Numerical studies on supersonic spray combustion in high-temperature shear flows in a scramjet combustor

Zhoxin REN,Bing WANG,Longxi ZHENG,Dn ZHAO

aSchool of Power and Energy,Northwestern Polytechnical University,Xi’an 710072,China

bSchool of Aerospace Engineering,Tsinghua University,Beijing 100084,China

cDepartment of Mechanical Engineering,College of Engineering,University of Canterbury,Private Bag 4800,Christchurch 8140,New Zealand

KEYWORDS Combustion;Flame kernel;Mixing layer;Numerical simulation;Shearing vortex;Spray flame;Supersonic flow

Abstract Numerical simulation is applied to detail the combustion characteristics of n-decane sprays in highly compressible vortices formed in a supersonic mixing layer.The multi-phase reacting flow is modeled,in which the shear flow is solved Eulerianly by means of direct numerical simulation,and the motions of individual sub-grid point-mass fuel droplets are tracked Lagrangianly.Spray combustion behaviors are studied under different ambient pressures.Results indicate that ignition kernels are formed at high-strain vortex braids,where the scalar dissipation rates are high.The flame kernels are then strongly strained,associated with the rotation of the shearing vortex,and propagate to envelop the local vortex.It is observed that the flammable mixtures entrained in the vortex are burned from the edge to the core of the vortex until the reactants are completely consumed.As the ambient pressure increases,the high-temperature region expands so that the behaviors of spray flames are strongly changed.An overall analysis of the combustion field indicates that the time-averaged temperature increases,and the fluctuating pressure decreases,resulting in a more stable spray combustion under higher pressures,primarily due to the acceleration of the chemical reaction.

1.Introduction

A comprehensive understanding of the physics of spray flames associated with the vortex dynamics in highly compressible flows is desired for the optimal design and operation of scramjet combustors.The high airstream velocity and hence the flow compressibility inside a combustor reduce the residence time available for evaporation of atomized droplets,fuel-oxidizer mixing,and combustion,and also preclude flame propagation.A stable combustion must depend on auto-ignition due to high-speed flow velocities,and ignition and flame propagation are usually influenced by the unsteady motions of turbulent eddies.

Most available literature concentrated on low-speed/subsonic combustion of fuel sprays1–13.Miller1investigated the influence of reacting/non-reacting fuel droplets/particles on combustion characteristics in a compressible shear flow via numerical simulations.Reveillon and Vervisch3numerically studied the flame structures of spray combustion and considered the effects of the equivalence ratio and dilution.Supersonic combustion has been investigated through experiments subjected to some certain conditions,but there is a lack of flow- field data due to the limitation of experimental measurement technology4–8.Stable combustion in supersonic flows pertains to thermal auto-ignition of flammable mixtures and flame anchoring by cavities or struts4.The influence of the combustor pressure on supersonic combustion has not been well studied,which can accelerate the chemical reaction and increase the heat release intensity.In addition,such effects should be associated with the vortex dynamics during spray combustion,since fuel droplets will be dispersed by the motions of vortices in the flow field.

The droplet size and equivalence ratio have influences on spray combustion characteristics.Previous research has shown that droplets change flame structures,in which a diffusion-like combustion occurs for large droplets due to a slow evaporation process,while a premixed-like combustion is evident9for small droplets.This is mainly because the ignition process of spay fuels is found to be mixing-controlled for small-sized droplets,but evaporation-controlled for large-sized droplets13.Later studies have found that the gaseous temperature is suppressed by the increase of the droplet mass flow rate due to the thermal cooling effect from droplet evaporation9.However,those studies were initiated from low-speed flows,and spray combustion in high-speed flows will be influenced by the high-speed flow compressibility.The compressibility will depress the large vortex spatiotemporal evolution.Large vortices coexist with small vortices in a supersonic flow field.Although small eddies could also influence the mixing process as well as the chemical reaction,it should be noted that their effects on combustion features are less obvious than those of large eddies4.Therefore,the influences of the turbulent motions of large eddies on auto-ignition and flame propagation should be studied considering the high-compressibility effects.

The aim of the present investigation is to study spray flames associated with the motions of highly compressible shearing vortices via a two-dimensional Direction Numerical Simulation(DNS).The liquid fuel is chosen as n-decane(C10H22),which is a promising surrogate fuel for a scramjet engine study,because its physicochemical properties are similar to those of practical jet fuels.A single evaporating droplet is modeled as a sub-grid point mass and tracked individually by means of the Lagrangian trajectory method.Interphase interactions including mass,momentum,and energy between the dispersed fuel droplets and the local carrier gas are modeled as source terms on the Eulerian gasphase.

2.Numerical models

2.1.Governing equations for the gas-phase

In the present study,the motion of evaporating fuel droplets in supersonic flows14,15is considered,and the governing equations of the gaseous flow,including the conservations of mass,momentum,energy as well as species transportation,supplemented by the ideal gas equation of state with multiple species,are given as

where ρ is the density.uiand ujare the velocities in the streamwise and transverse directions,respectively.P is the static pressure of gas. δijis the Kronecker delta function. τijis the Newtonian viscous stress tensor.qjis the heat flux term.Vk,jis the diffusion velocity of species k and Vcjis the correction velocity for the global mass conservation.T is the static temperature.R is the universal gas constant.Ykand Wkare the molecule weight and the mass fraction of species k,respectively.NSis the total number of species.etis the total energy,i.e.,kinetic energy combined with internal(containing chemical)energy,and is expressed as

where cp,kis the specific heat at a constant pressure of species k,and h0f,kis the specific chemical formation enthalpy at the reference temperature,Tref.The specific heat capacity and enthalpy are computed from the fifth-order polynomials16.The transport properties of the gaseous mixture are obtained based on the kinetic theory17.The Lennard-Jones potential model is applied to compute the inter-molecular forces.The modified Eucken model is used to calculate the thermal conductivity of each species.The Chapman-Enskog theory is utilized for the dynamic viscosity as well as the binary diffusivity.The thermal conductivity λ and the dynamic viscosity μ of the gaseous mixture are obtained from the semi-empirical mixing rule proposed by Wake and Wassiljewa.

The source terms,i.e.,Sm,SF,and SQ,on the right-hand sides of the above governing equations of the gas-phase model are the interphase interaction of mass,momentum,and energy,respectively.Scombustion,kis the source term,resulting from the combustion reaction.Based on the assumption of the Point-Source In Cell(PSIC)model18,these source terms are computed by counting the total number of droplets,Nc,in each computation cell,denoted as

where ΔV is the grid size for the gas-phase.mdis the droplet mass,and˙mdrepresents dmd/dt.ud,iis the droplet velocity.Fdand Qdare the drag force and the convective heat transfer,respectively,which will be described later.hV,sfis the vapor enthalpy at the droplet surface.The source term,Scombustion,k,is calculated by using the chemical reaction rate per unit volume,RF,shown as

where nFand nkare the molar stoichiometric coefficients of the fuel and the kth species,respectively.WFis the molecular weight of the n-decane fuel.

2.2.Governing equations for the droplet-phase

The trajectory of an evaporation spherical fuel droplet in a gaseous flow is usually tracked using the Lagrangian model.Recently,the model of a small rigid sphere in an incompressible flow19has been extended to compressible flows20.Because the density of liquid fuel is much higher than the gas density and the droplet size is sufficiently small,the unsteady force can be neglected compared to the drag force,which acts on droplets21.Sprayed droplets are sparsely dispersed in the present research;hence,droplet-droplet interactions are neglected.The temperature distribution inside a droplet is assumed to be uniform.

Under the above assumptions,the governing equations of a droplet for the spatial position(in the ith direction),xd,i,velocity(in the ith direction),ud,i,temperature,Td,and mass,md,are given as follows:

where uiand T are the velocity and temperature of the gasphase seen by the droplet at its position,respectively.cpis the specific heat of the mixture gas surrounding the droplet,and cLis the specific heat of the liquid fuel.The droplet acceleration time,τa,is represented as

where ddis the droplet diameter.The Prandtl and Schmidt numbers of the gas-phase are expressed as Pr= μcp/λ and Sc= μ/ρD,respectively.Here D is the mass diffusion coefficient of gas mixture.Based on the Ranz-Marshall correlations,the Nusselt and Sherwood numbers are given by

Here,the droplet Reynolds number,Red,based on the slip velocity between the local gas and the droplet,is defined as Red= ρ|ui-ud,i|dd/μ.f1is a correction22to the Stokes drag and is suitable for a large range of droplet Reynolds numbers in supersonic flows,which is given as

f2is a correction of heat transfer for the effect of droplet evaporation and is given as f2= β/(eβ-1), where β =-1.5Prτa(dmd/dt)/md.A Langmuir-Knudsen model considering the non-equilibrium effects on evaporation23is utilized.The mass transfer number,BM,is calculated as BM=(Ysf-YV)/(1-Ysf),where YVis the mass fraction of fuel vapor under the far- field condition,Ysfis the mass fraction of fuel vapor at the droplet surface obtained from the surface molar fraction χsf,and LVis the latent heat of evaporation at the droplet temperature Td.

2.3.Numerical methods

The governing equations of the present multiphase reacting flows are solved via using the in-house numerical simulation codes14.An overview of the numerical procedures is shown below.A finite difference methodology is utilized for solving the partial differential equations of the gas-phase.An explicit third-order Runge-Kutta scheme is applied for the time integration.The Roe-averaged matrix for the flux projection is calculated24for the multi-species.For the scalar conservation,a high-order accurate Maximum-Principle (MP)satisfying scheme25is applied.A sixth-order symmetric compact difference scheme is used to solve diffusion fluxes,and the discretization of convection fluxes is implemented by using the adaptive Central-Upwind Sixth-order Weighted Essentially Non-Oscillatory(WENO-CU6)scheme26to achieve a good resolution of flow properties around the shock waves in supersonic flows and to consider the low dissipation for simulating turbulent fields.

In the forward coupling(gas-to-droplet coupling),the physical quantities(velocity,temperature,pressure,viscosity and so on)of the gas-phase at the droplet center are calculated by using a fourth-order Lagrangian polynomial interpolation method.A third-order Adams-Bashforth scheme is used for the time integration of the ordinary differential equation of the droplet-phase.In the backward coupling(droplet-to-gas coupling),the PSIC model is used.

The numerical simulation codes have been previously utilized for a variety of compressible flows and combustion problems,and a series of validation studies has also been conducted14.The codes have been validated for non-reacting supersonic shear flow and single-droplet evaporation,and computational results are in good agreement with available experimental measurements14.

3.Numerical set-ups

3.1.Computational domain

A spatially-developing two-dimensional shear layer is studied in the present work,as given in Fig.1.The mixing layer is formed between a high-speed air stream with a velocity,UA1,and a low-speed air stream with a velocity,UA2.In Fig.1,letters x and y represent the streamwise and transverse directions,respectively.δ(x)is the shear layer thickness at streamwise position x,and δ0is the initial shear layer thickness.Liquid n-decane droplets are injected into the hot-air stream from the transverse centerline,with a width of 10δ0.Black dots refer to dispersed fuel droplets,and grey dots indicate droplets that are evaporating.Dash-dot lines represent large-scale vortices formed in the mixing layer,and dashed lines scale the edges of the mixing layer.

The fine-grid rectangle computational domain for the present investigation extends from-Ly/2 to Ly/2 in the transverse direction and from 0 to Lxin the streamwise direction.By setting Lx=4Ly,the transverse boundaries have slight effects on the main flow in the center of the mixing layer,as well as the dynamics of the flame kernel formed in the reactive multiphase flows.The computation domain is discretized by inhomogeneous Cartesian grids,and a fine resolution is set around the center of the mixing layer.There are 512 cells and 256 cells in the streamwise and transverse directions respectively after a grid independence test.

For the in flow boundary,the distribution of the streamwise velocity is given as a hyperbolic tangent profile by using a supersonic inlet boundary condition.To excite the flow instability in the highly compressible shear layers,random disturbances are added to the transverse velocity of in flow based on the most unstable frequency of the main flow field.Nonreflective boundaries are imposed for the upper and lower boundaries since the pressure waves formed in the combustion are undesirable to reflect from the boundaries to influence the main flows.A supersonic out flow condition is utilized to set the out flow boundary.

Fig.1 Schematic of fuel spray in two-dimensional shear layer.

3.2.In flow conditions

In this study,air is injected from the inlet boundary.We consider the in flow air as a mixture of nitrogen(N2)and oxygen(O2)with (in terms of mass fraction)YN2=0.77and YO2=0.23.The temperature of the in flow air,T0,is 1500 K.The streamwise velocity of the low-speed stream,UA2,is 1242 m/s,and the velocity ratio,UA1/UA2=2.The convective Mach number Mcfor the in flow,based on the streamwise velocity difference,ΔUA,as well as the sound velocities of the two streams,is 0.8.Spray spherical droplets injected from the inlet have the same diameter,and the injecting velocity of each droplet equals to the velocity of the local air around the droplet.The initial temperature of cold droplets is 298.15 K.

Simulation cases are summarized in Table 1,which lists the initial gaseous pressure,P0,the in flow Reynolds number,Re0,the initial droplet Stokes number,St0,and the spray equivalence ratio,Φ0.Here the subscript A means values of the airflow.andare mass flow rates of spray fuel and air,respectively,and( F/O)stis the stoichiometric fuel to air ratio.The following conventions are applied for naming the present simulation cases:Case LP=low initial pressure,Case MP=medium initial pressure,and Case HP=high initial pressure.Initial pressures increasing from 0.1 to 0.5 MPa are utilized to study the effects of elevated pressures on ignition and combustion.The spray equivalence ratio is 1.2 for the three simulations.Note that the localized reactive mixture could be either lean or rich due to the unsteady dispersion of fuel droplets as the discrete fuel evaporating sources.Moreover,an unladen gaseous flow case with a low initial pressure,i.e.,Case Unladen,is performed as a comparison.

A one-step irreversible reaction of n-decane with oxygen27is applied for the present study.In this chemical model,the oxidation reaction is given as

where niis the molar stoichiometric coefficient of each chemical species.The global combustion reaction constant,k(mol/(cm3·s)),is calculated as follows:

where A is the frequency factor,and E is the activation energy.Their values as well as the descriptions of n,a,and b can be found in Ref.27.The oxidation reaction rate per unit volume(g/(cm3·s))is given by RF=WVk × 103.

The one-step reaction mode has been suitably used to investigate the auto-ignition of a turbulent mixture28anddetonation dynamics29in previous research.Moreover,experiments of a shock tube30–32have indicated that the chemical reaction of n-decane from low to intermediate initial temperatures(＜1000 K)experiences a region of a Negative Temperature Coefficient(NTC),in which the ignition delay increases with an increase of the initial temperature,as shown in Fig.2(a).Here Φ is the equivalence ratio and 1 atm=1.013×105Pa.When the initial temperature increases to be higher than 1000 K,the ignition delay is found to decrease linearly with an increase of the initial temperature.The single-step combustion model could exhibit this dependence of ignition delay times on the initial temperature.For the ignition of n-decane at normal pressures(0.1 MPa)30,several experiments have shown that there seems no obvious NTC region during the ignition,and the ignition delay time has a linear inverse dependence on the initial temperature,as depicted in Fig.2(b).In general,the chemical reaction during the autoignition in a high-temperature environment has a much stronger temperature dependence than that of reactant concentration.Therefore,the present study is limited to the simplified chemistry model,but aims to illustrate the ignition behaviors associated with motions of shearing vortices,yielding to the thermal auto-ignition and combustion converged to a high temperature of air in flows.Referred to the practical operation of a scramjet combustor,the air temperature compressed by intakes of an aircraft ranges from 1200 to 1600 K at the combustor inlet.In the present study,the in flow temperature of air streams is hence specified as 1500 K.

Table 1 Summary of simulation cases.

Fig.2 Redrawn of n-decane ignition delay times in Ref.30.

4.Results and discussions

4.1.Overall characteristics

In this section,simulation results for the whole combustion field are outlined for a later analysis of spray flame dynamics;the focus is on overall quantities that indicate compressibility,reacting flow field,and time-averaged behavior.

Fig.3 indicates the overall features of spray combustion fields,which present the spatial distributions of instantaneous fuel droplets and the dimensionless gaseous temperature and mass fraction of fuel vapor for Case LP,Case MP,and Case HP at the same instant.Here,the grey dots indicate fuel droplets.The dispersion of fuel droplets in the upstream flow field expands transversely according to the in flow disturbance.It can be found that the dispersing droplets disappear in the downstream region prior to the roll-up of large eddies caused by droplet evaporation.Therefore,the distribution of fuel vapors is associated with the motions of eddies.The regime of the fuel vapor distribution corresponds to the pattern of large eddies,and the fuel mass accumulates in the cores of those shearing vortices.For Case LP with a normal ambient pressure,a flame firstly occurs in the high-strain vortex-braid regions,and the fuel-oxygen mixtures entrained in the large eddies are not ignited within the present computational domain due to the low gaseous temperature and pressure in the vortex cores,affected by the high compressibility.Here,it is regarded as successful ignition if continuous stable spatial combustion occurs.An increase of the ambient pressure accelerates the chemical reaction rates,and ignition occurs upstream obviously for Case MP and Case HP.For Case HP,the fuel vapors in the convective vortices are completely burned due to high oxidization rates,resulting in a stable flame formed in the shearing large eddies.

Fig.3 Instantaneous distributions of dimensionless temperature T/T0and fuel mass fraction,with blue dashed lines given by YF=0.05.

Fig.4 gives the instantaneous scatter plots of the dimensionless gaseous temperature against mixture fractions.Here,a description of the combustion structure is obtained by drawing all the data in the computation domain.Here,the definition of the mixture fraction,Z,is based on the mass fractions of the gas-phase as

where the subscript i represents the inlet values,and s is the stoichiometric mass of oxidizer per unit mass of fuel.For the n-decane fuel in the present study,the values are YF,i=1.0,YO,i=0.23,and s=3.49,and the value for the stoichiometric mixture fraction,Zst=0.0618.Moreover,the dashed lines refer to the theoretic adiabatic flame temperature against the mixture fraction.

It is found that the gaseous temperature for Case HP is generally higher than those for Case MP and Case LP at a fixed Z,especially for rich mixtures.For Case LP,the peak values of the gaseous temperature lie in the fuel-lean mixtures;it can be found in Fig.3(a)that only the fuel-oxidizer mixtures in the high-strain vortex-braid regions are burned in a narrow flame strained between the vortices.Fig.4(a)also shows that the temperature of fuel-rich mixtures is lower than the ambient gaseous temperature,because plenty of the fuel mass entrained in the large eddies cools the local gas,and the heat release from the chemical reaction cannot counterbalance the heat loss due to droplet evaporation.Comparisons between the scatter plots display that an increment of the ambient pressure shifts the peaks of the flame temperature from fuel-lean to stoichiometric conditions,since a pressure increase results in an acceleration of the exothermic reaction rate.

Followed by the above demonstrations of the instantaneous characteristics of the overall reacting flow field,the below discussion will display time-averaged features.Comparisons of the streamwise distributions of the time-averaged gaseous temperature,as well as standard deviations of temperature,σT,at y=0.5Ly,are shown in Fig.5.σTdenotes the intensity of temperature perturbation.These first-and second-order statistics are normalized by the initial gaseous temperature,T0.Compared with the unladen flow,the streamwise variations ofˉT for the droplet-laden cases display a decrease upstream due to the cooling effects from the evaporation of fuel spray droplets.After ignition occurs,the heat release from the exothermic reaction increases the gaseous temperature.It is observed that the values offor Case HP are higher than those of the other cases with lower ambient pressures.However,the streamwise variations of σTshow that the temperature fluctuations for Case HP are found to be much higher than those of the other cases;the discrepancy becomes marked downstream after the ignition occurs.During the droplet evaporation process,before an ignition kernel is established,the high values of σTfor Case HP are mainly attributed to the high turbulence intensities associated with high Reynolds numbers.After the exothermic chemical reaction starts to consume the fuel vapors,the increasing temperature from the chemical heat release enlarges the temperature fluctuations,resulting in an increase of σT.Moreover,the elevated pressures accelerate the chemical reactions and enhance the combustion,which lead to a high temperature and large temperature fluctuations.

Fig.4 Scatter plots of instantaneous dimensionless temperature T/T0vs mixture fraction Z.

Fig.5 Spatial variations of values along streamwise direction at y=0.5Ly.

The mixing layer thickness provides a length scale for the development of the mixing region and shows self-similarity characteristics.The streamwise evolution of the mixing layer thickness,δ,for the cases considered is shown in Fig.6.For Case Unladen,after a transient of the roll-up of eddies,a quasi-linear growth regime of δ is found,and the mixing layer becomes self-similar.From a comparison between Case LP and Case Unladen,it is observed that the modulation from evaporating fuel droplets accelerates the development of the mixing layer,and the droplet-laden flows achieve a linear growth of δ at an upper streamwise location.With an increment of the ambient pressure,the growth rate of the mixing layer thickness,dδ/dx,tends to be high.This is because turbulence becomes strong as the density(i.e.,the Reynolds number)increases.The streamwise location of ignition is marked in Fig.6.It is found that downstream combustion decreases the growth rate of δ,and the reduction of the growth rate becomes obvious with increasing pressures associated with enhanced combustion.The thickness of the mixing layer,δ,grows more slowly for the reacting cases than that for the inert case,which agrees with earlier research results of compressible flows with small convective Mach numbers33.The differences come from the effects of the decrease in Reynolds stresses due to the heat release from the exothermic reaction in reacting flows.

4.2.Instantaneous features

In this section,the ignition and dynamics of a spray flame with elevated pressures are investigated by analyzing the time-series data of the combustion field and by tracking the evolution of flame kernels with the turbulent motions of large eddies.

For evaluating the effects of elevated pressures on the instantaneous features of ignition as well as combustion,we record the maximum values of the fuel mass fraction and chemical reaction rate in the flow- field during one flow circle,since droplets are injected into the carrier flow,as depicted in Fig.7.Here,the dimensionless time is t*=t/tflow,in which tflowis the characteristic time of a flow circle.Cold fuel droplets enter the hot gaseous flow and evaporate,resulting in a fast increase of fuel vapors in the flow field.At the initial increase stage of YF,max,the effects of elevated pressures are not obvious.However,after time t*=0.1,an increase of the ambient pressure results in an increase of the growth rate of YF,max.Although the differences between the three cases are slight,the effects on the chemical reaction rates are found to be significant.As the fuel droplets evaporate to form vapors,a pressure rise obviously accelerates the chemical reactions.

The maximum gaseous temperature in the combustion field during five flow circles once fuel droplets are injected into the carrier flow are then recorded,as shown in Fig.8(a).It is found that the maximum gaseous temperature is enlarged by a factor of 1.1 compared with the initial gaseous temperature T0before the injection of fuel droplets.This is because that the high compressibility effect leads to a partial recovery of the kinetic energy of the free-stream,as discussed before.The distinction between the burning and non-burning conditions is arbitrarily specified at a temperature of Tig=T0+ΔT(=500 K).The initial droplet mass flow rate is the same for all three simulations,but the times of occurrence of flame kernels decrease as the ambient pressure increases.This is attributed to the fact that an increase of the ambient pressure accelerates the chemical reaction rate.For Case LP,judging from the time-series T*max,extinction occurs in the flow field,and a less stable combustion mode appears,compared with the high-pressure cases.Fig.8(b)shows the time evolutions of dimensionless fluctuating pressures for all three cases at an observed point(x/δ0=2000,y/δ0=0)in the combustion field.Pressure fluctuations are obtained based on the time-averaged pressure,at the observed point.It is found that pressure oscillations display synthesized-sinusoidal shapes for all simulations.Shearing vortices move coherently in the mixing layer.Low and high pressures appear at the centers of the vortexes and the strain regions between the vortices,respectively.Therefore,such a sinusoidal shape of pressure fluctuations caused by the periodic motions of coherent structures in the flow field could be observed at a fixed Eulerian point.Obviously,a high ambient pressure could lead to a large fluctuation amplitude,because the pressure undergoes strong oscillations during the turnover of vortices.Here,we normalize the pressure fluctuations to eliminate the effects of vortex dynamics;hence,we could focus on the influence of an elevated pressure on the combustion and further the pressure oscillations.It is observed that a disparity of pressure fluctuations among the three cases can be observed in the peak values of pressure,where the peaks are smoother with an increasing ambient pressure.As observed in Fig.3(a)for Case LP,a flame only occurs in the vortexbraid regions,which means that the combustion is spatially intermittent.Therefore,the unstable combustion results in relatively large oscillations of temperature and pressure.For Case HP, flame kernels are formed not only in the vortex-braids but also in the vortex-cores,and the more stable combustion leads to a reduction of the fluctuating pressures.

Fig.7 Maximum values in flow- field during one flow circle since droplets are injected.

Fig.8 Variations of temperature and pressure in flow field.

Fig.9 Scatter plots of dimensionless scalar dissipation rate vs chemical reaction rate at times t=0,t=0.4tL,and t=0.8tL.

Fig.9 depicts the scatter plots of the dimensionless scalar dissipation rate,χ*,versus the chemical reaction rate,RF,for a shearing vortex at instant times t=0,t=0.4tL,and t=0.8tL.tLis the time when the vortex completes an eddy turnover.The data is sampled at the computation grids in the domain for the vortex.The scalar dissipation rate,χ,is defined as

where DFis the mass diffusion coefficient of fuel in the gas mixture.χ*of the mixturefraction is normalized by χ*= χ/(|UA1-UA2|/δ0).For the three cases,it is found that incipient reaction spots originate in regions with high scalar dissipation rates at time t=0,corresponding to the high strain vortex-braid zones with high temperatures and pressures.Moreover,χ*for Case HP and Case MP are slower than that of Case LP.This is because the fast chemical reactions for Case HP and Case MP consume the local fuel-oxygen mixtures in a short time,resulting in a decrease of the local mixture gradient.Another reason is that the higher turbulence for Case HP and Case MP promotes scalar mixing.Subsequently,for the three cases,reaction spots form in the regions with a quite lower χ*at time t=0.4tL.The reason for this phenomenon is that the merging of two neighbor vortices contributes to the mixing of scalars,as well as the dissipation.For Case LP,the local flammable mixtures in the vortex-braids are burned continuously from t=0.4tLto t=0.8tLdue to the self acceleration of the combustion rate.Therefore,the local gradient of scalars is dissipated,and χ*is reduced.At time t=0.8tL,for Case HP,the reactants within the vortex are burned,and the chemical reaction rates decrease due to a lack of flammable mixtures.It is also found that the most reactive spots localize in regions with low dissipation rates.

The evolution of the spray flame in highly compressible shearing vortices can be summarized as three stages;the scenarios are depicted in Fig.10,in which the black circle means a rotating vortex entraining fuel-oxygen mixtures,and the arrow refers to the rotating direction.Flame fronts are sketched using red dashed lines.At first,an ignition kernel occurs at the braid structures due to the local high temperatures and pressures of the gas-phase,as shown by Fig.10(a),and braid flames are formed.Then,the flame propagates to envelop the vortex associated with the rotation process,and reactive pockets in the periphery of the vortex burn with enveloping flames(see Fig.10(b)).Subsequently,thermal diffusion from the periphery of the vortex heats the inner fuel–air mixtures,as shown by the blue arrows in Fig.10(c),and combustion finally occurs in the rich mixtures within the vortex core,resulting in a core flame.

5.Conclusions and remarks

The present paper numerically investigates the overall behavior and instantaneous characteristics of spray flames in highly compressible vortices formed in shearing flows laden with n-decane droplets.The gas-droplet reacting flow system is numerically captured by a hybrid Eulerian-Lagrangian method,in which the dispersion of evaporating dispersed droplets is tracked in the Lagrangian frame.The high-Mach number gaseous flow is simulated by DNS in the Eulerian frame.

The overall behavior of the flow field with high convective Mach numbers shows that the high-speed flow compressibility effect is obvious.High-and low-temperature regions are interlaced,formed in vortex braids and cores,respectively.For spray flames associated with shearing vortices,ignition kernels occur at high-strain vortex braids with high scalar dissipation rates.Then,the flame kernels are strained,associated with the rotation of the vortex,and propagate to envelop the vortex.The flammable mixtures entrained in the vortex are burned from the edge to the core of the vortex until the reactants within the shearing vortex are completely consumed.With an increment of the ambient pressure,the regime with a high temperature tends to be larger,the time-averaged temperature increases,and the fluctuating pressure decreases,resulting in a more stable combustion.

The influence of initial droplet sizes as well as spray equivalence ratios on supersonic combustion characteristics in mixing layers will be studied in details in the future for better understanding the physics of compressible spray flames.

Acknowledgements

The second author would like to thank the partial finical supports from the National Natural Science Foundation of China(No.51676111),the Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics(No.U1730104),and the Tsinghua University Initiative Scientific Research Program,China(No.2014Z05091).