Finite Element Models for Steady Flows of Viscous Incompressible Fluids
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1 Finit Elmnt Modls for Stad Flows of Viscous Incomprssibl Fluids Rad: Chaptr 10 JN Rdd CONTENTS Govrning Equations of Flows of Incomprssibl Fluids Mid (Vlocit-Prssur) Finit Elmnt Modl Pnalt Function Mthod - algbraic problm Pnalt Finit Elmnt Modl of Viscous Incomprssibl Fluids Numrical Rsults Closur Viscous Incompr. Flows: 1
2 JN Rdd Govrning Equations of Flows of Viscous incomprssibl Fluids Equations of motion Dv σ + f = ρ Dt σ σ Dv + + f = ρ Dt σ σ Dv + + f = ρ Dt Matrial tim drivativ D D = + v = + v + v Dt t Dt t Consrvation of mass v v = 0 v + = 0 Constitutiv rlations σ = P I + τ, τ = 2µ D Viscous Incompr. Flows: 2
3 Govrning Equations of Flows of Viscous incomprssibl Fluids Kinmatics rlations 1 T D = v + ( v) 2 v v v v D =, D =, 2D = + Strss-vlocit-prssur rlations v v v v σ = 2µ P, σ = 2µ P, σ = µ + Boundar conditions ( v, t ) and ( v, t ) JN Rdd t = σ n + σ n, t = σ n + σ n Viscous Incompr. Flows: 3
4 Govrning Equations in Trms of Vlocitis and Prssur (2D) Diffrntial quations v v v σ σ ρ + v + v f = t 0 JN Rdd v v v σ σ ρ + v + v f = 0 t v v + = 0 Boundar conditions ( v, t) and ( v, t) v v v v σ = 2µ P, σ = 2µ P, σ = µ + Viscous Incompr. Flows: 4
5 WEAK FORMS OF THE EQUATIONS 0 0 v v v σ σ = w1 ρ + v + v f dd t Ω v v v w1 w 1 = ρw1 + v + v + σ + σ w1f dd w1t ds, t Ω v v v σ σ = w2 ρ + v + v f dd t Ω v v v w w = ρw + v + v + σ + σ wf t Ω dd w2 t ds, t = σ n + σ n, t = σ n + σ n w Γ w Γ v 1 v 2 0 v v = w3 + Ω dd w3 P JN Rdd Viscous Incompr. Flows: 5
6 JN Rdd Mid Finit Elmnt Modl for th stad-stat cas m m n v = v, v = v, P = P, j j j j j j j= 1 j= 1 j= M 0 0 v K + G(v) K K v F M 0 v + K K + G(v) K v = F P K K K P F 11 i j j j j i Kij = 2 + dd, Gij( v, v ) = i v + v dd 12 i j K, i, i ij = dd Kij = j dd Kij = j dd, K 22 ij i j i = + 2 j dd, K = K, K = K, K = K, K = ij ji ij ji ij ji ij F = f dd + t ds, F = f dd + t ds 1 2 i i i i i i 0 Viscous Incompr. Flows: 6
7 Mid Finit Elmnt Modl (continud) v = u, v = v v 1 1 P u 1 v 3 3 u 3 v 2 2 u 2 Quadratic (u,v); linar P Quadratic (u,v); linar P Linar (u,v); constant P Linar (u,v); constant P JN Rdd Viscous Incompr. Flows: 7
8 Pnalt Function Mthod-algbraic Problm: Find th minimum of th function F (; ) subjct to th constraint G(; ) = Lagrang multiplir mthod d = 0 F L (; ; ) F (; ) + G(; ) df d d d = 0 d + G(; @ @ = 0; G(; ) = 0 Viscous Incompr. Flows: 8
9 Pnalt Function Mthod-algbraic (continud) Pnalt function mthod F P (; ) = F (; ) + [G(; ) 0]2 2 df d d = 0 = + G(; )@G @ + G(; d + G(; + G(; = 0 JN Rdd Viscous Incompr. Flows: 9
10 df d d d = 0 µ = function d + G(; F P (; ) = F (; ) [G(; ) 0]2 + = 0; G(; ) = df d d = 0 = + G(; )@G @ + G(; @ + G(; + G(; = 0 JN Rdd Approimation of th Lagrang multiplid can b computd in th pnalt mthod from λ γ = γ G( γ, γ ) Viscous Incompr. Flows: 10
11 Pnalt Function Mthod-algbraic (An Eampl) F (; ) = ; G(; ) 2 = 0 Lagrang Multiplir Mthod = 0; = 0; 2 = 0 Pnalt Function Mthod = 0:5; = 1:0; = 3: (2 ) = 0; (2 ) = 0 Clarl, as! 1, w hav = ; = lim!1 = 0:5 = ; lim!1 = 1:0 = JN Rdd Viscous Incompr. Flows: 11
12 Pnalt Function Mthod-algbraic (Eampl - continud) Tabl: A comparison of th pnalt solution with th act for various valus of th pnalt paramtr G( ; ) λγ = γ G( γ, γ ) JN Rdd Viscous Incompr. Flows: 12
13 0 PENALTY FINITE ELEMENT FORMULATION for th Stad-Stat Cas Considr th wak forms v v v w2 w 2 0 = ρw2 + v + v + σ + σ w2f dd w2tds, t 0 v v v w1 w 1 = ρw1 + v + v + σ + σ w1f dd w1t ds, t Ω Ω = 3 + Ω w v v dd, t = σ n + σ n, t = σ n + σ n Γ Γ JN Rdd Viscous Incompr. Flows: 13
14 Pnalt Finit Elmnt Formulation (continud) Now suppos that th vlocit fild satisfis th constraint v v v v w w = = 0 Thn adding th thr wak statmnts, w obtain w v w v 1 2 w w v v w w v v 1 2 P w w f w f dd v v v w v v v w v 1 2 v dd w t w t ds 1 2 JN Rdd Viscous Incompr. Flows: 14
15 JN Rdd Pnalt Finit Elmnt Formulation - continud Thus, th wak form of th problm, subjctd to th constraint is w1 u w2 v w w 1 2 u v w f w f dd w t w t ds v v v v w v v w v v dd 1 2 Thus, th variational problm is 0 I( v, v) v subjct to th constraint v + = 0 Viscous Incompr. Flows: 15
16 JN Rdd Pnalt Finit Elmnt Formulation - continud Thn, th modifid wak form with th constraint is 0 v v I p( v, v) I( v, v) dd 2 w1 u w2 v w w 1 2 u v w f w f dd w t w t ds v v v v w v v w v v dd 1 2 w1 w 2 v v dd 2 Pnalt prssion Viscous Incompr. Flows: 16
17 Pnalt Finit Elmnt Formulation - continud Th wak form of th problm can b sparatd into th following two statmnts: w v w v v 1 1 w v v w f dd 1 v v w t ds w v v dd 1 1 w v 2 w v v 2 w v v w f dd 2 v v w t ds w v v dd 2 2 Ths statmnts form th basis of th pnalt FE Modl. JN Rdd Viscous Incompr. Flows: 17
18 JN Rdd Pnalt Finit Elmnt Formulation continud Altrnativl, th prssur (ngativ of th Lagrang multiplir) in th govrning quations can b rplacd b v v P W obtain v v v v v v f 0 v v v v v v f 0 Th wak forms of ths quations ar prcisl th sam as thos on th prvious slid. Viscous Incompr. Flows: 18
19 JN Rdd Pnalt Finit Elmnt Modl m v = v (, ), v = v (, ) j j j j j= 1 j= 1 Substitution into th wak forms (adding inrtia trms) ilds th quations M 0 v K + Gv ( ) K v F M v = K K + Gv ( ) v F 11 j j j K 2 i i i ij = + dd + dd K = dd + dd, K = K 12 i j i j ij ij ji 22 i j j j K 2 i i ij = + dd + dd F = f dd + t 1 i i m ds, F = f dd + t ds 2 i i i i Viscous Incompr. Flows: 19
20 Elmnts Usd for Pnalt FE Modl v 3 v 2 2 v 2 3 v 3 v 1 1 v 1 Nods with v and v JN Rdd Viscous Incompr. Flows: 20
21 Computational Aspcts of th Pnalt FEM Gnral form of th Pnalt FEM: ¹[K 1 ] + ½[K 2 ] + [K 3 ] f g = ff g Elmnt `locking': lim! 0 ¹[K 1 ] + ½[K 2 ] + [K 3 ] f g = ff g! [K 3 ]f g = ff g Choic of th pnalt paramtr: = 10 4 ¹ to = ¹ Rducd intgration of th pnalt trms JN Rdd Viscous Incompr. Flows: 21
22 Numrical Eampls Viscous fluid squzd btwn paralll plats V 0 Computational domain Linar lmnt Quadratic lmnt v = V0, v = 0 2b 2a v t = 0 = 0 t t = 0 = 0 V 0 (a) v 0, t = 0 = (b) JN Rdd Viscous Incompr. Flows: 22
23 Viscous fluid squzd btwn paralll plats: Vlocit fild Distanc, v at = 4 Analtical solution v at = 6 0.0, v Horizontal vlocit JN Rdd Viscous Incompr. Flows: 23
24 Viscous fluid squzd btwn paralll plats: Prssur fild Prssur, P Analtical solution ( = 2) FEM solution ( = ) Prssur, P Analtical solution ( = 0) FEM solution ( = ) Distanc, Distanc, JN Rdd Viscous Incompr. Flows: 24
25 LID-DRIVEN CAVITY FLOW v = 1, v = 0 Vr long v = 0, v = 0 a a 0.5 a v = 0, v = 0 v 0, v = 0 = JN Rdd Viscous Incompr. Flows: 25
26 1.0 LID-DRIVEN CAVITY FLOW Distanc along vrtical cntrlin, Linar lmnt (88 msh) Quadratic lmnt (44 msh) Linar lmnt (88 msh) Quadratic lmnt (44 msh) Circls dnot pnalt paramtr of 10^8 Squars dnot pnalt paramtr of 10^ JN Rdd Horizontal vlocit, u Viscous Incompr. Flows: 26
27 Wall-drivn cavit flow vlocit profils Distanc, Pnalt FEM 8 8 msh msh Horizontal vlocit, v (0.5,) JN Rdd Viscous Incompr. Flows: 27
28 Wall-Drivn Cavit rsults (continud) Distanc, msh of bilinar lmnts ( R = 500, β = 0.5, ε = 10 2 ) R = 10,000 R = 5,000 R = 1,000 R = 500 JN Rdd Horizontal vlocit, v (0.5,) Viscous Incompr. Flows: 28
29 Stramlins 4 R = 10 Prssur contours Dilatation contours JN Rdd FluidMch LSFEM - 29 Viscous Incompr. Flows: 29
30 SUMMARY Th following topics wr covrd in this lctur: Govrning quations of flows of incomprssibl fluids Mid (vlocit-prssur) finit lmnt modl Pnalt function mthod - algbraic problm Pnalt finit lmnt modl of viscous incomprssibl fluids Numrical rsults JN Rdd Viscous Incompr. Flows: 30
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