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In hydrodynamics the applications range of incompressible flows is very wide. In this study, a robust, high order modeling approach is introduced, based on the MLPG meshfree method-based radial basis functions (RBF-MLPG) method, for solving the incompressible flow field. In other words a MLPG meshfree method based on an interpolation function is presented to solve the 2-D unsteady incompressible fully developed fluid flows. This meshfree method is based on the quartic (4 th order) spline. The method is then compared against the Finite Element Method on a test case of unidirectional fully developed incompressible fluid. The performance of this weight function proved that the quartic (4th order) spline gains the highest accuracy, convergence and efficiency. Finally, it can be concluded that the presented method is formidable for simulating fluid dynamics.

This article presents a numerical algorithm using the Meshless Local PetrovGalerkin (MLPG) method for numerical simulation of unsteady incompressible flows, governed by the Navier-Stokes equations via the stream function-vorticity (ψ-ω) formulation. The driven flow in a square cavity is used as the model problem. The present method is a truly meshless method based only on a number of randomly scattered nodes. The multiquadrics RBFs are employed for constructing trial functions in the local weighted meshless local Petrov-Galerkin method. The present numerical algorithm is based on a local weighted residual method with the Heaviside step function as the test function over a local subdomain. The efficiency, accuracy and robustness of the numerical algorithm are demonstrated by the standard driven cavity. It is observed that the obtained results agreed very well with the results of Ghia. Therefore the ability and accuracy of the present numerical algorithm was presented by solving the standard driven cavity flow problem with reasonable accuracy when compared to solutions obtained by Ghia. In other words the benchmark computations indicate that the MLPG Meshless method is very effective in the simulation of fluid flow problems.

Free surface flows are frequently encountered in hydraulic engineering problems including water jets, weirs and around gates. An iterative solution to the incompressible two-dimensional vertical steady Navier-Stokes equations, comprising momentum and continuity equations, is used to solve for the priori unknown free surface, the velocity and the pressure fields. The entire water body is covered by a unstructured finite element grid which is locally refined. The dynamic boundary condition is imposed for the free surface where the pressure vanishes. This procedure is done continuously until the normal velocities components vanish. To overcome numerical errors and oscillations encountering in convection terms, the SUPG (streamline upwinding Petrov-Galerkin) method is applied. The solution method is tested for different discharges onto a standard spillway geometries. The results shows good agreement with available experimental data.