Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-21T11:33:22.247Z Has data issue: false hasContentIssue false
  • English
  • Français

Fuzzy Solutions for Two-Dimensional Navier-Stokes Equations

Published online by Cambridge University Press:  23 November 2015

Y.-Y. Chen ,
R.-J. Hsiao  and
M.-C. Huang
Y.-Y. Chen*
Affiliation:
Department of Systems and Naval Mechatronic EngineeringNational Cheng Kung UniversityTainan, Taiwan
R.-J. Hsiao
Affiliation:
Department of Systems and Naval Mechatronic EngineeringNational Cheng Kung UniversityTainan, Taiwan
M.-C. Huang
Affiliation:
Department of Systems and Naval Mechatronic EngineeringNational Cheng Kung UniversityTainan, Taiwan
*
*Corresponding author (yungyuchen@mail.ncku.edu.tw)
Get access

Abstract

A new methodology via using an adaptive fuzzy algorithm to obtain solutions of “Two-dimensional Navier-Stokes equations” (2-D NSE) is presented in this investigation. The design objective is to find two fuzzy solutions to satisfy precisely the 2-D NSE frequently encountered in practical applications. In this study, a rough fuzzy solution is formulated with adjustable parameters firstly, and then, a set of adaptive laws for optimally tuning the free parameters in the consequent parts of the proposed fuzzy solutions are derived from minimizing an error cost function which is the square summation of approximation errors of boundary conditions, continuum equation and Navier-Stokes equations. In addition, elegant approximated error bounds between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors have also been proven. Furthermore, the error equations in mesh points can be proven to converge to zero for the 2-D NSE with two sufficient conditions.

Keywords

Navier-Stokes equationsAdaptive Fuzzy SolutionsApproximation error bounds
Type
Research Article
Information
Journal of Mechanics , Volume 34 , Issue 1 , February 2018 , pp. 1 - 10
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Trumble, K. A., Schauerhamer, D. G., Kleb, W. L., Carlson, J. and Edquist, K. T., “Analysis of Navier-Stokes Codes Applied to Supersonic Retro-Propulsion Wind Tunnel Test,” IEEE Aerospace Conference, pp. 1–13 (2011).Google Scholar
2. Chen, C. W., Kouh, J. S. and Tsai, J. F., “Modeling and Simulation of an AUV Simulator with Guidance System,” IEEE, Oceanic Engineering, 38, pp. 211–225 (2013).Google Scholar
3. Brower, R. W., Reddy, R. V. R. and Noordergraaf, A., “Difficulties in the Further Development of Venous Hemodynamics,” IEEE Trans, Biomedical Engineering, 16, pp. 335–338 (1969).Google Scholar
4. Li, H. and Rui, Z., “Stabilized Finite Element Method for Vorticity Velocity Pressure Formulation of the Stationary Navier-Stokes Equations,” IEEE, Computational and Information Sciences Conference, pp. 634–637 (2010).Google Scholar
5. De Angeli, J. P., Valli, A. M. P. Jr. Reis, N. C. and De Souza, A. F., “Finite Difference Simulations of the Navier-Stokes Equations using Parallel Distributed Computing,” IEEE,Computer Architecture and High Performance Computing, pp. 149–156 (2003).Google Scholar
6. Wang, B., Guo, Y., Liu, O. and Shen, M., “Higher Order Accurate and High-Resolution Implicit Upwind Finite Volume Scheme for Solving Euler/Reynolds-Averaged Navier Stokes Equations,” IEEE, Tsinghua Science and Technology, 5, pp. 47–53 (2000).Google Scholar
7. Narasimhan, S., Kuan, C. and Stenger, F., “The Solution of Incompressible Navier Stokes Equations Using the Sine Collocation Method,” IEEE, Thermal and Thermomechanical Phenomena in Electronic Systems Conference, 1, pp. 199–214 (2000).Google Scholar
8. Ji, L. and Jianxin, Z., “The Boundary Element Method for Boundary Control of the Linear Stokes Flow,” IEEE, Decision and Control Conference, 3, pp. 1192–1194 (1990).Google Scholar
9. Ding, S., Wen, H., Yao, L. and Zhu, C., “Global Weak Solution to One-Dimensional Compressible Isentropic Navier-Stokes Equations with Density-Dependent Viscosity,” American Institute of Physics, Journal of Mathematical Physics, 50, pp. 023101–023101–17 (2009).Google Scholar
10. Frank, M. W., Fluid Mechanics, McGraw-Hill, New York (2011).Google Scholar
11. Kincaid, D. and Cheney, W., Numerical Analysis, Brooks/Cole, Pacific Grove (1991).Google Scholar
12. Chen, Y. Y., Chang, Y. T. and Chen, B. S., “Fuzzy Solutions to Partial Differential Equations: Adaptive Approach,” IEEE, Transaction on Fuzzy Systems, 17, pp. 116–127 (2009).Google Scholar
13. Saatry, S. S. and Bodson, M., Adaptive Control: Stability, Convergence, and Robustness, Prentice-Hall, Englewood Cliffs (1989).Google Scholar