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8.4  Computer  Program  STP                                           257



            To  calculate  the  location  of  the  onset  of  transition  it  is  usually  sufficient
         to  perform  neutral  stability  calculations  at  four  x-locations  so  that  four  fre-
         quencies  can  be  computed  and  amplification  rates  determined  on  four  curves
         with  constant  dimensional  frequencies.  While  STP  is  general  and  can  be  used
         for  external  flows  with  suction  and  injection  as  well  as  free  shear  flows  such  as
         jets  and  wakes,  changes  are  required  to  accommodate  boundary  conditions  and
         provide  better  estimates  of  initial  eigenvalues  where  required.  STP  can  also  be
         used  for  flows  with  heat  transfer  as  discussed  in  [3].
            STP  consists  of MAIN and  four subroutines  -  VELPRO,  CSAVE,  NEWTON
         and  NEWTONI   -  and  are  described  in  the  following  subsections.

         8.4.1  M A I N

         In  MAIN  the  total  number  of  x-stations  (NXT)  and  the  x-station  where  the
         stability  calculations  begin  (NXO)  are  identified  together  with  the  requirement
         of neutral  stability  (IXT  =  0)  or  transition.  In  the  latter  case,  it  is necessary  to
         specify  the  number  of  frequencies,  and  this  is  done  by  setting  IXT  to  1,  2,  or
         any  number  other  than  zero  but  less  than  twenty  so  that,  for  example,  1  will
         lead  to  transition  calculations  for  one  frequency  and  2  for  two  frequencies,  etc.
            To calculate  the  neutral  stability  curves,  the  dimensionless  external  velocity
         UE(I)  (=  Ue/uoo)  is required  as  a  function  of  surface  distance  S(I)  (=  s/L),  to-
         gether  with  Reynolds  number  Ri  (=  u^L/v),  a reference  length  L  and  velocity
         UINF  (=  i^oo)- The  stability  Reynolds  number  REY(I)  (=  R)  is then  calculated
         from  Eq.  (8.1.7)  with  R x  given  by

                                     Rx  =  u e  ( | )  R L                 (8.4.5)

         The  calculation  of the  eigenvalues  a  and  UJ at  NX  =  NXO  (s  =  SQ)  also  requires
         the specification  of their  initial estimates  at the Reynolds number  corresponding
         to  its  value  at  SQ. For  Blasius  flow,  they  can  be  obtained  from  the  already
         constructed  stability  diagrams,  such  as  those  given  by  Fig.  8.1.  When  other
         figures  are  used,  care  should  be  taken  to  ensure  that  the  estimated  eigenvalues
         have  the  same  length  and  velocity  scales  as  those  used  in  STP.
            The  velocity  profiles  u  and  u"  needed  in  the  stability  equation  are  obtained
         by  calling  VELPRO,  and  the  calculations  for  neutral  stability  are  performed
         by  calling  NEWTON  and  for  amplification  rates  ai  and  n-factor  by  calling
         NEWTONI.


         8.4.2  Subroutine  VELPRO

         The  velocity  profiles  for  STP  can  be  generated  by  any  boundary-layer  method.
         Here  we  assume  that  they  are  provided  by  BLP  of  Section  7.4.  The  read-in
         values  of KX  and NP  in this subroutine  refer  to the  NX-station  and the j-points
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