Simulation of viscous fluid flows using a control volume finite element-multigrid method

Abstract

The formulation and implementation of an equal-order colocated control volume finite element-multigrid (CVFE-MG) method for steady, two-dimensional, viscous incompressible flows is presented in this thesis. In the proposed CVFE-MG, the calculation domain is discretized using three-node triangular elements. Each element is further divided in such a way that control volumes are formed around each node in the calculation domain. The proposed method is formulated using the velocity components and pressure as dependent variables, and interpolation functions for these dependent variables are all based on an elemental level. The pressure and the diffused scalars are interpolated linearly; the convected scalars are interpolated using mass weighted interpolation which guarantees positive contributions to the coefficients in the algebraic discretization equation; and the transporting velocities are interpolated using a linear interpolation of pseudo-velocities and pressure coefficients, in which the pressure gradients appear explicitly. This feature allows the formulation of an equal-order colocated method valid for incompressible flows. Using these interpolation functions, the discretized forms of the governing equations are obtained by deriving algebraic approximations to integral conservation equations for each control volume. These nonlinear, coupled, algebraic equations are then solved by a segregated solution algorithm. This solution method is implemented in the context of FMV- and V-cycle multigrid algorithms in an attempt to improve its convergence behaviour. -- The proposed CVFE-MG method was found to generate solutions that could capture the physical behaviour of the fluid flows used as test problems. The multigrid methods were found to accelerate the convergence rate 2.18 to 7.43 times for the outflow and recirculating flow test problems presented. The effectiveness of the multigrid algorithm was reduced for higher Reynolds numbers, due to the interpolation schemes used in the control volume finite element method (CVFEM). -- The successful implementation of multigrid algorithms in the context of a primitive variables, viscous flow CVFEM is very encouraging. Further research will be performed to improve the effectiveness of CVFE-MG implementations.