Shear stress in nonideal fluid lattice Boltzmann simulations

Modified phase-field-based lattice Boltzmann model for incompressible multiphase flows

Based on the phase-field theory, a multiple-relaxation-time (MRT) lattice Boltzmann model is proposed for the immiscible multiphase fluids. In this model, the local Allen-Chan equation is chosen as the target equation to capture the phase interface. Unlike previous MRT schemes, an off-diagonal relaxation matrix is adopted in the present model so that the target phase-field equation can be recovered exactly without any artificial terms. To check the necessity of removing those artificial terms, comparative studies were carried out among different MRT schemes with or without correction. Results show that the artificial terms can be neglected at low March number but will cause unphysical diffusion or interface undulation instability for the relatively large March number cases. The present modified model shows superiority in reducing numerical errors by adjusting the free parameters. As the interface transport coupled to the fluid flow, a pressure-evolution lattice Boltzmann equation is adopted for hydrodynamic properties. Several benchmark cases for multiphase flow were conducted to test the validity of the present model, including the static drop test, Rayleigh-Taylor instability, and single rising bubble test. For the rising bubble simulation at high density ratios, bubble dynamics obtained by the present modified MRT lattice Boltzmann model agree well with those obtained by the FEM-based level set and FEM-based phase-field models.

Interface tracking characteristics of color-gradient lattice Boltzmann model for immiscible fluids

We study the interface tracking characteristics of a color-gradient-based lattice Boltzmann model for immiscible flows. Investigation of the local density change in one of the fluid phases, via a Taylor series expansion of the recursive lattice Boltzmann equation, leads to the evolution equation of the order parameter that differentiates the fluids. It turns out that this interface evolution follows a conservative Allen-Cahn equation with a mobility which is independent of the fluid viscosities and surface tension. The mobility of the interface, which solely depends upon lattice speed of sound, can have a crucial effect on the physical dynamics of the interface. Further, we find that, when the equivalent lattice weights inside the segregation operator are modified, the resulting differential operators have a discretization error that is anisotropic to the leading order. As a consequence, the discretization errors in the segregation operator, which ensures a finite interface width, can act as a source of the spurious currents. These findings are supported with the help of numerical simulations.

Interaction pressure tensor on high-order lattice Boltzmann models for nonideal fluids

In this work we address the application of pseudopotentials directly on high-order lattice Boltzmann models. We derive a general expression for the pressure tensor on high-order lattices considering all nonideal interactions, including intra- and intermolecular interactions, following the discrete lattice theory introduced by X. Shan [Phys. Rev. E 77, 066702 (2008)PLEEE81539-375510.1103/PhysRevE.77.066702]. From the derived expression, a generalized continuum approximation, truncated at fourth-order isotropy, is obtained that is readily applicable to high-order lattices. With this, we demonstrate that high-order lattice models with pseudopotentials can satisfy thermodynamic consistency. The derived generalized expression and continuum approximation are validated for the case of a flat interface and compared against the standard definition available from the literature. The generalized expression is also shown to accurately reproduce the Laplace experiment for a variety of high-order lattice structures. This work sets the preliminary steps towards the application of high-order lattice models for simulating nonideal fluid mixtures.

Fluctuating multicomponent lattice Boltzmann model

Current implementations of fluctuating lattice Boltzmann equations (FLBEs) describe single component fluids. In this paper, a model based on the continuum kinetic Boltzmann equation for describing multicomponent fluids is extended to incorporate the effects of thermal fluctuations. The thus obtained fluctuating Boltzmann equation is first linearized to apply the theory of linear fluctuations, and expressions for the noise covariances are determined by invoking the fluctuation-dissipation theorem directly at the kinetic level. Crucial for our analysis is the projection of the Boltzmann equation onto the orthonormal Hermite basis. By integrating in space and time the fluctuating Boltzmann equation with a discrete number of velocities, the FLBE is obtained for both ideal and nonideal multicomponent fluids. Numerical simulations are specialized to the case where mean-field interactions are introduced on the lattice, indicating a proper thermalization of the system.

Pattern formation in liquid-vapor systems under periodic potential and shear

In this paper the phase behavior and pattern formation in a sheared nonideal fluid under a periodic potential is studied. An isothermal two-dimensional formulation of a lattice Boltzmann scheme for a liquid-vapor system with the van der Waals equation of state is presented and validated. Shear is applied by moving walls and the periodic potential varies along the flow direction. A region of the parameter space, where in the absence of flow a striped phase with oscillating density is stable, will be considered. At low shear rates the periodic patterns are preserved and slightly distorted by the flow. At high shear rates the striped phase loses its stability and traveling waves on the interface between the liquid and vapor regions are observed. These waves spread over the whole system with wavelength only depending on the length of the system. Velocity field patterns, characterized by a single vortex, will also be shown.

Sliding drops across alternating hydrophobic and hydrophilic stripes

We perform a joint numerical and experimental study to systematically characterize the motion of 30 μl drops of pure water and of ethanol in water solutions, sliding over a periodic array of alternating hydrophobic and hydrophilic stripes with a large wettability contrast and a typical width of hundreds of microns. The fraction of the hydrophobic areas has been varied from about 20% to 80%. The effects of the heterogeneous patterning can be described by a renormalized value of the critical Bond number, i.e., the critical dimensionless force needed to depin the drop before it starts to move. Close to the critical Bond number we observe a jerky motion characterized by an evident stick-slip dynamics. As a result, dissipation is strongly localized in time, and the mean velocity of the drops can easily decrease by an order of magnitude compared to the sliding on the homogeneous surface. Lattice Boltzmann numerical simulations are crucial for disclosing to what extent the sliding dynamics can be deduced from the computed balance of capillary, viscous, and body forces by varying the Bond number, the surface composition, and the liquid viscosity. Beyond the critical Bond number, we characterize both experimentally and numerically the dissipation inside the droplet by studying the relation between the average velocity and the applied volume forces.

Interaction pressure tensor for a class of multicomponent lattice Boltzmann models

We present a theory to obtain the pressure tensor for a class of nonideal multicomponent lattice Boltzmann models, thus extending the theory presented by X. Shan [Phys. Rev. E 77, 066702 (2008)] for single-component fluids. We obtain the correct form of the pressure tensor directly on the lattice and the resulting equilibrium properties are shown to agree very well with those measured from numerical simulations. Results are compared with those of alternative theories.

Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows

A phase-field-based hybrid model that combines the lattice Boltzmann method with the finite difference method is proposed for simulating immiscible thermocapillary flows with variable fluid-property ratios. Using a phase field methodology, an interfacial force formula is analytically derived to model the interfacial tension force and the Marangoni stress. We present an improved lattice Boltzmann equation (LBE) method to capture the interface between different phases and solve the pressure and velocity fields, which can recover the correct Cahn-Hilliard equation (CHE) and Navier-Stokes equations. The LBE method allows not only use of variable mobility in the CHE, but also simulation of multiphase flows with high density ratio because a stable discretization scheme is used for calculating the derivative terms in forcing terms. An additional convection-diffusion equation is solved by the finite difference method for spatial discretization and the Runge-Kutta method for time marching to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. The model is first validated against analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of a three-dimensional deformable droplet and bubble at various Marangoni numbers and density ratios, and satisfactory agreement is obtained between numerical results and theoretical predictions.

Effect of the forcing term in the multiple-relaxation-time lattice Boltzmann equation on the shear stress or the strain rate tensor

In this work, the effect of the forcing term (or external force) in the multiple-relaxation-time lattice Boltzmann equation (MRTLBE) on the shear stress or the strain rate tensor is studied theoretically and numerically. Through a Chapman-Enskog analysis and numerical simulations, we show that the shear stress (or the strain rate tensor) derived from the MRTLBE is second-order accurate in space. We then examine the influence of the forcing term on the shear stress or the strain rate tensor, and demonstrate that the forcing term effect must be included when the shear stress or the strain rate tensor is computed with the nonequilibrium part of the distribution function.

Forcing term in single-phase and Shan-Chen-type multiphase lattice Boltzmann models

Numerous schemes have been proposed to incorporate a bulk forcing term into the lattice Boltzmann equation. In this paper we present a simple and straightforward comparative analysis of five popular schemes [Shan and Chen, Phys. Rev. E 47, 1815 (1993); Phys Rev Lett. 81, 1618 (1998); He et al., Phys. Rev. E 57, R13 (1998); Guo et al., Phys. Rev. E 65, 046308 (2002); Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)] in which their differences and similarities are identified. From the analysis we classify the schemes into two groups; the behaviors of the schemes in each group are proven to be identical up to second order. Numerical test simulating the two-dimensional unsteady Taylor-Green vortex flow problem demonstrate that all five schemes are of comparable accuracy for single-phase flow. However, for two-phase flow the situation is different, which is demonstrated by incorporating these schemes into different Shan-Chen-type multiphase models. The forcing scheme in the original Shan-Chen (SC) multiphase model turns out to be inaccurate in terms of the resulting surface tension for different density ratios and relaxation times. In the numerical tests, a typical equation of state and interparticle interactions including next-nearest neighbors were incorporated into the SC model. Our results confirm that the surface-tension values obtained from the original SC lattice Boltzmann method (LBM) simulation depend on the value of the relaxation time τ. For τ<0.7Δt, the surface tension agree well with the analytical solutions. However, when τ>0.7Δt, the surface tension turns out to be systematically larger than the analytical one, exceeding it by more than a factor of 2 for τ=2Δt. In contrast, with the application of the scheme proposed by He et al., the SC LBM produces very accurate surface tensions independent of the value of τ. We also found that the densities of the coexisting liquid and gas can be adjusted to match those at thermodynamic equilibrium if the particle interaction term includes next-nearest-neighbor contributions. The obtained results will be useful for further studies of two-phase flow with high density ratios using the SC LBM approach.