Combination of Meshless Local Petrov-Galerkin and Finite Difference Methods for Analysis of Transient and Incompressible Navier–Stokes Equations

Authors

Sirjan University of Technology

Abstract

This paper presents a numerical algorithm for solving unsteady viscous incompressible two–dimensional (2D) Navier–Stokes equations. In the proposed method, for discretization of time derivatives and solving the Poisson equation of the pressure, Meshless Local Petrov-Galerkin (MLPG) and forward finite difference methods are employed, respectively. In the present analysis, the moving least-square (MLS) approximation is regarded for interpolation, and the Gaussian weight function is used as the test function. To satisfy the boundary conditions, the penalty approach is applied. In the numerical examples, the accuracy and efficiency of the method are compared with those of the exact solutions. The effects of the number of nodes, the size of time interval, as well as the nodes distribution (both regular and irregular) on the relative errors are investigated. Moreover, the Gaussian integral sub-domains with the circular and square shapes are considered, and the accuracy of the results is compared with each other. Analysis of these results for 2D benchmark geometries with different boundary conditions clearly displays that the accuracy of the suggested combined method for solution of the problem related to unsteady viscous incompressible 2D flows is high such that its differences with analytical solution is negligible. Since no limitations is considered on the design process of the regarded numerical algorithm; therefore, it is respected that this approach is successful and has sufficient efficiency to solve the governing equations.

Keywords

References

[1] Hoffmann KA, Chiang S (2000) Computational fluid dynamics engineering education system. Wichita, Kan, USA 1.
[2] Atluri SN, Zhu T (1998) A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics 22: 117-127.
[3] Atluri SN, Shen S (2002) The Meshless Local Petrov-galerkin (MLPG) method.
[4] Lin H, Atluri SN (2001) Meshless Local Petrov-Galerkin (MLPG) method for solving incompressible Navier-Stokes equations. CMES-Comp Model Eng 2: 117-142.
[5] Wu YL, Liu GR, Gu YT (2005) Application of Meshless Local Petrov-Galerkin (MLPG) approach to simulation of incompressible flow. Numer Heat Tr B-Fund 48: 459-475.
[6] Mohammadi M (2006) Meshless Local Petrov-Galerkin (MLPG) method for incompressible viscous fluid. Proceedings of European Fluids Engineering Summer Meeting 1-11.
[7] Mohammadi MH, Shamsai A (2006) meshless solution of 2d fluid flow problems by subdomain variational method using MLPG method with Radial Basis Functions (RBFS). In ASME 2006 2nd Joint US-European Fluids Engineering Summer Meeting Collocated With the 14th International Conference on Nuclear Engineering. 333-341.
[8] Wu XH, Tao WQ, Shen SP, Zhu XW (2010) A stabilized MLPG method for steady state incompressible fluid flow simulation. J Comput Phys 229: 8564-8577.
[9] شایان ا، دادوند ع، میرزایی ا (1394) شبیه­سازی عددی جریان خارجی لزج تراکم­ناپذیر با استفاده از روش        لاتیس بولتزمن بدون شبکه. مجله مکانیک سازه­ها و شاره­ها       189-175 :(4)4.
[10] Enjilela V, Arefmanesh A (2015) Two-step Taylor-Characteristic-Based MLPG method for fluid flow and heat transfer applications. Eng Anal Bound Elem 51: 174-190.
[11] Kovářík K, Masarovičová S, Mužík J, Sitányiová D (2016) A meshless solution of two dimensional multiphase flow in porous media. Eng Anal Bound Elem 70: 12-22.
[12] Musavi HS, Ashrafizaadeh M (2016) A Mesh-Free lattice boltzmann solver for flows in complex geometries. Int J Heat Fluid Fl 59: 10-19.
[13] Dehghan M, Abbaszadeh M (2016) Proper Orthogonal Decomposition Variational Multiscale Element Free Galerkin (POD-VMEFG) meshless method for solving incompressible Navier–Stokes equation. Comput Method Appl M 311: 856-888.
[14] Zhou Y (2017) A Sharp-Interface treatment technique for Two-Phase flows in meshless methods. Comput Fluids 147: 90-101.
[15] Benkhaldoun F, Halassi A, Ouazar D, Seaid M, Taik A (2017) A stabilized meshless method for Time-Dependent convection-dominated flow problems. Math Comput Simulat 137: 159-176.
[16] Bourantas GC, Loukopoulos VC, Chowdhury HA, Joldes GR, Millerc K, Bordasacd SPA (2017)  An implicit potential method along with a meshless technique for incompressible fluid flows for regular and irregular geometries in 2D and 3D. Eng Anal Bound Elem 77: 97-111.
[17] Chen J, Zhang X, Zhang P, Deng J (2017) Variational multiscale element free galerkin method for natural convection with porous medium flow problems. Int J Heat Mass Tran 107: 1014-1027.
[18] Kosec G (2018) A local numerical solution of a fluid-flow problem on an irregular domain. Adv Eng Softw 120: 36-44.
[19] Reséndiz-Flores EO, Saucedo-Zendejo FR (2018) Meshfree numerical simulation of free surface thermal flows in mould filling processes using the finite pointset method. Int J Therm Sci 127: 29-40.
[20] Talat N, Mavrič B, Belšak G, Hatić V, Bajt S, Šarler B (2018) Development of meshless phase field method for two-phase flow. Int J Multiphas Flow 108: 169-180.
[21] Mahmoodabadi MJ, Mahmoodabadi F, Atashafrooz M (2019) Development of the meshless local Petrov-Galerkin method to analyze three-dimensional transient incompressible laminar fluid flow. J Serbian Soc Comput Mech 12 (2):128-152.
[22] محمودآبادی م ج، شجاعی ف، آرسته ز (1397) تحلیل معادله شرودینگر وابسته به زمان سه بعدی به روش بدون المان پتروف-گالرکین محلی. پژوهش سیستم­های بس ذره­ای 51-58 :(17)8.
Journal of Solid and Fluid Mechanics
Volume 10, Issue 4
December 2021
Pages 209-223

Files

Share

How to cite

Statistics