A Semi-Implicit Meshless Method for Incompressible Flows in Complex Geometries

Abstract

We present an exponentially convergent semi-implicit meshless algorithm for the solution of Navier-Stokes equations in complex domains. The algorithm discretizes partial derivatives at scattered points using radial basis functions as interpolants. Higher-order polynomials are appended to polyharmonic splines (PHS-RBF) and a collocation method is used to derive the interpolation coefficients. The interpolating kernels are then differentiated and the partial-differential equations are satisfied by collocation at the scattered points. The PHS-RBF interpolation is shown to be exponentially convergent with discretization errors decreasing as a high power of a representative distance between points. We present here a semi-implicit algorithm for timedependent and steady state fluid flows in complex domains. At each time step, several iterations are performed to converge the momentum and continuity equations. A Poisson equation for pressure corrections is formulated by imposing divergence free condition on the iterated velocity field. At each time step, the momentum and pressure correction equations are repeatedly

Consistency and Convergence of a High Order Accurate Meshless Method for Solution of Incompressible Fluid Flows

“…The bottom and top boundaries are prescribed as walls with no-slip and nopenetration. For details of the implementation of periodicity in meshless method, please refer to our previous work [26]. We define the minimum width of the channel (L = 0.5) as the characteristic length and the density (ρ) is set to unity.…”

“…There are several types of meshless methods used for the solutions of heat transfer and fluid flow problems [14][15][16][17][18][19][20][21][22][23][24][25][26]. The intent of this work is to study the consistency and convergence characteristics of a recently developed high order meshless technique [25] that solves the incompressible Navier-Stokes equations using Polyharmonic Spline Radial Basis Function (PHS-RBF) interpolations of scattered data.…”

Shear-Driven Flow in an Elliptical Enclosure Generated by an Inner Rotating Circular Cylinder

“…In order to reduce the computational time, the flow field at a lower Reynolds number is used as an initial guess for a higher Reynolds number. Detailed analysis of the current semi-implicit procedure are presented in [54].…”

“…The two-dimensional Navier-Stokes equations in the complex enclosure are solved by a novel highaccuracy meshless method [52] utilizing radial basis function (RBF) interpolations [53] of scattered data. We recently developed a semi-implicit time marching algorithm [54] to integrate the time-dependent incompressible fluid flow equations. All the computations in this paper are performed using the open-source Meshless Multi-Physics Software (MeMPhyS) [55].…”

“…Collocating the governing equation at a given point gives an implicit equation connecting value at the local point with values at the points in the cloud. Further details of the derivation and properties of the derivative and Laplacian coefficients are given in ourprevious publications[52,54,57,58].…”