Journal of Applied Mathematics

Stochastic 2D Incompressible Navier-Stokes Solver Using the Vorticity-Stream Function Formulation

Mohamed A. El-Beltagy and Mohamed I. Wafa

Full-text: Open access

Abstract

A two-dimensional stochastic solver for the incompressible Navier-Stokes equations is developed. The vorticity-stream function formulation is considered. The polynomial chaos expansion was integrated with an unstructured node-centered finite-volume solver. A second-order upwind scheme is used in the convection term for numerical stability and higher-order discretization. The resulting sparse linear system is solved efficiently by a direct parallel solver. The mean and variance simulations of the cavity flow are done for random variation of the viscosity and the lid velocity. The solver was tested and compared with the Monte-Carlo simulations and with previous research works. The developed solver is proved to be efficient in simulating the stochastic two-dimensional incompressible flows.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 903618, 14 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808094

Digital Object Identifier
doi:10.1155/2013/903618

Mathematical Reviews number (MathSciNet)
MR3092003

Zentralblatt MATH identifier
06950928

Citation

El-Beltagy, Mohamed A.; Wafa, Mohamed I. Stochastic 2D Incompressible Navier-Stokes Solver Using the Vorticity-Stream Function Formulation. J. Appl. Math. 2013 (2013), Article ID 903618, 14 pages. doi:10.1155/2013/903618. https://projecteuclid.org/euclid.jam/1394808094


Export citation

References

  • O. H. Galal, W. El-Tahan, M. A. El-Tawil, and A. A. Mahmoud, “Spectral SFEM analysis of structures with stochastic parameters under stochastic excitation,” Structural Engineering and Mechanics, vol. 28, no. 3, pp. 281–294, 2008.
  • O. Galal, The solution of stochastic linear partial differential equation using SFEM through neumann and homogeneous chaos expansions [Ph.D. thesis], Cairo University, Cairo, Egypt, 2005.
  • S. Rahman and H. Xu, “A meshless method for computational structure mechanics,” International Journal for Computational Methods in Engineering Science and Mechanics, vol. 6, pp. 41–58, 2005.
  • M. Kamiński, “Stochastic perturbation approach to engineering structure vibrations by the finite difference method,” Journal of Sound and Vibration, vol. 251, no. 4, pp. 651–670, 2002.
  • G. Stefanou, “The stochastic finite element method: past, present and future,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 9-12, pp. 1031–1051, 2009.
  • M. Shinozuka and T. Nomoto, “Response variability due to spatial randomness of material properties,” Tech. Rep., Department of Civil Engineering, Columbia University, New York, NY, USA, 1980.
  • A. A. Henriques, J. M. C. Veiga, J. A. C. Matos, and J. M. Delgado, “Uncertainty analysis of structural systems by perturbation techniques,” Structural and Multidisciplinary Optimization, vol. 35, no. 3, pp. 201–212, 2008.
  • W. Lue, Wiener chaos expansion and numerical solutions of stochastic partial differential equations [Ph.D. thesis], California Institute of Technology, Pasadena, Calif, USA, 2006.
  • R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach, Dover, New York, NY, USA, 2003.
  • H. Panayirci, Computational strategies for efficient stochastic finite element analysis of engineering structures [Ph.D. thesis], Habil and Schuëller, Institute of Engineering Mechanics, University of Innsbruck, Innsbruck, Austria, 2010.
  • M. El-Beltagy, M. Wafa, and O. Galal, “Upwind finite-volume solution of Stochastic Burgers' equation,” Applied Mathematics, vol. 3, no. 11, pp. 1818–1825, 2012.
  • X. D. Wang and K. Shun, “Application of polynomial chaos on numerical simulation of stochastic cavity flow,” Science China Technological Sciences, vol. 53, no. 10, pp. 2853–2861, 2010.
  • T. Sapsis and P. Lermusiaux, “Stochastic responses of Navier-Stokes equations computed with dynamically orthogonal field equations,” in Applications of Statistics and Probability in Civil Engineering, M. Faber, J. Koehler, and K. Nishijima, Eds., Taylor & Francis, London, UK, 2011.
  • C. Lacor, “Intrusive polynomial chaos for the compressible Navier-Stokes equations,” in Proceedings of the ONERA Symposium on Accuracy and Uncertainty in Flow Simulations, Chatillon, France, December 2010.
  • O. Le Maitre and O. Knio, Spectral Methods for Uncertainty Quantification with Applications to Computational Fluid Mechanics, Springer, New York, NY, USA, 2010.
  • M. El-Beltagy, On the serial and parallel finite-volume solutions of the incompressible navier-stokes equations using unstructured grids [Ph.D. thesis], Cairo University, Cairo, Egypt, 2008.
  • D. Calhoun, A Cartesian grid method for solving the streamfunction vorticity equations in irregular geometries [Ph.D. thesis], University of Washington, Washington, DC, USA, 1999.
  • E. Sousa and I. J. Sobey, “Effect of boundary vorticity discretization on explicit stream-function vorticity calculations,” International Journal for Numerical Methods in Fluids, vol. 49, no. 4, pp. 371–393, 2005.
  • P. M. Gresho, “Some interesting issues in incompressible fluid dynamics, both in the continuum and in numerical simulation,” Advances in Applied Mechanics, vol. 29, pp. 615–639, 1992.
  • A. Sherif and M. Hussein, “Computational investigation of the backstep flow,” in Proceedings of the ASME 7th International Congress on Fluids Mechanics and Propulsion, Cairo, Egypt, 2001.
  • M. El-Beltagy, A. Sherif, and A. Seddik, “Unstructured finite-volume solver for the two-dimensional laminar backstep flow using Vorticity-Streamfunction,” in Proceedings of the ASME 8th International Conference of Fluid Dynamics and Propulsion, ICFDP8-EG-199, Sharm El-Shiekh, Egypt, 2006.
  • J. Menaldi and S. S. Sritharan, “Stochastic 2-D Navier-Stokes equation,” Applied Mathematics and Optimization, vol. 46, no. 1, pp. 31–53, 2002.
  • J. Lions and G. Prodi, “Une theoreme d'existence et unicite dans les equations de Navier-Stokes en dimension 2,” Comptes Rendus de l'Académie des Sciences, vol. 248, pp. 3519–3521, 1959.
  • R. Mikulevicius and B. L. Rozovskii, “Stochastic Navier-Stokes equations for turbulent flows,” SIAM Journal on Mathematical Analysis, vol. 35, no. 5, pp. 1250–1310, 2004.
  • R. Mikulevicius and B. L. Rozovskii, “Global L2-solutions of stochastic Navier-Stokes equations,” Annals of Probability, vol. 33, no. 1, pp. 137–176, 2005.
  • B. Birnir, “The existence and uniqueness of turbulent solutions of the Stochastic Navier-Stokes equationčommentComment on ref. [26?]: Please update the information of this reference, if possible.,” CNLS. In press.
  • U. Ghia, K. N. Ghia, and C. T. Shin, “High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method,” Journal of Computational Physics, vol. 48, no. 3, pp. 387–411, 1982.
  • O. Schenk and K. Gartner, “Solving unsymmetric sparse systems of linear equations with PARDISO,” Future Generation Computer Systems, vol. 20, no. 3, pp. 475–487, 2004.
  • O. Schenk and K. Gärtner, “On fast factorization pivoting methods for sparse symmetric indefinite systems,” Electronic Transactions on Numerical Analysis, vol. 23, pp. 158–179, 2006.