Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are... more Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are reported, which indicate that blow-up of inviscid flows is tamed by the emergence of collective dynamics of coherent structures. The simulations also suggest that this collective dynamics might lead to universal behaviour during the transient evolution of turbulence. In particular, simulations with three different initial conditions show evidence of a ${k}^{- 3} \log k$ spectrum in the transient stage, before the Kolmogorov ${k}^{- 5/ 3} $ asymptotic regime is attained. Such a universal transient might serve as a spectral funnel to the time-asymptotic Kolmogorov spectrum, which is invariably observed in the late stage of all three simulations presented in this work. The present work is entirely based on simulation evidence. However, the statistical analysis of the coherent structures suggests an analogy with population dynamics, which might be conducive to new mathematical models of tra...
A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice... more A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice Boltzmann method (LBM) and the multigrid (MG) technique is presented. Several benchmark elliptic equations are solved numerically with the inclusion of multiple grid-levels in two-dimensional domains at an optimal computational cost within the LB framework. The results are compared with the corresponding analytical solutions and numerical solutions obtained using the Stone's strongly implicit procedure. The classical PDEs considered in this article include the Laplace and Poisson equations with Dirichlet boundary conditions, with the latter involving both constant and variable coefficients. A detailed analysis of solution accuracy, convergence and computational efficiency of the proposed solver is given. It is observed that the use of a high-order stencil (for smoothing) improves convergence and accuracy for an equivalent number of smoothing sweeps. The effect of the type of scheduling cycle (V-or W-cycle) on the performance of the MG-LBM is analyzed. Next, a parallel algorithm for the MG-LBM solver is presented and then its parallel performance on a multi-core cluster is analyzed. Lastly, a practical example is provided wherein the proposed elliptic PDE solver is used to compute the electro-static potential encountered in an electro-chemical cell, which demonstrates the effectiveness of this new solver in complex coupled systems. Several orders of magnitude gains in convergence and parallel scaling for the canonical problems, and a factor of 5 reduction for the multiphysics problem are achieved using the MG-LBM.
International Journal for Numerical Methods in Fluids, 2015
Central moment lattice Boltzmann method (LBM) is one of the more recent developments among the la... more Central moment lattice Boltzmann method (LBM) is one of the more recent developments among the lattice kinetic schemes for computational fluid dynamics. A key element in this approach is the use of central moments to specify collision process and forcing, and thereby naturally maintaining Galilean invariance, an important characteristic of fluid flows. When the different central moments are relaxed at different rates like in a standard multiple relaxation time (MRT) formulation based on raw moments, it is endowed with a number of desirable physical and numerical features. Since the collision operator exhibits a cascaded structure, this approach is also known as the cascaded LBM. While the cascaded LBM has been developed sometime ago, a systematic study of its numerical properties, such as accuracy, grid convergence and stability for well defined canonical problems is lacking and the present work is intended to fulfill this need.
A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann... more A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier-Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.
Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are... more Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are reported, which indicate that blow-up of inviscid flows is tamed by the emergence of collective dynamics of coherent structures. The simulations also suggest that this collective dynamics might lead to universal behaviour during the transient evolution of turbulence. In particular, simulations with three different initial conditions show evidence of a spectrum in the transient stage, before the Kolmogorov asymptotic regime is attained. Such a universal transient might serve as a spectral funnel to the time-asymptotic Kolmogorov spectrum, which is invariably observed in the late stage of all three simulations presented in this work. The present work is entirely based on simulation evidence. However, the statistical analysis of the coherent structures suggests an analogy with population dynamics, which might be conducive to new mathematical models of transient decaying turbulence.
Coupling of lattice Boltzmann (LB) and phase-field (PF) methods is discussed for simulation of a ... more Coupling of lattice Boltzmann (LB) and phase-field (PF) methods is discussed for simulation of a range of multiphase flow problems. The local relaxation and shifting operators make the LB method an attractive candidate for the simulation of the single-phase as well as multiphase flows. For simulating interface dynamics, LB methods require to be coupled with an appropriate scheme representing interfacial dynamics. To this end, we have used a model based on the order parameter, which could be either an index function or a phase-field variable, and coupled it with a LB solver for the simulation of various classes of complex multi-physics and multiphase flows. The LB method is used to compute the flow-field, and, in the case of electrodeposition process modeling, the electro-static potential-field. The application of such a coupled LB-PF is illustrated by the solution of a variety of examples. Finally, fast simulation of such a coupled algorithm is achieved using the state-of-art numerical solution acceleration techniques involving preconditioning and multigrid approaches.
A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice... more A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice Boltzmann method (LBM) and the multigrid (MG) technique is presented. Several benchmark elliptic equations are solved numerically with the inclusion of multiple grid-levels in two-dimensional domains at an optimal computational cost within the LB framework. The results are compared with the corresponding analytical solutions and numerical solutions obtained using the Stone's strongly implicit procedure. The classical PDEs considered in this article include the Laplace and Poisson equations with Dirichlet boundary conditions, with the latter involving both constant and variable coefficients. A detailed analysis of solution accuracy, convergence and computational efficiency of the proposed solver is given. It is observed that the use of a high-order stencil (for smoothing) improves convergence and accuracy for an equivalent number of smoothing sweeps. The effect of the type of scheduling cycle (V- or W-cycle) on the performance of the MG–LBM is analyzed. Next, a parallel algorithm for the MG–LBM solver is presented and then its parallel performance on a multi-core cluster is analyzed. Lastly, a practical example is provided wherein the proposed elliptic PDE solver is used to compute the electro-static potential encountered in an electro-chemical cell, which demonstrates the effectiveness of this new solver in complex coupled systems. Several orders of magnitude gains in convergence and parallel scaling for the canonical problems, and a factor of 5 reduction for the multiphysics problem are achieved using the MG–LBM.
A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann... more A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier–Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.
In this paper, the lattice Boltzmann equation (LBE) method is applied for simulation of lid-drive... more In this paper, the lattice Boltzmann equation (LBE) method is applied for simulation of lid-driven flow in a two-dimensional, rectangular, deep cavity. First, the code is validated for the standard square cavity, and then the results of a deep cavity are presented. Steady results are presented for deep cavities with aspect ratios of 1.5–4, and Reynolds numbers of 50–3200. Several features of the flow, such as the location and strength of the primary vortex, and the corner-eddy dynamics are investigated and compared with previous findings from experiments and theory. Steady results for deep cavities show the existence of corner eddies at the bottom, which coalesce to form a second primary-eddy as the cavity aspect-ratio is increased above a critical value. However, at relatively high Reynolds numbers, the second primary-eddy is formed via a rapid transition of an unsteady wall-eddy. The predicted results from LBE simulations are shown to be consistent with experiments and theory.
We introduce a scheme which gives rise to additional degree of freedom for the same number of dis... more We introduce a scheme which gives rise to additional degree of freedom for the same number of discrete velocities in the context of the lattice Boltzmann model. We show that an off-lattice D3Q27 model exists with correct equilibrium to recover Galilean-invariant form of Navier-Stokes equation ͑without any cubic error͒. In the first part of this work, we show that the present model can capture two important features of the microflow in a single component gas: Knudsen boundary layer and Knudsen Paradox. Finally, we present numerical results corresponding to Couette flow for two representative Knudsen numbers. We show that the off-lattice D3Q27 model exhibits better accuracy as compared to more widely used on-lattice D3Q19 or D3Q27 model. Finally, our construction of discrete velocity model shows that there is no contradiction between entropic construction and quadrature-based procedure for the construction of the lattice Boltzmann model.
The classical Chapman-Enskog expansion is performed for the recently proposed finite-volume formu... more The classical Chapman-Enskog expansion is performed for the recently proposed finite-volume formulation of lattice Boltzmann equation (LBE) method [D.V. Patil, K.N. Lakshmisha, Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, J. Comput. Phys. 228 (2009) 5262–5279]. First, a modified partial differential equation is derived from a numerical approximation of the discrete Boltzmann equation. Then, the multi-scale, small parameter expansion is followed to recover the continuity and the Navier–Stokes (NS) equations with additional error terms. The expression for apparent value of the kinematic viscosity is derived for finite-volume formulation under certain assumptions. The attenuation of a shear wave, Taylor–Green vortex flow and driven channel flow are studied to analyze the apparent viscosity relation.
In this paper, the lattice Boltzmann equation (LBE)-based framework is used to obtain the solutio... more In this paper, the lattice Boltzmann equation (LBE)-based framework is used to obtain the solution for the linear radiative or neutron transport equation. The LBE framework is devised for the integrodifferential forms of these equations which arise due to the inclusion of the scattering terms. The interparticle collisions are neglected, hence omitting the nonlinear collision term. Furthermore, typical representative examples for one-dimensional or two-dimensional geometries and inclusion or exclusion of the scattering term (isotropic and anisotropic) in the Boltzmann transport equation are illustrated to prove the validity of the method. It has been shown that the solution from the LBE methodology is equivalent to the well-known P(n) and S(n) methods. This suggests that the LBE can potentially provide a more convenient and easy approach to solve the physical problems of neutron and radiation transport.
Recently, it was shown that energy conserving (EC) lattice Boltzmann (LB) model is more accurate ... more Recently, it was shown that energy conserving (EC) lattice Boltzmann (LB) model is more accurate than athermal LB model for high-resolution simulations of athermal flows. However, in the sub-grid (SG) domain, the behavior is found to be opposite. In this work, we show that via multi-relaxation model, it is possible to preserve the accuracy of the EC LB for both SG and direct numerical simulation (DNS) models. We show that by introducing a nonunit Prandtl number, under-resolved simulations can also be performed quite efficiently, a property which we attribute to the enhanced sound-relaxation.
In this paper, a coupled phase-field (PF) and lattice Boltzmann method (LBM) is presented to mode... more In this paper, a coupled phase-field (PF) and lattice Boltzmann method (LBM) is presented to model the multiphysics phenomenon involving electro-chemical deposition. The deposition (or dissolution) of the electrode is represented using variations of an order-parameter. The time-evolution of an order-parameter is proportional to the variation of a Ginzburg–Landau free-energy functional. Further, the free-energy densities of the two phases are defined based on a dilute or an ideal solution approximation. An efficient LBM is used to obtain the converged electro-static potential field for each physical time-step of the evolution of the PF variable. The coupled approach demonstrates the applicability of the LBM in a multiphysics scenario. The numerical validation for the coupled approach is performed by the simulation of the electrodeposition process of Cu from CuSO4 solution.
Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are... more Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are reported, which indicate that blow-up of inviscid flows is tamed by the emergence of collective dynamics of coherent structures. The simulations also suggest that this collective dynamics might lead to universal behaviour during the transient evolution of turbulence. In particular, simulations with three different initial conditions show evidence of a ${k}^{- 3} \log k$ spectrum in the transient stage, before the Kolmogorov ${k}^{- 5/ 3} $ asymptotic regime is attained. Such a universal transient might serve as a spectral funnel to the time-asymptotic Kolmogorov spectrum, which is invariably observed in the late stage of all three simulations presented in this work. The present work is entirely based on simulation evidence. However, the statistical analysis of the coherent structures suggests an analogy with population dynamics, which might be conducive to new mathematical models of tra...
A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice... more A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice Boltzmann method (LBM) and the multigrid (MG) technique is presented. Several benchmark elliptic equations are solved numerically with the inclusion of multiple grid-levels in two-dimensional domains at an optimal computational cost within the LB framework. The results are compared with the corresponding analytical solutions and numerical solutions obtained using the Stone's strongly implicit procedure. The classical PDEs considered in this article include the Laplace and Poisson equations with Dirichlet boundary conditions, with the latter involving both constant and variable coefficients. A detailed analysis of solution accuracy, convergence and computational efficiency of the proposed solver is given. It is observed that the use of a high-order stencil (for smoothing) improves convergence and accuracy for an equivalent number of smoothing sweeps. The effect of the type of scheduling cycle (V-or W-cycle) on the performance of the MG-LBM is analyzed. Next, a parallel algorithm for the MG-LBM solver is presented and then its parallel performance on a multi-core cluster is analyzed. Lastly, a practical example is provided wherein the proposed elliptic PDE solver is used to compute the electro-static potential encountered in an electro-chemical cell, which demonstrates the effectiveness of this new solver in complex coupled systems. Several orders of magnitude gains in convergence and parallel scaling for the canonical problems, and a factor of 5 reduction for the multiphysics problem are achieved using the MG-LBM.
International Journal for Numerical Methods in Fluids, 2015
Central moment lattice Boltzmann method (LBM) is one of the more recent developments among the la... more Central moment lattice Boltzmann method (LBM) is one of the more recent developments among the lattice kinetic schemes for computational fluid dynamics. A key element in this approach is the use of central moments to specify collision process and forcing, and thereby naturally maintaining Galilean invariance, an important characteristic of fluid flows. When the different central moments are relaxed at different rates like in a standard multiple relaxation time (MRT) formulation based on raw moments, it is endowed with a number of desirable physical and numerical features. Since the collision operator exhibits a cascaded structure, this approach is also known as the cascaded LBM. While the cascaded LBM has been developed sometime ago, a systematic study of its numerical properties, such as accuracy, grid convergence and stability for well defined canonical problems is lacking and the present work is intended to fulfill this need.
A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann... more A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier-Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.
Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are... more Numerical results from large-scale, long-time, simulations of decaying homogeneous turbulence are reported, which indicate that blow-up of inviscid flows is tamed by the emergence of collective dynamics of coherent structures. The simulations also suggest that this collective dynamics might lead to universal behaviour during the transient evolution of turbulence. In particular, simulations with three different initial conditions show evidence of a spectrum in the transient stage, before the Kolmogorov asymptotic regime is attained. Such a universal transient might serve as a spectral funnel to the time-asymptotic Kolmogorov spectrum, which is invariably observed in the late stage of all three simulations presented in this work. The present work is entirely based on simulation evidence. However, the statistical analysis of the coherent structures suggests an analogy with population dynamics, which might be conducive to new mathematical models of transient decaying turbulence.
Coupling of lattice Boltzmann (LB) and phase-field (PF) methods is discussed for simulation of a ... more Coupling of lattice Boltzmann (LB) and phase-field (PF) methods is discussed for simulation of a range of multiphase flow problems. The local relaxation and shifting operators make the LB method an attractive candidate for the simulation of the single-phase as well as multiphase flows. For simulating interface dynamics, LB methods require to be coupled with an appropriate scheme representing interfacial dynamics. To this end, we have used a model based on the order parameter, which could be either an index function or a phase-field variable, and coupled it with a LB solver for the simulation of various classes of complex multi-physics and multiphase flows. The LB method is used to compute the flow-field, and, in the case of electrodeposition process modeling, the electro-static potential-field. The application of such a coupled LB-PF is illustrated by the solution of a variety of examples. Finally, fast simulation of such a coupled algorithm is achieved using the state-of-art numerical solution acceleration techniques involving preconditioning and multigrid approaches.
A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice... more A new solver for second-order elliptic partial differential equations (PDEs) based on the lattice Boltzmann method (LBM) and the multigrid (MG) technique is presented. Several benchmark elliptic equations are solved numerically with the inclusion of multiple grid-levels in two-dimensional domains at an optimal computational cost within the LB framework. The results are compared with the corresponding analytical solutions and numerical solutions obtained using the Stone's strongly implicit procedure. The classical PDEs considered in this article include the Laplace and Poisson equations with Dirichlet boundary conditions, with the latter involving both constant and variable coefficients. A detailed analysis of solution accuracy, convergence and computational efficiency of the proposed solver is given. It is observed that the use of a high-order stencil (for smoothing) improves convergence and accuracy for an equivalent number of smoothing sweeps. The effect of the type of scheduling cycle (V- or W-cycle) on the performance of the MG–LBM is analyzed. Next, a parallel algorithm for the MG–LBM solver is presented and then its parallel performance on a multi-core cluster is analyzed. Lastly, a practical example is provided wherein the proposed elliptic PDE solver is used to compute the electro-static potential encountered in an electro-chemical cell, which demonstrates the effectiveness of this new solver in complex coupled systems. Several orders of magnitude gains in convergence and parallel scaling for the canonical problems, and a factor of 5 reduction for the multiphysics problem are achieved using the MG–LBM.
A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann... more A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier–Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.
In this paper, the lattice Boltzmann equation (LBE) method is applied for simulation of lid-drive... more In this paper, the lattice Boltzmann equation (LBE) method is applied for simulation of lid-driven flow in a two-dimensional, rectangular, deep cavity. First, the code is validated for the standard square cavity, and then the results of a deep cavity are presented. Steady results are presented for deep cavities with aspect ratios of 1.5–4, and Reynolds numbers of 50–3200. Several features of the flow, such as the location and strength of the primary vortex, and the corner-eddy dynamics are investigated and compared with previous findings from experiments and theory. Steady results for deep cavities show the existence of corner eddies at the bottom, which coalesce to form a second primary-eddy as the cavity aspect-ratio is increased above a critical value. However, at relatively high Reynolds numbers, the second primary-eddy is formed via a rapid transition of an unsteady wall-eddy. The predicted results from LBE simulations are shown to be consistent with experiments and theory.
We introduce a scheme which gives rise to additional degree of freedom for the same number of dis... more We introduce a scheme which gives rise to additional degree of freedom for the same number of discrete velocities in the context of the lattice Boltzmann model. We show that an off-lattice D3Q27 model exists with correct equilibrium to recover Galilean-invariant form of Navier-Stokes equation ͑without any cubic error͒. In the first part of this work, we show that the present model can capture two important features of the microflow in a single component gas: Knudsen boundary layer and Knudsen Paradox. Finally, we present numerical results corresponding to Couette flow for two representative Knudsen numbers. We show that the off-lattice D3Q27 model exhibits better accuracy as compared to more widely used on-lattice D3Q19 or D3Q27 model. Finally, our construction of discrete velocity model shows that there is no contradiction between entropic construction and quadrature-based procedure for the construction of the lattice Boltzmann model.
The classical Chapman-Enskog expansion is performed for the recently proposed finite-volume formu... more The classical Chapman-Enskog expansion is performed for the recently proposed finite-volume formulation of lattice Boltzmann equation (LBE) method [D.V. Patil, K.N. Lakshmisha, Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, J. Comput. Phys. 228 (2009) 5262–5279]. First, a modified partial differential equation is derived from a numerical approximation of the discrete Boltzmann equation. Then, the multi-scale, small parameter expansion is followed to recover the continuity and the Navier–Stokes (NS) equations with additional error terms. The expression for apparent value of the kinematic viscosity is derived for finite-volume formulation under certain assumptions. The attenuation of a shear wave, Taylor–Green vortex flow and driven channel flow are studied to analyze the apparent viscosity relation.
In this paper, the lattice Boltzmann equation (LBE)-based framework is used to obtain the solutio... more In this paper, the lattice Boltzmann equation (LBE)-based framework is used to obtain the solution for the linear radiative or neutron transport equation. The LBE framework is devised for the integrodifferential forms of these equations which arise due to the inclusion of the scattering terms. The interparticle collisions are neglected, hence omitting the nonlinear collision term. Furthermore, typical representative examples for one-dimensional or two-dimensional geometries and inclusion or exclusion of the scattering term (isotropic and anisotropic) in the Boltzmann transport equation are illustrated to prove the validity of the method. It has been shown that the solution from the LBE methodology is equivalent to the well-known P(n) and S(n) methods. This suggests that the LBE can potentially provide a more convenient and easy approach to solve the physical problems of neutron and radiation transport.
Recently, it was shown that energy conserving (EC) lattice Boltzmann (LB) model is more accurate ... more Recently, it was shown that energy conserving (EC) lattice Boltzmann (LB) model is more accurate than athermal LB model for high-resolution simulations of athermal flows. However, in the sub-grid (SG) domain, the behavior is found to be opposite. In this work, we show that via multi-relaxation model, it is possible to preserve the accuracy of the EC LB for both SG and direct numerical simulation (DNS) models. We show that by introducing a nonunit Prandtl number, under-resolved simulations can also be performed quite efficiently, a property which we attribute to the enhanced sound-relaxation.
In this paper, a coupled phase-field (PF) and lattice Boltzmann method (LBM) is presented to mode... more In this paper, a coupled phase-field (PF) and lattice Boltzmann method (LBM) is presented to model the multiphysics phenomenon involving electro-chemical deposition. The deposition (or dissolution) of the electrode is represented using variations of an order-parameter. The time-evolution of an order-parameter is proportional to the variation of a Ginzburg–Landau free-energy functional. Further, the free-energy densities of the two phases are defined based on a dilute or an ideal solution approximation. An efficient LBM is used to obtain the converged electro-static potential field for each physical time-step of the evolution of the PF variable. The coupled approach demonstrates the applicability of the LBM in a multiphysics scenario. The numerical validation for the coupled approach is performed by the simulation of the electrodeposition process of Cu from CuSO4 solution.
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Papers by Dhiraj V Patil