Hypersonic flow: introduction
Van Dyke: Hypersonic flow is flow past a body at high Mach number, where nonlinearity is an essential feature of the flow.
Also understood, for thin bodies, that if τ is the thickness-to-chord ratio of the body, M τ is of order 1.
Special Features
Thin shock layer: shock is very close to the body. The thin region between the shock and the body is called the Shock Layer.
Entropy Layer: Shock curvature implies that shock strength is differentfor different streamlines – stagnation pressure and velocity gradients -rotational flow
http://www.onera.fr/conferences/ramjet-scramjet-pde/images/hypersonic-funnel.gif
The “Hypersonic Tunnel” For Airbreathing Propulsion
Velocity-Altitude Map For Re-Entry
Velocity
Altitude
Typical re-entry case: Very little deceleration untilVehicle reaches denser air
(Deliberately so - to avoid large fluctuations in aerodynamicloads and landing point )
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 <z < 85 km NO+
85 <z < 140 km NO+, O2+
140 <z < 200 km NO+, O2+, O+
Z> 200 km N+, O+
A Simple Model for Variation of density with altitude
gdzdp !"=
M
TRp
ˆ
ˆ!=
Neglect dissociation and ionization – Molecular weight is constantAssume isothermal (T = constant) poor assumption
dzTR
Mg
p
dp
ˆ
ˆ!"
!"
#$%
&' z
TR
Mge ˆ
ˆlog0((
Non-lifting body moving at velocity V, which is inclined at angle θ to the x-axis:
!DCosdt
xdm "=
2
2
mgDSindt
zdm != "
2
2
mgSCUdt
zdm D != "# sin
2
1 2
2
2
!!"
#$$%
&
SC
m
D
is the “Ballistic Parameter”.
Assuming that the drag force is >> weight and that θ is constant because gravitational force istoo weak to change the flight path much
!"
#$%
& ''=!!
"
#$$%
&
RT
gMz
m
SC
U
ULog D
ee exp
sin2
1 0
(
)
U
D
θ
www.galleryoffluidmechanics.com/shocks/s_wt.htm
High Angle of Attack Hypersonic Aerodynamics
http://www.scientificcage.com/images/photos/hpersonic_flow.jpgy
Crocco’s 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 that specific 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.
http://www.aerospace-technology.com/projects/x43/images/X-43HYPERX_7.jpg
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 Cone”approach: 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.
Local Surface Inclination MethodsApproximate 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
!22SinCp =
!SinU" with nUrr
•!
2
2
2
!
! •=U
nUCp
r
Shadow region: 0=Cp
Shadow region is where 0>•! nUrr
Remarks on Newtonian Theory:
Poor in low speed flow. Predicts . 2!"lC
(1) Works well as Mach number gets large and specific heat ratio γ tends towards 1.0Why? Because shock is close to surface, and velocity across the shock is very large – most of the normal momentum is lost.
(2) Tends to overpredict cp and cd (CD) 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.
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
!MpC
maxpC
!2max
sinpp CC =
Here is the coefficient behind a Normal shock wave,at the stagnation point. That is,
maxpC pC
2
02
max
2
1!!
!"=
U
ppCp
#
Modified Newtonian
From Rankine-Hugoniot relations,
( )
( ) !!"
#
$$%
&
+
+'
!!"
#
$$%
&
''
+= (
'
(
(
( 1
21
124
121
2
2202
)
))
))
) )
)
M
M
M
p
p
(3.17)
Then
2
02
2
1
!
!
"
=
M
p
p
cp #
In the limit as ,!"!M We get
( )
( ) 1
4
4
1
1
1
2
+!!
"
#
$$
%
&+
=
'
'
((
(
((
((
pc
As ,4.1!" 839.1max
!pc
As ,1!" 2max
=pcProposed by Newton
Exercise: Compute cp values for configurations shown on Figures 3.8,3.6, 3.11 and 3.12 using Newtonian and Modified Newtonian theories.Biconvex Airfoil.
y/c = 0.05 -0.2 (x/c)2
Where does freestream Mach number appear in the above? Only in the dependence of downstream pressure, density, temperature.
As freestream Mach number becomes large,( )( )1
1
1
2
!
+"#
#
$
$
!"
#$%
&!"
#$%
&
+==
''
''
'
'''2
22
2
2
2
2 1sin
1
2
MM
U
p
p
p
U
p
()
(
(
**
!"
" 2sin1
2
+=
Why nondimensionalize by 2
!!U"
Because ( )22 ~ !!UOp " And it allows cancellation of Mach number
Examine other relations for properties downstream of the shock – freestream Mach number does not appear anywhere.
Mach Number Independence
The blast wave theory argues that the sudden addition of energy to thefluid by the body is equivalent to a high explosive of energy E beingexploded at time t=0.
A shock wave associated with the explosion spreads away from the originwith time
In 2-D problem: the shock wave is a plane wave:
!
=U
xt
Shock wave moves outward with tBlast wave origin
Hypersonic Shock & Expansion Relations
Why?
1. Simpler than exact expressions - for analysis2. Key parameter is seen to be Mθ where θ is the flow turning angle, for M>>1 and θ<<1
Oblique Shock Relations
( ) 2cos
1sincot2tan
22
1
22
1
++
!=
"#
""$M
M
M1 >>1, small β!"
#$%
&
+'
1
2
(
)*
Pressure jump:!
"
" 221
1
2sin
1
21 M
p
p
++=
M1 >>1!
"
" 221
1
2sin
1
2M
p
p
+#
( )1
sincot2tan
2
1
22
1
+!
"
##$M
M
M1 >>1, small β
!"
#$%
&'
++= 1sin
1
21
221
1
2 ()
)M
p
p
!"
#$%
&'
++( 1
1
21
22
1 )*
*M
( ) ( )2
2
22
1
2 1
4
1
4
11
KKK
p
p+!
"
#$%
& ++
++=
'''
Defining pressure coefficient
2
1
1
2
2
1
M
p
p
Cp !
""#
$%%&
'(
)
!!
"
#
$$
%
&+'
(
)*+
, ++
+=
''(
)**+
,-
.2
2
2
1
2
2
1
4
1
4
12
2
1
KK
p
p
Cp ///0
Next
( )( ) 2
1
221
1
2
1
1sin1
M
M
v
u
+
!!=
"
#
In the hypersonic limit,
1
sin21
2
1
2
+!"
#
$
v
u
Also
( )( ) 2
1
221
1
2
1
1sin2
M
CotM
v
v
+
!=
"
##
( )12sin
1
2
+!
"
#
v
v
Density Jump Across Shock
( )
( ) 2sin1
sin1
221
221
1
2
+!
+=
"#
"#
$
$
M
M
In the hypersonic limit, for large M1 >>1, finite β
( )( )1
1
1
2
!
+"#
#
$
$
Then
( )
( )2
221
1
2
1
2
1
2
1
sin12
+
!==
"
#"
$
$ M
p
p
T
T
2
1
1
2
2
1
M
p
p
Cp !
""#
$%%&
'(
)
1
42
+=
!
"SinCp 11 >>M
Hypersonic Shock Relations in the Limit of Large but FiniteMach number and small turning angle
We define a similarity parameter !1MK = which can be used to collapse avariety of data
( ) 2cos
1sincot2tan
221
221
++
!=
"#
""$M
M
For large but finite M, small θ and β
becomes
( )!!
"
#
$$
%
&+
++
+'
221
21
16
1
4
1
(
))
(
*
M
Works for finite values of M1θ = K
Hypersonic Expansion Wave Relations
From Prandtl-Meyer theory, 12 !!" #=
( ) ( )1tan11
1tan
1
1 2121 !!"#
$%&
'(()
*++,
-!
!
+
!
+= !!
MM.
.
.
./
For 11 >>M 2
1
2
1 1 MM !"
Also ( ) !"
#$%
&'= ''
xx
1tan
2tan
11 (
From Taylor series
..5
1
3
111tan
53
1 !+!="#
$%&
'!
xxxx
2
1
1
11
21
1 !
"
"!
"
"# $+%
&
'()
*
$
+$
$
++
MM
( ) 21
2
1
1
2
!
""
"!# $
$$
$
+=
M
Then
( ) !"
#$%
&'
'='=
21
12
11
1
2
MM())*
( )
( )
1
2
2
2
1
1
2
11
11 !
""#
$
%%&
'
++
++=
(
(
(
(
M
M
p
p1
2
2
1!
""#
$%%&
'(
)
)
M
M
1
2
1
2
1
1
2
2
11
2
11
!!
"#
$%&
' !!="#
$%&
' !!=
(
(
(
((
)(
KMp
p
!!!
"
#
$$$
%
&
'!"
#$%
& ''=
(()
*++,
-'
.'
12
11
2
2
11
2
22
1
2
2
/
//
//0K
KK
p
p
Cp ),(2
!"
KfCp
#
Note that
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
22
2
2
2
2 1sin
1
2
MM
U
p
p
p
U
p
()
(
(
**
!"
" 2sin1
2
+=
Why nondimensionalize by 2
!!U"
Because ( )22 ~ !!UOp " And it allows cancellation of Mach number
Examine other relations for properties downstream of the shock – freestream Mach number does not appear anywhere.
Mach Number Independence
This Mach number independence is also observed in experiments. Sphere drag coefficient, for example.
Hypersonic Aerodynamics Roadmap
SupersonicAero
Local Surface Inclination Methods
Blast Wave Theory
Newtonian Aerodynamics Newton
Buseman
Hypersonic Small Disturbance: Mach Number Independence
Full shock-expansion methodWith real gas effects
Stagnation Point: CFD
Conical Flow / Waveriders
Non-Equilibrium Gas Dynamics