11.the influence of the angle of inclination on laminar natural convection in a square enclosure
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: email@example.comAbstractThis 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
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
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
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
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
International Journals Call for PaperThe IISTE, a U.S. publisher, is currently hosting the academic journals listed below. The peer review process of the following journalsusually takes LESS THAN 14 business days and IISTE usually publishes a qualified article within 30 days. Authors shouldsend their full paper to the following email address. More information can be found in the IISTE website : www.iiste.orgBusiness, Economics, Finance and Management PAPER SUBMISSION EMAILEuropean Journal of Business and Management EJBM@iiste.orgResearch Journal of Finance and Accounting RJFA@iiste.orgJournal of Economics and Sustainable Development JESD@iiste.orgInformation and Knowledge Management IKM@iiste.orgDeveloping Country Studies DCS@iiste.orgIndustrial Engineering Letters IEL@iiste.orgPhysical Sciences, Mathematics and Chemistry PAPER SUBMISSION EMAILJournal of Natural Sciences Research JNSR@iiste.orgChemistry and Materials Research CMR@iiste.orgMathematical Theory and Modeling MTM@iiste.orgAdvances in Physics Theories and Applications APTA@iiste.orgChemical and Process Engineering Research CPER@iiste.orgEngineering, Technology and Systems PAPER SUBMISSION EMAILComputer Engineering and Intelligent Systems CEIS@iiste.orgInnovative Systems Design and Engineering ISDE@iiste.orgJournal of Energy Technologies and Policy JETP@iiste.orgInformation and Knowledge Management IKM@iiste.orgControl Theory and Informatics CTI@iiste.orgJournal of Information Engineering and Applications JIEA@iiste.orgIndustrial Engineering Letters IEL@iiste.orgNetwork and Complex Systems NCS@iiste.orgEnvironment, Civil, Materials Sciences PAPER SUBMISSION EMAILJournal of Environment and Earth Science JEES@iiste.orgCivil and Environmental Research CER@iiste.orgJournal of Natural Sciences Research JNSR@iiste.orgCivil and Environmental Research CER@iiste.orgLife Science, Food and Medical Sciences PAPER SUBMISSION EMAILJournal of Natural Sciences Research JNSR@iiste.orgJournal of Biology, Agriculture and Healthcare JBAH@iiste.orgFood Science and Quality Management FSQM@iiste.orgChemistry and Materials Research CMR@iiste.orgEducation, and other Social Sciences PAPER SUBMISSION EMAILJournal of Education and Practice JEP@iiste.orgJournal of Law, Policy and Globalization JLPG@iiste.org Global knowledge sharing:New Media and Mass Communication NMMC@iiste.org EBSCO, Index Copernicus, UlrichsJournal of Energy Technologies and Policy JETP@iiste.org Periodicals Directory, JournalTOCS, PKPHistorical Research Letter HRL@iiste.org Open Archives Harvester, Bielefeld Academic Search Engine, ElektronischePublic Policy and Administration Research PPAR@iiste.org Zeitschriftenbibliothek EZB, Open J-Gate,International Affairs and Global Strategy IAGS@iiste.org OCLC WorldCat, Universe Digtial Library ,Research on Humanities and Social Sciences RHSS@iiste.org NewJour, Google Scholar.Developing Country Studies DCS@iiste.org IISTE is member of CrossRef. All journalsArts and Design Studies ADS@iiste.org have high IC Impact Factor Values (ICV).