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Dispersion in fractured media impacts many environmental and geomechanical practices. It is mainly controlled by the structure of fracture networks and the Péclet number (Pe)$(Pe)$ , but predicting it remains challenging. In this study, numerous three‐dimensional stochastic discrete fracture networks (DFNs) were generated, where the density, size, and orientation vary significantly. The aperture and conductivity are proportional to the size, following power‐laws. Through flow and transport simulation, we evaluated the longitudinal dispersion coefficients DL$\\left({D}_{L}\\right)$ . We found that, as density increases, the tortuosity decreases and the first passage time distributions approximate bell‐shaped curves more closely, which suggests, but does not fully guarantee, that an asymptotic dispersion regime may emerge for denser DFNs, as solute particles traverse more fractures and the macroscopic inter‐fracture mixing is more homogeneous. We then determined the DL${D}_{L}$values for DFNs in which the time evolution of the variance of particle displacements becomes linear and hence asymptotic. The results show that both Pe$Pe$and fracture density affect DL${D}_{L}$ , but the former has a much stronger influence than the latter. A new Péclet number Pec$\\left(P{e}^{c}\\right)$was recalculated for all DFNs, where the characteristic length scale accounts for the influence of large fractures. Dimensionless DL${D}_{L}$values show a unique power‐law relationship with high Pec$P{e}^{c}$values. Furthermore, when advection dominates, the dimensionless DL${D}_{L}$can be described by a universal finite‐size scaling function depending on fracture density and domain sizes. The findings of this study enhance the understanding of transport in fracture networks and imply the potential for predicting DL${D}_{L}$in a broad range of scenarios using statistics on fracture parameters obtainable at the field scale. Key Points Asymptotic regimes emerge for sufficiently dense fracture networks A new definition of Péclet number is proposed considering the influence of large fractures The dependence of dispersion coefficients on Péclet number and the finite‐size scaling function are universal

This paper focuses on the laminar flame instability of three high molecular weight n-alkanes, namely n-hexane, n-octane, and n-decane. The experiment was carried out in a constant volume combustion bomb to get the flame images. The critical radius under different conditions was extracted using the image processing program. Combined with the existing critical Peclet number theory, the dominant factors of flame instability under current conditions for three n-alkanes can be figured out. Moreover, the average cell size (equivalent cell radius, R cell ) was extracted to provide quantitative analysis of the flame cellular structure, based on the method developed in this work. The theoretical R cell were also calculated and compared with the experimental results to validate the proposed method.

It is proposed to determine the main parameters of the structure of flows in objects by velocity profile without taking response curves using the indicator method in heat and mass transfer apparatuses and reactors. A formula is derived for calculating the variance (central moment of the second order) and the flow structure function for a half-open vessel. An algorithm is also proposed for calculating the average residence time, the Peclet number of longitudinal diffusion for a one-parameter diffusion model, the number of cells and the differential response function of a combined model with a sequential connection of ideal displacement and mixing zones for a cell model. Moreover, calculation of the distribution density of the differential and integral response curves corresponding to a given velocity profile is provided.

To assess the water quality within the distribution networks, simplified models are used, which adopt an advective–reactive approach and neglect diffusion–dispersion phenomena. Although such simplifications can be sufficiently accurate in complete turbulent uniform flow regimes, literature works demonstrated that they could produce wrong results in laminar and transitional regimes that are relevant when analysing low flows, dead-end pipes in looped distribution networks or service connections. On the other hand, advective simplification allows for considerable computational savings during the simulation of large networks. Therefore, a criterion is needed for better discriminate pipes in which the advective approach is sufficient or the diffusive approach is required. The present study aims to investigate the use of the Péclet number to discriminate the use of advective simplification both adopting the two-dimensional (2D) advection–dispersion equation and the one-dimensional (1D) cross-section averaged advection–dispersion equation. The numerical analysis was applied to a linear pipeline using the EPANET, 1D advective–dispersive–reactive, and EPANET-DD (Dynamic–Dispersion) models. The results showed the inadequacy of the Péclet number in discriminating the dominance of the advective–dispersive process in real systems, as it is linked to the pipe's length, regardless of the flow regime occurring on the pipeline.

Understanding the behavior of fluid flow and solute transport in fractured rock is of great significance to geoscience and engineering. The discrete fracture network is the predominate channel for fluid flow through fractured rock as the permeability of fracture is several magnitudes higher than that of the rock matrix. As the basic components of the fracture network, investigating the fluid flow in crossed fractures is the prerequisite of understanding the fluid flow in fractured rock. First, a program based on the successive random addition algorithm was developed to generate rough fracture surfaces. Next, a series of fracture models considering shear effects and different surface roughness were constructed. Finally, fluid dynamic analyses were performed to understand the role of flowrate and surface roughness in the evolution of flow field, concentration field, solute breakthrough, and solute mixing inside the crossed fractures. Results indicated that the channeling flow at the fracture intersection became more pronounced with the increasing Péclet number (Pe) and Joint Roughness Coefficient (JRC), the evolution of the concentration field was influenced by Pe and the distribution of the concentration field was influenced by JRC. For Pe < 10, the solute transport process was dominated by molecular diffusion. For 100 > Pe > 10, the solute transport process was in the complete mixing mode. In addition, for Pe > 100, the solute transport process was in the streamline routing mode. The concentration distribution was affected by the local aperture at the fracture intersection corresponding to different surface roughness. Meanwhile, the solute mixing equation was improved based on this result. The research results are beneficial for further revealing the mechanism of fluid flow and solute transport phenomenon in fractured rock.

The numerical modeling of transverse laminar flow past a new type of hollow-fiber membranes with external profiling has been performed. A model system of parallel fibers with symmetrical parallel protrusion obstacles or grooves is considered. The absorption of point particles (solute or gas molecules) from a laminar transverse flow of a viscous incompressible liquid (gas) is calculated for a row of fibers, and the dependences of the efficiency of retention of particles by fibers on the Peclet (Pe), Reynolds (Re), and Schmidt (Sc) numbers and on the distance between neighbor fibers in a row are determined. The flow velocity and concentration fields are calculated by numerical solution of the Navier–Stokes equations and the convective diffusion equation in a wide range of Peclet numbers Pe = 0.1 − 105 for Sc = 1, 10, 1000 and Re ≤ 100.

This article addresses the abilities and limitations of the Lattice Boltzmann (LB) method in solving advection-dominated mass transport problems. Several schemes of the LB method, including D2Q4, D2Q5, and D2Q9, were assessed in the simulation of two-dimensional advection-dispersion equations. The concepts of Single Relaxation Time (SRT), Multiple Relaxation Time (MRT), and linear and quadratic Equilibrium Distribution Functions (EDF) were taken into account. The results of LB models were compared to the well-known Finite Difference (FD) solutions, including Explicit Finite Difference (EFD) and Crank-Nicolson (CN) methods. All LB models are more accurate than the aforementioned FD schemes. The results also indicate the high potency of D2Q5 SRT and D2Q9 SRT in describing advection-controlled mass transfer problems. The numerical artificial oscillations are observed when the Grid Peclet Number (GPN) is greater than 10, 25, 20, 25, and 10 regarding D2Q4 SRT, D2Q5 SRT, D2Q5 MRT, D2Q9 SRT, and D2Q9 MRT, respectively, while the corresponding GPN values obtained for the EFD and CN methods were 2 and 5, respectively. Finally, several LB models were used to satisfactorily solve a coupled system of groundwater and solute transport equations. In terms of computational time, all LB models are much faster than CN method.

The aggregation kinetics of settling particles is studied theoretically and numerically using the advection–diffusion equation. Agglomeration caused by these mechanisms (diffusion and advection) is important for both small particles (e.g., primary ash or soot particles in the atmosphere) and large particles of identical or close size, where the spatial inhomogeneity is less pronounced. Analytical results can be obtained for small and large Péclet numbers, which determine the relative importance of diffusion and advection. For small numbers (spatial inhomogeneity is mainly due to diffusion), an expression for the aggregation rate is obtained using an expansion in terms of Péclet numbers. For large Péclet numbers, when advection is the main source of spatial inhomogeneity, the aggregation rate is derived from ballistic coefficients. Combining these results yields a rational approximation for the whole range of Péclet numbers. The aggregation rates are also estimated by numerically solving the advection–diffusion equation. The numerical results agree well with the analytical theory for a wide range of Péclet numbers (extending over four orders of magnitude).

Purpose – The purpose of this paper is to present a simple meshless solution method for challenging engineering problems such as those with high wave numbers or convection-diffusion ones with high Peclet number. The method uses a set of residual-free bases in a local form. Design/methodology/approach – The residual-free bases, called here as exponential basis functions, are found so that they satisfy the governing equations within each subdomain. The compatibility between the subdomains is weakly satisfied by enforcing the local approximation of the main state variables pass through the data at nodes surrounding the central node of the subdomain. The central state variable is first recovered from the approximation and then re-assigned to the central node to construct the associated equation. This leads to the least compatibility required in the solution, e.g. C0 continuity in Laplace problems. Findings – The authors shall show that one can solve a variety of problems with regular and irregular point distribution with high convergence rate. The authors demonstrate that this is impossible to achieve using finite element method. Problems with Laplace and Helmholtz operators as well as elasto-static problems are solved to demonstrate the effectiveness of the method. A convection-diffusion problem with high Peclet number and problems with high wave numbers are among the examples. In all cases, results with high rate of convergence are obtained with moderate number of nodes per cloud. Originality/value – The paper presents a simple meshless method which not only is capable of solving a variety of challenging engineering problems but also yields results with high convergence rate.

Nanodroplets on a solid surface (i.e., surface nanodroplets) have practical implications for high-throughput chemical and biological analysis, lubrications, laboratory-on-chip devices, and near-field imaging techniques. Oil nanodroplets can be produced on a solid–liquid interface in a simple step of solvent exchange in which a good solvent of oil is displaced by a poor solvent. In this work, we experimentally and theoretically investigate the formation of nanodroplets by the solvent exchange process under well-controlled flow conditions. We find significant effects from the flow rate and the flow geometry on the droplet size. We develop a theoretical framework to account for these effects. The main idea is that the droplet nuclei are exposed to an oil oversaturation pulse during the exchange process. The analysis shows that the volume of the nanodroplets increases with the Peclet numberPeof the flow as ∝Pe 3/4, which is in good agreement with our experimental results. In addition, at fixed flow rate and thus fixed Peclet number, larger and less homogeneously distributed droplets formed at less-narrow channels, due to convection effects originating from the density difference between the two solutions of the solvent exchange. The understanding from this work provides valuable guidelines for producing surface nanodroplets with desired sizes by controlling the flow conditions.