热流科学与工程西幸文交通大堂教育部重点实验室Numerical Heat TransferChapter3NumericalMethodsforSolvingDiffusionEquationandtheirApplications(2)(Chapter4ofTextbook)QInstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryof ThermalScience&EngineeringXi'anJiaotongUniversityXi'an,2022-Sept-281/56
1/56 Instructor Tao, Wen-Quan Key Laboratory of Thermo-Fluid Science & Engineering Int. Joint Research Laboratory of Thermal Science & Engineering Xi’an Jiaotong University Xi’an, 2022-Sept-28 Numerical Heat Transfer Chapter 3 Numerical Methods for Solving Diffusion Equation and their Applications (2) (Chapter 4 of Textbook)
热流科学与工程西幸文交通大堂E教育部重点实验室Contents3.11-D Heat Conduction Equation3.2FullyImplicit Scheme ofMulti-dimensionalHeat Conduction Eguation3.3TreatmentsofSourceTermand B.C3.4TDMA&ADIMethodsforSolvingABEs3.5FullyDevelopedHTinCircularTubes3.6FullyDevelopedHTinRectangleDucts2/56
2/56 3.1 1-D Heat Conduction Equation 3.2 Fully Implicit Scheme of Multi-dimensional Heat Conduction Equation 3.3 Treatments of Source Term and B.C. Contents 3.4 TDMA & ADI Methods for Solving ABEs 3.6 Fully Developed HT in Rectangle Ducts 3.5 Fully Developed HT in Circular Tubes
热流科学与工程西幸交通大堂E教育部重点实验室3.4TDMA & ADIMethodsforSolving ABEs3.4.1TDMAalgorithm（算法）for1-Dconductionproblem1.General form of algebraic equations of 1-Dconductionproblems2.Thomasalgorithm3.Treatment of 1st kind boundary condition3.4.2ADlmethodforsolvingmulti-dimensionalproblem1.Introduction tothematrixof 2-Dproblem2.ADliterationofPeaceman-Rachford3/56
3/56 3.4 TDMA & ADI Methods for Solving ABEs 3.4.1 TDMA algorithm (算法) for 1-D conduction problem 3.4.2 ADI method for solving multidimensional problem 1. Introduction to the matrix of 2-D problem 1.General form of algebraic equations of 1-D conduction problems 2. ADI iteration of Peaceman-Rachford 2.Thomas algorithm 3.Treatment of 1st kind boundary condition
热流科学与工程西幸文交通大堂E教育部重点实验室3.4TDMA&ADIMethodsforSolvingABEqs3.4.1TDMAalgorithmfor1-Dconductionproblem1.General form of algebraic equations.of 1-D conductionproblemsThe ABEqs fora,T +a,T +...+aT +...+amTm=b (i=l,Ml)steady and unsteady(f >0) problems takethe following form(I+1)(I-N)apT,=aT+awTw+bi=lThe matrix (矩阵)Threeofthecoefficients is atriunknownsdiagonal（三对角）one.4/56
4/56 3.4 TDMA & ADI Methods for Solving ABEqs 3.4.1 TDMA algorithm for 1-D conduction problem 1.General form of algebraic equations. of 1-D conduction problems The ABEqs for steady and unsteady (f >0) problems take the following form The matrix (矩阵) of the coefficients is a tridiagonal (三对角) one . P P E E W W a T a T a T b 1 1 2 2 1 1 .+ . = 1, 1) i i M M a T a T a T b i M a T （ Three unknowns i I (I-1) I (I+1)
热流科学与工程西幸文交通大堂E教育部重点实验室2. Thomas algorithm(算法The numbering method of W-P-E is humanized(人性化),but itcannotbeacceptedbyacomputer!Rewrite above equation:AT =BT+ +CT-, +D, i=1,2,....M1 (a)End conditions: i=1,C=C,=0; i=Ml,B,-Bm=0(1)Elimination（消元)-Reducingtheunknownsateachlinefrom3to2Assuming the eq.afterelimination asT,-, = P-,T, +Qi-↓(b)Coefficient has been treated tol5/56
5/56 2. Thomas algorithm(算法) Rewrite above equation： End conditions：i=1, Ci=C1 =0; i＝M1, Bi=BM1 =0 (1) Elimination (消元)－Reducing the unknowns at each line from 3 to 2 Assuming the eq. after elimination as 1 1 , 1,2,. 1 AT BT CT D i M i i i i i i i (a) T P T Q i i i i 1 1 1 Coefficient has been treated to 1. (b) The numbering method of W-P-E is humanized (人性化), but it can not be accepted by a computer!