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A NUMERICAL METHOD FOR SOLVING THE OLDROYD-B MODEL FOR 3D FREE SURFACE FLOWS

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Abstract. This work presents a numerical method,for solving three-dimensional viscoelastic free surface flows governed by the Oldroyd-B constitutive equation. It is an extension to three dimensions of the technique introduced by Tom´ e et al., The
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  A NUMERICAL METHOD FOR SOLVING THE OLDROYD-B MODELFOR 3D FREE SURFACE FLOWS Murilo F. Tom´e  ∗ , Antonio Castelo ∗ , Fernando M. Federson  ∗ and Jos´e A. Cuminato  ∗∗ Grupo de Mecˆanica dos Fluidos ComputacionalDepartamento de Ciˆencias de Computac¸˜ao e Estat´ıstica - ICMC - USPAv. do Trabalhador S˜ao-Carlense, 400 S˜ao Carlos - SP, Brazil - CEP 13560-970e-mail: (murilo, fernando, castelo, jacumina)@icmc.usp.br, web page: http://www.lcad.icmc.usp. Key Words:  Oldroyd-B, Viscoelastic Flow, Finite Difference, Marker-and-Cell, Free Surface. Abstract.  This work presents a numerical method for solving three-dimensional viscoelastic free surface flows governed by the Oldroyd-B constitutive equation. It is an extension to threedimensions of the technique introduced by Tom´ e et al. 1 The governing equations are solved bya finite difference method on a 3D-staggered grid. Marker particles are employed to describethe fluid providing the visualization and the location of the fluid free surface. As currently im- plemented, the numerical method presented in this work can simulate three-dimensional freesurface flows of an Oldroyd-B fluid. The numerical technique presented in this paper is val-idated by using an exact solution of the flow of an Oldroyd-B fluid inside a pipe. Numericalsimulation of the extrudated swell is given. Mec´ anica Computacional   Vol. XXIIIG.Buscaglia, E.Dari, O.Zamonsky (Eds.) Bariloche, Argentina, November 2004 1  1 INTRODUCTION The numerical treatment of free surface flows is an area that has attracted the attention of manyresearchers over the last two decades and still presents several challenges: the flow is transient,non-isothermal, non-Newtonian and possess multiple free surfaces. Nonetheless, a number of researchers have developednumerical methods capable of simulatingfree surface flows that canbe applied to the design and manufacture of many industrial processes. Among the numericaltechniques employed the finite difference method has been employed by various researchers. 2–5 The development of numerical methods for simulating viscoelastic flows has been in area of intense research. However, the majority of papers published can only cope with confined flowssuchas thenumerical simulationoftheflow ina abruptcontraction (seeYoo and Na, 3 Mompeanand Deville, 6 Xue et al., 7 Pinho, 8 Phillips and Williams 9 ) which has been investigated in 2 and3 dimensions. Numerical methods for viscoelastic flows with a free surface have also beeninvestigated. 10–13 However, due to the complexity of these flows, only problems having a smallfree surface deformation or steady state problems are usually treated. Recently, Tom´e et al. 1 developeda numerical method for solving time-dependent viscoelasticfree surface flows. Moreprecisely, Tom´e et al. 1 presented a numerical technique for simulating two-dimensional freesurface flows of a fluid described by the Oldroyd-B constitutiveequation. Numerical results forthe extrudate swell and the jet buckling for high Weissenberg numbers were obtained. In thiswork we use the ideas presented by Tom´e et al. 1 and develop a numerical method for solvingthe governing equations for the three-dimensional flow of an Oldroyd-B fluid. The techniqueemploys the finite difference method on a staggered grid and the fluid free surface is modelledby the Marker-and-Cell method. The numerical method presented in this paper is validated bysimulating the flow of an Oldroyd-B fluid in pipe and compared with its analytic solution. 2 GOVERNING EQUATIONS The governing equations of incompressible flows are the mass conservation equation and theequation of motion which can be written as ∂v i ∂x i = 0  ,  (1) ρDv i Dt  = − ∂p∂x i +  ∂σ ik ∂x k +  ρg i  ,  (2)where  t  is the time,  v i  = ( u,v,w ) T  is the velocity vector,  x i  = ( x,y,z  ) T  is the position vector,  p  is the pressure,  σ ik  is the extra-stress tensor,  ρ  is the fluid density and  g i  = ( g x ,g y ,g z ) T  is thegravity field. In this work the fluid is described by the Oldroyd-B model so that we employ thefollowing constitutive equation for  σ ik σ ik  +  λ 1 ▽ σ ik = 2 µ 0  d ik + ▽ d ik   ,  (3) M. Tom´e, A. Castelo, F. Federson, J. Cuminato2  where  d ik  is the rate-of-deformation tensor d ik  = 12  ∂v i ∂x k +  ∂v k ∂x i   ,  (4) µ 0  is the fluid viscosity,  λ 1 ,λ 2  are time constants defining the Oldroyd-B model and  ( ▽ • )  repre-sents the upper convected derivative defined by ▽ σ ik =  ∂σ ik ∂t  +  ∂v m σ ik ∂x m −  ∂v i ∂x m σ mk −  ∂v k ∂x m σ im  .  (5)To solve equations (1)-(3) we employ the change of variables (known as EVSS method 14 ) σ ik  = 2 µ 0  λ 2 λ 1  d ik  +  S  ik  (6)where  S  ik  represents the non-Newtonian contribution to the extra-stress tensor. Introducing (6)into equations (2) and (3) we obtain the followingequations (we employed the non-dimensionalform where  v i  =  U  ¯ v i , p  = ( ρU  2 )¯  p,S  ik  = ( ρU  2 )¯ S  ik ,t  = ( L/U  )¯ t,x i  =  L ¯ x i , the bars have beendropped for clarity) ρDv i Dt  = − ∂p∂x i + 1 Re  λ 2 λ 1   ∂ ∂x k  ∂v i ∂x k  +  ∂S  ik ∂x k +  ρg i  (7)and S  ik  +  We ▽ S  ik = 2 Re  1 −  λ 2 λ 1  d ik  .  (8)where  Re  =  ρULµ  ,  Fr  =  U  √  Lg  and  We  =  λ 1 U L  are the Reynolds, Froude and Weissenberg num-bers, respectively. We consider three-dimensional flows. Thus, the mass equation (1) togetherwith the momentum equation (7) and the Oldroyd-B constitutive equation (8), consist of a sys-tem of partial differential equations with 10 equations for the unknowns  u,v,w,p,S  xx ,S  xy , S  xz ,S  yy ,S  yz ,S  zz . By usingCartesiancoordinates, themassconservationequation(1)becomes ∂u∂x  +  ∂v∂y  +  ∂w∂z  = 0  ,  (9) while the  x -component of equations (7) and (8) can be written as ∂u∂t  +  ∂  ( u 2 ) ∂x  +  ∂  ( vu ) ∂y  +  ∂  ( wu ) ∂z  =  − ∂p∂x  + 1 Re  λ 2 λ 1  ∂  2 u∂x 2  +  ∂  2 u∂y 2  +  ∂  2 u∂z 2  + ∂S  xx ∂x  +  ∂S  xy ∂y  +  ∂S  xz ∂z  + 1 Fr 2 g x  ,  (10) S  xx +  We  ∂S  xx ∂t  +  ∂  ( uS  xx ) ∂x  +  ∂  ( vS  xx ) ∂y  +  ∂  ( wS  xx ) ∂z  − 2  ∂u∂xS  xx +  ∂u∂yS  xy +  ∂u∂zS  xz   =2 Re  1 −  λ 2 λ 1   ∂u∂x ,  (11)respectively. Similarly, the other components of (7) and (8) can be easily obtained. M. Tom´e, A. Castelo, F. Federson, J. Cuminato3  2.1 Boundary Conditions In order to solve equations (7)-(8) it is necessary to imposeboundary conditions for the velocityfield on mesh boundaries. For rigid boundaries we employ the no-slip condition  u  =  0  whileat fluid entrances (inflows) the normal velocity is specified by  u n  =  U  inf   and the tangential ve-locities are set to zero, namely,  u m 1  =  u m 2  = 0 , where  m 1  and  m 2  denote tangential directionsto the inflow. At fluid exits (outflows) the Neumann condition  ∂  u ∂n  = 0  is adopted. We considera viscous fluid flowing in a passive atmosphere so that if we neglect surface tension forces thenon the free surface the correct boundary condition is given by (see Batchelor, 15 page 153) n i · ( π ij · n  j ) = 0  ,  (12) m 1 i · ( π ij · n  j ) = 0  ,  (13) m 2 i · ( π ij · n  j ) = 0  ,  (14)where  n i  is the outward unit normal vector to the free surface and  m 1 i ,m 2 i  are unit tangentialvectors and  π ij  is the stress tensor  π ij  = −  pδ  ij  +  σ ij . 2.2 Computation of the non-Newtonian tensor  S  ik  on mesh boundaries When solving equations (8) we shall apply a high order upwind method to approximate theconvective terms. This will require the values of the non-Newtonian stress tensor on the meshboundaries. Following the ideas presented by Tome et al., 1 the values of   S  ik  are obtained asfollows. Computation of the non-Newtonian tensor  S  ik  on inflow boundaries:  On these type of boundaries the components of the non-Newtonian tensor  S  ik  are set to zero, namely,  S  ik  =0 ,i,k  = 1 , 2 , 3 . Computation of the non-Newtonian tensor  S  ik  on outflow boundaries:  Here we employthe homogeneous Neumann condition for  S  ik :  ∂S  ik ∂n  = 0 ,i,k  = 1 , 2 , 3 , where  n  represents thenormal direction to the outflow. Computation of the non-Newtonian tensor  S  ik  on solid boundaries:  As rigid boundariesmay be regarded as caracteristics the components of the non-Newtonian tensor  S  ik  can be com-puted as follows.First, by introducing the change of variables  ¯ S  ik  =  e − ( t/We )  S  ik  into (8) it reduces to ▽  S  ik = 2 Re  1 −  λ 2 λ 1  e ( t/We ) d ik  .  (15)If we consider rigid boundaries which are parallel to one of the coordinate axis then in 3 dimen-sions the rigid boundaries can be represented by 6 planes. These planes are easily identified asthe faces of the unit cubic. For instance, considering the plane shown in figure 1 we can seethat there are 2 planes corresponding to the  z  -axis, one has the normal vector pointing to thepositive z  -direction and the other is pointing to the negative z  -direction. The computation of thenon-Newtonian tensor  S  ik  on these planes can be easily calculated. For example, if we consider M. Tom´e, A. Castelo, F. Federson, J. Cuminato4  Figure 1: Rigid boundary parallel to the  xy -plane. the rigid boundary represented by the  xy -plane shown in figure 1, then the no-slip conditionapplied to the velocity field produces  ∂ ∂x  =  ∂ ∂y  = 0  and using the mass conservation equationimplies that  ∂w∂z  = 0 . Thus, only the terms  ∂u∂z  and  ∂v∂z  are non-zero. In this case, equation (15)reduce to the following equations ∂   S  xx ∂t  = 2 ∂u∂z   S  xz ;  ∂   S  yy ∂t  = 2 ∂v∂z   S  yz ;  ∂   S  zz ∂t  = 0 ;  (16) ∂   S  xy ∂t  =  ∂v∂z   S  xz +  ∂u∂z   S  yz ;  (17) ∂   S  xz ∂t  =  ∂u∂z   S  zz + 1 Re 1 Wee ( t/We )  1 −  λ 2 λ 1   ∂u∂z   ;  (18) ∂   S  yz ∂t  =  ∂v∂z   S  zz + 1 Re 1 Wee ( t/We )  1 −  λ 2 λ 1   ∂v∂z  .  (19)If we assume the initial condition  S  ik  = 0  then following the ideas of Tome et al., 1 equations(16)-(19) can be solved for the components of   S  ik  and are found to be S  xx ( t  +  δt ) =  e − ( δt/We ) S  xx ( t ) +  δt  S  xz ( t  +  δt ) ∂u∂z  ( t  +  δt ) +  e − ( δt/We ) S  xz ( t ) ∂u∂z  ( t )   , (20) S  xy ( t  +  δt ) =  e − ( δt/We ) S  xy ( t ) +  δt 2  S  xz ( t  +  δt ) ∂v∂z  ( t  +  δt ) +  S  yz ( t  +  δt ) ∂u∂z  ( t  +  δt )+  S  xz ( t ) ∂v∂z  ( t ) +  S  yz ( t ) ∂u∂z  ( t )  e − ( δt/We )   ,  (21) S  xz ( t  +  δt ) =  e − ( δt/We ) S  xz ( t ) + 1 Re  1 −  λ 2 λ 1   ∂u∂z  ( t ∗ )  1 − e ( − δt/We )   ,  (22) S  yy ( t  +  δt ) =  e − ( δt/We ) S  yy ( t ) +  δt  S  yz ( t  +  δt ) ∂v∂z  ( t  +  δt ) +  e − ( δt/We ) S  yz ( t ) ∂v∂z  ( t )   , (23) S  yz ( t  +  δt ) =  e − ( δt/We ) S  yz ( t ) + 1 Re  1 −  λ 2 λ 1   ∂v∂z  ( t ∗∗ )  1 − e − ( δt/We )   .  (24) M. Tom´e, A. Castelo, F. Federson, J. Cuminato5
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