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(c)l999 American Institute of Aeronautics & Astronautics

A994 6126

AIAA 9%0204 Numerical Simulation of Heterogeneous Flow wlith Homogeneous Reaction Farzad l’dlashayek Universitif of tiavvaii

37ith AIAA Aerospace Sciences IMeeting and Exhibit

January ii-14,1999 / Reno, NV

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AIAA 99-0204

NUMERICAL SIMULATION OF HETEROGENEOUS FLOW WITH HOMOGENEOUS REACTION

Farzad Mashayek* University of Hawaii at Manoa

2540 Dole Street, Honolulu, HI 96822

1Lbstract

Several important iysues pertaining to dispersion and polydispersity of; evaporating and reacting fuel droplets in forced tyrbulent flows are investigated via numerical simulation. The carrier phase is considered in the Eulerirln context and is simulated by direct numerical simulation (DNS). The dispersed phase is tracked in the Lagrangian frame and the interactions between’the phases are taken into account in a realistic t’lKo-way (coupled) formulation. It is assumed that combustion takes place in the v* por phase, resulting ~ in a “homogeneous” reaction described by fuel + o,xidizer + products + energy. The resulting schemeiis applied for extensive DNS of a forced, isotropic, lop Mach number turbulent flow laden with a large number (up to 5.5 x 105) of fuel droplets. Here, the r+ults are presented for different values of the mass lo$ding ratio and the heat release coefficient. It is sho&n that during the early times and/or for small valuls of the heat release coefficient, the combustion pro&s is significantly affected by the rate of evaporat\on. The normalized (with its initial value) mass of the droplets decreases faster at smaller mass 1oading:ratios during the initial times; an opposite trend isl observed in long times. The results are also used ~to discuss the temporal evolu- tion of the mean temperatures and the mean mass fractions.

No+enclature

B (T* - Td)/X t~ransfer number c C”” md/(pW) droplet concentration

Ce & [X - (1 + r) &] heat release f

*Assistant Professor, Qepartment of Mechanical Engineer- ing, Member. Copyright 01999 The American Institute of Aeronautics and Astronautics Inc. All rights reserved.

CP 4i Da EI EK

ET

fl

f2

f3

f4

3i h

hi k

co&cient specific heat of the carrier phase droplet diameter pfLfKfwd/Uf Damkijhler number internal energy of the carrier phase ipuiui kinetic energy of the carrier phase EI + EK total energy of the carrier phase (1 + 0.15Refs7)/(1 + B) Nuj3P~ p*ShX/SSc s(18/pd).5(p*Sh/Relf5Sc) a zero-mean solenoidal random force specific enthalpy enthalpy of formation of the product gas wavenumber

nfWd forward reaction rate constant

L f reference length

LV latent heat of vaporization of the liquid

md mass of the droplet

Mf Uf/,/m reference Mach number nd number of droplets within the cell volume N number of collocation points in each direction

$ total number of droplets (2 + 0.6Re2Pr.33)/(1 + B) Nusselt number

P pressure of the carrier phase PT &J/K Prandtl number T stoichiometric coefficient R gas constant

Red Refp*ddlu; - Vi( droplet Reynolds number

Ref pf Uf Lf /p reference Reynolds number Sij $(BUi/BXj + BUj/dZi) rate-of-strain tensor S,, Sui, S, coupling source/sink terms SC ~/PI’ Schmidt number Sh 2 + 0.6Re2Sc33 Sherwood number t time T temperature TB boiling temperature of the liquid

W velocity of the carrier phase in direction Zi (i = l-,2,3)

1 American Institute of Aeronautics and Astronautics


Uf reference velocity

‘ui velocity of the droplet in the direction xi Xi spatial coordinates Xi position of the droplet

Yfu fuel vapor mass fraction Y oz oxidizer mass fraction

Greek svmbols

Y ratio of the specific heats of the carrier gas I? binary msss diffusivity coefficient &j Kronecker delta function 6X node spacing 6V cell volume A &j/&j dilatation e dissipation rate K thermal conductivity of the carrier phase x L,/C,Tf normalized latent heat of

evaporation

P viscosity of the carrier phase

P density Td Refp&18 droplet time constant @J* mass loading ratio W p2 DaYfvYos reaction rate

Subscripts 0 initial value (at t = 0) d droplet properties

f reference parameters for normalization

iv

surface of the droplet fuel vapor

ox oxidizer

Sunerscrints I fluctuating quantity * carrier phase properties at the droplet

location

Svmbols < > Eulerian ensemble average over the number

of collocation points << >> Lagrangian ensemble average over the

number of droplets

Introduction

Numerical simulation of turbulent reacting flows has proven to be one of the most challenging problems within the general field of computational fluid dy-

namics.1~2 Turbulent flows are exemplified by their many length and time scales which require high resolution and/or accurate modeling for their simulation. On the other hand, the chemical reaction in- troduces a new set of variables, often accompanied with sharp gradients in small scales. The complexity of the turbulent reacting flow is obviously escalated by the presence of liquid droplets (or solid particles). Due to its nature, the multiphase flow is best analyzed in the Lagrangian frame by tracking a large number of droplets.3

With the advent of the supercomputers, it became possible to simulate the turbulent reacting flows via direct numerical simulation (DNS) without resort- ing to any modeling.4-13 DNS has also been used for simulations of turbulent flows laden with particles.14-23 The extent of contributions is too large to be discussed here-in detail; we refer to recent re- views.24-26 Most of the previous studies have considered dispersion of solid particles only. Recently, we2’ implemented DNS for the study of evaporating droplets in one-way coupling with an incompressible carrier phase. This study shows that, after an initial transient period, the droplet size distribution becomes nearly Gaussian. This result is, qualitatively, verified at low mass loading ratios by Mashayek2a~2g who considers the compressible flow with two-way coupling. The results in Refs.28~2g also indicate that evaporation is sij@iificantly affected by two factors, i.e. the gradient of the fuel vapor concentration in the vicinity of the droplets, and the heat transfer from the carrier phase to the dispersed phase.

In light of the previous studies, in this paper we extend the application of DNS to evaporating and reacting droplets dispersed in a compressible carrier phase. Since this is the first attempt in DNS of the reacting droplet dispersion in turbulent flows with two-way coupling, the problem is formulated based on models and correlations which are relatively well- established. It must be emphasized that the term ‘DNS’ is used here with the understanding that there are models involved in describing the effects of the droplets on the carrier phase. Due to the presence of the droplets the flow is ‘heterogeneous’ whereas the combustion is assumed to take place in the vapor phase only, thus rendering a ‘homogeneous’ reaction. For the range of the mass loading ratios considered, this assumption seems reasonable.30 The investiga-



tion is based on the, forced isotropic turbulent flow to avoid the extra complexity in the analysis of the results. A large nur$ber of parameters emerge from the formulation of the problem. While we have been able to perform extensive DNS to study the effects of these parameters, ,due to space limitation here we only consider the effects of the mass loading ratio and the heat release coefficient.

Formulatioiz and Methodolow

We consider the motion of a large number of droplets (dispersed phase) in a turbulent flow (carrier phase). The transport of the carrier phase is considered in the Eulerisn frame :,whereas the dispersed phase is treated in the Lagrsngian manner. Also, conservation equations (in the Eulerian frame) are considered for the mass fractions of the fuel vapor and the oxidizer. For sinrplicity, the fuel vapor, the oxidizer, and the products are assumed to have the same properties such, that their mixture (hereinafter referred to as the ‘carrier phase’ or the ‘fluid’) can be treated as one entity - the Eulerian continuity, momentum, and energy equations are solved for the mixture. The speci,fic enthalpies of these compo- nents, however, are considered to be different in order to satisfy the f&t law of thermodynamics.

The carrier phase is considered to be a compressible and Newtonian fluid with zero bulk viscosity, and to obey the perfect gas equation of state. The Eu- lerian forms of the n’on-dimensional continuity, momentum, and energy: equations for the carrier phase are expressed as:

~(PUi) + &(Pi”j) = -g I

+$& f J (

Sij - :Abij >

+ p3i + Sui, (2)

JET 8 x+z [(ET + py = (-/ - 1);&,q g 3

2 a +--

Ref 8xj Sij’- iA&j + CeDap2Yf,Yo,

followed by the conservation equations for the fuel

3

vapor and the oxidizer mass fractions:

$PYfv) +

-Dap2YfvXz + S,, (4)

-rDap2YfvYoz. (5)

The equation of state is p = pT/rMf2. All of the variables are normalized by reference length, density, velocity, and temperature scales. The coupling of the carrier phase with the droplets is through the terms S,.,.,, Sui, and S, which describe the mass, momentum, and energy exchange between the phases, respectively. The formulation of these terms and their calculation from the discrete droplet fields are described later in this section.

With the knowledge of the velocity field, the solution of (3) provides the internal energy of the mixture of the oxidizer, the fuel vapor, and the products. To determine the temperature of the mixture, the fuel vapor mass fraction is used to subtract the contribu- tion of the latent heat of evaporation. With the ideal gas assumption, the change in the enthalpy upon mixing is negligible and the enthalpy of the mixture can be described as the mass-weighted sum of the enthalpies of its constituents. After some algebraic manipulations, the following relation is obtained for the internal energy of the mixture:

which yields:

T = ‘(’ - l)M;Er - rXYfv, (7)

P

where we have assumed that the specific heat of the liquid is the same as that of the gas.

The liquid droplets are allowed to evaporate but are assumed to remain spherical with diameter smaller than the smallest length scale of the turbulence and to experience an empirically corrected Stoke- Sian drag force. Both interior motions and rota- tion of the droplets are neglected. The density of the droplets is considered to be constant and much

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larger than the density of the carrier phase such that only the inertia and the drag forces are significant to the droplet dynamics. The droplet-droplet interaction and the radiation heat transfer are also neglected. The droplets are tracked individually in a Lag-rang& manner, and the droplet position, velocity, temperature, and mass are determined from the following non-dimensional equations?

For the dilute two-phase flow considered here, it can be assumed that the chemical reaction is in the cax- rier phase where the fuel vapor and the oxidizer are mixed.30 Since this is the first DNS of reacting droplets, we consider the second-order and irre- versible reaction:

F+rO K,rd (l+r)P, (13)

dTd dt = rd- b,* - Td) - k(y. -Y,‘), (10)

and dmd -= dt

-f4Ty2(Y. - Yi”). (11)

The droplet variables are normalized using the same reference scales as those used for the gas phase variables. The function fr in (9) represents an empir- ical correction to the Stokes drag due to droplet Reynolds numbers of order unity and larger and is valid for Red 5 1000.32

The droplets are assumed “lumped”, so that there is no tempera&e variation within each droplet. The factor f2 in (10) represents a correlation for the con- vective heat transfer coefficient based on an empirically corrected Nusselt number (Nu).~~ The second term on the right hand side -of (10) represents the change in the thermal energy due to phase change. The correlation fs is a function of an empirically corrected Sherwood number (Sh).33 The fuel vapor mass fraction at the droplet surface is equal to the vaporization pressure of the droplet (for equiva- lent molecular weights of the gas and the liquid) and obeys the Clausius-Clapeyron equation:

x=p”ap=exP[(y^i;)Te (l-Z)], (12)

where the boiling temperature of the liquid is assumed to be constant. Finally, (11) governs the rate of mass transfer from the droplet due to phase change which is a function of the fuel vapor mass fraction difference at the droplet surface, the droplet time constant, and the Sherwood number dependent correlation, f4.

where, F, 0, and P represent the fuel vapor, the otidizer, and the product, respectively. The effects of the heat released by combustion and the phase change energy are accounted for in the description of the absolute enthalpies.

The source/sink terms S,, Z&i, and S, appear- ing in (l)-(4) p re resent the integrated effects of the droplets mass, m~omentum, and energy exchange with the carrier phase. These Eulerian variables are calculated from the Lagrangian droplet variables by volume averaging the contributions from all of the individual droplets residing within the cell volume centered around each grid point. The coupling terms are expressed as:2Q

(14)

Simulations are conducted within the domain 0 5 zi 5 27r, i = 1, 2, 3. The reference length (Lf) is conveniently chosen such that the normalized length of the computational box is 2n. The reference temperature (Tf) and density (pf) are set the same as the initial mean temperature and density of the carrier phase, and the reference velocity (Uf) is found by setting Mf = 1. A Fourier spectral collocation method with triply periodic boundary conditions is employed to simulate the transport equation of the carrier phase. The simulations are conducted on a domain discretized by IV equally spaced collocation points in each direction. Time advancement for both the Eulerian carrier phase equations and the Lagrangian droplet equations is performed using



the second order accurate Adams-Bashforth scheme. The magnitudes of, the Eulerian gas variables at the droplet locations are determined by a fourth order Lagrange polyno~mial interpolation scheme. The turbulence field pertaining to the carrier phase is isotropic and stationary. To emulate the stationary field, a low wavenumber forcing scheme is imposed by adding energy to, the large scales of the Aow.~~ The droplets are initially treated as solid particles and are allowed to iinteract with the flow for more than three eddy turnover times. Once the particle- laden flow becomes statistically stationary, the time is set to zero and the droplets are allowed to evaporate and react.

(Results

The formulation presented above involves a large number of parameters: the droplet time constant, the mass loading ratio, the initial droplet temperature, the latent hey of evaporation, the boiling temperature, the Damkijhler number, the heat release coefficient, and the stoichiometric coefficient. Extensive numerical ~ simulations have been carried out to study the effects of these parameters on various statistics of the reacting flow. Space limitation, however, does Inot allow to discuss all of the results in this paper; Therefore, the discussion be- low is limited to the,effects of the mass loading ra tio (for aPme = 0.25/0.5,1 with Ce = 24) and the heat release coefficient (for Ce = 4,8,16,24 with 0 mo = 1) which exhibit rather stronger effects on the combustion process. For all of the cases considered here, Ref = 500, SC = 0.7, y = 1.4, and Pd = 1000. Also, rd,, != 2.5, x = 2, TB = 5, Tdo = 1, Da = 0.5, and r = 1.: All of the simulations are performed on 643 collocation points using as many as 5.5 x lo5 droplets. A. typical run (for the case with 5.5 x lo5 droplets) requires about 60 hours of CPU time on the CRAY-C,90 supercomputer. The results of the simulations sre statistically analyzed; in the following, the notation < > indicates the Eulerian ensemble average and << >> denotes the Lagrangian average values.

The analysis of the DNS results indicates that the amount of energy added to the system by external forcing is very negligi,ble in comparison to the phase change energy and the heat released by combus-

tion.2Q Therefore, the statistics presented here are not affected by external forcing. The results also indicate that the mean turbulent Mach number is less than 0.12 for all of the cases and the flow is free of shocklets. This is important as a pseudospec- tral method has been used for the simulation of the carrier phase. The results show that the turbulent Mach number decreases with the increase of either the heat release coefhcient or the mass loading ratio. This is in agreement with previous observations in single phase reacting flows in that the heat release by combustion tends to decrease the compressibil- ity effects. A similar behavior with the increase of the mass loading ratio has also been observed in the simulations of reacting droplets dispersed in a homogeneous shear turbulent flo~.~~

lo-3

z 6

m

1O4

lo-’

10” I I I 1 10 loo

---Cd -

I (W 1 I 1 10 100

k

Figure 1: (a) Power spectra of the carrier phase density at t = 60. (b) Normalized power spectra of the fluctuating fuel vapor mass fraction time averaged for 5 5 t 5 65.

Since the grid resolution for these simulations is somewhat low, various statistics of the flow have



been carefully analyzed to ensure that all of the Eu- lerian fields are accurately resolved. In general, the results show that the carrier phase density field is more sensitive to the grid resolution than are the other fields. This is due mainly to the fact that the fuel vapor is added to the carrier phase at droplets location, which is not uniform in space. Since there is no diffusion term in the carrier phase continuity equation (l), sharp gradients in the density field may occur at small scales. This is illustrated in Fig. la which shows the power spectra of the carrier phase density for various mass loading ratios at a typical time t = 60. The increase of the energy observed at high wavenumbers is an indication of the relatively high density fluctuations at small scales. With the increase of the mass loading ratio (for the same droplet size) the number of droplets present in the flow increases. -This results in a more uniform production of the fuel vapor, and decreases the density gradients, as verified by the decrease in the magnitude of the high wavenumber ‘lift’ in Fig. la. It must be noted that this effect may not completely vanish with the increase of the mass loading ratio due to preferential distribution of the droplets.17l25 The power spectra of the carrier phase density was monitored throughout the simulations to ensure that the magnitude of the lift is never larger than a half- decade. As expected, this resolution problem is significantly alleviated in the solution of the conservation of mass equation (4) for the fuel vapor. This equation contains a diffusion term which tends to di- minish the gradients in the fuel vapor mass fraction and prevents the appearance of the high wavenumber lift in the power spectra as witnessed in Fig. lb. The spectra shown in Fig. lb are obtained by time averaging over a long period (5 5 t 5 65). Inter- estingly, a collapse of the spectra for various values of the heat release coefficient is observed when they are normalized by their respective variances.

The temporal variation of the Lagrangian average droplet mass (normalized with its initial value) is shown in Fig. 2. A close inspection of this figure re- veals that, during the initial times, the normalized mass decreases faster for smaller mass loading ratios. This is in qualitative agreement with the results of Mashayek 2g for evaporating (not reacting) droplets. In early times, combustion is not very effective as the amount of the fuel vapor is small. As the chemical reaction proceeds, more heat is released and the

1.0

3 0.8

V

0.6 I I I I I I 0 26 40 60 80

1.0

0.9

0.7

0.6

c . -mm -8 \

--Ce=l6 ‘\ ‘.

0 20 40 60 80 t

Figure 2: Temporal variations of the Lagrangian average mass of the droplets normalized with its initial value.

rate of evaporation increases which, in turn, results in higher rate of reaction. This is more noticeable at higher mass loading ratios as the absolute mass of the vapor is larger. Consequently, at longer times the normalized mass of the droplets decreases faster at higher mass loading ratios. For the same reason, i.e. low reaction rate, the variation of the heat release coefficient does not significantly affect the evaporation rate during the early stages. As expected, for longer times the increase of the heat release coefficient results in the increase of the heat generated by combustion, thus increasing the evaporation rate.

The heat released by reaction causes the temperature of both phases to rise as illustrated in Fig. 3. However, it must be noted that a portion of the heat is utilized to account for the latent heat of evaporation, and does not contribute to the change of



2.5 , I I I I I

t- -&=4 OJ)

2.2 --- tea, i I d

0 20 40 60 80 t

Figure 3: Temporal variations of the mean temperatures of the carrier ibhase (thick lines) and the dispersed phase (thin lines).

the sensible heat which is exhibited by a rise in the temperature. This is imore evident during the early stages as the fuel vapqr mass fraction is small and so is the heat released by combustion. For small values of the heat release coe’flicient the rate of evaporation is small and the rate of temperature increase remains small for all times (Fig. 3b). For large values of the heat release coefhcient;, however, more vapor is generated and the reaction rate increases. Therefore, a large amount of heat i:s generated while only a small fraction is consumed for the latent heat of evaporation. This results in significant increase in the mean temperature at long times. The increase of the mass loading ratio enhancei the amount of the fuel vapor and increasingly raised the temperature. It is noted in Fig. 3 that the difference in the temperatures of the two phases also increases with time. This in-

creases the heat transfer from the carrier phase to the droplets, and results in higher evaporation rates.

1.0

0.8

20.6 V

I (a) -

0.0 - <Y,> ---

I I I I

0 20 40 60 80

0 20 40 60 80 t

Figure 4: Temporal variations of the mean mass fraction of the oxidizer and the fuel vapor.

The discussion above on the rate of evaporation (cf. Fig. 2) does not identify the temporal behavior of the fuel vapor mass fraction as part of the generated vapor is continuously consumed by combustion. In fact, as shown by the plots of the mean fuel vapor mass fraction < YfV > in Fig. 4, for a large period of time the mean vapor mass fraction remains close to zero; for small values of the heat release coefficient, it is nearly zero throughout. Therefore, the fuel vapor participates in the chemical reaction al- most instantaneously after its production. The increase of the heat release coefficient and/or the mass loading ratio increases the amount of heat transfer from the carrier phase to the droplets, as discussed above. This may result in local accumulation of the vapor without access to enough oxidizer. Therefore,



the mean fuel vapor mass fraction becomes larger than zero. The decrease of the mass fraction of the oxidizer at long times (see Fig. 4) is also respon- sible for this phenomenon. fin general, it may be concluded that for small values of the heat release coefficient and/or during the initial times, the combustion process is mostly governed by evaporation. For large values of the heat release coefticient and during large times, the heat release mechanism becomes more significant.

-.-- 0 20 40 60 90

t

Figure 5: Temporal variations of the mean mixture fraction.

A convenient approach usually adopted to analyze a reaction, is via the concept of ‘Lmixture fraction” defined as?

I= %J - $Yo, + f l+f .

In single-phase flows, without any source of fuel or oxidizer, [ behaves like a ‘conserved scalar’. For the present case, however, < is not conserved and increases with time as shown in Fig. 5. To analyze the behavior observed in the figure, the transport equation for < is derived using (1)) (4)) and (5):

$PE) + &(Pb‘j) = &-& + S,, (18) 3 3

which resembles the equation for a conserved scalar except for the droplet mass source term. For evaporating droplets, this term is always positive and results in the increase of E. For homogeneous flows, a relationship between the-mean values of < and the

mean carrier phase density may be obtained by averaging (18). After some algebraic manipulations and by assuming that < p[ >z< p > < < > the following relation is obtained:

<E>- <p>-<PO>

<p~>~ ’

where pe is the initial value of the carrier phase density. Equation (19) is in agreement with the DNS results.

8

2

0

-CC=4

w-m m

- -- &=I6 / I

0 -1 2 3

Figure 6: Variations of the expected value of the droplet concentration conditioned on the reaction rate.

When analyzing the DNS results, special attention must be paid to various mechanisms influencing the mixing of the fuel vapor and the oxidizer. The extensive previous studies conducted on single phase turbulent reacting flows have revealed the important role of turbulence mixing in enhancing the rate of reaction. In two-phase reacting flows, the dispersion of the droplets may also be considered as an effective mechanism for improving and/or control- ling the mixing process. The most important issue in this consideration, is the preferential distribution of the droplets. 17125 Figure 6 shows the variations of < Clw > / < C > with w/ < w > for different values of the heat release coefficient. Here, w is the reaction rate and C is the droplet concentration defined in the Eulerian frame. It is observed that the droplets are mostly concentrated in the regions of the flow with high reaction rates. This may be explained by considering the fact that the droplets



are the source for th,e fuel vapor, therefore, there is a higher probability for having higher reaction rates in the vicinity of the droplets. However, it could also be (at least par$ally) due to the effects of reaction on the vorticity, field which plays a significant role in the preferential distribution of the droplets. This latter scenario :has not yet been explored and demands more future elaboration.

Cbnclusion

Results obtained via’ numerical simulation are used to investigate some iissues of relevance to dispersion and polydispersjty of evaporating and reacting droplets in forced isotropic compressible turbulence at low Mach numbers While we have been able to consider a reasonably wide parameter range, due to space limitation, here we only discuss the effects of the mass loading radio and tlre heat release coefficient. The decrease o!f the mass loading ratio results in a faster decrease iof the relative mass (normalized with its initial value) of the droplets during the initial times; an opposite trend is observed at long times. The increase of the heat release coefficient monotonically increases the rate of evaporation at all times. An analysis of the results indicates that evaporation plays a s’ignificant role during the early stages of combustion1 for all of the cases. At longer times, and for larger values of the heat release coefficient, the temperature of the mixture increases to large values which (by increasing the heat transfer between the phases) results in large evaporation rates.

Ackr@4edgment

Acknowledgment is made to the Donors of The Petroleum Research Fund, administered by the American Chemical gociety, for partial support of this research. Compu~tational resources for this work were in part provided by the San Diego Supercom- puting Center and thi College of Engineering Com- puter Facility at the University of Hawaii at Manoa.

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