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Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method
Applied Mathematical Modelling xxx (2015) xxx–xxx
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Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method Hao Zhou ⇑, Guiyuan Mo, Kefa Cen Zhejiang University, Institute for Thermal Power Engineering, State Key Laboratory of Clean Energy Utilization, Hangzhou 310027, PR China
a r t i c l e
i n f o
Article history: Received 2 September 2011 Received in revised form 16 April 2015 Accepted 1 June 2015 Available online xxxx Keywords: LBM MRT Gas–solid Turbulent jet Particle dispersion
a b s t r a c t The gas–solid two-phase turbulent plane jet flow with high Reynolds number of 4500 was numerical investigated by means of lattice Boltzmann method (LBM). The multiple relaxation time (MRT) was employed to deal with the high Reynolds number fluid flows, and the particles were traced by the Lagrangian method. The results show that the flow changes from initial symmetric mode to asymmetric mode with the development of the flow. And asymmetric pattern appears first at the position of x/d = 4, where the vortex structures begin to form. The dispersion of particles at different Stokes number shows various distributions. The MRT-LBM shows its good ability in simulating turbulent flow with the high Reynolds number. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction The lattice Boltzmann method is originated from Lattice Gas Automata (LGA) [1,2], and it can also be derived directly from the Boltzmann equation [3]. A direct connection between the Boltzmann equation and the incompressible Navier–Stokes equation can be obtained under the condition of near incompressible [4,5]. Compared with the traditional Computational Fluid Dynamics (CFD) such as Navier–Stokes equation, LBM shows distinct advantages of high efficiency of parallel implementation, high calculation efficiency and ability in dealing with moving and complex boundaries [6–9]. LBM is a relatively new computational fluid method, and it has attracted considerable attention in recent years. Over the past decade, LBM has been widely used in scientific research and engineering application. Shan and Chen [10] developed a lattice Boltzmann model to simulate flows containing multiple phases and components, and this model presented many applications in large-scale numerical simulations of various types of fluid flows. Park et al. [11] applied the lattice Boltzmann method to simulate the flow in the electrode of a PEM fuel cell. Succi [12] employed the LBM successfully to calculate a variety of configurations, even those with a complex coupling of physical and chemical processes. Chen et al. [13] carried out simulations of complex fluid physics such as cars and airplanes using an extended kinetic Boltzmann equation, analyzing different approaches based on extensions of the Boltzmann equation. Han et al. [14] and Feng et al. [15] presented a new solution strategy that coupled lattice Boltzmann (LB), large eddy simulation (LES), and discrete element (DE) methodologies to simulate particle–fluid systems at moderately high Reynolds numbers. Enhanced collision operator was presented to be an efficient strategy for the lattice gas Boltzmann equation by Higuera et al. [16]. Then the multiple relaxation time (MRT) is developed and used widely. McCracken and Abraham [17] developed a MRT lattice-Boltzmann model for multiphase flow and evaluated the accuracy in several test problems such as oscillating ⇑ Corresponding author. Tel.: +86 571 87952598; fax: +86 571 87951616. E-mail address: [email protected] (H. Zhou). http://dx.doi.org/10.1016/j.apm.2015.06.005 0307-904X/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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liquid cylinders and capillary waves. Pan et al. [18] conducted a comparative study of the LBE models with MRT and the BGK-SRT, and reported that the MRT–LBE model is superior to the BGK–LBE model. Guo and Zheng [19] simulated the Poiseuille flow in the slip flow regime by the lattice Boltzmann equation (LBE) with multiple relaxation times (MRTs). Yu et al. [20] carried out the application of multiple-relaxation-time (MRT) lattice Boltzmann equation (LBE) for large-eddy simulation (LES) of the turbulent square jet flow. The MRT-LBE model for LES was carried out in a surface mounted cube in a channel at Re = 40,000 by Krafczyk et al. [21], and their preliminary results agree well with experimental data. Dynamic subgrid scale (SGS) models were incorporated in the MRT-LBM of LES of fully-developed turbulent channel flow at two different shear Reynolds numbers of 180 and 395, reported by Premnath et al. [22]. General lattice Boltzmann equation (GLBE) using MRT with forcing term for eddy capturing simulation of wall-bounded turbulent flows was presented by Premnath et al. [23], they found that markedly stability characteristics and good agreements were achieved. Deep understanding of the flow coherent structures and particle dispersion in the turbulent jet is of great significance in efficient engineering applications in the fields of chemical reaction and jet combustion [24]. The particle dispersion is an important factor in the engineering design and efficient engineering applications. The turbulent jet has been studied for a long time. Bradbury used a hot-wire anemometry to measure the statistical quantities in the self-similar region of a plane jet, and observed that the initial and external conditions have a strong influence on the development of the plane jet flow fields [25]. Stanley and Sarkar [26] investigated the 2D single-phase flow in the plane jet by direct numerical simulation. There are also many experimental and numerical studies reported on the gas–solid two-phase turbulent jet flows. Melville and Bray [27] investigated the two-phase turbulent jet flows and presented a model to describe them. Tsuji et al. [28] measured axisymmetric particle-laden jets by three different devices, and particle and air velocities, particle concentration and air turbulence were obtained. Despirito and Wang [29] numerically studied the influence of the particle Stokes number on the flow stability in the two-way coupled particle-laden jet, and found that the particles at Stokes numbers on the order of 1 correspond to the maximum flow stability. Fan et al. [24] investigated the gas-particle two-phase flow in the 2D turbulent plane jet by solving the compressible flow fields using direct numerical simulation, and they focused their study on the evolution of coherent vortex structures and dispersion patterns of particles in the near field. Menon and Soo [30] developed a three-dimensional lattice Boltzmann equation solver to study flows associated with synthetic jets and turbulent-forced rectangular jets by large-eddy simulation (LES). Premnath and Abraham [31] investigated the transient and incompressible turbulent plane jets by the discrete lattice BGK Boltzmann equation, and a spatially and temporally dependent relaxation time parameter is used to represent the averaged flow field, satisfactory results is obtained. However, there are few studies reported about the particle dispersion and large-scale vortex structures in the gas–solid two-phase turbulent jet flow with the moderately high Reynolds numbers using LBM. In addition, most of the numerical studies mentioned above were based on Navier–Stokes equation. In this paper, we use the LB method to investigate the vortex characteristics and the particle dispersion patterns in a 2D gas–solid turbulent plane jet. The MRT-LBM was employed to deal with the high Reynolds number fluid flows. The main objective of this study is to study the evolution of the coherent structures and the dispersion of particles at different Stokes numbers by the means of MRT-LBM, and the particles were traced by the Lagrangian method. The Letter is organized as follows. In Section 2 we describe the lattice Boltzmann method. Section 3 introduces the governing equations for particles. Section 4 presents the computational conditions. The numerical results and discussions are given in Section 5. The last section is devoted to conclusions. 2. D2Q9 MRT-LBE model Lattice BGK (LBGK) model and DnQb (n is the dimension of the space and b is the discretized speed) model presented by Qian et al. [32]. On each node of the lattice, there are a set of symmetric discrete velocities fei ji ¼ 0; 1; ; b 1g and a set of a velocity distribution functions ff i ji ¼ 0; 1; ; b 1g. In this paper, direct simulation using LBM is carried out to study the turbulent jet flow, large eddy simulation (LES) is not considered. A nine velocities two dimensional (D2Q9) LBE model coupled with multiple relaxation time (MRT) [32,33] was employed. The evolution equation for D2Q9 LBM could be described as:
f ðx þ ei dt; t þ dtÞ f ðx; tÞ ¼ Sðf ðx; tÞ f eq ðx; tÞÞ;
ð1Þ
eq eq T eq T where S is the collision matrix, and S ¼ diagðs0 ; s1 ; . . . ; s8 Þ. f ¼ ðf 0 ; f 1 ; . . . ; f 8 Þ and f eq ¼ f 0 ; f 1 ; . . . ; f 8 are vectors of the distribution function and their corresponding equilibria, and the T donates the transpose operator. The MRT-LBE equation can be written as follows:
f ðx þ ei dt; t þ dtÞ f ðx; tÞ ¼ M 1 Sðmðx; tÞ meq ðx; tÞÞ;
ð2Þ
1
where M is a 9 9 transform matrix, and M 1 ¼ MT ðMM T Þ . M was used to transform the distribution function to the velocT
ity moments m ¼ ðm0 ; m1 ; . . . ; m8 Þ as follows.
m¼Mf
and f ¼ M 1 m:
ð3Þ
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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The matrix M can be written as:
0
1
1
1
1
1
1
1
B 4 1 1 1 1 2 2 B B B 4 2 2 2 2 1 1 B B 1 0 1 0 1 1 B 0 B 0 2 0 2 0 1 1 M¼B B B 0 1 0 1 1 1 B 0 B B 0 0 2 0 2 1 1 B B 0 1 1 1 1 0 0 @ 0 0 0 0 0 1 1
1 2 1 1 1 1 1 0 1
1
1
2 C C C 1 C C C 1 C C 1 C C C 1 C C 1 C C C 0 A 1
ð4Þ
In the D2Q9 MRT-LBE model, the nine discrete velocities are given by
8 ð0; 0Þ i¼0 > > > < ði1Þp ði1Þp i ¼ 1; 2; 3; 4 ei ¼ c cos 2 ; sin 2 > p ffiffiffi > > p p : ; sin ð2i1Þ i ¼ 5; 6; 7; 8 2c cos ð2i1Þ 2 2
ð5Þ
where c is the lattice velocity, and c ¼ dx =dt . dx is the lattice spacing and dt is the time step. In this present work, dx ¼ dt ¼ 1, pffiffiffi the speed of sound cs ¼ 1= 3 and the variables were given in non-dimensionless units, normalized by the spacing dx and the time step dt . For the D2Q9 MRT-LBE model [34], the nine components of the moment vector m can be defined as: T
m ¼ ðq; e; e; jx ; qx ; jy ; qy ; pxx ; pxy Þ . m0 ¼ q is the fluid density, m1 ¼ e is the energy mode, m2 ¼ e is related to energy square, m3 ¼ jx and m5 ¼ jy are the x and y components of momentum j ¼ qm ðu; v Þ, m4 ¼ qx and m6 ¼ qy are the x and y components of energy flux, m7 ¼ pxx and m8 ¼ pxy are related to the components of the stress tensor. The equilibrium moments for the non-conserved moments are [34]: 2
2
meq 1 ¼ 2q þ 3ðjx þ jy Þ; 2
2
meq 2 ¼ q 3ðjx þ jy Þ=qm ; meq 4 meq 6
¼ jx ;
ð6Þ
¼ jy ; 2
2
meq 7 ¼ ðjx jy Þ=qm ; meq 8 ¼ jx jy =qm ; where qm is the mean fluid density. The relaxation parameters si can be obtained by the linear stability analysis [34]. In this work, they were chosen as the following: s1 ¼ s2 ¼ 1:4, s4 ¼ s6 ¼ 1:2 and s7 ¼ s8 ¼ 2=ð1 þ 6tÞ. t is the kinematic fluid viscosity. The macroscopic fluid variables, such as velocity u and density q are obtained by:
q¼
X f i;
qu ¼
i X ei f i :
ð7Þ
i
The non-equilibrium extrapolation method developed by Guo et al. [35] was employed to deal with all the boundaries in this study. This boundary schemes shows advantages of second-order accuracy and convergence, with good numerical stability [36]. For each boundary node xb , the distribution function includes the equilibrium f eq ðxb ; tÞ and non-equilibrium f neq ðxb ; tÞ part as follows:
f ðxb ; tÞ ¼ f eq ðxb ; tÞ þ f neq ðxb ; tÞ:
ð8Þ
At boundary node, velocity ub is known, but the density qb is unknown. So the equilibrium part could be calculated approximately from:
f eq ðxb ; tÞ ¼ f eq ðqðxf ; tÞ; ub Þ:
ð9Þ
Fluid node xf is the adjacent node of xb . The non-equilibrium part could be determined approximately by:
f neq ðxb ; tÞ ¼ f neq ðxf ; tÞ ¼ f ðxf ; tÞ f eq ðxf ; tÞ:
ð10Þ
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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3. Particle motion Newton’s second law was employed to calculate the motion of each individual particle. In this presented simulation, the particles were assumed to be rigid spheres, and the density ratio of the particle to fluid was 2000. The effect of the particles on fluid and the interaction between particles were neglected since he gas–solid flow was considered to be dilute flow even for Stokes number of O(10). In this study, only drag force were considered which is the major force on the particles, while other forces such as the Saffman lift force, gravity force and the Basset force were neglected. Then, the non-dimensional governing equations for the particle motion and the position can be given as:
dv p f ¼ ðuf v p Þ; St dt
ð11Þ
dxp ¼ vp; dt
ð12Þ
where v p is the particle velocity, and xp is the particle position. uf is the fluid velocity. The fluid parameters of the particle position, involved in the particle calculation, are obtained by linearly interpolating the fluid parameters of the neighborhood (Rep is the particle nodes. f is the modified factor for the Stokes drag force, and it can be described as f ¼ 1 þ 0:15Re0:687 p Reynolds number) [37]. The particle Stokes number St is defined as: St ¼
ðqp d2p =18lÞ . ðL0 =U 0 Þ
qp is the particle density, dp is the particle
diameter, and l is the fluid dynamics viscosity. L0 is the characteristic length scale, and U 0 is the characteristic velocity scale.
Fig. 1. Schematic of the computational domain of the 2D gas–solid turbulent jet.
Fig. 2. Distribution of mean velocity profiles along the lateral direction (y/b).
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 3. Evolution of vortex structures at different non-dimensional time of t = 45, 100 and 120.
In this work, six different distribution of particles at Stokes number of 0.01, 0.1, 0.5, 1, 10 and 50 were studied. For each case, 26 particles were injected into the flow-field at the nozzle with even distribution every 12 time steps, and the inlet velocity was equal to the local fluid velocity.
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 4. Evolution of vortex structures at different non-dimensional time of t = 145, 157.5 and 175.
4. Flow conditions Fig. 1 shows the computational domain of the two-dimensional gas–solid two-phase turbulent plane jet. The nozzle located in the center of inlet boundary with the width of d = 0.04 m. The high-speed stream in the region of nozzle was calculated according to the following typical shape velocity profile [38].
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
H. Zhou et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx
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Fig. 5. Dispersions of particles at different Stokes number at the non-dimensional time of t = 45.
u¼
U2 þ U1 U2 U1 y ; tanh þ 2 2 2h0
ð13Þ
where U2 = 2.2 m/s, and U1 = 0.2 m/s. h0 is the initial momentum thickness set to be 0.05d: The co-flow stream velocity was U1. The lateral velocity in the inlet boundary was zero. Based on the velocity difference between two streams and the nozzle width, the flow Reynolds number was 4500. The length and height of the computational domain were 25d and 20d; respectively. Uniform grid of Dx ¼ Dy ¼ d=20 was employed for the whole domain, and a total of 400 500 grids were calculated in the simulation. Different grid sizes were tested, it turned out that the results were independent of the chosen grid resolution. The side and outlet boundary conditions are formulated as:
Lateral side : Outlet :
@u ¼ 0; v ¼ 0 @y @u @ v ¼ ¼0 @x @x
ð14Þ
where u and v are the streamwise and lateral velocity, respectively. In order to assess the simulation accuracy of LBM, the Mach number defined as Ma ¼ v max is calculated. v max is the maxC imum velocity in the flow field, and the C is the lattice speed. The Mach number is required to satisfy Ma < 1. But in practice, it should be smaller than 0.1 at least. In this work, the lattice speed is equal to 1, and the maximum fluid velocity is 0.071, then the maximum Mach number is equivalent to 0.071 smaller than 0.1. The maximum Mach number shows that the Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 6. Dispersions of particles at different Stokes number at the non-dimensional time of t = 125.
calculation results have a reasonable accuracy. So in LBM simulations, the choosing of the lattice speed is of importance to make sure the results are reasonable. The simulations were perform on a CPU platform of AMD (Athlon(tm) 64 X2, dual core processor 3800+, 2 GHz) standard PC with RAM of 2 GB. The running time for each cases was about 20 h.
5. Numerical results and discussions The validity of this present simulation and the profiles of mean streamwise velocity were illustrated in Fig. 2. DU c ¼ U n U 1 is the mean streamwise velocity excess of the centerline, U n is the velocity of jet centerline. DU e ¼ U U 1 is the mean streamwise velocity excess at any point of the flow domain. b is the distance between the jet centerline to the place with DU e ¼ 12 DU c . U m is non-dimensional mean streamwise velocity, defined as U e =U c . Fig. 2 shows the U m at the position of x=d ¼ 8, when the mean streamwise velocity became self-similar. As shown in the figure, the numerical results agreed well with the experimental data [39], and were also in good accordance with direct numerical simulation (DNS) results reported by Fan et al. [24]. The development of vortex structures was illustrated in Figs. 3 and 4. The coordinates of x and y were nondimensional, normalized by nozzle width d. As can be seen, at the non-dimensional time of t = 45, 100 and 120, two free shear layers appeared and roller vortex structures were formed. These roller vortex structures evolved symmetrically, that was to say, Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 7. Dispersions of particles at different Stokes number at the non-dimensional time of t = 157.5.
the vortex structure at each side of the shear layer showed the same mode. As shown in Fig. 3, with the evolution of the turbulent flow, the roller vortex rotated, and moved downstream to form a new bigger roller. Furthermore, the roller structures remained their symmetric mode. The vorticity can be calculated by:
x ¼ jr ~ vj
ð15Þ
x is the vorticity and ~ v is the velocity. Fig. 4 shows the fully developed profile of the turbulent flow, and the vortex structures had changed from symmetric to asymmetric mode. At t = 145, the symmetric mode of the vortex structure was destroyed, and asymmetric pattern appeared first at the position of x/d = 4, where the vortex structures began to form. This result is different form the previous result reported by Fan et al. [24]. They found that the flow field switch from symmetric to asymmetric mode occurs at the downstream position where the shear layers are merged, rather than the upstream position where the vortex structures starts. As the vortex developed from upstream to downstream, the asymmetric pattern occurred at the upstream first then moved towards downstream with the development of the flow. So asymmetric vortex should appears first where the vortex structures began to form. As the development of the asymmetric vortex structures, the distribution of vortices was more complicated and interesting. Single-roller vortex structure appeared, and the opposite-sign vortex structures were formed alternately. Interlaced vortex structures moved downstream, interacted with other vortex structures, and then merged to produce a larger vortex structure. This result coincided with the previous studies [40,41]. Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 8. Dispersions of particles at different Stokes number at the non-dimensional time of t = 175.
Figs. 5 and 6 shows the particle dispersion for different particles at St = 0.01, 0.1, 0.5, 1, 10 and 50 at the non-dimensional time of t = 45 and 125, respectively. When the flow field showed symmetric profile at this point of time, all the particles dispersed symmetrically along the jet centerline over the whole region. The particle Stokes number has an important effect on the particle dispersion. When the particle Stokes number was 0.01, the particles were able to follow the turbulent flow very well and respond quickly to fluid vortex structures, due to their small aerodynamic response time. Since particles at small Stokes number have good ability to trace the gas flow, they were often used as tracer for flow field in experiments. When the Stokes number increased to 0.5 and 1, the particle dispersion was very interesting. Most of the particles were located on the vortex structures boundaries and highly concentrated dispersions were formed. While in the vortex core region, there were few particles. However, particles at the Stokes number of 10 and 50 as illustrated in Fig. 6(e) and (f), showed less lateral dispersion. Most of the particles can go though the vortex structures and move downstream. The reason for the phenomenon was that their aerodynamic response time was more than the vortex structures characteristic time. These results demonstrated that the flow vortex structures produced less influence on the dispersions of particles at relatively large Stokes number. The particle dispersion patterns at t = 157.5 and t = 175 were depicted in Figs. 7 and 8. When the vortex structures of the flow field changed to asymmetric mode, the particle distribution became asymmetric. At the same time, particles at small Stokes number also showed close movement to the vortex structures, while the large particles respond slowly to the vortex structures.
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 9. Time history of the particle number in the flow field for particles at different Stokes number.
Fig. 10. Time history of the lateral dispersion function for particles at different Stokes number.
In order to investigate the particle number in the flow field for particles at different Stokes numbers, the time history of the particle number was showed in Fig. 9. As can be seen, there were more particles stayed in the flow field, when the particle Stokes number was 0.5 and 1. If the particle aerodynamic response time scale has the same order as the flow characteristic time scale, the particles will concentrate on the large-scale vortex boundaries, and more particles can stay in the flow field. These results agreed well with the particle distribution analysis mentioned above. But for particles with the Stokes number of 0.01, 10 or 50, the values of particle number became smaller, and there were fewer particles at these Stokes number stayed in the flow field. To study the particle dispersion along the lateral direction, the global dispersion function in the lateral direction was defined, and it can be given as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Nt uX DY ðtÞ ¼ t ðY i ðtÞ Y i ð0ÞÞ2 =Nt ;
ð16Þ
i¼1
where N t is the total number of particles in the flow field at the non-dimensional time t, Y i ðtÞ is the particle displacement in the lateral direction of the different particles. Y i ð0Þ is the initial lateral displacement for particle i, when it was injected into the flow field. Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 10 gives the time history of the lateral dispersion function for particles at St = 0.01, 0.1, 0.1, 0.5, 1, 10 and 50. It can be seen that the values of DY ðtÞ for particles at the Stokes number of 0.5 and 1 were higher than those of the particles at the other Stokes number. The results showed that the particles at the Stokes number of 0.5 and 1 had good ability to disperse in the lateral direction. However, for particle at large Stokes number of St = 10 and 50, the lateral dispersion was unremarkable. 6. Conclusions Numerical studies of the gas–solid two-phase flow with high Reynolds number in the turbulent plane jet were carried out by means of the MRT-LBM. Six kinds of particles with various Stokes numbers equal to 0.01, 0.1, 0.5, 1, 10 and 50 were investigated, and the particles were traced by the Lagrangian method. The results show that with the development of the flow, the flow field in the plane jet changes from initial symmetric mode to asymmetric mode. And asymmetric pattern appears first at the position of x/d = 4, where the vortex structures begin to form. Interlaced vortex structures move downstream, interact with the downstream vortex structures, and then merged to produce a larger vortex structure. These predicted results are coincided with the previous studies, and demonstrated that the MRT-LBM has good ability to simulate turbulent flow with the high Reynolds number. The particle Stokes number has great influence on the particle dispersion. Particles at Stokes number of 0.01 can respond to flow motion quickly, and show even distribution. The dispersion of particles at the Stokes number of 0.5 and 1 show strong concentrated distribution, and their dispersions along the lateral direction are remarkable. But particles with St = 50 show the even concentration distribution along the spanwise direction and hardly dispersed along the lateral direction. Acknowledgements This work was supported by National Basic Research Program of China (2009CB219802), Program for New Century Excellent Talents in University (NCET-07-0761), a Foundation for the Author of National Excellent Doctoral Dissertation of China (200747), Zhejiang Provincial Natural Science Foundation of China (R107532), and Zhejiang University K.P. Chao’s High Technology Development Foundation (2008RC001), the Fundamental Research Funds for the Central Universities. References [1] U. Frisch, B. Hasslacher, Y. Pomeau, Lattice-gas automata for the Navier–Stokes equation, Phys. Rev. Lett. 56 (14) (1986) 1505–1508. [2] J. Hardy, O. Depazzis, Y. Pomeau, Molecular-dynamics of a classical lattice gas-transport properties and time correlation functions, Phys. Rev. A 13 (5) (1976) 1949–1961. [3] X.Y. He, L.S. Luo, Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E 56 (6) (1997) 6811–6817. [4] R. Benzi, S. Succi, M. 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Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method Hao Zhou ⇑, Guiyuan Mo, Kefa Cen Zhejiang University, Institute for Thermal Power Engineering, State Key Laboratory of Clean Energy Utilization, Hangzhou 310027, PR China
a r t i c l e
i n f o
Article history: Received 2 September 2011 Received in revised form 16 April 2015 Accepted 1 June 2015 Available online xxxx Keywords: LBM MRT Gas–solid Turbulent jet Particle dispersion
a b s t r a c t The gas–solid two-phase turbulent plane jet flow with high Reynolds number of 4500 was numerical investigated by means of lattice Boltzmann method (LBM). The multiple relaxation time (MRT) was employed to deal with the high Reynolds number fluid flows, and the particles were traced by the Lagrangian method. The results show that the flow changes from initial symmetric mode to asymmetric mode with the development of the flow. And asymmetric pattern appears first at the position of x/d = 4, where the vortex structures begin to form. The dispersion of particles at different Stokes number shows various distributions. The MRT-LBM shows its good ability in simulating turbulent flow with the high Reynolds number. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction The lattice Boltzmann method is originated from Lattice Gas Automata (LGA) [1,2], and it can also be derived directly from the Boltzmann equation [3]. A direct connection between the Boltzmann equation and the incompressible Navier–Stokes equation can be obtained under the condition of near incompressible [4,5]. Compared with the traditional Computational Fluid Dynamics (CFD) such as Navier–Stokes equation, LBM shows distinct advantages of high efficiency of parallel implementation, high calculation efficiency and ability in dealing with moving and complex boundaries [6–9]. LBM is a relatively new computational fluid method, and it has attracted considerable attention in recent years. Over the past decade, LBM has been widely used in scientific research and engineering application. Shan and Chen [10] developed a lattice Boltzmann model to simulate flows containing multiple phases and components, and this model presented many applications in large-scale numerical simulations of various types of fluid flows. Park et al. [11] applied the lattice Boltzmann method to simulate the flow in the electrode of a PEM fuel cell. Succi [12] employed the LBM successfully to calculate a variety of configurations, even those with a complex coupling of physical and chemical processes. Chen et al. [13] carried out simulations of complex fluid physics such as cars and airplanes using an extended kinetic Boltzmann equation, analyzing different approaches based on extensions of the Boltzmann equation. Han et al. [14] and Feng et al. [15] presented a new solution strategy that coupled lattice Boltzmann (LB), large eddy simulation (LES), and discrete element (DE) methodologies to simulate particle–fluid systems at moderately high Reynolds numbers. Enhanced collision operator was presented to be an efficient strategy for the lattice gas Boltzmann equation by Higuera et al. [16]. Then the multiple relaxation time (MRT) is developed and used widely. McCracken and Abraham [17] developed a MRT lattice-Boltzmann model for multiphase flow and evaluated the accuracy in several test problems such as oscillating ⇑ Corresponding author. Tel.: +86 571 87952598; fax: +86 571 87951616. E-mail address: [email protected] (H. Zhou). http://dx.doi.org/10.1016/j.apm.2015.06.005 0307-904X/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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liquid cylinders and capillary waves. Pan et al. [18] conducted a comparative study of the LBE models with MRT and the BGK-SRT, and reported that the MRT–LBE model is superior to the BGK–LBE model. Guo and Zheng [19] simulated the Poiseuille flow in the slip flow regime by the lattice Boltzmann equation (LBE) with multiple relaxation times (MRTs). Yu et al. [20] carried out the application of multiple-relaxation-time (MRT) lattice Boltzmann equation (LBE) for large-eddy simulation (LES) of the turbulent square jet flow. The MRT-LBE model for LES was carried out in a surface mounted cube in a channel at Re = 40,000 by Krafczyk et al. [21], and their preliminary results agree well with experimental data. Dynamic subgrid scale (SGS) models were incorporated in the MRT-LBM of LES of fully-developed turbulent channel flow at two different shear Reynolds numbers of 180 and 395, reported by Premnath et al. [22]. General lattice Boltzmann equation (GLBE) using MRT with forcing term for eddy capturing simulation of wall-bounded turbulent flows was presented by Premnath et al. [23], they found that markedly stability characteristics and good agreements were achieved. Deep understanding of the flow coherent structures and particle dispersion in the turbulent jet is of great significance in efficient engineering applications in the fields of chemical reaction and jet combustion [24]. The particle dispersion is an important factor in the engineering design and efficient engineering applications. The turbulent jet has been studied for a long time. Bradbury used a hot-wire anemometry to measure the statistical quantities in the self-similar region of a plane jet, and observed that the initial and external conditions have a strong influence on the development of the plane jet flow fields [25]. Stanley and Sarkar [26] investigated the 2D single-phase flow in the plane jet by direct numerical simulation. There are also many experimental and numerical studies reported on the gas–solid two-phase turbulent jet flows. Melville and Bray [27] investigated the two-phase turbulent jet flows and presented a model to describe them. Tsuji et al. [28] measured axisymmetric particle-laden jets by three different devices, and particle and air velocities, particle concentration and air turbulence were obtained. Despirito and Wang [29] numerically studied the influence of the particle Stokes number on the flow stability in the two-way coupled particle-laden jet, and found that the particles at Stokes numbers on the order of 1 correspond to the maximum flow stability. Fan et al. [24] investigated the gas-particle two-phase flow in the 2D turbulent plane jet by solving the compressible flow fields using direct numerical simulation, and they focused their study on the evolution of coherent vortex structures and dispersion patterns of particles in the near field. Menon and Soo [30] developed a three-dimensional lattice Boltzmann equation solver to study flows associated with synthetic jets and turbulent-forced rectangular jets by large-eddy simulation (LES). Premnath and Abraham [31] investigated the transient and incompressible turbulent plane jets by the discrete lattice BGK Boltzmann equation, and a spatially and temporally dependent relaxation time parameter is used to represent the averaged flow field, satisfactory results is obtained. However, there are few studies reported about the particle dispersion and large-scale vortex structures in the gas–solid two-phase turbulent jet flow with the moderately high Reynolds numbers using LBM. In addition, most of the numerical studies mentioned above were based on Navier–Stokes equation. In this paper, we use the LB method to investigate the vortex characteristics and the particle dispersion patterns in a 2D gas–solid turbulent plane jet. The MRT-LBM was employed to deal with the high Reynolds number fluid flows. The main objective of this study is to study the evolution of the coherent structures and the dispersion of particles at different Stokes numbers by the means of MRT-LBM, and the particles were traced by the Lagrangian method. The Letter is organized as follows. In Section 2 we describe the lattice Boltzmann method. Section 3 introduces the governing equations for particles. Section 4 presents the computational conditions. The numerical results and discussions are given in Section 5. The last section is devoted to conclusions. 2. D2Q9 MRT-LBE model Lattice BGK (LBGK) model and DnQb (n is the dimension of the space and b is the discretized speed) model presented by Qian et al. [32]. On each node of the lattice, there are a set of symmetric discrete velocities fei ji ¼ 0; 1; ; b 1g and a set of a velocity distribution functions ff i ji ¼ 0; 1; ; b 1g. In this paper, direct simulation using LBM is carried out to study the turbulent jet flow, large eddy simulation (LES) is not considered. A nine velocities two dimensional (D2Q9) LBE model coupled with multiple relaxation time (MRT) [32,33] was employed. The evolution equation for D2Q9 LBM could be described as:
f ðx þ ei dt; t þ dtÞ f ðx; tÞ ¼ Sðf ðx; tÞ f eq ðx; tÞÞ;
ð1Þ
eq eq T eq T where S is the collision matrix, and S ¼ diagðs0 ; s1 ; . . . ; s8 Þ. f ¼ ðf 0 ; f 1 ; . . . ; f 8 Þ and f eq ¼ f 0 ; f 1 ; . . . ; f 8 are vectors of the distribution function and their corresponding equilibria, and the T donates the transpose operator. The MRT-LBE equation can be written as follows:
f ðx þ ei dt; t þ dtÞ f ðx; tÞ ¼ M 1 Sðmðx; tÞ meq ðx; tÞÞ;
ð2Þ
1
where M is a 9 9 transform matrix, and M 1 ¼ MT ðMM T Þ . M was used to transform the distribution function to the velocT
ity moments m ¼ ðm0 ; m1 ; . . . ; m8 Þ as follows.
m¼Mf
and f ¼ M 1 m:
ð3Þ
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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The matrix M can be written as:
0
1
1
1
1
1
1
1
B 4 1 1 1 1 2 2 B B B 4 2 2 2 2 1 1 B B 1 0 1 0 1 1 B 0 B 0 2 0 2 0 1 1 M¼B B B 0 1 0 1 1 1 B 0 B B 0 0 2 0 2 1 1 B B 0 1 1 1 1 0 0 @ 0 0 0 0 0 1 1
1 2 1 1 1 1 1 0 1
1
1
2 C C C 1 C C C 1 C C 1 C C C 1 C C 1 C C C 0 A 1
ð4Þ
In the D2Q9 MRT-LBE model, the nine discrete velocities are given by
8 ð0; 0Þ i¼0 > > > < ði1Þp ði1Þp i ¼ 1; 2; 3; 4 ei ¼ c cos 2 ; sin 2 > p ffiffiffi > > p p : ; sin ð2i1Þ i ¼ 5; 6; 7; 8 2c cos ð2i1Þ 2 2
ð5Þ
where c is the lattice velocity, and c ¼ dx =dt . dx is the lattice spacing and dt is the time step. In this present work, dx ¼ dt ¼ 1, pffiffiffi the speed of sound cs ¼ 1= 3 and the variables were given in non-dimensionless units, normalized by the spacing dx and the time step dt . For the D2Q9 MRT-LBE model [34], the nine components of the moment vector m can be defined as: T
m ¼ ðq; e; e; jx ; qx ; jy ; qy ; pxx ; pxy Þ . m0 ¼ q is the fluid density, m1 ¼ e is the energy mode, m2 ¼ e is related to energy square, m3 ¼ jx and m5 ¼ jy are the x and y components of momentum j ¼ qm ðu; v Þ, m4 ¼ qx and m6 ¼ qy are the x and y components of energy flux, m7 ¼ pxx and m8 ¼ pxy are related to the components of the stress tensor. The equilibrium moments for the non-conserved moments are [34]: 2
2
meq 1 ¼ 2q þ 3ðjx þ jy Þ; 2
2
meq 2 ¼ q 3ðjx þ jy Þ=qm ; meq 4 meq 6
¼ jx ;
ð6Þ
¼ jy ; 2
2
meq 7 ¼ ðjx jy Þ=qm ; meq 8 ¼ jx jy =qm ; where qm is the mean fluid density. The relaxation parameters si can be obtained by the linear stability analysis [34]. In this work, they were chosen as the following: s1 ¼ s2 ¼ 1:4, s4 ¼ s6 ¼ 1:2 and s7 ¼ s8 ¼ 2=ð1 þ 6tÞ. t is the kinematic fluid viscosity. The macroscopic fluid variables, such as velocity u and density q are obtained by:
q¼
X f i;
qu ¼
i X ei f i :
ð7Þ
i
The non-equilibrium extrapolation method developed by Guo et al. [35] was employed to deal with all the boundaries in this study. This boundary schemes shows advantages of second-order accuracy and convergence, with good numerical stability [36]. For each boundary node xb , the distribution function includes the equilibrium f eq ðxb ; tÞ and non-equilibrium f neq ðxb ; tÞ part as follows:
f ðxb ; tÞ ¼ f eq ðxb ; tÞ þ f neq ðxb ; tÞ:
ð8Þ
At boundary node, velocity ub is known, but the density qb is unknown. So the equilibrium part could be calculated approximately from:
f eq ðxb ; tÞ ¼ f eq ðqðxf ; tÞ; ub Þ:
ð9Þ
Fluid node xf is the adjacent node of xb . The non-equilibrium part could be determined approximately by:
f neq ðxb ; tÞ ¼ f neq ðxf ; tÞ ¼ f ðxf ; tÞ f eq ðxf ; tÞ:
ð10Þ
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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3. Particle motion Newton’s second law was employed to calculate the motion of each individual particle. In this presented simulation, the particles were assumed to be rigid spheres, and the density ratio of the particle to fluid was 2000. The effect of the particles on fluid and the interaction between particles were neglected since he gas–solid flow was considered to be dilute flow even for Stokes number of O(10). In this study, only drag force were considered which is the major force on the particles, while other forces such as the Saffman lift force, gravity force and the Basset force were neglected. Then, the non-dimensional governing equations for the particle motion and the position can be given as:
dv p f ¼ ðuf v p Þ; St dt
ð11Þ
dxp ¼ vp; dt
ð12Þ
where v p is the particle velocity, and xp is the particle position. uf is the fluid velocity. The fluid parameters of the particle position, involved in the particle calculation, are obtained by linearly interpolating the fluid parameters of the neighborhood (Rep is the particle nodes. f is the modified factor for the Stokes drag force, and it can be described as f ¼ 1 þ 0:15Re0:687 p Reynolds number) [37]. The particle Stokes number St is defined as: St ¼
ðqp d2p =18lÞ . ðL0 =U 0 Þ
qp is the particle density, dp is the particle
diameter, and l is the fluid dynamics viscosity. L0 is the characteristic length scale, and U 0 is the characteristic velocity scale.
Fig. 1. Schematic of the computational domain of the 2D gas–solid turbulent jet.
Fig. 2. Distribution of mean velocity profiles along the lateral direction (y/b).
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 3. Evolution of vortex structures at different non-dimensional time of t = 45, 100 and 120.
In this work, six different distribution of particles at Stokes number of 0.01, 0.1, 0.5, 1, 10 and 50 were studied. For each case, 26 particles were injected into the flow-field at the nozzle with even distribution every 12 time steps, and the inlet velocity was equal to the local fluid velocity.
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 4. Evolution of vortex structures at different non-dimensional time of t = 145, 157.5 and 175.
4. Flow conditions Fig. 1 shows the computational domain of the two-dimensional gas–solid two-phase turbulent plane jet. The nozzle located in the center of inlet boundary with the width of d = 0.04 m. The high-speed stream in the region of nozzle was calculated according to the following typical shape velocity profile [38].
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 5. Dispersions of particles at different Stokes number at the non-dimensional time of t = 45.
u¼
U2 þ U1 U2 U1 y ; tanh þ 2 2 2h0
ð13Þ
where U2 = 2.2 m/s, and U1 = 0.2 m/s. h0 is the initial momentum thickness set to be 0.05d: The co-flow stream velocity was U1. The lateral velocity in the inlet boundary was zero. Based on the velocity difference between two streams and the nozzle width, the flow Reynolds number was 4500. The length and height of the computational domain were 25d and 20d; respectively. Uniform grid of Dx ¼ Dy ¼ d=20 was employed for the whole domain, and a total of 400 500 grids were calculated in the simulation. Different grid sizes were tested, it turned out that the results were independent of the chosen grid resolution. The side and outlet boundary conditions are formulated as:
Lateral side : Outlet :
@u ¼ 0; v ¼ 0 @y @u @ v ¼ ¼0 @x @x
ð14Þ
where u and v are the streamwise and lateral velocity, respectively. In order to assess the simulation accuracy of LBM, the Mach number defined as Ma ¼ v max is calculated. v max is the maxC imum velocity in the flow field, and the C is the lattice speed. The Mach number is required to satisfy Ma < 1. But in practice, it should be smaller than 0.1 at least. In this work, the lattice speed is equal to 1, and the maximum fluid velocity is 0.071, then the maximum Mach number is equivalent to 0.071 smaller than 0.1. The maximum Mach number shows that the Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 6. Dispersions of particles at different Stokes number at the non-dimensional time of t = 125.
calculation results have a reasonable accuracy. So in LBM simulations, the choosing of the lattice speed is of importance to make sure the results are reasonable. The simulations were perform on a CPU platform of AMD (Athlon(tm) 64 X2, dual core processor 3800+, 2 GHz) standard PC with RAM of 2 GB. The running time for each cases was about 20 h.
5. Numerical results and discussions The validity of this present simulation and the profiles of mean streamwise velocity were illustrated in Fig. 2. DU c ¼ U n U 1 is the mean streamwise velocity excess of the centerline, U n is the velocity of jet centerline. DU e ¼ U U 1 is the mean streamwise velocity excess at any point of the flow domain. b is the distance between the jet centerline to the place with DU e ¼ 12 DU c . U m is non-dimensional mean streamwise velocity, defined as U e =U c . Fig. 2 shows the U m at the position of x=d ¼ 8, when the mean streamwise velocity became self-similar. As shown in the figure, the numerical results agreed well with the experimental data [39], and were also in good accordance with direct numerical simulation (DNS) results reported by Fan et al. [24]. The development of vortex structures was illustrated in Figs. 3 and 4. The coordinates of x and y were nondimensional, normalized by nozzle width d. As can be seen, at the non-dimensional time of t = 45, 100 and 120, two free shear layers appeared and roller vortex structures were formed. These roller vortex structures evolved symmetrically, that was to say, Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 7. Dispersions of particles at different Stokes number at the non-dimensional time of t = 157.5.
the vortex structure at each side of the shear layer showed the same mode. As shown in Fig. 3, with the evolution of the turbulent flow, the roller vortex rotated, and moved downstream to form a new bigger roller. Furthermore, the roller structures remained their symmetric mode. The vorticity can be calculated by:
x ¼ jr ~ vj
ð15Þ
x is the vorticity and ~ v is the velocity. Fig. 4 shows the fully developed profile of the turbulent flow, and the vortex structures had changed from symmetric to asymmetric mode. At t = 145, the symmetric mode of the vortex structure was destroyed, and asymmetric pattern appeared first at the position of x/d = 4, where the vortex structures began to form. This result is different form the previous result reported by Fan et al. [24]. They found that the flow field switch from symmetric to asymmetric mode occurs at the downstream position where the shear layers are merged, rather than the upstream position where the vortex structures starts. As the vortex developed from upstream to downstream, the asymmetric pattern occurred at the upstream first then moved towards downstream with the development of the flow. So asymmetric vortex should appears first where the vortex structures began to form. As the development of the asymmetric vortex structures, the distribution of vortices was more complicated and interesting. Single-roller vortex structure appeared, and the opposite-sign vortex structures were formed alternately. Interlaced vortex structures moved downstream, interacted with other vortex structures, and then merged to produce a larger vortex structure. This result coincided with the previous studies [40,41]. Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 8. Dispersions of particles at different Stokes number at the non-dimensional time of t = 175.
Figs. 5 and 6 shows the particle dispersion for different particles at St = 0.01, 0.1, 0.5, 1, 10 and 50 at the non-dimensional time of t = 45 and 125, respectively. When the flow field showed symmetric profile at this point of time, all the particles dispersed symmetrically along the jet centerline over the whole region. The particle Stokes number has an important effect on the particle dispersion. When the particle Stokes number was 0.01, the particles were able to follow the turbulent flow very well and respond quickly to fluid vortex structures, due to their small aerodynamic response time. Since particles at small Stokes number have good ability to trace the gas flow, they were often used as tracer for flow field in experiments. When the Stokes number increased to 0.5 and 1, the particle dispersion was very interesting. Most of the particles were located on the vortex structures boundaries and highly concentrated dispersions were formed. While in the vortex core region, there were few particles. However, particles at the Stokes number of 10 and 50 as illustrated in Fig. 6(e) and (f), showed less lateral dispersion. Most of the particles can go though the vortex structures and move downstream. The reason for the phenomenon was that their aerodynamic response time was more than the vortex structures characteristic time. These results demonstrated that the flow vortex structures produced less influence on the dispersions of particles at relatively large Stokes number. The particle dispersion patterns at t = 157.5 and t = 175 were depicted in Figs. 7 and 8. When the vortex structures of the flow field changed to asymmetric mode, the particle distribution became asymmetric. At the same time, particles at small Stokes number also showed close movement to the vortex structures, while the large particles respond slowly to the vortex structures.
Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 9. Time history of the particle number in the flow field for particles at different Stokes number.
Fig. 10. Time history of the lateral dispersion function for particles at different Stokes number.
In order to investigate the particle number in the flow field for particles at different Stokes numbers, the time history of the particle number was showed in Fig. 9. As can be seen, there were more particles stayed in the flow field, when the particle Stokes number was 0.5 and 1. If the particle aerodynamic response time scale has the same order as the flow characteristic time scale, the particles will concentrate on the large-scale vortex boundaries, and more particles can stay in the flow field. These results agreed well with the particle distribution analysis mentioned above. But for particles with the Stokes number of 0.01, 10 or 50, the values of particle number became smaller, and there were fewer particles at these Stokes number stayed in the flow field. To study the particle dispersion along the lateral direction, the global dispersion function in the lateral direction was defined, and it can be given as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Nt uX DY ðtÞ ¼ t ðY i ðtÞ Y i ð0ÞÞ2 =Nt ;
ð16Þ
i¼1
where N t is the total number of particles in the flow field at the non-dimensional time t, Y i ðtÞ is the particle displacement in the lateral direction of the different particles. Y i ð0Þ is the initial lateral displacement for particle i, when it was injected into the flow field. Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005
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Fig. 10 gives the time history of the lateral dispersion function for particles at St = 0.01, 0.1, 0.1, 0.5, 1, 10 and 50. It can be seen that the values of DY ðtÞ for particles at the Stokes number of 0.5 and 1 were higher than those of the particles at the other Stokes number. The results showed that the particles at the Stokes number of 0.5 and 1 had good ability to disperse in the lateral direction. However, for particle at large Stokes number of St = 10 and 50, the lateral dispersion was unremarkable. 6. Conclusions Numerical studies of the gas–solid two-phase flow with high Reynolds number in the turbulent plane jet were carried out by means of the MRT-LBM. Six kinds of particles with various Stokes numbers equal to 0.01, 0.1, 0.5, 1, 10 and 50 were investigated, and the particles were traced by the Lagrangian method. The results show that with the development of the flow, the flow field in the plane jet changes from initial symmetric mode to asymmetric mode. And asymmetric pattern appears first at the position of x/d = 4, where the vortex structures begin to form. Interlaced vortex structures move downstream, interact with the downstream vortex structures, and then merged to produce a larger vortex structure. These predicted results are coincided with the previous studies, and demonstrated that the MRT-LBM has good ability to simulate turbulent flow with the high Reynolds number. The particle Stokes number has great influence on the particle dispersion. Particles at Stokes number of 0.01 can respond to flow motion quickly, and show even distribution. The dispersion of particles at the Stokes number of 0.5 and 1 show strong concentrated distribution, and their dispersions along the lateral direction are remarkable. But particles with St = 50 show the even concentration distribution along the spanwise direction and hardly dispersed along the lateral direction. 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Please cite this article in press as: H. Zhou et al., Numerical investigation of a gas–solid turbulent jet flow with Reynolds number of 4500 using lattice Boltzmann method, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.06.005