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*"lijber das Spektrum bei Eigenschwi~g~ungen ebener Lamirnars tr~jmanlgen." Zeitschrift fiir angewandte Mathemiatik und Mechanik, vol. Sk, no. 89, AugustSe~ptember 1954, pp. 344557. TIATIOlJAL ADVISOBYI CO1MITTEE FOR: AEROlJAU~ITICS TECHNICAL MEMORANDUM 141, ON~ THE SPECTRUM1 OF 11ATURAL O)SCILLATIOIIS OF TWODIIJEIJSIONIAL WNADJAR FLOW1S* By D. Crohne 1. INTIRODUCTION~ AN'D STATERENPT OF THE: PR~OBLEP In the investigation of stability of a twodimensional la1ina3r flow with respect to small disturbances, we describe a disturbance of the stream function moving downstream (in the direction of the xaxis) by, the "partial wave formula" 9 = g(y)eia(xct) (1.1) and obtain then for the distribution of the disturbance amplitude Q(y) at right angles to the main flow the socalled stability differential equation of the fourth order (U c) 9" a29). U~IyF =a 1 c  222 n+ a 0) (1.2~) where U(y) designates the velocity profile of the basic larrinar flow In addition, we enforce certain boundaryr conditions,, in the specific case of the parallel channel 9(+1) = 0 9'(+1) = 0 (1.s) whiich express the fact that even the disturbed flow adheres to the bound ing walls. In these equations, the velocities U and c are referred to a velocity of reference UO; furthermore, the lengths x, y, and 1,13 to half the channel width b, and finally the time t to the tihe unit b/UO. The Reynolds number R is defined by UO" The boundaryvalue problem consisting of differential equation and boundary conditions determines, for each pair of parameters a and R, a spectrum of an infinite number of eigenvalues ep. The associated dIs~turbances (1.1) are damped when Emi(c,) < O, and are excited when Imly > ; ais assumed to be positive an~d real. A basic flow is called stable for a value of R when the entire eigenvalue spectrum n,t for all possible values of a, contains olnly damped disturbances. Thus the range of' thle Reynolds number R is divided up into a region of t ab ili~ty I < R < RR and a region of instability R > R", which are separated from one another by the stabilityr boundary R*. Since, in the literature published up till now almost exclusively neutral oscillstions at most, excited oscillations have been investi gated, we shall investigate in the present report, following a suggestion of Pro~f. Djr. W. Tollkien, the entire spectrum of the eigenvalues c, as a function of a and R; for simplification, we shall emphasize the dependence on ari. A general solution of this problem is possible in the fojllowinC two special cases: (1) in the case U = const. which is equivalent to U .We deal here with the "oscillations of a fluid at rest" already treated by Lord Rayrleigh. The solution is possible in thc domar~in of elementary and transcendental functions. The second special case co~nerns the rectilinear Couette flow U =y investigated by L Hof fef.5).The solution can be reduced to tabulated Bessel funci~t ions. For more general velocity profiles U(y), the eigenvalues on can be detersined app~rxima3tively' analytically in the following limiting cases: I. In the lirmitintg case aR 0( for arbitrary order n of the II. In the liimitin~g case n ,om for constant aR III. In the limiltintg case aR so for restricted order n. NACA TM 1417 NACA TMl 1417( A continuus transformation of the three cases into one another for constant subscripts is possible in the above named special cases U = O and U = y. The assignment of subscripts of the eigenva~lues cn can be made in the cases I and II according to increasing damping, that is, ac.cordingt to the rule Im en+1) Im en (1.4) Ho3wever, this rule is not always applicable to the case III whn the subt scripts used ar~e to remain constant for continuous variation of a and R. The boundaryvalue problem formlated in (1.,2), (1.3) is, generally, not selfadjoijnt; thus, th~e re~ducltion to the well3knw statements andl estimates of thec SturmLiouille theory is eliminated. The eigeat~une tions generally do not .formu an orthogonal system. They do form, however, as 0. Haupt (ref. 3) has shown, under certain assumptions, a system of functions that is complete with respect to each of th~e functions which. satisfyl the boLundary conditions (1_.3) and are four times continuously differenztia~ble. This system of functions can. be transformed into an orthogonal system. 2. THE LD11TITG CASES a;R >10 FOR ARBITRARY ORD3ER n OF" THIE EIE~NVALUESenAN n sm, FOR LIMITED aR( As already found by Lo~rd Raylieigh (ref. 8), the entire system of eigenftunctions and eigeEnvalues in th~e case of the basic flow U = 0, that is, for a mediumr at rest, can be given as a closed system. Since thee eigenvalues are suitable foir approximative representations in the ase of' mo~re general basic flows also, we shall. derive themn here briefly;. In the case of the basic flow U = O, the stability differential, equation (1.2) is simplified to ;(4) 2/f" +I a 4$ iaR C 4i" 4) = 0 (2.1) wherr wt tshall den;ote thel eigenv.alues by C, to dis~tinguishi them from~ thec eigenvalulea c ofr the; general stability differential. equation. The equation is solve~d by each of thre functions cosh as y inli; sinh a i~ O4I(;) = cajsh my cosh; Q, 4 y sn y ihm cosh 2.I SiT sr a (2.2) NACA TMr 1417 if we put m2 = a2 R C(2 .5) The part cp(4l) = 0 of the boundary conditions is .identically satisfied. The remaining boundary conditions #'(+l) = 0 lead to the related branches of the eigenvalue: equation: a tanh a = a> tanh a> in the case I a coth a = m coth nu in the case II 24 The equations (2.4) have, for positive a, no roots Lo outside the ima~ginary axis of the complex auplane. With my, also iu, is an eigen value associated with the same eigenfunction. Thus it is sufficient to consider only the positive imaginary eigenvalues a). If we designate even. eigeafunctions by even subscripts and odd eigeaf'unctions by odd subscripts, the equation (2.4) may have the solutions UjO, 02, my4, .. in the case I and cul, a3 ,5 up in the case II. The eigenvalues can always be made to form a monotonic sequence CUO n1 03 2L; 0<<< <... The associated eigenvalues Cn are according to equation (2.5) Cn iaR.(2.5) They are, therefore, arranged in the order of increasing damping. The nth eigenvalue may be estimated upward and downward by a2 +<2 2< iaR ., C< a.2+% 26 From the representation (2.5), it follows that the eigenvalues C become~ very large for aR >0 as well as for n sm. The same behavior occurs, also, for more general velocity profiles U(y) because the main parts of the stability differential equation (1.2) are then represented NACA TM 14171 by the3 equation (2.1). We shall, now express this train of thought more accurately by subjecting t~he difference c C of the eigenvalues c to a more accurate estimate compared to the eigeavatlue C of equa tion. (;2.1), for more general profiles. With introduction. of the differ ential operators NCrpl = (p(4) 2a~" + a4q~ LC~I= U((pl' a2~p> U"'4~ nLPT1=cp" a2~ (2.7) theF stability differential equation (1.2) may be written in the form cM 9 = LP N 9  Correspondingly, equation (2.1) reads (2.8) Utilizing thre .fact that the operators M and N ar~e selfazdjoint, we obtain from these two equations thie relationship 1 PM 4 dy; = + L day (2.9) If the normalization which is still open for Q, is fixed by the rule +1 +1 Ir~1 = (2.10) there follows, after introduction of the auxiliary quantities j_, +1 ld QM, m,. dy +1 +1 i Q M # dy (2._1) (PMCe dy NACA TM~J 1417 from equattion. (2.9) the representation c C = + q (2.12) In this equation, C an may be :regarded as known by virtue of the functions a, represented in equations (2.2). The eigenvalues Cn have already been delimited in expression (2.6). For Q we obtain directly the estimate +1 Q 1 (2.13) In connection with a simultaneous estimate of the function (9 4)" a2(9 (g), we obtain for q aR for  > a: If we substitute both inrto equation (2.12), we obtain, with consideration of equation (2.5) and expression (2.6), the two partial statements on Cn n ~+ 0  for CLR >0 and for arbitrary order n (2.15a) e, Cn 2 1 U *dy + O for n >mo for fixed aiP (2.15b) The latter estimate indicates thazt the eigenvalues o~n of the stability differential equation. for sufficientlyr high order n finally tend toward the eigenvalues C, of the "zero flow" withini the real part increased by the mean velocity of the basic flow). (Comp~are eq. (2.1).) A mutual coordination of the ei~cnvalues on to the eigenvalues Cn, however, is by virtue of equation (2.15b), meaningful only due to the fact that the difference Cen Cn+1l of the appro~ximation eigenovalues comes out consl~cscideral larger than the estimated remiainder in equation (2.15b): ~For, b~ea~use ::' (2.5) and (2.6i) U y + 0 @ >> 1 Q = O  NIACA TM 1417T SCn Cn+1 ; ostn (2.16) is valid. It should be mentioned that F. Noether (ref. 7, p. 239, for mula (28)) has already indicated an asymptotic representation for slightly differently defined eigentvalues for unlimitedly increasing order, although only intimating an argument which, leads one to expect con siderabDle difficulties. We mention, furthermre, an estimate for the eigenvalues e indi cated by C. S. Morawetz (ref. 6, p. 580) lc e~n <:6 *(aR)1/2 where on is an approx:imative eigenvalue which (in our notation) is determined by the equation +1 and corresponds more or less to our approximate eigenvlalue C, intro duced in. equattion (2.1). In the above estimae of Morawets, neither aR nor n. ma become arbitrarsily large; in the first case, the eigenvalus would shift into the excluded neighborhood of a = v yl vw = U; yl des ignates the wall); in the other case the estimate would become meaning lejs .;ince the behavior of the quantity a for nlimitedly increasing n is not given. 5. RECTILIIREAE CGUETT'E FLOW TH~E LIT4ITTING CASE aR >=c FORi FI'IBITE ORDER n In the special case of the basic flow at rest, U = 0, the behavior of the eigenavalues on for un~limiftedly increasing aR~ is described by the formulla (2.S) in. which the quantities an, no longer depend on aR. InI deviation from this lawz, there results for more general velocity pr~ufiles a behavior like NACA TM 141'1 c (1  for aR tm (3.1) where the complex valued qutntities Pn n~o longer depend on aRi. If these eigenvalues are adjoined to the eigenvlues in equation (2.15a), by continuous transition of a and aR, the ordering principle (1.4) according to increasing damnping is lost even in special cases like the rectilinear Couette flow. If we therefore desire that the subscripts of Lthe eigenvalues on remain. unchaned for coninuous transformation of the limiting cases aR1 >0O and aR ,am into one another, we must actually carry out this procedure which presupposes a general solution of the eigenvalue problem or a solution which is a~pproxim~ate only inso far as the individual eigenvalues still remain distinguishable from one another. We succeeded in, obtaining a solution in this sense only in the special case of the rectilinear Couette flow. It will, therefore, form the subject of the following section. After in~sertionl of the velocity profile U = y of the rectilinrear Couette flow into the stability differential equation (1.2), the latter can be reduced, by means of the substitution to the Bessel differential equation in the auxiliary form v" iaR(y c)C +~V sq= (3.5) In order to arrive, through the boundary conditions, at the eigenvalues, we must invert equation (3.2) in the form cp(y) = : t(?) sinh a(y ?) d j4 The boundary conditions 9(1 ')=Othen are identically satis fied; the remaining boundary conditions cP(+1) = '9'(+1) = 0 require that the two eluationrs .+1 +1 j 1 J( sinh ay dy = 0 1 4(y)cosh amy dy = 0 (3.5) hold. NACA TM 1417~ By means of' the substitution and yk =c + =El with E = GR, (.6) y yk equation (73.) mayr be transformed into the differential the differential equation i a;k+ qq~ = o d12 (3.7) *I(s) and q,(n) (J.8) are assumed to be two suitable fundamental solutions of this equation, and 41 81 is assumd to designate the values which are, because of equations (3.6;), associatted with the walls y = 1 and y = +1 '11 = 1  c  1 1= 1 c ~2 (5.9) there follows from equations (35.) the eigenvalue equation 1 1 +1 11 1 . 1 +1  E1 smn acq dq .I a 4 I(?)COSh aE d?  Q (?)cosh ae? d'l = 0 (3.10) For further treatment of sequence of functions Ani 1) this equation, the introduction of a by the Laplace integral 2 .z qz+1  e .n1 z A,() = 1 (3.11) 10 NACA TM 1417 is advis~able in which the path of integration Ai runs from infinity to infinity in the manner drawn in figure 1. The functions Ant?) ~stisfy the differential formla =( An+( q ( 5.12) and the recursion formla i A +3 q *I An+1 + n An = 0 (3.13) by means of which. all1 the functions An(4) and their integrals and derivatives may be constructed recursively from the three b!asic functions AO( r) Al( ?) A2( q) (5.14) The significance of LhZEe functions An(4T) for the stability prob lem lies in the f'act thatt the two particular solutions of the differential equation (3.7) needed in the eigenvalue equation (5.10) can be repre sented in the form 2xi 2n'i (?)n() = AlII('I) =e *A e(.5 Th~at the differential equation (3.7) is satisfied follows from the formlas (3.12) and (3.13). The linear independence of the two functions follows frocm the fact that the Wronski determinant, which is constant of course, does not disappear at the point n = 0. The basic functions Al( 1) and A2(?) = Al'(Bl) have been numeri cally tabulated (in somewhat different notation) for a quadratic point grid with the mesh width 0.1 within the circle /T 6 of the complex r)plane by H. H. Aiken (ref. 1). The basic functionAO)= Algd can be determined from it by a numrical integration. Outside of this table, the b~ehav.ior of th~e functions Ar,(?) may be infe~rredl from the s3:rmp~totic series representation v=0 NACA TM4 1417 11 which is valid for Ir em ini the ang~le space 1 + 8~ arg il9 8 with arbitrarily small 5 > 0. AcTCOrding to H., Holstein (ref. 4), the first coefficient of the series i3 G~d (3.16a) a 1 e TIhe asympto~ti c eries are obtainable directly from the Lalace integral (3.11) bjy means of Riemann~'s saddlepoint method. For the representation of the eigenvalues, the zeros rlN of the function AO(?) are necessary,. An ajsymptotic calculation of these seros for I ip> s not possible directly by means of expression (7.16), since the zeros would move ouit of' the range of validity of this repre sentation. We avoid this difficulty by applying the second relations obtainable from the integral ($.11) i An ie e i . A( / e (3.17) (* = conjugateconplex values) 4a= Pu(l) . n + e i n ' n al . Ann Ange Anfl?) + e (3.18) P (g) o~f the degree n, satisfy the same recur where the polynomriias sion fo~rmula (3r.15) (3.19) 1 Pn,+) n+,1 + nFn = O with the initial elements PO = 1 P1 O P2 = 0 Cor.binatiion of fo~rmulas (j.16~) and (3.18) then yields the asymp.totic, representati:n valid fo~r I_, in the angle space larg zl < n 6 NACA TM 1417 cosj z5/2 x ~ i 6 (3.20) Hence, there follows, for the zeros 5N of AO ~N) = 0 which, according to equation (3.17)r lie in pairs symmetrically with respect to the straight line arg = =ix/6, the asymnptotic representation ?N (jEn)2/ 5 6 3n  n = 1, 2, 3..... (j.21) The value of the lowest pair of zeros was calculated to be (5.21a) according: to the table of Aiken (ref. 1). For further treatment of the eigenvalue equation. (5.10), it is advisable to expand the functions sinlh asy1 and cosh ac q into their Taylor series, and then to interchange the summtion with integration which is justified by a theorem of Bromwich. (Compare ref. 2, p. 398.) The series obtained converge, according to theorems of the Laplace trans formation, for each value of arE. If these series are broken off after the first terms, provided with residual terms, and substituted into the eigenvalue equ~ation (3.10), the latter is, for this reason, and with conslideat ic~n of equation (3.1rl), simplified to A ~?f A Y]*1l = O a2E ) fOr I ? constant (J.24) What happens now when aR increases beyond all limits, that is, when E tenyds toward 0? Because of the relationship following from equation (jt.9), pl 1 = 2/E, at least one of the two quantities ~11 1 must te~nd toward infinity for E 40t, on a parallel to the real axis. It is sufficient to assume~ this regarding 71, because in the other case everything would form a mirror image with respect to the iaginrsly alxis of the rlplane (ss essentiallyr occurs in. the Couette flow ?N = 4.257 ei x 0.270 ] AO 91 ) AO 1 ) A2 T1 A2 1 NACA TM 1417 where with C, also, CX. is an eigenvalue). With consideration of the asymptotic behavior (eq. (3.16)), the eigenvalue equattion (3.24) then is simplified to iAO ~1) E OE2 For 6 << 1 (3.25) A2 r1) 2 From this formula, we recognize that q1 for E +0 must tend toward the zeros qN of the function AO('1) estimated in expres sionls (3.21). We thus obtain for the eigenvalues c, with considera tion of equations (3.9), the asymrptotic representation c + 1 Eq+ O( E (3.26) with qN from AO N)= O Thuts we hav:e proved the previously given eigenvalue formula (3.11) for the special case of the Couette flow. In order to follow the variation of the eigenvalues c over the entire range i0 4 aR < I~, we must go back to the eigen,alue equa tion (5.10) or its approxtimate for in equation (3.24), with the fune tions AO(1), Al( 1), A2('1) to be assumed as known. We have accord ingly calculated the 1;2 lowest eigenvalues c as :functions of aLR for a fixed value a = 1 an represented themo in figure 2. The variability of the eigenvalue curves with a is only slight and becomes, for instance, for aLR >mn with E Small Of thne order 0 (a2 E ) . The eigenvalue spectrum of the rectilinear Couette flow has been discussed already by L. Ho~pf (ref. 5). 'Hopf replaced, more or less on the level of our eigenvalue equation (3.10), the solutions $ JII represented by him by :Hank~el functions of the order 1/3 by the first terms of their asymrptotic series (3.16), whereby the eigenvalue equa tiojn was simplified to an algiebraic equation of auxiliary arguments and circular and hyp~erbolic functions. However, since Hopf committed certain errors in the asymptotic representations of the Hankel functions, his results require partial corrections. Although these changes are hardly significant for small values of arR, the vatlues of, for instance, qJ in a folrm:ula correspondilng to (3.26) undergo a considerable change. Th~e topological connection of the eigenva~lue curves c = e(adaR) a~lso appears different to us from what it appeared to Hopf1. However, the NACA TMI 1417 qualitative p~icture of the eigerafunctions, the physical conclusions drawn from it, and the main result that all2 o~scillatio~ns are damped  remain the samne. 4. THE L~llTIllC CASE aR >= FOR FINITE ORDER( n FOR SYlelIETRICAL BASIC F`LGIS For a basic flow with symmtrical velocity profile U(y) = U(y) (4.1) the stability differential equation (1.2)ua~lwayrs has a f~undamnental system of four solutions (1, I21 (P3 so th~at $2 y 1 (y and (4.2) If a linear combination, of these solutions is to satisfy~ the boun~dary, .jnditions of equl~ations (1.5) in thle sequence W)=0 ')=O 9(+1) = O, Q'(+1) = 0, the following determinant, simplified with considraction of the s:,mme~tries (4.2t), mst disappear: c$1(1) 1 i( 1) i2(1) C(1) I Since this determinant ma be written as the products ofT the twoco~lumnJ (4.~) $(y) are even functions ofy 5(y) are odd function; o~f Q)1(1) 1() cp (1) (1) 6 (1 9( 1) 1) 1) NcACA TMr 1417 one obtains, by equstinig one of the two factors to zero, one branch of the eigen.alue equation each time. For this reason the~ eigeznfulnctions can be either only even or only odd, with the respective ei~genvalues c 6enerslly being different. In order to arrive fromr these equations at asymnptotic eigenvalue formula;, we shall determine the: four fu.ndamental solutions (4.2) 91 Sl in such a manner that they are available for appropiriate asymptotic expansions. We find that the fundamental solutions desccribed by W. Tollkien (ref. 12), "Asymnptatic Inegrati~on of the Stability Differ enltial. EquaLtion", the asymrptotic representations of which are provided with residualterm estimates, are suited to the problem. In order to establish the connection of these fundamental solutions with ours, it is indispensable to discuss first the concept of "friction less approxiimatiorn." The quest for solutions of the complete stability differential equatio;Cn (1.2) which for aR +, my together with their derivatives with respect to y, tend toward a limiting function lim cP(y,aR) = X(y) (4.4) aR , leads to the socalled "frictionless differential equation"' (U c)(X" a.2X) U'"X = 0 (4.5) which must necessarily be satisfied by such limiting functions. If we want to use the solutions of this frictionless differential equation fr'_r th~e approximation of the solutions of thre completed~ differential equationr for aR my, we must not disregard the orange of validity of the ti~Lraundayvalue statements in, equation (4.4) in the comrplex yplane. A3ccor~ding to W. Wasow (ref. 15), the following theorem is valid with respect toi this: "lOfr t~he four~2 fundamental solutions (4.;2), one even and one odd solution can be determined in, each. case so that with two appropriate frictri unless solutions 1i(y) anid j2(;y) the approximations 9 1 1()+OX1(y) = odd function of 1 () =12() +0 2(y) = even function of y 46 NACA TM 1417 in each fixed interior of a io~uble region (I + II) or (II + III) or (III + I) are valid and become invalid, in each case, in the comple mentary third region. III or I or II. The sam is true for the derivra tives with respect to y." (Compare fig. 3.) The boundaries between the regions I, LII, and III satisfy the equation Re 1l~ JU e = 0 if yki denotes the "critical point" defined by U Yk) = c Re yTk < O (6.7) For more details regarding the regions I, II, and III see W~asow (ref. 15). The frictionless differential equation (4.5) has at thle critical point, U yk) = c, a singular point with regard to dete~rminateness. Two fundamental solutions take the form X1tY) = (v k) 1 Y Yh) U; X2 7) = Pl ;Yk) U1 7 ~ k) ( Ik)1n~! y Yk) 49 if P and P denote power series with the begiruning 1 2 Pl(z) = 1. + z + O z2) P (z) = 1 + O iz2) (4.9a) 1 2Uk  (Co~r.pare W. Tcolrie~n, ref. 12, p. 35.) The commn radius of convergence of these power ;Frles is limited either by the radius of convergencee of a corresponindl series for U c or by the nextadjacent zero of U c as a cingulzr point of the differential equastion. NACA TM 1417 For the further developmnt it is a~dvisable to introduce a sequence of factions Bn1(T) by the Laplace integral By(D)~~ ~~ =e .E1(In a + ^)dz (.0 which is comparable to equation (3.11). In it, 8 = 0.5772 .. denotes the Euler constant. The path of integration B runs, in the manner indicated in figure 4, in the complex zplane cut open along (O,ioo) from infinity to infinity. TIhe functions Bl(tl) satisfy the differential formula n = Bu1(B)(4.11) an the recursion formla i Bn+3 + '1 Bn+1 + n3 Bn= n (4.12) in which Pn(rl) are the polynomials defined in equation (3.19). By means of these two formulas all the3 fuctions B (rl) and their derivatives anrd integrals can be constructed recursivel~y from the three basic functions BO(A) Bl(4) B2( t) (4.13) By means of the representation B (l) = 2ni AOt7) (1n~)A(1) A ~(I*) (4.13a) (* = conjugatecomplex value), the basic functions B1 and B2 can be reduced to the funtions An. (Comare W. Tollmien, ref'. 10, p. 27.) The significance of the functions En(q) for our stability eigen value problems lies in the fact that the function Bl(?), because of equations (4.11) and (4.12), satisfies the differential equation NACA TM 1417 d B, d2 B1 i + p 1 du dq2 (4.14) which, with the designation "differential equation for the friction correction," has been introduced as an essential constituent into the asymptotic integration of the stability differential equation bjy W. Tollkien refss. 10 and 12). After these preparations, we turn to the four funmdam~ental solutions 97,I r7 (P777, CPV of the complete stability differential Iquation constructed by Tollmien, regarding its ability to be expanded asymp totically. According to W. Tollmien (ref. 12, p. 77) these four solu tions ma~y be determined, with. use of the substitution with 6 = anRd y frOm U Yk) = c (b.16) y ;Yk = E' in such a manner that they have in a fixed interior of the qplaLne (com plex for reasons of analytic continuation) as well as in every fixed interior of the region II of the frictionless approximation (compare eq. (4.6a)) the following asymptotic representations: cp (y) = X1(Y) + 0 6 ) (4.17a) cp,,() = P2 E ) or in every fixed + 6) G (4) p+ q n e +E In id interior of II(.1b rII(Y) = x2(Y) + o0e5) Fuirthermorelf. 'SIII(Y) = A1(;1) + O(E) (6.17e) is valid. Finally, there applies, according to W. Wasow (ref. 15), quotient asymp~totically in every fixed interior of II ( compare eqs. (4.6)) in ) I constant NJACA ?TM 1417 777 (y), Wy(y) constant 5c) (4.ia) Corresponding formlas are valid for the derivatives. For further treatment of the eigenvalue equation (4.5), we must express the fundamental solutions $1 ''4 used in it by the above fundamental solutions ... If for the latter, the represen~ta tions (4.17) are used~ immediately, and with the residual terms in each circle @constant for a >0 being valid, the result reads '1,21Y d1,2 PIII(y) = 1+E *~ InE +8, B()+0e1 U~ X2(0) U; X (0) with S S = X i Ukl 1 x O U 2, (4.;21a) X1 and X2. (Compare eqs. (4.9)) fromr the frietrionlless solutions ur~the~rmore, consta E = A1(?) + 0(E) $ l(y) = cPI(y) + OEi e (4.21b) is valid. Corresponding formulas are valid for the derivatives. If we nowcl write the two eigenvalue equations obtained in equa tion (4.5) as a product of three factors, for instance, *exp idR re U e iak cp (1) L3 ;20 NACA TM 1417 (9 = arbitrary constant), the zeros of the two first factors do not make a contribution to the eigenvalue configuration since the; are compensated gy corresponding poles of the third factor unless the derivative rp (1) should disappear simultaneously. It is therefore sufficient to find only the zeros of the third factor. After insertion of the approximations (4.21_) we thus obtain f UIn t6+S1,+81,2 +BO U1 1+e.~ 1 n+1+S1,2) +Bd1 il ~III  dr + O(E In E 1 III ( = p1 0 (E n E) e n E + 1+ 81,2 + BO 1)] 1 f ln + c1 E + 1 + 1,2) 1L ~ AO( ) with n = q1 = 1 k)/e and S1, from the frictionless solutions. The next~;igher approximation in equation 82 according to equations (4.21a) fun~ct ion JI(1) (4.21b) reads stermming from a (4.22a) A,1 (4) and ma be reduced, by means of the formulas (3j.12) and three tabulated basic functions AO0, A1, A2' (5.15), to the How do the eigenvalueis c behave if in the eigenvalue equa tion (4.22) we let aR sm, that is E >0O? Evidentlyr r1 then tends towi~ardj the zeros ?[ of the function AO(?). By Tayrlor series expanded about these zeros, there follows more exac:ly AQ Eq~O A1 I C + U ~ n = ] + 1 11 NACA TM 1417 The eigenvalues then behave asymptoticall~y like c U(1) = U~ EN 1 1 6186+ 6 4. with r from AO(6 N = As a supplement to equations (5.21) we shall give here a few zeros rlN and values iA2 Al: 5N ) A?l N 4.1;22 + i 1.065 1.686 1 1.222 For >> ]1 2.983 + i .037 +1.902 + 1 0.851 6.8 + i .2.5 2.2 1.5 there applies 5.5 + i. 4.5 +2.4 + i .1.14 A Al UN) V The remarkble fact about the asymrptotic eigenvalue formula (4.23) is that it is transformd into the corresponding :formula (3.;26) after substitution of the velocity p~rofile U = y of the Couette flow, although the two formuilas were derivred under completely different assum~ptions. The asymtotic eigenlvalue formula (4.23) is already so greatly reduced that it no longer permits a distinction of the eigenvalues c which are associated with even or odd eigen functions. For this, we crust go back to the moe exact formlat (4.22) in which the character istics "even"" or "odd"" of thre eigenfunctions are taken into consider ation by rreans of the constants S1 atnd 82' to be determined "without friction." We. have used the eigenvlu equation in. the form (4.22) also for the n~umerical. calculation of the eigenvalues c in the examples treated. We selected as examples the twdimenrsional Poiseuille flow and a flo with an inf'lect~ionpoin~t profile. We represented the variation of the four lowest eigenva~lues as funtions of R for a fixed value of a in figures 5 and 6. The numericlal calculation itself is after redue tiocn of the nonalgebraic elements contained in. the eigenvlue equation to the three tabuclated basic functions AO, Al," A2 and to the fric tionless solutionss a prob~len involving numerical methods, the details of which cannot be discussed here. We shall mention only the following approx>icate representation of the frictionless constant S2 NACA TM 1417 U.S2 A .a2 + O(1) for a << 1 with A = 0~ U' I (U c) dy. (Compare W. Tollmien, ref. 11, p. 100), which may be applied ad:anta geously for small values of a. The subscripts for the eigenvalues c obtained frcm equation (4.22), in. the sense of a continuous connection with the limiting5 case aR 0,O remain an open problem here. In the range of validity of equation (4.22) alone, a generally valid choice of subscripts according to the rule Im en+1) Lm en), that is, according to increasing damping, cannot in principle be carried out, either. The zeros in equation (4.25) can be ordered according to the increasing imaginary part, but the~ Im 0 ) curves may penetrate one another if a is changed. 5. THE ~FRICTIONLESS EIGENVAIDES WITHIN TH LORITING CASE aR sm Determination of the Extcited Eigenvalues Let the approximation (4.6) by means of thle frictiornless solutions be suited either to the double region I + II (compare fig. 3) or to the double region II + III whereby the logarithmic term is alwayse uniquely determined in the frictionless solutions. Applyring the approximations (4.17), we then obtain, by way of the eigenvalue equations (4.3), eigen value s c which, for aLR , tend toward the socalled "frictioniess eigen values" c(0)(a) which are defined by the boundary co~ndition X1(1) Oor ji2(1) = 0 of the odd or even frictionless solutions X1' X2. Thne following general statements may be made .regarding these frictionless eigenvalues, limited by the range of validity; of the bioulndryvalue expressions (4.6), according .to W. Tol~mien (ref. 11), partlly on the basis of the "RayleighTlollmien theojrems:" "For velocity profiles without turning points, no excited friction less eigen.alu~es are possible. The approximation (4.6), associated with the damed frictionless eigenvaluies, must always take place in the interior of the double region I + II." "For inflectionpoin~t profiles, there always exist excited friction less leienvalues associated with an even eigerafunction." NACA TM 1417 Beyond these general statements, frictionless eigenvalues associ ated with an oIddl eigenfunction were not fou~ind in any of the examples; neither did we find eigenvalues such that the associated approxima tion (4.6i) would have taken Iplace in the interior of the double region II + III. As examples, we chose the twodimensional Poiseuille flow as representative of a profile without an inflection point, and the inflection point profile U =(2e 1 + 2 (E) xos 3T~y. The frictionless eigenvalues c, fouLnd only associated with an even eigenfunction, are represented in figures 7 and 8. The range of existence of these eigen values is always given by an interval O < osat o h rc tionless eigenvalue c, T~ollien (ref. 11, p. 100) has set up the following ap~proximte formulas: crU'( 1) JtU" a~~~adC r 1iq (cr)l =@  with U yK)==cr O (5.1) We now seek the connection between the frictionless eigenvalues and those discussed up till now. The closed solution, in the case of the Couette flow, canot give an answer to this problem because the frictiolnless eigenvalues in question do not exist there at all. :How ever, it is po3ssle to insert the frictionless eigenvalues into the equation (4.22) and thus to interpret them as a limiting case within the eigenval~ues (eq. (4.23)). Let us, therefore, perform on the eigenrvalue equa~tiojn (4.22) the limiting process aR! smo, that is, E >0 for constant (1 yk) = Eq Jl thle ,Istificiation of this procedure is based on the equation preceding (6.2).By, means of the asymptotic formlas (3.16) there then. follows for aR o I + 1 yk) I 1y , These are, however, precisely the first Taylor term of thet frictionless eigenvalue equation X1)=Oor X2(1) = O which would be obtained, according to the significance (equation (4.21a)) of .S1, S2J in thei case of Taylor expansion in the sense of the series (4.9). 24 NACA TM 1417 On the basis of this finding, the determination of the excited eigenvalues (Ime > 0) can be simplified. Since, according to the results of the second section, excited eigenvalues can1 appear only within the first eigen~values of finite number and for sufficiently large values of R, it suffices to examine equation (4.22) with respect to excited eigenvalues. For this, the following alternative is valid: Excited eigenvalues can be (a~pproxi~mately) d~etermined either by the friction less boundaryvalue problem in combination with sufficiently large values of R, or they lie in a neighborhood of c = U(1) and can be determined by means of one of the equations (4.22) or (4.23) as associated with finite values of ql' As follows from this for sufficiently large values of R, but as was confirmed in the examples for th~e smaller values of R also, the greatest excitation for inflectionpoint profiles is always combined with the frictionless eigenvalue or its continuation toward smaller R values. Hence, there follows the wellknown fact that, in the case of turningpoint profiles, the stability behavior mayr be concluded even from the frictionless differential equation alone. Let us compare to this the calculation of the frictionzless eigenvalues of' G. Rosenbrook (ref. 9) for an inflectionpoint profile which he had measured in a diver gent channel. 6. FORM OF THE EICEHFUN~CT~rION. THE IllJIER FRICTION LAYER. THE 'VAR~IATION OF A DISTURiBAllCE WITH TIME. In order to Judge the variation, with. time of a disturbance, we shall 3ecomposeju the latter into partial waves of the type in equation (1.1). It is then. necessary to know the variation of the amplitude cF(y) over the channel ~widthi. We consider here ornly the case of very large values of a~R. For the Couette flowr, there follows, by~ equation (5.4), in the notation of equations (3.6), (3.9)t and (3.15) for E << 1, that is, aR >> 1, as approximaste expression for ~the eigenfunction iW(y) = F(7) F qe (6.1) F(7) = (6.2) NACA TM 14c17 The boundary conditions 9(p(1) = O are identically satisfied; the remuaining boundary conditions are identical with the eigenivalue equa tion. (3.24). As follows even from the differential equation alone, ?*~( y) is an eigenfunction associated with the eigenvalue c0. We calculated accordingly for a = 1 and R = 103 the eigenfunction associated with the eigenvalue e = 0.7 iO.3 and represented it in figure 9. It is striking in this figure that the essential changes of the eignfncionocurin a layr 1 y yO which could be defined perhaps by the angle space arg nr = x6 of the strong increase of A1(?). In the variable y this "inner friction layer" is, according to equations (3.6) and (3.26), approximately 1 y 61+Ee]+ 1 <165 This representation shows that the width of the lawyer increases with grow ing order n. of the eigerafuLnctions; the magnitude~ of damping increasing simultaneously. The velocity of the associated disturbance wae is approximately equal to the velocity of the basic flow in the center of the layer. Furthermore, the thickness of the layer tends with a towad zero. The physicaEl interpretation of this situation signifies according to Hopf (ref. 5, p. 57) "that any arbitrary disturbance for lartIe v.aluesJ of R is damped in such a manner that, finally, distances seem; to manate only from the walls, without mutual interference a behavior which reminds one of frictionless fluids." For more general basic flows, Tollkien (ref. 12) set up an approxi F~ate expression for the eigenfunction; in it, one~ can recognize again, in the case of damping, an "inner friction layer" which would have to be defined by the angle space %/6 I arg rl 6 5Jrl6 of the great changes in increase of B1(q) or A().In the variable ;y, this layer is, accolrdinlg to equations (4.16), or~ +\ Soi r Sci 1 + =y =1 +  E >0 (6.9) U'(1) U'(1) whence we obtain. for the higher eigenvalues, according to equation (4.23), again the formula (6.3) for the C~ouette flow. If, however, frictionless dam~ped e~igenva~lues e in the sense of section 5 exist, the inner fric tion lawyer, expressions. (6.9), retains also for the limting process S a finite thickness and a finite distance from the wall. We calcu lated, for this latter case, the even eigeafunction in the example of the Poilseuiille flow, for a = 1 anld R = 7.7 x 105, and represented it 26 NACA TM 1417 in. figue 10. The eigenvalue c = 0.1178 i x 0.049 h~ardlyi deviates from the frictionless eigenvalue associated with a = 1. Comparing the inner friction layer with the boundary lawyer, we muay say that the boundary layer represents that flow region in which the behavior of the lamninar basic flow is decisively influenced b; the inner friction, whereas the inner friction layer indicates the region where the disturbance is decisively subject to the influence of the f'rictionr, since outside this layer the disturbance can be determined without friction. Translated by Mary L. Mahler National Advrisory Committee for Aeronautics NACA TM 1417 REFERENCES 1., Aiken, H. H.: Tables of the Modified Hankel Functions of Order One Third and of Their Derivatives. Harvard University Press, Candbridge, rlass., 19345. 2. Doetsc~h, G.: Thleorie und Anwendung der La~placeTransformation . (Berlin), 19537. 3. Haut, 0.: Uber die E~ntwicklung einer wilkiirlichen Funkrtion nach den Eienfun~kt~ionen des Turbulenzprojblem~s Sitzber. d. M~inchener .Akad., Mathem. phys. K1. 1912. 4. Holstein, H.: Uber die aii~ere und innere Reibungschicht bei Stijrungen lamninarer Strijmungen. Z.1 angev. Mah. Mech. 1950, pp. 2=149. 5. Hopf, L.: D~er Verlauf kleiner Schwringnen auf einer Stridmung reiblender Fltasigk~eit. Ann. d. Phys. 1914, pp. 160. 6. l!orawetz,J G. S.: Thne Eigenvalues of Somet Stability Problems In~volvingt Viscosity. J~ourn. of Ration. M~echan. and Analysis. Vol. 1, 1952, PP. 579603. 7. Wdoeth~er, F.: Zur .Asymptotischen Behandlung; der statiolEiren Lb'sungen im Turbulenloproblem.m Z. angew. Math. Mech., 1926, pp. 232243. 8. Losrd Rayleigh: Scient. Papsers III, pp. 575584. 9. Rocsenbrrook, G.: Instabilitait der Gleitschicht imn schwach divergenrten Kanal. Z. ang~ew. MYath. Miiech., 1937, pp. 824. 10. olkenW.:Uber die Entstehung de~r Turbulenz. Nachr. d. Ges. d. Wissensch., Gottingen 1929. 11. TolLnien, WJ.: Ein allgemeines Kriteriumn der lInstatbilitait lampina~rer Ge schwindigkeitsverteilunen. Nachr. d. Ges. d. Wissensch. Cittingen 12. Toltr.ie~n, W.: Asymtotische Integration. der Storungsdifferential gleichunzg ebener laninarer Stromungen bei hohen Reynoldssehen Zahlen. Z. angew. Miath. Mech. 1987, pp. 3750 and 7083. 28 NACA TM 1417 15. Wasow, W.: The Complex Asymptotic Theo~crie of a Fourth O~rder Differential Equation of H~ydrodynamics. Ann. of flathi. 49,J 1968, pp. 852871. NACA TM 11417 A Figure 1. Path of intgratIon A in the complex zplane. Figure 2. Riectillnear Cojuette flow. The twelve lowest eigen values c as functions of R for a = 1. NACA TM 1417 Fig~iure 3. The regions I, II, and III in the complex. yplane. Figure 4. Path of integration B in the complex zplane cut open along (0, ioo. NACA TM 1417r 0.4 t O 3 0.2 Ci O 4 105 105 106 R 0. 1 Umb R v Figure 5. Twodimensional Poiseuille flow. The four lowest eigen vaues c as functions of RZ for as = 0.87. 32 NACA TMI 141[ Cr 2b ~Um 0.2~ Umb tUs O 105 106 R 0. 1 Figure 6. Inflectionpointi profile. U=(r 1) + i2 !cos yr .T The four lowest eigenla.1ues c as fu~nctions of Rt for o = 0.5. NACA TM 1417 r= 0.5 0.4 Cr 0.1  0.2 ` Ci Figure 7. Two~3d~ilen pension P~oiseuille flowv. The frictionless eigen Ialuez c associated wit~h an even~ ilna Mucti:n, as a function CI 0 2 0 4 0.6 Cr 0.05 a 0. 1 U = J 1) + (2 \J)cos y ~. Figure 8. Inflectionpoint profile. The frictionless eigenvalue c associated wit an even eigen functio~n, as a fun~ct~ion of a. NACA TM 1417 Figure 9. Redziinea~r Couette~ Gowl. El aen fun~et in at = 1 and R= 10o associated wiithi the eigen;value c = 0.70 ix 0.30. Y'i) 10 01 01 P '(yr) for 2 y Figure 10:. Tw~odimensioinal Poiseuille flow. Eigenlunction Ip'(y) for a = 1 and R = 7i.7 x 10" associated with the eigenvalue c = 01.178 i x 0.049. UNIVSITY OF FlOIDAl 
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