1 Introduction

Particle–fluid and particle–particle interactions can be widely seen in natural and industrial process and one particle and two particles models were usually investigated in experiments [1, 2] or simulations [3,4,5,6,7,8,9,10]. Though two particle model can be regarded as the simplest model for multi-particles flow which can reflect the interactions between particles to some extent, it also has great limitations. In order to understand the interactions between the particles and particles/fluid deeply, two-dimensional fluid flow around multiple particles should be studied.

Some scholars have analyzed the mechanical behavior of the particle clusters and the variation of drag coefficient. For example, O’Brien and Syamlal [11] carried out the experiments to consider the influences of particle clusters. The gas and particle drag law is modified by the experiments results. Sandeep and Zuritz [12] also conducted experiments to study the drag of single and multiple sphere assemblies in non-Newtonian fluid. The results show that when the particle concentration increases, the drag exerted on each sphere in assembly increases. And one equation was established to calculate the drag correction factor for the drag on a sphere in the assembly. Beetstra et al. [13] used the lattice Boltzmann method (LBM) to investigate the drag and flow characteristics of irregularly shaped (spherical, star-shaped, H-shaped, etc.) clusters. It is found that the drag coefficient of the particle clusters increases as the inter-particle distance increases for these shapes. And when a particle in the particle cluster is shielded by other particles in the flow direction, the particle’s drag decreases. When the particles clusters become dispersed, the drag greatly increases. Shah [14] also used LBM model to investigate the influences of clusters on drag. They found that compared with random distribution, the particles in the cluster have a lower drag under the same simulation condition. With the total voidage increasing, the drag decreases and the decrease is evident for voidage higher than 0.7. Wei et al. [15] numerically quantified the interaction between the gas and cluster and an empirical correlation was established. They discovered the clusters can significantly influence the flow field.

In above researches, the voidage/distance between the particles are considered, but the voids inside the spheres or particles are neglected. However, porous particles are general in industry such as gasification and there is little research about multiple porous particles. Compared with solid particles, porous particles allow fluid penetrating into them and the flow characteristics are very different. So it is necessary to investigate the flow characteristics around and through multiple porous particles. In addition, the lattice Boltzmann method was applied in these numerical simulations which indicates it is a good method to study the interactions between the particles and particles/fluid. And compared with traditional method, LBM is easy to handle complex geometries and it is widely used in porous media [16, 17]. Therefore, two-dimensional fluid flow around and through multiple porous particles was numerically investigated based on the LBM in this paper. Uniform spatial distribution and random spatial distribution were considered and the effects of Re, Da and the distance between the particles on streamlines and drag coefficient were analyzed in detail.

2 Numerical method

2.1 Governing equations

Darcy-Brinkman-Forchheimer model [18] can fully reflect the linear and nonlinear resistance term in the porous media and therefore was employed in this paper. And the following macroscopic governing equations [19] were used to describe fluid flow in the porous particles.

Continuity equation:

$$\nabla \cdot \left\langle {u_{f} } \right\rangle^{f} = 0$$
(1)

Momentum equation:

$$\frac{{\partial \left\langle {u_{f} } \right\rangle^{f} }}{\partial t} + \left\langle {u_{f} } \right\rangle^{f} \cdot \nabla \left\langle {u_{f} } \right\rangle^{f} = - \frac{1}{{\rho_{f} }}\nabla \left\langle {p_{f} } \right\rangle^{f} + \upsilon \nabla^{2} \left\langle {u_{f} } \right\rangle^{f} + F_{n}$$
(2)

where \(\left\langle {u_{f} } \right\rangle^{f}\) is the intrinsic phase average velocity and can be defined as \(\left\langle {u_{f} } \right\rangle^{f} = \frac{1}{{V_{f} }}\int_{{V_{f} }} {u_{f} {\text{d}}V}\), and \(u_{f}\) is the velocity of the fluid, V is the representative volume and \(V_{f}\) is fluid volume within V. Similarly, \(\left\langle {p_{f} } \right\rangle^{f}\) is intrinsic phase average pressure which can be defined as the same way of \(\left\langle {{\varvec{u}}_{f} } \right\rangle^{f}\). \({\varvec{F}}_{n}\) is the total force, \(\upsilon\) and \(\rho_{f}\) are the kinematic viscosity and density of the fluid, respectively. In order to simplify the writing and read easily, following we will use u to represent \(\left\langle {{\varvec{u}}_{f} } \right\rangle^{f}\).

2.2 Lattice Boltzmann method

2.2.1 Lattice Boltzmann (LB) governing equations

$$m_{\alpha } \left( {{\varvec{x}} + {\varvec{e}}_{\alpha } \delta_{t} ,t + \delta_{t} } \right) - m_{\alpha } \left( {{\varvec{x}},t} \right) = - \frac{1}{\tau }\left[ {m_{\alpha } \left( {{\varvec{x}},t} \right) - m_{\alpha }^{{{\text{eq}}}} \left( {{\varvec{x}},t} \right)} \right] + \delta_{t} F_{\alpha }$$
(3)

where \(m_{\alpha } \left( {{\varvec{x}},t} \right)\) is the density distribution function at position x and time t along \(\alpha\) direction. D2Q9 lattice model [20] was employed as shown in Fig. 1 and \({\varvec{e}}_{\alpha }\) can be written as follows:

$${\varvec{e}}_{\alpha } = \left\{ {\begin{array}{*{20}l} {\left( {0,0} \right)} \hfill & {\alpha = 0} \hfill \\ {c\left( {\cos \left[ {\left( {\alpha - 1} \right)\frac{\pi }{2}} \right],\,\sin \left[ {\left( {\alpha - 1} \right)\frac{\pi }{2}} \right]} \right)} \hfill & {\alpha = 1,2,3,4} \hfill \\ {\sqrt 2 c\left( {\cos \left[ {\left( {2\alpha - 1} \right)\frac{\pi }{4}} \right],\,\sin \left[ {\left( {2\alpha - 1} \right)\frac{\pi }{4}} \right]} \right)} \hfill & {\alpha = 5,6,7,8} \hfill \\ \end{array} } \right.$$
(4)

here c is the lattice speed and is defined by \(c = \delta_{x} /\delta_{t}\), \(\delta_{x}\) and \(\delta_{t}\) are lattice step and time step, respectively.

Fig. 1
figure 1

D2Q9 model

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On the right side of Eq. (3), the equilibrium density distribution function \(m_{\alpha }^{{{\text{eq}}}} \left( {{\varvec{x}},t} \right)\) and forcing term \(F_{\alpha }\) are defined by

$$m_{\alpha }^{{{\text{eq}}}} \left( {{\varvec{x}},t} \right) = \rho_{f} w_{\alpha } \left[ {1 + \frac{{{\varvec{e}}_{\alpha } \cdot {\varvec{u}}}}{{c_{s}^{2} }} + \frac{{\left( {{\varvec{e}}_{\alpha } \cdot {\varvec{u}}} \right)^{2} }}{{2c_{s}^{4} }} - \frac{{{\varvec{u}}^{2} }}{{2c_{s}^{2} }}} \right]$$
(5)
$$F_{\alpha } = \rho_{f} w_{\alpha } \left( {1 - \frac{1}{2\tau }} \right)\left[ {\frac{{{\varvec{e}}_{\alpha } \cdot {\varvec{F}}_{n} }}{{c_{s}^{2} }} + \frac{{{\varvec{e}}_{\alpha } \cdot {\varvec{u}}}}{{c_{s}^{4} }}\left( {{\varvec{e}}_{\alpha } \cdot {\varvec{F}}_{n} } \right) - \frac{{{\varvec{u}} \cdot {\varvec{F}}_{n} }}{{c_{s}^{2} }}} \right]$$
(6)

where \({w}_{\alpha }\) is the weighting factor and in D2Q9 model, w0 = 4/9, w1−4 = 1/9, w5−8 = 1/36. \(c_{s} = c/\sqrt 3\) is sound speed.

The total force \({\varvec{F}}_{n}\) in Eqs. (2) and (6) is expressed as

$${\varvec{F}}_{n} = - \frac{\varepsilon \nu }{K}{\varvec{u}} - \frac{{\varepsilon^{2} F_{\varepsilon } }}{\sqrt K }{\varvec{u}}\left| {\varvec{u}} \right|$$
(7)

where \(\varepsilon\) and K are porosity and permeability, respectively. Geometry function \({\varvec{F}}_{\varepsilon }\) and kinematic viscosity \(\nu\) can be calculated as follows:

$${\varvec{F}}_{\varepsilon } = \frac{1.75}{{\sqrt {150\varepsilon^{3} } }}$$
(8)
$$\nu = c_{s}^{2} (\tau - 0.5)\delta_{t}$$
(9)

here \(\tau\) is relaxation time.

Furthermore, the density and velocity can be calculated from the density distribution function \(m_{\alpha } \left( {{\varvec{x}},t} \right)\):

$$\rho_{f} = \sum\limits_{\alpha } {m_{\alpha } } \left( {{\varvec{x}},t} \right)$$
(10)
$$\rho_{f} {\varvec{u}} = \sum\limits_{\alpha } {{\varvec{e}}_{\alpha } m_{\alpha } \left( {{\varvec{x}},t} \right)} + \frac{1}{2}\delta_{t} \rho_{f} {\varvec{F}}_{n}$$
(11)

Combined Eqs. (7) and (11), u can be calculated by

$${\varvec{u}}{ = }\frac{{\varvec{v}}}{{q_{0} + \sqrt {q_{0}^{2} + q_{1} \left| {\varvec{v}} \right|} }}$$
(12)

where temporary velocity ν is defined as

$$\rho_{f} {\varvec{v}} = \sum\limits_{\alpha } {e_{\alpha } m_{\alpha } } \left( {{\varvec{x}},t} \right)$$
(13)

The parameters q0 and q1 in Eq. (12) are defined by

$$q_{0} = \frac{1}{2}\left( {1 + \frac{{\delta_{t} }}{2}\frac{\varepsilon \nu }{K}} \right),q_{1} = \frac{{\delta_{t} }}{2}\frac{{\varepsilon^{2} F_{\varepsilon } }}{\sqrt K }$$
(14)

2.2.2 Dimensionless parameters

  1. (1)

    Reynolds number

    The Reynolds number (Re) used in this study is defined by:

    $${{Re}} = \frac{{u_{\infty } D}}{\nu }$$
    (15)

    where \({u}_{\infty }\) is the velocity of the fluid at the inlet and D is the diameter of the porous particles.

  2. (2)

    Darcy number

    The Darcy number (Da) can be evaluated from the Carman-Kozeny relation [21],

    $${{Da}} = \frac{K}{{D^{2} }} = \frac{1}{180}\frac{{\varepsilon^{3} d_{p}^{2} }}{{D^{2} (1 - \varepsilon )^{2} }}$$
    (16)

    here dp is the diameter of a particle in the porous aggregate.

  3. (3)

    Drag coefficient and lift coefficient

    The momentum exchange method [22] is applied to obtain the total fluid force \({{\varvec{F}}}_{T}\) acting on one porous particle:

    $${\varvec{F}}_{T} = \sum\limits_{{all{\varvec{x}}_{b} }} {\sum\limits_{\alpha \ne 0} {{\varvec{e}}_{\alpha } } } \left[ {m_{\alpha } ({\varvec{x}}_{b} ,t) + m_{{\overline{\alpha } }} ({\varvec{x}}_{b} + {\varvec{e}}_{{\overline{\alpha }}} \delta t,t)} \right] \times \left[ {1 - w\left( {{\varvec{x}}_{b} + {\varvec{e}}_{{\overline{\alpha }}} \delta t} \right)} \right]$$
    (17)

    where xb is the boundary node of the porous particle and \({m}_{\overline{\alpha }}\) is the density distribution function with \({\varvec{e}}_{{\overline{\alpha }}} \user2{ = }{\mathbf{ - }}{\varvec{e}}_{\alpha }\). There are two values of w(i,j): zero at fluid nodes and one at porous particle nodes.

    The drag coefficient (CD) and lift coefficient (CL) are obtained from the total fluid force.

    $$C_{D} = \frac{{F_{x} }}{{0.5\rho_{f} U_{{{\text{ref}}}}^{2} L_{{{\text{ref}}}} }}$$
    (18)
    $$C_{L} = \frac{{F_{y} }}{{0.5\rho_{f} U_{{{\text{ref}}}}^{2} L_{{{\text{ref}}}} }}$$
    (19)

    here Fx is x-component and Fy is y-component of the force FT. Uref and Lref are the characteristic velocity and characteristics length, respectively. The characteristic velocity we used here is the inlet velocity of the fluid and the characteristic length is the diameter of the porous particle.

  4. (4)

    Average drag coefficient and average lift coefficient

    In addition, the average drag coefficient (CDave) and average lift coefficient (CLave) can also be calculated by:

    $$C_{{{\text{Dave}}}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {C_{Di} }$$
    (20)
    $$C_{{{\text{Lave}}}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {C_{Li} }$$
    (21)

    where N is the quantity of all porous particles. CDi and CLi are the drag coefficient and lift coefficient of the i-th porous particle.

2.3 Computational domain

Steady flow around and through multiple porous particles was studied numerically. Uniform and random spatial distributions were considered in the paper and here we took uniform spatial distribution as an example to illustrate the computational domain as depicted in Fig. 2. The diameters of the particles were D. The quantity of the porous particles (N) was nine and the column (p) and row (q) of the particles are both three. The distance between the first column and the inlet and outlet were LF and LR and the distance between the particles in x direction and y direction were dx and dy. The particles were symmetrically distributed in y direction. Fluid flowed into the computational domain with a uniform velocity \(U_{\infty }\) and was fully developed at the outlet boundary. Besides, non-slip boundary condition was imposed on the top and bottom wall. For the convenience of analysis, the particles’ serial numbers were also shown in Fig. 2.

Fig. 2
figure 2

Computational domain and particles’ serial numbers

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2.4 Validation

Because the grids numbers can greatly affect the accuracy of the simulation results, several meshes were chosen to test the independence of the mesh. Fluid flow around nine (p = q = 3) particles with the same diameter and permeability as shown in Fig. 2 was simulated and the distance between the particles was 2D (dx = dy = 2D). The test Reynolds number and Darcy number were 20 and 10–3, respectively. The average drag coefficients and relative errors of the meshes are shown in Table 1. From the table, we can see the relative error between Mesh 1 and Mesh 2 was 0.22% and the relative error between Mesh 2 and Mesh 3 was 0.14%. Therefore, considering the computation resources and accuracy, Mesh 2 was chosen to carry out all simulations in this paper. Under this grid size, the particles’ diameter D expressed by the number of grids is 40.

Table 1 Test of the mesh independence at Re = 20, Da = 10–3 and dx = dy = 2D
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Because there is little research about multiple porous particles, the method was applied to simulate fluid flow around one particle to test the accuracy. Let p = q = 1 and there is only one particle in the computational domain. The porous particle was at the center of y-axis with distances of LF and LR to the inlet and outlet boundary. The comparison of drag coefficient at different values of Re and Da is depicted in Fig. 3. We can see that the simulated results were in good agreement with the results in the previously published literatures [2, 3, 6, 9, 10]. Therefore, it is concluded that the method can be used to simulate fluid flow around and through porous particles accurately.

Fig. 3
figure 3

The comparison of drag coefficient of one particle

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3 Results and discussions

In the present study, fluid flow around and through nine porous particles was investigated numerically within the following parameter ranges: 10 ≤ Re ≤ 40, 10–6 ≤ Da ≤ 10–2, D ≤ dx ≤ 4D, D ≤ dy ≤ 4D.

Several situations were considered and the effects of above parameters on the flow characteristics were analyzed in detail.

3.1 Particles with uniform spatial distribution

3.1.1 The streamlines analysis with dx = dy

Fluid flowing around nine (p = q = 3) identical (same diameter and permeability) porous particles was simulated to analyze the influences of Re, Da and the distances between the particles (dx and dy). Firstly, the particles were uniformly distributed and the distance between the particles in the x and y direction were the same (dx = dy).

Figure 4 shows the streamlines for the flow around and through nine identical porous particles at Re = 20 for dx = dy = D and dx = dy = 3D. It is observed that when Da = 10–6, there is little fluid penetrating through the particles and almost all fluid flow around the particles. In other words, the flow patterns of the porous particles at Da = 10–6 are similar to those of impermeable solid particles and the porous particles can be regarded as solid particles. When Da increases, the resistance of the particles to the fluid decreases and the fluid can flow through the permeable particles. Besides, the larger the Darcy number, the smaller the degree of fluid deviation in the particles.

Fig. 4
figure 4

The streamlines for the flow around and through nine identical porous particles at Re = 20 for a dx = dy = D and b dx = dy = 3D

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For dx = dy = D, the distance between the particles are small and the fluid is difficult to enter the gap between the particles. Besides, the particles form a cluster which has a great interference to the fluid and two recirculating symmetrical wakes are formed behind the cluster at different values of Da. When the distance increases, the particles are dispersed and can’t be regarded as a cluster. As depicted in Fig. 4b, when dx and dy increases to 3D, most fluid flows through the space between the particles with low resistance due to the larger inter-particle distance, especially when the Darcy number is low. This is because at high Darcy number, the resistance of porous particles to the fluid is small. And the distribution of the particles is dispersed which no longer exerts an overall influence on the fluid like the particle cluster. The flow characteristics of the fluid around each particle are different. At Da = 10–6, two wakes are formed behind the particles at p = 1. For #4, because it is situated at the center of y-axis, two symmetrical wakes are formed. But for #1 and #7, two asymmetrical wakes are formed behind them. Furthermore, the particles at p = 1 have significant effects on the particles at p = 2 and p = 3. Similarly, #4 is situated at the symmetric axis in y direction and when fluid flow around it, the fluid keep symmetrical. As a result, two symmetrical wakes are also formed behind #5 and #6. Compared with #4, #1 and #7 cause a larger fluid deflection when fluid flow around them and therefore, no wake is formed behind #2, #3, #8 and #9. When Da increases, the fluid can penetrate into the particles and the interference to the fluid is reduced. At Da = 10–2, the wakes behind all particles vanish totally.

In order to better understand the influences of Reynolds number, the streamlines at Re \(=\) 40 and dx = dy = D are plotted as shown in Fig. 5. Compared with Fig. 4a, it is observed as Re increases, the symmetrical wakes behind the cluster increase obviously due to the increase in inertial force and the phenomenon can also been seen in the situation where fluid flow around one porous particle.

Fig. 5
figure 5

The streamlines for the flow around and through nine identical porous particles at Re = 40 and dx = dy = D

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3.1.2 The average drag coefficient analysis with dx = dy

Figure 6 shows the average drag coefficient of the nine identical porous particles at Re = 20 and 40 for different values of dx(dy). It is observed that the average drag coefficient increases with dx(dy) increases at Re = 20. Because as the distance between the particles increases, the shading effect of the front particles on the rear particles reduces. Besides, we can also see that when dx(dy) increases, the growth rate of CDave slows down which indicates the influences of dx(dy) gradually decrease. It is expected when the distance increases to a certain value, the average drag coefficient will reach a maximum value and then remains unchanged. At Re = 40, CDave increases for D ≤ dx(dy) ≤ 2D and almost keeps constant for 2D ≤ dx(dy) ≤ 4D. That indicates the effects of dx(dy) decrease as the inertial force increases. Furthermore, under the same Da and dx (dy), CDave is in a reduction with Re increasing.

Fig. 6
figure 6

The average drag coefficient for the flow around and through nine identical porous particles at Re = 20 and 40 for different values of dx (dy)

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When Darcy number increases, the resistance of the particles on the fluid reduces, and therefore, the average drag coefficient decreases for 2D ≤ dx(dy) ≤ 4D. At dx = dy = D, like the analysis in Sect. 3.1, the particles form a cluster and the influences of multiple particles on the fluid are presented as the overall effect of the cluster on the fluid. Therefore, for 10–6 ≤ Da ≤ 10–3, the average drag coefficient of the particles almost remains unchanged with Da increasing and only at Da = 10–2, CDave increases a little. However, compared with the increase resulted from Darcy number at other dx(dy), this increase can be ignored.

At Re = 20, the average drag coefficient shows different tendency to Da when dx = dy = D and dx = dy = 3D. To better understand the phenomenon, Fig. 7 shows the drag coefficient of each particle at dx = dy = D and dx = dy = 3D. i is the serial number of the particles as shown in Fig. 2.

Fig. 7
figure 7

The drag coefficient of the nine identical porous particles at Re = 20 for a dx = dy = D and b dx = dy = 3D

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For dx = dy = D, it can be seen that the drag coefficient of each particle at Da = 10–2 is basically the same as Da = 10–4 and for dx = dy = 3D, the drag coefficient of all particles at Da = 10–4 are evidently larger than those at Da = 10–2. As a result, at dx = dy = 3D, the average drag coefficient at Da = 10–2 is significantly smaller than that at Da = 10–4. In addition, CD of the particles decreases as p increases. Comparing Fig. 7a with b, when dx (dy) increases to 3D, CD of particles at p = 1 decreases and CD at p = 2 increases. It is noticed that positive or negative of the drag coefficient means that the direction of the force is different. Therefore, for p = 3, the force exerted on the particles changes in direction and the magnitude decrease.

Because the particles are uniformly distributed and symmetric about the y axis, the average lift coefficient of the particles equals zero. But for the lift coefficient of every particle, not all equals zero. Figure 8 indicates the lift coefficient of the nine identical porous particles at Re = 20 for dx = dy = D and dx = dy = 3D. The particles at q = 2 is at the symmetric axis in y direction, the lift coefficients equal zero. At dx = dy = D, the lift coefficients of #1 and #7 for Da = 10–2 is a little larger than those for Da = 10–4. CL of #2, #3, #8 and #9 at Da = 10–2 is the same as that at Da = 10–4. In addition, the lift forces exerted on particles at the q = 1 and q = 3 are opposite. At q = 1, the direction of the lift force is downward. Therefore, the cluster has the tendency to gather together more. When dx (dy) increases to 3D, CL of the particles at q = 1 and q = 3 for Da = 10–2 is smaller than those for Da = 10–2. Furthermore, the lift forces exerted on particles at the q = 1 and q = 3 are also opposite, but the direction of the lift force is upward at q = 1, which indicates the particles tend to disperse.

Fig. 8
figure 8

The lift coefficient of the nine identical porous particles at Re = 20 for a dx = dy = D and b dx = dy = 3D

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3.1.3 The streamlines analysis with dx ≠ dy

In order to discuss the influences of the distance between the particles in x and y direction, following we consider the case of dx ≠ dy. The range of dx(dy) is from 0.25 to 4.0. When dx = D, dx(dy) varies from 0.25 to 1.0; when dy = D, dx(dy) varies from 1.0 to 4.0. Figure 9 depicts the streamlines of nine identical porous particles at Re = 20 and Da = 10–3 for dx(dy) = 0.5 and dx(dy) = 2.0. Compared with Fig. 4a, it is observed under the same Re and Da, when dy increases to 2D (Fig. 9a), the wakes behind the cluster vanish but wakes appear behind the particles of p = 3. Most fluid flow through the gap between the particles in y direction. When dx increases to 2D (Fig. 9b), the flow field resembles to Fig. 4a to some degree. And at the gap between p = 2 and p = 3, there exist two vortices. Besides, area covered by the wakes extends to the inside of the particles.

Fig. 9
figure 9

The streamlines of nine identical porous particles at Re = 20 and Da = 10–3 for a dx(dy) = 0.5 and b dx(dy) = 2.0

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3.1.4 The average drag coefficient analysis with dx ≠ dy

At the same time, the average drag coefficient for different values of Da and dx(dy) is shown in Fig. 10. As dx(dy) increases, the average drag coefficients increase for 0.25 ≤ dx(dy) ≤ 0.33; then it drop dramatically and reach a minimum when dx(dy) increases to 1.0; for 1.0 ≤ dx(dy) ≤ 4.0, increase slightly. That indicates the distance change in the y direction has more significant impact on the average drag coefficient. For instance, for Da = 10–4 and Re = 20, at dx = D, when dy increases from 2D to 3D, CDave increases by 5.79%; at dy = D, when dx increases from 2D to 3D, CDave increases by 2.61%. When dx(dy) increases to 1.0, the flow pattern changes dramatically and the average drag coefficient drop greatly. Under the same dx(dy), with Da increasing, CDave decrease for 0.25 ≤ dx(dy)≤ 0.5; and increase for 1.0 ≤ dx(dy) ≤ 4.0.

Fig. 10
figure 10

The average drag coefficient of nine identical porous particles at Re = 20 for different values of Da and dx(dy)

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3.2 Particles with random distribution within a certain region

3.2.1 The particles with the same Darcy number and diameter

In reality, particles are rarely uniformly distributed as discussed above, so following we will discuss the situation that particles are randomly distributed. Firstly, fluid flow around and through nine randomly distributed porous particles with the same diameter and Darcy number was considered. In order to compare with the results of the uniform distribution (dx = dy = 4D), the distribution of particles is restricted within a certain range. Figure 11 shows two kinds of streamlines under random spatial distributions. It can be seen that despite under the same Darcy number and Reynolds number, the distribution of porous particles is quite different. Therefore, the characteristics of the flow pattern are also completely different: for Fig. 11a, there is a large vortex at a certain distance behind the porous particles; but for Fig. 11b, there are only two small vortices behind the two porous particles.

Fig. 11
figure 11

The streamlines of nine identical porous particles with random distribution at Re = 20 and Da = 10–3

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First of all, the particles have a specific random distribution (Fig. 11a) and we analyze the relationship between the average drag/lift coefficient and Darcy number. The average drag coefficient and average lift coefficient of nine identical porous particles with random distribution and uniform distribution (dx = dy = 4D) at Re = 20 for different values of Da are depicted in Fig. 12. It is observed that for a specific random distribution, when Da increases, the average drag coefficient decreases which is the same as the uniform distribution. CDave of the random distribution is larger than that of the uniform distribution and with Da increasing, the difference between them becomes larger. Because the particles’ distribution is irregular, the average lift coefficients don’t equal zero. As indicated in Fig. 12b, for the specific random distribution, the direction of the total lift force exerted on the nine particles is downward and CLave decreases with the increase of the Da.

Fig. 12
figure 12

The a average drag coefficient and b average lift coefficient of nine identical porous particles for a specific random distribution and uniform distribution at Re = 20

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Figure 13 shows the average drag coefficient and average lift coefficient of the particles for random distributions. Four random distribution simulations are performed for each Darcy number (Da = 10–4, 10–3 and 10–2). It is observed that the average drag and lift coefficient change irregularly when Da increases and although under the same Da, the four random simulations of CDave and CLave are very different. Because the randomly distributed particles cause irregular flow field characteristics shown in Fig. 11.

Fig. 13
figure 13

The a average drag coefficient and b average lift coefficient of nine identical porous particles for four times random distributions at Re = 20

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3.2.2 The particles with different Darcy numbers or diameters

In the section, fluid flow around nine randomly distributed particles with random Da or D was investigated. The fluid has the same inlet velocity as discussed above. The particles’ diameter varied from 30 to 50 and the Darcy number varies from 10–6 to 10–2. Figure 14 shows the streamlines for a specific random distribution. The porous particles have the same D and Da (D = 40, Da = 10–3) in Fig. 14a, the same Da (Da = 10–3) and different diameters in Fig. 14b and different Da and different diameters in Fig. 14c. It is also noted that the diameters of porous particles in Fig. 14c are the same as those in Fig. 14b. Comparing Fig. 14a, b with c, when the diameters or Darcy number of porous particles change, the overall flow pattern is similar-one vortex is formed at a certain distance behind the particles. However, the position of the vortex has changed. Comparing Fig. 14a with b, when the diameters change, the vortex moves to the upper left. Comparing Fig. 14b with c, when the Darcy numbers change, the vortex moves to the upper left further.

Fig. 14
figure 14

Fluid flow around and through nine porous particles for a specific random distribution under different particles conditions

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The average drag and lift coefficient of nine particles for a specific random distribution with different D is depicted in Fig. 15. In order to compare with the results of particles with the same Da and D, CDave and CLave of the particles with the same Da and D are also shown in Fig. 15. It is observed that despite the particles’ diameter are different, for a specific random distribution, the CDave and CLave were in a reduction when Da varies from 10–4 to 10–2 which is the same as the simulation results of the particles of the same Da and D. However, under the same Da, the CDave and CLave of the particles with the same Da and different D is larger than those of the particles with the same Da and D.

Fig. 15
figure 15

The a average drag coefficient and b average lift coefficient of nine particles for a specific random distribution with different D

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3.3 Particles with random distribution within the whole computation region

In Sect. 3.2, in order to compare with the results of the uniform distribution, the particles which are randomly distributed in a certain region were studied, and in Sect. 3.3, we expand the distribution range to the whole computation region except the wall boundary to avoid the wall effect.

One simulated flow field is shown as Fig. 16. The particles’ diameter and Darcy number are different. It is observed that when the fluid flow through the porous particles, the velocity of the fluid decreases. When particles are close, the behind particle are severely influenced by the low velocity resulting from the front particle. And when the distance between the particles becomes larger, the velocity of the fluid passing through the front particles gradually recovers as it flows to the behind particles.

Fig. 16
figure 16

The flow field of nine particles with different D and Da for a larger random distribution

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Figure 17 shows the average drag and lift coefficient for several times random distributions. It can be seen for the seven times random distribution, although every time the particles’ diameter and Darcy number are different, there is not much difference between the average drag coefficient. But the average lift coefficient. changes greatly. This is because when the region where the particles are randomly distributed is large enough, the porous particles can be completely dispersed in the computational domain. The random spatial distribution of the particles and the random diameter and Darcy number generated by the computer system may have a little effect on the drag coefficient. However, since the distribution of particles is not symmetrical about the y-axis, it will inevitably lead to a large difference in the lift coefficient.

Fig. 17
figure 17

The a average drag coefficient and b average lift coefficient of the particle with different D and Da for a larger random distribution

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4 Conclusions

Two-dimensional steady flow around and through nine porous particles was investigated numerically. Uniform and random spatial distribution were both considered and the influences of Reynolds number, Darcy number and the distance between particles on the flow characteristics were analyzed in detail. Some important conclusions are summarized as follows:

  1. (1)

    When dx = dy = D, the particles form a cluster which exerts an overall influence on the fluid, and therefore, two recirculating symmetrical wakes are formed behind the cluster. When dx (dy) increases, most fluid flows through the space between the particles with low resistance and this phenomenon is more obvious at low Da.

  2. (2)

    The average drag coefficient increases as dx(dy) increases, but the growth rate gradually slows down which indicates the influences of dx(dy) decrease.

  3. (3)

    The distance change in y direction has more significant impact on the average drag coefficient. For instance, for Da = 10–4 and Re = 20, at dx = D, when dy increases from 2D to 3D, CDave increases by 5.79%; at dy = D, when dx increases from 2D to 3D, CDave increases by 2.61%.

  4. (4)

    The flow field and drag coefficient are closely related to the distribution of particles. For a specific random distribution, the change of average drag coefficient with Da is similar to the uniform spatial distribution. When the region where the particles are randomly distributed is large enough, the random distribution may have little effect on the drag coefficient.