finite element analysis of transient free convection flow over vertical plate

DOI 10.1007/s10010-006-0039-z ORIGINALARBEITEN · ORIGINALS Forsch Ingenieurwes (2006) 70: 263–268 Finiteelement analys is of tra nsi ent free con ve ction flow over vertical pla te Ibrahim Abbas Received: 18 June 2006 / Published online: 26 August 2006 © Springer-Verlag 2006 Abstract A finite element method is used to study the heat- transfer response of an incompressible, laminar, transient free convection flow over a semi-infinite vertical plate. The non-dimensional governing equations are solved by the finite element method (FEM). The resulting non-linear integral equations are linearized and solved using the Newton– Raphson iteration. The resulting first-order ordinary differential equations with respect to time are solved using the implicit Euler scheme. Numerical results for the details of the velocity and temperature contours and profiles as well as heat transfer rate in terms of Nusselt number which are shown on graphs have been presented. Finite Elemente Analyse der instation ¨ aren, freien konvektiven Str ¨ omung ¨ uber eine vertikale Plate Zusammenfassung Eine Finite Elemente Methode wird f ¨ ur die Unte rsuchung der W¨ arme ¨ ubertra gung bei der lamin aren, instation¨ aren freik onv ekti ven Str ¨ omung eines inkompres- sible n Fluids ¨ uber eine halb-un endlic he, vertikale Platte eingesetzt. Die das Problem beschreibenden nichtlinearen und dimensionsfreien Gleichungen werden linearisiert und unter Einsat z des Newton–Raphson Iterat ionsv erfahr ens gel ¨ ost. Die erhaltenen, zeitabh ¨ angigen Differentialgleichun- gen 1. Ordnung werden unt er Verwendung des impl izi- ten Eulerverfahrens numerisch ausgewertet. Ergebnisse f ¨ ur Geschwindigkeit- und Temperaturprofile und auch f ¨ ur die Nusseltzahl werden dargestellt und diskutiert. I. Abbas (u) Department of Mathematics, Faculty of Science, 82524 Sohag, Egypt e-mail: [email protected] List of Symbols g acceleration due to gravity Gr l Grashof number Pr Pr an dt l numb er t time U , V velocity components in the x and y directions, respectively X coordinate along the vertical direction Y coordinate axis normal to the plate T temperature l length of the plate β coefficient of volume expansion ν kinematic viscosity θ temperature in non-dimensional form x dimensionless coordinate along the vertical direction y dimensionless coordinate axis normal to the plate u, v components in the x and y directions, respectively τ dimensionless time 1 Int rodu ction Free convection flows past different types of vertical bodies are studied because of their wide application. The study of natural convection flow over a semi-infinite vertical plate, at normal temperature, was first made by Pohlhausen [1] using an integral method, whereas the similarity method was first used to study this problem by Ostrach [2] and the resulting non-linear ordinary differential equations were solved comput ationa lly . Lighthill [3] was the first to study the un- steady forced flow of a viscous incompressible fluid past both a flat plate and a circular cylinder with the free stream havi ng small amplitud e oscillatio ns. The simil arity sol utions for free convection flow from a vertical plate with a uni- 1 3

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