THE DEVELOPMENT OF MESHLESS LOCAL PETROV GALERKIN (MLPG) METHOD FOR COMPLEX FLUID FLOW SIMULATION

Abstract

Over the past decades, mesh reducing techniques (meshfree or meshless) have considered as e ective numerical techniques for solving engineering problems. Meshing di culties and consumption of more time for speci c engineering problems, in the existing numerical methods (FDM,FEM,FVM), which produces serious numerical errors, were the initial thrust for the development of the meshfree methods. The prede ned numerical methods(FDM, FEM, FVM) is not well suited to those problems which having complex geometries, large deformation of the materials, encountering moving discontinuities such as crack propagation along arbitrary and complex paths, involving meshing or remeshing in structural optimization problems and moving material interfaces etc (Liu, 2003a). Therefor it is impossible to completely overcome to these di culties by a well known mesh-based method. Due to overcome the above mentioned di culties by the mesh reducing techniques, the meshfree or meshless methods achieved great attentions from past few decade. There are several meshfree techniques have been developed by the previous researchers, in which main attentions achieved by Lucy (1977) and Gingold and Monaghan (1977) by his rst developed meshfree technique to solve the astrophysical phenomena, which is well known as SPH method now these days. Due to inaccurate or having large numerical errors at the boundary in the SPH method, further various types of meshfree techniques have been developed such as di use element method(DEM), element free Galerkin(EGG), hp Cloud method, reproducing kernal particle method, nite point method (FPM), partition of unity method (PUM), boundary node method (BNM), local boundary integral equation (LBIE) method, meshless local Petrov- Galerkin method (MLPG) etc. Between these meshfree techniques having there advantages and disadvantages, not considered as complete meshfree technique because to need a background mesh for integration iii ABSTRACT iv purpose of the computational domain except LBIE and MLPG method. The MLPG and LBIE is the only complete meshfree methods which acts as a locally and does not need any background mesh for integration. Atluri and Zhu (1998a) were the rst who developed the MLPG method for solid mechanics problems. But for the time being MLPG achieved great attentions to solve structural mechanics, uid ow (convection-di usion) phenomena and other engineering eld problems successfully. Hence Meshless Local-Petrov Galerkin (MLPG) is selected as a numerical tool for our objectives because of its truly meshfree characteristics for the various types of bench mark engineering problems having one dimensional and two dimensional space, including di usion problems and uid ow examples to check its e ciency and accuracy. Indeed non-Newtonian uid ow behavior is encountered in almost all the chemical as well as allied processing industries. Therefor this method is extended for the one dimensional and two dimensional complex uid ow simulations, because of all the natural rehology follows the non-Newtonian behaviors and are very useful such as sewage industries, paint industry, water treatment plants and other process industries etc. Due to very limited or negligible research has been done in the complex uid ow simulations by using this method. Therefor In this study, a numerical investigation is done to observe the complex uid ow behavior and characterize the non-Newtonian uid properties. Based on considering available literature in chapter 2, the essential features and its numerical formulation of MLPG method, basic numerical techniques to impose the boundary conditions are explained in chapter 3. According to study the MLPG needs three basic steps to formulate on a local sub-domain s in to entire computational domain , as (1) a weight function w(x), to convert the PDE in to weaker consistency, (2) a basis function P(x) and (3) a shape function, which is formulated with the help of basis function P(x) by using non-element interpolation technique such as moving least square (MLS) approximations (Lancaster and Salkauskas, 1986). It is clearly distinguished between support domain J tr and test domain I te, and to select the suitable domain sizes for multidimensional spaces. The types of MLPG and their applications are explained for di erent eld problems as given in Atluri and Shen (2005). Due to the complex nature of MLPG integrational technique described in Atluri et al. (1999b), the Gauss integration technique for quadrature domain q are considered, which is well explained in the book "Meshfree Methods, moving beyond the nite element method" (Liu, 2003a). A computational tool (C++) for MLPG has been developed to validate the basic bench mark CFD problems for 1D space di usion problems given in the text book Versteeg and Malalasekera (2007) and 2D di usion problems. For one dimensional problem the ABSTRACT v numerical solutions are validated with exact solution, and for 2D space the numerical solution are validated with commercial CFD package FLUENT 6.3. The results shows very close agreement, thus it is expected that MLPG method (which is a truly meshless) is very promising in solving engineering heat conduction problems in a great extent. Afterwords 1D convection-di usion problems (Lin and Atluri, 2000) at high Peclet number by using suitable upwind scheme to avoid its wiggle nature at high Pe, is validated, and found very good agreement for convection dominated problems, which is discussed in detailed as in chapter 4. In the rst attempts to validate and verify the use of proposed in-house MLPG based solver for complex uid ow simulations for multidimensional space. In order to ful ll this objective, the one-dimensional incompressible steady ow of non-Newtonian power-law

uids through rectangular channel ow problem has been considered. It is governed by the mass continuity and Navier-Stokes equations. The uid rheology is governed by the non-Newtonian power-law uid viscosity model. The computational solutions using in-house solver has been compared and validated with the exact solution. The in uence of computational nodes, N=21, 41 and 51 is also analyzed. In particular, the velocity pro le, shear-stress pro le and the values of pressure drop have been obtained a compared for a range of power-law index (0:4 n 1:8). Since the shear-dependent viscosity, special attempts was required to handle the oscillations in numerical solutions due to shear-dependent nature of viscosity and shear-stress. The present results are corresponding well with the exact solution over the ranges of conditions considered herein. Hence, the present work builds up a new numerical algorithm for their reliable use in the understanding of complex uid ow phenomena and lastly summarizes the present work and opens up the scope of future work. Subsequently, the two dimensional incompressible steady ow of non-Newtonian power-law

uids through square channel ow problem has been considered at di erent Reynolds number(Re). In the non-Newtonian uid rheology is also governed by the power-law

uid viscosity model which is governed by continuity and momentum equations in both the directions. The space domain for N=11 11 and 21 21 is chosen. The numerical solutions has been compared with exact solution. The power law index between 0.6 n 1.4 is varied at di erent Re. Since due to shear-dependent viscosity, special attempts was required to handle the oscillations in numerical solutions. A good agreement has been observed for di erent power-law index (n) at Re = 1 and 20. But at high Reynolds number Re 1, the well established upwind scheme for Newtonian ow is not enough to avoid its wiggle nature due to shear-dependent nature of viscosity and shear-stress. Therefore, more ABSTRACT vi investigation is needed for non-Newtonian uid rheology at higher Reynolds number ow of shear-dependent viscous ow. Finally, the last chapter summarizes the ndings of the present thesis and provides the possibilities of future scope of the research.