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

High Angle of Attack Hypersonic Aerodynamics

www.galleryoffluidmechanics.com/shocks/s_wt.htm

http://www.scientificcage.com/images/photos/hypersonic_flow.jpg

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.

http://www.aerospace-technology.com/projects/x43/images/X-43HYPERX_7.jpg

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

From Rankine-Hugoniot relations,

211 2122

02

MMp

1124 2

02Mp

pThen

02 1

pp

cp 22 M

p

In the limit as ,M We get

41 12

c 14 1

pc

As ,4.1 839.1max pc

As ,1 2max pc Proposed by Newton

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