Keywords: meshless, radial basis function, lid-driven, biharmonic.

The lid-driven cavity flow problem is considered as a classical test to assess and validate Navier-Stokes codes. In this paper we address its solution using a radial basis functions (RBF) meshless method. We formulate the steady state incompressible Navier-Stokes equations for a two-dimensional laminar flow in terms of the stream function only. This approach reduces the problem to the solution of a nonlinear biharmonic equation describing the stream function. In the context of finite-difference methods, the numerical solution of the biharmonic equation is cumbersome, since high order stencils are required to solve the resulting equations satisfactorily. However, using the RBF collocation method, it is essentially as complex to use the biharmonic formulation compared with the stream function-vorticity formulation or the velocity-vorticity formulation. Furthermore, the biharmonic formulation has the great advantage that continuity is exactly enforced in the whole domain, thus avoiding one of the main drawbacks of the solution of the lid-driven cavity problem with the RBF method in terms of the velocity-vorticity formulation.

In the RBF collocation method, the stream function is considered to lie in the subspace spanned by the RBFs centered at N chosen nodes. The RBF used is the multiquadric, which is often used for its excellent convergence properties. The coordinates of the numerical solution in the RBFs subspace are determined by collocation of the biharmonic equation at interior nodes, and collocation of the boundary conditions at boundary nodes. The non-slip boundary condition implies that both the stream function and its normal derivative are constant along the boundary. Thus, two boundary conditions have to be enforced at boundary nodes thus leading to more equations than unknowns. To avoid this problem, we include a set of as many ghost RBF nodes located outside the domain, as boundary nodes.

We will describe the results obtained with this method, both for the Stokes problem (Re=0), and for the nonlinear Navier-Stokes problem (Re>0). As could be expected, the accuracy decreases near the singularities in the two upper corners, resulting from to the inability of the smooth RBF basis to capture those singularities. Also, the residual of the biharmonic operator in boundary nodes, where the biharmonic equation is not enforced, is rather large, specially in the vicinity of the singular points. In order to avoid this degradation, we enlarge the space spanned by the RBF basis functions by including additional basis functions, which satisfy the biharmonic equation and which are able to capture the singular behavior of the solution near the corners. With this approach, the spectral convergence of the RBF method is restored.

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