Hypersonic Nonequilibrium Flow Simulation Based on Kinetic Models

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Frontiers in Aerospace Engineering Vol 1 Iss 1 November 2012

1

Hypersonic Nonequilibrium Flow SimulationBased on Kinetic Models JS Shang 1 ST Surzhikov 2 H Yan 3 Wright State University USA Russian Academy of Science Moscow Russia

Northwestern Polytechnical University China1 josephshangwrightedu 2surgipmnetru 3yanhongnwpueducn

Abstract

The nonequilibrium hypersonic flowfield over a blunt body is computationally simulated by self-consistentkinetic models The conservation governing equationis based on the continuum formulation by the kinetictheory of a diluted gas for transport properties and agroup of chemical-physical kinetic models of chemicalkinetics for nonequilibrium chemical reactions andenergy transfer among internal degrees of excitationIn this regard the present analysis summarizes thecurrent progress illuminating efforts for aninterdisciplinary computational fluid dynamicsendeavour Further progress of this computationalsimulation discipline has to be built on the newlyfound knowledge base

Keywords

Interdisciplinary CFD Nonequilibrium Hypersonic Flow

Introduction

Nonequilibrium thermodynamic state and chemicalreactions are common features of hypersonic flowsThe flow field around a typical blunt body at analtitude above 60 km with a flight speed exceeding 2kms cannot maintain a chemical equilibrium state Inthe framework of continuum mechanics the bridgelinking the Boltzmann equation and conservative lawsof aerodynamics is the kinetic theory of gases throughthe Chapman-Enskog expansion [1] Since the kinetictheory does not consider the internal structure ofatoms and molecules the higher degrees of excitation beyond translational motion must be resolved bymodelling

Traditionally the connection between molecular andatomic structures and thermodynamic behaviour ofhigh-temperature gas is described by statisticalthermodynamics and augmented by quantum physicsTherefore the internal degrees of freedom of atomic

and molecular species were studied via the partitionfunction and the gas mixture was assumed to be inlocal equilibrium [2] A substantial amount of researchresults have been derived from this approach [3] and

more recently even extended to include the ablationphenomenon [4] However significant discrepancyhas been noted for this group of results whencompared with a limited amount of flight data A partof the discrepancy is due to inaccurate computationsand experimental measurements but a major portionof this disparity is incurred by the modelling of thephysics [5-7]

At the present a thorough understanding of theextremely complex physical-chemical phenomena ofnonequilibrium hypersonic flows is still beyond ourreach All the interactions of the internal degrees offreedom occur at the molecular and atomic scalestherefore must be modelled and can be verified only by a sparse validating data base for physical fidelityFor this reason the present investigation is focused onthe blunt-body problem which is dominated by thestrong bow shock In this flowfield the energy contentin the translational mode is initially much greater thanthe vibrational mode which mostly occupies mostlythe first few quanta above the ground state Therefore

the classic theory of Landau-Teller provides acreditable approximation [8] Meanwhile a separatetreatment for nonequilibrium rotational degrees offreedom of gas component may not be necessary Infact the rotational excitation can reach its classicequilibrium state by about 10 kinematic collisions toelevate its energy level by one κ T Therefore it is fully justified to assume the rotational excitation to be inequilibrium with the translational mode [6]

On the other hand the vibrational relaxation of air becomes significant in the temperature range above1000 K and beyond that the molecule dissociated intoatoms At the lower limit of the temperature range it




2

is estimated that the vibrational excitation requiresfrom 1000 to 10000 collisions to reach thermalequilibrium with the translational mode [6] Underthis circumstance the energy exchange betweendifferent molecular species has a higher probability

than the vibration-translation transfer of samemolecule [67] In addition the adiabatic collision has alow probability of vibrational excitation and will not be taken into consideration

For the electronic excitations the vibrational energy ofthe molecule plays a main role in the dissociation Thisprocess will be carefully examined using a detailedanalysis for the generation and depletion balanceFinally when the energy of the excitation is greaterthan 10 eV the cross sections of inelastic collisions ofheavy particles are several orders of magnitudesmaller than the cross sections of elastic electronimpact [236] Therefore the modelling of ionization by electrons impact will be emphasized

All the accumulative knowledge in treating thedetailed physical-chemistry modelling has guided thedevelopment of the widely used model equations fornonequlibrium hypersonic flow simulations [3-12] Itwould be valuable to summarize the current state-of ndashthe-art in kinetic modelling and to project future basicresearch needs for better understanding

Governing Equation

By examining the governing equations in thecontinuum domain the effects of internal atomic andmolecular structures are explicitly contained in thespecies and energy conservation equations but onlyimplicitly in the conservation law of momentumthrough the transport property of the gas mixture Thecoupling of conservation laws has offered a widerange of options on how this interdisciplinaryequation system can be efficiently solved On the other

hand the quantum levels of rotational excitation aredensely packed and readily reach equilibrium withtranslational excitation but this fundamental mode isnot necessarily equilibrated with the vibrational andelectronic excitations [2] In the present approach theenergy exchangecascade between vibration-translation and electronic-translation interactions will be explicit simulated by the separated energyconservation equations for vibrational and electronicexcitations [7] In other words the present approach is built on a refined energy balance from the internal

structure of the atom and molecule The detailedmechanisms are developed by models of the gas

kinetics The conservation equations in a continuummedium can be given as [57]

[ ( )]i ii i

dwu u

t dt ρ

ρ part

+ + =part

(2-1)

( ) 0u

uu pI t ρ

ρ τ part

+ + minus =part

(2-2)

[ ( )] 0i i i rad

eeu T u h q u pI

t ρ

ρ κ ρ τ part

+ minus + + + minus =part

sum

(2-3)

Vibrational energy conservation equations fordifferent species are

[ ( ) )]i iv ii i iv iv iv v

e dwu u e q e Q

t dt ρ

ρ Σ

part+ + + = +

part (2-4)

Electronic energy conservation equation can be given

as

[ ( ) )]i e ii i e e e e e

e dwu u e u p I q e Q

t dt ρ

ρ Σ

part+ + + + = +

part (2-5)

where Q vΣ and Q eΣ are the net energy transfer between translational vibrational and electronicexcitations A key addition to the present formulationis that the rates of energy released from thevibrational-translational and electronic-translationalinteractions will be examined and modelled [5] Thissubject shall remain as a sustained research focus into

the future and will also be extended further to includeablation

For calculating the equilibrated translational-rotationaltemperature and gas mixture density the total internalenergy is defined as

( ) ( )2 2

o e ei v i i v i i i e v e e

i e i e i e

u uu ue c T e h c T ρ ρ ρ ρ ρ

ne ne ne

= + + + ∆ + +sum sum sum

(2-6)

where o

ih∆ is the standard heat of formation The

vibrational energy of different species is obtained fromthe harmonic oscillator

[ exp( 1)]v iu

v i v ii

Re

M t κ

Θ= Θ minus (2-7)

The pressure of the gas mixture is calculated withDaltonrsquos law including the contribution from theelectrons collision

ui

i

R p T

M ρ = sum (2-8)

where Θv is the characteristic vibrational temperatureof species i R u is the universal gas constant 8314x10 7




3

FIGURE 1 COMPARISON OF BOW SHOCK DIMENSIONS OFEQUILIBRIUM AND NONEQUILIBRIUM SIMULATIONS

FIGURE 2 TEMPERATURE DISTRIBUTIONS ALONGSTAGNATION STREAMLINE FOR PERFECT GASEQUILIBRIUM AND NONEQUILIBRIUM FLOWS

FIGURE 3 PRESSURE DISTRIBUTIONS ALONG THESTAGNATION STREAMLINE FOR PERFECT GASEQUILIBRIUM AND NONEQUILIBRIUM FLOWS

ergg-Mol K and Mi is the molecular weight ofchemical species i

In the present formulation a further breakdown of theinternal energy in highly excited electron states is notincluded [2 6 7] For the weakly ionized air theelectrostatic force ρ eE work done by this force on thecontrol volume and Joule heating JE are not takeninto consideration

Kinetic Modelling For Internal EnergyTransfer

The significance of relative time scales of the fluiddynamics and chemical kinetics can be easilydemonstrated by the traditional demarcation forchemically reacting interactions as the equilibrium and

nonequilibrium flows The numerical resultsgenerated by a bench mark (GASP) for hypersonicflow are presented in the following three figures toillustrate the distinctive features of the flowfieldinvolving chemical reaction All these computationalresults are obtained in an axisymmetric frame with adimension of (163x93) for a spherical cone (Ram-C-II)The flow conditions can be summarized by the Machnumber of 2709 Reynolds number of 138x10 4 basedon the length of the model and a static temperature of1986 K

The energy distribution among different internaldegrees of excitation transport properties andchemical composition of the gas can changesubstantially in the thin shock layer of the blunt bodyflowfield The comparison of bow shock structuresreveals a significant difference in the standoffdistances of the two limiting conditions as it ispresented in Figure 1 For the equilibrium limit thedecreased density ratio across the bow shock due tothe instantaneously redistributed thermal energy to allinternal degrees of freedom reduces the shock standoff

distance to a value of 0075 of the nose radius Fromthe computed temperature contours the temperatureof the gas mixture is also much lower than the morerealistic nonequilibrium simulation

The detailed comparison of the translationaltemperature distributions for perfect gasnonequilibrium and equilibrium conditions aredepicted in Fig 2 There is no need to describe theresult of the perfect gas model but it is inserted in hereas a frame of reference The difference in the statictemperatures is substantial the peak temperatures ofthe nonequilibrium and the equilibrium computationsare 2078746 K and 585544 K respectively In all the

comparing results the maximum temperature isconsistently located immediately downstream of theshock It can be easily anticipated that the conductiveheat transfer rates at the stagnation region will besubstantially different




4

Fig 3 presents the comparison of the static pressurealong the stagnation streamline for the same threecases studied There is only a slight difference betweenresults of the perfect gas and nonequilibrium modelfrom the equilibrium condition on the body surface

This small difference is due to the existence of a largerfraction of free electrons for the equilibrium conditionElectronic excitation is also the only internal mode ofthe high-temperature gas that can contribute to thestatic pressure of the mixture All other internaldegrees of excitations are quantum phenomena thatwill not contribute to any net exchange of momentumwith the ambient In fact the equilibrium computationgenerates the highest mass fraction of the freeelectrons which attains a value of 716x10 -8 incomparison with the nonequilibrium limit

The energy transfer among the internal degrees offreedom of high temperature air consists of two majormechanisms energy exchange within the excitedinternal modes and the chemical kinetic processes Thechemical kinetic process can be further broke downinto seven main mechanisms for energy transfer byPark [7] They are ionization electron impactdissociation elastic collision between electrons andheavy particles interaction between electrons andvibrational mode energy exchange by the formationof species associative ionization and finally radiationlosses These processes are complex and ourunderstanding of the physical chemical interaction inthe molecular and atomic scales is extremely limited

The chemical kinetic models of the present analysis areadopted based on some unique flow featuresimmediately downstream of a strong shock In thisenvironment the initial level of vibrational energy isfar lower than that of the translational energy and theenergy exchange among the internal modes is

dominated by the adjacent quantum level (ladderclimbing) Especially the vibrational quantum statesare limited to the first few above the ground stateTherefore the anharmonic correction may not benecessary Most important from the principle ofdetailed balance the classic result of Landau-Teller [8] becomes the backbone for the present kineticmodelling The following kinetic models have guidedthe development of the practical applications [3-511-15]

The energy exchange between translational andvibrational modes is approximated by

1 2 1 2

( )( )

1 1( )

[1 ( )]

V V V

V V V V TV V V

T T TV

v

e T e R RQ e T e

e eT Exp T

p Exp T

ρ τ

κ κ τ

Θ Θ

minus Θ Θ= = =

minus minus

=minus minusΘ

(3-1)

The energy transfer among the vibrational modes ofdifferent species has also been modelled by Masterequation via a single quantum level However therelaxation time scale is provided by the experimentalcorrelation of Millikan and White [9]

4 113 1 2 1 23 32

1 2 1 2

116 10 ( ) [( 015( ) 184TV V

M M M M T

M M M M τ

minusminus= times Θ minus minus+ +

(3-2)

The preferential dissociation model by Treanor andMarrone [10] was adopted as the research progressed

to the energy transfer between electronic andvibrational excitations

( ) [( ( ) ]k ET e k i

dnQ Exp D E kT

dt ε κ ν = Σ = minus minus (3-3)

The energy transfer between elastic electron-ioncollisions is approximated by the formulation bySurzhikov et al [11]

2032121 10 lne

Ei e ie

T T Q x x

T

minus= times Λ (3-4)

In this formulation xe and xi denote the molar fractionsof electron and positively charged ions ln Λ is theCoulomb logarithm

As a base for comparison a global approximation forthe energy transfer between heavy particle andelectronic collisions formulated by Lee [12] has beenused [131415]

12

2

83 ( )( )e k

ET e e ek k ee k

RT N Q R T T

M M ρ

ρ σ π ne

= minus sum (3-5)

A more recently developed on chemical-physicalmodel that describes the electronic-vibrational kineticscan be found in the theory of Chernyi et al [16]

Transport Properties

A landmark of the kinetic theory of a diluted gasmixture is its ability in describing the transportproperties of any gaseous mixture To be consistentwith the kinetic model of the internal structure of a gastransport properties of the gas mixture for thermaldiffusion viscosity and thermal conductivity arecalculated from Boltzmann equation by using theChapman-Enskog expansion [17]




5

FIGURE 4 PRODUCTS OF COLLISION INTEGRAL Ω(11) AND CROSS SECTION FOR AIR MIXTURE 200 ltTlt 28000 K

FIGURE 5PRODUCTS OF COLLISION INTEGRAL Ω (22)AND CROSS SECTION FOR AIR MIXTURE 200 ltTlt 28000 K

Thermal diffusivity is

3 3 2 (11) 1858 10 i j

i j i ji j

M M D T

M M σ minus

+= times Ω (4-1)

The molecular viscosity is given as5 2 (22)267 10 i i M T micro σ minus= times Ω (4-2)

and the thermal conductivities are

Mono-atomic gas

4 2 (22) 1989 10 i m i

i

T M

κ σ minus= times Ω (4-3)

Poly-atomic gas

4 2 (22) 2519 10 i p i

i

T M


All collision integrals and cross sections are obtainedfrom either the Lenard-Jones potential for non-polarmolecules or a polarizability model for ion-neutralnon-resonant collisions by Capitelli et al [18] andLevin et al [19]

The Chapman-Enskog theory gives the viscosity andthermal conductivity for a gas mixture these formulasare also known as the Wilkersquos mixing rule [17]

12 12 14 2

1(1 ) [1 ( ) ( ) ]

8

i ii j

i i j

ji ii j

j j i

x

x M M

M M

micro micro

φ micro

φ micro

minus

=

= + +

sumsum (4-5)

and

14 2 12 [1 ( )( ) ] [8(1 )]

i ii j

i i

ji ii j

j i j

x x

M M M M

κ κ

φ

κ φ

κ

=

= + +

sum sum (4-6)

The transport properties of the present analysis are

derived from kinetic theory via the collision integralsand collision cross sections [18 19] For purpose ofunderstanding a very wide range of temperaturesfrom 200 to 28000 K for the eleven species of airmixture are calculated According to the study ofCapitelli et al [18] the calculated collision integrals arenot dramatically dependent on the specific form of thepotential models

Figure 4 depicts the products of collision integral Ω (11)and its associated collision cross sections for all elevenspecies of air The collision cross section is given inAngstrom squared (A 2) [18 19] The collision integralis a dimensionless double integral whose value is

depended on the dynamics of binary collisions thus iscontrolled by the intermolecular force law From thecalculated results the magnitudes of the collisionintegral have the greatest value for the charged carriedmolecules N2+ O2+ and NO+ over all other species in

the high-temperature domain Therefore the thermaldiffusion of these species should have the lowest valueMeanwhile the collision integrals of the polarizedatomic species O+ N+ and the dissociated species Nand O have a similar magnitude in the temperatureregime calculated

The product of collision integrals Ω (2 2) and itsassociated cross sections for calculating thermalconductivity and molecular viscosity of all air speciesin the investigated temperature domain are presentedin Fig 5 The variation of these products is lesspronounced in the high temperature region to reflect a




6

FIGURE 6 HEAT RELEASE RATE ALONG STAGNATIONSTREAMLINE

FIGURE 7 DAMKOHLER NUMBER DISTRIBUTION ALONGSTAGNATION STREAMLINE

linear dependence to the local temperature andmolecular weight of the species Although there issubstantial progress in the kinetic theory of gas forcalculating transport properties the specificcomparison of the calculated transport properties still

yields a difference of 163 to 242 in viscosity andthermal conductivity for high-temperature air usingdifferent sets of collision integrals in the temperaturerange from 300 to 30000 K [19]

Chemical Kinetics

The high-temperature air mixture is considered toconsist of 11 species O N O2 N2 NO O + N+ O2+ N2+ NO + and e - For a non-equilibrium chemicalreaction the production rate of chemical species andheat release are calculated by the rate constants for theforward and reverse chemical reactions The energyconversion in chemical reaction becomes anincreasingly important process for energy transfer inhypersonic flows However based on the accumulatedexperience for hypersonic flow simulations thechemical reactions and aerodynamics can still beloosely coupled to achieve the most efficientcomputation and this approach is adopted by thepresent study [3-5 7 20]

The rate constant is determined by the Arrhenius

equation

[ ] f r in f r i f r i f r i

E k A T Exp

kT = minus (5-1)

The forward reaction rate is frequently obtained fromexperimental observations and the reverse reactionrate can be determined by the Sharma approximation[3 5 7]

f je a

r j

T κ

κ κ

asymp (5-2)

The two most widely adopted non-equilibriumchemical reaction models are developed by Dunn andKang [21] and Park [22] Parkrsquos model is chosensimply because it has been frequently updated In hismodel (48 elementary reactions are included) thereverse reaction rates are not provided For the presentanalysis the reverse reaction rate constants calculated by Surzhikov are used [5]

Several interesting features of nonequilibriumchemical reactions in the shock layer are easilyillustrated by the heat release rate and the Damkohlernumber distribution generated by the GASP benchmark Figure 6 gives the energy exchange rate

along the stagnation streamline to the surroundingsIt is well known that in order to excite internal degreesof freedom of a gas mixture an energy input isrequired to proceed The behaviour of an endothermicreaction is clearly observed immediately downstreamof the bow shock However when the gas mixturemoves adjacent to the cooler blunt body then thechemical radicals begin to recombine the localchemical reaction becomes exothermic This fact leadsto an important observation that the catalytic effect is

an open issue for the chemically reacting flowfieldaround aerospace vehicles [3 4 7 12 15] In thestagnation region the chemical reactions and thechemical-physical processes are complex and arereflecting by the rapid rate of chemical reactionslocally which demands additional attention to better




7

FIGURE 8 MASS FRACTION CALCULATIONS USING MIN-MOD LIMITER

understand the energy exchange processes with asolid surface

Figure 7 presents the calculated Damkohler numberdistribution along the stagnation streamline In themajor portion of the chemically reacting domain thereaction rate is slightly faster or on the comparableorder of the diffusion speed However the relativetime scales drastically changes in the near-wall regionthe Damkohler number shows multiple extremes Themaximum at the solid surface in fact exceeds a valueof 2014 From the chemical reaction viewpoint this behaviour can be interpreted as the mixing speed ismuch slower than the chemical reaction rateNevertheless the most complex chemical reactionprocess is located near the fluid-solid interface rather

than on the shock front This result also points out thechallenge in studying the ablation phenomenon

Numerical process

The interdisciplinary partially differential equations(2-1) through (2-5) are solved by a combination ofimplicit cell-centered finite-volumefinite-differenceschemes commonly adopted for solving thecompressible Navier-Stokes equations [51113-15] Thegoverning equations can be cast into the flux vectorform

U F R

t

part+ =

part

(6-1)

where U is the dependent variables ( )U U u e ρ ρ ρ =

and F is the flux vector component [232425]

Although multiple numerical procedures wereadopted as the solving scheme the basic approach isthe second-order implicit method [23 24] Especiallythe NERAT [5 23] uses three difference processes forsolving the momentum energy conservation and the

chemical species conservation equations As acommonly accepted computational approach theupwind-biasing approximation is applied to theconvective and pressure terms and centraldifferencing is used for the diffusion and heat transferterms In addition a flux-difference splittingprocedure for shock capturing is implemented forsolving the governing equations The flux vectors atthe control surface are written as the solution to theRoe-type approximate Riemann problem [25]

12

12

1 2[ ( ) ( )] ( )]1 2[ ( ) ( )] ( )]

i L R inv R L i

L R inv R L i

U F U F U M U U F U F U M U U

δ +

minus

= + minus minus

minus + minus minus(6-2)

where U L and U R are interpolated values of thedependent variables ρ ρ u and ρe at the interface ofthe control volume and M inv is the Jacobian matrix ofthe inviscid or convective terms [2223]

A slope limiter is also used to control thediscontinuous pressure jumps at the shock front Timeadvancement is implicit to solve the flows that have asteady state asymptote A min-mod limiter is adoptedfor some of the present computations

12

12 1 1

( ) 1 4[(1 ) (1 ) ]

( ) 1 4[(1 ) (1 ) ] L i i i

L i i i

U U U U

U U U U

κ κ

κ κ +

+ + +

= + minus + + ∆

= minus minus ∆ + + (6-3)

where the one- side difference operators are ΔU = Ui+1 ndash U i and U = Ui ndash U i-1 respectively The val ue of κ isassigned a value of -1 to yield the second-order fully-upwind differencing approximation

The effectiveness of the slop limiter for controllingoscillations around the strong bow shock is revealed by the calculated species concentration for thechemical reactions In one of the present solvingschemes the min-mod operator was applied to controlthe excessive numerical oscillation at the shock front

Figure 8 gives the mass fractions of atomic nitrogenand oxygen along the stagnation streamline The sloplimiter suppresses completely the numericaloscillations but also increases the number of gridpoints required to capture the shock jump This issuewill be addressed again later in the discussion

In order to concentrate on the rather complex physicsthe numerical simulation is limited to an axisymmetricconfiguration For the same reason the importantradiation-gas-dynamic interaction in hypersonic flowis also deferred from the present investigation but will be included in the continuing effort Specificallyradiation heat transfer will be treated by the multi-




8

FIGURE 9 MACH NUMBER CONTOUR M infin=2709RE=1385times104 Tinfin =1986 K (163times93) MESH

group spectra approximation in which the radiatingenergy transfer is grouped according to the emittingfrequencies [5 23]

Numerical Results

Computational results of hypersonic flow over theRam-C-II vehicle [26] are carried out The basicconfiguration is a spherical cone with a nose radius of1524 cm and a semi-conical angle of 9 degree Theoverall length of the vehicle is 12954 cm The flighttest data were collected at altitudes from 533 to 853km and at flight speed of 762 ms with a small angleof attack The flight test report consists of a detaileddescription of the data collection technique flighttrajectory information and the data of the relatively

rare ionized air concentration at a small angle of attackFor now two groups of computational simulations arelimited to axisymmetric flow The benchmark resultfrom GASP is included together with the most recentresults from NERAT [5 11] The latter represents themost recent and self-consistent kinetic modelformulation for nonequilibrium hypersonic flow andis the key starting point of the present research

For the purpose of providing a reference forunderstanding the chemical composition in massfraction along the stagnation streamline generated bythe benchmark GASP is presented in Figure 9 Thecomputation is obtained for a freestream Machnumber of 2709 with the freestream and walltemperatures of 1986 and 1000 K respectively The body-oriented grid system (163times93) is also closelyaligned with the shock envelope to facilitate shockcapturing scheme by an analytical matching ofspherical and hyperbolic functions On this meshsystem the well-known ldquocarbunclerdquo artefact near theaxis of symmetry is completely absent which isfrequently considered to be associated with theapproximate Riemann formulation

In this Figure molecular oxygen is nearly completelydepleted and at the wall has a mass fraction of 817times10 -

4 which is the same order of magnitude as the chargedspecies N+ and O+ The charged carried molecularcomponents N2 + O2+ and the free electron e - have thelowest mass fraction in the shock layer among alleleven species and the former two species exhibit astrong dependence to the local temperature

In the following the simulated chemical compositionsand the translational and vibrational temperaturesfrom NERAT are presented The numerical simulation

is conducted at an altitude of 61 km with a freestreamMach number of 239 on a mesh system of (105times51)The freestream temperature according to the flightdata is 244 K but the surface temperature is unknownFor most numerical simulations this temperature has

been prescribed in the range from 1000 to 1500 K andit is assumed that the surface is fully catalytic [13-15]In the present investigation the surface temperature isassigned a value from 300 to 1500 K extending therange of uncertainty According to commonlyaccepted practices the vibration temperatures arecalculated from the model of harmonic oscillator [513-15] The species concentration however will be givenin terms of number density per unit volume to betterdescribe the chemical composition The numericalresults will be validated by comparing with earlier

simulations and available data [5 13-15 26]Interpretation of new findings will be speciallyemphasized

The calculated translational temperatures of the airmixture and the vibration temperatures of themolecular oxygen and nitrogen in the shock layer ofRAM-CII along the stagnation streamline are given inFigure 10 For the lower surface temperature of 300 Kthe peak translational temperature has a value of14754 K the calculated vibrational temperature formolecular nitrogen and oxygen are 11712 and 10873K respectively These values are lower than thesimulations using a much higher assumed surfacetemperature [13-15] This discrepancy may beanticipated because at a lower surface temperature theincreased heat transfer to the spherical conesignificantly reduces the thermal energy in the shocklayer More importantly this type of discrepancy hasalso been noticed from earlier simulations and has




9

FIGURE 10 TEMPERATURES ALONG STAGNATIONSTREAMLINE M infin =239 T infin =244 K T W=300K (105X51) MESH

FIGURE 11 CHEMICAL COMPOSITION ALONG THE

STAGNATION STREAMLINE M infin =239 T infin =244K T W=300K(105 51)

FIGURE 12 TEMPERATURES ALONG STAGNATIONSTREAMLINEM infin =239 T infin =244 K T W=1500K (55X41) MESH

even been pointed out by the fundamental researchusing ab initio approach [27] Due to the lowertranslational temperature the bow shock standoffdistance is also shorter than other simulations 009 cmversus 0099 by Josyula et al [15] and 0106 cm by

Candler et al [13] However the main feature of thetemperature profiles remains in that the energy istransfer from the translational mode to the vibrationalmode and the temperatures of internal degrees offreedom are equilibrated at a short distancedownstream of the bow shock

The corresponding chemical composition in numberdensity per unit volume along the stagnationstreamline is depicted in Figure 11 In this presentationthe particle number density is given in number percubic centimetre ncm 3 or ncc In this graph themolecular oxygen under goes a wide range ofvariation from the freestream value of 112times10 15cm 3 dips to a value less than 10 13cm 3 in the shock layerand finally due to recombination of atomic oxygenreaches a value higher than 11times10 16cm 3 on the conesurface The concentrations of O + and N + consistentlyhave the lowest value less than 10 13 cm 3 over theentire domain The electron number density on thestagnation point has a value of 32times10 13 cm 3 In thenumber density per unit volume format the partiallyionized nonequilibrium air exhibits the classic behaviour of plasma the flow medium is globallyelectrical neutral from the shock front until a thinsheath immediately adjacent to the surface This

behaviour fully justifies the omission of theelectrostatic force from the present formulation

In order to demonstrate the importance of gridtopology for shock capture and highly clustered griddensity near the solid surface a redistribution gridsystem is implemented for the following computationsThe rest of the simulations are performed on a coarser(55times41) mesh but at an optimal distribution to focuson the kinetic modelling

Figure 12 depicts the translational and vibrationaltemperatures of molecular nitrogen and oxygen alongthe stagnation streamline with adjustment to thekinetic models For the higher surface temperature of1500 K the peak translational temperature still has avalue of 14490 K the calculated vibrationaltemperature for molecular nitrogen and oxygen are




10

FIGURE 13 COMPOSITION ALONG STAGNATIONSTREAMLINE M infin =239 T infin =244 K T W=1500K (55times41) MESH

FIGURE 14 COMPARISON OF ELECTRON NUMBER DENSITYDISTRIBUTION ALONG RAM-CII WITH DATA

11411 and 11396 K respectively The calculatedresults display very small change from the results ofthe lower assumed surface temperature (300 K) andeven a less effect due to the redistribution of gridpoints in the shock layer For the present simulations

using two different surface temperatures the mainfeature of the temperature profile is unaltered in thatthe equilibrated temperature for vibrational rotationaland translational modes has been reached at thedistance of one-third of the shock-layer thicknessdownstream of the bow shock This feature is shared by all numerical simulations [13-15] to revealvibrational excitation requires a sufficient number ofcollisions to attain the same thermodynamic state asthe translational and rotational modes A minordifference in the result using the (55times41) mesh from

the previous calculation is also observable a nearlyconstant temperature region exists shortly after theequilibrated temperature of the internal modes isreached Whether this feature is physically significantor not will be determined in future investigations

The corresponding species concentration along thestagnation streamline is depicted in Figure 13 Incomparison with the previous calculation by the(105times51) mesh two significant differences stand outFirst the molecular components including thedissociated components N and O are nearly identicalto the early simulation by an unrefined grid Themolecular oxygen still exhibits the most drastic changefrom the shock front to the stagnation point but therecombined number density at the surface is muchlower than the lower surface temperature calculationA value of 53times10 12cm 3 is noted at the stagnation point

in contrast to a value of 11times10 16cm 3 The second majordifference is the concentrations of the charge carriedatomic and molecular species O + N+ O2+ and N2 + have now fallen below a value less than 10 12cm 3Therefore the number density of electrons is mostly

balanced by the species NO+

as it is expected to behaveas the globally neutral partially ionized plasma

The only validation of the present results is derivedfrom the comparison of electron number densitieswith the flight data by Jones and Cross [26] The major

portion of their electron number density data ismeasured by a reflectometer and a checking point byan electrostatic probe at the distance around 8 noseradii from the stagnation point The data by thereflectometer represents the averaged peak value atthe sample location and the probe data is the time-averaged value among the innermost and theoutermost probes The estimated error bar of the probedata covers a range of the peak-to-peak densityfluctuation from 30times10 11cm 3 to 12times1012cm 3 due the body motion The comparison of the present

computational result with flight data is depicted inFigure 14 In spite of a noticeable difference in thepredicted temperature profiles in the shock layer bythe present study from other computationalsimulations [13-15] the agreement with the flight datais reasonable and is comparable with earlier numericalresults by Candler et al [13] Josyula et al [14]

However a better understanding of the quantumenergy transition among internal degrees of excitationof a high-temperature gas can only be achievedthrough an ab-initio approach of the computationalchemical-physics This computational simulation is based on the first principle of the collision process




11

from classic mechanics and requires knowledge ofinter-atomic forces exerted by the potential energyThe computational resource requirement of thisapproach is extremely intensive At the present thisresearch is limited to simulation of a single gas species

[27]

Conclusion

An internally consistent formulation for simulating thehypersonic nonequilibrium flow based on the kinetictheory of gas has been initiated The presentinvestigation combines the physically based kineticmodels for internal degrees of freedom and the mostrecent advance in collision integrals research fortransport properties of high-temperature air

Different implicit algorithms are adopted to solveseparately the conservation equations for speciesconcentration vibrational electron energy equationand finally coupled with the compressibleinterdisciplinary equations The solving scheme has been proven to be computational stable and accuratein comparison with parallel efforts

The present progress built on the luminary pioneeringefforts in chemical-kinetic models reveals just the firststep in developing an accurate modelling andsimulating tool for nonequilibrium hypersonic flow

simulations The high fidelity descriptions of thecomplex quantum chemical-physical phenomena for better understanding still requires a combined andsustained effort by ab-initio computations and non-intrusive experimental observations for analysingablation including the pyrolysis and radiationphenomena

ACKNOWLEDGMENT

The authors are sincerely appreciated the sponsorship by Dr J Schmisseur and Dr F Frahoo of the Air ForceOffice of Scientific Research Air Force ResearchLaboratory

REFERENCES

[1] Chapman S and Cowling TG The Mathematical

theory of non-uniform Gases Cambridge University

Press 1964 pp134-150

[2] Clarke JF and McChesney M The Dynamics of Real

Gases Butterworth Inc Washington DC 1964 pp 66-

86[3] Park C Chemical-kinetic Parameters of Hypersonic

Earth Entry J of Thermophysics and heat Transfer

Vol 15 No 1 2001 pp76-90

[4] Chen YK and Milo FS Navier-Stokes Solutions with

Finite Rate Ablation for Planetary Mission Earth

Mission Earth Reentries J Spacecraft and Rockets Vol42 No 6 2005 pp961-970

[5] Surzhikov ST and Shang JS Kinetics Models

Analysis for Super-Orbital Aerophysics AIAA 2008-

1278 Reno NV 2008

[6] Zelrsquodovich Ya B and Yu P Raizer Physics of Shock

Waves and High-temperature Hydrodynamic

Phenomena Dover Publications Inc Mineola NY

2002 pp 177-232

[7] Park C Nonequilibrium Hypersonic aerodynamics

John Wiley amp Sons New York NY 1989 pp89-118

[8] Landau L and Teller E Zur Theore der

Schalldispersion Physik Z Sowjetunion b 10 h 1

1936 p34

[9] Millikan RC and White DR Systematics of

vibrational Relaxation J Chemical Physics Vol 39 No

12 1963 pp3209-3213

[10] Treanor CE and Marrone PV Effect of Dissociation

on the rate of Vibrational Relaxation Physics of Fluids

Vol5 No 9 1962 pp1022-1026

[11] Surzhikov S Sharikov I Capitelli M Colonna G

Kinetic Models of nonequilibrium Radiation of Strong

Air Shock Waves AIAA 2006-0586 Reno NV 2006

[12] Lee JH Basic Governing Equations for the Flight

Regime of Aeroassisted Orbital Transfer Vehicles

Progress in Aeronautics and Astronautics Thermal

Design of Aeroassisted Orbital Transfer Vehicles Vol

96 AIAA New York NY 1985 pp3-53

[13] Graham GV and MacCormack RW Computation

of Weakly ionized Hypersonic Flows in

Thermochemical Nonequilibrium J Thermophysics

Vol 5 No 3 1991 pp 266-272

[14] Scalabrin LC and Boyd ID Numerical Simulation of

Weakly Ionized Hypersonic Flow of Reentry

Configurations AIAA 2006-3773 San Francisco June

2006

[15] Josyula E and Bailey WF Governing Equations for

Weakly ionized Plasma Flow field of Aerospace

Vehicles J Spacecraft and Rockets Vol 40 No 6 2003




12

pp 845-857

[16] Chernyi GG Losev SA Macheret SO Potapkin

BV Physical and chemical processes in Gas Dynamics

Physical and chemical kinetics and thermodynamics of

gases and plasmas Vol 2 Progress in Astronautics andAeronautics Ed P Zarchan Vol 197 2004

[17] Bird RB Stewart WE and Lightfoot EN Transport

phenomenon 2 nd edition John Wiley amp Sons New

York 2002 pp 25 274 526

[18] Capitelli M Gorse C and Longo S J Collision

Integrals of High-temperature Air Species

Thermophysics and Heat Transfer Vol 14 No 2 2000

pp259-268

[19] Levin E and Wright MJ Collision Integrals for Ion-

Neutral interactions of Nitrogen and Oxygen J

Thermophysics and Heat Transfer Vol18 No 1 2004

pp143-147

[20] Shang JS Numerical Simulation of Hypersonic Flows

Computational Methods in Hypersonic Aerodynamics

Computational Mechanics Publications South

Hampton UK 1992

[21] Dunn MG and Kang SW Theoretical and

experimental Studies of reentry Plasma NSA CR 2232

April 1973

[22] Park C Review of Chemical Kinetics Problems of

Future NASA Missions I Earth Entries J


[23] Surzhikov ST TC3 Convective and Radiative Heating

of MSRO for Simplest Kinetic Models Proceedings ofthe International Workshop on Radiation of High

Temperature Gases in Atmospheric Entry Part II30

Sept-1 Oct 2005 Porquerolles France (ESA SP-583

May 2005 pp55-62)

[24] Rumsey C Sanetrik M Biedron R Melson ND

and Parlette E Efficiency and Accuracy of Time-

Accurate Turbulent Navier-Stokes Computations

Computer amp Fluids Vol 25 No 2 1996 pp217-236

[25] Roe P Approximate Riemann Solvers Parameter

Vectors and Difference Scheme J Computational

Physics Vol 48 1981 pp357-372

[26] Jones LJ and Cross AE Electrostatic Probe

Measurements of Plasma Parameters for Two Reentry

Flight Experiments at 25000 feet per second NASA TN

D 66-17 1972

[27] Chaban G Jaffe R Schwenke DW and Huo W

Dissociation Cross Sections and Rate Coefficients for

Nitrogen from Accurate Theoretical Calculations

AIAA 2008-1209 Neo NV Jan 2008




2

is estimated that the vibrational excitation requiresfrom 1000 to 10000 collisions to reach thermalequilibrium with the translational mode [6] Underthis circumstance the energy exchange betweendifferent molecular species has a higher probability

than the vibration-translation transfer of samemolecule [67] In addition the adiabatic collision has alow probability of vibrational excitation and will not be taken into consideration

For the electronic excitations the vibrational energy ofthe molecule plays a main role in the dissociation Thisprocess will be carefully examined using a detailedanalysis for the generation and depletion balanceFinally when the energy of the excitation is greaterthan 10 eV the cross sections of inelastic collisions ofheavy particles are several orders of magnitudesmaller than the cross sections of elastic electronimpact [236] Therefore the modelling of ionization by electrons impact will be emphasized

All the accumulative knowledge in treating thedetailed physical-chemistry modelling has guided thedevelopment of the widely used model equations fornonequlibrium hypersonic flow simulations [3-12] Itwould be valuable to summarize the current state-of ndashthe-art in kinetic modelling and to project future basicresearch needs for better understanding

Governing Equation

By examining the governing equations in thecontinuum domain the effects of internal atomic andmolecular structures are explicitly contained in thespecies and energy conservation equations but onlyimplicitly in the conservation law of momentumthrough the transport property of the gas mixture Thecoupling of conservation laws has offered a widerange of options on how this interdisciplinaryequation system can be efficiently solved On the other

hand the quantum levels of rotational excitation aredensely packed and readily reach equilibrium withtranslational excitation but this fundamental mode isnot necessarily equilibrated with the vibrational andelectronic excitations [2] In the present approach theenergy exchangecascade between vibration-translation and electronic-translation interactions will be explicit simulated by the separated energyconservation equations for vibrational and electronicexcitations [7] In other words the present approach is built on a refined energy balance from the internal

structure of the atom and molecule The detailedmechanisms are developed by models of the gas

kinetics The conservation equations in a continuummedium can be given as [57]

[ ( )]i ii i

dwu u

t dt ρ

ρ part

+ + =part

(2-1)

( ) 0u

uu pI t ρ

ρ τ part

+ + minus =part

(2-2)

[ ( )] 0i i i rad

eeu T u h q u pI

t ρ

ρ κ ρ τ part

+ minus + + + minus =part

sum

(2-3)

Vibrational energy conservation equations fordifferent species are

[ ( ) )]i iv ii i iv iv iv v

e dwu u e q e Q

t dt ρ

ρ Σ

part+ + + = +

part (2-4)

Electronic energy conservation equation can be given

as

[ ( ) )]i e ii i e e e e e

e dwu u e u p I q e Q

t dt ρ

ρ Σ

part+ + + + = +

part (2-5)

where Q vΣ and Q eΣ are the net energy transfer between translational vibrational and electronicexcitations A key addition to the present formulationis that the rates of energy released from thevibrational-translational and electronic-translationalinteractions will be examined and modelled [5] Thissubject shall remain as a sustained research focus into

the future and will also be extended further to includeablation

For calculating the equilibrated translational-rotationaltemperature and gas mixture density the total internalenergy is defined as

( ) ( )2 2

o e ei v i i v i i i e v e e

i e i e i e

u uu ue c T e h c T ρ ρ ρ ρ ρ

ne ne ne

= + + + ∆ + +sum sum sum

(2-6)

where o

ih∆ is the standard heat of formation The

vibrational energy of different species is obtained fromthe harmonic oscillator

[ exp( 1)]v iu

v i v ii

Re

M t κ

Θ= Θ minus (2-7)

The pressure of the gas mixture is calculated withDaltonrsquos law including the contribution from theelectrons collision

ui

i

R p T

M ρ = sum (2-8)

where Θv is the characteristic vibrational temperatureof species i R u is the universal gas constant 8314x10 7




3
















4







1 2 1 2

( )( )

1 1( )

[1 ( )]

V V V

V V V V TV V V

T T TV

v

e T e R RQ e T e

e eT Exp T

p Exp T

ρ τ

κ κ τ

Θ Θ

minus Θ Θ= = =

minus minus

=minus minusΘ

(3-1)


4 113 1 2 1 23 32

1 2 1 2

116 10 ( ) [( 015( ) 184TV V

M M M M T

M M M M τ


(3-2)



( ) [( ( ) ]k ET e k i

dnQ Exp D E kT



2032121 10 lne

Ei e ie

T T Q x x

T




12

2

83 ( )( )e k

ET e e ek k ee k

RT N Q R T T

M M ρ

ρ σ π ne

= minus sum (3-5)







5




3 3 2 (11) 1858 10 i j

i j i ji j

M M D T

M M σ minus

+= times Ω (4-1)



Mono-atomic gas

4 2 (22) 1989 10 i m i

i

T M


Poly-atomic gas

4 2 (22) 2519 10 i p i

i

T M




12 12 14 2

1(1 ) [1 ( ) ( ) ]

8

i ii j

i i j

ji ii j

j j i

x

x M M

M M

micro micro

φ micro

φ micro

minus

=

= + +

sumsum (4-5)

and

14 2 12 [1 ( )( ) ] [8(1 )]

i ii j

i i

ji ii j

j i j

x x

M M M M

κ κ

φ

κ φ

κ

=

= + +

sum sum (4-6)










6





Chemical Kinetics



equation


E k A T Exp

kT = minus (5-1)


f je a

r j

T κ

κ κ

asymp (5-2)








7





Numerical process


U F R

t

part+ =

part

(6-1)





12

12

1 2[ ( ) ( )] ( )]1 2[ ( ) ( )] ( )]

i L R inv R L i

L R inv R L i


δ +

minus

= + minus minus




12

12 1 1

( ) 1 4[(1 ) (1 ) ]

( ) 1 4[(1 ) (1 ) ] L i i i

L i i i

U U U U

U U U U

κ κ

κ κ +

+ + +

= + minus + + ∆









8



Numerical Results














9














10

















11


[27]

Conclusion





ACKNOWLEDGMENT


REFERENCES



Press 1964 pp134-150





Vol 15 No 1 2001 pp76-90






1278 Reno NV 2008




2002 pp 177-232





1936 p34



12 1963 pp3209-3213



Vol5 No 9 1962 pp1022-1026












Vol 5 No 3 1991 pp 266-272




2006







12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972








3
















4







1 2 1 2

( )( )

1 1( )

[1 ( )]

V V V

V V V V TV V V

T T TV

v

e T e R RQ e T e

e eT Exp T

p Exp T

ρ τ

κ κ τ

Θ Θ

minus Θ Θ= = =

minus minus

=minus minusΘ

(3-1)


4 113 1 2 1 23 32

1 2 1 2

116 10 ( ) [( 015( ) 184TV V

M M M M T

M M M M τ


(3-2)



( ) [( ( ) ]k ET e k i

dnQ Exp D E kT



2032121 10 lne

Ei e ie

T T Q x x

T




12

2

83 ( )( )e k

ET e e ek k ee k

RT N Q R T T

M M ρ

ρ σ π ne

= minus sum (3-5)







5




3 3 2 (11) 1858 10 i j

i j i ji j

M M D T

M M σ minus

+= times Ω (4-1)



Mono-atomic gas

4 2 (22) 1989 10 i m i

i

T M


Poly-atomic gas

4 2 (22) 2519 10 i p i

i

T M




12 12 14 2

1(1 ) [1 ( ) ( ) ]

8

i ii j

i i j

ji ii j

j j i

x

x M M

M M

micro micro

φ micro

φ micro

minus

=

= + +

sumsum (4-5)

and

14 2 12 [1 ( )( ) ] [8(1 )]

i ii j

i i

ji ii j

j i j

x x

M M M M

κ κ

φ

κ φ

κ

=

= + +

sum sum (4-6)










6





Chemical Kinetics



equation


E k A T Exp

kT = minus (5-1)


f je a

r j

T κ

κ κ

asymp (5-2)








7





Numerical process


U F R

t

part+ =

part

(6-1)





12

12

1 2[ ( ) ( )] ( )]1 2[ ( ) ( )] ( )]

i L R inv R L i

L R inv R L i


δ +

minus

= + minus minus




12

12 1 1

( ) 1 4[(1 ) (1 ) ]

( ) 1 4[(1 ) (1 ) ] L i i i

L i i i

U U U U

U U U U

κ κ

κ κ +

+ + +

= + minus + + ∆









8



Numerical Results














9














10

















11


[27]

Conclusion





ACKNOWLEDGMENT


REFERENCES



Press 1964 pp134-150





Vol 15 No 1 2001 pp76-90






1278 Reno NV 2008




2002 pp 177-232





1936 p34



12 1963 pp3209-3213



Vol5 No 9 1962 pp1022-1026












Vol 5 No 3 1991 pp 266-272




2006







12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972








4







1 2 1 2

( )( )

1 1( )

[1 ( )]

V V V

V V V V TV V V

T T TV

v

e T e R RQ e T e

e eT Exp T

p Exp T

ρ τ

κ κ τ

Θ Θ

minus Θ Θ= = =

minus minus

=minus minusΘ

(3-1)


4 113 1 2 1 23 32

1 2 1 2

116 10 ( ) [( 015( ) 184TV V

M M M M T

M M M M τ


(3-2)



( ) [( ( ) ]k ET e k i

dnQ Exp D E kT



2032121 10 lne

Ei e ie

T T Q x x

T




12

2

83 ( )( )e k

ET e e ek k ee k

RT N Q R T T

M M ρ

ρ σ π ne

= minus sum (3-5)







5




3 3 2 (11) 1858 10 i j

i j i ji j

M M D T

M M σ minus

+= times Ω (4-1)



Mono-atomic gas

4 2 (22) 1989 10 i m i

i

T M


Poly-atomic gas

4 2 (22) 2519 10 i p i

i

T M




12 12 14 2

1(1 ) [1 ( ) ( ) ]

8

i ii j

i i j

ji ii j

j j i

x

x M M

M M

micro micro

φ micro

φ micro

minus

=

= + +

sumsum (4-5)

and

14 2 12 [1 ( )( ) ] [8(1 )]

i ii j

i i

ji ii j

j i j

x x

M M M M

κ κ

φ

κ φ

κ

=

= + +

sum sum (4-6)










6





Chemical Kinetics



equation


E k A T Exp

kT = minus (5-1)


f je a

r j

T κ

κ κ

asymp (5-2)








7





Numerical process


U F R

t

part+ =

part

(6-1)





12

12

1 2[ ( ) ( )] ( )]1 2[ ( ) ( )] ( )]

i L R inv R L i

L R inv R L i


δ +

minus

= + minus minus




12

12 1 1

( ) 1 4[(1 ) (1 ) ]

( ) 1 4[(1 ) (1 ) ] L i i i

L i i i

U U U U

U U U U

κ κ

κ κ +

+ + +

= + minus + + ∆









8



Numerical Results














9














10

















11


[27]

Conclusion





ACKNOWLEDGMENT


REFERENCES



Press 1964 pp134-150





Vol 15 No 1 2001 pp76-90






1278 Reno NV 2008




2002 pp 177-232





1936 p34



12 1963 pp3209-3213



Vol5 No 9 1962 pp1022-1026












Vol 5 No 3 1991 pp 266-272




2006







12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972








5




3 3 2 (11) 1858 10 i j

i j i ji j

M M D T

M M σ minus

+= times Ω (4-1)



Mono-atomic gas

4 2 (22) 1989 10 i m i

i

T M


Poly-atomic gas

4 2 (22) 2519 10 i p i

i

T M




12 12 14 2

1(1 ) [1 ( ) ( ) ]

8

i ii j

i i j

ji ii j

j j i

x

x M M

M M

micro micro

φ micro

φ micro

minus

=

= + +

sumsum (4-5)

and

14 2 12 [1 ( )( ) ] [8(1 )]

i ii j

i i

ji ii j

j i j

x x

M M M M

κ κ

φ

κ φ

κ

=

= + +

sum sum (4-6)










6





Chemical Kinetics



equation


E k A T Exp

kT = minus (5-1)


f je a

r j

T κ

κ κ

asymp (5-2)








7





Numerical process


U F R

t

part+ =

part

(6-1)





12

12

1 2[ ( ) ( )] ( )]1 2[ ( ) ( )] ( )]

i L R inv R L i

L R inv R L i


δ +

minus

= + minus minus




12

12 1 1

( ) 1 4[(1 ) (1 ) ]

( ) 1 4[(1 ) (1 ) ] L i i i

L i i i

U U U U

U U U U

κ κ

κ κ +

+ + +

= + minus + + ∆









8



Numerical Results














9














10

















11


[27]

Conclusion





ACKNOWLEDGMENT


REFERENCES



Press 1964 pp134-150





Vol 15 No 1 2001 pp76-90






1278 Reno NV 2008




2002 pp 177-232





1936 p34



12 1963 pp3209-3213



Vol5 No 9 1962 pp1022-1026












Vol 5 No 3 1991 pp 266-272




2006







12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972








6





Chemical Kinetics



equation


E k A T Exp

kT = minus (5-1)


f je a

r j

T κ

κ κ

asymp (5-2)








7





Numerical process


U F R

t

part+ =

part

(6-1)





12

12

1 2[ ( ) ( )] ( )]1 2[ ( ) ( )] ( )]

i L R inv R L i

L R inv R L i


δ +

minus

= + minus minus




12

12 1 1

( ) 1 4[(1 ) (1 ) ]

( ) 1 4[(1 ) (1 ) ] L i i i

L i i i

U U U U

U U U U

κ κ

κ κ +

+ + +

= + minus + + ∆









8



Numerical Results














9














10

















11


[27]

Conclusion





ACKNOWLEDGMENT


REFERENCES



Press 1964 pp134-150





Vol 15 No 1 2001 pp76-90






1278 Reno NV 2008




2002 pp 177-232





1936 p34



12 1963 pp3209-3213



Vol5 No 9 1962 pp1022-1026












Vol 5 No 3 1991 pp 266-272




2006







12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972








7





Numerical process


U F R

t

part+ =

part

(6-1)





12

12

1 2[ ( ) ( )] ( )]1 2[ ( ) ( )] ( )]

i L R inv R L i

L R inv R L i


δ +

minus

= + minus minus




12

12 1 1

( ) 1 4[(1 ) (1 ) ]

( ) 1 4[(1 ) (1 ) ] L i i i

L i i i

U U U U

U U U U

κ κ

κ κ +

+ + +

= + minus + + ∆









8



Numerical Results














9














10

















11


[27]

Conclusion





ACKNOWLEDGMENT


REFERENCES



Press 1964 pp134-150





Vol 15 No 1 2001 pp76-90






1278 Reno NV 2008




2002 pp 177-232





1936 p34



12 1963 pp3209-3213



Vol5 No 9 1962 pp1022-1026












Vol 5 No 3 1991 pp 266-272




2006







12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972








8



Numerical Results














9














10

















11


[27]

Conclusion





ACKNOWLEDGMENT


REFERENCES



Press 1964 pp134-150





Vol 15 No 1 2001 pp76-90






1278 Reno NV 2008




2002 pp 177-232





1936 p34



12 1963 pp3209-3213



Vol5 No 9 1962 pp1022-1026












Vol 5 No 3 1991 pp 266-272




2006







12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972








9














10

















11


[27]

Conclusion





ACKNOWLEDGMENT


REFERENCES



Press 1964 pp134-150





Vol 15 No 1 2001 pp76-90






1278 Reno NV 2008




2002 pp 177-232





1936 p34



12 1963 pp3209-3213



Vol5 No 9 1962 pp1022-1026












Vol 5 No 3 1991 pp 266-272




2006







12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972








10

















11


[27]

Conclusion





ACKNOWLEDGMENT


REFERENCES



Press 1964 pp134-150





Vol 15 No 1 2001 pp76-90






1278 Reno NV 2008




2002 pp 177-232





1936 p34



12 1963 pp3209-3213



Vol5 No 9 1962 pp1022-1026












Vol 5 No 3 1991 pp 266-272




2006







12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972








11


[27]

Conclusion





ACKNOWLEDGMENT


REFERENCES



Press 1964 pp134-150





Vol 15 No 1 2001 pp76-90






1278 Reno NV 2008




2002 pp 177-232





1936 p34



12 1963 pp3209-3213



Vol5 No 9 1962 pp1022-1026












Vol 5 No 3 1991 pp 266-272




2006







12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972








12

pp 845-857







York 2002 pp 25 274 526




pp259-268




pp143-147




Hampton UK 1992



April 1973








May 2005 pp55-62)











D 66-17 1972





Documents

Hypersonic Nonequilibrium Flow Simulation Based on Kinetic Models