Hydrodynamic dispersion at stagnation points: Simulations and experiments.

<p indent=0mm>In the past three decades, the lattice Boltzmann (LB) method has been developed into an efficient numerical method for simulating fluid flow and heat transfer. It is a mesoscopic numerical approach, sitting in the middle between the molecular dynamic method and conventional numerical methods based on the continuum assumption. Unlike conventional numerical methods, the LB method is built on the mesoscopic kinetic equation. It tracks the evolution of a particle distribution function and then accumulates the particle distribution function to obtain the macroscopic properties. Owing to its kinetic nature, the LB method has exhibited many advantages over conventional numerical methods. For example, in the LB equation the convective operator is linear, whereas the convective terms of the Navier-Stokes equations are nonlinear. Moreover, in the LB simulations the complex boundary conditions can be formulated with the elementary mechanical rules such as the bounce-back rule according to the interaction of the “LB particles” with the solid walls. Furthermore, the LB method is ideal for parallel computing because of its explicit scheme and local interactions. Particularly, the intermolecular interactions of fluids can be easily incorporated into the LB method. As a result, the interface between different phases can arise, deform, and migrate naturally in the LB modeling without using any techniques to track or capture the interface, which is often required in the VOF or Level Set method. The existing multiphase LB models can be generally classified into four categories, i.e., the color-gradient model, the pseudopotential model, the free-energy model, and the phase-field model. In the color-gradient LB model, two distribution functions are introduced to represent two different fluids and a color-gradient-based perturbation operator is employed to generate surface tension. Moreover, a recoloring step is used in the color-gradient model so as to separate different phases. The pseudopotential LB model is the simplest and the most popular multiphase LB model. In this model, the fluid interactions are mimicked by an interparticle potential, through which the separation of different phases or components can be achieved naturally. Accordingly, the interface between different phases can arise, deform and migrate naturally. The free-energy LB model was developed based on thermodynamics considerations, which modifies the second-order moment of the particle equilibrium distribution function so as to include a non-ideal thermodynamic pressure tensor. The phase-field LB model is based on the phase-field theory, in which the interface dynamics is described by an order parameter that obeys the Cahn-Hilliard equation or a Cahn-Hilliard-like equation. In this paper we briefly review some recent advances in these multiphase LB models. Particularly, recent progress in the pseudopotential LB model is highlighted. It is shown that the thermodynamic consistency of the pseudopotential model can be achieved by adjusting the mechanical stability condition via an improved forcing scheme. Furthermore, an alternative approach is presented to tune the surface tension of the pseudopotential model and an improved contact angle scheme is introduced, which retains the advantages of the original virtual-density scheme but does not suffer from an unphysical thick mass-transfer layer near the solid boundary. Moreover, the applications of the pseudopotential LB model to boiling and condensation heat transfer are outlined.

Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows

We use two pore-scale approaches, lattice-Boltzmann (LB) and pore-network modeling, to simulate single-phase flow in simulated sphere packings that vary in porosity and sphere-size distribution. For both modeling approaches, we determine the size of the representative elementary volume with respect to the permeability. Permeabilities obtained by LB modeling agree well with Rumpf and Gupte's experiments in sphere packings for small Reynolds numbers. The LB simulations agree well with the empirical Ergun equation for intermediate but not for small Reynolds numbers. We suggest a modified form of Ergun's equation to describe both low and intermediate Reynolds number flows. The pore-network simulations agree well with predictions from the effective-medium approximation but underestimate the permeability due to the simplified representation of the porous media. Based on LB simulations in packings with log-normal sphere-size distributions, we suggest a permeability relation with respect to the porosity, as well as the mean and standard deviation of the sphere diameter.

Recently, kinetic theory-based lattice Boltzmann (LB) models have been developed to model nonequilibrium gas flows. Depending on the order of quadratures, a hierarchy of LB models can be constructed which we have previously shown to capture rarefaction effects in the standing-shear wave problems. Here, we further examine the capability of high-order LB models in modeling nonequilibrium flows considering gas and surface interactions and their effect on the bulk flow. The Maxwellian gas and surface interaction model, which has been commonly used in other kinetic methods including the direct simulation Monte Carlo method, is used in the LB simulations. In general, the LB models with high-order Gauss-Hermite quadratures can capture flow characteristics in the Knudsen layer and higher order quadratures give more accurate prediction. However, for the Gauss-Hermite quadratures, the present simulation results show that the LB models with the quadratures obtained from the even-order Hermite polynomials perform significantly better than those from the odd-order polynomials. This may be attributed to the zero-velocity component in the odd-order discrete set, which does not participate in wall and gas collisions, and thus underestimates the wall effect.

We conduct a comparative study to evaluate several lattice Boltzmann (LB) models for solving the near incompressible Navier-Stokes equations, including the lattice Boltzmann equation with the multiple-relaxation-time (MRT), the two-relaxation-time (TRT), the single-relaxation-time (SRT) collision models, and the entropic lattice Boltzmann equation (ELBE). The lid-driven square cavity flow in two dimensions is used as a benchmark test. Our results demonstrate that the ELBE does not improve the numerical stability of the SRT or the lattice Bhatnagar-Gross-Krook (LBGK) model. Our results also show that the MRT and TRT LB models are superior to the ELBE and LBGK models in terms of accuracy, stability, and computational efficiency and that the ELBE scheme is the most inferior among the LB models tested in this study, thus is unfit for carrying out numerical simulations in practice. Our study suggests that, to optimize the accuracy, stability, and efficiency in the MRT model, it requires at least three independently adjustable relaxation rates: one for the shear viscosity ν (or the Reynolds number Re), one for the bulk viscosity ζ, and one to satisfy the criterion imposed by the Dirichlet boundary conditions which are realized by the bounce-back-type boundary conditions.

Droplet freezing not only is of fundamental interest but also plays an important role in numerous natural and industrial processes. However, it is challenging to numerically simulate the droplet freezing process due to its involving a complex three-phase system with dynamic phase change and heat transfer. Here we propose an axisymmetric lattice Boltzmann (LB) model to simulate the freezing process of a sessile water droplet with consideration of droplet volume expansion. Combined with the multiphase flow LB model and the enthalpy thermal LB model, our proposed approach is applied to simulate the sessile water droplet freezing on both hydrophilic and hydrophobic surfaces at a fixed subcooled temperature. Through comparison with the experimental counterpart, the comparison results show that our axisymmetric LB model can satisfactorily describe such sessile droplet freezing processes. Moreover, we use both LB simulations and analytical models to study the effects of contact angle and volume expansion on the freezing time and the cone shape formed on the top of frozen droplets. The analytical models are obtained based on heat transfer and geometric analyses. Additionally, we show analytically and numerically that the freezing front-to-interface angle keeps nearly constant (smaller than 90°).

Lattice Boltzmann (LB) method models have been demonstrated to provide an accurate representation of the flow characteristics in rarefied flows. Conditions in such flows are characterized by the Knudsen number (Kn), defined as the ratio between the gas molecular Mean Free Path ( MFP, λ) and the device characteristic length (L). As the Knudsen number increases, the behavior of the flow near the walls is increasingly dominated by interactions between the gas molecules and the solid surface. Due to this, linear constitutive relations for shear stress and heat flux, which are assumed in the Navier-Stokes-Fourier (NSF) system of equations, are not valid within the Knudsen Layer (KL). Fig. 1 illustrates the characteristics of the velocity field within the Knudsen layer in a shear-driven flow. It is easily observed that although the NSF equations with slip flow boundary conditions (represented by dashed line) can predict the velocity profile in the bulk flow region, they fail to capture the flow characteristics inside the Knudsen layer. Slip flow boundary conditions have also been derived using the integral transform technique [1]. Various methods have been explored to extend the applicability of LB models to higher Knudsen number flows, including using higher order velocity sets, and using wall-distance functions to capture the effect of the walls on the mean free path by incorporating such functions on the determination of the local relaxation parameters. In this study, a high order velocity model which contains a two-dimensional, thirteen velocity direction set (e.g., D2Q13), as shown in Fig. 2, is used as the basis of the current LB model. The LB model consists of two independent distribution functions to simulate the density and temperature fields, while the Diffuse Scattering Boundary Condition (DSBC) method is used to simulate the fluid interaction with the walls. To further improve the characterization of transition flow conditions expected in nano-scale heat transfer, we explored the implementation of two wall-distance functions, derived recently based on an integrated form of a probability distribution function, to the high-order LB model. These functions are used to determine the effective mean free path values throughout the height of the micro/nano-channel, and the resulting effect is first normalized and then used to determine local relaxation times for both momentum and energy using a relationship based on the local Knudsen number. The two wall-distance functions are based on integral forms of 1) the classical probability distribution function, ψ(r) = λ0−1e−r/λ0, derived by Arlemark et al [2], in which λ0represents the reference gas mean free path, and 2) a Power-Law probability distribution function, derived by Dongari et al [3]. Thus, the probability that a molecule travels a distance between r and r+dr between two successive collisions is equal to ψ(r)dr. The general form of the integral of the two functions used can be described by ψ(r) = C − f(r), where f(r) represents the base function (exponential or Power Law), and C is set to 1 so that the probability that a molecule will travel a distance r+dr without a collision ranges from zero to 1. The performance of the present LB model coupled with the implementation of the two wall-distance functions is tested using two classical flow cases. The first case considered is that of isothermal, shear-driven Couette flow between two parallel, horizontal plates separated by a distance H, moving in opposite directions at a speed of U0. Fig. 3 shows the normalized velocity profiles across the micro-channel height for various Knudsen numbers in the transition flow regime based on our LB models as compared to data based on the Linearized Boltzmann equation [4]. The results show that our two LB models provide results that are in excellent agreement with the reference data up to the high end of the transition flow regime, with Knudsen numbers greater than 1. The second case is rarefied Fourier flow within horizontal, parallel plates, with the plates being stationary and set to a constant temperature (TTop &gt; TBottom), and the Prandtl number is set to 0.67 to match the reference data based on the Direct Simulation Monte Carlo (DSMC) method [5]. Fig. 4 shows the normalized temperature profiles across the microchannel height for various Knudsen numbers in the slip/transition How regime. For the entire Knudsen number range studied, our two LB models provide temperature profiles that are in excellent agreement with the non-linear profile seen in the reference data. The results obtained show that the effective MFP relationship based on the exponential function improves the results obtained with the high order LB model for both shear-driven and Fourier flows up to Kn∼1. The results also show that the effective MFP relationship based on the Power Law distribution function greatly enhances the results obtained with the high order LB model for the two cases addressed, up to Kn∼3. In conclusion, the resulting LB models represent an effective tool in modeling non-equilibrium gas flows expected within micro/nano-scale devices.

Because the standard lattice Boltzmann (LB) method is proposed for Cartesian Navier-Stokes (NS) equations, additional source terms are necessary in the axisymmetric LB method for representing the axisymmetric effects. Therefore, the accuracy and applicability of the axisymmetric LB models depend on the forcing schemes adopted for discretization of the source terms. In this study, three forcing schemes, namely, the trapezium rule based scheme, the direct forcing scheme, and the semi-implicit centered scheme, are analyzed theoretically by investigating their derived macroscopic equations in the diffusive scale. Particularly, the finite difference interpretation of the standard LB method is extended to the LB equations with source terms, and then the accuracy of different forcing schemes is evaluated for the axisymmetric LB method. Theoretical analysis indicates that the discrete lattice effects arising from the direct forcing scheme are part of the truncation error terms and thus would not affect the overall accuracy of the standard LB method with general force term (i.e., only the source terms in the momentum equation are considered), but lead to incorrect macroscopic equations for the axisymmetric LB models. On the other hand, the trapezium rule based scheme and the semi-implicit centered scheme both have the advantage of avoiding the discrete lattice effects and recovering the correct macroscopic equations. Numerical tests applied for validating the theoretical analysis show that both the numerical stability and the accuracy of the axisymmetric LB simulations are affected by the direct forcing scheme, which indicate that forcing schemes free of the discrete lattice effects are necessary for the axisymmetric LB method.

Lattice Boltzmann (LB) Method is a relatively new method for flow simulations. The start point of LB method is statistic mechanics and Boltzmann equation. The LB method tries to set up its model at molecular scale and simulate the flow at macroscopic scale. LBM has been applied to mostly incompressible flows and simple geometry.

Characterization of immiscible phase displacement in heterogeneous pore structures: Parallel multicomponent lattice Boltzmann simulation and experimental validation using three-dimensional printing technology

Modeling moving contact-line with surfactant has become a widely sought methodology due to its scientific relevance and extensive applications. Within the phase field framework, we present an improved lattice Boltzmann (LB) model for simulating moving contact-line dynamics with soluble surfactant. In this model, a LB equation is used to solve the Navier–Stokes equations, and another two LB equations are utilized to solve the two Cahn–Hilliard-like equations. The modified chemical potentials are incorporated in the LB model by using an equivalent variant of the free energy functional and the corresponding equilibrium distribution functions are also amended. These modifications could circumvent the degraded accuracy of previous LB models in capturing the interfacial behavior and surfactant distribution, and also improve the well-posedness of the LB model. In addition, a dynamic contact angle formulation is introduced to account for the surfactant effect on surface wettability and the resulting contact angle is further implemented in the LB model via a popular geometrical wetting approach. We comprehensively evaluate the numerical performance of the LB model by simulating some benchmark problems. It is found that the LB model achieves a higher accuracy than previous LB models in solving the phase field and surfactant profiles, and also numerical prediction of moving contact-line dynamics with surfactant shows good agreement with the analytical solution. Finally, the LB model is applied to investigate droplet shearing dynamics on solid substrate. The influences of capillary number and solid wetting property on droplet deformation and breakup are analyzed in detail.

The lattice Boltzmann (LB) method can be formulated directly from the Boltzmann equation with the Bhatnagar–Gross–Krook assumption. This kinetic origin stimulates wide interest in applying it to simulate flow problems beyond the continuum limit. In this article, such a thought is examined by simulating Couette flows from the slip to free molecular flow regimes using the LB models equipped with different discrete velocity spaces, derived from the half-range Gauss Hermite (HGH), Gauss Legendre (GL), Gauss Kronrod (GK), and Gauss Chebyshev first and second quadrature rules. It is found that the conventional HGH-based LB models well describe noncontinuum Couette flows in the slip and weak transition flow regimes. Nonetheless, they suffer from significant errors with the further increasing Knudsen number, even if a large number of discrete velocities have been employed. Their results contrast with those by the LB models derived from the other Gaussian quadrature rules, which have far better accuracy at large Knudsen numbers. In particular, the GL- and GK-based LB models well capture the velocity fields of Couette flows in the strong transition and free molecular flow regimes. These numerical simulations in this article highlight the importance of velocity discretization for the LB simulations at different Knudsen numbers. They reveal that the LB models based on the Gauss Hermite (GH) quadrature rule are not always the best choice for simulating low-speed bounded flows at moderate and large Knudsen numbers; under strong noncontinuum conditions, those non-GH-based LB models proposed in this article have yielded more accurate results.

SummaryWe present results from simulations of two-phase flow directly on digitized rock-microstructure images of porous media using a lattice Boltzmann (LB) method. The implemented method is performed on a D3Q19 lattice with fluid/fluid and fluid/solid interaction rules to handle interfacial tension and wetting properties. We demonstrate that the model accurately reproduces capillary and wetting effects in pores with a noncircular shape. The model is applied to study viscous coupling effects for two-phase concurrent annular flow in circular tubes. Simulated relative permeabilities for this case agree with analytical predictions and show that the nonwetting-phase relative permeability might greatly exceed unity when the wetting phase is less viscous than the nonwetting phase.Two-phase LB simulations are performed on microstructure images derived from X-ray microtomography and process-based reconstructions of Bentheimer sandstone. By imposing a flow regulator to control the capillary number of the flow, the LB model can closely mimic typical experimental setups, such as centrifuge capillary pressure and unsteady- and steady-state relative permeability measurements. Computed drainage capillary pressure curves are found to be in excellent agreement with experimental data. Simulated steady-state relative permeabilities at typical capillary numbers in the vicinity of 10−5 are in fair agreement with measured data. The simulations accurately reproduce the wetting-phase relative permeability but tend to underpredict the nonwetting-phase relative permeability at high wetting-phase saturations. We explain this by pointing to percolation threshold effects of the samples. For higher capillary numbers, we correctly observe increased relative permeability for the nonwetting phase caused by mobilization and flow of trapped fluid. It is concluded that the LB model is a powerful and promising tool for deriving physically meaningful constitutive relations directly from rock-microstructure images.

In the pseudopotential lattice Boltzmann (LB) model, the physical behaviors of fluids are modeled through interparticle forces, which are closely tied to the equation of state (EOS). Existing simulations mainly rely on cubic EOS, which significantly lags behind modern multiparameter EOS in terms of the prediction of thermodynamic properties. However, there have been no reports on the application of such a high-precision EOS in LB simulations. In this study, a method for implementing fundamental equations of state in Helmholtz energy form (HEOS) in the LB framework is proposed. A novel unit conversion approach is developed, which enables the appropriate conversion of all information between lattice and physical units, overcoming the limitations of existing methods that fail to correctly convert energy information. This approach allows the direct conversion of the pressure between the lattice and physical units without the need to specify the lattice unit values for each parameter in the equation of state. The HEOS of water is used as an example to validate the feasibility of the proposed method and unit conversion approach. The average error of liquid–vapor coexistence densities obtained from the LB simulations using the HEOS is 0.46%, significantly lower than 22.5% by using the typical cubic Peng–Robinson (PR) EOS. Although the computational resource consumption tripled that of the PR EOS, the incorporation of HEOS demonstrated much stronger capabilities in simulations with phase-change phenomena, accurately predicting the specific latent heat of water in film evaporation from 100 to 341.6 °C where the one with PR EOS failed.

The water transport in a gas diffusion layer (GDL) has been investigated using two-dimensional lattice Boltzmann (LB) simulation. The LB model is developed to simulate the dynamic behavior of liquid water and enables to visualize the water-invasion process through micro-pores in GDL. To investigate the effect of rib structure on water invasion process in GDL, two different cases (i.e., with and without rib structure) are compared. The numerical model is verified by the comparison of the flow permeabilities in GDL. The validation result indicates that the LB model can properly predict the permeability of GDL and enables to simulate the water transport behavior in the GDL. The reconstruction of GDL is established by randomly placing the particles in GDL and ignoring the GDL deformation due to clamping force. The results of LB simulation confirm that the liquid water transport inside GDL is strongly governed by capillary force and the rib structure greatly impacts on the water transport behavior. The rib structure influences on the location of water breakthrough by comparing the simulation results of two different cases. This is due to the higher resistance force underneath the rib, resulting in the change of flow path which preferentially selects the lower resistance force. The water saturation level under the channel is higher than that under the rib caused by the suppression of growth of water cluster. After water breakthrough, the liquid water distribution under the channel has little change, whereas that under the rib keeps stretching for a while. The result indicates that a careful control of rib structure would enhance the water removal from the GDL. Therefore further studies for the optimum design of rib structure are needed. In an operating PEMFCs, the mechanism of water transport and wetting characteristics play an important role on flooding behavior. Therefore the results of the present study would contribute to the novel design for better water removal and flooding alleviation from the GDL.