Multigrid Computation of Steady and Time-Dependent Incompressible Viscous Flows

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The present thesis deals with the development of multigrid-assisted incompressible viscousflow solvers for both steady and unsteady flows and their application to several problems of varying complexities to throw light on various aspects of the working of the multigrid method. Some of the applications deal with problems to which multigrid method was not used earlier. Multigrid methods, which are based on many levels of grids, have the ability to overcome the slow convergence characteristic of the single-grid methods for large-scale problems. This method has been established as a powerful tool for accelerating numerical convergence and, thus reducing the computational time. The two versions of multigrid methods are linear multigrid method, known as the correction scheme applicable to linear equations and nonlinear multigrid method also known as the full-approximation storage (FAS) multigrid method applicable to nonlinear equations. In the computation of transient viscous flows the pressure-Poisson equation has to be solved accurately at every time-step and it is imperative that this computation is carried out with good time wise efficiency. Use of multigrid here to compute the solution of the pressure-Poisson equation is justified at the beginning by showing that for a heat conduction problem, linear multigrid is better than the point successive overrelaxation (PSOR) method when Gauss-Seidel method is used to solve the discretized algebraic system. In the transient solvers the convective terms are discretized to third-order accuracy and the viscous terms are discretized to fourth-order accuracy, and a fractional-step technique is used to obtain second-order temporal accuracy. Finite-difference discretization with transformation from the physical space to the computational space is carried out on graded Cartesian meshes to obtain better scale resolution in shear layers, so that the transient computations enjoy the advantages of good grid economy provided by the graded mesh and good time efficiency provided by the multigrid. Some of the transient computations include (i) computation of unsteady and asymptotically obtained steady flow in a single-sided lid-driven square cavity including periodic flows that develop at higher Reynolds numbers like Re = 8200 and 10000 as shown here. (ii) Computation of transient flow in a hitherto-unexamined configuration of two-sided lid-driven square cavity that involves gradual development of a free Dshear layerD with associated vortices. (iii) Computation of steady flow in the backward-facing step in a channel through time-marching, that shows that 2D flows at Re = 800 and 1000 appear to be steady. (iv) Computation of flow past a square prism confined in a channel, which - beyond a certain Reynolds number - involves periodic vortex shedding giving rise to the so-called von KDarmDan vortex street. Periodicity of the flow is established through phase plots, power spectrum analysis and temporal plots of lift and drag coefficients. A computer code having similar features as the 2D code described above is also developed to compute 3D flows especially in geometries that allow graded Cartesian meshes to resolve sharp gradients in shear layers by locally clustering the grid there. First, computations of transient and steady flows are carried out for the single-sided lid-driven cubical cavity. The flow patterns are studied at differ.................
Supervisor: Anoop Kr. Dash & Anupam Dewan