An Efficient Localized Radial Basis Function Meshless Method for Fluid Flow and Conjugate Heat Transfer

Abstract

A localized radial basis function (RBF) meshless method is developed for coupled viscous fluid flow and convective heat transfer problems. The method is based on new localized radial-basis function (RBF) expansions using Hardy Multiquadrics for the sought-after unknowns. An efficient set of formulae are derived to compute the RBF interpolation in terms of vector products thus providing a substantial computational savings over traditional meshless methods. Moreover, the approach developed in this paper is applicable to explicit or implicit time marching schemes as well as steady-state iterative methods. We apply the method to viscous fluid flow and conjugate heat transfer (CHT) modeling. The incompressible Navier–Stokes are time marched using a Helmholtz potential decomposition for the velocity field. When CHT is considered, the same RBF expansion is used to solve the heat conduction problem in the solid regions enforcing temperature and heat flux continuity of the solid/fluid interfaces. The computation is accelerated by distributing the load over several processors via a domain decomposition along with an interface interpolation tailored to pass information through each of the domain interfaces to ensure conservation of field variables and derivatives. Numerical results are presented for several cases including channel flow, flow in a channel with a square step obstruction, and a jet flow into a square cavity. Results are compared with commercial computational fluid dynamics code predictions. The proposed localized meshless method approach is shown to produce accurate results while requiring a much-reduced effort in problem preparation in comparison to other traditional numerical methods.

An RBF Interpolation Blending Scheme for Effective Shock-Capturing

“…It is often found that a higher valued shape parameter causing the interpolation matrix to become close to ill-conditioned gives the best accuracy [1,2]. The issue of the ill-conditioning does not allow for global interpolation as solutions are not accurate, but researchers have found that locally interpolating using RBF methods resolves these issues [2,4,7,8,10].…”

“…These techniques originate from spectral methods based on Legendre or Chebyshev polynomial, which require uniform point distribution [8][9][10]. However, the meshless methods using radial basis functions (RBF) can be used on non-uniform distributions of points.…”

“…Current numerical methods make use of the Finite Difference Methods (FDM), Finite Element Method (FEM) and more commonly in CFD the Finite Volume Method (FVM). These mesh based methods are well developed numerical techniques and are the methods used today for solving PDEs [1][2][3][4][5][6][7][8][9][10]. Mesh based methods require a mesh, grid, or connectivity to be defined within the domain so the governing PDEs can be discretized and solved.…”

A Coupled Localized RBF Meshless/DRBEM Formulation for Accurate Modeling of Incompressible Fluid Flows

“…The localized RBF collocation meshless methods (LRC-MM) address the issues that arise from the global RBF approach, see [1,6]. LRC-MM have been implemented in the solution of incompressible fluid flows effectively adapting upwinding schemes for convective-dominated flows, see [1,3,5,6]. However, in decoupling the governing equations for incompressible fluid flows, a Poisson-like equation arises for the solution of the pressure field at each time step.…”

“…These are being used to resolve complex problems, such as flow transport problems, see [3]. There are several advantages to them such as accuracy control, since additional nodes made be added where needed, and a more accurate representation of the objects through meshfree discretization, see [4].…”