hypersonic ae3021_f06_10
TRANSCRIPT
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Hypersonic flow: introduction
Van Dyke: Hypersonic flow is flow past a body at high Mach number, wherenonlinearity is an essential feature of the flow.
Also understood, for thin bodies, that if is the thickness-to-chord ratio of the body,M is of order 1.
Special Features
Thin shock layer: shock is very close to the body. The thin region between the shock andthe body is called the Shock Layer.
Entropy Layer: Shock curvature implies that shock strength is different
for different streamlines stagnation pressure and velocity gradients -rotational flow
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http://www.onera.fr/conferences/ramjet-scramjet-pde/images/hypersonic-funnel.gif
The Hypersonic Tunnel For Airbreathing Propulsion
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Velocity-Altitude Map For Re-Entry
Velocity
Altitude
Typical re-entry case:Very little deceleration untilVehicle reaches denser air
(Deliberately so - to avoidlarge fluctuations in aerodynamicloads and landing point )
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Atmosphere
Troposphere: 0 < z < 10km
Stratosphere: 10 < z < 50km
Mesosphere: 50 < z < 80km
Thermosphere: z > 80km
Ionosphere 65 < 365 km Contains ions and free electrons
60
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A Simple Model for Variation of density with altitude
gdz dp !
M T R
p
=
Neglect dissociation and ionization Molecular weight is constant Assume isothermal (T = constant) poor assumption
dz T R
M g
p
dp
!
!"
#$%
&' z
T R M g
e
log0
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Non-lifting body moving at velocity V, which is inclined at angle to the x-axis:
!DCosdt
x d m "
2
2
mg DSindt
z d m !"2
2
mg S C U dt z d
m D !"sin21 2
2
2
!"
#$%
&S C
m
Dis the Ballistic Parameter.
Assuming that the drag force is >> weight and that is constant because gravitational force istoo weak to change the flight path much
!"#
$%&''!
"
#$%
&RT gMz
mS C
U U
Log De
e expsin21 0
(
)
U
D
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www.galleryoffluidmechanics.com/shocks/s_wt.htm
High Angle of Attack Hypersonic Aerodynamics
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http://www.scientificcage.com/images/photos/hpersonic_flow.jpgy
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Croccos Theorem:
!rr
"uhsT 0
Viscous Layer:
Implies vorticity in the shock layer.
Thick boundary layer, merges with shock wave to produce a merged shock-viscous layer.Coupled analysis needed.
High Temperature Effects:
Very large range of properties (temperature, density, pressure) in the flowfield, so thatspecific heats and mean molecular weight may not be constant.
Low Density Flow:
Most hypersonic flight (except of hypervelocity projectiles) occurs at very high altitudes
Knudsen No. =
L
!= ratio of Mean Free Path to characteristic length
Above 120 km, continuum assumption is poor. Below 60 km, mean free path is less than 1mm.
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http://www.aerospace-technology.com/projects/x43/images/X-43HYPERX_7.jpg
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Summary of Theoretical Approaches
Newtonian Flow: Flow hits surface layer, and abruptly turns parallel to surface.
Normal force decomposed into lift and drag.Modified Newtonian Flow: Account for stagnation pressure drop across shock.
Local Surface Inclination Method : Cp at a point is calculated from static pressure behind an obliqueshock caused by local surface slope at freestream Mach number.
Tangent Coneapproach: similar to local surface slope arguments.
Mach number independence: Shock/expansion relations and Cp become independent of Machnumber at very high Mach number.
Blast wave theory: Energy of Disturbance caused by hypersonic vehicle is like a detonation wave.Hypersonic similarity: Allows developing equivalent shock tube experiments for hypersonicaerodynamics.
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Local Surface Inclination Methods Approximate methods over arbitrary configurations, in particular,
where Cp is a function of local surface slope.Newtonian Aerodynamics
Newton (1687) concept was that particles travel along straight lines withoutInteraction with other particles, let pellets from a shotgun. On striking a surface,they would lose all momentum perpendicular to the surface, but retain all tangential momentum i.e., slide off the surface.
In 3D flows we replace
ASinU !" #22Net rate of change of momentum
!22 SinCp =
!SinU " with nU
rr
!
2
2
2!
!
=
U
nU Cp
r
Shadow region: 0Cp
Shadow region is where 0! nU r
r
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Remarks on Newtonian Theory:
Poor in low speed flow. Predicts .2
!l C
(1) Works well as Mach number gets large and specific heat ratio tends towards 1.0Why? Because shock is close to surface, and velocity across the shock is very large most of thenormal momentum is lost.
(2) Tends to overpredict c p and c d (C D) see figure 3.11
(3) Works better in 3-D than in 2-D(4) In 3-D, works best for blunt bodies; not good for wedges, cones, wingsetc.
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Was proposed by Lester Lees in 1955, as a way of improving Newtoniantheory, and bringing in Mach Number and dependence on
. He proposed replacing 2 with
!M pC
max pC
!2
maxsin p p C C =
Here is the coefficient behind a Normal shock wave,at the stagnation point. That is,
max pC pC
2
02
max
21
!
!=
U
p pC
p#
Modified Newtonian
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From Rankine-Hugoniot relations,
( )( ) !
!"
#
$$%
&
+
+
!!"
#
$$%
&
'
+= (
(
(
( 1
21
124
1 21
2
2202
)
)
)
) ))
M
M
M p p
(3.17)
Then
2
02
2
1
!
!
"
=
M
p p
c p #
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In the limit as ,!M We get
( )
( )1
4
4
1
1
12
+!
"
#$$
%
& +=
'
'
((
(
((
((
pc
As ,4.1 839.1max ! pc
As ,1 2max=
pc Proposed by Newton
Exercise: Compute c p values for configurations shown on Figures 3.8,3.6, 3.11 and 3.12 using Newtonian and Modified Newtonian theories.Biconvex Airfoil.
y/c = 0.05 -0.2 (x/c) 2
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Where does freestream Mach number appear in the above?Only in the dependence of downstream pressure, density, temperature.
As freestream Mach number becomes large,( )( )1
1
1
2
!
+"
#
#
$
$
!"#$
%&
!"#
$%&
+
=
''
'
'
'2
222
22
2 1sin1
2
M M
U p
p p
U p
()
((
*
!"
" 2sin1
2
+=
Why nondimensionalize by2
!
U
Because ( )22 ~ !U O p " And it allows cancellation of Mach number Examine other relations for properties downstream of the shock freestream Machnumber does not appear anywhere.
Mach Number Independence
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The blast wave theory argues that the sudden addition of energy to thefluid by the body is equivalent to a high explosive of energy E being
exploded at time t=0.
A shock wave associated with the explosion spreads away from the originwith time
In 2-D problem: the shock wave is a plane wave:
!
=
U
x t
Shock wave moves outward with tBlast wave origin
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Hypersonic Shock & Expansion Relations
Why?
1. Simpler than exact expressions - for analysis2. Key parameter is seen to be M where is the flow turning angle, for M>>1 and
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!"
#$%
&'
+
+ 1sin1
21 221
1
2(
)
)M
p p
!"
#$%
&'
+
+ 11
21 221 )
*
*M
( ) ( )2
222
1
2 1
4
1
4
11
K K K
p p
+"#
$%& ++
++
'
Defining pressure coefficient
21
1
2
2
1
M
p p
C p!
"#$
%&' (
)
!!"
#
$$%
&+
()
*+, +
++
=
'()
*+, -
. 2
2
2
1
2
2
1
4
1
4
12
2
1
K K
p p
C p //0
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Next
( )( ) 21
221
1
2
1
1sin1
M
M
v
u
+
!!"
#
In the hypersonic limit,
1
sin21
2
1
2
+
!
#
$v
u
Also
( )( ) 21
221
1
2
1
1sin2
M
Cot M
v
v
+
!
= "
#
( )12sin
1
2
+!
"
#v
v
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Density Jump Across Shock
( )( ) 2sin1
sin122
1
221
1
2
+
+=
"
"
$
$
M
M
In the hypersonic limit, for large M 1 >>1, finite
( )( )1
1
1
2
!
+"
#
#
$
$
Then
( )( )222
112
12
12
1
sin12
+
!=
"#
$$ M p pT T
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21
1
2
2
1
M
p p
C p !
"#
$%&
'(
)
1
4 2
+=
!
"SinC p 11 > >M
Hypersonic Shock Relations in the Limit of Large but FiniteMach number and small turning angle
We define a similarity parameter !1M K = which can be used to collapse avariety of data
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( ) 2cos1sin
cot2tan22
1
221
+!="
""
M
M
For large but finite M, small and
becomes
( )!!"
#$$%
&+
++
+' 22
1
2 1
16
1
4
1
(
)
(
*
M
Works for finite values of M1 = K
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Hypersonic Expansion Wave Relations
From Prandtl-Meyer theory, 12 !#
( ) ( )1tan11
1tan
1
1 2121!"
#
$%&
'()
*+,
-!
!+
!+
= ! M M .
.
.
./
For 11 > >M 212
1 1 M M !
Also ( ) !"#
$%&' '
x x
1tan
2tan 11
(
From Taylor series
..5
1
3
111tan
531
!#$
%&'
x x x x
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2
1
1
11
21
1 !
"
"
"
"# $%
&
'()
*$+
$$+
+M M
( ) 212
11
2!
"
"# $$$+= M
Then
( ) !"#
$%&
'' 21
12
11
1
2
M M )
( )( )
1
22
21
1
2
11
11 !
""#
$
%%&
'
+
+=
(
(
(
(
M
M p p
1
2
2
1 !"#$%
&'( )
)
M M
1
2
1
2
11
2
2
11
2
11 !
"#$
%&' !!"#
$%&' !!
(
(
(
(
()
(K M
p p
!!!
"
#
$$$
%
&'!"
#$%& ''
()*
+,- '
.' 1
2
11
2
2
11
2
22
1
2
2/
/
/
/0K
K K
p p
C p ),(2 !"K f
C p#
Note that
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Consider flow over a blunt body:
Where does freestream Mach number appear in the above?Only in the dependence of downstream pressure, density, temperature.
As freestream Mach number becomes large,( )( )1
1
1
2
!
+"
#
#
$
$
!"#$
%&!
"#$
%&
+
=
''
'
'
'2
222
22
2 1sin1
2
M M
U p
p p
U p
()
((
*
!"
" 2sin1
2
+=
Why nondimensionalize by2
!U
Because ( )22 ~ !U O p " And it allows cancellation of Mach number Examine other relations for properties downstream of the shock freestream Machnumber does not appear anywhere.
Mach Number Independence
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This Mach number independence is also observed in experiments. Sphere drag coefficient,for example.
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Hypersonic Aerodynamics Roadmap
Supersonic Aero
Local SurfaceInclinationMethods
Blast WaveTheory
Newtonian Aerodynamics Newton
Buseman
HypersonicSmall Disturbance:Mach Number Independence
Full shock-expansion methodWith real gas effects
Stagnation Point: CFD
Conical Flow /Waveriders
Non-Equilibrium Gas Dynamics