on air-chemistry reduction for hypersonic external flow applications

International Journal of Heat and Fluid Flow xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

International Journal of Heat and Fluid Flow

journal homepage: www.elsevier .com/ locate/ i jhf f

On air-chemistry reduction for hypersonic external flow applications

http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.10.0210142-727X/� 2014 Published by Elsevier Inc.

⇑ Corresponding author.

Please cite this article in press as: Ibrahim, A., et al. On air-chemistry reduction for hypersonic external flow applications. Int. J. Heat Fluid Flowhttp://dx.doi.org/10.1016/j.ijheatfluidflow.2014.10.021

Ashraf Ibrahim a,⇑, Sawan Suman b, Sharath S. Girimaji a

a Aerospace Engineering Department, Texas A&M University, College Station, TX 77843, USAb Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110016, India

a r t i c l e i n f o

Article history:Available online xxxx

Keywords:Hypersonic flowsSlow manifoldDirect numerical simulations

a b s t r a c t

In external hypersonic flows, viscous and compressibility effects generate very high temperatures leadingto significant chemical reactions among air constituents. Therefore, hypersonic flow computationsrequire coupled calculations of flow and chemistry. Accurate and efficient computations of air-chemistrykinetics are of much importance for many practical applications but calculations accounting for detailedchemical kinetics can be prohibitively expensive. In this paper, we investigate the possibility of applyingchemical kinetics reduction schemes for hypersonic air-chemistry. We consider two chemical kineticssets appropriate for three different temperature ranges: 2500 K to 4500 K; 4500 K to 9000 K; and above9000 K. By demonstrating the existence of the so-called the slow manifold in each of the chemistry sets,we show that judicious chemical kinetics reduction leading to significant computational savings is pos-sible without much loss in accuracy.

� 2014 Published by Elsevier Inc.

1. Introduction

Hypersonic boundary layers encounter high temperature condi-tions triggering vibrational excitation, chemical reactions and ion-ization of air constituents. These thermochemical processes cansubstantially modify the transition and subsequent breakdown toturbulence in hypersonic flows. Therefore, it is critical that hyper-sonic external flow calculations accurately account for flow-ther-mochemical interactions. In many problems of interest, accuratecalculations of flow-thermochemistry interactions are rendereddifficult by a wide range of thermochemical and flow timescales.Fig. 1 shows the range of fluid and chemistry timescales in a typicalcombustion environment. In general, chemistry timescales span awider range than those of the flow. If all of the chemistry scalesare slow than that of the flow then it is reasonable to invoke thefrozen-chemistry approximation. At the other extreme, if all ofthe chemistry scales are faster than the flow scales, then equilib-rium-chemistry is the appropriate simplification. In many practicalflows, the fluid and thermochemical timescales can potentiallyoverlap ruling out the simple assumptions of frozen or equilibriumthermochemistry. Such situations call for detailed nonequilibriumcalculations, which would add substantial computational burdento the parent computational fluid dynamic (CFD) simulations. Inhypersonic flows, these computations could prove to be even morechallenging as this overlap can substantially change in time andspace owing to a wider range of flow velocity and temperatures

prevalent across the flow field. The highly disparate thermochem-ical timescales introduce severe computational stiffness into theflow solver. It is highly desirable to develop appropriate chemistryreduction methods which can reduce the computational burdenwithout significantly compromising the accuracy of the flow solu-tion (Peeters and Rogg, 1993; Smooke, 1992; Maas and Pope,1992). Much progress toward chemistry reduction has been madein the field of combustion. Quasi-steady state assumption (QSSA,(Maas and Pope, 1992)), intrinsic low-manifold method (ILDM,(Maas and Pope, 1992)) and the locally linear assumption (LLA,(Girimaji and Ibrahim, 2014)) are some of the reduction techniquesthat have been developed and applied to air–fuel mixtures (Maasand Pope, 1992; Skrebkov and Karkach, 2007) and atmosphericpollutant chemistry (Tomlin et al., 2001). However, application ofthese reduction methods has the pre-condition that a slow lower-dimensional manifold must inherently exist in the chemical kineticsystem under consideration.

Presence of vibrational non-equilibrium and endothermic reac-tions make a hypersonic external flow air chemistry fundamentallydifferent from an internal combustion flow chemistry, whereinvibrational nonequilibrium is less important and reactions aremainly exothermic. Thus the existence of slow manifolds in non-equilibrium reacting air mixture cannot be taken for granted, andtheir existence must be clearly demonstrated before any reductionmethod can be applied. With this motivation, the objective of thiswork is to examine if the same combustion chemical kinetics strat-egies can be utilized for reducing the computational burden –stemming from thermochemical non-equilibrium – of large scale

(2014),

http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.10.021


http://www.sciencedirect.com/science/journal/0142727X

http://www.elsevier.com/locate/ijhff


ns total number of speciesm0ji stoichiometric coefficient of the ith species in the jth for-

ward reactionm00ji stoichiometric coefficient of the ith species in the jth re-

verse reaction1i chemical symbol of the ith speciesnr total number of reactionst time, swi mass production-rate of the ith species, kg m�3 s�1

Mi molecular weight of the ith speciesKfj forward reaction rate coefficient for the jth reactionq density, kg m�3

Yi mass fraction of the ith speciesKbj backward reaction rate coefficient for the jth reaction

T temperature, KKeqj

equilibrium constant for the jth reactioncpi

specific heat of the ith species at constant pressure,J kg�1 K�1

h enthalpy, JP pressure, atmospherehi specific enthalpy of the ith species, J kg�1 K�1

Ri gas constant, J kg�1 K�1

hoi zero-point specific enthalpy of the ith species, J kg�1 K�1

R universal gas constant, J mol�1 K�1

eeqvib equilibrium vibration energy

�h Planck’s constantm fundamental vibration frequencys vibration relaxation time

2 A. Ibrahim et al. / International Journal of Heat and Fluid Flow xxx (2014) xxx–xxx

hypersonic boundary layer transition and turbulence calculations.Toward this objective we (i) first demonstrate the existence ofattracting slow manifolds in the hypersonic reacting air mixture,and subsequently (ii) evaluate the performance of the QSSAmethod in approximating these reduced slow manifolds at differ-ent temperature ranges.

This paper is organized into five sections. In Section 2, we dis-cuss the detailed chemical kinetic sets and describe the flow-ther-modynamic system of high temperature air mixture examined inthis study. In addition, we introduce the perfectly-stirred reactorsimplification and the governing equations of chemical speciesevolution. In Section 3, the reduction of general dynamical systemsis described and applied to homogeneous perfectly-stirred reactorsystem. In Section 4, we establish the existence of slow manifold ofthe air-chemistry kinetics and demonstrate that QSSA performsadequately in capturing this manifold. The final section providesa brief summary.

Preliminary results of the reduction of simple air-chemistryreaction set valid at low-to-moderate temperatures have been pre-sented in an AIAA conference paper – Suman et. al. (Suman et al.,2011). In the present paper we present a more detailed theoreticaldevelopment and investigate the reduction of realistic detailedreaction sets at three different temperature regimes encounteredin hypersonic flight.

2. Chemical kinetics

Atmospheric air undergoes different thermochemical eventsdepending on the resident temperature (Fig. 2). Below 800 Krotational and translational modes are the only major modes ofinternal energy of air. Above 800 K diatomic molecules start tovibrate which, depending on the flow conditions, may or may notbe in equilibrium with local thermodynamic conditions. Above2500 K oxygen molecules start to dissociate and this phenomenais complete by 4000 K. Around the same temperature N2 beginsto dissociate and by 9000 K all Nitrogen molecules are fully disso-ciated into the constituent atoms. Around 9000 K, ionization com-mences. In this work, we consider thermochemical (chemical and

9 8 7 6 5

Trturbu

910− 810− 710− 610− 510− 10−

Fast chemistry time scales(partial equilibrium)

Intermed

Fig. 1. Typical fluid and chemistry timescales in a combustion environment (adap

Please cite this article in press as: Ibrahim, A., et al. On air-chemistry reductiohttp://dx.doi.org/10.1016/j.ijheatfluidflow.2014.10.021

vibrational) non-equilibrium air mixture over a temperature rangeof 2500–10,000 K. We categorize our study into various sub-rangesof temperature (see Table 5). In the low and mid temperatureranges (2500–4500 K and 4500–9000 K), we use an air mixturemodel comprising of 5 species and 17 reactions. On the other hand,in the high temperature range (above 9000 K) we use an air mix-ture model comprising of 7 species and 24 reactions to appropri-ately include the ionization physics. Further, to include thevibrational non-equilibrium effects, we employ the Landau-Tellerrelaxation model (Anderson, 1989).

2.1. Flow and thermodynamic equations

For a chemically reactive flow, the species mass conservationequation is:

@Yi

@tþ uj

@Yi

@xj¼ wi

qþ Di; ð1Þ

where Yi is the mass fraction of species i;uj is the j component of thevelocity field, wi is the mass production-rate of species i and Di isdiffusion. The velocity field uj is governed by compressibleNavier–Stokes equations.

The total enthalpy of the mixture h and pressure P are related tospecies mass fractions ðYiÞ through the following relationships:

h ¼Xns

i

hiYi ð2Þ

and the ideal gas law

P ¼ qRTXns

i

Yi

Mi: ð3Þ

The calculations of the species mass production-rate wi involvecomputing reaction rates which, in turn, are highly dependent ontemperature. Therefore, the chemistry timescales are influencedby temperature and they cover a wider range relative to the flowtimescales. When the flow timescales and the chemistry timescalesare of the same order of magnitude, then there is a need for detailed

Time scale (s)

4 3 2 1 0

ansition / lence time scales

U

4 310− 210− 110− 010

iate chemistry time scales

Slow frozen chemistry time

scalesT

ted from Maas and Pope (1992)). U and T represent velocity and temperature.

n for hypersonic external flow applications. Int. J. Heat Fluid Flow (2014),


Fig. 2. Ranges of vibrational excitation, dissociation and ionization for air at 1-atmpressure (adapted from Anderson (1989)).

A. Ibrahim et al. / International Journal of Heat and Fluid Flow xxx (2014) xxx–xxx 3

nonequlibrium computations. The detailed chemistry calculationswould add significant computational stiffness and chemistry reduc-tion techniques could be an appropriate approach to reduce thecomputational burden. In order to investigate the possibility ofreduction, we consider a simpler case of a spatially homogeneous,adiabatic and isobaric system (or a perfectly-stirred reactor).

In a perfectly-stirred reactor, the velocity field is neglected andthe species mass fraction evolution equation is given by

dYi

dt¼

_wi

q; ð4Þ

where _wi is the net mass-production rate of the ith species. In Eqs.(2) and (3), vibrational energy ðevibÞ and mass fractions ðYiÞ areunknowns that need to be computed for flow under thermochemi-cal non-equilibrium conditions. Under thermochemical equilibriumconditions the unknowns are temperature ðTÞ and the species massfractions ðYiÞ.

2.2. Enthalpy and internal energy states

To account for the evolution of vibrational energy, we employthe Landau-Teller model (Anderson, 1989):

devib

dt¼ 1

sðeeq

vib � evibÞ; ð5Þ

where eeqvib is the equilibrium vibrational energy and s is the vibra-

tional relaxation time. From statistical thermodynamics consider-ations, the equilibrium vibrational energy is given by

eeqvib ¼

�hmi=kTe�hmi=kT � 1

RT; ð6Þ

where �h is Planck’s constant and mi is the fundamental vibration fre-quency. The vibration relaxation time s depends on local tempera-ture and pressure and for diatomic species it is given by

s ¼ a1 expða2=TÞ=P; ð7Þ

where a1 and a2 are constants (Anderson, 1989) (and P is pressurein atm). The vibrational temperature Tvib is defined from the valueof nonequilibrium vibrational energy evib by

evib ¼�hm=kTvib

e�hm=kTvib � 1RTvib: ð8Þ


Park’s two-temperature model (Park, 1990) suggests the use of thegeometric average of temperature and vibrational temperature tocompute the dissociation reaction rates. Hence, the effective tem-perature Teff is given by

Teff ¼ffiffiffiffiffiffiffiffiffiffiffiTTvib

p: ð9Þ

The total enthalpy of the mixture is given by Eq. (2) and the spe-cies enthalpy hi can be computed using the formula

hi ¼ CiRiT þ evib;i þ Dh�f ;i; ð10Þ

where Ci ¼ 3=2 for monoatomic molecules and 5=2 for diatomicmolecules.

2.3. Chemistry equations

In general, a chemical kinetic set can be represented by:Xns

i¼1

m0ji1i�Xns

i¼1

m00ji1i ð11Þ

for j ¼ 1; . . . ;nr , where nr represents the total number of reactions,and ns represents the total number of species. The symbol 1i

denotes the chemical symbol of the ith species and m0ji and m00ji arethe stoichiometric coefficients of the ith species in the jth reaction.Now using the above notation, the net mass-production rate of theith species, wi, can be determined by:

wi ¼ Mi

Xnr

j¼1

m00ji � m0ji� �

Kfj

Yns

n¼1

qYn

Mn

� �m0jn

� Kbj

Yns

n¼1

qYn

Mn

� �m00jn

" #; ð12Þ

where Mi and Yi represent the molecular weight and mass fractionof the ith species, respectively. The symbol q represents mixturedensity. Further, the symbols Kf j

and Kbjdenote the forward and

reverse reaction rate coefficients of the jth reaction. The forwardreaction rate coefficient is expressed in its Arrhenius form as:

Kf j¼ AjT

bj expð�Taj=TÞ ð13Þ

the quantities Aj; bj and Taj are Arrhenius parameters. The backwardreaction rate coefficient ðKbj

Þ can be determined using the followingrelationship:

Kbj¼ Kf j

=Keqj; ð14Þ

where Keqjrepresents the equilibrium constant of the jth reaction.

Note that Park’s two-temperature (Park, 1990) model suggeststhe use of the geometric average ðTeff Þ of temperature ðTÞ andvibrational temperature (Tvib, defined in Eq. (8)) to compute thedissociation reaction rates:

Kfj ¼ AjTbj

eff expð�Taj=Teff Þ: ð15Þ

On the other hand, the backward reaction rates are computed usingthe local temperature T. In the next two subsections, we providedetailed analysis of the above quantities for the two chemicalkinetic sets used as air mixture models.

2.4. 5-species air mixture model

At low and mid temperature ranges (Table 5), the air chemistrycan be adequately represented by a chemical kinetic system com-prising of two elements (Nitrogen and Oxygen) and five species(N2;O2;NO;N and O) along with the following reaction set:

N2 þM�2NþM; ðR1ÞO2 þM�2OþM; ðR2ÞNOþM�Nþ OþM; ðR3ÞN2 þ O�NOþ N; ðR4ÞNOþ O�O2 þ N; ðR5Þ

ð16Þ




where M is any collision partner (N2;O2;NO;N or O). The first threereactions - ðR1Þ; ðR2Þ and ðR3Þ - actually represent 15 dissociationreactions, whereas the last two are exchange reactions. This reac-tion set has been used in several previous studies to model hightemperature reacting air mixtures (Hudson, 1996; Hudson et al.,1997; Pimentel and Hetem, 2011; Duan and Martin, 2009; Duanand Martin, 1994). The net rates of these reactions can be expressedas a sum of the forward and backward rates as follows

R1 ¼X

m

Kb1mqYNMN

� �2 qYmMm� Kf 1m

qYN2MN2

qYmMm

� �;

R2 ¼X

m

Kb2mqYOMO


qYO2MO2

qYmMm

� �;

R3 ¼X

m

Kb3mqYNMN

qYOMO

qYmMm� Kf 3m

qYN2MN2

qYmMm

h i;

R4 ¼ Kb4qYNOMNO

qYNMN� Kf 4

qYN2MN2

qYOMO

;

R5 ¼ Kb5qYO2MO2

qYNMN� Kf 5

qYNOMNO

qYOMO

:

The chemical source terms can be written as a linear combination ofthe individual reactions rates Rj in the forms

wN2 ¼ MN2 ðR1 þR4Þ;wO2 ¼ MO2 ðR2 �R5Þ;wNO ¼ MNOðR3 �R4 þR5Þ;wN ¼ MNð�2R1 �R3 �R4 �R5Þ;wO ¼ MOð�2R2 �R3 þR4 þR5Þ:

Various coefficients required to compute the Arrhenius reactionrates (13) are shown in Table 1 (Duan and Martin, 1994). The equi-librium constants for various reactions (14) are computed using thefollowing curve fit, see (Duan and Martin, 1994):

Keqj¼ C expðA1j=Z þ A2j þ A3j lnðZÞ þ A4jZ þ A5jZ

2Þ; ð17Þ

where Z � 10;000T and the numerical values of various constants are

shown in Table 2.

2.5. 7-species air mixture model

For the high temperature range (Table 5), we consider sevenmajor constituents – N2;O2;NO;NOþ;N;O and e� – to representthe mixture. This mixture includes ionized species and free elec-trons to capture the ionization physics that exists in high temper-ature air. The chemical system comprises of the followingindividual reactions.

N2 þM�2NþM; ðR1ÞO2 þM�2OþM; ðR2ÞNOþM�Nþ OþM; ðR3ÞN2 þ O�NOþ N; ðR4ÞNOþ O�O2 þ N; ðR5ÞNþ O�NOþ þ e�; ðR6Þ

ð18Þ

where M is, again, any collision partner (N2;O2;NO;NOþ;N;O or e�).The above six reactions can be written as

Table 1Arrhenius parameters used by Duan and Martin (1994) (Eq. (13)).

Reaction A b Ta

R1 7.00e+18 �1.60e+00 113,200R2 2.00e+13 �3.82e+00 59,500R3 5.00e+12 +0.00e+00 75,500R4 6.40e+14 �1.00e+00 38,400R5 8.40e+09 +0.00e+00 19,450


R1 ¼X

m

Kb1mqYNMN


qYN2MN2

qYmMm

� �;

R2 ¼X

m

Kb2mqYOMO


qYO2MO2

qYmMm

� �;

R3 ¼X

m

Kb3mqYNMN

qYOMO

qYmMm� Kf 3m

qYN2MN2

qYmMm

h i;

R4 ¼ Kb4qYNOMNO

qYNMN� Kf 4

qYN2MN2

qYOMO

;

R5 ¼ Kb5qYO2MO2

qYNMN� Kf 5

qYNOMNO

qYOMO

;

R6 ¼ Kb6qYNOþMNOþ

qYeMe� Kf 6

qYN2MN2

qYOMO

:

Similar to the 5-species kinetic set, the chemical source terms canbe written as a linear combination of the individual reactions ratesRj which take the forms

wN2 ¼ MN2 ðR1 þR4Þ;wO2 ¼ MO2 ðR2 �R5Þ;wNO ¼ MNOðR3 �R4 þR5; ÞwN ¼ MNð�2R1 �R3 �R4 �R5Þ;wO ¼ MOð�2R2 �R3 þR4 þR5Þ:

The Arrhenius parameters for these reactions are given in Table 3(Park, 1990). The equilibrium constants for this reaction sets arecomputed using the following the expression (Park, 1990):

Keqj¼ expðA1j þ A2jZ þ A3jZ

2 þ A4jZ3 þ A5jZ

4Þ; ð19Þ

where Z ¼ 10;000=T (T in K), and the corresponding constants aregiven in Table 4.

3. The slow manifold: mathematical description

A chemical kinetic set, like any dynamical system, can bereduced in a mathematically rigorous manner if and only if itexhibits an inherent slow manifold in its phase space. In this sec-tion, we first provide a brief overview of the inherent slow mani-folds of a dynamical system and how the manifolds can beutilized to reduce the parent dynamical system. Then we explainthe QSSA method of reduction and evaluate its accuracy for com-puting the slow manifolds for high temperature air mixtures in dif-ferent temperature ranges.

As mentioned before, we consider a closed, homogeneous, adi-abatic and isobaric reacting system. The system consists of ns num-ber of species, ne elements and nr reactions. The elemental matrixE ¼ ½Eij� is an ns � ne matrix with component Eij indicates the num-ber of atoms of element j in a molecule of species i. The vector Mcontains the molecular weights of the species, N is the number ofmoles vector, Y is the mass fraction vector and z is the specificmole vector of the species which is defined by

z ¼ ½Yi=Wi�:

The full composition space C ¼ Rnsþ1, which is R5 in the case of 5-spe-cies set with canonical basis e1; e2; e3; e4 and e5. Note that z>W ¼ 1.We use ze to denote the vector of the specific moles of atoms of theelements, i.e.

ze ¼ E>z:

If a is the atomic weights vector of the elements, then

a>ze ¼ 1:

Note that the ne element specific moles ze is conserved, i.e. ze is con-stant. If the total enthalpy h ¼ H is constant, then the reactive sys-tem can be described in the (ns � ne þ 1)-dimensional reactive spacedefined by



Table 2Curve-fit coefficients for Keqj

(Duan and Martin, 1994) (Eq. (17)).

Reaction C A1 A2 A3 A4 A5

R1 1.e6 1.6060e+0 �1.5732e+0 1.3923e+0 �1.1553e+1 �4.5430e�3R2 1.e6 6.4183e�1 2.4253e+0 1.9026e+0 �6.6277e+0 3.5151e�2R3 1.e6 6.3817e�1 6.8189e�1 6.6336e�1 �7.5773e+0 �1.1025e�2R4 1.0 9.6794e�1 8.9131e�1 7.2910e�1 �3.9555e+0 6.4880e�3R5 1.0 �3.7320e�3 �1.7434e+0 �1.2394e+0 �9.4952e�1 �1.46341e�1

Table 3Arrhenius parameters for the 7-species model.

Reaction Partner A b Ta

R1 N2 3.700e+18 �1.60e+00 113,200O2 3.700e+18 �1.60e+00 113,200NO 3.700e+18 �1.60e+00 113,200NOþ 3.700e+18 �1.60e+00 113,200N 1.110e+09 �1.60e+00 113,200O 1.110e+09 �1.60e+00 113,200e� 1.110e+09 �1.0e+00 113,200



R4 – 3.180e+10 0.10e+00 37,700R5 – 2.160e+5 1.290e+00 19,220R6 – 6.50e+8 0.0e+00 32,000

Table 4Curve-fit coefficients for Keqj

used by Candler (1988) (Eq. (19)).

Reaction A1 A2 A3 A4 A5

R1 3.898 �12.611 0.683 �0.118 0.006R2 1.335 �4.127 �0.616 0.093 �0.005R3 1.549 �7.784 0.228 �0.043 0.002R4 2.349 �4.828 0.455 �0.075 0.004R5 0.125 �3.652 0.843 �0.136 0.007R6 �6.234 �5.536 0.494 �0.058 0.003


Cðze;HÞ :¼ fðz; TÞ : E>z ¼ ze;hðTÞ ¼ H; z 2 Cg:

The non-negativity of the species confines the composition to therealizable region of this reactive affine space, defined by

Cðze;HÞ :¼ fðz; TÞ : zi P 0;E>z ¼ ze;hðTÞ ¼ H; z 2 Cg;

which is a bounded convex polytope.

3.1. The slow manifold

The chemical kinetic dynamical system (Eq. (4)) takes the form:

dzdt¼ FðzÞ: ð20Þ

The above system represents a homogeneous perfectly-stirred reac-tor and the composition vector of the mixture z ¼ ðziÞ representsthe chemical species concentration. In mathematical terms, thisequation set constitutes an autonomous, stable dynamical systemwith one globally attracting hyperbolic equilibrium point. The


evolution of the system occurs at different timescales in form ofslow and fast processes represented by zs and zf respectively. Ifwe assume that z is of the form

z ¼ ðzs; zf Þ;

where zs are slow variables and zf are the fast variables, then we canassume Eq. (20) takes the form

dzs

dt¼ f ðzs; zf Þ; zs 2 Rn

dzf

dt¼ gðzs; zf Þ; zf 2 Rm

ð21Þ

where ðzsð0Þ; zf ð0ÞÞ ¼ z0s ; z

0f

� �2 RN;N ¼ nþm. The chemical sys-

tem we are considering is an adiabatic and isobaric system wherethe total enthalpy and pressure are constant in time. In this case,the dimension of the composition space is N ¼ ns þ 1.

For any subset S of the composition space RN , if we start withinitial condition z0 2 S and the trajectory zðtÞ stays in S all the time,we say S is an invariant set under the evolution of the system Forexample, the realizability region which contains all the points sat-isfying the physical conditions of the chemical reaction (e.g. theconservation of mass and boundedness of temperature and pres-sure) is an invariant subset of the composition space. The existenceof invariant manifolds allows us to reduce the parent system to asubspace with smaller dimension. This reduction can be done pro-vided that it preserves the main characteristics of the full systemand it does not compromise accuracy. We say a subset S � RN isattracting if all trajectories in a neighborhood of S tend to it. Theslow manifold is an invariant attracting manifold of dimensionsmaller than N, the dimension of the composition space.

3.2. Modeling the slow manifold

In the quasi-steady state assumption (QSSA) approach, the spe-cies are categorized into slow and fast based on their rate of evolu-tion. Slow species are the ones that evolve at the fluid timescalesand those species that evolve more rapidly are considered fast. Inthis two-timescale analysis, QSSA says that the time derivative ofthe concentration of the fast species is approximately zeros, i.e.

dzf

dt� 0: ð22Þ

From Eqs. (21) and (22), we have

gðzs; zf Þ � 0; ð23Þ

which can be rewritten in the form

zf � hQSSAðzsÞ: ð24Þ

The last Eq. (24) represents the QSSA approximation of the system’sslow manifold. The QSSA is one of the simplest reduction methodsand widely used due to its computational ease and conceptual clar-ity. The QSSA can be improved by considering higher-order approx-imation of the Jacobian of the dynamical system (Girimaji andIbrahim, 2014). The reduced system must ensure the physical andkinetic realizability in addition to mathematical consistency.



Table 5The chemical systems and the thermodynamic assumptions.

Low-temperature range (2500–4500 K) Mid-temperature range (4500–9000 K) High-temperature range (>9000 K)

Vibrationalequilibrium

Vibrational non-equilibrium





5 Species Case A No Case B Case C Case D Case E7 Species No No No No Case F Case G


4. The slow manifold: computational approach

In this section we (i) establish the existence of the slow mani-fold in the air-chemistry kinetics set and then (ii) proceed to exam-ine if (QSSA) can capture the slow manifold.

4.1. The slow manifold of full chemistry

Eq. (4) is the dynamical system which represents the evolutionof the chemical species mass fraction which is a special case of thegeneral Eq. (20). A pragmatic approach to finding the slow-mani-fold of a dynamical system is now given. Multiple solution trajec-tories of the system are generated from different realizable initialconditions. As the trajectories go from various initial conditionsto the unique equilibrium point, we notice that they ’bunch

Fig. 3. The trajectories of YN2 ;YO2 , YNO and YO (the dashed lines) tend rapidly to th


together’ closer to the equilibrium point. A system with intrinsicslow-manifold will feature such ’bunching’ whereas ones withoutsuch manifold will approach equilibrium from different directions.To examine if the full air-chemistry kinetic set exhibits a slow-manifold, we compute several trajectories of the system of Eqs.(4) and (5) in addition to the conservation of enthalpy and pres-sure, i.e.

@h@t¼ @P@t¼ 0: ð25Þ

Such computations are performed at different temperatureregimes for vibrational equilibrium and non-equilibrium condi-tions. We generate multiple solution trajectories correspondingto different initial conditions. The sets of initial conditions arechosen for the system variables ðT;YiÞ in the cases of vibrational

e slow manifold (the solid line) on their way to the unique equilibrium point.



Fig. 4. The QSSA approximation of the one-dimensional slow manifold for case A. The trajectories of YN2 ;YO2 , YNO and YO are the dashed lines.

Fig. 5. The time evolution of the species mass fractions for temperature range2500 K to 4500 K (case A). The letters (A, B, C and D) represent the interval oftimescales of different chemistry regimes: A for frozen chemistry, B for fullchemistry, C for reduced chemistry and D for equilibrium chemistry. The species inthe figure are N2ð�Þ;O2ð�Þ;NOð}Þ;NðÞ and Oð.Þ.


equilibrium and (cases A, B, D and F in Table 5) or ðYi; evibÞ in thecases of vibrational non-equilibrium and (cases C, E and G in Table5) subject to the following constraints:

(i) The total enthalpy is constant, i.e.

Plehtt

h ¼X

i

YihiðTÞ ¼ H: ð26Þ

(ii) The elemental mass is constant, i.e.

E>diagð1=WiÞY ¼ ze: ð27Þ

ase cite this article in press as: Ibrahim, A., et al. On air-chemistry reductiop://dx.doi.org/10.1016/j.ijheatfluidflow.2014.10.021

(iii) The boundedness of temperature, 0 6 T and boundednessof mass fraction 0 6 Yi 6 1.

For example, for the 5-species kinetic set, these constraints aremathematically expressed as:

h ¼ Hðconst:Þ; YO2 þ YNO1630þ YO ¼ /ðconst:Þ;

YN2 þ YNO1430þ YN ¼ 1� /; ð28Þ

where / represents mass fraction of oxygen. Similar procedure isemployed to choose the initial values for experiments in whichvibrational energy is assumed to be in equilibrium with instanta-neous temperature (Table 5) except the initial values of vibrationalenergy, evib which are constrained to be eeq

vib initially (Eq. (6)). Eqs.(4),(5) and (25) are numerically solved using a pre-defined function(ODE15s) available with the MATLAB� package. This function usesnumerical differential formulas (NDF) for integration.

Now we present the construction of one-dimensional QSSAmanifold. Following similar works in combustion literature, thetranslational temperature T is taken to be the slow variable. Vibra-tional energy is assumed to be in quasi-steady state with respect tothe translational temperature value. The constraints in Eq. (28)provide the first set of nc conditions for determining the variousspecies concentrations. Next, ne(¼ ns � nc) major species areassumed to be in quasi-steady state with respect to the specifiedtranslational temperature. The following procedure is used to con-struct the QSSA manifold:

1. Start with an initial value of translational temperature.2. The vibrational energy value as a function of translation tem-

perature is given by Eq. (6).3. Invoke nc conditions given in Eq. (28).



Fig. 6. The trajectories of YN2 ;YO2 , YNO and YO (the dashed lines) tend rapidly to the slow manifold (the solid line) on their way to the unique equilibrium point.


4. Select neð¼ ns � ncÞ major species and subject them to theQSSA condition wi ¼ 0 for i ¼ 1; . . . ;ne. Clearly, the other nc

species are not enforced to be in equilibrium.5. Then the QSSA values of various species concentration is

obtained algebraically from the equation resulting fromsteps 3 and 4. We employ subroutines from GNU scientificlibrary (GSL) to find the roots of non-linear equations.

6. Now, the initial state of the gas mixture is fully specified.7. Update the translational temperature to the next time-step

using the energy equation derived as follows. Starting fromEq. (2) we can write:

Plehtt

dhdt¼ d

dt

Xns

i¼1

hiYi ¼ 0: ð29Þ

Upon differentiation we get

Xns

i¼1

Yidhi

dt¼ �

Xns

i¼1

hidYi

dt: ð30Þ

Note that the main contribution to the right-hand-side of theabove equation comes from the out of equilibrium species. Thensubstituting the individual species enthalpies to obtain:

Xns

i¼1

Yiddt

CiRiT þ eeqvib;i þ Dh�f ;i

� �¼ �

Xns

i¼1

wihi; ð31Þ

ase cite this article in press as: Ibrahim, A., et al. On air-chemistry reductiop://dx.doi.org/10.1016/j.ijheatfluidflow.2014.10.021

leading to

Xns

i¼1

YiCiRi

!dTdtþXns

i¼1

Yideeq

vib;i

dt¼ �

Xns

i¼1

wihi: ð32Þ

The above equation is solved numerically to obtain the new tem-perature value.8. Steps 2 through 5 are then repeated until the final equilib-

rium state is reached.

4.2. Temperature range I (2500 K to 4500 K)

For the low-temperature range, we consider Case A (Table 5),which employs the 5-species air mixture model and assumed allthe vibrational energy to be in equilibrium with translational tem-perature (Eq. (6)). To determine the existence of a slow manifoldfor this kinetic system, we need to examine the transient behaviorof various system variables (Yi; T) starting with different sets of ini-tial conditions and subject to constraints given in (28)(H ¼ 13� 106 kJ;/ ¼ 0:21 and P ¼ 1 atmosphere). The governingequations for Case A simulation are (4), (5) and (25). We studythe behavior of solution trajectory in phase space of Yi and T. InFig. 3a (mass fraction of N2 vs. T), each solution trajectory evolvesin two stages. In the first stage, evolution is strongly influenced byits initial choice of Yi and T. However, in the second stage, all



Fig. 7. The QSSA approximation of the one-dimensional slow manifold for case B. The trajectories of YN2 ;YO2 , YNO and YO are the dashed lines.

Fig. 8. The time evolution of the species mass fractions for temperature range4500 K to 9000 K (case B). The letters (A, B, C and D) represent the interval oftimescales of different chemistry regimes: A for frozen chemistry, B for fullchemistry, C for reduced chemistry and D for equilibrium chemistry. The species inthe figure are N2ð�Þ;O2ð�Þ;NOð}Þ;NðÞ and Oð.Þ.

Fig. 9. The equilibrium values of mass fraction for different for the five species(N2;O2;NO;N and O) for temperature range 4500 K to 9000 K.


trajectories seem to collapse onto a single curve (bold dashed curveon Fig. 3) and then eventually attain the common equilibriumpoint as dictated by the thermodynamic constraints of the kineticsystem. Our computations further reveal that for each solution tra-jectory, the total time elapsed during the first stage is much shorterthan the total time elapsed during the second stage of each com-puted trajectory. Exactly the same dynamical behaviors and trendsare observed for other species as well (Fig. 3b–d). Based on theseobservations, we conclude that the considered kinetic system (CaseA) (Eq. (16)) indeed has a strongly attracting slow manifold.


The highlighted curves on Fig. 3a–d are actually the projectionsof this manifold on the planes of individual species andtemperature.

In this temperature range, we compute the QSSA approxima-tions of the slow manifolds. In Fig. 4, we show the QSSA estimationof the slow manifold along with same exact trajectories as shownin Fig. 3. Clearly, the QSSA computation shows excellent agreementwith the exact slow manifold of the kinetic system shown earlier inFig. 3. The different timescales of frozen, full, reduced and equilib-rium chemistry are shown in Fig. 5. For flow applications with



Fig. 10. The time evolution of the species mass fractions for temperature rangeabove 9000 K (case D). The letters (A, B, C and D) represent the interval of timescalesof different chemistry regimes: A for frozen chemistry, B for full chemistry, C forreduced chemistry and D for equilibrium chemistry. The species in the figure areN2ð�Þ;O2ð�Þ;NOð}Þ;NðÞ and Oð.Þ.

Fig. 11. The time evolution of the species mass fractions for temperature rangeabove 9000 K (5-species and assuming thermal nonequilibrium, i.e. case E). Theletters (A, B, C and D) represent the interval of timescales of different chemistryregimes: A for frozen chemistry, B for full chemistry, C for reduced chemistry and Dfor equilibrium chemistry. The species in the figure are N2ð�Þ;O2ð�Þ;NOð}Þ;NðÞand Oð.Þ.

Fig. 12. The time evolution of the species mass fractions for temperature rangeabove 9000 K (7-species and assuming thermal nonequilibrium i.e. case G). Theletters (A, B, C and D) represent the interval of timescales of different chemistryregimes: A for frozen chemistry, B for full chemistry, C for reduced chemistry and Dfor equilibrium chemistry. The species in the figure are N2ð�Þ;O2ð�Þ;NOð}Þ;NðÞand Oð.Þ.


timescale around Oð10�8Þ, we should consider full chemistry. Onthe other hand, the timescale of the reduced chemistry in the tem-perature range is of order 10�2 seconds which is relatively a largevalue. It takes the mass fraction trajectories few seconds to reachequilibrium (see Fig. 6).

4.3. Temperature range II (4500 K to 9000 K)

For the mid-temperature range (cases B and C), we again usethe 5-species air mixture model. We assume the vibrational energyequilibrium for case B and the vibrational energy nonequilibriumfor case C. We choose again different initial conditions satisfyingthe constrains (Eqs. (4), (5) and (25)). Similar to Range I, the trajec-tories tend rapidly (in time of order Oð10�6Þ) to the slow manifold(in both Case B and Case C) and most of the evolution of the chem-ical species takes place on the slow manifold. The QSSA approxima-tion is quite adequate for this Range II (see Fig. 7) and the reductioncan be proved useful in many hypersonic applications. If the flowtimescale is large (of order of 10�2, see Fig. 8) then equilibrium


values are relevant. Fig. 9 shows the equilibrium values of differentspecies at different temperature values. In this range, the effects ofthermal nonequilibrium are not very significant. The nonequilibri-um assumption only perturbs the values of chemical equilibrium(mass fractions and temperature values).

Fig. 8 shows the different timescales for frozen, full, reducedand equilibrium chemistry. In this temperature range, the typicaltimescale of full chemistry is about Oð10�8Þ and the timescale ofreduced chemistry is Oð10�6Þ as shown in the figure. As for equilib-rium chemistry, the timescale is relatively large (about Oð10�3Þ). Inthis case the equilibrium chemistry assumption could not be ade-quate for some hypersonic flow applications. On the other hand,we notice that most of evolution time of the species is spent onthe manifold on the way to the equilibrium state. Hence, thereduced chemistry is appropriate and the slow manifold computa-tions could be practical in many applications.

4.4. Temperature range III (above 9000 K)

For the high temperature range (cases D, E, F, and G), again theslow manifold exits for all the cases. When we use the 5-speciesmodel of the air-mixture, we notice the temperature evolutiontakes longer time on the manifold compared to the 7-speciesmodel. This observation is also true if we assume the vibrationalenergy to be in non-equilibrium with translational temperature.In this temperature range, we observe that the time required forthe chemical species to reach equilibrium is around Oð10�8Þ whichis relatively fast for many hypersonic flow applications. The laststatement is true for the 5-species model (with vibrational equilib-rium and vibrational nonequilibrium) as well as the 7-speciesmodel (with vibrational equilibrium and vibrationalnonequilibrium).

Fig. 10 shows the different timescales of frozen, full, reducedand equilibrium chemistry of temperature range above 9000 Kfor 5-species kinetic set assuming vibrational equilibrium. The typ-ical timescale of full chemistry is around Oð10�9Þ. On the otherhand, the timescale of the reduced chemistry is about Oð10�7Þ.The evolution trajectories reach equilibrium state in less that10�6 seconds and hence in this case the equilibrium assumptioncould be suitable for many applications. Similar to Fig. 10, thetimescales of frozen, full, reduced and equilibrium chemistry areshown in Fig. 11 but assuming vibrational non-equilibrium. The



Table 6The timescales of different chemistry simplifications for the three temperature ranges (the values of timescales are in seconds).

Temperature range Timescale for appropriate chemistry simplification (s)

Frozen Full Reduced Equilibrium

Range I (2500–4500 K) Oð10�9Þ Oð10�7Þ Oð10�2Þ Oð10þ1ÞRange II (4500–9000 K) Oð10�9Þ Oð10�8Þ Oð10�6Þ Oð10�2ÞRange III (above 9000 K) Oð10�9Þ Oð10�8Þ Oð10�7Þ Oð10�5Þ


full chemistry timescale is of order 10�8 and the timescale ofreduced chemistry is about Oð10�7Þ. In this range, the evolutiontrajectories reach equilibrium in more than 10�5 seconds but like-wise the equilibrium chemistry assumption is still appropriate.

For the 7-species kinetic set, Fig. 12 shows the time evolutiontrajectories of the major species (N2;O2;NO;N;O). The amount ofthe ionized species NOþ and the electron are too small to affectthe general characteristics of the flow (for this reason we do notinclude the time evolution curves of NOþ and e� in Fig. 12). Thetimescale of the full chemistry is in the range 10�9 to 10�8 andthe timescale of reduced chemistry is of order 10�7. Similar tothe 5-spices set in this temperature range, the timescales of fulland reduced chemistry are relatively small. The trajectories attainequilibrium in less that 10�6 seconds and for applications withflow timescale of this order, we could assume equilibriumchemistry.

5. Conclusion

In this work, we demonstrate the existence of the slow manifoldfor the air-mixture chemical system. We examine seven differentchemical kinetics depending on temperature ranges and numberof species and in all of them the slow manifold exists. In additionto that, we have computed the quasi-steady state assumption(QSSA) estimate of the slow manifold and it has been shown thatthe one-dimensional slow manifold is approximated fairly accu-rately for two different temperature ranges. This shows the reduc-tion mechanism is applicable to many hypersonic flow problems.In the low-temperature range, considering the full chemistry couldbe the more appropriate for the hypersonic flow computations butthe reduction is also possible. For mid-temperature range thereduction mechanisms could be a more optimal choice in termsof accuracy and computational demand. In the high-temperaturerange the chemical reactions time-scales could be too small formany hypersonic applications and in this case, the equilibrium


chemistry values of the species concentrations would be very use-ful. Table 6 shows approximate values of timescales for the appro-priate chemistry simplifications: frozen chemistry, full chemistry,reduced chemistry and equilibrium chemistry. The timescales val-ues are given for the three temperature ranges: low-temperaturerange (2500 K to 4500 K), mid-temperature range (4500 K to9000 K) and high-temperature range (above 9000 K).

References

Anderson, J.D., 1989. Hypersonic and high temperature gas dynamics. McGraw-Hill,New York.

Candler, G., 1988. The Computation of Weakly Ionized Hypersonic Flows in Thermo-Chemical Nonequilibrium. PhD thesis, Stanford University.

Duan, L., Martin, M.P., 1994. Effective approach for estimating turbulence-chemistryinteraction in hypersonic turbulent boundary layers. J. Aircraft 31 (2).

Duan, L., Martin, M.P., 2009. Procedure to validate direct numerical simulations ofwall-bounded turbulence including finite-rate reactions. AIAA J. 47 (1).

Girimaji, S.S., Ibrahim, A.A., 2014. Advanced quasi-steady state approximation forchemical kinetics. J. Fluids Eng. 136 (3), 031201.

Hudson, M.L., 1996. Linear stability of hypersonic flows in thermal and chemicalnon-equilibrium. PhD thesis, North Carolina State University.

Hudson, M.L., Chokani, N., Candler, G.V., 1997. Linear stability of hypersonic flow inthermochemical non-equilibrium. AIAA 35 (6).

Maas, U., Pope, S.B., 1992. Simplifying chemical kinetics: intrinsic low-dimensionalmanifolds in composition space. Combust. Flame 88 (3), 239–264.

Park, C., 1990. Non-equilibrium Hypersonic Aerothermodynamics. Wiley.Peeters, N., Rogg, B. (Eds.), 1993. Reduced Kinetic Mechanisms for Applications in

Combustion Systems. Springer-Verlag, New York.Pimentel, C.A.R., Hetem Jr., A., 2011. Computation of air chemical equilibrium

composition until 300,000 K Part 1. J. Aerosp. Technol. Manag. 3, 111–126.Skrebkov, O.V., Karkach, S.P., 2007. Vibrational non-equilibrium and electronic

excitation in the reaction of hydrogen with oxygen behind a shock wave. Kinet.Catal. 48, 367–375.

Smooke, M. (Ed.), 1992. Reduced Kinetic Mechanisms and Flames. Springer-Verlag,Berlin.

Suman, S., Ibrahim, A.A., Girimaji, S.S., 2011. On the use of reduced chemical kineticsfor hypersonic transition and breakdown to turbulence computations. In: AIAAPaper 2011-3715. 41st AIAA Fluid Dynamics Conference and Exhibit, Honolulu,Hawaii, 27–30 June 2011.

Tomlin, A.S., Whitehouse, L., Lowe, R., Pilling, M.J., 2001. Low-dimensionalmanifolds in tropospheric chemical systems. Roy. Soc. Chem. 120, 125–146.


http://refhub.elsevier.com/S0142-727X(14)00146-5/h0005


























on air-chemistry reduction for hypersonic external flow applications

Documents