wfta^M-/v/7 ^ < NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1417 ON THE SPECTRUM OF NATURAL OSCILLATIONS OF TWO-DIMENSIONAL LAMINAR FLOWS By D. Grohne Translation of "Uber das Spektrum bei Eigenschwingungen ebener Laminarstromungen. " Zeitschrift fur angewandte Mathematik und Mechanik, vol. 34, no. 8-9, August-September 1954. UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY P.O. BOX 117011 GAINESVILLE.FL 32611-7011 USA Washington December 1957 I ^ \p 1 ^ ' ^ ^ ' NATIONAL ADVISORY COI^ITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM U+l? ON THE SPECTRUM OF NATURAL OSCILLATIONS OF TWO-DIMENSIONAL LAMINAR FLOWS* Ely D. Grohne 1. INTRODUCTION AND STATEMENT OF THE PROBLEM In the investigation of stability of a two-dimensional laminar flow with respect to small disturbances, we describe a disturbance of the stream function moving downstream (in the direction of the x-axis) by the "partial wave formula" ^ = cp(y)ei^(^-^t) (1.1) and obtain then for the distribution of the disturbance amplitude 9(y) at right angles to the main flow the so-called stability differential equation of the fourth order (U - c)(cp" - a^cp). - U"cp = — (q)^^'' - 2aV' + a%) (l.2) where U(y) designates the velocity profile of the basic laminar flow In addition, we enforce certain boundary conditions, in the specific case of the parallel channel cp(±l) = cp'(±l) = (1.3) It** ** n * Uber das Spektrum bei Eigenschwingungen ebener Laminarstromungen. Zeitschrift fur angewandte Mathematik und Mechanik, vol. jU, no. 8-9, August -September 195^, pp. 5^^-357. NACA TM ikYJ which express the fact that even the distiirbed flow adheres to the bo\inding walls. In these equations, the velocities U and c are referred to a velocity of reference Uqj f \irthermore , the lengths x, y, and l/a to half the channel width b, and finally the time t to the time unit b/UQ. The Reynolds number R is defined by Uob The bo\indary -value problem consisting of differential equation and boundary conditions determines, for each pair of parameters a and R, a spectrum of an infinite niimber of eigenvalues c^^. The associated disturbances (l.l) are dajnped when Im(cj^) < 0, ajid are excited when Im(cj^) > 0; a is assumed to be positive and real. A basic flow is called stable for a value of R when the entire eigenvalue spectrum Cjj, for all possible values of a, contains only damped disturbances. Thus the range of the Reynolds number R is divided up into a region of stability < R < R* and a region of instability R > R*, which are separated from one another by the stability boundary R*. Since, in the literature published up till now almost exclusively neutral oscillations - at most, excited oscillations - have been investi- gated, we shall investigate in the present report, following a suggestion of Prof. Dr. W. Tollmien, the entire spectrum of the eigenvalues c^ as a function of a and R; for simplification, we shall emphasize the dependence on oR. A general solution of this problem is possible in the following two special cases; (l) in the case U = const, which is equivalent to U = 0. We deal here with the "oscillations of a fluid at rest" already treated by Lord Rayleigh. The solution is possible in the domain of elementary and transcendental functions. The second special case concerns the rectilinear Couette flow U = y investigated by L. Hopf (ref. j) . The solution can be reduced to tabulated Bessel functions. For more general velocity profiles U(y), the eigenvalues c^ can be determined apprcximatively analytically in the following limiting cases: I. In the limiting case oR -^ for arbitrary order n of the eigenvalues Cr-^ II. In the limiting case n -♦ » for constant oR III. In the limiting case oR -> oi- for restricted order n. NACA TM 1417 5 A continuous trajisformation of the three cases into one another for constant subscripts is possible in the above named special cases U = and U = y. The assignment of subscripts of the eigenvalues Cj^ can be made in the cases I and II according to increasing damping, that is, according to the rule Im(c^^J < Im(c^) (l.M However, this rule is not always applicable to the case III when the sub- scripts used are to remain constant for continuous variation of a and R. The bo\indary-value problem formulated in (1.2), (1.3) is, generally, not self-adjoint; thus, the reduction to the well-known statements and estimates of the Sturm-Liouville theory is eliminated. The eigenfunc- tions generally do not form an orthogonal system. They do form, however, as 0. Haupt (ref . 5) has shown, under certain assumptions, a system of functions that is complete with respect to each of the functions which satisfy the boundary conditions (1.3) and are four times continuously differentiable. This system of functions can be transformed into an orthogonal system. 2. THE LIMITING CASES oE -^ FOR ARBITRARY ORDER n OF THE EIGENVALUES c^, AND n -> « FOR LIMITED oR As already found by Lord Rayleigh (ref. 8), the entire system of eigenf unctions and eigenvalues in the case of the basic flow U = 0, that is, for a medium at rest, can be given as a closed system. Since these eigenvalues are suitable for approximative representations in the case of more general basic flows also, we shall derive them here briefly. In the case of the basic flow U = 0, the stability differential equation (1.2) is simplified to I' (^^ - 2a%" + ah + ioR • c(i" - a^o) = (2.1) where we shall denote the eigenvalues by C, to distinguish them from the eigenvalues c of the general stability differential equation. The equation is solved by each of the functions ^ I \ , , cosh ca ^ r \ . - . , sinh oj ^Tw ) - cosh a^y - cosh ay — ^jjly) = sinn coy - sinh by the equation (2.1). We shall now express this train of thought more accurately by subjecting the difference c - C of the eigenvalues c to a more accurate estimate compared to the eigenvalue C of equa- tion (2.1), for more general profiles. With introduction of the differ- ential operators l[cp] = u(cp" - a^L [$Jdy n+1 $L Cp - ij) dy ,+1 $M| I dy ,+1 (2.11) $M $ dy 5 NACA TM 1417 from equation (2.9) the representation c - C = Q + q. (2.12) In this equation, C and Q may be regarded as known by virtue of the functions $ represented in equations (2.2). The eigenvalues C^^ have already been delimited in expression (2.6), For Q we obtain directly the estimate Q = | J U • dy + O^i^ |a)|»l (2.13) In connection with a simultaneous estimate of the function (cp - $)" - a^(cp - 0), we obtain for q q = 0(2^) for S£ _o (2.14) If we substitute both into equation (2.12), we obtain, with consideration of equation (2.5) and expression (2.6), the two partial statements for qlR ^ and for arbitrary order n (2.15a) U • dy + 0' — 1 for n -; °o for fixed oH \n J (2.15b) The latter estimate indicates that the eigenvalues c^ of the stability differential equation for sufficiently high order n finally tend toward the eigenvalues C^, of the "zero flow" (with the real paort increased by the mean velocity of the basic flow). (Compare eq. (2.1).) A mutual coordination of the eigenvalues Cj^ to the eigenvalues C^, however, is by virtue of equation (2.15b), meaningful only due to the fact that the difference I C^^ - C^^-^ I of the approximation eigenvalues comes out considerably larger than the estimated remainder in equation (2.15b): For, because of (2.5) and (2.6) NACA TM 1417 '-'n " '-'n+1 ^ constant • f-^j (2.l6) 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 eigenvalues for unlimitedly increasing order, altnough only intimating an argument - which leads one to expect con- siderable difficulties. We mention, furthermore, an estimate for the eigenvalues c indi- cated by C. S. Morawetz (ref. 6, p. 580) - Cj^l < A • (aR)"V2 where c^^ is an approximative eigenvalue which ( in our notation) is determined by the equation Im r^ ^i(u - cj . dy = ^ • 1 ajid corresponds more or less to our approximate eigenvalue C^^ intro- duced in equation (2.1). In the above estimate of Morawetz, neither oE nor n may become arbitrarily large; in the first case, the eigenvalues would shift into the excluded neighborhood of c = w/y ^ /w = U; y, des- ignates the wall); in the other case the estimate would become meaning- leas since the behavior of the quantity A for iinliraitedly increasing n is not given. 5. RECTILIKEAR CCUETTE FLOW. THE LII4ITING CASE clR -^ » FOR FINITE ORDER n In the special case of the basic flow at rest, U = 0, the behavior of the eigenvalues c^^ for unlimitedly increasing oR is described by the formula (2.5) in which the quantities cd^ no longer depend on oR. In deviation from this law, there results for more general velocity profiles a behavior like 3 NACA TM 1417 r 2^ - -Ui-l) = — — for oR -4 00 (5-1) where the complex valued quantities P no longer depend on oR. If these eigenvalues are adjoined to the eigenvalues in equation (2.15a), by continuous transition of a and aR, the ordering principle (1.4) according to increasing damping is lost even in special cases like the rectilinear Couette flow. If we therefore desire that the subscripts of the eigenvalues c^^ remain unchanged for continuous transformation of the limiting cases olR -^ and aR -* ^ into one another, we must actually carry out this procedure which presupposes a general solution of the eigenvalue problem or a solution which is approximate 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 insertion of the velocity profile U = y of the rectilinear Couette flow into the stability differential equation (1.2), the latter can be reduced, by means of the substitution i|; = cp" - CL^cp (3.2) to the Bessel differential equation in the aioxiliary form r iaR(y - c) + a ^ = (3.3) In order to arrive, through the boundary conditions, at the eigenvalues, we must invert equation (3-2) in the form q>( y)= /%(n) ^'-^-[y-iK r, (3.4) The boundary conditions 9(-l) = 9'(-l) = then are identically satis- fied; the remaining boundary conditions 9(+l) = cp'(-(-l) = require that the two equations ,>+l g^j^ ^ p+1 / *^y^ ■ ""^ a °"^ ^y = ^ / ^l'(y)cosh ay dy = (3-5) hold. NACA TM lii-lT 9 By means of the substitution y - y, = 6T1 with e = (oR)" '^ and y = c + i^ (5-6) '^ ^ aR the differential equation (j.j) may be transformed into the differential equation i^+ n^y = (5.7) If ^^{^]) and ^jj(ri) (5.8) are assumed to be two suitable fiindamental solutions of this equation, and r\ -, , n -, is assumed to designate the values which are, because of equations (3-6), associated with the walls y = -1 and y = +1 there follows from equations (5-5) the eigenvalue equation r "^^^ sinh ae F "^^ Jri_^ *I^^) ^^^T^ '^T • Jn_^ ^^Il(n)cosh aeil drj - r "^"-^ sinh ae T / \|f^j(n)^^ — -2-3dTi • / ^^(Ti)cosh aeii dri = (3.10) For further treatment of this equation, the introduction of a sequence of functions k^{ \) by the Laplace integral 2 qz+i ^ ^ . z^-1 . dz (3.11) A j_Q NACA TM li+l? is advisable in which the path of integration A rions from infinity to infinity in the manner drawn in figure 1. The functions Aq(t]) satisfy the differential formula and the recursion formula i • ^1+5 + H • ^+1 + n • A^ = (3.15) by r.eans of which all the functions Aj^(il) and their integrals and derivatives may be constructed recursively from the three basic fiinctions Aq(ti) Ai(n) AgCri) (3.14) The significance of these functions A^(t]) for the stability prob- lem lies in the fact that the two particular solutions of the differential equation (5-7) needed in the eigenvalue equation (3.10) can be repre- sented in the form 2iri / 2jti\ ^^{t^) = A^(n) ij/jj(ti) = e 5 . A^L • e 5 1 (3.15) That the differential equation (5-7) is satisfied follows from the formulas (3-12) and (3.13). The linear independence of the two functions follows from the fact that the Wronski detenninant, which is constant of course, does not disappear at the point t] = 0. The basic functions A3_(ri) and A2(t)) = A]_'(t)) have been numeri- cally tabulated (in somewhat different notation) for a quadratic point grid with the mesh width 0.1 within the circle I t^ I = 6 of the complex ■q-planc by H. H. Aiken (ref. 1). The basic function AQ{r\) = / A-i_(Ti)dTi can be determined from it by a numerical integration. Outside of this table, the behavior of the functions A^(t) may be inferred from the asymptotic series representation v=0 ' ' NACA TM 1417 11 which is valid for I t] | -> °° in the angle space - -LS + 6 = arg r| = •Sil - 6 ' ' 6 6 with arbitrarily small S > 0. According to H. Holstein (ref . ^), the first coefficient of the series is a^0.-^e"'A"^^8^ (5.16a) The asymptotic series are obtainable directly from the Laplace integral (5.11) by means of Riemann's saddle-point method. For the representation of the eigenvalues, the zeros t\ of the function A^(ti) are necessary. An asymptotic calculation of these zeros for I T]-, I » 1 is not possible directly by means of expression (5.l6), since the zeros would move out of the range of validity of this repre- sentation. We avoid this difficulty by applying the second relations obtainable from the integral (3. 11) / _i rt\ lis / _ m\ A^\rie ^j= e 5 . A^l^.,* . e ^) (5.I7) (* = conjugate-complex values) and i 22m / i 2zr\ . W / i itZl\ An(-l) + e 5 A^l^'ie ^J+e ^ A^\^e ^ j ^ F^{r^) (5.I8) where the polynomials Pn^^) °^ ^^^ degree -n satisfy the same recur- s ion formula ( 3 • 13 ) i • Pn+3 + fn+1 + "Pn = (5-19) with the initial elements Pq = 1 P-L = Pg = Combination of formulas (3-16) and (3.l8) then yields the asymptotic representation valid for I z [ -> 00 in the angle space I arg z 1 < jt - 6 12 NACA TM Ii(-1T 2 3/2 Ti cos — z^ + — \| jr :w^ (3.20) A ■o(%) which, according Hence, there follows, for the zeros tl^ of to equation {J).YJ), lie in pairs symmetrically with respect to the straight line arg = 5it/6, the asymptotic representation (5rtn) 2/3 ^jr^ ln(n-2rt\f3) _6 5«n n = 1, 2, 3 (3.21) The value of the lowest pair of zeros was calculated to he 4.257 e ■- ^ ± . 2708 6 (3.21a) according to the table of Aiken (ref. l). For further treatment of the eigenvalue equation (3. 10), it is advisable to expand the functions sinh aer] and cosh aerj into their Taylor series, and then to interchange the summation 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 ae. If these series are broken off after the first terms, provided with residual terms, and substituted into the eigenvalue equation (3-10), the latter is, for this reason, and with consideration of equation (3-17)^ simplified to *oK) - *o(i-i' *■ H\) - '-Axi) H) - '-1 A«{-lj) A- (-to = o(aV) for I T I = constant (3.24) What happens now when oE increases beyond all limits, that is, when € tends toward 0? Because of the relationship following from equation (3.9)> T] - T) = 2/€, at least one of the two quantities li ^ must tend toward infinity for e 0, on a parallel to the real axis. It is sufficient to assume this regardine T\ , because in the other case everything would form a mirror image with respect to the imaginary axis of the rj-plane (as essentially occurs in the Couette flow NACA TM IhYJ 13 where with C, also, -C* is an eigenvalue). With consideration of the asymptotic behavior (eq. (3.l6)), the eigenvalue equation (5.24) then is simplified to i Mllil _ ^ = ofe^) for €«l (5.25) From this formula, we recognize that r] for e ^ must tend toward the zeros t^ of the function A (t]) estimated in expres- sions (5-21). We thus obtain for the eigenvalues c, with considera- tion of equations (5'9)j the asymptotic representation C + 1 = -€TL. + \ .(e2) (5.26) with T^j from A^^t^) = Thus we have proved the previously given eigenvalue formula (5-1) for the special case of the Couette flow. In order to follow the vaxiation of the eigenvalues c over the entire range ^ oR < ^, we nust go back to the eigenvalue equa- tion (5-10) or its approximate form in equation (5-24), with the func- tions A-,(t]), Ai(t), Ap( t) to be assumed as known. We have accord- ingly calculated the 12 lowest eigenvalues c as functions of oR for a fixed value a = 1 and represented them in figure 2. The variability of the eigenvalue curves with a is only slight and becomes, for instance, for oE -> >=" with e. small of the order O'^a^e ). The eigenvalue spectrum of the rectilinear Couette flow has been discussed already by L. Hopf (ref. 5). Hopf replaced, more or less on the level of our eigenvalue equation (5. 10), the solutions ^y , \(( represented by him by Hankel functions of the order 1/5 - by the first terms of their asymptotic series (5-16), whereby the eigenvalue equa- tion was simplified to an algebraic equation of auxiliary arguments and circular and hyperbolic 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 oR, the values of, for instance, n in a formula corresponding to (5-26) undergo a considerable change. The topological connection of the eigenvalue curves c = c(a,aR) also appears different to us from, what it appeared to Hopf. However, the 11^ NACA TM li+17 qualitative picture of the eigenfunctions, the physical conclusions drawn from it, and the main result - that all oscillations are damped remain the same. h. THE LIMITING CASE oR -♦ oo FOR FINITE ORDER n FOR SYMMETRICAL BASIC FLOWS For a basic flow with symmetrical velocity profile U(y) = U(-y) (4.1) the stability differential equation (1.2) always has a fundamental system of four solutions $ , $_, 9 , $. so that and ^oiy) $j,(y) aJ^s even functions of y •? (y) 9 (y) are odd functions of y (1..2) If a linear combination of these solutions is to satisfy the boundary conditions of equations (1.3) in the sequence 9(-l) =0, cp'(-l) = 0, cp(+l) = 0, cp'(+l) = 0, the following determinant, simplified with consideration of the symmetries {k.2), must disappear: D = 9i(-l) $5(-l) $2(-l) $i,(-l) ?{(-!) %i-l) $^(-1) 9;(-l) \{-l) -93(-l) %(.-!) \i-l) 9[(-l) 9^(-l) -9;(-i) -?j;(-i) = Since this determinant may be written as the product of the two-column deterrrdnants D 4 $i(-l) 9|(-1) ^jC-i) q;(-i) 3 9,'(-1) = (4.3) NACA TM Ihl-J 15 one obtains, by equating one of the two factors to zero, one branch of the eigenvalue equation each time. For this reason the eigenfunctions can be either only even or only odd, with the respective eigenvalues c generally being different. In order to ajrrive from these equations at asymptotic eigenvalue formulas, we shall determine the four fundamental solutions (^.2) $...$, in such a manner that they are available for appropriate asymptotic expansions. We find that the fundamental solutions described by W. Tollmien (ref. 12), "Asymptotic Integration of the Stability Differ- ential Equation", the asymptotic representations of which are provided with residual-term 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 approximation." The quest for solutions of the complete stability differential equation (1.2) which for clR -^ 0°, together with their derivatives with respect to y, tend toward a limiting function lim cp(y,aR) = x(y) aE-»<» (^.M leads to the so-called "frictionless differential equation" (U - c)(x" - a^x) - U"X = (^•5) which must necessarily be satisfied by such limiting functions. If we want to use the solutions of this frictionless differential equation for the approximation of the solutions of the complete differential equation for clR -.■ =", we must not disregard the range of validity of the boundary-value statements in equation (^.^) in the complex y-plane. According to W. Wasow (ref. 13), the following theorem is valid with respect to this: "Of the four fundamental solutions (^.2), one even and one odd solution can be determined in each case so that with two appropriate frictionless solutions X-]^(y) and X2(y) the approximations .ofi^ ^^(y) = ^i(y) + 0[^j \{y) = odd function of y ■■P^^y) = ^2^^'^ ^ ^&) ^2^^^ " ^^^^ function of y > (^.6) 16 NACA TM ikYJ in each fixed interior of a double region (l + II) or (ll + III) or (hi + l) are valid and become invalid, in each case, in the comple- mentary third region III or I or II. The same is true for the deriva- tives with respect to y." (Compare fig. 3-) The boundaries between the regions I, II, and III satisfy the equation Re U - c dy = if y denotes the "critical point" defined by ufy Re yv < (^.7) For more details regarding the regions I, II, and III see Wasow (ref. 15)' The frictionless differential equation (^.5) has at the critical point, U/y \ = c, a singular point with regard to determinateness . Two fundamental solutions take the form Xi(y) = (y - yj,) • Pi(y - y^ u" My) = p(y - Yk) + t# IM - yk) • ^y - y},)My - y^) (^-9) if P and P denote power series with the beginning P.(2) = 1 + -1 ' U" 2Uk ^ P (z) = 1 + o(z^ (^.9a) (Cor.pare W. Tollmien, ref. 12, p. 55-) The common radius of convergence of these power series is limited either by the radius of convergence of a corresponding series for U - c or by the next-adjacent zero of U - c as a singular point of the differential equation. NACA TM li+17 IT For the further development it is advisable to introduce a sequence of functions B (ri) by the Laplace integral which is comparable to equation (3-ll). In it, c = 0.5772 . . . denotes the Euler constant. The path of integration B runs, in the manner indicated in figure h, in the complex z-plane cut open along (0,-ioo) from infinity to infinity. The functions Bj^(t]) satisfy the differential formula (ill and the recursion formula = Bn+i(ri) (^.11) i . Bj,^5 + r, . B^+i + n . B^ = P^ (4.12) in which P {r\) are the polynomials defined in equation (3.19)» By means of these two formulas all the functions Bj^(ti) and their derivatives and integrals can be constructed recursively from the three basic functions Bq(ii) b^(t]) BgCn) (4. 13) By means of the representation B^(ti) = 2rti (n=l,2) (* = conjugate-complex value), the basic functions B-, and Bp can be reduced to the functions A^. (Compare W. Tollmien, ref. 10, p. 27.) The significance of the functions Bj^(r|) for our stability - eigen- value problem lies in the fact that the f\inction B-,(t]), because of equations (4.11) and (4.12), satisfies the differential equation 18 NACA TM 1417 i^. dT] d%. + 11 dT]^ (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 by W. Tollmien (refs. 10 and 12). After these preparations, we turn to the four fundamental solutions ^1 9 II' 9 III' 9 IV of the complete stability differential equation constructed by Tollmien, regarding its ability to be expanded asymp- totically. According to W. Tollmien (ref . 12, p. 77) these four solu- tions may be determined, with use of the substitution er] with e (oEU- -1/5 and y from U(y. = c (4.16) in such a manner that they have in a fixed interior of the -q-plane ( 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: 9j(y) = X^(y) + o(e5) (^.17a) q'jj(y) = Pg^eri) + ^ . P^(€Ti) . efB_^(n) +T1+ n In e] +o(e2lji e) or in every fixed interior of II cp (y) = X2(y) + o(e5) Furthermore , 'PlIl(y) = A_;j_( q) + 0(e) in U= constant ,(4. 17b) (^.17c) is valid. Finally, there applies, according to W. Wasow (ref. 15), quotient-asymptotically in every fixed interior of II (compare eqs. (4.6)) NACA TM 11+17 19 ^jlliy)) being valid, the result reads K ^i,2(y) --'i,2^iii(y) = i + e . -f Tjfln e+ l + S3_^2) + ^-lH) "'"'^V^^l'^ ^) with -^ S x^(o) XgCo) u' r^ S. (lj-.21a) from the frictionless solutions Xt and X2- (Compare eqs . (14-.9)) Furthermore , constant ?3^4(y) = 9jjj(y) + o^e^/i^ .3 € v/F j^ ^^^^ ^^ ^ ^^^^ is valid. Corresponding formulas are valid for the derivatives, (i+.21b) If we now write the two eigenvalue equations obtained in equa- tion (I+.3) as a product of three factors, for instance. = D = $3(-l) ^l(-l) - ^ ' 93(-l)] 9^(-l) ${(-1) - ^9^(-l) I 9 (-1) ' 9i(-l) - ^95(-l) J 20 NACA IM 11+17 (•3 = arbitrary constant), the zeros of the two first factors do not make a contribution to the eigenvalue configuration since they are compensated by corresponding poles of the third factor - unless the derivative cp'(-l) should disappear simultaneously. It is therefore sufficient to find only the zeros of the third factor. After insertion of the approximations (^.21) we thus obtain dcp III U" k (ri)] 9. Ill dq l+€. yf |ri(ln e+l+Si ,2)+B_i(ri]] + o(e2ln e b '-1 or AqCq) A_i( l) + ejf(ri) In e + 1 + S 1.2 + Bo(ti)] 1 + € T](ln e + 1 + S-j^^g) + P-l(n)| (e^ln e) + e^ln e (i^.22) with yj^)/€ and S^, Sp according to equations (4.21a) from the frictionless solutions. The function tC^) stemming from a next-higher approximation in equation (4.21b) reads ^(n)=^|^ A_p(a) +1 A.(ri) - ^ A,(i) 10 "4^ A_i(ti) (4.22a) and may be reduced, by means of the formulas (5. 12) and (3. 13), to the three tabulated basic functions A_, A,, A„. How do the eigenvalues c behave if in the eigenvalue equa- tion (4.22) we let oE ^ «, that is e -> 0? Evidently t] then tends toward the zeros t] of the function A (r^). By Taylor series expanded about these zeros, there follows more exactly ''-I "^N u; ^2(% '1 M e In e + 0(e) NACA TM 1417 2^ The eigenvalues then behave asymptotically like c-U(-l)=-Ui.c,„-|.i^Anc.0(.2) (^.2,) with n from A fn] = 0. As a supplement to equations (3.21) we shall give here a few zeros tl^ and values iA^/A-^: -i|.122 + i . 1.065 -1.686 - i . 1.222 For In I » 1 -2.983 + i . 3.037 +1.902 + i . 0.851 ^^ ' ' . -6.8 + i . 2.5 -2.2 - i . 1.5 ^^^^^ applies -5.5 + i . 4.5 +2.4 + i . I.l4 . ^2{\) _ \\ (4.24) The remarkable fact about the asymptotic eigenvalue formula (4.23) is that it is transformed into the corresponding formula (3.26) after substitution of the velocity profile U = y of the Couette flow, although the two formulas were derived under completely different assumptions. The asymptotic eigenvalue 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 must go back to the more exact formula (4.22) in which the character- istics "even" or "odd" of the eigenf unctions are taken into consider- ation by means of the constants S-, and Sp, to be determined "without friction." We have used the eigenvalue equation in the form (4.22) also for the numerical calculation of the eigenvalues c in the examples treated. We selected as examples the two-dimensional Poiseuille flow and a flow with an inflection-point profile. We represented the variation of the four lowest eigenvalues as functions of R for a fixed value of a in figures 5 and 6. The numerical calculation itself is - after reduc- tion of the nonalgebraic elements contained in the eigenvalue equation to the three tabulated basic functions A^, A,, Ap and to the fric- ticnless solutions - a problem involving numerical methods, the details of which cannot be discussed here. We shall mention only the following approxiiTiate representation of the frictionless constant So 22 NACA TM li+l? "k 2 . ^ N^ — . So = A . a""^ + 0(1) for a « 1 with A = — ^ u; ^ r (U - c)^dy (^.25) (Compare W. Tolljiiien, ref. 11, p. 100 ), which may be applied advanta- geously for small values of a. The subscripts for the eigenvalues c obtained from equation (4.22), in the sense of a continuous connection with the limiting case oR ^ 0, 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/'c -,') ^ Im(Cj^'), that is, according to increasing damping, cannot in principle be carried out, either. The zeros in equation (4.23) can be ordered according to the increasing imaginary part, but the Im(Cj^^ curves may penetrate one another if € is changed. 5 . THE FRICTIONLESS EIGENVALUES WITHIN THE LIMITING CASE oE -^ °o Determination of the Excited Eigenvalues Let the approximation (4.6) by means of the frictionless 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 always uniquely determined in the frictionless solutions. Applying the approximations (4.17), we then obtain, by way of the eigenvalue equations (4.3), eigen- values c which, for clR -^ <», tend toward the so-called "frictionless eigen values" c^ '(a) which are defined by the boundary condition X-|(-l) =0 or X2(-l) =0 of the odd or even frictionless solutions X-i, X2- The following general statements may be made regarding these frictionless eigenvalues, limited by the range of validity of the boundary-value expressions (4.6), according to W. Tollmien (ref. 11), partly on the basis of the "Rayleigh-Tollmien theorems:" "For velocity profiles without turning points, no excited friction- less eigenvalues are possible. The approximation (4.6), associated with the dajnped frictionless eigenvalues, must always take place in the interior of the double region I + II." "For inflection-point profiles, there always exist excited friction- less eigenvalues associated with an even eigenfunction. " NACA IM li+17 25 Beyond these general statements, frictionless eigenvalues associ- ated with an odd eigenfunction were not found in any of the examples; neither did we find eigenvalues such that the associated approxima- tion (^.6) would have taken place in the interior of the double region II + III. As examples, we chose the two-dimensional Poiseuille flow as representative of a profile without an inflection point, and the inflection- point profile U = N2 - ij + f 2 - vfi" j x cos ^ y. The frictionless eigenvalues c, found only associated with an even eigenf unction, are represented in figures 7 and 8. The range of existence of these eigen- values is always given by an interval = a = constant. For the fric- tionless eigenvalue c, Tollmien (ref. 11, p. lOO) has set up the following approximate formulas : ^ = '^N=— 3 — ^i = =i(^r)==r • — ^ ""^ U(yj^)=S (U - c^)-dy ("kI (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, cannot give an answer to this problem because the frictionless eigenvalues in question do not exist there at all. How- ever, it is possible to insert the frictionless eigenvalues into the equation (4.22) and thus to interpret them as a limiting case within the eigenvalues (eq. {k-.23)). Let us, therefore, perform on the eigenvalue equation (if. 22) the limiting process oR -> 00, that is, e -4 for constant ( -1 - yi.) = ^ri ; the Justification of this procedure is based on the equation preceding (4.22). By means of the asymptotic formulas (3.16) there then follows for clR -^ '^ u;: = i + (-i - yk) ^\M-^ - yk) + Si^2J = These are, however, precisely the first Taylor terms of the frictionless eigenvalue equation X-|_(-l) = or X2(-l) = which would be obtained, according to the significance (equation (4.21a)) of S-^, 82^ in the case of Taylor expansion in the sense of the series (4.9). 24 NACA TM Il+IT On the basis of this finding, the determination of the excited eigenvalues ( Imc > O) can be simplified. Since, according to the results of the second section, excited eigenvalues can appear only within the first eigenvalues of finite number and for sufficiently large values of R, it suffices to examine equation (^.22) with respect to excited eigenvalues. For this, the following alternative is valid: Excited eigenvalues can be (approximately) determined either by the friction- less boundary-value problem in combination with sufficiently large values of R, or they lie in a neighborhood of c = U(-l) and can be determined by means of one of the equations (4.22) or (U.25) as associated with finite values of q . As follows from this for sufficiently large values of R, but as was confirmed in the examples for the smaller values of R also, the greatest excitation for inflection-point profiles is alvrays combined with the frictionless eigenvalue or its continuation toward smaller R values. Hence, there follows the well-known fact that, in the case of turning-point profiles, the stability behavior may be concluded even from the frictionless differential equation alone. Let us compare to this the calculation of the frictionless eigenvalues of G. Rosenbrook (ref. 9) for an inflection-point profile which he had measured in a diver- gent channel . 6. FORM OF THE EIGENFUNCTION. THE INKER FRICTION LAYER. THE VARIATION OF A DISTURBANCE WITH TIME. In order to judge the variation with time of a disturbance, we shall decompose the latter into partial waves of the type in equation (l.l). It is then necessary to know the variation of the amplitude 9(y) over the channel width. We consider here only the case of very large values of coR. For the Couette flow, there follows, by equation (j.^), in the notation of equations (5-6), {3-9), and (5.15) for e « 1, that is, aE » 1, as approximate expression for the eigenf unction 2rti\ with ^(y) = F('i) - F\^]e 5 J (6.1) A .(^) - A (ti ; NACA TM li+lT 25 The boundary conditions *(-y) is an eigenfunction associated with the eigenvalue -c*. We calculated accordingly for a = 1 and R = 10-^ the eigenfunction associated with the eigenvalue c = -O.7 - iO.5 and represented it in figure 9- It is striking in this figure that the essential changes of the eigenfunction occiir in a layer -1 = y = y which could be defined perhaps by the angle space arg t] = jt / 6 of the strong increase of A -.(t]). In the variable y this "inner friction layer" is, according to equations (5'6) and (5.26), approximately -i^y^ 1 + e f-Re rij^ + J5 Im Ti J €«1 (6.5) This representation shows that the width of the layer increases with growing order n of the eigenfunctions; the magnitude of damping increasing simultaneously. The velocity of the associated disturbance wave is approximately equal to the velocity of the basic flow in the center of the layer. Fiirthermore , the thickness of the layer tends with e toward zero. The physical interpretation of this situation signifies according to Hopf (ref. 5^ P- 57) "that axiy arbitrary disturbance for large values of R is daiaped in such a manner that, finally, disturbances seem to emanate only from the walls, without mutual interference - a behavior which reminds one of frictionless fluids." For more general basic flows, Tollmien (ref. 12) set up an approxi- mate 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 jt/6 '^ arg r) % 5^/6 of the great changes in increase of B_-|_( q) or A_2_(ri). In the variable y, this layer is, according to equations {k.l6), U'(-l) U'(-l) ^ ^ whence we obtain for the higher eigenvalues, according to equation (4.25), again the formula (6.3) for the Couette flow. If, however, frictionless dcunped eigenvalues c in the sense of section 5 exist, the inner fric- tion layer, expression (6.9), retains also for the limiting process e ->0 a finite thickness and a finite distance from the wall. We calcu- lated, for this latter case, the even eigenfunction in the example of the Poiseuille flow, for a = 1 and R = 7»7 x 10-^, and represented it 26 NACA TM li^lT in, figure 10. The eigenvalue c = O.I78 - i x 0.0^9 hardly deviates from the frictionless eigenvalue associated with a = 1. Comparing the inner friction layer with the boundary layer, we may say that the "boundary layer represents that flow region in which the behavior of the laminar basic flow is decisively influenced by the inner friction, whereas the inner friction layer indicates the region where the disturbance is decisively subject to the influence of the friction, since outside this layer the disturbance can be determined without friction. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM li+lT 27 REFERENCES 1. Aiken, H. H. : Tables of the Modified Hankel Functions of Order One- Third and of Their Derivatives. Harvard University Press, Cambridge, Mass., 19^5. 2. Doetsch, G. : Theorie und Anwend\ing der Laplace-Transformation. (Berlin), 1957- 3. Haupt, 0.: Uber die Entwicklimg einer wilkiirlichen Funktion nach den Eigenfunktionen des Turbulenz -problems. Sitzber. d. Munchener Akad., Mathem. phys. Kl. 1912. h. Holstein, H. : Uber die aiiBere und innere Reib\angschicht bei Stbrungen laminarer Strbmujigen. Z. angew. Math. Mech. 1950, pp. 25-i+9. 5. Hopf, L. : Der Verlauf kleiner Schwingungen auf einer Strbmung reibender Flussigkeit. Ann. d. Phys. 191^^ pp. I-60. 6. Morawetz, G. S.: The Eigenvalues of Some Stability Problems Involving Viscosity. Journ. of Ration. Mechaxi. and Analysis. Vol. 1, 1952, pp. 579-605. 7. Noether, F.: Zur Asymptotischen Behandlung der stationaren Lbsungen im Turbulenzproblem. Z. angew. Math. Mech., I926, pp. 232-2^3' 8. Lord Rayleigh: Scient. Papers III, pp. 575-584. 9. Rosenbrook, G.: Instabilitat der Gleitschicht im schwach divergenten Kanal. Z. angew. Math. Mech., 1937^ PP- 8-24. 10. Tollmien, W. : Uber die Entstehung der Turbulenz. Nachr. d. Ges. d. Wissensch., Gbttingen 1929- 11. Tollnien, W. : Ein allgemeines Kriterium der Instabilitat laminarer Geschwindigkeitsverteilungen. Nachr. d. Ges. d. Wissensch. Gbttingen 1935. 12. TolLr.ien, W. : Asyrnptotische Integration der Storungsdifferential- gleichung ebener laminarer Strbmungen bei hohen Reynoldsschen Zahlen. Z. angew. Math. Mech. 19^7, PP- 57-50 and 7O-83. 28 NACA TM 1417 13. Wasow, W. : The Complex Asymptotic Theorie of a Fourth Order Differential Equation of Hydrodynamics. Ann. of Math, kg, 19^+8, pp. 852-871. NACA TM IUI7 29 Figure 1. - Path of integration A in the complex z-plane. Figure 2. - Rectilinear Couette flow. The twelve lowest eigen- values c as functions of R for a = 1. 50 NACA TM ikYJ Figure 3. - The regions I, n, and IE in the complex y-plane. 0^ B Figure 4. - Path of integration B in the complex z-plane cut open along (0, -ioo). NACA TM ikYJ 51 c, , :o^____ I05 I06 R 1 0/ '^ ^~ — ______ ^^^— ' "^^^ ^ jX ^^^ -0.1 / ////////////// r 2b i ~.^ \ N ) y /// ////////// Figure 5. - Two-dimensional Poiseiiille flow. The four lowest eigen- values c as functions of R for a = 0.87. 32 NACA TM 1^4-17 Figure 6. - Inflection-point profile. U = (\/2 - l) + 1^2 - \/2)co3 y The four lowest eigenvalues c as functions of R for a = 0.5. 3tt NACA TM 11+17 53 a =0.5 0.2 0.3 0.4 Or Figure 7. - Two-dimensional Poiseuille flow. The frictionless eigen- value c associated with an even eigenfuncticn , as a function of o. c, 0.05 0.05 -0.1 , a=0/ /-^='l^ X, -- 1 02 0.4\ a = 2\ 1 0.6 \ Cr - \ Figure 8. - Inflection-point profile. U = fjl - ij + f 2 - \/2 jcos y The frictionless eigenvalue c associated with an even eigen- function, as a function of a. ^h NACA TM ikYJ -y Figure 9.- Rectilinear Couette flow. Eigenf unction cp '(y) for a = 1 and R = 10^ associated with the eigenvalue c- -0.70 - ix 0.30. -10 Figure 10.- Two-dimensional Poiseullle flow. Eigenfunction *'(y) for a = 1 and R= 7.7 x 10^ associated with the eigenvalue c = 0.178 - ix 0.049. «Ba 0-5 g am c u .2 Qt C9 ^ - bo-o . m , c u • ^ g|?"a c q o o> [5 c« 3 q 'hI::^ -jaa ■° a- o . •o 1^ '^ c ■ 0) »; o o « ?! ° S > O 0) ■ .So -n CO "< ■d T3 a CO t; « £ I a " 5 2: a E 2 = o ■ c< o -a i — " 0) CO :2 o cj tio£ ■ a a o !z <- o §3 JJ o -c £ o j: c« 3 aii a » 2 ° o- i3 j= — q CD g C c - - •" ca bD— o 3 tt- J2 ■'^ -»-' j:: a- £■§ . •-. ca CO * CO g ■§ 5a S.« ° c« C T3 c o .a 0) c4 u CO (4 o 0) C4 CO ^ m q >>^ 2 -^ i3 -c — q » t; g .2 s a ?, c -q 0) Qi ■" 0) ui^ a £ 'S 13 "13 £ c CO < i* CO m » c < CO !* o z <" t. 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