A Comparison of Stochastic Separated Flow Models for Particle

Energy & Fuels 2000, 14, 95-103

95

A Comparison of Stochastic Separated Flow Models for Particle Dispersion in Turbulent Flows E. A. Hennick† and M. F. Lightstone* Department of Mechanical Engineering, McMaster University, 1280 Main Street W., Hamilton, Ontario, Canada, L8S 4L7 Received March 23, 1999

This paper investigates stochastic separated flow (SSF) models used to predict particle dispersion in turbulent flows. A new model is presented that incorporates a temporal and spatial autocorrelation in the description of the fluctuating component of the turbulent gas-phase velocity. This model and four SSF models available in the literature are evaluated by comparing predictions with the grid-generated turbulence experiments of Snyder and Lumley (Snyder, W. H.; Lumley, J. L. J. Fluid Mech. 1971, 48, 41-71) and Wells and Stock (Wells, M. R.; Stock, D. E. J. Fluid Mech. 1983, 136, 31-62). Results are discussed and deficiencies in the models are explored. The new model presented herein accounts for the crossing trajectory effect with the inclusion of a spatial correlation based on the relative velocity of the particle and the time step employed. Promising results are obtained using this method.

Introduction The dispersion of particulate in turbulent flows occurs in many engineering applications. Spray combustion systems and combustors that use pulverized coal for fuel are the most common examples of dilute two-phase flows.3 In gas turbines, diesel engines, and industrial furnaces employing spray combustion, the behavior of liquid fuel droplets plays an integral role in determining the combustion characteristics and efficiency of the process. Flame stability in coal-fired power stations is also strongly dependent on the motion of the coal particles. Stochastic separated flow (SSF) models use a Lagrangian framework to solve the trajectory equations of a particle as it interacts with a succession of discrete turbulent eddies. This Lagrangian framework treats the particles as distinct entities within the fluid phase. This approach is well suited to applications that require detailed information about the interaction between the particulate and the continuous phase. The solution is stochastic since the particle interacts with the instantaneous velocities of the gas-phase turbulence. The model incorporates the turbulent fluctuations by random sampling to determine the instantaneous velocity of the continuous gas-phase. This paper will describe and compare various SSF models. The models will be validated using the benchmark experiments of Snyder and Lumley1 and Wells and Stock2 where particles were released into a uniform * Author to whom correspondence should be addressed. Fax: (905) 572-7994. E-mail: [email protected]. † E-mail: [email protected]. (1) Snyder, W. H.; Lumley, J. L. J. Fluid Mech. 1971, 48, 41-71. (2) Wells, M. R.; Stock, D. E. J. Fluid Mech. 1983, 136, 31-62. (3) Shirolkar, J. S.; Coimbra, C. F. M.; McQuay, M. Q. Prog. Energy Comb. Sci. 1996, 22, 363-399.

flow of grid-generated turbulence. Results will be discussed, and deficiencies in the models will be explored. SSF Models SSF models use a Lagrangian formulation of the particle equation of motion to follow the trajectory of a particle as it enters a series of discrete turbulent eddies. For the case of a gas-solid particle flow, the density of the particle is much greater than the density of the gasphase, Fg/Fp , 1. If gravity is the only body force acting on the particle, the equation of motion is given by

mp

dup 1 bg - b up||(ug - up) + mpg (1) ) FgApCd||u dt 2

where Cd is the drag coefficient given by

24 f Rep

(2)

Fgdp||u bg - b up|| µ

(3)

Cd ) where Rep is defined as

Rep )

and f is the ratio of the drag coefficient to the Stokes drag given by

f ) 1 + 0.15Rep0.687

(4)

The expression for the acceleration of a particle given in eq 1 can be simplified even further by introducing a particle relaxation time, τp. Thus, eq 1 becomes

dup f ) (ug - up) + g dt τp where τp is given by

10.1021/ef990045p CCC: $19.00 © 2000 American Chemical Society Published on Web 12/02/1999

(5)

96

Energy & Fuels, Vol. 14, No. 1, 2000

τp )

Fpdp2 18µ

Hennick and Lightstone

(6)

The particle relaxation time is a weighting factor that reflects the inertia and size of the particle. The inertia of a particle plays a significant role in its dispersion. If a particle is small and light, it can follow the highfrequency fluctuations of the turbulence, essentially acting like a fluid tracer particle. If a particle is heavy, it has a reduced response to the turbulence and a reduced root-mean-square particle velocity. The time interval over which particle velocities are correlated will increase as the particle inertia increases. Since dispersion is related to the product of u′p2 and the Lagrangian integral time scale, τLI,4 the dispersion can increase or decrease as the particle inertia increases. Indeed, in the absence of drift, the dispersion of particles can increase to exceed that of fluid particles.5 This is known as the inertia effect.3 In the SSF approach the instantaneous gas velocity is decomposed into a mean velocity plus a random fluctuation, ug ) ug + u′g. Each particle is exposed to a new instantaneous velocity every time it enters an eddy. The fluctuating velocity is determined using a random number generator. In most models this velocity is kept constant during the particle-eddy interaction time. The differences between particle dispersion models lie in the treatment of the gas-phase turbulence and the manner in which it is thought to interact with particle motion. The conventional stochastic dispersion model of Gosman and Ioannides6 computes the gas-phase fluctuating components using random sampling of a Gaussian probability density function. The instantaneous velocity is assumed to be a random variable with a variance given by

2 σ2 ) k 3

(7)

where k is the turbulent kinetic energy of the flow as determined by the k - model7 for the turbulent gas phase. Here, each orthogonal component of the fluid velocity is assumed to be statistically independent. The time interval over which the particle interacts with the randomly sampled velocity field is determined by two time scales: the eddy lifetime, te, and the transit time, ttr. The latter time scale is the time required for the particle to cross an eddy. If the particle is moving at nearly the same velocity as the fluid, it may be captured by the eddy and remain in the eddy for the entire eddy lifetime. However, if the particle has a velocity very different from that of the fluid, it can traverse the eddy before the eddy dies. This phenomenon is known as the “crossing trajectory” effect.8 Thus the interaction time, tint is determined by taking the minimum of the two scales, that is

tint ) min(te, ttr)

(8)

(4) Taylor, G. I. Proc. London Math. Soc. 1921, 20, 196-211. (5) Squires, K. D.; Eaton, J. K. J. Fluid Mech. 1991, 226, 1-35. (6) Gosman, A. D.; Ioannides, E. J. Energy 1983, 7, 482-490. (7) Launder, B. E.; Spalding, D. B. Comput. Methods Appl. Mech. Eng. 1974, 3, 269-289. (8) Csanady, G. T. J. Atmos. Sci. 1963, 20, 201-208.

These times are estimated by assuming that the characteristic size of a randomly sampled eddy is the dissipation length scale, le, given by

le ) C0.5 µ

k1.5

(9)

where Cµ is equal to 0.09 and is the dissipation rate of turbulence. The eddy lifetime is given by

te )

le |u′g|

(10)

where u′g is the gas-phase velocity fluctuation. The transit time of a particle is found from the solution of the linearized form of the equation of motion of a particle in uniform flow

(

ttr ) -τp ln 1 -

le τp|u bg - b u p|

)

(11)

where b ug - b up is the relative velocity at the beginning of the interaction time and τp is the particle relaxation time given by eq 6. When le > τp|u bg - b up|, Equation 11 has no solution. In this case, the particle is captured by the eddy, and the interaction time is the eddy lifetime. Shuen, Chen, and Faeth9 proposed modifications to the expressions for characteristic eddy size and eddy lifetime. Since the turbulent kinetic energy is related to the fluctuating turbulent velocity component, they defined the eddy lifetime as

te )

le

x(2/3)k

(12)

In addition, the characteristic eddy size is defined similarly to that of Gosman and Ioannides6 in eq 9 but a different exponent is used, that is Cµ0.75. However, the value of the power that Cµ is raised to in eq 9 is not agreed on by researchers. Analytical work by Lightstone and Raithby10 indicate an exponent of Cµ0.63. Using the Gosman and Ioannides methodology, the turbulent gas-phase velocity fluctuation in each orthogonal direction is sampled only once per eddy and the gasphase velocity is held constant over the particle/eddy interaction time. With this approach, the particle interaction time is chosen to be large, such that the time correlation between fluctuating velocities in subsequent time steps will be effectively zero. Zhou and Leschziner11 argue that within an eddy, a particle will, in fact, see a series of time-correlated gas-phase velocities. The gasphase velocity fluctuations in the Zhou and Leschziner model are therefore expressed as

u′t ) R(δt)u′t-δt + et

(13)

where u′t is the gas-phase velocity fluctuation at the current time t, R(δt) is a temporal autocorrelation, u′t-δt is the velocity fluctuation at the previous timestep, and (9) Shuen, J. S.; Chen, L. D.; Faeth, G. M. AIChE J. 1983, 29, 167170. (10) Lightstone, M. F.; Raithby, G. D. Comb. Flame 1998, 113, 424441. (11) Zhou, Q.; Leschziner, M. A. Proc. 1st Joint JSME-ASME Fluids Engineering Conf., 4th Symp. on Gas-Solid Flow June, 1991.

Particle Dispersion in Turbulent Flows

Energy & Fuels, Vol. 14, No. 1, 2000 97

et is a random component independent of previous velocities. It is important to note that δt is much smaller than the particle interaction times employed using the Gosman and Ioannides approach. For any given δt, the expected value of et is given by

u′t ) R(δt, δx)u′t-δt + et

et ) u′t - R(δt)u′t-δt ) 0

(14)

and the variance is 2 e2t ) (u′t - R(δt)u′t-δt)2 ) u′2t - R2(δt)u′t-δt (15)

A functional form for the time correlation

( )

δt R(δt) ) exp τLI

is12

(16)

The Lagrangian integral timescale, τLI, used in Zhou and Leschziner’s model is given by

(

τLI ) βte ) β C0.75 µ

)( )

k1.5 2 / k 3

(18)

where R(δt, δx) is an autocorrelation based on time and position. The autocorrelation is a function of the change in time and position experienced by a particle over a single time step δt. As shown by Lu, Fontaine, and Aubertin,13 the autocorrelation appearing in eq 18 can be written as the product of a temporal (Lagrangian) and spatial (Eulerian) correlations

R(δt, δx) ) Rt(δt)Rx(δx)

(19)

The distance δx, used for evaluation of the spatial correlation, is the distance traveled by the particle over a time step δt relative to a fluid point. This quantity is calculated as δx ) uRELδt where uREL is the relative velocity of the particle. For any given δt, the expected value of et is zero and the variance is given by

0.5

(17)

Here, β is based on analytical and experimental considerations. Zhou and Leschziner11 report the optimum value of β to be 0.8 based on their simulation of the Wells and Stock experiment. Zhou and Leschziner’s model improves the conventional SSF model by incorporating the temporal autocorrelation into the definition of the turbulent gas-phase velocity fluctuation. As a result, a particle may see more than one instantaneous velocity while in an eddy, more closely modeling the physics of the flow. As noted by the authors, the model as presented above does not account for the crossing trajectory effect. Equation 13 describes the velocity fluctuation using a time-correlation coefficient, R(δt), derived for fluid particles. Unless a particle has a small particle relaxation time, it will not be able to follow the high-frequency fluctuations of the turbulence, and in general, will not see the same velocity fluctuations as a fluid point. Thus, a particle falling through an eddy will see a velocity fluctuation at time t that should have a much lower correlation. Using Zhou and Leschziner’s model, however, a heavy particle sees velocity fluctuations that are highly correlated. Hence, the model is unable to capture the crossing trajectory effect. In this paper a new model is proposed that builds on Zhou and Leschziner’s time-correlated stochastic model. The crossing trajectory effect is incorporated by introducing a spatial correlation coefficient based on the relative velocity of the particle and the time step δt. In addition, the magnitude of the Lagrangian integral time scale is determined independent of the model’s performance by relating the variance of particle position to Taylor’s result for fluid particles.10 Similar to conventional SSF models, the instantaneous gas-phase velocity is decomposed into a mean velocity plus a random fluctuation, ug ) ug + u′g. Here, the subscript “g” is dropped for simplicity and the subscript employed refers to the velocity fluctuation with respect to time. The gas-phase velocity fluctuations are expressed as (12) Hinze, J. O. Turbulence, 2nd ed.; McGraw-Hill: New York, 1975.

2 e2t ) u′2t - R2t (δt)R2x (uRELδt)u′t-δt

(20)

The functional form of the time correlation is the same as that used by Zhou and Leschziner

( )

δt Rt(δt) ) exp τLI

(21)

Determination of the Lagrangian integral time scale appearing in eq 21 follows from an extension of Taylor’s work. Lightstone and Raithby10 relate the variance of particle position to Taylor’s result for fluid particles released into a uniform flow of homogeneous isotropic turbulence. In contrast to Zhou and Leschziner’s model, this result is obtained solely on the basis of analytical considerations. The Lagrangian integral time scale suggested by Lightstone and Raithby10 is given by

τLI ) 0.135

k

(22)

The spatial autocorrelation is approximated by

(

)

uRELδt LI

Rx(uRELδt) ) exp -

(23)

where uREL ) |u bg - b up| the instantaneous relative velocity of the particle, and LI is an integral length scale given by

LI ) 2τLI

x23k ) C

0.63k µ

1.5

(24)

Incorporating the approximations for the temporal and spatial autocorrelations, the gas-phase velocity fluctuation becomes

((

u′t ) exp -

))

δt uRELδt + u′t-δt + et τLI LI

(25)

The behavior of the new model can be explored by considering the limiting cases of relative velocities. For (13) Lu, Q. Q.; Fontaine, J. R.; Aubertin, G. Int. J. Multiphase Flow 1993, 347-367.

98



example, for dyed fluid particles the relative velocity is zero. Thus the spatial correlation goes to unity and the velocity fluctuation is based solely on the temporal correlation as required. In contrast, if the relative velocity of the particle is large, then the spatial correlation becomes important and the correlation between old and new fluid velocities is reduced. Particle Dispersion in Grid Generated Turbulence Models were evaluated using the measurements of Snyder and Lumley1 and Wells and Stock2. These experiments involved the dispersion of single particles released from a point source into a uniform flow of gridgenerated turbulence. A simulation of each experiment was created using a commercial CFD code.14 Measurements of particle dispersion and velocity variance were compared to results generated by each model described above. Snyder and Lumley, 1971. Four types of particles were considered in Snyder and Lumley’s experiments: hollow glass beads, solid glass beads, corn pollen, and copper beads. The particle relaxation time of the hollow glass beads was small allowing the particle to essentially follow the turbulent gas; the hollow glass beads therefore acted like “fluid” or “tracer” particles. Corn pollen had more inertia and therefore did not follow the turbulence exactly. The solid glass beads and the copper particles were the heaviest with similar relaxation times. Since the behavior of the solid glass beads was nearly identical to the copper beads, the experiments involving the former were not considered and will not be included in further discussion. Predictions of the turbulent kinetic energy and dissipation rate calculated by the CFD code were compared to the analytical solutions for grid-generated turbulence. Excellent agreement was obtained. The particle tracking module was also verified by comparing code calculations to independent hand calculations. Implementation of the Gosman and Ioannides model revealed ambiguity in the definition of the eddy lifetime. It was found that using an eddy lifetime described by

te )

le

x

(26)

u′2 + v′2 + w′2 3

produced results closest to those reported by the researchers as seen in Figure 1. It was not possible to exactly reproduce Gosman and Ioannides computed results for this flow. Researchers Shuen, Chen, and Faeth,9 and Chen and Crowe15 reported the same difficulty. Figure 1 presents the particle dispersion of each particle type with respect to time using the model of Gosman and Ioannides.6 Particle dispersion is given by the variance of the y-position [m2] in a coordinate system where the main flow direction is in the positive axial (x) direction and the gravity force is in the negative axial (14) Advanced Scientific Computing Ltd. Version 2.4 Theory Documentation; Waterloo, Ontario, Canada, 1995. (15) Chen, P. P.; Crowe, C. T. Proc. Int. Symp. on Gas-Solid Flow 1984, 37-41.

Figure 1. Predicted and measured particle dispersion for the model of Gosman and Ioannides.6

(x) direction. As seen in the figure, the dispersion of both the corn pollen and copper beads is significantly overpredicted, while the hollow glass beads are somewhat underpredicted. The predicted dispersion of the corn pollen compared to that of the hollow glass beads is interesting. From Figure 1 it is seen that the predicted dispersion of the corn pollen is very close to that predicted for the hollow glass beads. This can be explained as follows: since the hollow glass beads essentially follow the turbulent gasphase, the particle interaction time for the hollow glass beads is dominated by the lifetime of the eddy as given in eq 26. The hollow glass beads are considered to be captured by the eddies and follow the gas-phase turbulent fluctuations while traveling along the duct. The corn pollen particles are slightly heavier and therefore have a greater particle relaxation time; examination of the interaction time of the corn pollen revealed that the particle interaction time is dominated by the transit time of the particle for the initial physical time step and then dominated by the eddy lifetime until the particle exits the duct. Thus the corn pollen particles remain in the first eddy for the duration of the transit time and are then captured by the subsequent eddies similar to the hollow glass beads. These predictions actually illustrate the “inertia effect” since the predicted drift does not allow for crossing trajectories. Predicted results for particle dispersion using Shuen, Chen, and Faeth’s model are shown in Figure 2. The model underpredicts the dispersion for all three particle types with significant underprediction for the hollow glass beads. Note that the predicted dispersion of the corn pollen is close to the predicted dispersion of the hollow glass beads similar to the predictions made using the model of Gosman and Ioannides. Predicted dispersion results for all particle types using the model of Shuen, Chen, and Faeth do not agree with those reported by the researchers. The computed results underpredict the reported results by as much as 14% for the case of the hollow glass particles. However, it can be shown, using analysis similar to that of Lightstone and Raithby,10 that these results are a correct implementation of the model for the case of hollow glass particles. This analysis is given below. Consider a particle undergoing a random walk in a uniform flow of grid-generated turbulence. For a particle



Figure 2. Predicted and measured particle dispersion for the model of Shuen, Chen, and Faeth.9

Figure 3. Using the slope to show the correct implementation of the model of Shuen, Chen, and Faeth.9

initially at y ) 0 undergoing discrete steps, the yposition of the particle after N steps, yp,N, is given by N

yp,N )

N

∆yp,i ) ∑vp,i∆ti ∑ i)1 i)1

(27)

where ∆yp,i is the size of the ith step, vp,i is the velocity of the particle, and ∆ti is the time step. An equation for the variance of the particle position can be found by squaring both sides of eq 27 and taking the expected value N

2 yp,N

)(

vp,i∆ti)2 ∑ i)1

(28) Figure 4. Predicted and measured particle dispersion for the model of Lightstone and Raithby.10

Now consider modeling a fluid particle using the approach outlined by Shuen, Chen, and Faeth. A fluid particle experiences no relative velocity and, therefore, the particle interaction time is always the eddy lifetime. Hence, eq 28 can simplified by substituting the eddy 2 lifetime given by eq 12 into eq 28 and setting v′g,i ) 2k/ 3. Further, noting that vp,ivp,j ) 0 for i * j, and that k2/ is roughly constant for grid-generated turbulence yields

y2p ) C0.75 µ

x

2 k2 t 3

(29)

Equation 29 provides an analytical solution describing the dispersion of fluid particles using the modeling approach of Shuen, Chen, and Faeth. This expression is used to verify the hollow glass bead predictions. Figure 3 shows the analytical solution given by eq 29 plotted with the hollow glass dispersion curve computed using Shuen, Chen, and Faeth’s model, and a linear regression of the hollow glass prediction. The predicted dispersion of the hollow glass particles is seen, quantitatively, to have the same slope determined by eq 29. This result provides confidence in the correct implementation of the model of Shuen, Chen, and Faeth. Additional confirmation was provided by verifying a single-particle trajectory for this model and predictions for particle velocity and displacement were compared to the analytical solution for both physical and computational time steps.

Figure 4 shows results obtained using a model modified as suggested by the work of Lightstone and Raithby. These results show trends similar to those of Shuen, Chen, and Faeth and Gosman and Ioannides in that corn pollen and hollow glass beads are predicted to have similar dispersions. As seen in the figure, the dispersion of the hollow glass and the copper beads is well predicted, whereas, the corn pollen dispersion is overpredicted. Note that the prediction of the hollow glass particles appears as a “wavy” line in Figures 2 and 4. This behavior results from the discrete eddy concept and can be explained as follows. A particle is injected into the flow where it enters a discrete eddy and sees an instantaneous turbulent velocity that is held constant for the lifetime of the eddy. The light particle is captured by the eddy and quickly approaches the instantaneous velocity of the turbulence. After interacting with the eddy over the eddy lifetime, the particle enters a new eddy whose lifetime is determined once again by the turbulent kinetic energy and dissipation at the current axial position of the particle. When considering the statistics of many particles, it is seen that each particle will enter a new eddy at roughly the same axial position, corresponding to the valleys of the dispersion curve. The corn and copper particles do not exhibit this behavior since the interaction time of each particle is not dominated by the eddy lifetime; the interaction time of the corn particle is distributed between the eddy lifetime

100



Figure 5. Predicted and measured particle velocity variance for the model of Lightstone and Raithby.10

Figure 7. Predicted and measured particle dispersion using Zhou and Leschziner’s model; time step δt ) 0.005.

Figure 6. Velocity decay predicted by theory for particle interaction with a single eddy.

Figure 8. Predictions of Snyder and Lumley’s experimental data for particle dispersion using the model of Hennick and Lightstone, δt ) 0.005 s.

and the transit time while the copper particle is dominated by the transit time. The velocity decay of the hollow glass particles shown in Figure 5 displays even more prominent dips and is consistent with the prediction of the particle dispersion. This behavior can be verified by considering the analytical solution for the particle equation of motion for the y-direction.

dvp 1 ) (vg - vp) dt τp

(30)

Solving eq 30 with initial condition vp(0) ) vp,o and t ∈ [0,tint], squaring the result and time averaging with vgvp,o ) 0 yields

v2p(t) ) v2g - 2v2g exp

( )

( )

-t -2t + (v2p,o + v2g) exp τp τp

(31)

Figure 6 was created using the analytical solution given in eq 31. This curve was generated by setting v2p,o ) 0.022 m2/s2, v2g ) 0.018 m2/s2, and τp ) 1.7 × 10-3 s. This analytical result provides confidence in the predictions shown in Figure 5. These “waves” are an inherent flaw in an approach where the eddy lifetime and characteristic eddy size are essentially deterministic and where the particle interacts with a constant fluid

velocity over the particle interaction time. These results provide motivation for an alternate method of describing the gas-phase turbulence, for example, the model of Zhou and Leschziner. Results using Zhou and Leschziner’s model with a time step of δt ) 0.005 are shown in Figure 7. This value of the time step is small relative to the integral time scale (δt e 0.2 τLI) and is sufficient for time step independence.13 The hollow glass particles, corn pollen, and copper particles are predicted to have roughly the same dispersion. The poor predictions for the corn pollen and copper bead experiments are attributed to the inability of Zhou and Leschziner’s model to capture the “crossing trajectory” effect. Similar predictions are obtained by Picart et al.16 using an Eulerian model without crossing trajectories effects. The model of Hennick and Lightstone accounts for the crossing trajectory effect by incorporating both a temporal and spatial autocorrelation in the definition of the gas-phase velocity fluctuation. Figure 8 presents the particle dispersion of each particle type for the time step of δt ) 0.005 s. Unlike the other models considered, the model of Hennick and Lightstone is able to capture the reduced dispersion of the corn pollen particles relative to that for the hollow glass beads. Overall, this model (16) Picart, A.; Berlemont, A.; Gousebet, G. Int. J. Multiphase Flow 1986, 12, 237-261.


Figure 9. Hollow glass bead predictions using Zhou and Leschziner’s model and Hennick and Lightstone’s model with and without the spatial correlation.

provides good predictions for both the hollow glass and the copper beads. The corn pollen dispersion is somewhat underpredicted. It is interesting to compare the predicted dispersion of the hollow glass beads using both Zhou and Leschziner’s model and Hennick and Lightstone’s model. While Hennick and Lightstone’s model incorporates a spatial correlation and a temporal correlation, the spatial correlation is based on the relative velocity of the particle. Since the hollow glass beads are small and light, the particles can follow the high-frequency fluctuations of the turbulence and therefore have a negligible relative velocity. To quantify the impact of including a spatial correlation, the model of Hennick and Lightstone is used to predict the dispersion of the hollow glass beads including and excluding the spatial correlation. Figure 9 presents the dispersion predictions for the hollow glass beads using Zhou and Leschziner’s model, Hennick and Lightstone’s model without the spatial correlation, i.e., Zhou and Leschziner’s model but using the value for the Lagrangian integral time scale derived from an extension of Taylor’s analysis given in eq 22, and the new model of Hennick and Lightstone including both the spatial and temporal correlations. All models use a time step of δt ) 0.005 s. Using Hennick and Lightstone’s model, the inclusion of a spatial correlation results in decreased dispersion prediction with a maximum decrease of approximately 10%. However, dispersion results predicted using a time correlated model with two different values for the Lagrangian integral time scale indicate a strong sensitivity to the correlation function. Dispersion decreases by as much as 13% with a decrease in the Lagrangian integral time scale of 18%. This sensitivity to the choice of the Lagrangian integral time scale provides further motivation for determining this value independent of the model’s performance. Wells and Stock, 1983. The experiment of Wells and Stock investigated the effects of “crossing trajectories” and inertia on particles dispersed from a point source in grid-generated turbulence. To separate the crossing trajectories and inertia effects, particles were exposed to a potential field where the strength of the field could be varied. Particles were charged and a uniform electric


Figure 10. Particle dispersion coefficients for the small particles from Y2 predictions using SSF models.

field was set up between two parallel plates in the wind tunnel. The electric field strength was sufficient to counter gravitational effects and to intensify the potential force. Two different sizes of solid glass beads were used in the experiment: 5 and 57 µm in diameter. In the simulation of the small particles, single particles were released from the inlet at their drift velocity. Since the particles are small and light, they were able to follow the high-frequency fluctuations of the turbulence and exhibit the appropriate velocity variance at the position of the first measurement, x/M ) 20. In the simulation of the large particles, single particles were released as close as possible to the position of the first measurement. At the injection point, particles were released with an instantaneous velocity comprised of a mean plus a fluctuating component. The fluctuating component was determined by random sampling of a Gaussian probability density function with zero mean and variance equal to the reported experimental value at the position of the first measurement. The experimental results of Wells and Stock can be summarized and compared to the predicted dispersion results by considering the particle dispersion coefficient, Γp, defined as

Γp )

1dy2 2 dt

(32)

Dispersion coefficients are presented in two sections; the first section presents predictions for the small particle, the second presents predictions for the large particle. The predictions of the SSF models of Gosman and Ioannides, Shuen, Chen, and Faeth and Lightstone and Raithby are presented together for each particle type. Predictions using the models of Zhou and Leschziner and Hennick and Lightstone are presented together. Particle dispersion coefficients for each drift velocity for the small particles are presented in Figure 10 for the SSF models. Linear curves were fit through the y2 data, and the slope was then used to calculate the dispersion coefficients as given in eq 32. The dispersion coefficients for the experimental data were digitized from Wells and Stock’s results.

102



Figure 11. Particle dispersion coefficients for the small

Figure 12. Particle dispersion coefficients for the large

particles from Y2 predictions using models involving correlations.

particles from Y2 predictions using SSF models.

The dispersion coefficients for the small particles indicate that the SSF models are not sensitive to the increase in gravity for the range of drift velocities considered. This result is supported by Wells and Stock’s experimental data. Wells and Stock2 report that “the crossing trajectory effect appears to be negligible when the drift velocity is less than the rms fluid velocity.” Further, the SSF models significantly underpredict the dispersion coefficients and, in general, underpredict the dispersion of the small particles for all gravity cases. Figure 11 presents particle dispersion coefficients for the small particle using the models of Zhou and Leschziner and Hennick and Lightstone. These simulations were performed using time steps δt of 0.005 and 0.01 s for both models. Dispersion coefficients are calculated for each model for each time step employed. The dispersion coefficients for the small particles indicate that the model of Zhou and Leschziner is not sensitive to the increase in gravity. Again, this result is supported by Wells and Stock’s experimental data. Zhou and Leschziner’s model gives good predictions of the dispersion of the small particles for all gravity cases. This result is expected since the value of β used in the description of the Lagrangian integral time scale given in eq 17 is determined on the basis of the researchers’ simulation of the Wells and Stock experiment. Dispersion coefficients for the model of Hennick and Lightstone show that dispersion is predicted to decrease with increasing gravitational force. This is due to the modeling of the crossing trajectory effect through the inclusion of a spatial correlation that is based on the relative velocity of the particle. The spatial correlation is employed even when the drift velocity is less than the rms fluid velocity, accounting for the difference between model predictions and experimental results. In general, the model of Hennick and Lightstone underpredicts the dispersion of the small particles for each drift velocity. Particle dispersion coefficients for each drift velocity for the large particles are presented in Figure 12 for the SSF models. Predictions indicate that dispersion decreases with increasing drift velocity. This result is consistent with the physical premise for the crossing trajectory effect. Wells and Stock2 report that “the

Figure 13. Particle dispersion coefficients for the large particles from Y2 predictions using models involving correlations.

dispersion coefficient was found to decrease approximately as the drift velocity squared within the range of velocities tested.” While the SSF models correctly capture the trend of decreasing dispersion with increasing gravitational force, all models underpredict the dispersion of the large particles for each drift velocity considered. Dispersion coefficients presented in Figure 13 show that the predictions of Zhou and Leschziner’s model did not decrease with increasing gravitational force. The similar predictions of Zhou and Leschziner’s model for all gravity cases is due to the inability of the model to capture the crossing trajectory effect. This inability results in overpredicted dispersion of the large particle for all drift velocities considered. Hennick and Lightstone’s model incorporates the crossing trajectory effect by introducing a spatial correlation coefficient in the description of the fluctuating component of the instantaneous turbulent velocity. As the gravitational force increases, therefore increasing the relative velocity of the particle, the predictions of the model improve. Dispersion coefficients for the model of Hennick and Lightstone presented in Figure 13 decrease with increasing drift velocity capturing the crossing trajectory effect as expected. However, large particle dispersion is underpredicted for all drift velocities considered.


Conclusion The paper has been concerned with assessing and comparing SSF models for particle dispersion in turbulent flows. A new model has been presented that incorporates a temporal and spatial autocorrelation in the description of the fluctuating component of the turbulent gas-phase velocity. This model and four SSF models available in the literature have been evaluated by comparing predictions with the grid-turbulence experiments of Snyder and Lumley1 and Wells and Stock.2 The conventional SSF model of Gosman and Ioannides was the first model considered. Variations on this model as suggested by Shuen, Chen, and Faeth, and Lightstone and Raithby were also evaluated. None of these models was able to predict the reduced dispersion of the medium-sized corn pollen particles relative to that for the light hollow glass beads. Further, using the latter two models, the predicted dispersion of the small, light particles exhibited waves; this was shown to be an inherent flaw in an approach where the eddy lifetime and characteristic eddy size are essentially deterministic. Alternative methods of describing the gas-phase turbulence were explored by considering the models of Zhou and Leschziner and Hennick and Lightstone. While the model of Zhou and Leschziner produced satisfactory results for the predictions of small, light particles, the dispersion of heavier particles was greatly overpredicted. This result was partly attributed to the inability of Zhou and Leschziner’s model to capture the “crossing trajectory” effect. The model of Hennick and Lightstone compensates for the crossing trajectory effect by introducing a spatial correlation in the description of the gas-phase velocity fluctuation. This model predicted the dispersion of all particle types with the greatest accuracy of all models considered in this paper in the simulation of Snyder and Lumley’s experiment.


In the Wells and Stock simulations, the SSF models of Gosman and Ioannides, Shuen, Chen, and Faeth, and Lightstone and Raithby, captured the correct trends for both the small and large particle simulations but, in general, underpredicted particle dispersion. Zhou and Leschziner’s model accurately predicted small particle dispersion; this result was expected since the value of β used in the description of the Lagrangian integral time scale was determined on the basis of the researchers’ simulation of the Wells and Stock experiment. For the large particle case, the model was unable to capture the trend of decreasing dispersion with increasing drift velocity and overpredicted particle dispersion for each drift velocity. This result is due to the inability of the model to capture the crossing trajectory effect. Although underpredicting particle dispersion, the model of Hennick and Lightstone showed sensitivity to increasing gravitational force for both the large and small particle predictions. This is a direct result of the inclusion of a spatial correlation in the definition of the fluctuating component of the gas-phase velocity. While promising results are obtained using the model of Hennick and Lightstone, further research is necessary to validate the model over a wide range of flows including pipe flows, jet flows, boundary layers, and wakes. Acknowledgment. The support of AEA Technology/ Advanced Scientific Computing and the provision of the CFD code TASCflow is gratefully acknowledged. The support of the Natural Sciences and Engineering Research Council of Canada (NSERC), through an operating grant to the second author, is gratefully acknowledged. EF990045P

A Comparison of Stochastic Separated Flow Models for Particle

Recommend Documents