15. Numerical Simulation of The Motion of Pendula in an Incompressible Viscous Fluid by Lagrange Multiplier/Fictitious Domain Methods

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1 Foureenh Inernaional Conference on Domain Decomposiion Mehods Ediors: Ismael Herrera, David E. Keyes, Olof B. Widlund, Rober Yaes c 23 DDM.org 5. Numerical Simulaion of The Moion of Pendula in an Incompressible Viscous Fluid by Lagrange Muliplier/Ficiious Domain Mehods L. H. Juárez, R. Glowinski 2. Inroducion. Lagrange Muliplier/Ficiious Domain Mehods have proved o be effecive in he direc numerical simulaion of he moion of rigid bodies in incompressible viscous fluids [7]. In his work we discuss he applicaion of his mehodology, combined wih finie elemen approximaions and operaor spliing, o he numerical simulaion of he moion of pendula in a Newonian incompressible viscous fluid. The pendula are circular cylinders consrained o move in a circular rajecory. The moion of he cylinders are driven only by he hydrodynamical forces and graviy. In he presen calculaions we allowed solid surfaces o ouch and penerae, conrary o wha was done in previous work where hese mehods were applied [7]. In fac, a good feaure of his mehodology is ha he numerical soluion does no break down when he rigid bodies overlap. On he oher hand, numerical mehods in which he compuaional domain is remeshed may break down when collision occurs, because his would break he laice modeling of he fluid [8]. Hence a repulsive force beween he paricles need o be incorporaed when hey are close o each oher o preven conac beween surfaces. In he numerical simulaions in his work we did no inroduce hese arificial repulsive forces, in par because we waned o invesigae he soluions when he rigid bodies are near collision or when hey acually collide and overlap. The mechanics of how solid paricles in viscous liquids sick or rebound has no been fully undersood and is sill subjec of curren research. I has been demonsraed heoreically ha when a perfec rigid sphere approaches a rigid wall is kineic energy is dissipaed by non-conservaive viscous forces. The rae of close approach is asympoically slow and he sphere do no deform or rebound [2]. By simulaneously accouning for elasic deformaion of he body and viscous fluid forces, Davis e al. [3] showed ha par of he incoming paricle kineic energy is dissipaed by fluid forces and inernal solid fricion, and he res is ransformed ino elasic-srain energy of deformaion. Depending on he fracion of he kineic energy ha becomes sored as elasic-srain energy, he deformaion of he spheres may be significan and rebound may occur. The relevan parameer for he bouncing ransiion, which is ofen obained experimenally [9], is he Sokes number, which characerize he paricle ineria relaive o viscous forces. Numerical resuls of colliding bodies in viscous fluids may help o undersand he mechanics of individual collisions in solid-liquid flows, which is an imporan issue in pariculae muli-phase flow modeling and in he acual numerical compuaions of hese flows. The numerical experimens in his work include he moion of a single pendulum, and he moion of wo pendula. The wo pendula case include differen numerical experimens where he disks may have differen densiies and iniial posiions. An ineresing sudy of pendula in viscous fluids wih some applicaions can be found in [2] and references herein. In Secion 2 we describe he model for a single pendulum. A Lagrange Muliplier/Ficiious Domain equivalen formulaion is presened in Secion 3. The discreizaion of he resuling problem is discused in Secions 4 and 5. Numerical resuls and conclusions are given in Secion Fluid-Rigid Body Ineracion Model. We describe he model for he case of one pendulum in a viscous fluid. Is generalizaion o several pendula is sraighforward. Le IR 2 be a space region wih boundary Γ, filled wih an incompressible viscous fluid, Universiy of Houson and UAM-I, hecor@mah.uh.edu, and hec@xanum.uam.mx 2 Universiy of Houson, roland@mah.uh.edu

2 86 JUAREZ, GLOWINSKI O Φ l n B G Γ f Figure 2.: Pendulum in a viscous fluid of densiy ρ f, and ha conains a rigid body B, wih cener of mass G. The rigid body is consrained o move in a circular rajecory around he axis of roaion defined by a poin O, as shown in Figure 2.. The posiion of he rigid body is known a any ime hrough he angle Φ = Φ(). We denoe by n he uni normal vecor poining ouward o he flow region f () =\ B(). Assuming ha he only exernal force acing on he mixure is graviy (denoed by g), in he verical negaive direcion, he fluid flow is modeled by he Navier-Sokes equaions ρ f [ u ] +(u )u = ρ f g + σ in f (), (2.) u =in f (), (2.2) where u denoes he velociy of he fluid, p is he fluid pressure, and σ = τ pi is he sressensor, wihτ = µ f ( u + u ) for a Newonian fluid wih viscosiy µ f. These equaions are compleed by some iniial condiions and by he following no-slip boundary condiions: u = on Γ, and u(x,)=v() +ω() G()x, x B(). Here V() andω() arehe ranslaional and angular velociies of he rigid body, respecively. The moion of he rigid body B is modeled by he Newon-Euler equaions: M dv = Mg + F, (2.3) d I dω = T, (2.4) d where M and I are he mass and ineria ensor of he rigid body, respecively. F is he resulan of he hydrodynamical forces acing on B, andt is he orque a G of he above hydrodynamical forces acing on B. The previous equaions are compleed by he kinemaic equaions dg/d = V, dφ/d = ω, and by imposing iniial condiions on G, Φ, V, andω. Here we use he noaion ω =(,,ω), and Φ =(,,φ). Finally, he above equaions are simplified by using he consrain relaion V = ω GO = lω(cosφ, sinφ). 3. Ficiious Domain Formulaion wih Disribued Lagrange Mulipliers. To obain his formulaion we fill he rigid bodies wih he surrounding fluid, and compensae he above sep by inroducing an aniparicle of mass Mρ f /ρ B and ineria Iρ f /ρ B. Finally we impose he rigid body moion on B() via a Lagrange muliplier λ suppored by B(). We obain, hen, a flow problem over he enire region, for which he

3 SIMULATION OF THE MOTION OF PENDULA 87 global variaional formulaion is: For >, find u() (H ()) 2, p() L 2 (), ω() R, φ() R, λ() Λ() =(H (B())) 2 such ha [ ] u ρ f +(u )u vdx p vdx + µ f u : vdx+ ( ρ f )(Ml 2 + I) dω θ λ, v θ Ox Λ() = (3.) ρ B d ρ f g v dx ( ρ f )Mg lsinφθ, v (H ()) 2, θ R, ρ B q u()dx =, q L 2 (), (3.2) µ, u() ω() Ox Λ() =, µ Λ(), (3.3) dφ = ω, d (3.4) ω() = ω, φ() = φ, (3.5) u(x, ) = u (x) in, wih u (x) =ω Ox inb(), (3.6) A naural choice for <, > is defined by < µ, v >= (µ v + δ 2 µ : v)dx, µ, v Λ(), (3.7) B() wih δ a characerisic lengh (he diameer of B, for example). 4. The Finie Elemen Discreizaion. We assume ha is a polygonal domain in R 2.Leh(= h )beaspace discreizaion sep, T h a finie elemen riangulaion of, and P s he space of polynomials in wo variables of degree s. The funcional spaces (H ()) 2 for velociy, and L 2 () for pressure, are approximaed by he following finie dimensional spaces V h = {v h (C ()) 2 : v h T P 2 P 2, T T h }, (4.) L 2 h = {q h C () : q h T P, T T h }, (4.2) respecively. The space (H ()) 2 is hen approximaed by V h = {v h V h : v h = on Γ}. This is he Taylor-Hood finie elemen approximaion [3]. For he discreizaion of he Lagrange mulipliers λ(), we can approximae he funcional spaces Λ() =(H (B())) 2 by a finie elemen on a grid defined on he rigid body B(). However we prefer he following alernaive ha is easier o implemen: le {x i} N i= be a se of poins from B() whichcover B(). We define Λ h () ={µ h : µ h = N µ iδ(x x i= i), µ i IR 2, i =,..., N}, (4.3) where δ( ) is he Dirac measure a x =. Then, insead of he scalar produc of (H (B h ())) 2 we use <, > h defined by < µ h, v h > h = N µ i v h (x i= i), µ h Λ h (), v h V h. (4.4) This approach makes lile sense for he coninuous problem, bu is meaningful for he discree problem; i amouns o forcing he rigid body moion of B() via a collocaion mehod. A similar echnique has been used o enforce Dirichle boundary condiions by Berrand e al. [].

4 88 JUAREZ, GLOWINSKI 5. Time Discreizaion by Operaor Spliing. Afer space discreizaion of (3.) (3.6) by he finie elemen mehod, we obain an iniial value problem of he form dϕ d + 4 Ai(ϕ, ) =f, ϕ() = ϕ, (5.) i= where he operaors A i can be mulivalued, and are associaed o each of he following numerical difficulies: (i) he incompressibiliy condiion and he relaed unknown pressure, (ii) an advecion erm, (iii) a diffusion erm, (iv) he rigid body moion of he B h () andhe relaed mulipliers λ h (). The following fracional sep mehod à la Marchuk-Yanenko [] is used o solve his problem: Given ϕ = ϕ,forn, compue ϕ n+ from ϕ n via ϕ n+i/4 ϕ n+(i )/4 + A i(ϕ n+i/4, (n +) ) =f n+ i, i =,..., 4, (5.2) wih 4 f n+ i = f n+,and aime discreizaion sep. An applicaion of his scheme i= o he finie elemen formulaion of (3.) (3.6) resuls in he following equaions: Given u = u h, φ, ω, B,forn, knowing u n, φ n, ω n, B n, compue u n+/4 V h,and p n+/4 L 2 h via he soluion of u n+/4 u n ρ f vdx p n+/4 vdx =, v V h, (5.3) q u n+/4 dx =, q L 2 h. Compue u n+2/4 = u( n+ ) V h,whereu() is he discree soluion of he following pure advecion problem on ( n, n+ ) u vdx + (u n+/4 )u vdx =, v V h, (5.4) u( n )=u n+/4. Nex, find u n+3/4 V h by solving he diffusion problem u n+3/4 u n+2/4 ρ f vdx + µ f u n+3/4 : vdx = ρ f g vdx, v V h. (5.5) Now, predic he posiion and velociy of he rigid body by solving dω d = Mglsinφ (M l 2 + I), and dφ = ω, (5.6) d on n << n+,wihφ( n )=φ n,andω( n )=ω n. Then se φ n+3/4 = φ( n+ ),and ω n+3/4 = ω( n+ ). Finally, we enforce he rigid body moion in he region B( n+3/4 ) by solving for u n+, λ n+,andω n+ he following equaion u n+ u n+ 3 4 ρ f vdx +( ρ f )(M l 2 + I) ωn+ ω n+ 3 4 θ = ρ B < λ n+, v θ Ox n+ 34 (5.7) > v V h, θ IR, < µ j, u n+ ω n+ Ox n+ 34 >=, µ j Λ n+ 3 4 h. Problems (5.3) and (5.7) are finie dimensional linear saddle-poin problems which are solved by an Uzawa/conjugae gradien algorihm [6]. The pure advecion problem (5.4) is solved by he wave-like equaion mehod discussed in Dean e al. [4] and [5]. Problem (5.5) is a discree ellipic sysem whose ieraive or direc soluion is a quie classical problem. In his work all he linear sysems are solved by a sparse marix algorihm based on Markowiz mehod []

5 SIMULATION OF THE MOTION OF PENDULA Numerical Experimens and Conclusions. We consider a wo-dimensional recangular domain = ( 3, 3) (, ) filled wih a viscous fluid of densiy ρ f =. The axis of roaion of he pendula is fixed a O =(, ), and he diameer of he circular rigid bodies is.25 in all cases bellow. As a es case we consider one pendulum wih a circular rigid body of densiy ρ B =3 released from res a φ =.4 radians in a liquid of viscosiy µ f =.5. We solved his problem using wo meshes: an unsrucured mesh (Fig. 6.) which akes advanage ha we know in advance he possible rajecory of he rigid body, and a uniform mesh wih space discreizaion sep h = /64. Figure 6.2 shows he comparison of he ime hisory of he angle and of he angular velociy obained wih he wo meshes. The agreemen is saisfacory. As expeced, he pendulum exhibis damped oscillaions around he verical posiion, and i goes o a seady posiion as ime increases. The maximum Reynolds number obained (based on he maximum falling velociy and diameer of he circular rigid body) was 835. Since he unsrucured mesh has much less velociy degrees of freedom han he regular mesh (3823 versus 49665), we used he unsrucured mesh in he subsequen calculaions. As a second example we consider wo pendula. One pendulum wih a circular rigid body of densiy ρ =. is iniially hold in he verical posiion φ =, and he oher pendulum wih densiy ρ 2 = 5 is released from res a φ 2 =.4radians in a liquid of viscosiy µ f =.5. Figure 6.3 shows ha, afer a shor ime, he heavier cylinder collides wih he ligher fixed body. Afer collision he wo bodies move ogeher as a single body all he ime. This is more eviden in in Figure 6.4 where he ime hisory of he angle, angular velociy, and separaion disance is shown. The maximum Reynolds number in his case was,85. We expeced he wo bodies o separae afer hey reach he maximum negaive angle since he heavier rigid body is below o he ligher one a ha posiion, and he acion of graviy is sronger on he heavier body. However hey never separae afer collision. The only forces in our model problem ha can preven separaion afer collision are he viscous forces which in his case seem o dominae. To corroborae his srong dependence from viscous effecs, we reduced µ f from.5 o. and repeaed he numerical calculaion. Figure 6.5 shows ha his ime, afer he wo bodies collide, hey sick ogeher unil hey reach he maximum negaive angle (where he angular velociy is close o zero), and hen separae when hey sar o move in he counerclockwise direcion by he acion of graviy. This is clearly shown in Figure 6.6 where we plo he ime hisory of angle, angular velociy, and separaion disance. The maximum Reynolds number his ime was 5,8. I is eviden ha a more deailed sudy of his and relaed phenomena is needed in order o beer undersand he mechanics of paricle collision in viscous liquids and o generae models ha simulae more accuraely solid-liquid pariculae flows which are very imporan in applicaions. REFERENCES [] F. Berand, P. A. Tanguy, and F. Thibaul. A hree-dimensional ficiious domain mehod for incompressible fluid flow problems. In. J. Num. Meh. Fluids, 997. [2] R. E. Cox and H. Brenner. The slow moion of a sphere hrough a viscous fluid owards a plane surface II. Small gap widhs including inerial effecs. Chem. Engng. Sci., 22:753, 967. [3] R. A. Davis, J. M. Serayssol, and E. J. Hinch. The elasohydrodynamic collision of wo spheres. J. Fluid Mech., 63: , 986. [4] E. J. Dean and R. Glowinski. A wave equaion approach o he numerical soluion of he Navier-Sokes equaions for incompressible viscous flow. C.R. Acad. Sc. Paris, 325(Serie ):783 79, 997. [5] E. J. Dean, R. Glowinski, and T. W. Pan. A wave equaion approach o he numerical simulaion of incompressible viscous flows modeled by he Navier-Sokes equaions. In J. A. D. Sano, edior, Mahemaical and Numerical Aspecs of Wave Propagaion, pages 65 74, Philadelphia, PA, 998. SIAM.

6 9 JUAREZ, GLOWINSKI [6] R. Glowinski and P. Le Tallec. Augmened Lagrangian and operaor spliing mehods in nonlinear mechanics. In T. F. Chan, R. Glowinski, J. Périaux, and O. B. Widlund, ediors, Proc. 2nd Inernaional Symposium on Domain Decomposiion Mehods, Philadelphia, 989. SIAM. [7] R. Glowinski, T.-W. Pan, T. I. Hesla, D. D. Joseph, and J. Periaux. A ficiious domain approach o he direc numerical simulaion of incompressible viscous flow pas moving rigid bodies: Applicaion o pariculae flow. J. Compu. Phys., 69: , 2. [8] H. H. Hu, D. D. Joseph, and M. Croche. Direc simulaion of fluid paricle moions. Theore. Compu. Fluid Dynamics, 3:285 36, 992. [9] G. Joseph, R. Zeni, M. Hun, and A. Rosenwinkel. Paricle-wall collision in a viscous fluid. J. Fluid Mech., 433: , 2. [] G. I. Marchuk. Handbook of Numerical Analysis, Spliing and alernaing direcion mehods, volume I. Elsevier Science Publishers B.V., Norh-Holland, Amserdam, 99. [] S. Pissanezky. Sparse Marix Technology. Academic Press, 984. [2] Y. Roux, E. Rivoalen, and B. Marichal. Vibraions induies d un pendule hydrodynamique. C. R. Acad. Sci. Paris, 328(Série II b): , 2. [3] C. Taylor and P. Hood. A numerical soluion of he Navier-Sokes equaions using he finie elemen mehod. Compu. & Fluids, :73, 973. Figure 6.: The unsrucured mesh φ.5 5 ω Figure 6.2: Comparison of he ime hisory of he angle (lef) and of he angular velociy (righ) obained wih he unsrucured mesh (dashed line) and he regular mesh (coninuous line) for one pendulum

7 SIMULATION OF THE MOTION OF PENDULA = = = =.3 Figure 6.3: Velociy vecor field and pressure a differen imes for he wo pendula wih µ f =.5, ρ =., and ρ 2 =5. φ ω dis Figure 6.4: Time hisory of he angle (op lef), angular velociy (op righ), and separaion disance (boom) of he wo pendula wih µ f =.5, ρ =., and ρ 2 =5.

8 92 JUAREZ, GLOWINSKI = = = =.25 Figure 6.5: Velociy vecor field and pressure a differen imes for he wo pendula wih µ f =., ρ =., and ρ 2 =5. φ ω dis Figure 6.6: Time hisory of he angle (op lef), angular velociy (op righ), and separaion disance (boom) of he wo pendula wih µ f =., ρ =., and ρ 2 =5.