Transcript Ch 4
Ch4 Oblique Shock and Expansion Waves
4.1 Introduction
Supersonic flow over a corner.
4.2 Oblique Shock Relations
1
sin
M
1
…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
t
s
f
d
pd
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
s
Energy eq :
Q W shaft Wviscous 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
1Mn12
1 1Mn12 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
1Mn12
1M12 sin 2
2
tan u2 1 1Mn1 2 1M12 sin 2 2
M 12 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
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
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 M1
M2
M * 1
M * 1
2
1
1
*
2
M
1
*
M
2.45,
1
for
1 .4
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 polarthe maximum lefleation anglemax for a given M 1*
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
sin 1
a1
V1
intersection
1
Point C
sin 2
a2
V2
sin
u2
V2
Subsonic
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 M 2 )
A straight
oblique shock
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 M 1
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
1
V
1 d tan
sin 1
dV
V
t an
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
a
T
2
2
1 2
a a0 1
M
2
1
2
dV
1
dM
V 1 1 M 2 M
2
2
M
1 dM
2
M2
d
0
2
1
M1 1 2 M
1
M
2
let
vM
M 2 1 dM
M
1 2 M
1
M
2
1 1 1 2
tan
M 1 tan1 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