Mechanism of Particle Transport in a Fully Developed Wake Flow

Article pubs.acs.org/IECR

Mechanism of Particle Transport in a Fully Developed Wake Flow Jun Yao,*,†,‡ Yanlin Zhao,*,§ Ning Li,† Youqu Zheng,∥ Guilin Hu,∥ Jianren Fan,‡ and Kefa Cen‡ †

School of Energy Research, Xiamen University, Xiamen, People’s Republic of China Institute of Thermal Power Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China § Institute of Particle Science and Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom ∥ Department of Mechanical Engineering, Zhejiang University of Science and Technology, Hangzhou 310023, People’s Republic of China ‡

ABSTRACT: In this paper, the time-dependent Navier−Stokes equations were integrated in time using a mixed explicit-implicit operator splitting rules. The spatial discretization was processed using the spectral-element method. Nonreflecting conditions were employed at the outflow boundary. Particles were traced by the Lagrangian approach based on one-way coupling between the continuous and the disperse phases. The simulation results of the flow field agree well with experimental data. Particles (St = 4) are found to concentrate in the regions between adjacent vortex structures (RAVS) together with other particles dispersed outside of vortex outlining the boundaries of the large-scale vortex structures, which is independent of the flow Reynolds number. Due to U-velocity component difference, at the entrance of RAVS the flow does “compact” on particles causing particle concentration, and it increases with the flow Reynolds number and particle size. The mechanism of particle transport in the wake flow mostly depends on the strong interactions between two alternative and successive shedding vortex structures with opposite sign. Particle transport in the wake flow could be briefly described as follows: moving around vortices and going ahead with vortices. studied by numerical8−12 and experimental techniques.13,14 Crowe et al.15 introduced the Stokes number of particles to predict the effect of organized vortex structures on the particle dispersion process. It says that articles with small Stokes number closely follow the turbulent flow and that, conversely, particles with intermediate Stokes number would be dispersed significantly faster than the fluid motion due to the centrifugal effects created by the organized vortex structures. By quoting the concept, the different dispersion patterns for particles with different orders of Stokes numbers were found in the twodimensional experimental and numerical studies.16,17 Noticeably, the mechanism for the dispersion of intermediate Stokes number particles was first established by Wen et al.17 i.e. suggesting a mixing layer to consist of a stretching and folding process. The stretching process primarily involves a focusing of the particles into regions and a stretching of the distances between particles. This process occurs primarily on the highspeed boundaries of the large-scale structures. The folding process involves a rotating of the particle sheet down on top of each other, which appears to be connected to vortex pairing interactions. Moreover, Tang et al18 presented experimental and numerical results concerning solid particle motion in a plane wake to demonstrate the particle dispersion process in the wake flow differing considerably from the mixing layer. Since vortex pairing interactions are rarely observed, there appears to be no obvious folding process. Recently, Ling et al.19 traced particles with different Stokes numbers in a temporal

1. INTRODUCTION Laminar and turbulent flows such as bluff body wakes laden with solid particles or liquid droplets are a common occurrence in both nature and technology. The interaction between the particles and the wake flows often plays an important role in determining the performance of many producing processes, such as coal combustion systems, coal liquefaction-gasification pipelines, pin-type heat exchangers, oil droplet fueled gas combustors, and pneumatic processing of food particles. In addition to the foregoing pragmatic significance, there is an intrinsic theoretical interest in such model flows to further our understanding of the underlying physical processes. Therefore, particulate bluff body flows constitute an important class of problems within the domain of fluid mechanics. The work of particulate flow over a bluff body can be mainly divided into two parts. The first part includes the nature of the solid surface impact and the material erosion. Considerable research has been done on the prediction of erosion of bluff body due to particle impacts.1−4 The second part includes the particle trajectories in a bluff body flow. Brandon and Aggarwal5 numerically simulated particle transport and deposition in an unsteady particle laden flow over a square cylinder placed in a channel using a staggered-grid control volume approach. Richmond-Bryant et al.6 estimated concentration of particles with low St in the wake of circular cylinder using the steady state Reynolds Averaged Navier−Stokes (RANS) renormalized k∼ε model. Salmanzadeh et al.7 investigated particle trajectories and deposition rates on the obstruction and duct walls in a laminar unsteady channel flow with a rectangular obstruction using a finite volume method. The prediction of particle dispersion and the associated effects of the large-scale structures have also been extensively © 2012 American Chemical Society

Received: Revised: Accepted: Published: 10936

November 26, 2011 May 13, 2012 June 7, 2012 June 7, 2012 dx.doi.org/10.1021/ie202749a | Ind. Eng. Chem. Res. 2012, 51, 10936−10948

Industrial & Engineering Chemistry Research

Article

2.1. Governing Equations. For an incompressible viscous fluid flow past a circular cylinder, Navier−Stokes equations are considered as the governing equations. Variables considered in the physical motion are normalized as follows

mixing layer by the Lagrangian approach based on one-way coupling between the continuous and the dispersed phases. With the progress of computer technology, the research of particle dispersion in turbulent fluid has achieved great improvement and appeared perfected in discovering its mechanism. Recent direct numerical simulation of particle dispersion in turbulent channel flow20−24 and in a threedimensional mixing layer (Fan et al.)25 confirm that the dynamics of turbulent dispersion phenomena are strongly influenced by coherent structures and the related sweepejection events cycle. On the other hand, experimental measurements have been developed for studying the instantaneous preferential concentration of particles in turbulent eddies (Eaton and Fessler26). Many researchers13,17 (Wen et al.; Longmire and Eaton) examined simple free shear flows which are dominated by large-scale, two-dimensional or axisymmetric vortices. All of these researchers have observed highly organized particle concentration fields for Stokes numbers on the order of one. Particles are flung away from the vortex cores and in many cases collected in rings surrounding the vortices. In the jet and wake flows the particles are concentrated in the highly strained saddle region between successive vortices. So far, particle transport in a circular cylinder wake has less been reported. Present work studies a particle-laden flow around a circular cylinder using a numerical simulation solution of the Navier−Stokes equations coupled with Langrangian particle tracking. The major objective is to study particle preferential concentration by analyzing the fluid-particle characteristics and examining the availability of two-dimensional numerical simulation method in two-phase coherent vortex structures. The outline of the paper is listed as follows: in section 2 the governing equations for the fluid and the particle phase are described and explained together with the numerical details of solution of governing equations and that of the particle computations. In section 3 the simulation of the fluid field is evaluated and the instantaneous U- and V- velocity component of the fluid is analyzed. After then particle preferential concentration (PDF, probability density distribution) together with particle-fluid relative velocity is studied. In the end relevant physical modeling is developed for particle transport in the wake flow.

u=

ũ , U0

P=

v=

P̃ , P0

ṽ , U0

T0 =

R , U0

x̃ , R

y=

P0 =

1 ρ U02 2 0

x=

ỹ , R

t=

t̃ , T0 (1)

with R being the cylinder’s radius, Uo being the uniform stream velocity, and ρ0 being the fluid density. Then the continuity equation and the time-dependent Navier−Stokes equations are written in conservative form as ∇·V = 0

(2)

∂u 1 ∂P + ∇·(uV ) − ∇·(∇u) = − ∂t Re ∂x

(3)

∂v 1 ∂P + ∇·(vV ) − ∇·(∇v) = − ∂t Re ∂y

(4)

where V is the velocity (u, v), and Re is the flow Reynolds number defined as Re = (U02R)/υ with υ being the kinematic viscosity. Boundary conditions are specified for four lines enclosing present computation area. In the physical domain the flow is not confined. Nevertheless, fictitious external boundaries are necessary far from the cylinder. The solution of the above system will be obtained on a two-dimensional domain. The choice of the boundary conditions is an important problem in order to not confine the calculation domain of the present exterior flow. In nondimensional units, the cylinder diameter is 2. The inflow boundary with an initial condition is set as follows: u = 1, v = 0. For the top and bottom boundary at the ydirection, in order to not confine the flow, a Neumann-type boundary condition is adopted for the u-velocity component and a Dirichlet-type boundary condition is taken for the vvelocity component: ∂u/∂y = 0, v = 0. These conditions are consistent with the domain size and do not confine the flow, as shown in the results. It is tested that Neumann-type and Dirichlet boundary conditions do not much affect the final results provided that the boundary is far from the cylinder (|y|/r > 5). In the end, the best boundary condition, i.e. the Neumann-type boundary condition, is finally chosen for the ucomponent and the Dirichlet-type boundary condition for the v-component. For the outflow boundary, nonreflecting conditions are employed. It conducts in the following manner. A buffer is set up close to the outflow boundary and flow viscidity herein is divided into two parts: x-component and ycomponent. Viscidity at the x-component decreases progressively in the buffer and reaches zero at the outflow boundary, which ensures the governing equation being parabolic and prevents the upper flow from the diffusing effect by the perturbing wave behind. On the contrary, viscidity of the ycomponent increases gradually at streamwise and introduces a diffusing effect on the flow. In principle, the buffer is assumed so long that the following conditions meet at the outflow boundary: ∂u/∂x = 0, ∂v/∂x = 0, p = 0. Moreover, the buffer is connecting with the physical domain smoothly to avoid the reflecting effect. 2.2. Numerical Procedure. 2.2.1. Time-Discretization. The time-discretization of the governing equations was to

2. NUMERICAL METHOD A schematic diagram of the cylinder and computational domain and coordinate system used is given in Figure 1. The flow considered was two-dimensional and described using a Cartesian coordinate system (x, y). The corresponding velocity components in the (x, y) directions are (u, v), respectively.

Figure 1. A sketch of the computational domain. 10937

dx.doi.org/10.1021/ie202749a | Ind. Eng. Chem. Res. 2012, 51, 10936−10948


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2.3. Lagrangian Particle Tracking. Particle motion was modeled using a Lagrangian approach (Fan et al.;4 Yao et al.28) in which the particles are followed along their trajectories through the unsteady, nonuniform flow field. To simplify the analysis, the following assumptions were made: the particleladen flow is dilute; interactions between particles are negligible; the flow and particles are one-way coupled, i.e. the effect of particles on the fluid is neglected; all particles are rigid spheres with the same diameter and density; and particle-wall collisions are elastic. The Lagrangian motion of a rigid, spherical particle suspended in a flow is governed by a force balance equation

employ the mixed explicit/implicit time-integration rules, which can be implemented in three steps V ̂ − ∑qe= 0 αqV⃗ J −1

n−q

Δt

Ji − 1

=

∑ βqN⃗ (V⃗

n−q

)

q=0

⎯ →̂ ⎯ ̂ →̂ V −V = −∇p ̅ n + 1 Δt

γ0V⃗

n+1

Δt

→̂ ⎯̂ −V

(5)

(6)

= Re−1∇2 V⃗

n+1

dVp

(7)

dt

→̂ ⎯̂ →̂ ⎯ where V and V are intermediate velocity fields defined in eqs 5 and 6, Je and Ji are the parameters characterizing the accuracy of the overall scheme denoting the individual accuracy of the explicit integration (nonlinear terms) and the implicit integration (viscous terms), respectively, αq and βq are implicit/explicit weight-coefficients for the stiffly stable scheme, and γ0 is a weight coefficient of the backward differentiation scheme. In the present research, the process of timediscretization was implemented by third order with the parameters listed as follows: γ0 11/6, α0 3, α1 −3/2, α2 1/3, β0 3, β1 −3, β2 1. Taken the divergence of eq 6 and combined with the continuity eq 2, a Poisson equation for the pressure can be obtained 2 n+1

∇ p̅

⎛ →̂ ⎯ ⎞ V = ∇·⎜ ⎟ ⎜ Δt ⎟ ⎝ ⎠

((V − Vp) × ω) |ω|

(10)

where V is the fluid velocity, ρ is the fluid density, Vp is the particle velocity, ρp is the particle density, dp is the particle diameter, and g is gravity. ω = ∇ × V is the fluid rotation, Res = ρ·d2p|ω|/μ is the particle Reynolds number of the shear flow, and cls = Fls/Fls,Saf f represents the ratio of the extended lift force to the Saffman force, with ⎧ 0.0524(βRe )0.5 ·Re > 40 p p ⎪ ⎪ cls = ⎨(1 − 0.3314β 0.5)e−Rep /10 + 0.3314β 0.5·Re p ⎪ ⎪ ⎩ ≤ 40

(11)

Here, β is a parameter given by β = 0.5Res/Rep (for 0.005 < β < 0.4). CD is the Stokes coefficient for drag, with CD = (1 + 015 Re0.687 ) · 24/Rep, where Rep is the particle Reynolds number, p with Rep = dp|V − Vp|/v. The third term on the right-hand side of eq 10 is the slip-shear force that is based on the analytical result of Saffman29 and extended for higher particle Reynolds numbers according to Mei.30 Even though a number of possible forces can act on a particle, many of these may be neglected without any appreciable loss of accuracy, depending on the particle inertia. The most important force acting on a particle is the Stokes drag force, with gravity also significant depending on the orientation of the flow. In this study, Stokes drag, gravity, buoyancy, and lift forces were considered. Due to particle-wall impaction, electrostatic charge can be generated at the particle surface and the cylinder wall.31−33 However, in this work the cylinder area is limited, such that the electrostatic force never acts on a particle after the cylinder and consequently can be neglected. Other forces acting on a particle, such as the hydrostatic force, Magnus effect, Basset history force, and added mass force, were not taken into account due to their being orders of magnitude smaller than the effects considered (Armenio and Fiorotto34). A fourth-order Runge−Kutta scheme was used to solve the equation of motion, given the initial particle location and velocity. The initial particle positions were distributed randomly throughout the computation region, corresponding to an initially uniform wall-normal particle number density profile. The initial particle velocity was set equal to the fluid velocity, interpolated to the particle position. Particles were assumed to interact with eddies over a certain period of time, that being the lesser of the eddy lifetime and the transition time. For particles that moved out of the simulation region in

(8)

J−1 ⎡ J−1 ∂p ̅ n + 1 n−q −1 ⎢ ⃗ ⃗ = −n ⃗ · [ ∑ βqN (V ) − Re ∑ βq∇ ⎢⎣ q = 0 ∂n q=0

⎤ )]⎥ ⎥⎦

⎛ ⎞ ρ 3 ρ CD (V − Vp)|V − Vp| + ⎜⎜1 − ⎟⎟g 4 ρp dp ρp ⎠ ⎝ + 1.615dpμRes0.5cls

along with the consistent high-order pressure boundary condition

× (∇ × V ⃗

=

n−q

(9)

which ensures that the divergence is equated to zero on the solid surface. As a result, equations were translated to the Hemholtz form eqs 7 and 8 in the term of V and P. In this work, the time step was set at a constant value of 0.01. 2.2.2. Spatial Discretization. The spatial discretization of eqs 7 and 8 as Hemholtz form was processed by the method of static condensation for the system matrix. In summary, the standard spectral element discretization the computational domain was broken up into a series of quadrangular elements in two dimensions, which were mapped isoparametrically to canonical squares. Field unknowns and data with the geometry, velocity, and pressure were then expressed as tensorial products in terms of high-order Lagrangian interpolants through Chebyshev collocation points. The final system of discrete equations was then obtained via a Galerkin variational statement by Korczak and Patera.27 Further details of the mathematical model employed, and the numerical algorithm and its implementation, may be found in the work of Yao et al.28 10938



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the streamwise direction, they were ignored and the same amount of particles were introduced at the inlet. The total number of particles used within that domain ranged from 10,000 to 1000,000, with the precise number employed being sufficient to ensure statistical independence of the results which was tested according to the method of Yao et al.28

3. RESULTS AND ANALYSIS 3.1. Fluid Field. In this work, the computational domain with eighth order polynomial basis, all simulations are divided into two groups, one is for steady flow (Re ≤ 60) and the other is for two-dimensional shedding flow (Re ≤ 175). In order to shorten each case compute period, all simulations are started from an initial conditions by the solution of Re = 30 reaching at steady state. The computational domain is shown in Figure 1. Through tests with various grids and other conditions, it is found that the inflow length does an important role in determining the simulation results. Here, the case 1 with 50 elements, location of the inflow boundary x = −14, the total length at x- and y-direction 40, 28 respectively, is found to be the best one. To study the grid effect on simulation results, cases 2 and 3 were chosen with all the same parameters as case 1 except grid (60 and 80 elements, respectively). The results (drag coefficient, pressure coefficient, Strouhal number, and ucomponent velocity) of cases 2 and 3 are found that they are almost the same as those of case 1. To save computation, we retained the coarser grid (case 1) for the final computations. Figure 2 shows the characteristic coefficient (viscous drag coefficient, pressure drag coefficient, mean drag coefficient, and base pressure coefficient) of the wake in comparison with the measurements of Henderson,35 Grove et al.,36 and Tritton37 at Re = 25−60. The agreement between the simulation and the experiments is good. The corresponding Strouhal number reaches at 0.174 with a discrepancy of approximately 4%. The discrepancy may be caused by the inflow length of computation region. Based on a large amount of simulation, the simulation accuracy is found to be mainly determined by the inflow length of computation region, the longer inflow length the better simulation result achieved, but much larger computation capacity required. As the Reynolds number is less than 50, the flow is steady and symmetric about the centerline of the wake. When the Reynolds number is beyond a certain value (Re = 46 ± 1, Henderson35), the circular cylinder flow becomes unstable and the process of vortex shedding begins, resulting in the wellknown Karman vortex street. As the flow Reynolds number increases, the transition state keeps linear at two-dimension, until it is beyond the critical value of 188.5 ± 1 (Henderson35), where three-dimensional perturbation becomes strong enough to make an obvious effect on the vortex structures. The present research on the wake at the interval Reynolds numbers below 175, where the three-dimensional no-linear effect can be ignored. 3.1.1. U-Velocity Component. In Figure 3, it is seen that the amplitude of the U-velocity component at Re = 175 (−0.38− 0.71) is larger than that at Re = 100 (−0.4−0.52) saying that the amplitude of the U-velocity component increases with the flow Reynolds number. Figure 3 shows that the frequency of the oscillations for the U-velocity component increases with the flow Reynolds number. The U-velocity component instantaneous contours combining with the vortices field at t = 100, Re = 100 is shown in Figure 4a. It is seen that the value of the U-velocity component

Figure 2. Coefficient versus Reynolds number (a) viscous drag coefficient Cdf; (b) pressure drag coefficient Cdp ; (c) mean drag coefficient Cd; and (d) base pressure coefficient CPb.

is positive (as x > 4) and fluctuates in a narrow range (shown in Figure 4b,c). To observe the character of the U-velocity component at different spots in the wake, three cases of the Uvelocity component of spots along the axis of Y = 0,1,−1 are shown in Figure 4b,c. It is clear that the U-velocity component of the three cases appear no-identical status at t = 95 and t = 100, for example, as the U-velocity component value reaches the top level at Y = 1 while the value decreases to the bottom level at Y = −1, and vice versa. It seems that the U-velocity component value along the axis Y = 1 appears opposite to that along the axis Y = −1, which is independent of time development. Moreover, the U-velocity component of spots 10939



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seen that the U-velocity component velocity decreases/ increases along the y axis, and the trend is against each other for the two adjacent vortices. 3.1.2. V-Velocity Component. The evolution of the Vvelocity component is illustrated in Figure 5. It demonstrates

Figure 3. Time-dependent evolution of the U-component at x/R = 2, y/R = 0.5: (a) Re = 100 and (b) Re = 175.

on the axis of Y = 0 is obviously less active than that of Y = 1 and −1. Therefore, it suggests that the U-velocity component trend alters with Y-axis. Figure 4b,c shows that the U-velocity component status along Y-axis appears different with time, which indicates that the U-velocity component of spots along Y-axis is changeable with time-dependent. Figure 4d presents a scheme of the U-velocity component velocity distribution along the y-direction for two adjacent vortices (fourth and fifth). It is

Figure 5. Time-dependent evolution of the V-component at x/R = 2, y/R = 0.5: (a) Re = 100 and (b) Re = 175.

that with the increase of the flow Reynolds number, the amplitude of the V-velocity component becomes larger with the Reynolds number (Re = 100: −0.67−0.53; Re = 175: −0.8−

Figure 4. (a) Instantaneous superimposed U-component contours and vortices field (ωmin = −0.5, ωmax = 0.5, 150 step; Re = 100; t = 100); (b), (c) instantaneous superimposed U-component value along the axis of Y = 0,1, −1 and vortices field at Re = 100 (ωmin = −0.5, ωmax = 0.5, 150 step) (b) t = 95; (c) t = 100; (d) scheme of U-component velocity distribution along the y-direction for two adjacent (fourth and fifth) vortices, t = 100. 10940



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Figure 6. (a) Instantaneous superimposed V-component contours and vortices field (ωmin = −0.5, ωmax = 0.5, 150 step; Re = 100, t = 100); (b), (c) instantaneous superimposed V-component value along the axis of Y = 0,1,−1 and vortices field at Re = 100 (ωmin = −0.5, ωmax = 0.5, 150 step) (b) t = 95; (c) t = 100; (d) scheme of V-component velocity distribution along the x-direction for two adjacent (fourth and fifth) vortices, t = 100.

3.2.1. Particle Preferential Concentration Analysis. Probability density distributions of the particle concentration p(f) were calculated from 8000 measurements at each spatial location using 50 bins equally spaced over the 3-σ limits of the data. The distributions have been normalized so that the following relation is valid: ∫ 10p(f)df = 1. For analysis of the wake flow above, it is known that the wake flow structure appears different with the flow Reynolds number. For example, with Re less than 50, the flow is steady and symmetric about the centerline of the wake. With the Re number beyond 100, vortex coherent structures are in shape behind the circular cylinder. Under the effect of flow structures, particles concentration varies with flow Re number, which can be further identified by the PDFs of particle concentration as shown in Figure 7. At low Re number 40, the profile fluctuates little indicating a rare particle concentration in the wake flow.

0.71). The frequency of the oscillations of the V-velocity component increases with the flow Reynolds number. V-velocity component contours superimposed the instantaneous vortices field at t = 100, Re = 100 is shown in Figure 6. It is found that the V-velocity component (absolute) value in the vortex structure from edge to center decreases and reaches the minimum level at the center of vortex where the direction of the V-velocity component turns to be opposite (shown in Figure 6d). In the region between adjacent vortexes (RAVS) the V-velocity component velocity direction is constant, while the (absolute) value increases first and then decreases. The state of the V-velocity component in the flow field can be further examined by spots along the axis of Y = 0,1,−1 as shown in Figure 6b,c. It is interesting to find that the V-velocity component does have an identical trend for the three positions. The value is at the top level (positive/negative) in the region between adjacent vortex structures (RAVS) but low level inside the vortex structures, which is constant for the three Y positions at two times (t = 95, 100). Figure 6d presents the scheme of Vvelocity component velocity distribution along the x-direction for two adjacent vortices (fourth and fifth). In the region of RAVS, the V-velocity component velocity becomes dominant but decreases inside of the vortices structure. Particularly, Vvelocity component velocity alters its direction inside of the vortices. The minimum level of V-velocity component velocity occurs near the center of the vortices. Based on the above analysis, it is clear that the V-velocity component appears regular and identical independent of Y-axis and time. The V-velocity component has a high level (positive/ negative) in the region between adjacent vortexes structures (RAVS) but a low level inside the vortex structures. The direction of the V-velocity component alters near the center of the vortex structure. 3.2. Particle Study. To make statics of the particle temporal position in an easy way, a uniform structured grid system (nx = 100, ny = 50) is set behind the circular cylinder (starting at X = 0).

Figure 7. PDF of particle dispersion in the wake flow t = 100. 10941



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Figure 8. Particle dispesion in the wake flow Re = 100, t = 100, (a) superimposed particle dispersion and vortices field (ωmin = −0.5, ωmax = 0.5, 150 step); (b) PDF of particle dispersion; and (c) particle concentration contour.

At medium Re number 100, the profile of particle PDF concentration becomes more fluctuated. At high Re number 150, the PDF concentration has the highest value. Therefore, it could be concluded that particle concentration in the wake flow increases with the flow Re number. Particles instantaneous dispersion (St = 4, Re = 100) in the wake at t = 100 is shown in Figure 8a where vortices contours clearly show vortex structure. The figure illustrates explicitly the connections between the vortex structure development and the particle dispersion patterns in the region. It is clear that particles with St = 4 are focused in the region between adjacent vortex structures (RAVS) together with other particles dispersed outside of vortex outlining the boundaries of the large-scale vortex structures. Quantitatively, Figure 8b demonstrates that the high concentration density does happen in the region RAVS with a number of 1, 2, 3, 4 as labeled in Figure 8a. The periodic undulations in the concentration can be more confirmed by the contour configuration of particle dispersion concentration in Figure 8c. The concentration contours were obtained by numerically counting particles over descretized area elements. To examine the working mechanism of particle dispersion in the wake flow, the linkage between flow velocity and particle dispersion could be found in Figure 9 where instantaneous

particle patterns are superimposed on local U- (Figure 9a) and V- (Figure 9b) component contours. In Figure 9a, particle dispersion pattern is superimposed on local U-velocity component contour. Particle-dispersion may rely on the local U-velocity component. For example, around the fourth vortex after the cylinder, the U-velocity component is found to decrease with the y-axis from upward to downward (Figure 4d). Correspondingly, upward particle dispersion is seen to go ahead, but downward section particles turn back. The fifth vortex after the cylinder U-velocity component increases with the y-axis from upward to downward (Figure 4d). Accordingly, particle dispersion trend is seen to be against that around the fourth vortex: return back in the upward section but go ahead in the downward vortex. In short, it could be concluded that the U-velocity component may affect the particle dispersion pattern (moving direction) in the xdirection. Figure 9b superimposes the particle dispersion pattern on local V-velocity component contour. It is found that particles do concentrate in the regions between adjacent vortex structures (RAVS). For example, between the fourth and fifth vortices particles have high concentration in the RAVS with the moving trend from upward to downward, which is in line with V-velocity component velocity distribution in this region. It is 10942



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It is noted that particle high concentration in RAVS may be relevant to U-velocity component distribution as well. For example, as shown in Figure 4d, at the top of the fourth vortices U-velocity component velocity is very high, while at the same height the U-velocity component value is very low in the fifth vortices. Once a particle enters in this region, on one hand, high drag force from the fourth vortices promotes the particle to go ahead; on the other hand, low drag force from the fifth vortices degrades such a trend due to much low U-velocity component velocity. Such difference between fourth and fifth vortices decreases from upward to downward in the RAVS. As a result, particle concentration does occur in this area from upward to downward. More mechanism is analyzed in the 3.3 section. Figure 10 shows the flow Reynolds number effect on particle concentration in the wake flow. It is clear that particle concentration increases with the flow Reynolds number. This could be explained by the velocity qualities in this flow. Either with the U-velocity component or the V-velocity component, the amplitude (Figure 3, 5) increases with the flow Reynolds number indicating the flow effect on particles increases with the flow Reynolds number. In addition, the wake vortices oscillation frequency increases with the flow Reynolds number so it is seen that in Figure 10 particle concentration appears earlier with the flow Reynolds number. 3.2.2. Particle-Fluid Relative Velocity (PRVF). To find the following-ship of particles with the wake flow, a parameter is introduced as Vrm(x), a mean function of particle relative velocity with local fluid (PRVF) at the vertical direction. It could be used to account the mean level of PRVF at a different x-position and evaluate the level of wake flow effect on particle dispersion

Figure 9. Instantaneous (t = 100) superimposed particle pattern and (a) U-component contour and (b) V-component contour.

Ncp

known that (as shown in Figure 6d), V-velocity component velocity increases to the highest level with a negative direction in the middle of RAVS between the fourth and fifth vortices. Due to the flow element having a high V-velocity component value (negative) between two adjacent vortexes structures (RAVS), once particles enter the region, under a strong vertical effect, the particles move downward in the area.

Vrm(x) =

∑ Vr(x)/Ncp i=1

(12)

where Vr(x) is the relative slip velocity, which represents the difference between the particle and fluid velocity at the same time. Ncp is the total number of particles locating in the strip between x = xi‑1 and x = xi, and the value of independent

Figure 10. PDFs of particle instantaneous dispersion: (a) t = 90; (b) t = 100 (labeled 1, 2, 3, 4 as shown in Figure 7). 10943



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variable x assumes to be the middle value of xi‑1 and xi, which can be acceptable due to the dense grid division. The temporal Vrm(x) at t = 100 is quantitatively figured as Figure 11a, and its qualitative figure is shown in Figure 11 (b).

In Figure 11, the distribution of relative slip velocity vectors is extraordinary in the region of RAVS where exist a series of saddles in the distribution of relative slip velocity vectors at the boundaries that link two opposite-sign vortex structures. In these saddles regions, the magnitude of fluid velocity is very high, but the magnitude of vorticity is quite low. The relative slip velocity vector is concentrated on the vortex braid regions that connect two roller vortex structures with opposite sign. In this case, the local focusing process of particles happens due to the folding phenomena of particles into the vortex braid regions from the adjacent vortex structures. It seems that the relative slip velocity vector can well reflect the concentration mechanism of particles. From these results obtained, the dispersion patterns of particles at the intermediate Stokes numbers in the circular cylinder wake is found to be different from those in the plane mixing layers19,25 and jets.8,9 The mechanism depends mostly on the strong interactions between two alternative and successive shedding vortex structures with opposite sign. Based on relative slip velocity, drag force and gravity could be calculated38 using eq 10 and listed in Table 1. Particle and fluid Table 1. Forces Acting on Particles and Relevant Parameters St(τp) 0.01 0.1 1 10

dp (μm) 36.37 115.08 363.7 1150.8

gravity/N 4.84 1.53 4.84 1.53

× × × ×

−9

10 10−7 10−6 10−4

drag force-x/N 3.50 8.91 1.23 5.99

× × × ×

−11

10 10−10 10−8 10−7

drag force-y/N 4.52 1.05 2.18 3.06

× × × ×

10−11 10−9 10−8 10−7

densities were set to ρp = 2450 kg m−3 and ρ = 1.205 kg m−3, respectively, with the kinematic viscosity of the fluid υ = 1.494 × 10−6 m2 s−1. The particle relaxation time is τp= ρpd2p/18ρυ, and the nondimensional particle response time is defined as the particle Stokes number St = τpu2τ /v, where the uτ is the shear velocity (defined as uτ = (τw/ρ)1/2, with τw the wall shear stress). Four Stokes numbers were considered, namely St = 0.01, 0.1, 1, and 10, with corresponding particle diameters and forces acting on particles given in Table 1. It shows that the drag force is less than the gravity by at least 2 orders of the magnitude, which may be due to low Re of the flow. It can be concluded that, at the x-direction, drag force dominates particle behavior, and, at the y-direction, gravity plays the dominant role. 3.3. Physical Modeling Particle Transport in the Wake. In Figure 11b, it is noted that as particles pass through the region between adjacent vortex structures (RAVS) particle relative velocity with local fluid (PRVF) alters its direction near the central line of the wake (y = 0). It could be further described in the scheme of a particle going through RAVS as shown in Figure 12a. Two adjacent vortex structures (labeled 4 and 5 shown in Figure 12d,e) with counter-rotating directions, the direction of the V-velocity component of the flow between them is downward. At t = t1, a particle approaches the top of vortex 4, under the stretch effect (Wen et al.17) of local fluid, the particle is drawn downward having a fairly high relative velocity with local flow (PRVF). Under the flow effect for a while (at t = t2), the particle moves downward with flow direction, and its V-velocity component value increases having smaller PRVF. Sequentially (at t = t3), the particle approaches the bottom of vortices structure having fairly higher PRVF, but its direction is opposite to the original one (at t = t1). The procedure is the same for a particle moving through RAVS

Figure 11. Instantaneous particle relative velocity at t = 100 (a) Vrm(x) at Re = 100; (b) velocity vector at Re = 100; and (c) Vrm(x) at Re = 60−175.

It is clear that particles’ relative velocity with local flow appears obviously large in the region between adjacent vortex structures (RAVS). Figure 11c represents the instantaneous Vrm(x) of the wake with different Reynolds numbers (Re = 60, 100, 150, 175). It shows that the peak value of Vrm(x) increases with the Reynolds number of the wake. It may be due to the fact that the amplitude of the V-velocity component increases with the flow Reynolds number. 10944



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upward or downward direction as shown in Figure 6. Therefore, it is reasonable to believe that the fluid (V-velocity component) does much affect the particle’s motion in this region. More than one particle, the scheme of particles motion in the region RAVS is shown in Figure 12b. As particles approach the exit of RAVS they turn left/right and move around the vortices under the effect of the nearest vortices rotating direction. Once particles are drawn to the entrance of neighbor RAVS, they then repeat the procedure of going through RAVS as described in Figure 12a. It is noted that as the particle reaches the exit of the neighbor RAVS it would be drawn back to re-enter the original RAVS. For example, a particle in fourth-fifth RAVS turns left and enters third-fourth RAVS and then re-enters fourth-fifth RAVS. Therefore, it suggests that particles transporting in the wake flow actually move around vortices and go ahead with wake flow. It is noted that as particles approach the exit of RAVS, under the effect of vortex structures (fourth and fifth) with countrotating direction, particles cluster burst and separate into two branches, which makes particle shaped as “mushroom” dispersion (shown in Figure 12c,d). Figure 12c presents the particle “mushroom” dispersion pattern with local flow velocity distribution and the working mechanism could be analyzed as follows. At the up-half part, the local flow velocity at the horizontal direction from left to right decreases. The gradient decreases from top to downward. Once particles enter this region, two adjacent vortices do “compact” effect on them causing high concentration herein. However such a trend of flow variation becomes against in the below-half region. The flow velocity at the horizontal direction from left to right increases, and the gradient increases to downward. In this part, two adjacent vortices do “spread” effect on particles causing them to spread out. Therefore, particle “mushroom” dispersion shapes at the exit, which can be also found in Figure 12d,e with simulation results. Figure 12d shows particle temporal distribution in the wake flows, and the corresponding particle relative slip velocities for each particle at the same conditions are shown in Figure 12e. In the whole region RAVS, whether up-half or blow-half part, V-velocity component velocity directs downward/upward, and it reaches the highest level at the middle position, which enables particles to go through and move out of the region. It is noted that such a “mushroom” dispersion pattern just occurs at the exit of flow region between the two adjacent vortices (RAVS). In addition, “mushroom” is unable to shape near the cylinder due to time limitation particles unable to pass through the region. The “mushroom” pattern is found at the exit of RAVS. Typical “mushroom” structures which are caused by pairs of counter-rotating streamwise vortices are evident here. The “mushroom” shapes of particle distribution were also observed by Ling et al.19 and Fan et al.25 The “mushroom” shapes here are not as regular as those in the study of Ling et al.19 and Fan et al.25 because the mechanism depends mostly on the strong interactions between two alternative and successive shedding vortex structures with opposite sign. 3.4. Particle Size Effect. Particle size (Stokes number) is one of the important factors effect on particle dispersion. The Stokes number is defined as the ratio of the particle response time τp to the characteristic time scale of turbulence τf. The particle response time is defined as τp = ρp·dp2/18 μ, where μ is the fluid viscosity. The fluid characteristic time scale τf in the present study is based on the initial Kolmogorov time scale τη

Figure 12. Scheme of particle dispersion within the region between adjacent (fourth and fifth) vortices structures (RAVS): (a) a particle relative velocity; (b) particles’ cycle motion around vortices structures; (c) particle “mushroom” dispersion; at t = 100, St = 1, Re = 100; (d) particles temporal dispersion in the wake with the presence of “mushroom” dispersion; and (e) distribution of relative velocity.

from downward to upward. It should be noted that in comparison with other regions, the V-velocity component is always at the highest level in the region RAVS with either 10945



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Figure 13. Particle dispersion pattern in the wake flow Re = 150, t = 100: (a) St = 0.01; (b) St = 0.1; (c) St = 1.0; (d) St = 10; and (e) PDF of particle (St = 0.01−10) dispersion in the wake flow.

(0). When St ≪ 1, particles act as fluid tracers and follow fluid motion closely; when St ≫ 1, particle motion is almost independent of fluid turbulence. In Figure 13a small particles (St = 0.01) not only enter the core region of vortex structures but also distribute in the region between adjacent vortexes structures (RAVS). The particle dispersion PDF shown in Figure 13e indicates that this particle has well distribution in the wake flow with less particle concentration. Such a dispersion pattern is associated with the smaller aerodynamic response time of these particles. For the intermediate particles (St = 0.1, 1, Figure 13b,c), most of them are prevented from entering the vortex core region but are concentrated on the vortex boundaries to form highly organized distribution. This is because the aerodynamics response time of these particles is almost the same as the characteristic time scale of the wake flow. Figure 13e shows that the intermediate particles have obvious concentration in the wake flow with higher PDF concentration than small particle St = 0.01. For the large particles (St = 10, Figure 13d), particles are still prevented from entering the vortex core regions, but the conglomeration of the particles in the boundary regions of the vortex structures becomes less, especially in the vortex braid region between two adjacent vortexes. Such particles dispersion along the lateral direction is very little, and most of the particles disperse outside of the vortex street and move toward the downstream just a slight waviness detectable in their paths. Figure 13e shows that this large particle has the highest concentration PDF in comparison with other particles. Such a pattern of particle

distribution indicates that particle motion does not depend much on the vortex structure in the flow field when the Stokes number is relatively large. The reason is that the aerodynamic response time of these particles is quite longer than the characteristic time scale of the large-scale vortex structures in the flow field. Compared with small particle (St = 0.01) and intermediate particle (St = 0.1, 1), large particle (St = 10) has a longer aerodynamic response time so it takes a longer time to respond to large-scale vortex motion. This statement could be verified by the fact of the concentration PDF of large particle appears later than those of small particle and intermediate particle as shown in Figure 13e. Similar patterns of particle dispersion have been often observed in the gas-particle twophase flow in the free shear flows, such as the mixing layer17,25 and the plane wake.18,39 From the above analysis, it could be concluded that particle concentration increases with particle size although the concentration occurs slower due to the larger aerodynamic response time of larger particles.

4. CONCLUSIONS In the wake flow around a cylinder, for the U-velocity component, both the amplitude and frequency of its oscillations increase with the flow Reynolds number. Along the vertical (yaxis) direction, its value decreases/increases, and the trend is against each other for the two adjacent vortices. Therefore, there is an obvious gradient of the U-velocity component between the two adjacent vortices. For the V-velocity 10946



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component of the wake flow, both the amplitude and frequency of its oscillations increase with the flow Reynolds number. It is at high level (positive/negative) in the region between adjacent vortexes structures (RAVS) but low level inside the vortex structures. The properties of the wake flow dominate particle transport in it. Particles (St = 4) are found to concentrate in RAVS together with other particles dispersed outside of vortex outlining the boundaries of the large-scale vortex structures. Such a trend (concentration) in the wake flow increases with the flow Reynolds number. This could be explained by the velocity characters in the wake flow. At the entrance of two adjacent vortices, due to the U-velocity component decrease between them, the flow does “compact” effect on particles. On the other hand, the high V-velocity component leads particles to enter this region. Either with the U-velocity component or the V-velocity component, the amplitude increases with the flow Reynolds number indicating the flow effect on particles increases with the flow Reynolds number. In addition, the wake vortices oscillation frequency increases with the flow Reynolds number so it is seen that particle concentration appears with higher frequency with the flow Reynolds number increasing. In addition, particle concentration increases with particle size. In the wake flow, at the exit of the region between adjacent vortexes structures (RAVS) due to the U-velocity component increase between them, the flow does “spread” effect on particles causing particle dispersion shaping as “mushroom” pattern. The mechanism of particle transport in the wake flow mostly depends on the strong interactions between two alternative and successive shedding vortex structures with opposite sign. Particle transport in the wake flow could be briefly described as follows: moving around vortices and going ahead with them. Based on this work, it is suggested to extend the study with the following points. First, consider particle−particle interaction for the two-phase wake flow with high particle concentration. Second, study particle-flow interaction in the two-phase wake flow and see the effect on the Karman vortex street. Third, extend the study to 3-dimension wake flow at high Reynolds number.

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Je, Ji = parameters characterized the accuracy of the overall scheme, dimensionless P = general pressure, N/m2 p = pressure, N/m2 P̃ = normalized pressure, dimensionless P0 = uniform pressure, N/m2 R = cylinder’s radius, m Re = flow Reynolds number, dimensionless Rep = particle Reynolds number, dimensionless Res = particle Reynolds number of the shear flow, dimensionless St = particle stokes number, dimensionless t = time, s t ̃ = normalized time, dimensionless T0 = uniform time, s u, v = fluid velocity components in x, y directions, m/s V = fluid velocity (u, v), m/s ũ, ṽ = normalized fluid velocity components in x, y directions, dimensionless U0 = uniform stream velocity, m/s uτ = shear velocity, m/s ⎯̂ →̂ ⎯ →̂ V , V = intermediate velocity fields, m/s Vp = particle velocity, m/s x, y = Cartesian coordinate system, m x̃ , ỹ = Cartesian coordinate system in wall units, dimensionless Greek Letters

AUTHOR INFORMATION

Corresponding Author

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*J.Y.: e-mail [email protected], Y.Z.: e-mail [email protected]. uk.

ρ0 = fluid density, kg/m3 αq, βq = implicit/explicit weight-coefficients for the stiffly stable scheme, dimensionless γ0 = a weight coefficient of the backward differentiation scheme, dimensionless β = ratio of Res and Rep, dimensionless ρ = fluid density, kg/m3 ρp = particle density, kg/m3 τf = characteristic time scale of turbulence, s τp = particle relaxation time, s τw = wτσ all shear stress, kg·s‑2·m‑1 τσ = Kolmogorov time scale, s ω = fluid rotation, dimensionless υ = kinematic viscosity, m2·s‑1 μ = the fluid viscosity, m2·s‑1

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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Natural Science Foundation of Fujian Province (Grant No. 2012J01235) and National Natural Science Foundation of China (51176172; 50806068).

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NOMENCLATURE cls = ratio of the extended lift force to the Saffman force, dimensionless CD = Stokes coefficient, dimensionless dp = particle diameter, m Fls = extended lift force, N Fls,Saf f = Saffman force, N g = gravity, N/kg 10947



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Mechanism of Particle Transport in a Fully Developed Wake Flow

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