Finite Difference Lattice Boltzmann Method Research Articles

Differential quadrature lattice Boltzmann method for simulating fluid flows

In this study, a novel method for simulating incompressible flows using the lattice Boltzmann method is presented, aiming to improve robustness. This method is enhanced by the Qadyan numerical method. In the Qadyan method, Partial differential equations are solved using semi-discrete schemes combined with the differential quadrature method. To discretize the spatial derivatives of the lattice Boltzmann equation, the upwind differential quadrature method is employed, while the first-order forward difference scheme and/or fourth-order Runge-Kutta method handle the temporal term. The decision to exclude more complex methods stems from their requirement for refined computational grids. This work focuses on achievable results with similar types of uniform grids. The proposed scheme introduces and evaluates a novel method for solving both steady and unsteady problems. The present numerical method is validated by solving five benchmark problems, including heat diffusion on a slab, Stokes’ first problem, Taylor-Green vortex flow, flow around a flat plate, and flow in a lid-driven cavity. Reasonable agreements are obtained between the solutions obtained using this method and those from analytical/other numerical approaches. The Qadyan formulation delivers more accurate results compared to the finite-difference lattice Boltzmann method, (FDLBM), without requiring an increase in the number of grids. Notably, the novel Qadyan method achieves this increased accuracy without introducing additional complexity to the standard lattice Boltzmann method or resorting to non-uniform grids. This is possible due to the nature of the differential quadrature method, which utilizes more combinations of candidate stencils. The Qadyan method has a higher computational cost in comparison with the FDLBM.

Hybrid multiscale computational approach for heat transfer in GaN semiconductors

Abstract With ongoing advancements in semiconductor technology, understanding thermal transport in semiconductor devices—where phonons are the primary charge carriers—is crucial. This involves a multiscale process in which phenomena at different scales are governed by distinct control equations, necessitating various numerical methods. In this context, this paper introduces a novel multiscale simulation method called the Second-Order Reconstruction Operator Finite Difference Lattice Boltzmann Method (ROsFDLBM). This method effectively combines the Lattice Boltzmann Method (LBM) with the Finite Difference Method (FDM). ROsFDLBM employs reconstruction operators to accurately map the relationship between distribution functions and macroscopic physical quantities, significantly reducing errors caused by the transmission of interface information. The efficacy of this coupling scheme has been confirmed through an interface-coupled thermal diffusion model. Comparative analyses have revealed that the relative error of ROsFDLBM does not exceed 1.2%, demonstrating its superior accuracy. Further simulations of GaN devices have confirmed that traditional macroscopic methods tend to underestimate the temperature in the heat source area at micro-nano scales, with discrepancies reaching up to 60.9 K. Overall, the ROsFDLBM aligns with current research findings and offers an accurate and efficient simulation strategy for addressing complex heat transfer problems.

Computation of three-dimensional incompressible flows using high-order weighted essentially non-oscillatory finite-difference lattice Boltzmann method

In the present work, an accurate and robust solution methodology based on the high-order weighted essentially non-oscillatory (WENO) finite-difference lattice Boltzmann method (LBM) in the three-dimensional generalized curvilinear coordinates is presented and applied for simulating the three-dimensional incompressible flows over complicated configurations with curved boundaries. Here, the incompressible form of the lattice Boltzmann equation in three dimensions is considered and the discretization of the spatial derivative terms is performed with the fifth-order WENO finite-difference method and the temporal derivative term is discretized with the fourth-order Runge–Kutta scheme to ensure the accuracy and stability of the solution method for both the steady and unsteady problems. The three-dimensional lattice Boltzmann equation applied here is based on a nineteen discrete velocity model for transforming the microscopic properties to the macroscopic ones. To assess the accuracy and robustness of the present three-dimensional high-order finite-difference LBM solver, different incompressible flow benchmarks and practical test cases are studied that are the cavity flow, the Beltrami flow, the flow in the curved ducts of rectangular cross sections, and the flow over a sphere for different flow conditions. The decay of the homogeneous isotropic turbulence is also computed to examine the suitability of the present solution method to be applied as the direct numerical simulation of turbulent flows. It is demonstrated that the solution methodology presented based on the high-order WENO finite-difference LBM in the three-dimensional generalized curvilinear coordinate can be used for accurately and effectively computing the three-dimensional practical incompressible flow problems.

Investigation of natural convection and entropy generation of non-Newtonian flow in molten polymer-filled odd-shaped cavities using finite difference lattice Boltzmann method

This study examines the natural convection heat transfer (NCHT) and entropy generation (EG) of a non-Newtonian (NN) flow inside an odd-shaped cavity filled with molten polymer. The cavity configuration comprises hot internal walls, cold external walls, and insulated remaining walls. The finite difference lattice Boltzmann method (FDLBM) is employed to solve the governing equations involved. The input parameters span a range of values, with Rayleigh number (Ra) varying from 104 to 105, power-law index (n) covering a range from 0.5 to 1.5, and width ratio (WR) ranging from 0.2 to 0.4, while maintaining Prandtl number (Pr) at a constant value of 10. The results revealed that at Ra values of 104 and 105, the overall entropy increases as the WR increases from 0.2 to 0.4 and decreases as the n index increases. In contrast to dilatant fluids, where heat transfer (HT) decreases as the n index increases from 1 to 1.5, pseudo-plastic fluids show an opposite trend with HT increasing as the n index decreases from 1 to 0.5. This trend is attributed to the general decrease in the average Nusselt number (Nu) as the n index increases. Additionally, it is observed that the n index has no significant impact on the average Nu due to low buoyancy force at Ra = 104 and WRs of 0.2 and 0.3. However, it notably influences the Bejan number (Be) across all WRs. Overall, the present model demonstrates excellent agreement with previous numerical results, affirming the FDLBM as a superior, reliable, and well-suited technique for relevant applications. These results suggest the potential to extend the application of the FDLBM approach to various cavity shapes, allowing for a comprehensive exploration of NCHT and EG under various conditions.

Implementing vorticity–velocity formulation in a finite difference lattice Boltzmann method for two-dimensional incompressible generalized Newtonian fluids

A finite difference lattice Boltzmann approach is introduced to address the two-dimensional macroscopic equations of velocity–vorticity for generalized Newtonian fluids (GNFs). The study involves equations governing macroscopic momentum, energy, and concentration, along with constitutive models applicable to Newtonian, power-law, and viscoplastic fluids. Subsequently, the lattice Boltzmann method, which recovers these macroscopic equations, is detailed, along with proof of its capability to reproduce the aforementioned equations. In order to evaluate the effectiveness and time efficiency of the method, it is validated against various benchmarks. The results demonstrate the efficacy of the proposed method in successfully solving isothermal, thermal, and solutal problems of GNFs, while significantly reducing computational time compared to our previously suggested approach in this domain.

Finite difference lattice Boltzmann method for modeling dam break debris flows

A finite difference lattice Boltzmann method (FDLBM) for the simulation of mud and debris flows for one-dimensional cases has been introduced. The proposed FDLBM recovers the generalized equations of mud and debris flows, that is, an unsteady one-dimensional Saint-Venant equation, including the effects of the non-Newtonian behavior of the mixture of water and soil, contraction–expansion losses (or large eddy loss), wind force, various geometries, and lateral inflow or outflow. The proposed FDLBM can be implemented for various non-Newtonian viscoplastic constitutive models of the studied mud and debris flows. The method is validated against previous studies for several benchmark cases, including steady-state problems, hydraulic jump tests, dam breaks with dry and wet beds, and slope dam break floods. Finally, the Anhui debris dam failure flood was investigated by this approach, and the results demonstrated a good agreement with the observed computational and field tests.

Simulation of high-Mach-number inviscid flows using a third-order Runge-Kutta and fifth-order WENO-based finite-difference lattice Boltzmann method

A discrete-velocity Boltzmann equation(DVBE) with Bhatnagar-Gross-Krook (BGK) approximation is discretized in time and space using a third-order Runge-Kutta (RK3) and fifth-order weighted essentially nonoscillatory (WENO) finite-difference scheme to simulate benchmark inviscid compressible flows. The implementation of the WENO ensures that solutions behave with minimal or no oscillations, narrowing the gap between the exact and the numerical results. Discrete-velocity sets given by Kataoka and Tsutahara [Phys. Rev. E 69, 056702 (2004)10.1103/PhysRevE.69.056702] are used. The additional dissipation terms as well as artificial viscosity are incorporated in the formulation to solve the compressible flows at high Mach number. Further, the flows which are subjected initially to a high density ratio are effectively simulated. In this article, one-dimensional benchmarks are simulated at initial Mach number up to 30 and density ratio up to 1000, whereas, the benchmarks in two dimensions are simulated with a Mach number up to 10. The algorithm is assessed by simulating numerous benchmarks, namely, (i) one-dimensional Riemann problem for various shock waves combinations [namely (a) shock-shock waves in the case of different Mach numbers, (b) rarefaction-shock waves for various density ratios, (c) sudden contact shock discontinuity, and (d) shock-rarefaction waves for density ratio 5], (ii) isentropic vortex convection test, (iii) regular shock reflection (RR) for Mach numbers 2.9 and 10, (iv) double Mach reflection (DMR) for inflow Mach numbers as 6 and 10, and (v) forward-facing step for inflow Mach numbers 2 to 5. Further, the effect of change in Mach numbers and wedge angles on the flow structures in the case of DMR are detailed. In the case of a forward-facing step, the variations of flow structure (e.g., the Mach stem height, triple points location, and shock standoff distance) are detailed with respect to Mach number, step height, and specific-heat ratios. Finally, the numerical stability of the proposed formulation is carried out to assess the behavior of the free parameters.

Hydro-thermal performance on the two phase nanofluid convection heat transfer using the LBM-FD method

This paper is dedicated to the investigation of nanofluid natural convection inside a 2D cavity with various kinds of cold walls. To consider the different nanoparticle distribution in the cavity, Buongiorno’s two-phase model is used. The equations are solved by the homemade hybrid Lattice Boltzmann-Finite Difference (LBM-FD) code. It found that although the nanoparticle concentration is almost uniform inside the cavity, the apparent difference can be observed around the walls and edges of the vortices. The nanoparticles have higher concentrations around the cold wall and the lower concentration adjacent to the hot wall. For all cases, the local and average Nusselt number increases by increasing the solid volume fraction of nanofluid. The configuration of cold walls affects the flow pattern and distribution of temperature significantly. By applying the combination of right-angle and inclined walls, the best heat transfer performance for the present system.

A phase field-finite difference lattice Boltzmann method for modeling dendritic growth solidification in the presence of melt convection

In this study, an alternative numerical method is proposed to predict the two-phase flow during dendritic growth under melt convection. It consists of two main parts. The phase-field method (PFM) is used for modeling the solid phase and tracking the morphological changes during dendritic solidification, and the finite-difference lattice Boltzmann method (FDLBM) is used for predicting the fluid flow of the liquid phase. The no-slip boundary condition to take into account the interaction between liquid and solid phases at the interface is modeled as a diffusive drag force. This method is referred to as phase field-finite difference lattice Boltzmann method (PF-FDLBM). In the numerical formulation, a collision term with single relaxation time (SRT) is used for simplicity, and a negative viscosity term is included to improve numerical stability. The proposed method can be implemented on regular and non-regular grids. Two-dimensional benchmark computations of flows past a circular cylinder show that the no-slip boundary condition is well satisfied. The flow patterns and drag coefficients are appropriately predicted even for flows at low Reynolds numbers, in which viscous forces commonly deteriorate numerical stability. Then, simulations of dendritic growth under melt convection are carried out with the present method. The effect of melt convection on the solidification patterns is well predicted for an alloy with its actual physical properties. Finally, the efficiency of the proposed method is evaluated considering an increase of the relaxation time, and its influence related to the computational resources. As a result, faster and stable computations are possible by appropriately setting the relaxation time and the negative viscosity term in PF-FDLBM formulation.

Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation.

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) is developed for the nonlinear convection-diffusion equation (NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.e., single-relaxation-time finite-difference lattice Boltzmann method (SRT-FDLBM). Then, a simplified version of MRT-FDLBM (SMRT-FDLBM) is also proposed, which can save about 15% computational cost. In addition, a series of real and complex-value NCDEs, including the isotropic convection-diffusion equation, Burgers-Fisher equation, sine-Gordon equation, heat-conduction equation, and Schrödinger equation, are used to test the performance of MRT-FDLBM. The results show that both MRT-FDLBM and SMRT-FDLBM have second-order convergence rates in space and time. Finally, the stability and accuracy of five different models are compared, including the MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method [H. Wang, B. Shi et al., Appl. Math. Comput. 309, 334 (2017)10.1016/j.amc.2017.04.015], and the lattice Boltzmann method (LBM). The stability tests show that the sequence of stability from high to low is as follows: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method, and LBM. In most of the precision test results, it is found that the order from high to low of precision is MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, and the previous finite-difference lattice Boltzmann method.

Hybrid lattice-Boltzmann finite-difference simulation of ternary fluids near immersed solid objects of general shapes

In this paper, a hybrid lattice-Boltzmann finite-difference method is developed for the simulation of ternary fluids near immersed solid objects of general shapes. The flow equations are solved by the lattice-Boltzmann method and the coupled Cahn–Hilliard equations for interface evolutions are solved by the finite-difference method. A special implementation of the wetting boundary condition on a surface of general shapes immersed inside the domain was extended for ternary fluids within the phase-field framework with no need to use complicated interpolations. Several two and three dimensional problems with three immiscible fluids were studied by using the proposed method and the results agree well with analytical predictions and/or previous numerical and experimental studies. In particular, the inclusion of properly chosen free energy to handle total spreading enabled us to numerically reproduce the encapsulation of a small droplet by another bigger one of different component on a round fiber. The proposed method is expected to be useful to investigate a variety of multiphase problems involving ternary fluids and surfaces with different configurations, including the challenging total spreading regime.

Computational modeling of three-dimensional thermocapillary flow of recalcitrant bubbles using a coupled lattice Boltzmann-finite difference method

This study analyzes the thermocapillary flow of recalcitrant bubbles within thin channels using a hybrid finite difference lattice Boltzmann method (LBM). It extends a recently developed phase-field LBM to account for temperature effects by coupling the scheme with a fourth-order Runge–Kutta algorithm to solve the governing energy equation. The LBM makes use of a weighted-multiple relaxation-time collision scheme, which has been previously shown to capture high density and viscosity contrasts. This paper makes contributions in two fundamental areas relating to thermocapillary flow. First, it presents and verifies a novel, three-dimensional model to resolve thermocapillary dynamics for practical applications. The verification was undertaken via comparison with analytical solutions for the flow of immiscible fluids in a heated microchannel and for the migration of a droplet in a temperature field. Second, it provides new insight into the inherently three-dimensional nature of recalcitrant bubbles. It was found that the competing inertial and thermal effects allow these bubbles to propagate against the bulk motion of the liquid toward regions of low surface tension.

THE IMPROVEMENT AND REALIZATION OF FINITE-DIFFERENCE LATTICE BOLTZMANN METHOD

The Lattice Boltzmann Method (LBM) is a numerical method developed in recent decades. It has the characteristics of high parallel efficiency and simple boundary processing. The basic idea is to construct a simplified dynamic model so that the macroscopic behavior of the model is the same as the macroscopic equation. From the perspective of micro-dynamics, LBM treats macro-physical quantities as micro-quantities to obtain results by statistical averaging. The Finite-difference LBM (FDLBM) is a new numerical method developed based on LBM. The first finite-difference LBE (FDLBE) was perhaps due to Tamura and Akinori and was examined by Cao et al. in more detail. Finite-difference LBM was further extended to curvilinear coordinates with nonuniform grids by Mei and Shyy. By improving the FDLBE proposed by Mei and Shyy, a new finite difference LBM is obtained in the paper. In the model, the collision term is treated implicitly, just as done in the Mei-Shyy model. However, by introducing another distribution function based on the earlier distribution function， the implicitness of the discrete scheme is eliminated, and a simple explicit scheme is finally obtained, such as the standard LBE. Furthermore, this trick for the FDLBE can also be easily used to develop more efficient FVLBE and FELBE schemes. To verify the correctness and feasibility of this improved FDLBM model, which is used to calculate the square cavity model, and the calculated results are compared with the data of the classic square cavity model. The comparison result includes two items: the velocity on the centerline of the square cavity and the position of the vortex center in the square cavity. The simulation results of FDLBM are very consistent with the data in the literature. When Re=400, the velocity profiles of u and v on the centerline of the square cavity are consistent with the data results in Ghia's paper, and the vortex center position in the square cavity is also almost the same as the data results in Ghia's paper. Therefore, the verification of FDLBM is successful and FDLBM is feasible. This improved method can also serve as a reference for subsequent research.

A finite-difference lattice Boltzmann method with second-order accuracy of time and space for incompressible flow

In this paper, a kind of finite-difference lattice Boltzmann method with the second-order accuracy of time and space (T2S2-FDLBM) is proposed. In this method, a two-stage time-accurate discretization approach is applied to construct time marching scheme, and the spatial gradient operator is discretized by a mixed difference scheme to maintain a second-order accuracy in space. It is shown that the previous finite-difference lattice Boltzmann method (FDLBM) (Guo and Zhao, 2003) is a special case of the T2S2-FDLBM. Through the von Neumann analysis, the stability of the method is analyzed and two specific T2S2-FDLBMs are discussed. The two T2S2-FDLBMs are applied to simulate some incompressible flows with the non-uniform grids. Compared with other high order FDLBMs, this two T2S2-FDLBMs keep the simplicity of standard lattice Boltzmann method. Compared with the previous FDLBM and lattice Boltzmann method, the two T2S2-FDLBMs are more accurate and more stable. The value of the Courant–Friedrichs–Lewy condition number in our method can be up to 0.9, which also significantly improves the computational efficiency.

High-order upwind compact finite-difference lattice Boltzmann method for viscous incompressible flows

In this work, a high-order upwind compact finite-difference lattice Boltzmann method (UCDLBM) is developed to efficiently solve viscous incompressible flow problems. A fifth-order upwind compact difference scheme is adopted to discretize the spatial derivatives of the lattice Boltzmann equation, and the third-order total-variation-diminishing Runge–Kutta scheme is utilized for the discretization of the temporal term. Compared to the existing central compact finite-difference lattice Boltzmann method (CFDLBM), the present UCDLBM can prevent non-physical oscillations without filtering due to the natural dissipative property of upwind schemes. Three benchmark problems involving the Taylor–Green vortex problem, the doubly periodic shear layer flow problem and the lid driven square cavity flow problem are numerically solved to demonstrate the accuracy and efficiency of the present method. Numerical results computed are in good agreement with the analytical solution or other available numerical results. And, the present UCDLBM is less time-consuming than the CFDLBM without degenerating the order of accuracy of the numerical solutions.

Single recalcitrant bubble simulation using a hybrid lattice boltzmann finite difference model

In this paper, 2D simulations of a single recalcitrant bubble using a hybrid finite difference lattice Boltzmann method are presented. To solve the flow field, the conservative phase field lattice Boltzmann model of Geier et al. (2015) is employed with a multi-relaxation-time (MRT) scheme which makes it capable of handling multiphase flows with high density and viscosity contrasts. A finite difference approach is applied to obtain the temperature field with a second-order spatial discretization (central) which marches in time with the 2nd order Runge-Kutta marching technique. The well-known continuum surface tension force (CSF) is implemented to define the thermocapillary force which depends on the temperature field. Before delving into the main aim of this work, we first assess the accuracy of the present model by verifying the Laplace law for a static bubble with four different values of surface tension. To further validate the model, thermocapillary migration of a deformable drop, and thermocapillary flow with two superimposed planar fluids are studied as our second and third benchmarks. Comparing the analytical solutions for both velocity and temperature fields with the simulation ones, good agreements are found. We then propose a simple theory with some simplification for the study of the migration of a single recalcitrant bubble, and after that weperform a series of 2D simulations to analyze the effects of density ratios, viscosity ratios, entrance average velocities of the flow, and the values of second derivative of surface tension with respect to the temperature.

Investigation of the entropy generation during natural convection of Newtonian and non-Newtonian fluids inside the L-shaped cavity subjected to magnetic field: application of lattice Boltzmann method

In the present paper, free heat convection and entropy generation of Newtonian and two types of non-Newtonian fluids, shear-thickening and shear-thinning, inside an L-shaped cavity subjected to a magnetic field have been investigated by the finite difference lattice Boltzmann method. The power-law model was used for modeling the rheology of the fluids. The bottom and left walls of the cavity have been kept at a uniform high temperature. Internal walls are also kept cold. The remaining walls have been insulated against heat and mass transfer. The Boussinesq approximation is used to take the temperature dependency of density into account. The distribution functions of energy and density are modeled through the use of the nine-velocity two-dimensional scheme. The effects of Hartmann number (Ha), aspect ratio, power-law index, and Rayleigh number (Ra), on the flow field, temperature distribution, and entropy distributions are studied. The results show that the magnetic field and the power-law index have an ever-decreasing effect on the heat transfer rate and the entropy generation, while the Ra number has an ever-increasing effect. The maximum heat transfer enhancement of 71% happens at the lowest and the highest values of power-law index and Ra number, respectively, for the case with no magnetic field. The maximum heat transfer deterioration of 77% happens at the highest and lowest values of power-law index and Ra number, respectively, in the presence of the highest magnetic field strength. It is interesting that the sensitivities of heat transfer rate and the entropy generation to the Ha number become significant for shear-thinning fluids. It is found that there is an everlasting interplay between conduction and convection contributions to the irreversibilities, so that, for the Newtonian and shear-thinning fluids, thermal irreversibilities, characterized by Be number, increase with Ha number, reaching to a maximum, and then decline. The heat transfer rate and the total entropy generation for the Newtonian and shear-thinning fluids increase, monotonically, by raising the aspect ratio, while the figure for the shear-thickening case is different. It is decreased first and then increased.

Numerical analysis of free convection and entropy generation in a cavity using compact finite-difference lattice Boltzmann method

Purpose The purpose of this paper is to investigate the compact finite-difference lattice Boltzmann method is used to simulate the free convection within a cavity. Design/methodology/approach The finite-difference discretization method enables the numerical simulations to be run when there are non-uniform and curvilinear grids with a finer near-wall grid resolution. Furthermore, the high-order method is applied in the numerical approach, which makes it possible to go with relatively coarse mesh in respect to simulations, which used classical lattice Boltzmann method. The configuration of the cavity is set to sine-walled square. In addition, the cavity is filled with Al2O3-water nanofluid, and the Koo–Kleinstreuer–Li model is used to estimate the properties of nanofluid. Findings The nanoparticle (Al2O3) concentration in the base fluid (water) is considered in a range of 0-0.04. The nanofluid flow and heat transfer are investigated in laminar regime with Rayleigh number in the range of 103-106. The second law analysis is used to study the effects of different governing parameters on the local and volumetric entropy generation. The Rayleigh number, configuration of the cavity and nanoparticle concentration are considered as the governing parameters. The results are mainly focused on the flow structure, temperature field, local and volumetric entropy generation and heat transfer performance. Originality/value The originality of this study is using of a modern numerical method supported by an accurate prediction for nanofluid properties to simulate the flow and heat transfer during natural convection in a cavity.

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.

A high-order compact finite-difference lattice Boltzmann method for simulation of natural convection

In the present work, compact finite-difference lattice Boltzmann method (CFDLBM) is extended to simulate natural convection in square and concentric-circular-annulus cavities, in two-dimensions, for Rayleigh numbers in the range of 103 to 109. Owing to the finite-difference discretization of the method, non-uniform and curvilinear grids with a finer near-wall grid resolution could be used in the present simulations. The use of high-order methods further enabled the simulations to be run on relatively coarse meshes when compared with simulations that used classical lattice Boltzmann method. The simulation results are analyzed with the help of streamlines, isothermal contours and temperature, velocity, Nusselt number variations along the walls. The average Nusselt number on the hot wall shows less than 1% error for natural convection in the square cavity study. The circular-annular-cavity is found to have laminar flow for Rayleigh numbers in the range of 103–106 and it becomes unsteady and chaotic from 107 onwards. The instantaneous and time-average contours of isotherms and streamlines were used to study the unsteady behavior of the flow. A universal near-wall velocity profile is found to exist in all the unsteady natural convection studies in the annular cavity case. The location of the maximum velocities occurring in the annulus is analyzed for all the Rayleigh numbers. The spectra of velocity magnitude fluctuations and temperature fluctuations indicate that the unsteadiness in the cavity is dominant around 0.001 Hz. Natural convection in sine-walled and concentric star shaped cavities are simulated to demonstrate the ability of the present solver to handle complex geometries. The present results indicate that CFDLBM can be successfully used to simulate steady and unsteady chaotic natural convection flows in both rectangular and curvilinear geometries accurately.