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A9993371 1

AMA 99-3760 Numerical Simulation of Polydisperse Two-Phase Turbulent Jets J.M.M. Barata, P.S.N.D. Lopes and N.F.F. Perestrelo Universidade. da Beira Interior Coviih2, Portugal

30th AMA Fluid Dynamics Conference ’ .28 June - 1 July, 1999 / Norfolk, VA

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AMA-99-3760

Numerical Simulation of Polydisperse Two-Phase Turbulent Jets Jorge M. M. Barata”, Pedro S. N. D. Lopes and Nuno F. F. Perestrelo

Aerospace Sciences Department Universidade da Beira Interior

Rua Marques d’Avila e Bolama 6200 Covilha, Portugal

Abstract

This paper describes the application of a computational method to the problem of a polydisperse two-phase turbulent jet. An Eulerian iiame for the gas phase was used in conjunction with a Lagrangian approach to describe both interphase slip and turbulence on particle motion. Turbulence modulation and anisotropy effects were introduced and yere found to be very important to the successful performance of the computational method. Good agreement of the computations with the experimental data is obtained for both the gas and the particles.

cp Cd, = Gz, c&3 D Z= dP = g = k = L zz P = R = Rt? = s = t = tc = u zz V =

X =

Greek symbols A = CD =

l- = E z.z

P zz

Nomenclature

coefftcients in the turbulence model

diameter of the jet particle diameter gravitational acceleration turbulent kinetic energy eddy size pressure radial coordinate Reynolds number source term time transit time axial velocity, u = U + 24’ radial velocity, v = V + v’ axial coordinate

time interval turbulent kinetic energy production term transport coefficient dissipation rate of turbulent kinetic energy dynamic viscosity

*Associate Professor. Senior member of AIAA. Copyright 0 1999 The American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

V zz

P =

01 =

z =

TFL =

zp =

G =

01 =

Subscripts =

; = i =

j = 0 =

i = =

m = P ZZ rms =

Superscripts OLD zz NEW =

kinematic viscosity density turbulent PrandtVSchmidt number turbulent time scale eddy lifetime particle relaxation time residence time of the particle in the eddy turbulent Prandtl/Schmidt number

eddy property fluid property ith time interval; ith coordinate

jth coordinate jet-exit conditions conditions at the jet centerline Lagrangian maximum value particle property root mean square

present time step after one time step

1. Introduc!

Describing the motion of a dispersed phase is complex and is of great interest in a very wide range of practical situations. The applications that require the solution of this problem are as varied as the dispersion of passive pollutant particles in the atmosphere to combustion systems with dispersing fuel particles. At the present, considerable effort is being made to lower the emissions of oxides of nitrogen (NOx) and other minor species from aircraft gas turbine engines. However, advanced technology for lowering NOx emissions, such as the Lean Burn, Premixed and Prevaporised, has not been incorporated into current production engines, and more research in particle dispersion is needed to improve gas turbine design methods.

A comprehensive undeistanding of the spray dispersion processes requires a detailed knowledge of

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the interaction mechanism between the droplets id the surrounding turbulent fluid. Earlier theofetical and experimental investigations of two-phase jets have recently been reviewedt6, and in general they have

-shown a strong influence of operating parameters such as size distribution, mass loading and droplet/gas aerodynamics. Many of the experimental studies have been undertaken in the absence of the complexities introduced by droplet- evaporation, and for simple flow configurations, which reproduce the main features found in practical systems. They have selected two main fundamental cases: the particle- laden plane shear layer7-’ and the particle-laden jet”- “. Most of the above mentioned works focused on the effects of the dispersed phase on the gas-flow propertiesI and very little information regarding particle quantities is reported. Others” have made measurements of both phases but do not report sufficient information concerning initial conditions and structure parameters for the modelling of the flow. More detailed measurements were performed to provide a more complete data set about the two phases12’2’, but they also concentrated on the interaction mechanisms between the two phases, and consequently they used monosize, particle distributions. To provide a better insight into the spray drop behaviour is necessary to study the effect of size distribution22. Experimental measurements in polydisperse two-phase flows are sparsely reported in literature23-27 and do not provide enough infdrmation for modelling purposes. Recently, detailed measurements were reported for particle size distributions concerning complex geom@ries28*2g, but also for the particle-laden je?“.3’, which is of primary interest for modelling purposes.

w-90 pm normal distribution

50 pm mean diameter

$5 Mm standard deviation

Fig. 1 Flow configuration.

The present paper reports a numerical study of the particle dispersion -in polydisperse two-phase turbulent jets based on the measurements of Heitor and Moreira3’. The flow configuration is shown in Figure 1 and consists in a jet with glass beads issuing vertically downward from a pipe. The glass beads used in the experiments follow closely a normal size distribution in the range of lo-9Opm with a munber- average particle diameter of 50ym and a standard deviation. of 15pm. The corresponding Reynolds number at the jet exit is 15,000 based on a bulk velocity of IO&s.

The next section gives the details of the mathematical model. Section 3 presents a quantitative comparison of numerical results with the measurements of Heitor and Moreira3’. The final section summarises the main findings and conclusions of this work.

2. Mathematical Model

This section describes the mathematical model for turbulent particle dispersion assuming that the particles are sufficiently dispersed so that particle- particle interaction is negligible. It is also assumed that the mean ,flow is steady and the material properties of the phases are constant.

The particle phase is described using a Lagrangian approach while an Eulerian frame is used to describe the effects of both interphase slip and turbulence on particle motion using random-sampling techniques (Monte Carlo).

In the following paragraphs the governing equations, solution procedure and boundary conditions are summarized.

2.1 Continuous Phase

The method to solve the continuous phase is based on the solution of the conservation equations for momentum and mass. Turbulence is modeled with the “k-a” turbulence modeP2. A similar nietbod has been used for three-dhnensionaP3 or axisymmetric flows ‘2s34s35 and only the main features are summarized here.

The governing equations constitute a set of coupled partial differential equations that can be written in a general form as

where 4 may stand for any of the velocities, turbulent kinetic’enerb, dissipation, or any scalar property, and r+ &d S4 take on different values for each particular 4 as shown in Table 1.

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Table 1. The fluid phase differential equations.

where

The turbulent diffusion fluxes are approximated with the two-equation “k-d’ model described in detail by Launder and Spalding32. The Reynolds stresses are related to the rate of strain by the following equation:

where ,f+ is the turbulent viscosity, which is derived from the turbulence model and expressed by

pT = C,PC E

The turbulence model constants that are used are those indicated by Launder and Spalding32 that have given good results for a large number of flows, and are summarized in the next table.

Table 2. Turbulence model constants.

0.09 1.44 1.92 1.0 1.3

2.2 Dispersed phase

The dispersed phase was treated using the Lagrangian reference frame. Particle trajectories were obtained by solving the particle momentum equation through the Eulerian fluid velocity field, for 1,000 groups of 2,000 particles that were found to be statistically representative of the measured distribution.

The equations used to calculate the position and velocity of each particle were obtained considering the usual simplification for dilute particle-laden flowP2. Static pressure gradients are small, particles can be assumed spherical and particle collisions can be neglected. Since pp/p,)200, the effects of Basset, virtual mass, Magnus, Saffinan and buoyancy forces are negligible5,12. Under these conditions the simplified particle momentum equation is:

aupi _ 1 b dt -i-, Uf;i -"p;i ( 1 +gi

The mathematical expression for the relaxation time, zp, is

zp = 24ppd; 1%~~ CD Rep

where Re, is the particle Reynolds number,

Re, =

and CD is the drag coeffrcient5,

for Rep< 1 03.

The particle momentum equation can be analytically solved over small time steps, At, and the particle trajectory is given by

NEW up;i -A”rp

x;

The critical issues are to determine the instantaneous fluid velocity and the evaluation of the time, At, of interaction of a particle with a particular eddy.

The time step is obviously the eddy-particle interaction time, which is the minimum of the eddy lifetime, zFL , and the eddy transit time, t,. The eddy lifetime is estimated assuming that the characteristic size of an eddy is the dissipation length scale in isotropic flow:

& &

where A and B are two dependent constants’.

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The transit time, to is the minhmmr time a particle would take to cross an eddy with characteristic dimension, I,, and is given by

where < is the relative velocity between the particle and the fluid (drift velocity).

A different expression for the transit time is also recommended in the literature5’36’37, and was used in the present work:

where the drift velocity is also estimated at the beginning of a new iteration.

This equation has no solution when le ’ zp I”f;i - ‘p;i 1 9 that is, when the linearized stopping distance of the particle is smaller than the eddy size. In such a case, the particle can be assumed to be trapped by the eddy, and the interaction time will be the eddy lifetime.

The instantaneous velocity at the start of a particle- eddy interaction is obtained by random sampling from an isotropic Gaussian pdf having standard deviations of m and zero mean values.

The above isotropic model was extended in the present work to account for cross-correlation’s and anisotropy. To obtain the fluctuating velocities u> and v> two fluctuating velocities u’~ and uf2 are sampled independently, and then are correlated using the correlation coefficient R,:

U’f = d1

v; =RJtl+ JT, l-z+,

u’, v; where R,,, = _ _ was obtained from the

g/z measurements.

The interaction between the continuous and dispersed phase is introduced by treating particles as sources .of mass, momentum and energy to the gaseous phase. The source terms due to the particles are calculated for each Eulerian cell of the continuous phase12 and are summarized in Table 3.

4

Table 3. Particle source terms.

4 s PI

k -- USPU,” - =pu,,

& E aspu,v

-2uJu, -- k &

where Sti = 2 [Aii~~u~;~]k I Vj and Cd is an k=l

empirical constant12~15.

2.3 Solution Procedure

The solution procedure for the continuous phase is based on the SIMPLE algorithm3’ widely used and reported in the literature.

A solution for the gaseous field assuming no particles is initially obtained, and the particle trajectories and source terms are calculated. The gas field is then recomputed with the contribution of the particle source terms. This process is repeated until convergence is achieved.

The particle trajectories are obtained using the SSF model described in the previous section. The sizes of the particles are generated randomly so that their distribution corresponds to the experimental data. The initial position at the jet exit is also generated randomly and the initial velocity and turbulent kinetic energy are obtained horn the measurements for the corresponding location, The fluctuating velocity components are generated and added to the local velocity components. Using the particle equation the next particle velocity is obtained. The new position of the particle is calculated- by integrating the velocity equation of the particle over the time step given by the minimum of the local eddy transit time and the local eddy lifetime. The corresponding particle source terms are stored, and the process is repeated until the particle leaves the calculation domain.

2.4 Boundary Conditions

The computational domain has four boundaries ivhere dependent variables are specified: an inlet and outlet radius, a symmetry axis and a free boundary parallel to the axis. The sensivity of the solutions to the location of the boundaries was investigated, and their final position is sufficiently far away from the jet so that the influence on the computed results

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0.25 -

OO 5 10 15

Fig. 2 Nozzle exit conditions for the (a) mean axial velocity, U, and the (b) fluctuating axial velocity component, lJ,m: 0, measurements3’ at XZI=O.15; -, boundary values used at x/D=O.

is negligible. At the inlet boundary a l/5 power law velocity profile obtained from the measurements was used for the axial velocity (see Fig2a). The k profile -- at the jet exit was prescribed assuming u” = v12 and - w12 = 0, and by extrapolating the measured values at

x/D=O.2 (see Fig.2b). On the symmetry axis, the normal velocity vanishes, and the normal derivatives of the other variables are zero. At the outflow boundary, the gradients of dependent variables in the axial direction are set to zero.

3. Results and Discussion

In this section, the numerical predictions are compared with the measurements of Heitor and Moreira3’.

The predicted spread of the jet and axial velocity decay is used to test the grid dependency of the computation. Figure 3 compares the axial velocity decay for different grid sizes and shows that the

results are independent of numerical influences for grids finer than the 3 1 x3 1 mesh.

The numerical results presented in this section are compared with the Laser Doppler measurements of Heitor and Moreira3’ obtained for a jet with glass beads characterized by a normal size distribution in the range of lo-9Opm with a number-average-particle diameter of 50~ and standard deviation of 15pm. Figures 4 and 5 illustrate the influence of the particles in the mean properties of the carrier phase, while figure 6 and 7 show mean and turbulent velocities for different classes of particle diameters.

The predicted mean velocities are illustrated in figure 4, and the results are compared with the

Fig. 3 Grid size dependency test based on the axial velocity decay.

measurements3 ‘. Calculated velocities of the gaseous flow are presented together with values for each individual particle. From figure 4a is evident that the larger particles are moving slower than the smaller particles at the center of the jet, but they are moving faster near the edge of the jet. The differences in the axial velocities of the particles are larger near the edge of the jet.

The calculated values for the pure gas jet are also plotted in figures 4 and 5 and revealed that the exchange of momentum and turbulence energy between the two phases is very important for the case of a polydisperse two-phase-jet. In the outer region of the jet the values of the mean ‘axial velocity of the gas with or without particles are similar, but this doesn’t mean that the particles follow more closely the gas in the outer part of the shear layer. This may be explained by the dependency of the efficiency of momentum transfer between phases on the particle mass flux, and this decreases radially outwards at a much larger rate than the velocity.

Figure 4b shows radial profiles of the radial velocity and reveals very high velocity fluctuations, although they are smaller in absolute value than those of the axial velocity. The calculated radial velocities for x/D>6.5 show negative values for the

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a) Mean axial velocity, U b) Mean radial velocity, V 3

2.5 2

1.5

I v 5. 0.5 >

0

4.5

-I

-1.5

XID=0.2

particles around the centerline. This is probably associated with an inward shear induced lift force acting more effkiently on small particles due to their lower response time40~41, but it still requires further investigation.

A more detailed analysis on the response of the particles to the air flow can be done with the help, of figure 5. The calculations for the gaseous phase of a single and two-phase jets are compared with the

X/D-10.9 : .

.*.. 0

X/D=10.9

r(mm) r(mml

Fig. 4 Radial profiles of the mean velocity characteristics. @xperiments3’: A, 30-35pm; n ,40-45pm; + , 60-65pm; 0, gaseous phase. Predictions: o , particles; 7 , gaseous phase; -. - , gaseous phase without particles).

measurements of Heitor and Moreira3’. Near the jet exit at x/D=O.2 the particle. axial velocity fluctuations are larger than those of the air at the center of the jet, but further downstream, the smaller beads respond faster. However, the small particles always exhibit larger axial velocity fluctuations.

The predictions of 7 show that the axial J-’ velocity fluctuations of the gas phase increase due to

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- a) Axial velocity fluctuations, J- U I2

2- :

1.75 -

X/D=0.2

1. X/D=&5

1.2

g

d 0.7

X/D-10.9

0.25 -

0 .b.““, 1 Q” ‘Q -,sb r(mm)

- b) Radial velocity fluctuations, J- v12

1.7 XID=0.2

Fig. 5 Radial profiles of the fluctuating velocity characteristics. (Experiments31: II ,30-35pm; n ,40-45pm; + ,60-65pm; 0, gaseous phase. Predictions: o , particles; - , gaseous phase; -. - , gaseous phase without pkticles),

the presence of the particles and agree reasonably velocity fluctuation increase 14% of the single-phase well with the measurements. However, the turbulence value. - intensity J-- a” /U is reduced due to the relative velocity between particles and the gas, which increases the dissipation rate of turbulent kinetic energy of the gas. At X/D = 10.9 the local turbulence intensity is reduced from 24% to 19.5% at the centerline, although the absolute values of the

The results for @(figure 5b) agree qualitatively well with the measurements, but in the region far away Corn the jet exit the velocity fluctuations are overestimated. They are smaller than those of the gas are, but they correspond to values generally larger than the mean local values, which are close to zero.

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a) Mean axial velocity, U 1

b) Mean radial velocity, V

,:m

Fig. 6 Radial profiles of the mean velocity characteristics of different classes of particle diameters at X/D= 0.2, A ; 6.5, H ; and 10.9, + . (Open symbols: present work; Closed symbols: experiments31)

a) Axial velocity tluctuations, -

b) Radial velocity fluctuations, d-- v12

rtmm)

Fig. 7 Radial profiles of the mean velocity characteristics of different classes at X/D= 0.2, A ; 6.5, n ; and 10.9, +. (Open symbols: present work; Closed symbols: experiments3’).

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- . - . _--. . -,. b I 0 , I *. , ,. I ..I ,a.> . - I.., I .P , ,- ,..I .,.- , ., ,. J , ,

0 5 10 15 20 25 X/D

Fig. 8 Numerical visualization of the flow. (a) Gaseous particles and LHF; (b) Particles (SW) and polydisperse two-phase jet (lo-90pm).

Figures 6 and 7 show radial profiles of the mean and fluctuating velocity characteristics. For comparison with the measurements, the predictions were number averaged over the same size classes used in the experiments (30-35, 40-45 and 60-65pm). The axial flow velocity fluctuations of the particles are shown in figure 7 , and are much larger than those in the radial direction confirming the large values of turbulence anisotropy, I@ /p observed experimentally. This particular result is a consequence of the use of the anisotropic approach, and the agreement with the experiments is very good.

Figure 8 shows velocity vectors and particle tracks or streaklines of (a) the gaseous phase with particles using a local homogeneous flow method (LHF), and (b) the particle tracks obtained with the present method (SSF).

The results of Figure 8a were obtained neglecting the interphase slip slip and turbulence modulation (LHF) and reveal that the particles accelerate the flow and move close to the axis of symmetry with a

straight trajectory. In contrast, the SSF calculations including turbulence modulation and anisotropy effects (Figure 8b) exhibit a completely different pattern. Due to the radial velocity fluctuations some particles can leave the main direction of the gaseous flow for X/D>15, and there are particles in a circular region with a diameter of 20D, while the single phase jet would have a width of only 1.5D. This shows clearly that the anisotropy effects need to be included in a SSF model if particle dispersion needs to be calculated.

4. Conclusions

A computational method has been used to study a polydisperse two-phase turbulent jet, for an exit Reynolds number of 15,000 based on a bulk velocity of lOm/s and a particle distribution in the range of lo- 9Opm with a number average particle diameter of 5Opm and a standard deviation of 15pm.

An Eulerian frame for the gas phase was used in conjunction with a Lagrangian approach to describe

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the effects of both interphase slip and turbulence on particle motion. The computational method yielded good results and revealed great capabilities to improve knowledge of the particle dispersion phenomena and to extend the experimental studies to more complex practical configurations.

The exchange of momentum and turbulence energy between the two phases was shown to play a decisive role on the flow development. The predictions confirmed the measurements and showed that the particles do not follow the turbulence gas flow, but they affect it significantly.

Considerably high values of the radial velocity were detected around the centerline far from the jet exit where the velocity of the particles lags that of the fluid. This seems to be associated with the effect. of shear induced lift forces but further research is still needed.

The predictions obtained for different particle size classes show a good agreement with the experiments and confirmed the large values of anisotropy observed experimentally, which are higher than those of the carrier phase.

Acknowledeements

The present work has been performed in the scope of the activities of the Aero-Thermodynamics Group of the Center for Aerospace Sciences and Technology, and was sponsored by the Portuguese Ministry of Science under Contract Praxis XXI n”CTAE/3/3.1/1917/95 and Grant n”BICJ/3666/95.

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