Modeling of Supercritical Vaporization, Mixing, and Combustion

925

Proceedings of the Combustion Institute, Volume 28, 2000/pp. 925–942

INVITED TOPICAL REVIEW

MODELING OF SUPERCRITICAL VAPORIZATION, MIXING, AND COMBUS-TION PROCESSES IN LIQUID-FUELED PROPULSION SYSTEMS

VIGOR YANGThe Pennsylvania State University

104 Research Building East

University Park, PA 16802 USA

This paper addresses the physiochemical mechanisms involved in transcritical and supercritical vapori-zation, mixing, and combustion processes in contemporary liquid-fueled propulsion and power-generationsystems. Fundamental investigation into these phenomena poses an array of challenges due to the obstaclesin conducting experimental measurements and numerical simulations at scales sufficient to resolve theunderlying processes. In addition to all of the classical problems for multiphase chemically reacting flows,a unique set of problems arises from the introduction of thermodynamic nonidealities and transport anom-alies. The situation becomes even more complex with increasing pressure because of an inherent increasein the flow Reynolds number and difficulties that arise when fluid states approach the critical mixingcondition. The paper attempts to provide an overview of recent advances in theoretical modeling andnumerical simulation of this subject. A variety of liquid propellants, including hydrocarbon and cryogenicfluids, under both steady and oscillatory conditions, are treated systematically. Emphasis is placed on thedevelopment of a hierarchical approach and its associated difficulties. Results from representative studiesare presented to lend insight into the intricate nature of the problem.

Introduction

Liquid droplet vaporization and spray combustionin supercritical environments have long been mat-ters of serious practical concern in combustion sci-ence and technology, mainly due to the necessity ofdeveloping high-pressure combustion devices suchas liquid-propellant rocket, gas-turbine, diesel, andpulse-detonation engines. Liquid fuels and/or oxi-dizers are usually delivered to combustion chambersas a spray of droplets, which then undergo a se-quence of vaporization, mixing, ignition, and com-bustion processes at pressure levels well above thethermodynamic critical points of the fluids. Underthese conditions, liquids initially injected at subcrit-ical temperature may heat up and experience a ther-modynamic state transition into the supercritical re-gime during their lifetimes. The process exhibitsmany characteristics distinct from those in an at-mospherical environment, thereby rendering con-ventional approaches developed for low-pressureapplications invalid.

Modeling supercritical mixing and combustionprocesses numerically poses a variety of challengesthat include all of the classical closure problems anda unique set of problems imposed by the introduc-tion of thermodynamic nonidealities and transportanomalies. From the classical point of view, reacting,multiphase flows introduce the complicating factorsof chemical kinetics, highly nonlinear source terms,and a variety of sub-grid scale (SGS) velocity and

scalar-mixing interactions. Flowfield evolution is af-fected by compressibility effects (volumetricchanges induced by changes in pressure) and vari-able inertia effects (volumetric changes induced byvariable composition and/or heat addition). The re-sultant coupling dynamics yield an array of physi-ochemical processes that are dominated by widelydisparate time and length scales, many being smallerthan can be resolved in a numerically feasible man-ner. The situation becomes more complex with in-creasing pressure because of an inherent increase inthe flow Reynolds number, which causes a furtherdecrease in the scales associated with SGS interac-tions and difficulties that arise when fluid states ap-proach the critical conditions. Near the critical point,propellant mixture properties exhibit liquid-likedensities, gas-like diffusivities, and pressure-depen-dent solubilities. Surface tension and enthalpy of va-porization approach zero, and the isothermal com-pressibility and specific heat increase significantly.These phenomena, coupled with extreme local prop-erty variations, have a significant impact on the evo-lutionary dynamics exhibited by a given system.

This paper attempts to provide an overview of re-cent advances in theoretical modeling and numericalsimulation of supercritical vaporization, mixing, andcombustion processes in liquid-fueled propulsionsystems. Three subject areas are considered: (1)droplet gasification and combustion, (2) spray fielddynamics, and (3) multiphase mixing and combus-tion processes. A variety of liquid propellants and

926 INVITED TOPICAL REVIEW

propellant simulants, including hydrocarbon andcryogenic fluids, at both steady and oscillatory con-ditions, are treated systematically.

Supercritical Droplet Vaporization andCombustion

We first consider the vaporization and combustionof an isolated liquid droplet when suddenly placedin a combustion chamber. This configuration allowsus to focus on the effects of pressure on the ther-modynamic and transport processes involved. As aresult of heat transfer from the surrounding gases,the droplet starts to heat up and evaporate due tothe vapor concentration gradient near the surface.Two scenarios, subcritical and supercritical condi-tions, must be considered in order to provide a com-plete description of various situations encountered.If the chamber condition is in the thermodynamicsubcritical regime of the injected liquid, the dropletsurface provides a well-defined interfacial boundary.To facilitate analysis, physical processes in the drop-let interior and ambient gases are treated separatelyand then matched at the interface by requiring liq-uid-vapor phase equilibrium and continuities ofmass and energy fluxes. This procedure eventuallydetermines the droplet surface conditions and evap-oration rate. The situation, however, becomes qual-itatively different in the supercritical regime. Then,the droplet may continuously heat up, with its sur-face reaching the critical mixing point prior to theend of the droplet lifetime. When this occurs, thesharp distinction between gas and liquid disappears.The enthalpy of vaporization reduces to zero, and noabrupt phase change is involved in the vaporizationprocess. The density and temperature of the entirefield as well as their gradients vary continuouslyacross the droplet surface. The droplet interior, how-ever, remains at the liquid state with a subcriticaltemperature distribution. For convenience of dis-cussion, the droplet regression is best characterizedby the motion of the surface which attains the criticalmixing temperature of the system. The process be-comes totally diffusion controlled.

Attempts to study supercritical droplet vaporiza-tion and combustion have been made for more thanfour decades. The earliest development of a predic-tive model was initiated by Spalding [1] and subse-quently refined by Rosner [2]. Both studies consid-ered an isolated fuel droplet in a stagnantenvironment by approximating the droplet as a pointsource of dense gas with constant physical proper-ties. The same problem was re-examined by otherresearchers [3–5] in order to investigate the influ-ence of convection, density variation, and finite ratechemical kinetics.

Systematic treatment of droplet vaporization atnear-critical conditions was initiated by Manrique

and Borman [6], based on a quasi-steady model.They concluded that the effects of thermodynamicnonidealities, property variations, and high-pressurecorrections for phase equilibrium modify the vapor-ization mechanisms significantly. In light of thesefindings, Lazar and Faeth [7] and Canada and Faeth[8] conducted a series of experimental and theoreti-cal studies on droplet combustion in both stagnantand forced convective environments, with special at-tention focused on the high-pressure phenomena ofphase equilibrium. The effects of forced convectionin the gas phase were treated by conventional mul-tiplicative corrections. The assumptions of quasi-steadiness and uniform property distributionsadopted in Refs. [6–8] were later relaxed. In worksby Rosner and Chang [9] and Kadota and Hiroyasu[10], the effects of transient processes, natural con-vection, and the conditions under which a dropletmay be driven to its critical point were examined.

More recently, several researchers [11–17] haveemployed numerical techniques to simulate high-pressure droplet vaporization and combustion withconsiderable success. All of these models, however,adopted certain rudimentary assumptions and em-pirical formulas for fluid properties that were ex-trapolated from low-pressure cases, with their ac-curacy for high-pressure applications subject toquestion. Furthermore, no effort was made to treatthe thermodynamic phase transition through thecritical point. In order to remedy these deficiencies,a series of fundamental studies [18–28] were con-ducted using state-of-the-art treatment of thermo-dynamic and transport phenomena. Of particularimportance is the unified analyses of thermophyscialproperties employed in Refs. [19–20] and [25–28]based on fundamental thermodynamic theories.These approaches allow for a self-consistent solutionfrom first principles, thereby enabling a systematicinvestigation into underlying mechanisms involvedin supercritical droplet gasification and combustion.The effect of non-equilibrium phase transition ondroplet behavior was further addressed by Harstadand Bellan [25–28] using Keizer’s fluctuation theory.In addition, Umemura and Shimada [29–31] con-structed approximate analyses to elucidate many in-triguing characteristics of supercritical droplet gasi-fication. Extensive reviews of this subject wererecently given Givler and Abraham [32] and Bellan[33].

Thermodynamic and Transport Properties

Owing to the continuous variations of fluid prop-erties in supercritical environments, classical tech-niques dealing with liquids and gases individually of-ten lead to erroneous results of droplet dynamics.The problem becomes even more exacerbated whenthe droplet surface approaches the critical condition.

SUPERCRITICAL VAPORIZATION, MIXING, AND COMBUSTION 927

The thermophysical properties usually exhibit anom-alous variations and are very sensitive to both tem-perature and pressure in the vicinity of the criticalpoint, a phenomenon commonly referred to as near-critical enhancement. Thus, an essential prerequisiteof any realistic treatment of supercritical droplet be-havior is the establishment of a unified propertyevaluation scheme capable of treating thermophys-ical properties of the system and its constituent spe-cies over the entire fluid thermodynamic state, fromcompressed liquid to dilute gas.

All the thermodynamic properties can be derivedfrom a modified Benedict–Webb–Rubin (BWR)equation of state proposed by Jacobsen and Stewart[34] due to its superior performance over conven-tional cubic equations of state [35]. This equation ofstate has been extremely valuable in correlating bothliquid and vapor thermodynamic and volumetricdata; however, the temperature constants involvedare available only for a limited number of pure sub-stances [36]. To overcome this constraint, an ex-tended corresponding-state (ECS) principle [37,38]is used. The basic idea is to assume that the prop-erties of a single-phase fluid can be evaluated viaconformal mappings of temperature and density tothose of a given reference fluid. As a result, only theBWR constants for the reference fluid are needed.For a multicomponent system, accounting forchanges in properties due to mixing is much morecomplicated. A pseudo pure-substance model isadopted to evaluate the properties of a mixture,treating the mixture as a single-phase pure substancewith its own set of properties evaluated via the ECSprinciple. This method improves prediction accu-racy and requires only limited data (i.e., criticalproperties and Pitzer’s acentric factor) for each con-stituent component.

Successful application of the corresponding-stateargument for the evaluation of fluid p-V-T propertiesalso encourages similar improvement in the predic-tion of thermophysical data. In the following, a briefsummary of the corresponding-state method in con-junction with the mixture combining rule is firstgiven, followed by the BWR equation of state for thereference fluid. Finally, the techniques for evaluat-ing thermodynamic and transport properties are ad-dressed.

Extended corresponding-state principleThe ECS model of Ely and Hanley [37,38] is used

to evaluate volumetric and transport properties of amixture over its entire thermodynamic fluid state.The scheme assumes that the configurational prop-erties (such as temperature, density, viscosity, ther-mal conductivity, etc.) of a single-phase mixture canbe equated to those of a hypothetical pure fluid,which are then evaluated via corresponding-stateprinciples with respect to a given reference fluid.For example, the viscosity of a mixture, lm, can be

related to that of a reference fluid, l0, at the corre-sponding thermodynamic state as

l (q,T) � l (q ,T ) F (1)m 0 0 0 l

where Fl represents the mapping function. The cor-respondence of temperature and density betweenthe mixture of interest and the reference fluid canbe characterized by the following two scaling factors:

T q0f � ; � � (2)m m

T q0

The former represents the conformation of potentialdistribution of energy, while the latter characterizesthe effect of mixture molecular size.

Equation of stateUnder the assumption of the ECS principle, the

density of a mixture can be evaluated by

q (T , p )0 0 0q (T, p) � (3)m � (T, p)m

where q0, T0, and p0 denote the corresponding den-sity, temperature, and pressure of the referencefluid, respectively. Since the temperature at the con-formal state is calculated by equation 2, the corre-sponding pressure can be derived based on the gen-eral compressibility theory:

�mp � p (4)0 � �fm

To ensure the accuracy of the density prediction, ageneralized BWR equation of state [34] is adoptedfor the reference fluid:

9np (T, q) � a (T)q0 � n

n�1

1522n�17 �cq� a (T)q e (5)� n

n�10

where c is 0.04, and the temperature coefficientsai(T) depend on the reference fluid used.

Although this equation of state must be solved it-eratively for density at given pressure and tempera-ture, the prediction covers a wide range of thermo-dynamic states and as such promotes theestablishment of a unified evaluation scheme of ther-mophysical properties. Fig. 1 shows the comparisonof oxygen density between experimental data [39]and the prediction by the BWR equation of state inconjunction with the ECS principle. The referencefluid is selected to be propane due to the availabilityof sufficiently reliable data correlated over a widerange of experimental conditions for this substance.The result shows excellent agreement over the entirefluid state. Fig. 2 presents the relative errors of den-sity prediction based on three commonly used equa-tions of state, namely, BWR, Peng–Robinson (PR),


Fig. 1. Comparison of oxygen density predicted by theBWR equation of state and measured by Sychev et al. [39].

Fig. 2. Relative errors of density predictions by threedifferent equations of state. Experimental data fromSychev et al. [39].

and Soave–Redlich–Kwong (SRK). The ECS prin-ciple is embedded into the evaluation procedure ofthe BWR equation of state and shows its superiorperformance with the maximum relative error of1.5% for the pressure and temperature ranges underconsideration. On the other hand, the SRK and PRequations of state yield maximum errors around 13%and 17%, respectively.

Thermodynamic PropertiesThermodynamic properties such as enthalpy, in-

ternal energy, and specific heat can be expressed asthe sum of ideal-gas properties at the same tem-perature and departure functions which take into ac-count the dense-fluid correction. Thus,

v �p0h � h � T � p dv�� T v

� RT(Z � 1) (6)�v �p0e � e � T � p dv (7)��

� �T v

2v � p0C � C � T dvp p �� 2�� T v

2�p �p� T � R (8)� � � � ��T �vv T

where superscripts 0 refer to ideal-gas properties.The second terms on the right sides of equations 6–8 denote the thermodynamic departure functionsand can be obtained from the equation of state de-scribed previously.

Transport PropertiesEstimation of viscosity and thermal conductivity

can be made by means of the ECS principle. Thecorresponding-state argument for the viscosity of amixture can be written in its most general form as

l (q, T) � l (q , T )F v (9)m 0 0 0 l l

where Fl is the scaling factor. The correction factorvl accounts for the effect of non-correspondenceand has the magnitude always close to unity basedon the modified Enskog theory [40].

Because of the lack of a complete molecular the-ory for describing transport properties over a broadregime of fluid phases, it is generally accepted thatviscosity and thermal conductivity can be dividedinto three contributions and correlated in terms ofdensity and temperature [41]. For instance, the vis-cosity of the reference fluid is written as follows:

0 excl (q , T ) � l (T ) � Dl (q , T )0 0 0 0 0 0 0 0

crit� Dl (q , T ) (10)0 0 0

The first term on the right-hand side represents thevalue at the dilute-gas limit, which is independent ofdensity and can be accurately predicted by kinetic-theory equations. The second term is the excess vis-cosity which, with the exclusion of unusual variationsnear the critical point, characterizes the deviationfrom l0 for a dense fluid. The third term refers tothe critical enhancement which accounts for theanomalous increase above the background viscosity(i.e., the sum of and ) as the critical point is0 excl Dl0 0approached. However, the theory of nonclassicalcritical behavior predicts that, in general, propertiesthat diverge strongly in pure fluids near the criticalpoints diverge only weakly in mixtures due to thedifferent physical criteria for criticality in a pure fluidand a mixture [42]. Because the effect of critical en-hancement is not well-defined for a mixture and islikely to be small, the third term is usually notcritDl0considered in the existing analyses of supercriticaldroplet gasification.

Evaluation of thermal conductivity must be care-fully conducted for two reasons: (1) the one-fluidmodel must ignore the contribution of diffusion toconductivity, and (2) the effect of internal degrees


Fig. 3. Vapor–liquid phase equilibrium compositions forO2/H2 system at various pressures.

Fig. 4. Pressure–temperature diagram for phase behav-ior of O2/H2 system in equilibrium.

of freedom on thermal conductivity cannot be cor-rectly taken into account by the corresponding-stateargument. As a result, thermal conductivity of a puresubstance or mixture is generally divided into twocontributions [38]:

k (q, T) � k� (T) � k� (q, T) (10)m m m

The former, , arises from transfer of energy viak� (T)mthe internal degrees of freedom, while the latter,

, is due to the effects of molecular collisionk� (q, T)mor translation which can be evaluated by means ofthe ECS method. For a mixture, can be eval-k� (T)muated by a semi-empirical mixing rule.

Estimation of the binary mass diffusivity for a mix-ture gas at high density represents a more challeng-ing task than evaluating the other transport proper-ties, due to the lack of a formal theory or even atheoretically based correlation. Takahashi [43] sug-gested a simple scheme for predicting the binarymass diffusivity of a dense fluid by means of a cor-responding-state approach. The approach appears tobe the most complete to date and has demonstrated

moderate success in the limited number of tests con-ducted. The scheme proceeds in two steps. First, thebinary mass diffusivity of a dilute gas is obtained us-ing the Chapman-Enskog theory in conjunction withthe intermolecular potential function. The calcu-lated data is then corrected in accordance with ageneralized chart in terms of reduced temperatureand pressure.

Vapor-Liquid Phase Equilibrium

The result of vapor-liquid phase equilibrium is re-quired to specify the droplet surface behavior priorto the occurrence of critical conditions. The analysisusually consists of two steps. First, an appropriateequation of state as described in the section on ther-modynamic and transport properties is employed tocalculate fugacities of each constituent species inboth gas and liquid phases. The second step lies inthe determination of the phase equilibrium condi-tions by requiring equal fugacities for both phases ofeach species. Specific outputs from this analysis in-clude (1) enthalpy of vaporization, (2) solubility ofambient gases in the liquid phase, (3) species con-centrations at the droplet surface, and (4) conditionsfor criticality. The analysis can be further used inconjunction with the property evaluation scheme todetermine other important properties such as sur-face tension.

As an example, the equilibrium compositions fora binary mixture of oxygen and hydrogen is shownin Fig. 3, where pr represents the reduced pressureof oxygen. In the subcritical regime, the amount ofhydrogen dissolved in the liquid oxygen is quite lim-ited, decreasing progressively with increasing tem-perature and reducing to zero at the boiling point ofoxygen. At supercritical pressures, however, the hy-drogen gas solubility becomes substantial and in-creases with temperature. Because of the distinctdifferences in thermophysical properties betweenthe two species, the dissolved hydrogen may appre-ciably modify the liquid properties and, subse-quently, the vaporization behavior. The phase equi-librium results also indicate that the critical mixingtemperature decreases with pressure.

The overall phase behavior in equilibrium is bestsummarized by the pressure-temperature diagramin Fig. 4, which shows how the phase transition oco-curs under different thermodynamic conditions. Theboiling line is made up of boiling points for subcrit-ical pressure. As the temperature increases, an equi-librium vapor-liquid mixture may transit to super-heated vapor across this line. The critical mixing lineregisters the variation of the critical mixing tem-perature with pressure. It intersects the boiling lineat the critical point of pure oxygen, the highest tem-perature at which the vapor and liquid phases of anO2/H2 binary system can coexist in equilibrium.


Fig. 5. Time variations of droplet surface temperatureat various pressures. T� � 1000 K, T0 � 90 K, D0 � 100lm.

Fig. 6. Instantaneous distributions of mixture specificheat in the entire field at various times. T0 � 90 K, T� �

1000 K, p � 100 atm, D0 � 100 lm.

Droplet Vaporization in Quiescent Environments

Several theoretical works have recently been de-voted to the understanding of droplet vaporizationand combustion under high-pressure conditions.Both hydrocarbon droplets in air [11–13,18,27,28]and liquid oxygen (LOX) droplets in hydrogen[14,17,19–26,44] were treated comprehensively,with emphasis placed on the effects of transient dif-fusion and interfacial thermodynamics. The works ofLafon et al. [22,44], Hsiao and Yang [20,45], andHarstad and Bellan [25–28] appear to be the mostcomplete to date because of the employment of aunified property evaluation scheme, as outlined pre-viously.

We first consider the vaporization of LOX dropletin either pure hydrogen or mixed hydrogen/waterenvironments [22,44] due to its broad applicationsin cryogenic rocket engines using hydrogen and ox-ygen as propellants [46]. Fig. 5 shows the time var-iations of droplet surface temperature at various

pressures. The ambient hydrogen temperature is1000 K. Three different scenarios are noted. First,at low pressures (i.e., p � 10 atm), the surface tem-perature rises suddenly and levels off at the pseudowet-bulb state, which is slightly lower than the oxy-gen boiling temperature because of the presence ofhydrogen on the gaseous side of the interface. Forhigher pressures (i.e., p � 50 atm), the surface tem-perature rises continuously. The pseudo wet-bulbstate disappears, and the vaporization process be-comes transient in nature during the entire dropletlifetime. For p � 100 atm, the droplet surface evenreaches its critical state at 1 ms. Fig. 6 shows thedistributions of the mixture specific heat at varioustimes. The weak divergence of the specific heat nearthe droplet surface (i.e., defined as the surface at-taining the critical-mixing temperature) is clearly ob-served.

An extensive series of numerical simulations wereconducted for a broad range of ambient tempera-tures (500 � T� � 2500 K) and pressures up to 300atm. The calculated LOX droplet lifetime in purehydrogen can be well correlated using an approxi-mate analysis that takes into account the effect oftransient heat diffusion in terms of the reduced criti-cal temperature, . TheT* � (T � T )/(T � T )c � c � 0resultant expression of the droplet lifetime takes theform

2R0 ls � [0.0115 � 0.542 (1 � T*)] f (� /� ) (11)c � 0l�0

where the correction factor is chosen as

lf(� /� )�� 0

l1�3.9 [1�exp(�0.035(� /� �1))] (12)� 0

The expression of may be related to the SpaldingT*ctransfer number, which has more physical signifi-cance, as follows:

T � T T*� c cB � � (13)T

T � T 1 � T*c 0 c

This correlation clearly shows that pressure affectsthe droplet lifetime through its influence on the mix-ture critical temperature and ambient thermal dif-fusivity.

The behavior of hydrocarbon fuel droplets basi-cally follow the same trend as cryogenic droplets interms of their vaporization characteristics. In gen-eral, the droplet lifetime decreases smoothly withincreasing pressure, and no discernible variation oc-curs across the critical transition. In the subcriticalregime, the decrease in enthalpy of vaporization withpressure facilitates vaporization process and conse-quently leads to a decrease in droplet lifetime. Thesituation becomes different at supercritical condi-tions, at which the interfacial boundary disappears


Fig. 7. Time variations of droplet surface temperatureat various pressures, n-pentane/air system.

Fig. 8. Effect of pressure on milestone times associatedwith droplet gasification and burning processes, n-pentane/air system. T0 � 300 K, T� � 1000 K.

with the enthalpy of vaporization being zero. In thiscase, the decrease in droplet lifetime is mainly at-tributed to the increase in thermal diffusivity nearthe droplet surface. As critical conditions are ap-proached, the divergence of the specific heat com-pensates for the effect of vaporization enthalpy re-duction, so that no abrupt change occurs during thecritical transition.

Droplet Combustion in Quiescent Environments

Much effort was devoted to the study of super-critical droplet combustion (e.g., Refs. [17,18,22]).In spite of the presence of chemical reactions in thegas phase, the general characteristics of a burningdroplet are similar to those involving only vaporiza-tion. We consider here the combustion of a hydro-carbon (e.g., n-pentane) fuel droplet in air [18]. The

droplet initial temperature is 300 K, and the ambientair temperature is 1000 K. Fig. 7 shows the timehistory of the droplet surface temperature at variouspressures for D0 � 100 lm. Once ignition isachieved in the gas phase, energy feedback from theflame causes a rapid increase in droplet surface tem-perature. At low pressures (p � 20 atm), the surfacetemperature varies very slowly following onset offlame development and almost levels off at thepseudo wet-bulb state. As the ambient pressure in-creases, the high concentrations of oxygen in the gasphase and the fuel vapor issued from the dropletsurface result in a high chemical reaction rate, con-sequently causing a progressive decrease in ignitiontime. Furthermore, the surface temperature jumpduring the flame-development stage increases withincreasing pressure. Since the critical mixing tem-perature decreases with pressure, the dropletreaches its critical condition more easily at higherpressures, almost immediately following establish-ment of the diffusion flame in the gas phase for p �110 atm.

Figure 8 presents the effect of pressure on variousmilestone times associated with droplet gasificationand burning process for D0 � 100 lm [18]. Here,gasification lifetime is the time required for com-plete gasification; droplet burning lifetime is the gas-ification lifetime minus ignition time; single-phasecombustion lifetime is the time duration from com-plete gasification to burnout of all fuel vapor; com-bustion lifetime is the sum of single-phase combus-tion lifetime and droplet burning lifetime. Thegasification lifetime decreases continuously withpressure, whereas the single-phase combustion life-time increases progressively with pressure due to itsadverse effect on mass diffusion. More importantly,the pressure dependence of combustion lifetime ex-hibits irregular behavior. This phenomenon may beattributed to the overlapping effect of reduced en-thalpy of vaporization and mass diffusion with in-creasing pressure.

Since the time scales for diffusion processes areinversely proportional to the droplet diametersquared, it is important to examine the effect ofdroplet size on the burning characteristics. In thisregard, calculations were conducted for large drop-lets having an initial diameter of 1000 lm, which iscomparable to the sizes considered in most experi-mental studies of supercritical droplet combustion[47–49]. Furthermore, the ignition transient occu-pies only a small fraction of the entire droplet life-time for large droplets. The uncertainties associatedwith the ignition procedure in determining the char-acteristics of droplet gasification can be minimized.Consequently, a more meaningful comparison withexperimental data can be made. The combustion be-havior of a large droplet reveals several characteris-tics distinct from those of a small droplet. First, ig-nition for large droplets occurs in the very early stage


Fig. 9. LOX droplet gasification in supercritical hydrogen flow. p � 100 atm, U� � 2.5 m/s.

of the entire droplet lifetime. The influence of gas-ification prior to ignition on the overall burningmechanisms appears to be quite limited. Second, thecombustion lifetime decreases with increasing pres-sure, reaching a minimum near the critical pressureof the liquid fuel. As the pressure further increases,the combustion time increases due to reduced massdiffusivity at high pressures. The gasification lifetimedecreases continuously with pressure, whereas thesingle-phase combustion lifetime increases progres-sively with pressure. At low pressures, the gasifica-tion of liquid fuel primarily controls the combustionprocess, whereas in a supercritical environment, thetransient gas-phase diffusion plays a more importantrole.

Droplet Vaporization in Convective Environments

When a droplet is introduced into a cross-flow, theforced convection results in increases of heat andmass transfer between the droplet and surroundinggases, which consequently intensifies the gasificationprocess. Although many studies have been con-ducted to examine droplet vaporization in forced-convective environments, effects of pressure andfreestream velocities on droplet dynamics, especiallyfor rocket engine applications which involve super-critical conditions, have not yet been addressed indetail. Hsiao et al. [20,45] developed a comprehen-sive analysis of LOX droplet vaporization in a super-critical hydrogen stream, covering a pressure range

of 100–400 atm. The model takes into account mul-tidimensional flow motions and enables a thoroughexamination of droplet behavior during its entirelifetime, including dynamic deformation, viscousstripping, and secondary breakup. Detailed flowstructures and thermodynamic properties are ob-tained to reveal mechanisms underlying droplet gas-ification as well as deformation and breakup dynam-ics.

Figure 9 shows six frames of isotherms and iso-pleths of oxygen concentration at a convective ve-locity of 2.5 m/s and an ambient pressure of 100 atm.The freestream Reynolds number Re is 31, based onthe initial droplet diameter. Soon after the introduc-tion of the droplet into the hydrogen stream, theflow adjusts to form a boundary layer near the sur-face. The gasified oxygen is carried downstreamthrough convection and mass diffusion. The evolu-tion of the temperature field exhibits features dis-tinct from that of the concentration field due to thedisparate time scales associated with thermal andmass diffusion processes (i.e., Lewis number � 1).The thermal wave penetrates into the droplet inte-rior with a pace faster than does the surroundinghydrogen species. Since the liquid core possesseslarge momentum inertia and moves slower com-pared with the gasified oxygen, at t � 0.79 ms, thedroplet (delineated by the dark region in the tem-perature contours) reveals an olive shape, while theoxygen concentration contours deform into a cres-cent shape with the edge bent to the streamwise di-rection. At t � 1.08 ms, the subcritical liquid core


Fig. 10. LOX droplet gasification in supercritical hydrogen flow at p � 100 atm. (a) Spherical mode. U� � 0.2 m/s,t � 0.61 ms. (b) Deformation mode. U� � 1.5 m/s, t � 0.61 ms. (c) Stripping mode. U� � 5 m/s, t � 0.17 ms. (d)Breakup mode. U� � 15 m/s, t � 0.17 ms.

disappears, leaving behind a puff of dense oxygenfluid which is convected further downstream withincreasing velocity until it reaches the momentumequilibrium with the ambient hydrogen flow.

Figure 10 summarizes the streamline patterns andoxygen concentration contours of the four differentmodes commonly observed in supercritical dropletgasification. The droplet may either remain in aspherical configuration, deform to an olive shape, oreven break up into fragments, depending on the lo-cal flow conditions. Unlike low-pressure cases inwhich the large shear stress at the gas–liquid inter-face induces internal flow circulation in the liquidcore [50], no discernible recirculation takes place inthe droplet interior, regardless of the Reynolds num-ber and deformation mode. This may be attributedto the diminishment of surface tension at supercrit-ical conditions. In addition, the droplet regresses sofast that a fluid element in the interphase region maynot acquire the time sufficient for establishing aninternal vortical flow before it gasifies. The rapid de-formation of the droplet configuration further pre-cludes the existence of stable shear stress in the liq-uid core and consequently obstructs the formationof recirculation.

The spherical mode shown in Fig. 10a typicallyoccurs at very low Reynolds numbers. Although flowseparation is encouraged by LOX gasification, no re-circulating eddy is found in the wake behind thedroplet. The vorticity generated is too weak to formany confined eddy. When the ambient velocity in-creases to 1.5 m/s, the droplet deforms into an oliveshape with a recirculating ring attached behind it, asshown in Fig. 10b. Owing to the droplet deformationand gasification, the threshold Reynolds numberabove which the recirculating eddy forms is consid-erably lower than that for a hard sphere. Figure 10cdepicts the flow structure with viscous stripping atan ambient velocity of 5 m/s, showing an oblatedroplet with a stretched vortex ring. The flattenededge of the droplet enhances the strength of the re-circulating eddies and as such increases the viscousshear stress dramatically. Consequently, a thin sheetof fluid is stripped off from the edge of the dropletand swept toward the outer boundary of the recir-culating eddy. At a very high ambient velocity of 15m/s, droplet breakup takes place, as clearly shownin Fig. 10d. The hydrogen flow penetrates throughthe liquid phase and divides the droplet into two


parts: the core disk and surrounding ring. The vor-tical structure in the wake region expands substan-tially as a result of the strong shear stress.

The effect of ambient pressure and Reynoldsnumber on droplet lifetime can be correlated withrespect to the reference value at zero Reynolds num-ber. The result takes the form

s 1f� (14)

1.1 �0.88s 1 � 0.17Re (p )f •Re�0 r,O2

where Re and refer to the Reynolds numberpr,O2based on the initial droplet diameter and the re-duced pressure of oxygen, respectively. This corre-lation bears a resemblance to the popular Ranz-and-Marshall correlation [51] for droplet heat-transfercorrection due to convective effect. The Ranz-and-Marshall correlation applies only to low pressureflows and has a weaker Reynolds number depen-dency.

Drag coefficient has been generally adopted as adimensionless parameter to measure the drag forceacting on a droplet. Chen and Yuen [52] found thatthe drag coefficient of an evaporating droplet issmaller than that of a non-vaporizing solid sphere atthe same Reynolds number. Several researchers[53,54] numerically analyzed evaporating dropletmotion by solving the Navier–Stokes equations andproposed the following correlation:

0CDC � (15)D b(1 � B)

where denotes the drag coefficient for a hard0CDsphere, and b is a constant which has a value of 0.2for Renksizbulut’s model and 0.32 for Chiang’s cor-relation. A transfer number B is adopted to accountfor the effect of blowing on momentum transfer tothe droplet. For droplet vaporization at low to mod-erate pressures (pr � 0.5), the Spalding transfernumber is widely used to characterize the vapori-zation rate:

C (T � T )p � sB � (16)

Dhv

The enthalpy of vaporization Dhv becomes zero atthe critical point, rendering an infinite value for thetransfer number. This deficiency may be remediedby introducing a transfer number suited for super-critical droplet vaporization [20,45].

T � T� cB � (17)D � �T � Tc l

where Tl is the instantaneous average temperatureof droplet, and Tc the critical mixing temperature.Since BD diverges at Tc � Tl at the end of dropletlifetime, the calculation of drag force was terminatedwhen Tc � Tl becomes less than 1 K, at which the

droplet residual mass is usually less than one thou-sandth of the initial mass. The influence on the ac-curacy of data reduction is quite limited. Followingthe procedure leading to equation 15, a correlationfor LOX droplet drag coefficient is obtained:

0CDC � (18)D b(1 � aB )D

where a and b are selected to be 0.05 and 1.592( )�0.7, respectively. The data clusters along thepr,O2classical drag curve (equation 15) in the low Rey-nolds-number region, but deviates considerably athigh Reynolds numbers (i.e., Re � 10). Although ashape factor may be employed to account for thisphenomenon arising from the increased form dragdue to droplet deformation, the difficulty of calcu-lating this factor and conducting the associated dataanalysis precludes its use in correlating the drag co-efficient herein. Instead, a simple correction factorRe0.3 is incorporated into equation 18 to provide thecompensation. The final result is given below:

0 0.3C ReDC � 0 � Re � 300 (19)D �0.71.592(P )r,O2(1 � aB )D

Droplet Response to Ambient Flow Oscillation

Although unsteady droplet vaporization and com-bustion have long been recognized as a crucialmechanism for driving combustion instabilities inliquid-fueled propulsion systems [55], they are ex-tremely difficult to measure experimentally. In par-ticular, the measurement of the effect of transverseoscillations on instantaneous evaporation and/orburning rates is quite formidable. The droplet vol-ume dilatation arising from rapid temperature in-crease may overshadow the surface regression as-sociated with vaporization and thereby obscure thedata analysis, especially in the early stage of thedroplet lifetime. Furthermore, conventional sus-pended-droplet experiments may not be feasible inthe presence of gravity due to reduced or diminishedsurface tension. In view of these difficulties, it is ad-vantageous to rely on theoretical analyses to studythe responses of droplet vaporization and combus-tion to ambient flow oscillations. The model is basedon the general approach described in Refs. [20] and[44], but with a periodic pressure oscillation super-imposed in the gas phase. Both cases involving hy-drocarbon droplets in nitrogen [20] and LOX drop-lets in hydrogen [22] are examined carefully. Thepurpose of these studies is to assess the effect of flowoscillation on vaporization process as a function offrequency and amplitude of the imposed oscillationas well as its type.

The most significant result is the enhanced dropletvaporization response with increasing ambient meanpressure. Among the various factors contributing to


this phenomenon, the effect of pressure on enthalpyof vaporization plays a decisive role. At high pres-sures, the enthalpy of vaporization decreases sub-stantially and becomes sensitive to variations of am-bient pressure and temperature. Any smallfluctuations in the surrounding gases may consider-ably modify the interfacial thermodynamics and con-sequently enhance the droplet vaporization re-sponse. This phenomenon is most profound whenthe droplet surface reaches its critical condition atwhich a rapid amplification of vaporization responsefunction is observed. The enthalpy of vaporizationand related thermophysical properties exhibit ab-normal variations in the vicinity of the critical point,thereby causing a sudden increase in the vaporiza-tion response. On the other hand, the effect of meanpressure on the phase angle of response functionappears quite limited. The phase angle decreasesfrom zero at the low-frequency limit to �180� athigh frequency, a phenomenon which can be easilyexplained by comparing various time scales associ-ated with fluid transport and ambient disturbance.

Supercritical Spray Field Dynamics

Modeling high-pressure, multiphase combustiondynamics in a fully coupled manner poses a varietyof challenges. In addition to the classical closureproblems associated with turbulent reacting flows,the situation is compounded with increasing pres-sure due to the introduction of thermodynamic non-idealities and transport anomalies. A comprehensivediscussion of this subject is given by Oefelein andYang [56–58].

Experimental efforts to characterize propellant in-jection, mixing, and combustion processes at near-and super-critical conditions have only recently ledto a better qualitative understanding of the mecha-nisms involved [59–65]. The current database, how-ever, is not adequate with respect to quantitative as-sessments, and theoretical efforts are similarlydeficient because of a lack of validated theories, dif-ficulties associated with numerical robustness, andlimited computational capacities. Depending on theinjector type, fluid properties, and flow characteris-tics, two limiting extremes may be deduced [58]. Atsubcritical chamber pressures, injected liquid jetsundergo the classical cascade of processes associatedwith atomization. For this situation, dynamic forcesand surface tension promote the formation of a het-erogeneous spray that evolves continuously over awide range of thermophysical regimes. As a conse-quence, spray flames form and are lifted away fromthe injector face in a manner consistent with thecombustion mechanisms exhibited by local dropletclusters. When chamber pressures approach or ex-ceed the critical pressure of a particular propellant,however, injected liquid jets undergo a “transcritical”

change of state as interfacial fluid temperatures riseabove the saturation or critical temperature of thelocal mixture. For this situation, diminished inter-molecular forces promote diffusion-dominated pro-cesses prior to atomization and respective jets va-porize, forming a continuous fluid in the presenceof exceedingly large gradients. Well-mixed diffusionflames evolve and are anchored by small but inten-sive recirculation zones generated by the shear layersimposed by adjacent propellant streams. Theseflames produce wakes that extend far downstream.

Figure 11, excerpted from Ref. 58, illustrates thebasic phenomena just described for the case of aLOX/gaseous hydrogen shear-coaxial injector ele-ment. The optical diagnostic studies conducted byMayer et al. [59,60] and Candel et al. [63,64] dem-onstrated the dramatic effect of pressure on mixingand combustion processes within this type of injec-tor. When the liquid oxygen is injected at low-sub-critical pressures, jet atomization occurs, forming adistinct spray similar to that depicted in Fig. 11a.Ligaments are detached from the jet surface, form-ing spherical droplets, which subsequently break upand vaporize. As the chamber pressure approachesthe thermodynamic critical pressure of oxygen, thenumber of droplets present diminishes, and the sit-uation depicted in Fig. 11b dominates. For this sit-uation, thread-like structures evolve from the liquidcore and diffuse rapidly within the shear layer in-duced by the co-flowing jets. At a downstream dis-tance of approximately 50 diameters, the dense fluidcore breaks into “lumps,” which are of the same or-der of magnitude as the diameter of the LOX jet.Results of optical measurements [59–64] reveal thatflame attachment occurs instantaneously after igni-tion in the small but intensive recirculation zone thatforms just downstream of the annular post. A well-mixed diffusion flame forms within this region, pro-ducing a wake that separates the oxygen stream fromthe hydrogen-rich outer flow. This wake persists atleast 15 jet diameters downstream. Fig. 12 shows theresultant flame structure and corresponding flow-field. Here, the injected jet exhibits a pure diffusionmechanism at a pressure of 4.5 MPa, which isslightly below the thermodynamic critical pressureof oxygen, but significantly above that of hydrogen.Hutt and Cramer [66] have reported similar findingsusing a swirl-coaxial configuration.

The trends outlined in the preceding text coincidewith the results of high-pressure liquid nitrogen/he-lium cold-flow tests reported in Refs. [60] and [65].From a qualitative standpoint, these experimentshave demonstrated the effect of mixture propertieson injected liquid jets and the prevalence of onelimit over the other, as illustrated in Fig. 11. Thecutoff associated with these two limits, however, isnot distinct and does not necessarily coincide withthe critical point properties of either propellant. The


Fig. 11. Schematic diagram illustrating the basic phenomena associated with (top) low and (bottom) high chamberpressures for the case of a LOX/gaseous hydrogen shear-coaxial injector element [58].

dynamic forces associated with a swirl-coaxial injec-tor, for example, would most likely dominate evenunder supercritical chamber conditions. For this sit-uation, a combination of dense transcritical spray dy-namics and diffusion-dominated processes wouldneed to be considered. A shear-coaxial element,however, would be more prone to exhibit a pure dif-fusion mechanism. Transient conditions would alterthe dynamics associated with either extreme. Thus,depending on the situation, a combination of twofundamental theoretical approaches is warranted[58]. The first is the classical Lagrangian–Eulerianimplementation in which a particulate phase is as-sumed to exist in the flowfield. The second employsa pure Eulerian implementation to treat a dense

multicomponent fluid. Both approaches require de-tailed representations of mixture properties and mi-croscale interactions over the full range of thermo-physical regimes encountered in contemporarysystems.

The complexity of the system outlined earlier isnumerically demanding, and several tradeoffs are re-quired with respect to the hierarchy of processes andcharacteristic scales resolved. A variety of uncertain-ties exist with regard to the closure problem, and inmany cases computational capacity is in direct con-flict with the accuracy of the SGS models employed.To address these difficulties in a manner consistentwith current experimental efforts, Oefelein [56–58]focused on detailed representations of fluid dynamic


Fig. 12. Near injector region of a LOX/gaseous hydro-gen shear-coaxial injector. (a) Flame. (b) Correspondingflowfield. Oxygen and hydrogen velocities are 30 and 300m/s, respectively. Oxygen and hydrogen injection tempera-tures are 100 and 300 K. Oxygen jet diameter is 1 mm, andthe chamber pressure is 4.5 MPa [59].

and physicochemical processes within planar mixinglayers. The hydrogen/oxygen system is employed inall cases. The approach follows four fundamentalsteps: (1) development of a general theoreticalframework, (2) specification of detailed propertyevaluation schemes and consistent closure method-ologies, (3) implementation of an efficient and time-accurate numerical framework, and (4) simulationand analysis of a systematic series of case studies thatfocus on model performance and accuracy require-ments.

The theoretical framework developed by Oefeleinaccommodates dense multicomponent fluids in theEulerian frame and dilute particle dynamics in theLagrangian frame over a wide range of scales in afully coupled manner. This system is obtained by fil-tering the small-scale dynamics from the resolvedscales over a defined set of spatial and temporal in-tervals. Beginning with the instantaneous system, fil-tering is performed in two stages. First, the influ-ences of SGS turbulence and chemical interactionsare taken into account, yielding the well-known clo-sure problem for turbulent reacting gases. The sec-ond filtering operation incorporates SGS particle in-teractions. The resultant system of Favre-filteredconservation equations of mass, momentum, total

energy, and species concentration for a compressi-ble, chemically reacting fluid composed of N speciesare derived.

Modeling SGS phenomena poses stringent nu-merical demands, and robust models are currentlybeyond the state of the art. Because of the uncer-tainties associated with the current models and theintensive numerical demands that these modelsplace on computational resources, the current workneglects the effects of SGS scalar-mixing processesand focuses on detailed treatments of thermody-namic nonidealities and transport anomalies. To ac-count for these effects over a wide range of pressureand temperature, the extended corresponding statesprinciple outlined in the section on thermodynamicand transport properties is employed using two dif-ferent equations of state. The 32-term BWR equa-tion of state is used to predict the fluid pressure–volume–temperature (PVT) behavior in the vicinityof the critical point. The SRK equation of state isused elsewhere. Having established an analyticalrepresentation for the real mixture PVT behavior,explicit expressions for the enthalpy, Gibbs energy,and constant pressure specific heat are obtained asa function of temperature and pressure using Max-well’s relations to derive thermodynamic departurefunctions. Viscosity, thermal conductivity, and effec-tive mass diffusion coefficients are obtained in asimilar manner.

Turbulence quantities are modeled using thelarge-eddy simulation (LES) technique and the SGSmodels proposed in Ref. [67]. Spray dynamics aretreated by solving a set of Lagrangian equations ofmotion and transport for the life histories of a sta-tistically significant sample of particles. Here, thestochastic separated flow (SSF) methodology devel-oped by Faeth [68] is employed, with transcriticaldroplet dynamics modeled using a set of correlationssummarized previously.

The analyses developed were applied to studytranscritical spray dynamics in a manner consistentwith the phenomena depicted in Fig. 11a. Effort wasfirst devoted to assessing the effect of pressure onthermophysical properties such as kinematic viscos-ity. This quantity is particularly significant and has adirect impact on the characteristic scales associatedwith the turbulence field. For both oxygen and hy-drogen, an increase in pressure from 1 to 100 atmresults in a corresponding reduction in the kinematicviscosity of up to three orders of magnitude. Thisimplies a three order of magnitude increase in thecharacteristic Reynolds number. Based on Kolmo-gorov’s universal equilibrium theory, the order ofmagnitude of the Kolmogorov microscale, denotedhere as mt, and the Taylor microscale, kt, are relatedto the Reynolds number by equations of the form

g kt t�3/4 �1/2� Re , � Re (20)t tl lt t

Here, the Reynolds number is defined as Ret �


qtlt/m, where . The term qt representsq � 2k /3t tthe turbulence intensity, kt the SGS kinetic energy,and lt the integral length scale. The relations givenby equation 20 indicate that a three order of mag-nitude decrease in the kinematic viscosity results in2.25 and 1.5 order of magnitude decreases in theKolmogorov and Taylor microscales, respectively.These reductions have a direct impact on the overallgrid density required to resolve key processes. Thetrend associated with the effective mass diffusivity isalso of concern. When the pressure is increased from1 to 100 atm, oxygen and hydrogen exhibit a twoorder of magnitude decrease in the mass diffusionrate over the full range of temperatures considered.Oxygen exhibits a decrease of up to four orders ofmagnitude at temperatures below the critical mixingpoint. The diminished mass diffusion rates coupledwith the liquid-like densities at high pressures sig-nificantly alter the coupling dynamics associatedwith local fluid dynamic and physicochemical inter-actions.

A sequence of representative calculations wereconducted which highlight dominate processes as-sociated with high-pressure spray field dynamics inturbulent mixing layers [56–58]. In contrast to thelow-pressure case, high-pressure mixing character-istics are much more sensitive to SGS fluctuationsand the large-scale coherency associated with the gasphase. Gas-phase dynamics are also affected by thespray as a result of particle damping, transport be-tween phases, and induced variations in mixtureproperties. Such changes alter the evolutionarystructure of the flowfield. Momentum exchange be-tween particles and the gas phase tends to damp thevelocity field and reduce the level of turbulent dif-fusion. Elevated pressures, on the other hand, cansignificantly increase turbulent diffusion processes.Over the ranges considered, the change in densitythat accompanies increasing pressure causes the mo-lecular kinematic viscosity to decrease significantly.This produces an overall increase in the local Rey-nolds number, and the normalized turbulent viscos-ity ratio. These effects amplify the dispersion attrib-utes associated with the spray and markedly altermixing and combustion processes at high pressures.

Supercritical Mixing and CombustionProcesses

Extensive effort has been made by Oefelein andYang [58] to study supercritical mixing and combus-tion of cryogenic propellants, motivated by the flowvisualization experiments conducted by Mayer andTamura [59–62] (see Fig. 12). Much insight was alsoachieved by Candel and his co-workers using laser-based optical diagnostic techniques [63,64]. Empha-sis was placed on the near-field physiochemical pro-cesses that occur in the vicinity of the splitter plate,

the impact of pressure on these processes, and thedownstream flow dynamics induced as a conse-quence. The computational domain consists of co-flowing hydrogen and oxygen streams separated bya 0.5 mm splitter plate, as shown in Fig. 13. A heatconduction model was applied to the splitter plateto provide a realistic energy flux distribution at thewalls. Conditions were specified using a referencepressure of 100 atm, with the injection temperaturesselected to produce an optimal matrix of transcriticaland supercritical conditions.

Two sets of inlet flow conditions were examinedat p � 100 atm, as detailed in Ref. [58]. The firstset provides thermodynamic conditions similar tothose employed by Mayer and Tamura [59]. Hydro-gen and oxygen injection temperatures of T1 � 150K and T2 � 100 K are specified to produce a su-percritical hydrogen stream and a liquid oxygenstream that undergoes a transcritical change of statewithin the mixing layer. The corresponding oxygento hydrogen density ratio is q2/q1 � 72.7. The sec-ond set of conditions provides supercritical stateswithin both streams. For these cases hydrogen andoxygen injection temperatures of T1 � T2 � 300 Kare specified to produce an oxygen to hydrogen den-sity ratio of q2/q1 � 17.4.

Representative results are given in Figs. 14 and15. Fig. 14 shows the instantaneous density and tem-perature fields for U1 � 125 m/s and U2 � 30 m/s. The corresponding plots for the supercritical caseare presented in Fig. 15. The solutions shown pro-vide a comparison between transcritical and super-critical mixing in the near field when respective hy-drogen and oxygen streams are injected at identicalvelocities.

Transcritical mixing induces a vortical structurewithin the injected hydrogen stream that is analo-gous to that produced by a backward-facing step.This structure emanates from the upper corner ofthe splitter plate and coalesces downstream with ad-jacent vortices. The oxygen stream, on the otherhand, exhibits no noticeable structure and proceedsunimpeded in an essentially straight line. Because ofthe liquid-like characteristics of the oxygen stream,an extremely large-density gradient exists within thisregion. Diminished mass diffusion rates are also evi-dent. The combined effect produces a flame thatbehaves in a manner qualitatively similar to that de-picted in Fig. 11. Combustion occurs at near stoi-chiometric conditions and produces a wake that ef-fectively separates the hydrogen and oxygen streamsas the flow evolves downstream.

A direct comparison between Figs. 14 and 15highlights the wide disparities that exist betweentranscritical and supercritical fluid dynamic andphysicochemical interactions. In contrast to the tran-scritical situation, relatively strong vortical interac-tions are prevalent between the oxygen and hydro-gen streams at pure supercritical conditions. The


Fig. 13. Physical model employed for the analysis of high-pressure hydrogen/oxygen mixing and combustion processes.

Fig. 14. Contours of density and temperature in the near-field region for transcritical mixing [58].

resultant interactions cause disturbances that growand coalesce immediately downstream of the splitterplate on a scale that is of the same order of magni-tude as the splitter plate thickness. Combined effectsinduce increased unsteadiness with respect to theflame-holding mechanism and produce significantoscillations in the production rates of H2O and OHradicals. The temperature within the recirculationzone also fluctuates about the stoichiometric value,with relatively cooler temperatures observed im-mediately downstream.

Conclusions

Fundamentals of high-pressure transport andcombustion processes in contemporary liquid-fueled

propulsion and power-generation systems were dis-cussed. Emphasis was placed on the development ofa systematic approach to enhance basic understand-ing of the underlying physiochemical mechanisms.Results from representative studies of droplet va-porization, spray-field dynamics, and mixing andcombustion processes were presented to lend insightinto the intricate nature of the various phenomenaobserved. In addition to all of the classical issues formultiphase chemically reacting flows, a unique setof challenges arises at high pressures from the intro-duction of thermodynamic nonidealities and trans-port anomalies near the critical point. The situationbecomes even more complex with increasing pres-sure because of an inherent increase in the flow Rey-nolds number. The resulting flow dynamics andtransport processes exhibit characteristics distinct


Fig. 15. Contours of density and temperature in the near-field region for supercritical mixing [58].

from those observed at low pressures. In spite of theencouraging, but limited, results obtained to date,the current database is not adequate with respect toquantitative assessments due to the difficulties inconducting experimental diagnostics. The theoreti-cal efforts are similarly deficient because of a lack ofvalidated theories and limited computational re-sources. Much effort needs to be expended to over-come these obstacles.

Acknowledgments

The author owes a large debt of gratitude to his col-leagues and former students, especially Joe Oefelein,George Hsiao, Patric Lafon, Norman Lin, and J. S. Shuen.The paper would not have been completed without theirhard work, support, and intellectual stimulation. The workwas sponsored partly by the Pennsylvania State University,partly by the U.S. Air Force Office of Scientific Research,and partly by the NASA Marshall Space Flight Center.

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COMMENTS

Josette Bellan, Jet Propulsion Laboratory, USA. My firstquestion regards your terminology of supercritical vapor-

ization. Since, under supercritical conditions, the latentheat is null, there cannot be vaporization. Please explain.My second question regards the calculation of transportproperties such as the diffusivity and the thermal diffusionfactors for multicomponent mixtures at high pressures. Areyou aware of established methodologies for the reliable cal-culation of these transport properties for multicomponentmixtures at high pressures? We should note that a singlespecies fuel burning in air constitutes already at least a five-component mixture under the conservative assumptionthat combustion is complete and thus its products are waterand carbon dioxide.

Author’s Reply. For simplicity, supercritical vaporizationin the present paper refers to vaporization of liquid drop-lets in supercritical environments. Calculations of ther-modynamic and transport properties for multicomponentmixtures at high pressures are discussed in great detail in

the paper. In general, thermodynamic and transport prop-erties except mass diffusivity can be treated reasonably wellby means of the corresponding-state principles. Estimationof mass diffusivity remains a challenging task due to thelack of a formal theory or even a theoretically based cor-relation. Only moderate success was achieved by Takahashi(Ref. [43] in this paper) for the limited number of casesstudied.

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Yaakov Timnat, Techion, Israel. Could you comment onthe type of equation of state you recommend to use forobtaining the best results?

Author’s Reply. As discussed in the paper, the Benedict-Webb-Rubin equation of state in conjunction with the cor-responding-state principles provides the best result of fluidp-V-T properties over all the fluid thermodynamic states,from compressed liquid to dilute gas.

Documents

Modeling of Supercritical Vaporization, Mixing, and Combustion