Chinese Science Bulletin

© 2007 Science in China Press

Springer-Verlag

Transitional phenomenon of particle dispersion in gas-solid two-phase flows

LUO Kun†, FAN JianRen & CEN KeFa

State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, China

Without using any turbulent model, direct numerical simulation of a three-dimensional gas-solid two-phase turbulent jet was performed by finite volume method. The effects on dispersion of particles with different Stokes numbers by the transitional behavior of turbulent structures were investigated. To produce high-resolution results and reduce the computation and storage, the fractional-step projection algorithm was used to solve the governing equations of gas phase fluid. The low-storage, three-order Runge-Kutta scheme was used for time integration. The governing equations of particles were solved in the Lagrangian framework. These numerical schemes were validated by the good agreement be-tween the statistical results of flow field and the related experimental data. In the study of particle dis-persion, it was found that the effects on particle dispersion by the spanwise vortex structures were prominent. The new behaviors of particle dispersion were also observed during the evolution of the flow field, i.e. the transitional phenomenon of particle dispersion occurs for the particles with small and intermediate Stokes numbers.

direct numerical simulation, gas-solid two-phase flows, coherent structures, particle dispersion, transition

1 Introduction

The gas-solid two-phase flow is a new booming disci-pline with only about a 30-year history, but it has very important applications in many engineering technologic fields. Taking the energy power industry as an example, a series of industrial processes, ranging from the prepa-ration, dryness, transport, mixing and separation process of the pulverized coal to the fluidization, burning, re-burning of the pulverized coal and the emission of the pollutants, are all the important research subjects of the gas-solid two-phase flows. The sandstorm phenomenon that frequently happens in some cities and areas of north China in recent years is also the typical gas-solid two-phase.

In gas-solid two-phase flows, the motion of the dis-persed solid particle in the gas phase flow-field is called particle dispersion. Particle dispersion is an important factor which influences the efficiency and the stability of

the system. Predicting and controlling the particle dis-persion in gas-solid two-phase flows are of great sig-nificance for optimizing designing and efficient applica-tions.

Crowe and his colleagues[1―5] conducted a series of numerical simulation and experimental studies, and first proposed the particle Stokes number, i.e. the ratio of the aerodynamics time constant of the particles to the char-acteristic time of large-scale vortex structures, to de-scribe the dispersion property of different particles in the free shear flows. They reported that the particles with smaller Stokes numbers can be easily affected by the fluid, and the concentration distribution in the flow-field is uniform. The particles with intermediate Stokes num-ber of the order of unity are influenced more by the cen-trifugal effect and concentrate in the outer boundary re- Received July 7, 2006; accepted October 13, 2006 doi: 10.1007/s11434-007-0055-x †Corresponding author (email: zjulk@zju.edu.cn) Supported by the National Natural Science Foundation of China (Grant No. 50506027)

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gions of the large-scale vortex structures. Their disper-sion is the highest and the concentration is the most non-uniform. However, for the larger Stokes number, the dispersion is not obvious due to the larger particle inertia. Thereafter, many related studies appeared and the similar conclusions were reached[6―8]. These research results are very helpful for understanding the dispersion patterns of different particles influenced by the large-scale vortex structures. However, the above studies mainly touch upon the effect on particle dispersion by the large-scale vortex structures. There is little study about the effect on particle dispersion by turbulence transition. Turbulence transition is an important phenomenon, which indicates the transitional process from laminar flow to turbulent flow and from large-scale vortex structures to small-scale vortex structures. During the transition, be-sides the large-scale vortex structures, there also coexist multi-scale vortex structures. These vortex structures with different scales have different coupling effects with the particles. To deeply understand the dispersion be-haviors of particles in multi-scale turbulent flows, direct numerical simulation (DNS) method is used to investi-gate a gas-solid two-phase turbulent transitional jet. The dispersion characteristics of particles with different Stokes numbers during the transitional process of turbu-lent vortex structures from large-scale to small-scale are specifically examined.

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2 Mathematic description

2.1 Flow configuration

In the present study, a three-dimensional gas-solid two-phase planar turbulent jet is chosen as the research object. For the typical flow configuration, please refer to Figure 1 in ref. [9]. In order to observe the transitional process of the flow, the Reynolds number based on in-flow velocity U0 and nozzle width d is set as

0 /Re U d ν= =5000. The extension of the computational

domain in the streamwise (x), lateral (y) and spanwise (z) direction has been set to 25d × 20d × 4d. The fluid is injected into the domain through the whole slot nozzle, but the particles are just injected through a square region located in the center of the slot with side length d. Ini-tially, a shear layer with the velocity profile as follows is given in the region with the nozzle width d in the

flow-field:

0 0

0tanh ,

2 2 20,0,

U U yu

vw

θ⎛ ⎞

= + ⎜ ⎟⎝ ⎠

==

(1)

where 0θ is the initial momentum thickness, selected as 0.05d. u, v and w represent the streamwise, lateral and spanwise velocities, respectively. Outside the region of the nozzle width d, all the fluid velocities are set as zero. At the streamwise outflow boundary, Neumann bound-ary conditions for velocity and pressure are used and the pressure is also corrected. At the lateral boundaries, the pressure is set as zero. In the spanwise direction, the periodic boundary conditions are applied.

2.2 Governing equations

Regarding that the gas phase is an incompressible New-tonian fluid, and the gas-solid two-phase jet is dilute flow. When the effect on fluid by particles is neglected based on one-way coupling, the non-dimensional gov- erning equations for the fluid can be expressed as

continuum equation: 0,i

i

ux∂

=∂

(2)

momentum equation:

1 1 ,i j ji i

j j j

u u uu upt x x Re x x xiρ

⎛ ⎞∂ ∂∂ ∂∂ ∂+ = − + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠i

(3)

where the characteristic velocity and length scales are U0 and d, respectively. The reference density is the den-sity of the air under the normal temperature and pres-sure.

The particles are traced in the Lagrangian framework. In each case, the particles entering the computational domain are supposed to be regular sphere with the same diameter pd and density p.ρ The material density

ratio of particles to fluid equals 2400. Thus, the minor force terms such as the virtual mass, the buoyancy, the Basset and the Magnus forces can be neglected. Consid-ering only the Stokes drag force and the gravity, the non-dimensional governing equation for particle motion can be written as below according to the second law of Newton [10]:

(4)

LUO Kun et al. Chinese Science Bulletin | February 2007 | vol. 52 | no. 3 | 408-417 409

where V is the particle velocity vector and U is the fluid velocity vector at the position of the particle. f is the modified factor for the Stokes drag force, and

[11]0.687p1 0.15f Re= + when Re ≤1000. The particle

Reynolds number

p

p | |Re dp / .ν−＝ U V The particle

Stokes number St is defined as ( )2

p p

0

18/

dSt

d Uρ μ

= . Fr is

the Froude number, defined as Fr = U02/g d. g is the unit

vector of gravity acceleration, and g is the standard gravity acceleration.

b

b

To demonstrate dispersion patterns of different parti-cles, five kinds of typical particles with the Stokes numbers of 0.01, 0.1, 1, 10 and 100 are traced. For each case, 100 particles are injected into the flow-field every 10 time intervals when considering the dilute assump-tion. The largest volume fraction of the particles is about 1×10−10. Initially, the particles are distributed uniformly in the nozzle and the velocity equals the velocity of the local fluid. The particles that leave the computational domain are not traced any longer.

3 Numerical algorithms

The finite volume method[12] is used to discretize the governing equations of the fluid. Central differences are used for the spatial discretization to ensure the second- order precision of the solutions in space. The governing equations are solved using the fractional-step projection technique[13].

It is shown in the previous study[14] that when the grid scale used in DNS is smaller than or in the same order of the Kolmogorov micro scale ,η then one can get the solutions with enough precision. In the present study, the Kolmogorov micro scale η is estimated as 0.043d.

Therefore, the uniform staggered grid 116

x dΔ = =

1.45η and 132

z dΔ = = 0.725η are arranged along

the streamwise (x) and the spanwise (z) directions. To capture the small-scale vortex structures in the strong

shear core region of the jet, the grid width 125

y dΔ = =

0.93η is set in the region of 4.5d < y < 4.5d. Outside this area, the grid is stretched in the lateral direction. The total grid points 400×300×128 = 15.36×106 are used

along x, y and z directions. In time integration, an explicit low-storage, third-

order Runge-Kutta scheme[15] is used to reduce the computation and storage and ensure the high-precision results. The non-dimensional time step is set to Δt = 0.02 based on the requirements of the time resolution, the computational stability and the statistics of the flow filed. For detailed numerical strategies, please refer to refs. [16,17]. The velocity and displacement of the particle can be obtained by integrating eq. (4). Third-order La-grangian interpolating polynomials are adopted to get the fluid velocity at the position of the particle. In each case, about 400000 particles are traced to satisfy the sta-tistical requirement.

4 Results and discussion

4.1 Validation with experimental data

To validate the numerical algorithms used in the present study, the profiles of mean velocities are statistically calculated and compared with the related experimental data. Figure 1 shows the profiles of the streamwise and lateral mean velocities of the planar jet. Here, b is the velocity half-width, and Um and Vm are the streamwise and the lateral mean velocity, respectively. Ucl denotes the streamwise mean velocity in the centerline. It is found that both the streamwise and the lateral mean ve-locities reach self-similar status in the downstream re-gions. The numerical streamwise mean velocity is in good agreement with the experimental data of Gutmark & Wygnanski[18], Ramaprian & Chandrasekhara[19] and Namer & Ötügen[20]. The predicted lateral mean velocity is also consistent with the experimental data in the re-gion of y/b < 2. In addition, the second-order statistical variables agree well with the related experimental data too, but the comparisons are not given in the present paper due to the length limitation. It should be pointed out that the effects on fluid by particles are neglected in the present study. In other words, the one-way coupling method is employed. Therefore, the data of the gas phase can be used to compare with the previous experi-mental data of single-phase jets to validate the reliability of the numerical algorithms adopted in the present paper. Although the Reynolds numbers in previous experi-ments are obviously higher than that of the present simulation, the comparisons are feasible based on the self-similar characteristics of the jet flows.

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Figure 1 Distribution of streamwise and lateral mean velocities in self-similar regions of the plane jet. (a) Streamwise mean velocity profile and com-parison with experimental data; (b) lateral mean velocity profile and comparison with experimental data.

4.2 Lateral dispersion of particles

Figure 2 shows the distribution of particles with differ-ent Stokes numbers in the spanwise plane z/d = 0 at the non-dimensional time t = 41.6. To compare particle dis-tribution with coherent structures, the corresponding streamwise, spanwise and lateral vorticities are also shown. By comparison, one can observe the effects on dispersion of the particles with different Stokes numbers by coherent structures in detail. Obviously, the lateral dispersion of particles is mainly influenced by the span-wise vortex structures. In the up stream of the jet, there primarily are the large-scale vortex structures. For St = 0.01, the particles can disperse into the vortex core re-gions and form a similar uniform distribution to the vor-tex structures. Most particles with the Stokes number of 0.1 also distribute uniformly, but part of them concen-trates in the inner boundaries of the large-scale struc-tures. For St = 1, the particles congregate largely in the outer boundaries of the large-scale structures, and the local-focusing phenomenon[21] happens in the conver-gence regions of multiple vortex structures. With the

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increasing of the Stokes number, the particles are influ-enced less and less by vortex structures. These results agree well with previous related studies. However, in the down stream of the jet, the dispersion of particles be-comes more complicated due to the transition of vortex structures from large-scale to small-scale. Due to the coexistence of various vortex structures with different time scales and space scales in the flow field, the parti-cle dispersion exhibits selective characteristic, i.e. the local particle concentration correlates well with the ratio of the response time of particles to the characteristic time of the local vortex structures. The transition of par-ticle dispersion from uniform to non-uniform occurs for

the particles with small Stokes numbers, but the reverse trend occurs for the particles with intermediate Stokes numbers. Due to the larger inertia, particles with large Stokes numbers are not influenced by turbulence transi- tion and disperse less along the lateral direction. At the same time, it is found that particles with small and in- termediate Stokes numbers move slowly towards down stream because of the large lateral dispersion. However, the large particles with Stokes numbers of 10 and 100 move fast towards down stream and can easily leave the computational domain. In general, all particles tend to assemble in the low-vorticity and high-strain regions.

Figure 3 shows the transitional dispersion process for particles with the Stokes number of 0.01 in the spanwise plane of z/d = 0. In the initial stage of the jet, the large-scale vortex structures are dominant, and the parti-cles are strongly dragged. Due to the smaller aerody-namic response time scale, this kind of particles can re-sponse quickly to fluid motion and are distributed uni-formly in the flow field. They can even directly disperse into the vortex core regions of large-scale vortex struc-tures, as shown in Figure 3(a) and (b). This dispersion is consistent with the results of previous studies[2,21]. However, with the further development of coherent structures, the distribution of particles becomes non- uniform when the vortex structures are translated from large-scale into small-scale, as shown in Figure 3(e) and (f). The reason is that for large-scale vortex structures, the characteristic time scale is substantially larger than the response time of the particles, but when large-scale structures change to small-scale, the characteristic time scale of vortex structures may be in the same order of the response time scale of particles; then, the preferential concentration effect[22] happens, and the particles are

LUO Kun et al. Chinese Science Bulletin | February 2007 | vol. 52 | no. 3 | 408-417 411

Figure 2 Distribution of vorticities and particles with different Stokes number in the plane z/d = 0 at the time t = 41.6. (a) Streamwise vorticity; (b) span-wise vorticity; (c) lateral voriticity; (d) St = 0.01; (e) St = 0.1; (f) St = 1; (g) St = 10; (h) St = 100. dispersed to the boundary regions of small-scale struc-tures to form the non-uniform particle distribution pat-tern. This indicates that the local particle concentration is closely related to the characteristics of the local vortex structures. For the multi-scale vortex structures in transi-

tional flows, their time scales are various, but the re-sponse time of certain kind of particles is fixed, which makes the particle dispersion become selective. It also reflects the coupling mechanism among multi-scale structures in gas-solid two-phase flows.

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Figure 3 Distribution of particles with St = 0.01 in the plane z/d = 0 at different times. (a) t = 13.6; (b) t = 29.6; (c) t = 45.6; (d) t = 61.6; (e) t = 77.6; (f) t = 93.6.

Contrary to particles with Stokes numbers of 0.01 and

0.1, particles with the Stokes number of 1 are thrown out of the vortex core regions and concentrate largely in the outer boundaries to form a dense-dilute-dense non-uni- form distribution pattern during the evolution and development process of the large-scale vortex structures in the flow field. When the large-scale vortex structures change to small-scale, the particle concentration is still higher in some regions, but becomes relatively uniform in other regions. This is because the characteristic time scale of the particles is close to the characteristic time scale of large-scale vortex structures, which easily leads to the preferential concentration effect. While after the

transition of the large-scale vortex structures, the relaxa-tion time scale of the particles can exceed the character-istic time scale of the small-scale vortex structures, and thus the particle concentration in these regions becomes uniform.

As to the larger particles with Stokes numbers of 10 and 100, the dispersion of particles is not influenced by the transition of vortex structures from large-scale to small-scale. The reason is that the characteristic time scale of the particles is always higher than that of the vortex structures during the turbulence transition. Then, the congregating of particles cannot happen, and the transitional phenomenon of particle dispersion does not

LUO Kun et al. Chinese Science Bulletin | February 2007 | vol. 52 | no. 3 | 408-417 413

occur.

4.3 Spanwise dispersion of particles

To examine the effects on particle dispersion by the streamwise vortex structures, Figure 4 depicts the dis-tribution of both the streamwise vorticity and the parti-cles with different Stokes numbers in the spanwise plane of z/d＝0. By comparison, it can be observed that the particles with Stokes numbers of 0.01 and 0.1 can dis-perse into the streamwise vortex core regions. Their dis-tribution is uniform near the large-scale structures and changes to non-uniform status near the small-scale structures due to preferential concentration. Particles

with the Stokes number of 1 seldom disperse into vortex core regions, but assemble in the common boundary regions connecting multiple vortex structures. While near the small-scale vortex structures, the particles tend to distribute uniformly due to the transitional dispersion. For large Stokes numbers of 10 and 100, the particles are influenced less by streamwise vortex structures and the spanwise dispersion is lower. Compared with Figure 2, it can be concluded that the spanwise vortex struc-tures have a more significant effect on particle disper-sion than the streamwise vortex structures. The effect on particle dispersion along the lateral direction by span-wise structures becomes obvious in the upstream region

Figure 4 Distribution of particles with different Stokes numbers in the lateral plane y/d = 0 at t = 81.6. (a) St = 0.01; (b) St = 0.1; (c) St = 1; (d) St = 10; (e) St = 100.

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of x/d < 5, but the effect on particle dispersion along the spanwise direction by streamwise structures becomes distinguishable until the downstream region of x/d > 7.5.

4.4 Statistical properties of particle dispersion

In order to quantitatively examine the dispersion char-acteristics of particles with different Stokes numbers along the lateral and the spanwise directions, the lateral dispersion function and the spanwise dispersion function are defined as

( )

12 2( )

01 ( )( ) ,

( )

N ti i

y

y t yD t

N t

⎡ ⎤−⎢ ⎥⎣=∑

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⎦ (5)

( )

12 2( )

01 ( )( ) ,

( )

N ti i

z

z t zD t

N t

⎡ ⎤−⎢ ⎥⎣=∑

⎦ (6)

where N(t) is the total number of particles in the flow field at time t. and are respectively the

lateral and spanwise displacements of the i-th particle. y

)(tyi )(tzi

i0 and are the initial displacements of the i-th par-

ticle when it is injected into the jet. 0iz

Figure 5 shows the time history of the lateral disper-sion function for particles with different Stokes numbers. Apparently, the lateral dispersion of particles with the Stokes number of 100 is the lowest, and the variation is also smaller, which means that the effect on dispersion of this kind of particles by spanwise vortex structures is very slight. Due to the fact that the preferential concen-tration occurs for particles with the Stokes numbers of 0.01, 0.1 and 1 early or late, the lateral dispersion func-

Figure 5 Time history of lateral dispersion function for particles with different Stokes numbers.

tion values are higher and the three curves are very close to each other. It indicates that the spanwise vortex struc-tures impose great effect on these three kinds of particles. For particles with the Stokes number of 10, the lateral dispersion is prominent at first under the effect of the two formed spanwise large-scale structures, but decreases fast after turbulence transition towards small-scale structures and finally maintains a lower level.

The time history of the spanwise dispersion function for particles with different Stokes numbers is given in Figure 6. Similarly, it can be seen that the spanwise dis-persion of particles with the Stokes number of 100 is very low. The spanwise dispersion for particles with Stokes numbers of 0.01 and 0.1 is obvious during the initial stage of the jet, and maintains a high value ap-proaching that of the particles with the Stokes number of 1 after turbulence transition. This suggests that these three kinds of particles are all remarkably influenced by the streamwise vortex structures. The spanwise disper-sion for particles with the Stokes number of 10 smoothly increases at first and maintains intermediate level at last. Compared with Figure 5, it can be found that the lateral dispersion function of particles increases faster and is substantially higher than the spanwise dispersion func-tion during the initial stage of the flow. The reason is that the development of the spanwise vortex structures is faster than that of the streamwise vortex structures. Thus, the particle dispersion is influenced more by the span-wise vortex structures than by the streamwise vortex structures.

Figure 6 Time history of spanwise dispersion function for particles with different Stokes numbers.

Figure 7 shows the time history of the total particle number in the flow field for each kind of particles. It can

LUO Kun et al. Chinese Science Bulletin | February 2007 | vol. 52 | no. 3 | 408-417 415

Figure 7 Time history of total particle number in the flow field for each kind of particle. also reflect the dispersion and concentration distribution characteristics of different particles. During the first mean convection period, the total particle number in the flow field increases linearly with time. After that, the particle numbers for St = 10 and 100 first become stable and maintain a small value because these particles dis-perse less along the lateral and spanwise directions, move towards down stream with higher velocity and can easily leave the computational domain. However, the numbers of particles with St = 0.01, 0.1 and 1 in the flow field are larger because the preferential concentration effect occurs for these particles, which results in the higher lateral and spanwise dispersions and the longer stay time of the particles in the flow field. Comparing the curves for the three kinds of particles with each other, a turning point is found at about the non-dimensional time t = 90. Before this point, the total particle number in the flow for St = 1 is larger than those for St = 0.01 and 0.1, but after this point, the trend is reverse, i.e. the total particle number in the flow for St = 0.01 and

0.1 exceed that for St = 1. This indicates that the transi-tional phenomena happens going with the jet transition from large-scale to small-scale, i.e. the particles at the intermediate and the small Stokes numbers adjust their dispersion patterns from non-uniform to uniform and from uniform to non-uniform, respectively.

5 Conclusions

The finite volume method and fractional-step projection scheme are applied to directly simulating a gas-solid two-phase turbulent jet, and the dispersion behaviors of particles with different Stokes numbers during turbu-lence transition are studied. The good agreement be-tween statistical results and related experimental data validates the reliability of the numerical algorithms. When the large-scale vortex structures are dominant, the dispersion behaviors of particles with different Stokes numbers are consistent with previous results. However, when the coherent structures transit from large-scale to small-scale, the particle dispersion changes from uni-form pattern to non-uniform pattern for particles with Stokes numbers of 0.01 and 0.1, and changes from non-uniform pattern to uniform pattern for intermediate particles with the Stokes number of 1. The transitional behaviors make the dispersion function of these three kinds of particles approach each other in most of the time framework. In addition, it is found that the span-wise vortex structures have a more significant effect on particle dispersion than the streamwise vortex structures. The effects on gas-solid two-phase flow system by the gravity, the flow Reynolds number, the initial particle conditions and the particle-particle collision will be in-cluded in future research.

Thanks are due to Dr. M. Klein for his great help and to the anonymous referees for their comments which have improved this work.

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EDITOR

YAN Luguang Institute of Electrical Engineering Chinese Academy of Sciences Beijing 100080, China

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