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Ch4 Oblique Shock and Expansion Waves. 4.1 Introduction. Supersonic flow over a corner. 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. Given : . Find : . or. Given : . Find : .

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Ch4 Oblique Shock and Expansion Waves

4.1 Introduction

Supersonic flow over a corner.


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


Given :

Find :


Given :

Find :

Oblique shock wave geometry


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

Energy eq :

The changes across an oblique shock wave are governed by the normal

component of the free-stream velocity.


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 &

Same algebra as applied to the normal shock equction

For a calorically perfect gas






For =1.4


or Handout)


, there are two values of β for a given M1

strong shock solution (large )

2. If

M2 is subsonic

weak shock solution (small )

M2 is supersonic except for a small region near

Note :

1. For any given M1 ,there is a maximum deflection angle


no solution exists for a straight oblique shock wave

shock is curved & detached,


5. For a fixed M1


Shock detached


(weak shock solution)

4. For a fixed

→Finally, there is a M1 below which no solutions are possible

→shock detached

Ex 4.1


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)

4.3 Supersonic Flow over Wedges and Cones

  • Straight oblique shocks

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



Point A in the hodograph plane represents the entire flowfield

of region 1 in the physical plane.

4.4 Shock Polar –graphical explanations


Increases to

Shock polar

(stronger shock)

Locus of all possible velocities behind the oblique shock

Nondimensionalize Vx and Vy by a*

(Sec 3.4, a*1=a*2 adiabatic )

Shock polar of all possible for a given


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

4. Shock wave angle

5. The shock polars for different mach numbers.


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

(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

Shock – expansion

Shock – free boundaries

Expansion – expansion

Wave interaction




(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.

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


2. and have

(the same direction.

In general they differ in magnitude. )

  • Dividing streamline EF
  • (slip line)
  • If
  • coupletely sysmuetric
  • no slip line

Assume and are known & are known

if solution

if Assume another


4.8 Intersection of Shocks of the same family

Will Mach wave emanate from A & C

intersect the shock ?

Point A supersonic


Point C




(or expansion wave)

A left running shock intersects

another left running shock


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


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

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 :

  • M2>M1


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.


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.


…tangential component

is preserved.


…governing differential equation for prandtl-Meyer flow

general relation holds for perfect, chemically reacting gases

real gases.


Specializing to a calorically perfect gas

--- for calorically perfect gas

table A.5 for

Have the same reference point


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