A Spectral Element Method for Viscoelastic Fluid Flow

This thesis establishes a new spectral element technique for solving incompressible Newtonian flows and viscoelastic flows of non-Newtonian fluids. In the numerical simulation of the incompressible Newtonian fluid in two dimen­sions, the spectral element approximations of the Stokes and Navier-Stokes equations are established, based on the primitive variables: velocity and pres􀀉ure. An Uzawa algorithm is introduced to decouple the original saddle point problem into two sym­metric positive definite forms for the velocity and pressure, then a preconditioned conjugate gradient iteration is developed to solve the system of equations. The appro­priate approximation spaces for the velocity and pressure are discussed. The numerical simulations for a planar channel and a planar 2 : 1 contraction channel flows show the high accuracy of the current scheme and excellent agreement with other numerical studies. In order to improve the continuity of the spectral element approximation in the element interface, a smoothing technique is developed to calculated the first deriva­tive of the approximations by using only the adjacent sub-element information. The nonlinear convective terms are calculated by the smoothing method in the governing equations. A new algorithm, which combines the spectral element method with an elastic vis­cous split stress method, is developed for solving viscoelastic fluid flows in the planar contraction channel. In this algorithm, a new variable, the rate of deformation tensor, is introduced into the spectral element discretization formulation in order to maintain the mathematical elliptic property of the momentum and continuity equations. The system of spectral element approximations to the velocity, pressure, extra-stress and the rate of deformation variables is solved by the preconditioned conjugate gradient method. The choice of the approximation function spaces for the extra-stress and the rate of deformation are discussed. In the thesis, the extra-stress approximation space is chosen to be the same as the velocity space and the rate of deformation approximation space is the same as the pressure space. The numerical approach is implemented on the planar 4: 1 contraction channel for a fluid governed by an Oldroyd-B constitutive equation. The behaviour of the Oldroyd-B fluids in the contraction channel is inves­tigated for various Weissenberg numbers. The numerical simulations show that the spectral element method with the EVSS approach is efficient for computing flows of the Oldroyd-B fluids in the planar contraction channel. Numerical solutions show the size of the corner vortex compares well with other numerical predictions as the elas­ticity parameter We increases for both creeping flow and inertial flow. The influence of the Weissenberg numbers on the stress and vorticity is also discussed.

Name dc:type.qualificationname

PhD

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

De Montfort University

Year dc:date.issued

2001

Author dc:creator

• Meng, Sha

Licence

• https://dora.dmu.ac.uk/bitstreams/77f13826-ba8a-4c00-ab9c-291d7d70f218/download