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