[en] This paper presents a self-similar solution of the coupled problem of magneto-hydrodynamic free convection flow of an electrically conducting fluid arising from melting of a semi-infinite solid substrate. At steady state, buoyancy induced free convection of the electrically conducting fluid is influenced by the Lorentz force. A set of governing PDEs is developed for a two dimensional boundary layer problem including phase change which is simplified to a set of ODEs using a similarity transformation and are solved iteratively using an implicit Keller-box method. An asymptotic analytical solution for melting and heat transport rates is also presented for the case of small Prandtl numbers. The effect of each of the three characteristic parameters, viz., the Prandtl number, the melting parameter and the Lykoudis number on the similarity velocity and temperature profiles in the boundary layer over melting substrate is studied. It is observed that increasing the Lykoudis number or decreasing the Prandtl number lowers the melting rate and heat transfer at the substrate-melt interface. The use of magnetic field in controlling the free convection heat transfer, the melting rate and the thickness of the velocity and thermal boundary layers over melting substrate is elucidated and discussed. (authors)

[en] The wall-adapting local eddy-viscosity (WALE) and Vreman subgrid scale models for large eddy simulations are compared within the framework of a generalised lattice Boltzmann method. Fully developed turbulent flows near a flat wall are simulated with the two models for the shear (or friction) Reynolds number of 183.6. Compared to the direct numerical simulation (DNS), damped eddy viscosity in the vicinity of the wall and a correct velocity profile in the transitional region are achieved by both the models without dynamic procedures. The turbulent statistics, including, e.g., root-mean-square velocity fluctuations, also agree well with the DNS results. The comparison also shows that the WALE model predicts excellent damped eddy viscosity near the wall. (authors)

[en] Highlights: • A cascaded central moment based lattice Boltzmann (LB) method is developed for solving thermal convective flows. • A double distribution function framework is used to solve for the temperature and velocity fields in a cascaded formulation using a D2Q9 lattice. • Heat sources are consistently implemented using a variable transformation. • A Chapman–Enskog analysis is performed to show consistency with the convection diffusion equation for the temperature field. • The method is demonstrated to be second order using various benchmark problems. - Abstract: A cascaded central moment based lattice Boltzmann (LB) method for solving low Mach number thermal convective flows with source terms in two-dimensions in a double distribution function framework is presented. For the passive temperature field, which satisfies the convection diffusion equation (CDE) with a source term to represent internal/external local heat source, a new cascaded collision kernel is presented. Due to the use of a single conserved variable in the thermal energy equation, the cascaded structure in its collision operator begins from the first order moments and evolves to higher order moments. This is markedly different from the collision operator for the fluid flow equations, constructed in previous work, where the cascaded formulation starts at the second order moments in its collision kernel. A consistent implementation of the spatially and temporally varying source terms in the thermal cascaded LB method representing the heat sources in the CDE that maintains second order accuracy via a variable transformation is discussed. The consistency of the thermal cascaded LB method including a source term for the D2Q9 lattice with the macroscopic convection–diffusion equation is demonstrated by means of a Chapman–Enskog analysis. The new model is tested on a set of benchmark problems such as the thermal Poiseuille flow, thermal Couette flow with either wall injection or including viscous dissipation and natural convection in a square cavity. The validation study shows that the thermal cascaded LB method with source term is in very good agreement with the analytical solutions or numerical results reported for the benchmark problems. In addition, the numerical results show that our new thermal cascaded LB model maintains second order accuracy.

[en] Highlights: • Cascaded lattice Boltzmann (LB) scheme using a pressure formulation for solution of two-phase flows at high density ratios. • Central moments of equilibria and sources of the method obtained by matching with their continuous counterparts. • Method capable to handle both uniform surface tension and variable surface tension (Marangoni) stress effects. • Interface capturing using a conservative phase field model solved using another cascaded LB scheme with modified equilibria. • Simulation results demonstrate accuracy and improvements in numerical stability. Simulation of multiphase flows, which are ubiquitous in nature and engineering applications, require coupled capturing or tracking of the interfaces in conjunction with the solution of fluid motion often occurring at multiple scales. In this contribution, we will present unified cascaded LB methods based on central moments for the solution of the incompressible two-phase flows at high density ratios and for capturing of the interfacial dynamics. Based on a modified continuous Boltzmann equation (MCBE) for two-phase flows, where a kinetic transformation to the distribution function involving the pressure field is introduced to reduce the associated numerical stiffness at high density gradients, a central moment cascaded LB formulation using multiple relaxation times for computing the fluid motion will be constructed. In this LB scheme, the collision step is prescribed by the relaxation of various central moments to their equilibria that are reformulated in terms of the pressure field obtained via matching to the continuous equilibria based on the transformed Maxwell distribution. Furthermore, the differential treatments for the effects of the source term representing the change due to the pressure field and of the source term due to the interfacial tension force and body forces appearing in the MCBE on different moments are consistently accounted for in this cascaded LB solver that computes the pressure and velocity fields. In addition, another cascaded LB scheme will be developed to solve for the interfacial dynamics represented by a phase field model based on the conservative Allen-Cahn equation that evolves interfaces by advection and under the competing effects due to a diffusion term and a phase segregation flux term. The latter is introduced into the cascaded LB scheme via a modification to the moment equilibria. Based on numerical simulations of a variety of two-phase flow benchmark problems at high density ratios and involving the effects of surface tension and its tangential gradients (Marangoni stresses), we will validate our unified cascaded LB approach and also demonstrate improvements in numerical stability.

[en] Lattice Boltzmann method (LBM) is a kinetic based numerical scheme for the simulation of fluid flow. While the approach has attracted considerable attention during the last two decades, there is a need for systematic investigation of its applicability for complex canonical turbulent flow problems of engineering interest, where the nature of the numerical properties of the underlying scheme plays an important role for their accurate solution. In this paper, we discuss and evaluate a LBM based on a multiblock approach for efficient large eddy simulation of three-dimensional external flow past a circular cylinder in the transitional regime characterized by the presence of multiple scales. For enhanced numerical stability at higher Reynolds numbers, a multiple relaxation time formulation is considered. The effect of subgrid scales is represented by means of a Smagorinsky eddy-viscosity model, where the model coefficient is computed locally by means of a dynamic procedure, providing better representation of flow physics with reduced empiricism. Simulations are performed for a Reynolds number of 3900 based on the free stream velocity and cylinder diameter for which prior data is available for comparison. The presence of laminar boundary layer which separates into a pair of shear layers that evolve into turbulent wakes impose particular challenge for numerical methods for this condition. The relatively low numerical dissipation introduced by the inherently parallel and second-order accurate LBM is an important computational asset in this regard. Computations using five different grid levels, where the various blocks are suitably aligned to resolve multiscale flow features show that the structure of the recirculation region is well reproduced and the statistics of the mean flow and turbulent fluctuations are in satisfactory agreement with prior data. (paper)