NUMERICAL SOLUTIONS OF 2-D UNSTEADY
INCOMPRESSIBLE FLOW IN A DRIVEN SQUARE CAVITY
USING STREAMFUNCTION-VORTICITY FORMULATION
Vusala Ambethkar1, Manoj Kumar2, Mohit Kumar Srivastava3
1,2Department of Mathematics
Faculty of Mathematical Sciences
University of Delhi
Delhi, 110007, INDIA
3Ambedkar Institute of Advanced Communication
Technologies and Research
Delhi, 110031, INDIA

Abstract

In this paper, we have used a streamfunction-vorticity $(\psi-\xi)$ formulation to investigate the problem of 2-D unsteady viscous incompressible flow in a driven square cavity with moving top and bottom walls. We used this formulation to solve the governing equations along with no-slip and slip wall boundary conditions. A general algorithm was used for this formulation in order to compute the numerical solutions for the flow variables: streamfunction $\psi$, vorticityfunction $\xi$ for low Reynolds numbers $Re \leq 50$. We have executed this with the aid of a computer programme developed and run in C++ compiler. We have proved the stability and convergence of the numerical scheme using matrix method. Following this, the stability conditions obtained for the time and space steps have been used in numerical computations to arrive at the numerical solutions with desired accuracy. Also investigated was the variation of $u$ and $v$ components of velocity at points closed to left side, top and bottom walls of the square cavity at different time levels for Reynolds number 50. Based on the numerical computations of $u$-velocity, we found: (i) the $u$-velocity is almost same near the top and bottom wall of the square cavity above and below the geometric center for $Re$=15 and 30; (ii) the absolute value of the $u$-velocity first decreases, then increases, and finally decreases, near top and bottom wall of the square cavity. Based on the numerical computations of the $v$-velocity, we found: (i) the absolute value of the $v$-velocity increases uniformly as time level increases' (ii) the $v$-velocity at the right wall of the square cavity attains a maximum value, and then, decreases towards the geometric center. The numerical solutions of the vorticity vector $\xi$ at $Re$=50 for different times along the horizontal line through geometric center of the square cavity indicate the following: (i) the vorticity contours decreased in between the left wall boundary to the midpoint of the square domain; (ii) it, then, increased in between the midpoint to the right wall boundary. Based on the numerical solutions of streamfunction $\psi$, we found: (i) the size of streamline contours decreased when the time increased; (ii) two streamline contours are generated, one above the geometric center, and the other, below the geometric center in clockwise direction.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 29
Issue: 6
Year: 2016

DOI: 10.12732/ijam.v29i6.6

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References

  1. [1] 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, 48 (1982), 387-411.
  2. [2] R. Schreiber and H.B. Keller, Driven cavity flow by efficient numerical techniques, Journal of Computational Physics, 49 (1983), 310-333.
  3. [3] G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, New York, U.S.A. (1985).
  4. [4] T. Fusegi and Bakhtierfarouk, Predictions of fluid flow and heat transfer problems by the vorticity-velocity formulation of the Navier-Stokes equa tions, Journal of Computational Physics, 65 (1986), 227-243.
  5. [5] S.C. Chapra and R.P. Canale, Numerical Methods for Engineers, McGraw Hill Co., New York (1989).
  6. [6] C.H. Bruneau and C. Jouron, An efficient scheme for solving steady in compressible Navier-Stokes equations, Journal of Computational Physics, 89 (1990), 389-413.
  7. [7] T.E. Tezduyar, J. Liou, D.K. Ganjoo and M. Behr, Solution techniques for the vorticity-streamfunction formulation of two-dimensional unsteady in compressible flows, International Journal for Numerical Methods in Fluids, 11 (1990), 515-539.
  8. [8] C.K. Aidun, N.G. Triantafillopouios and J.D. Benson, Global stability of a lid-driven cavity with through flow: flow visualization studies, Phys. Fluids A, 3 (1991), 2081-2091.
  9. [9] D.A. Zumbrunnen, K.C. Miles and Y.H. Liu, Auto-processing of very fine scale composite materials by chotic mixing of melts, Composites Part A, 27A (1995), 37-47.
  10. [10] P.S. Ghoshdastidar, Computer Simulation of Flow and Heat Transfer, Tata McGraw-Hill Publishing Co. Ltd., New Delhi (1998).
  11. [11] P.N. Shankar and M.D. Deshpande, Fluid mechanics in the driven cavity, Annual Review of Fluid Mechanics, 32 (2000), 93-136.
  12. [12] J.C. Kalita, D.C. Dalal and A.K. Dass, Fully compact higher order com putation of steady-state natural convection in a square cavity, Physical Review E, 64 (2001), 1-13.
  13. [13] J.C. Kalita, D.C. Dalal and A.K. Dass, A transformation-free HOC scheme for steady convection-diffusion on non-uniform grids, International Journal for Numerical Methods in Fluids, 44 (2004), 33-53.
  14. [14] H. Shokouhmand and H. Sayehvand, Numerical study of flow and heat transfer in a square driven cavity, International Journal of Engineering Transactions A: Basics, 17 (2004), 301-317.
  15. [15] E. Erturk, T.C. Corke and C. Gokcol, Numerical Solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, International Journal for Numerical Methods in Fluids, 48 (2005), 747-774.
  16. [16] C.H. Bruneau and M. Saad, The 2-D lid-driven cavity problem revisited, Computers and Fluids, 35 (2006), 326-348.
  17. [17] A. Salem, On the numerical solution of the incompressible Navier-Stokes equations in primitive variables using grid generation techniques, Mathe matical and Computational Applications, 11 (2006), 127-136.
  18. [18] J.C. Kalita and S. Sen, The (9, 5) HOC formulation for the transient Navier-Stokes equations in primitive variable, International Journal for Numerical Methods in Fluids, 55 (2007), 387-406.
  19. [19] J.M. McDonough, Lectures on Computational Numerical Analysis of Par tial Differential Equations, Available et: http://www.engr.uky.edu/∼acfd (2007).
  20. [20] K. Muralidhar and T. Sundararajan, Computational Fluid Flow and Heat Transfer, 2nd ed., Narosa, New Delhi (2008).
  21. [21] E. Erturk, Discussions on driven cavity flow, International Journal for Numerical Methods in Fluids, 60 (2009), 275-294.
  22. [22] T.Y. Hou, Stable fourth-order stream-function methods for incompressible flows with boundaries, Journal of Computational Mathematics, 27 (2009), 441-458.
  23. [23] S. Biringen and C. Y. Chow, An Introduction to Computational Fluid Me chanics by Examples, John Wiley and Sons, Inc., Hoboken, New Jersey (2011).
  24. [24] K. Poochinapan, Numerical implementations for 2-d lid-driven cavity flow in stream function formulation, International Scholarly Research network ISRN Applied Mathematics, 2012 (2012), Article ID 871538, 17 pp., doi:10.5402/2012/871538.
  25. [25] K.M. Salah Uddin and L.K. Saha, Numerical solution of 2-d incompressible driven cavity flow with wavy bottom surface, American Journal of Applied Mathematics, 3, No 1-1 (2015), 30-42.
  26. [26] D.W. Peaceman and H.H. Rachford, The numerical solution of parabolic and elliptic differential equations, Journal of the Society for Industrial and Applied Mathematics, 3 (1955), 28-41.