Turbulence modulation in a particle-laden flow over a hemisphere-roughened wall

International Journal of Multiphase Flow 87 (2016) 250–262

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Turbulence modulation in a particle-laden flow over a hemisphere-roughened wall Xiaofei Liu, Kun Luo, Jianren Fan∗ State Key Laboratory of Clean Energy Utilization, Zhejiang University, 38 Zheda Road, 310027 Hangzhou, PR China

a r t i c l e

i n f o

Article history: Available online 20 September 2016 Keywords: Direct numerical simulation Immersed boundary method Rough wall Turbulence modulation

a b s t r a c t Turbulence modulation by the inertia particles in a spatially developing turbulent boundary layer flow over a hemisphere-roughened wall was investigated using the direct numerical simulation method. The Eulerian and Lagrangian approaches were used for the gas- and particle-phases, respectively. An immersed boundary method was employed to resolve the hemispherical roughness element. The hemispheres were staggered in the downstream direction and arranged periodically in the streamwise and spanwise directions with spacing of px /d= 4 and pz /d= 2 (where px and pz are the streamwise and spanwise spacing of the hemispheres, and d is the diameter). The effects of particles on the turbulent coherent structures, turbulent statistics and quadrant events were analyzed. The results show that the addition of particles significantly damps the vortices structures and increases the length scales of streak structures. Compared with the particle-laden flow over the smooth wall, the existence of the wall roughness decreases the mean streamwise velocity in the near wall region, and makes the peaks of Reynolds stresses profiles shift up. In addition, the existence of particles also increases the percentage contributions to Reynolds shear stress from the Q4 events, however, decreases the percentage contributions from other quadrant events. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The physics of particle transfer in a turbulent boundary layer appears in many environmental and industrial processes such as atmospheric haze, electrostatic precipitators and gas duct, etc. It is known that, at high volume fraction, the particles have a significant effect on the turbulence production, dispersion and dissipation, which may change the behavior of natural or engineering systems. Thus, there has been a continuing interest on the turbulent modulation by the particles in recent decades (Vreman, 2015; Zhao et al., 2010). Despite many numerical and experimental studies on it, the mechanism of the turbulence modulation are poorly understood due to the wide range of relevant parameters. The earlier investigations on jet and pipe flows (Gore and Crowe, 1991) have demonstrated that the particle diameter/fluid length scale ratio determines whether the particles will cause an increase or decrease of the turbulent intensity. However, the turbulence modification also depends on other factors such as the Stokes number (St), particle Reynolds number (Rep ) and volume fraction (Balachandar and Eaton, 2010). Kulick et al. (1994) studied the interactions between

∗

Corresponding author. Fax: +86 571 87991863 E-mail address: [email protected] (J. Fan).

http://dx.doi.org/10.1016/j.ijmultiphaseflow.2016.09.012 0301-9322/© 2016 Elsevier Ltd. All rights reserved.

the particles and fluid turbulence in a downflow fully developed channel. The particles were smaller than the Kolmogorov length scale of the flow, and their results showed that the degree of turbulence attenuation increased with particle Stokes number, particle mass loading and distance from the wall. And the turbulence in the spanwise direction was more strongly attenuated than the turbulence in the streamwise direction. In addition, Nasr and Ahmadi (2007) found that the inter-particle collision was also an important factor in the turbulence modulation. In the studies of Portela and Oliemans (2003), they found that the particles damped the intensity of the streamwise vortices, however, without any significant change in their shape and size. This damping also leaded to a weakening of the near-wall streaky-pattern and a reduction in the accumulation of particles at the wall. Li et al. (2016) investigated the particle-laden flow in a laminar-turbulent transitional boundary layer using direct numerical simulation. They found that the presence of particles displaced the quasi-streamwise vortices towards the wall, which, in turn, enhanced the mean streamwise fluid velocity. It is known that many engineering and environmental conditions are always bounded with rough surfaces. For single-phase flow, a number of works have shown that the effects of roughness mainly concentrate in the roughness sublayer (Flack et al., 2005; Jimenez, 2004; Wu and Christensen, 2007), and the

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turbulent coherent structures are locally destroyed by the wall roughness (Guala et al., 2012; Volino et al., 2011). The effect of wall roughness on the particle-laden channel flow has also been addressed in various simulations studies (Cheng and Zhu, 2015; Konan et al., 2009; Mando and Yin, 2012; Sommerfeld, 2003). In these works, a stochastic approach for the particle-wall collisions was employed to analyze the effect of roughness without determining the real rough wall structure, however this model as mentioned by Mallouppas and van Wachem (2013) can reproduce the particle transverse dispersion enhancement. Different from previous studies that focused on the effects of roughness on the particle distributions in a channel or pipe flow, in present works, a direct numerical simulation of the particle-laden flow with hemispherical wall roughness was performed to investigate the turbulence modulation of particles. In order to model the roughness elements, an immersed boundary method proposed by Li et al. (2015) was adopted to resolve the roughness elements. The particle velocity and angular velocity can be directly calculated from the collisions, thus it can be categorized to the deterministic modeling. The paper is organized as follows. In Section 2, the numerical methods that describe the physical problem have been addressed. In Section 3, the results obtained in our simulations are showed and discussed. Finally, the main conclusions are summarized in Section 4.

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the particle-wall collisions in the rough-wall boundary layer, the slip-shear lift force and slip-rotation lift force are also taken into account. Other forces such as buoyancy force, pressure gradient force, Basset force and virtual mass force as mentioned by Li et al. (2016) are order of magnitude smaller than the drag force and can be neglected. In additions, similar to the numerical studies of Portela and Oliemans (2003); Squires and Simonin (2006) and De Marchis et al. (2016), the gravity is also neglected, in order not to mix the gravitational effects with the particle-turbulence interaction. Finally, the governing equations for the particles flow can be wrote as,

dx p = up dt mp Ip

(3)

du p = FD + FLS + FLR dt

dω p =T dt

(4) (5)

2. Numerical method

where xp , up and ωp are the particle position vector, velocity vector and angular velocity vector, respectively. mp = π ρ p dp 3 /6 is the mass of the particle and Ip = mp dp 2 /10 is the moment of inertia. FD , FLS , FLR and T represent the drag force, the slip-shear lift force, the slip-rotation lift force and the torque acting on the particle, respectively. The formulations that are used to calculate the forces can be found in the reference of Li et al. (2016).

2.1. Governing equations for fluid

2.3. Immersed boundary method and the two-way coupling

In present studies, the fluid is assumed to be incompressible, and its governing equation can be wrote in a dimensionless form as follow:

In order to solve the Eq. (1), the external force frw and fp must be solved in advance. Here, the immersed boundary method proposed by Li et al. (2015) is used which is efficient to describe the hemispherical rough elements under the Cartesian coordinates. The force frw is calculated explicitly in time using previous velocity field to satisfy the no-slip condition on the immersed boundary. A moving least-square reconstruction method was adopted to interpolate the no-slip velocity at the immersed boundary. The detail description and validation can be found in the reference of Li et al. (2015). The interaction between the particles and fluid is implemented by the addition of the feedback force fp . The feedback force can be calculated by,

∇ •u=0

(1)

∂u 1 2 + u • ∇ u = −∇ p + ∇ u + frw + f p ∂t Re

(2)

where u is the fluid velocity vector and p is the pressure term. Re = U∞ θ in /ν is the Reynolds number based on the freestream velocity (U∞ ), the inlet momentum thickness (θ in ) and the kinematic viscosity (ν ). frw and fp represent the external body forces imposed by the rough wall and particles, respectively. The Eqs. (1) and (2) are solved with the fractional step method described by Perot (1993). The diffusion terms in the wall-normal direction are treated implicitly and other terms are treated explicitly. For time advancement, the low-storage three-step Runge– Kutta scheme is used for the explicit terms and the secondorder Crank–Nicolson scheme is used for the implicit terms. A fourth order accurate finite difference is adopted to discretize the convective terms on the staggered grids, whereas a highorder Lagrange polynomials are employed to discretize the viscous terms (Desjardins et al., 2008; Shukla et al., 2007). To enhance the efficiency and accuracy of the numerical scheme, a transformation from physical space to computational space is performed (Desjardins et al., 2008). The BiConjugate Gradient Stabilized (BICGSTAB) solver with multigrid preconditioner is applied (Saad, 2003; Wesseling, 1991) to solve the pressure Poisson equation. 2.2. Governing equations for particles The particles are treated as point-particles with identical diameter dp = 147 μm and density ρ p = 1807.5 kg/m3 . Because the ratio of the particle and fluid density (ρ p /ρ f = 1500) is far larger, the drag force is the main force that must be considered firstly. Moreover, due to the existence of the large velocity gradients and

fp = −

ρ pnVpn n F W ( xn , xm ) ρ Vm f n=1

Np 

(6)

where Vm is the virtual control volume of the fluid grid m. Vp is the volume of a particle and Np is the particle number in the control volume. W(xn , xm ) is the weight function that locally distributes the interaction force Fn to the Eulerian point via a fourth-order Lagrange interpolation scheme (Li et al., 2016). 2.4. Particle-wall collision The elastic collisions between particles and rough wall take place when the distance between the mass center of particle and rough wall is smaller than the particle radius. The method of Yamamoto et al. (2001) is used to calculate the particle velocity and angular velocity after the collision as shown below,

ˆ p = up − u

J mp

(7)

ωˆ p = ω p +

dp n×J 2

(8)

where n is the unit vector. J is the impulse force of the particle and can be expressed as,

J = Jn n + Jt t

(9)

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X. Liu et al. / International Journal of Multiphase Flow 87 (2016) 250–262 Table 1 Domain size and mesh resolutions. Lx /θ

Ly /θ

Lz /θ

Nx

Ny

Nz

x+

y+ min

y+ max

z+

128

60

32

1025

129

257

4.25

0.5

63

4.25

Fig. 1. Schematic diagram of the computational domain. The small black dots represent the omitted hemispherical roughness elements and the large red dots represent the particles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Jn = (1 + e )m p c · n

 2   Jt = min −μJn , m p c f c 

(10)



7

(11)

where Jn and Jt are the normal and tangential components of the impulse force. e and μ are the restitution and friction coefficients. c and cfc are the particle velocity and the particle slip velocity relative to the rough wall (Liu et al., 2015). 2.5. Computational detail The direct numerical simulation of the particle-laden flow over the rough wall was performed by means of a MPI parallel computation using 256 CPUs. The domain sizes in the streamwise, wallnormal and spanwise directions are 128θ in × 60θ in × 32θ in , and the corresponding grid sizes are 1025 × 129 × 257. The domain size and the grid resolutions are summarized in Table 1. Fig. 1 shows a schematic diagram of the computational domain and the threedimensional hemisphere-roughened wall. The small black dots represent the omitted hemispherical roughness elements and the red dots represent the particles in the flow field. The hemispherical roughness elements were arranged in a staggered manner in the streamwise direction with pitches of px /d= 4.0 and pz /d= 2.0 (where px and pz are the streamwise and spanwise spacing, d is the diameter of the hemisphere). The roughness diameter is d= 1.0θ in , and the height is k= 0.5θ in . The first row of the roughness elements was located at xstart = 10θ in downstream from the inlet. As for the inlet boundary, a turbulent inlet plane extracted from a previous simulation of transitional boundary layer flow was adopted. The detail computational parameters can be found in the reference of Li et al. (2016). In order to trigger the laminar boundary layer to turbulence, an isotropic turbulence box of the turbulent intensity 8.3% was used. The time series of the y–z plane

Table 2 Detail parameters for the particle-laden flow. St

Zm

Фv

Np ,total

dp +

10

1.0

6.67 × 10−4

3.4 × 107

0.47

flow data were extracted at Reθ = 800 where the flow had become fully developed turbulence. The Reynolds number based on the momentum thickness was varied in the range Reθ = 800–1130 and the corresponding ratio of roughness height k to boundary layer thickness δ is 0.048–0.053. Periodic boundary conditions were imposed in the spanwise directions and a convective outflow condition was used at the outflow boundary. The no-slip boundary condition in applied at the solid wall and the top boundary condition was u= U∞ , v=VBlasius , w= 0. At first, a fully developed single-phase turbulent boundary layer field was used as the initial field of the particle-laden flow, and the particles were randomly distributed in flow field below y= 12θ in in the wall-normal direction. This height is slightly higher than the boundary layer thickness at the outflow boundary. The initial velocity of each particle was set equal to the local fluid by the fourthorder Lagrangian polynomials. For updating the particle velocity and angular velocity, an explicit third-order Runge–Kutta scheme was used. The second-order Crank–Nicolson scheme was employed for updating the particle position. The computational time step was set to be t+ = 0.073 which made the particle would not travel more than half grid cell per iterative step. When the particles moved outside of the computational domain in the streamwise direction, they were randomly injected into the domain from the inlet boundary below y=12θ in in the wall-normal direction. In the spanwise direction, periodic boundary conditions were imposed for the particles. For the upper boundary, it was high enough so that the particles were hardly moving outside. The detail parameters for the two-phase flow are showed in Table 2. Because of the spatially

X. Liu et al. / International Journal of Multiphase Flow 87 (2016) 250–262

developing in the streamwise direction for the turbulent boundary layer, the turbulent statistics were obtained at x/θ in = 108 in the streamwise direction where the equilibrium state had been established. The equilibrium states were identified by applying a 5% tolerance to the velocity defect form and Reynolds stresses in outer layer (Ahn et al., 2013). At this point, the Reynolds numbers based on the momentum thickness are 1094 and 1039 for the single- and two-phase flow, respectively. In addition, selecting an appropriate scaling parameter for the normalization of the boundary layer variables is also an important issue. In the present work, the friction velocity (uτ /U∞ = 0.0643) of the single-phase TBL flow over the rough wall was used. It was calculated directly from the total drag of the skin-frictional and form drags, and then averaged temporally and spatially along the spanwise direction and one spacing in the streamwise direction (Lee et al., 2011). 3. Results and discussion 3.1. Instantaneous visualizations The instantaneous turbulent vortex structures colored by the streamwise velocity and particles distributions for the single- and two-phase flow are showed in Fig. 2. The hemispherical roughness elements and particles are represented by grey and white color, respectively. It can be seen that the large scale vortex packet structures are the common features in the turbulent boundary layer flow (Wu and Christensen, 2010), although these vortex structures are significantly disturbed by the surface roughness. Compared with the single-phase flow, the particle-laden flow shows an obvious reduction of the vortex structures as shown in Fig. 2(b) where a partial enlarged drawing is also extracted from the flow filed for a better comparison. The same phenomenon can be also observed from Fig. 3 which shows the instantaneous velocity field with the spanwise vortices in the x–y plane. The abscissa is presented as a new coordinate that the origin is defined in terms of s, and s= 0 indicates the location of x= 106θ in . For a better visualization, a Galilean decomposition has been applied, with a uniform convection velocity 0.8U∞ subtracted from the velocity field (Volino et al., 2011). The hairpin vortices in a packet can be visible if the convection velocity is subtracted from the instantaneous velocity field. These hairpin packets are usually accompanied by the large scale ejection- and sweepevents which can extend from the near wall region into the entire width of the boundary layer. The addition of particles produce a large decrease of the hairpin packets structures as shown in Fig. 3(b) which is consistent with previous researches in the particleladen pipe flow (Portela and Oliemans, 2003; Vreman, 2007). The instantaneous streamwise fluctuating velocity and particles distributions in the x–z plane are showed in Fig. 4. The position in the wall-normal direction is selected at y/δ = 0.08 which is slightly higher than the roughness height. The red dash lines represent the initial location of the first row roughness elements and the dark points represent the particles. As shown in Fig. 4(a), the high- and low-speed streaks are torn when they pass from the rough surface, and their scales are obviously reduced by the wall roughness. However, the reduction effects are significantly weaken when the inertia particles are added in. These high- and low-speed streaks become highly elongated in the streamwise direction and extend beyond 40θ in (near 4δ ) downstream. In the instantaneous contour of the streamwise velocity component given by Zhao et al. (2010), it was also shown that the coherence of the near wall flow structures was considerably increased by the particles and the alternating high- and low-velocity bands appeared wider and more regular than the unladen flow. This effect can be related to the damping of vortices structures, since the sweep- and ejection-events associated with the vortices structures are responsible for the generation of

253

high- and low-speed streaks (Portela and Oliemans, 2003; Robinson, 1991). Moreover, the particles gradually tend to accumulate in the instantaneous locations of the low-speed streaks and avoid the high-speed regions, even though the inertial particles are randomly injected from the inlet boundary, which is consistent with previous results for the particle-laden flow over the smooth wall (Eaton and Fessler, 1994; Li et al., 2016; Marchioli and Soldati, 2002). 3.2. Two-point correlations The average extent and shape of the hairpin packets and streaky structures can be quantified through the two-point correlations of the fluctuating velocity. The two-point spatial correlations at the position (xref , , yref , zref ) can be defined as





RAB xre f , yre f , zre f =

 





A xre f , yre f , zre f B(x, y, z )

 

A2 xre f , yre f , zre f

 

(12)

B2 (x, y, z )

where A and B are the quantities of interest at the locations of (xref , yref , zref ) and (x, y, z), respectively. Fig. 5 shows the contours of the auto-correlations Ruu and Rvv in the x–z plane. The reference position is selected at x/θ in = 108 which is behind the hemisphere and at the center of two adjacent hemisphere rows. The wall-normal position is located at y/δ = 0.08 which is slightly higher than the roughness height. Even though the correlations for the two-phase flow show a similar shape to the single-phase flow, they has a noticeably different extent in the streamwise and wall normal directions. If the streamwise extents of Ruu and Rvv (Lx(Ruu ) and Lx(Rvv )) are defined as twice the distance from the self-correlation peak to the most downstream location on the contour of Ruu = 0.5 (Christensen and Wu, 2005), the values of the two-phase flow are over twice as that of the single-phase flow. And the spanwise extents of the correlations Lz(Ruu ) and Lz(Rvv ) are 54% and 37% larger that of the single-phase flow. The modification in the wall-normal direction can be observed from Fig. 6 which shows the contours of the auto-correlations Ruu and Rvv in the x–y plane. If the wallnormal extents Ly(Ruu ) and Ly(Rvv ) are determined based on the distance between points closest and farthest from the wall on the Ruu = 0.5 contour (Volino et al., 2009), the extents of the two-phase flow are 31% and 21% larger than that of the single-phase flow. It seems that the particles affect the auto-correlations of streamwise velocity fluctuation more strongly than the auto-correlations of wall-normal velocity fluctuation, and the effects are more obvious in the streamwise direction than in the wall-normal and spanwise directions. It is known that the angle of inclination of Ruu can be related to the average inclination of the hairpin packets (Volino et al., 2009). In present cases, the inclination angles are approximate to 6.8°and 5.0° for the single- and two-phase flow. It seems that the addition of particles also slightly decreases the inclination of hairpin packets in the turbulent boundary layer. 3.3. Roughness sublayer It is known that the introduction of the roughness elements onto the wall surface significantly affects the turbulent variables and the transmission properties of particles. Previous works of the unladen flow over the rough wall generally assumed that the effects of surface roughness mainly concentrated in the roughness sublayer which usually had a height of 3–5 times the roughness height. Here, the attention is also paid on the roughness sublayer where the roughness disruption strongly exists. Fig. 7 displays the iso-contours of some mean quantities in the roughness sublayer. The location of the contour in the spanwise direction is passing through the center of the hemisphere, and the hemispheres behind the fluid are represented by the dashed lines at s/k= 8.0. The streamline iso-contours for present hemisphere-roughened wall as

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Fig. 2. Instantaneous vortex structures and particles distribution in the turbulent boundary over hemispherical rough walls: (a) single-phase, (b) two-phase. The vortex structures are colored by the instantaneous streamwise velocity.

shown in Fig. 7(a) is very similar to the cubic rough wall as reported by Lee et al. (2011). The flow between the hemispheres moves parallel to the streamwise direction for both of the single and two-phase flow. There is only a slight change of the reattachment point for the addition of particles which moves from s/k= 4.2 to 5.5. Because of the narrow passages between the spanwise hemispheres at s/k= 8.0, the flow appears a higher mean streamwise velocity at this location. For the particle-laden flow, this phenomenon is apparently weakened. The wall-normal mean velocity increases in front of the leading edge of the roughness and becomes negative at the center of the cavity because of the flow entrainment (Lee and Sung, 2007). In the middle of the two ad-

jacent hemisphere roughness elements, the wall-normal mean velocity also shows a positive values indicating the ejection of fluid into the outer layer. Even though the iso-contours of wall-normal mean velocity for the two-phase flow shows a similar distribution to the single-phase flow, the magnitudes are apparently lower due to the damping effects of the particles. The isocontours of the velocity fluctuations normalized by the friction velicity in the roughness sublayer are displayed in Fig. 8. For the single-phase flow, the streamwise velocity fluctuation increases above the crest, extends to the next hemisphere and reaches the maximum values at s/k= 3. Due to the interaction between the upstream flow and the adjacent hemisphere, there is

X. Liu et al. / International Journal of Multiphase Flow 87 (2016) 250–262

255

Fig. 3. Instantaneous velocity vector field with spanwise vortex strength: (a) single-phase, (b) two-phase.

also another peak value forming at s/k= 13. However, the broad peak values disappear when the particles are added in. The wallnormal and spanwise velocity fluctuations increase at the leading edge of the hemisphere and generate the maximum values at the cavity of the hemispheres. The existence of particles also reduces the formation of these region and decreases their magnitudes. The same effects can be also observed from the distributions of the Reynolds shear stress as shown in Fig. 8(h). 3.4. Mean velocity and Reynolds stresses The choice of the grid resolution is based on the works of Ashrafian et al. (2004) and Lee et al. (2011). There are 30 grid points within the distance y= 3k from the wall which is larger than that of Ashrafian et al. (2004). The grid resolutions in each direction are x+ = z+ = 4.25 and y+ min = 0.5 which corresponds to 13 points in the wall-normal direction and 8 points in other directions for the discretization of a hemispherical roughness el-

ement. The number of grid points used for the discretization of roughness in the present study is larger than that of Ashrafian et al. (2004) and Lee et al. (2011). In addition, in order to estimate the grid resolution, we also conducted another simulation of the two-phase turbulent boundary layer flow over the hemisphereroughened wall with the grid number of 1025 × 257 × 257 in the streamwise, wall-normal and spanwise directions. The comparisons of the mean velocity and Reynolds stresses profiles are displayed in Figs. 9 and 10. It can be seen that the differences between two grid systems are very small, which also indicates that the present grid system can resolve the flow efficiently. Fig. 11 shows the mean fluid and particle velocity profiles in the inner and outer coordinates for the single- and two-phase turbulent boundary layer over the hemisphere-roughened wall. For a best comparison, the results of Lee et al. (2011) for the smooth and cube-roughened walls have also been given and the virtual origin is removed from their results. The logarithmic laws is depicted by the red dashed lines. It can be seen that our results show an agree-

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X. Liu et al. / International Journal of Multiphase Flow 87 (2016) 250–262

Fig. 4. Instantaneous streamwise velocity fluctuations and particle distributions at y/δ = 0.08 in the x–z plane: (a) single-phase, (b) two-phase. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 5. Contours of Ruu and Rvv centred at y/δ = 0.08 in x–y plane, outermost contour Ruu = 0.5, contour spacing 0.1: (a) and (b) single-phase, (c) and (d) two-phase.

X. Liu et al. / International Journal of Multiphase Flow 87 (2016) 250–262

Fig. 6. Contours of Ruu and Rvv centred at y/δ = 0.08 in x–z plane, outermost contour Ruu = 0.5, contour spacing 0.1: (a) and (b) single-phase, (c) and (d) two-phase.

Fig. 7. Isocontours of the mean quantities in the roughness sublayer. (a)–(c): single phase; (d)–(f): two-phase. The dashed lines denote regions of negative value.

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Fig. 8. Isocontours of Reynolds stresses in the roughness sublayer. (a)–(d): single phase; (e)–(h): two-phase.

ment with their results for the smooth wall case. Similar to their cube case, the mean velocity profiles with hemispheres show a downward shift from the smooth wall. The downward shift in the logarithmic layer named roughness function U+ was estimated to be 6.7 which quantifies the increase of local drag due to the surface roughness (Mishra and Bolotnov, 2015). Mishra and Bolotnov (2015) and Chatzikyriakou et al. (2015) investigated the effects of hemispherical roughness elements with a smaller size and showed that the roughness function were significantly dependent on the roughness density, Reynolds number and roughness height. In present work, it can be seen that the addition of particles also modified the roughness function. Compared with the single-phase flow, the mean fluid velocity in the logarithmic layer is obviously increased. A similar increment was also observed in previous studies of particle-laden flow in a flat plate boundary layer (Li et al., 2016), which was associated to the displacement of the streamwise vortices towards the wall. Differently, in the near wall region (y+ < 44), the mean fluid velocity is significantly reduced due to the existence of the roughness elements. This difference can be attributed to the modification of the recirculation region after the hemispherical roughness element. It can be seen from Fig. 7(b) and (e) that the reattachment point after the roughness element is increased by particles, which leads to a lower fluid velocity at the center of the two adjacent hemispheres than that for the singlephase flow in the near wall region. Moreover, it can be also seen

that the particle velocity is larger than the fluid velocity in the near wall region which is consistence with the experimental results of Kulick et al. (1994). This may be caused by the large particle inertia that makes them move toward the wall, hit the wall and rebound with much of their momentum. The Reynolds stresses in the outer coordinates normalized by the friction velocity are showed in Fig. 12. It can be seen that the values of , , and <−u+ v+ > are obviously lower than that for the single-phase flow, which indicates that the turbulent kinetic energy is substantially attenuated by the inertial particles. In the works of Li et al. (2016), a similar reduction was also observed in the wall-normal and spanwise velocity fluctuations. As for the streamwise fluctuating velocity, they found a slight increment in the near wall region (y+ < 11) and a decrement in the outer layer (y+ > 11). For the single-phase flow over a rough wall as shown in Fig. 8(a), the shear layer generating at the crest of the hemisphere makes a great contribution to the production of the Reynolds stresses in the near wall region. The presence of particles significantly reduces the effect of hemisphere on the formation of the Reynolds stresses in the near wall region and makes the maximum values of Reynolds stresses generate above the height of hemisphere as shown in Fig. 8(b). Thus, the profile of the streamise velocity fluctuation decrease along the whole wall-normal direction, and the peaks of the Reynolds stress profiles shift up comparing with the single-phase flow.

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Fig. 9. The comparison of the mean fluid and particle velocity profiles for two grid systems.

259

Fig. 10. The comparison of the mean fluid Reynolds stresses profiles for two grid systems.

3.5. Quadrant analysis The instantaneous velocity field has shown that the particles have a significant effect on the frequent large eruptions of fluid. The turbulence modulation can be also quantified by the probability density function (PDF) of the instantaneous Reynolds shear stress as shown in Fig. 13. It can be seen that both of the negative and positive profiles are significantly reduced by the particles. It is known that particles are transferred by the sweep event into the wall region and accumulated in the low-speed streak region, whereas the ejection event transfers particles from the near wall into the outer flow. Previous works of Marchioli and Soldati (2002) have confirmed that the sweep and ejection events are efficient transfer mechanisms for particles. In order to inves-

tigate the effects of particles on the Reynolds shear stress producing events, the quadrant decomposition (Lu and Willmarth, 1973) was adopted. With this technique the Reynolds shear stress can be sorted into four quadrant events in the u–v plane: outward motion (Q1), u> 0 and v> 0; ejection event (Q2), u< 0 and v> 0; inward motion (Q3), u< 0 and v< 0; sweep event (Q4), u> 0 and v< 0. The negative tails of the probability density function in Fig. 11 embody contributions from both the ejection (Q2) and sweep (Q4) events, while the positive tails contains contributions from both the outward (Q1) and inward (Q3) motions (Wu and Christensen, 2007). As it can be seen from Fig. 13, the ejection and sweep events are obviously stronger than the inward and outward motion which is

Fig. 11. Mean fluid and particle velocity profiles in the inner and outer coordinates for single- and two-phase turbulent boundary layer flow. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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Fig. 12. Reynolds stresses in the outer coordinates, normalized by the friction velocity.

tion:

< uv>i (x, y, H ) = lim

T →∞

1 T



T 0

ui (x, y, t )vi (x, y, t )Si (x, y, t, H )dt (13)

where the subscript i refers to the ith quadrant events and

Si (x, y, t, H ) =

⎧ ⎪ ⎨1 ⎪ ⎩

0

Fig. 13.. Probability density function of instantaneous Reynolds stress at y/δ = 0.08.

similar to previous studies of unladen and laden flow (Kaftori et al., 1998; Krogstad et al., 2005). Following the methodology proposed by Lu and Willmarth (1973), the contributions from the four quadrants events at a given wall-normal location can be calculated from the following equa-

i f |ui (x, y, t )vi (x, y, t )| ≥ H urms (x, y )vrms (x, y )

(14)

otherwise

H is the hyperbolic hole size. Fig. 14 shows the percentage contributions to the Reynolds shear stress from four quadrant events obtained with H= 0. The data is presented as the percentage contribution to the Reynolds shear stress from each quadrant so that none of the inaccuracy scaling variables is used such as the friction velocity (Krogstad et al., 2005). The DNS results of Krogstad et al. (2005) in a fully turbulent channel flow with a rod-roughened wall are also given for a better comparison. Present unladen flow results show a good agreement with their results in the log region. In the near wall region, the larger differences are mainly induced by the different wall conditions. The distinct near-wall differences extend nearly to y/δ = 0.1 for Q1 and Q3 events, and y/δ = 0.3 for Q2 and Q4 events. This result indicates that the Q2 and Q4 events are more susceptible to the surface conditions than the Q1 and Q3 events. Compared with the unladen flow, a general reduction of Q1 and Q3 events along the wall-normal direction is found for the laden flow. The particle effects are more pronounced for the Q2 and Q4 events. As observed in Fig. 14(b) and (d), a significant

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Fig. 14. Contributions to the quadrants for single- and two-phase turbulent boundary layer. —, single-phase; - - -, two-phase; , Rough wall DNS of Krogstad et al. (2005).

near-wall reduction in Q2 activities are found for y/δ < 0.7, however the Q4 activities show an increment below y/δ = 0.7.

Acknowledgments This work is supported by The National Natural Science Foundation of China (no. 51136006). We are grateful to that.

4. Conclusion Direct numerical simulation was performed to investigate the effect of wall roughness on the particle-induced turbulence modulation in a spatially developing turbulent boundary layer. The wall roughness was presented by a series of hemispherical roughness elements and resolved using the immersed boundary method. The fluid and particles were resolved in the Eulerian and Lagrangian frame, respectively. The modification of the coherent structures due to the addition of particles was examined using the instantaneous flow field and two-point correlations. It is found that the existence of particles significantly damps the generation of the coherent structures and increases their scale in three directions. In the roughness sublayer, the peak value regions of Reynolds stresses generated by the roughness elements are also reduced. Different from the laden flow over a smooth wall, the mean streamwise velocity is reduced in the inner layer and the peaks of Reynolds stresses profiles shift up for present hemisphere-roughened wall case. The results from the quadrant analysis show that the effects of particles on the ejection and sweep events are more obvious than the inward and outward motions. Moreover, the addition of particles reduces the percentage contributions of Q1, Q2 and Q3, but increase the percentage contributions of Q4.

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