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258                                                8.  Stability  and  Transition



         (J)  where  the  profiles  are  computed.  With  the  specification  of  U(J)  (= ')  and
                                                                           /
         V(J)  (= ")  as  a  function  of  ETA(J)  (=  nj),  UUDP(J)  (=  u'<) are  calculated,
                 /
         by  differentiating  V(J)  with  respect  to  77.

         8.4.3  Subroutine  CSAVE
         This  subroutine  is used  to  obtain  the  solutions  of the  Orr-Sommerfeld  equation
         for  a given  set  of  a  and  u  when the  neutral  stability  curve  is required  or  a r  and
         a,i for the  determination  of the  location  of the  onset  of transition.  The  standard
         problem  refers  to the  solution  of Eq.  (4.4.29)  subject  to the  boundary  condition
               ,f
         that  (f) (0)  (=  s 0)  is  equal  to  1.  The  definitions  of  {c\)j  to  (04)j  in  the  Orr-
         Sommerfeld  equation,  as  well  as  the  edge  definitions,  are  given  by  Eqs.  (8.2.4)
         and  (8.2.6), respectively,  and the  fj  terms  by Eqs.  (8.2.3)  and  (8.2.7). All  values
         of  fj  are  zero  except  for  (r2)o  which  is  equal  to  1 because  of  the  requirement
         that  s 0  =  1.
            This  subroutine  also  contains  the  coefficients  of the  variational  equations  of
         the  standard  problem,  Eq.  (4.4.29),  with  respect  to  a,  u  and  i?, together  with
         the  right-hand  sides  of  these  equations  as  given  by  Eq.  (8.2.16)  for  those  with
         respect  to  a,  Eq.  (8.2.17)  for  CJ, and  Eq.  (8.2.21)  for  R.
            The  variational  equations  are  written  with  respect  to  a,  a;  and  i?,  but  the
         coefficient  matrix  A  in  Eq.  (4.4.29)  is  the  same.  For  this  reason,  it  is  only
         necessary  to  define  those  (fj)  terms  that  are  not  zero.
            This  subroutine  also  contains  the  block-elimination  algorithm  to  solve  Eq.
         (4.4.29)  with  the  procedure  described  in  subsection  4.4.3.  Note  that  AA(1,1,1)
                             (
         and  AA(2,2,1)  denote a n ) 0  and  (0:22)0 and  correspond  to the  "wall"  boundary
         conditions.
         8.4.4  Subroutine  N E W T O N

         For  initial  estimates  of  ALFA(=  a r)  and  OMEGA(=  CJ),  this  subroutine  com-
         putes  the  eigenvalues  a  and  u  according  to  the  procedure  described  in  Sec-
         tion  8.2. The  perturbation  quantities  DALFA(EE  8a r)  and  DOMEGA(=  Sou)  are
         computed  according  to  Eq.  (8.2.14). Upon  convergence  of the  iterations,  the  di-
         mensional  frequencies  WSO(IX)(=  a;*)  are  calculated  and  printed  for  that  NX-
         station,  and  for  each  corresponding  frequency,  IX, together  with  the  values  of  a
         and  UJ which  serve as initial estimates  for the next  NX-station  (or i?), are  stored.
         Also  stored  are  the  values  of  UM(=  / 0 ) ,  UMA(=  df 0/da),  UMO(=  df 0/du),
         UMR(=   dfo/dR),  UE  and  REY.


         8.4.5  Subroutine  N E W T O N I
        In  the  calculation  of  transition,  the  amplification  factor  n  is  computed  for  a
        dimensional  frequency  ou*  according to  Eq.  (8.3.3)  which  in terms  of the  dimen-
        sionless  quantities  used  in  the  solution  of  the  Orr-Sommerfeld  equation  with
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