Lattice Boltzmann methods for two-phase flow modeling

Investigation of the Different Regimes Associated with the Growth of an Interface at the Exit of a Capillary Tube into a Reservoir: Analytical Solutions and CFD Validation

The emergence of a droplet from a capillary tube opening into a reservoir is an important phenomenon in several applications. In this work, we are particularly interested in this phenomenon in an attempt to highlight the physics behind droplet appearance. The emergence of a droplet from a tube opening into a reservoir under quasi-static conditions passes through three stages. The first stage starts when the meniscus in the tube reaches the exit. At this moment, the meniscus intersects the wall of the tube at the equilibrium contact angle. The interface then develops until its radius of curvature becomes equal to the tube radius. During this stage, the capillary pressure increases. In the second stage, the interface continues to evolve with its radius of curvature increasing until the static contact angle with respect to the surface of the reservoir is achieved. This marks the end of the second stage and the start of the third in which the contact line (CL) starts to depart the tube opening along the reservoir surface and the contact angle remains constant. Analytical models for the three stages have been derived based on the law of conservation of linear momentum. The models account for pressure, gravitational, capillary, and viscous forces, while inertia force is ignored. The model can predict the profiles of the mean velocity in the tube, the capillary pressure, and the evolution of the contact angle. In addition, a computational fluid dynamics (CFD) simulation has been conducted to provide a framework for validation and verification of the developed model. The CFD simulation shows qualitative behavior in terms of snapshots of the emerging droplet with time similar to that speculated by the analytical model. In addition, quantitative comparisons with respect to velocity, pressure, and volume profiles of the droplet show very good agreement, which builds confidence in the modeling approach.

Lattice Boltzmann solution of the transient Boltzmann transport equation in radiative and neutron transport

Applications of the transient Boltzmann transport equation (BTE) have undergone much investigation, such as radiative heat transfer and neutron transport. This paper provides a lattice Boltzmann model to efficiently resolve the multidimensional transient BTE. For a higher angular resolution, enough transport directions are considered while the transient BTE in each direction is treated as a conservation law equation and solved independently. Both macroscopic equations recovered from a Chapman-Enskog expansion and simulated results of typical benchmark problems show not only the second-order accuracy but also the flexibility and applicability of the proposed lattice Boltzmann model. This approach may contribute a powerful technique for the parallel simulation of large-scale engineering and some alternative perspectives for solving the nonlinear transport problem further.

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.

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.

Pore-scale studies of spontaneous imbibition into oil-saturated porous media

Spontaneous imbibition phenomenon was modeled using the lattice Boltzmann method (LBM). The model was validated using silicon-etched micromodel and sandpack visualization experiments. The strongly water-wet model saturated with oil (kerosene or mineral oil) was exposed to distilled water in order for a capillary interaction to take place under static conditions. These experiments mimic the transfer between rock matrix and fracture during any wetting phase flow in fractures while the matrix contains a nonwetting phase as encountered in oil, gas, and geothermal reservoirs as well as during the application of subsurface CO2 sequestration or waste disposal reservoirs. Despite the vast amount of research work on this common process, pore-scale investigations and modeling are limited especially at small time scales. The results showed that the LBM captures the physics of the process at pore scale for low viscosity values of nonwetting phase for any type of (cocurrent or countercurrent imbibition) interaction.

Analysis and reduction of the spurious current in a class of multiphase lattice Boltzmann models

We show that the spurious current present near a curved interface in a class of multiphase lattice Boltzmann (LB) models is due to the insufficient isotropy of the discrete gradient operator. A method of obtaining highly isotropic gradient operators on a lattice is given. Numerical simulations show that both the magnitude and the spatial extent of the spurious current are significantly reduced as gradient operators of increasingly higher order of isotropy is adopted in multiphase LB models.

Two-fluid model based on the lattice Boltzmann equation

A two-fluid model for dispersed two-phase flows based on the lattice Boltzmann equation (LBE) is proposed. Two sets of LBEs are used to describe the two phases. The continuity and Navier-Stokes equations for the continuum-based two-fluid model are obtained from the LBEs by adding a pressure term. The phase hold-up is readily calculated from the partial pressure of each phase because the two phases are described by the ideal gas equation of state. A simulated laminar gas-liquid two-phase flow gave good agreement with the results calculated by the conventional continuum-based two-fluid model.

Accuracy of the lattice Boltzmann method based on analytical solutions

In this paper, a simple method is proposed to obtain steady analytical solutions for the lattice Boltzmann method. Based on such analytical results, it is demonstrated how the accuracy of the lattice Boltzmann method can depend on the relative orientation of the lattice and the flow field. It is also demonstrated that the method can be useful to obtain a general class of analytical solutions for the lattice Boltzmann method. Finally, a simple relation is given between the compressibility error and the velocity field.

Three-dimensional lattice-Boltzmann model of van der Waals fluids

A three-dimensional lattice-Boltzmann model is developed for the simulation of nonideal fluids under static and flow conditions. The van der Waals formulation of quasilocal thermodynamics for nonuniform fluids is used, and the interfacial stress tensor for nonideal fluids appears explicitly in the hydrodynamic equations. The continuity and flow equations are fully recovered, and Galilean invariance is restored through appropriate manipulations of the pressure tensor. Although applied here to the D3Q15 lattice, the methodology of Galilean restoration can be easily modified for use with other three-dimensional lattices as well. The Laplace law and Gibbs-Thomson equations are satisfied with excellent accuracy by the model, as demonstrated by droplet equilibrium simulations. Spinodal decomposition and droplet coalescence simulations are also carried out, revealing a direct proportionality of the characteristic times to the viscosity, as expected. A wettability adjustment was made possible through the prescription of a chemical potential profile along the fluid-wall interface, and used for the simulation of droplet formation from a conical orifice.