Numerical studies of the flow around a circular cylinder by a finite element method

Numerical solutions of the steady, incompressible, viscous flow past a circular cylinder are presented for Reynolds numbers R ranging from 1 to 100. The governing Navier-Stokes equations in the form of a single, fourth order differential equation for stream function and the boundary conditions are replaced by an equivalent variational principle. The numerical method is based on a finite element approximation of this principle. The resulting non-linear system is solved by the Newton-Raphson process. The pressure field is obtained from a finite element solution of the Poisson equation once the stream function is known. The results are compared with those determined by other numerical techniques and experiments. In particular, the discussion is concerned with the development of the closed wake with Reynolds number, and the tendency of R ≥ 40 flow toward instability.     

New finite element results for the square cavity

New results for the recirculating flow inside a square cavity obtained by a finite element method are presented. The full Navier-Stokes equations in the form of a single, fourth order equation for stremfunction is recast into arestricted variational principles, which form finite element discretization. A triangular element with Hermitian interpolation is used, such that the velocities are continuous and the incompressibility is satisfied exactly. The resulting nonlinear system is solved by Newton-Raphson iteration. Calculations are carried out with several gridworks of progressive show: (1) the convergence of solutions with refinement for fixed Reynolds number $\text{R}$; and (2) the loss of accuracy with $\text{R}$ for fixed gridwork. The range of $\text{R}$ covered is from $\text{10}{}^{\mathrm{?4}}$ to 3450. The features illustrated include the enlargement of the inviscid core of rigid rotation, the intensification of primary and secondary vortices and the appearance of a third secondary vortex near the upper upstream corner at $\text{R =1500}$.     

A transient finite element solution method for the Navier-Stokes equations

This paper is concerned with the discrete finite element formulation and numerical solution of transient incompressible viscous flow in terms of the primitive variables. A restricted variational principle is introduced as equivalent to the momentum equations and the Poisson equation for pressure. The latter is introduced to replace the continuity equation, and thus the incompressibility condition is realized only asymptotically; i.e. through the iterative process. An incomplete cubic interpolation function is used for both the velocities and pressure within a triangular finite element. The discrete equations are integrated in time with backward finite differences. We illustrate the similarity between the (ψ,ζ) finite difference method and the ($\text{u}\text{,p}$) finite element method by calculations on the driven square cavity problem.