Ghostscript wrapper for D:\Digitalizacja\MTS81_t19z1_4_PDF_artyku³y\mts81_t19z3.pdf M E C H A N I K A TEORETYCZNA 1 STOSOWANA 3, 19 (1981) ON THE CALCULATION OF THE VELOCITY INDUCED BY A VORTEX-SOURCE CONE K. V A R S A M O V , K. Y O S I F O V , A. H A I M O V (WARNA) The present paper describes the transition from numerical quadrature to linearized expressions for calculation of the velocity induced by unit vortex cone at points near the vortex sheet. A criterion is obtained by comparing the results with the exact solution for singularity distribution of constant strength. I, The method of singularities is often used for solving axisymmetric potential flow problems in doubly connected regions (ring aerofoils, bodies in ducts, etc.). The bodies- are represented by a vortex or source-vortex distribution along the so called camber sur- faces. The strength of the singularities gives the velocity at any point of the flow by nume- rical quadrature. The numerical integration for the velocity at a control point near a vortex, sheet becomes inaccurate due to the singularity of the integrand at points on the sheet. Consider a continuous vortex or source-ring distribution of strength y(s'), q(s'), res- pectively, along a camber line s (see fig. I) and the control point M located near s. Any component of the velocity induced by such a sheet may be written as N 0) C - F(S, S')ds' - 456 K. VARSAMOV I I N N I where A( are the weight coefficients of the quadrature. It was mentioned in [I] that the increasing error of quadrature formula for small values of b forces a special consideration of the interval A - £ j—fj_i. Eq. (I) can be expressed in the form i- l  N (2)  C =  £ where ( 3 ) c* The exact solution of the integral (3) can be found by using linearization of the camber line and the vortex strength and setting J= i (4 ) atfO - ««+/•  *', where a, e , / a r e the corresponding derivatives and gń ,, y,„, jfm—- the  mean  values. F or  the  axial  and  radial  velocity  components  from  (I) in  [I] and  [2]  we  obtain: - d A+ a- \ b\ ON  THE  CALCULATION  457 where  the  notation  is  expedient  from  fig. b  , b2 c = - A ~ c 2.  The  above  equations  for  the  induced  velocity  have  been  applied  to  the  numerical solutions  [1]  [3] of the  inverse  hydrodynamic  problem  for  propelling  complex  in  partially nonlinear  formulation.  They  have  been  used  not  only  for  calculation  of  the  camber (p  =  0)  lines, but  also  for  the  construction  of  duct's  profile  in  case  of  control  points  si- tuated near the camber line. For the points sufficiently distant from the camber line the contribution of the vortex-source cone containing in its interval the control point is ob- tained from (1). The trapezoidal rule is used in both solutions. It is obvious that the boun- dary value of b, which gives more accurate result using eq. (5-̂ -8) than the trapezoidal rule must be obtained for the considered interval. The purpose is to obtain a smooth tran- sition in accuracy for both formulas. 3. The solution of the problem in [1] is obtained empirically by numerical tests in quite a narrow interval along the duct profile (~ 0.5) and about 20 points used in the case of large relative thickness. The results of computations [3] show that it is not convenient to use a fixed value of b like a transition criterion from (1) to (5-^8). Obviously, the varia- tion of the number of points with other parametres fixed changes the length of the inter- val A (fig. 1) and the relative position of the control point M towards the influencing cone. That is why the comparison of the accuracy of (5 + 8) and the quadrature formulae will be made for normalized values of b, q = —,-, p = —7-. Only the main components 11* Qm Qm and v* will be discussed. 4. Consider the influence of vorticity distributon of constant strength placed along a cylindrical surface. Hence, assuming that y = const, q = const, where a = 0, e = 0, / = 0, c = 0, d — b2, from eq. (5) and (8) we obtain the expressions: do) 5?-—«; y y v (11) 0* = — v t where P 1 458 K.  VARSAMOV  I  I N N I I t  is  sufficient  to  analizę   eq.  (10). The  simple  chosen  scheme  gives  the  possibility  to calculate  the  exact  values  of  the  velocity,  substituting  the  vorticity  distribution  by two equivalent  source  disks  (fig.  2).  This  suggests  the  solution (12) u*  = u Dl +u Dx , Rys.  2 where (13) »D = - f -̂W . * ) ] } . K(k), E(k),ll(m 2 , k) are the complete elliptic  integrals  of first,  second and third kind with - V-  "argen ts K ,   »"  - F ig.  3, 4, 5 and 6 show  certain results  of calculations obtained by  the use  of  the three methods  (eq.  (1), (10)  and  (12)  for  which  computer  programs  were  written. The results linearized  method — —  trapezoidal rule o  exact  values - 0.025 - 0.050 - 0.075 - 0.100 Rys.  3 trapezoidal  rule o  exact  values 0.025 0.050 Rys.  4 0.075 0.100 linearized  method — —  trapezoidal  rule o  exact  values -0.1 -0.2 -0.3 -0.4 -05 q Rys.  5 [459] -0.6 -0.7 -0.8 460 K .  VARSAMOV  I  INNI -1 1 1 f 9>9m linearized  method trapezoidal  rule o  exact  values 0.1 0.2 Rys.  6 1 I - - 1 1 - • 1 p \ \ B i i V X i i i A 0.5 0./. 0.3 0.2 0.1 o - P - - —-cr i i i o exact values — — approx.-polynomial y^ B \ ^ i i i i i / /o - - 1 0.6 0.5 0.i 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 q Rys.  7 of  (10) and  (12)  are in  good  agreement,  specially  for  small values  of \q\,  where the  error of the  quadrature  formula  increases rapidly. Increasing the value of q we  obtain  that, the accuracy  of  both  methods  becomes  equal  and  next  the  results  from  (1)  are  better. The  behaviour  of  the  solution  permits  to  obtain  the  curve p  = f(q)  plotted  in  fig. 7 which,  gives  the  same  accuracy  in  using  both  formulae.  In  the  interval  — 0,6  < q < 0,6 the  function  is  approximated  by  the  polynomial (14) p  = - 0 , 5 3 9 3 • q4-0,0852 - g 3 + 1 , 4 9 4 - q 2 - 0 , 4 4 5 - 1 0 ~ 3 g + 0 , 3 8 6 • 1 0 ' O N   THE  CALCU LATION   461 The  resulting  formula  defines  two  regions A and B which correspond to the  sufficiently accurate  use of eq.  (10)  and (I),  respectively. As  a concluding  remark it should  be mentioned  that  eq.  (14)  permits  the automatic transition from  numerical  quadrature to linearized  expressions  in calculating  the  velocity,, induced  by a vortex- source  come  in points  of its neighbourhood. References 1.  M. POPOV, K.  VARSAMOV, Ducted Propeller Design Method,  Theor. and Appl.  M echanics, N r. 2, 1 1970- (in  Bulgarian). 2.  K.  YOSIFOV, K.  VARSAMOV,  G . G EN TCH EV, On the Ducted Propeller Design, R eports  of the F irst N a - tional  Conference  on F luid  M echanics  and F luid  M achinery,  Varna.  1975 (in Bulgarian). 3.  N .  LUDSKAN OV,  L.  PAN OV, Potential Flow of Incompressible Fluid around Arbitrary Profile,  Annual Journal  Appl.  M echanics,  N r.  2, vol. I ll, 1967 (in Bulgarian). P  e 3 w  M e B t r a H C J I E H H E  C K OP OC TH   I I OP O)KJl, E H H Oń BH P OBŁ IM   K O H YC O M B  pa6oTe  paccMaTpjraaeicH   nepexofl  OT BbraHCHHTenbuoft  KBaflpaiypti flo jmHeapH30BaH£ix  4>°P- CKOpOCTb n o p O WfleH yK )  eflH H IWH BIM   BH pOBblM   KOHyCOM  B TOIKaX  B OKpeCTHOCTH  BH XpeBOrO  CJIOH . H 3  cpaBH eH H H   p e 3 yn b T a T 0 B  c  TO^JHbiM   pein eH H eiw  n o ji y ^ e H   KpH TepH H   fljia  c H H r y ji a p H o r o  p a c n p e - Buxpa  o  n ocT om m oH   H H TCH CH BH OCTH . Streszczenie WYZN ACZAN IE  P R Ę D KOŚ CI PRZEPŁ YWU   WYWOŁ AN EJ  WIROWYM   STOŻ KI EM W  pracy  przedyskutowaliś my  przejś cie  od  kwadrtury  numerycznej  do  zlinearyzowanych  wzorów obliczeniowych,  z których  wyznaczona  został a  prę dkość  przepł ywu  wywoł ana  jednostkowym  stoż kiem wirowym  w punktach  w pobliżu  warstwy  wirowej.  Przez  porównanie  wyników z rozwią zaniem  ś cisł ym otrzymano  kryterium  n a rozkł ad  osobliwoś ci  o stał ej  wydajnoś ci. Praca został a zł oż ona w Redakcji dnia 11 listopada 1980 roku