Hypersonic flow: introductionadl.gatech.edu/research/extrovert/classes/hispd/hispd_notes08.pdf ·...
Transcript of Hypersonic flow: introductionadl.gatech.edu/research/extrovert/classes/hispd/hispd_notes08.pdf ·...
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 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 and the 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 flowrotational flow
The “Hypersonic Tunnel” For Airbreathing Propulsion
http://www.onera.fr/conferences/ramjet-scramjet-pde/images/hypersonic-funnel.gif
Velocity-Altitude Map For Re-Entry
AltitudeT i l t Typical re-entry case: Very little deceleration untilVehicle reaches denser air
(Deliberately so - to avoid large fluctuations in aerodynamicloads and landing point )
Velocity
loads and landing point )
Atmosphere
Troposphere: 0 < z < 10km
Stratosphere: 10 < z < 50km
Mesosphere: 50 < z < 80km
Thermosphere: z > 80kmp
Ionosphere 65 < 365 km Contains ions and free electrons
60 <z < 85 km NO+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
TRp ˆˆ
M
p
Neglect dissociation and ionization – Molecular weight is constantg gAssume isothermal (T = constant) poor assumption
dMgdp ˆdz
TRg
pp
ˆ
ˆ
z
TRMg
e ˆˆ
log0 TR
Crocco’s Theorem:
uhsT 0
Implies vorticity in the shock layer
Viscous Layer:
Implies vorticity in the shock layer.
Thick boundary layer, merges with shock wave to produce a merged shock-viscous layer. y y , g p g yCoupled 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: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.
Summary of Theoretical ApproachesNewtonian Aerodynamics: Flow hits surface layer, and abruptly turns parallel to surface. Momentum
normal to the surface is transferred to normal force on the body. Normal force on body = drag of normal flow component normal flow component. Normal force is decomposed into lift and drag.
Modified Newtonian Aerodynamics: Account for stagnation pressure drop across shock.
Local Surface Inclination Method : Cp at a point is calculated from static pressure behind an oblique h k d b l l f l t f t M h b shock 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 Mach Mach number independence: Shock/expansion relations and Cp become independent of Mach number 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 hypersonic aerodynamics.
Hypersonic Aerodynamics Roadmap
SupersonicAero Stagnation Point: CFD
Non-Equilibrium Gas Dynamics
Aero
Full shock-expansion methodWith real gas effects
Stagnation Point: CFD
Blast Wave Theory
Hypersonic Small Disturbance:
With real gas effects
TheoryMach Number IndependenceConical Flow /
Waveriders
Local Surface Inclination Methods
N t i Newtonian Aerodynamics
NewtonBuseman
Newtonian Aerodynamics: Flow hits surface layer, and abruptly turns parallel to surface.
Momentum normal to the surface is transferred to normal force on the body Momentum normal to the surface is transferred to normal force on the body.
Normal force on body = drag of normal flow component. Normal force is decomposed into lift and drag.
N i f h k i d
D
•No info on shock. or viscous drag•No influence of body shape
NL
Local Surface Inclination MethodsApproximate methods over arbitrary configurations, in particular, where Cp is a function of local surface slope.
Newtonian AerodynamicsNewtonian AerodynamicsNewton (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 they would lose all momentum perpendicular to the surface, but retain all tangential momentum – i.e., slide off the surface.
ASinU 22Net rate of change of momentum
In 3D flows we replace
22SinCp
SinU with nU
2
22
U
nUCp
Shadow region: 0CpShadow region: 0Cp
Shadow region is where 0 nU
Remarks on Newtonian Theory:
Poor in low speed flow. Predicts . 2lC
(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 the normal momentum is lost.
(2) Tends to overpredict c and c (C ) see(2) Tends to overpredict cp and cd (CD) see
(3) Works better in 3-D than in 2-D(4) In 3-D, works best for blunt bodies; not good for wedges, cones, wings etc.(4) In 3 D, works best for blunt bodies; not good for wedges, cones, wings etc.
Was proposed by Lester Lees in 1955, as a way of improving Newtonian theory, and C
Modified Newtonian
bringing in Mach Number dependence . He proposed replacing 2 with maxpC
2max sinpp CC
Here is the behind a Normal shock wave, at the stagnation point That is
maxpC pCpoint. That is,
202
max 1
U
ppCp
2 U
Hypersonic Shock & Expansion Relations
Why?
1. Simpler than exact expressions - for analysis2 K t i t b h i th fl t i l f M 1 d 1 2. Key parameter is seen to be where is the flow turning angle, for M>>1 and <<1
Oblique Shock Relations22
1MK
M1
2cos1sincot2tan 22
1
221
MM
22
2
1sincot2tan 2
1
221
MM
M1 >>1, small
M1 >>1, small
1
2
Pressure jump: 2pPressure jump:
22
11
2 sin1
21 Mpp
M1 >>1 22
12 sin2 Mp 1
1sin
1M
p
1sin
121 22
12 Mp
1 11
p
1
121 22
1 M 1 1
2
2222 1111 KKp
21 44
1K
KKp
Defining pressure coefficient 2 1pp
g p
2 1p
21
1
2M
pCp
2
2
21
2
21
41
412
2
1
KK
pCp 2
14 2
SinCp 11 M
1p
Next M 2 i 2 1 Nextu2u1
1M1
2 sin2 1 1 M1
2
In the hypersonic limit, u2 1 2sin2 1u1 1
Also u22 M1
2 sin2 1 Cot2
u1
1 M12
u2u1
sin2 1 1
Density Jump Across Shock
22 2sin1
sin122
1
221
1
2
MM
In the hypersonic limit, for large M1 >>1, finite
1 1
11
2
Then the temperature jump is:
221222 sin12 MpT
21
1
2
1
2
1
21
pp
T
p
21
2 1
M
pp
Cp
14 2
SinCp
11 M
12M
For large but finite M, small and
21sincot2tan 22
221
M
M 2cos22
1 M
becomes
2 111
221164 M
Works for finite values of M1 = K
Hypersonic Expansion Wave Relations
From Prandtl-Meyer theory, 12
1tan11tan1 2121
MM 1tan11
tan1
MM
For 11 M 22 1 MM For 11 M 11 1 MM
Also
xx 1tan
2tan 11
x2
From Taylor series
..5
13
111tan 531
xxxx
21
111
211
MM 2121
MM
212
11
2
M
Then
2112
111
2MM
12
2
21
1
21111
MM
pp 1
2
2
1
MM
21 2 M
12
12
12 1111
KMp1
1 22 p
1 22 pC )(Kf
Cp
Note that
1
2112
2
11
221
2
KKK
pCp ),(2
Kfp
2
Mach Number Independence
As freestream Mach number becomes large, 1
11
2
2
222
22
2 1sin1
2M
MUp
pp
Up
2 2sin
12
2UWhy nondimensionalize by 2U
Because p2 ~ O U2
And it allows cancellation of Mach number
2
Examine other relations for properties downstream of the shock – freestream Mach b d t h number does not appear anywhere.
Non-lifting body moving at velocity U, which is inclined at angle to the x-axis:
DCosdt
xdm 2
2D
dt2
mgDSindt
zdm 2
2U
dt
mgSCUdt
zdm D sin21 2
2
2
SC
mD
is 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
RTgMz
mSC
UULog D
ee exp
sin21 0