Discrete Lattice Boltzmann Equation Research Articles

Stability and bifurcation of annular electro-thermo-convection

Abstract We numerically investigated the global linear instability and bifurcations in electro-thermo-convection (ETC) of a dielectric liquid confined in a two-dimensional (2-D) concentric annulus subjected to a strong unipolar injection. Seven kinds of solutions exist in this ETC system due to the complex bifurcations, i.e. saddle-node, subcritical and supercritical Hopf bifurcations. These bifurcation routes constitute at most four solution branches. Global linear instability analysis and energy analysis were conducted to explain the instability mechanism and transition of different solutions and to predict the local instability regions. The linearized lattice Boltzmann method (LLBM) for global linear instability analysis, first proposed by Pérez et al. (Theor. Comput. Fluid Dyn., vol. 31, 2017, pp. 643–664) to analyse incompressible flows, was extended here to solve the whole set of coupled linear equations, including the linear Navier–Stokes equations, the linear energy equation, Poisson's equation and the linear charge conservation equation. A multiscale analysis was also performed to recover the macroscopic linearized Navier–Stokes equations from the four different discrete lattice Boltzmann equations (LBEs). The LLBM was validated by calculating the linear critical value of 2-D natural convection; it has an error of 1.39% compared with the spectral method. Instability with global travelling wave behaviour is a unique behaviour in the annulus configuration electrothermohydrodynamic system, which may be caused by the baroclinity. Finally, the chaotic behaviour was quantitatively analysed through calculation of the fractal dimension and Lyapunov exponent.

Global linear instability analysis of thermal convective flow using the linearized lattice Boltzmann method

Modal global linear stability analysis of thermal convection is performed with the linearized lattice Boltzmann method (LLBM). The onset of Rayleigh–Bénard convection in rectangular cavities with conducting and adiabatic sidewalls and the instability of two-dimensional (2-D) and three-dimensional (3-D) natural convection in cavities are studied. The method of linearizing the local equilibrium probability distribution function that was first proposed by Pérez et al. (Theor. Comp. Fluid Dyn., vol. 31, 2017, pp. 643–664) is extended to solve the coupled linear Navier–Stokes equations together with the linear energy equation in this work. A multiscale analysis is also performed to recover the macroscopic linear Navier–Stokes equations from the discrete lattice Boltzmann equations for both the single and multiple relaxation time models. The present LLBM is implemented in the framework of the Palabos library. It is validated by calculating the linear critical value of 2-D natural convection that the LLBM with the multiple relaxation time model has an error less than 1 % compared with the spectral method. The instability mechanism of the flow is explained by kinetic energy transfer analysis. It is shown that the buoyancy mechanism and inertial mechanism tend to stabilize the Hopf bifurcation of the 2-D natural convection at Pr < 0.08 and Pr > 1, respectively. For 3-D natural convection, subcritical bifurcation of the Hopf type is found for low-Prandtl-number fluids (Pr < 0.1).

The immersed boundary-lattice Boltzmann method for solving solid-fluid interaction problem with Navier-slip boundary condition

To solve the solid-fluid interaction with Navier-slip boundary condition, we propose a new method in this paper. In this approach, the solid body is modeled by the Newton’s equations and the fluid is assumed to satisfy the discrete lattice Boltzmann equations. The Navier-slip boundary condition is applied as the velocity boundary condition for the solid-fluid interface. We discretize the Newton’s equations by a prediction/correction approach and the discrete lattice Boltzmann equations by a second-order finite difference scheme in space together with a second-order Range–Kutta scheme in time. The Navier-slip boundary condition and the immersed boundary body force are almost implicitly updated by using the implicit velocity correction-based immersed boundary-lattice Boltzmann method. Numerical simulations are carried out to verify the new presented method.

A Lattice Boltzmann-Immersed Boundary-Finite Element Method for Nonlinear Fluid–Solid Interaction Simulation with Moving Objects

A nonlinear finite element method (FEM) was introduced into the immersed boundary (IB) — lattice Boltzmann method (LBM) framework to simulate the nonlinear fluid–solid interactions for moving deformable objects in incompressible fluid flow. The fluid motion is obtained by solving the discrete lattice Boltzmann equation, the moving boundaries of the solids are handled by the IB method, and the nonlinear dynamics of the deforming/moving objects are calculated by nonlinear FEM solvers in which the structural nodes are also set as Lagrangian markers to track the moving boundaries of the structures in IB method. The simulation results indicate that the proposed IB-LBM-FEM simulation framework not only satisfies the nonslip boundary condition well at the boundary points, but also has better accuracy for capturing nonlinearity because of its mature nonlinear FEM solver.

A novel geometry-adaptive Cartesian grid based immersed boundary–lattice Boltzmann method for fluid–structure interactions at moderate and high Reynolds numbers

A novel computational framework which combines the lattice Boltzmann method (LBM) and an improved immersed boundary method (IBM) based on a dynamic geometry-adaptive Cartesian grid system is introduced for the fluid–structure interaction (FSI) problems at moderate and high Reynolds numbers. In this framework, the fluid dynamics is obtained by solving the discrete lattice Boltzmann equation. The boundary conditions at the fluid–structure interfaces are handled by an improved IBM based on a feedback scheme, which drives the predicted flow velocity (calculated after the LBM stream process without the IBM body force) near the immersed boundaries to match the solid velocity. In the present IBM, the feedback coefficient is mathematically derived and explicitly approximated. The Lagrangian force density is divided into two parts: one is the traction caused by the predicted flow velocity, and the other is caused by the acceleration of the immersed boundary. Such treatment significantly enhances the numerical stability for modelling FSI problems involving small structure-to-fluid mass ratios. A novel dynamic geometry-adaptive refinement is applied to provide fine resolution around the immersed geometries and coarse resolution in the far field. The overlapping grids between two adjacent refinements consist of two layers. In order to enhance the numerical stability, two-layer “ghost nodes” are generated within the immersed body domain which is a non-fluid area. The movement of fluid–structure interfaces only causes adding or removing grids at the boundaries of refinements and consequently a high mesh-update efficiency is guaranteed. Finally, large eddy simulation models are incorporated into the framework to model turbulent flows at relatively high Reynolds numbers. Several validation cases, including an impulsively started flow over a vertical plate, flow over stationary and oscillating cylinders, flow over flapping foils, flexible filaments in a uniform flow, turbulent flow over a wavy boundary, flow over a stationary sphere and a dragonfly in hovering flight, are conducted to verify the accuracy and fidelity of the present solver over a range of Reynolds numbers.

Numerical investigation of nonlinear fluid–structure interaction dynamic behaviors under a general Immersed Boundary-Lattice Boltzmann-Finite Element method

A numerical approach based on the immersed boundary (IB), lattice Boltzmann and nonlinear finite element method (FEM) is proposed to simulate hydrodynamic interactions of very flexible objects. In the present simulation framework, the motion of fluid is obtained by solving the discrete lattice Boltzmann equations on Eulerian grid, the behaviors of flexible objects are calculated through nonlinear dynamic finite element method, and the interactive forces between them are implicitly obtained using velocity correction IB method which satisfies the no-slip conditions well at the boundary points. The efficiency and accuracy of the proposed Immersed Boundary-Lattice Boltzmann-Finite Element method is first validated by a fluid–structure interaction (F-SI) benchmark case, in which a flexible filament flaps behind a cylinder in channel flow, then the nonlinear vibration mechanism of the cylinder-filament system is investigated by altering the Reynolds number of flow and the material properties of filament. The interactions between two tandem and side-by-side identical objects in a uniform flow are also investigated, and the in-phase and out-of-phase flapping behaviors are captured by the proposed method.

Unified Lattice Boltzmann Method for Electric Field–Space Charge Coupled Problems in Complex Geometries and Its Applications to Annular Electroconvection

A lattice Boltzmann method (LBM) is developed to solve the electric field–space charge coupled problems. Instead of solving the macroscopic charge conservation equation and the Poisson's equation for electric potential, two discrete lattice Boltzmann equations are formulated and solved. A nonequilibrium extrapolation scheme is used to treat the boundary conditions with complex geometry. Our method is validated with several test cases for which analytical solutions and/or reference numerical results exit. An attractive feature of this methodology lies in its natural coupling with the LBM for the fluid flow. As a demonstration and also an application, the unipolar injection induced electroconvection of dielectric liquids in annular geometries is considered. The different flow patterns of the concentric and eccentric configurations are highlighted.

Lattice Boltzmann modelling of electro-thermo-convection in a planar layer of dielectric liquid subjected to unipolar injection and thermal gradient

In this paper, the electro-thermo-convective phenomena induced by the simultaneous action of a unipolar injection of ions and a thermal gradient in a dielectric liquid lying between two parallel plates are studied. We develop a lattice Boltzmann model (LBM) to solve the whole set of coupled governing equations, including the Navier–Stokes equations, the conservation equation of charge density, the Poisson’s equation for electric potential and the energy equation. In this method, four different particle distribution functions are used to calculate the flow field, electric potential, charge density distribution and temperature field, respectively. A multi-scale analysis is also performed to recover the macroscopic equations from the discrete lattice Boltzmann equations (LBEs). The present method is validated with several carefully chosen test cases, and all LBM results are found to be highly consistent with available analytical solutions or other numerical works. Besides, the typical subcritical bifurcations and the hysteresis loops in electro-thermo-convective are clearly presented and analyzed.

Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I: Mathematical Derivation of Adjoint Lattice Boltzmann Equations

AbstractThe significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector. Moreover, this approach was not limited to flow fields and could be employed for steady as well as unsteady flows. Initially, the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time, respectively. Meanwhile, new adjoint concepts in lattice space were introduced. Finally, the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.

Immiscible multicomponent lattice Boltzmann model for fluids with high relaxation time ratio

An immiscible multicomponent lattice Boltzmann model is developed for fluids with high relaxation time ratios, which is based on the model proposed by Shan and Chen (SC). In the SC model, an interaction potential between particles is incorporated into the discrete lattice Boltzmann equation through the equilibrium velocity. Compared to the SC model, external forces in our model are discretized directly into the discrete lattice Boltzmann equation, as proposed by Guo et al. We develop it into a new multicomponent lattice Boltzmann (LB) model which has the ability to simulate immiscible multicomponent fluids with relaxation time ratio as large as 29.0 and to reduce ‘spurious velocity’. In this work, the improved model is validated and studied using the central bubble case and the rising bubble case. It finds good applications in both static and dynamic cases for multicomponent simulations with different relaxation time ratios.

Lattice Boltzmann model for the volume-averaged Navier-Stokes equations

A numerical method, based on the discrete lattice Boltzmann equation, is presented for solving the volume-averaged Navier-Stokes equations. With a modified equilibrium distribution and an additional forcing term, the volume-averaged Navier-Stokes equations can be recovered from the lattice Boltzmann equation in the limit of small Mach number by the Chapman-Enskog analysis and Taylor expansion. Due to its advantages such as explicit solver and inherent parallelism, the method appears to be more competitive with traditional numerical techniques. Numerical simulations show that the proposed model can accurately reproduce both the linear and nonlinear drag effects of porosity in the fluid flow through porous media.

Lattice Boltzmann investigation of acoustic damping mechanism and performance of an in-duct circular orifice.

In this work, three-dimensional numerical simulations of acoustically excited flow through a millimeter-size circular orifice are conducted to assess its noise damping performance, with particular emphasis on applying the lattice Boltzmann method (LBM) as an alternative computational aeroacoustics tool. The model is intended to solve the discrete lattice Boltzmann equation (LBE) by using the pseudo-particle based technique. The LBE controls the particles associated with collision and propagation over a discrete lattice mesh. Flow variables such as pressure, density, momentum, and internal energy are determined by performing a local integration of the particle distribution at each time step. This is different from the conventional numerical investigation attempting to solve Navier-Stokes (NS) equations by using high order finite-difference or finite-volume methods. Compared with the conventional NS solvers, one of the main advantages of LBM may be a reduced computational cost. Unlike frequency domain simulations, the present investigation is conducted in time domain, and the orifice damping behavior is quantified over a broad frequency range at a time by forcing an oscillating flow with multiple tones. Comparing the numerical results with those obtained from the theoretical models, large eddy simulation, and experimental measurements, good agreement is observed.

IB-LBM simulation on blood cell sorting with a micro-fence structure

A size-based blood cell sorting model with a micro-fence structure is proposed in the frame of immersed boundary and lattice Boltzmann method (IB-LBM). The fluid dynamics is obtained by solving the discrete lattice Boltzmann equation, and the cells motion and deformation are handled by the immersed boundary method. A micro-fence consists of two parallel slope post rows which are adopted to separate red blood cells (RBCs) from white blood cells (WBCs), in which the cells to be separated are transported one after another by the flow into the passageway between the two post rows. Effected by the cross flow, RBCs are schemed to get through the pores of the nether post row since they are smaller and more deformable compared with WBCs. WBCs are required to move along the nether post row till they get out the micro-fence. Simulation results indicate that for a fix width of pores, the slope angle of the post row plays an important role in cell sorting. The cells mixture can not be separated properly in a small slope angle, while obvious blockages by WBCs will take place to disturb the continuous cell sorting in a big slope angle. As an optimal result, an adaptive slope angle is found to sort RBCs form WBCs correctly and continuously.

An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments

We have introduced a modified penalty approach into the flow-structure interaction solver that combines an immersed boundary method (IBM) and a multi-block lattice Boltzmann method (LBM) to model an incompressible flow and elastic boundaries with finite mass. The effect of the solid structure is handled by the IBM in which the stress exerted by the structure on the fluid is spread onto the collocated grid points near the boundary. The fluid motion is obtained by solving the discrete lattice Boltzmann equation. The inertial force of the thin solid structure is incorporated by connecting this structure through virtual springs to a ghost structure with the equivalent mass. This treatment ameliorates the numerical instability issue encountered in this type of problems. Thanks to the superior efficiency of the IBM and LBM, the overall method is extremely fast for a class of flow-structure interaction problems where details of flow patterns need to be resolved. Numerical examples, including those involving multiple solid bodies, are presented to verify the method and illustrate its efficiency. As an application of the present method, an elastic filament flapping in the Kármán gait and the entrainment regions near a cylinder is studied to model fish swimming in these regions. Significant drag reduction is found for the filament, and the result is consistent with the metabolic cost measured experimentally for the live fish.

Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion

Cancer invasion is the process of cells detaching from a primary tumor and infiltrating the healthy tissue. Cancer invasion has been recognized as a complex system, since a tumor’s invasive behavior emerges from the combined effect of tumor cell proliferation, tumor cell migration and cell–microenvironment interactions. Cellular automata (CA) provide simple models of self-organizing complex systems in which collective behavior can emerge out of an ensemble of many interacting “simple” components. Here, we introduce a lattice-gas cellular automaton (LGCA) model of tumor cell proliferation, necrosis and tumor cell migration. The impact of the tumor environment on tumor cells has been investigated in a previous study. Our analysis aims at predicting the velocity of the traveling invasion front, which depends upon fluctuations that arise from the motion of the discrete cells at the front. We find an excellent agreement between the velocities measured in simulations of the LGCA and an analytical estimate derived in the cut-off mean-field approximation via the discrete Lattice Boltzmann equation and its linearization. In particular, we predict the front velocity to scale with the square root of the product of probabilities for mitosis and the migration coefficient. Finally, we calculate the width of the traveling front which is found to be proportional to the front velocity.

From reactive lattice gas automaton rules to its partial differential equations

We present a general statistical mechanical theory of multispecies lattice gas automata (LGA) for reaction-diffusion systems. We describe how reactive and elastic collisions and free streaming of molecules between collisions are modeled by LGA chemical transformation, mixing and propagation steps. We characterize these steps in terms of microdynamical variables and transition probabilities. From these steps we construct diffusion and reaction-diffusion automaton dynamics. We describe these dynamics in terms of LGA microdynamical equations and discrete lattice Boltzmann equations. From the LGA discrete Boltzmann equations we derive the corresponding LGA macroscopic diffusion and reaction-diffusion PDEs. By comparing these equations with those arising from phelattice gas research J.P. Boon, D. Dab, K. Diemer, D. Gruner, B. Hasslacher, R. Kapral, P. Masiar, S. Ponce-Dawson and J.P. Voroney.