Weakly viscous two-dimensional incompressible fluid flows using efficient isogeometric finite element method

Abstract

This paper presents a straightforward and efficient numerical simulation method for solving the Navier–Stokes equations for weakly viscous incompressible fluids describing steady flow. Our approach utilizes isogeometric finite elements to handle higher-order partial differential operators associated with weakly viscous incompressible flow problems. Specifically, our numerical formulation employs a principle of virtual power (PVP)-based weak formulation that utilizes a stream-function field, which distinguishes it from the more commonly used bi-harmonic type formulations. The usage of a stream-function field ensures a pointwise divergence-free velocity condition, making the present method suitable for low to moderately high Reynolds number flow problems. In contrast to the bi-harmonic formulation, which is typically used for describing internal flow and requires special treatment of outflow boundary conditions, the PVP-based formulation is more general and does not require special treatment at the outflow boundary. It is also demonstrated that both bi-harmonic and PVP-based weak formulations yield identical results for internal flow problems. Our method employs non-uniform rational B-spline basis functions, and we present a simple stitching technique for imposing no-slip Dirichlet boundary conditions. Finally, we solve Poisson's equation to recover the pressure field. Furthermore, we use an appropriate Gaussian quadrature that is exact for splines to speed up the computation of various element matrices, especially for high polynomial degrees. The proposed formulation is evaluated by solving a set of numerical problems, including internal flow and channel flow problems.