Comparative Study of Particle-Resolved and Point-Particle Simulations of Particle–Bubble Collisions in Homogeneous Isotropic Turbulence

Abstract

Bubble–particle collisions in turbulent flows are fundamental to flotation processes, yet their complex dynamics remain challenging to characterize accurately. In this study, a comparison study of a particle–bubble collision system in homogeneous isotropic turbulence was performed using the particle-resolved method and point-particle method. Direct numerical simulations of turbulent flows were achieved using the lattice Boltzmann method (LBM). The effects of hydrodynamics on the collision particles were compared between Lagrangian tracking and directly resolving the disturbance flows around finite-size solid particles using an interpolated bounce-back scheme. The differences between point-particle and particle-resolved simulations are evaluated, highlighting their respective strengths and limitations. These findings enhance the understanding of turbulence-driven bubble–particle interactions and provide guidance for improving the accuracy of flotation modeling and process optimization.

1. Introduction

Flotation is a separation process in which target particles are collected by bubbles in a liquid-phase flow field, and it has wide applications in industrial, pharmaceutical, and other processes. Taking typical mineral processing as an example, ores are usually ground finer to increase the yield of minerals. Various flotation devices have been developed to meet different needs, and one of their design objectives is to provide the necessary bubbles and an appropriate turbulent environment to achieve optimal flotation results [1]. Obviously, in-depth research and understanding of the flotation process mechanism, especially the particle bubble collision process, is an important prerequisite for developing efficient processing equipment.

In the literature, most of the studies on collisions among dispersed phases are about droplets in the turbulent cloud [2,3], solid particles in turbulent air or liquid [4,5], and particle–bubble collisions in flotation [6,7]. The turbulence within flotation machines should be optimized to enhance flotation performance by maximizing particle–bubble collisions and attachments while simultaneously minimizing particle detachment [8]. Abrahamson [4] proposed a new collision rate model for the problem of particle collisions under high-intensity turbulent conditions in industrial applications and provided the critical conditions for particle size in these conditions. Schubert [1] analyzed the fluid dynamics of bubble particle flow in the flotation process controlled by micro turbulence, revised the collision rate model of Abrahamson, and established a quantitative relationship between turbulence intensity and collision rate, providing a theoretical–experimental pattern for particle collision dynamics. The underestimation of the turbulent dissipation rate was systematically analyzed as a function of spatial resolution relative to the Kolmogorov length scale [9]. This emphasizes the critical importance of accurately capturing turbulence effects on particle–bubble interactions, which is essential for enhancing the predictive accuracy of flotation models and optimizing process efficiency.

The essence of collisions between two different species of particles in turbulent flow lies in the transport mechanism and the accumulation effects, which are represented by the radial distribution function at contact and the radial relative velocity between particles, respectively [10]. The relative velocity between colliding particles is primarily influenced by large-scale energetic eddies, whereas small-scale dissipative eddies, which contribute to the accumulation effect, serve to increase the collision rate [11]. The phenomenon of particle accumulation, also referred to as the preferential concentration of particles, was initially observed by Maxey [12,13], who found that inertial particles tend to concentrate in regions characterized by low vorticity and high strain rates. Sundaram and Collins [14] further reported that this preferential effect could enhance the collision frequency by one to two orders of magnitude.

Due to the limitations of the experimental study of particle–bubble collision in a turbulent flow field, researchers have used computational fluid dynamics for detailed modeling of particle–bubble collisions [7]. A comprehensive literature review was conducted to summarize previous studies, with a particular focus on an in-depth analysis of the impact of hydrodynamics on the flotation process [7]. Studies employing direct numerical simulation (DNS) to explore the effects of turbulence on particle–bubble interactions are relatively scarce. Fayed and Ragab [15] employed DNS with the point-particle method to investigate the particle–bubble collision kernel in homogeneous isotropic turbulence. Their study revealed preferential concentrations of both particles and bubbles within the turbulent flow. The segregated behavior of these phases resulted in a reduced collision frequency between particles and bubbles. Wang et al. [16] compared the analytical model with CFD simulations and found that the analytical model tended to overestimate the collision efficiency while underestimating the 90% collision radius. Chan et al. [17] investigated bubble–particle collisions in turbulent flow using a point-particle approach with turbulence fully resolved through DNS. Their findings indicate that the influence of finite particle density, nonlinear drag, and lift force on the bubble–particle collision rate is relatively limited. DNS with a particle-resolved method has appeared to be a useful tool to study particle-laden turbulent flows [18]. In particle-resolved DNS, the particle–fluid interaction force is directly calculated by integrating the fluid stress along the particle boundaries. Additionally, turbulent structures are fully resolved across all scales.

This limitation highlights the need for further research using DNS to gain a deeper understanding of the intricate dynamics between turbulence and particle–bubble behavior, which could significantly improve the optimization of flotation processes. In the current work, particle–bubble collisions in a homogeneous isotropic turbulence are studied using point-particle and particle-resolved direct numerical simulations. A statistical analysis of particle–bubble collisions in forced turbulent flows is conducted to investigate the impact of turbulence on collision dynamics and to explore the underlying mechanisms governing these interactions in a turbulent environment.

2. Materials and Methods

The solid particles and bubbles are treated using both point-particle and particle-resolved methods, which have been reported in our previous work [19,20,21]. These two methods are only described briefly in this paper.

Due to limited computing resources, it is difficult for DNS to directly simulate scales higher than centimeters. However, for flow fields far from device walls, their typical characteristic is three-dimensional homogeneous isotropic turbulence, which provides simplified conditions for DNS. The boundary condition used in the simulation is “periodic boundary in all three directions”. After the solid particles settle out of the lower boundary of the calculation domain, they re-enter the upper boundary at the same speed. All simulations in this paper were conducted under such conditions.

2.1. Point-Particle Method

A simplified and modified form of the motion equation derived by Maxey and Riley [22] is applied to solve the motion of particles:

$${\displaystyle \frac{d{V}_i}{dt}}={\displaystyle \frac{\left(U_i+u_i{-V}_i\right)f\left({Re}_{\alpha }\right)}{{\tau }_{\alpha }}}+\beta {\displaystyle \frac{D{U}_i}{Dt}}+\left(1-\beta \right)g_i$$

(1)

where $V_i$ is the particle velocity (m/s), $U_i$ is the fluid velocity at the location of the particles (m/s) and is determined numerically from the values of its neighboring grids using a six-point Lagrange interpolation scheme, $u_i$ is added to embed the hydrodynamic interactions (HDI) among particles and represents the disturbance flow velocities from surrounding particles (m/s), $g_i$ is the gravity force (m/s²), $\beta =3{\rho }_f/\left({\rho }_f+2{\rho }_{\mathsf{\alpha }}\right)$ with ${\rho }_f$ is the fluid density, and ${\rho }_{\mathsf{\alpha }}$ is the density of solid particles or bubbles (g/cm³).

When conducting numerical simulations, two sets of unit systems are involved, namely physical scenarios and numerical systems.

Table 1. Basic parameter settings and corresponding relationships in point-particle method.

Parameter	Physics	DNS
Domain size	6 mm	256
Fluid density	1.0000 g/cm3	1
Fluid kinematic viscosity	0.01 cm2/s	0.0058
Time step	$3.2\mathsf{\mu }$s	1

shows the conversion between these two systems.

For point-particle simulation, the discrete phase parameters are solid particle density of 5.0000 g/cm³, diameter of 60 $\mathsf{\mu }$m, number of 190,986, volume fraction of 10%; bubble density of 0.001 g/cm³, diameter of 200 $\mathsf{\mu }$m, number of 5157, volume fraction of 10%. The fluid field parameters are selected as turbulent kinetic energy dissipation rate, which, on the one hand, can well reflect the changes in turbulence intensity, and on the other hand, it is easy to directly calculate from the turbulent flow field.

2.2. Particle-Resolved Method

In this study, an interpolated bounce-back scheme designed by Lallemand and Luo [23] was adopted to treat the interaction of the fluid with moving solid surfaces as the default scheme for the no-slip boundary condition. The translational and rotational velocities of a single particle within a suspension are based on Newton’s equation of motion. The particle translational velocity V, position Y, angular velocity $\mathsf{\Omega },$ and displacement $\mathsf{\Theta }$ are updated as

$$\begin{gathered} V^{t+\delta t}=V^t+{\displaystyle \frac{1}{M_p}}\left({\displaystyle \frac{F^{(t+\delta t)/2}{+F}^{(t-\delta t)/2}}{2}}+F_{\mathrm{l}\mathrm{u}\mathrm{b}}^t+F_{\mathrm{V}\mathrm{D}}^t+F_{\mathrm{s}}^t\right)\delta t, \\ {Y}^{t+\delta t}=Y^t+{\displaystyle \frac{1}{2}}\left(V^{t+\delta t}+V^t\right)\delta t, \\ {\mathsf{\Omega }}^{t+\delta t}={\mathsf{\Omega }}^t+{\displaystyle \frac{1}{I_p}}\left({\displaystyle \frac{{\mathsf{\Gamma }}^{(t+\delta t)/2}+{\mathsf{\Gamma }}^{(t-\delta t)/2}}{2}}\right)\delta t, \\ {\mathsf{\Theta }}^{t+\delta t}={\mathsf{\Theta }}^t+{\displaystyle \frac{1}{2}}\left({\mathsf{\Omega }}^{t+\delta t}+{\mathsf{\Omega }}^t\right)\delta t \end{gathered}$$

(2)

where $Mp$ (kg) and $Ip$ (m⁴) are the mass and moment of inertia of the particle. It should be noted that the hydrodynamic force F and torque Г (kg $·$m²/s⁻²) are expressed at half time since the momentum exchange actually happens somewhere between t and t + δt during the streaming process.

When conducting particle-resolved numerical simulations, the fluid density and kinematic viscosity remain the same while the length system changes. Due to the need for a sufficient number of grids to analyze the boundaries of solid particles in fully resolved simulations, Motta [24] indicates that a minimum number of 8 grids is required to analyze the length of particle diameter in order to ensure a certain degree of accuracy, which was used in this work.

Table 2. Basic parameter settings and corresponding relationships in point-resolved method.

Parameter	Physics	DNS
Solid particle diameter	$60\mathsf{\mu }$m	8
Fluid density	1.0000 g/cm3	1
Fluid kinematic viscosity	0.01 cm2/s	0.0058
Time step	$0.3\mathsf{\mu }$s	1

shows the basic parameter settings and the conversion.

The discrete phase parameters are solid particle density of 5.0000 g/cm³, diameter of 60 $\mathsf{\mu }$m, number of 3755, volume fraction of 6%; bubble density of 0.100 g/cm³, diameter of 200 $\mathsf{\mu }$m, number of 101, volume fraction of 6%. The fluid field parameters are still at a turbulent kinetic energy dissipation rate.

3. Results and Discussion

From the perspective of a real flotation process, both point-particle simulations and particle-resolved simulations involve certain assumptions and limitations. As a result, neither approach can provide data with the same level of accuracy as real experimental results. However, given the difficulty of conducting direct experimental measurements in such complex systems, both simulation methods can still offer valuable insights into understanding the intricate dynamics of the process from different perspectives. This section uses data from the point-particle model to predict the results of the particle-resolved simulation and perform a comparative analysis. It can be anticipated that both methods are reasonable, as they study the same system, and thus, their results should be relatively comparable.

3.1. Fitting and Prediction of the Point-Particle Model

In this section, the fitting and prediction capabilities of the point-particle model are explored. The point-particle model simplifies the representation of particles in the flotation process by treating them as discrete points, which significantly reduces the computational complexity. Despite this simplification, the model aims to capture the essential dynamics of particle–bubble interactions within a turbulent flow. The model parameters are carefully calibrated using simulation data. These calibrated parameters are then used to predict the behavior of the particles in different turbulence conditions.

When fitting the results of the point-particle simulation, the complexity of the raw data must be considered, particularly given the fluctuations observed in weak turbulence conditions. To improve the accuracy of the prediction, the fitting process is performed using only the subset of data where $\epsilon >0.24{\mathrm{m}}^2·{\mathrm{s}}^{-3}$. This selection ensures a more stable dataset, reducing the influence of turbulence-induced variability and enhancing the reliability of the model. By applying this approach, the fitting correlation coefficient is effectively maintained close to 1, as shown in the following figures, thereby improving the predictive capability of the point-particle model for particle–bubble interactions in flotation processes. Figure 1 illustrates the relationship between the Stokes number and the turbulent kinetic energy dissipation rate for both solid particles and bubbles. As shown, the Stokes number follows a power–law scaling with epsilon, expressed as y = 1.0941x^0.4993 for solid particles and y = 1.1074x^0.4993 for bubbles, both with a coefficient of determination (R² = 1.0), indicating an excellent fit. The exponent of approximately 0.5 suggests that the Stokes number scales with the square root of the turbulence dissipation rate, highlighting the strong influence of turbulence on particle–bubble dynamics. The minor difference in the prefactors between solid particles and bubbles suggests a slight variation in their response to turbulence.

Figure 2 presents the fitted formulas for the radial relative velocity and the radial distribution function under different turbulence intensities. Figure 2a illustrates the relationship between the radial relative velocity and the turbulent kinetic energy dissipation rate, where three interaction types—particle–bubble (PB), particle–particle (PP), and bubble–bubble (BB)—follow distinct power–law trends. The fitted equations indicate that the BB interactions exhibit the highest scaling exponent (0.7222), followed by PB interactions (0.6099) and PP interactions (0.6791), all with coefficients of determination (R²) exceeding 0.99, demonstrating excellent agreement. This suggests that the influence of turbulence on radial relative velocity is more pronounced in BB interactions compared to PB and PP interactions. Figure 2b shows the variation in the radial distribution function with epsilon. The negative exponents in the fitted power–law equations indicate a decreasing trend with increasing turbulence intensity. The PP interactions exhibit the steepest decline (−0.2733), followed by BB (−0.1244) and PB interactions (−0.0471), suggesting that turbulence has a stronger dispersive effect on PP interactions, reducing the clustering tendency of different particles. The relatively lower reduction in BB interactions implies that bubble clustering is less sensitive to turbulence compared to particle clustering. These findings highlight the fundamental differences in turbulence-driven collision dynamics among different interaction types.

Figure 3 presents the fitted power–law relationships between the collision kernel and the turbulent kinetic energy dissipation rate for different interaction types. The results are categorized into kinetic (K) and dynamic (D) collision kernels, denoted, respectively, in the figure. The fitted equations demonstrate that the collision kernel follows a power–law dependence on epsilon, with exponents ranging from 0.4932 to 0.6204. The BB interaction exhibits the highest exponent (0.6204), indicating a stronger dependence on turbulence intensity, while the PB and PP interactions follow lower scaling trends (0.5583 and 0.4932, respectively). The high coefficients of determination (R² > 0.98) confirm the robustness of these relationships. These findings suggest that increasing turbulence intensity enhances the collision frequency for all interaction types, with BB collisions being the most sensitive to turbulence variations.

Based on the characteristics of different interaction types in Figure 2 and Figure 3, a gradient turbulence field can be designed in the flotation device to improve the flotation process. For example, firstly, set up high turbulence intensity zones: utilizing the high sensitivity of BB interaction to turbulence (with the highest BB exponent of 0.6204 in Figure 3), prioritizing the aggregation of bubble collisions, promoting the merging of microbubbles into stable large bubbles, and improving the carrying capacity for mineral particles. Secondly, set up medium to low turbulence zones: by reducing the turbulence intensity, the dispersion effect of PP interaction is weakened (the exponent of PP in Figure 2b is the highest at −0.2733), maintaining a moderate local enrichment effect between particles and avoiding excessive dispersion of fine particles leading to a decrease in recovery rate.

To predict particle-resolved simulation results, fitting formulas derived from point-particle simulations can be applied. These formulas, which are developed through regression analysis of point-particle simulation data, provide a means to extrapolate results for more complex simulations involving detailed particle dynamics in turbulent flows. Once the predicted data for particle-resolved simulations are obtained through the fitting formulas from point-particle simulations, a comparison between the predicted and actual particle-resolved simulation results can be made. Based on these fitted formulas, the corresponding predicted data for particle-resolved simulations under various turbulence conditions are obtained, as shown in

Table 3. Prediction of particle-resolved simulation results based on point-particle model fitting formulas.

Project	Group 1	Group 2	Group 3	Group 4	Group 5	Group 6
Turbulent kinetic energy dissipation rate	0.56	2.26	7.84	16.63	21.13	36.06
Stokes number of solid particles	0.82	1.64	3.06	4.45	5.02	6.55
Stokes number of bubbles	0.83	1.66	3.10	4.51	5.08	6.63
PP radial distribution function	4.04	2.75	1.96	1.60	1.49	1.29
PB radial distribution function	1.16	1.09	1.02	0.99	0.98	0.95
BB radial distribution function	1.61	1.35	1.16	1.05	1.02	0.96
PP radial relative velocity	1.06	2.76	6.41	10.69	12.57	18.08
PB radial relative velocity	5.75	13.53	28.86	45.67	52.84	73.21
BB radial relative velocity	4.81	13.24	32.49	55.94	66.50	97.82
PP collision kernel	0.001	0.002	0.003	0.004	0.005	0.006
PB collision kernel	0.01	0.02	0.03	0.05	0.05	0.07
BB collision kernel	0.02	0.04	0.10	0.15	0.18	0.25

. Since the kinematic collision kernel and dynamic collision kernel exhibit a high degree of consistency in the point-particle simulations (shown in Figure 3), the predicted results are not distinguished between these two types of kernels in this analysis.

3.2. Comparison and Analysis of Results from the Two Models

In this section, we compare the results obtained from the point-particle model and the particle-resolved model and analyze the differences in their predictions. By examining the performance and suitability of each model in simulating particle–bubble interactions under different turbulence intensities, we aim to better understand the strengths and limitations of each approach.

To quantitatively assess the discrepancy between the point-particle model (PPS) and the fully resolved particle simulation (PRS), the relative error is defined as follows:

$$\gamma ={\displaystyle \frac{y_{\mathrm{P}\mathrm{P}\mathrm{S}}-y_{\mathrm{P}\mathrm{R}\mathrm{S}}}{y_{\mathrm{P}\mathrm{R}\mathrm{S}}}}$$

where $y_{\mathrm{P}\mathrm{P}\mathrm{S}}$ represents the predicted value from the point-particle model, and $y_{\mathrm{P}\mathrm{R}\mathrm{S}}$ corresponds to the result obtained from the fully resolved simulation. This metric indicates how much the point-particle model overestimates the fully resolved simulation results. The computed values are presented in

Table 4. The factor by which the point-particle simulation predictions exceed the fully resolved particle simulation results.

Project	Group 1	Group 2	Group 3	Group 4	Group 5	Group 6
Turbulent kinetic energy dissipation rate	0.56	2.26	7.84	16.63	21.13	36.06
Stokes number of solid particles	−0.005	−0.004	−0.003	−0.005	−0.003	−0.005
Stokes number of bubbles	−0.17	−0.17	−0.17	−0.17	−0.17	−0.17
PP radial distribution function	0.43	−0.47	−0.69	−0.76	−0.77	−0.81
PB radial distribution function	−0.52	−0.65	−0.73	−0.75	−0.74	−0.76
BB radial distribution function	−0.27	−0.45	−0.57	−0.62	−0.68	−0.64
PP radial relative velocity	14.46	16.17	20.49	21.18	20.07	20.29
PB radial relative velocity	40.77	44.34	49.30	52.79	53.83	58.08
BB radial relative velocity	18.06	18.24	21.47	31.94	35.22	39.21
PP kinematic collision kernel	17.73	7.78	6.05	5.01	4.55	3.95
PB kinematic collision kernel	19.17	14.75	12.30	12.90	12.92	12.91
BB kinematic collision kernel	12.26	9.34	8.70	13.28	10.84	14.16
PP dynamic collision kernel	−0.20	−0.25	−0.24	−0.13	−0.06	−0.12
PB dynamic collision kernel	0.36	0.58	0.63	1.21	1.31	1.26
BB dynamic collision kernel	0.62	1.37	1.33	2.60	2.63	2.61

. The results reveal that the prediction of the Stokes number for solid particles is highly accurate. However, in the fully resolved simulations, the bubble density does not correspond to its true physical value, leading to an overestimation of the Stokes number by approximately 17% compared to the predicted values. This deviation is consistent with theoretical estimates of computational errors arising from density assumptions, where the pressure gradient term is underestimated by 16.5%, and the Stokes drag term is underestimated by 15.4%. For the radial distribution function, the fully resolved simulation results are systematically 40% to 70% higher than the predictions from the point-particle model. In terms of radial relative velocity, the fully resolved simulation demonstrates that particle interactions reduce relative velocity when particles are in close proximity. In contrast, the point-particle model, being a fully one-way coupled approach, does not account for particle–particle interactions, meaning that relative velocities remain unaffected by the presence of nearby particles. Consequently, the predicted radial relative velocity values are 20 to 50 times higher than those observed in the fully resolved simulations. As a direct consequence of these discrepancies, the predicted kinematic collision kernel is overestimated by a factor of 10 to 20 compared to the statistical results from the fully resolved simulations. This discrepancy originates from differences in the radial distribution function. These findings further substantiate the inadequacy of using kinematic-based estimations to evaluate collision kernels within the framework of fully resolved simulations.

Notably, the relative error of the dynamic collision kernel, derived from the direct detection of collision events, exhibits a certain level of validity in both simulation approaches. As shown in Table 4, the predicted values for particle–particle collisions are approximately 10% to 20% lower than those obtained from fully resolved particle simulations. Theoretically, point-particle simulations represent an idealized scenario where the collision kernel should reach its maximum value. However, due to the increased complexity of real-world conditions, the actual collision efficiency cannot reach 100%, leading to lower collision kernel values in fully resolved simulations. The occurrence of collision efficiencies exceeding 100% in this study is clearly a non-physical phenomenon. A possible explanation for this discrepancy lies in the lubrication force model. Under the current framework, once two particles collide, they may be difficult to separate. As a result, post-collision particles experience frequent interactions and may oscillate around their equilibrium position like a spring. However, the current collision detection algorithm is unable to effectively exclude repeated detections of such interactions. Further systematic investigations are required to identify the exact cause and verify these findings.

The collision kernels for solid particle–bubble and bubble–bubble interactions fall within a reasonable range. This can be attributed to the larger volume of bubbles, which makes them more susceptible to shear effects in the flow field. As a result, post-collision adhesion, which commonly occurs between solid particles, is less likely for bubbles, preventing repeated collision detections in statistical analyses. According to the data in the table, the predicted values are generally 1 to 2 times larger than those obtained from fully resolved particle simulations. This overestimation is due to the simplification of the fluid dynamic effect between particles in the point-particle model, but it can still capture the overall impact pattern of turbulence intensity on the change in the collision kernel. Although the dynamic collision kernel is overestimated in the point-particle model (Table 2), its power–law growth trend with increasing turbulence intensity is consistent with the particle-resolved simulation results (Figure 3). This indicates that the point-particle model is suitable for analysis of the trend changes in the collision kernel, but in prediction, correction factors (such as calibration coefficients based on PRS results) need to be introduced to compensate for errors caused by simplified assumptions.

Using collision efficiency, the influence of turbulence intensity on collision efficiency can be determined, as illustrated in Figure 4. There are multiple ways to define collision efficiency. Karakashev et al. [25] define it as the time required for particle–bubble attachment within the time of collision occurrence. The shorter the attachment time, the higher the efficiency. In this study, we defined it as the ratio of real collision kernel to ideal collision kernel, which can also be represented by the ratio of kernel of PPS to PRS:

$$CollisionEfficiency={\displaystyle \frac{CollisionKernelinPPS}{CollisionKernelinPRS}}$$

Within the investigated turbulence range, collision efficiency decreases as turbulence intensity increases at relatively low turbulence levels. However, at higher turbulence intensities, collision efficiency remains nearly constant. Nevertheless, the universality of this trend requires further validation through additional simulations.

The results indicate that both the radial distribution function and radial relative velocity exhibit significant discrepancies compared to the point-particle simulation results. Consequently, the kinematic collision kernel can no longer maintain consistency with the dynamic collision kernel, rendering the estimation of the collision kernel based on kinematic quantities ineffective in particle-resolved simulations. However, the dynamic collision kernel closely matches the predictions of the point-particle simulation and exhibits an increasing trend with turbulence intensity. Furthermore, the results from both simulation methods can be used to estimate collision efficiency, providing a more accurate methodological foundation for future research on collision efficiency.

To better understand the differences between the point-particle model and the particle-resolved model, as well as their respective applicability, it is essential to analyze the advantages and limitations of both approaches and comprehend the discrepancies in their simulation results. The comparison helps assess the accuracy and reliability of using point-particle fitting formulas as a means of predicting more complex dynamics in particle-resolved simulations. This approach allows for faster predictions and can be particularly useful in scaling up or optimizing flotation processes where detailed particle behavior in turbulent flows is essential. It also provides a way to bridge the computational complexity gap between point-particle and particle-resolved simulations.

The primary limitations of the point-particle model stem from the assumptions regarding particle size, which restrict the accuracy of particle–fluid and particle–particle interactions, leading to lower overall precision. However, this model offers several advantages: it can handle particles of any density, supports a larger computational domain, is time-efficient, allows for tracking a greater number of particles, provides more statistically reliable results, and requires relatively fewer computational resources. In contrast, the particle-resolved model has its own set of drawbacks: the particle density is constrained by the computational method, particle–particle interactions are limited by the accuracy of the lubrication force model, the solvable computational domain is small, simulations are time-consuming, fewer particles can be tracked, statistical noise is higher, and significantly more computational resources are required. Nevertheless, this approach offers key advantages: particle size is unrestricted, particle–fluid coupling is solved with high accuracy, and the overall precision of the simulation is superior.

4. Conclusions

Due to the extreme difficulty in experimentally measuring the dynamic characteristics of solid particles and bubbles in the flotation of turbulent flows, numerical simulations have become the primary method for studying such phenomena. In this study, the authors employed DNS to investigate the effects of different turbulence conditions on the dynamic characteristics of solid particles and bubbles in the flotation process. The study systematically explores how different turbulence states influence the distribution characteristics and collision dynamics of solid particles and bubbles in turbulent flow fields.

The results compare the particle-resolved simulation with the point-particle simulation, highlighting the differences in their predictive accuracy and the level of detail they provide in modeling particle–bubble interactions. The particle-resolved simulation offers a more comprehensive representation of the physical processes, capturing the detailed motion and interaction of individual particles and bubbles, while the point-particle simulation, though computationally more efficient, tends to simplify these interactions by treating particles as discrete points. The comparison underscores the strengths and limitations of both approaches, providing valuable insights into their respective applicability in studying turbulence effects on flotation dynamics.

The analysis of the results indicates that many factors may affect the dynamic behavior of solid particles and bubbles in the actual flotation process. The turbulent kinetic energy dissipation rate changes the local enrichment of particles by affecting the Stokes number of the discrete phase. In addition, the increase in turbulence intensity will enhance the transport effect of discrete phases, leading to an increase in collisions. These two mechanisms exhibit a certain degree of competitiveness. By examining the radial relative velocity, radial distribution function, collision kernel, and collision efficiency, possible optimization methods for flotation processes have been proposed, providing in-depth insights and theoretical basis for flotation process optimization.