Axisymmetric Lattice Boltzmann Model Research Articles

Axisymmetric lattice Boltzmann model for three-phase fluids and its application to the Rayleigh-Plateau instability

Based on the multi-component phase field theory, in this paper we propose an axisymmetric lattice Boltzmann model for three-phase fluids. The proposed model takes advantage of two particle distribution functions for capturing phase interface among three different fluids, and another particle distribution function for solving the hydrodynamic equations for flow field. In order to describe the axisymmetric effect arising from the coordinate transformation, we elaborately design the equilibrium distribution function and forcing distribution function in the evolution equation, which ensures that the model can accurately recover the macroscopic governing equation for three-phase fluids. Also, the introduced source terms accounting for the axisymmetric effect contain no additional gradient term, which makes it be simpler than the existing lattice Boltzmann model for axisymmetric three-phase fluids. To validate the proposed model, a series of axisymmetric multiphase benchmark examples are performed, including the static double droplets, the spreading of liquid lens, and the binary-fluid Rayleigh-Plateau instability. It is reported that the present model can accurately capture the phase interface, and the predicted steady shapes of the liquid lens agree well with the analytical profiles. Then, the proposed model is used to study the three-phase Rayleigh-Plateau instability and the effects of the wavenumber and the radius ratio of liquid column on the interfacial dynamic behaviour, the breakup time of liquid threads and the size of daughter droplet are investigated in detail. It can be found that the compound liquid thread at a high wavenumber could break up into one main droplet and one satellite droplet, but the multiple satellite droplets can be produced at a low wavenumber, which leads to that the sizes of main and satellite droplets increase with the wavenumber at first and then decrease with it. Besides, we can observe that the inner fluid undergoes the breakup at earlier time than the middle fluid, and the breakup time for both inner and middle fluids increases with the decrease of the wavenumber. Finally, we can find that increasing the radius ratio of liquid column accelerates the breakup of inner-fluid thread, but prevents the breakup of the middle-fluid thread. In addition, the size of the compound main droplet increases with the radius ratio of liquid column, while the size of the compound satellite droplet doest not change much with it.

Axisymmetric lattice Boltzmann model for liquid flows with super-hydrophobic cylindrical surfaces

The lattice Boltzmann (LB) method has been extensively applied to the simulation of various fluid flows. Nonetheless, the standard LB model in rectangular coordinates faces some challenges when attempts are made to use it to simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length. Therefore, in this paper, boundary conditions for an axisymmetric LB model are proposed for the simulation of liquid slip at both convex and concave cylindrical surfaces. Based on the Navier slip boundary conditions, an equilibrium–nonequilibrium boundary scheme is developed to obtain the liquid slip in the azimuthal direction, with a combined bounce-back and specular-reflection boundary condition being developed to obtain the liquid slip in the axial direction. These boundary schemes are verified by simulating several established cases, namely, cylindrical Couette flow, harmonic oscillatory cylindrical Couette flow, Hagen–Poiseuille flow, and annular Poiseuille flow. The results are in excellent agreement with analytical results. Furthermore, some more complex flows involving super-hydrophobic cylindrical surfaces, such as stepwise oscillatory cylindrical Couette flow, are also simulated and studied using the proposed LB model. The axisymmetric LB model presented here overcomes the inability of standard LB models to accurately and efficiently simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length.

Modified radius-weighted lattice Boltzmann model to address singularities in axisymmetric multiphase flows

The radius-weighted lattice Boltzmann model has achieved great success in the simulation of axisymmetric flows. However, severe spurious currents near the axis are observed when this model is extended to simulate axisymmetric multiphase flows. In this study, to determine the origin of this singularity, we conducted a truncation error analysis based on high-order Taylor series expansion and identified the leading error terms through dimensionless analysis. By neglecting the error terms in proportion to the radius, we obtained the final forms of the singular terms in the axisymmetric lattice Boltzmann model. We proposed a modified model by including an additional correction term, to remove the singularity at the third order. We validated the proposed model using numerical tests for flat and spherical interfaces. Results showed that the present modified model reduced the spurious currents near the axis by two orders of magnitude compared with the original model. This modified model also has been successfully applied to predict bubble dynamics in an air-water system. Our numerical results are in excellent agreement with available experimental observations in terms of bubble shapes and terminal velocities.

Double Diffusive Convection of Power Law Fluids Through Taylor–Couette Flow

The proposed study focuses on the effect of double diffusive convection on the flow structures of power-law fluids between two coaxial cylinders of finite aspect ratios with end caps. We adopted a D2Q9 axisymmetric lattice Boltzmann model to predict the power-law fluid flow and D2Q4 (Lattice Boltzman method) model for azimuthal velocity, concentration, and the temperature. For ( is the solutal and thermal buoyancy forces ratio), fixed Taylor number with increasing Rayleigh number , the thermal base flow regime is more pronounced for the pseudoplastic fluid compared to the dilatant fluid. In addition, the thermal base flow in the annulus is delayed when increases. Thus, the increase of the behavior index has a stabilizing effect on the flow structure. On the other hand, when , the solute buoyancy forces in the annulus prevail over the centrifugal and thermal buoyancy forces, and the flow becomes ultimately a solute base flow for . For , the flow becomes a thermal base flow for . Thus, the mass transfer with negative values has a greater stabilizing effect on the flow structures than with positive .

Axisymmetric lattice Boltzmann model with slip boundary conditions for liquid flows in microtube

Lattice Boltzmann (LB) method, which is a mesoscopic numerical method, has been considered as a powerful tool to study microscale gas and liquid flows. Boundary slip phenomenon, which is a significant feature of both microscale gas and liquid flows, has been extensively studied by the LB method in the past decade. However, most of the previous works have focused on the microchannel flows and studies on the microtube flows are very rare. In this paper, we investigate the widely used slip boundary conditions, i.e., combined bounce-back and specular-reflection (BS) scheme, combined Maxwell-diffusion and specular-reflection (MS) scheme, and combined Maxwell-diffusion and bounce-back (MB) scheme, for the axisymmetric LB model with multi-relaxation-time (MRT) in detail. In order to realize the Navier slip boundary condition for liquid flows, we put forward to a reasonable strategy for determining the combination coefficients and the relaxation time. The proposed boundary schemes are validated by some numerical tests including the Hagen–Poiseuille flow, axisymmetric Womersley flow, Poiseuille flow in a circular annulus, and Womersley flow in a circular annulus. Numerical results are consistent with the analytical solutions, which demonstrate that the proposed boundary schemes are suitable for microscale liquid flows in microtube.

Evaporation of a sessile droplet on flat surfaces: An axisymmetric lattice Boltzmann model with consideration of contact angle hysteresis

Droplet evaporation is a widely encountered heat and mass transfer phenomenon. Due to its 3D in nature, it is challenging to simulate the complex 3D droplet evaporation process. Here, we propose an axisymmetric lattice Boltzmann (LB) model to simulate the dynamics of contact line motion during sessile droplet evaporation on a flat heated surface, taking contact angle hysteresis into consideration. We demonstrate that this LB model can numerically simulate the evaporation processes of sessile droplet evaporation processes undergoing constant contact angle (CCA), constant contact radius (CCR), and mixed modes sequentially. The classical D2-law for droplet surface mass transfer is confirmed numerically by using the case of a floating droplet evaporation, thereby proving that the axisymmetric LB model is capable of 3D simulation. Our simulated results on temporal variations of contact radius, contact angle are found in good agreement with two different sets of literature experimental data. Specifically, we compute the local mass transfer rate on the droplet surface from the simulated velocity field to show that the evaporation rate near the triple-phase contact region is much higher than that at the droplet apex region, with the peak heat flux occurring near the location of the dynamic contact line. The contact angle fluctuates with stick-slip motion are found during the CCA stage of the evaporation process at higher substrate temperatures. Additionally, we analyse the non-quasi-equilibrium evaporation mode and contact line slip velocity on substrates surface at higher temperatures.

Axisymmetric lattice Boltzmann model for simulation of ternary fluid flows

In this paper, a lattice Boltzmann model based on the phase-field Cahn–Hilliard-based approach is proposed to deal with ternary fluid flows in the axisymmetric coordinate system. The present model can handle both high density and viscosity ratios for partial and total spreading scenarios. A variety of numerical tests, including a static droplet test, a breakup of a liquid thread test, a spreading of a liquid lens test, and a droplet collision test, were performed to verify the model. It is found that the proposed model produces accurate results which agree well with the available data.

Axisymmetric lattice Boltzmann model for simulating the freezing process of a sessile water droplet with volume change.

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°).

A kinetic theory-based axisymmetric lattice Boltzmann flux solver for isothermal and thermal swirling flows

In this study, an alternative axisymmetric lattice Boltzmann flux solver (LBFS) is proposed within the framework of the kinetic theory-based axisymmetric lattice Boltzmann (LB) model. The macroscopic equations recovered from the axisymmetric LB model are globally resolved by the finite volume method, while numerical fluxes on the cell interface are locally reconstructed from solutions to the axisymmetric LB equation with the second-order of accuracy. Unlike the previous fractional step – axisymmetric LBFS which is based on the standard LB model and contains complicated source terms of various space derivatives, the present scheme originates from the kinetic theory-based lattice Boltzmann equation defined on the cylindrical coordinates which naturally incorporates the axisymmetric effects into the distribution functions and greatly simplifies the source terms involved. As a result, no space derivatives emerge in the source terms, and the corrector step in the previous LBFS model is removed, which yields more compact formulations. Moreover, previous inconsistency between the local reconstruction and the global governing equations is fixed in the present solver, which helps to nurture the computational results. By consolidating the evolutions of the azimuthal velocity and the temperature into the present axisymmetric LBFS, it is particularly suitable for simulations of isothermal and thermal swirling flows. The accuracy and robustness of the proposed scheme are validated through various benchmark tests. Comprehensive comparisons with the results from the previous solver demonstrate that the present solver is quantitatively more accurate than the previous solver, particularly in predicting the pressure distributions.

A simplified axisymmetric lattice Boltzmann method for incompressible swirling and rotating flows

A simplified axisymmetric lattice Boltzmann method (SALBM) is developed in this paper for effective simulation of incompressible swirling and rotating flows. This model explores an alternative approach of reconstructing an axisymmetric model within the lattice Boltzmann (LB) framework. Essentially, SALBM reconstructs solutions to the macroscopic governing equations recovered from the axisymmetric LB equation through the Chapman-Enskog expansion analysis. Two variations of schemes, which bear different orders of temporal accuracy and are, respectively, suitable for the steady and the unsteady axisymmetric flow problems, can be evolved from SALBM. The proposed schemes reflect direct evolution of macroscopic variables instead of distribution functions, which could reduce the cost in virtual memory. Meanwhile, analytical interpretation of physical boundary conditions is available in SALBM which avoids tedious transformations as required in conventional LB models. Numerical tests further reveal that SALBM performs better than the existing axisymmetric LB models in numerical stability. These merits endow the present SALBM with advantages over previous models and forge its prospect in engineering applications.

Axisymmetric lattice Boltzmann model for multiphase flows with large density ratio

In this paper, a novel lattice Boltzmann (LB) model based on the Allen-Cahn phase-field theory is proposed for simulating axisymmetric multiphase flows. The most striking feature of the model is that it enables to handle multiphase flows with large density ratio, which are unavailable in all previous axisymmetric LB models. The present model utilizes two LB evolution equations, one of which is used to solve fluid interface, and another is adopted to solve hydrodynamic properties. To simulate axisymmetric multiphase flows effectively, the appropriate source term and equilibrium distribution function are introduced into the LB equation for interface tracking, and simultaneously, a simple and efficient forcing distribution function is also delicately designed in the LB equation for hydrodynamic properties. Unlike many existing LB models, the source and forcing terms of the model arising from the axisymmetric effect include no additional gradients, and consequently, the present model contains only one non-local phase field variable, which in this regard is much simpler. In addition, to enhance the model’s numerical stability, an advanced multiple-relaxation-time (MRT) model is also applied for the collision operator. We further conducted the Chapman-Enskog analysis to demonstrate the consistencies of our present MRT-LB model with the axisymmetric Allen-Cahn equation and hydrodynamic equations. A series of numerical examples, including static droplet, oscillation of a viscous droplet, breakup of a liquid thread, and bubble rising in a continuous phase, are used to test the performance of the proposed model. It is found that the present model can generate relatively small spurious velocities and can capture interfacial dynamics with higher accuracy than the previously improved axisymmetric LB model. Besides, it is also found that our present numerical results show excellent agreement with analytical solutions or available experimental data for a wide range of density ratios, which highlights the strengths of the proposed model.

Numerical analyses and optimization of tubular thermochemical heat storage reactors using axisymmetric thermal lattice Boltzmann model

Axisymmetric lattice Boltzmann (LB) model has gained great development in the field of fluid flow and heat transfer in recent decades. Compared with the standard LB method, the axisymmetric LB method can solve 3D axisymmetric problems with minimal meshes. This numerical method will be helpful in research on thermochemical heat storage, especially considering the profound understanding and optimization of reactor performance. In this study, we extended a previous axisymmetric LB model to simulate the charging/discharging processes of a MgO/Mg(OH)2 thermochemical reactor. We derived a new evolution function by including an unstable heat source term out of a chemical reaction. New isothermal and heat flux boundary treatment approaches were proposed to maintain the precision of this axisymmetric LB model in the presence of the heat source term. In addition, a local method for transient conjugate heat transfer simulation without the utilization of temperature gradient information at the fluid-solid interface was developed to predict heat transfer between the reactor bed and heat transfer fluid. With the proposed axisymmetric thermal LB model, the charging/discharging processes of a tubular MgO/Mg(OH)2 thermochemical reactor unit with a heat exchanger were simulated. Results showed that the average power output was maximized with a bed thickness of 1.0 cm. The length of the reactor unit should be considered by balancing power output and cost.

Axisymmetric lattice Boltzmann method for non‐Newtonian flows

SummaryA new axisymmetric lattice Boltzmann model is proposed for non‐Newtonian fluid flows. The model incorporates the additional axisymmetric contributions into the evolution equation by introducing a force term. The precise form of the Navier‐Stokes equations in the cylindrical polar coordinate system is derived through Chapman‐Enskog analysis, and the consistency in dimension with the lattice Boltzmann equation is retained in the introduction of force terms. The model is used to simulate the flows in a straight pipe and an annulus channel for various power‐law indexes and Reynolds number. The velocity profiles in the straight pipe and annulus channel are presented and shown to be in excellent agreement with analytical results. The 4:1 axisymmetric contraction flow was studied as well. The results show that the reattachment length increases as the power‐law indexes for a shear‐thickening fluid. The opening angle increases substantially with the increase of the power‐law index for both a shear‐thinning fluid and a shear‐thickening fluid. The velocity profiles and vortex sizes of contraction flows agree very well with those of the finite element analysis. Results demonstrate that the proposed method is very effective to analyze the axisymmetric flows of a non‐Newtonian fluid.

An alternative implementation of the kinetic theory based axisymmetric lattice Boltzmann model

The semi-implicit centered scheme, as an approximated implementation of the trapezium rule based scheme, has proved to be capable of avoiding the discrete lattice effects in our recent forcing scheme analysis for the axisymmetric LB method, and is applied here for an alternative implementation of the kinetic theory based axisymmetric model by Guo et al. In particular, the external force terms and the first-order part of the source terms accounting for the axisymmetric effect are discretized with the semi-implicit centered scheme, while the rest are represented with the trapezium rule. The proposed axisymmetric LB scheme has the advantage of simpler calculations of the macroscopic variables since the zeroth- and the first-order moments of the distribution function are independent of the complicated source terms. Chapman–Enskog analysis in the Appendix demonstrates that axisymmetric Navier–Stokes (N–S) equations are recovered from the proposed model, and the error terms caused by the semi-implicit centered scheme do not affect the overall accuracy of the present model. Moreover, the accuracy and applicability of the proposed axisymmetric LB model are verified by the adopted numerical tests, and the complicated macroscopic variables computations in the Guo et al. model and its derivative models are avoided by the present model.

Investigation of solid-liquid phase change in the spherical capsule using axisymmetric lattice Boltzmann model

The solid-liquid phase change process is important to phase change material (PCM). In this paper, in order to investigate the solid-liquid phase change process in a spherical capsule, the axisymmetric lattice Boltzmann (LB) for phase change is proposed firstly. The problems of one-region phase change in cylindrical coordinate system and solid-liquid phase change by convection in cylindrical enclosure have been solved to verify the present LB model. The distributions of outer wall temperature of spherical capsule are linear. The results show that more heat transferred through the upper region of outer wall may enhance the natural convection and accelerate the process of heat transfer. However, the ratio of energy consumed by latent heat decreases with the slope, leading to more non-uniform temperature distribution. Furthermore, when the slope is larger than 0.2547, more heat is applied to rise the temperature of PCM, resulting in the slower melting rate.

A comparative study of the axisymmetric lattice Boltzmann models under the incompressible limit

A comparative study on four axisymmetric lattice Boltzmann (LB) models, namely, the kinetic theory based model by Guo et al. (2009), the consistent model by Li et al. (2010), the centered scheme model by Zhou (2011), and our model (based on applying the centered scheme to the Guo et al. (2009) model), is conducted both theoretically and numerically. The finite difference interpretation of the LB method by Junk (2001) is applied to evaluate the accuracy of the models under the incompressible limit. Particularly, the finite difference stencils adopted for the spatial gradient terms in the macroscopic axisymmetric Navier–Stokes (N–S) equations are compared. Besides, the numerical performance (i.e., the numerical accuracy, stability and the convergence efficiency) of the models is compared by two benchmark tests, namely, the unsteady-state Womersley flow and the cylindrical cavity flow. The numerical results accord well with the theoretical analysis. Additionally, it is also found that the numerical stability of the axisymmetric LB models is effectively improved by removing the effects from the non-hydrodynamic variables.

Alternative extrapolation-based symmetry boundary implementations for the axisymmetric lattice Boltzmann method.

In this study, alternative symmetry boundary implementations for the axisymmetric lattice Boltzmann (LB) method are proposed based on the nonequilibrium extrapolation and the direct extrapolation schemes. The proposed boundary schemes are directly implemented on the symmetry axis, and the postcollision distribution function and the macroscopic variables at the boundary nodes are extrapolated from the inner fluid nodes; thereby, the singularities arising at the symmetry axis (r=0) during the collision and the macroscopic variable calculations are completely avoided. The accuracy of the present schemes is consistent with the well-established axisymmetric LB model. Moreover, in comparison with previous symmetry boundary schemes, the present implementations are slightly more accurate than the symmetry scheme by Guo etal. [Phys. Rev. E 79, 046708 (2009)10.1103/PhysRevE.79.046708] and numerically more stable than the specular reflection-based schemes.

Forcing scheme analysis for the axisymmetric lattice Boltzmann method under incompressible limit.

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.

Consistent lattice Boltzmann methods for incompressible axisymmetric flows

In this work, consistent lattice Boltzmann (LB) methods for incompressible axisymmetric flows are developed based on two efficient axisymmetric LB models available in the literature. In accord with their respective original models, the proposed axisymmetric models evolve within the framework of the standard LB method and the source terms contain no gradient calculations. Moreover, the incompressibility conditions are realized with the Hermite expansion, thus the compressibility errors arising in the existing models are expected to be reduced by the proposed incompressible models. In addition, an extra relaxation parameter is added to the Bhatnagar-Gross-Krook collision operator to suppress the effect of the ghost variable and thus the numerical stability of the present models is significantly improved. Theoretical analyses, based on the Chapman-Enskog expansion and the equivalent moment system, are performed to derive the macroscopic equations from the LB models and the resulting truncation terms (i.e., the compressibility errors) are investigated. In addition, numerical validations are carried out based on four well-acknowledged benchmark tests and the accuracy and applicability of the proposed incompressible axisymmetric LB models are verified.

Axisymmetric multiphase lattice Boltzmann method for generic equations of state

We present an axisymmetric lattice Boltzmann model based on the Kupershtokh et al. multiphase model that is capable of solving liquid–gas density ratios up to 103. Appropriate source terms are added to the lattice Boltzmann evolution equation to fully recover the axisymmetric multiphase conservation equations. We validate the model by showing that a stationary droplet obeys the Young–Laplace law, comparing the second oscillation mode of a droplet with respect to an analytical solution and showing correct mass conservation of a propagating density wave.