Powiadomienia systemowe
- Sesja wygasła!
Identyfikatory
Warianty tytułu
Języki publikacji
Abstrakty
This paper proposes a method to numerically study viscous incompressible two-dimensional steady flow in a driven square cavity with heat and concentration sources placed on its side wall. The method proposed here is based on streamfunction-vorticity (Ψ-ξ) formulation. We have modified this formulation in such a way that it suits to solve the continuity, x and y-momentum, energy and mass transfer equations which are the governing equations of the problem under investigation in this study. No-slip and slip wall boundary conditions for velocity, temperature and concentration are defined on walls of a driven square cavity. In order to numerically compute the streamfunction Ψ, vorticityfunction ξ , temperature θ, concentration C and pressure P at different low, moderate and high Reynolds numbers, a general algorithm was proposed. The sequence of steps involved in this general algorithm are executed in a computer code, developed and run in a C compiler. We propose that, with the help of this code, one can easily compute the numerical solutions of the flow variables such as velocity, pressure, temperature, concentration, streamfunction, vorticityfunction and thereby depict and analyze streamlines, vortex lines, isotherms and isobars, in the driven square cavity for low, moderate and high Reynolds numbers. We have chosen suitable Prandtl and Schmidt numbers that enables us to define the average Nusselt and Sherwood numbers to study the heat ad mass transfer rates from the left wall of the cavity. The stability criterion of the numerical method used for solving the Poisson, vorticity transportation, energy and mass transfer has been given. Based on this criterion, we ought to choose appropriate time and space steps in numerical computations and thereby, we may obtain the desired accurate numerical solutions. The nature of the steady state solutions of the flow variables along the horizontal and vertical lines through the geometric center of the square cavity has been discussed and analyzed. To check the validity of the computer code used and corresponding numerical solutions of the flow variables obtained from this study, we have to compare these with established steady state solutions existing in the literature and they have to be found in good agreement.
Słowa kluczowe
Rocznik
Tom
Strony
17--30
Opis fizyczny
Bibliogr. 30 poz., rys.
Twórcy
autor
- Department of Mathematics Faculty of Mathematical Sciences University of Delhi, Delhi-110007, India
autor
- Department of Mathematics Faculty of Mathematical Sciences University of Delhi, Delhi-110007, India
Bibliografia
- [1] Torrance, K.E. (1968). Comparison of finite-difference computations of natural convection. Journal of Research of the National Bureau of Standards, 72B, 281-301.
- [2] Torrance, K.E. & Rockett, J.A. (1969). 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, 33-54.
- [3] Kopecky, R.M. & Torrance, K.E. (1973). Initiation and structure of axisymmetric eddies in a rotating stream. Computers and Fluids, 1, 289-300.
- [4] Bozeman, J.D. & Dalton, C. (1973). Numerical study of viscous flow in a cavity. Journal of Computational Physics, 12, 348-363.
- [5] Ghia, U., Ghia, K.N. & Shin, C.T. (1982). High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics, 48, 387-411.
- [6] Davis, D.V. (1983). Natural convection of air in a square cavity; a benchmark numerical solution. Int. J. Numer. Methods in Fluids, 3(4), 249-264.
- [7] Li, M., Tang, T. & Fornberg, B. (1995). A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 20, 1137-1151.
- [8] Ghoshdastidar, P.S. (1998). Computer Simulation of Flow and Heat Transfer. New Delhi: Tata McGraw-Hill Publishing Company Limited.
- [9] Tian, Z. & Ge, Y. (2003). 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, 495-518.
- [10] Zhang, J. (2003). Numerical simulation of 2-D square driven cavity using fourth-order compact finite difference schemes. Computers and Mathematics with Applications, 45, 43-52.
- [11] Oztop, H.F. & Dagtekin, I. (2004). Mixed convection in two sided lid-driven differentially heated square cavity. International Journal of Heat and Mass Transfer, 47, 1761-1769.
- [12] Deng, Q.-H., Zhou, J., Mei, C. & Shen, Y.M. (2004). Fluid, heat and contaminant transport structures of laminar double-diffusive mixed convection in a two-dimensional ventilated enclosure. International Journal of Heat and Mass Transfer, 47, 5257-5269.
- [13] Erturk, E., Corke, T.C. & Gokcol, C. (2005). Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. International Journal for Numerical Methods in Fluids, 48, 747-774.
- [14] Erturk, E. & Dursun, B. (2007). Numerical solutions of 2-D steady incompressible flow in a driven skewed cavity. Journal of Applied Mathematics and Mechanics, 87, 377-392.
- [15] Nithyadevi, N., Kandaswamy, P. & Lee, J. (2007). Natural convection in a rectangular cavity with partially active side walls. Int. J. of Heat and Mass Transfer, 50 (23-24), 4688-4697.
- [16] Muralidhar, K. & Sundararajan, T. (2008). Computational Fluid Flow and Heat Transfer. Second ed., New Delhi: Narosa.
- [17] Kandaswamy, P., Nithyadevi, N. & Ng, C.O. (2008). Natural convection in enclosures with partially thermally active side walls containing internal heat sources. Phys. of Fluids, 20, 097104-097123.
- [18] Teamah, M.A. & El-Maghlany, W.M. (2010). Numerical simulation of double-diffusive mixed convective flow in rectangular enclosure with insulated moving lid. Int. J. of Thermal Sciences, 49, 1625-1638.
- [19] Tian, Z.F. & Yu, P.X. (2011). An efficient compact difference scheme for solving the streamfunction formulation of the incompressible Navier-Stokes equations. Journal of Computational Physics, 230, 6404-6419.
- [20] Oztop, H.F., Salem, K.A. & Pop, I. (2011). MHD mixed convection in a lid-driven cavity with corner heater. International Journal of Heat and Mass Transfer, 54, 3494-3504.
- [21] Alam, P., Kumar, A., Kapoor, S. & Ansari, S.R. (2012). Numerical investigation of natural convection in a rectangular enclosure due to partial heating and cooling at vertical walls. Commun. in Nonlinear Sci. Numer. Simulat., 17(6), 2403-2414.
- [22] Zaman, F.S., Turja, T.S. & Molla, Md.M. (2013). Buoyancy driven natural convection in an enclosure with two discrete heating from below. Procedia Eng., 56, 104-111.
- [23] Qarnia, H.E., Draoui, A. & Lakhal, E.K. (2013). Computation of melting with natural convection inside a rectangular enclosure heated by discrete protruding heat sources. Appl. Math. Modelling, 37(6), 3968-3981.
- [24] Ambethkar, V. & Kumar, M. (2017). Numerical solutions of 2-D steady incompressible flow in driven square cavity using streamfunction vorticity formulation. Turk. J. Math. Comput. Sci., 6, 10-22.
- [25] Alleborn, N., Raszillier, H. & Durst, F. (1999). Lid-driven cavity with heat and mass transport. Int. J. Heat and Mass Transfer, 42, 833-853.
- [26] Ambethkar, V., Kumar, M. & Srivastava, M.K. (2016). Numerical solutions of 2-D incompressible flow in a driven square cavity using streamfunction-vorticity formulation. Int. J. Applied Maths., 29(6), 727-757.
- [27] Nithyadevi, N., Divya, V. & Rajarathinam, M. (2017). Effect of Prandtl number on natural convection in a rectangular enclosure with discrete heaters. J. Applied Science and Engineering, 20(2), 173-182.
- [28] Ambethkar, V. & Kumar, M. (2018). Numerical solutions of 2-D steady incompressible viscous flow with heat transfer in a driven square cavity using stream function-vorticity formulation. Int. J. Nonlinear Science, 25(2), 87-101.
- [29] Lax, P.D. & Richtmyer, R.D. (1956). Survey of the stability of linear finite difference equations. Communications on Pure Applied Mathematics, 9, 267-293.
- [30] Biringen, S. & Chow, C.-Y. (2011). An Introduction to Computational Fluid Mechanics by Examples. Hoboken, New Jersey: John Wiley and Sons, Inc.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-16d09b37-1ba3-4997-96e8-698f7e2f028b