Ch4 Oblique Shock and Expansion Waves
4.1 Introduction
Supersonic flow over a corner.
4.2 Oblique Shock Relations
  sin
1
 
1
M
…Mach angle
(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



(

u
)




Momentum eq : ( u  ds )u  
d



f
d


pd
s

 t


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   P1  P2 
P1  1u12  P2   2u22
Energy eq :
 


Q  Wshaft  Wviscous   Pu  ds    ( f  u )d
s




u2
u2  
  [  (e  )]d    (e  )u  ds
2
2
 t
s
 2
 2
u1
u2
 ( P1u1  P2u2 )   1 (e1 
)u1   2 (e2 
)u2
2
2
2 
2

u
u

2
2
h  1   h  2 
u

u


1
1
1
 1 2   2 2 

 


u12
u22
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  M1 sin 
For a calorically perfect gas

2
  1Mn12

1   1Mn12  2
P2
2

 1
Mn12  1
P1
 1

Mn12   2
   1
Mn22 
2
 Mn 2  1
   1 1
M2 
and
T2 P2 1

T1 P1  2
Mn2
sin    
Special case  


2
normal shock
Note：changes across a normal shock wave the functions of M1 only
changes across an oblique shock wave the functions of M1 & 
tan  
and
u1
1
tan    
u2
2


tan 
u1  2
  1Mn12
  1M12 sin 2 

 


2
tan    u2 1   1Mn1  2   1M12 sin 2   2
 M12 sin 2   1 
tan   2 cot   2



M


cos
2


2
 1

  M
relation
For  =1.4
(transparancy
or Handout)
Note :
1. For any given M1 ，there is a maximum deflection angle
If
   max
 max
no solution exists for a straight oblique shock wave
shock is curved & detached,
2. If
   max
, 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   
4. For a fixed


2
or  
M1   
(weak shock solution)
M1   
→Finally, there is a M1 below which no solutions are possible
→shock detached
5. For a fixed M1
Ex 4.1
   , P2 , T2 and  2 , M 2 
   max  Shock detached
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,
The flow streamlines behind the shock are
straight and parallel to the wedge surface.
The pressure on the surface of the wedge
is constant = P2
Ex 4.4 Ex 4.5 Ex4.6

2 , T2 are lower

Integration (Taylor &
Maccoll’s solution, ch 10)
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
 V2 
C
(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 M 1
M2 
M * 1
  1
    1
*
2

M 
 1
M* 
 2.45,
 1
for
  1 .4
M * 1
2
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 max for a given M1*
For 0   max  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
 1
 

a *   a *  1 


2
2
V 
M 1*   x*  M 1*  1
 1
a 
2
 Vy 
 * 
a 
2
 
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  M1   2     1
(i.e. the reflected shock wave is not specularly reflected)
Ex 4.7
4.6 Pressure – Deflection Diagrams
Wave interaction
-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
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
1.
2.
P4  P4'

V4

V4'
and
have
(the same direction.
In general they differ in magnitude. )
 s4  s4'
•Dividing streamline EF
(slip line)
•If  2   3 
coupletely sysmuetric
 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 ?
 supersonic
Point A
sin  
u1
V1
 u1  a1
a1
V1
 intersection
sin 1 
   1
Point C
sin  2 
a2
V2
sin     
 Subsonic
u2
V2
u 2  a2
       2
 intersection
(or expansion wave)
A left running shock intersects
another left running shock
4.9 Mach Reflection
(  max for M 1 )
   (  max for
A straight
oblique shock
M2 )
A regular reflection is
not possible
Much reflection
   max for M2
Flow parallel to the upper
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  M1i  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
2.
P2
P1
 1,  2
1
 1, T2
T1
1
3. The expansion fan is a continuous expansion region. Composed of an infinite
number of Mach waves.
Forward Mach line : 1  sin 1  1 M1 
1

Rearward Mach line : 2  sin  M 2 
4. Streamlines through an expansion wave are smooth curved lines.
1
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 
1
 d 
dV
dV
1

V
1  d tan 
  sin 1
V
tan 
 d  M 2  1
tan  
dV
V
as
1
M
1
M 2 1
d  0
…governing differential equation for prandtl-Meyer flow
general relation holds for perfect, chemically reacting gases
real gases.
2
d
1

M
 M 2 M 2  1
V  Ma
1
dV
V
dV
?
V
dV  Mda  adM

dV da dM


V
a
M
da
?
a
Specializing to a calorically perfect gas
 1 2
a  T
  0   0  1
M
T
2
a
2

  1 2 
a  a0 1 
M 
2



1
2
dV
1
dM

 1 2 M
V
1
M
2
2
M
 1 dM
2
M2
d




0

2
 1
 M1   1 2 M
1
M
2
let
vM  
M 2  1 dM
 M   
 1 2 M
1
M
2
  1 1   1 2

tan
M  1  tan 1 M 2  1
 1
 1
2   M 2   M1 
Have the same reference point
--- for calorically perfect gas
table A.5 for   1.4
  M 
• procedures of calculating a Prandtl-Meyer expansion wave
1.  M1  from Table A.5 for the given M1
2.
 M 2   2  M1 
3. M2 from Table A.5
4.  the expansion is isentropic
1
 1
M 22
T1
2
 
T2 1    1 M 2
1
2
  1 2 
1
M2 
P1 
2


P2 1    1 M 2 
1
2



 1
T0 , P0 are constant through the wave