Non-Stationary Lattice Theory Citation Shieh, An Huh (1982) Non-Stationary Lattice Theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/xknf-r113. https://resolver.caltech.edu/CaltechETD:etd-09142006-112813 Abstract No abstract submitted. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Aeronautics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 16 April 1982 Record Number: CaltechETD:etd-09142006-112813 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-09142006-112813 DOI: 10.7907/xknf-r113 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3537 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 03 Oct 2006 Last Modified: 24 Apr 2025 21:54 Thesis Files Preview PDF (Shieh_ah_1982.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

NON-STATIONARY LATTICE THEORY Thesis by A. H. Shieh In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technolgoy Pasadena, California 1982 (Submitted April 16, 1982) ii Part ff APPLICATION OF PRANDTL'S ACCELERATION POTENTIAL TO LATTICE THEORY Ie IT. Itt. IV. Vo Lii Contents Introduction The Acceleration Potential The Method of Acceleration Potential The Stationary Lattice 1. Couformal Representation of the Lattice 2. The Complex Potential Punction 3. Determination of the Lift and Moment Results References Page ew NY OD fF Pe OR EF pH | ed bi Page l APPLICATION OF PRANDTL'S ACCELERATION POTENTIAL fO LATTICE THEORY [. Introduction The lattice theory has important applications, especially to the design of airscrews and turbine blades. The problem of a stationary lattice in a uniform flow has been treated by several authors using different methods. Karman and Burgers used the method of conformal transformation (Ref. 5, p.- 91), while Pistolesi used the method of vortex distribution (Ref. 4). This article deals with the combined use of conformal transformation and acceleration potential for the solution of the problem. In the following treatment, the basic idea of the acceleration potential, as it was introduced by L. Prandtl, is first reviewed, and then shown how it can be applied to the solution of steady-state two-dimensional problems. The method is then applied to the determination of the lift and moment of a stationary lattice in a uniform flow. A new expression for the moment is obtained by this method. It. The Acceleration Potential The equation of motion for a non-viscous incompressible fluid, if the body force is neglected, is given by Sf e-d op (1) Page 2 where q, p and p are the velocity, pressure and density of the fluid respectively. For the acceleration vector 4h » & scalar function ¢ can be defined such that _ ag VP = at (2) Then Eq. (1) becomes pvp=-vP (3) where is called the acceleration potential (Prandtl, ref.2). In applications, both ¥ and p can be taken as zero at infinity, then Eq. (5) gives ree -P (4) Hence there exists a very simple relation between the pressure field and the acceleration potential. In the so-called linearized theory where the perturbation velocity q'of the fluid is assumed to be everywhere small as compared with the uniform velocity U in the x-direction, so that Then, by neglecting higher order terms, Eq-e (2) may be written as it ax (5) fhe equation of continuity in this case is veg’ s o (6) faking the divergence on both sides of Eq. (5) gives Page 3 Hence the acceleration potential satisfies the Laplace's differential equation for the linearized theory. TIT. The Method of Acceleration Potential The method of solving two-dimensional problems by the use of acceleration potential is to find a complex potential function. Fixtiy) = (xy) te wey) (8) to satisfy the boundary conditions. In Eq. (8), ~ is the acceleration potential and y is the conjugate function of y , The two functions ¥ and yY satisfy the so-called Cauchy-Riemann relations: ay ay ax 2 j (9) ay oe ay gx ) which give the accelerations of the fluid in the x- and y= direction respectively. It is evident from Eq-(9) that both ¢ and Y satisfy the Laplace's differential equation. To obtain the boundary condition for the: complex function (8), consider a stationary thin airfoil. The equation of the airfoil can be represented by y= y(x) (10) Then the vertical velocity of the fluid adjacent to the airfoil will be vy, oy U dx (12) Page 4 From Eq.(9), for steady state; ener 2) Integrating both sides of Eq-(12) with respect to x and assuming that both y and v are zero at infinity gives, on the boundary: a-Uyv:=-U ax (15) fhe restriction now imposed on the complex function (8) is that the imaginary part must satisfy the boundary condition (135). In addition, the function must also satisfy the condition at Z=io which will be considered later... Iv. The Stationary Lattice The above method is now applied to the problem of a stationary lattice in a uniform flow field, as shown in Pig.l, where the uniform velocity 0 is at a small inclination « to the lattice. In treating this problem, the conformal representation of the lattice will be first considered. 1. Conformal Representation of the Lattice. Using the idea of "hydrodynamic analogy” (see Ref. 5, p. 84 and p. 92), a mapping function between a unit circle and a non- staggered lattice composed of straight airfoils (see Pig.1) can be constructed ag follows: aa h z= x { bog pt by ock<l (14) where k is the parameter of conformal transformation. In the expression (14), it is to be noted that for keigfod », the term Page 5 as S*K as positive, while Ce is negative, the negative behavior f-k S K of the latter term can be eliminated by writing (14), with the introduction of an unimportant additive constant, as: 7 | *xty¥ | Ste. log X o<ke| (15) K 2 } z= 55 bboy y-k Jays a K — ——* “tL S zZ-plane ¥-plane Fig. | On the mit circle y¥-.°°, the transformation equation (15), owing to the many-valued nature of the logarithmic function, # gives h 1+ 2kce 8 +k? x= — log an 1-2k tg +k (16) y= nh n is any integer. % Here the value of 2n7m has been added only to the imaginary part i of Log ant » for the reason that only the exterior region of the unit circle lyJ=1 is to be considered. Page 6 which are the equations of the boundaries of the non-staggered lattice composed of straight airfoils. fhe chord of each airfoil can be easily found from the first equation of (16) by calculating the maximum and minimum x-coordinates and by subtracting one from the other, there results: c= Bh log te (a7) Which may be expressed as k = tanh 3S 18) For wery small k, Eq.(17) can be approximated, by taking only the first term of the series expansion, as? c= ohk (19) T It is to be observed that in the expression (19), when k becomes infinitely small, h will become infinitely large, if ¢ is fixed. The product hk, however, will remain to be finite and is equal to ae - Hence it appears that the first-term approximation for very small k is really the limiting case k~O which corresponds to the case of a single flat plate. 2. The Complex Potential Punction The complex potential function for the flow past the none staggered lattice will be of the form: tops Ul(inew) + peep (20) where prey! represents the disturbance function; «~ is the angle between the direction of the uniform velocity U0 and the chord line Page 7 of each airfoil of the lattice, and this angle is assumed to be small. The conditions which determine the function (19) are the following: (a) On the boundary, eo, yso, according to Eq.(135). Hie UM (b) Condition at z:sie«, when expressed in terms of ¥, will be aaa aroren = finite + 0, ask +0 The meaning of the second condition is that the disturbance func- tion ¥# oy! has to be finite at the finite points of the ¢ -plane; but as k>0O, these points approach to infinity, and the problem reduces to that of a single flat plate, accordingly y+,’ has to vanish. fhe complex potential function which satisfies the conditions (a) and (b) is found to be z2eyie it kiy . 2 s 4o% = lesk) + ——— props U (Ines) + Te (21) On the unit circle y-n', the real part of (21) is 2 pe kt wu im C0s¢ Y,2u'+ Sux ae (22) where the subscript b denotes the value of the quantity on the boundary. 3. Determination of the Lift and Moment The lift is calculated by L=$ dx (23) Page 8 By substituting (4) and the series expansion of x from (16), there is obtained: L [Epa 2 5 k "sin ne ) dé (24) a may. Eq-(24), after substituting 7, from (22), becomes 1 = teh jek ux f" /- wit) s s kh" Sin HO dé@ (25) GT tek* Sia lye, 3. é To evaluate the integral in Eq.(25), expand aoe into the ae following series; sie hb = 6 Jem os (H] tn GHAI” roy) (-1) if A= fi- nei og are i(ne2) ots we 8 “et ep ar aad 8 where ni is odd. Then integrating gives . od £2 Wt @ ,@ Siahé yk ° t - A" de FR CU TY, (e) Ju wr) dO Sin @ Aside e Le Li y(aes') ! . Awd ad LC Rae A Galle a where U, = I B 2 ° “ #4 § 3 fa... TE? t to n terms. Substituting (26) into (25), the result is tH y = 4ph LA ye 5 k Mo) mel, de which can be simplified to uphk ot = ra ts 4+? (27) Page 9 Tf the lift is expressed in the form: Pt £ = ~ 28 LeC,sue (28) Then the lift coefficient is given by 2¢ttex (29) =F r¢h? which, in wiew of (18), may be expressed as te fhm tank (30) The moment is calculated by m= § px dx (31) By substituting (4) and the series expansion of x from (16), there is obtained: ow 7 & o@ os . M | Cpa oh 5 x = cor (2m-i)@ Sin (2-1) 6} dé (32) -1 ms] aay 3 Using the identity: cos(2m-1)8 sin(2n-1)96= # [-ain2 (m-n) 0+ sin? (mn=1) 6] the 6xpression (32) can be written in the forme ou . Mz 2ph "y, X Cy Sie 8 dé (33) w* La yz, Y- ne mprmen pmtiane where Cm = 2 pre +E (a mene] Eq-(33), after introduc ing %, from (22), becomes @ . , Sin MO u's [ (eet) E 2 fm TG de (54) more @ gy M = ae It kt Page 10 The integral in Eq.(34) can be evaluated by expanding same into the following series: m(a*2') con 8 : . +] Smee 2-4) * m (650 < Sin 8 eb Bei med 4 in( in) (re = f°) cor 8 a (-1) (2 eos 8) | & where m is even. fhen integrating gives +g 6 Sty j (1- 0086) § Om oe dg= - E be (<1)? V., (35) , 23, atl po memset) 32d where Va = 5° iD 4g 2 ry a iy (n= 2) m #) 5 32 4... to m terms. fs 6 ¢ 2 Substituting (35) into (34), the result is A+] 2 HERI Oe Fey V, W ke moe, Y- which can be simplified to afk us Ux 1 atk: =e 36 M of TK (56) If the moment is expressed in the form: P22 Mz Coz Uc (37) Then the moment coefficient is given by G,e- the fog f-k* (58) which, in view of (18), may be expressed as Gy ae Ag con (39) Page 11 It is interesting to note that in the limiting case k~0, which corresponds to the case of a single flat plate; the expressions (29) and (38), in view of (19), are reduced respectively to C, = 21% (40) C,, 2 -7% (41) which are the well-known results for a flat-plate airfoil. Ve Results fhe expressions for the lift and moment are = ¢,fu'e (28) Mos c.f ure (37) where ¢ and C,, are given by Eqs.(50) and (59) C2 ftw tend (30) yee SE a deg ook (39) It is better to re-write Eqsa.(50) and (39) as C= Qn re (42) C, = -Dax (43) where © and A are interference factors which are given by ce 8 tank FE (44) ne 2h Page 12 ht The relations (44) and (45) may be represented in graphical forms by plotting cr and A against the chord-gap ratio => as shown in Fig .2. Prom the figure, it is seen that as the chord-gap ratio ® increases, the values of ¢ and »* decrease, that is, the interference effect increases, 10 08 WN AN o 96 ~~ = 04 PJ ~~ 0.2 9 O85 10 45 20 25 a h Fig.2 1. 4 Page 15 REFERENCES Prandtl, £L.: Beitrag zur Theorie der tragenden Fldche. ztschr. f. angew. Math. und Mech., Bd. 16, Heft 6, s. S60, 1936 Prandtl, £.: Uber nevere Arbeiten zur Theorie der tragenden Flache. Proc. Sth. Int. Gong. App. Mech., Cambridge, Mass., TeSeAcy Pe 478, 19358. English Translation, N-A-CAe Teche MemoOo NO. 962 Biot, M. A.: Some Simplified Methods in Airfoil Theory. gour. Aero. Sci., Vole 9, Woe &, po 185, 1942 Pistolesi, E.: On the Calculation of Flow Past an Infinite Screen. W.A.C.A. feche Memo. No. 968 Durand, W. Fe: Aerodynamic Theory, Vol. II, 19354 Page 14 Part II NON=STATIONARY LATTICE THEORY I. IT. Tit. Iv. Vo VI- Contents List of Symbols Introduction The Method of Acceleration Potential The Non-Stationary Lattice 1. Conformal Representation of the Lattice 2. The Complex Potential Function 5. Determination of the Lift and Moment 4. Determination of the Function A 5S. The Limiting Case k =» 0 6. Evaluation of the Integrals 7. Formulas for the Lift and Moment Numerical Calculation Discussion of Resulta Summary of Equations References Page o Oo mm & 1O 16 19 21 24 25 52 3S o7 Page 15 P i Page 16 List of Symbols gap between airfoils of the lattice chord of each airfoil of the lattice parameter of conformal transformation Te k= tanh 7 O<k<l 1 k+—= b(t) s+/g'=1 @b tanh—= xee & h 2 imaginary unit, used in connection with the complex potential function imaginary unit, used in comnection with the harmonic functions of time amplitude of the oscillation of each airfoil of the lattice time, also used as the variable of integration ve locity acce leration uniform velocity in the x-direction Le” = 1irt fo) jut uM. @ = moment and My = quasi-steady lift and moment respectively = a function, but not of z=, as given by Hq-(68) and Q = functions of k and s, as defined by Rys.(54) and (51) respectively Page 17 G, and G,+ functions of b and jy, as defined by Eqs.(74) and (76) respectively G R, and I, = real and imaginery parts of G+, E and F i functions of b and jy, as defined by Eqs.(80) and (81) respectively f = density of fluid y = acceleration potential, also used as the phase angle v= conjugate function of w = circular frequency of the oscillation i GF = Te tanh. » 3 ; log coshz 2K P 2 TL nT vez —— =re duced :- frequency 2mu 4 Note ¢ fhe subscript o has the meaning, such as jut jut $27, 2 ¥=¥%24 The subscript b denotes the value of a quantity on the boundary. Page 18 WON-STATIONARY IATTICE THEORY A. H- Shieh Ie Introduction The problem of non-stationary lattice composed of straight airfoils in a uniform flow is treated by the combined use of conformal transformation and acceleration potential. The given conditions in the physical zeplane are first transformed to those in the § -plane (the transformed plane). Then the complex potential function is expressed directly in the ¢ -plane in terms of the boundary conditions in this plane. This procedure is quite similar to the expressing of the complex velocity directly in terms of ¥ (see Ref. 35), and it greatly simplifies the solution of certain airfoil problems. In the following treatment, the method of acceleration potential used to solve two-dimensional problems is first introduced. The method is then applied to the determination of the lift and moment of a non-stationary lattice in a uniform flow. fhe results obtained involve Gamma-functions together with hyperge ometric functions of the complex variable. For numerical calculation, the expressions for the lift and moment are reduced to simpler forms (without approximation). ‘The calculated results are represented graphically by curves with different values of chordegap ratio = and the reduced frequency YY. It is found that in order to keep the amplitudes of the periodic lift and moment at their lowest values, the optimum Page 19 conditions are: = 1.5 and Y= 0.5 . Pollowing this, the physical interpretation of the lift and moment equations is given. The @quations for calculating the amplitudes of the periodic lift and moment at any values of _ and Y are finally summarized. Page 20 IIe The Method of Acceleration Potential In the steady flow of a non-viscous incompressible fluid, if the body force is neglected, it can be shown that there exists a very simple relation between the pressure field and the acceleration potential PY =-p (1) where p and p are the pressure and density of the fluid respectively, and / is the acceleration potential. In the so-called linearized theory where the perturbation velocity of the fluid is assumed to be everywhere small as compared with the uniform velocity 0 , the acceleration potential satisfies the Laplace's differential equation, v’y=o. The method of acceleration potential used to solve two - dimensional problems is to find a complex potential function P(x+iy)= ~(x,y) + Ly(x,y) (2) to satisfy the boundary conditio. In Eqe(2), ¥ is the acceleration potential and yy is the conjugate function of 7 . The two functions ¥ and ¥ satisfy the so-called Cauchy — Riemann relations: ay , oF ax ~ Oy dy ~ %x which give the accelerations in the x- and y-direction respec- tively. It is evident from (3) that both ¥ and ¥ satisfy the Page 21 faplace's differential equation. To obtain the boundary condition for the complex function (2), consider a non-stationary thin airfoil. The equation of the airfoil may be represented by y= y(x,t) (4) then the vertical velocity and vertical acceleration of the fluid adjacent to the airfoil will be - oY 24 “%* 54 7 U yx (S) ayy OV; aa tU Tx (6) respectively, where the subscript b denotes the value on the boundary. The restrictions imposed on the complex function (2) may now be stated as follows: The complex function (2) must satisfy the following three conditions: (a) On the boundary, P) (b) In the fluid, xy. dv ae where v is the velocity of the fluid which hag to satisfy the condition of continuity at the edge of the airfoil. This corresponds to the so-called "Kutta-Joukowski condition" Page 22 in the classical theory that the velocity is to be finite at the trailing edge. (c) The condition at z= £0 , which will be considered later. Page 23 XII. The Nén-Stationary Lattice fhe above method is now applied to the problem of a non = stationary lattice in a uniform flow field, as shown in Fig.l, where the fluid is flowing at a uniform velocity U to the lattice. In treating this problem, the conformal representa- tion of the lattice will be first considered. 1. Conformal Representation of the Lattice fhe equation for the conformal transformation of the non = staggered lattice composed of straight airfoils to a unit circle (see Fig.1) may be expressed as 2 oh [lag SE 4 boy od ‘o<ks| (9) K On the unit circle, fen"? » the transformation equation (9), owing to the many-valued nature of the logarithmic function,” gives . ho it ake +k Xe 409 z 2H (4 Jr2zk wGtk | (10) 9 F nh n is any integer [ which are the équations of the boundaries of the non-staggered lattice composed of straight airfoils. The chord of each airfoil can be easily found from the first equation of (10) * Here the value of 2n7 has been added only to the imaginary } woe part of tog » for the reason that only the exterior K region of the unit circle |4%{/= 1 1s to be considered. Page 24 by calculating the maximum and minimum x-coordinates and by subtracting one from the other. There results: b 1+k which may be expressed as We = 12 k tanh th {12} J Po (| A U 1 f°) x f Llc -% VK ip z Z~plane ¥- plane Fig. i On solving Eq-(10) for , there is obtained vo NZ / 2 2 nz 2 a > tanh—— ty b tanh (= )-1 (13). whore be k(k+ —) (14) It is to be observed that the inverse function (15) is a two-valued function of z2. The sign before the radical must be chosen carefully. In later developments, the region Page 25 outside the unit circle [4¥/= 1 and along the negative real axis will be considered. For this case, yea'™, the proper expression to be used is WX 2 8, WK ¢ = = ft (SE) = (15) The Complex Potential Function fet the oscillation of each airfoil of the lattice be a harmonic function of time; thus yey," (16) where Vo and w are the amplitude and circular frequency of the oscillation respectively. Then the velocity and acceleration of the fluid adjacent to each airfoil will be at jaye” (17) Y% 3 a, = 77,8 (18) Substituting (18) into (7) and integrating gives, on the lattice boundary, the following expression for ¥: wt ¥,= (aly,x + Ajo” (19) where A is a function which will be determined later. wt Let 7 ¥ ye!” Then Eq.(19) gives 2 ty? tw) Jor tA (20) Page 26 Introducing the series expansion of x from (10) into (20), the expression becomes Rael k Bhel 2h = | Ye4 2 = WY, Ps cos(2n-1)6 +A (21) which gives the condition on the boundary of the wit circle in the {-plane. The more complicated problem of fluid flow in the physical z-plane has now been reduced to a much simpler one in the 4-plane with the condition (21) together with the condition at yea which correspondsto z:=ie according to the transfor- mation equation (9). fhe next step is to find a complex potential function of the form: p00 =§ -y. 2h * fF k - ‘Pes ficpe = Tod ay cos(2n-1)8 +A(¥% +*¥) (22) where ¥%+:i% represents the disturbance function which has to satisfy the following two conditions: (a) Wire, = 1, according to Eq.(21) ) (23) Ba (b) eK ys, = finite ~ 0, ask20O f a ae The meaning of the second condition is that the disturbance function ¥+/¥ has to be finite at the finite points of the ¥ -plane; but as k -—» 0, these points approach to infinity, and the problem reduces to that of a single flat plates; accordingly the disturbance function ¥+.«¥ has to vanish. One more remark which should be made here is that the mere points fe tea on the ¢ -plane have nothing to do with the boundary-condition consideration, as there are no corresponding Page 27 points on the physical z-plane. fhe complex potential function which satisfies the conditions (25) is found to be wo. @ Re é 24A It k*g amd Hee% = Ge Ole 2 sa Pet Ig (24) of 24 . ke : 8 jak” fo cos 8 =: woe of “i + et a fa 7 = W to 2. or tn (29-0) A jek' sin 8 (25) Before going to the determination of the function A, it is expedient to consider the lift and moment first. - 3. Determination. of the Lift and Moment The lift is of the form: wit j# Ls Loe (26) where Ly is given by L, ¢ has dx (27) By substituting (1) and the series expansion of x from (10), there is obtained: \ rt | (- ~p tu) - 225 k" gin(en-1) 6) de (28) where ¥, is given by Eqg-(25). Since the integrand is an even function; Eq.(28), after substituting 74 from (25), becomes Page 28 w af me net) . ve 7* ° meal wel zmel ° (29) , uph i- k* [’ j= cos @ 5 kam ne do nN eg k* Sin 8 ast,}- 8 The first integral in (29) is readily evaluated: v almenei) : 5 5 k fon (rm-1) 8 Sto (un-i] 8 dé mci az 2 mj 6 (50) _2¢ gree oe yp tk | 2 pet dae} 4 / 1-k* ] To evaluate the second integral in (29), expand —<*? into the following series: . fe! ais he od ) sin 00 (-t) 2 pe fl costg 4 is 3 two ty Sin 8 12 uv ares Tee ee a fet ad ) _ ayia PL ee (1) ® (x28) } when n is odd. Then integrating gives wel 7m Oo bn Sind oe ae “z ° J Gr ewe) RARE dee DR CYS (31) 6 wst,3e- “0 met3e, where Wee pe Ey OE) 3d . ez 2 if ¢ 2 ve » - ,t PINs) os 2 by, ton terms. i 6 ¢ 2 Page 29 Substituting (50) and (31) into (29), the result is raph tk>. Lemp Ode 7: rk dekh” k* (-1) = yf, ruphae J¢&* 7 Ek which can be simplified to b , . 2h ink ¥PhkA Ly = — Jo log rk” jek? (52) which, in view of (12), may be expressed as 2Ph 2 we re L,= - a7, log cosh~5- +2Pna tanh-—- (53) where A remains to be determined. fhe moment is also of the form: put M= Me (54) where M, is given by = $ hs x dx (35) By substituting (1) and the series expansion of x from (10) into (35), there is obtained: aes 08 af[mener) ( ) 7 . YY 2 fotos fam-t) 6 Sin [un-t) 8 | da 3 on] “epni(-4 a 22, ime] / ( 6) Using the identity: cos(2m-1)$ sin(2n-1)0= %/-sin2 ( men)6+sin 2 (min-1) 6) the expression (56) can be written in the form: me ts, $y I Com same do (37) where in Pee wo, ufaera-t) premeens Cra zs tae] - 4, ( mo rr-| * tae / Bs pad Page 30 and ¥4 is given by Eqe (25). Since the integrand is an even function, Eq. (37), after introducing ¥, from (25), becomes pend 3 M, oe oy, [i py a som (2m-1) 6 7 Crm Sin tng dé ney bho] mst (58) b , Seas to O + Ph ae | "he cot 5 Cn <a 48 ke 2 fe tin 6 is m ow way B00 E grrr ‘ k FO where C,, = 3 A-1 t Z ( Aol mene / 42,4 ehere The first integral in (38) is zero because of the orthogonality or the trigonometric functions. The second integral in (38) sin 8 can be evaluated by expanding “sine into the following series: xa . a oi B® sem 2 (-;) * j mers nlm) costa Sind B ® & 4 el + ial de A ATTY a aw et (aes 8) fia ; when m is even. .fhen integrating gives ci Sram = Zt! ) (1-9 6) iS C., = d@=- EF C1) nV (39) henge 108 wen where eg mime 2S er ne yo 2 ey » L L + mlamVle=e) £34 ~ oe to m terms. “ @ # < Substituting (39) into (38), the result is 2 ke fon) adel +] Miz- sh, eR FS cl) ¥, mod ge ° {tk which can be simplified to Page 31 (40) which, in view of (12), may be expressed as 2Ph" rid Tc A log cosh; (41) “Mg? = where A remains to be determined. 4. Determination of the Function A. fhe function A will be determined by the condition (8). By substituting vot wt tad Gs +=¥ a” vey, 2! into (8) and after cancelling out the term e 8 there is obtained: ax (42) Ke & K i Gu ‘T dy or er (ee) Integrating the above by using the conditions: Yo? Lfinite at Ks «ce . c Vos [Ye at X= = o one obtains: mle 2 uae “pW, & -4/ 2 © & dx (43) ~ oY where Y, is given by Eq. (24) write : A A¥+y Page 32 Then Eq- (43) becomes jw -£ jux al av s - U pat + De) dy , he he dy dx (44) ~where . 2 H le k* + ¢ oe Bee] rho k te We Foy de 2 oo pe (48) Notice that there is a singularity at = -l for ¥ and in order to make the first integral in (48) integrable, it is convenient to integrate this by parts, using the following conditions; ¥ = finite at x= -o %= 1 at X= - = 2 ‘Thus . . -£ Jur 7 ho a 4 2 ey) U fis % Basa? 2p a 8 od (47) x ; d By substituting (47) into (44) and after transposing, there is obtained: . uv AY, “jp Hy, U2 U a & — dx A= =o (48) Cr ere w 2 The integrals in Eq. (48) will now be considered. On the negative real axis, {= roe” , Eq. (45) gives 2 tek’r = ad ! ipk* oper Now, fpom Eq- (15) r= st/stq=| WX c e Ss TT = cum where F} b tanh Ke 3 After some calculation, ¥ can be expressed as Yet-Q where Q = ink” os + i-k* tek? Joy 14h” fgreot -£ sux _ dae “gga Then 2 > ; U qu 2 7 J A Ydx 2 - ) Q dx : wy J -ol ~a@ Similarly, from Eq.(46), with / =< rai” awed ¥ 2- thu 5 ‘OO oak a "7 Jee, Rhel pen where _ ike os P “ i+ k* g-] Then c SM jpwe ~£ jut FT at , IT ty fia % Pdx 2 4 SO dx =-jWyU2 -4 fo} 4 TY } Substituting (52) and (55) into (48) gives Page 33 (49) (50) (51) (52) (53) (54) (55) Page 34 | Jer f. é ° Pdx A OY,U se (56) J -€ jax [ ¢ Q dx Using the following substitutions: g=z=b tanh—= bh nods (57) = Fb yes Kw@2 == g =z ' | R= @ o s=b Then Eq-.(56) becomes ae hop its [- ve 7 me Pgs WyeU re As (58) gaih 14 J fs “FH! tt , 3 Fe ds i “FP where P and Q are given by Eqs. (54) and (51) respectively. Before proceeding further, it is advisable here to examine the limiting case k = 0. 5» The Limiting Case k = 0 Gonsider the limiting case k=» 0 which corresponds to a single flat plate. Por k<<1, Eq-(11) can be approximated, by taking only the first term of the series expansion, as 4hk (59) VT c= Page 35 ‘and for eel e b nt 25 Log —F = as (60) b iL where b= &(k+ r) Substituting (60) into (58) gives . Ak b - abe mY rm) i as Shi _* : * (61) J 6 .i@ thk | 2 7 mhsk) ods s* 2 / ue For k =~ 0 » »D =—& oa » Eqe(61), in virtue of (59), becomes oS 1S s 2¢ i ds . . J if ‘ Vs* | 62 A =~) Yo i ~ 196 5 [= ( ) a“ f SH! ds &-{ which can be expressed in terms of the modified Bessel functions of the second kind, thus f= -) “YU (65) Kj) + KY) where ) awe LU yy is called the reduced frequency. For k = 0, the lift and moment expressions (33) and (41) become respectively Page 36 (64) (65) The results obtained, as given by Eqs. (63), (64) and (65) check with those obtained by Karman and Sears (Ref 6). 6. Evaluation of the Integrals The integrals in Eq.(58) will now be evaluated. Using the notations: A" Th 20 ap Then Eq.(61) becomes -)v a)” P ds _ 4, U af ‘3 s s*-s* / yt 1 | bes bes” By substituting (51) and (54) into (67), me obtains: G, s ejuy U J°¥g G, +G, where a | (bey? == ds sé yt ; . b+s ! I d «,-[ (#24) jugs fer ; i These integrals (69) and (70) can be evaluated by means the following substitutions: of (66) (67) (68) (69) (70) Page 3/7 begs _ 1 bel ‘ be-s t bel b(b+l)=b(be1l)t bel¢(bel)t a5 Tpele(b-1)t]' 7) Pirst, consider the integral (69). By substituting (71) ‘into (69) and after reducing, there is obtained: . i . h-i wat 14) % j- 2 f) 2/h i #1 ; i eA EYE) . Pd . G ' (£2! ay 5 idee (bart) 9 gen peg df cee {| -— —_— en H "a leya § wr} nso U4 aa bet / / v}-t \ | ernie i 7 isl of Mae . op te “ty = (4) JTS ge Unt) (ot [ to af 2/he bti aco U4 tA bel Viet ] 4 Using the formulas: Page 36 if peur dt (dd Plast ty) ao «vine | P (mez +jv) eee (4) « P (aed) Lede ah P 2) 7 ihe Eq-(75) can thus be expressed in terms of the Gamma-functions. J [4S P (ned Pl nejy) : bos . \ . | a i Ce oe a ( 74) This can also be expressed in terms of the hypergeometric functions (see Ref. 7, p.288). iy 2 ! bet \) iL fa tty. [beh ) G-—— /——) 8 Liv] F jij 44 (FS | 7g (ei) PGT Poa bad) . . (75) ityy -} ! fa-|\ 4 ai (z fey, sey; (4p | where dyer (2) rig¢jt) The integral (70) can be evaluated in a similar manner by using the substitutions (71). The result is i (22 jv 2 rinzd) Piney), G,--— (- { 2 2b / he) 7) Aro P(nstejy) \ 1y¥ 2 rinad) platteyy) pen ye" +. io P(ntssjr) bei] J ne (4) or expressed as: Page 39 t b-1 . i (se! oe te (YY "a (4j7) F(z yi ptyyi (Ga) (77) i b-! ray jp pos H ° 2 oc. be! *| , —_———_ |— om, 1 =-; Yj s¢jyy; (-—/] !} se (et aan) F (3 myn 7) Sal y 7. Formulas for the Lift and Moment fhe formulas for the lift and moment are now summarizod below: ‘wt a L= te (26) o jut Mo= Me (34) ZPh* > Ly = —— Yo log coshie+2PhA tanh am (33) . 2Ph Moe oe =fh 4 Log cosh 24 (41) A= -jwy_U S (68 ~ J“Vo" “Gea, where G, and G, are given by (74) and (76), or by (75) and (77) respectively. Page 40 TV. Numerical Calculation Let G, Ratjla* SSar (78) Substituting (74) and (76) into (78) and after cancelling out the term 4 (= Jy r(z)ruyy) ree atl Fidaj) gives the following expression; 1+E-F R.+)1 79) avd L6g-P-—= (1+E+P) where [met) of LL el] on iL . & , bel (ned) (nev) —E-y (= — (80) IZ O hso dapi)(aegt jr) mat, bt Vt mM ened) (nts) Fel 25 (ta) | yma) (81) mro mtz +1 h20 (nti) (neg d The next step is to substitute R,+ / I, into the lift and moment equations and then to express them in non-dimensional forms. This is done as follows: By substituting (68), (78) into (335, (43) and then into (26), (34), one obtains: = [EE uy, dey cok sate? _ jut > (82) h Ratpl,) tanh ae | o j -2f oie aty le A | Me i 2Phiy A fo (Rat 4a | dag ood 3p 8” (85) Page 41 In the stationary lattice theory, the lift and moment have been shown to be LF 2Phu« tanh2# (84) . 2Phy 2 a, Mo= 7 = Ux logeosh2 , (85) . put # Now replace Us by =)0F 52 in the above equations L = “2Phu Wye tanh aM (86) v “wt efh : J My = 7 Uy4yL,e log cosh 24 (87) where L, and M, may be called the quasi-steady lift and moment respectively. Introducing the notations: T= tanh 24 (88) As 1 Bae logcosh uy (89) which have been called the interference factors in the stationary lattice theory. Then Eqs.(82) and (8&3) can be expressed in the following non-dimensional forms : aa ot Here the negative sign means that when Vp = jeye" is in the upward direction, the perturbation velocity 0 is downward. . > at 8 s M ; Ml} jt Henin Hil Mz 8 i “a Ms roy. /p Ayy where ral = [Ro + (I, + #2”) RE L 2 er = / Rot Ty 8 » of It+ee” ? = tan Re o2 : i tan” & Ra ¥Y, and Y, are the phase angles relative to the directions of L, and M, respectively. The calculation were performed at various values = and y . The results of calculation are plotted in Pigs. 2-5. Page 42 (90) (91) (92) (95) (94) (95) Page 43 0.50 0.75 1.0 0.25 tt > 9 1 = aa evs} " “i = vs Y/ N rv) |. § vl<s n ws © — al Ne > w A\? 3 n = rR a a ° = ~ oS coy o Fig. 2 Page “44 125 100 0.75 050 0.25 0.50 Qls 10 Fig. 3 025 Page 45 LTS l 150 - 7 a2 | YL 125 fe 100 7 Vf Ete —T 7 ~ sin ° O75 050 025 Page 46 u liq 4A ed NE D4 |e vj£= } LTS 1.50 125 Ms 1.00 0.75 0.50 025 Page 47 Ve Discussion of Results Since complex quantities may be represented graphically by points in a plane or by vectors drawn from the origin to these points, in Figs. 2 and 3 are plotted the vector diagrams which give the complete information about the vectors —- and Ss a L ML | - i with their varying magnitudes Fes | and ity and vary ing phase angles ¥, and /,,at different values of = and - . In Figs. 4 and 5 are plotted the diagrams which give the values of “el and || at different values of — and ¥. Prom 3 Mg h Fig.5, it 1s seen that fo =? 1.5, the value of , remains j FS} practically constant at the lowest value 0.5 for all values of Y 3 while from Fig. 4, it is seen that for v= 0.5, the value of | remains practically constant at the approximate ig lowest value 0.60 for all values of =: For discussion of the above results, write Eqs. (86) and (87) in the following forms: | Q(t * fis) L,= [D, (96) - j(ot+ fin) M,=]M,| ¢@ tM (97) where }L)= mpure- ary (98) 2 a [u|= mPu of tay (99) ¢.= tan”- a= -2 (100) ~,42 tan ~ = > (101) Page 48 Then Eqs. (90) and (91) can be written as ° WT be HEI ing of Ot 79% (102) . rT M = [HE] ag) of COEF 2 th) (105) It has been seen that for a fixed value of = 1-5, the lowest values of i! and |_M_| occur at a low value of ¥=0.5 . Lg! lif! At this low value of ¥Y , it is evident from Eqs- (98) and (99) that for = fixed, the values of |£,| and |Mg) will also be low. Thus it may be concluded that the lowest amplitudes of LZ and M occur approximately at ~ = 1.5 and ¥=0.5 . In practical applications, the amplitudes of the periodic lift and moment G& and M should be kept at their lowest values. Hence may be considered as the optimum conditions. At these conditions, Li. =) ° 0.60 (105) Mi. “Hi = 0.50 (106) |t5| = 0.415 pu’ c- (107) |[Mg| = 0.1507/y"c?. % (108) Page 49 The next discussion is to give the physical meaning to the two terms in Eq-(82) In Eq.(82), the lift may be considered as consisting of two parts: 2 jut L,= ( 2 £8 Wy, log cosh zp)e (109) L,= - |2phjay, (Ry j I,) tanhys]e (110) It should be noted here that the two terms L, and L, correspond to the two terms in the complex potential expression (24). The physical meaning of L, and L, may now be stated as follows: the lift £, is produced due to the reaction of the accelerated masses, while the lift £, 1s produced due to the circulation about each airfoil of the lattice. The lift L, which may be called the apparente-mass lift, acts through the center of each airfoil, while the lift L, produces the moment about the center of each airfoil, and its line of action can be obtained from Eqs- (83) and (110). ee meer e pte (121) M 1 For k = 0, ns anny ah (112) This result checks again with that obtained by Karman and Sears (Ref.6). Page 50 VI-e Summary of Equations The equations for calculating the amplitudes of the periodic lift and moment at any values of = and Y are now summarized below: . ra 3 uw = HE] |Mel o/ (tet +H) (103) 3 1h. [pt aAyy? LE |-/nis (htop) (92) ae|=/ Ret I, (93) |n,|="pu'e- 2. acy (98) [M,| = PU". #.Lyy | (99) . 1+B-F Ri ty I, (79) 1+R-P- + 1 +E+F) oy B= 5 (=) : (80) mz o b+] heo (atl nagar) Pei met (+ 7 3 tata) otstt (81) may 2 +] neo (ntij(nttaj™) bel f tek \* 14 De) ~ rE) (14) k= tanh (12) TC 4h } : s «ee $unh 2 e i i = «mem log cosh 7 2p S J 2vU Page 51 (88) (89) (66) Le Page 52 VI-e References Ge Prandtl: Beitrag zur Theorie der tragenden Fl&che. Ztschr. f. angew. Mathe und Mech., Bde 16, Heft 6, Pp. 360, 1936 Le Prandtl: Uber nevere Arbeiten zur Theorie der tragenden Fliche. Proc. 5th. Int. Cong. App. Mech., Cambridge, Mass., U.SeA.e, Pe 478, 1938. English Translation, N.A.C.A. Tech. Memo. No. 962 Mo Ae Blot: Some Simplified Methods in Airfoil Theory. Jour. Aero. Sci., Vol. 9, No. 5, pe 185, 1942 He Jo Stewart: A Simplified Two-Dimensional Theory of Thin Airfoils. Jour. Aero. Sci., Vol. 9, Noe 12, pe 452, 1942 C. B. Millikan: Perfect Fluid Lecture Notes. Cal. Inst. Tech., Pasadena, Calif., U.S.A. W. F. Durand: Aerodynamic Theory, Vol. II, 1934 Th. v- Karman and W. R. Sears: Airfoil Theory for Non-uniform Motion. Jour. Aero. Sci., Vol. 5, Wo. 10, p. 379, 1938 E. Pistolesi: On the Calculation of Flow Past an Infinite Screen. N.A.C.A. Tech. Memo. NO. 968 (rep. 1948) Ge Klingemann: Verfahren zur Berechnung der theoretischen Kennlinien von Turbomaschinen. Ing. - Arc., Bde ll, Heft 3, p. 151, 1940 E. T. Whittaker and G. N. Watson: Modern Analysis, 4th. Edition, 1927