### Comparison of the Coupled Solution of the Species, Mass, Momentum, and Energy Conservation Equations by Unstructured FVM, FDM, and FEM

Received: Sep. 21, 2022    Accepted: Oct. 25, 2022    Published: Oct. 28, 2022

Share

Abstract

In order to simulate fluid flow, heat transfer, and other related physical phenomena, it is necessary to describe the associated physics in mathematical terms. Nearly all the physical phenomena of interest to us in this book are governed by principles of conservation and are expressed in terms of partial differential equations expressing these principles. In this research paper, is a summary of conservation equations (Continuity, Momentum, Species, and Energy) that govern the flow of a Newtonian fluid. In particular, this paper studied the solution of two-dimensional (2D) Navier-Stokes (N-S) equations using the finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) on a test problem of Methane combustion in a laminar diffusion flame. First, the computational domain was decomposed into grids in FDM and elements in FEM, later the Navier-Stokes equations, Energy, and, Species conservation equations were solved at the grid points and a MATLAB code has been written to check the consistency, stability, and accuracy for finer meshes. Following this step, the discretized equations for each sub-domain will be developed using the finite difference and finite element method, resolved using an iterative solver-Gauss Seidel technique. The MATLAB code is written for 2D geometries for science and engineering applications. The focus of this research paper is the development of physical models using numerical methods like FDM, and FEM for modeling science and Engineering applications by using Navier-stokes equations, Energy equations, and Species conservation equations.

 DOI 10.11648/j.ijamtp.20220803.11 Published in International Journal of Applied Mathematics and Theoretical Physics ( Volume 8, Issue 3, September 2022 ) Page(s) 52-57 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2024. Published by Science Publishing Group
Keywords

Coupled Solution, Conservation Equations, FDM, FEM, FVM

References
 [1] Coupled solution of the species conservation equations using the unstructured finite-volume method: Ankan Kumar and Sandip Mazumder, 2009. [2] Mazumder S. Critical assessment of the stability and convergence of the equations of multi-component diffusion. Journal of Computational Physics 2006; 212 (2): 383–392. [3] Kuo KK. Principles of Combustion. Wiley: New York, 1986. [4] Bird RB, Stewart WE, Lightfoot EN. Transport Phenomena (2nd ed). Wiley: New York, 2001. [5] Toselli A, Widlund O. Domain Decomposition Methods — Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer: New York, 2004. [6] Sung K, Shirley P. Ray tracing with a BSP tree. In Computer Graphics Gems III, Kirk D (ed.). AP Professional: San Diego, CA, 1992; 271–274. [7] Karypis G, Kumar V. A fast and high-quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing 1998; 20: 359–392. [8] Henk Kaarle Versteeg, Weeratunge Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Pearson Education Limited, 2007. [9] Petrila, Titus, Trif, Damian, Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics, Springer Publication, 2012. [10] Hassan Khawaja (2021). SIMPLE (https://www.mathworks.com/matlabcentral/fileexchange/66129-simple), MATLAB Central File Exchange. Retrieved November 29, 2021. [11] R Surya Narayan (2022). Navier-Stokes Solver using SIMPLE (https://www.mathworks.com/matlabcentral/fileexchange/74996-navier-stokes-solver-using-simple), MATLAB Central File Exchange. Retrieved October 19, 2022. [12] B. R. Baliga and S. V. Patankar. A control-volume finite element method for two-dimensional fluid flow and heat transfer. Numerical Heat Transfer, 6: 245–261, 1983. [13] C. Hsu. A Curvilinear-Coordinate Method for Momentum, Heat and Mass Transfer in Domains of Irregular Geometry. Ph.D. thesis, University of Minnesota, 1981. [14] C. M. Rhie and W. L. Chow. A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation. AIAA Journal, 21: 1525–1532, 1983. [15] M. Peric. A Finite Volume Method for the Prediction of Three-Dimensional Fluid Flow in Complex Ducts. Ph.D. thesis, University of London, August 1985. [16] S. Majumdar. Role of under relaxation in momentum interpolation for calculation of flow with non-staggered grids. Numerical Heat Transfer, 13: 125–132, 1988. [17] J. Y. Murthy and S. R. Mathur. Computation of anisotropic conduction using unstructured meshes. Journal of Heat Transfer, 120: 583–591, August 1998.
• APA Style

Desta Sodano Sheiso, Saket Apparao Kuchibhotla. (2022). Comparison of the Coupled Solution of the Species, Mass, Momentum, and Energy Conservation Equations by Unstructured FVM, FDM, and FEM. International Journal of Applied Mathematics and Theoretical Physics, 8(3), 52-57. https://doi.org/10.11648/j.ijamtp.20220803.11

ACS Style

Desta Sodano Sheiso; Saket Apparao Kuchibhotla. Comparison of the Coupled Solution of the Species, Mass, Momentum, and Energy Conservation Equations by Unstructured FVM, FDM, and FEM. Int. J. Appl. Math. Theor. Phys. 2022, 8(3), 52-57. doi: 10.11648/j.ijamtp.20220803.11

AMA Style

Desta Sodano Sheiso, Saket Apparao Kuchibhotla. Comparison of the Coupled Solution of the Species, Mass, Momentum, and Energy Conservation Equations by Unstructured FVM, FDM, and FEM. Int J Appl Math Theor Phys. 2022;8(3):52-57. doi: 10.11648/j.ijamtp.20220803.11

• ```@article{10.11648/j.ijamtp.20220803.11,
author = {Desta Sodano Sheiso and Saket Apparao Kuchibhotla},
title = {Comparison of the Coupled Solution of the Species, Mass, Momentum, and Energy Conservation Equations by Unstructured FVM, FDM, and FEM},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {8},
number = {3},
pages = {52-57},
doi = {10.11648/j.ijamtp.20220803.11},
url = {https://doi.org/10.11648/j.ijamtp.20220803.11},
abstract = {In order to simulate fluid flow, heat transfer, and other related physical phenomena, it is necessary to describe the associated physics in mathematical terms. Nearly all the physical phenomena of interest to us in this book are governed by principles of conservation and are expressed in terms of partial differential equations expressing these principles. In this research paper, is a summary of conservation equations (Continuity, Momentum, Species, and Energy) that govern the flow of a Newtonian fluid. In particular, this paper studied the solution of two-dimensional (2D) Navier-Stokes (N-S) equations using the finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) on a test problem of Methane combustion in a laminar diffusion flame. First, the computational domain was decomposed into grids in FDM and elements in FEM, later the Navier-Stokes equations, Energy, and, Species conservation equations were solved at the grid points and a MATLAB code has been written to check the consistency, stability, and accuracy for finer meshes. Following this step, the discretized equations for each sub-domain will be developed using the finite difference and finite element method, resolved using an iterative solver-Gauss Seidel technique. The MATLAB code is written for 2D geometries for science and engineering applications. The focus of this research paper is the development of physical models using numerical methods like FDM, and FEM for modeling science and Engineering applications by using Navier-stokes equations, Energy equations, and Species conservation equations.},
year = {2022}
}
```
• ```TY  - JOUR
T1  - Comparison of the Coupled Solution of the Species, Mass, Momentum, and Energy Conservation Equations by Unstructured FVM, FDM, and FEM
AU  - Desta Sodano Sheiso
AU  - Saket Apparao Kuchibhotla
Y1  - 2022/10/28
PY  - 2022
N1  - https://doi.org/10.11648/j.ijamtp.20220803.11
DO  - 10.11648/j.ijamtp.20220803.11
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 52
EP  - 57
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20220803.11
AB  - In order to simulate fluid flow, heat transfer, and other related physical phenomena, it is necessary to describe the associated physics in mathematical terms. Nearly all the physical phenomena of interest to us in this book are governed by principles of conservation and are expressed in terms of partial differential equations expressing these principles. In this research paper, is a summary of conservation equations (Continuity, Momentum, Species, and Energy) that govern the flow of a Newtonian fluid. In particular, this paper studied the solution of two-dimensional (2D) Navier-Stokes (N-S) equations using the finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) on a test problem of Methane combustion in a laminar diffusion flame. First, the computational domain was decomposed into grids in FDM and elements in FEM, later the Navier-Stokes equations, Energy, and, Species conservation equations were solved at the grid points and a MATLAB code has been written to check the consistency, stability, and accuracy for finer meshes. Following this step, the discretized equations for each sub-domain will be developed using the finite difference and finite element method, resolved using an iterative solver-Gauss Seidel technique. The MATLAB code is written for 2D geometries for science and engineering applications. The focus of this research paper is the development of physical models using numerical methods like FDM, and FEM for modeling science and Engineering applications by using Navier-stokes equations, Energy equations, and Species conservation equations.
VL  - 8
IS  - 3
ER  - ```
Author Information
• Department of Mathematics, College of Natural and Computational Sciences, Wolkite University, Wolkite, Ethiopia

• Department of Chemical Engineering, Indian Institute of Technology, Guwahati, India

• Section