# 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
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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