Compressible flow-Aerothermodynamics

Hypersonic Educational Initiative

Introduction toHypersonic Aerothermodynamics

Iain D. BoydDept. Aerospace Eng.University of Michigan

Ann Arbor, MI

Graham V. CandlerDept. Aerospace Eng. & Mech.

University of MinnesotaMinneapolis, MN


1. Hypersonic Gas Dynamics

1.1 Introduction and ExamplesOutline (1)

1. Hypersonic Gas Dynamics (1.5 hours)1.1 Introduction and Examples1.2 Post-shock conditions: perfect gas vs. equilibrium gas

Iteration approach for post-shock conditionsExamples

1.3 Reacting gas effects:Finite-rate reactions – nonequilibrium vs. equilibriumIonizationRadiation

1.4 Transport phenomena


Outline (2)

2. Hypersonic Aerodynamics: Pressure (1.0 hours)2.1 Exact and approximate equilibrium gas solutions:

Stagnation pointsCones and wedges

2.2 Mach number independence2.3 Newtonian and Modified Newtonian aerodynamics2.4 Examples


Outline (3)

3. Hypersonic Aerothermodynamics: Heat Transfer (1.0 hours)3.1 Introduction:

role of aerodynamic heatinghypersonic boundary layers

3.2 Boundary layer equations, Lees-Dorodnitsyn transformation3.3 Flat plate / wedge / cone solutions3.4 Stagnation point solution3.5 Transition to turbulence3.6 Wall catalysis3.7 Examples


Outline (4)

4. Viscous Interactions (1.0 hours)4.1 Leading edge interactions4.2 Effect on high-altitude L/D; scaling for vehicles4.3 Shock-BL interactions, shock-shock interactions

5. Thermal Protection Systems (1.0 hours)5.1 Passive:

re-radiative cooling, equilibrium wall boundary conditionrole of wall temperature, material propertiesexamples

5.2 AblativeSurface ablatorsPyrolyzing ablators


Outline (5)

6. Aerothermodynamics of Hypersonic Vehicles (1.0 hours)Ballistic entryLifting capsule re-entry: ApolloHigh-lift re-entry: ShuttleAerocapture / AerobrakingAirbreathing scramjets


What is Hypersonic Flow?

• Hypersonic aerothermodynamic phenomena:– strong shock waves with high temperature– not calorifically perfect (variable γ)– 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


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


Hypersonic Examples:II. Space Planes

• Mission: orbital re-entry• Aerodynamics: gliders with thermal protection• Propulsion: none (except small control thrusters)• Examples: Space Shuttle, Buran, Hermes


Hypersonic Examples: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


Hypersonic Examples:IV. Planetary Entry

• Missions: EDL, aero-braking, aero-capture• Aerodynamics: very blunt, thick heat shield• Propulsion: none (sometimes RCS)• Examples: Apollo, MSL, CEV (Orion)


• 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 VehicleHistorical Overview


Some CurrentHypersonic Programs

Falcon (DARPA)

Orion(NASA)X51

(AFRL)

HyBoLT (NASA/ATK)


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


• 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 (γ=const)

Re-entry Trajectories

• Trajectory equations for Earth centered system:

WD

LT, U

γ

!

U ˙ "

g=L

W# 1#

U2

gR

$

% &

'

( ) cos(" )

!

T

W"

˙ U

g=

D

W+ sin(# )


• 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, γ~0 (shallow entry)

Re-entry Trajectories

!

1

2"U

2


Flight Velocity


Stagnation Point Heating


Stagnation PointTemperature


Deceleration Levels


1.2 Post-Shock Conditions

• Perfect-gas shock relations:

• Density ratio asymptotes to:

• Pressure and temperature are quadratic in M

– Makes sense: energy is conserved


Post-Shock Conditions

• Post-Shock Temperature:

Temperatures rapidlybecome huge!



• Variation of air internal energy with T:

10% departure fromcalorically perfect gasequation of state =onset of hypersonic flow



• More fundamentally – 1D gas dynamics:

• Plus equations of state:

• No exact solutions

Thermally perfect,calorically imperfect

General equilibriumgas mixture



• Hypersonic limit:

• Note that post-shock enthalpy and pressure onlydepend on upstream conditions in hypersonic limit.

Can solve for thethermodynamic state



• Iterative solution to shock relations:

• Guess a value of ε = εi and iterate:

Use tables, NASA CEA, etc.


Equilibrium Air

Temperature (K) Z = Compressibility



• Example: M = 12 at 30 km altitude:

Imperfect Perfect



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

Difference is due toenergy storage ininternal energymodes + chemistry



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

• Post-shock temperature and density have strong Machnumber (free-stream speed) dependence– Density ratio > (γ + 1)/(γ - 1) = 6– Temperature decreases significantly

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

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


1.3 Reacting Gas Effects

• Analysis of Earth hypersonic vehicles at U<8km/s:– 5-species air model sufficient: N2, O2, NO, N, O

• Reactions:– Dissociation-recombination:

– Zeldovich exchange:!

N2

+ M" N + N + M

!

N2

+O" NO+ N

!

NO+O"O2

+ N!

O2

+ M"O+O+ M

!

NO+ M" N +O+ M


Finite Rate of Reactions

• For illustration, consider:– 2-species: N2, N

• Each reaction proceeds at a finite rate:

• Forward rate coefficients measured experimentally, kf (T)

• Backward rate coefficients from equilibrium constant:

partition functions Q from quantum+statistical mechanics

!

N2

+ M" N + N + M

!

N2

+ N2"kb1

k f 1

N + N + N2

!

N2

+ N"kb 2

k f 2

N + N + N

!

Ke =k f

kb="Qproducts

"Qreactants



• 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:

– α=mass fraction, m=atom mass, ρ=density, V=volume,θd=dissociation temperature

!

d[N2]

dt= "k f 1[N2

][N2]" k f 2[N2

][N]+ kb1[N][N][N2]+ kb2[N][N][N]

!

" 2

1#"=m

$V

QN

2

QN2

exp(#%d /T)



• Chemical equilibrium:– O2 dissociates before N2 (has lower θd)– fewer atoms at high pressure (more recombination)


Finite Rate of Reactions• Chemical nonequilibrium:

– equilibrium end state reached only after finite time– in a flow field, this translates as finite distance


Nonequilibrium

• Impact of chemical nonequilibrium:– chemical composition mainly affects energy of flow

• exothermic reactions consume energy• catalysis: fraction of atoms reaching the vehicle

surface may recombine releasing heat– scaling:

• nonequilibrium flow occurs at lower densityand/or smaller body length scales

!

large Kn "#$L%

1

&$L

!

small Re "#$U$L

µ$


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:

!

N2

+ M" N + N + M

!

N2

+O" NO+ N

!

N + N" N2

++ e

#

!

N + e"# N

++ e

"+ e

"


Ionization

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

– φ=ion mole fraction,– C=constant,– p=pressure,– θi=ionization temperature

!

" 2

1#" 2= C

T5 / 2

pexp(#$i /T)


Ionization• Significance:

– plasma causes communications blackout– highly catalytic ions are source of heating


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

"

!

˙ q rad "RNU8.5#1.6


1.4 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):κ = thermal conductivity coefficient

!

JA = "#DAB

dCA

dy

!

" = µdu

dy

!

q = "#dT

dy


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)


Viscosity

• Affects momentum and energy equations

• Influences surface shear stress

• Coefficient evaluation:– for simple gas

– various mixing rules

– are viscosity collision integrals!

µi=5

16

"mikT

"#ii

(2,2)

!

µ = µ("ij

(1,1),"ij

(2,2))

!

"ij

(2,2)


Thermal Conductivity

• Affects energy equations

• Influences surface convective heat flux

• Coefficient evaluation:– 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

Mi

Cv

+9

4Ru

%

& '

(

) *

!

" ="(#ij

(1,1),#ij

(2,2))

!

"ij

(2,2)

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Compressible flow-Aerothermodynamics

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