Some Potential Problems Arising in the Theory of Axial Turbomachines Citation Mandelbrot, Benoit B. (1949) Some Potential Problems Arising in the Theory of Axial Turbomachines. Engineer's thesis, California Institute of Technology. doi:10.7907/QGD5-YD25. https://resolver.caltech.edu/CaltechETD:etd-11122003-085608 Abstract Some potential problems raised by the trailing-vortex theory of axial turbomachines are solved in the cases of cylindrical semi-infinite and infinite helicoidal vortices and of trailing vortices in a cone. The analysis is carried out for the cylindrical doubly infinite case and the dynamical problems are set up. The results are in a form where further applications to the physical problems may be undertaken and actual computations worked out. It is hoped that this work will be completed in the future. Item Type: Thesis (Engineer's thesis) Subject Keywords: Aeronautical Engineer Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Marble, Frank E. Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1949 Record Number: CaltechETD:etd-11122003-085608 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-11122003-085608 DOI: 10.7907/QGD5-YD25 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4517 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 12 Nov 2003 Last Modified: 20 Nov 2020 21:57 Thesis Files Preview PDF (Mandelbrot_bb_1949.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

SOME POTENTIAL PROBLEMS ARISING IN THE THEORY OF AXIAL TURBOMACHINES Thesis by Benoit B. Mandelbrot In Partial Fulfillment of the Requirements For the Degree of Aeronautical Engineer California Institute of Technology Pasadena, California 1949 ACKNOWLEDGEMENT The author wishes to express his sincere appreciation to Dr. Frank E, Marble who suggested the present investigation and aided in the preparation of the thesis by devoting generously his time to the discussion of the problems arising in the course of the research, SUMMARY Some potential problems raised by the trailing-vortex theory of axial turbomachines are solved in the cases of cylindrical semi= infinite and infinite helicoidal vortices and of trailing vortices in a cone, The analysis is carried out for the cylindrical doubly infinite case and the dynamical problems are set up. The results are in a form where further applications to the physical problems may be undertaken and actual computations worked out. It is hoped that this work will be completed in the future. TABLE OF CONTENTS Page Acknowledgement le Introduction 1 Chepter Ie Two-dimensional Problems foraCylindrical Turbomachine 4 2e Velocities in the Trefftz plane: Green's function 4 3e "Vorticity wake" method of approximation 7 4. Nicholson's approximation on total tangential velocity 8 5e Formulation of dynamic problems in terms of the circulation 1 Chapter II. Behavior of the Tangential Velocity in the Neighbor- 20 hood of the Blades 6. Velocities at finite distance downstream; (Rigorous influ- 20 ence function) 7 Approximate solution for the solid body rotation 2h Chapter III. Trailing Vortices on a Conical Turbomachine 28 8. A preliminary study of the influence function 28 Appendix. Details of derivations 35 References. 42 I. INTRODUCTION The central problems of the theory of axial-flow turbomachines subject to treatment by the methods of non-viscous, incompressible fluid theory are essentially thefollowing twos The “inverse problem''. Given: blade loading, fluid state far upstream, rotation speed of the machine, Find: the velocity field, blade shape, distribution of energy. The "direct problem". Given: blade shape, rotation speed, fluid state far upstream. Find: velocity field, blede loading, distribution of energy. Because of the great geometrical and therefore analytical com- plication of the problem, as formulated in three dimensions, two dif- ferent methods of reduction to two dimensions have been developed, answering to different and complementary needs; 1. The airfoil lattice theory has been applied in each isolated "in- finitesimal" cylindrical layer, neglecting their interaction. eo The actual shape of the blades has been heglected, and the shed vor- ticity assumed distributed continuously downstream of the blade row. References: Meyer ?); Marble”), — the purpose of the following treat- ment is to formulate a vortex theory of axial flow turbomachines, based on the principles of Prandtl's wing theory and Joukowski's screw propeller theory, end to solve a few of the potential problems thus formated. The axial motion of the incoming fluid is assumed uniform, as well as the rotation. The problem is linearized by assuming that the vorticity is carried downstream along the undisturbed streamlines, -2- (and not by its own induced velocity). Hach blade being represented by a radial bound vortex of strength varying with radial distance, the shed vortices fill a screw surface of constant pitch, Such a vortex sheet is highly unstable, and it is uninteresting to attempt to refine the treatment by assuming the trailing vortices to be other than strictly helicoidal, But in the case of blades impart— ing a solid body rotation to the fluid one may compute the pitch of the helix on the basis of the angular velocity downstream (which makes our theory applicable to the stator as well). In the first chapter, the method of expansion of the potential ¢ in a series of harmonics (Kawada >); Florine ‘*?) is used to derive the influence function of an infinite helicoidal vortex between two cylinders; the series expansion for d is approximated by an easily summable series, and ¢ is finally given by an integral, the kernel of which reproduces Florine's result for screw propellers with an additional term representing the influence of the boundaries. This investigation gives an insight into the behavior of axial and tangential velocities at a blade and permits one to set up the dynamical relations between circulation, thrust, torque and the in= duced velocities; it does not give, however, any indication as to radial induced velocity near the blade; this last is fortunately of little interest in the study of the interaction between successive rows of blades because one could not anyway take account of its var- iation around the axis. A mean value, such as determined by Marble‘? as function of radius and distance from blade row gives a sufficient -3= estimate.- In the second chapter, the behavior of the velocity near the blade is investigated by rigorous and then by approximate methods. — Finally the third chapter gives a preliminary mathematical investi- gation of the three-dimensional field of velocities induced by a trailing vortex between two cones. In no case does the treatment go beyond mathematical general- ities and it will have to be completed by a numerical study of a few significant cases for a comparison with previous theories, and a check on all the successive approximations. It is to be remarked that many aspects and methods of this theory are closely related to the study of the electromagnetic field due to a loose helicoidal conducting wire. See H. Lamb ‘>? Pe Jacottet who Limit themselves to the case where the coordinate Z does not enter ex- plicitly in the results. However the methods also apply when the cur- rent propagates along the wire, and to more complicated cases. mdno CHAPTER I TWO-DIMENSIONAL PROBLEMS FOR CYLINDERS 2e Velocities in the Trefftz plane: Green's function. Simple symmetry considerations show that, similarly to what hap- pens in wing theory, the axial and tangential velocities induced on the blades,and on the bisecting lines of these blades,by the shed vortices are exactly equal to one-half of the velocities induced by a doubly in- finite vortex, There lies the interest of the following investigation. Biot-Savart Law cannot be conveniently used, because there are no well-defined images in a cylinder, One applies therefore the method of separation of variables to Laplace's equation to find directly the per- turbation velocity potential p » as the solution of this equation which satisfies the required condition of continuity, and one-valuedness or many~valuedness and the boundary conditions, A return vortex is added along the axis to make easier the passage to three~dimensional considerations. (This axial vortex is then part of the shed vortex system). The purely formal operations which follow will be justified later on when the actual series development on which they had been performed will be obtained. Notations p= number of blades Iu) P U= advance velocity = strength of each shed vortex filament @= angular velocity Zor cylindrical coordinates fixed with respect to the blade row If the new variables 7; $-0- = j= at are introduced into the problem, the equations of the helicoidal vortices are 5= Zk (k-91,.... pt) 5 Mako the equations of the cylindrical boundaries are AL= 4, j b= fea ® end the potential p is the solution of Laplace's equation, depending only on the geometrical position of the point relative to the helices. That is: p depends on /}t and 6 , but is independent of Z. The equation for the velocity potential is : Ao - b+ +9. r er + - 06 =0 or in terms of the new variables ;: The boundary and periodicity conditions are: for Mos Mb < ky , p is a single valued, continuous function of (4,5) odd and periodic in g, Qn with period p ’ its ~derivative a = a¢ Y = uy, v mast vanish ye ara &) for K=Hy, « For ,<K< to, g will be a multiple-valued function of ¢, increasing by Yp when $ increases by ay, e Its normal /X -deriv- ative vanishes for K:y,, For 1W=jto » ot ak > g and %, must be con= tinuous in u »- Separating the variables, let b(4,5)- MwWZ(t) — » ~6= Then MZ + 2 M'Z © (le blue" =0 fe 7 “ / u and MA MY 2-4- r* mast be independent of wu and ¢ (I+ Luz )M Z Z"entZ2=0 gives 2Z=AS+B af n=0 ZL=Acsn$+BsinnS af n40 3 from periodicity conditions, ? must then be an integer multiple of p 1 n=Mp u = ives M= A lo B if n-4 or le M =0 g A + M's Bont (tee) MzAI, (na) + BKy(ne) if 240, where In, Knare the Bessel functions of purely imaginary argument. (See Watson‘), ) Keep only the terms which satisfy our periodicity conditions, ‘The solution is then, with undeterminate coefficients | rT 2.28 54 =), Sin mps Qn Kimp py) + by Limp ‘hu )f Mx fo ( ) In={ bd, = P Sim mps fan Ko (mpu) + 6m L mp (mpy)} ron ae p/ AA Le (2.20) Develop: ( g-% )= “Ls ap sin mp and write that the Me four boundary conditions are rotted identically in go; this deternines the four coefficients of each harmonic (see Appendix p35. for details). If n=mp (m=1,2.... ) and eXpressions such as J'(i) stand for a L. (x ) one obtains the expressions: n UA Mz, (2.3a) S$, -- “2, [MaK- Kajt [[ROIrM-KWMTOT sm mp$ 4 Klar). K'WI'e) mp on uw 5d; = Heslte )' Wk - KO TIER eo - KWL] si mps os It K(2) TO) - Kt) T(z) mp ¥( Mo) mel (2,3b) To obtain the actual linearized velocities, integrate these con- tributions along the blade, with for //o) the (unknow) distribution ef vorticity along the blades - oT) (i.e. the sum of vorticities Ye cpa around all blades): fe $= [56D ps po [on] Ede HA, Hz lad + [Stele polonpe] SE ye (204) Jf& There is of course no question of hendling these whole infinite series. _fwo methods have been considered to obtain more tractable expressions: a "vorticity wake" method, and a method of approximate estimation of the successive coefficients by terms of a directly summable series. 3. "“Vorticity wake" method of approximation (8) Ackeret points out that a sharp trailing vortex is anyway an idealized concept, and that the vorticity is actually distributed in a wake of finite "width", This width might for instance be assumed = a,a constant along the blade, and the following scheme, intermediate between (4) the strict vortex approach, and Marble's theory , might be adopted: 2 ar Writing %= ia ; %G= Ea os pga i teeo)- Wie ~ ~ J '\ Ki\ "eles —. Lo to! 1? ] E Ss S ot 22 p { of = &G Uo ++ te oC S-3. Let (ho) = be foo) + 20% (He) copa (T5_) (3.1) Five terms of the series give an excellent approximation. The corres- ponding Bessel functions are found in tables, and one has to integrate numerically: HOI 1p! . fe 4 — ft 0) dhe — kK Mm, old [5 np (mprel de 1 Jy, Ro (ntoa ls (and same thing for T}) for m= 1,2,3,4,5 only. This procedure is ex- tremely convenient when a distribution of I’ is preassigned; and it may then be used to bridge the gap between the present theory and the lat- tice theory. However it is not suitable for handling any inverse prob- lem, 4, Summation of the series for ds with the aid of Nicholson's approxima- tion on Jn, Xn functions. The essential components of reduced velocity will be the tangential velocity, and the velocity perpendicular to the screw surface. Both are linked to ¢ e By differentiation of the expression for ¢ » the following series is obtained: he = ial d, Len T4 Ho ° ~ K(2)1'G)-K (YT 2) POR) (OER OLEO) ML) (4.1) i like KWL COK-KWL ay + sa oft ——___~ kp) '0) - — 2 Ta) KD) ¥ z the Diem NU) KU) I (2) Oe) KOT () LO) (JRL N The lowest order for the Bessel functions in d is already p- There- fore the approximate forma derived by Nicholson?) can be used, and gives errors which are very reasonable (see Florine *); Annexe 2), By a series of integrations by parts (details of which are given in Appen- dix p. 36) the following result is derived: fy) smh mp (2-1) _ Pr ce mp § of 1 OT coah mp (€-€2) smh mb (to -6,) ng = —— + [soos co3h mp (E-b,) sah mp (to- te) (4.2) + = Wo Do JMh mp (€2-&) E where b (A) = [(4pn2 + log ae is tabulated in Chart I and [0,72 are the total circulations atu:%.,u, . [2 can be different from O (contrary to what is true for propellers) because for the purpose of this theory, the tips of the blades are assumed to join the outer wall, The next step is to write the trigonometric and hyperbolic func- tions as sums of exponentials. One obtains in this way a geometric series (which shows a posteriori that the manipulations on series were justified). The summation of this series, if one can assume that ee eee vie es ee fn aed ieerte ene eal Lay, at i aot hath : Weil. : ae -ll- eb lbe-ti) cy » and therefore replace the denominator by $ e pl Ee bs) » gives as remit: w (allo (Wee) 8) (229, -T )(l+x?) =f aa SH (to, t/b,, te ) deo (4.3) fe) é, where the kernel X has the expression: &p(to-t) eo KE (bo, &/ by, be) = +t bo-L1) - e 2pl ? Scospt e plto-b) - o- Lb (b+bo- 2s) (4,4) e f(t +é te) e p( @ _/ + ~_ PLE Ko- Ba) 26, be @ Pls te Hel e E+ bo (E+bo-26: tp (e+ plé+ ) 2) gos hte / This kernel is of the nature of a Green's function and has the same symmetries, For $=0, the kernel becomes: x eff e P(E + bo - 262) e ~PlE+ bo - &6,) 2. = e hte hte + [-e P(E +bo- 262) 7 [ -e7 PLE t Fo 2) : (4.5) In the absence of walls, this reduces to the kernel given by Florine ‘*) ’ if one takes into account the fact that the lowest Bessel function is here already of order p » rather large compared to the number of blades in a propeller. One can therefore use an approximation for Jn, <nsimp- ler than the approximation used by Florine, | ~ 4p (t2-€1) If e ~ 4p (Fe-b) is not </ but @€ is, other terns may be added to the kernel. (The details are given in Appendix p. 39 ), ~-12- Behavior of this kernel, especially for p large Consider the case 5=70 , ieee to the velocity on the blade. t _ eP — has a pole of order 1 for «-=4% , where it behaves ev -@ like oie « The two branches, for >‘o , where , 70 » and—---+] - CO as b+oco 4 ieee “ +,05 and “cto , where K,<0 , and ---0 ag t--@7 , Lees u ~0O », are geometrically identical. iw For f-co , &, tends toward the step function, aw Kact approximate uniformly for /¢-¢0/7€ , If oT an prt ». end >, is very ‘TA - pres "smooth", ieee its second a | &y é derivative very small, one ~ 4 4 may neglect the infinities in taking the Cauchy princi- pal value, ieee integrate from (, to to-€ and from fo+s to €, « But for /¢-f0o/>€ and forf»/, this kernel can be excellently approximated by: lé-€0) lee? for E> bo -@ ~plé-to) Ky, = fn b < bo The rest St of the kernel is always regular, the value is large (- é épltz-b,) ) only where ¢=¢5-€, or E=t.=€, 3 otherwise it is small, of the order of p (b+ bo-2tz) _p(t + bo -2t,) é e P -13- The result obtained for p>>/ means simply that when is inde- pendent of ¢ , the vorticity in layers which envlop eo is the only one to act: this is well-known in electromagnetic theory. if S#0 the kernel is a regular function of ¢ (or x) ~lh- 5e Formulation of the dynamic problems of the rotor in terms of the circulation function So far the circulation function - wes assumed known sround each blade, idealized as a bound radial vortex . This function lu) has now to be related to the actual geometrical properties of the blades, and to the forces acting on the bledes.e This leads to a series of integro-differential relations yielding the equations appropriate for each problem. Assume that the variation of the blade shape with fr is slow enough to permit to neglect altogether its reaction on axial and tangential induced velocities through the radial velocity. The axial and tangential velocity in Trefftz plane are Ua=-2 Ri we. 2 4 « The resultant of the velocities at the blade is rp" UK We -! ¢ w fier perpendicular to the resultant viru? of U “zh DHE and Wr. Gall Ci) the chord, m(u) the two-dimensional-lift-curve slope, (4) the angle of the zero-lift curve of the blade with the axis; Yu being as before the circulation around any individual blade, the “strip theory*® gives the relation; oY: shi vliurc = 3 (ted (H)[ (5 toate) llc -w / or TU) (A- ft uh) lt 2v / 1 (51) po mle) — Gaye Ue elt Mle tl, This equation, linear in |'(u) , relates lu) to the geometry of the blade, to @ and U0. The lift and drag formulas written for the blade strip are: or rd ry “oh Ww al _ ch - / ~ tay Mh Ww 2 2 V qu de £ m(weleys (Al) Qn A W Tse de (144) w dd ads u a _~ de eo Ww _ 7 we an 3 clerv (4 ) Oe fi Y vfiene and, Since the thrust and the torque are given by we obtain the following equalities w A BS om (Ateara)pvpidiia + fe(eviVige® Cou + 2 4 @ +p mltle(W) (A tan ju ) 25 u Ih ye ra [us) cf) @* (Mya Ie 4, (502) w dQ Gra Emel) ( A lew 'b eV ut # 2 poll ese cle) Coo + + 3 rola) elu) (4- fan ec Ze) W¢ (up _ ; 2 ; 2 (503) sre led SS fy =l6= These relations are unfortunately not linear and the second order terms do not seem a priori much smaller than the first order onese All the preceding relations permit the formulation of the follow- ing dynamical problem: 1) Find the circulation which would induce a given tangential vel- ocity in Trefftz plane. tf [2 K dbo - (eng - Th) (Iyer). ( ia “ue Ue - Pe) liye)" where Clu) = Ma) (lr) e This will be solved by successive approximations 7 er on the kernel, /z being given directly as [ (Me) My AS e First keep oO -€ efé ere. peo to the coordinate 6d defined by only Ky end neglect the rest Ko of the kernel K. Fass [> t { Z ‘ E . = eM As Bond 5 Ae ZleMveh) , Bf (eM- era) ,E--A Florine (4: Annexe 4) shows that the equation becomes y AG n ; eo Pee pee [ Co AO YE ue Te)(nyd)* -€ Z (8) 2, wb,wb | © Pert (54) and its solution is 6’ Mt “ , ie] : [ @(6") 22 tb sim g E- cos 8 (55) C (8) = - ag’ aa Cor 8- CO” This expression was obtained by looking for a Fourier series expansion of C() , and Florine gives other,equivalent forms. To © can be added an arbitrary function 4+6(6) which gives the value zero to the integral, but serves to obtain the desired values for 77 ana fl . ~l7- The value of C(9) is substituted into%z and the corresponding correction to ~ (4 gives by inversion of the Fredholm equation (5e) a correction to C(@). A particular case is investigated in more detail: the solid body rotation blades: the rotation far downstream is a solid body rotation of velocity ©, . There is a priori a question as to the practicality of such a flow: it is of course not realisable rigorously, and C, will be only the average of angular rotation, defined such that at a Z far downstream: an Tr)- [rte dt - tar (a,-2.)r (5.6) therefore the problem is actually again a simple "inverse problem" leading to no integral equation, but to an integral. Now in this in- tegral, one shall be much closer to reality if @ is replaced by ©, in the definition of iL *. To obtain reasonably close values of Ue as a function of 4 » one may compute Ue only for s-O and f= ~ ° and then interpolate al . by smoothing the step curve built on these Ue (5-0) _ two values which gives the average (W,-9) Wr Me (S-E) | _—>> The most important question remains to find the distribution of Ue 3 % ar with Z,near the blade; *This will also permit one to treat in the same theory the case of a stator inducing a solid body rotation. -18= this is left to the second chapter. 2) Given the blade geometry, find rye) ° Relation (501) is the integral equation for /", once we agree on a certain /2 . It is a Fredhola equation of second kind and there is little hope in finding an exact solution numerically tractable. For p large, the following approximate procedure is suggested: a) Neglect the X2 part of K and assimilate XK) +to the step function it becomes for pre « Then Oy ema mhe Vine View [_ (A (e) - tanh tye? miu )fu) v 4 2 I is the O-order approximation of /%. a) b) Write the equation as [ ore Ki, do = S(O) - S(L) is t, 60 obtained by replacing /” by’) in i and [xe K,dto., This is of Florine's type and gives ve, To estimate the importance of terms where the approximate value is used, compare e to me VEE « Take as typical values: mr mv6; End j fev; pw 30 - Then the first expression is roughly 1/6 of the second. ‘The approximation procedure appears as justified. c) Repeat procedure b) with /” to get /”. 3) Functioning off designe If Cu) and Al) have been designed to give a certain distribution of velocity at a given 2 » Which modification in t¢ and [" will be the consequence of a change of 2 « The following treatment applied to eqs (501) is suggested: The change in /’ is first of all neglected in i and the new value of d. is deduced by equalling the right hand ~19< side of both old and new (501). This gives a Plorine's type equation, for the new /" . . This new /’ is placed in the left hand side of (5-1) and the equa~ tion solved for a better value of Ue, and eventually of /” by another operation. 4) Determine the shape of blades giving minimum wake less (Betg's probleme This problem is not as crucial as in propeller theory, because of the presence of downstream stages, the energy is not definitely lost. However, the present method gives to it elegant solution. Betz's theory requires that _ Ww K a with W const. This leads to: 2M (u) = angle of attack d/y) - aoa a (507) pm(u) ep) | vo tyat and if las determined by (say) the condition of solid body rotation downstream, this is an all solved equation for 4/«)as a function of W, W, (the rotation downstream) and ¢(«/. It may be noted that S. Goldstein's classical approach to this problem in propellers allows itself to be applied with little difficulty to the case of a turbomachine. =20~ CHAPTER II BEHAVIOR OF THE TANGENTIAL INDUCED VELOCITY IN THE NEIGHBORHOOD OF THE BLADES. The velocity on a blade and on a blade bisector induced by the shed vortices was seen to be one-half of the velocity at infinity. However this result leaves aside the velocity induced by the bound vortices. Section 6 gives a derivation of the total induced velocity (infinite on the blade); the problem is enormously more complicated than in 1, and an approximate study is desirable. It would be tempting, because of the exponential nature in of the correction terms, to try to fit at best an exponential "decay" coefficient, the same for all i 's. This however is impossible, because the circulation must remain constant along the z-axis for Z70. In 87 these questions are discussed in detail 6- Velocities induced by p helicoidal semi-infinite vortex filaments, starting at Z=0 Consider the potential field composed of the solution of (2.3) for 270 and O for Z<0. This field satisfies Laplace's equation in three variables, the boundary conditions, continuity conditions for jaa s and perlodicity conditions. However it is discontinuous for X=-0. Therefore add to it a solution of Laplace's equation in three dimensions, continuous in fs satisfying the B.C. for fe=-4, fea» everye where one=valued and periodic in 0 of period = 3; and heving for z-0 precisely as jump the , or b: given by (203). This solution will have four different analytic expressions. < In coordinates KB, Q and Z= we (where xis the natural coordinate) the equation is: Py. + var: att fy * In the harmonic of order om in ¢ » the variables are separated (m) imp gr”. Mp) 2) for n-0, take J=/Z, — must be one-valued. Differentiate: gm, Za 2mm yz" so / aad / Mi, Me Mme’ big 0} 2 RIL =O # v7 where k is some fixed number to be determined This gives as solutions. ( Imp (Kp) - (e = ke. l Yinp (KH) c Write f= Dre? Sef Lit Inp (Re) © Enup (4) Yop (ex) f Rk m+#o + oe (k) Snap (ku) + Omp (£) nef (fu)f é- die im ) € OL (k) Imp (ku) + Crep (€) Yop (ku) | (6.1) Mto 0) ip (6) Imp (x) + np (4) Yap (4 )} = 2Qu0 and analogous expressions for 2, _ db; j g 37 ¢ « These must be continuous for Ho and their derivation must vanish for M4 and “2 for any Z .- However this condition is not realizable for any value of ke The calculations given in Appendix p4/ show that if non= trivial solutions for Fsand G's are to be found, k must satisfy the following characteristic equations (602) Call be the characteristic valuese Meyer (l:Anhang) indicates a series of methods for their approximate determination. Por instance for p=10, M=1 5 U,=! » u,-.8 4 the first root is already 11.089, the second 19.35 and their orders of magnitude increase proportionally to Mm, and to c&e The solutions dd, j.- May now be expressed in double series of these Bessel functionse Considering that all these functions must vanish when [Z]---o , the four series are {- Die impo y Age fe Cruyy. ON TOVO)- TOV m#o Mz-09 t= ),¢ ony (Am ein (ren - 1° Q@)) (TM Ye) - Fa) VO) ef ¢- ~ TG «km ’ / yyy Bae (TY-TYW) (Te) ¥O)- 7/2) com 2,8 ein? Y" DB ye fe (aYayy. TY(2))(IW Yo) - rv) 6&3) The last operation is to determine the coefficients An, B,, by the condition of smooth joining for 7-0 . First write the continuity of the derivative DTD (4g +82) BS (a) 0 aes (LTE TVOTLTAVO)- TOPO] ee te, where m ~ LIMA) V-IVA@CTO VO)- IOV I Mo fhe, is precisely by definition of kg ene of the set of eigensolutions of Bessel equation, with vanishing derivatives for Ae hy » 2 ; they are complete and orthogonal. Therefore: The last condition is continuity at 2-0 ; write Gm bm, @mbm for the known coefficientsof (202), that is: g=- 2 7 (é€ mS ot) Op Knp (mpy) t by, Lap (mp) f (bah) Met of : . cmpo im ' 7 ‘ y few) (¢ rs, Vf a Knp (mp) + bi, L mp [mpu) - mp f For Z-0 , 6-0 ; and therefore Aj-9, and for m#0, ; ' ror Ww Pasas [an Komp (mt) + bin Lng (map <O m Wy A= EL dem Kp (mp) + ban Imp (mpy)$ which finally determines An ‘s as coefficients of the development in . : : : : . a | 9 me generalized Fourier series in the eigenfunctions Bb, [5 Bry afr wy, | of another function F with bee Feo at ahh, ue » There is of course no question of actually computing all these sa : ca coefficients. However given the orders of magnitude of Kn, the =2he computation of the first coefficients would give excellent precision, as soon as Z leaves a close neighborhood of O , which is physicelly of no interest. 7. Approximate solution. Following the idea already used in 5, part 1, investigate sep- arately the behavior of 6 for $=0 and ¢- a end then fair in between by wee curve arranged to give the correct average for d, « The u, lr ~ Total I. Bound vortex contribution _ Trailing vortex contribution aa x ae contributions of the bound and trailing vortices are investigated sep- arately for the case of solid body rotation blades. frailing vortex. In this case introduce a single exponential decay same for all § 's, and for this purpose use the average of "4 » say a which gives the correct circulation at z-®@. At zZ=0O ,. the corres- ponding contribution to velocity will be 3 e The following is a very rough approximation which has however the merit of great simplicity. Thus P is defined by Uy - 6 ~- (w,-w)r < y= (% A) That is P= (rt) “20 B= Hurd He (uw “25= One wishes to haves § = P(l-fe”) Z>0 6- $ (ze") £<0 Apply Laplace's operator A to the exponential term only: Ad- (30 + 4 wr 26 )H fe’ It is impossible to have Ag=O . Therefore the mean weighted square only will be minimized: Me fk 2 Vaal (4) fo) where Z* has for purpose to favor large values of Z. Integrate in < and 0 ’ At t hee [G+ gi) de 7 fi) 3 t At ve SAME (ae Me) Be B (ad wd) Differentiate with respect to As AN 18 (Me-fe) 2a Ve eS) Be - Aa = 43. + ( a1 ) = S(re-n) |m This gives A = -( Ben (Ce ) (71) ier? This value of A is seen to be independent of = ° ~Pbheo It may be noted that the procedure is also applicable for dif- ferent forms of D. A more refined approach would result if we keep for ? the actual expression for the velocity potential far downstreame The expression for / depends considerably on the weighting factor (z*in our case). For a reasonable result, this weighting factor should give for/] an expression which is infinite for J-0 and 1-62. This in particular limits the possible exponents of a factor of type Z™% An incorrect choice of & would result in a variation of type A A (a) (b) (a) or (b) or more complicated, with more than one root ind for 1-0 » and other phenomenae These facts make difficult the critical q exemination of the whole approximation procedure. Bound vortex. As an approximation, for Z small, consider the action of the two nearest vortices only. The tangential component iss Z an Zee 2G zie rt (-9)© -27= with the assumption, usual in the study of the vorticities near a wing, that the velocity can be considered as induced by only the immediately close section of the bound vortex. This part of the induced velocity will cive another contribution to the total circuletion around the axis: this contribution is "fo for 2-0 « CHAPTER IIT TRAILING VORTICES IN A CONICAL TURBOMACHINE A great simplification resulted in Chapter I from the possibility in the case of a cylindrical turbomachine of assuming in a first study that the field is essentially independent of the exial coordinate. Chapter II gave a refinement of this theory to take the axial coordinate into accounte Such a two-step procedure is unfortunately impossible in the case of a conical turbomachine and a mathematical investigation must take three coordinates into account even if the vortex is assumed "infinite". The following section gives only a preliminary mathematical investigation of the influence function. 8e Preliminary study of the influence function. Let the polar coordinates be reo fixed with respect to the blade row, the walls of the machine being ?* 4)% + Prom the origin flows en incompressible source of strength Q; the emitted fluid rotates like a solid body with angular velocity W . From the circle r’=-4,¢2% are shed p vortices of strength Yo > they are continued by the bound vortices 72% , 0<?<% and the axis 9-0, r>Kh « Neglect all self-induced displacements. The differ- ential equation of the shed vortices is: db grr? ; _ 4h0 oe dr Therefore 6.6 - GRO ns Kd where K= “30 38 Let the origin of O" be defined in such a fashion that the p vortices heave the equation: ¥=6- kr3 = 22k k = 0,1, ... pel (861) The potential V of induced velocities is a solution of Laplace's equation r BWV / ! P) iat t a —_— tat —_—_—__ooOO" OT , v7 =¢ Bo2 satisfying the following boundary and other conditions: For r’<f it is a finite, one-valued, smooth function of r’¢’d ; its P-derivative vanishes for ~-=¢,% For ’'>% it mst be infinite for =%, oo Tk » on any of the vortices; it is otherwise smooth and finite; its Y-=derivative vanishes for %-y 4%; for f7%, it is one-valued, for ¥7<®% , it increases by Yp whenever Ce increases by 7 ° Define C= oO” Kri f= cor p/ rer Four different regions, where V has to be investigated separately, are shown below. =30- The solutions in 1, and 4 are called d, bu « The solutions in 2 and 3 are divided into two parts: ¢, ds satisfying the same conditions ag ?, 4 and d, d, taking care of the circulation. Separating the var- iables, define R and M through imp Vee? Rr) Mw) The equations obtained for R and M are classicale They give, if we introduce a number 2 which is such that “(n+!) is a real, R- p™ or po (ned mp mp if nis not en integer or hh (F ) or Qn (+ ) if nis an integer 2 mp PA" (k) or ar” le) ifn is an integer < mp - . -@)1] (notations of Hobsons p(x) = lim (n-@); RP“ (x) ) +m (nse jyt Case of b,. Pn, d, 2 py In % , $, are to be kept only the terms in re (nzo) -(ne) In g, * bs ere to be kept only the terms in . (n+lz0 One expects N not to be an integer. Therefore write for instance: - = ump mA, , Pp B! Q,”- Ms-co n (863) f= Jem oe” (Aaly Pai) + Bam en (A) Ms ~09 The boundary conditions and the continuity at /te lead to four homogeneous linear equations in A, B's for every m, and M. These equations are compatible only if A pin) ¢ aml) — 2 pla) 4 9™lu,) <0 (Sel) je Pol) ff (be) — FRM) Z (- The same equation being also obtained from 4, $, 3; and the coef= ficients are then defined only in their relative values. The absolute values will be determined later by the conditions of continuity at r= @ Case of dp P de. One wishes to have these as functions of ¢, r fs so as to have the required divergence for rf =O, fe foe Therefore write the solutions ass $, = yt eames De meee (arts arr? \(y pK) . 50,7" (4) je dle eft 7 eink (aap Pr) Oe Pop) 6 Ge" (1) Mz-0F Y 3 bi: - The domain of variation of 1 and the various coefficients are to be determinede For this purpose write the circulation term as + (- 2) + tke’ as ent tke? % yo cmp ZR - Case of m= +; one needs sta 9 gp - ); (Ar 4 Bret )(C Palp) + DAn(«)) oo = )! (Ar, Bir "dic! Fr (@) + D’Qn ()) + Ser 3 for any "#3 , one finds again the homogeneous conditions studied for p, ps : the corresponding terms have already been included in these parts of the potential. for nz3 2 & non-homogeneous set of equations is obtained which gives: go - vK FE iy ud) [Pa'(2)- QP (2) JlPijat)- VOPOS P(2) QU) -A"2) PC) oO (805) Pr? 4 - xk (1c) (2°) P- Pa I (Pl) (2) -2 (0) PJ ps, 7k ° Peja) a2) PW) 2h ; da where ((2)for instance stands for — @, (x) yu 3 lice, =33- -~Gase of N#0, one wants the boundary conditions to be satisfied by: gimme On) (Ann PZ + Bye, act ) pinh, —emp Kr3 P cr Brin Un) + e a ie d. (x ar (aw) +f ”) (Ania Pn iP The solution will have to be smooth at Ho and its ft -derivative to vanish at hk, kz for any 7. For this purpose, develop¢g again the circulation term in series: , _tmp Kr : ; Z e uy e mp 7 iy | YK r3 iyk?mpr on mp 2n mp ee an moe = Co +esr* t &« rote: x - By identification one obtains 1) that the coefficientof (av) o tmp mp are zero - 2) that the coefficients Amn : Brin ° Anin ) Brin of Pr Pan m in ’,” 4, must for each m,n satisfy a set of four non-homogeneous linear equationse Remember that: m 40," mn AO gem p( meet) (meet? J Pa qa - Wy, "ye ~ Jeu? [ (momel ) (name? ) 2 2 m ~ In (Say ) /-42 (Magnus-Oberhettinger, pe 83) Then the final result is: =3he a rm Seite bag pi [PQ(2)- OP (2) [(P(2)0(1)-9'() PW) i _ z (' 3) f, yee [oq PI) OU)- PO)@® “a m (2) -9(0) Pi2)) (8-8) ne Dy 2 imeke bq pq [ PAG)- OPO [feo a'(2? - ue) eS Ce Prejart:)- P(00'e) | q + tx 27 Wp tm where P(2) stands for a P "/ de "3g Lye + ££ 39% mp — f 39.< mp _ The complete formal solution of the problem requires as a final step the development of the above function teken for ‘= in a Fourier series of the eigen-functions. corresponding to the different solutions of the characteristic equation, for fixed m. The result would have to be integrated over foe The corresponding integrals will not be written down explicitly. EXPLICIT DETAILS OF DERIVATIONS Page ( 6 ) The four boundary conditions are: am Ko + bmI, — aim Ko - bm Io = 0 Qm K2 + bm 15 -o - A'm Ki - by 1 = 0 r A Where Ky du K mp Cp, for instance The determinant of the system is Ko [2 -Ko -lo Ko J, O O Ko To -K' -I5 Ko Io O O Ko fo Ky I, A= = = x Kho O O Ko I, Ky 12 Ko Io KY, 14 ) Oo -ki - 0 o -ki -l A-- Kl - Kile fe and ee am = I, (Kg UM - Kile) er bm = - Ki (Koll 4 Is) x eee am = Mi (Ko ls - Ke Is) Zp b= ~Ki (Ko - Kb 10) BR BR B&B B&B Page (9) By integration by parts, we obtain fies Rothe = Tye mp py, hing i, (uo K fe J, Ie uta Vans qe A Wh pm Je Jo) Kp (eppo/ Yo f2 nt ay To SE a yty Lit pn [0 (aoe. Jt rp a) ee [he qin fuc a? +2 fy J Pp Ip ( xe) mp (mp 00) Ger and same formulas are valid with Jn [u) replacing Kn |): The integrated terms summed over m yield: fi cm mps (Ta) § UR K- Kel )(L/K” Kil’) Ke KG 12 + (UK -KiI)(Kel'- K'F2 ) c2 = 7) copm $ Tu) met And the remaining terms yield: Xf - 2? copms for. Fm, Ki, oy -K'iN, [ [ler hae -Ki1) (I Ko- Kb lo) dec t (Mt + [ts (oor Zig. Ki 1) (ly Ko- KS to) quo | f ta -37= Nicholson's approximation is (see 9) Kn (yu) v - ny aS Y fs (treet) e 7h) lax (ryt) MA /+ J teed b= a + 40 In (44) By one of the recurrence formulas on functions /,/( we obtain then for I he and Kp the following approximations: A Kinyo + 2H pg thd y + WI) e 7) - Ely) + fu) Ad ay Jw e+e ) qe n (Ye on x (yet) ( and therefore: (td) (ny) (ke) t(Ke) Kilt -K¢l ; Carh [n(é-t)/ € +e lk (1402) (14 4e) "% It results from an oral communication of Re Se Phillips that precise upper and lower bounds can be found for the ratios 4/7 and Kix « The first integration by parts has left the derivatives only in terms of type /'(/) or Ki), and they enter to the seme degree in both numerator and denominator. Therefore the additional uncertainty #38- on [' kK’ (coming from the arbitrariness of the choice of recurrence formula giving MK’ ) is immaterial insofar as it is translated by a factor independent of t, and very close to 1 for p not small. Substitute into d. 3 the expressions derived for the Bessel func~ tion combinations: ng. - (les JI copms fr - pm [75 £ z (Inui) Crh mp (E-€2) cork mp (6,- bi) fut eo (tury Jinh mp (Ee -&) pm [I (/+ so) % coh mp (t-b.) Cooh mp oO yy y fo (144) sah mp (r-b1) Switch to variable é: 27, 2 Vy y de = Cea j Gas) afte = (i+uc) dk (sj and integrate by parts again &t [ [la lyads I pm coh pm (bo-b) dto = 4 é z Me (ly) “sink mp (eb) - f sEP opi P) Aids pm (bobs) oto &y a) and similarly for the other integrals, and the final result is: - rt , st “TE OM Tay mer ainh mp (Ee-& ) “[" sD lnas®) coh pm (t-b,) sah mp — t abo Junk hap (2-&) ff 211 eas ay, cao pn [b-b2) vinch mp (to-t) Ie Page (7 / ) Summation of the series on the kernel -2p (€2-& Develop the cosh in sums of exponentials and neglect e pl(E2-E) é , oe 2nd -f he + — d[To(leaBa] TD ¢ . mp (evbo - the eit) t “2s Lalo tte) © , Wat oMbltet 6S) e mp (t-bo + ees ~ Me +S), mpll + 26,- bo +05) e mp (&+ bo - 262 ~ 5) mp (bo-& 0S) mp (L-bo + Uy - Bg 0S) mp (~b-bo + 2 ye +@ -@ —~€ é Ye 7 mp (E+bo - Ure tt ee Lay) fe p Creed) E mee ako os Erbe - 2g SF ¢erl (Eb +bo + Ly Bez 4d) e mp(E-bo #05) _ @ mp (-6-bo + 4, 3), 9 MPC 2-65) All the moduli of the geometric series are</ »« Dropping the terms in e 4 l&-%) which still subsist , and summing , } tg IE Ts he] g Piette te +03) oO (+ ote ts < xia | Ib /-e plE+ bo- r+) t ne ~Plbtbs - Uy - es) . é (-bo t&-/y) ff a7, ee teS! (2 or7 ne, £ P "ne Pet be UB) | ote ° [-¢ plto-€+ W) dto Joe Phbe te -5 (+ same terms with -— instead of +{.) las Now the sum of the /two terms of the second line equals & gf] e7 Plbo-b +S) / - [ [ ako _ [ ('4pa)ty t, ot [-@ “Plo b+) The T’ cancels the initial 1’. Finally the first approximation kernel is / St =— Ice +3) a je Pleeb-1T) [-e ~PlE +bo- ez 410) / / / je plese © /-e PlE+ bo - %i 415) “a Je Poet bo - 2; -¢y) Now tf + fk ABI ia -/ AetS _ 3 Ae 5 _) AY] — 2 cops For f=0 i, Lb eg 2 A-I A-| A-! For §-4 fg ! ~~ 2. p -A-i -A-t A+! And the kernel becomes: (the 1 drops out by integration) Ke lee lb lto-t) in e ble tte- &2)_, e@ *Plte- 7) +/- 2 coops e Pleo -t) erhlé tte He) 7 cape -Plé tbo ~ 2) e 2p (6 + bo - er) / ErPlEt bo), 2eapye p (Et bo - 21) for t-0 K ; / e plbx bo - 22) er blest te - 2) = + oe xz /-e Pleo /-e@ Pll tho - ze) [— el (C+ bo Ley) or with terms of order e / (62-6) dropping out: xK e Pe ge Plb tbo - Ue) ef let bo Ce) — = _ If one may neglect oe “plete but not ef (Ee-b) one must add to the kernel for J » the terms: - €+bo- ~bo - é eplte-ts) [e PC re ee) e —plb+bo -t,) C ~ &p (é2-ti) [e ple 0 6 plE-bo}/ [-e Pe Je ple bo te) ~plbsto- ey /-e -Uplbe- “le plé- fe) e Pb ee) ~ile Pege (22) The Boundary equations are mp Jo + 6 mp Yo - Emp Jo - Omp “=O Frip Jo + Gmp Ya ~ Emp Jo ~ Ouf 5-0 where VK = Ze Tnep (bun The determinant is Jo Vo -Jy -Y% Jo Yo oO Oo J re a Jo Yo O O diy 7 8 7} ’ = 2 (I-10) yHYy O ° Joo, TL 0 o -3, -¥ 00 SI, -¥; and if it is 0, the coefficients are: Fup = Vi (To V3 TE 953 Guy = Il (To¥b Th 45) Finp =~, (Iv) -T) vs) \\ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) =l12Q= REFERENCES Re Ee Meyers Beitrag zur Theorie festgtehender Schaufelgitter Mit. Inst. Aeroe E. T. H. Zurich 1916. Fe Ee Marble. The Flow of a Perfect Fluid Through an Axial Turbomachine with Prescribed Blade Loading. Je Aes Sce August 1948. Se Kawada- Induced Velocity by Helical Vortices. Institute Tokyo, January 1939. N. Florine: Sur quelques problémes de la théorie tourbillonnaire de 1*hélice propulsive et de l'aile d'avion. Service Technique de l*Aeronautiquey Belgique.Bulletin Noe 19, July 194. Also results of computations in Bulletin Noe 20, 1946. He Lambe Magnetic Field of a Helix. Proce Camb. Phil. Soce 2h (1934) pe 477 Pe Jacottet: Hlektromagnetische wirbelfelder. Archiv fir Elek- trotechnik 39 (1948) p. 8 Ge N. Watsone A Treatise on the Theory of Bessel Functions. J» Ackeret: Zum Entwurf dithtstehender Schaufelgitter. Schweitz- erische Bauzeitung. 120. (1942) p. 103. Je We Nicholsons The Approximate Calculation of Bessel Functions of Imaginary Argumente Phil. Mag. 20 (1910) We Magnus. F. Oberhettinger. Formeln und S&tze ftir die Speziellen Funktionen der Mathematischer Physike Berlin 1948.