Development of a flow-condition-based interpolation 9-node element for incompressible flows

Abstract

The Navier-Stokes equations are widely used for the analysis of incompressible laminar flows. If the Reynolds number is increased to certain values, oscillations appear in the finite element solution of the Navier-Stokes equations. In order to solve for high Reynolds number flows and avoid the oscillations, one technique is to use the flow condition-based interpolation scheme (FCBI), which is a hybrid of the finite element and the finite volume methods and introduces some upwinding into the laminar Navier-Stokes equations by using the exact solution of the advection-diffusion equation in the trial functions in the advection term. The previous works on the FCBI procedure include the development of a 4-node element and a 9-node element consisting of four 4-node sub-elements. In this thesis, the stability, the accuracy and the rate of convergence of the already published FCBI schemes is studied. In addition, a new FCBI 9-node element is proposed that obtains more accurate solutions than the earlier proposed FCBI elements. The new 9-node element does not obtain the solution as accurate as the Galerkin 9-node elements but the solution is stable for much higher Reynolds numbers (than the Galerkin 9-node elements), and accurate enough to be used to find the structural responses in fluid flow structural interaction problems. The Cubic-Interpolated Pseudo-particle (CIP) scheme is a very stable finite difference technique that can solve generalized hyperbolic equations with 3rd order accuracy in space.

(cont.) In this thesis, in order to solve the Navier-Stokes equations, the CIP scheme is linked to the finite element method (CIP-FEM) and the FCBI scheme (CIP-FCBI). From the numerical results, the CIP-FEM and the CIP-FCBI methods appear to predict the solution more accurate than the traditional finite element method and t;he FCBI scheme. In order to obtain accurate solutions for high Reynolds number flows, we require a finer mesh for the finite element and the FCBI methods than for the CIP-FEM and the CIP-FCBI methods. Linking the CIP method to the finite element and the FCBI methods improves the accuracy for the velocities and the derivatives. In addition, when the flow is not at the steady state and the time dependent terms need to be included in the Navier-Stokes equations, or in the problems when the derivatives of the velocities need to be obtained to high accuracy, the CIP-FCBI method is more convenient than the FCBI scheme.