MAE 4230/5230: Introduction to CFDMy CoordinatesIntroduction to CFDIntroduction to CFDIntroduction to CFDGoverning Equations for a FluidIntroduction to CFDCFD ApplicationsCFD ApplicationsCFD ApplicationsStrategy of CFDNumerical Solution of Model EquationNumerical Solution of Model Equation1D Solution on 4-point gridGrid Convergence of 1D SolutionError on Different GridsError on Different GridsNumerical Solution: Second-Order AccuracyComparison of First and Second-Order SolutionsError on Different GridsDealing with NonlinearityDealing with NonlinearityDealing with NonlinearityDealing with NonlinearityLinearization ExampleMatrix inversionMAE 4230/5230: Introduction to CFDRajesh BhaskaranSibley School of Mechanical & Aerospace EngineeringCornell University Ithaca, New YorkMy Coordinates• Dr. Rajesh BhaskaranSwanson Director of Engineering SimulationSibley School of Mechanical & Aerospace Engineering• E-mail: [email protected]• Office: 102 Rhodes• Office hours: – Come with questions about FLUENT– Held in Swanson Lab (163 Rhodes)– Time to be announcedIntroduction to CFDIntro to CFD handout on blackboardComputations (CFD)FlowfieldV, P, T etc.Governing equationsAnalysisIntroduction to CFD• Approach:– Go through a series of case studies in the use of CFD to analyze flow problems– Case studies to be performed using FLUENT• Goals of the CFD case studies:– Build an understanding of the foundations of CFD.– Use hands-on learning to develop better physical feel for fluid flows and reinforce theory.Introduction to CFD• Before embarking on the CFD case studies, need to understand the rudiments of the CFD solution procedure operating under the hood of the software.• In order to understand the CFD solution procedure, we will apply it to a simple model problem.• Notes will be posted on course website.Governing Equations for a FluidIntroduction to CFD• Not possible to solve the governing equations analytically for most engineering problems.• However, it is possible to obtain approximate computer-based solutions for many engineering problems. • This is the subject matter of CFD.CFD ApplicationsPressure distribution for airplane configurationPressure distribution for helicopter configurationCFD ApplicationsTemperature distribution in a mixing manifold (Boeing 767)CFD ApplicationsPressure contours and velocity vectors in a blood pumpStrategy of CFDExample: Finite-Difference Approximation for du/dxNumerical Solution of Model Equation• Model equation (with m=1):• Finite-difference approximation:Numerical Solution of Model Equation• System of four simultaneous algebraic equations:• Discrete solution:• Exact solution:1D Solution on 4-point grid00.20.40.60.810.30.40.50.60.70.80.91xuNumerical solutionExact solutionGrid Convergence of 1D Solution00.20.40.60.810.30.40.50.60.70.80.91xuN=4N=8N=16Exact solutionError on Different Grids• Would like to know the error introduced by discretization on a given grid.• In general, not possible to determine the actual values of the discretization error.• However, we can estimate the rate at which error would decrease on refining the grid.• One measure of error:Error on Different Gridsε = C Δxα10-110-2∆xεActual errorLeast squares fitα = 0.92 from least squares fitNumerical Solution: Second-Order Accuracy• Model equation (with m=1):• Second-order finite-difference approximation:Comparison of First and Second-Order SolutionsError on Different Gridsε = C Δxαα = 2.18α = 0.92Dealing with Nonlinearity• Model non-linear equation• Finite-difference approximation• Linearize about guess value ug• Linearization error = O[(u- ug)^2]Dealing with Nonlinearity• Matrix system on four-point grid• Iterate until |u- ug|/ | u | < Tolerance• The difference |u- ug|/ | u | is called the ResidualDealing with Nonlinearity• Unscaled residual:• Scaled residual:Dealing with NonlinearityLinearization ExampleMatrix inversion• In each iteration, one can:1. Form the matrix and invert2. Sweep across the mesh updating each point in turn• Use guess value for those values in difference eq. that have not been updated• No need to form the matrix• Easier to code and faster• See Intro to CFD handout on BB for