particle dispersion in highly swirling, turbulent flows

182 Part. Part. Syst. Charact. 10 (1993) 182-190

Particle Dispersion in Highly Swirling, Turbulent Flows

Erich Bliimcke *, Michael Brandt **, Heinrich Eickhoff **, Christoph Hassa ** Dedicated to Professor Friedrich Loyfler on the occasion of his 60th birthday

(Received: 25 September 1991 ; resubmitted: 18 May 1993)

Abstract

Three-dimensional transport processes of monosized droplets in a turbulent swirling shear layer were investigated experimentally and theoretically. A model experiment was designed that represents the spray dispersion produced by airblast atomizers. Based on the experimental results, a stochastic dispersion model was developed in the frame of a Lagrangian formulation. Com-

parison with these experimental data for the dispersion of monosized droplets were made and proved to be satisfactory. The analysis of the remaining differences between calculation and measurement emphasized the importance of an accurate description of gas-phase turbulence characteristics.

1 Introduction

In recent years, it has become more feasible to design combustion chambers of modern gas turbine engines with the aid of numerical computations [l]. For this purpose, a suitable description of the transport of the liquid phase in the two-phase flow is essential. The modelling of two-phase flow phenomena such as turbulent particle dispersion can be done by two different approaches, the Eulerian formulation [2-41 or, more frequently found in the literature, the Lagrangian approach [5 - 81. Thereby, individual particles are tracked through a given turbulent gas flow field. Statistics over a significant number of trajectories deliver mean values characterizing the behaviour of the dispersed phase. The major problem in modelling the turbulent particle dispersion, i. e. the influence of gas turbulence on particle transport, in a Lagrangian frame is the determination of the actual gas velocity experienced by the particle. DesjonquPres [9] proposed a model accounting not only for a probability density function of the fluctuating velocities of the gas phase but also for a predefined spectral distribution of the turbulent kinetic energy of the gas phase. The main purpose of this investigation was the development of a so-called “spectral dispersion model” (following the ideas of DesjonquPres) and its evaluation with detailed experimental results. For this purpose, a model experiment was designed that exhibits some essential features of the two-phase flows found in combustors with airblast atomizers. In order to ensure well defined boundary and initial conditions for the liquid phase, this experiment clearly separates transport phenomena from effects originating from the atomization process. In Section 2 the Lagrangian dispersion model is described in detail. The measurement diagnostics used to analyse the liquid flow in the model experiment are covered in Section 3. The model experiment is presented in Section 4 and the com-

* Dr.-Ing. E. Biumcke, Audi AG, 85045 Ingolstadt (Federal Republic of Germany).

** Dip1.-Phys. M. Bran& Prof. Dr.-Ing. H. EickhofJf Dip1.-Ing. C. Hassa, DLR - Institute for Propulsion Technology, Linder Hohe, 51147 Kdln (Federal Republic of Germany).

parisons between predicted and measured data are discussed in Section 5.

2 Two-Phase Flow Model

The numerical calculations of particle transport processes in a Lagrangian frame are based on a measured or calculated mean gas flow field, where the available information on gas turbulence is limited owing to the averaging process. In the present approach, the axisymmetric gas flow field is described by the time mean equations for conservation of mass and momentum supplemented by a standard k-&-turbulence model. The numerical predictions of the gas flow were performed using a finite volume method [lo]. From the turbulent kinetic energy and the dissipation of turbulence, Lagrangian scales in time and space of large turbulent eddies can be evaluated. Following DesjonquPres [9 ] , a spectral distribution of the turbulent kinetic energy is additionally assumed. This means that the model accounts for gas velocity fluctuations connected with eddies of different sizes. The particle motion is described by an instantaneous momentum equation:

24 C,, = -(l + 0.15 x Re$687)

Re,

(3)

The assumption of a dilute spray is justified by the low mass loading and by the large droplet spacings in the experiment considered here. For the simulation of the particle motion in a turbulent environment, the instantaneous gas velocities experienced by the particle must be determined. In accordance with the averaging procedure of the Navier-Stokes equations, the components of the gas velocity are subdivided into mean and fluctuating parts:

0 VCH Verlagsgesellschaft mbH, D-69469 Weinheim, 1993 0934-0866/93/0407-0182 $5.00 + .25/0

Part. Part. Svst. Charact. 10 (1993) 182-190 183

(4)

The time mean velocity field is known from the analysis of the gas flow, either experimentally or numerically. In the present approach it is assumed that for the calculation of stochastic droplet trajectories the gas turbulence can be characterized by a Gaussian probability density function and a predefined Lagrangian autocorrelation function R of the velocity fluctuations. Thereby, the turbulent kinetic energy is distributed over a range of frequencies (hence the name “spectral dispersion model”):

u; ( to) x u; (to + A t ) R, ( A t ) =

URMS,i URMS,i + (5 )

As a first step, a pure exponential form of the autocorrelation function in time and space is assumed:

(7)

where 5L and LL denote the Lagrangian integral time and length scales, respectively. Various values for the coefficient CT can be found in the literature. Here the expression suggested by Cdabrese and Middleman [ll] for isotropic turbulence was used:

The Lagrangian integral length scales were described by Eq. (9), which assumes a link between the time and length scales [12]. The coefficient C, was found by computer optimization when simulating the experiment of Snyder and Lumley [13], where all parameters are known except the length scale. The values of the two coefficients remained unchanged for all predictions.

(9)

The generation of the velocity fluctuations is effected by a digital filtering procedure, as proposed by Ofivari and Benocchi [14]. The transfer function of such a filter is identical with the autocorrelation function of the Lagrangian velocity fluctuations and therefore requires as input parameters the Lagrangian integral time and length scales. The transfer function is now z- transformed and the discrete transform is multiplied with a ran- dom number sequence having a Gaussian probability density distribution and a “white noise” spectrum. That means that the convolution theorem in its discrete form is used to generate step by step a number sequence of output signals that exhibits a Gaussian PDF as well as a predefined spectral function. The digital filter is represented in the physical domain by a recursive equation:

1 m - 1 1 ” y ( l ) = - x [ c b j x v ( l + j - n ) - c c i x y ( l + i - r n ) (10)

where y is the output number sequence and v is the input number sequence. The coefficients and the order of this recursive equation are determined via the transfer function H(z) :

where Y and I/ denote the number sequences in the z-space. For the autocorrelation function used in the present approach [Eq. (6)], the z-transform is given by

Z H(z ) =

z - exp[ -el ‘ Eq. (10) then gives

as the recursive equation in the time space. The digital filtering procedure does not conserve the kinetic energy of the velocity fluctuations. Therefore, the output number sequence has to be renormalized by a constant factor:

Y,,,, = V i - exp [ -2 x c] . This renormalized filter output sequence is then multiplied by the rms value of the velocity component. Each of the three velocity components is assigned to one digital filter. This technique, which can easily be extended to account for anisotropic turbulence [15], permits the calculation of the trajectory of a fluid particle. Owing to the inertial force, the spectral manipulation technique, working solely in time, is not sufficient for the provision of velocity fluctuations along the path of a droplet (see Figure 1 a). Therefore, a fluid particle and a discrete particle are always followed simultaneously in the dispersion model. To that effect, the spatial evolution of velocity fluctuations between the location of the fluid particle and that of the discrete particle also has to be resolved by the spectral manipulation technique, which is now applied in the space domain as long as the droplet stays in a correlation domain. A correlation domain is defined by the volume in which there is a correlation between the velocity fluctuations at the location of the fluid point and that of the particle. For the autocorrelation function used, the dimension of the correlation domain is fixed by (see also Figure 1 b)

Leaving its correlation domain, the particle experiences a crossing trajectory effect. The initial position in the new correlation domain, shown in Figure 1 b, is then re-established while the digital filters are restarted. The remaining task of the proper matching of the two digital filters, working in time, Figure 1 bII, and space, Figure 1 bIII, requires that the filter input signals have to be stored. The matching instruction originates from the assumed link of Lagrangian time and length scales, Eq. (9). As a further conse- quence of Eq. (9), the same shape of the transfer functions of the digital filters working in time and space is assumed. The dimensionless step widths of the digital filters are fixed:

184 Part. Part. Syst. Charact. I0 (1993) 182-190

- Q - fluid particle + discrete par-

ticle

Fig. 1: a) Trajectories of a fluid and a discrete particle, resp. b) Link of the two digital filters.

Spatial evolution of gas velocity fluctuations.

I: Trajectory of a discrete particle 11: Correlation domain in time

111: Correlation domain in space

A t AS - - 0.01 .

TL LL

The storage problem was overcome by storing only one very long sequence of input signals. On this very large vector each particle is assigned to a separate start index and a separate incre- ment for every velocity component. Thereby, it is guaranteed that each velocity component of every particle experiences an individual filter input sequence.

3 Diagnostics

For the measurement of gas and particle velocities, a Model 9100-7 laser-Doppler-anemometer from TSI was used. Gas velocities were measured in a forward-scattering set-up, particle velocities under a 90" scattering angle, to limit accurately the length of the measuring volume in the direction of the optical axis. The size of the l/e ** 2 measuring volume was varied from 50 x 50 x 200 pm near the nozzle to 150 x 150 x 600 pm further downstream in order to adapt to the different droplet con- centrations. The form of the spatial filter used was rectangular, so that the duration of signals from particles entering the beam would not be limited by it. The Doppler signals were Bragg shifted, downmixed and processed by counters. For the measurement of particle fluxes, the counters were triggered by the axial velocity signal; for the other quantities, they were operated in the coincident mode. The computer evaluation was performed with an interface and software from Zech Elec- tronics. Velocity and transit time were transmitted to the com-

puter. With these data, burst length distributions could be assembled, which are the starting point of Suffmunns algorithm [16], giving an online value for the effective measuring volume. The software was modified accordingly, and the method of measuring the particle fluxes is the same as that implemented in commercial phase Doppler anemometers. The accuracy of the measurement is 3.5 Yo for the velocity and 20% for the turbulent kinetic energy. The main sources of error for the particle flux measurement are - incidents when two particles are in the measurement volume

at the same time, causing too small burst-length recordings; - particles crossing the measurement volume with a high

velocity component in the direction of the optical axis, causing the burst length to be limited by the rectangular slit width, again giving too small burst lengths.

When all three mean velocity components are known, the second error can be quantified, and a correction at the measurement plane at z = 49 mm yielded differences of up to 5 %, with the distribution of the relative error giving low values at the maximum flux positions and higher values at the outer boundary of the distribution. Integrating the measured local flux densities over the measurement planes and making a comparison with the production rate of the droplet generator provide a check of the overall accuracy of the technique. The result is that the measured fluxes are 20% too high, which is in accord with the sign of the discussed systematic errors.

4 Model Experiment

The purpose of the model experiment was the quantitative evaluation of models for the turbulent dispersion of particles. In order to simulate these transport processes in gas turbine combustors, the flow of that experiment should exhibit the rele- vant features of such flows like a swirling recirculating shear layer, which itself is a strong source of turbulent kinetic energy. The drops will prevail there, before they are vaporized in a combusting environment. Consequently, the gas flow in the model configuration, Fig- ure 2a, consists of a swirling air jet (1) discharged in a low-velocity co-flowing, non-swirling air stream (2) (mean velocity 4 m/s).

j w x r x d m R x i i x m

S =

R = radial swirler outlet radius u = mass-averaged velocity m = mass flow

The swirl number [defined in Eq. (17)] for the nozzle flow calculated from the measured values is 1.667. The expansion ratio of the nozzle is 1 : 4. Owing to the co-flowing air stream, the swirl flow is only weakly expanding with the streamlines diverging with a 30 O angle. The ratio of swirling to non-swirling flow is 1 : 2. The lower hatched line in Figure 2b gives the extent of the recirculation zone. At x/d = 2.5 the flow is forced through a radial slot, prohibiting any inflow from the downstream boundary. Great care was taken to ensure axial symmetry of the flow, which was achieved by using a radial swirler (3) with 32 con- toured vanes [17]. The use of a miniaturized droplet generator (9, described in more detail in [18], leads to well defined initial conditions and eliminates the need for a combined particle size and velocity measurement for the investigation of the particle dispersion.

Part. Part. Syst. Charact. I0 (1993) 182-190 185

1 inner airflow 2 coaxial airflow 3 swirler 4 swirl channel 5 droplet generator 6 annulus 7 flow straightener

Fig. 2: a) Schematic layout of the experiment. b) Calculated isothermal flow field.

Model experiment, "turbulent particle dispersion"

z Imml

The monodisperse particles were injected at a low velocity into the shear layer and disperse in the hatched region, where detailed measurements of the gas and the dispersed phase properties have been made. The initial conditions for the three flow cases investigated (dispersion of water droplets of either 23, 30 or 44 bm in diameter) are given in Table 1. The maximum of the measured relative size distribution width, 0, was 1.02, giving a maximum difference of about 20% for the particle relaxation times. By positioning the generator off-axis in the annulus between swirling and non-swirling flow - the analogue to the atomizing lip of an atomizer nozzle - the liquid phase flow becomes fully three-dimensional, whereas the underlying gas flow field is still axially symmetric. This means that the model experiment allows the analysis of three-dimensional aspects of particle dispersion process in an axisymmetric gas flow field, thus minimizing the experimental effort.

Table 1 : Initial conditions of the liquid phase of the model experiment "turbulent particle dispersion". Position of the droplet ejection point: axial at z = 0.0 mm; radial at r = 27.25 mm.

droplet- diameter (v)

initial velocities (m/s) RMS velocities (m/s) production axial radial tangential axial radial tangential rate (kHz)

44 10.5 0.15 -0.20 0.3 0.2 0.2 100 30 11.0 0.10 -0.15 0.3 0.2 0.2 130 23 12.0 0.15 -1.50 0.3 0.2 0.2 170

After crossing the short wake of the annulus (6), the particles are accelerated on entering the shear layer, resulting in particle Reynolds numbers of around 100. The ultimate prerequisite for the evaluation of the particle dispersion model is a knowledge of the correct gas flow field

11: 2 = 8 8 . 0 mm I: 2 = 2.0 mm

(a) axial velocity component

(b) radial velocity component 20

(m ls ) Y 10 0 (mYs.1 '!_' -10 -10

-20 -20 0 50 100 50 100

(c) tangential velocity component

1 60 I 1

::I 1 " ; . , , _ , 1 i;J,/-J 0 0 50 100 0 50 100 ~~

(d) turbulent kinetic energy 500 >

300

50 100

- calculation D measurement

Fig. 3: Comparison between measured and calculated data for the gas phase in two cross-sections of the model.

(a) axial velocity component (b) radial velocity component

I 60

- i 0 1 ! 22 30 40 50 60 22 30 40 50 60

(c) tangential velocity component ( d ) turbulent kinetic energy

I

r (mm) r (mm)

- calculation measurement

Fig. 4: Comparison between measured and calculated data for the gas phase at z = 37.5 mm.

including the Reynolds stress tensor and the dissipation. The comparison of calculated and measured data, shown in Figure 3 a and b in three cross-sections, showed a reasonable, but not yet sufficient, agreement for the mean gas flow field. The comparisons are made for the axial, radial and tangential


velocity components and the turbulent kinetic energy. A comparison between measured and calculated data for the gas phase is shown in Figure 4 for the axial distance z = 37.5 mm, where most of the results of the dispersed phase calculations are analysed. A more detailed investigation of the turbulent normal stresses made obvious the anisotropy of especially the radial normal stress, as was also observed by others [19]. In order to provide the correct gas flow field for the evaluation of the newly developed particle dispersion model, measured data, such as the mean gas velocity and the turbulent kinetic energy, replaced the calculated data of the mean gas flow field. As described in Section 2, the turbulent input parameters of the Lagrangian dispersion code are formed by a combination of the k and E values. While E could not be measured, the calculated Lagrangian integral time scales were judged by the evaluation of measured power spectral density distributions of the gas velocity fluctuations [20].

5 Results

The three drop sizes considered in the experiment are within the range of interest for airblast atomizers. The particle Reynolds numbers were kept similar to those in a combustor in order to provide a rough check on the applicability of the chosen particle drag description. Moreover, mean and turbulent particle Stokes numbers of the model experiment are within a range which guarantees that the droplets react on the gas flow in about the same way as in a combustor. The turbulent Stokes number is defined as

TL

5, s, = -

and characterizes the ability of the particles to follow the turbulent movements of the gas flow. Here, tL is the integral time scale of the gas-phase turbulence and tp is the particle relaxation time calculated with the mean relative velocity. An overall impression of the mean dispersion process of 44 pm diameter droplets is given in Figure 6, where calculated number flux density distributions over three successive cross-sections of the experimental facility are shown. In order to resolve graphically the three-dimensional dispersion in the radial- tangential plane, only a rectangular sector of the whole plane is sketched, as indicated in Figure 5 .

-t- Fig. 5: shown. x = Position of the drop ejection.

Position of the Cartesian sector where drop quantities are

low + high particle flux density

Fig. 6: the 44 V r n drops in three cross-sections. Note that the scale splits.

Perspective view of the calculated flux density distribution of

The local flux density was calculated according to the equation

ti flux density = -

Across

where Across is the cross-sectional area of the local volume cell and n is the number flux of particles found in that volume cell per unit time. On entering the turbulent shear layer, the droplets are strongly accelerated by the mean gas flow field. Thereby, the center of gravity of the flux density distribution also moves in the cir- cumferential direction. Owing to the large differences in the mean velocities of the gas phase and dispersed phase, the acceleration rates experienced by the drops were uniformly directed in space, thus leading to particle velocity distribution having a low standard deviation. In the numerical calculations a large number of “crossing trajectory effects” are observed, i. e. the particles experience there an Eulerian turbulence with smaller integral scales of turbulence compared with the original Lagrangian integral scales. After being accelerated to the mean gas velocities, the droplets experience acceleration rates only by being subjected to the turbulent gas velocity fluctuations, thus now “seeing” a Lagran- gian turbulence and thereby the flux density distribution widens more strongly. This behaviour of the particle dispersion in the model experiment can more clearly be recognized in Figure 7 a and b, where the variances of the number flux density distribution with respect to their center of gravity over the axial distance z are shown. These two diagrams have to be understood in much the same way as those of Snyder and Lumley (131. At a distance z = 27.5 mm after the ejection point, the 44 pm drops are accelerated to the mean gas velocity and the curve of


0

D p = 23 urn b)

15

n

E 1 0 - v

w

LD E

5 -

0 0 20 30 40 50

Z ( m m )

--?--4-. tleasurernent ( i n t e r p o l a t e d ) +--e C a l c u l a t i o n (EL= .40)

Comparison between measurement and calculation for the Fig. 7 : variances of the number flux density distributions along the z-axis.

SRMS from there on shows a steeper gradient, as long as almost all of the particles remain in the shear layer. The qualitative differences at the end of the dispersion process between the 44 pm (Figure 7a) and 23 pm droplets (Figure 7 b) can be explained by the well known fan spreading mechanismen of the 44 pm droplets, caused by higher inertia effects, and also by the fact that no longer d o all 23 pm droplets remain in the shear layer, thus experiencing a lower turbulence intensity level. The comparison between measurement and calculation is done for two droplet sizes (44 and 23 pm in diameter). As can be seen in Figure 7, there is good agreement between the measured and calculated data as far as the position of the maximum number flux density and the extent of the flux density distributions are concerned. This represents a considerable improvement in comparison with the predictions for a commonly used dispersion model [21]. The fact that there is good agreement for both particle size classes, featuring different particle inertias, different turbulent Stokes numbers and thus different particle reactions on the turbulent gas movements, corroborates the theory used for modelling the spectral energy transfer from the gas phase to the dispersed phase. It is expected that simulations of particle dispersion in turbulent gas flow fields exhibiting higher turbulence intensities as investigated will deliver the same good results, because higher turbulence intensities themselves d o not change the characteristics of the turbulent spectra. Nevertheless, there remain some discrepancies in the spectral dispersion model, as can be seen in Figure 8. In contrast to the measured flux velocity distribution of Figure 8 a, the calculated data (Figure 8 b) exhibit a preference for dispersion in the cir- cumferential direction, leading to a divergence between the measured and calculated values of SRMs, where the local number flux densities are shown at a height of z = 37.5 mm for the 44 pm droplets.

Fig. 8: Comparison between the measured and calculated flux densities of the 44 pm drops at z = 3 7 3 mm. a) Measured flux density distribution. b) Calculated flux density distribution. c) Relative error of the calculated flux density.

The analysis of the comparison between the measured and calculated mean particle velocities at height z = 37.5 mm (Fig- ure 9 ; I and 11) reveals the reason for this behaviour of the numerical simulations. The area in which the measurements were made was not as large as the evaluated area of the numerical results, because the data rate a t the outer boundary of the distribution was extremly low. Therefore, the areas where the velocities are compared are different. Calculated data represent the average of at least 20 events. In the measurements a different gradation of the mean velocity distributions of the three components can be seen, resulting mainly from the history of individual droplet trajectories, i.e. from inertia effects. Only in the radial velocity component does the gradation qualitatively resemble that of the underlying gas flow, while there is a very smooth distribution of the axial velocity component, contrary


[ mls 1

= ABOVE 4000 3500-4000 = 2800- 3100 2500 - 2800 2 2 0 0 - 2 5 0 0 1900 - 22.00 1600- 1900 a00 1600 1000 am 7 0 0 - x)00 4 m - 700 100- 400

BELOW loo

rn 3100- 3500

[ m/s 1

800 Do0 1 800- 800 4 0 0 - 600 = 200 400 = 0 0 0 - 2 0 0 -200- 000 -400 - 2 0 0 -600 -400

0 BECOW -600

[ mls 1

I: Calculated velocities

r +sin(?) [rnml

I t50-

~3 7 . 5 mm 44 um particle

r osin(y1 [mml 15 25 35 45

60

Po- 30 zs 20 f

z=37.5 mm

t 5 0 _

+

z=37.5 mm 44 um particle

11: Measured velocities

r ~ s i n ( y ) [mml

50 - 140 1 axial velocity component

F37.5 mm

r c sin($ [mml 5 15 25 35 45

60

50 radial velocity component

2-37.5 mm 44 um particle

0

40

30 E+ v) 0

0

r +sinCp) [mml -5 5 15 25 35 45

50

140E

L

~ 3 7 . 5 mm

tangential velocity component

Fig. 9: Comparison between calculated and measured mean velocities of the 44 pm drops at z = 37.5 mm.

to the results of the simulation. This latter fact is primarily due to the differences observed in the radial component: the dispersion model does not reproduce the range of the radial velocities of the droplets (from -8 up to 10 m/s) on the basis of the measurements of the gas-flow - where in the region under con- sideration only positive radial velocities exist - under the assumption of Gaussian probability density distributions of the velocity components. As already mentioned, some deviations from the Gaussian shape of the probability density distribution were observed in the experiment on the radial velocity component combined with a high degree of anisotropy of the Reynolds normal stresses [22] ; the radial normal stress component dominates the other

two compontents in the shear layer. The measured spatial radial velocity distribution of the droplets causes an extensive exchange in the radial direction. Owing to the inertia of the droplets, this increased exchange results in a smooth spatial distribution of the mean axial velocity component and in a high turbulence intensity of that component. Additionally, owing to the steep gradients of the underlying mean gas flow, initially small errors made in the numerical simulation are strongly amplified, as can be seen in the tangential velocity distribution. Therefore, it can be concluded that the wrong prediction of the radial velocity component of the droplets is the main reason for the discrepancies observed in the calculated drop dispersion.


Another difficulty in the calculations arises from the uncer- tainty concerning the empirical correlations used for the determination of the Lagrangian integral time and length scales. These uncertainties are mainly reflected in the differences between calculated and measured variances of the droplet velocity distributions.

6 Conclusions

It has been shown that the model experiment is suitable for the quantitative evaluation of dispersion models. A theoretical model for the description of liquid-phase transport processes in turbulent swirling flows has been developed. Good agreement between measurement and calculation was achieved for all three drop sizes investigated. Therefore, it can be concluded that the spectral energy transfer from the gas phase to the dispersed phase is correctly modelled. The analysis of the deviations of measured from calculated data for the dispersion process in the model experiment emphasized the importance of the correct description of gas-phase turbulence properties in the vicinity of the nozzle.

V

xi Y Y

YnorIIl Z Z E

I(

e L P

Indices g P L t, s

V

-

transformed input sequence of the digital filter spatial coordinate output sequence of the digital filter transformed output sequence of the digital filter normalization factor of the digital filter transform coordinate spatial coordinate dissipation rate viscosity kinematic viscosity density Lagrangian integral time scale particle relaxation time

Lagrangian temporal, spatial turbulent fluctuating property averaged property

7 Symbols and Abbreviations 8 References

cross-sectional area of the control volume polynomal coefficients relative velocity between gas and drop drag coefficient constants of the spectral dispersion model drop diameter gravitational constant z-transformed filter transfer function turbulent kinetic energy temporal and spatial coordinate of the digital filter Lagrangian integral length scale in coordinate direction i extent of the correlation domain in coordinate direction i mass flow polynomial degree number of particles per unit time radius autocorrelation function characteristic radius particle Reynolds number swirl number turbulent Stokes number variance of the number flux density distribution with respect to the centre of gravity spatial discretization width difference length between the location of the particle and that of the fluid point time coordinate temporal discretization width axial, radial and tangential velocity components, respectively velocity component in coordinate direction i mass flow weighted axial velocity root mean square (RMS) velocity in coordinate direction i input sequence of the digital filter

(11 D. L. Burrus: Application of numerical models for predictions of turbine engine combustor performance. ASME 89-GT-251, 1989.

[2] 0. Simonin: Second-moment prediction of dispersed phase turbulence in particle-laden flows. 8th Symp. on Turbulent Shear Flows, 1991.

[3] A. Picurt: Le code DISCO-2 pour la prtdiction du comportement de particules dans un Ccoulement turbulent. These de 3eme cycle, Rouen, 1984.

[4] S. Elgobushi, I: W Abou-Arab: A two-equation turbulence model for two-phase flows. Phys. Fluids 26 (1983).

[5] A. D. Gosrnun, E. Iounnides: Aspects of computer simulations of liquid-fueled combustors. AIAA-81-0323, 1981.

[6] D. Milojevic: Lagrangian stochastic-deterministic (LSD) predictions of particle dispersion in turbulence. Part. Part. Syst. Charact.

[7] J . 3 Shuen, L.-D. Chen, G. M. Fueth: Evaluation of a stochastic model of particle dispersion in a turbulent round jet. AIChE J. 29 (1983).

[8] A. Berlemont, I! Desjonqueres, G. Gouesbet: Particle Lagrangian simulation in turbulent flows. Int. J. Multiphase Flow I6 (1990).

[9] I? Desjonqu2res: Modelisation Lagrangienne du comportement de particules discrttes en tcoulement turbulent. Thtse de 3 eme cycle, Rouen, 1987.

[lo] M. Rachner: Flow computation in combusting chambers using zonal nonstaggered grids. AGARD-PEP-Meeting, USA, 1991.

[ll] R. I! Culubrese, S. Middleman: The dispersion of discrete particles in a turbulent fluid. AIChE J. 25 (1979).

[12] J. 0. Hinze: Turbulence. McGraw-Hill, New York 1975. [13] H. W Snyder, 1 L. Lumley: Some measurements of particle

velocity autocorrelation functions in a turbulent flow. J. Fluid Mech. 48 (1971).

[14] D. Olivuri, C. Benocchi: Modelling of scalar transport in turbulent shear flows with a spectral manipulation technique. Proc. 6th Symp. on Turbulent Shear Flows, 1987.

I151 E. Bliimcke, H. EickhofJ C. Hassa: Evaluation of spectral dispersion model with experimental results of the dispersion of monosized droplets in a turbulent swirling flow. Proc. 5th Int. Workshop on no-Phase Flow Predictions, 1990.

[16] M. Suffmun: Automatic calibration of LDA measurement volume size. Appl. Opt. 27 (1987).

[I71 C. Hussu, E. Bliimcke, H. EickhofJ: M. Rachner: The influence of a radial swirl generator on the flow field from a fuel nozzle model. Proc. 4th Int. Symp. on Application of Laser Techniques to Fluid Mechanics, 1988.

7 (1990) 181-190.

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[IS] E. Bltirncke, H. EickhofJ: C. Hassa: Control and use of a miniaturized droplet generator. Part. Part. Syst. Charact. 8 (1991)

[19] J. I? Sislian, R. A . Cusworth: Measurements of mean velocity and turbulent intensities in a free isothermal swirling jet. AIAA J. 24 (1986).

[20] C. Hassa, E. Blurncke, H. Eickhoff: Measurements of Eulerian macro timescales in highly swirling flows and comparison with a

297-300.

computational model. Proc. 5th Int. Symp. Application of Laser Techniques to Fluid Mechanics, 1990.

[21] E. Bliirncke, H. EickhofJ C. Hassa: Dispersion of monosized droplets in a turbulent swirling flow. Proc. 4th Int. Conf. on Li- quid Atomization and Spray Systems, 1988.

[22] C. Hussa: Experimentelle Untersuchung der turbulenten Par- tikeldispersion in Drallstromungen. Thesis, Ruhr-Universitat Bochum, in preparation.

particle dispersion in highly swirling, turbulent flows

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