The Discretization Process

Transcription

1 FMIA F Moukalled L Mangan M Darwsh An Advanced Introducton wth OpenFOAM and Matlab Ths textbook explores both the theoretcal foundaton of the Fnte Volume Method (FVM) and ts applcatons n Computatonal Flud Dynamcs (CFD) Readers wll dscover a thorough explanaton of the FVM numercs and algorthms used n the smulaton of ncompressble and compressble flud flows, along wth a detaled examnaton of the components needed for the development of a collocated unstructured pressure-based CFD solver Two partcular CFD codes are explored The frst s ufvm, a three-dmensonal unstructured pressure-based fnte volume academc CFD code, mplemented wthn Matlab The second s OpenFOAM, an open source framework used n the development of a range of CFD programs for the smulaton of ndustral scale flow problems Moukalled Mangan Darwsh Flud Mechancs and Its Applcatons 113 Seres Edtor: A Thess The Fnte Volume Method n Computatonal Flud Dynamcs Wth over 220 fgures, numerous examples and more than one hundred exercses on FVM numercs, programmng, and applcatons, ths textbook s sutable for use n an ntroductory course on the FVM, n an advanced course on CFD algorthms, and as a reference for CFD programmers and researchers Flud Mechancs and Its Applcatons F Moukalled L Mangan M Darwsh The Fnte Volume Method n Computatonal Flud Dynamcs The Fnte Volume Method n Computatonal Flud Dynamcs An Advanced Introducton wth OpenFOAM and Matlab Engneerng ISBN The Dscretzaton Process Chapter 04

2 The Dscretzaton Process

3 The Process Physcal Doman Physcal Phenomena Doman Modelng Physcal Modelng Set of Governng Equatons Defned on omputatonal Doman Doman Dscretzaton Equaton Dscretzaton Structured Grds Cartesan, Non-Orthogonal) Block Structured grds Unstructured Grds Chmera Grds System of Algebrac Equatons Fnte Dfference Fnte Volume Fnte Element Boundary Element Numercal Solutons Soluton Method Combnatons of Multgrd Methods Iteratve Solvers Coupled-Uncoupled 3

4 Doman Modelng Physcal Modelng heat snk heat spreader base Tsnk Insulated ( k T ) = Q Q smk mcroprocessor Tmcroprocessor Q source Doman Dscretzaton Equatons Dscretzaton ρφ + ( ρvφ) t = ( Γ φ ) + Q transent term convecton term φ C + dffuson term NB( C) source term a NB φ NB = b P Soluton Method = 4

5 Doman Dscretzaton T snk nsulated T mcroprocessor 5

6 Numberng F 2 F 3 C F 4 F 1 Element 9 Connectvty Neghbours Faces Vertces [ ] [ ] [ ] E S 1 f E f 2 owner neghbour Face 12 Connectvty Element1 9 Element2 10 Vertces [19 12] Vertex Connectvty Elements [ ] Faces [ ] 6

7 Sparse Matrx Equaton Element 3 Neghbors 1 local element equaton s assemble for element 3 element connectvty s used to transform local ndces to global ndces T C + a F1 T F1 + a F2 T F2 + a F3 T F3 = b C a F 1 F 2 F 3 33 T 3 + a 32 T 2 + a 35 T 5 + a 31 T 1 = b a N3 a N1 a N2 element equaton s assembled nto global matrx 3 = 7

8 Element Felds Element Feld nteror patch#1 patch#2 patch#

9 Equaton Dscretzaton x F 2 f 2 F F 3 C f 3 f 1 F f 1 4 δ y f 2 S f F 4 F 3 f 3 C S f1 F 1 y S f3 f 1 S f4 f 4 F 4 δ x ( k T ) = 0 ( k T ) ds ( k T ) f S f = 0 S C! = 0 f C S f1 δ x f1 T f1 = y f1 = x N1 x C = T x + T y j ( k T ) f1 S f1 ( k T ) f2 S f2 ( k T ) f3 S f3 ( k T ) f4 S f4 = 0 9

10 Coeffcents x F 2 T f1 S f1 = T x + T y j y f1 f 1 δ y F 3 f 3 S f3 f 2 C S f2 S f1 f 1 F 1 y = T x y f1 f 1 S f4 F 4 f 4 δ x T x f 1 = T N1 T C δ x f1 T f1 S f1 = T F 1 T C y f1 ( k T ) δ x f1 S f1 = a F1 T F1 T C f1 a F1 = k y f 1 δ x f1 10

11 Algebrac Equaton F 2 a F2 = k x f 2 δ y f2 f 2 4 S f2 a F3 = k y f 3 δ x f3 F 3 8 f 3 S f3 C 9 S f1 f 1 10 F 1 S f4 f 4 a F4 = k x f 4 δ y f4 F = ( k T ) f S f = a NB T NB T C f C NB C = ( a F1 + a F2 + a F3 + a F4 )T C + a F1 T F1 + a F2 T F2 + a F3 T F3 + a F4 T F4 T C + a NB T NB = 0 NB C = NB C a NB = ( a F1 + a F2 + a F3 + a F4 ) 11

12 System of Equatons Element 3 Neghbors 1 local element equaton s assemble for element 3 element connectvty s used to transform local ndces to global ndces T C + a F1 T F1 + a F2 T F2 + a F3 T F3 2 = b C a F 1 F 2 F 3 33 T 3 + a 32 T 2 + a 35 T 5 + a 31 T 1 = b 3 a N3 a N1 a N2 element equaton s assembled nto global matrx 3 = 12

13 Solvng the Equatons 13 T C = a NB T NB NB(C ) + b C =

14 Face Feld Face Feld nteror patch#1 patch#2 patch# S f d CF g CF S f 14

15 Node feld Vertex Feld nteror patch#1 patch#2 patch#