Executive Summary : | The discovery of high-speed computers has fortified a distinctive place for computational fluid dynamics (CFD). With the advent of high-performance computers (HPC), CFD has become a popular approach for researching/exploring complex engineering problems. Generally, the computational approaches can be distinguished into three zones, viz., macroscopic simulation (based upon continuum approach), mesoscopic (based upon collective behavior of a group of particles), and microscopic approach (involves the tracking of molecule or coupling of a couple of molecules). Due to the advantages of both macroscopic and microscopic approaches, the mesoscopic approach has proven to be efficient in simulation for a large number of applications. The mesoscopic approach is based upon the solution of the discretized form of the Boltzmann equation, and the method is called as lattice Boltzmann method (LBM). In this work, the development of a meshless form of LBM (in C++ programming language) will be targeted. It is observed that the original LBM is constructed on the rectangular lattice, resulting in uniformly structured grids. Even if there are several grid refinement techniques, the chances of meshing failure can not be neglected in highly complex problems. Therefore, the proposed project proposal is aimed to develop a meshless LBM solver for the magnetohydrodynamics convection phenomenon. The difference between the lattice-based LBM scheme with meshless one is that the separate solution of the advection equation. Space and time discretization of the advection equation is achieved by meshless local Petrov Galerkin method (MLPG) and Lax-Wendroff schemes, respectively. In the MLPG scheme, first, arbitrarily distributed nodal points are considered in the computational domain. A nodal point is assumed to be central to the quadrature (which may be rectangular, circular, or any shape). Once the quadrature is determined, then the support domain is estimated by using the multi-quadric-radial function. Then particle distribution functions (PDFs) are calculated by the addition of parameters over the nodal points in the local support domain. Once the PDFs are calculated, the macroscopic properties such as density, velocity, temperature, etc., can be calculated. First, the meshless solver will be tested with the standard benchmark problems, such as lid-driven cavity flow, channel flow, etc. After a thorough validation, the developed meshless solver shall be applied three problems, viz., MHD mixed convection in a porous lid-driven cavity containing square and circular block, and flow past a porous square cylinder subjugated to the external magnetic field. The general assumptions will be steady, laminar, incompressible, and non-Newtonian power-law fluids with negligible radiation for all considered problems. First, two-dimensional problems shall be simulated, then an extension to three-dimensional flows shall be entertained. |