A Hybrid Mesh Free Local RBF- Cartesian FD Scheme for Incompressible Flow Around Solid Bodies A
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World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:7, No:10, 2013 A Hybrid Mesh Free Local RBF- Cartesian FD Scheme for Incompressible Flow around Solid Bodies A. Javed, K. Djidjeli, J. T. Xing, S. J. Cox (FV) and Finite Element (FE) methods and significantly Abstract—A method for simulating flow around the solid bodies reduce the computational cost associated with grid generation. has been presented using hybrid meshfree and mesh-based schemes. Due to these features, meshless methods have become one of The presented scheme optimizes the computational efficiency by the hottest areas of research in computational fluid dynamics combining the advantages of both meshfree and mesh-based [12]-[19]. However, despite their stated advantages, meshless methods. In this approach, a cloud of meshfree nodes has been used in the domain around the solid body. These meshfree nodes have the methods possess overall lesser computational efficiency than ability to efficiently adapt to complex geometrical shapes. In the rest traditional mesh based methods like FD, FV and FD [21]. This of the domain, conventional Cartesian grid has been used beyond the in fact, is the major limitation which impedes the wide meshfree cloud. Complex geometrical shapes can therefore be dealt applicability of meshless methods [11]. efficiently by using meshfree nodal cloud and computational Radial Basis Functions (RBFs) have been used for many efficiency is maintained through the use of conventional mesh-based years for multivariate data and function interpolation. In 90s, scheme on Cartesian grid in the larger part of the domain. Spatial discretization of meshfree nodes has been achieved through local use of RBFs was proposed for the solution of Partial radial basis functions in finite difference mode (RBF-FD). Differential Equations (PDEs) [25]. Lately there has been Conventional finite difference scheme has been used in the Cartesian great interest of researchers to use RBFs for solution of PDEs ‘meshed’ domain. Accuracy tests of the hybrid scheme have been on irregular domains by collocation approach [12], [25]-[27]. conducted to establish the order of accuracy. Numerical tests have Local RBF methods have also been proposed to overcome the been performed by simulating two dimensional steady and unsteady ill-conditioning problem associated with dense, large sized incompressible flows around cylindrical object. Steady flow cases have been run at Reynolds numbers of 10, 20 and 40 and unsteady matrices. Local RBFs compromise on spectral accuracy and flow problems have been studied at Reynolds numbers of 100 and produce well-conditioned and sparse linear systems which are 200. Flow Parameters including lift, drag, vortex shedding, and capable of efficiently handling the non-linearities[17]. These vorticity contours are calculated. Numerical results have been found methods are therefore considered well suited for fluid to be in good agreement with computational and experimental results dynamics problems which involve large number of data points available in the literature. in meshfree domains. RBF in Finite Difference Mode (RBF- FD) is a local RBF method which was independently proposed Keywords—CFD, Meshfree particle methods, Hybrid grid, Incompressible Navier Strokes equations, RBF-FD. by Tolstykh et al. [29] and Wright et al. [30] for different set of applications. RBF-FD allows the use of Finite Difference I. INTRODUCTION method on randomly distributed data points (or nodes). The method has been successfully applied to various fluid ESHFREE or meshless methods have emerged as a new dynamics problems involving incompressible Navier Strokes Mclass of numerical techniques during past few years. equations [27], [32]. One of the common characteristics of these methods is that Like other Meshfree methods, RBF based methods also they eliminate, at least, the structure of the mesh by suffer from high computational cost compared with approximating the solution over a set of arbitrarily distributed conventional mesh based methods. For example, calculation of data points (or nodes) [4]-[9]. There is no requirement pre- RBF weights corresponding to the neighbouring particles of a selected connectivity or relationship amongst the nodes of data point requires expensive square root and matrix inversion meshless domain. Meshfree methods can also deal with processes. Moreover, calculation of derivative approximation complex geometries and irregular boundaries more efficiently. at a given order of accuracy usually requires much more Moreover, computational ease of adding or removing nodes number of neighboring particles (or nodes) for meshfree from the domain is another attractive advantage of meshless International Science Index, Mathematical and Computational Sciences Vol:7, No:10, 2013 waset.org/Publication/16932 methods on irregular grid than for finite difference methods on methods. Meshless methods therefore have the potential to Cartesian grid. As a result, the bandwidth of matrices alleviate the mesh generation complexities arising in representing the governing algebraic equations greatly traditional methods like Finite Difference (FD), Finite Volume expands in case of meshfree methods [11]. Therefore, the iteration process is slowed down due to relatively dense matrix A. Javed is with CED Group, Faculty of Engineering and Environment, equations and the computational efficiency is reduced. At the University of Southampton, SO171BJ, Southampton, United Kingdom(Phone same time, meshfree methods have the advantage to efficiently +44-7534354285, e-mail: [email protected]). K. Djidjeli, J. T. Xing, and S. J. Cox are with CED Group, Faculty of deal with complex geometry. Engineering and Environment, University of Southampton, SO171BJ, Conventional mesh-based methods (e.g. FD, FV and FE Southampton, United Kingdom (e-mail: [email protected], jtxing@ methods) offer better computational efficiency over soton.ac.uk, [email protected]). International Scholarly and Scientific Research & Innovation 7(10) 2013 1494 scholar.waset.org/1307-6892/16932 World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:7, No:10, 2013 meshfreemethods. However, generation of an efficient mesh, computational efficiency over its meshfree counterpart. Values which could ensure accurate results, is generally a tedious and of field variables at collocation points on meshfree boundary time consuming task. Cartesian grid is generally considered to serve as boundary conditions for Cartesian grid zone. be much more efficient to generate but it cannot be applied to Therefore, an overall efficient solution scheme is achieved complex geometries. In order to cope with this problem, which can effectively handle irregularly shaped solid coupled multi-grid methods have been proposed [33]-[35]. boundaries. However these methods rely on interpolation approximations at the grid interchanges and suffer from inaccuracies TABLE I COMMON RADIAL BASIS FUNCTIONS especially when high gradients exist at these junctures. Some Re St researchers have also proposed the use of Cartesian grids on Multiquadric (MQ) irregular shapes using boundary-fitted methods [36]-[38]. Inverse Multiquadric (IMQ) 1/ However, these methods necessitate special treatments close Inverse Quadric (IQ) 1/ to the boundary and suffer from time step restrictions due to Gaussian (GA) exp small cut cell to accurately embed irregular boundaries in the Cartesian grid. Another approach is to use non-body II. FORMULATION OF PROBLEM conformal methods [39], [40] in which a background structured mesh is defined behind the solid boundary. The A. Governing Equations interface between fluid and solid is traced using ‘marker Incompressible, transient, viscous Navier-Strokes equations nodes’. These schemes however suffer from inaccuracies in non-dimensional pressure-velocity form are expressed as: coming from accurate tracing of the boundary which is limited by the grid resolution. 0 (1) In view of the foregone, it is logical to come up with a / . 1/ (2) technique which combines the inherent strengths of both meshfree and mesh-based methods. Here, a hybrid technique is presented to simulate incompressible flow around arbitrarily whereis the velocity vector, is the pressure, and is the shaped solid objects. The proposed technique couples RBF- Reynolds number. The equations are discretized using time FD method on meshfree zone around the solid body and implicit RBF-FD method proposed by Javed et al [41]. The conventional Finite difference method applied to the Cartesian method has been shown to be stable and produce accurate grid zone in rest of the domain. Meshfree nodes efficiently results in the regions of gradients high field variables. Detail adapt to the irregular shape of the solid boundary. Outer of temporal and spatial discretization is described below. boundary of the meshfree zone is a regular rectangular or B. Time Splitting squared shaped. A Cartesian mesh can therefore be generated Time splitting of the governing equations has been achieved and coupled to the outer boundary of meshfree zone as shown using implicit fractional step method [41]. In this method, in Fig. 1. Computationally expensive RBF-FD method is nd convective term of momentum equation is treated with 2 therefore limited to only small meshfree zone of the order explicit Adam-Bashforth scheme whereas viscous term computational domain where it