AN APPROXIMATE PROJECTION SCHEME FOR INCOMPRESSIBLE FLOW USING SPECTRAL ELEMENTS

Abstract

SUMMARY An approximate projection scheme based on the pressure correction method is proposed to solve the NavierStokes equations for incompressible flow. The algorithm is applied to the continuous equations; however, there are no problems concerning the choice of boundary conditions of the pressure step. The resulting velocity and pressure are consistent with the original system. For the spatial discretization a high-order spectral element method is chosen. The high-order accuracy allows the use of a diagonal mass matrix, resulting in a very efficient algorithm. The properties of the scheme are extensively tested by means of an analytical test example. The scheme is Mer validated by simulating the laminar flow over a backward-facing step. The solution of the Navier-Stokes equations for unsteady incompressible fluid flow is still a major challenge in the field of computational fluid dynamics. An overview of the most important aspects with respect to the solution of the incompressible Navier-Stokes equations can be found in References 1-5. The Navier-Stokes equations form a set of coupled equations for both velocity and pressure (or, better, the gradient of the pressure). One of the main problems related to the numerical solution of these equations is the imposition of the incompressibility constraint and consequently the calculation of the pressure. The pressure is not a thermodynamic variable, as there is no equation of state for an incompressible fluid. It is an implicit variable which instantaneously 'adjusts itself' in such a way that the velocity remains divergence-free. The gradient of the pressure, on the other hand, is a relevant physical quantity: a force per unit volume. The mathematical importance of the pressure in an incompressible flow lies in the theory of saddle-point problems (of which the steady Stokes equations are an example), where it acts as a Lagrangian multiplier that constrains the velocity to remain divergence-free. There are numerous approaches to solve the Navier-Stokes equations. For the solution of unsteady Navier-Stokes flow perhaps one of the most successful approaches to date is provided by the class of projection methods.&' Projection methods have been developed as a useful way of obtaining an efficient solution algorithm for unsteady incompressible flow. In this paper, projection methods are considered that are applied to the set of continuous equations, yielding very efficient and simple-toimplement algorithms. By decoupling the treatment of velocity and pressure terms, a set of easier-tosolve equations arises: a convectiondiffusion problem for the velocity, yielding an intermediate velocity which is not divergence-free; and a Poisson equation for the pressure (or a related quantity).