FINITE ELEMENT METHOD IN FLUID MECHANICS & HEAT

FINITE ELEMENT METHODIN FLUID MECHANICS & HEAT TRANSFERAERSP-560Department : Aerospace EngineeringInstructor : Dr. Cengiz CamciTime and Place : T & R , 4:15 - 5:30 pm173 Willard BuildingPrerequisites : Fluid Mechanics, Thermodynamics, advanced calculus,matrix algebra, computer programming (Fortran or C )Grade : Computer assignments and a final examCredits : three creditsThis course is intended to provide a thorough introduction to the basic ideas employed in the application offinite element techniques to especially fluid flow and heat transfer problems. A student who successfullycompleted this course should be able to perform quick analysis of small problems using the finite elementmethod and write full sized application codes for analyzing fluid flow and heat transfer problems. Additionally,mastery of the material provided in this course will enable the student to more intelligently use commerciallyavailable codes based on the finite element method.PLEASE CONTACT CENGIZ CAMCI IF YOU HAVE INTEREST IN THIS COURSE

TYPICAL SEQUENCE OF LECTURESIntroduction to finite element method, interpolationsNumerical integration, iso-parametric elementsVariational principles, method of variationsMethod of weighted residualsPotential flow solutionsTransient heat conduction, parabolic diffusion problemsFull potential equation solutionsIncompressible viscous flow through the solution of Navier-Stokes equationsCoupled heat transfer/flow solutionsNatural/forced convectionDensity dependent convective diffusionPenalty methodFlow/Heat transfer solutions for turbomachinery internal flow configurations

AEROSPACE 560Finite Element Method in Fluid Mechanics and Heat TransferA.Bulletin Listing1.2.3.4.5.6.B.Designation: AERSPNumber: 560Title: Finite Element Method in Fluid Mechanics and Heat TransferAbbreviated title : Finite Elements in Thermo-fluids EngineeringCredits,class periods, practicum periods: 3,3,0Description: A thorough introduction to the basic ideas employed in theapplication of finite element techniques to especially fluid flow and heattransfer problems encountered in aerospace engineering and mechanicalengineering .Course OutlineIntroductionMathematical ToolsFoundations of Fluid Mechanics and Heat TransferAn Introduction to Finite Element Analysis Using "Galerkin Weak Statement"A Model One Dimensional ProblemThe Weak StatementDerivation of a Symmetric Weak FormulationThe Galerkin ProcedureRemoval of the ArbitrarinessThe Galerkin Procedure and Finite Element Discretization Construction of the Trial Space SetFinite Element Matrix CalculationsDevelopment of the Local Coordinate SystemElement Conductivity MatrixElement Load MatrixAssembly ProcedureA Solution for the Model ProblemA Higher Order FormulationFlow Between Two Parallel Plates (Poiseville Flow)A Galerkin Weak Statement Solution1

Couette FlowA Galerkin Weak Statement SolutionCubic Basis FunctionsA Comparison of Linear, Quadratic and Cubic Basis FunctionsAnother Finite Element Strategy Based on Euler's Equation of Variational CalculusA Direct Minimization TechniqueEuler's Theorem of Variational CalculusApplication of Euler's Equation in Finite Element AnalysisGeneralized Form of Euler's Equation in Three Dimensional SpaceExample problem: Steady State 3D Heat ConductionGeneralization of the Direct Minimization Technique to a Multi-DimensionalProblemExample Problem : u" x.u 1 , u(x 0) 0, u(x 1) 1Example Problem: Couette Flow Problem Using Direct Minimization TechniqueExample Problem: Heat Conducting Bar with Convection on the Lateral SurfaceUsing Direct MinimizationDetermination of the Energy Functional for the Equations Belonging to a Sturm-LiouvilleSystemAccuracy and Error Considerations in One-Dimensional ProblemsFinite Element Analysis in Multi-DimensionsLinear Triangular BasisQuadratic Triangular BasisTwo Dimensional Finite Element Algorithm DevelopmentGalerkin Weak Statement (GWS) for 2D Heat Conduction ProblemThe Discrete Weak StatementImplementation of the Linear BasisBasis Derivatives in Local CoordinatesCoordinate TransformationElement Matrix SystemBoundary Conditions(GWS) for Steady Heat Conduction with Boundary ConvectionAerodynamic Incompressible Potential Flow, A GWS solutionSteady Potential Flow over an Elliptic Cylinder2

Compressible Subsonic Potential FlowLinearized Theory For Thin AirfoilsA (GWS) Solution to Prandtl-Glauert EquationUnsteady Transport with Fluid MotionUnsteady Energy Equation An Imposed 1D Velocity FieldGalerkin Weak Statement and Discretized (GWS)Family of Single Step Time Iterative AlgorithmsExplicit Euler, Trapezoidal, Backward EulerViscous Incompressible Unsteady Flow (Laminar) in 2DA Stream-function/Vorticity Formulation of 2D Navier Stokes Equations(GWS) for the Equation L2ψ -ωRemoval of Arbitrariness(GWS) for the Vorticity Transport EquationRemoval of ArbitrarinessElement Matrix Formulation for the Stream Function/Vorticity FormulationDevelopment of the Time Marching ProcedureCalculation of Stream Function Values at Nodal PointsCalculation of Velocity ComponentsRemarks on the Recovery of the Pressure FieldExamples of Finite Element Solutions of 2D Unsteady Viscous FlowUnsteady Viscous Flow Around a CylinderUnsteady Viscous Flow Facing a Back StepUnsteady Viscous Flow Around a Cylindrical Obstacle on its bedWind Driven Flow in a Rectangular CavityFlow in a Channel of Finite Width with a Rectangular ObstructionThermal Fluid Flow (2D, Viscous, Incompressible, Unsteady)Natural Convection in an EnclosureA Stream Function Vorticity Formulation and Thermal Energy Equation withBuoyancy Force Terms(GWS) and Interpolations on Stream Function, Vorticity and TemperatureAn Incompressible Turbulent Boundary Layer Solution using the Finite Element Method(GWS) for x Momentum Equation,Removal of Arbitrariness, Boundary ConditionsCalculation of Cross Stream VelocitiesA (GWS) Solution for the Continuity EquationComputer Programming Aspects of Finite Element Analysis in 2D3

Finite Element Grid GenerationTypical Data Structure of A Finite Element ProgramAssembly ProcedureImplementation of Boundary ConditionsSolution of a System of Linear Algebraic Equations2D Quadrilateral Elements (Bi-linear and Quadratic Elements)Pure Rectangular Element (Bi-Linear)Generic Quadrilateral Element (Bi-Linear)Implementation of Bi-Linear Basis in Steady State Diffusion EquationTransformation of Differential Line Element into Local CoordinatesTransformation of Differential Area Element into Local CoordinatesNumerical Integration in 2D, Gausssian QuadratureFinite Element Analysis in Three Dimensional SpaceExample Problem : 3D Steady Heat Conduction with Boundary ConvectionProblem Statement(GWS) and Global Matrix SystemIsoparametric/Quadratic 3D ElementsConduction Matrix in 3DCoordinate TransformationNumerical Integration over a Three Dimensional CubeConvection Contribution in the Conduction MatrixNumerical Integration over a Curved Convective SurfaceRight Hand Side Column Matrix Resulting From Convective BoundariesAssembly Procedure for the 3D Global SystemPrescribed Boundary TemperaturesSolution of The Global SystemPenalty Formulation, Fractional Step Method and A Solution Method in PrimitiveVariablesFinite Element Strategies for the Flow Field Analyses in Rotating Machinery4

C.Justification Statement1.Instructional Objectives:The objectives of this course are to cover the basic ideas employed in theapplication of finite element techniques to especially fluid flow and heattransfer problems. Although there are many existing courses dealing withfinite element analysis, non of the current offerings specifically deal withfluid flow and heat transfer problems in depth. Almost all of the existingcourses are focussed on structural mechanics and dynamics applicationswith minimal coverage on viscous flow and heat transfer. The techniquesdiscussed in this course have wide application potential in AerospaceEngineering, Mechanical Engineering, Nuclear Engineering (thermohydraulics) and Chemical Engineering. A student who succesfullycompleted this course will be able to perform quick analysis of small scaleproblems using the finite element method and to write full sizedapplication codes for analyzing fluid flow and heat transfer problems.Additionally, mastery of the material provided in this course will enablethe student to more intelligently use commercially available codes basedon the finite element method.2.Evaluation Methods of the Course:Homework assignments, computer projects, midterm and final exams3.Relationship of Course to Other Courses :This is the only finite element course specifically dealing with fluidmechanics and heat transfer including viscous unsteady flow andconvective heat transfer . Other course offerings in the area of FiniteElement Method at the College of Engineering are as follows :ME 461Emch 560Emch/ME 563Applied Finite Element AnalysisAn Introduction to the FiniteElement Method of AnalysisNonlinear Finite Element AnalysisThe course outlines of the courses listed above are enclosed to thisdocument.4.Relationship of Course to Major/Minor/Option or General Education:This course would be required for Aerospace Engineering Studentsspecializing in Thermal Turbomachinery research and Heat Transfer. It5

would be optional for Aerospace and Mechanical Engineering Studentswho desire some background in the basics of Finite Element Method ofSolution in Fluid Mechanics, Aerodynamics and Heat Transfer.D.7.Consultation with other Departments and Academic Support Units:Faculty in Mechanical Engineering and Engineering Science and Mechanicswho offer courses in related areas were consulted to ensure there would beno significant overlap of material. A memo of support and suggestionsfrom Dr. H.R.Jacobs is attached. This memo was obtained in 1993 duringthe first offering of the course.6.Frequency of Offering and Enrollment:The course will be offered every two years during the spring semester.This course was offered four times (Fall 1993, Spring 1995 , Spring1997 and Fall 1998). The enrollment was 12, 10, 12 and 12 respectively.Effective DateSpring Semester 19996

Finite Element Method in Fluid Mechanics and Heat Transfer A. Bulletin Listing 1. Designation: AERSP 2. Number: 560 3. Title: Finite Element Method in Fluid Mechanics and Heat Transfer 4. Abbreviated title : Finite Elements in Thermo-fluids Engineering 5. Credits,class periods, practicum periods: 3,3,0 6.