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SPECIAL SECTION ON CFD METHODS

# A Meshless Local Petrov-Galerkin Method for Fluid Dynamics and Heat Transfer Applications

[+] Author and Article Information
Ali Arefmanesh

Islamic Azad University, Science & Research Division, Mechanical Engineering Department, Tehran 14155-4933, IRAN, Tel: 4817170, Fax: 4817175a̱aref32@yahoo.com

Islamic Azad University, Science & Research Division, Mechanical Engineering Department, Tehran 14155-4933, IRAN, Tel: 4817170, Fax: 4817175m̱njafi36@yahoo.com

Hooman Abdi

Islamic Azad University, Science & Research Division, Mechanical Engineering Department, Tehran 14155-4933, IRAN, Tel: 4817170, Fax: 4817175hoomanabdi@softhome.net

J. Fluids Eng 127(4), 647-655 (Apr 18, 2005) (9 pages) doi:10.1115/1.1949651 History: Received August 18, 2004; Revised April 18, 2005

## Abstract

The meshless local Petrov-Galerkin method has been modified to develop a meshless numerical technique to solve computational fluid dynamics and heat transfer problems. The theory behind the proposed technique, hereafter called “the meshless control volume method,” is explained and a number of examples illustrating the implementation of the method is presented. In this study, the technique is applied for one- and two-dimensional transient heat conduction as well as one- and two-dimensional advection-diffusion problems. Compared to other methods, including the exact solution, the results appear to be highly accurate for the considered cases. Being a meshless technique, the control volumes are arbitrarily chosen and possess simple shapes, which, contrary to the existing control volume methods, can overlap. The number of points within each control volume and, therefore, the degree of interpolation, can be different throughout the considered computational domain. Since the control volumes have simple shapes, the integrals can be readily evaluated.

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## Figures

Figure 1

Domain Ω with two circular control volumes

Figure 2

One-dimensional control volumes for (a) uniform point distribution with second-order interpolation function and (b) nonuniform point distribution

Figure 3

Temperature variation with time at x=0.5 for the one-dimensional transient conduction with uniform point distribution

Figure 14

MCVM solution of the one-dimensional steady advection-diffusion equation for Pe=50 with uniform point distribution

Figure 15

MCVM solution of the two-dimensional advection-diffusion equation for uniform point distribution

Figure 4

Temperature distribution in the domain at three different times for the one-dimensional transient conduction with uniform point distribution

Figure 5

Temperature distribution in the domain at three different times for the one-dimensional transient conduction with nonuniform point distribution

Figure 6

Domain and a typical control volume for the two-dimensional problems with (a) uniform point distribution (b) nonuniform point distribution

Figure 7

Temperature distribution in the domain at t=0.025s for the two-dimensional transient conduction with uniform point distribution

Figure 8

Temperature distribution along the y-axis at x=0.5 for the two-dimensional transient conduction with uniform point distribution

Figure 9

Temperature distribution along the y-axis at x=0.5 for the two-dimensional transient conduction with nonuniform point distribution

Figure 10

Domain and nonuniform point distribution for the potential flow over a square block

Figure 11

MCVM solution of the streamlines for a potential flow over a block

Figure 12

MCVM solutions of the one-dimensional steady advection-diffusion equation for uniform point distribution

Figure 13

MCVM solutions of the one-dimensional steady advection-diffusion equation for uniform point distribution

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