Aerosol Science and Technology, 39:319–332, 2005 c American Association for Aerosol Research Copyright ISSN: 0278-6826 print / 1521-7388 online DOI: 10.1080/027868290930961
Numerical Simulation and Validation of Dilute Gas-Particle Flow Over a Backward-Facing Step Z. F. Tian,1 J. Y. Tu,1 and G. H. Yeoh2 1 2
School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Vic., Australia Australian Nuclear Science and Technology Organisation (ANSTO), Menai, NSW, Australia
The physical characteristics of dilute gas–particle flows over a backward-facing step geometry are investigated. An investigation is performed to assess the performance of two computational approaches—the Lagrangian particle-tracking model and Eulerian two-fluid model—to predict the particle phase flow parameters under the influence of different particle inertia. Particles with corresponding diameters of 1 µm (small-size particles) and 70 µm (large-size particles) are simulated under the flow condition of two Reynolds numbers (based on the step height): Re = 15000 and Re = 64000, for which the model predictions are compared against benchmark experimental measurements. Among the various turbulence models evaluated, the RNG κ-ε and realizable κ-ε models provided better agreement with the experimental data for the range of particles considered. The Eulerian approach used in this study combines an overlapped technique with a particle–wall collision model to better present the particle–wall momentum transfer than traditional Eulerian models. In comparing to the Lagrangian and Eulerian approaches for particle flow predictions, the latter was shown to yield closer agreement with the measured values.
INTRODUCTION Dilute gas–particle flows are commonly found in many engineering applications. Contaminant particle flow in buildings (Fogarty and Nelson 2003), spray combustion (Sirignao 1999), and inhalers for delivering particle medicines (Gemci et al. 2003) are some typical examples. Interest in particle information such as the mean particle velocity and concentration in the turbulent gas flow field is of great significance to practicing engineers. With the development of the increasing computer power and advancement of commercial modeling software, computational fluid dynamics (CFD) technique to study dilute gas–particle flow problems is gradually becoming an attractive investigative tool. Here, more insights into the gas–particle behaviors in regards to
Received 17 September 2004; accepted 2 February 2005. The financial support provided by the Australian Research Council (project ID LP0347399) is gratefully acknowledged. Address correspondence to Jiyuan Tu, School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, P.O. Box 71, Bundoora, Vic. 3083, Australia. E-mail:
[email protected]
its interaction with turbulence or other flow effects can be better understood. There are two basic categories of computational approaches that are currently used to predict the particle phase flows: the Eulerian two-fluid model and the Lagrangian particle-tracking model. In the Eulerian model, the particle flow is treated as continuous fluid flow and regarded as interacting with the fluid phase. For the particle-tracking approach, the Lagrangian model considers the motion of a single particle and relevant variables along the particle trajectory. The Eulerian model simulates the particle flow via fluid-like equations. This enhances the ease of implementation of the approach in CFD codes. It can be handled efficiently through current state-of-the-art solvers, resulting in relatively less computational time of mean parameters for the particle flow. These advantages make the Eulerian model very attractive, and lately there are possible considerations of extending the two-fluid model by adopting the large eddy simulation (LES) approach (Pandya and Mashayek 2002) to circumvent the problems associated with current turbulence modelling. Nevertheless, there are some inherent problems in the use of Eulerian model for gas–particle flows. The first difficulty arises in the modelling of the surface boundary conditions for the particle phase. The current Eulerian formulation is still deficient in correctly describing the aerodynamics drag force on the particle phase in the vicinity of a solid wall. Properly quantifying the incident and reflected particles during the process of particle–wall impaction in a control volume at the boundary surface is still far from adequate resolution. More information regarding the particle behavior is required to develop suitable models to better represent the particle–wall impaction process (Tu 2000; Tu et al. 2004). Another problem with the Eulerian model is the justification of the continuum assumption as the particles equilibrate with neither local fluid nor each other when flowing through the flow field (Shirolkar et al. 1996). The Eulerian model describes the averaged local particle parameters instead of the individual particle trajectory. This introduces the problem of “crossing trajectories” (Slater and Young 2001), and the lost particle properties could be significant when considering the local reaction rate of a particle that is essential for reacting flow systems (Shirolkar 319
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et al. 1996). To overcome the problem of modelling particle–wall collision, Tu and Fletcher (1995) established a set of Eulerian formulation with generalized wall boundary conditions and developed a particle–wall collision model to represent better the particle–wall momentum transfer. Here good agreement with the experimental data was achieved using this model to investigate the particle-laden flow in an in-line tube bank. Later, Tu (1997) employed this model with overlapped grids to simulate the particle gas flow over a backward-facing step and two-phase flow in a T-junction channel. The model also yielded encouraging results, where good agreements between the predictions and experimental data were found. The Lagrangian model tracks the individual particle motion and therefore overcomes the difficulties associated with the Eulerian model. Here it provides detailed description of the particle motion including the effects of history and takes into consideration all the forces acting on the particles (Zhang et al. 2002). Nevertheless, one major concern of the Lagrangian approach is the large computational expense that may be experienced because of the requirement to track substantial number of the particles to attain successfully good statistical information of the particle phase. Lengthy computational times have prevented the use of the Lagrangian model in the past. However, with current computational resources, it has been demonstrated in this article that the Lagrangian computations carried out on the workstation computer platform resulted in reasonable computation times and good turnaround of a number of investigative studies. It is therefore expected that with the progress of computer speeds, the time expense will be significantly reduced and the Lagrangian model will become a popular tool among engineers. In the framework of the Lagrangian model, the discrete random walk (DRW) model is widely used. The DRW model describes how when a particle travels through the turbulent flow field it interacts with a succession of discrete stylized fluid-phase turbulent eddies. These eddies can be characterized with the mean velocity, the instantaneous turbulent velocity fluctuations, and the eddy lifetime. The model obtains the mean fluid-phase velocities from the Reynolds-averaged Navier-Stokes (RANS) equations; it uses the Gaussian distributed random tracking to include the effect of instantaneous turbulent velocity fluctuations on the particle trajectory. The eddy lifetime is calculated from the integral time that describes the time spent in turbulent motion along the particle path. The interaction time between the particle and eddies is the smaller of the eddy lifetime or the eddy crossing time that accounts for the crossing trajectory effect. During one interaction time, the particle velocity is kept constant and its trajectory is calculated. In the next interaction time the particle is at the new location, and the local fluid-phase turbulent fluctuations are used to calculate the current particle velocity and trajectory. To the best of the authors’ knowledge, no in-depth investigations have been performed on the comparison of the Lagrangian particle tracking model and Eulerian two-fluid model performance with measurements on benchmark problems such as gas–
particle flow over a backward-facing step problem. Hallman et al. (1995) focused on the computation of turbulent evaporating sprays using the Eulerian and Lagrangian approaches. Recently, Lee et al. (2002) simulated the gas–particle flow in a 45◦ ramp and an isolated single tube using Lagrangian and Eulerian methods, and both approaches have been found to yield good agreement with measured results. Bourloutski et al. (2002) later verified both approaches in turbulent gas–particle flows with heat transfer in a pipe. The objective of this article is therefore to simulate the dilute gas–particle flow on the benchmark problem of flow over backward-facing step using both a Lagrangian particle-tracking model (DRW model) and an Eulerian two-fluid model with overlapped grid system (Tu 1997). The backward-facing step is one basic geometry of the many engineering applications where the presence of turbulent flow over the backward-facing step allows the determination and fundamental understanding of important flow features such as flow separation, flow reattachment, and free shear jet phenomena. The simulation results are validated against experimental data obtained by Ruck and Makiola (1988), and the relative performance of the two approaches will be discussed. COMPUTATIONAL METHODS Lagrangian Particle-Tracking Model (DRW Model) A generic CFD commercial code, FLUENT, is utilized to predict the continuum gas phase of the velocity profiles under steady-state conditions through solutions to the conservation of mass and momentum. The gas-phase Reynolds-averaged conservation equations with the gas kinetic turbulence and its dissipation rate can be cast in a general form: ∂ ∂φ ∂ B = S, [1] (Ai φ) − ∂x j ∂x j ∂x j continuity equation (φ = 1), g
Ai = ρg u j ,
B = 0,
S = 0;
B = ρg νg , eff ,
S=−
[2]
g
momentum equation (φ = u i ), g
Ai = ρg u j ,
∂P ; ∂x j
[3]
turbulent kinetic energy equation (φ = κ g ), g
Ai = ρg u j ,
B = ρg νg , t /σκ ,
S = Pk − ρg ε g ;
[4]
dissipation equation (φ = ε g ), g
Ai = ρg u j ,
ε (Cε1 Pk − Cε2 ρg ε g ), κg εg = g (Cε1 Pk − Cε2 ρg ε g ) − R, κ
Sstandard = SRNG
B = ρg νg , t /σε ,
g
[5]
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1/2 g Srelizable = C1 ρg 2Si2j ε − C 2 ρg Si j =
g g ∂u j 1 ∂u i . + 2 ∂x j ∂ xi
2
εg , √ κ g + νε g
where
The realizable κ-ε model proposed by Shih et al. (1995) was intended to address deficiencies experienced in the standard and renormalization group (RNG) κ-ε models. The term “realizable” means that the model satisfies certain mathematical constraints on the normal stresses, consistent with the physics of turbulent flows. The development involved the formulation of a new eddy-viscosity formula involving the variable Cµ in the turbu2 lent viscosity relationship, νg,t = Cµ κ g /ε g , and a new model for the ε-equation based on the dynamic equation of the meansquare vorticity fluctuation. The κ-equation is the same as that in the standard and RNG κ-ε models except for model constants. The variable constant C1 can be expressed as 1/2 η κg C1 = max 0.43, . [6] and η = g 2Si2j η+5 ε The variable Cµ , no longer a constant, is computed from Cµ =
1 ∗ , A0 + As κU εg
˜ i j = i j − 2 εi jk ωk ,
U∗ ≡
˜ i2j , Si2j +
¯ i j − εi jk ωk , i j =
[7]
while the model constant, A0 and As , are determined by A0 = 4.04, W =
Si j S jk Ski ; S˜ 3
AS =
√
6 cos ϕ, S˜ = Si2j .
ϕ=
√ 1 cos−1 ( 6W ), 3 [8]
Other constants in the turbulent transport equations are given to be C2 = 1.9, σ κ = 1.0, and σ ε = 1.2. In the RNG turbulent transport, the presence of the rate-of-strain term R in the ε-equation is expressed by 2
Cµ η3 (1 − η/ηo ) ε g R= , 1 + βη3 κg
[9]
where β and ηo are constants with values of 0.015 and 4.38, respectively. The significance of the inclusion of this term is its responsiveness towards the effects of rapid rate strain and streamline curvature, which cannot be properly represented by the standard κ-ε turbulence model. According to the RNG theory (Yakhot and Orzag 1986), the constants in the turbulent transport equations are given by σ κ = 0.718, σ ε = 0.718, Cµ = 0.0845, Cε1 = 1.42, and Cε2 = 1.68. For the standard κ-ε model, the model constants have the respective default values (Launder and Spalding 1972): σ κ = 1.0 and σ ε = 1.3, Cµ = 0.09, Cε1 = 1.44, and Cε2 = 1.92. A Lagrangian-formulated particle equation of motion is solved using FLUENT. The trajectory of a discrete particle phase
is determined by integrating the force balance on the particle, which is written in a Lagrangian reference frame. This force balance equates the particle inertia with the forces acting on the particle. Appropriate forces such as the drag and gravitational forces were incorporated into the equation of motion. The equation can be written as g ∂u i
p = FD u i − u i + g(ρ p − ρg ). ∂t p
[10] p
g
The drag force per unit particle mass is FD (u i − u i ), and FD is given by p g 18ρg C D d p u i −u i FD = . ρ p d 2p 24
[11]
The drag coefficient, C D , is evaluated from an experimentally fitted expression of Morsi and Alexander (1972). A fourth-order Runge-Kutta method is employed to predict the particle velocity and trajectories for the particle phase. The CFD code, FLUENT, handles the turbulent dispersion of particles by integrating the trajectory equations for individual particles using the instantag neous fluid velocity, u i +u i (t), along the particle path during the integration process. Here a stochastic method, discrete random walk or “eddy lifetime” model, is utilized where the fluctuating velocity components, u i , that prevail during the lifetime of the turbulent eddy are sampled by assuming that they obey a Gaussian probability distribution, so that u i = ζ u i 2 ,
[12]
where ζ is a normally distributed random number, and the remaining right-hand side is the local root mean square (RMS) velocity fluctuations that can be obtained (assuming isotropy) by u i 2 = 2κ g /3.
[13]
The interaction time between the particles and eddies is the smaller of the eddy lifetime, τe , and the particle eddy crossing time, tcross . The characteristic lifetime of the eddy is defined as τe = −TL log(r ),
[14]
where TL is the fluid Lagrangian integral time, TL ≈ 0.15 κ/ε. The variable r is a uniform random number between 0 and 1. The particle eddy crossing time is given by tcross = −t p ln 1 −
Le g p t p ui − ui
,
[15]
where t p is the particle relaxation time (= ρs d 2p /18ρg vg ), Le g p is the eddy length scale, and u i − u i is the magnitude of the
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relative velocity. The particle interacts with the fluid eddy over the interaction time. When the eddy lifetime is reached, a new value of the instantaneous velocity is obtained by applying a new value of ζ in Equation (12).
the slip velocity between the two phases is given by
Eulerian Two-Fluid Model The Eulerian two-fluid model developed by Tu and Fletcher (1995) and Tu (1997) is employed in this present study. Here, a two-way coupling is achieved between the continuum gas and particle phase. This is in contrast to the use of the Lagrangian particle model where only one-way coupling is adopted in this present investigation. The validity on the use of one-way coupling approach for the Lagrangian model will be discussed later. Because of the small-size particles considered in this article, effects of the viscous stress, interparticle collision, and pressure for the particle can be safely neglected. Detail formulation of the transport equations for the particle phase can be found in Tu (1997). For the confined two-phase flow, the two-way coupling effects of the particle phase on the turbulence of the gas phase are accounted through additional source terms in the κ g and ε g equations:
The turbulent viscosity of the particle phase is computed through the following relationship:
κ Sadd =
2f ρ p (κ g − κ gp ) tp
[16]
FDi
ν p,t = l p,t
l p,t
2f ρ p (ε g − ε gp ) tp
[17]
in the ε g equation, where κ gp and ε gp are provided below from the particle turbulence model and f is the correction factor selected from Schuh et al. (1989). The particle phase conservation equations, casting in the general form of Equation (1), are expressed by the following: continuity equation (φ = 1), p
Ai = ρ p u j ,
B = 0,
S = 0;
[18]
p
momentum equation (φ = u i ), B = ρ p ν p , eff ,
S = FDi + FGi + FWMi ; [19]
particle turbulent fluctuating energy equation (φ = κ p ), p
Ai = ρ p u j ,
B = ρ p ν p , t /σκ ,
S = Pkp − Igp ;
The characteristic length of the system L s provides a limit to the characteristic length of the particle phase. In Equation (24), θ denotes the angle between the velocity of the particle and velocity of the gas to account for the crossing trajectories effect (Huang et al. 1993), while Bgp denotes a constant determined experimentally, which yields a value of 0.01. The relative fluctuating velocity in Equation (24) can be written as
2 u = u − 2u u + u 2 = g p g p r
2 g (κ − 2κ gp + κ p ). 3
[26]
The turbulent Prandtl-Schmidt number σ p for the particle turbulent fluctuating energy (Equation (21)) has a value of 0.7179. The turbulence production of the particle phase is formulated as:
p p p ∂u i 2 ∂u ∂u i − ρ p δi j κ + ν p,t k , ∂x j 3 ∂ xk ∂ x j [27] while the turbulence interaction between the gas and particle phases is Pkp = ρ p ν p,t
p
p ∂u j ∂u i + ∂x j ∂ xi
2f ρ p (κ g − κ gp ). tp
[28]
The turbulence production by the mean velocity gradients of two phases is given by
gas–particle correlation equation (φ = κ gp ). g p Ai = ρ p u j + u j , B = ρ p (νg , t /σg + νg , t /σ p ), S = Pkgp − ρε gp − gp .
[25]
Applying the modulus on Equation (25) yields the following:
Igp = [20]
[23]
u l g,t 2 = (1 + cos θ) exp −Bgp r sign(κ g − κ p ) . [24] u 2 g
u r = u g − u p .
ε Sadd =
p
2 k p. 3
[22]
The characteristic length, l p,t , is obtained by l p,t = min(l p,t , L S ), with l p,t given as
in the κ g equation and
Ai = ρ p u j ,
g p f ui − ui = ρp . tp
[21]
The forces FDi , FGi , and FWMi are the Favre-averaged aerodynamic drag, gravity, and wall-momentum transfer due to particle– wall collision, respectively. The aerodynamic force, FDi , due to
p g ∂u j ∂u 2 Pkp = ρ p νg,t i + ν p,t − ρ p δi j κgp ∂x j ∂ xi 3 p p p g ∂u j ∂u k ∂u k ∂u k 1 , − ρ p δi j νg,t + ν p,t + ν p,t 3 ∂ xk ∂ xk ∂x j ∂ xi [29]
NUMERICAL SIMULATION AND VALIDATION OF DILUTE GAS-PARTICLE
FIG. 1.
Schematic drawing of the backward-facing step geometry (h = 0.025 m).
while the interaction term between the two phases yields gp =
f ρ p [(1 + m)2κ gp − 2κ g − m2κ p ]. 2t p
[30]
Here, m is the ratio of particle to gas density, m = ρ p /ρ g . The dissipation term due to the gas viscous effect is given as εg = ε exp −Bε t p g , κ
ε
gp
g
[31]
where Bε = 0.4. Numerical Procedure The predicted results from both the Lagrangian particle tracking and Eulerian two-fluid models are compared against the benchmark experimental data of Ruck and Makiola (1988) for a gas–particle flow in a backward-facing step geometry (see Figure 1). Particles with corresponding diameters of 1 µm (ρ s = 810 kg/m3 ) and 70 µm (ρ s = 1500 kg/m3 ) are simulated under the flow condition of two Reynolds numbers (based on the step height h): Re = 64000 and Re = 15000. The numerical exercise is performed in a two-dimensional (2D) environment because only 2D representative measurements are available. The governing transport equations are discretized using the finite-volume approach. Third-order QUICK scheme is used to approximate the convective terms, while second-order accurate central differencing scheme is adopted for the diffusion terms.
The pressure–velocity coupling is realized through the SIMPLE method. The simulations are marched towards steady state. For the Lagrangian particle-tracking model, a computational domain has a size of 12 h × 1 h before the step and 50 h × 2 h after the step to ensure the flow was fully developed at the exit. Within the length of 12 h before the step, 120 (in the streamwise direction) × 20 (in the lateral direction) uniform grid points have been allocated. Further downstream, the mesh is with 500 uniform grid points in the streamwise direction and 40 uniform grid points in the lateral direction. Grid independence was checked by refining the mesh system through doubling the number of grid points along the streamwise and 50 uniform grid points in the lateral directions. Simulations using RNG model revealed that the difference of the reattachment length between the two mesh schemes is less than 3%. The coarser mesh was therefore applied in order to embrace the increase of computational efficiency towards achieving the final results. The convergence criteria for the gas-phase properties (velocities, pressure, k and ε) were achieved when the iteration residuals reduced by six orders of magnitude. The nonequilibrium wall function was employed for the gas-phase flow because of its capability to better handle complex flows where the mean flow and turbulence are subjected to severe pressure gradients and rapid change, such as separation, reattachment, and impingement. The distance y+
TABLE 1 y+ of boundary cells along the top wall, bottom wall, and inlet bottom wall Wall
RNG κ-ε model
323
Realizable κ-ε model
Standard κ-ε model
Top wall 35 < y + < 81 31 < y + < 81 37 < y + < 81 Bottom wall 2 < y + < 45 2 < y + < 47 2 < y + < 45 Inlet bottom 59 < y + < 73 59 < y + < 79 75 < y + < 59
FIG. 2.
Distribution of y+ of boundary cells (RNG κ-ε model).
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TABLE 2 Normalized CPU times of Lagrangian and Eulerian approaches Model Lagrangian (20,000 particles) Lagrangian (50,000 particles) Eulerian
Case
Normalized CPU time
Re = 15000, Dp = 1 µm Re = 15000, Dp = 70 µm Re = 15000, Dp = 70 µm Re = 64000, Dp = 70 µm Re = 15000, Dp = 1 µm Re = 15000, Dp = 70 µm Re = 64000, Dp = 1 µm Re = 64000, Dp = 70 µm
1.75 1.7 3.3 3.2 1.15 1 1.15 1
Grid 500 × 40 × 1 + 120 × 20 ×1 = 22400 162 × 60 × 3 = 29160
(a)
(b) FIG. 3. 1 µm).
Comparison between the experimental data and numerical simulation of (a) particle velocities and (b) particle turbulent fluctuations (Re = 64000, d p =
NUMERICAL SIMULATION AND VALIDATION OF DILUTE GAS-PARTICLE
FIG. 4. Maximum negative velocity U-max (normalized with free stream velocity Uo) in the recirculation zone of particles of different κ-ε models (Re = 64000, d p = 1 µm). ρ u y
is a dimensionless parameter defined as y + = g µTg P . Here, √ u T = τω /ρω is the friction velocity, y p is the distance from point P to the wall, ρ g is the fluid density, and µg is the fluid viscosity at point P. The ranges of y + of boundary cells along the top wall, bottom wall, and inlet bottom wall (see Figure 1) investigated are listed in Table 1. Figure 2 depicts the distribution
(a)
(b) FIG. 5. (a) the computed turbulent kinetic energy κ (normalized by the inlet mean κ) at x/H = 5, (b) the computed turbulent dissipation rate ε (normalized by the inlet mean ε) at x/H = 5.
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of y + obtained by the RNG κ-ε model. It was observed that in the recirculation region and recovering region, low values of y + were found along the bottom wall. The particle transport using the DRW model was computed from the converged solution of the gas flow. For the DRW model, a total of 20,000 particles were released from 10 uniformly distributed points across the inlet and they were individually tracked within the backward-facing step geometry. The independence of statistical particle-phase prediction from the increase of particle number was tested using 10,000, 20,000, and 50,000 particles for 70 µm. The difference of the maximum positive velocities of 20,000 and 50,000 particles was less than 1%. For the Eulerian two-fluid model, computations were performed using an overlapped grid system with three computational zones; 100 (in the streamwise direction) and 60 (in the lateral direction) uniform grid points are allocated for the region of x/h = 10 behind the step. There are a total of 162 grids in the streamwise direction, and three grids in the spanwise direction comprise two symmetric walls. All of the transport equations for both phases were discretized by a finite-volume formulation using an overlapped grid system. The QUICK scheme was used for the convective terms, while three-point symmetric formulas were used for the second-order derivatives. The SIMPLE algorithm was used for the velocity–pressure coupling. More details regarding the solution process and the overlapped grid system can be found in Tu (1997). All the calculations were performed on a PC workstation that had the following hardware specifications: 2.66 GHz CPU and 512 Mega-bytes of RAM. To investigate the computational performance for Lagrangian and Eulerian approaches, CPU times for both of the methods were recorded and normalized with the CPU time for Eulerian model in the case of Re = 15000, D p = 70 µm. Table 2 illustrates the normalized CPU time for Lagrangian model using 20,000 particles and 50,000 particles accompanied by the CPU time for the Eulerian model. The CPU times described here comprise both the convergence time for the gas phase and the calculation time for the particles phase. It clearly demonstrated that more computational time is required for the Lagrangian approach than the Eulerian approach. RESULTS AND DISCUSSION For gas–particle flows, the dimensionless number–Stokes Number, which is defined as the ratio between the particle relaxation time and system response time, i.e., St = t p /t s , presents an important criteria towards understanding the state of the particles whether they are in kinetic equilibrium with the surrounding gas. The system relaxation time, ts , in the Stokes number definition was determined from the characteristic length (L s ) and the characteristic velocity (Vs ) of the system under investigation, i.e., ts = L s /Vs . For the case of particles with a diameter size of 1 µm (categorized as small particles) with gas flows of Reynolds numbers of 15,000 and 64,000, the Stokes numbers evaluated for these
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FIG. 6.
The turbulent viscosity profiles predicted by three models.
particles were found to be very much less than unity. This particle flow could be considered to act more like fluid traces in the gas flow, which provided the necessary means of assessing the prediction accuracy of the gas-phase turbulent models because they played an important role in affecting the particle flow across the backward-facing step geometry. A parametric
study was performed using three available turbulent κ-ε models in FLUENT, standard κ-ε, realizable κ-ε, and RNG κ-ε, while employing the Lagrangian particle-tracking model. Figure 3a presents the computed particle velocity profiles against measurements for a Reynolds number of 64,000, at locations of x/H = 0, 1, 3, 5, 7, and 9 behind the step. The velocity profiles
NUMERICAL SIMULATION AND VALIDATION OF DILUTE GAS-PARTICLE
327
(a)
(b) FIG. 7.
Comparison between the experimental data and numerical simulation of particle velocities: (a) Re = 64000 and (b) Re = 15000 (d p = 70 µm).
are normalized by the free stream velocity, u o , that has a value of 40 m/s. Good agreement was achieved with the realizable κ-ε and RNG κ-ε models, while significant deviations were found employing the standard κ-ε model at downstream locations of x/H = 5, 7, and 9. Figure 3b gives the comparison of predicted and measured particle turbulent fluctuation. The predicted particle turbulent fluctuations downstream of x/H = 3 are lower than the experiment data. Under the same flow condition, the maximum negative velocity profiles of the particles in the recirculation zone are illustrated in Figure 4. All three turbulent models yielded lower maximum values of the maximum negative velocities. Nevertheless, the realizable κ-ε model gave the best prediction of the particle reattachment length of x/H = 8.2
compared to the measured value of x/H = 8.1, while the standard κ-ε model severely underpredicted the reattachment length by value of x/H = 6.9. The RNG κ-ε model predicted the particle reattachment length of x/H = 8.5. The standard κ-ε model generally overpredicts the gas turbulence kinetic energy, κ, in the recirculation region, which leads to a high turbulent viscosity, ν g,t . Figure 5a demonstrates the profile of the turbulent kinetic energy, κ, normalized by the mean κ at inlet at location of X/h = 5, and Figure 5b provides the profile of the turbulent dissipation rate, ε, normalized by the mean ε at inlet at the identical location. It is evidently clear that the standard κ-ε model yielded excessive normalized turbulent kinetic energy κ values, while the predicted normalized
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(a)
(b) FIG. 8. Comparison between the experimental data and numerical simulation results with Lagrangian and Eulerian models: (a) particle velocities and (b) particle turbulent fluctuations (Re = 64000, d p = 1 µm).
turbulent dissipation rate ε values at the same locations were, just marginally higher below the step height of 0.025 m. This therefore resulted in the overprediction of ν g,t and the production of excessive mixing in the standard κ-ε model, which significantly reduced the recirculation zone (confirmed through Murakami 1993). Another possible cause could be the ε transport equation of the standard κ-ε model. For example, in the RNG κ-ε model, an additional term, such as a strain-dependent term R, is included to aid the model in dealing with flows that experience large rates of deformation (Wright and Easom 2003). In Equation (9) it can be observed that in regions where η < ηo , R is positive. When η > ηo , R turns negative. The negative value of R decreases both the rate of production of κ and rate of destruc-
tion of ε, leading to smaller eddy viscosity (Orzag 1994). For rapidly strained flows, the RNG κ-ε model yields a lower turbulent viscosity than the standard κ-ε model. Figure 6 illustrates the turbulent viscosity profiles obtained by three models. It is clearly seen that the standard κ-ε model predicts much higher ν g,t than the other two models in the region from X/H = 3.5 to X/H = 7.3 and in the region downstream X/H = 9. Figures 7a and b present the velocity profile of particles with diameter size of 70 µm (categorized as large particles) in the flows of Re = 64000 and Re = 15000. Here the larger particle results were obtained via the Lagrangian model. Since the Stokes number is much greater than unity, the fluency from the fluid-phase fluctuations onto the particles was negligible. In
NUMERICAL SIMULATION AND VALIDATION OF DILUTE GAS-PARTICLE
329
Figure 7a all three turbulent κ-ε models gave identical velocity profile. The predictions are close to the measured data upstream of x/H = 5. At x/H > 5, the models clearly overpredicted the particle velocity. Similar results can be found in Figure 7b; three models gave predictions that were higher than the experimental data downstream of x/H = 5. However, the standard κ-ε model did not predict the particles in the recirculation region, while the other two models successfully predicted the particles. Figures 8a and b illustrate the particle-velocity profiles and particle turbulent fluctuation, respectively, using the Lagrangian particle-tracking and Eulerian two-fluid models for the flow condition of Re = 64000 and particles of 1 µm diameter. The model predictions have been attained through both methodologies employing the RNG κ-ε turbulent model. Good agreement was achieved between the model results and measured data. A closer investigation revealed that the Eulerian model was, however,
FIG. 9. Maximum negative velocity U-max (normalized with Uo) in the recirculation zone of particles; Lagrangian and Eulerian models (Re = 64000, d p = 1 µm).
(a)
(b) FIG. 10. Comparison between the experimental data and numerical simulation results with Lagrangian and Eulerian models: (a) particle velocities and (b) particle turbulent fluctuations (Re = 15000, d p = 70 µm).
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seen to provide better results than the Lagrangian approach, especially near the geometry exit. Comparison of the numerical and measured profiles of the particle maximum negative velocity is presented in Figure 9. It is evidently clear that better prediction was achieved through the Eulerian model by the excellent matching of its predicted profile shape and reattachment length with the experimental profile. For small particles, the discrete random walk (DRW) model used in this study has a deficiency. In strongly inhomogeneous diffusion-dominated flows where small particles should become uniformly distributed, DRW will predict high concentration in low-turbulence regions of the flow (Macinnes and Bracco 1992). Figure 10a shows the mean particle velocity profiles using the Lagrangian particle-tracking and Eulerian two-fluid models for the flow condition Re = 15000 and particles of 70 µm diameter. For the above case of lower inertial particles, particle penetration into the recirculation zone was identified. For larger size particles in high Reynolds number flow, Lagrangian model predicted particle-free zone in the recirculation zone (also see Figure 7), while the Eulerian two-fluid model revealed particles entering the recirculation zone. Slater et al. (2001) explained that the cause is derived from the ill posedness of the particle equation of the Eulerian method, where it saves all the information in the whole domain including the particlefree zone. However, it should be noted that the reported particle concentration using the Eulerian approach in the particle zone was decreasing with increasing particle size. When the particles were taken to 45 µm, the particle concentration at the center of the recirculation zone dropped to about 1 × 10−5 . When the particle sample number was increased to 5 × 104 , several particles were found to exist in the recirculation zone. Hence, the predictions of the Eulerian model were consistent with the Lagrangian model inasfar as the particles were flowing in the particle-free zone. Figure 10b presents the simulation and experimental results of particle turbulent fluctuation velocity profiles for flow condition Re = 15000 and particle size of 70 µm. The Lagrangian model significantly underpredicted the particle
FIG. 11. Maximum positive velocity U+max (normalized with Uo) in the streamwise portion of the velocity profile of particle (Re = 15000, d p = 70 µm).
turbulent fluctuation. Fessler and Eaton (1997) drew the conclusion that when κ-ε turbulence computations were performed alongside with the Lagrangian particle tracking, the methodology was unable to properly account for the interactions of the particle movement with the large-scale instantaneous flow structure. Another possible cause of error could be the absence of accurately quantifying the particle inlet fluctuation for the Lagrangian model. Comparison of the maximum positive velocity profile between measurements and computations is demonstrated in Figure 11. Here both models yielded identical results that were marginally higher than those measured. CONCLUSION The physical behavior of a dilute particle–gas flow over backward-facing step geometry was numerically investigated via a Lagrangian particle tracking model (DRW model) and an Eulerian two-fluid model combining with an overlapped grid technique (Tu 1997). A substantial amount of work was undertaken in this article to elucidate further the understanding of the particle flow under the influence of particle inertia (Stokes number). For the case where the particles acted like fluid traces (St 1), the prediction accuracy of the gas-phase turbulence was strongly affected by the application of turbulent κ-ε models for the particular geometry concerned. For a backward-facing geometry, the RNG κ-ε and realizable κ-ε models gave the better prediction, especially for the particle reattachment length, than the standard κ-ε model. Similarly, for the flow of heavier particles (St 1), the RNG κ-ε model and realizable κ-ε were still seen to provide marginally better performance than the standard κ-ε model. The computational results obtained by both the Lagrangian model and the Eulerain model were compared against benchmark experimental data. The comparison revealed that both approaches provided reasonably good predictions of the velocities and turbulent fluctuations for the gas and particle phases. Nevertheless, the closer numerical investigation showed that this Eulerian model gave marginally better performance than the Lagrangian model. It was also demonstrated that more computational time is required for the Lagrangian approach than Eulerian approach. In engineering applications, the Lagrangian approach is recommended when the detailed physics for the particle phase are required. For complex flows that require large computational resources for the Lagrangian model, the Eulerian model is an attractive alternative. NOMENCLATURE Ai convective flux A0 , As model constants for realizable κ-ε turbulence model B diffusion coefficient Bgp , Bε model constants for the Eulerian two-fluid model CD particle drag coefficient Cµ coefficient in the κ-ε turbulence model
NUMERICAL SIMULATION AND VALIDATION OF DILUTE GAS-PARTICLE
C1 , C2 Cε1 , Cε2 dp f FDi FGi FWMi
g h Igp
l Ls Le m Pkgp Pkp r R Re S Si j , S jk , S ki St tcross tp ts TL ui , u j uo Vs xi , x j , xk
model constants for realizable κ-ε turbulence model model constants for standard and RNG κ-ε turbulence models particle diameter correction factor aerodynamic drag force gravity force wall–momentum transfer due to particle–wall collision force gravitational acceleration step height turbulence interaction between the gas and particle phases for the particle-phase turbulent fluctuating energy length scale of energetic turbulent eddies characteristic length of the system eddy length scale ratio of particle to gas density turbulence production by the mean velocity gradients of two phases production term of the particle fluctuating energy uniform random number strain rate Reynolds number source term strain rates Stokes number eddy crossing time particle relaxation time system response time fluid Lagrangian integral time velocity free stream velocity characteristic velocity of the system Cartesian coordinate system
Greek Letters β model constant for RNG κ-ε turbulence model ε dissipation rate of turbulent kinetic energy φ governing variable η function defined in Equation (6) ηo model constant for RNG κ-ε turbulence model κ turbulent kinetic energy ν kinematic viscosity θ angle between velocities of the particle and gas ρ density σ turbulence Prandtl number ω fluctuating vorticity ζ normally distributed random number gp turbulence interaction between the gas and particle phases vorticity
Subscripts add eff g gp p s t
additional effective gas phase gas–particle particle phase solid phase turbulent phase
Superscripts g gp p ε κ (¯) ( )´
gas phase gas–particle particle phase dissipation rate of turbulent kinetic energy turbulent kinetic energy average fluctuation
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