v=o 
 
 (24) 
 
10 
 
 NACA TM 13 k-3 
 
 In the transformation of equation (21) into equation (23) the 
 relation 
 
 e l* = e l( X - if l) 
 
 (25) 
 
 is put to use; it is easily obtained from the perturbation equation (5! 
 if U(y) and cp-^ for c^ = 0, and for c^ = in the neighborhood 
 
 of the singularity U = c are developed into a Taylor series, since 
 
 2e ? = (a - y c )V' 
 
 V 
 
 :26) 
 
 and 
 
 2en* = (a - y c ). 
 
 U " + ici-° 
 
 Un' 
 
 U ' + i Cl -2- 
 
 V 
 
 ~ fa 
 
 ( a - y c ) 
 
 V 
 
 Un' 
 
 1 - 
 
 icWUo" 
 
 u o'\ u o' 
 
 = (* - y c )-^— (1 - ifi] 
 
 u ' 
 
 (27: 
 
 If we express in equation (27) the exact value 
 
 V U '" s 
 
 V V 
 
NACA TM 13^3 
 
 11 
 
 again by the approximate value fj_, this step is motivated by the fact 
 that due to the small value of c ± , one deals here only with a correction 
 
 term for the value of 
 
 Tl ' ' 
 ^0 
 
 U ' 
 
 For y, < 0, cp p * is represented by the expression 
 
 cp 2 * = 1 + £ b/y^ + 2e 1 * I *£-Ln 
 
 V =2 " y ° \ 
 
 yi 
 
 + 1 ar 
 
 g yi 
 
 cp 2 
 
 ICj 
 
 f^ 
 
 3 * ^ « 
 
 2 jr < arg y* < - 
 
 (28) 
 
 with 
 
 ov> = 
 
 a - y c 
 
 2e 1 In 
 
 2e 1 cp ] _/i 
 
 2ijre, 
 
 yi 
 
 CPl 
 
 V=0 
 
 g v y l 
 
 i* - y c v=0 
 
 Wl v )> 
 
 (29) 
 
 The term with i arg y-,* in equation (28) was obtained by 
 
 H. Schlichting (ref . k) and W. Tollmien (ref . 5) by a discussion of the 
 general perturbation equation (3) in the neighborhood of the singularity 
 U = c r + icj_ in a similar manner as the "transition substitution'' in 
 
 the critical layer for purely real c by taking the friction effect 
 into consideration. 
 
 Since we are, in the calculation of the transition substitution, 
 concerned with a representation of e-i of maximum accuracy, we shall 
 
 replace e-i by the expression (26) of the Taylor development of the 
 
 exact Hartree profile. 
 
12 NACA TM 13^3 
 
 For the linear and the parabolic approximate distribution 
 [n = 1, n = 2 in equations (10) and (l6)] H. Schlichting (ref. k) 
 has already given the solutions qv*, qW*. This calculation was con- 
 tinued for n = 3 and n = h. The coefficients 3v*> &v* > W* > 
 s, hy are compiled numerically in table 1. For the convergence of 
 the power series, the reflections made in reference 3 are valid. 
 
 (b) Sinusoidal Velocity Distribution 
 (Pressure Increase) 
 
 In the region of pressure increase, the Hartree velocity distri- 
 butions in the boundary layer (see ref. 3) were approximated by 
 formulation in terms of a sine formula introduced by W . Tollmien 
 (ref. 5). 
 
 U =U S + (1 - U s )sin[?-=-||] (30) 
 
 where s denotes the wall distance of the inflection point. 
 With the new real variable 
 
 IL^z (31) 
 
 and with 
 
 y 2 2 a- s 
 
 CC2 = -(a - s)a (32] 
 
 the perturbation equation (5) for the indifferent perturbations then 
 read 
 
 sin y 2 - y 2s + sin y 2s 
 
 d 
 
 2, 
 
 2., 
 
 — - o^rpj + sin |y 2 - y 2s )9 = (33) 
 
NACA TM 13^3 
 
 13 
 
 where y 2 was put equal to 
 
 ?2i 
 
 n s - y c 
 
 2 a - s 
 
 (3*0 
 
 We now introduce the complex variable 
 
 *2 
 
 * _ « y - y c 
 
 2 a-s 
 
 y2 + o 
 
 * y c - yc 
 
 2 a-s 
 
 = y2 
 
 ic- 
 
 U n ' 2(a - s) 
 
 !35) 
 
 With the abbreviation 
 
 U ' 2(a - s) (1 - U s )cos y 2£ 
 
 (36) 
 
 the frictionless differential equation, describing the perturbation 
 for non-vanishing excitation, then is transformed into 
 
 sin 
 
 v*yp* 2 
 
 - y2s 
 
 a 2 2 cp' 
 
 s , - cos y 2s 
 
 d 2 9* 
 
 + sin y 2g - if^cos fy 2 - y 2 
 
 (y2* - y2s)- if 2 cos (y2* - y2s) 
 
 sin 
 
 rp* = 
 
 (37) 
 
 If (37) is written like equation (l6) in the form 
 
 y 2 *2 *^L + y*f: PiW 1 = 
 
 dy 2 
 
 *2 
 
 i = l 
 
 (38) 
 
 the solutions rp-j. j <P2* are determined by analogy with the equations 
 (18), (23), and (28). One has 
 
 cpi = y 2 z_ e v y 2 
 
 2(a - s) v=0 
 
 IC-i 
 
 CPl 
 
 2(a - s)l ri U 
 
 7^1 
 
 (39) 
 
11+ 
 
 NACA TM 13^3 
 
 with 
 
 CO 
 
 i = 2ZZ * 
 
 v=0 
 
 V J 2 
 
 (to) 
 
 For y ? > 0, the solution cp 2 is obtai 
 
 ned 
 
 q> 2 * = 1 + ) b v *y 2 * v + 2e^ 
 v=2 
 
 ircp^ 
 
 2(a - s) 
 
 In 
 
 y2 7 
 
 v < arg y * < _ 
 
 (hi) 
 
 If one puts 
 
 
 *ir_*V = \ K „ V 
 
 v=2 
 
 b v ^2 + if 2 Z_ ^2 
 
 v=l 
 
 (to) 
 
 cpo* for y 2 > may be represented in the form 
 
 00 00 . 
 
 cp 2 * = 1 + Yl V2 V + if 2 XI V2 V + ^lf 1 + 
 v=2 v=l V 
 
 if 2 \ 
 
 sin y 2s cos y 2s 
 
 jrcp 1 
 
 2(a - s) 
 
 + i*2 2Z Sv y 2 V )( ln y 2 + 
 
 V=0 
 
 if 2 ' 
 
 lc i 
 Po «>p 
 
 V 
 
 (to) 
 
 with 
 
 C0 o = 1 
 
 2 2(a - s) 
 
 i h^ + 2fleiCPl /JL + ln ^ 
 
 V=l 
 
 2(a - s) \y 2 sin y 2s cos y 2 < 
 
 2e i ln y 2 H s v y 2 ' 
 
 (to) 
 
 v=0 
 
NACA TM 13^3 
 
 15 
 
 In transforming (kl) into (U3), one makes use of the relation 
 
 e-L* = ei fl + 
 
 Ifr 
 
 sin y 2s cos y 2s 
 
 (V>) 
 
 which follows from the two equations 
 
 . a - b.Uq 1 
 
 JT U, 
 
 
 
 (1*6) 
 
 and 
 
 'a - s) V + ici 
 
 ei' 
 
 V" 
 
 U ' 
 
 U Q - + i Ci 
 
 u o" 
 V 
 
 (a - s) U " 
 n Uo' 
 
 t t TT T I ! 
 
 1 - 
 
 i Ci /U " u 
 
 Uo'\u ' u ' 
 
 (a - s) U " / if 
 
 tt U r 
 
 1 + 
 
 sin y 2s cos y 2 
 
 (hi) 
 
 Cfu* for y 2 < 0, is represented by the expression 
 
 cp 2 * = 1 + Z_ b v *y 2 
 v=2 
 
 *„ *v * 
 
 (a - s) 
 
 9 * In 
 
 y 2 + i arg y g 
 
 9 2 
 
 ICh 
 
 Ur 
 
 OV 
 
 I n < arg y 2 * < 2 \(kQ] 
 
16 
 
 NACA TM 13^3 
 
 with 
 
 > av 
 
 2 a - s 
 
 $Ti ^ a - sly 2 sin y 2s cos y 2s I 
 
 2e l ln |y2 Z_ 6 v y 2 
 " v=0 
 
 Svy 2 
 
 v=0 
 
 + 2irr ei J 
 
 Jtrp 1 
 
 2(a - s) sin y 2£ 
 
 cos 
 
 *2t 
 
 (*9) 
 
 For the calculation of the transition substitution, we shall 
 replace the term ej_, as in section III (a), by the accurate value 
 
 (equation (k6)) of the Taylor development of the exact Hartree profile. 
 
 In a comparison of the relations (2U) and (hh) or (27) and (h-'j) or 
 (29) and (U9), it is striking that, in the expressions for e^* and o^ 
 
 for the sinusoidal boundary -layer profiles in case of pressure increase, 
 the product sin y 2s cos y 2 appears in the denominator of several 
 
 terms. The sign of this product is negative when the inflection point 
 of the velocity distribution lies more closely to the wall than the 
 critical layer (V 2s < 0); the critical layer thus lies in the part of 
 
 the velocity profile showing concave curvature. The sign of the expres- 
 sions divided by sin y Q cos y Q then is the same as the sign of the 
 
 corresponding terms in the solutions for the velocity distributions in 
 case of pressure decrease which, as is known, have concave curvature at 
 every wall distance. If, in contrast, the critical layer is located 
 between point of inflection and wall (V 2s > j, then, in the part of the 
 
 velocity profile having convex curvature the signs discussed are reversed. 
 In that case, when the critical layer shifts to the point of inflection 
 itself /Vps = 0\ , the behavior of e^* and cop is regular since then 
 
 ic- 
 
 ifr 
 
 2 1 
 
 Us) 
 
 (50) 
 
NACA TM 13^3 
 
 17 
 
 and 
 
 03 
 
 2 (y C = s ) 2(a - s) 
 
 v=l 
 
 hvy 2 A 
 
 2(a - s 
 
 i— - In y 2 
 
 (* 
 
 > 
 
 (51) 
 
 u>, 
 
 2 (y C = s ) " 2(a - s) 
 
 JS. 
 
 V=l 
 
 V 
 V^2 + 
 
 "Pi 
 
 2(a - s) 
 
 In 
 
 y 2 
 
 . ? 
 
 2 a - s 
 
 1 
 
 y 2 < o^ (52) 
 
 The series coefficients B v *, e v *, b v * , g v , h v for the 
 
 sinusoidal basic velocity are given numerically in table 2. For the 
 convergence of the series developments, the explanations of the earlier 
 report (ref. 3) are again valid. 
 
 IV. THE FRICTION SOLUTION cp^* FOR 
 FINITE EXCITATION 
 
 Besides playing a role in the critical layer U = c, the friction 
 of the fluid is of importance also in the neighborhood of the wall where 
 it occasions two more solutions cp^*, cph* of the general differential 
 
 equation describing the perturbation (3)- 
 If we introduce the variable 
 
 y - y c 
 
 (53) 
 
 with 
 
 e* = 
 
 [U ' + ic- 
 
 1 u - 
 
 •1/3 
 
 (5M 
 
18 
 
 NACA TM 13^3 
 
 we obtain from equation (3) in the limiting process e — »0 for large 
 Reynolds numbers the differential equation 
 
 iaS> 
 
 dT) 
 
 
 dT] 
 
 (55: 
 
 The solutions of this differential equation are just as independent of 
 the form of the velocity distribution U(y) as was the case for the 
 solutions cpo, cpii for the excitation zero (ref. 1) ; they read 
 
 corresponding to reference 2: 
 
 V 
 
 *3,V 
 
 dV 
 
 ^ * 
 
 1/2 
 T] H 
 
 (D,(2) 
 
 1/3 
 
 3V"> J 
 
 \3/2 
 
 dTi 
 
 (56) 
 
 where H ' signifies the Eankel function of the first and second 
 kind, respectively. 
 
 Since the Hankel function of the first kind increases for large 
 wall distance beyond all limits, it cannot be contained in the general 
 integral (k) so that we there may put C^ = 0. 
 
 V. STATEMENT AND SOLUTION OF 
 THE EIGENVALUE PROBLEM 
 
 After having found the particular solutions of the differential 
 equation describing the perturbation (3)> w e now state the eigenvalue 
 problem, (which results from the boundary conditions of the perturbation 
 equation) for investigation of the excitation of the unstable pertur- 
 bations. 
 
 At the wall, the tangential as well as the normal component of the 
 perturbation velocity disappear; thus one has 
 
 Cl*^!** + c 2* c P2*w + c 3*<P3*w = ° 
 
 (57) 
 
 c lW + c 2* r P 2 V + c 3*?3*v' = 
 
 (58) 
 
NACA TM 13^3 
 
 19 
 
 At the point of juction y = a to the region of constant velocity, 
 
 U" is zero and therewith cp*" - a 2 <p* = 0, that is cp* = e" a y, so that 
 the third boundary condition reads 
 
 c l**l*a + c 2**2*a = 
 
 (59) 
 
 with 
 
 V a r v a 
 
 *»' + °Va' V = 1 > 2 > 
 
 (60) 
 
 A term *o * does not appear because the particular friction solution 
 cpo* has already been damped at the point of junction. 
 
 In order that the three homogeneous equations (57)? (58), and (59) 
 may have a solution different from zero, the determinant must disappear 
 
 ¥w ^2 w <P 3 w 
 <PlV <P&V <P3*v' 
 
 *l*a *2*a ° 
 
 = 
 
 (6i: 
 
 The solution of this determinant yields the equation 
 
 ^V <feV*l*a " *lV*2* £ 
 
 (62) 
 
 or with Schlichting' s abbreviations 
 
 1 93V 
 
 e no ^3 w 
 
 n * 
 ^0 
 
 = F^q*) = F* 
 
 (63) 
 
20 NACA TM 1343 
 
 <P2*v*l*a " <PlV2ja 
 
 E*(a,c r ,ci) (6k) 
 
 Vc* <P*v'®l*a " ?l*w V 
 finally 
 
 F* = E*(a,c r ,ci) (65) 
 
 Therein 
 
 no = - — ( 66 ) 
 
 The complex equation (65) is equivalent to two real equations in which 
 the parameters a, Re*, c r , and c± are contained. If one limits 
 
 oneself to the case of indifferent perturbations (cj_ = 0), one obtains 
 from them after elimination of c r a relation between the Reynolds 
 
 number Re and the wave length — of the perturbation. That is the 
 
 equation of the neutral curve or the curve of constant zero excitation 
 by which the stable and unstable perturbation states are separated and 
 the lowest Reynolds number of which is the so-called critical Reynolds 
 number Re* cr ; when this critical Reynolds number is exceeded, the 
 excitation of the unstable perturbations begins. 
 
 These neutral curves were determined in a previous report (ref . 3) 
 for the velocity profiles for decreasing and for increasing pressure 
 calculated by Hartree. 
 
 In the present investigation, we shall assume the excitation 
 quantity c^ to be different from zero and investigate the curves of 
 constant excitation enclosed by this neutral curve. 
 
 For this purpose, we shall solve the complex equation (65) in a 
 somewhat simpler manner than Schlichting (ref. k) , by determining not 
 
 B C-! 3 C,' 
 
 the differential quotients and — at the location of the 
 
 d a d Re* 
 neutral curve, but the curves of constant excitation cj > directly 
 
 by a method similar to that applied for the neutral curve c^ = 0; 
 
 however, without making use of this curve itself. We consider first 
 the left side of equation (65). 
 
NACA TM 13^3 
 
 21 
 
 Since we intend to limit ourselves to small values of excitation, 
 we develop F* in the form 
 
 '(v) = f M + (^ - 
 
 ^0 
 
 dF 
 
 VV/V^o 
 
 (67) 
 
 where t\~ as before in reference 1 is defined by the relation 
 
 ,\ +1/3 
 
 tio = - y c (oRe*U '^ 
 
 (68) 
 
 According to equations (9), (5*0, (66), and (68) one now has 
 
 1/3 
 10* 
 
 c \ u 'y C ; 
 
 o^^Yi + ffiSalT 
 
 - y c faRe*U lN )l/3 
 
 ^0 
 
 V 2 
 
 ic i /-, L y c u ' 
 
 1 + i- 1 + 
 
 U 'y c \ 3 U ' 
 
 1+ ^L/ 1 + ^ u o"' 
 
 u 'y c 
 
 3 u o' 
 
 Thus equation (67) becomes 
 
 ^^~*M^o(-r^ 
 
 (69) 
 
 = G ( T l0' c r' c i) 
 
 (70) 
 
22 
 
 NACA TM 13^3 
 
 One sees therefore that the form of velocity distribution which does 
 not enter into the exact solution F(t} * j according to equations (63) 
 
 and (56) does appear explicitly in the development of F^tjq* \ with 
 
 respect to c^. 
 
 The differential quotient 
 
 dF 
 
 dF 
 
 dFi 
 + 1- 
 
 &T\ d.T\Q &T\Q 
 
 was determined by graphic differentiation of the function f/t] ). The 
 
 numerical values of its real and imaginary part are given numerically 
 in table 3« Since real and imaginary parts nowhere disappear simultane- 
 ously, the function Ff^o*) will be free from singularities in region 
 
 around the function F(t)q) and the development (67) will be thus per- 
 missible. 
 
 We now consider the right side of equation (65) which is defined 
 by the relation (6k). 
 
 Remembering the splitting of the frictionless solutions 
 <P v *(v = 1,2) in the form of equation (k) , we first put 
 
 with 
 
 furthermore 
 
 * icj_ 
 
 V a = *va " ~7~T ^Va 
 u 
 
 ^va = % a + c 
 
 Va 
 
 ICi 
 
 (TD 
 
 Vv w = ^vw - — r «\<w 
 
 K V W 
 
 = 9- 
 
 vw 
 
 IC-i 
 
 U r 
 
 OJ 
 
 VW 
 
NACA TM 13^3 
 
 23 
 
 For the binomial velocity distribution U = 1 - (a - y) n the 
 
 values of the additional functions co... co,, and their derivatives 
 
 v a v w 
 
 co v , ao v ' are given by the equations 
 a w 
 
 co 
 
 CO 
 
 v=0 
 
 CO, 
 
 '2a 
 
 a - y r 
 
 v=i a - y c 
 
 CD 
 
 la 
 
 £ 
 
 a - yc v=i 
 
 Vg, 
 
 CO, 
 
 '2a 
 
 vhvi + 
 
 2ei 
 
 2 9 l£ 
 
 (a - y c )2 v=1 ( a - y c )2\ a - y c 
 
 + cpia + co l£ 
 
 COn 
 
 iw = 2^ Sv^ 
 
 v=0 
 
 CO, 
 
 2w 
 
 a - y c 
 
 2e In 
 
 V=l 
 
 lw 
 
 V 
 
 v L Sei^iw/ 1 
 
 lw 
 
 a - y c Uiw 
 
 - ln |yiw|^iw\ + 
 
 CO 
 
 lw 
 
 + ifl ^0_/_Plw_ 
 
 u ' \ a - y c 
 
 - CO 
 
 lw 
 
 >(72) 
 
 CO 
 
 lw 
 
 a - y 
 
 00 
 
 c v =l 
 
 CO, 
 
 2w 
 
 00 
 
 y-K v=I 
 
 2ei 
 
 (a - y c )2 v 
 
 ,/ 1_ 
 
 lw 
 
 (a - y c ) 2 
 
 <Plw 
 
 'iviy 
 
 In 
 
 yiw 
 
 CD 
 
 lw 
 
 + _Ji + in 
 
 yiw 
 
 yiw 
 
 a - y c \ yiw" 
 
 co lw '(a - y c ; 
 
 yiw/ 
 
 I I / „ ' 
 
 i)r HQ_/_flw_ 
 
 U ' la - y c 
 
 co 
 
 lw 
 
2k 
 
 NACA TM 13^3 
 
 For the sinusoidal velocity distribution 
 
 U = U s + (1 - U s )sin( y ' s *' 
 
 la - s 1 
 
 in contrast, the corresponding formulas read 
 
 *la = *— ^2 
 V=o 
 
 u 2a 
 
 2(a - s) 
 
 H V2. 
 
 v + "ei^Pl a / 1 
 
 v=l 
 
 In y 2s 
 
 2e l ln y2a m la 
 
 u la 
 
 2(a - s) v =l 
 
 2 DO 
 
 vg v y 2£ 
 
 v-l 
 
 v-l + 
 
 if^ei 
 
 = MlTT^ fe vh v y 2 a- - ^^r^ 
 
 "qpi a 
 
 2(a-s)l y 2a 2 
 
 y 2a sin y 2s cos y 2s 
 
 sin yo = cos 
 
 y2E 
 
 y 2s cos y 2 . 
 
 ">la 2 ( a - s ) -, 
 
 — „ + ln ^a^la 
 
 y2a * 
 
 £ 
 
 u lw = Z_ S-/2w 
 V=0 
 
 2w 2(a - s) 
 
 2e i ln |y2w| a> i 
 
 t "-" * ^fe ♦ = 
 
 In y2w 
 
 U ' 
 
 y 2s cos y 2s 
 [<p lw 
 
 U ri '\2(a - s)sin y 2s cos y 2E 
 
 ^lw 
 
 u lw 
 
 u 2w 
 
 2 ( a " s ' v=0 
 
 ,2 
 
 X! vgvy 2w 
 
 v-l 
 
 Ma - s)2^TX 
 
 Hvhvy^- 1 .-^! 
 
 fl 2 e 
 
 2(a - s)< 
 
 " r Plw 
 
 2(a - s) \ y 2w : 
 
 y 2u sin y 2g cos y 2s 
 ^lw 2(a - s) 
 
 + »,'(+-+ ln M \ + 
 
 y2w 
 
 In y 2w U lw ' 
 
 y 2w sin y 2s cos y 2g/ 
 ._ U "/ ■ "9lw 
 
 f (73) 
 
 U ' l2(a - s)sin y 2s cos y 2E 
 
 ">lw 
 
NACA TM 13^3 
 
 25 
 
 If the critical point shifts to the inflection point of the velocity 
 distribution (y c = s, y2 S = CM, the equations for u>>a> ^2& > °^2\i> 
 
 ^2w' (equation (7*0) are simplified to 
 
 CD, 
 
 2a 
 
 2(a - s) 
 
 ncp l£ 
 
 CD, 
 
 ff 2 
 
 2a 
 
 k(a - 
 
 YL h vy2a V + , 
 
 V=l 2(a - s. 
 
 'Voa V_1 + 
 
 in y 2a 
 
 jrcp 1 
 
 a 
 
 V=l 
 
 8U •M^ + ^ ln ^ 
 
 U3, 
 
 ' 2(a - s) 
 J2. 
 
 CD, 
 
 2v 
 
 Ma - s)2 
 
 i7r3 ?lw' 
 Ma - s) 2 
 
 5 W * sit^t ta ^ 
 
 V 2 - Vh y V - X + ^Iw 
 
 fci V2w 2(« " 3)y 2w 
 
 1 tt3 
 
 tPlw 
 
 >(lh) 
 
 Ma - s)2 
 
 + <Plv' ln |y2w| 
 
 We now split cp 2w , cp 2w ' , °>>w> ^w' into real and imaginary 
 parts according to the formulas 
 
 cp 2w = A 1 + iB L 03 2w = M 1 + iN x 
 
 P 2w = A 2 + iB 2 co 2w = M 2 + iN 2 
 
 (75) 
 
 then one obtains 
 
 ^ = A i + U7 N i + iB i-U7 M i, 
 
 > 
 
 qfeV = A 2 + ^ N 2 + i B 2 " ^r M 2 
 
 J 
 
 (761 
 
26 
 
 NACA TM 13 U3 
 
 Then we put the expression (6k) into the form 
 
 E*(a,c r ,c i ) = E(a,c r ) 
 
 ( z l + iz 2]j 
 
 1 + c^z-^ + iz 
 
 (77! 
 
 where E, as previously defined in reference 1, is defined by 
 
 E = - 
 
 1 ^w^la " c Plw*2a 
 y c <P2v'*la " ^lw'^c 
 
 (78) 
 
 First, limiting oneself to. the terms linear in Cj_ one has 
 
 <P 2 Vl*a ~ Via + ^( N l°la + Via) 
 
 Via " ^(Vla + M l*la 
 
 cp l*w*2*a ~ <p lw*2a " fj-r( m 1^2a + cPi^a 
 
 ^vVa - Via + ^Vla + ^la) + 
 
 B 2*la " -^r( A 2%a + Via) 
 
 Vw'Va ~ r Plw' $ 2a " ^(^lw'*2a + ^lw'^2 
 
 > (79) 
 
NACA TM 13^3 
 
 27 
 
 Hence, there results after a short intermediate calculation 
 
 E* ~ E 
 
 1 - 
 
 ic< 
 
 u ' 
 
 u y c 
 
 (Ni$ia + Bjflia) + i(cuiw<&2a + r Plw^2a - Ajflia ~ Mi*i a ) 
 
 Ai*la " c Piw $ 2a + i*l 9 lt 
 
 (Ng1>ia + B2 n la) + iftlw'^a + ?lw'"2a - A 2 ^l a - M 2 ^l £ 
 
 A 2 $ la " ( Pl w '*2a + iB 2$la 
 
 (80)' 
 
 We put for abbreviation 
 
 N l°la + B l%a = m l 
 
 a) lw*2a + ^lw^a - A l fi la - Mi*i a = ni 
 
 N 2*la + B 2%a = m 2 
 
 03 
 
 lv'*2a + "Plv'^a " A 2%a - M 2 *i a = n 2 
 
 Via " f Plw°2a = K l 
 
 A~<J> 
 
 2°la " <Plw °2a = *2 
 
 "1 
 
 > 
 
 (81) 
 
28 
 
 NACA TM 13^3 
 
 Then there follows from equation (80) 
 
 m-L + in-[_ 
 
 E*~ E 
 
 = E 
 
 1 - 
 
 VYc U o' 
 
 
 
 m2 + m 2 
 
 L 
 
 K l + iB l°la K 2 + iB 2*la 
 
 1C 
 
 ^ + ^r 
 
 Uq yc u 
 
 m l K l + n l B l*la n^Kg + n 2 B 2 $]_, 
 
 K X 2 + B! 2 $ la 2 k 2 2 + B 2 2 3>la 2 
 
 Id 
 
 UO 
 
 n]_K]_ - m^B^ia n 2 K 2 - mjjB^ia 
 
 Ki 2 + Bi 2 $ia 2 K 2 2 + 
 
 B 2 2 $ia £ 
 
 (82) 
 
 If we finally introduce the designations 
 
 U r 
 
 m l^l + n l B l*la m 2 K2 + n 2 B 2 ( I ) ]_ a 
 
 Kl 2 + B^ct^ 2 
 
 K 2 2 + B 2 2 d, la 2 
 
 ^- + ^ 
 
 u y c 
 
 u, 
 
 n l K l - m l B l*la "2*2 ~ ^2° la 
 
 Kl 2 + B^Oia 2 K 2 2 + Bp^la 2 
 
 (83) 
 
 the representation (77) is attained. 
 
 By means of the equations (70), (11), and (83) we thus have divided 
 the two functions F* and E* (a,c r ,ci) appearing in equation (65) into 
 real and imaginary parts and can now graphically solve this equation. 
 For this purpose, we plot for constant c i for several values y c the 
 imaginary part of the function F* 
 
 l(F*) =1 F^o) 
 
 1)0 
 
 u 
 
 1 + ^VW 
 
 y c 
 
 3 U '/dT) 
 
 (8U) 
 
 against its real part 
 
 b (f*) = b [f(^ )] -no 
 
 + y c U Wi 
 
 u o'yd 3 Uo'idno 
 
 (85) 
 
NACA TM 13^3 29 
 
 and plot in this polar diagram to the same values c^ and y c , the 
 imaginary part of the function E*(co,c r ,ci) 
 
 ife*) = I(E)(1 + c iZl ) + ciz 2 R(E) (86) 
 
 against its real part 
 
 r(e*) = R(E)(1 + c iZl ) - ciz 2 l(E) (87) 
 
 The points of intersection of the functions F* and E* connected one 
 to the other by the same pairs of values c r , Ci yield first the values 
 t]q pertaining to the corresponding a-values, and then with (68) the 
 curve a(Re*) of the constant excitation c^. This plotting and cal- 
 culation then are repeated for other values c^ . 
 
 VI. RESULTS OF THE CALCULATION 
 
 As immediate result of the rather extensive calculations, the polar 
 diagrams with the curves E*(a,ci,cr) and F* = F(t|q*) for several 
 
 Hartree velocity distributions U(y) in the laminar boundary layer are 
 represented in figures 2 to 21. These boundary -layer profiles belong 
 to the special class of pressure distributions 
 
 U a = const. x m (88) 
 
 and are rigorous solutions of Prandtl's boundary-layer equations which 
 were obtained with the aid of a Bush apparatus (ref. 6). We use, 
 according to Hartree, for characterization of these boundary-layer pro- 
 files the parameter 
 
 2m (89) 
 
 1 + m 
 
 The profiles with m > 0, 3 > occur in case of decreasing pressure; 
 the profiles with m < 0, 3 < in case of increasing pressure. The 
 profile 3=1 is the exact Hiemenz profile at the stagnation point of 
 a cylindrical body, the profile 3=0 is the exact Blasius profile at 
 the flat plate in longitudinal flow, and the profile 3 = -0.198 is a 
 separation profile (wall shearing stress zero). 
 
30 
 
 NACA TM 13^3 
 
 
 1 
 
 «H 
 
 
 
 
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 0) 
 
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 CJ 
 
 •H 
 
 
 
 • 
 
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 0) 
 
 c 
 
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 •H 
 
 p 
 
 
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 cd 
 
 
 
 
 
 
 a 
 
 
 
 <m 
 
 r*> 
 
 •H 
 
 
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 £> 
 
 X 
 
 
 OJ 
 
 
 
 
 
 
 • 
 
 <u 
 
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 m 
 
 cu 
 
 ft 
 
 
 
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 ft 
 
 
 •\ 
 
 cd 
 
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 cd 
 
 
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 cu 
 
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 cd 
 
 
 
 
 U 
 
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 cc 
 
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 cd 
 
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 ft 
 
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 s 
 
 CU 
 
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 ca 
 
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 •H 
 
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 c 
 
 cu 
 
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 r»s 
 
 cu 
 
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 T) 
 
 •\ 
 
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 p 
 
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 c 
 
 
 c 
 
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 cd 
 
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NACA TM 13^3 31 
 
 is the dimensionless wall distance used by Hartree and the coordinate 
 a of the point of junction to the potential velocity U a is connected 
 with the displacement thickness by the relation 
 
 a_ = £ aV |2 - 3 
 5 k 6 
 
 (92) 
 
 the quantity kg*(3) may be found plotted graphically in figure 2 of 
 reference 3- 
 
 The wall distance £ c of the critical layer indicated in figures 2 
 to 21 is obtained if in equation (91) y is replaced by y c . 
 
 The following basic remarks should be made concerning the graphic 
 solution of the eigenvalue problem in figures 2 to 21: 
 
 Since c-j has been presupposed so small that only the variations 
 
 of the particular solutions cp v * linear in c-? need to be taken into 
 
 consideration compared to the solutions cp v for c^ =0, the curves E* 
 
 and F appear for equal c^-interval as "equidistant" curve families 
 
 in the sense of equations (77) and (70). Actually, this "equidistance" 
 will be lost in case of higher values of c^; however, the calculation 
 
 expenditure would increase intolerably even if only, for instance, the 
 terms quadratic in c± were to be taken into consideration. In this 
 sense, the curves 
 
 Pi 8* c ± ab* 
 
 ~vT = u a 
 
 against a& represented in figures 22 to 26 (from now on we use dimen- 
 sional quantities) which result from the evaluation of the polar diagrams 
 also are to be interpreted as approximations (in this and the following 
 figures, the value 3i which is, according to equation (l), physically 
 
 more important and which characterizes the logarithmic increment of the 
 excited perturbation amplitude has been plotted instead of c±) . Here, 
 as in Schlichting's report (ref. k) mentioned before, only the derivative 
 
 Pi 8* 
 
 d(cc&*) \ U 
 
32 NACA TM 13^3 
 
 at the location = 0, that is, the slope of the -rr — curve at 
 
 TT a 
 
 u a 
 
 the ccS*-axis is rigorously correct. H. Schlichting interpolated 
 
 between the base points at which he had determined the slope directly, 
 
 although with a much higher calculation expenditure, with a curve of the 
 
 third degree 
 
 — — = a Q + a 1 (aS*) + a 2 (aS*) 2 + a^aS*) 3 
 u a 
 
 where the four constants are fixed by the coordinates of the base 
 points and the values of the curve slope in them. Thereby, 
 
 Pi 5* 
 Schlichting obtained for the Blasius profile higher values — - — than 
 
 u a 
 
 those occurring in figure 25 whereas the values of the slope are in 
 
 3-5* 
 agreement. Actually, the curves — - — (ct5*) in the center part of the 
 
 Ua 
 
 (aS*) -region enclosed in each case will run somewhat higher or lower 
 
 than indicated by figures 22 to 26. At any rate, however, they may be 
 
 interpreted as a first approximation in the usual sense. 
 
 3iS* * 
 These curves — — over ct& represent sections through the 
 
 U a 
 
 "excitation mountain range" enclosed by the neutral curve as base curve, 
 
 along the lines y c = Const, or c r = Const. By interpolation one 
 
 obtains from them the maps of excitation represented in figures 27 to 32, 
 
 3 n -S* 
 in which the lines of equal excitation --= — = Const., can be interpreted 
 
 Ua 
 as "contour lines" of the "excitation mountain range." Instead of the 
 lines of intersection c r = Const., we plotted in figures 27 to 32 the 
 lines 
 
 p v c r ao* 
 
 —r- = — = Const. 
 
 U a 2 U a Re* 
 
 the significance of which will be discussed later in section VII. Even 
 at first consideration of these excitation maps, a fundamental difference 
 in the shape of the excitation mountain range is conspicuous according 
 to whether the velocity profiles of the laminar friction layer lie in 
 the region of decreasing pressure (3 > 0) or of increasing pressure 
 (3 < 0). 
 
 In the region of decreasing pressure, the "excitation mountain 
 range" has the form of a mountain with pronounced peak which is steeply 
 ascending for a small Reynolds number Re*, slowly flattens after a 
 larger Re* and shrinks to zero width and height for Re* — ><». The 
 absolute height of the peak, that is, the maximum excitation increases 
 with decreasing 3- 
 
NACA TM 13 V3 
 
 33 
 
 In the region of increasing pressure, in contrast, the excitation 
 mountain range changes behind the peak with growing Re* into a 
 "mountain ridge" of constant width and constant contour profile. 
 
 The properties of this "ridge" have been thoroughly investigated 
 by W. Tollmien (ref. 5)- Since we made use of Tollmien's theory, we 
 must now briefly represent its results. 
 
 Searching for a general instability criterion, W. Tollmien estab- 
 lished that the frictionless perturbation equation (5) possesses for the 
 laminar velocity profiles in the region of pressure increase - which 
 have a point of inflection in contrast to those of the region of 
 decreasing pressure - for Re* — ► °° aside from the neutral solution 
 existing for all profiles with the parameters 
 
 a = 
 
 <PnI 
 
 U 
 
 (93) 
 
 in addition, the neutral solution with the parameters 
 
 a = etc 
 
 cr = U £ 
 
 f PnII = 9s 
 
 (9*0 
 
 the subscript s therein denotes the point of inflection. 
 
 W. Tollmien calculated the eigenvalue a s and the eigenfunction 
 qp s for the sinusoidal velocity distribution (30) which we also used 
 from the frictionless perturbation equation (5) 
 
 (aa). 
 
 p cot p 
 
 sin p 2(1 - £ 
 
 (95) 
 
 9 S = 
 
 y 
 
 sin p — 
 
 ^ a 
 
 sin 
 
 ?* = 
 
 ps 
 
 sin p aafl - — 
 
 e \ a 
 
 sin 
 
 ps 
 
 (0 < y < a) 
 
 (y>a) 
 
 (96) 
 
3^ 
 
 NACA TM 13^3 
 
 For this second neutral eigensolution, the phase velocity c r therefore 
 equals the velocity of the basic flow at the point of inflection U s . 
 
 For the excitation $ ± and the circular frequency 3 r in the 
 neighborhood of the neutral frequencies a&* = and aS* = (aS*) s , 
 W. Tollmien has derived the following formulas 
 
 where the subscript w signifies that the values have to be taken at 
 the wall y = 0, and that 
 
 a = a s : — — = B 
 
 [aS*) 3 - (a s &*) 2 a5*|, ^ Re 
 
 Us R * BE Us 
 
 = ul a& " ~ uT 7 ^" 
 
 (a5*) 3 - (a s 5*) 2 ao*j 
 
 (98) 
 
 with 
 
 where 
 
 B = n 
 
 lim 
 
 € — »0 
 
 Uc 
 
 Uc 
 
 <Ps d y 
 
 u a & 
 
 *2 
 
 U 
 
 (U - U s ) 
 
 E^^L 
 
 9o 2 dy + 
 
 2 ^s 
 
 Uc 
 
 +e (U - Us) 2 " S 
 
 s+e 
 
 cp 2 dy 
 
 (99) 
 
 (100) 
 
 We have calculated the "ridge contour profiles" of the excitation 
 and the circular frequency according to the formulas (97) and (98) for 
 the Hartree velocity distributions 3 = -0.10, -O.lU, -0.l6, -0.18, 
 -O.I98; the values for the derivatives of the velocity were not taken 
 from the sinusoidal approximation distribution but from the exact 
 velocity distribution. These "ridge contour profiles" of the excitation 
 and the circular frequency have been plotted in figures 33 and 3k. The 
 curves corresponding to the formulas just mentioned have been drawn in 
 solid lines j the transitions between the two curve arcs, interpolated 
 somewhat arbitrarily, have been drawn in dashed lines. In the range 
 
NACA TM 13^3 35 
 
 of very large Re"*, the lines of constant excitation and constant cir- 
 cular frequency were transferred from figures 33 and 3^ into figures 31 
 and 32. 
 
 Figure 35 represents the variation of the maximum excitation 
 
 \ Ua /max 
 
 against the form parameter 3> separately for the excitation for finite 
 Re* (peak of the excitation mountain range) and for Re* > °o (ridge 
 of the excitation mountain range). For 3 < -0.10 these two values 
 seem more and more to approach one another so that with decreasing 3 
 the peak becomes less and less pronounced, and the excitation starting 
 from small Re* monotonically increases to the values of the "ridge 
 contour profile." 
 
 This figure and the preceding ones show clearly that the maximum 
 excitation and accordingly the excitation in general is considerably 
 larger in the region of pressure increase than in that of pressure 
 decrease for smaller Re* as well. 
 
 After thus having estimated the magnitude of excitation in the 
 entire instability range of all velocity distributions occurring in the 
 laminar friction layer, we shall discuss the physical conclusions 
 resulting from our calculations for the position of the transition point. 
 
 VII. DISCUSSION OF THE RESULTS 
 
 Let the pressure distribution U a (x) against the arc length of 
 the cross section profile, and the oscillation of circular time frequency 
 3 r be prescribed. Then this oscillation superposed on the boundary- 
 layer flow travels downstream on a curve 
 
 3 r v 
 
 ■^P = f(x) 
 
 U a 2 
 
 As we mentioned at the beginning, there pertains to every point x of 
 the profile in the flow a fixed value of the Pohlhausen parameter X 
 which characterizes by way of approximation the velocity distribution 
 in the boundary layer at this point. According to figure 3^ i- n refer- 
 ence 3 one may coordinate to this parameter X the Hartree parameter 3 
 
36 
 
 NACA TM 13^3 
 
 and therewith one of the excitation maps calculated in the present 
 report (figs. 27 to 32) . In order to obtain the excitation of the 
 perturbation 3r at a certain point x of the profile, one has there- 
 
 3-&* 
 
 fore to read off the excitation 
 
 Uc 
 
 'number of the "contour line") 
 
 on the corresponding excitation map at the point of the map determined 
 by the pair of values of the Reynolds number 
 
 Re 
 
 * U a S* 
 
 ;(x) 
 
 and the dimensionless circular frequency 
 
 3 r v 
 
 = f x 
 
 U s 
 
 We shall start the discussion of the results of our above excitation 
 
 Uoot 
 
 calculations with the limiting case of a small Reynolds number Re = . 
 
 According to the explanations in section I (compare fig. 1) the stability 
 limit lies, for small Reynolds numbers only, at the point where the 
 laminar boundary layer separates. If the perturbation waves are so 
 long that the curve 
 
 v 
 
 U 2 
 
 f(x) 
 
 intersects the instability region for the separation profile (3 = -0.198), 
 it is very violently excited when entering this zone and leads quickly 
 to transition to the turbulent flow pattern. The transition point then 
 practically coincides with the separation point which we had denoted 
 its rearward limit. If, on the other hand, the perturbation waves are 
 very short so that the curve 
 
 3 r v 
 
 f(x) 
 
 does not intersect the instability region, the laminar layer separates 
 without transition. 
 
NACA TM 13^3 37 
 
 If the Reynolds number Re = — ^- increases, the stability limit 
 
 shifts forward in the direction toward the pressure minimum (fig. 1), 
 the perturbation enters a region of instability further upstream for a 
 larger value of the parameter (3 or the Pohlhausen parameter A., and 
 is there initially excited to a degree which decreases the more the 
 stability limit shifts forward but which increases due to the fact that 
 the perturbation downstream (with decreasing p or A.) reaches insta- 
 bility zones with rapidly increasing excitation. The transition point 
 shifts frontward corresponding to the excitation which started earlier 
 and is still strong. 
 
 If we finally increase the Reynolds number Re = -^— so that the 
 
 stability limit shifts ahead of the pressure minimum, the excitation in 
 turn starts accordingly sooner; however - and this must be regarded as 
 the most important result of our calculations for the time being - in 
 the region of decreasing pressure, the excitation is so slight that it 
 generally attains amounts equalling the excitations produced in the 
 cases treated just now only after having passed the pressure minimum. 
 
 In this manner, one may easily give the theoretical explanation 
 for the fact proved by many experiments, that the transition point even 
 in case of very high Reynolds numbers rarely ever shifts ahead of the 
 pressure minimum. Only in cases of very long acceleration sections 
 and perturbations with very long waves where the perturbation does not 
 too soon leave the instability region again, the limited excitation of 
 the region of decreasing pressure will be sufficient to induce the 
 transition still in the region of pressure decrease. However, these 
 cases are rare in technical application. 
 
 A detailed calculation of the degree of excitation which causes 
 the transition is meaningful only in connection with corresponding 
 experiments. Therefore, it will be postponed until these experiments, 
 now in the preparatory stage, have been carried out. Probably one will 
 have to regard the form of the pressure distribution in the region of 
 pressure increase as the most important test condition since, according 
 to our theoretical deliberations, the contributions to the excitation 
 of perturbations in the region of pressure decrease are insignificant. 
 The aim of this experimental investigation and of the excitation cal- 
 culation to be performed simultaneously on the basis of the present 
 report will be to find a connection between the pressure-distribution 
 form and the degree of excitation attained at the measured transition 
 point so that it will be possible to calculate, inversely, the transition 
 point for a prescribed pressure distribution from this relation. 
 
38 NACA TM 13^3 
 
 VIII. SUMMARY 
 
 As a contribution to the solution of the important problem of the 
 calculation of the transition point of a plane laminar flow, we had 
 first determined (in an earlier report, according to Tollmien's method 
 of small oscillations for the Hartree velocity distributions appearing 
 in the boundary layer in case of decreasing and of increasing pressure) 
 only the critical Reynolds number beyond which the perturbations super- 
 posed on the laminar flow are excited. In connection with those cal- 
 culations now, the excitation itself in the entire instability range 
 of the perturbations was calculated. The excitation in the narrow 
 instability range of decreasing pressure turns out to be very much 
 smaller than the excitation in the more extensive instability range of 
 increasing pressure; thus the known fact that the transition point 
 generally does not shift ahead of the pressure minimum even in case of 
 high Reynolds numbers may be explained on a theoretical basis, as shown 
 in tables 1, 2, and 3- 
 
 Systematic experimental measurements of the transition point, 
 together with calculations to be performed on the basis of the results 
 given here, are to establish the connection between the variation of 
 the pressure gradient and the degree of excitation which produces the 
 transition and thereby a basis for determination of the transition point 
 for prescribed pressure variation by calculation. 
 
 Translated by Mary L. Mahler 
 National Advisory Committee 
 for Aeronautics 
 
NACA TM 13^3 39 
 
 REFERENCES 
 
 1. Tollmien, W. : Uber die Entstenhung der Turbulenz. Nachr. d. Ges. 
 
 d. Wiss. zw Gottingen Math. -Phys . Kl., 1929, pp. 21-i+U . (Available 
 as NACA TM 609-) 
 
 2. Schlichting, H., and Bussmann, K.: Zur Berechnung des Umschlages 
 
 laminar = turbulent (Preisausschreiben 19^0 der Lilienthal- 
 Gesellschaft flir Luftfahrtforschung) , dieses Jahrbuch. 
 
 3. Pretsch, J.: Die Stabilitat einer ebenen Laminarstromung bei 
 
 Druckgefalle und Druckanstieg (Preisausschreiben 19^+0 der Lilienthal- 
 Gesellschaft fur Luftfahrtforschung) . Jahrbuch 19^1 der Deutschen 
 Luftfahrtforschung, p. 58. 
 
 h. Schlichting, H.: Zur Entstehung der Turbulenz bei der Plattenstromung. 
 Nachr. Ges. Wiss. Gottingen Math. Phys. Klasse, 1933, pp. 181-208. 
 
 5. Tollmien, W.: Ein allgemeneines Kriterium der Instabilitat laminarer 
 
 Geschwindigkeitsverteilungen. Nachr. d. Ges. d. Wiss. zw Gottingen, 
 Math. -Phys. Kl., Neue Folge, Vol. 1, No. 5, 1935, PP- 79-H^. 
 (Available as NACA TM 792.) 
 
 6. Hartree, D. R.: On an Equation Occurring in Falkner and Skan's 
 
 Approximate Treatment of the Equation of the Boundary Layer. Proc. 
 Phil. Soc. Cambridge, Vol. 33, 1937, PP- 223-239. 
 
Uo 
 
 NACA TM I3I4-3 
 
 TABLE 1 
 SERIES COEFFICIENTS OF THE SOLUTIONS <Pj*, tp * FOR THE BASIC FLOW 
 
 U - 1 - (a - y)°; n = 1, 2, 3, It 
 
 7 = L^£ y. . J - arc* 
 
 y c yc 
 
 V = s sin ( a »*) s « Bln 
 
 <[^* = cob (a 7*^ * cos 
 
 Bin 
 
 T " "o y J 
 
 , — *— — , — ' 
 
 4 - ^ 
 
 V "o yc/ 
 
 * q 
 
 <• <Pi - j^J- cos Co. y) = 0^ - — fi q^ 
 
 "0 
 
 ic^a 
 
 1>2 r- Bin (ay) = <p p 
 
 u 
 
 "0 
 
 lcja^ 
 
 "O 
 
 |(1 - Ifi) - ai 2 
 
 (1 - lvfi) (v >3) 
 
 -±-(1 - 21fi) + -±_ 
 720 ' 120 
 
 e^r* = ±_(1 - Ulfii - x (1 - 2ifl) + 1 
 
 b 7200* iJ 1,53600* 1) 50to 
 
 (1 - 21f!) ♦ -£- 
 
 
 i ' U32 
 
 610-2 m* 
 
 U32 l i; 2lt 
 
 2 
 
 Bl * = - §U - lfi) 
 
 =3 
 
 18 
 
 (l - ifi; 
 
 23a, 1 * 
 — (1 
 
 v . j_d . 6lfl) , 2aS(i . u lfl) . s^j . Buy + sf 
 
 6to ' 72000 sstooo x ' 720 
 
 3l2O0O (1 - 51fl » * 7938ofe (1 " 31f1 ' + 5i9^ (1 * lfl > 
 
 
 2 
 
 b 3 * - 1(1 - 3 lfi) ♦ ^-(1 - lfi) 
 
 ^ » 1 109ai2 a.,h 
 
 b 5 ' I^t 1 " 5ifl) ♦ -gi^l - 31fi) ♦ ^1 - lfi) 
 
 V = 1(1 - 71fi) - 79 ° lg 
 1 1920* XJ 1512000 
 
 80 - 1 
 
 «1 
 
 1 a x 2 
 ■B-B* 2 
 
 «3 = - 9 "l 2 
 
 *► = to a i 2 + k *i k 
 
 2h 
 
 _ 1 g 23ai' t 
 
 g 5 " 3600 ai " 1800 
 
 3600 
 
 p 101 k oi° 
 
 6W00 
 
 720 
 
 1 o , 1M k lln l b 
 
 ^ 6300 x 1587600 x 33075 
 
 ho = 
 
 _ 19 a^ 
 "2 "T ~ 
 
 „ 7 67 2 "1* 
 
 h 3 " " Bf " 158 ai + — 
 
 192 lUto x * U5 
 
 k 1 5I111 2 2291 k a i 6 
 
 h«= = - — - -^ ' 0.1*= - — - — ai H + -=— 
 
 3 60 36OOOO 5l<000 120 
 
 h, . - ^_ . J2_ 2 + 6959_ fc t n 6 
 ^ 1920 U3200 i 567000 L 75600 x 
 
NACA TM 13^3 ^1 
 
 TABLE 1 
 SEBIES COEFFICIENTS OF THE SOLUTIONS (p 1 # , rp^' FOB THE BASIC FLOW 
 
 U = 1 - (a itf) n ; n - 1, 2, 3, k - Continued 
 
 n ■ 3 
 
 6 * - ° »1* ■ 2 ( l " lf D 
 
 S 2 * - - *i 2 »3*ov* " I" »"" p3v[^ " (3 * 6v)if^j 
 
 w ■ ( - i)V " ^E - < 6 * 6v » ir ] w ■ «• »** ^E - " * 6v, "3 
 
 W - 0| V . 0,1,2, . . . 
 
 e * -1 «j* - - (1 - Ifl) 
 
 . 1 "l 2 . "l 2 
 
 «2* - |(1 - 2lft) - -i- « 3 - - -5"<1 - 'fll 
 
 . "I 2 a,* . »1 2 23a, 1 ' 
 
 e lt = -Ml - 21fi) + !ll- e= = -f-(l - 31fi) =-(1 - ifil 
 
 h 36 1 1J 120 5 5«l l ■" 5«00 l 1J 
 
 . °1 2 7 a l 2 , ai 6 
 
 <* = -^(1 - '•lfil * T^-(l - 21ft) + — h- 
 6 1260* ' 8100 ' 50W 
 
 . 2a, 2 23a, 1 * LUx, 6 
 
 'T " 66l5 (1 " 51ri ' * 22W 1 - " 31fl) " l325oo (1 " lfl ' 
 
 » a l 2 ,• **l k 1409a, 6 a, 8 
 
 '8* ■ iok' 1 " 6lri » * I5S66I' 1 " Ulfl > * Soyrofe' 1 " 21fl ' * 3555S5 
 
 ,* - - 3(1 - 2lft) ♦ ^- b 3 * - |(1 - 31ft) ♦ ^i-(l - lfl) 
 
 .* • - §U - '•lfl) - ^ »1 2 (1 - 2 'ft) * ^ V - - jgU - 5<fl> ♦ 55 «i 8 (l - 3»1> ♦ Jj «i»(l - "I) 
 
 -• ,1(1. 6lf!) - &2- „,2(1 - 1.1ft) - JE. .,*(! . 21 fl ) 4 ^ 
 
 1 5IH 1) lkl7 5 1 l *' 10 125 1 v Ll 720 
 
 - . - 3^,1 - 7lfi, - life ^(1 - 51ft) . g=£(l - 31ft) * ^ ^6(1 - «1, 
 
 „„• =1(1- 8lft) - 2 i5?9- ^2(1 . 6111) * -W_(l - tafl) - 39727<116 (1 - 21ft, . ^4 
 8 567 1 1; 617koo x k l ' 13608000 * 1 ' 11 11 32000* !» I»320 
 
 »6 
 
 ■v 
 
 80-I Si 
 
 "l 2 
 
 2 U 
 3--J-,' 
 
 «. ' T^ ♦ ^T fc = - ^ a,2 . J5. a, 11 
 
 It 24 °5 U5 1 900 l 
 
 », - -ii " tt i> £5=1 
 
 "° 16200 x 720 ~ 1323 " HT75 l " 33075 
 
 ft,- "1° , U ■>, 673 "l 6 , a l° 
 ^ 1512 10O800 ! 3175200 40320 
 
 ^.-6.0^ 
 
 w 53 V> „ 2 . •I* 
 
 "o 
 
 • 
 
 
 
 
 »! 2 
 
 
 
 
 "2 
 
 ~ 
 
 
 + 
 
 3 
 
 
 
 
 
 
 11 
 
 
 \T 
 
 
 2 
 
 
 K 
 
 " 
 
 18 
 
 
 108 
 
 "1 
 
 ' »5 
 
 "1 
 
 3 - - y - 57 "i T" 
 
 ^ , 1 21737<li 2 _ 37«_ k t ^lf 
 " 6 " 18900 3375 120 
 
 „ 11 1063 , 2987 k 71 6 h.. 11 193 2 ""! 2 , 'Wl U 86909»i 6 , ^ e 
 
 ^ " I6i " is^o" ° l * iow al 37600 l ^ 378 * 2778300 1701000 * 27783000 50kO 
 
k2 
 
 NACA TM I3U3 
 
 TABLE 1 
 SERIES COEFFICIENTS OF THE SOLUTIONS <Pj*, <j> 2 * FOR THE BASIC FLOW 
 
 U = 1 - (a - y-) n ; n = 1, 2, 3, "» - Concluded 
 
 n = U 
 
 "..• 
 
 =. 
 
 »2 # 
 
 - - |(i - 21ft) - ai : 
 
 'k 
 
 ' - |(1 - urn 
 
 H' 
 
 27 
 • 35(1 - 6lft) 
 
 6B* 
 
 - ^(1 - Sift) 
 
 B 10* 
 
 * " ^2 (1 " 10iri) 
 
 3(1 - ift) 
 
 B 3 * ■-!(!- 31fl) 
 
 B, - A<1 - 51ft) 
 
 8 9* = 25SI 1 " 91f D 
 
 -•o' ■ 1 
 
 a, 2 
 e s * - (1 - 21ft) * — 
 
 . a,2 143ai4 a,6 
 
 «fi " - -r— 1 " k lft) * 1 1 - 21ft) ♦ — 4- 
 
 b • 2)400 ' 504O0 v ' 501*0 
 
 5(1 " 1ft) 
 
 i ( l . 31fl) . -i-,1 . lfl) 
 
 ^'--^^-w-wf 1 ^ 1 - 1 '" 
 
 I.31"! 2 
 2 s x 3 2 x 5 2 x 7 
 
 253o l : 
 
 lO^ 1 * 
 
 (' " 31fl > " BBlool 1 " lf D 
 
 (1 - 6lft) 
 
 2 10 x 3 2 x 5 2 x T 2 
 
 <1 - 41ft) 
 
 61^6 
 
 2 10 x 3 3 „ 5 x 7 
 2 8 x 3 5 x 5 3 x 7 s 
 
 12749a, ' 
 
 (1 - 71ft) - — -— i- -(1 - 5lft) 
 
 2 11 x 3 3 x 5 3 x 7 2 
 
 2? x 3" x f x 7 2 
 a.6 
 
 -(1 - 31ft) 
 
 1 - 1ft) 
 
 t>2 - - - 6(1 - 21ft) 
 
 . 47 1 
 
 3 ■ -£(1 - 31ft) ♦ -r-(l - 1ft) 
 
 H' - - g(i - **U - nnr' 1 - 21f D ♦ — k 
 
 2Q 3997a, 2 1151a! *» a^ 
 
 163»i ; 
 
 71^° 
 
 h ■ 643(1 - Tift) - j^fl • 5lft) . jg^l - 31ft) . j^fl - 1ft 
 
 39 , . 11573a, 2 , 1503559 32377a, b n, 
 
 224o" (1 " 8lfl) " 3135oo5- (l " 6lfl) " 395136ooo < 1 " Ulf H " 49392O0OI 1 " 21f D * TO355 
 
 9 a, 2 
 
 %"5* 2 
 
 8U 4 48 ' 24 
 
 43 2 21 4 „ °i 
 
 ^> 1200 A BOO ' 720 
 
 111 2 149a 1 1 ' 15»n6 ai B 
 gQ " " 89600 a l + 250880 + 28221*0 * 40320 
 
 >0 
 
 ■= 
 
 
 "2 
 
 237 
 
 8 
 
 2 
 
 
 701 
 
 2171 
 
 h, , . Ji , iSi!22 a, 2 . 2*§i_ .4 . -2L- a ,6 
 
 ^ 640 24000 ! 25200 - 1 25200 
 
 «! ■ - 3 
 
 g 3 - - 3 - § . L 2 
 
 101 2 23 4 
 "5 = " to 01 to ai 
 
 17 2 13LL! 1 * 11°! 6 
 87 4200 **! " 17640 " 11025 
 
 hi « - 22 + a^ 2 
 
 223 13 , ai 4 
 
 "3 ' " T ' T " 1 * 6 
 
 *, . - -52 . «?Z a,2 . I! 3 ! „4 , Si 
 ~ 160 250 6000 120 
 
 160 250 
 
 307 14) 
 4480 " 98OO0 
 
 307 1467 ? 4737223 t 1 7 l '31 a 6 , °1 
 I " " 4480 " 98OO0 "1 ' 49392000 *! " 3087000 "1 5040 
 
NACA TM I3I+3 ^3 
 
 TABLE 2 
 SERIES COEFFICIENTS OF THE SOLUTIONS cp-,*, <P 2 * F0R TEE BASIC FL0W 
 
 U = 0. + (1 - U 8 ) sin f^Ll- 8 . 2). ABBREVIATION: p = 
 
 u a v 8 ' la - a 21 aln y 2a coa y 2a 
 
 3 * =0 P x * = - tan y 2a (l + ip) P 2 * = 1 + | tan 2 y 2a (l + 2ip) - og 2 
 
 B3* = - i tan y 2a (l + ip) - I tan3y 2B (l + 3ip) 
 
 B u * = i tan 2 y 2B (l + 2ip) + | tanV 2s (l + Itlp) 
 
 p 5 * = - 7 tan y 2s (i + i p ) . -^ tan3y 2s (l + 3ip) - jg tan5y 2a (l + 5ip) 
 
 B 6 * = ^ tan^U + 21p) ♦ ^ tan\ 28 (l + Mp) + i tan6y 28 (l - 6lp) 
 
 3 7 * = - j^g tan y 2B (l + ip) - Ji ta n 3y 2B (l + 3ip) - i ta n 5y 23 (l + 5 ip) - Jj ta n Ty 2a (l + 71p) 
 
 B 8 * = _i7_ ta D 2y 2s (l ♦ 2ip) + ^ tanSr 2a U ♦ Uip) + ^ tan6y 2a (l + 6ip) ♦ jig tan^U + Sip) 
 
 e * = 1 n* = I tan ^a* 1 + iP) e 2* = - I + I ^ 
 
 e, - 
 
 Jjj tan y 2a (l + Ip) + ^- tan y 2s (l + Ip) 
 
 ^ - & " ^ + 35 tan2y Ss<l ♦ 21 P>] ♦ 3S 
 
 e 5* = 7S5 tan ^s' 1 + iP' " E§0-[t * an y 2= U + ip) " \ tan3y 2s (1 + 3ip)J + jj^o "e^an y 2a (l - ip) 
 *6* = - &> * i^E " » ^^ * 21P) - To -'Wi ♦ *>] " jfeE * ^ ^2sd ♦ -P)] * ^ 
 e 7* = " 55150 tSn y 2s (1 + 1P) + 6t1o"[j^ tSn y 2B (1 + ip) + % ^W 1 + 3ip) * 1 tan5y 2a (l ♦ 5ipTJ- 
 ^[» tan y 2a (l ♦ ip) - || tan^l ♦ 3 ip)j ♦ ^ tan y 2a (I ♦ ip) 
 
 V - 362k - iSo| + 555? tan2y * a(1 + 21p) + m ^^ + Up) + h tan6y2a(1 + 6lp !l + 
 
 ^_fl" - ili tan 2 y 2a (l ♦ 2ip) - i^ tan>*y 2a (l * Up)] - -^!_[l + -^ tan 2 y 2a (l ♦ 2ip^l ♦ °lo 
 60W0Q 315 J2s F/ 29160 2s ^J 90720 Q 12 60 2s _] 362880 
 
 *2* = " [I + tan 2 y 2s (l + 2ip7] + i c^ 2 
 
 b 3* = fi tan y2s<1 * ip) " 8 ^^s' 1 + 3ip)| - ^ tan y 2a (l + ip) 
 
 V = ft + 8 tan2y2 B (1 + 2ip) + 4 f"^.* 1 + U1 P^| " °^% * % tan 2 y 2a (l ♦ 21p)] + °£ 
 
 b 5* = " [lHo ^ y2s<1 + 1P) + 1§2 tan52 s (1 * 5ip |] + °? 2 \^5 tM y2 = (1 + lp) " 2T6OO tan3 y2B<l + 3ip7] - 
 
 ^5 tan y2s (1 + iP> 
 
 b 6* ■= [" 750 " 2^0 ^^Bt 1 + 21 P) + ^ tanSr 2a (l + Uip) t ^ ta n 6y 28 (l + 6ip)] + 
 
 2 ri + 20^1 t an 2 y as (l * 2ip71 4- ^ 
 " [2to 321)000 ds J 720 
 
 laiwo tan y2a<1 + lp) - 1^0 tan3y2 s (1 + 31p) - irk tan5y2 s (1 + 51p) - isk tan7 y2s(i * 7ip) 
 
 ^ [- glfSoo tan y2a(1 + ip » + I5& Wy2a(1 + 31p) + I5I000 tan5y2a(1 + 51p 3 + 
 
 ^Ifiw tan y2a(i + ip) + iSo tan3y2a(i + 3ip il - 3^ tan y2a(i + ip) 
 
 to320 + 72^60 tan2y2 B<i + 21 P> + 38^0 tanl Wl + ***) * ^m tan6 Wl * SiP) + 
 Jjg tan8y 2sU + Sip, + a, 2 Q ^ - ^^35 tan 2 y 2B (I ♦ 2ip, - 55 |W_ tanSgi(1 + , lp) . 
 Ig |3_ tan6y2s(1 + 6lp T] + ^ugL, + _1||_ tan 2 y 28 (l + 2ip) - 3™^ tanV 2B (I ♦ .ip)] + 
 026 [; Io55o - TOgSoo tan2y2 B (1 + 21p ] + 1^0 
 
 b 7 
 
uu 
 
 NACA TM 1314-3 
 
 SERIES COEFFICIENTS OF TEE SOLUTIONS tjlj* , <p g * FOR THE BASIC FLOW 
 
 U = U. + (1 - Us) sin [ J " ° i| ABBREVIATION: p 
 
 sin y 2a coa y 2 s 
 
 - Concluded 
 
 gg = 1 g x = tan y 2fl gg = + I tan 2 y 2s + °\- gj = i tan y 2s * | 
 
 ta " yo= ♦ I cn> 2 
 
 g u = - i- ta n 2y 28 ♦%-(-!♦ Jtan 2 y 2 \ ♦ ^ 
 
 2U 
 
 85 " ife tan y 2 s * -50 ( - tan *2s - ^ ^iy s \ ♦ -gL. tan y 2 
 
 B6 
 
 - * ^ * a ^2s ♦ StA - H tan 2 y 2a ♦ 1 tanV J ♦ %^( - U + ^ wO ♦ £ 
 
 720 
 
 I800I3 3 
 
 s=y 5too^ " 12 
 
 720 
 
 3150 \i68 
 
 *l = " ^0 ^ y2D + twlTffl *" y 2= - 51 tan3y SB - I tan5y 2=l 
 
 + — ^ tan3y, \ + — — — ap° 
 
 ^ (- SS3 tan y 2a ♦ _i tan^j ♦ ^ a," tan y 2s 
 
 * = " Hobo tan2y2s + feL" » + »° tan2y2s + ^ tan ' V2s + § tan6y2 ^| + 
 
 307 "I 
 + Ti" tan y 2s 
 
 ag"* 13 883 . 2 U73 . k 
 25200 T " 557 ta " y2B + 7775 tanV2s 
 
 1058 too 
 
 to320 
 
 1-2 tan 2 y2 8 +02 t^ = - - tan y 2a 
 
 - ^ tan3 y2a - ^ °2 2 tan y 2a 
 
 X 329 . 3 \ 
 * "I- tan:iy 2B J " ^ °2 
 
 ho = h x 
 
 »3 " | * £ ^s " £ *« »S. " 5 °2<f * I ^Ss) ♦ I 02" 
 
 ^ = Js" tSn Y2e + I tan3y 2= + SS tan5y 2s + JogQ- °2 2 ( - 227 tan y 2| 
 
 5 23 . 2 1 ^ k. 1^6. 1 
 
 he = - — ^r- - tan yp R - -rr- tanWo H - -r- tan"yo R + — - — 
 
 J 288 720 2s 6k dB 60 ds 1800 
 
 h 6 " " |S *« y2a " lifi- -^2b ♦ ^ tan5y 2a ♦ jB_ tanTy Sa ♦ JL ^2^1 tan ^ f lOJ* ^3^ 
 
 1 2 
 
 — 05^ 
 
 60 
 
 — 02 U tan yg s 
 ^ tan2y 2a - g tan^TJ- 
 
 g tan5y 2 "^ 
 
 321*000 " 
 
 iZ599 ten + 321883 tan3y "; 
 
 7 2s 81 d 
 
 *n ■ „i^„„ ^ — tan2y P „ ii- tan^po 7— tan 6 y 0= ?2— tanSy,. + 
 
 " 25920 362880 2s 2880 2B Hkto 2s 13W>0 2B 
 
 1 2 l~~ 3253 1069 ,.2 221 ^ k. 1^5." 
 
 535» "8 [^ IT " ut55 ten y2a + TT taD ^ 2a - I ta " y2 * 
 
 ^_ kfe U239877 tanSy . 111|22 tan^ssl * -*- ^ 
 
 25200 ^ [TT 59535 J2s 1U580 2b 33075 
 
 ^ - ii^l tan2y 2a 1 ♦ sL 
 16 105 2o 50to 
 
NACA TM I3U3 
 
 45 
 
 TABLE 3 
 
 VALUES OF THE DIFFERENTIAL QUOTIENTS OF THE REAL AND OF THE IMAGINARY 
 
 PART OF THE FUNCTION ^U \ WITH RESPECT TO t\ q 
 
 
 c3F r 
 
 *1 
 
 
 5F r 
 
 u F i. 
 
 ^0 
 
 St >0 
 
 Sl io 
 
 ^0 
 
 ^0 
 
 St io 
 
 -2 
 
 0.135 
 
 -0.226 
 
 -3-6 
 
 0.396 
 
 -0.066 
 
 -2.1 
 
 .121 
 
 -.230 
 
 -3-7 
 
 • 399 
 
 -.030 
 
 -2.2 
 
 .117 
 
 -.235 
 
 -3.8 
 
 • 395 
 
 -.010 
 
 -2.3 
 
 .118 
 
 -.239 
 
 -3-9 
 
 • 379 
 
 -.055 
 
 -2.4 
 
 .121 
 
 -.242 
 
 -4.0 
 
 .360 
 
 -.100 
 
 -2.5 
 
 .128 
 
 -.245 
 
 -4.1 
 
 .3^0 
 
 -.155 
 
 -2.6 
 
 .135 
 
 -.246 
 
 -4.2 
 
 .297 
 
 -.205 
 
 -2.7 
 
 .145 
 
 -.246 
 
 -4.3 
 
 .250 
 
 -.270 
 
 -2.8 
 
 •157 
 
 -.243 
 
 -4.4 
 
 .180 
 
 -.338 
 
 -2.9 
 
 .172 
 
 -.237 
 
 -4.5 
 
 .080 
 
 -.341 
 
 -3.0 
 
 .191 
 
 -.226 
 
 -4.6 
 
 .020 
 
 -.282 
 
 -3.1 
 
 .219 
 
 -.210 
 
 -4.7 
 
 -.026 
 
 -.226 
 
 -3.2 
 
 .256 
 
 -.188 
 
 -4.8 
 
 -.045 
 
 -.182 
 
 -3.3 
 
 .296 
 
 -.162 
 
 -4.9 
 
 -.057 
 
 -.150 
 
 -3-4 
 
 .340 
 
 -.133 
 
 -5-0 
 
 -.062 
 
 -.118 
 
 -3-5 
 
 .380 
 
 -.102 
 
 
 
 
46 
 
 NACA TM 13^3 
 
 
 IO 
 
 _ 
 
 
 
 
 
 b 
 
 
 
 
 
 
 X 
 
 
 
 
 d 
 
 
 cr 
 i 
 
 
 
 
 o 
 
 ———*>.. 
 
 f 
 
 * 
 
 
 
 u 
 
 ^^fc 
 
 | 
 
 
 
 a 
 
 \ 
 
 1 
 
 
 a 
 
 \ 
 
 
 .1 ^ 
 
 " 
 
 CD 
 
 \ 
 
 
 
 
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 \ 
 
 
 CM <W 
 
 
 \ T 
 
 .£ o 
 
 \ 
 
 
 ~~ ',/ 
 
 I 
 
 \ 
 
 - 1 X5 
 
 \ 
 
 
 °}J 
 
 1 
 
 \ 
 
 T3 ^i 
 
 \ 
 
 
 i V 
 
 
 
 \ 
 
 £h nj 
 
 
 
 °K 
 
 
 
 
 IO 
 
 oj O 
 
 *«> \ 
 
 
 ID 
 
 
 
 
 °"\ 
 
 r *-• 
 
 tr \ 
 
 
 O 
 
 _ <M- 
 
 
 
 CJ 
 
 - 
 
 CD .£ 
 
 \ 
 
 
 a 
 
 8 
 
 >> 
 
 \ 
 
 
 
 
 J 
 
 3 
 
 
 T3 CJ 
 
 \ 
 
 
 \ 
 
 J ^ 
 
 ) 
 
 
 \ 
 
 
 A 
 
 
 J 
 
 £ o 
 
 \ 
 
 
 IO 
 
 — o 
 
 \ 
 
 
 ■ 
 
 .§1 
 
 \ 
 
 
 ; " 
 
 
 
 ^—m^ 
 
 
 / O 
 
 
 ^■h 
 
 o 
 
 ^_ 
 
 
 / ■*•" 
 
 
 4_ 
 
 >> a, 
 
 
 
 / 
 
 
 
 .*-> 
 
 E 
 
 
 / » 
 
 
 E 
 
 rf « 
 
 
 
 -< 
 
 
 — 
 
 ^a o 
 
 ■o 
 
 
 1 
 
 
 ■a 
 
 
 k. 
 
 
 » 8 
 
 3 
 
 
 
 
 o 
 
 $ 
 
 u 
 o 
 
 +i w 
 
 o 
 c 
 
 \ <*> / 
 
 \ - / 
 
 
 
 £5 u 
 
 o 
 
 4»- 
 
 \ " / 
 
 \ *a> / 
 
 
 
 k- 
 
 5 CD 
 
 
 \ K / 
 
 
 
 * — ' 
 
 . — i X. 
 
 •*- 
 
 
 •+. 
 
 
 - C - 
 
 'o 
 
 "S k 
 
 E 
 
 
 W 
 
 
 g ° 
 
 — 
 
 
 
 
 Q. 
 
 £ 
 
 >. 
 
 
 
 
 c 
 o 
 
 o .h 
 
 r^ 
 
 
 
 
 
 Sh o 
 
 O 
 
 /l\ *y> 
 
 / 
 
 
 o 
 
 h a 
 
 
 1 \ 
 
 
 
 
 
 CO 
 
 
 OJ 
 
 
 
 
 
 
 
 ft) / 
 
 
 
 
 "o 
 
 
 «/>/ 
 
 T— 1 
 
 
 
 ^^^ 
 
 
 -/ 
 
 CD 
 
 
 
 r/ i" 
 
 
 
 £h 
 
 ' < * m ~^ 
 
 
 \x^2 
 
 
 J< 
 
 9 
 
 •r-t 
 
 2 u 
 
 ■> c 
 
 5 <« 
 
 < h 
 
 
 
 
 
 
 
 ■ 
 
 
 1 
 
 
 i 
 
NACA TM 13^3 
 
 UT 
 
 0.6 
 
 l(F*),l(E*) t : .05 
 
 t ^ = l -° C r = 0.06 
 
 -0.4 -0.2 
 
 0.2 0.4 0.6 OS 
 
 -^R(F*),R(E*) 
 
 Figure 2 
 
 
 0.6 
 
 no. 
 
 '(F*),l 
 
 T 
 
 (E*) 
 
 ( 
 
 = 0.10 
 
 : r - 0.118 
 
 
 0° 
 
 A 8 * 
 
 IOXIO-3 
 
 >9 = i.o 
 
 
 
 ( 
 
 — — — — < 
 
 >• 
 
 
 \^a\ 
 
 
 -0.4 -0.2 
 
 Figure 3 
 
 0.2 0.4 0.6 0-8 
 
 -**R(F*),R(E*j 
 
 -04 -0.2 
 
 0.2 04 0.6 0.8 
 
 — V R (F*) , R (E*) 
 Figure 4 
 
 Figures 2, 3, and 4.- Polar diagrams for determination of the curves of con- 
 stant excitation. 
 
U8 
 
 NACA TM 13^3 
 
 
 0.6 
 04 
 
 
 
 
 n? 
 
 
 1 
 
 -0.4 -0.2 
 
 R(F*),R(E*) 
 
 -0.4 -0.2 
 
 Figure 6 
 
 0.2 0.4 0-6 0.8 
 
 — ►r(f*),r(e*) 
 
 
 0.6 
 04 
 
 |(F*)i(E*) 
 
 ♦ ' -3 
 1 2 4X10 a 
 
 1 
 £ c =0.20 
 C r =0-187 
 
 
 Cj=0- 
 
 
 £ = 0.6 
 
 
 
 ( 
 
 >• 
 
 
 
 
 -0.4 -Q2 0.2 0.4 0.6 0.8 
 
 — -**R(f*),r(e1 
 
 Figure 7 
 
 Figures 5,6, and 7.- Polar diagrams for determination of the curves of con- 
 stant excitation. 
 
NACA TM I3U3 
 
 h 9 
 
 
 0.6 
 04 
 
 i(f*),.(e*) 
 
 1 
 £ c = 0.05 
 C r = 0.033 
 
 4X10 
 
 -3 
 
 l>^0.2 
 
 ^J 
 
 
 0=0.2 
 
 
 Ci = 
 
 J 
 
 
 
 
 
 -0.4 -0.2 0.2 0.4 0-6 0.8 
 
 — - R (f*),R (e*) 
 
 Figure 8 
 
 -0.4 -0.2 
 
 Figure 9 
 
 0.6 0.8 
 
 R(F*),R(E*) 
 
 -0.4 -0.2 
 
 0.2 0.4 0.6 0.8 
 
 -^ R (F*), R (E*) 
 Figure 10 
 
 Figures 8,9, and 10.- Polar diagrams for determination of the curves of con- 
 stant excitation. 
 
50 
 
 NACA TM 1314-3 
 
 -0.4 -0.2 
 
 0.4 0.6 0.8 
 
 ^r(f*),r(e*) 
 
 Figure I I 
 
 -04 -0.2 
 
 R(F*),R(E) 
 
 Figure 12 
 
 Figure 13 
 
 R(F),R(n 
 
 Figures 11, 12, and 13.- Polar diagrams for determination of the curves of 
 
 constant excitation. 
 
NACA TM 13^3 
 
 51 
 
 I6X 
 Ci=0\X 
 
 ——0.6 
 10- 3 
 
 \V 04 
 
 l(F*),l(l 
 
 t 
 
 •*) 
 
 r I 
 
 b c =0.60 
 C r = 0.281 
 
 
 0? 
 
 
 
 ,8 = 
 
 
 
 '( 
 
 > 
 
 
 
 S;^ 
 
 -0.4 -0.2 
 
 -0.4 -0.2 
 
 0.2 0.4 0.6 0.8 
 
 R(F*),R(E*) 
 
 Figure 14 
 
 
 0.6 
 
 Ol 
 
 l(F*),l(E*) 
 
 .6// 12X10 3 
 4 \L<\^- 
 
 C 
 
 = 070 
 r =0.327 
 
 
 °2j 
 
 
 
 yQ=o 
 
 
 
 ( 
 
 ■ 1 
 
 > 
 
 
 
 
 0.2 
 
 Figure 15 
 
 0.4 0.6 0-8 
 
 R(F*),R(E*) 
 
 0.4 0.6 0.8 
 
 ^ R (F*),R(E*) 
 
 Figure 16 
 
 Figures 14, 15, and 16.- Polar diagrams for determination of the curves of 
 
 constant excitation. 
 
52 
 
 NACA TM 13^3 
 
 -0.4 -0.2 
 
 r(f*),r(e*) 
 
 Figure 17 
 
 r(f*),r(e*) 
 
 Figure 18 
 
 Figures 17 and 18.- 
 
 Polar diagrams for determination of the curves of con- 
 stant excitation. 
 
NACA TM 13^3 
 
 53 
 
 
 | 0.6 
 
 04 
 
 <(F*), 
 
 r 
 
 (E*) 
 
 c 
 
 = 1.12 
 
 r =0.410 
 
 Cj = 40xl 
 
 3-3 
 
 
 
 /3=-o.i 
 
 
 2 5- 
 20- 
 15- 
 10- 
 
 
 
 
 
 
 
 
 
 
 
 -0.4 -0.2 
 
 0.2 04 0.6 
 
 0.8 
 
 r(f*),r(e*J 
 
 Figure 19 
 
 -0.4 -0.2 
 
 r(0,r(e*) 
 
 Figure 20 
 
 -0.4 -02 
 
 0.4 0.6 0.8 
 
 r(f*),r(e*) 
 
 Figure 21 
 
 Figures 19, 20, and 21.- Polar diagrams for determination of the curves of 
 
 constant excitation. 
 
* 
 
 NACA TM I3U3 
 
 ^3 
 
 5xlO' 
 
 -4 
 
 A 8* 
 
 /3--i.o 
 
 0.170 
 
 0.05 0.060 
 0.10 0.1 18 
 0.15 Q.I70 
 
 Uo 
 
 C r = 0.060 
 
 A l 
 
 
 
 0.05 0.10 0.15 
 
 -^►aS* 
 
 0.20 
 
 Figure 22 
 
 10" 
 
 5x10 
 
 -4 
 
 A s* 
 
 Ua 
 * 
 
 0.15/ 
 
 ^-x^ = ' 
 
 6 
 0.20 
 
 0.10 0.097 
 0.15 0.142 
 0.20 0.187 
 
 ^c=o-'y 
 
 
 
 \ \ 
 
 
 
 0.05 0.10 0.15 
 
 0.20 0.25 
 
 Figure 23 
 
 10" 
 
 5x10 
 
 .4 
 
 As* 
 
 
 °\ 3 /~\ 
 
 1 
 
 Cr 
 
 U 
 4 
 
 £ = 0.2 
 
 1 \ 
 
 0.05 
 
 0.033 
 
 
 / \ 
 
 0.4 0I ° 
 
 0.068 
 
 
 0.2 ^-v / 
 
 ' ' 
 
 \ 0. zo 
 
 0,133 
 0.197 
 
 
 
 I / 
 
 A \ °* 
 
 to 
 
 0.258 
 
 £ c =0.05 
 
 
 \ / 
 
 
 
 
 \J 0.1 j 
 
 
 / 
 
 
 \ 
 
 
 0.05 0.10 0.15 
 
 ^ a 
 
 s* 
 
 0.20 
 
 0.25 
 
 Figure 24 
 
 Figures 22, 23, and 24.- The excitation £i — as a function of the reciprocal 
 
 U a 
 
 perturbation wave length a = — for constant critical velocity c . 
 
 A r 
 
NACA TM 13^3 
 
 55 
 
 4x10 
 3x10' 
 
 2x10 
 
 0.05 
 
 0.10 
 
 0.15 
 
 0.20 
 
 S* 
 
 Figure 25 
 
 0.25 
 
 t. 
 
 C r 
 
 0.4 
 
 0.188 
 
 0.6 
 
 0.281 
 
 0.7 
 
 0.327 
 
 0.8 
 
 0.372 
 
 0.30 
 
 2x10 
 
 Figure 26 
 
 Asymptotic value 
 for Re -*- oo 
 - Interpolated 
 
 Figures 25 and 26.- The excitation 
 
 2tt 
 
 <3jS 
 U Q 
 
 as a function of the reciprocal 
 
 perturbation wave length a = — for constant critical velocity Cj 
 
56 
 
 NACA TM 1314-3 
 
 Rel^ 
 
 Figure 27 
 
 Figure 28 
 
 Figures 27 and 28.- The curves of constant excitation in the instability regions 
 of a few laminar boundary-layer profiles. 
 
NACA TM 13^3 
 
 57 
 
 
 
 / °J 
 
 
 
 / * 
 
 
 CM 
 
 
 
 
 o" 
 
 
 // v 
 
 
 CQ. 
 
 
 // v 
 11 T i 
 
 k2 
 
 
 
 II fi\ 1 /<» 
 
 
 
 M / M / / ' 
 
 
 
 // / M / o 
 
 
 
 /// \ \ \ — 
 
 
 
 11/ \ / »" i 
 
 
 
 // / <a x / / in 
 
 
 
 /// /\ /A/® 
 
 
 
 7/ ) /// N 2 
 
 * 
 
 //// 
 
 / CO / // /<° 
 
 o 
 
 
 
 
 
 
 
 X 
 
 X 
 
 
 — 1 rUl tf> 
 
 10 
 
 
 Of / / fir 
 
 ^" 
 
 
 J I Hi 9 
 
 II 
 
 
 c ////// 
 
 * 
 oo 
 
 o 
 
 3 
 
 ////////y "A 
 
 FUJI 1*- 
 
 1 1 1) jCZ— i 
 
 Qa. 
 
 
 /////////// 
 
 '7// u ? <« 
 
 
 ///////ry//A 
 
 
 f~/J i/ff toyyoy 
 
 1// Q 
 
 
 4^— : -£^>yy\ 
 
 
 
 'vV^^^^^^IB 
 
 
 
 ^ ===:: *> : ^ g 
 
 
 
 b * 
 
 
 
 — in 
 
 
 
 M 
 
 
 
 A 
 
 CJ 
 
 
 
 q£ 
 
 Z> 
 
 
 *co ^ 
 
 
 
 CJ 
 
 
 
 O 
 
 O 
 
 o 
 
 3 
 
 m 
 O 
 
 ^0) 
 
 O 
 
 O 
 
 CO 
 
 O 
 
 O" 
 
 CvJ 
 O 
 
 a 
 
 o 
 
 •r-l 
 
 h0 
 CD 
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 d 
 
 +-> 
 w 
 
 a 
 
 CD 
 
 « 
 
 
 crl 
 
 ra 
 
 a 
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 o 
 
 
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 CTJ 
 
 I — I 
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 >> 
 
 o 
 
 cti 
 
 2 
 o 
 
 CD 
 
 En 
 
 Oi 
 
 
 
 •rH 
 
 ■ — i 
 
 
58 
 
 NACA TM I3I+3 
 
 
 
 /°J 
 
 
 
 
 
 / / 
 
 11 
 
 
 
 / 
 
 01 
 
 
 
 / 10 / 
 / °" / 
 
 to 
 
 
 
 /C/^~\ 1 
 
 1 
 
 
 
 
 
 
 S V I / 
 
 
 
 
 
 ' ° N 1 / 
 
 X 
 
 
 
 , r- \/ / 
 
 m 
 
 
 
 \ aa* 
 
 * 
 
 
 z< "■ 
 
 ! I / / O 
 
 (O 
 
 
 / ^£" 
 
 
 .i, 11 
 
 
 
 1 ^ / / / 
 
 *« 
 
 
 
 
 "6. 
 
 1 iU *~ 
 
 cb. 
 
 z> 
 
 /////' n ~ 
 
 ^cJ v 
 
 / v* / / — 
 
 
 y/y// ** 
 
 
 / /* * 
 
 
 
 
 / //V ) m 
 
 
 // s' ' — • 1 
 
 
 / // O 
 
 
 // ~7^ 
 
 
 
 
 <<*»"/ 
 
 
 
 
 ^>* ^>( / j 
 
 
 
 
 
 
 2 x 
 
 /// [r"v~ -^ «-* 
 
 
 
 \n 
 
 
 s.- — --^^ 
 
 <* 2 
 
 
 
 
 
 
 
 
 ~ 
 
 
 
 
 — VO 
 
 
 
 * 
 
 II 
 
 
 
 <X3 ^ 
 
 
 
 
 
 O*^ 
 
 ft 
 
 
 
 
 TO 
 O 
 
 C\J 
 
 O' 
 
 t-- 
 
 
 
 
 cd 
 
 
 MH 
 
 
 O 
 
 
 w 
 
 
 c! 
 
 
 
 
 
 • ^ 
 
 
 bO 
 
 
 CD 
 
 
 U 
 
 CD 
 
 >> 
 
 O 
 
 -i-> 
 
 ~~ 
 
 . — 1 
 
 
 • i-H 
 
 
 X3 
 
 
 nj 
 
 
 ->-> . 
 
 
 .5 © 
 
 
 0) £i 
 
 
 -C O 
 
 
 -^ fH 
 
 10 
 
 
 
 
 
 (3 CD 
 
 
 O >, 
 
 * 1 
 
 «3 
 
 t3 7 1 
 
 Oft 
 
 ..H >> 
 
 3 
 
 O *H 
 
 1 
 
 X OJ 
 
 ^■11 
 
 CD XS 
 
 1 
 
 rt 
 
 — 4 
 
 -*J ^2 
 
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 W . 
 
 l 
 
 jtf 
 
 
 s 
 
 
 w ■— ' 
 
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 O 
 
 0*" 
 
 
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 1 
 
 8 
 
 CJ 
 
 
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 3 
 
 
 hJD 
 
 
 •rH 
 
 
 fr 
 
NACA TM I3I4-3 
 
 59 
 
 rt 
 
 w 
 
 a 
 o 
 •t-i 
 
 a) 
 
 ^2 
 +-> 
 
 a 
 
 •rH 
 
 CD 
 .3 
 
 o 
 
 X 
 
 CD 
 
 ra 
 
 o 
 o 
 
 w 
 
 CD 
 . — I 
 
 m 
 
 o 
 
 & 
 ^1 
 
 CD 
 
 ■ — 1 
 I 
 
 >> 
 
 T3 
 
 o 
 
 
 o 
 
 CD 
 
 Eh 
 
 CO 
 CD 
 
60 
 
 NACA TM I3I+3 
 
 1.2 
 
 -* 
 
 A 
 
 1.0 
 
 as 
 
 Q6 
 
 04 
 
 Q2 
 
 10 10' 
 
 £=-0.198 
 
 P T v I 
 
 2^10- 
 
 u a j 
 
 1 
 
 / 
 
 ^° 
 
 -7 
 
 f 
 
 10' 
 
 AS 
 
 001 
 
 0.02 
 
 0.03 
 0.04 
 0.05 
 
 0.06 
 0.07 
 
 0.08 
 
 -0.07 
 0.06 
 
 -0.05 
 0.04 
 
 I0 H 
 
 * u a S 
 
 Re = —— 
 
 I0 6 
 
 I0 C 
 
 Figure 32.- The curves of constant excitation in the instability regions of a 
 few laminar boundary -layer profiles. 
 
NACA TM 13^3 
 
 61 
 
 
 
 
 
 
 
 CO 
 
 
 O 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 hO 
 CD 
 
 
 
 
 
 
 
 
 H 
 
 
 
 fn 
 
 
 
 
 
 
 
 
 QO-V 
 
 
 *<SO 
 
 CD 
 
 
 
 
 
 
 
 
 
 
 O 
 
 
 
 
 
 
 
 
 
 
 — / 
 
 i 
 
 G 
 
 
 
 
 
 
 
 
 
 1 7 
 
 »■ T 
 
 ■r-i 
 
 CO 
 
 
 
 
 
 
 
 
 ri 
 
 
 ■ 
 
 CD 
 
 . — t 
 
 
 
 
 
 
 
 / 
 
 
 / S£ 
 
 
 ■4-1 
 
 
 
 
 
 
 
 / 
 
 
 / , 
 
 
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 f-< 
 
 
 
 
 
 
 
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 Q, 
 
 
 
 
 
 
 
 / 
 
 
 
 
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 8 
 
 
 
 
 
 
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 >> 
 
 T 
 
 
 
 
 
 
 / 
 
 1 
 
 
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 . — < 
 
 1 
 >> 
 
 
 
 
 
 
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 i\ 
 
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 / 
 
 
 1 
 
 / 
 
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 3 
 
 
 
 
 
 
 
 
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 / 
 
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 w 
 
 
 
 
 
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 oa 
 
 cd 
 
 
 
 
 
 
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 1 
 
 / / 
 
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 bo 
 
 
 
 
 
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 .1-1 
 
 
 
 
 
 
 J 
 
 
 1 
 
 
 
 
 
 crt 
 
 
 
 
 
 
 
 1 
 
 1 
 
 CM 
 
 a 
 
 •rH 
 
 CD 
 
 
 
 
 
 
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 1 
 
 1 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 CJ 
 
 
 
 
 
 ^ 
 
 \ 
 
 U 
 
 \ 
 
 
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 X 
 CD 
 
 CD 
 XI 
 
 Eh 
 
 •rH 
 
 
 
 
 
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 ^s 
 
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62 
 
 NACA TM 13^3 
 
 /3 r i/ * 
 
 — - — Re 
 
 U, 
 
 j 
 
 04 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fi=-0f- 
 
 
 R- - n i-a 
 
 
 
 P. n ic 
 
 
 
 n.i 
 
 
 
 /J- U.IH 
 
 / 
 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ni 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 V 
 
 
 
 
 
 ^ 
 
 
 
 
 
 A 
 
 ? 
 
 
 
 
 
 
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 _h^ 
 
 vr 
 
 
 
 
 
 
 
 0.2 0.4 0.2 0.4 
 
 0.2 0.4 0.6 0.8 
 
 a8 
 
 
 
 
 
 ft- 
 
 - / 
 
 ■^ IQO 
 
 
 
 
 
 
 
 1" v*.i-»a 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 • 
 
 
 
 
 
 
 
 
 
 
 0.2 0.4 0.6 0.8 02 0.4 0.6 OS 1.0 
 
 P r v 
 
 Figure 34.- The circular time frequency — - Re* for boundary -layer pro- 
 files in the region of increasing pressure for Re* — > *> . 
 
NACA TM 13^4-3 
 
 63 
 
 0.08 
 
 Ridge contour peak of the 
 excitation mountain range 
 
 -0.198 
 
 0.5 
 
 1.0 
 
 /3 
 
 Figure 35.- The maximum excitation 
 
 ^8" 
 
 CuT 
 
 as a function of the form 
 
 a /max 
 parameter 3 of the laminar boundary -layer profiles. 
 
 NACA-Langley - 9-5-S2 -1000 
 
~ « - a 
 
 a 
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 xi 
 
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 J= CO 
 
 cj ^ 
 
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 cu bo ti< 
 
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 o r*i *j +j 
 
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 bo 
 
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 bfl 
 
 co W co 
 
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 > - 
 
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 kt eg 
 
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 53 05 
 
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 *-h bo 
 
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 H-aa 
 
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 3=3' 
 
 <. cu 
 
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