A solution to the time-dependent two-dimensional Navier-Stokes equation in a rectangular domain using the Adomian decomposition method and theory of Gröbner Basis

Authors

DOI:

https://doi.org/10.5902/2179460X87459

Keywords:

Navier-Stokes equation, Gröbner basis, Two-dimensional interpolation, Polynomial of smallest degree

Abstract

In the present work we propose a modified decomposition method to derive approximate solutions for non-linear problems. Depending on the type of non-linearity, the source terms of the differential equations to be solved in each recursion step may result in extensive expressions, impractical for computational implementations and applications. This shortcomings are circumvented by the present methodology, which contemplates as a solution procedure in each recursion step a combined variable separation method together with Duhamel's principle, where the non-linearity appears as inhomogeneity. The source terms of the equation in each step of recursion are interpolated by polynomials and, using the Gröbner basis of the set points, the polynomial of reduced degree is obtained so that the integration may be carried out easily. As an application we considered a simplified version of the Navier-Stokes equation, which was used to simulate the wind field making use of the micrometeorological data from the Copenhagen experiment. The derived solution was evaluated against these experimental data from the field experiments showed that the computed results are acceptable and thus the solution may be considered an acceptable one and may be used as a simulation device for these type of field experiments. For almost all experiments twenty eigenvalues and ten recursion steps were sufficient. As results the wind speed at certain positions was simulated and compared to the measured values. The results obtained allow us to affirm that the presented methodology works satisfactorily and, therefore, can be considered a promising tool for solving non-linear problems, which are not tractable with the conventional decomposition method.

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Author Biographies

Luciana Rossato Piovesan, Universidade Federal de Pelotas

She holds a bachelor's degree in Mathematics from the Federal University of Santa Maria (2006) and a master's degree in Applied Mathematics, with an emphasis on Biomathematics, from the Federal University of Rio Grande do Sul (2009) and a PhD in Engineering, with an emphasis on Transport Phenomena, from the Federal University of Rio Grande do Sul (2022). She is currently an Adjunct Professor at the Federal University of Pelotas.

Alexandre Sacco de Athayde, Universidade Federal de Pelotas

He holds a bachelor's degree in Applied Mathematics from the Federal University of Rio Grande do Sul (2006), a master's degree in applied mathematics and a doctorate in mechanical engineering from the same university. He has experience in the area of Mathematics, with an emphasis on Applied Mathematics. He is currently an adjunct professor at the Federal University of Pelotas.

Bardo Ernst Josef Bodmann, Universidade Federal do Rio Grande do Sul

He holds a bachelor's degree in Physics from Johannes Gutenberg Universität Mainz (1984), a master's degree in Physics from Friedrich Alexander Universität Erlangen Nürnberg (1988) and a PhD in Physics from Friedrich Alexander Universität Erlangen Nürnberg (1992). He has experience in the area of Physics, with emphasis on Fields and Particles, Strong and Weak Interaction, Simulation and Computational Modeling.

Marco Túllio Menna Barreto de Vilhena, Universidade Federal do Rio Grande do Sul

He holds a degree in Civil Engineering from the Federal University of Rio Grande do Sul (1968), a master's degree in Mathematics from the Federal University of Rio Grande do Sul (1982) and a PhD in Mechanical Engineering from the Federal University of Rio Grande do Sul (1988). He was a full professor and is currently a visiting professor at the Federal University of Rio Grande do Sul. He has experience in the area of Mechanical Engineering, with emphasis on Transport Phenomena, working mainly on the following topics: laplace transform, LTSN method, transport equation and pollutant dispersion simulation.

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Published

2024-11-07

How to Cite

Piovesan, L. R., Athayde, A. S. de, Bodmann, B. E. J., & Vilhena, M. T. M. B. de. (2024). A solution to the time-dependent two-dimensional Navier-Stokes equation in a rectangular domain using the Adomian decomposition method and theory of Gröbner Basis. Ciência E Natura, 46(esp. 1), e87459. https://doi.org/10.5902/2179460X87459

Issue

Vol. 46 No. esp. 1 (2024): ERMAC e ENMC

Section

Special Edition 1

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