Abstract
We present a strong form, meshless point collocation explicit solver for the numerical solution of the transient, incompressible, viscous Navier-Stokes (N-S) equations in two dimensions. We numerically solve the governing flow equations in their stream function-vorticity formulation. We use a uniform Cartesian embedded grid to represent the flow domain. We discretize the governing equations using the Meshless Point Collocation (MPC) method. We compute the spatial derivatives that appear in the governing flow equations, using a novel interpolation meshless scheme, the Discretization Corrected Particle Strength Exchange (DC PSE). We verify the accuracy of the numerical scheme for commonly used benchmark problems including lid-driven cavity flow, flow over a backward-facing step and unbounded flow past a cylinder. We have examined the applicability of the proposed scheme by considering flow cases with complex geometries, such as flow in a duct with cylindrical obstacles, flow in a bifurcated geometry, and flow past complex-shaped obstacles. Our method offers high accuracy and excellent computational efficiency as demonstrated by the verification examples, while maintaining a stable time step comparable to that used in unconditionally stable implicit methods. We estimate the stable time step using the Gershgorin circle theorem. The stable time step can be increased through the increase of the support domain of the weight function used in the DC PSE method.
Highlights
In this paper, we describe the development of a strong form meshless point collocation method for numerically solving the two-dimensional, non-stationary, incompressible Navier-Stokes (N-S)equations in their stream function-vorticity formulation
In the previous example, we considered the flow past a in aninunbounded this example, we study the vortex behind cylinder a rectangulardomain
We further demonstrate the accuracy of the proposed scheme, by comparing our results with those computed using the finite element method (FEM)
Summary
We describe the development of a strong form meshless point collocation method for numerically solving the two-dimensional, non-stationary, incompressible Navier-Stokes (N-S). Our method relies on the ability of meshless methods to accurately compute spatial derivatives, and on their flexibility to solve partial differential equations (PDEs) in complex geometries. The proposed numerical method can efficiently handle uniform Cartesian point clouds embedded in a complex geometry, and/or irregular sets of nodes that directly discretize the flow domain. To compute the spatial derivatives that appear in the governing flow equations, we apply an interpolation meshless scheme, which computes spatial derivatives on both Cartesian embedded grids and irregular nodal distributions. The mathematical formulation of the N-S equations, depending on the choice of dependent variables, can be classified as (i) primitive variables, (ii) stream function-vorticity, and (iii) velocity-vorticity. The majority of numerical methods are expressed in the primitive variables
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