Control volume-radial basis function solution of 2D driven cavity flow in terms of the velocity vorticity formulation

The two-dimensional Navier Stokes system of equations for incompressible flows is solved in the velocity vorticity formulation by means of the Control Volume-Radial Basis Function (CV-RBF) method. This method is an improvement to the Control Volume Method (CVM) based on the use of Radial Basis Function (RBF) Hermite interpolation instead of the classical polynomial functions. The main advantages of the CV-RBF method are the approximation order, the meshless nature of the interpolation scheme and the presence of the PDE operator in the interpolation. Besides, the vorticity boundary values are computed in terms of the values of the velocity field at the neighbouring nodal points according to its definition by applying the curl operator to the local velocity interpolation function. Several interpolation strategies are tested for both the velocity and vorticity fields. A Newton type algorithm is implemented to solve the coupled system of non linear equations. As test example, the proposed numerical scheme is used to solve the lid driven cavity flow problem up to Re = 5000, where high Reynolds number solutions are achieved by using a Conservative and Hermitian interpolation for the velocity field.