Advances in Physics Theories and Applications                                                     www.iiste.orgISSN 2224-7...
Advances in Physics Theories and Applications                                                   www.iiste.orgISSN 2224-719...
Advances in Physics Theories and Applications                                          www.iiste.orgISSN 2224-719X (Paper)...
Advances in Physics Theories and Applications                                www.iiste.orgISSN 2224-719X (Paper) ISSN 2225...
Advances in Physics Theories and Applications                                www.iiste.orgISSN 2224-719X (Paper) ISSN 2225...
International Journals Call for PaperThe IISTE, a U.S. publisher, is currently hosting the academic journals listed below....
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11.the influence of the angle of inclination on laminar natural convection in a square enclosure

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11.the influence of the angle of inclination on laminar natural convection in a square enclosure

  1. 1. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 4, 2012 The Influence of the Angle of Inclination on Laminar Natural Convection in a Square Enclosure Kumar Dookhitram1* Roddy Lollchund2 1. Department of Applied Mathematical Sciences, School of Innovative Technologies and Engineering, University of Technology, Mauritius, La Tour Koenig, Pointe-aux-Sables, Republic of Mauritius 2. Department of Physics, Faculty of Science, University of Mauritius, Réduit, Republic of Mauritius * E-mail of the corresponding author: kdookhitram@umail.utm.ac.muAbstractThis paper discusses the results obtained by the numerical modeling of natural convection in a water-filledtwo-dimensional square enclosure inclined to the horizontal. Here, the top and bottom walls of the cavityare considered adiabatic, left vertical wall is maintained at a constant low temperature and the right verticalwall is maintained at a constant high temperature. The aim is to investigate the effects of angle ofinclination on the flow patterns. We use the Krylov subspace method, GMRES, to solve the discretizedformulation of the governing equations. At the validation stage, our results are in good agreement withthose reported in the literature. Results are presented in the form of velocity vector and isotherm plots aswell as the variation of the average Nusselt number for different angles of inclination.Keywords: natural convection, GMRES, Nusselt number1. IntroductionNatural convection is governed by the conservation laws of mass, momentum and energy. For atwo-dimensional unsteady flow in rectangular coordinates (x, y), these conservation laws can be expressedin the dimensionless form as (Bejan 1993): ∂u/∂x+∂v/∂x= 0, (1) ∂u/∂t+∂u2/∂x+∂uv/∂y = -∂p/∂x+ (1/R e)(∂2u/∂x2+∂2u/∂y2)+(Gr/R e2)T sinθ, (2) ∂v/∂t+∂v2/∂y+∂uv/∂x = -∂p/∂y+ (1/R e)(∂2v/∂x2+∂2v/∂y2)+(Gr/R e2)T cosθ, (3) ∂T/∂t+u∂T/∂x+v∂T/∂y = (1/R e Pr)(∂2T/∂x2+∂2T/∂y2). (4)In the above equations, u and v are fluid velocity components along x and y axes respectively, p is thepressure, T is the temperature and θ is the inclination angle. Re, Gr and Pr are the Reynolds, Grashof andPrandtl numbers. The system we study is illustrated in Fig. 1. It is a square cavity Ω( x, y ) = [0 , 1] × [0, 1],which is completely filled by the fluid, with no-slip boundary condition for velocity components (u = v = 0)on all its walls and adiabatic conditions for temperature on its top and bottom walls. The left vertical wallof the cavity is maintained at a constant low temperature (Tc) and the right vertical wall is maintained at aconstant high temperature (Th). The cavity is tilted from the horizontal with angle θ and the working fluid ischosen as water with Prandtl number, Pr = 5.96.2. ProcedureWe employ the finite difference method (Ozisik 1994) on a uniform staggered grid to discretize (1)-(4). Theresulting set of algebraic equations can be cast as the linear system ,which is solved using the Krylov subspace method, GMRES (Saad & Schultz 1986). The GMRESalgorithm aims at projecting the coefficient matrix A onto the Krylov subspace and henceforth solves aleast square system of lower dimension, which is less expensive. The solutions are assumed to converge 23
  2. 2. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 4, 2012when the following convergence criterion is satisfied for every dependent variable at every point in thesolution domain: | (Ψnew - Ψold)/Ψold | < 10-6,where Ψ represents a dependent variable u, v, p and T.3. ExperimentsThe simulation is performed for Gr = 167.8 with different angle of inclination θ, which is varied from 0 to900. Results are obtained for the average Nusselt number (Nu) with respect to θ as shown in Fig. 2. Theminimum average Nusselt number Numin is 0.7426 occurring at θ = 53 .60 and the maximum Numax is θ =38.80 with value 1.1144.Figs. 3 to 6 display the velocity vector plots and isotherms for different angle of inclination, calculated on a21×21 grid. It can be observed that near the hot wall the temperature of the fluid becomes positive, causinga pronounced movement of the fluid against the gravity direction. In a similar manner, the fluid gets coldernear the opposite wall. A negative buoyancy effect is produced, causing the fluid to move in the direction ofgravity. Hence, an overall anticlockwise movement of fluid is produced.The isotherm patterns are uniformly distributed, which indicate that the heat transfer mechanism is mainlydriven by conduction and the temperature field is only marginally distorted by the convective flows. Theeffect of cavity inclination is clearly visible on both the flow patterns and isotherms.4. ConclusionsThe natural convection in a two-dimensional rectangular enclosure has been analyzed numerically using thefinite difference method. The top and bottom walls of the cavity were considered adiabatic and the twovertical walls were maintained at constant low and hot temperatures. The resulting discretized formulationof the governing equations was solved using the GMRES algorithm. The results obtained demonstratepromising avenues for further research.ReferencesBejan A. (1993), Heat Transfer, Wiley, New York, NY.Ozisik M. N. (1994), Finite Difference Methods in Heat Transfer, CRC Press.Saad Y. & Schultz M. H. (1986), “GMRES: A generalised minimal residual algorithm for solvingnonsymmetric linear systems”, SIAM J. Sci. Stat. Comput. 7, 856–869. 24
  3. 3. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 4, 2012 Figure 1. Geometry of the problem in the domain Ω , Figure 2. The average Nusselt number versus tilt angle for Gr = 167.8 25
  4. 4. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 4, 2012 Figure 3. Velocity vectors and isotherms for θ = 00 Figure 4. Velocity vectors and isotherms for θ = 350 Figure 5. Velocity vectors and isotherms for θ = 550 26
  5. 5. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol 4, 2012 Figure 6. Velocity vectors and isotherms for θ = 750 27
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