‘P

/

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

AIAA 2000-0183 Velocity and Temperature Statistics in Reacting Droplet-Laden Homogeneous Shear Turbulence Farzad Mashayek University of Hawaii Honolulu, HI

38th Aerospace Sciences Meeting & Exhibit

lo-13 January 2000 / Rem, NV For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

AIAA2000-0183

VELOCITY AND TEMPERATURE STATISTICS IN REACTING DROPLET-LADEN HOMOGENEOUS SHEAR TURBULENCE

Farzad Mashayek* University of Hawaii at Manoa

2540 Dole Street, Honolulu, HI 96822

Abstract

The results of numerical simulations are used to investi- gate the effects of the mass loading ratio and the droplet time constant on the velocity and temperature statistics in a two-phase homogeneous shear turbulence. The fuel vapor generated by the evaporation of the droplets re- acts with the oxidizer carrier gas through a single-step, second-order reaction. The carrier phase is simulated in the Eulerian frame using direct numerical simulation (DNS) whereas the droplets are tracked in a Lagrangian manner. It is shown that the turbulence kinetic energy and its small-scale dissipation increase with the decrease of either the mass loading ratio or the droplet time con- stant. The opposite trend is observed ,for the early varia- tions of the dissipation due to drag of the droplets. It ap- pears that the velocity and temperature fluctuations of the dispersed phase can be reasonably described by Gaussian distribution. Both the mean temperature of the carrier phase and the Lagrangian mean temperature difference between the two phases increase with the increase of the mass loading ratio. The increase of the droplet time con- stant however decreases the former while increasing the latter.

Nomenclature

I

B (T* - Td)/A transfer number

Ce &)M: [A - ( 1 + r) &] heat release

coefficient C, , C, specific heats of the carrier phase dd droplet diameter Da pfLfKf,,,d/Uf Damkohler number eT pC,T + JpUiUi total energy of the carrier

phase (

*Assistant Professor, Department of Mechanical Engineering, Se- nior Member of AIAA. Copyright 02000 The American Institute of Aeronautics and Astronautics Inc. All rights reserved.

1

fl A

A

f4

6 Kfwd

Lf L md M

Mf nd NU P PB

(1 +0.15Refg7)/(1 +B) Nu/3Pr p*Shh/3Sc ?‘c(18/pd)*5(p*Sh/Re;5Sc) enthalpy of formation of the product gas forward reaction rate constant reference length latent heat of vaporization of the liquid mass of the droplet (uiui/yRT)“’ turbulence Mach number Uf/m reference Mach number number of droplets within the cell volume (2 + 0.6Re2Pr33)/( 1 + B) Nusselt number pT/yM; pressure of the carrier phase pressure at which the boiling temperature is defined

Pr C&c Prandtl number Qij 6ij - St8i182j transformation tensor r stoichiometric coefficient R gas constant Red Refp*ddjuf - vi( droplet Reynolds number Ref p$lfLf/p reference Reynolds number s a < Ut > 18x2 mean velocity gradient &, &, ,S, coupling source/sink terms SC

Sh t T TB

4

Uf ui

Vi

~/PI’ Schmidt number 2 + 0.6ReiSc33 Sherwood number time temperature boiling temperature of the liquid fluctuating velocity of the carrier phase in direction Xi (i = 1,2,3) reference velocity mean velocity of the carrier phase fluctuating velocity of the droplet in the direction xi mean velocity of the dispersed phase spatial coordinates position of the droplet fuel vapor mass fraction

American Institute of Aeronautics and Astronautics

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

Y ox oxidizer mass fraction

Greek svmbols ratio of the specific heats of the carrier gas binary mass diffusivity coefficient Kronecker delta function node spacing cell volume Qi&/&j dilatation dissipation rate thermal conductivity of the carrier phase L/C,Tf normalized latent heat of evaporation viscosity of the carrier phase density Refp&fj/ 18 droplet time constant mass loading ratio

Subscrims 0 initial value (at t = 0) B boiling condition d droplet properties f reference parameters for normalization

;Y surface of the droplet fuel vapor

OX oxidizer

SuDerscfiDts * carrier phase properties at the droplet

location

Symbols < > Eulerian ensemble average over the number

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

number of droplets

Introduction

In a previous work,’ simulations were presented of re- acting droplets dispersed in a homogeneous shear turbu- lent flow. The results of those simulations were used to study the evolution of the oxidizer, the fuel vapor, and the droplet size distribution. The results indicated that the droplets are organized in the direction of the mean shear. Further, a strong correlation was found, for the range of parameter values considered, between the re- action zone and the droplet distribution. It was shown

that the reaction zone is mainly structured in the regions with high droplet concentration, thus featuring both the effects of large-scale mean shear as well as small-scale preferential distribution2e3 of the droplets in high strain rate regions of the flow.

In this work, we utilize the same simulations to analyze the statistics of the velocity and temperature which were not discussed in Mashayek.’ In recent years, a wide vari- ety of configurations have been considered by various in- vestigators to study two-phase turbulent flows, including homogeneous isotropic,4-6 homogeneous anisotropic,7-9 and inhomogeneous. tM3 Among these configurations, the homogeneous shear flow provides a very convenient means for the study of anisotropy effects. An attrac- tive feature of this flow is the possibility of calculating statistics using the entire flow. This is made possible in our study by solving all the governing equations in a deforming coordinate which renders both phases homo- geneous. Another useful feature offered by this flow is its transient nature which allows the presentation of the time-dependent statistics. These features make the statis- tics generated by simulations of homogeneous shear flow very valuable for model validation.‘* I4

In the following section we provide a brief description of the formulation and methodology. Here only the fi- nal form of the equations in the deformed coordinates is presented and the reader is referred to Mashayekt** for more information.

Formulation and Methodolow

We consider the motion of a large number of Lagrangian droplets in an Eulerian carrier phase. The droplets are allowed to evaporate and react but are assumed to re- main spherical. It is also assumed that the chemical re- action takes place in the carrier phase via the single-step, second-order irreversible reaction:

F+rO K* (1 +r)P, (1)

where F, 0, and P refer to the fuel vapor, the oxidizer, and the product, respectively. For simplicity, the fuel va- por and the product are assumed to have the same molec- ular weight, viscosity, mass diffusivity, and specific heat as those of the oxidizer gas. The specific enthalpies of the vapor and the product, however, are considered to be dif- ferent than that of the gas. The carrier phase (composed of the oxidizer, the fuel vapor, and the product) is con-

2 American Institute of Aeronautics and Astronautics

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

sidered to be a compressible and Newtonian fluid with zero bulk viscosity, and to obey the perfect gas equation of state.

In order to study anisotropy features a homogeneous shear flow is considered. To configure this flow for DNS, a linear mean velocity profile is applied to a zero mean turbulent velocity field and the carrier phase instanta- neous velocity is expressed as: Vi = Sx&r + Ui. The magnitude of the imposed shear is given with the ampli- tude of the mean velocity gradient, S = 3 < lJr > 13x2 = const. For homogeneous flows which either start from isotropic initial conditions or develop to become inde- pendent of the initial conditions, Blaisdell et ~1.‘~ show that the Favre average fluctuating quantity is the same as the Reynolds average one. In this study we indicate the fluctuating quantity with the same notation for both averaging; the type of averaging is understood from the context.

In order to employ the Fourier spectral method, periodic boundary conditions must be imposed. This is accom- plished by solving the governing equations for fluctuat- ing velocities on a grid which deforms with the mean flow. This transformation has been discussed in litera- turei5T l6 and here we only present the governing equa- tions in the deforming coordinates. For the carrier phase these equations are expressed as:’

i(PUi) + Qkj$(Puiuj) = -Pu$% - Q/cig

+&i, (3)

aeT at f Qkj&[uj(P+-W)] = -PUIWS

+CeDbp2Yf;Y, + Se, f4)

-Dap2Y~vL + S,

-rDap2Y&x, (6)

where all the variables are normalized by reference length, density, velocity, and temperature scales.

The droplets are assumed to be smaller than the smallest length scale of the turbulence and to exhibit an empiri- cally corrected Stokesian drag force. The density of the droplets is constant and much smaller than the density of the carrier phase such that only the inertia, the drag, and the gravity forces are significant to droplet dynamics. Due to restrictions imposed by the numerical methodol- ogy implemented here, the buoyancy effects cannot be considered. In addition, the droplet volume fraction is assumed to be small and droplet-droplet interactions are neglected. In the absence of gravity or other external body forces and by assuming that the droplets start from the same initial velocities as those of their surrounding fluid elements, the droplet instantaneous velocity is de- scribed as:14 & = Sx2&r + v’ By performing ensem- ble averaging on the droplet mstantaneous equations, it can be shown that the dispersed phase is homogeneous within the deforming domain used to simulate the car- rier phase; thus, periodic boundary conditions can be ap- plied to the dispersed phase as well. The droplet position and momentum equations in the transformed coordinates

and

dvi .fl - = G(l4f -Vi) -lQSGil. dt

dTd dt - zd

- f”(T* - Td) - e(Ys - Y&y,,

dmd dt- - -,b7:5(Y, -$A

(9)

(10)

where Y,, the vapor mass fraction at the surface of the droplet, is calculated using the Clausius-Clapeyron rela- tion:

Y”‘$exp[(yJ~)TB (l-$1. (11)

The source/sink terms &, &i, and Se appearing in (2)- (5) represent the integrated effects of the droplets-mass, momentum, and energy exchange with the carrier phase. These Eulerian variables are calculated from the La- grangian droplet variables by volume averaging the con-

3 American Institute of Aeronautics and Astronautics

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

tributions from all of the individual droplets residing within the cell volume @‘I’= (&x)~, where 6x is the node spacing) centered around each grid point. In the deform- ing coordinates, these terms are expressed as:

&i=-$~[~(U~-Vi)+~Vi], (13)

A?=-&$[ (Y- :,q $(mdGJ - (y-:)M; 2

+y(UT_.i)Vi+d2 (:ViVi)]. (14)

The computational methodology and the initialization are the same as those employed in Mashayek’q’ and will not be detailed here. All the Eulerian fields are calcu- lated using a pseudospectral method and the Lagrangian droplet equations are integrated using a second-order ac- curate Adams-Bashforth method. To evaluate the carrier phase variables at the droplet location a fourth-order ac- curate Lagrange polynomial interpolation scheme is em- ployed. The density and velocity fields are initialized as random Gaussian, isotropic, and solenoidal fields in Fourier space. The initial temperature field has no fluctu- ations and the initial pressure field is calculated using the equation of state. The droplets are randomly distributed in the flow at r = 0 with the same velocity and tempera- ture as those of their surrounding fluid. The initial mean gas density and mean gas temperature are used as the ref- erence scales for density and temperature, respectively. The reference Mach number is Mf = 1; therefore, the speed of sound based on the initial mean gas temperature is the reference scale for the velocity. For all of the sim- ulations Ref = 500, Pr = SC = 0.7, y = 1.4, S = 2, and f&f = 500.

Results

All the simulations are performed on 963 collocation points, using as many as 1.24 x lo6 droplets, and are con- tinued till the nondimensional time St = 14. Although a wide variety of cases have been considered,’ due to space limitations, here we only discuss the effects of the mass loading ratio and the droplet time constant on velocity and temperature fields. A listing of the simulations con- sidered in this study is provided in table 1, with subscript

Case Ref. Coup. Evap. Reac. @d r,jo

one-way l-way No No - 1.0 base case 2-way Yes Yes 0.2 1.0 %nOb 2-way Yes Yes 0.4 1.0 ‘%nOb 2-way Yes Yes 0.8 1.0 o.%fflh 2-wav Yes Yes 0.2 0.5

Table 1: Cases considered in the study.

‘0’ referring to initial value of a variable at t = 0. Here, we consider a ‘base case’, indicated by boldface in the table, and then change the value of one of the parame- ters in each of the following simulations. The results of the reacting simulations are compared (wherever possi- ble) with the data from a one-way coupling case. The abbreviation used to refer to each case in the presenta- tion of the results, is shown in the first column of table 1, where the subscript ‘b’ refers to the value of a param- eter in the base case. The droplets begin to evaporate and react at the normalized time St = 2 when the car- rier phase turbulence kinetic energy starts to grow (see Fig. la). The condition of one- or two-way coupling, however, is imposed from t = 0 for corresponding cases. To avoid very small droplets, a criterion was set in the code to remove the droplets that their diameters fall be- low a certain Value. only for the Case with r&j = 0.%&b the application of this criterion resulted in the removal of some of the droplets for Sr >_ 12. This has affected some of the statistics pertaining to this case (for St 2 12) as can be seen in the following figures. For all of the cases, A = 0.8, TB = 4, Da = 0.5, Ce = 20, r = 1, and PB is cho- sen to be the same as the initial pressure of the oxidizer gas.

The temporal variation of the turbulence kinetic energy of the carrier phase, k = 3 < puiui >, is shown in Fig. la. It is observed that for all of the cases there is an initial (St < 2) decay of the kinetic energy. This is due to the isotropic initial velocity field which lacks the shear Reynolds stress component and sets the production term in the transport equation for k equal to zero. Once the shear stress component is generated by the imposed mean velocity gradient, the homogeneous shear flow is established and the kinetic energy starts to grow, approx- imately at St = 2. This time is chosen for initiating the evaporation and reaction of the droplets. Our previous results8 show that the presence of the droplets decreases the turbulence level by introducing an extra dissipation

.

4 American Institute of Aeronautics and Astronautics

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

0 2 4 6 8 10 12 14

M base case

0 2 4 6 8 10 12 14 St

Figure 1: Temporal variations of (a) the turbulence ki- netic energy and (b) the velocity variance of the carrier phase.

(see below). This is in agreement with early variations of the kinetic energy in Fig. la. At long times, however, an increase in k is observed in reacting cases as com- pared to the one-way coupling case. As pointed out by Mashayek,8 there are two mechanisms involved in this increase of the turbulence level: (i) the decrease of the size of the droplets which decreases the drag dissipation, and (ii) the transfer of the kinetic energy from the dis- persed phase to the carrier phase by the evaporated mass. To separate the effects of the density variations, in Fig. lb, we consider the temporal variations of ‘half of the carrier phase velocity variance’, 1 < UiUi >. The trends are similar to those observed in Fig. la, however, the modifications of 1 < UiUi > are in general smaller than those of k. The decrease of either the mass loading ratio or the droplet time constant results in the increase of the turbulence kinetic energy at all times.

0.8

0.0

0 2 4 6 8 10 12 14 St

Figure 2: Temporal variations of the dissipation rate (a) of the carrier phase. Also shown on the figure, is the vari- ation of the drag contribution (&) to the carrier phase turbulence kinetic energy. Both of the parameters are normalized with the initial value of the dissipation rate (Eg).

More insight into the evolution of the turbulence kinetic energy is gained by examining the variations of its dissi- pation rate, E. The temporal evolution of E, normalized with its initial value (a), is shown in Fig. 2. In general, the presence of the droplets in two-way coupling with the carrier phase results in the decrease of the dissipation rate.* However, when the droplets are evaporating and reacting, the decrease in the total dissipation rate is less pronounced. This is more noticeable in the case with smaller initial droplet time constant which results in E values larger than those in the one-way coupling case to- wards the end of the simulation. Also shown in Fig. 2 is the temporal variation of

Dd = - $2 %(U;-Vi)U!])) ( [.

(15)

which represents the contribution of the droplets drag to the variation of the turbulence kinetic energy - a posi- tive value for Dd indicates a decrease of the turbulence kinetic energy by drag. Figure 2 shows that Dd increases during the early stages as the droplets, due to their inertia, cannot follow the rapid changes in the carrier phase and the relative velocity between the two phases increases. After the effects of the initial conditions are diminished, Dd starts to decrease and takes small values at long times. For. the .case with c&c = 4@,cb, Dd assumes negative

5 American Institute of Aeronautics and Astronautics

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

values towards the end of the simulation, which indi- cates an energy transfer ‘from’ the droplets to the carrier phase.

0.8

8= 0.6 2

(3---E) base case

% +a 0.4 > Y

0.2

0.0

0 2 4 6 8 10 12 14 St

Figure 3: Anisotropy of the dispersed phase.

One of the most important features of homogeneous shear flow is the anisotropy of the normal Reynolds stress components which makes this flow very valuable for the assessment of various Reynolds stress models.‘yt4 The anisotropy of the dispersed phase is portrayed in Fig. 3. In the figure, kd = i < YiVi > is the turbu- lence kinetic energy of the dispersed phase and no sum- mation is assumed on Greek indices. A comparison with Reynolds stress components of the carrier phase (not shown) indicated that the dispersed phase is more anisotropic than the carrier phase. With the neglect of the droplet-droplet interaction there is no direct mecha- nism similar to the fluid pressure (within the dispersed phase) for the exchange of energy among various direc- tions. This significantly enhances the anisotropy of the Reynolds stress components in the dispersed phase dur- ing the early stages of the simulations. At longer times, the indirect effects of the fluid pressure on the droplets (through drag) diminishes the level of anisotropy in the dispersed phase. The results in Fig. 3 indicate that the anisotropy of the dispersed phase decreases with the de- crease of either the mass loading ratio or the droplet time constant.

The effects of evaporation and reaction on the alignment of the droplet velocity with the carrier phase velocity can be investigated by considering the temporal evolution of cos(ut,vi) =< urvj >> /[< (UT)’ >><( v: >I:. This

-n Ai-- -A-- f

M base case

0 2 4 6 8 10 12 14 St

Figure 4: Alignment of the velocity of the droplets with the velocity of the carrier phase.

alignment affects the magnitude of the correlation be- tween the droplet and fluid velocities, < Urvi >>, which is a key element in the modeling of two-phase flows. Fig- ure 4 shows that, during the early times, the alignment decreases rapidly as the droplets are unable to adjust to fast evolution of the turbulence. At long times, however, the turbulence is more established and it is easier for the droplets to follow the fluid motion. As expected, two- way coupling improves the alignment by modifying the velocity of the surrounding fluid to mimic some of the features of the droplet velocity. Since smaller droplets have less inertia and more easily follow the fluid mo- tion, a larger alignment is observed for the case with r&J = 0.5rdcb. It is also noted that the change in the mass loading ratio has very little effect on the alignment at long times.

In the stochastic modeling of two-phase flows, of great interest is the form of the probability density function (p.d.f.) of the droplet velocity fluctuation. In Fig. 5 the temporal evolution of the skewness and kurtosis of the droplet velocity in the streamwise direction are pre- sented; similar trends were observed in the other direc- tions. The general observation is that the p.d.f. of the droplet velocity is not significantly affected by evapora- tion and chemical reaction. The skewness is very close to zero throughout the simulation and, except for a short time around St = 3, the kurtosis values remain close to 3. These values are very similar to those for a normal distri- bution and suggest that the droplet velocity fluctuations

6 American Institute of Aeronautics and Astronautics

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

skewness

o--f) base case

kurtosis

0 2 4 6 8 10 12 14 St

Figure 5: Skewness and kurtosis of the droplet fluctuat- ing velocity in the streamwise direction.

may be accurately modeled by a Gaussian p.d.f. This is in particular true at long times when the turbulence and the droplets reach some quasiequilibrium states.

Figure 6a shows the temporal variations of the mean tem- perature of the carrier phase. A slight growth in the tem- perature is observed in the absence of chemical reaction (i.e. for the one-way case) due to the energy added to the system by the mean shear. As expected, the growth rate is much larger in the presence of chemical reaction. It must be noted however that part of the heat released by reaction is consumed for the latent heat of evaporation and does not contribute to the increase of the tempera- ture. To investigate the heat transfer between the two phases, in Fig. 6b we demonstrate the temporal variation of the ‘Lagrangian mean temperature difference,’ calcu- lated based on the gas temperature at the droplet location, T’. It is observed that this difference increases in time for all the reacting cases. One exception is the case with ‘rdo = 0.5TdOb at long times due to the full evaporation of a portion of the droplets as was discussed earlier in this section. For this case the temperature difference at long time is smaller than the base case despite a higher mean temperature of the carrier phase. This is due to smaller heat capacity of the smaller droplets. The increase of the mass loading ratio results in the increase of both < T > .and << T+ -Td > at long times.

The effects of evaporation and chemical reaction on the temperature fluctuation can be further studied by con- sidering the temporal variation of the root mean square

1.0

(3--E) base case

2 4 6 8 10 12 14

0.35 1 ’ I r I ’ I a 1 ’ M base case Cb)

0.28

0.00 2 4 6 8 10 12 14

St

Figure 6: Temporal variations of (a) the mean carrier phase temperature and (b) the Lagrangian mean temper- ature difference.

(r.m.s.) of this quantity. Figure 7a shows the variations Of Tdrm = [< (Td- < Td XZ-)~ >>]i for the droplets. It

is observed in Fig. 7a that, in the absence of the chem- ical reaction, the curve for the one-way coupling case approaches an asymptotic value at long times. The r.m.s. temperature for cases with chemical reaction, however, do not reach asymptotic values. The results show that the tm.s. values are smaller for temperature fluctuations of the droplets than those of the fluid (not’ shown). This is due mainly to the larger heat capacity of the droplets. For the same reason, the r.m.s. temperature is larger for the case with smaller initial droplet time constant. The trend of variation of r.m.s. temperature with the mass loading ratio is not as clear, although it appears that at long times Tdrms decreases with the increase of the.mass loading ratio. To investigate the effects of reaction on the p.d.f. of the droplet tempersiture fluctuations, in Fig. 7b the temporal evolution of the skewness and kurtosis of

7 American Institute of Aeronautics and Astronautics

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

Q--O base case

0 2 4 6 8 10 12 14

-1 0 2 4 6 8 10 12 14

St

Figure 7: Temporal variations of (a) r.m.s. and (b) skew- Figure 8: Temporal variations of (a) the mean and (b) the ness and kurtosis of the droplet temperature. r.m.s. of the turbulence Mach number.

.

Td is shown. It appears that the skewness and kurtosis values for various cases are close to those for a Gaussian p.d.f. and suggest that, in the absence of more accurate information, the p.d.f. of the droplet temperature fluctu- ation may be approximated by a normal distribution. An interesting feature noted in Fig. 7b is that chemical reac- tion- results in negative skewness, i.e. when the droplets- are reacting, a larger number of the droplets exhibit tem- peratures lower than the mean temperature.

The effects of the droplets on the turbulence Mach num- ber, M = [Uiui/(@T)] 1, are investigated by considering the temporal variations of the mean and the r.m.s. of this quantity. A comparison with the one-way coupling case indicates that both the mean and the r.m.s. values of the Mach number decrease in the presence of chemical reac- tion, This is in agreement with previous studies in sin- gle phase flows in that chemical reaction diminishes the compressibility effects. The decrease of the Mach num-

.

2 4 6 8 10 12 14

0.08 .._...... ~.-.-----..~ . . . . . . . . q .___-_.._

0.07

0.06 0 2 4 6 8 10 12 14

St

ber is despite the fact that the velocity fluctuations may increase when the droplets are allowed to react (cf. Fig. lb for rdc = 0.%&b). It is seen in Fig. 6a that the tem- perature of the carrier phase significantly increases for cases with chemical reaction. This results in the increase of the speed of sound and the decrease of the Mach num- ber. The results in Fig. 8 also show that the Mach irum- - .- -- ber is small for various cases and the carrier phase is free of shocklets. This is an important issue in this study as a Fourier pseudospectral method has been used for the simulation of the carrier phase.

Conclusion

Results of numerical simulations are used to investigate the effects of the mass loading ratio and the droplet time constant on statistics of temperature and velocity in a re- acting droplet-laden homogeneous shear turbulence. The

8 American Institute of Aeronautics and Astronautics

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

chemical reaction takes place in the carrier phase and is described by a single-step, second-order irreversible scheme. The statistics presented here may be used for development and/or assessment of statistical/stochastic models for two-phase turbulent flows. It is shown that the turbulence kinetic energy and its dissipation rate both in- crease with the decrease of either the mass loading ratio or the droplet time constant. The drag of the droplets acts as a dissipation for the kinetic energy during the early stages but assumes very small values at long times - in the case with the highest mass loading ratio the droplets tend to act as a source for the kinetic energy of the car- rier phase at long times. The anisotropy of the dispersed phase is decreased with the decrease of the mass load- ing ratio or the droplet time constant. The alignment of the droplets velocity with the carrier phase velocity is in- creased with the decrease of the droplet time constant. The change in the mass loading ratio does not show any significant effect on the alignment at long times. The results suggest that the droplet velocity and temperature fluctuations may be reasonably approximated with Gaus- sian distribution - this is more strongly true for velocity fluctuations. Both the mean temperature of the carrier phase and the Lagrangian mean temperature difference increase with the increase of the mass loading ratio. The r.m.s. of the droplet temperature fluctuations increases in time in the presence of chemical reaction, and shows an increase with the decrease of the droplet time constant.

Acknowledpment

The support for this work was provided by the U.S. Of- fice of Naval Research under Grant N00014-99-1-0808 with Dr. G. D. Roy as Technical Monitor, and by the National Science Foundation under Grant CT%9874655 with Dr. M. C. Roco as Program Director. Computational resources were provided by the San Diego Supercomput- ing Center.

References

[l] Mashayek, F., “Numerical Investigation of React- ing Droplets in Homogeneous Shear Turbulence,” J, Fluid Mech., 1999. in press.

[2] Wang, L.-P and Maxey, M. R., “Settling Velocity and Concentration Distribution of Heavy Particles in Isotropic Turbulence,” J. Fluid Mech., Vol. 256, 1993, pp. 27-68.

[31

141

iI51

I.61

171

181

PI

[W

r111

WI

Eaton, J. K. and Fessler, J. R., “Preferential Con- centration of Particles by Turbulence,” Znt. J. Mul- tiphase Flow Suppl., Vol. 20, 1994, pp. 169-209.

Riley, J. J. and Patterson, G. S., “Diffusion Exper- iments with Numerically Integrated Isotropic Tur- bulence,” Phys. Fluids, Vol. 17,1974, pp. 292-297.

Squires, K. D. and Eaton, J. K., “Particle Response and Turbulence Modification in Isotropic Turbu- lence,” Phys. Fluids, Vol. 2, 1990, pp. 1191-1203.

Elghobashi, S. and Truesdell, G. C., “On the Two- Way Interaction Between Homogeneous Turbu- lence and Dispersed Solid Particles. I: Turbulence Modification,” Phys. Fluids, Vol. 5, No. 7, 1993, pp. 1790-1801.

Mashayek, F., Taulbee, D. B., and Givi, P., “Model- ing and Simulation of Two-Phase Turbulent Flow,” in Propulsion Combustion: Fuels to Emissions (Roy, G. D., ed.), ch. 8, pp. 241-280, Washington, D.C.: Taylor&Francis, 1998.

Mashayek, F., “Droplet-Turbulence Interactions in Low-Mach-Number Homogeneous Shear Two- Phase Flows,” J. Fluid Mech., Vol. 376, 1998, pp. 163-203.

Mashayek, F., Barre, C., and Taulbee, D. B., “Di- rect Numerical Simulation of Particle-Laden Ho- mogeneous Plain Strain Turbulent Flow,” in the Proceedings of the First Znt. Symp. on Turbulence and Shear Flow Phenomena, (Santa Barbara, CA), 1999. To appear.

McLaughlin, J. B., “Aerosol Particle Deposition in Numerically Simulated Channel Flow,” Phys. Flu- ids, Vol. 1,1989, pp. 1211-1224.

Samimy, M. and Lele, S. K., ‘Motion of Particles with Inertia in a Compressible Free Shear Layer,” Phys. Fluids, Vol. 3, No. 8, 1991, pp. 1915-1923.

Pan, Y. and Banerjee, S., “Numerical Simulation of Particle Interactions with Wall Tkubulence,” Phys. Fluids, Vol. 8, NO. 10, 1996, pp. 2733-2755.

[13] Miller, R. S. and Bellan, J., “Direct Numerical Simulation of a Confined Three-Dimensional Gas Mixing Layer with One Evaporating Hydrocarbon- Droplet Laden Stream,” J. Fluid Mech., Vol. 384, 1999, pp. 293-338.

[14] Simonin, O., Deutsdh, E., and Boivin,. M., “Large Eddy Simulation and Second-Moment Closure Model of Particle Fluctuating Motion in Two-Phase Turbulent Shear Flows,” in Turbulent Shear Flows

9 American Institute of Aeronautics and Astronautics

(c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization.

9 (Durst, E, Kasagi, N., Launder, B., Schmidt, F., and Whitelaw, J., eds.), pp. 85-115, Springer- Verlag, 1995.

[15] Blaisdell, G. A., Mansour, N. N., and Reynolds, W. C., “Numerical Simulation of Compressible Ho- mogeneous Turbulence,” Department of Mechani- cal Engineering Report ‘IF-50, Stanford University, Stanford, CA, 1991.

[16] Rogallo, R. S., “Numerical Experiments in Homo- geneous lkbulence,” NASA TM 81315,198l.

10 American Institute of Aeronautics and Astronautics