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10.10  Model  Problem  for  the  MacCormack  Method:  Unsteady  Shock  Tube  313



         not.  In  fact,  it  will  be  shown  in  Problem  P10.4  that  the  first  procedure  is  best
         suited  for  discontinuities  moving  in the  — x  direction,  while the  second  approach
         is best  suited  for  discontinuities  moving  in the  +x  direction.  It  can  be  seen  that
         a  numerical  boundary  scheme  is  necessary  on  one  side  of  the  predictor  step,
         while  a  numerical  boundary  scheme  is  necessary  on  the  opposite  side  on  the
         corrector  step.  This  is  true  for  both  forward-backward  and  backward-forward
         procedures  described  above.



         10.10  Model   Problem    for  the  MacCormack      Method:
         Unsteady    Shock   Tube


         To  demonstrate  the  solution  of  the  Euler  equation  for  a  one-dimensional  com-
         pressible  flow  with  the  MacCormack  method,  the  model  problem  of  a  unsteady
         shock  tube  is  used  as  shown  in  Fig.  10.12.



                   Diaphragm

              P                P
               \         J      5


         Fig.  10.12.  Shock  tube  model  problem:  Initial  conditions.



            The  tube  is  filled  with  a  gas  at  different  states  on  the  left  and  right  side  of
         a  diaphragm.  The  gas  states  have  different  densities  and  pressures  and  are  at
         rest.  At  time  t  =  0,  the  diaphragm  is  broken  and  if  it  is  assumed  that  viscous
         effects  are negligible and the tube  is of infinite  length  (reflection  waves are zero),
         then  the  unsteady  Euler  equations  for  a  one-dimensional  flow  can  be  solved
         analytically  with  a  family  of  characteristics  travelling  to  the  left  and  right  of
         the  diaphragm.  If  the  left  side  contains  the  gas  at  the  highest  pressure,  the
         right  state  will  expand  in  the  left  side  region  through  expansion  waves  (region
         2),  whereas  a  compression  wave  will  travel  in  the  right  direction.  This  will  be



                                1

         p                3     |   4      P
          l                                 5
                                1
                  2
         Fig.  10.13.  Shock  tube  model  problem:  gas states  at  time  t  >  0.
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