Equation of the xz shear stress component of the Reynolds stress:
Equation of the yz shear stress component of the Reynolds stress:
where are the coefficients characterizing the entrainment of particles into the fluctuating motion of the flow  for x, y, and z directions, respectively.
The wall conditions are set for the gas at the side walls of the channel based on the control volume method by [27, 28] in a similar way as in the case of the coincidence of shear of the mean flow velocity and gravity.
The numerical simulation considers the turbulent dispersion of solid particles in horizontal channel uniform shear turbulent flow for two different cases: i) shear of the mean flow velocity is along the direction of gravity (Figure 1a) and ii) shear of the mean flow velocity is directed normally to gravity (Figure 1b). Therefore, two sets of the boundary conditions are used for the calculations.
The boundary conditions for the particulate phase are set at the flow axis as follows:
Case 1 for z = 0:
Case 2 for y = 0:
The boundary conditions for the particulate phase are set at the channel walls according to :
Case 1 for y = 0.5hy:
and applying the expression , where is the coefficient of friction between the particles and the wall,
Case 2 for z = 0.5hz:
ex is the coefficient of restitution in the axial direction, which is modeled as:
Here, the parameter , where is the coefficient of restitution of the particle velocity normal to the wall; is the angle of attack between the trajectory of the particle and the wall; is the reflection coefficient, which is the probability of the particles recoiling off the boundaries and back to the flow. The coefficient of restitution reflects the loss of the particle momentum as the particle hits the walls. In the given model, = 1/3, = 1 and = 0.39 .
The conditions for the transverse and spanwise components of the gas velocity are set at the channel walls in terms of impenetrability and no-slip.
The set of boundary conditions for gas and particulate phase at the exit of the channel is written, respectively, as follows:
The control volume method was applied to solve the 3D partial differential equations written for the unladen flow and the particulate phase (Eqs. (1)–(11)), taking into account the boundary conditions (Eqs. (12)–(21)). The governing equations were solved using the implicit lower and upper (ILU) matrix decomposition method with the flux-blending-differed correction and upwind-differencing schemes by . This method is utilized for the calculations of the particulate turbulent flows in channels of the rectangular and square cross sections. The calculations were performed in the dimensional form for all the flow conditions. The number of the control volumes was 1120000.
The obtained numerical results have been verified and validated in comparison with the data obtained by the experimental facility of Tallinn University of Technology.
The experimental method for the determination of the particle dispersion was based on recording the particle trajectories by means of a high-speed video camera on separate regions of a flow that locate at various distances from a point source of particles, and the subsequent processing of the frames .
The experimental setup for the investigations of particle dispersion (Figure 2) allowed to generate the shear flow similarly to  by means of flat plates installed with a varied pitch. The test section was 2 m long with 400 × 200 mm cross section.
Two cases of spatial orientation of shear of the mean flow velocity were investigated. Figure 2 shows the top view of the setup for the case when shear is along the direction of gravity (Figure 1a). For investigations of the particle dispersion when shear is directed normally to gravity (Figure 1b), the setup was turned sideways as a whole at an angle of 90° around the axis of the flow.
The mean flow velocity was 5.1 m/s. Glass spherical particles (physical density of 2500 kg/m3) with an average diameter of 55 μm were used in the experiment runs. The root-mean-square deviation of the diameter of particles did not exceed 0.1. The particles were entered into the flow through the source point which was the L-shaped tubule of 200 μm inner diameter.
All measurements and data processing were carried out at the flow location x = 1212 mm.
The data processing technique  was applied to determine the particle spatial displacement along the y-axis, namely Dy, which characterizes quantitatively the particle turbulent dispersion. Dy is calculated as the axial displacement of the maximum value of distribution of the particle mass concentration determined at the location x = 1212 mm relative to the initial flow location that disposes near the exit of the source point.
The numerical results presented below have been obtained at two locations of the flow: initial location signed “ini” and disposed at the exit of the particle source point and the location 2x/hy = 12.63 from the exit of the particle point source. The turbulent dispersion of 55-μm glass spherical particles was examined. The flow mass loading was about 10−6 kg dust/kg air.
Figures 3–15 show the numerical data obtained by the presented model for two cases of spatial orientation of shear of the mean flow velocity: shear is along the direction of gravity (case 1), and shear is directed normally to gravity (case 2).
Figure 3 shows the transverse distributions of axial velocities of gas and particles for case 1. It is evident that the linear profiles of the averaged axial velocity components of gas and particulate phase across the flow are almost preserved starting from the initial cross section till the pipe exit. Besides, they occupy almost the whole turbulent core of the flow with slight increase of the values in the turbulent core and decrease near the walls due to the effect of a viscous dissipation. The similar profiles are observed with respect of distribution of the same averaged axial velocity components for gas and particulate phase along the spanwise direction (Figure 4).
Since the axial velocity increases toward the bottom wall, the profiles of a turbulence kinetic energy have their higher values near the bottom wall area (Figure 5). However, along the spanwise direction, the profiles of the turbulence kinetic energy are symmetrical, since there is no change of the axial velocity along this direction (Figure 6).
The profiles of the Reynolds shear stresses of gas and particulate phase are shown in Figures 7 and 8. Here it is evident that there is some kind of plateau in the turbulent core. This confirms that we deal with the shear flow; hence, it must be the constant value of the Reynolds shear stresses observed for cases 1 and 2, i.e., for the xy-plane (case 1) and xz-plane (case 2) Reynolds shear stress components. Here, the linear distributions of the averaged axial velocity components across the flow take place along the spanwise direction.
Figure 9 show the transverse distributions of x-normal components of the Reynolds stress of gas and particulate phase obtained for case 1. It can be seen that unlike , the maximum value of distribution located near the channel top wall is larger than the one near the bottom wall. This is due to the effect of particle inertia and their crosswise motion that cause different axial particle accelerations near the top and bottom walls (Figure 3).
Figures 10–13 present the transverse and spanwise distributions of the particle mass concentration c/c0 across the flow at the initial location and the location 2x/hy = 12.63 for both the cases of spatial orientation of shear of the mean flow velocity. Here c0 is the value of the particle mass concentration at the initial location at the flow axis. These distributions reflect the character of the particle turbulent dispersion that occurs in the given channel shear flow. It is obvious that a) due to gravity the particles go down, and thus the mass concentration profile shifts toward the bottom wall (case 1) and b) the profiles become wider relative to their initial distributions due to the particle turbulent dispersion (Figure 10).
Since in case 1 there is symmetrical distribution of parameters along the spanwise direction (Figures 6 and 11), the symmetrical distribution of the mass concentration along this direction (Figure 12) can be observed, both at the initial and exit cross sections.
A similar situation is observed for case 2, when the linear change of the axial velocity takes place along the spanwise direction. Here the particles go down due to gravity (see Figure 13), and simultaneously there is no shift of the distribution of the mass concentration along the spanwise direction (Figure 14).
Table 1 presents the values of the particle spatial displacement Dy obtained experimentally and numerically for two cases of spatial orientation of shear of the mean flow velocity. This displacement characterized quantitatively the particle turbulent dispersion. It is evident that the numerical values of displacement fit satisfactory with the experimental ones that validate the reliability of the presented model.
|Case 1||Case 2|
Table 1 shows that the particle dispersion in case 1 is smaller than in case 2. This fact can be explained by the particle axial velocity taking place in case 2 is smaller than the one for case 1 in the same y location (Figures 10, 13, and 15).
The 3D Reynolds stress turbulence model (RSTM) based on the 3D RANS and statistical PDF approaches has been elaborated for the turbulent dispersion of solid particles in particulate horizontal channel shear flow domain.
The main distinctive feature of the given model is in use of the same closure for both the carrier flow and particulate phase, namely the Reynolds differential equation.
The presented model has several important advantages over the Lagrangian approach:
direct simulation of the particle concentration;
direct simulation of the particles influence on a carrier flow;
there is no basic limit for the parameters of a particulate flow, namely the flow Reynolds number and value of the particle concentration.
Based on the given model, two cases of spatial orientation of shear of the mean flow velocity have been examined. It has been obtained that the effect of orientation of shear appears through decrease of the particle dispersion in case of directional coincidence between shear and gravity as compared with the case of their mutual perpendicularity.
The validity of the elaborated model has been confirmed by experimental investigations of effect of shear of the mean flow velocity on the turbulent particle dispersion.
The authors are grateful for the technical support of Texas Advanced Computing Center (TACC) in Austin, USA. The authors are grateful for the fulfilled research to the Estonia-Norway project EMP230 support. This study is related to the activity of the European network action COST MP1106 “Smart and green interfaces—from single bubbles and drops to industrial, environmental, and biomedical applications.”
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