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

2. 4.2 Oblique Shock Relations
…Mach angle
(stronger disturbances)
A Mach wave is a limiting case for oblique shocks.
i.e. infinitely weak oblique shock

3. Oblique shock wave geometry
Given :
Find : or

4. Galilean Invariance :
The tangential component of the flow velocity is preserved.
Superposition of uniform velocity does not change static variables.
Continuity eq :
Momentum eq :
parallel to the shock
The tangential component of the flow velocity is
preserved across an oblique shock wave
Normal to the shock

5. Energy eq :
The changes across an oblique shock wave are governed by the normal
component of the free-stream velocity.

6. Same algebra as applied to the normal shock equction
For a calorically perfect gas
and
Special case
normal shock
Note：changes across a normal shock wave the functions of M1 only
changes across an oblique shock wave the functions of M1 &

7. and
relation

8. For =1.4
(transparancy
or Handout)

9. Note :
1. For any given M1 ，there is a maximum deflection angle If
no solution exists for a straight oblique shock wave
shock is curved & detached,
2. If
strong shock solution (large )
M2 is subsonic
weak shock solution (small )
M2 is supersonic except for a small region near
, there are two values of β for a given M1

10. 3.
(weak shock solution)
4. For a fixed
→Finally, there is a M1 below which no solutions are possible
→shock detached
5. For a fixed M1
and
Shock detached
Ex 4.1

11. 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, , T2 are lower
Integration (Taylor &
Maccoll’s solution, ch 10)
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

12. 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.

13. Shock polar
Increases to
(stronger shock)
Locus of all possible velocities behind the oblique shock
Nondimensionalize Vx and Vy by a*
(Sec 3.4, a*1=a* adiabatic )
Shock polar of all possible for a given

15. Important properties of the shock polar
For a given deflection angle , there are 2 intersection points D&B
(strong shock solution) (weak shock solution)
tangent to the shock polarthe maximum lefleation angle for a given
For no oblique shock solution
3. Point E & A represent flow with no deflection
Mach line
normal shock solution
Shock wave angle
5. The shock polars for different mach numbers.

16. ref：1. Ferri, Antonio, “Elements of Aerodynamics of Supersonic Flows” , 1949.
2. Shapiro, A.H., ”The Dynamics and Thermodynamics of Compressible
Fluid Flow”, 1953.

17. 4.5 Regular Reflection from a Solid Boundary
(i.e. the reflected shock wave is not specularly reflected)
Ex 4.7

18. 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
Shock – expansion
Shock – free boundaries
Expansion – expansion
Wave interaction

21. (+)
(-)
(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.

22. diagram for sec 4.5

23. 4.7 Intersection of Shocks of Opposite Families
C&D:refracted shocks
(maybe expansion waves)
Assume
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.
and have
(the same direction.
In general they differ in magnitude. )
Dividing streamline EF
(slip line) If
coupletely sysmuetric
no slip line

24. Assume and are known & are known
if solution
if Assume another

25. 4.8 Intersection of Shocks of the same family
Will Mach wave emanate from A & C
intersect the shock ?
Point A supersonic
intersection
Point C
Subsonic

26. (or expansion wave)
A left running shock intersects
another left running shock

27. 4.9 Mach Reflection
( for )
( for )
A straight
oblique shock
A regular reflection is
not possible
Much reflection
Flow parallel to the upper
wall & subsonic
for M2

28. 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

29. 4.11 Three – Dimensional Shock Wave
Immediately behind the shock at point A
Inside the shock layer , non – uniform variation.

30. 4.12 Prandtl – Meyer Expansion Waves
Expansion waves are the
antithesis of shock waves
Centered expansion fan
Some qualitative aspects :
M2>M1 2.
3. The expansion fan is a continuous expansion region. Composed of an infinite
number of Mach waves.
Forward Mach line :
Rearward Mach line :
4. Streamlines through an expansion wave are smooth curved lines.

31. 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.

32. …tangential component
is preserved. as
…governing differential equation for prandtl-Meyer flow
general relation holds for perfect, chemically reacting gases
real gases.

34. Specializing to a calorically perfect gas
--- for calorically perfect gas
table A.5 for
Have the same reference point

35. procedures of calculating a Prandtl-Meyer expansion wave
from Table A.5 for the given M1 2.
M2 from Table A.5
the expansion is isentropic are constant through the wave