Ch4 Oblique Shock and Expansion Waves 4 by P7q0qv

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									Ch4 Oblique Shock and Expansion Waves
4.1 Introduction
  Supersonic flow over a corner.
4.2 Oblique Shock Relations




                                        1
                                  sin1
                                                …Mach angle
                                        M

                                         (stronger disturbances)


                  A Mach wave is a limiting case for oblique shocks.

                                    i.e. infinitely weak oblique shock
Oblique shock wave geometry




                              Given :   V1 , 1 , 

                               Find :   V2 , 2 ..., 


                                 or

                              Given :   V1 , 1 , 

                              Find :    V2 , 2 ..., 
Galilean Invariance : 1  2  
The tangential component of the flow velocity is preserved.
Superposition of uniform velocity does not change static variables.
Continuity eq :
                      1u1 A1   2u2 A2  0        A1  A2

                   1u1   2u2
                                    
                   
Momentum eq : ( u  ds )u    ( u )                     
                           t      d   fd   pds
           s                                           s

                      • parallel to the shock
                       1u1 1  2u2 2  0  1  2
                      The tangential component of the flow velocity is
                         preserved across an oblique shock wave
                      • Normal to the shock
                      ( 1u1)u1  2u2 u2   P  P2 
                                                    1

                      P  1u12  P2   2u2
                       1
                                           2
Energy eq :                                               
               
              Q  Wshaft  Wviscous   Pu  ds    ( f  u )d
                            
                                      s             
                                                   
                            u2                     u2  
                [  (e  )]d    (e  )u  ds
                  t          2           s        2
                                              2                  2
                                             u1                  u2
               ( Pu1  P2u2 )   1 (e1 
                    1                            )u1   2 (e2      )u2
                                              2                    2
                     2             2
               h    u1  
                           h     u2         
                                                 u 1  u1  1
                                                          2      2
                1 2   2 2 
                                       
                            

                      u12      2
                              u2
                  h1   h2 
                       2      2

      The changes across an oblique shock wave are governed by the normal
      component of the free-stream velocity.
 Same algebra as applied to the normal shock equction

                   Mn1  M 1 sin 
For a calorically perfect gas

          2
             
                 1Mn12
          1   1Mn12  2
                   2
          P2
          P
              1
                   1
                          
                       Mn12  1     
           1

                 Mn12   2        
                        
                             1
                                                      T2 P2 1
          Mn2 
            2
                                                         
                2         Mn 2  1         and      T1 P  2
                   1 1
                                                         1


                   Mn2
         M2 
                sin    

                           
     Special case                    normal shock
                              2
Note:changes across a normal shock wave the functions of M1 only
     changes across an oblique shock wave the functions of M1 & 
                    u1
          tan  
                    1
  and
                           u2
          tan     
                           2



     tan      u1  2
                    
                          1Mn12    1M12 sin 2 
  tan     u2 1   1Mn1  2   1M1 sin 2   2
                                2            2




                  M 12 sin 2   1 
 tan   2 cot   2                          M   relation
                  M 1   cos 2   2 
For  =1.4
(transparancy
or Handout)
Note :
         1. For any given M1 ,there is a maximum deflection angle         m ax
           If         m ax        no solution exists for a straight oblique shock wave
                                    shock is curved & detached,

           2. If      m ax   , there are two values of β for a given M1
                                    strong shock solution (large  )
                                        M2 is subsonic
                                     weak shock solution (small  )
                                        M2 is supersonic except for a small region near
                                        max
                     
3.   0              or  
                     2
4. For a fixed           M 1      (weak shock solution)
                          M 1   

    →Finally, there is a M1 below which no solutions are possible
                      →shock detached
5. For a fixed M1       , P2 , T2 and  2 , M 2 
                        m ax  Shock detached

Ex 4.1
4.3 Supersonic Flow over Wedges and Cones

                                               •Straight oblique shocks

                                               •3-D flow, Ps  P2.
                                               •Streamlines are curved.
                                               •3-D relieving effect.
                                               •Weaker shock wave than
                                                a wedge of the same  ,

                                                            
                                               •P2,    2 , T2 are lower
                                                            
 The flow streamlines behind the shock are
                                               Integration (Taylor &
 straight and parallel to the wedge surface.
                                               Maccoll’s solution, ch 10)
 The pressure on the surface of the wedge
 is constant = P2


 Ex 4.4 Ex 4.5 Ex4.6
4.4 Shock Polar –graphical explanations




                                      c.f




                                          Point A in the hodograph plane
                                          represents the entire flowfield
                                          of region 1 in the physical plane.
Shock polar




                                        B   Increases to   C

                                      V2       (stronger shock)




   Locus of all possible velocities behind the oblique shock       max 

                                Nondimensionalize Vx and Vy by a*
                               (Sec 3.4, a*1=a*2  adiabatic )

                                  *              *
    Shock polar of all possible M 2 for a given M1
                                   2
                  M2 
                        1      
                                *2     1
                              M 
                          1
                  M*           2.45,
         M * 1           1
                  for
                    1 .4
M * 1
                  if
                  M 


                  M1  1  M  1
                   *


                  M1  1  M  1
                   *


                  M * 1 M 1
Important properties of the shock polar

1. For a given deflection angle      , there are 2 intersection points D&B

                                 (strong shock solution)  (weak shock solution)
2. OC tangent to the shock polarthe maximum lefleation angle m ax for a given M 1
                                                                                  *




       For 0   m ax  no oblique shock solution

3. Point E & A represent flow with no deflection
                   Mach line
   normal shock solution

4.   OH  AB  HOA  Shock wave angle 

5. The shock polars for different mach numbers.
                                                           M *  Vx  Vx  M *  1
                                                                        2

                                                           1                 
                                                                    a*   a*  1 
                                                    2
                                            Vy                                      
                                            *
                                           a          
                                                             2
                                                              1
                                                                   * 2   V 
                                                                   M 1   x  M 1*  1
                                                                           a 
                                                                             *




ref:1. Ferri, Antonio, “Elements of Aerodynamics of Supersonic Flows” , 1949.
    2. Shapiro, A.H., ”The Dynamics and Thermodynamics of Compressible
       Fluid Flow”, 1953.
4.5 Regular Reflection from a Solid Boundary




   M 2  M 1     2     1

  (i.e. the reflected shock wave is not specularly reflected)

 Ex 4.7
4.6 Pressure – Deflection Diagrams    -locus of all possible static pressure
                                       behind an oblique shock wave as a
                                       function deflection angle for given
                                       upstream conditions.




                             Shock wave – a solid boundary
                             Shock – shock
Wave interaction             Shock – expansion
                             Shock – free boundaries
                             Expansion – expansion
(+)
               (-)




      (downward  consider negative)



      •Left-running Wave :
       When standing at a point on
      the waves and looking
      “downstream”, you see the wave
      running-off towards your left.
P   diagram for sec 4.5
4.7 Intersection of Shocks of Opposite Families
                                                   •C&D:refracted shocks
                                                   (maybe expansion waves)


                                                   •Assume  2  1
                                                    shock A is stronger
                                                    than shock B
                                                    a streamline going through
                                                    the shock system A&C
                                                    experience or a different
                                                    entropy change than a
                                                    streamline going through the
                                                    shock system B&D
                                                   s4  s4 '
 1.     P4  P4'                                  •Dividing streamline EF
                                                (slip line)
 2.   V4    and   V4'  have                        •If  2   3 
      (the same direction.                          coupletely sysmuetric
      In general they differ in magnitude. )         no slip line
Assume      4'   and   4   are known
                                                    '
                                            P4 & P4 are known

 if      P4  P4'        solution


 if      P4  P4'        Assume another    
4.8 Intersection of Shocks of the same family



                                           Will Mach wave emanate from A & C
                                           intersect the shock ?

                                           Point A                     supersonic
                                                sin  
                                                            u1         u1  a1
                                                            V1
                                                                         1
                                                            a1
                                                sin 1                intersection
                                                            V1
                                           Point C

                                                            a2
                                                sin  2 
                                                            V2           Subsonic
                                                                        u 2  a2
                                                sin     
                                                                 u2
                                                                 V2            2
                                                                         intersection
(or expansion wave)

A left running shock intersects
another left running shock
                      (  m ax for M 1 )    (  m ax for M 2 )
4.9 Mach Reflection


                      A straight                   A regular reflection is
                      oblique shock                not possible

                                                     Much reflection




                                               Flow parallel to the upper
     m ax for M2                            wall & subsonic
4.10 Detached Shock Wave in Front of a Blunt Body




                                    From a to e , the curved shock goes
                                    through all possible oblique shock
                                    conditions for M1.




                                     CFD is needed
4.11 Three – Dimensional Shock Wave




                             
             Mn1  M 1i   n  P2 ,  2 , T2 , h2 , Mn2
              Immediately behind the shock at point A
              Inside the shock layer , non – uniform variation.
4.12 Prandtl – Meyer Expansion Waves



                                                              Expansion waves are the
                                                              antithesis of shock waves




  Centered expansion fan
  Some qualitative aspects :
 1. M2>M1
      P2         1,  2         1, T2        1
 2.        P1              1             T1

 3. The expansion fan is a continuous expansion region. Composed of an infinite
    number of Mach waves.
    Forward Mach line : 1  sin 1  1 M 1 
                                    
                                    
                                            
                                            
                                                     1   
    Rearward Mach line : 2  sin  M 2 
                                                1

                                       
 4. Streamlines through an expansion wave are smooth curved lines.
5. ds  0    i.e. The expansion is isentropic.     (  Mach wave)
   Consider the infinitesimal changes across a very weak wave.
                                          (essentially a Mach wave)



               An infinitesimally small flow deflection. d
           V cos   V  dV  cos  d  …tangential component
                                                    is preserved.

                      V  dV      cos 
                            
                        V      cos  d 

                           dV         1
                      1      
                           V    1  d tan 

                                                1
         dV                          sin 1
                                                M
 d         V
         tan                      tan  
                                                    1
                                                M 2 1
                  dV
 d  M 2  1              as   d  0
                  V
   …governing differential equation for prandtl-Meyer flow
    general relation holds for perfect, chemically reacting gases
    real gases.
2        M              dV      dV

1
   d    M 2 M 2  1              ?
            1            V       V


V  Ma          dV  Mda  adM

                          

                 dV da dM         da
                                   ?
                 V   a  M          a
Specializing to a calorically perfect gas
                                                                 1
                                                             
                  1 2                           1 2 
              2
  a  T                                 a  a0 1 
                                                                 2
  0   0  1     M                                 M 
   a   T         2                                2   

    dV        1    dM
      
    V        1 2 M
         1     M
              2

                                        M 2  1 dM
            d   2  0  
        2                      M2
        1                      M1
                                         1 2 M
                                     1      M
                                          2
                                M 2  1 dM
        let        M   
                                 1 2 M
                             1      M
                                  2
                     1 1   1 2
        vM           tan      M 1  tan 1 M 2 1              --- for calorically perfect gas
                    1      1                                         table A.5 for   1.4
                                                                                        M 
     2   M 2   M 1 

                           Have the same reference point
• procedures of calculating a Prandtl-Meyer expansion wave
1.  M 1  from Table A.5 for the given M1

2.     M 2    2  M1 
3. M2 from Table A.5
4.  the expansion is isentropic                 T0 , P0 are constant through the wave

                      1
                1               2
                                M2
      T1       2
      
      T2 1    1 M 2
                    1
               2
                                     
          1 2                         1
          1       M2 
     P1       2
                     
     P2 1    1 M 2 
                      
                    1
               2

								
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