Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme

Siele Charles; Rotich John; Adicka Daniel

doi:doi:10.11648/j.acm.20211001.12

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Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme

Siele Charles, Rotich John, Adicka Daniel

Published in Applied and Computational Mathematics (Volume 10, Issue 1)

Received: 18 January 2021 Accepted: 26 January 2021 Published: 12 March 2021

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Abstract

The Navier-Stokes (N-S) equations for incompressible fluid flow comprise of a system of four nonlinear equations with five flow fields such as pressure P, density ρ and three velocity components u, v, and w. The system of equations is generally complex due to the fact that it is nonlinear and a mixture of the three classes of partial differential equations (PDEs) each with distinct solution methods. The N-S equations fully describe the unsteady fluid flow behaviour of laminar and turbulent types. Previous studies have shown existence of general solutions of fluid flow models but little has been done on numerical solution for velocity of flow in N-S equation of incompressible fluid flow by Crank-Nicolson implicit scheme. In practice, real fluid flows are compressible due to the inevitable variations in density caused by temperature changes and other physical factors. Numerical approximations of the general system of Navier-Stokes equations were made to develop numerical solution model for incompressible fluid flow. Adequate solutions of the latter produce numerical solutions applicable in numerical simulation of fluid flows useful in engineering and science. Non-dimensionalization of variables involved was done. Crank-Nicolson (C.N) implicit scheme was implemented to discretize partial derivatives and appropriate approximation made at the boundaries yielded a linear system of N-S equations model. The linear numerical system was then expressed in matrix form for computation of velocity field by Computational fluid dynamics (CFD) approach using MATLAB software. Numerical results for velocity field in two dimensional space, u(x,y,t)and v(x,y,t) generated in uniform 32×32 grids points of the square flow domains, 0≤x≤1.0 and 0≤y≤1.0 were presented in three dimensional figures. Results showed that the velocity in two dimensional space does not change suddenly for any change in spatial levels, x and y. Therefore, C-N implicit Scheme applied to solve the N-S equations for fluid flow is consistent.

Published in	Applied and Computational Mathematics (Volume 10, Issue 1)
DOI	10.11648/j.acm.20211001.12
Page(s)	10-18
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

Navier- Stokes Equation, Nonlinear System, Incompressible Fluid, Vorticity, Coriolis Force, Discretization

References

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[2]	Christov C. I. & Marinova R.S (2001). Implicit Vectorial Operator Splitting for Incompressible Navier- Stokes Equations in Primitive Variables. Vol 6, No. 4
[3]	Cohen N. & Pijush K Kundu. 4^th Edition. (2008). Fluid mechanics. (Elsevier)
[4]	Fadugha S. & Zelibe S.C. (2015). Crank-Nicolson method for solving parabolic partial differential equations. Adoekiti University (Nigeria). c/oi http//www.researchgate.net/pub/ 287195203.http://doi.org./10.1016/jcp.2006.01.036
[5]	Fernandes Enrique-Cara & Irene Marin I-Gayte. (2018) A New Proof of Existence of Suitable Weak Solution and Other Remarks for Navier-Stokes Equations. Applied Mathematics, 9,383-402. http://doi.org./10.4236/am.2018.94029
[6]	Jiyuan Tu, Guan Heng Y & Chaoqun Lui. 2^nd Edition. (2013). Computational fluid dynamics: a practical approach. (b) (Elsevier)
[7]	Kanti Pandey. & Verma Lijja.(2011). A note on Crank-Nicolson for Burger’s Equation. Applied Mathematics. 2011, 2,883-889. http://doi.org./10.4236./am.2011.27118
[8]	Kweyu C.M. Nyamai B. M., Wahome J. N., Kandie J. N. & Bundotich J. K. (2013). Modified Crank-Nicolson scheme for Numerical solutions of the 2D coupled Burger’s system. Moi university (Kenya) c/oi http//www.researchgate.net/pub/ 263100789
[9]	Loukopoulos V. C, Messaris G. T & Bourantas G. C. (2011). Numerical solution for incompressible Navier -Stokes Equations in primitive Variables and velocity- vorticity formulations: Journal of Applied Mathematics and computations, 222 (2013) 575-588.
[10]	Munson B.R, Donald F.Y, Theodore H.O. & Wade W.H (2009). Fundamentals of Fluid Mechanics. (Wiley)
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[12]	Panton L. Ronald 4^th Edition (2013) Incompressible flow. (Wiley).
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[14]	Maritim S., Rotich J. K, & Bitok J.K. (2019). Hybrid Hopscotch Method for Solving Two Dimensional System of Burgers’ equations. International Journal of Science and Research. http://doi.org./10.21275/ART.2020.186 ISSN: 2319-7064
[15]	Saqib M., Shahid H. & Daoud S. M (2017).Computational solutions of Two-dimensional Convection-Diffussion Equations Using Crank-Nicolson Implicit and Time Efficient ADI. American Journal of Computational Mathematics. 2017, 7, 208-227. http://doi.org./10.4236/ajcm.2017.73019
[16]	Schezt J.A & Rodney D.W.P. (2011) boundary layer analysis. (AIAA Education series)
[17]	Stampolidis P & Guosiduo-Kouttita M.C. (2008). A computational Study with Finite Difference Methods for second order Quasilinear Hyperbolic Partial Differential Equations in Two Independent Variables. Applied mathematics, 2018, 9, 1193-1224.
[18]	Vineet Kumar S., Muhamed T., Utkarsh B. & Sanyasiraju YUSS (2011). Crank-Nicolson Scheme for Numerical Solution of Two Dimensional Coupled Burger’s Equations. International journal for scientific Engineering and Research. http://www.ijser.org
[19]	Zikanov Oleg (2010). Essentials of computational Fluid Dynamics. (Wiley)

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APA Style

Siele Charles, Rotich John, Adicka Daniel. (2021). Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme. Applied and Computational Mathematics, 10(1), 10-18. https://doi.org/10.11648/j.acm.20211001.12

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ACS Style

Siele Charles; Rotich John; Adicka Daniel. Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme. Appl. Comput. Math. 2021, 10(1), 10-18. doi: 10.11648/j.acm.20211001.12

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AMA Style

Siele Charles, Rotich John, Adicka Daniel. Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme. Appl Comput Math. 2021;10(1):10-18. doi: 10.11648/j.acm.20211001.12

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@article{10.11648/j.acm.20211001.12,
  author = {Siele Charles and Rotich John and Adicka Daniel},
  title = {Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme},
  journal = {Applied and Computational Mathematics},
  volume = {10},
  number = {1},
  pages = {10-18},
  doi = {10.11648/j.acm.20211001.12},
  url = {https://doi.org/10.11648/j.acm.20211001.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211001.12},
  abstract = {The Navier-Stokes (N-S) equations for incompressible fluid flow comprise of a system of four nonlinear equations with five flow fields such as pressure P, density ρ and three velocity components u, v, and w. The system of equations is generally complex due to the fact that it is nonlinear and a mixture of the three classes of partial differential equations (PDEs) each with distinct solution methods. The N-S equations fully describe the unsteady fluid flow behaviour of laminar and turbulent types. Previous studies have shown existence of general solutions of fluid flow models but little has been done on numerical solution for velocity of flow in N-S equation of incompressible fluid flow by Crank-Nicolson implicit scheme. In practice, real fluid flows are compressible due to the inevitable variations in density caused by temperature changes and other physical factors. Numerical approximations of the general system of Navier-Stokes equations were made to develop numerical solution model for incompressible fluid flow. Adequate solutions of the latter produce numerical solutions applicable in numerical simulation of fluid flows useful in engineering and science. Non-dimensionalization of variables involved was done. Crank-Nicolson (C.N) implicit scheme was implemented to discretize partial derivatives and appropriate approximation made at the boundaries yielded a linear system of N-S equations model. The linear numerical system was then expressed in matrix form for computation of velocity field by Computational fluid dynamics (CFD) approach using MATLAB software. Numerical results for velocity field in two dimensional space, u(x,y,t)and v(x,y,t) generated in uniform 32×32 grids points of the square flow domains, 0≤x≤1.0 and 0≤y≤1.0 were presented in three dimensional figures. Results showed that the velocity in two dimensional space does not change suddenly for any change in spatial levels, x and y. Therefore, C-N implicit Scheme applied to solve the N-S equations for fluid flow is consistent.},
 year = {2021}
}

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TY  - JOUR
T1  - Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme
AU  - Siele Charles
AU  - Rotich John
AU  - Adicka Daniel
Y1  - 2021/03/12
PY  - 2021
N1  - https://doi.org/10.11648/j.acm.20211001.12
DO  - 10.11648/j.acm.20211001.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 10
EP  - 18
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20211001.12
AB  - The Navier-Stokes (N-S) equations for incompressible fluid flow comprise of a system of four nonlinear equations with five flow fields such as pressure P, density ρ and three velocity components u, v, and w. The system of equations is generally complex due to the fact that it is nonlinear and a mixture of the three classes of partial differential equations (PDEs) each with distinct solution methods. The N-S equations fully describe the unsteady fluid flow behaviour of laminar and turbulent types. Previous studies have shown existence of general solutions of fluid flow models but little has been done on numerical solution for velocity of flow in N-S equation of incompressible fluid flow by Crank-Nicolson implicit scheme. In practice, real fluid flows are compressible due to the inevitable variations in density caused by temperature changes and other physical factors. Numerical approximations of the general system of Navier-Stokes equations were made to develop numerical solution model for incompressible fluid flow. Adequate solutions of the latter produce numerical solutions applicable in numerical simulation of fluid flows useful in engineering and science. Non-dimensionalization of variables involved was done. Crank-Nicolson (C.N) implicit scheme was implemented to discretize partial derivatives and appropriate approximation made at the boundaries yielded a linear system of N-S equations model. The linear numerical system was then expressed in matrix form for computation of velocity field by Computational fluid dynamics (CFD) approach using MATLAB software. Numerical results for velocity field in two dimensional space, u(x,y,t)and v(x,y,t) generated in uniform 32×32 grids points of the square flow domains, 0≤x≤1.0 and 0≤y≤1.0 were presented in three dimensional figures. Results showed that the velocity in two dimensional space does not change suddenly for any change in spatial levels, x and y. Therefore, C-N implicit Scheme applied to solve the N-S equations for fluid flow is consistent.
VL  - 10
IS  - 1
ER  -

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Author Information

Siele Charles

Department of Mathematics and Computer Science, School of Science and Technology, University of Kabianga, Kericho, Kenya
Rotich John

Department of Mathematics and Computer Science, School of Science and Technology, University of Kabianga, Kericho, Kenya
Adicka Daniel

Department of Mathematics and Computer Science, School of Science and Technology, University of Kabianga, Kericho, Kenya

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Siele Charles, Rotich John, Adicka Daniel. (2021). Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme. Applied and Computational Mathematics, 10(1), 10-18. https://doi.org/10.11648/j.acm.20211001.12

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ACS Style

Siele Charles; Rotich John; Adicka Daniel. Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme. Appl. Comput. Math. 2021, 10(1), 10-18. doi: 10.11648/j.acm.20211001.12

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AMA Style

Siele Charles, Rotich John, Adicka Daniel. Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme. Appl Comput Math. 2021;10(1):10-18. doi: 10.11648/j.acm.20211001.12

Copy | Download

@article{10.11648/j.acm.20211001.12,
  author = {Siele Charles and Rotich John and Adicka Daniel},
  title = {Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme},
  journal = {Applied and Computational Mathematics},
  volume = {10},
  number = {1},
  pages = {10-18},
  doi = {10.11648/j.acm.20211001.12},
  url = {https://doi.org/10.11648/j.acm.20211001.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211001.12},
  abstract = {The Navier-Stokes (N-S) equations for incompressible fluid flow comprise of a system of four nonlinear equations with five flow fields such as pressure P, density ρ and three velocity components u, v, and w. The system of equations is generally complex due to the fact that it is nonlinear and a mixture of the three classes of partial differential equations (PDEs) each with distinct solution methods. The N-S equations fully describe the unsteady fluid flow behaviour of laminar and turbulent types. Previous studies have shown existence of general solutions of fluid flow models but little has been done on numerical solution for velocity of flow in N-S equation of incompressible fluid flow by Crank-Nicolson implicit scheme. In practice, real fluid flows are compressible due to the inevitable variations in density caused by temperature changes and other physical factors. Numerical approximations of the general system of Navier-Stokes equations were made to develop numerical solution model for incompressible fluid flow. Adequate solutions of the latter produce numerical solutions applicable in numerical simulation of fluid flows useful in engineering and science. Non-dimensionalization of variables involved was done. Crank-Nicolson (C.N) implicit scheme was implemented to discretize partial derivatives and appropriate approximation made at the boundaries yielded a linear system of N-S equations model. The linear numerical system was then expressed in matrix form for computation of velocity field by Computational fluid dynamics (CFD) approach using MATLAB software. Numerical results for velocity field in two dimensional space, u(x,y,t)and v(x,y,t) generated in uniform 32×32 grids points of the square flow domains, 0≤x≤1.0 and 0≤y≤1.0 were presented in three dimensional figures. Results showed that the velocity in two dimensional space does not change suddenly for any change in spatial levels, x and y. Therefore, C-N implicit Scheme applied to solve the N-S equations for fluid flow is consistent.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Numerical Solution of the Navier-Stokes Equations for Incompressible Fluid Flow by Crank-Nicolson Implicit Scheme
AU  - Siele Charles
AU  - Rotich John
AU  - Adicka Daniel
Y1  - 2021/03/12
PY  - 2021
N1  - https://doi.org/10.11648/j.acm.20211001.12
DO  - 10.11648/j.acm.20211001.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 10
EP  - 18
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20211001.12
AB  - The Navier-Stokes (N-S) equations for incompressible fluid flow comprise of a system of four nonlinear equations with five flow fields such as pressure P, density ρ and three velocity components u, v, and w. The system of equations is generally complex due to the fact that it is nonlinear and a mixture of the three classes of partial differential equations (PDEs) each with distinct solution methods. The N-S equations fully describe the unsteady fluid flow behaviour of laminar and turbulent types. Previous studies have shown existence of general solutions of fluid flow models but little has been done on numerical solution for velocity of flow in N-S equation of incompressible fluid flow by Crank-Nicolson implicit scheme. In practice, real fluid flows are compressible due to the inevitable variations in density caused by temperature changes and other physical factors. Numerical approximations of the general system of Navier-Stokes equations were made to develop numerical solution model for incompressible fluid flow. Adequate solutions of the latter produce numerical solutions applicable in numerical simulation of fluid flows useful in engineering and science. Non-dimensionalization of variables involved was done. Crank-Nicolson (C.N) implicit scheme was implemented to discretize partial derivatives and appropriate approximation made at the boundaries yielded a linear system of N-S equations model. The linear numerical system was then expressed in matrix form for computation of velocity field by Computational fluid dynamics (CFD) approach using MATLAB software. Numerical results for velocity field in two dimensional space, u(x,y,t)and v(x,y,t) generated in uniform 32×32 grids points of the square flow domains, 0≤x≤1.0 and 0≤y≤1.0 were presented in three dimensional figures. Results showed that the velocity in two dimensional space does not change suddenly for any change in spatial levels, x and y. Therefore, C-N implicit Scheme applied to solve the N-S equations for fluid flow is consistent.
VL  - 10
IS  - 1
ER  -

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