Lattice Boltzmann equation hydrodynamics

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.

Unsteady melt heat flow coupling optimization method for sapphire crystal seeding growth by Kyropoulos method

High-quality seeding is the key to ensure the preparation of high-quality sapphire single crystal. The strength and direction of melt flow control the crystal seeding shape, bubble movement and will...

Axisymmetric compact finite-difference lattice Boltzmann method for blood flow simulations

An axisymmetric compact finite-difference lattice Boltzmann method is proposed to simulate both Newtonian and non-Newtonian flow of blood through a lumen. The curvature of the arteries could be accurately resolved using body-fitted mesh owing to the proposed finite-difference formulation. The axisymmetric nature of the flow, as well as the non-Newtonian nature of blood, are incorporated into the lattice Boltzmann equation using separate source terms. Using Chapman-Enskog expansion it is shown that the resulting lattice Boltzmann equation with these additional source terms recovers the macroscopic axisymmetric hydrodynamic equations. The solver is verified for (1) steady inflow of a Newtonian fluid through a stenosed lumen, (2) temporally developing pulsatile flow (Womersley flow) through a straight lumen with Newtonian fluid, and (3) steady inflow of a non-Newtonian fluid through a straight lumen. The solver is then applied to simulate the steady flow of a non-Newtonian fluid through a stenosed lumen, and it was found that a smaller recirculation zone and lower WSS values are obtained when compared with the flow of a Newtonian fluid. The capability of the solver to simulate spatially developing (velocity-driven) pulsatile flow is then demonstrated by simulating physiological pulsatile flow through an axisymmetric abdominal aortic aneurysm. From this simulation, the cycle-averaged wall shear stress is observed to have a steep gradient going from a minimum (negative) to a maximum (positive) value towards the distal end of the aneurysm, which is prone to the risk of rupture. An iterative procedure to select the geometric and flow parameters for unsteady inflow condition in the lattice Boltzmann method framework is demonstrated that accurately resolves all the timescales to achieve incompressibility. Overall, the present solver seems to be promising to simulate axisymmetric flow of blood with steady and pulsatile inflows while considering the blood rheology.

Axisymmetric lattice Boltzmann simulation of droplet impact on solid surfaces

Droplet-solid interaction is a ubiquitous fluid phenomenon that underpins a wide range of applications. To further the understanding of this important problem, we use an axisymmetric lattice Boltzmann method (LBM) to model the droplet impact on a solid surface with different wettability. The method applies a popular free-energy LBM developed by Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010)10.1016/j.jcp.2010.07.007] to simulate incompressible binary fluids with physical density and viscosity contrasts. The formulation is recast in cylindrical coordinates for modeling the normal impact of a three-dimensional (3D) droplet in the no-splashing regime, in which an axisymmetric flow is considered. The droplet deposits on or rebounds from the surface, governed by three key parameters: Weber number, Ohnesorge number, and equilibrium contact angle, which quantifies the surface wettability. We elucidate the distinct impact dynamics by probing droplet morphology and contact line behavior in great detail, which are quantitatively characterized by spreading factor, droplet aspect ratio, and dynamic contact angle. The simulations also resolve fluid velocity field inside and outside the droplet, which provides additional insight into the morphological evolution and mass-momentum transfer during impact. Explicit comparison between axisymmetric and conventional 2D LBM highlights the importance of axisymmetric terms in governing equations for reproducing physical impact behavior. The axisymmetric LBM significantly reduces computational cost as compared with 3D LBMs and offers an effective means to study droplet impact in applicable conditions.

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 et al. [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.

Influence of magnetohydrodynamic viscous flow on entropy generation within porous micro duct using the Lattice Boltzmann Method

In this work entropy generation and heat transfer for magnetohydrodynamic (MHD) forced convection flow in a micro duct filled with a porous medium are investigated using a modified axisymmetric Lattice Boltzmann Method.

Lattice Boltzmann model for incompressible axisymmetric thermal flows through porous media

The present work proposes a simple lattice Boltzmann model for incompressible axisymmetric thermal flows through porous media. By incorporating forces and source terms into the lattice Boltzmann equation, the incompressible Navier-Stokes equations are recovered through the Chapman-Enskog expansion. It is found that the added terms are just the extra terms in the governing equations for the axisymmetric thermal flows through porous media compared with the Navier-Stokes equations. Four numerical simulations are performed to validate this model. Good agreement is obtained between the present work and the analytic solutions and/or the results of previous studies. This proves its efficacy and simplicity regarding other methods. Also, this approach provides guidance for problems with more physical phenomena and complicated force forms.

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.

Phase-field-based lattice Boltzmann model for axisymmetric multiphase flows

In this paper, a phase-field-based lattice Boltzmann (LB) model is proposed for axisymmetric multiphase flows. Modified equilibrium distribution functions and some source terms are properly added into the evolution equations such that multiphase flows in the axisymmetric coordinate system can be described. Different from previous axisymmetric LB multiphase models, the added source terms that arise from the axisymmetric effect contain no additional gradients, and therefore the present model is much simpler. Furthermore, through the Chapmann-Enskog analysis, the axisymmetric Chan-Hilliard equation and Navier-Stokes equations can be exactly derived from the present model. The model is also able to deal with flows with density contrast. A variety of numerical experiments, including planar and curve interfaces, an elongation field, a static droplet, a droplet oscillation, breakup of a liquid thread, and dripping of a liquid droplet under gravity, have been conducted to test the proposed model. It is found that the present model can capture accurate interface and the numerical results of multiphase flows also agree well with the analytical solutions and/or available experimental data.

Polar-coordinate lattice Boltzmann modeling of compressible flows

We present a polar coordinate lattice Boltzmann kinetic model for compressible flows. A method to recover the continuum distribution function from the discrete distribution function is indicated. Within the model, a hybrid scheme being similar to, but different from, the operator splitting is proposed. The temporal evolution is calculated analytically, and the convection term is solved via a modified Warming-Beam (MWB) scheme. Within the MWB scheme a suitable switch function is introduced. The current model works not only for subsonic flows but also for supersonic flows. It is validated and verified via the following well-known benchmark tests: (i) the rotational flow, (ii) the stable shock tube problem, (iii) the Richtmyer-Meshkov (RM) instability, and (iv) the Kelvin-Helmholtz instability. As an original application, we studied the nonequilibrium characteristics of the system around three kinds of interfaces, the shock wave, the rarefaction wave, and the material interface, for two specific cases. In one of the two cases, the material interface is initially perturbed, and consequently the RM instability occurs. It is found that the macroscopic effects due to deviating from thermodynamic equilibrium around the material interface differ significantly from those around the mechanical interfaces. The initial perturbation at the material interface enhances the coupling of molecular motions in different degrees of freedom. The amplitude of deviation from thermodynamic equilibrium around the shock wave is much higher than those around the rarefaction wave and material interface. By comparing each component of the high-order moments and its value in equilibrium, we can draw qualitatively the main behavior of the actual distribution function. These results deepen our understanding of the mechanical and material interfaces from a more fundamental level, which is indicative for constructing macroscopic models and other kinds of kinetic models.

Axisymmetric multiphase lattice Boltzmann method

A lattice Boltzmann method for axisymmetric multiphase flows is presented and validated. The method is capable of accurately modeling flows with variable density. We develop the classic Shan-Chen multiphase model [Phys. Rev. E 47, 1815 (1993)] for axisymmetric flows. The model can be used to efficiently simulate single and multiphase flows. The convergence to the axisymmetric Navier-Stokes equations is demonstrated analytically by means of a Chapmann-Enskog expansion and numerically through several test cases. In particular, the model is benchmarked for its accuracy in reproducing the dynamics of the oscillations of an axially symmetric droplet and on the capillary breakup of a viscous liquid thread. Very good quantitative agreement between the numerical solutions and the analytical results is observed.

Axisymmetric lattice Boltzmann method revised

A reformulated lattice Boltzmann model is described for incompressible axisymmetric flows with or without swirling. It is a further development and improvement on the author's axisymmetric lattice Boltzmann method. The main features of the revised scheme are (a) all the macroscopic variables such as velocities are determined in the same formulas as those in the conventional lattice Boltzmann approach to the Navier-Stokes equations and (b) the added sink or source and force terms are simple with no calculation for a derivative. Such features distinguish the present method from the other existing simplified schemes, leading to a simple and efficient model. The scheme is naturally suitable for generic incompressible axisymmetric rotational flows involving more physical phenomena. The numerical solutions to Womersley and cylindrical cavity flows are presented to demonstrate its accuracy and capability.

Simultaneous incorporation of mass and force terms in the multi-relaxation-time framework for lattice Boltzmann schemes

This paper presents an analysis of the simultaneous incorporation of force and mass source terms into the multi-relaxation-time (MRT) collision operator. MRT force incorporation was obtained through Chapman-Enskog analysis. The numerical scheme was tested on different benchmark problems, including the decay of a shear wave with different bulk and kinematic viscosities and axisymmetric flow.

Improved axisymmetric lattice Boltzmann scheme

This paper proposes an improved lattice Boltzmann scheme for incompressible axisymmetric flows. The scheme has the following features. First, it is still within the framework of the standard lattice Boltzmann method using the single-particle density distribution function and consistent with the philosophy of the lattice Boltzmann method. Second, the source term of the scheme is simple and contains no velocity gradient terms. Owing to this feature, the scheme is easy to implement. In addition, the singularity problem at the axis can be appropriately handled without affecting an important advantage of the lattice Boltzmann method: the easy treatment of boundary conditions. The scheme is tested by simulating Hagen-Poiseuille flow, three-dimensional Womersley flow, Wheeler benchmark problem in crystal growth, and lid-driven rotational flow in cylindrical cavities. It is found that the numerical results agree well with the analytical solutions and/or the results reported in previous studies.

Rectangular lattice Boltzmann method

A set of rectangular lattice Boltzmann methods for fluid flows is developed. It is shown that reformulating local equilibrium distribution functions can result in the rectangular lattice Boltzmann models without the aid of an interpolation for shallow water equations, Navier-Stokes equations, and axisymmetric flow equations. In addition, schemes for correct incorporation of force terms into the models are proposed for simulations of flows involving forces in practice. The methods completely retain the innate kinetic features and the simple procedure of the standard lattice Boltzmann method with an additional benefit of being suitable for rectangular lattices at little extra computational cost. The methodology is illustrated and validated through its application to different flow problems, demonstrating the potential power of the models.

Lattice Boltzmann model for axisymmetric thermal flows

A thermal lattice Boltzmann (LB) model is presented for axisymmetric thermal flows in the incompressible limit. The model is based on the double-distribution-function LB method, which has attracted much attention since its emergence for its excellent numerical stability over the multispeed LB method. Compared with the existing axisymmetric thermal LB models, the present model is simpler and retains the inherent features of the standard LB method. Numerical simulations are carried out for the thermally developing laminar flows in circular ducts and the natural convection in an annulus between two coaxial vertical cylinders. The Nusselt number obtained from the simulations agrees well with the analytical solutions and/or the results reported in previous studies.

Simple lattice Boltzmann subgrid-scale model for convectional flows with high Rayleigh numbers within an enclosed circular annular cavity

Natural convection within an enclosed circular annular cavity formed by two concentric vertical cylinders is of fundamental interest and practical importance. Generally, the assumption of axisymmetric thermal flow is adopted for simulating such natural convections and the validity of the assumption of axisymmetric thermal flow is still held even for some turbulent convection. Usually the Rayleigh numbers (Ra) of realistic flows are very high. However, the work to design suitable and efficient lattice Boltzmann (LB) models on such flows is quite rare. To bridge the gap, in this paper a simple LB subgrid-scale (SGS) model, which is based on our recent work [S. Chen, J. Tölke, and M. Krafczyk, Phys. Rev. E 79, 016704 (2009); S. Chen, J. Tölke, S. Geller, and M. Krafczyk, Phys. Rev. E 78, 046703 (2008)], is proposed for simulating convectional flow with high Ra within an enclosed circular annular cavity. The key parameter for the SGS model can be quite easily and efficiently evaluated by the present model. The numerical experiments demonstrate that the present model works well for a large range of Ra and Prandtl number (Pr). Though in the present study a popularly used static Smagorinsky turbulence model is adopted to demonstrate how to develop a LB SGS model for simulating axisymmetric thermal flows with high Ra, other state-of-the-art turbulence models can be incorporated into the present model in the same way. In addition, the present model can be extended straightforwardly to simulate other axisymmetric convectional flows with high Ra, for example, turbulent convection with internal volumetric heat generation in a vertical cylinder, which is an important simplified representation of a nuclear reactor.

Numerical simulation of fluid flow and heat transfer inside a rotating disk-cylinder configuration by a lattice Boltzmann model

A simple lattice Boltzmann model for numerical simulation of fluid flow and heat transfer inside a rotating disk-cylinder configuration, which is of fundamental interest and practical importance in science as well as in engineering, is proposed in this paper. Unlike existing lattice Boltzmann models for such flows, which were based on "primitive-variable" Navier-Stokes equations, the target macroscopic equations of the present model for the flow field are vorticity-stream function equations, inspired by our recent work designed for nonrotating flows [S. Chen, J. Tölke, and M. Krafczyk, Phys. Rev. E 79, 016704 (2009); S. Chen, J. Tölke, S. Geller, and M. Krafczyk, Phys. Rev. E 78, 046703 (2008)]. The flow field and the temperature field both are solved by the D2Q5 model. Compared with the previous models, the present model is more efficient, more stable, and much simpler. It was found that, even though with a relatively low grid resolution, the present model can still work well when the Grashof number is very high. The advantages of the present model are validated by numerical experiments.

Theoretical and numerical study of axisymmetric lattice Boltzmann models

The forcing term in the lattice Boltzmann equation (LBE) is usually used to mimic Navier-Stokes equations with a body force. To derive axisymmetric model, forcing terms are incorporated into the two-dimensional (2D) LBE to mimic the additional axisymmetric contributions in 2D Navier-Stokes equations in cylindrical coordinates. Many axisymmetric lattice Boltzmann D2Q9 models were obtained through the Chapman-Enskog expansion to recover the 2D Navier-Stokes equations in cylindrical coordinates [I. Halliday, Phys. Rev. E 64, 011208 (2001); K. N. Premnath and J. Abraham, Phys. Rev. E 71, 056706 (2005); T. S. Lee, H. Huang, and C. Shu, Int. J. Mod. Phys. C 17, 645 (2006); T. Reis and T. N. Phillips, Phys. Rev. E 75, 056703 (2007); J. G. Zhou, Phys. Rev. E 78, 036701 (2008)]. The theoretical differences between them are discussed in detail. Numerical studies were also carried out by simulating two different flows to make a comparison on these models' accuracy and tau sensitivity. It is found all these models are able to obtain accurate results and have the second-order spatial accuracy. However, the model C [J. G. Zhou, Phys. Rev. E 78, 036701 (2008)] is the most stable one in terms of tau sensitivity. It is also found that if density of fluid is defined in its usual way and not directly relevant to source terms, the lattice Boltzmann model seems more stable.

Theory of the lattice Boltzmann equation: Lattice Boltzmann model for axisymmetric flows

A lattice Boltzmann equation (LBE) for axisymmetric flows is proposed. The model has some distinct features that distinguish it from existing axisymmetric LBE models. First, it is derived from the Boltzmann equation so that it has a solid physics base and is easy for generalization; second, the model can describe the axial, radial, and azimuthal velocity components in the same fashion; and third, the source terms of the model contain no velocity gradients and are much simpler than other LBE models. Numerical tests, including steady and unsteady internal and external flows, demonstrate that the proposed LBE can serve as a viable and efficient method for low speed axisymmetric flows.

Validation of multicomponent lattice Boltzmann equation simulations using theoretical calculations of immiscible drop shape

Quantitative comparison between the measured deformation of a neutrally buoyant drop, obtained with an appropriately conceived three-dimensional, multicomponent lattice Boltzmann equation simulation methods for continuum multicomponent hydrodynamics [Phys. Rev. E 76, 026708 (2007); 76, 026709 (2007)], are shown to be in agreement with the theoretical predictions of Taylor and Acrivos [J. Fluid. Mech. 18(3), 466 (1964)].

Simulation of buoyancy-driven flows in a vertical cylinder using a simple lattice Boltzmann model

Axisymmetric thermal flow is of fundamental interest and practical importance. However, the work to design suitable and efficient lattice Boltzmann models on axisymmetric thermal flows is quite rare. In order to bridge the gap, a simple lattice Boltzmann model for axisymmetric thermal flow is proposed in this paper. In the present study, we show how to transform the governing equation for temperate field in the cylindrical coordinate system to the pseudo-Cartesian representation in the same way as that for the flow field. Therefore the flow field and the temperature field both are solved by the two-dimensional five-speed (D2Q5) lattice Boltzmann model. The treatment of the "geometrical forcing" due to the coordinate transformation and the physical forcing due to the temperature field is simpler than that in all existing models. Thanks to its intrinsic features, the present model is more efficient, more stable, and much simpler than the existing models. In this paper, several kinds of nontrivial thermal buoyancy-driven flows in vertical cylinders, which are of interest from the standpoint of both basic fluid dynamics and practical applications, are simulated by the present model.

Lattice Boltzmann model for incompressible axisymmetric flows

A lattice Boltzmann model for incompressible axisymmetric flow is proposed in this paper. Unlike previous axisymmetric lattice Boltzmann models, which were based on "primitive-variables" Navier-Stokes equations, the target macroscopic equations of the present model are vorticity-stream-function formulations. Due to the intrinsic features of vorticity-stream-function formulations, the present model is more efficient, more stable, and much simpler than the existing models. The advantages of the present model are validated by numerical experiments.

Axisymmetric lattice Boltzmann method

A lattice Boltzmann method is developed for incompressible axisymmetric flows. Both force and source or sink terms are incorporated into the lattice Boltzmann equation in a natural way, which is consistent in dimension with the lattice Boltzmann equation. The correct macroscopic equations for incompressible axisymmetric flows are recovered through the Chapman-Enskog expansion. It turns out that the added terms are nothing but the additional in the governing equations for the axisymmetric flows compared with the Navier-Stokes equations, resulting in a simple and efficient model. This provides an additional unique advantage that the proposed scheme is naturally suitable for general axisymmetric flows involving more physical phenomena. Two numerical simulations have been presented to verify the method.

Multicomponent lattice Boltzmann method for fluids with a density contrast

We present and verify a multicomponent lattice Boltzmann simulation scheme for two immiscible and incompressible fluids with a large density contrast. Our method is constructed from a continuum approximation description of a single inhomogeneous, and essentially incompressible, fluid. The equations that arise from this analysis are mapped onto an established multicomponent lattice Boltzmann method. The approach avoids the computational expense of a numerical solution of the fluid pressure field in a separate step. We present results obtained with our model which validate the initial assumptions and verify correct static and dynamic operation of the model up to a fluid density contrast ratio of more than 500. The paper concludes with an example that illustrates the potential utility of the approach by modeling a gas bubble rising under gravity and breaking through a free surface.

Numerical validation of a consistent axisymmetric lattice Boltzmann model

A recently derived axisymmetric lattice Boltzmann model is evaluated numerically. The model incorporates a spatially and temporally varying source term into the evolution equation for the momentum distribution function on a two-dimensional Cartesian lattice. The precise form of the source term is derived through a Chapman-Enskog analysis so that the additional axisymmetric contributions in the Navier-Stokes equations are furnished when written in the cylindrical polar coordinate system. The validity of the model is confirmed by simulating Hagen-Poiseuille flow. Numerical predictions for the drag coefficient in Stokes' flow over a sphere are presented and shown to be in excellent agreement with analytical results. At larger Reynolds numbers the numerical predictions are shown to approach an empirically derived formula for the drag coefficient.

Lattice Boltzmann algorithm for continuum multicomponent flow

We present a multicomponent lattice Boltzmann simulation for continuum fluid mechanics, paying particular attention to the component segregation part of the underlying algorithm. In the principal result of this paper, the dynamics of a component index, or phase field, is obtained for a segregation method after U. D'Ortona [Phys. Rev. E 51, 3718 (1995)], due to Latva-Kokko and Rothman [Phys. Rev. E 71 056702 (2005)]. The said dynamics accord with a simulation designed to address multicomponent flow in the continuum approximation and underwrite improved simulation performance in two main ways: (i) by reducing the interfacial microcurrent activity considerably and (ii) by facilitating simulational access to regimes of flow with a low capillary number and drop Reynolds number [I. Halliday, R. Law, C. M. Care, and A. Hollis, Phys. Rev. E 73, 056708 (2006)]. The component segregation method studied, used in conjunction with Lishchuk's method [S. V. Lishchuk, C. M. Care, and I. Halliday, Phys. Rev. E 67, 036701 (2003)], produces an interface, which is distributed in terms of its component index; however, the hydrodynamic boundary conditions which emerge are shown to support the notion of a sharp, unstructured, continuum interface.

Modified lattice Boltzmann model for axisymmetric flows

A modified lattice Boltzmann model based on the two-dimensional, nine-velocity lattice-Bhatnagar-Gross-Krook fluid is presented for axisymmetric flows. A spatially and temporally varying source term is incorporated into the evolution equation for the momentum distribution function on a two-dimensional Cartesian lattice. The precise form of the source term is derived through a Chapman-Enskog analysis, so that the additional axisymmetric contributions in the Navier-Stokes equations are furnished when written in the cylindrical polar coordinate system.

Lattice Boltzmann simulations of two-phase flow with high density ratio in axially symmetric geometry

In this paper, a two-phase lattice Boltzmann (LB) model, developed for simulating fluid flows on a Cartesian grid at high liquid-to-gas density ratios, is adapted to an axisymmetric coordinate system. This is achieved by incorporating additional source terms in the planar evolution equations for the density and pressure distribution functions such that the axisymmetric mass and momentum conservation equations are recovered in the macroscopic limit. Appropriate numerical treatment of the terms is performed to obtain stable computations at high density ratio for this axisymmetric model. The particle collision is modeled by employing multiple relaxation times to attain stability at low viscosity. The model is evaluated by verifying the Laplace-Young relation for a liquid drop, comparing computed frequency of oscillations of an initially ellipsoidal drop with analytical values and comparing the behavior of a spherical drop impinging on a wet wall with prior results. The time evolution of the radial distance of the tip of the corona, formed when the drop impinges, agrees well with prior data.

Improved simulation of drop dynamics in a shear flow at low Reynolds and capillary number

The simulation of multicomponent fluids at low Reynolds number and low capillary number is of interest in a variety of applications such as the modeling of venule scale blood flow and microfluidics; however, such simulations are computationally demanding. An improved multicomponent lattice Boltzmann scheme, designed to represent interfaces in the continuum approximation, is presented and shown (i) significantly to reduce common algorithmic artifacts and (ii) to recover full Galilean invariance. The method is used to model drop dynamics in shear flow in two dimensions where it recovers correct results over a range of Reynolds and capillary number greater than that which may be addressed with previous methods.

Lattice Boltzmann model for axisymmetric multiphase flows

A lattice Boltzmann model is presented for axisymmetric multiphase flows. Source terms are added to a two-dimensional standard lattice Boltzmann equation for multiphase flows such that the emergent dynamics can be transformed into the axisymmetric cylindrical coordinate system. The source terms are temporally and spatially dependent and represent the axisymmetric contribution of the order parameter of fluid phases and inertial, viscous, and surface tension forces. A model which is effectively explicit and second order is obtained. This is achieved by taking into account the discrete lattice effects in the Chapman-Enskog multiscale analysis, so that the macroscopic axisymmetric mass and momentum equations for multiphase flows are recovered self-consistently. The model is extended to incorporate reduced compressibility effects. Axisymmetric equilibrium drop formation and oscillations, breakup and formation of satellite droplets from viscous liquid cylindrical jets through Rayleigh capillary instability, and drop collisions are presented. Comparisons of the computed results with available data show satisfactory agreement.

A many-component lattice Boltzmann equation simulation for transport of deformable particles

We review the analysis of single and N-component lattice Boltzmann methods for fluid flow simulation. Results are presented for the emergent pressure field of a single phase incompressible liquid flowing over a backward-facing step, at moderate Reynolds Number, which is compared with the experimental data of Denham & Patrick (1974 Trans. IChE 52, 361-367). We then access the potential of the N-component method for transport of high volume fraction suspensions of deformable particles in pressure-driven flow. The latter are modelled as incompressible, closely packed liquid drops. We demonstrate the technique by investigating the particles' transverse migration in a uniform shear ('lift'), and profile blunting and chaining.