The Finite Volume Method in

Transcription

1 FMIA F Moukalled L Mangani M Darwish An Advanced Introduction with OpenFOAM and Matlab This textbook explores both the theoretical oundation o the Finite Volume Method (FVM) and its applications in omputational Fluid Dynamics (FD) Readers will discover a thorough explanation o the FVM numerics and algorithms used in the simulation o incompressible and compressible luid lows, along with a detailed examination o the components needed or the development o a collocated unstructured pressure-based FD solver Two particular FD codes are explored The irst is ufvm, a three-dimensional unstructured pressure-based inite volume academic FD code, implemented within Matlab The second is OpenFOAM, an open source ramework used in the development o a range o FD programs or the simulation o industrial scale low problems Moukalled Mangani Darwish Fluid Mechanics and Its Applications 113 Series Editor: A Thess The Finite Volume Method in omputational Fluid Dynamics With over 220 igures, numerous examples and more than one hundred exercises on FVM numerics, programming, and applications, this textbook is suitable or use in an introductory course on the FVM, in an advanced course on FD algorithms, and as a reerence or FD programmers and researchers Fluid Mechanics and Its Applications F Moukalled L Mangani M Darwish The Finite Volume Method in omputational Fluid Dynamics The Finite Volume Method in omputational Fluid Dynamics An Advanced Introduction with OpenFOAM and Matlab Engineering ISBN The Finite Volume Method hapter 05

2 The Process Physical Domain Physical Phenomena Domain Modeling Physical Modeling Set o Governing Equations Deined on a omputational Domain Domain Discretization Equation Discretization Structured Grids artesian, Non-Orthogonal) Block Structured grids Unstructured Grids himera Grids System o Algebraic Equations Finite Dierence Finite Volume Finite Element Boundary Element Solution Method Numerical Solutions ombinations o Multigrid Methods Iterative Solvers oupled-uncoupled 2

3 heat sink microprocessor heat spreader base Domain Modeling Physical Modeling heat sink heat spreader base T sink insulated ( k T ) =!q microprocessor T microprocessor Domain Discretization Equations Discretization Patch2 Patch1 ( ρφ) + ( ρvφ) t = ( Γ φ ) + Q transient term convection term a φ + diusion term F NB source term a F φ F = b Patch3 Solution Method = 3

4 Balance Form onsider the steady state conservation equation o a scalar ρvφ! " $ =! " Γ φ $ +! Qφ convectionterm diusionterm sourceterm FdV =! F ds V V ( ρvφ)dv = ( Γ φ)dv + QdV V V V Diusion onvection Transient! ( ρvφ) ds =! ( Γ φ) ds + QdV V V V Source/ Sink ( ρvφ) ds = ( Γ φ) ds + QdV = aces( V ) = aces( V ) V 4

5 Face Flux Integration = aces( V ) ( ρvφ) ds J ds = ( J S ) ip = J ip w ip S ip ip ( ρuφ) S FluxT = Flux φ + FluxF φ F1 + FluxV FluxT = Flux φ + FluxF φ F1 + FluxV FluxT = Flux φ + FluxF φ F1 + FluxV F 1 F 1 F 1 F S 1 F S 1 F S 1 d N1 6 F 6 6 F 6 6 F 6 d N1 d N F 3 F 3 F F 4 F 5 one integration point 4 5 F 4 F 5 two integration points 4 5 F 4 F 5 three integration points ( ρuφ) 1( ) w 1( ) S + ( ρuφ) 2( ) w 2( ) S 5

6 Source Integration FluxT = QdV = Q V = Q ip ip ip ω ip V ip V = Fluxφ + FluxV ip Q V Q 1 V 1 + Q 2 V 2 + Q 3 V 3 + Q 4 V 4 FluxT = Fluxφ + FluxV FluxT = Fluxφ + FluxV FluxT = Fluxφ + FluxV F 1 F 1 F 1 F 2 F 2 1 F 2 2 F F 6 F 6 F 3 F 3 F 3 F 4 F 5 one integration point 4 5 F 4 F 5 our integration points F 4 F 5 nine integration points 6

7 Flux Linearization F 1 F J φ S = FluxT!" = Flux!" $ φ + FluxF!" φ + FluxV F!" total lux or ace lux linearization coeicient or lux linearization coeicient or F non linearized part F F ( J φ S ) = FluxT nb nb = Flux φ + FluxF φ F + FluxV nb F 4 F 5 Q φ V = FluxT = Fluxφ + FluxV a φ + F NB a F φ F = b a = nb a F = FluxF b = Flux Flux nb FluxV + FluxV 7

8 Boundary onditions Value Speciied (Dirichlet Boundary ondition) t n b S b φ b,speciied J b φ S b = J b φ, S b = ρvφ b S b = Flux b φ + FluxV b Flux b = 0 = ( ρ b v b S b )φ b =!m φ b,speciied FluxV b =!m φ b,speciied e b Flux Speciied (Neumann Boundary ondition) J φ b S b =! J φ b n S b b Speciied lux Flux b = 0 = q b, speciied S b FluxV b = q b, speciied S b n t b e b S b q b,speciied 8

9 FVM Accuracy

10 Spatial Variation φ( x) x φ φ x φ(x) = φ P + ( x x P ) ( φ) P + 1 x x 2 ( P ) 2 : ( φ ) P + 1 3! ( x x P ) 3 :: ( φ) P + n ::: n + 1 n! x x P!( " $ φ) + P n φ φ(x) = φ P + ( x x P ) ( φ) P + O( Δx 2 ) 10

11 Mean Value Theorem φ φ ell Value lux φ = 1 V φ dv V = 1 φ + ( x x ) φ V = φ V V + O x x 2 dv dv + 1 ( x x ) ( φ) V V dv O x x V V dv V = φ + O x x 2 11

12 Mean Value Theorem convective lux S J φ,d S = Γ φ φ! " $ Diusion lux F S J φ, S = ρvφ!" $ onvection lux ( ρ v S )φ = ( ρvφ) ds = ρ v φ + ( x x ) φ + O x x 2 ds 2 = ρ v φ ds + ( x x ) ( φ) ρ v ds + O x x = ρ v S 2 φ + O x x ρ v ds diusion lux ( Γ φ φ) ds = Γ φ φ ds = Γ φ φ + x x + O x x 2 ( Γ φ φ) + ( x x )ds = Γ φ 2 ( φ) S + O x x : Γ φ 2 ( ( φ) ) + O x x ds 12

13 Properties o the Discretized Equations

14 onservation Flux φ + FluxF φ F + FluxV FluxF φ F + Flux φ + FluxV F 1 FluxT = FluxT S Flux φ F 1 S 14

15 Transportiveness Pe = 0 Pe > 0 W E 15

16 Other Properties onsistency Stability Economy Boundedness o the Interpolation Proile 16

17 Variable Arrangment

18 Variable Arrangement ell-centered Vertex-centered 18

19 Other Issues S F S θ e t n d F F 19

20 Matrix onnectivity

21 uvm onnectivity 21!!!!!!!!!!!!!!!!! " $ %! " $ % =! " $ % [3] 2 [3,7] 3 [ ] 4 [3 6] 5 [3 6 7] 6 [4 5] 7 [2 5] onnectivity A B

22 OpenFoam onnectivity 22!!!!!!!!!!!!!!!!! " $ %! " $ % =! " $ % ldumatrix diag() lower() upper()