Numerical Solutions of 2-D Steady Incompressible Flow in a Driven Square Cavity Using Stream Function-Vorticity Formulation

Yıl 2017, Cilt: 6 , 10 - 22, 30.06.2017

Vusala Ambethkar Manoj Kumar

Öz

In this study, a streamfunction-vorticity $(\psi-\xi)$ method is suitably used to investigate the problem of 2-D steady viscous incompressible flow in a driven square cavity with moving top and bottom wall. We used this method to solve the governing equations along with no-slip and slip wall boundary conditions at low Reynolds number. A general algorithm was used for this method in order to compute the numerical solutions for streamfunction $\psi$, vorticityfunction $\xi$ for low Reynolds numbers $Re \leq 100$. We have executed this with the aid of a computer programme developed and run in C++ compiler. We have also proposed the stability criterion of the numerical scheme used. Streamline, vorticity and isobar contours have been depicted at different low Reynolds numbers. For flows at Reynolds number $Re$=100, our numerical solutions are compared with established steady state results and excellent agreement is obtained.

Anahtar Kelimeler

Stream function-vorticity formulation, numerical solutions, geometric center of square cavity, u-velocity, v-velocity, pressure, streamline, isobars, low and moderate Reynolds number

Kaynakça

• Anthony, O.O., lyiola, O.O., Miracle, O.O., Numerical simulation of the lid driven cavity flow with inclined walls, International Journal of Scientific and Engineering Research, 4(5)(2013).
• Biringen, S., Chow, C. Y., An Introduction to Computational Fluid Mechanics By Examples, JohnWiley and Sons, Inc., Hoboken, New Jersey, 2011.
• Bozeman, J.D., Dalton, C., Numerical study of viscous flow in a cavity, Journal of Computational Physics, 12(1973), 348–363.
• Bruneau, C.H., Jouron, C., An ecient scheme for solving steady incompressible Navier-Stokes equations, Journal of Computational Physics, 89(1990), 389–413.
• Chamkha, A.J., Nada, E.A., Mixed convection flow in single and double-lid driven square cavities filled with water-Al2O3 nanofluid: Effect of viscosity models, European Journal of Mechanics B/Fluids, 36(2012), 82–96.
• Demir, H., Erturk, V.S., A numerical study of wall driven flow of a viscoelastic fluid in Rectangular cavities, Indian Journal of Pure and applied Mathematics, 32(10)(2001), 1581–1590.
• Erturk, E., Corke, T.C., Gokcol, C., 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.
• Erturk, E., Dursun, B., Numerical solutions of 2-D steady incompressible flow in a driven skewed cavity, Journal of Applied Mathematics and Mechanics, 87(2007), 377–392.
• Ghia, U., Ghia, K.N., Shin, C.T., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, Journal of Computational Physics, 48(1982), 387–411.
• Ghoshdastidar, P.S., Computer Simulation of Flow and Heat Transfer, Tata McGraw-Hill Publishing Company Limited, New Delhi, 1998. 3
• Gupta, M.M., Manohar, R.P., Boundary approximations and accuracy in viscous flow computations, Journal of Computational Physics, 31(1979), 265–288.
• Gustafson, K., Halasi, K., Vortex dynamics of cavity flows, Journal of Computational Physics, 4(1986), 279–319.
• Ismael, M.A., Pop, I., Chamkha, A.J., Mixed convection in a lid-driven square cavity with partial slip, International Journal of Thermal Sciences, 82(2014), 47–61.
• Kopecky, R.M., Torrance, K.E., Initiation and structure of axisymmetric eddies in a rotating stream, Computers and Fluids, 1(1973), 289–300.
• Lax, P.D., Richtmyer, R.D., Survey of the stability of linear finite difference equations, Communications on Pure Applied Mathematics, 9(1956), 267–293.
• Li, M., Tang, T., Fornberg, B., A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 20(1995), 1137–1151.
• Oztop, H.F., Dagtekin, I., Mixed convection in two sided lid-driven dierentially heated square cavity, International Journal of Heat and Mass Transfer, 47(2004), 1761–1769.
• Oztop, H.F., Salem, K.A., Pop, I., MHD mixed convection in a lid-driven cavity with corner heater, International Journal of Heat and Mass Transfer, 54(2011), 3494–3504.
• Perumal, D.A., Dass, A.K., Simulation of incompressible flows in two-sided lid-driven square cavities, Part I - FDM, CFD Letters, 2(1)(2010).
• Schreiber, Keller, Spurious solutions in driven cavity calculations, Journal of Computational Physics, 49(1983), 165–172.
• Smith, G.D., Numerical Solution of Partial Differential Equations: Finite Dierence Methods, Oxford University Press, New York, U.S.A., 1985.
• Spotz, W.F., Accuracy and performance of numerical wall boundary conditions for steady 2-D incompressible streamfunction vorticity, International Journal for Numerical Methods in Fluids, 28(1998), 737–757.
• Tian, Z., Ge, Y., A fourth-order compact finite difference scheme for the steady stream function-vorticity formulation of the Navier-Stokes/Boussinesq equations, International Journal for Numerical Methods in Fluids, 41(2003), 495–518.
• Tian, Z.F., Yu, P.X., An effcient compact difference scheme for solving the streamfunction formulation of the incompressible Navier-Stokes equations, Journal of Computational Physics, 230(2011), 6404–6419. 1
• Torrance, K.E., Comparison of finite-difference computations of natural convection, Journal of Research of the National Bureau of Standards, 72B (1968), 281–301.
• Torrance, K.E., Rockett, J. A., Numerical study of natural convection in an enclosure with localized heating from below-creeping flow to the onset of laminar instability, Journal of Fluid Mechanics, 36(1969), 33–54.
• Wahba, E.M., Multiplicity of states for two sided and four sided lid driven cavity flows, Computers and Fluids, 38(2009), 247–253.
• Yu, P.X., Tian, Z.F., A compact streamfunction-velocity scheme on nonuniform grids for the 2-D steady incompressible Navier-Stokes equations, Computers and Mathematics with Applications, 66(2013), 1192–1212.
• Zhang, J., Numerical simulation of 2-D square driven cavity using fourth-order compact finite difference schemes, Computers and Mathematics with Applications, 45(2003), 43–52.