Dipankar Chatterjee

An enthalpy-based hybrid lattice-Boltzmann method for modelling solid-liquid phase transition in the presence of convective transport

An extended lattice Boltzmann model is developed for simulating the convection–diffusion phenomena associated with solid–liquid phase transition processes. Macroscopic hydrodynamic variables are obtained through the solution of an evolution equation of a single-particle density distribution function, whereas, the macroscopic temperature field is obtained by solving auxiliary scalar transport equations. The novelty of the present methodology lies in the formulation of an enthalpy-based approach for phase-change modelling within a lattice-Boltzmann framework, in a thermodynamically consistent manner. Thermofluidic aspects of phase transition are handled by means of a modified enthalpy–porosity formulation, in conjunction with an appropriate enthalpy-updating closure scheme. Lattice-Boltzmann simulations of melting of pure gallium in a rectangular enclosure, Rayleigh–Benard convection in the presence of directional solidification in a top-cooled cavity, and crystal growth during solidification of an undercooled melt agree well with the numerical and experimental results available in the literature, and provide substantial evidence regarding the upscaled computational economy provided by the present methodology.

Lattice Boltzmann Simulation of Incompressible Transport Phenomena in Macroscopic Solidification Processes

A lattice Boltzmann (LB) simulation strategy is proposed for the incompressible transport phenomena occurring during macroscopic solidification of pure substances. The proposed model is derived by coupling a passive scalar-based thermal LB model with the classical enthalpy–porosity technique for solid–liquid phase-transition problems. The underlying hydrodynamics are monitored by a conventional single-particle density distribution function (DF) through a kinetic equation, whereas the thermal field is obtained from another kinetic equation which is governed by a separate temperature DF. The phase-changing aspects are incorporated into the LB model by inserting appropriate source terms in the respective kinetic equations through the most formal technique following the extended Boltzmann equations along with an appropriate enthalpy updating scheme. The proposed model is validated extensively with one- and two-dimensional solidification problems for which analytical and numerical results are available in the literature, and finally, it is used for solving a benchmark problem, the Bridgman crystal growth in a square crucible.

The Effect of Aiding/Opposing Buoyancy on Two-Dimensional Laminar Flow Across a Circular Cylinder

The effect of thermal buoyancy on the upward flow and heat transfer characteristics around a heated/cooled circular cylinder is studied. A two-dimensional finite-volume model is deployed for the analysis. The influence of aiding/opposing buoyancy is studied for the range of parameters −0.5 ≤ Ri ≤ 0.5, 50 ≤ Re ≤ 150, and the blockage ratios of B = 0.02 and 0.25. The flow shows unsteady periodic nature in the chosen range of Reynolds numbers for the forced convective cases (Ri = 0), and the vortex shedding stops completely at some critical values of Richardson numbers.

An enthalpy-based thermal lattice Boltzmann model for non-isothermal systems

An enthalpy-based thermal lattice Boltzmann model is introduced for simulating a class of strongly coupled thermo-hydrodynamic problems. The novelty of the model lies in the formulation of an enthalpy density distribution function to simulate the temperature field, in place of the existing internal energy density distribution function. The proposed model has a clear advantage over the earlier internal energy density distribution function based thermal lattice Boltzmann model, in a sense that it can simulate certain classes of thermofluidic transport problems without facing mathematical difficulties in handling with additional energy source terms.

Hydromagnetic Mixed Convective Transport in a Vertical Lid-Driven Cavity Including a Heat Conducting Rotating Circular Cylinder

Two-dimensional numerical simulation is performed for the hydromagnetic mixed convective transport in a vertical lid-driven square enclosure filled with an electrically conducting fluid in the presence of a heat conducting and rotating solid circular cylinder. Both the top and bottom horizontal walls of the enclosure are considered thermally insulated, and the left and right vertical walls are kept isothermal with different temperatures. The left wall is moving in the upward direction at a uniform speed, while all other walls are stationary. A uniform magnetic field is applied along the horizontal direction normal to the moving wall. A heat conducting circular cylinder is placed centrally within the outer enclosure. The cylinder is made to rotate in its own plane about its centroidal axis. Both the clockwise and counterclockwise rotations of the cylinder are considered. All solid walls are assumed electrically insulated. Simulations are performed for various controlling parameters, such as the Richardson number (1 ≤ Ri ≤ 10), Hartmann number (0 ≤ Ha ≤ 50), and dimensionless rotational speed (−5 ≤ Ω ≤ 5) keeping the Reynolds number based on lid velocity fixed as Re = 100. The flow and thermal fields are analyzed through streamline and isotherm plots for various Ha, Ω, and Ri. Furthermore, the pertinent transport quantities such as the drag coefficient, Nusselt number, and bulk fluid temperature are also computed to understand the effects of Ha, Ω, and Ri on them. It is observed that the heat transfer greatly depends on the rotational speed of the cylinder.

MODELING OF TURBULENT TRANSPORT IN LASER SURFACE ALLOYING

A three-dimensional, transient model is developed in order to address the turbulent transport in a typical laser surface alloying process. Turbulent melt-pool convection is taken into account by using a suitably modified high-Reynolds-number k–ϵ model in the presence of a continuously evolving phase-change interface. The phase-change aspects of the problem are addressed using a modified enthalpy-porosity technique. This newly developed mathematical model is subsequently utilized to simulate a typical high-power laser surface alloying process, where effects of turbulent transport can actually be realized. In order to investigate the effects of turbulence on laser molten pool convection, simulations with laminar and turbulent transport are carried out for some problem parameters. Significant differences in the molten pool morphology are observed on comparing the laminar and turbulent simulation results. It is also revealed that turbulent simulations yield a much better match with the experimentally obtained species concentration distribution within the alloyed layer, compared to that in corresponding laminar simulations.

MHD Mixed Convection in a Lid-Driven Cavity Including a Heated Source

Numerical simulations are performed to understand the thermo-magneto-convective transport of fluid and heat in a vertical lid-driven square enclosure following a finite volume approach based on the SIMPLEC algorithm. The enclosure is filled with an electrically conducting fluid and having a heated source on the right vertical wall. Two different types of sources, such as a semicircular and a rectangular one, are considered. Both the top and bottom horizontal walls and the right vertical wall, except the source of the enclosure, are assumed insulated and the left vertical wall and the sources are kept isothermal with different temperatures. The left vertical wall is also translating in its own plane at a uniform speed, while all other walls are stationary. Two cases of translational lid motion, viz., vertically upward and downward are considered. A uniform magnetic field is applied along the horizontal direction normal to the translating wall. Shear forces due to lid motion, buoyancy forces as a result of differential heating, and magnetic forces within the electrically conducting fluid act simultaneously. Heat transfer due to forced flow, natural convection, and Joule dissipation are taken into account. Simulations are conducted for various controlling parameters, such as the Rayleigh number (103 ≤ Ra ≤ 105), Hartmann number (0 ≤ Ha ≤ 100), and Joule heating parameter (0 ≤ J ≤ 5), keeping the Reynolds number based on lid velocity fixed as Re = 100. The flow and thermal fields are analyzed through streamline and isotherm plots for various Ha and J. Furthermore, the pertinent transport quantities such as the drag coefficient, Nusselt number, and bulk fluid temperature are also plotted to show the effects of Ha, J, and Ra on them.

Forced Convection Heat Transfer From Tandem Square Cylinders for Various Spacing Ratios

This article presents a two-dimensional numerical study on the fluid flow and forced convection heat transfer around two equal isothermal square cylinders placed in a tandem arrangement and subjected to the cross flow of a Newtonian fluid at low Reynolds numbers. The spacing between the cylinders is varied by changing the gap to cylinder size ratio as S/d = 1, 2, 3, 4, 5, 7, and 10. The flow is considered in an unbounded medium; however, fictitious confining boundaries are chosen to make the problem computationally feasible. Numerical calculations are performed by using a PISO algorithm based finite volume solver in a collocated grid system. The Reynolds number is considered in the range 50 ≤ Re ≤ 150 and the Prandtl number is chosen constant as 0.71. The instantaneous vorticity and isotherm patterns are presented and discussed at various Reynolds numbers and spacing ratios for the flow and thermal transport visualization. The critical spacing ratios are predicted for the above range of Re. Additionally, the global flow and heat transfer quantities such as the overall drag and lift coefficients, local and surface average Nusselt numbers, and Strouhal number are calculated and discussed for various Reynolds numbers and spacing ratios.

Mixed Convection Heat Transfer from Tandem Square Cylinders in a Vertical Channel at Low Reynolds Numbers

The fluid flow and heat transfer characteristics around two isothermal square cylinders arranged in a tandem configuration with respect to the incoming flow within an insulated vertical channel at low Reynolds number range (1 ≤ Re ≤ 30) are estimated in this article. Spacing between the cylinders (S) is fixed at four widths of the cylinder dimension (d) and, the blockage parameter (B) is set to 0.25. The buoyancy-aided/opposed convection is examined for the Richardson number (Ri) ranges from −1 to 1 with a fixed Prandtl number (Pr) of 0.7. The transient numerical simulation for this two-dimensional, incompressible, laminar flow and heat transfer problem is carried out by a finite volume code based on the PISO algorithm in a collocated grid system. The results suggest that the flow remains steady for the entire range of parameters chosen in this study. The representative streamlines, vorticity, and isotherm patterns are presented to interpret the flow and thermal transport visualization. Additionally, the time average drag coefficient (C D ) as well as time and surface average Nusselt number (Nu) for the upstream and downstream cylinders are determined to elucidate the effects of Re and Ri on flow and heat transfer phenomena.

Mixed Convection Heat Transfer from Tandem Square Cylinders for Various Gap to Size Ratios

This article presents a two-dimensional numerical study on the fluid flow and mixed convection heat transfer around two equal isothermal square cylinders placed in a tandem arrangement and subjected to the cross flow of a Newtonian fluid at moderate Reynolds numbers. The spacing between the cylinders is varied by changing the gap to cylinder size ratio as S/d = 1, 2, 3, 4, 5, 7, and 10. The Reynolds number is considered in the range 50 ≤ Re ≤ 150. The mixed convection effect is studied for Richardson number range of 0–2, and the Prandtl number is chosen constant as 0.71. The flow is considered in an unbounded medium; however, fictitious confining boundaries are chosen to make the problem computationally feasible. Numerical calculations are performed by using a PISO algorithm-based finite volume solver in a collocated grid system. The effect of superimposed thermal buoyancy on flow and isotherm patterns are presented and discussed. The global flow and heat transfer quantities such as overall drag and lift coefficients, local and surface average Nusselt numbers, and Strouhal number are calculated and discussed for various Reynolds and Richardson numbers and spacing ratios. The notable contribution is the quantification of the critical spacing ratio which is observed to decrease with increasing thermal buoyancy effect for a specific Reynolds number.