4.1 Introduction

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# 4.1 Introduction - PowerPoint PPT Presentation

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 :

or

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

and

and

relation

For =1.4

(transparancy

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

If

no solution exists for a straight oblique shock wave

shock is curved & detached,

5. For a fixed M1

and

Shock detached

3.

(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

c.f

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*

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

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.

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

intersection

Point C

Subsonic

intersection

(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

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.

• Consider the infinitesimal changes across a very weak wave.
• (essentially a Mach wave)

An infinitesimally small flow deflection.

…tangential component

is preserved.

as

…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