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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1343 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 twodimensional perturbations, the excitation of the unstable perturbations for the Hartree velocity distributions occurring in plane boundarylayer 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, c2* OF THE DIFFERENTIAL EQUATION DESCRIBING THE FRICTIONLESS PERTURBATION FOR FINITE EXCITATION (al Binomial Velocity Distribution (Pressure Decreasel (b) Sinusoidal Velocity Distribution (Pressure Increase) IV. THE FRICTION SOLUTION p3* FOR FINITE EXCITATION V. STATEMENT AND SOLUTION OF THE EICENVALUE PROBLEM VI. RESULTS OF THE CALCULATION VII. DISCUSSION OF THE RESULTS VIII. SUMMARY IX. REFERENCES *"Die Anfachung instabiler Stbrungen in einer laminaren Reibungs schicht." Bericht der Aerodynamischen Versuchsanstalt Gottingen E. V., Institute fur Forschungsflugbetrieb und Flugwesen, Jahrbuch der deutschen Luftfahrtforschung, August 1942, p. 154171. NACA TM 1343 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 = U0t (U, = velocity of air flow, t = chord of the body). V One may calculate it according to the wellknown 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 (\ = 12 separation. As forward limit for the transition point, one may take the stability limit of the laminar layer with respect to small twodimensional 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 socalled critical Reynolds number Re*cr = ) (Ua = local potential velocity, 'cr 6* = local displacement thickness) below which all perturbations are damped; its value increases as the form of the velocity profile becomes fuller (with increasing X). Where the Reynolds number Re* = Uab formed with the actual potential velocity Ua and the actual displace ment thickness 6* 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. 1). 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 1343 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. 4) 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), we 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 boundarylayer flow; let x denote the length of the arc and y the normal to the profile contour. The stream function of the twodimensional perturbation motion which we superpose on this basic flow is assumed to be t(x,y,t) = cp(x,y)ei(xct) = c(x,y)e itei(xrt) c = cr + ici pi = cia Pr = cr( here t denotes the time, a the spatial circular frequency of the perturbation, the real part cr of c its phase velocity, and the imaginary part ci a measure for its excitation (ci > O) or damping (ci < 0); besides pi is the logarithmic increment of the excitation of the perturbation amplitude, and Or the circular frequency in time of the perturbation. NACA TM 1343 If we substitute the motion originating by superposition of the boundarylayer flow U, V, and the perturbation motion u = q'eia(xct) u = e  ay S(2) v = = i + acp(xt) into NavierStokes' differential equations, we obtain as was proved in reference 3 in detail the differential equation describing the perturbation in the form (U UpIV 2a2p, + (p (3) Re* } Here the prime (') denotes differentiation with respect to the wall distance y; the velocities are referred to the local velocity Ua at the boundarylayer limit, and the wall distance as well as the wave 2ir length A = are referred to the local displacement thickness b*. 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 xdependence of the perturbation amplitude q. 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 boundarylayer 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 1343 excitation quantity ci; thus, the general solution of the differential" equation describing the perturbation (3) may be represented in the form cp* = Cv*qv; = C V ici UO' = U' (4) v=1 v=l UO There p, denotes the particular solutions for ci = 0 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 wy signifies additional functions for ci > 0. We turn first to calculating the integrals cpl* CP2* and the additional functions ai 2*. III. THE SOLUTIONS qpl*, C2* OF THE DIFFEREIITIAL EQUATION DESCRIBING THE FRICTIOIILESS PERTURBATION FOR FII!ITE EXCITATION If aRe* is assumed to be very large, the differential equation (3) is simplified to the socalled frictionlessperturbation equation (U c)(q" a2p) U"c = 0 (5) This differential equation has a pole of the first order at the location U = c = cr + ici to which we coordinate the point yc* of the complex yplane. 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 yc* and the wall distance yc of the "critical" layer U = cr. From U(yc) = Cr 6 NACA TM 1343 and U(yc*) = r + ici (7) there follows U(yc) U(yc) = ici = (* Y)UO (8 and, with limitation to the terms linear in ci, therefore ici Y* = Yc + (9) U ' We shall now indicate the construction of the solutions pl", * of equation (5) for those special velocity distributions U(y' by which re 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 = 2,3,4, (10) where a denotes the coordinate of the point of junction to the potential velocity. With the new variable Y Yc Yl = (11) a Yc NACA TM 1343 and with al = a(a Yc) the perturbation equation (5) for indifferent perturbations then reads 1 (1 iyni( 0129 > I + n(n 1)(1 yl)n2c = 0 Using the relations (9' and (11), we now introduce the complex variable Sy Yc Y1 = = Y a Yc c Y Yc ici a Yc Uo'(a Yc) With the abbreviation ci ci fl i =i Uo'(a yc) n(cr 1) (15) the differential equation describing the frictionless perturbation for nondisappearing excitation then is transformed into the form 1 yl n + nif 1l (1 y)nl 12 *) _~ L dyl n(n 1) 1 y1*)"2 + (n 2)ifl1( yl)n3 = 0 (16) in which, according to our presupposition, only the terms linear in fl have been taken into consideration. (12) (13) NACA TM 1343 If one writes equation (16) after multiplication by *2 1n1 1 (i yln + nifl (I y and after division in the form yl.2 dq* dyl*2 + oi iy*i = O i=l the first solution 'ji* is given by v=0 VO a Y 1 a ye ~ (19) CO = o g1V V=o The series coefficients ev to the recursion formula v(v + The solution 2 p are obtained from the B * v=1 r is f t for all y1 is found to be = 1 + by *=2 V + v =2 2e* al In y7 a Yc (17) a  a ye with ici U0 (18) according (20) eo = 1 2 1 dy 2 =T1IP * NACA TM 13h3 In this equation, the logarithms are, however, ambiguous; one must cut the complex yl*plane in such a manner that 3 * Sit < arg y < 2 1 2 If we put bv*l*v=2 v=2 byy + if1 hyy1V v=l (22) 2 may be represented for form 92* = 1 + V=2 yl = real part of yl* > 0 byV + if! hvy + V=1 ifl) / Yc 00 + if, Z V=0 ici = qp2 ,2 UO a2 Yc hVy + a V t1 "elc l in y + 2el In yl VOgy1Y V=o at first in the 2el(l  (n yl gVYl) with (23) (24) NACA TM 1343 In the transformation of equation (21) into equation (23) the relation el = e(l ifl) (25) is put to use; it is easily obtained from the perturbation equation (5) if U(y) and 'l* for ci = 0, and for ci = 0 in the neighborhood of the singularity U = c are developed into a Taylor series, since 2e = (a yc) (26) Uo' and UO''' 2el = (a yc) UO' + ici O Uo' U'" ici/U0" Uo'__ ~ (a yc)o 1 __ UOK' O' UO' Uo" = (a yc)O (1 ifl) (27) U0' If we express in equation (27) the exact value UO"' UO'' UO' UO"/ IIACA TM 13143 again by the approximate value that due to the small value of term for the value of term for the value of  fl, this step is motivated by the fact ci, one deals here only with a correction For y1 < D, 'p2 is represented by the expression r9* = 1 + bv y* V =2 ici UI"I + 2el*a l ln yl* + i arg y* (28) w2 = y hyV + I 2 Iy a yc +a yel 1) (29) The term with i arg yl* in equation (28) was obtained by H. Schlichting (ref. 4) and W. Tollmien (ref. 5) by a discussion of the general perturbation equation (3) in the neighborhood of the singularity U = cr + ici 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 el of maximum accuracy, we shall replace el by the expression (26) of the Taylor development of the exact Hartree profile. with NACA TM 1343 For the linear and the parabolic approximate distribution n = 1, n = 2 in equations (10) and (16)] H. Schlichting (ref. 4) has already given the solutions p1*, 2* This calculation was con tinued for n = 3 and n = 4. The coefficients Bv*, ev*, by*, g hb 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 = Us + (1 Us)sin( (30) where s denotes the wall distance of the inflection point. With the new real variable y2 = y (31) Y2 2 a s and with a2 = (a s)a (32) the perturbation equation (5) for the indifferent perturbations then read sin y + sin y2]( _2 2 + sin y2 y2)s = 0 (33) S \ (Y2 Ysdy22 NACA TM 1343 where y2s was put equal to Y2s 2 as (34) We now introduce the complex variable y Yc* Yc Yc* ici 2 2 a s 22(a s) 2 as 2 as U0' 2(a s) (35) With the abbreviation f2 Ci _1 ci UO' 2(a s) (1 Us)cos y2s t h e f r i c t i o n l e s d i f r e t al e q a i n d e c i i g h e t u b t o (36) the frictionless differential equation, describing the perturbation for nonvanishing excitation, then is transformed into [in 2* Y2s) + sin y2s f2cos Y2* 2s cos y2s (d2 *  c2*) + [sin 62* Y2s)if2 cos (y2 Y2s) P* = O (37) If (37) is written like equation (16) in the form 2*2 d2p* dy2*2 + i i=l the solutions pl*, q1 are determined (18), (23), and (28). One has Bii*Y2*i = 0 (38) by analogy with the equations 2 iv* ( (* I ici ) n(a 1 s Y2 e, *y2 2(a s) 1 2(a s) v=0 2a ) U (39) NACA TM 1343 with 00 all v 2 v v = V =0 (40) For y2 > 0, the solution cp2 is o p2 = 1 + > b *y2* bt i ned + 2el* 2(a n y2 2(a s 3 < arg y < 2 2 If one puts v= bv*y2*v v =2 00 = bVy2 V=2 00 + if2 h 'y2 v=l for y2 > 0 may be represented in the form o00 1 + bI bvy2 + if2 v=2 0o 11 hy2v V=1 + 2el + if2 sin y2s cos Y2s 2(a s) ici = P2 u 0 "2 U if2 vY2 I1n Y2 + V=0 V) ( S+ 21ell (1 2(a s) y2 in Y2 + + sin Y2s cos Y2s, 2el In Y2 v Y2 v=0 (41) CP2 = (42) if2 y2 / with (43) (44) NACA TM 1343 In transforming (41i into (43'), one makes use of the relation el* = el 1 + 2 sin Y2s cos Y2s eif (+ if Y which follows from the two equations a s Uo'' el UO' S+ ic Us el 1T U0 f L + U " lUO + ici0 UUO (a s) if UO'', U0O (a s) Uo =  n UO + if2 s sin Y2s cos Y2s (47) for y2 < O, is represented by the expression S= + b y2* + eln y + i v=2 + 1 (a s) Tl (n Y2* + i ici 2 UO I 2 U0', 3 2 n arg Y2*) < arg Y2 15 (45) (46) V < F)(48 16 NACA TM 1343 with Sel1 CP Y12 In Y2 2 2(a s) 1 h2 a s y sin y2s cos y2s 2e, In y2 gYy2 + 2iel 1 pl 21 Y + 2 (a s) sin y2s COs Y2s 5 VY2 >1 (49) V=O For the calculation of the transition substitution, we shall replace the term el, as in section III(a), by the accurate value (equation (46)) of the Taylor development of the exact Hartree profile. In a comparison of the relations (24) and (44) or (27) and (47) or (29) and (49), it is striking that, in the expressions for el* and a2 for the sinusoidal boundarylayer profiles in case of pressure increase, the product sin y2s cos y2s 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 (y2 < 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 y2s cos y2s 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 (y2s > 0), 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 (Y2s = 0O, the behavior of el* and m2 is regular since then el ici if2 (50) 2(1 Us) 2 NACA TM 1343 and 2(yc = s) a2(yc (a s) hY2 +In y21 2(a s =1 2(a s) (2 > 0) (51) = s) hy2 2(~ s) In  YV + 1Pl i 1nlY2_ hVY2 2(a s) 2 < 0) (52) The series coefficients PV*, ev*, bv*, sinusoidal basic velocity convergence of the series report (ref. 3) are again are given numerically in table developments, the explanations valid. 2. For the of the earlier IV. THE FRICTION SOLUTION cP3* 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*, 'p of the general differential equation describing the perturbation (3). If we introduce the variable Y Yc* = E* ( U 1/3 e* = R + ici 1 Uo I (53) (54) gv, hv for the with NACA TM 1343 we obtain from equation (3) in the limiting process E 0 Reynolds numbers the differential equation id43,4* d * CL114 for large d2 * + 9 3,4 = 0 dT1*2 (55) The solutions of this differential equation the form of the velocity distribution U(y) solutions (P3, P4 for the excitation zero corresponding to reference 2: P3,4* = are just as independent of as was the case for the (ref. 1); they read dT.1* *1/2 1/3 where H(1),(2) signifies the Hankel 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 (4) so that we there may put C4 = 0. V. STATEMENT AND SOLUTION OF THE EIGENVALUE PROBLEM After having found the particular solutions of the differential equation describing the perturbation (3), we 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* l w + C2 2 w + C3* 3 w = 0 Cl 1lw' + c2 (2 w + C3*'3*w' = 0 (57) (58) : L 2 i*3/2) * j ii d NACA TM 1343 At the point of auction y =a to the region of constant velocity, U" is zero and therewith cp9" a29 = O, that is cp* = eay, so that the third boundary condition reads C1 *D a + C2 2 *a = 0 (59) with v a = a' + cP a' V = 1, 2, . (60) A term 03a* does not appear because the particular friction solution , has already been damped at the point of junction. J In order that the three homogeneous equations (57), may have a solution different from zero, the determinant Pl"w 2 w 3 w l 2' 3w' = 0 la 2a 0 The solution of this determinant yields the equation 3w 3* ' 3 w 2*w '1a  "'*w lw 2a ilw2a or with Schlichting's abbreviations 1 93L D((O*) = F = F 0*nO CP3*w 1 *0 (58), and (59) must disappear (61) (62) (63) I NACA TM 1343 _1 2*wla Pl*w2*a E*(a,cr,cij (64) Yc* wP'rDi*a P1 w 2*a finally F* = E*(,cr,ci) (65) Therein 0 Ye* *IO (66) The complex equation (65) is equivalent to two real equations in which the parameters a, Re*, Cr, and ci are contained. If one limits oneself to the case of indifferent perturbations (ci = 0), one obtains from them after elimination of cr a relation between the Reynolds number Re* and the wave length 2 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 socalled 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 ci 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. 4), by determining not the.differential quotients i and at the location of the 8a 6 Re* neutral curve, but the curves of constant excitation ci > 0 directly by a method similar to that applied for the neutral curve ci = 0; however, without making use of this curve itself. We consider first the left side of equation (65). NACA TM 1343 Since we intend to limit ourselves to small values of excitation, we develop F* in the form F F ( = F(0o) o 0/ dF \ ) 0 dTl 0 1 + . (67) where aO as before in reference 1 is defined by the relation 0= yc(Re*Uo,) +1/3 (68) According to equations (9), (54), (66), and (68) one now has = Yc1 + ~ YycRe*Uo L Yc UO 3 UO UO (69) Thus equation (67) becomes F* = F (O*) F(T0) ici U UO Yc U_ dF uo '?0 + Y 3 = G(TD0,crci) O( * (70) NACA TM 1343 One sees therefore that the form of velocity distribution which does not enter into the exact solution (?q 0 according to equations (63) and (56) does appear explicitly in the development of F (O*) with respect to ci. The differential quotient dF dFr dFi = 4 1 d0IO d0i dTl0 was determined by graphic differentiation of the function F q). 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 F(TO*) will be free from singularities in region around the function F(qO) and the development (671 will be thus per missible. We now consider the right side of equation (65. which is defined by the relation (64). Remembering the splitting of the frictionless solutions (pvV = 1,2) in the form of equation (4), we first put 0 (D ici V a va "va UO with Iva = Uva + ava furthermore (71) Pvw t ici 'V* ici ' v vw  vw U0 NACA TM 1343 For the binomial velocity distribution values of the additional functions mVa WVw mv' Ua,' are given by the equations a w U = 1 (a y)n the and their derivatives Wla = gv V =0 pa 1LF hv + a yc v= 2el'1 a Yc 1 I 1 J2a ( (a yc,)2 v=1 w1w =T gVY1wv V=O 2w 1= b iwa yc V=1 w 2el vhv + 2 (a yc)2 + 2elqlw/ a Yc Ylw 2 la a Yc + Cla + W  in yiw wi) + 2e In y lwl + il UO  w_ lw U1 a Yc 00 lw a Yc V=l gvYlwy1 3 vhVYl V0 + w (a yc)2 VVI (a Yc 1 \" In y 1 Ul y UO a Yc lw >(72) + 0w Ylw + In IYlw l (a ( 1 11 S Ylw2 yw/ Yc) + NACA TM 1343 For the sinusoidal velocity distribution U = Us + (1 Us) sin( s in contrast, the corresponding formulas read la = gVY2a v=O V hv f $elqj1a + 2a= ah  2a 2(a s) v=1 a s \2a 2e1 In y2amla la ~n Vl 2aV1 (la a s) v=1 I2 iw V + r2e1 (a a 2 v h2a 2(a s Y2a sin y2s co y2s + 2a 0la 2(a s) + in Y2a li' Y2a . al = ; g2wV v=O In Y2a + Bin y2s cos y2s )2L(a s)( y In Y2a sin y2s cos Y2s _ n t h.v V+ elPlwl 1 In y2w _s \Y2w in cos 2 2(a s) 1 2w Y2w sin y2s cos y2s) + U0. _____________ I UO lw 2e1 In Y2j Uw + in  ... I d Uo' 2(a s)sin Y2s cos Y2s WIw) UIw = Z_ VgVY 2wVg 1 S2(a s) 0 032 vhvY2wv1 + 2(a2el a + 2 (a s)2 v=l 2(a s)2(a ) 2w2 S, 1 In Y2w + Y2w sin 2s cos Y2s) 2w sin Y2s cos y2 "lw 2(a s) + UO cp' y2w+ n y2w i + 2(a s)sin l lw y2v f 1 '1 UO 2(a s sin y2, Cos yVsa (731 NACA TM 1343 If the critical point shifts to the inflection point distribution (Yc = s, Y2s = 0), the equations for w2a, m2w' (equation (74)) are simplified to 2a i" hVY2a V 2(a s) L =il la n y2aI 2(a s) w2a Va2 S)2 vb y2a 2w (a s)2 hvy2v 4(a S)L v= inkl/ 4(a s)2 Vl + Pla S2(a s)Y2a ~rPlw in Y2w 2(a s) 2+ sylw 2(a s)Y2w + la n y2a i IT3Ciw We now split P2w, 92w', )2w, parts according to the formulas Cp2w = A + iB1 P2w = A2 + iB2 '2w into real and imaginary )2w = M + iN1 W2w = M2 + iN2 then one obtains p2* = A1 Ci + '7 N1 + i(BI '2 w A2+ N2 of the "2a , velocity "2w, S(74) (75) U M  M c i j 0 / + iB2 (76) NACA TM 1343 Then we put the expression (64) into the form E*(C,cr,ci) = E(a,cr) 1 + ci(zl + iz2) (77) where E, as previously defined in reference 1, is defined by 1 E  yc cP2wcla ~lw 2a P2w' la ~lw 2a First, limiting oneself to.the terms linear in ci Al la + 1 [l la (78) one has c . U (N L la + Bl la + rii~i~j ici ~ lw 2a 1ulv a ~ A2 la + 1(N2la + B21l2a) a C2 ) i 2 la  ,(A2 U0 + M2la)J P1 2 "a ~ cPlw 2a  ci/ ,P 1 r(alw"I'2a + ;Plw'^al U * 'P1 w 2 a 2 w la (79) UCi (Alj U0 NACA TM 1343 Hence, there results after a short intermediate calculation E* ~E 1 ci + UO 'Yc Ci (Illa + Bli2la) + i(wlw'f2a + tPlw2a Alila Ml0la) UO0 Al la cPlwo2a + iBl1la (N2'Tla + B2lal + i(clw' 2a + q(lw'2a Ala M2la) (80)" A20la 'Plw'2a + iB2 la We put for abbreviation Nl1la + Blnla = m Wlw 2a + P1wI2a Al2la Mlla = nI N2 la + B2ala = m2 (81) alw'2a + Tlw22a A2nla M2 la = n2 Al la clw2a = K1 A2 la qlw' 2a = K2 NACA TM 1343 Then there follows from equation (80) ci ml + in1 "2 + in +  UJ' l + iB]Yl K2 + iB2 la ci ilKl + nlB1"la ++ B12a2 Uo [Kl + Bl2la2 m2K2 + n2B2la K22 + B22"la2j n2K2 m2B2laj K22 + B22la2 j (82) icilnlK1 mlBlla UO' Kl2 + B12.la2 If we finally introduce the designations S1 mlK + nlBlla UO' 0K2 + Bi2 la2 1 Z2 . m2K2 + n2B2Dla K22 + B2 (la + 1 nlK mlBl la UO' Ki2 + B12la2 n2K2 mB2 la K22 + B22 la2] the representation (77) is attained. By means of the equations (70), (77), and (83) we thus have divided the two functions F* and E* (a,cr,ci) appearing in equation (65) into real and imaginary parts and can now graphically solve this equation. For this purpose, we plot for constant ci for several values ye the imaginary part of the function F* I (F) = (I o0 + T Y1 UO Ic against its real part F*) = R[F( O] 0 1 UYC( (84) (85) I + U 'd 3 Ug ydio ici 1 UO'Yc ici 1 UoYc UO IYc E* E E (83.1 +3 Uo'"' 0 ddjo NACA TM 1343 and plot in this polar diagram to the same values ci and yc, the imaginary part of the function E*(M,,cr,ci) I(E*) = I(E)(1 + cizl) + ciz2R(E) (86) against its real part R(E*) = R(E)(1 + ciZl) ciz2I(E) (87) The points of intersection of the functions F* and E* connected one to the other by the same pairs of values cr, ci yield first the values TI pertaining to the corresponding avalues, and then with (68) the curve a(Re*) of the constant excitation ci. This plotting and cal culation then are repeated for other values ci. 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(lO*) for several Hartree velocity distributions U(y) in the laminar boundary layer are represented in figures 2 to 21. These boundarylayer profiles belong to the special class of pressure distributions Ua = const. xm (88) and are rigorous solutions of Prandtl's boundarylayer equations which were obtained with the aid of a Bush apparatus (ref. 6). We use, according to Hartree, for characterization of these boundarylayer pro files the parameter P= 2m (89) 1 + m The profiles with m > 0, P > 0 occur in case of decreasing pressure; the profiles with m < O, 0 < 0 in case of increasing pressure. The profile 0 = 1 is the exact Hiemenz profile at the stagnation point of a cylindrical body, the profile P = 0 is the exact Blasius profile at the flat plate in longitudinal flow, and the profile 0 = 0.198 is a separation profile (wall shearing stress zero). NACA TM 1343 I 0 0 o ,  0 r0 S El a? 4* *o 'd 0g4)4 4 94 0 i 4. + Pl oi O M 04A 0 z c 4 ca lol 4 a, 4) a P " 0 A 4 (a 0 P P4 ori l > r4 o r 4 P RA  0 A A $4 a 0 cc 0 ,c 3 T*H o S 4) D o H 'd P4 0 4 r4 40 4 4 H H 4)o 1 0 10 44 Pb D < 'd02 sem! ( +' H O4 +K A 4) r m S)1 0 M LH 4 4: 0 ri 0 U W cU 0 H 0 4)4) *HIs +)n 02 h4$ CC ri 1 ON *i 0 F H O r\ FL 4 r .O *1 co 4P 4 0 r CO CkP *H H O 4 0 1 4.) 4 ' \0 r4 r4 0 CO1 II II cl cd an An 0 ( ) a, U . C'J 0 II >i 4$ UJ' II aA Ca CO O 0 C(I I II a II 111 COL .4 CU bid r0 II *~I II C 4A 'Y) t m II (11 I Cl iI II dl  I I Ca II NL ai 0\ ON CI 0 II 81 4$ cIa 0 II O II .r Cu 0 II 4$ b;l .D r 0 0 O C cl 04 0 II Ir 0 I 0 + r II D NACA TM 1343 is the dimensionless wall distance used by Hartree and the coordinate a of the point of junction to the potential velocity Ua is connected with the displacement thickness by the relation a _ = ** (92) kg the quantity k6*(B) may be found plotted graphically in figure 2 of reference 3. The wall distance Cc of the critical layer indicated in figures 2 to 21 is obtained if in equation (91) y is replaced by yc. The following basic remarks should be made concerning the graphic solution of the eigenvalue problem in figures 2 to 21: Since ci has been presupposed so small that only the variations of the particular solutions (p* linear in ci need to be taken into consideration compared to the solutions pV for ci = 0, the curves E* and F* appear for equal ciinterval as "equidistant" curve families in the sense of equations (77) and (70). Actually, this "equidistance" will be lost in case of higher values of ci; however, the calculation expenditure would increase intolerably even if only, for instance, the terms quadratic in ci were to be taken into consideration. In this sense, the curves Pi5* cia&* Ua Ua against a6b 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 Pi which is, according to equation (1), physically more important and which characterizes the logarithmic increment of the excited perturbation amplitude has been plotted instead of ci). Here, as in Schlichting's report (ref. 4) mentioned before, only the derivative d d(a*)\Ua / rACA TM 1343 Pi5* i5* at the location = O, that is, the slope of the Ua curve at Ua the a&*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 2.i6* (e*\2 U = a0 + a1(ab*) + a2(a5*)2 + a (a*13 Ua where the four constants are fixed by the coordinates of the base points and the values of the curve slope in them. Thereby, Schlichting obtained for the Blasius profile higher values than Ua those occurring in figure 25 whereas the values of the slope are in agreement. Actually, the curves (a&*) in the center part of the Ua (a8*)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. These curves i* over ab represent sections through the Ua "excitation mountain range" enclosed by the neutral curve as base curve, along the lines yc = Const. or cr = Const. By interpolation one obtains from them the maps of excitation represented in figures 27 to 32, in which the lines of equal excitation i = Const., can be interpreted Ua as "contour lines" of the "excitation mountain range." Instead of the lines of intersection cr = Const., we plotted in figures 27 to 32 the lines prV cra5* rV = c = Const. Ua2 UaRe* 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 (P > 0) or of increasing pressure (< <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*>o. The absolute height of the peak, that is, the maximum excitation increases with decreasing B. NACA TM 1343 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. Tollmfen 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 = 0 cr = 0 9,nI = U (93) in addition, the neutral solution with the parameters a = as cr = Us Pnll = Ps (94) the subscript s therein denotes the point of inflection. W. Tollmien calculated the eigenvalue as and the 4qs for the sinusoidal velocity distribution (30) which from the frictionless perturbation equation (5) (aa)s = p cot p sin p y Ts  sin ps a sin p aa ( Ps= pe a sin  a eigenfunction we also used sp 1 ) sin p 2( s \ a) (0 (>a) (95) (96) NACA TM 1343 For this second neutral eigensolution, the phase velocity cr therefore equals the velocity of the basic flow at the point of inflection Us. For the excitation Pi and the circular frequency Or in the neighborhood of the neutral frequencies a5* = 0 and a5* = (aB*),, W. Tollmien has derived the following formulas a = 0 U' Ua3 )Re*  (a*)2 (97) Us Uw' 1 2 Ua2 U 5 where the subscript w signifies that the values have to be taken at the wall y = 0, and that c = a: B [ *)3 Re* Us U a = aUS BE U 2 a6*)3 (as*26 (98) B = f Jo (99)dy U12 Ua*2 2 + .(99 where E = lim 2dY 2 E urn U 200 Ulf 2 d (100) E O (U Us2 S s+6 (U Us'2 s 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 P = 0.10, 0.14, 0.16, 0.18, 0.198; 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 34. The curves corresponding to the formulas just mentioned have been drawn in solid lines; the transitions between the two curve arcs, interpolated somewhat arbitrarily, have been drawn in dashed lines. In the range TACA TM 1343 of very large Re*, the lines of constant excitation and constant cir cular frequency were transferred from figures 33 and 34 into figures 31 and 32. Figure 3 represents the variation of the maximum excitation \ Ua max against the form parameter P, separately for the excitation for finite Re* (peak of the excitation mountain range) and for Re* ) o ridgee of the excitation mountain range). For 0 < 0.10 these two values seem more and more to approach one another so that with decreasing P 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 Ua(x) against the arc length of the cross section profile, and the oscillation of circular time frequency r be prescribed. Then this oscillation superposed on the boundary layer flow travels downstream on a curve U2 = f(x) Ua2 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 34 in refer ence 3 one may coordinate to this parameter X the Hartree parameter 0 NACA TM 1343 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 pr at a certain point x of the profile, one has there 0i8* fore to read off the excitation (number of the "contour line") Ua on the corresponding excitation map at the point of the map determined by the pair of values of the Reynolds number Re* = g(x) V and the dimensionless circular frequency prv = f(x) Ua2 We shall start the discussion of the results of our above excitation Umt 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 S= f(x) Ua2 intersects the instability region for the separation profile (i = 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 Orv S= f(x) Ua2 does not intersect the instability region, the laminar layer separates without transition. NACA TM 1343 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 p or the Pohlhausen parameter X, 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 X) reaches insta bility zones with rapidly increasing excitation. The transition point shifts frontward corresponding to the excitation which started earlier and is still strong. Umt 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 pressuredistribution 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. NACA TM 1343 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 1343 REFERENCES 1. Tollmien, W.: Uber die Entstenhung der Turbulenz. Pachr. d. Ges. d. Wiss. zw Gattingen Math.Phys. Kl., 1929, pp. 2144. (Available as NIACA TM 609.1 2. Schlichting, H., and Bussmann, K.: Zur Berechnung des Ums.hlages laminar = turbulent (Preisausschreiben 1940 der Lilienthal Gesellschaft fir Luftfahrtforschung), dieses Jahrbuch. 3. Pretsch, J.: Die Stabilitat einer ebenen Laminarstr6mung bei Druckgefalle und Druckanstieg (Preisausschreiben 1940 der Lilienthal Gesellschaft fur Luftfahrtforschung). Jahrbuch 1941 der Deutschen Luftfahrtforschung, p. 58. 4. Schlichting, H.: Zur Entstehung der Turbulenz bei der Plattenstromung. rlachr. Ges. Wiss. Gottingen Math. Phys. Klasse, 1933, pp. 181208. 5. Tollmien, W.: Ein allgemeneines Kriterium der Instabilitat laminarer Geschwindigkeitsverteilungen. Nachr. d. Ges. d. Wiss. zw G6ttingen, Math.Phys. Y1., fleue Folge, Vol. 1, No. 5, 1935, PP. 79114. (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. 223239. NACA TM 1343 TABLE 1 SEIES COFCICIEWTS OF THE SOUIONS 9 2, F* FOR TH BASIC FIlD U =1 (a y)n n=1, 2, 3, 4 n=l Yc y* Ye Y W s ain 5^ asin tCo o(aY)=1 tV = = cos (a a cos sin (gy) = ! n=2 oo0 = l (1 l) X =* (1 ivf) (v 3) e1 = +1 = 2I eC*= ( 2l) e )  e2 >* m 1212 192 23 = (1 2. fl) 21) + f= (1 51) 0 1 ) L b21 = 1 61) + 31f2 11 20if b (1 7fl) 37;1 2fl) + 3 b5 51 ) + 1 31 fi) 1 6 6 to b2 (1 fl) 2000, 1( 0(1 33f1) + 1 if,) 1920 1512000 9D = 1 ao=O 1 a212 g2 = 2 .2 * ll 1 + I 1 8 1 1 2 +101 a 4 __ 6=36001 64800 720 ho =0 4 IL+ 431M1i2 t U14 192 1440 1 5 S T 1 27 2 6959 1 71 6 = o1920 432002 7 7 gl= 1 23= 12 S 1 23a14 5 3600 1800 1 2 141 .4 la16 = 6300 1587600 o 33075 h1 = 2 + a12 1 511 3 2 2291 1 a!.6 h560 36Doo C' 54000 Mi i NACA TM 1343 TABiE 1 IE M2 LC EFICIE2T OF THE SOUITIOS I*, q* FOR THE BASIC FIDO U 1 (a .y )"n;a 1, 2, 1, Contied n=3 ." 0 2 2( ifl) " "_ : ( 1) (3 6v)it ea,.' :. L : .2 :e i L *e (1 t1) k.lf (1fi)  l i l e n 1 3 l) 1 ) l, l _. 1 I 6 Z rli (1 .a1 2 _3 1 .* 1 i 2" I 5 I 3il) 1 ( 21fl) * 7f rfLE. 1)+ (1311)+ 7 61 i r I 1'4 _____1)+ 71_ 8 .. +' I 41Y, a f1)4 1 4i1f) + 1 7640320 Aj = I S .14 t 4 8 L I "W7 6 W1 L69 o 37x''. _1 &6 ,<87 i , C 190 1 .'.O TC S= 2 2 22 1 13 42 2 16 h 6 + a? 3 7 o n 7 L +  b5 6 6 7, l'89nao 3 12 ^ 378 2 ayc3 E 1707: + P1 T76.:'00 5040 NACA TM 1343 TABZ 1 s8BR CCsOICISm OF T28E aOnWS ', w' FOR I BASIC FOW U ( y.); y 1, 2, 3, 4 Concluded 0 0 1* 3(1 Itl) .* = (1 2tfl) I2 (1 ") (1 lifl) 1) * (1= ) 9 1 fi) S10* k _1 =0I) e2 (1 21fi) * ( 31f) _(1 if) ,t 5 l5 1 21fl) + e l (1 3_1 1) alk 2 14 1?1, e 2 1 41fl) + 31 f'1 l))  =7 I 51 fl) 31 BB5sB T1I if) 21 x33 x 5 x 7 2 x 33 x 53 2 2 3 53 7 563 8 28 x 35 53 x 72 S 1 ) 1 21 ) 1 1) (1 31 ) 1 . o2k 111 T b ~ 1 61t1) 1 1 i 43 2 2, ) S 1 T f 711) s (1 51tl) 3t1) 1761(1 itl) b 1 U76 1 6417) 3 51 if1 16 f) b _e 39 1 15731 20 35 29 32 Pi1 6 go 1 h 3 3k 9 2 I +21 9 23 k 16+. +8 .k 1 2 2 1 k 2 1 3OT10 17 0 I 6 2Ul ll9o14 +1516 '18 bo 0 hl 12 *o12 237 12 23 13 k T01 21712o 69k7 h.o,0 120 23 I 64 48o 15 W L 1 1231 1 I + O6 6 15"9 2+ 561 4+ 71 6 307 1*67 274 i k "'.31 6 6*0 2W=00 259LW1 '250o'1 hr W&980l 3 067D 375 A 3 NACA TM 1343 TABLE 2 .EUIE COXEFFICIENTS OF THE SOLUTIONS I1*, f*9 FOR THE BASIC FLOW U = U l Us) sin ). ABBREIATION: p = cs \ sin y2. COB Y25 B = 0 8" = tn yi t* = 1 + I tanS2ys(l + ip) a2 tar .r j L lP' lar3 I'(l + 31p) B = tanyarys p i .p I S .'(1 + 4ip) e*  t n y y lp n tan3y2s(l + 3ip) tan5e(l1 + 51p) 8 t&O ?'ll Ip ) nrly2s(1 + 4 p + ki tan m 2 (1 61p) 87 L l11 p l tanY2(1 + 3ip) 1 tan5y,(1+ 5ip) tanTy2s(1 + Tip) 6= tanydl lp canya(l + + ip) + tan6y2(1 + + i tan8y2s(1 + 8ip) e' el = tar I, p e2* = + as2 3 Il e'= t n yt 1 Ipl n ran y2s(l + ip) es 2 = 5 ,al 1 Ip tan ys(1 + ip) tan"25(1 + 31i + 24tan y2(1 ip) e 4 2* _1 ta 1 + 2ip) ta+y2s(1 + ip "2 + tan2 + 2 e7 t l + p tan y27(1 + ip) + 2 t3 1 + ip) + 2 ta5 + tan y+ 1 1pi 2 ta[12 (1 + 1i8 + tn ta y2 1 + ipp e = y=li a _(1 + ,2p) + 3P1 + 0 P) + ia 2(1 + 6ipj + 6  ta Is + 4 p) n 2(1 + Y 2 t s(1 + + o = ++ ta21p) 3 + 270p + a+2 be 2 [67 tan a (1 + ip) 9 tan3ytany(l+ 31p tan yt y(1 + ip)+ c2 = + 5 2s(1 + 21 p) + s + p + a s1 + 2ip] + ( b = 3tan y2s1 + Lp) + tan 2s(1 + 5ip + a22 tan y2a(1 + 1p) t tan^y2s(1 + 36p8  _2 t tan r1 yia1 + pp) b 1 t n yp t85 2eip(1 + ti y + tani +a(1 + 0ip) + tan y2(1 + 6ip51 + 2a3 + 164tan260(0 + p+2 1 = o tan y2(1 +) tanS21 + 3ip) ay1 + 16) 3 tany2s(1 + Tit) ta. ta7 2(1 + ip) + tan3ya(1 + 2ip) + t5y( + i + 2 t.1 y<1 + ip) 2 + tay( + 3 tan y2s( + ip) S= tany2 a(1 + +) + i) + Sta2Y(l1 + 81p) + p22 tan22(1 + .ip) ta2y+(1 +p)  b 4. tan y2(1l + ip) + ta 32o + +1P ]) tan 2s(1 + ip) + 02 tan tan1 1 + + 583 tan2Y2,(l + 21p)1 + 1Fi +a ( 13589 41p) + Ltan(2sp(l + 17173 + NACA TM 1343 TABLE 2 SERIES COEFFICIENTS OF THE SOLUTIONS *, V2 OBR THE BASIC FLCW 0 = U *+ (1 Ue) in L. ABBREVIATION: p Sin 2 ConciuQed a a Sjsin y2, coo ;,a 2 g= 1 g1 = tan y2 g = + tan2y 3 = tan y + tan g + tan, Y2, 1 + tan y2. 94 = tan y2s + tan y tan3ya 1 + tan y22 S tan y 4 tan3y t +24) tan y2. tan3y, + 23 tan Y2 120 2a 50\ 72 1800 6 5 = + l tan2y5 2s + t? + t2 Io t oo + 4 11 + 101tan y +  720 n3 2 5400E 12 72 o = 50 1 tan ya + 2 +a22 1 2 9 tan Yy2 a +tan3 a 2 tan yt S( 12 3 tan 2.e + 3 2 + 11 2 tan + 2) t = I tan y 2s + tan~yL2! a + +a 27 tant y2 + 451 j a2 t+ 1 g6 53a + a_ ++ 4 306720 35 84o0 h ^ ta y t anBy2 + 61tany + ttan am + %s n yB 1 a rB tn a22 = 0 h+ 222 1 a26 tan yy2 S72 28 tan 72856 any2 n + 4 81 6 4 25 121 18 6 )6 hi,6= 89 5 a 0 ^ t + tan+I yB + 1 e 2 tan Y + 1 a 1 2884 3 64 108 4 6) 30 h5_0_23 h12ta.2Y22.+aY2. t6 Y2 +an 6019 ta192.253 n t2tnsy2 4288 720 60 h3 an2Y(53 3 tan26 y7 tan2Y2a) + 1., c 6 157 tan y 5 ta 3yp2, + 59 t 2 ( an 2a + 1 a 22 [6 5y 720 c2 tan y ta + 321883 1800 L Y 6. 1 81 291 IiT 97 t_2y2. I +.tan4y2 7 ta.6y2 29 t8y2. + 1 25920 362880 2880 1440 134o0 1 2 [_ 3253 1069 t ,+ 221 tan1y2s tan6 Y2d + 23520 90 4725 tan y26 14 3 j 1 6 9 t2y2 111899 tan4y2 + 1 IM 6 [ 49 t91 +!8 25200 2 914580 33075 IT 10. a 50 NACA TM 1343 TABLE 3 VALUES OF THE DIFFERENTIAL QUOTIENTS OF THE REAL AND OF THE INAGIIIARY PART OF THE FUlICTICOI F(T0) WITH RESPECT TO Fr Fi Fr 6Fi o 0 "no ' 0 a0o n j o ao 0.226 .230 .235 .239 .242 .245 .246 .246 .243 .237 .226 .210 .188 .162 .133 .102 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 0.396 .399 .395 .379 .360 .330 .297 .250 .180 .080 .020 .026 .045 .057 .062 0.066 .030 .010 .055 .100 .155 .205 .270 .338 .341 .282 .226 .182 .150 .118 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 ?.3 3.4  .5 0. 135 .121 .117 .118 .121 .128 .135 .145 157 .172 .191 .219 .38 . 34'1' . i3 NACA TM 1343 0 4 (I 0r 0 o i r' Coa)  a / NACA TM 1343 0.4 0.4 0.2 Figure 3 ).4 0.6 0.8  R (F), R (E) 0.2 0.2 0.4 0.6 0.8 R (F*), R (E*) Figure 4 Figures 2, 3, and 4. Polar diagrams for determination of the curves of con stant excitation. Figure 2 = 0.10 C,= 0.118 u  0.4 6 8 o0xl0'3 S2 9 Z 1.0 Ci=o 02 0.2 0 (F (E*E NACA TM 1343 OA 0.2 0.2 0.4 06 0.8 Figure 5 R (F),R (E ) 1 (F ),(= o0.15 Ci = O = 0.142 04 0246 I03 S.8=0.6 0.2 0.4 0.2 0.2 0.4 0.6 0.8 Figure 6 F), (E) Figure 6 0.6 (F'),, (Ej I tc= 0.20 S04 4x3 Cr, =0.187 Ci= o 0.6 02 0.4 Q2 0.2 0.4 0.6 0.8 w R(F),R (E*) Figure 7 Figures 5, 6, and 7. Polar diagrams for determination of the curves of con stant excitation. NACA TM 1343 0.2 0.4 0.6 U.8 R (F,R (Ee Figure 8 Figure 9 Figure 10 Figures 8, 9, and 10. Polar diagrams for determination of the curves of con stant excitation. 96 I I(F), (E*) t 0.05 SCr= 0.033 4 X 103j90 Ci=o   0.4 0.2 A NACA TM 1343 Figure II  R (F ),R(E*) Figure 12 I(F),I(E) Sc 0.40 Cr= 0.188 20 x 103 3 = 0 Ci= o 0.4 0.2 0.2 0.4 0.6 0.8 R(F)), R(E) Figure 13 Figures 11, 12, and 13. Polar diagrams for determination of the curves of constant excitation. NACA TM 1343 R (F),R(E*) Figure 14 0.6 I (Fi*), (E*) c=0.70 8 l0o Cr0.327 6 12 x103IO' P=o Ci= 0.4 0.2 0.2 0.4 0.6 0.8 SR (F ), R(E*) Figure 15 '(F*), I(E ) c=0.80 Ci= Cr= 0.372 04 3xio 3 0.2 0.4 2 0.2 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. NACA TM 1343 V R (F ),R (E') Figure 17 Figure 18 Figures 17 and 18. Polar diagrams for determination of the stant excitation. curves of con 0R (F*), R (E*) NACA TM 1343 0.6 0.4 Gi=40xIO3 0.2 I(F*), (E) tc =.12 Cr=0.410  =0.1 0.2 0.4 0.6  R (F*) ,R (E*) Figure 19 Figure 20 Figures 19, 20, and 21. ) R(F) ,R(E*) Figure 21 Polar diagrams for determination constant excitation. of the curves of i NACA TM 1343 5xI00 Cr= 0.060 0 0.05 0.10 0.15 0.20 Figure 22 IO0 i 8* =0.6 c ,r U 0.15 0.20 0.10 0.097 0.15 0.142 50.20 0.187 5xl104 0 0.05 0.10 0.15 0.20 0.25 _ a8 Figure 23 as* 0.3 C Cr Ua 9=0.2 0.05 0.033 3 0.4 0.10 0.068 O  0.20 0.133  0.2 0.30 0.197 0.40 0.258 5x104 C=0.05 0.I 0 0.05 0.10 0.15 0.20 025 a 8 * Figure 24 Figures 22, 23, and 24. The excitation i as a function of the reciprocal Ua perturbation wave length a = for constant critical velocity cr, A NACA TM 1343 0.15 0.20 Fig a u Figure 25 0.25 0.: tc 0.80 1.00 1. 1 2 1.20 1.30 .6 0.2 0.3 0.4 0.5 F.u 6* Figure 26 Figure 26 Cr 0.285 0.363 0.410 0.442 0.462 * Asymptotic value for Re  oo  Interpoloted Figures 25 and 26. The excitation as a function of the reciprocal 2pV Ua perturbation wave length a = for constant critical velocity cr. 4xl0"3 3xl03 2x103 0 0 .05 Cr 0.188 0.281 0.327 0.372 0.10 2 x 10 102 0 0 U 1.12 B / I .1I I ` i 8S* 0=o 0.6 U"a  0.7  0 Y4 /^,~04. A ^ ; 0 NACA TM 1343 103 10 4 15 106 1 Re=p Figure 27 Q3 ,3 0.6 a1 Figure 28 Figures 27 and 28. The curves of constant excitation in the instability regions of a few laminar boundarylayer profiles. pie=; 9.1 X04 103 104 105 106 107 Re Rz eU Figure 28 Figures 27 and 28. The curves of constant excitation in the instability regions of a few laminar boundarylayer profiles. NACA TM 1343 57 o f o0 cti II 0 00J 0 0//" ,d 0 o 00 C Sa 0 0 uIP" 0 o . I_ a Q1Ic a 0 o 1 0 M.. ^ 58 NACA TM 1343 Co O . 0 o o *J a . O 4"1 2 0 C .o 2 e 0O OI 0 / b c "' S*~ *" ^/^ A/ A .2s I\ __________ *0. NMo 0 0r o *' M NACA TM 13:, 59 0 Cd 00 110 \D S. ... c t s .'50 If 7 1 " O S Cd 4 SCL o >  a i oa) ro c A 4 mgo o "'4 5x103 ,/ S=0.198 ai*  = 0 aU 9 1 I I (1 f 11 a8 t 10 10I 4 5 6 7  0.0, 1021 I I j o.o S I I0.0. o0.6I 0. I. I I 0.0 041 1 0 o.o f SI I S1 I o.61 1 o. o I I I I I I I ///, 0.07 0.06 0.05 0.04 Uo*  Re 7, Figure 32. The curves of constant excitation in the instability regions of a few laminar boundarylayer profiles. NACA TM 1343 3 I mI I 103 108 NACA TM 1343 cL 8  *" _o c?_ _ $I 3 0 0 bD 4l ci) 0) ci 4 o 0 4 r * i nci 3o 4 mi "  0 , 0D ro ii) NACA TM 1343 ow 8* Re U02 t~ 0.2 0.4 0.6 0.8 02 04 0.6 0.8 1.0 Sa* ^Q0 Figure 34. The circular time frequency Re* for boundarylayer pro Ua' files in the region of increasing pressure for Re* . NACA TM 1343 \ Uo/max 0.06 Figure 35. The maximurr excitation \ as a function of the form parameter of he ainr bondarylayer profiles. parameter 0 of the laminar boundarylayer profiles. NACALngley 9552 1000  N II e 0 * .P S S  > .0 3' 3 S n .c > 5 *. w! M CJ A .2g~ g ^ L 0. .0 CM 4 \ I.w 440  0 ^d a  S14 u 0 on C 4 6 d~d O~O 0 ~u ,. cU14 0 'L~m4 .' Z. c.ld .o w a o Zb 0 o u w< O OLz gii< 3 C rt cl "[wo6 s . S .!52^iS E 0<< 0c E So . eM gSin 'Sg ^*^ %,am1=*, <=z  0  "f A5 r^ P0u^^ ~^ y j 0, aw 5 0 0" .0 4" O ) o44 C r. L. 0.* ed s 0 0 00 m MO 0 'a o rL e U 8 S.O w . ^^?S?L~ L br, fLo u  0 :3 o4 4 C 0 la *a 9 O^ < 4 4 q d : a = z w 0 4 . S O 1 S3 U ^9U .0 2 **4* S HM B 44 .^ s e 'S r C B 30 P 6.^ .1 o~~~; ~ LU g 41, C c Cd 0 c, L) I'MmI ot 0,4 > JO 0" r3 t Ln.  ie   l I Ci 0 a B9 & i t U ,0 0m ~ 0400 m0 < 0r ~UU~''P ~Cd Lk z ui V6 3 Li ;s ~ Z4 0 U i .0 4 0 L' r' . , r ,.  c75< 0 o s. 4 w 0 F ., . W0 w. 0 o 0 zm = IvU "C  i S^J^Ss^ S sgzPigi. 3 S < Isy. d Ii 'id.iio EMeI II^S Z^Lsaria 0 4 c 04 M r U5 < 4a .0 .0 14r~.4ci "a. 0 O e *^ c OMiliS ^ SMB 0U Iiw1 1 .o E m c G m .C ; uwE o 1 V0 0 0 c Z u !2 Z 0. 0 I u , o44a ; 4 a S 32 gi 0 .0 44 14 @3 C..O l c bD 0 14 oc ^ 1.. 4 0~c@ w. rxd:,0 f~~. 4W9 Q454 O OU d g ".o 1 ll Q > ro o4 i i .00 gcilVeeoa . wh~ .a 3^:1 E4 S .U g i o0 .a 6 0 usaco el Z.MImu R"se 5^sl gcass !!! ^lauga lsl" a U E a 'C 0 8* U j 0 0 0. 0 *a *E e br.0 St "3 D U 5g *H.0 OL4U S 0 01 t 0. 0 79 Eb~ Ut > .I to.. 0 n B0 Z k... *to 0 a 0 V. s * a U 0 0 (0 0 SK e *R 0 9. 0 bD 0 > U t e 0.8 a 0 .. *o 0 .0 0~ ca'o 1 0i S3 *R es a'. 0 b*2 SS S .0 III0 5 *5 i* S I0 I ' 0 0 3 " 3 4. * a  * eq i 0 0 ctC0 r0 .5 i LoS ^I "a t 002 JZ^.0 3.j <>3^ S W 0r ,0u0) Bi00gla . .5 *3 SZ Li S i S a C ' o zut < 42 ; ci d 0 . s0 b s t g u0 mm 0. z z 2 z. i s .0 mW o Li 2 w :0 2Aii. ; S2 S .0 S c w Cl C:" oa o, R a, Li o3 LDZ.0 Ofl 2  r 0 Li u W w f g 5 U .2 a,  .c ua :a U* r_ E C. E] (U 0 aw 0W W 005.0 W L. '0 z 'iicu  eq i 0 lOf4~0 0 02 Cv) LocE .0 ~'* 4r(i H d o St 0 0 w eo I uW u ,3 US * 0 *?* 000. < 05.0 8 "'00 0 0' . .^t Uy'3 I 010  '5 fc83 fr..., U S 0 3 = 'r . 5 > .0 = m a, 0 l 0 0 ~ E: e ~ L =g toT 1, i Li 4 = = S w aut' r 44 i L.W o.0^ .y tEO,  eJ  (fl .000 4 ._0 00.. 1 ..J Lii 2 "! 3: .0 U. , .W 0 ,.. r. I; g.. L' EMis u g Gn.^ Su g in L.S 00 LL.  3.U L S So n 025 fra ^ = s rLo z.C J S A^ d z ) EfiiJ. n^E rilgg r == W o :: ci T *:2 o 8 0 S, d m C 0L w. Cc S cc 0 Cc C 0, o at A 0 n c' 5 S 0 a' L' .5 0 oa 0 0 0.  M2 W 00 El~ E. M awm .0w 0 0 0 ^ S Li 2. g Wa. 600 M o aj v .5 1 t 0 0 us g a S Q > m S'a a *ss s a 3= So SL. 81 0. C1 Oa L Cu .2  ;g0 to c w 14 bD C. i' 014a 01 * a 41.i 1.J L a .0 l0~ . 0 S a U'  0 a * a. I.J l UNIVERSITY OF FLORIDA 3 1262 08105 483 4 