Chp 1 Hypersonics

8/17/2019 Chp 1 Hypersonics

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Hypersonic Educational Initiative

Advanced

Hypersonic Aerothermodynamics

Iain D. Boyd

Dept. Aerospace Eng.

University of Michigan

Ann Arbor, MI

Graham V. Candler

Dept. Aerospace Eng. & Mech.

University of Minnesota

Minneapolis, MN


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1. Hypersonic Gas Dynamics

1.1 Introduction

Outline (1)

1. Hypersonic Gas Dynamics (4.0 hours)1.1 Introduction and Examples

Vehicle trajectories and properties

1.2 Hypersonic gas dynamic processes

Shock waves / boundary layers wakes

1.3 Fundamental equations of gas dynamicsBoltzmann / Euler / Navier-Stokes Equations


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Outline (2)

1. Hypersonic Gas Dynamics (4.0 hours)1.4 Real gas effects:

Quantum mechanics + statistical mechanics

Perfect gasVibrational activation

Chemical reactions – nonequilibrium vs. equilibrium

Ionization, radiation

1.5 Shock wave analysis

Perfect gasIterative approach for equilibrium gas

1.6 Transport phenomena


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Outline (3)

2. Hypersonic Aerodynamics: Pressure (1.0 hour)

2.1 Exact and approximate equilibrium gas solutions:

Stagnation points

Cones and wedges2.2 Mach number independence

2.3 Newtonian and Modified Newtonian aerodynamics


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Outline (4)

3. Hypersonic Aerothermodynamics: Heat Transfer

(2.0 hours)

3.1 Introduction:

role of aerodynamic heating

hypersonic boundary layers

3.2 Boundary layer equations, similarity transformation

3.3 Flat plate / wedge / cone solutions

3.4 Stagnation point solution and scaling

3.5 Laminar and turbulent boundary layers; transition

3.6 Wall catalysis


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Outline (5)

4. Viscous Interactions (1.0 hour)

4.1 Leading edge interactions

4.2 Effect on high-altitude L/D; scaling for vehicles

4.3 Shock-BL interactions, shock-shock interactions

5. Thermal Protection Systems (1.5 hours)

5.1 Passive:

Re-radiative cooling, equilibrium wall boundary condition

Role of wall temperature, material properties5.2 Ablative

Surface ablators

Pyrolyzing ablators


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Outline (6)

6. Computational Analysis (2.0 hours)6.1 Continuum CFD methods

Requirements for aeroheating predictions

Approaches for real gas effects

Current code capabilities6.2 Non-continuum method - DSMC

7. Experimental Testing (1.0 hour)7.1 Types of facilities:

Blow-down tunnelsVitiated-air facilities

Impulse facilities

7.2 Overview of measurement techniques


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Outline (7)

8. Aerothermodynamics of Hypersonic Vehicles(1.5 hours)

Ballistic entry

Lifting capsule re-entry: Apollo

High-lift re-entry: Shuttle Aerocapture / Aerobraking

Airbreathing scramjets


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What is Hypersonic Flow?

• Hypersonic aerothermodynamic phenomena: – strong shock waves with high temperature

– not calorifically perfect (variable !)

– chemical reactions

– significant surface heat flux – several different types of vehicles:

• missiles, space planes, capsules, air-breathers

• Working definition of hypersonic flow:

M = (U / a) >> 1


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Hypersonic Examples:

I. Missiles

• Mission: high-speed delivery of explosives• Aerodynamics: slender body with blunt nose

• Propulsion: rockets, ramjets

• Examples: AMRV, SCUD, Patriot, Hy-Fly


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II. Space Planes

• Mission: orbital re-entry• Aerodynamics: gliders with thermal protection

• Propulsion: none (except small control thrusters)

• Examples: Space Shuttle, Buran, Hermes


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III. Air-breathing Systems

• Missions: launch, cruise, orbital re-entry

• Aerodynamics: slender with integrated engines

• Propulsion: ram/scram-jets, rockets, turbojets

• Examples: X-15, NASP, X-43, X-51


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IV. Planetary Entry

• Missions: EDL, aero-braking, aero-capture• Aerodynamics: very blunt, thick heat shield

• Propulsion: none (sometimes RCS)

• Examples: Apollo, MSL, CEV (Orion)


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• Flight vehicles:

– WAC Corporal missile (1949, M~8)

– Vostok I (1961, M~25)

– X-15 (1963-1967, M~7)

– Space Shuttle (1981-???, M~25)

– HyShot (2002, M~8)

– X43 (2004, M>7)

– Hy-CAUSE (2007)

• Recent programs without flight:

– NASP, Hermes, AFE, AOTV (1990)

– VentureStar-X33 (2000)

Hypersonic Vehicle

Historical Overview


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Some Current

Hypersonic Programs

Falcon (DARPA)

Orion

(NASA)

X51

(AFRL)

HyBoLT (NASA/ATK)


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Hypersonic Tales of Woe

• Hypersonics produces unexpected phenomena

• X15 test flight with dummy scramjet installed:

– unexpected shock interactions generated

– burned holes in connection pylon

• First re-entry of Space Shuttle (STS-1):

– larger than expected nose-up pitch generated

– required near-maximum deflection of body flap

• Shock-shock interactions:

– heating amplified significantly

– leading edges, cowl lips,

engine flow paths


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• Ballistic missiles:

– mission: short flight, fast impact

– rocket launch, ballistic entry – no thrust or lift during entry (T=0, L=0)

– fixed flight path at large angle (!=const)

Re-entry Trajectories

• Trajectory equations for Earth centered system:

W

D

LT, U

!

U ̇"

g=

L

W # 1#

U 2

gR

$

% &

'

( ) cos(" )

T W

"

˙Ug= DW

+ sin(# )


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• Air-breathing vehicle:

– missions: cruise, orbital return

– completely reusable

– powered take-off and entry

– constant for engine efficiency

• Space Shuttle:

– mission: orbital return

– rocket launch

– equilibrium glide entry – no thrust, L/D~1, !~0 (shallow entry)

Re-entry Trajectories

1

2" U

2


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Flight Velocity


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Stagnation Point Heating


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Stagnation Point

Temperature


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Deceleration Levels


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1.2 Hypersonic Gas Dynamics

• Goal of hypersonic gas dynamics analysis:

– predict flow field around a hypersonic vehicle

– predict properties on vehicle surface (pressure,

shear stress, heat transfer)

– hence predict aerodynamic and thermal

performance of a vehicle

• Three main gas dynamic flow phenomena:

– shock waves

– boundary layers

– wakes


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• Formed when an object is placed in supersonic flow:

– Mach number M = U / a (a~330 m/s at STP)

– supersonic: M > 1

• A shock wave is an extremely thin layer of fluid acrosswhich there are significant changes in properties:

– at STP, thickness ~ 10-7 m

– T, p, ! all increase; M, U decrease

– variable-! = compressible; high-T=real gas

• Several different types:

– normal (strongest), oblique, bow

1.2.1 Shock Waves


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• Thin fluid layer formed next to vehicle surface:

– created by friction between gas and wall

• Boundary layer (BL) determines surface properties:

– pressure (nearly constant across BL) – shear stress and heat transfer

– flow separation

• Different types with very different properties:

– laminar

– turbulent

– transition between the two

1.2.2 Boundary Layers


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• Region of flow in the rear of the vehicle:

– smaller influence on aerothermodynamics

• Often involves strong expansion processes:

– T, p, ! all decrease; M, U increase

• Recompression:

– leads to increased heating on a capsule backshell

– base pressure effects on missiles

– flow may become turbulent

– complex vortex dynamics

1.2.3 Wakes


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1.3.1 Boltzmann Equation

• Consider gas flow at the molecular level:

– molecule has position x, velocity C

– enormous volume of information (at STP, there

is about 1026 air molecules per cubic meter)

– adopt a statistical approach

– velocity distribution function, VDF:

f(t, x, C) dx dC

– probability of finding a particle at time t, in

physical space element dx, and in velocityspace element dC


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1d VDF’s


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• VDF is affected by:

– particle transport across dx due to C

– particle transport across dC due to a

– scattering in/out of dC due to molecular collisions

• Leads to the Boltzmann Equation:

– n=number density,a

=acceleration (force), g=relativevelocity, !=collision cross section, "=solid angle,

C’=particle velocity after collision, Z=velocity of

particle with which C-particle collides

" (nf )

" t +C #

" (nf )

" x+ a #

" (nf )

" C= n

2{ f (C') f (Z') $ f (C) f (Z)}g% d &dZ ''



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• Equilibrium solution:

– no net change in VDF, f , requires LHS = RHS = 0

– RHS = 0 satisfied by

– thus, log[f] is a linear combination of collision

invariants (mass, momentum, energy)

– leads to Maxwellian (equilibrium) VDF

f (C) f (Z) = f (C') f (Z')


log[ f (C)]+ log[ f (Z)]= log[ f (C')]+ log[ f (Z')]

f (C)d C =m

2" kt

#

$ %

&

' ( 3 / 2

exp ) m

2kT C 1 ) u1( )

2+ C 2 ) u2( )

2+ C 3 ) u3( )

2( )*

+ , -

. / d C


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• Very difficult to obtain solutions:

– seven-dimensional space

– non-linear collision integral on RHS

– expensive computations required

• Solution only required for strongly nonequilibrium flow:

– Knudsen number: Kn = # / L

– at Kn < 0.01, f is close to Maxwellian

– for L=1 m, Kn~0.01 at h=90 km in atmosphere

– for small Kn, Boltzmann equation can be used to

derive sets of macroscopic transport equations



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• Take moments of Boltzmann Eq (Maxwell):

– Q=Q(C) is a particle property, e.g. momentum mC

– $[Q]=rate of change of Q due to collisions

–

• Assumptions for f , Q lead to sets of partial differential

equations describing macroscopic flow

1.3.2 Method of Moments

" n Q

" t +

" n CQ

" x= # Q[ ]

Q = Qf (C )dC "


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• Assumptions:

– local thermodynamic equilibrium

– VDF’s are Maxwellian everywhere

– use Q=m, mC, 0.5mC2 for which $[Q]=0

– leads to 5 PDE’s: the Euler equations

– no viscosity, no thermal conductivity

1.3.2 Euler Equations

"#

" t +

"# u

" x= 0

"# u

" t +

"# CC

" x= 0 = #

" u

" t + # u $ %u+% p

" 0.5

# C

2

" t + "

0.5# C

C

2

" x= 0 = 3

2

" p" t

+ 32$ % pu+ p$ % u


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• Chapman-Enskog distribution:

– VDF is small perturbation from equilibrium

– where %(C)=1+F(Kn)=1+F(heat flux, shear stress)

• Navier-Stokes equations from transfer equation:

– same five relations as Euler Equations

– also: Q=mCiC j, mCiC jCk (15 further relations)

– replace heat flux tensor by heat flux vector

– derive and substitute linear transport relations

1.3.3 Navier-Stokes Eqns.

f (C)d C = "(C) f M (C)d C


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• Again, a set of five PDE’s:


"

" t U+

"

" xF +

"

" yG +

"

" zH = 0

U =

"

" u

" v

" w

" E

#

$

% % %

&

% % %

'

(

% % %

)

% % %

F =

" u

" u2+ p# $

xx

" uv # $ xy" uw # $ xz

" E + p# $ xx( )u # $ xyv # $ xzw + q x

%

&

' ' '

(

' ' '

)

*

' ' '

+

' ' '

G =

" v

" uv # $ yx

" v2+ p# $ yy

" vw # $ yz

" E + p# $ yy( )v # $ yxv # $ yzw + q y

%

&

'

' '

(

' ' '

)

*

'

' '

+

' ' '

H =

" w

" uw # $ zx" vw # $

zy

" w2+ p # $ zz

" E + p# $ zz( )w # $ zxu # $ zyv + q z

%

&

'

' '

(

' ' '

)

*

'

' '

+

' ' '


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• Stress tensor modeled using Stokes’ hypothesisfor Newtonian fluid:

–

µ = viscosity coefficient

• Heat flux vector modeled using Fourier’s Law:

– ! = coefficient of thermal conductivity


" xx

= 2µ # u

# x $ 1

3% &u

'

( ) *

+ , " xy

= " yx

= µ # u

# y+

# v

# x

$

% &

'

( )

q x = "# $ T

$ x


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1.4 Real Gas Effects

• In low-speed flows, air is a perfect gas:

– simplified thermodynamics, no chemistry

• In most hypersonic flows, air is a real gas:

– mainly due to high temperatures – complex thermodynamics, chemical reactions

• Requires us to consider:

– quantum+statistical mechanics

– vibrational activation, chemistry – equilibrium and finite rate processes

– ionization and radiation


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1.4.1 Quantum Mechanics

• An atom / molecule has several energy modes:

– translation (due to 3D motion)

– rotation (gyration of atoms in a molecule)

– vibration (oscillation of atoms in a molecule)

– electronic (arrangement of electrons in orbits)

• Quantum mechanics provides energy states:

– solution of Schrodinger equation

– separate analysis for each energy mode – each atom / molecule may only occupy certain

quantized energy states in each energy mode


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1.4.1 Quantum Mechanics

• Translational energy:

– h=Planck’s constant, L=dimension, m=mass

– n1, n2, n3 are translational quantum numbers

– !"t ~10-38 J so continuum can be assumed

• Rotational energy (rigid rotor):

– k=Boltzmann const, #r =char. temp. for rotation – J=rotational quantum number

– for air, !"r ~10-23 J so continuum OK

" t =

h2

8mL2 n

1

2+ n

2

2+ n

3

2( )

" r= k #

r $ J ( J +1)


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1.4.2 Statistical Mechanics

• Each atom / molecule has an enormous number of

possible energy levels:

– degenerate energy states for each energy level

• Large number of atoms/molecules in a gas: – huge volume of information

• Statistical mechanics:

– distributions of particles across energy states

– macroscopic thermodynamics viaintegration/summation of atomic/molecular

behavior


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• Boltzmann energy distribution:

– gi=degeneracy of energy level i

– summation in denominator is partition function, Q

• Overall partition function:

(diatomic)

f (" i) =gi exp #" i /kT ( )gi exp #" i /kT ( )$

Qt =V 2" mkT

h2

#

$ %

&

' (

3 / 2

Qr=

T

" r

Qv =

1

1" exp "# v /T ( )

Qe = g0 + g1exp "# 1 /T ( ) $ g0

Q = Qt "Q

r "Q

v "Q

e


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• Internal energy (thermodynamics):

• Specific heats:

cv=

" e

" T

#

$ %

&

' ( v

=

3 R

2+ R + R

) v /2T

sinh() v /2T )

*

+ ,

-

. /

2

et =

3 RT

2

er= RT

ev=

R" v

exp " v /T ( )#1

ee " 0

e = et + e

r + e

v+ e

e

c p = cv + R

" =c

p

cv


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• Specific heats (non-reacting):


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1.4.3 Perfect Gas

• Assumption often invoked in hypersonics:

– no vibrational activation, no chemistry

– cp, cv, $ are all constants

• For air:

p=

" RT

e=

c vT

h=

c pT

c p=

7 R

2

" =7

5

cv=

5 R

2

c p=

" R

" #1

cv=

R

" #1


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• At high temperatures (T>1,000K) air molecules

become vibrationally activated:

– equilibrium (infinite rate) results provided by

quantum mechanics + statistical mechanics

– finite rate activation modeled using

– where e*v=equilibrium value at temperature T

– %v=vibrational relaxation time (Millikan & White)

dev

dt =

ev

*(T ) " e

v

# v

1.4.4 Vibrational Activation


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• Analysis of Earth hypersonic vehicles at U


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Rate Processes

• For illustration, consider:

– 2-species: N2, N

• Each reaction proceeds at a finite rate:

• Forward rate coefficients measured experimentally, k f (T)

• Backward rate coefficients from equilibrium constant:

partition functions Q from quantum+statistical mechanics

N 2+ M " N + N + M

N 2 + N 2"k b1

k f 1

N + N + N 2

N 2 + N "k b 2

k f 2

N + N + N

K e =k f

k b=

"Qproducts

"Qreactants


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Rates of Change

• Net rate of change in concentration of a species:

– contributions from forward and backward directions

• Chemical equilibrium: – final state reached instantaneously

– production of each species balanced by its destruction

– analytical solution for our system:

– &=mass fraction, m=atom mass, '=density, V=volume,

#d=dissociation temperature

d [ N 2]

dt = "k f 1[ N 2][ N 2]" k f 2[ N 2][ N ]+ k b1[ N ][ N ][ N 2]+ k b2[ N ][ N ][ N ]

"

2

1#" = m$ V

Q N 2

Q N 2exp(#% d /T )


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Infinite Rate of Reactions

• Chemical equilibrium:

– O2 dissociates before N2 (has lower #d)

– fewer atoms at high pressure (more recombination)


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Finite Rate of Reactions

• Chemical nonequilibrium: – equilibrium end state reached only after finite time

– in a flow field, this translates as finite distance


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Nonequilibrium

• Impact of chemical nonequilibrium:

– chemical composition mainly affects energy of flow

• endothermic reactions consume energy

• catalysis: fraction of atoms reaching the vehicle

surface may recombine releasing heat – scaling:

• nonequilibrium flow occurs at lower density

and/or smaller body length scales

large Kn " # $ L%

1

& $ L

small Re " #

$U $ L

µ $


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Ionization

• Very high temperature reacting air (U>8km/s): – N2, O2, NO, N, O, N2+, O2+, NO+, N+, O+, e-

• Reactions:

– dissociation-recombination:

– exchange:

– associative Ionization:

– direct Ionization:

N 2 + M " N + N + M

N 2+O" NO+ N

N + N " N 2

+

+ e#

N + e"# N

+

+ e"+ e

"


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Ionization

• Equilibrium solution (Saha) for [N, N+, e-] system:

– (=ion mole fraction,

– C=constant, – p=pressure,

– #i=ionization

temperature

" 2

1#" 2= C

T 5 / 2

pexp(#$ i /T )


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Radiation

• Another important process at high temperature: – activation-deactivation:

– spontaneous emission:

– analysis is complex, no closed form expressions – research area, e.g. NEQAIR (NASA-ARC)

• Radiative heating important at U>12km/s:

– e.g. stagnation point heating correlation (Martin)

– also proportional to shock layer thickness

– Stardust: radiation provides 10% of total heating

N *" N + h#

N + e"# N

*+ e

"

q̇rad " R N U 8.5

# 1.6


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1.5 Post-Shock Conditions

• Perfect-gas shock relations:

• Density ratio asymptotes to:

• Pressure and temperature are quadratic in M

– Makes sense: energy is conserved


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Post-Shock Conditions

• Post-Shock Temperature:

Temperatures rapidly

become huge!


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• Variation of air internal energy with T:

10% departure from

calorically perfect gas

equation of state =

onset of hypersonic flow


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• More fundamentally – 1D gas dynamics:

• Plus equations of state:

• No exact solutions

Thermally perfect,

calorically imperfect

General equilibrium

gas mixture


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• Hypersonic limit:

• Note that post-shock enthalpy and pressure only

depend on upstream conditions in hypersonic limit.

Can solve for the

thermodynamic state

!


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• Iterative solution to shock relations:

• Guess a value of ! = !i and iterate:

Use tables, NASA CEA, etc.


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Equilibrium Air

Temperature (K) Z = Compressibility


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• Example: M = 12 at 30 km altitude:

Imperfect Perfect


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• Perfect-gas vs. equilibrium post-shock conditions:

Difference is due to

energy storage ininternal energy

modes + chemistry


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• Post-shock pressure has weak dependence on non-ideal gas effects (just through (1- !))

• Post-shock temperature and density have strong Mach

number (free-stream speed) dependence

– Density ratio > (" + 1)/(" - 1) = 6 – Temperature decreases significantly

• Concept of " no longer has much meaning; if:

• Matlab code:

ftp://ftp.aem.umn.edu/users/candler/HEI/mollier.m


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1.6 Transport Phenomena

• Generated by gradients in flow properties:

– diffusion (Fick’s Law):

DAB

=diffusion coefficient

– viscosity (Newtonian fluid):

µ = viscosity coefficient

– thermal conduction (Fourier’s Law):

! = thermal conductivity coefficient

J A = "# D ABdC A

dy

" = µ du

dy

q = "# dT

dy


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Diffusion

• Affects continuity and energy equations

• Influences transport of species to surface

• Coefficient evaluation:

– for simple gas (self diffusion)

– for gas mixture

– are diffusion collision integrals

– averaged binary coefficient D1m often used

Dii =3

8"

# mikT

# $ii

(1,1)

Dij "kT

p

(mi + m j )kT

mim j

1

# $ij (1,1)

"ij (1,1)


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Diffusion

Depends on temperature, pressure, species


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Viscosity

• Affects momentum and energy equations

• Influences surface shear stress


– for simple gas

– various mixing rules

– are viscosity collision integrals

µ i =5

16

" mikT

" #ii

(2,2)

µ = µ ("ij (1,1)

,"ij (2,2)

)

"ij (2,2)


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Viscosity

• Sutherland law kg/m/s

– depends on pressure at high T due to chemistry

µ air =1.458"10#6 T

1.5

T +110.4


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Thermal Conductivity

• Affects energy equations

• Influences surface convective heat flux


– for simple gas (Eucken)

– various mixing rules

– are again viscosity collision integrals – curve fits for collision integrals from the literature

" i

=

5

16

# mikT

# $ii(2,2)1

M i

cv

+

9

4 R

u

%

& '

(

) *

" =" (#ij (1,1)

,#ij (2,2)

)

"ij (2,2)


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Thermal Conductivity

• Sutherland law W/m/K

– complex behavior due to chemistry

" air =1.993#10$3 T

1.5

T +112


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Prandtl Number

• Often used to evaluate thermal conductivity:

• Eucken’s relation:

– monatomic gas: Pr=0.67

– diatomic gas ("=1.4): Pr=0.737

Pr =c

pµ

"

Pr =4"

9" # 5


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Prandtl Number

• Real gas:

– again, complex behavior due to chemistry

Documents

Chp 1 Hypersonics