The fluid flow in the sink is governed by the principles of mass conservation, momentum conservation, and energy conservation. Given the negligible impact of heat exchange and temperature changes on the water tank, and considering the fluid medium as incompressible, it is sufficient to focus on the simplified mass conservation (Eq. (1)) and momentum conservation (Eq. (2)):(1) ∂ui∂xi=0, (2) ∂(ρui)∂t+∂(ρuiuj)∂xj=∂τij∂xj−6πμa(u−up), where t is time, ρ is the fluid density, u is the fluid velocity, up is the particle velocity, a is the particle radius, and μ is the fluid dynamic viscosity. The variable τ is the total stress tensor, which is expressed as:(3) τij=−pδij+μ(∂ui∂xj+∂uj∂xi), where p is the fluid pressure.

To better analyze the forces acting on particles in a cylindrical flow, the forces can be categorized based on their distinct modes of action into three groups: forces unrelated to the relative motion between the fluid and particles (e.g., gravity, inertia force, and pressure gradient); forces caused by the relative motion between particles and fluids in the direction of particle motion velocity (e.g., drag force, virtual mass force, and Basset force); and forces arising from the relative motion between particles and fluids, but acting perpendicular to the relative motion direction (e.g., Magnus forces and Safhmn forces).

Due to the small velocity gradient in the mainstream region of the flow field and the particle size less than 5 mm, the virtual mass force, Safhmn lift, and Magnus lift forces acting on solid particles are excluded from consideration in this simulation. It is assumed that particles primarily experience gravity, inertial forces, pressure gradient forces, drag, and Basset forces, as well as contact forces between particles. The resulting motion equation for solid particles is:(4) mpd2xpd2t=Fc+Fp+Fg+Fin+Fr+FB, where mp is the mass of the particle and xp is the translational displacement. Fc ,  Fp , Fg , Fin , Fr , and  FB represent contact force, pressure gradient force, gravity, inertial forces, drag, and Basset forces, respectively.

Unlike the rigid particle two-phase flow model, the flexible impurity two-phase flow model comprises multiple particle models arranged in a row, with contact between particles. Therefore, it is necessary to consider the contact force between particles, primarily consisting of normal and tangential contact forces (Fig. 2a). The following section provides mainly a detailed introduction to the contact force between particles.

The Hertz-Mindlin non-slip model can be used to calculate the particle contact force. This contact model is a variant of the nonlinear spring-damper contact model [17], as shown in Fig. 2a. The contact force is determined as follows:

(5) Fc=Fn+Ft, where Fn denotes the normal contact force, and the normal contact is replaced by the combined action of a spring and damper. As depicted in Fig. 2a, the spring reflects the stiffness of the material in the normal direction (Young’s modulus), while the damper represents the normal recovery. Thus, the normal contact force Fn is defined as:

(6) Fn=−Kndn−Nnυn, where Kn is the normal elastic coefficient, dn represents the normal displacement between particles, Nn is the normal damping coefficient, and vn denotes the normal relative velocity between particles.

The tangential contact force Ft  is replaced by the combined action of a spring, damper, and slider. As shown in Fig. 2a, the spring reflects the stiffness of the material in the tangential direction (Young’s modulus), while the damper represents the tangential recovery and the slider represents the friction. The normal contact force Ft is defined as follows:

(7) Ft=−Ktdt−Ntυt, where Kt represents the tangential elastic coefficient, dt is the tangential displacement between particles, Nt is the tangential damping coefficient, and υt indicates the tangential relative velocity between particles.

The particles are connected by massless bonds, as illustrated in Fig. 2b.

The soft sphere model simplifies the normal force between particles into springs and dampers, and the tangential force into springs, dampers, and sliders. Accordingly, its application allows for the creation of more intricate particle models, including shapes like bending and folding. In this simulation, the particle cluster model is employed to group particles and transform them into complex shapes, as shown in Fig. 3. Displacement coordinates are recorded at points located at the top, middle, and bottom of the model.

The particles constituting the rag have a radius of 1 mm, and the contact radius is 1.2 mm. The size of the rag used in the simulation is 100 mm × 100 mm, resulting in a configuration with 50 particles × 50 particles. The density of spherical particles is 2500 kg/m³, Poisson’s ratio is 0.25, and the shear modulus is 1 × 10⁸ Pa.

Interactions between flexible 2D rags and fixed 3D cylindrical obstacles, ignoring gravity, are taken into account in this simulation. The rag is transported in a rectangular channel flow and initially located perpendicular to the flow direction. The initial shape of the sink is shown in Fig. 6. The size of the sink is 0.45 m × 0.2 m × 0.2 m, the diameter of the cylinder is 0.05 m, the distance from the center of the cylinder to the upstream area is 0.2 m, the distance to the downstream area is 0.2 m, the distance to the lower wall is 0.08 m, and the distance to the upper wall is 0.12 m.

The computational fluid domain uses hexahedral structured mesh. Grid independence verification was conducted on the model to reduce the influence of the number of grids on the numerical solution. Table 2 shows the variations in drag coefficient (C_d) and lift coefficient (C_l) with the number of grids. The drag coefficient reflects the degree of obstruction of the fluid to the motion of an object, while the lift coefficient reflects the vertical upward force generated by the fluid on the object. The hydrodynamic effects on a cylinder can be evaluated through these two coefficients. In this case, increasing the number of grids beyond 1001286 has a negligible imapact on the calculation results. Therefore, the chosen grid count is 1001286. The grid division is depicted in Fig. 7.

The trajectory of a rag in a water tank and the influence of different factors on its trafficability was simulated with the CFD-DEM method. Table 1 outlines seven schemes established for this analysis, each with distinct conditions. In FLUENT, the velocity inlet and pressure outlet were selected as the boundary conditions for calculation. The SIMPLE method in pressure-velocity coupling was chosen as the numerical calculation scheme. The structure is relatively simple, so the relevant requirements were satisfied by setting the momentum, turbulent kinetic energy, and turbulent dissipation rate as first-order upwind. All walls were set as non-slip boundaries.

Real-time data transmission is essential for accurate coupling calculation results. To satisfy this requirement, FLUENT was operated with transient calculation processes. The time step in FLUENT, in accordance with the necessary integer multiple of 10–100 with the time step in EDEM, was set to 1.0 × 10⁻⁴ s with an EDEM time step of 5.0 × 10⁻⁶ s. The Reynolds numbers were 75000, 100000, and 125000 for groups with velocities of 1.5, 2, and 2.5 m/s, respectively.

The trajectory of Group 1 (2 m/s, 90°, 100 mm) was analyzed to begin the analysis. The evolution at six consecutive instants is illustrated in Fig. 8. Initially, before contact with the cylinder, the rag moves linearly in the fluid. Upon contact, the middle part of the rag halts on the cylinder’s surface while its ends continue to move in the fluid direction. Subsequently, propelled by the upper end, the rag ascends along the cylinder as a whole. Finally, after passing over the cylinder, the rag continues to move downstream in the channel propelled by the fluid.

The time point at 0.09 s was isolated for a closer examination of pressure and velocity distributions during rag-cylinder contact. The velocity and pressure distributions at 0.09 s are shown in Figs. 9 and 10, respectively. As shown in Fig. 9, the fluid velocity is lower on the left side of the cylinder due to contact with the rag, creating smaller vortices on the right side. Velocity gradually increases with distance. The distance between the upper and lower regions of the cylinder influences fluid velocity; as it decreases, the fluid velocity in these regions increases. Simultaneously, the velocity in the upper region of the rag surpasses that in the lower region, propelling the rag forward. As illustrated in Fig. 10, the pressure is mainly concentrated in the contact area between the rag and the cylinder. Negative pressure forms in the upper and lower areas of the cylinder, with lower pressure in the lower area compared to the upper area.

The displacement of three points (marked A, B, and C) on the edge of the rag (Fig. 3) is shown in Fig. 11. The collision process can be divided into four consecutive stages [19].

(1) In the first stage, the rag is displaced before colliding with the cylinder. The trajectory remains essentially linear from the release point.

(2) In the second stage, collision occurs. The middle point of the rag lags behind the stagnation point, while its tips continue translating along the flow direction as the rag wraps around the obstacle.

(3) In the third stage, the rag slides slowly on the upper part of the cylinder. The lower point (C) is pulled up while the other two points move slowly downstream.

(4) In the final stage, the rag is released behind the cylindrical obstacle.

The simulation results obtained in this study align with the four stages of rag movement summarized in the literature, validating the feasibility of the proposed method.

Table 3 shows three sets of experiments with the same angle and entry position, namely, Group 1 (2 m/s inlet velocity), Group 2 (1.5 m/s inlet velocity), and Group 3 (2.5 m/s inlet velocity), with the same angle and entry position, are shown in Fig. 12. The overall motion trends of the three experimental groups are similar, as are the relative changes in the displacement of the three points corresponding to each group. The motion process in Group 2 and Group 3 conforms to the four stages of rag collision mentioned above. However, speed has a marked effect on the rag’s movement. Although higher velocities result in faster overall movement and larger corresponding values, distinctions persist among the different groups. Prior to 0.18 s, the value of Point A in Group 1 exceeds that of Point C, signifying normal movement. However, after that time point, the value of Point A falls below that of Point C, indicating the flipping of the rag. Coincidentally, Group 3 undergoes a similar transition and the rag flips at 0.14 s, while no such phenomenon was observed in Group 2.

Fig. 13 presents three corresponding streamline maps of fluid velocity, with similar overall distributions. The fluid velocity in the frontal and rear regions of the cylinder is low, whereas the fluid velocity in the upper and lower regions of the cylinder is relatively high. The maximum velocity roughly 1.5 times the inlet velocity, indicating a similar velocity variation. Furthermore, as speed increases, the vortices produced by the fluid behind the cylinder become more pronounced. Additionally, the maximum and minimum fluid velocity values intensify as the inlet velocity escalates.

Fig. 13 depicts the relationship between pressure distribution and velocity for three fluid sets. The various colors in the pressure contour map indicate that pressure fluctuations become more pronounced as the entry velocity rises. The fluid pressure on the left side of the cylinder increases, while the negative pressure in the upper and lower areas of the fluid diminishes. Simultaneously, the area of negative pressure expands, and the overall distribution remains analogous. As the speed increases, the maximum and minimum fluid pressure values also increase. Furthermore, due to the presence of the rag, the fluid pressure at the upper end of the cylinder surpasses the pressure at the lower end.

Fig. 14 shows the particle velocity distribution at 0.09 s. The position of the moving rag and the velocity of each area are relatively large at the upper end. In effect, higher speed is conducive to the rag’s passage through the cylinder.

Table 4 shows three sets of experiments with the same speed and entry position. The displacements of three points located on the edge of the rag in Groups 1, 2, and 3 at various time intervals are illustrated in Fig. 15. The entry inclinations for these three groups, relative to the horizontal direction, are 90°, 60°, and 45°, respectively. Three entry angles of the rag were examined (45°, 60°, and 90°) while keeping the entry speed and position constant. Examining Fig. 15 reveals a consistent overall motion trend among the three groups, with similar relative changes in the displacement of the three points corresponding to each group. Groups 4 and 5 also adhere to the four stages of the collision process mentioned above.

Due to the angle inclination, the upper initial positions of the three groups are roughly at the same distance, while the middle and bottom positions of the 60° and 45° groups are farther behind. However, in the subsequent movement, the two groups increase steadily while the 90° group demonstrates a relatively flat area and a reversal. The last 45° group existed the tank promptly, but there is no substantial overall gap between the last three groups. The group that entered at 45° bends after passing through the cylinder. At 0.19 s, Point C at the bottom exceeds Point B at the middle, but when leaving the channel, the bottom (Point C) does not exceed the top (Point A). The 60° group also bends after passing through the cylinder. At 0.18 s, the bottom (Point C) exceeds the middle (Point B), and the bottom (Point C) surpasses the top (Point A) at 0.2 s. The 90° group completes a rag flip in 0.19 s, with the bottom (Point C) in the front and the top (Point A) at the back.

Fig. 16 shows velocity streamline diagrams of the three corresponding fluid groups, revealing a striking similarity in their overall distributions. The fluctuation of fluid velocity from the inlet to the left side of the cylinder is relatively small, approximately 2 m/s. As the fluid flows through the cylinder, its velocity decreases while increasing in the upper and lower regions. Notably, due to the smaller distance between the lower end of the cylinder and the channel boundary compared to the upper end, the fluid velocity on the lower side of the cylinder exceeds that on the upper side, leading to the formation of vortices of similar size on the right side of the cylinder. As these vortices gradually move away from the cylinder, velocity increases as both the maximum and minimum values exhibit similar distributions. This suggests that changing the angle does not significantly impact velocity, with the three groups experiencing similar changes. The overall magnitude and range of velocity changes are essentially identical, confirming the limited impact of angle on velocity.

The distributions of the three pressure groups are illustrated in Fig. 16. The pressure changes and distributions of the three groups appear to be roughly the same with varying angles. The pressure on the left side of the cylinder is higher than the pressure on the right side, and the pressure in the contact area between the fluid and the left side of the cylinder is the highest. Clear boundaries and similar positions are present in all three groups, with relatively low pressure in the upper and lower areas. Moreover, the maximum and minimum pressure values are similar across these groups. However, as the angle increases, the area where the fluid pressure on the lower side of the cylinder is less than −2000 Pa gradually expands.

From Fig. 17, it can be seen that these three groups exhibit similar positions at 0.15 s, with higher breaking speeds in the area near the upper boundary of the channel. The maximum velocity values are also comparable, all measuring 1 m/s. All three groups exhibit a trend of the rag bending. Based on these results, the incidence angle has a minimal impact on the rag’s trafficability. However, a 45° inclination is advantageous for the rag to pass through the cylinder.

Table 5 shows three sets of experiments with the same angle and speed. Fig. 18 depicts the displacement over time for Groups 1, 2, and 3 at different entry positions. The velocities and entry angles of the three experiments are identical, with distances of 90, 100, and 110 mm from the entrance boundary of the channel, respectively. From Fig. 18, it is evident that the motion trends of the three groups of experiments are similar, and the relative displacement changes for the three points corresponding to each group are similar as well. The motion process of Groups 6 and 7 conforms to the four stages of the collision process described above.

Prior to the collision with the cylinder at 0.08 s, the slopes of the three groups are essentially the same. The numerical differences at the corresponding points at the same time mirror those at the initial position. This indicates that the motions are comparable across the three groups of experiments. After 0.08 s, the positions of the three lower points (C) decrease, mainly due to rebounding of the lower area of the model after the collision, causing the position to move backward. After 0.12 s, the rag begins to leave the cylinder and resumes rightward movement, where the values began to increase. Subsequently, all three sets of rags begin to bend and fold after leaving the cylinder until flipping at 0.19 s. The displacement differences in the corresponding points are close to the initial differences for all three experimental groups.

The velocity streamlines of three sets of experiments with similar overall distributions are shown in Fig. 19. From the entrance to the left side of the cylinder, the fluid velocity fluctuates slightly at around 2 m/s. In the contact area between the fluid and the left side of the cylinder, the fluid velocity decreases while the velocity in the upper and lower areas increases. The maximum velocity of the three groups of experiments is also similar, measuring 3.5 m/s. As the fluid moves away from the cylinder, the velocity gradually decreases to 2 m/s, while the velocity distributions of the three groups are nearly identical. Therefore, changes in distance from the entrance boundary of the channel when the model enters have a negligible impact on speed. However, as the distance from the entrance boundary of the channel increases, the vortex on the right side of the cylinder gradually intensifies.

Pressure cloud maps of three groups of experiments are also provided in Fig. 19, where the images reveal close similarities in the distributions of pressure among the three groups. Pressure is consistently higher on the left side of the cylinder and lower on the right side. The pressure in the upper and lower regions is also lower than the mid-section. The maximum and minimum pressure values are similar, and the pressure distributions are very close, indicating that changes in distance from the cylinder when the model enters have a negligible impact on pressure.

Fig. 20 displays a rag velocity cloud map as the rag moves in the channel at various entry distances for 0.09 s. The image reveals that the rag has a more forward-facing trajectory and higher top velocity when it enters the cylinder farther away from the inlet boundary.