On a Wavenumber Optimized Streamline Upwinding Method for Solving Steady Incompressible Navier-Stokes Equations

Abstract

A stabilized upwind finite-element model is developed to solve the three-dimensional incompressible steady Navier-Stokes equations. The test function is constructed to have a larger weight on the upstream side. This has been achieved by adding a stabilized term to the shape function so as to optimize the numerical wavenumber for convection terms. To avoid Lanczos or pivoting breakdown while solving the resulting unsymmetric and indefinite mixed finite-element matrix equations iteratively, the finite-element equation has been modified by pre-multiplying it with its transpose. The resulting normalized matrix equation becomes symmetric and positive-definite. We can therefore apply a computationally efficient conjugate gradient Krylov iterative solver to get an unconditionally convergent solution. However, the condition number of the new system becomes the square of the original unsymmetric indefinite system. To fully exploit excellent convergence nature of the conjugate gradient iterative solver, an element-by-element strategy is adopted to avoid assembling of all the stiffness matrices obtained at element level. We alleviate the drawback of slower convergence of the conjugate gradient method due to the increased condition number by preconditioning the positive-definite matrix. The resulting preconditioned matrix equation is solved in a matrix-free manner using the preconditioned conjugate gradient iterative solver. The developed finite-element code is first verified by solving a problem amenable to analytical solution. The benchmark lid-driven cavity problem is also solved in a cube for assessing the three chosen iterative solvers.