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
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
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
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
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 2 2 2 2 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
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= 1 11 11 12 13 1 M 0 0 v K + G(v) K K v F 22 21 22 23 2 0 M 0 v + K K + G(v) K v = F 31 32 33 3 0 0 0 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 13 23 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 = 21 12 31 13 32 23 33 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
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
Pnalt Function Mthod-algbraic Problm: Find th minimum of th function F (; ) subjct to th constraint G(; ) = 0 df @F @ Lagrang multiplir mthod d + @F @ d = 0 F L (; ; ) F (; ) + G(; ) df L @F L @ d + @F L @ d + @F L @ d = 0 µ @F = @ + @G µ @F d + @ @ + @G d + G(; )d @ JN Rdd @F @ + @G @ = 0; @F @ + @G @ = 0; G(; ) = 0 Viscous Incompr. Flows: 8
Pnalt Function Mthod-algbraic (continud) Pnalt function mthod F P (; ) = F (; ) + [G(; ) 0]2 2 df P @F P @ d + @F P @ d = 0 µ @F = + G(; )@G d + @ @ µ @F @ + G(; )@G @ d @F @ + G(; )@G @ = 0; @F @ + G(; )@G @ = 0 JN Rdd Viscous Incompr. Flows: 9
df L @F L @ d + @F L @ d + @F L @ d = 0 µ Pnalt @F = function mthod @ + @G µ @F d + @ @ + @G d + G(; )d @ F P (; ) = F (; ) + @F [G(; ) 0]2 2 @ + @G @ = 0; @F @ + @G = 0; G(; ) = 0 @ df P @F P @ d + @F P @ d = 0 µ @F = + G(; )@G d + @ @ µ @F @ + G(; )@G d @ @F @ + G(; )@G @ = 0; @F @ + 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
Pnalt Function Mthod-algbraic (An Eampl) F (; ) = 2 2 + 2 8 + + 1; G(; ) 2 = 0 Lagrang Multiplir Mthod 4 8 + 2 = 0; 2 + 1 = 0; 2 = 0 Pnalt Function Mthod = 0:5; = 1:0; = 3:0 4 8 + 2 (2 ) = 0; 2 + 1 (2 ) = 0 Clarl, as! 1, w hav = 8 + 3 4 + 6 ; = 3 1 2 + 3 lim!1 = 0:5 = ; lim!1 = 1:0 = JN Rdd Viscous Incompr. Flows: 11
Pnalt Function Mthod-algbraic (Eampl - continud) Tabl: A comparison of th pnalt solution with th act for various valus of th pnalt paramtr. 1.0 10.0 25.0 50.0 100.0 1000.0 1.1 0.5938 0.5390 0.5197 0.5099 0.5010 0.4 0.9063 0.9610 0.9803 0.9901 0.9990 G( ; ) 1.8 0.2813 0.1169 0.0592 0.0298 0.0030 1.8 2.8125 2.9221 2.9605 2.9801 2.9980 λγ = γ G( γ, γ ) JN Rdd Viscous Incompr. Flows: 12
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
Pnalt Finit Elmnt Formulation (continud) Now suppos that th vlocit fild satisfis th constraint v v v v w w 1 2 + = 0 + + = 0 Thn adding th thr wak statmnts, w obtain w v w v 1 2 w w v v 1 2 0 2 2 w w v v 1 2 P w w f w f dd 3 1 2 0 0 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
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 0 2 2 w f w f dd w t w t ds 1 2 1 2 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
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 0 2 2 w f w f dd w t w t ds 1 2 1 2 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
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 1 0 2 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 2 0 2 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
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
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 11 11 12 1 M 0 v K + Gv ( ) K v F 22 + 21 22 2 0 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 21 12 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
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
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 = 10 12 ¹ Rducd intgration of th pnalt trms JN Rdd Viscous Incompr. Flows: 21
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
Viscous fluid squzd btwn paralll plats: Vlocit fild Distanc, 2.0 1.5 1.0 0.5 v at = 4 Analtical solution v at = 6 0.0, v 0 1 2 3 4 Horizontal vlocit JN Rdd Viscous Incompr. Flows: 23
Viscous fluid squzd btwn paralll plats: Prssur fild Prssur, P 10 9 8 7 6 5 4 3 2 1 Analtical solution ( = 2) FEM solution ( = 0.1875) Prssur, P 8 7 6 5 4 3 2 1 0 Analtical solution ( = 0) FEM solution ( = 0.0625) 0 0 1 2 3 4 5 6 Distanc, -1 0 1 2 3 4 5 6 Distanc, JN Rdd Viscous Incompr. Flows: 24
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
1.0 LID-DRIVEN CAVITY FLOW Distanc along vrtical cntrlin, 0.8 0.6 0.4 0.2 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 10^2 0.0-0.2 0.0 0.2 0.4 0.6 0.8 1.0 JN Rdd Horizontal vlocit, u Viscous Incompr. Flows: 26
Wall-drivn cavit flow vlocit profils 1.0 0.9 0.8 Distanc, 0.7 0.6 0.5 0.4 0.3 Pnalt FEM 8 8 msh 16 20 msh 0.2 0.1 0.0-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 Horizontal vlocit, v (0.5,) JN Rdd Viscous Incompr. Flows: 27
Wall-Drivn Cavit rsults (continud) 1.0 0.9 0.8 Distanc, 0.7 0.6 0.5 0.4 0.3 0.2 0.1 16 20 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 0.0-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Horizontal vlocit, v (0.5,) Viscous Incompr. Flows: 28
Stramlins 4 R = 10 Prssur contours Dilatation contours JN Rdd FluidMch LSFEM - 29 Viscous Incompr. Flows: 29
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