Free (open) boundary condition: some experiences with viscous flow simulations

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2012; 68: Published online 6 June 2011 in Wiley Online Library (wileyonlinelibrary.com) Free (open) boundary condition: some experiences with viscous flow simulations Evan Mitsoulis 1, *, and Nikolaos A. Malamataris 2,3 1 School of Mining Engineering & Metallurgy, National Technical University of Athens, ografou, , Athens, Greece 2 Department of Mechanical Engineering, TEI of Western Macedonia, GR-50100, Kila-Kozani, Greece 3 Department of Computational and Data Sciences, George Mason University, 22030, Fairfax, VA, USA SUMMARY The free (or open) boundary condition (FBC, OBC) was proposed by Papanastasiou et al. (A new outflow boundary condition, International Journal for Numerical Methods in Fluids, 1992; 14: ) to handle truncated domains with synthetic boundaries where the outflow conditions are unknown. In the present work, implementation of the FBC has been tested in several benchmark problems of viscous flow in fluid mechanics. The FEM is used to provide numerical results for both cases of planar and axisymmetric domains under laminar, isothermal or non-isothermal, steady-state conditions, for Newtonian fluids. The effects of inertia, gravity, compressibility, pressure dependence of the viscosity, slip at the wall, and surface tension are all considered individually in the extrudate-swell benchmark problem for a wide range of the relevant parameters. The present results extend previous ones regarding the applicability of the FBC and show cases where the FBC is inappropriate, namely in the extrudate-swell problem with gravity or surface-tension effects. Particular emphasis has been given to the pressure at the outflow, which is the most sensitive quantity of the computations. In all cases where FBC is appropriate, excellent agreement has been found in comparisons with results from very long domains. The formulation for Picard-type iterations is given in some detail, and the differences with the Newton Raphson formulation are highlighted regarding some computational aspects. Copyright 2011 John Wiley & Sons, Ltd. Received 15 February 2011; Revised 18 April 2011; Accepted 24 April 2011 KEY WORDS: free (open) boundary condition (FBC, OBC); viscous flow; Newtonian fluid; inertia; gravity; compressibility; pressure dependence of viscosity; wall slip; surface tension 1. INTRODUCTION The free (open) boundary condition (FBC, OBC) was proposed by Papanastasiou et al. [1] to handle truncated domains with synthetic boundaries where the outflow conditions are unknown. This came in response to a concerted effort some 20 years ago to solving the problem of boundary conditions at outflows, in a symposium entitled Minisymposium on Outflow Boundary Conditions for Incompressible Flow, which took place on July 14, 1991 at the University of California at Davis [2]. In the résumé of the minisymposium, the OBC proposed by Papanastasiou et al. [1] was given plaudits for impressively accurate results for the benchmark backward-facing step (BFS) problem [3] and stratified BFS (SBFS) problem [4], and the method was also implemented by others for the same problem with equally good results. The original idea by Papanastasiou et al. [1] has been implemented also for free surface flows by Malamataris [5] and Malamataris and Papanastasiou [6]. Since then, several papers have appeared *Correspondence to: Evan Mitsoulis, School of Mining Engineering & Metallurgy, National Technical University of Athens, ografou, , Athens, Greece. mitsouli@metal.ntua.gr Copyright 2011 John Wiley & Sons, Ltd.

2 1300 E. MITSOULIS AND N. A. MALAMATARIS in the literature on the subject [7 11], sometimes referring to the FBC as an open boundary condition [4] or a synthetic outflow boundary condition [1] or a no boundary condition [7, 8]. The mathematics behind its workings has also been explained [1, 7, 8]. It seems that the method has been implemented in several computational codes for fluid mechanics, and in many papers, there is a passing mentioning of its application as part of the solution, for example, [12]. However, details about its implementation in other linear-system solution schemes (like Picard iterations) and a more careful analysis of the results, especially regarding individual effects of various fluid mechanics parameters, as well as the outflow pressures, which is a sensitive quantity, appear missing from the literature. It is therefore the purpose of the present work to re-examine the FBC and test it in several benchmark problems of viscous flow in fluid mechanics, namely, the original two problems of BFS and SBFS, and the extrudate-swell problem [13, 14]. It should be noted that the fiber-spinning process, which is related to the extrudate-swell problem, has been mentioned in the work by Papanastasiou et al. [1] as a possible application for the free boundary. However, no attempt has been made so far in the literature, and this is the first effort to examine the usefulness of FBC for this type of processes. The FEM is used to provide and compare numerical results with the previous solutions [1 4]. The extrudate-swell problem is studied with the FBC for both cases of planar and axisymmetric domains under laminar, isothermal or non-isothermal, steady-state conditions for Newtonian fluids. The effects of inertia, gravity, compressibility, pressure dependence of the viscosity, slip at the wall, and surface tension are all considered individually, for long and truncated domains. The formulation is given in some detail for fixed-point iterative schemes (Picard iteration) as opposed to Newton Raphson schemes, with regard to computational aspects. 2. MATHEMATICAL MODELING 2.1. Governing equations Benchmark problems for FBC. In the present work, we first revisit the two benchmark problems studied by Papanastasiou et al. [1] to validate our numerical scheme, and then we embark on the free-surface extrudate-swell benchmark problem to study the implementation of the FBC for various fluid mechanics parameters with free boundaries. The flow geometries for the two previous benchmark problems are shown in Figure 1, along with the boundary conditions. In Figure 1(a), an incompressible Newtonian fluid enters the upper half of a rectilinear channel (BFS) [1, 3]. As the flow progresses, the fluid is rearranged to a parallel flow at the outflow of the long domain after some eddy formation in the interior of the channel due to inertia. In the first problem, the flow is isothermal, and the energy equation is not considered. In the second problem (SBFS; Figure 1(b)) [1 4], the flow is non-isothermal, with inertia, and under gravity acting downwards to the direction of flow; the temperature of the fluid is kept constant along the floor and along the ceiling of the channel but at different levels. In both cases, the fluid is considered flowing with an average velocity, U, from left to right. The governing equations of the SBFS problem at steady state are [1, 4] as follows: Mass conservation rnu D 0. (1) Momentum conservation (including effects of inertia, pressure, viscous, and gravity forces due to buoyancy) Nu rnu D rp C 1 Re r2 Nu 1 Fr N kt. (2) Energy conservation (including effects of convection and diffusion) Nu rnu D 1 Pe r2 T. (3)

3 FREE BOUNDARY CONDITION 1301 y E A v x =24y(0.5-y) v r =0 x v x =v y =0 H D OBC (a) 0.5 B v x =v y =0 v x =v y =0 L C y E +0.5 T=2y T u =1 D 0 A x g H OBC (b) 0.5 B dt/dx=0 T l =0 L C Figure 1. Schematic diagram of flow domain and boundary conditions for the two benchmark problems: (a) backward-facing step and (b) stratified backward-facing step. OBC, open boundary condition. Here, Nu D.u, v/ is the velocity vector of the fluid, with u and v as its components in the x- and y-directions, respectively, p is the pressure, T is the temperature, and N k is the unit vector in the direction of gravity (opposite to the y-direction). The dimensionless numbers appearing in the above equations are Re, Fr,andPe, and their definitions follow Extrudate-swell problem. Extrudate swell refers to a simple exit flow problem from a tube or a slit whereupon the fluid acquires a free surface [13]. It constitutes a benchmark problem in the field of Newtonian and non-newtonian fluid mechanics [13 15]. Many flowing materials are non-newtonian, exhibiting either pseudoplastic (shear thinning or shear thickening) or viscoplastic (presence of a yield stress) behavior or viscoelastic behavior [13], and extrudate-swell studies for these have been abundant [13]. Here, we concentrate on the Newtonian fluid model and study the effect that various fluid mechanics parameters have on the FBC. For viscous, compressible or incompressible, isothermal, steady-state flows, the conservation equations are written as follows: Mass conservation (including density changes for compressible fluids) r. Nu/ D 0. (4) Momentum conservation (including effects of inertia, pressure, viscous, and gravity forces) For compressible fluids, we have Re Nu rnu D rp CrN C St N k. (5) N D rnu CrNu T 2 3.rNu/ N I. (6) In the above, St is the Stokes number (see succeeding equations), and N I is the unit tensor. Here, we have assumed that we deal with dense fluids, which have a zero dilatational (bulk) viscosity [16].

4 1302 E. MITSOULIS AND N. A. MALAMATARIS Also, for compressible fluids, density and pressure are related via a simple thermodynamic equation of state [16]. Its linear form is given by D 1 C Bp, (7) where B is the dimensionless isothermal compressibility coefficient. Similarly, the viscosity can be a function of pressure [17]. Its exponential form is given by D exp.b p p/, (8) where B p is the dimensionless pressure-shift coefficient. Along the wall, a slip boundary condition may be occurring [18]. Its linear form is given by Nt Nu D B sl.nt Nn W N/, (9) where B sl is the dimensionless slip coefficient, and Nn and Nt are the normal and tangential unit vectors to the wall. Along the free surface, surface-tension effects may be also occurring [18, 19]. A force balance gives. N Nn/ Nn D 2R c =Ca, (10) where N D pin C N is the total stress, Ca is the capillary number, and 2R c is the mean curvature of the free surface given by [20]: 2R c D h 1 C h 2 3=2 r p. (11) 1 C h 2 In the above, the subscripts and denote first-order and second-order differentiation of the free surface location h with respect to, andr is the local radius. The parameter is an auxiliary one, being 0 for planar flows and 1 for axisymmetric flows. Thus, the second term is 0 in planar flows Dimensionless numbers The relevant dimensionless numbers of the Navier Stokes equations need a characteristic length and a characteristic velocity. In the present case, these are the channel gap, H, and the average incoming velocity, U, respectively. All lengths are made dimensionless by H and all velocities by U Benchmark problems for FBC. The pressures and stresses are scaled with U 2. The following dimensionless numbers are then defined: (1) The Reynolds number, Re, is a measure of inertia over viscous forces: Re D UH, (12) where is the density and is the viscosity. (2) The Froude number, Fr, is a measure of inertia over gravity forces: U 2 U 2 Fr D D U B ˇg. T /H, (13) where U B is the buoyancy velocity, ˇ is the volumetric expansion coefficient, g is the acceleration due to gravity, and T is the maximum temperature difference between the walls (T u T l /, with T u as the temperature of upper wall and T l as the temperature of lower wall. (3) The Peclet number, Pe, is a measure of fluid thermal convection over diffusion: Pe D c puh, (14) where c p is the heat capacity and is the thermal conductivity.

5 FREE BOUNDARY CONDITION 1303 (4) The Prandtl number, Pr, related to the Pe and Re numbers, is a measure of fluid viscous and thermal properties: Pr D Pe Re D c p. (15) The governing equations of the BFS problem (Figure 1(a)) are Equations (1) and (2), with the Froude number set to infinity in Equation (2). Hereafter, short domain refers to a domain truncated to seven units of width from the inlet (7H/, and long domain refers to a truncated domain of at least 15 width units (15H/or higher Extrudate-swell problem. The pressures and stresses are scaled with U/R, wherer is the tube radius. The density is scaled with a reference density 0 and the viscosity with a reference viscosity 0. The following dimensionless numbers are then defined as follows: (5) The Stokes number, St, is a measure of gravity over viscous forces: St D gr2 U. (16) (6) The compressibility coefficient, B, is a measure of fluid compressibility: B D ˇU R. (17) (7) The pressure-shift coefficient, B p, is a measure of pressure dependence of the fluid viscosity: B p D ˇpU R. (18) Also, the following dimensionless numbers arise because of boundary flow. (8) The slip coefficient, B sl, is a measure of fluid slip at the wall: B sl D ˇsl R. (19) (9) The capillary number, Ca, is a measure of surface tension over viscous effects: Ca D U, (20) where is the surface tension. The base case is for Re D St D B D B p D B sl D 0, Ca! Boundary conditions Benchmark problems for FBC. The boundary conditions are shown in Figure 1(a and b). In both benchmark problems at the outlet, the FBC is applied by simply evaluating the surface integrals of the momentum and energy equations in terms of the, as of yet, unknown outflow nodal values of u, v, p, andt along with the volume integrals. This is equivalent to extending the weighted momentum and energy equations to the synthetic exit, as explained by Papanastasiou et al.[1].the specification of a datum pressure p D 0 by means of the continuity equation at a unique node of the synthetic outflow was found to give incorrect results and was removed. The pressure values were correspondingly given by subtracting the pressure value at the origin (point A) Extrudate-swell problem. Figure 2 shows the solution domain and boundary conditions for the axisymmetric geometry. There is symmetry along AB, no slip along the walls DS, a fully developed Poiseuille velocity profile corresponding to a unit average velocity U along the inflow boundary DA, along the free surface SC (becoming SC 0 ), vanishing tangential and normal stresses, and no flow through the surface. In the case of slip at the wall, the tangential velocity obeys the slip law, whereas the normal velocity is zero. In the case of surface tension on the free surface SC 0,there

6 1304 E. MITSOULIS AND N. A. MALAMATARIS D u z = sl w, u r =0 S n =( 2R c Ca)n n u=0 C' C p=free u z =f(r) u r =0 A R zr=u r =0 r 0 g z h(z) zr=u r =0 h f u r =0 or OBC zz= /Cah f cl B L 1 L 2 Figure 2. Schematic diagram of flow domain and boundary conditions for the benchmark problem of extrudate swell with free surface. OBC, open boundary condition. are still vanishing tangential stress and no flow through the surface, but the normal stresses satisfy a force equilibrium, and at the outlet BC (becoming BC 0 ), there is a force balance. Previous numerical experiments [21] for setting a reference pressure as a boundary condition showed that the best solution is obtained by not specifying anywhere a reference pressure. The results for the pressure are then obtained by subtracting the pressure value at point C (becoming C 0 ) from each nodal pressure. 3. METHOD OF SOLUTION The numerical solution is obtained with the FEM, using two different programs, which employ as primary variables the two velocities, pressure, temperature, and free surface location (u v p T h formulation). The first one is based on the Newton Raphson (N-R) iterative scheme [1,5,6,19] and the second on the Picard (P) (direct substitution) scheme [22 24]. The latter (called uvpth) has been developed and used mainly for non-newtonian (viscoelastic and viscoplastic) problems, for which it has been found more suitable. In the present work, the uvpth program was modified to account for implementing the FBC. The details of the FEM formulation are given in Appendix A. The simulations for the two benchmark problems have been run with the FEM meshes shown in Figure 3. Mesh M1 consists of 960 elements and 3977 nodes. Mesh M2 is obtained from M1 by subdivision of each element into four sub-elements (3840 elements). For all the simulations, we have used both M1 and M2, and we show the influence of the mesh on the results. Mesh M3 (8320 elements and 33,705 nodes) has been used with the Newton Raphson (N-R) scheme to check the results against those of Papanastasiou et al. [1], who have used this scheme to obtain their original results, albeit with fewer elements (in the order of 1000). As shown, many elements are concentrated near the entry because of the step AB there and at the outlet to make sure that the FBC is accurately calculated. The meshes for the extrudate-swell problem are also shown in Figure 3(d and e). Similar strategies for mesh arrangement and density are followed. Mesh M4 has 1200 elements whereas M5 has 4800 elements for the short domain ( 5, C6). For the long domain ( 5, C40), the meshes are M6 with 1450 elements and M7 with 5800 elements. The criteria for termination of the iterative process were, for N-R, a norm-of-the-residuals jjejj <10 6 and, for P, both the norm-of-the-error and the norm-of-the-residuals <10 4 and for the maximum free surface change < RESULTS AND DISCUSSION 4.1. Backward-facing step problem Validation of our results for the FBC was obtained for Re D 800. The standard solution was derived by Gartling [3] by using 8000 elements and a domain length of 30H, whereas Papanastasiou et al.

7 FREE BOUNDARY CONDITION 1305 M1 (a) M2 (b) M3 (c) M4 M5 (d) (e) M6 M7 Figure 3. Finite element meshes used in the computations. (a) Mesh M1 (P), (b) mesh M2 (P), (c) mesh M3 (N-R), (d) mesh M4 (P, N-R), (e) mesh M5 (P, N-R). P, Picard scheme; N-R, Newton Raphson scheme Gartling [3] This work (N-R) This work (P) M1 This work (P) M2 y Coordinate Velocity, u x Figure 4. Axial velocity profile at the outlet of the truncated domain (L 2 D 7H ) by applying the free boundary condition to the backward-facing step problem. Results compared with those of reference [3]. Results obtained using M3 (N-R), and M1 and M2 (P). N-R, Newton Raphson scheme; P, Picard scheme.

8 1306 E. MITSOULIS AND N. A. MALAMATARIS Gartling [3] This work (N-R) This work (P) M1 This work (P) M2 y Coordinate Velocity, u y Figure 5. Transverse velocity profile at the outlet of the truncated domain (L 2 D 7H ) by applying the free boundary condition to the backward-facing step problem. Results compared with those of reference [3]. Results obtained using M3 (N-R), and M1 and M2 (P). N-R, Newton Raphson scheme; P, Picard scheme Gartling [3] This work (N-R) This work (P) M1 This work (P) M2 y Coordinate Pressure, P Figure 6. Pressure profile at the outlet of the truncated domain (L 2 D 7H ) by applying the free boundary condition to the backward-facing step problem. Results compared with those of reference [3]. Results obtained using M3 (N-R), and M1 and M2 (P). N-R, Newton Raphson scheme; P, Picard scheme. [1] used 1044 elements and a truncated domain of 7H to compare the results. The same was performed here with the meshes of Figure 3. The results are presented at the synthetic outflow boundary of 7H for the three works. In this work, we have used both the Newton Raphson (N-R), originally used in [1], and the Picard (P) scheme. The N-R scheme starts with the solution at Re D 1 and proceeds with increments of Re D 50 up to 800, using zero-order continuation. Quadratic convergence (four iterations) was attained in all computations up to Re D 800. A similar continuation strategy was used with the Picard scheme. The number of Picard iterations needed to achieve the convergence criteria (Section 3) from start.re D 0/ to finish.re D 800/ were 288. The results are shown for the primary variables u v p at the synthetic outflow boundary of 7H in Figures 4 6, respectively. Overall, the agreement is very good, both qualitatively and quantitatively. For the u x profile, the results with Picard and M1 and M2 converge towards the results

9 FREE BOUNDARY CONDITION Gartling [3] This work (N-R) This work (P) M1 This work (P) M2 outflow Pressure, P top wall 4 bottom wall 0 inflow x Coordinate Figure 7. Axial pressure distribution along the boundary of the truncated domain (L 2 D 7H ) by applying the free boundary condition to the backward-facing step problem. Results compared with those of reference [3]. Results obtained using M3 (N-R), and M1 and M2 (P). N-R, Newton Raphson scheme; P, Picard scheme y Coordinate Leone [4] Papanastasiou et al. [1] (N-R) This work (P) M Velocity, u x Figure 8. Axial velocity profile at the outlet of the truncated domain (L 2 D 7H ) by applying the free boundary condition to the stratified backward-facing step problem. Results compared with those of references [4] and [1] (open boundary condition, N-R). Results obtained using M2 (P). N-R, Newton Raphson scheme; P, Picard scheme. with N-R and M3. This is more evident for the u y profile in Figure 5, where the results with P/M2 and N-R/M3 are virtually identical despite the small values of u y compared with the u x values. Similar trends are found for the pressure in Figure 6, which is the most sensitive quantity, and which obviously depends on the mesh used and its arrangement at the outflow. Still, it is remarkable that the behavior is correctly captured by the FBC at the outflow. This is also a good test for a correct implementation of FBC in Picard schemes, because although the velocities may be about right, the pressure is not so forgiving! A further corroboration of the above is given by plotting the overall pressure distribution along the perimeter of the domain (upper and lower walls, inflow and outflow) as shown in Figure 7. All pressure results have been obtained by subtracting the pressure at the origin, point A(x, y)=(0,0). It becomes obvious that the pressures are more dependent on the mesh arrangement in the axial

10 1308 E. MITSOULIS AND N. A. MALAMATARIS Leone [4] This work (P) M2 0.2 y Coordinate Velocity, u y Figure 9. Transverse velocity profile at the outlet of the truncated domain (L 2 D 7H ) by applying the free boundary condition to the stratified backward-facing step problem. Results compared with those of reference [4]. Results obtained using M2 (P). P, Picard scheme y Coordinate Leone [4] Papanastasiou et al. [1] (N-R) This work (P) M Pressure, P Figure 10. Pressure profile at the outlet of the truncated domain (L 2 D 7H ) by applying the free boundary condition to the stratified backward-facing step problem. Results compared with those of references [4] and [1] (open boundary condition, N-R). Results obtained using M2 (P). N-R, Newton Raphson scheme; P, Picard scheme. direction and that this is perhaps the most severe test of the meshes used and the FBC. It should be noted that these results have been obtained with no pressure level set at the outlet (usually one node is set to a reference pressure of 0, but this does not work with the FBC because the pressures at the synthetic outflow are not zero) Stratified backward-facing step problem We continue with the second benchmark problem of SBFS, for which results were obtained for Re D800, Fr D16=9, andpr D1. The standard solution was derived by Leone [4] by using 38,400 elements and a domain length of 15H, whereas Papanastasiou et al. [1] used 1044 elements and a truncated domain of 7H to compare the results. The same was performed here with the meshes of Figure 3.

11 FREE BOUNDARY CONDITION y Coordinate Leone [4] Papanastasiou et al. [1] (N-R) This work (P) M Temperature, T Figure 11. Temperature profile at the outlet of the truncated domain (L 2 D 7H ) by applying the free boundary condition to the stratified backward-facing step problem. Results compared with those of references [4] and [1] (open boundary condition, N-R). Results obtained with M2 (P). N-R, Newton Raphson scheme; P, Picard scheme. 0.5 This work (P) M1 This work (P) M2 top wall 0.3 Pressure, P inflow outflow bottom wall x Coordinate Figure 12. Axial pressure distribution along the boundary of the truncated domain (L 2 D 7H ) by applying the free boundary condition to the stratified backward-facing step problem. Results obtained using M1 and M2 (P). P, Picard scheme. The results are presented at the synthetic outflow boundary of 7H for the three works. The Newton Raphson (N-R) has been used in [1] and the Picard scheme in the present work. The same strategy as before was followed. This problem was easier to solve and required a total of 167 P iterations. The results are shown for the primary variables u v p T at the synthetic outflow boundary of 7H in Figures 8 11, respectively. Again, overall the agreement is very good, both qualitatively and quantitatively. For the u x profile (Figure 8), the results with Picard and M2 are very similar with the results with N-R [1], but there are some small discrepancies with the results by Leone [4]. This is more evident for the u y profile in Figure 9, but again, the values of u y are very small compared with the values of u x. Similar trends are found for the pressure in Figure 10, where the present results with Picard and M2 are very similar with the results with N-R [1], but there are again small

12 1310 E. MITSOULIS AND N. A. MALAMATARIS centerline 1.2 Velocity, u x / U free surface 0.2 Re=100 (L =100) Re=200 (L =200) Re=100 (L =6, OBC) Re=200 (L =6, OBC) (a) Re=100 (L =100) Re=200 (L =200) Re=100 (L =6, OBC) 0.96 Re=200 (L =6, OBC) Free Surface, h / H (b) Figure 13. (a) Axial velocity profile and (b) free surface profile along the boundary of the truncated domain (L 2 D 6H ) by applying the free boundary condition to the extrudate-swell problem. The results coincide with those from the long domain (inertia flow). OBC, open boundary condition. discrepancies with the results by Leone [4]. As we pointed out above, the pressure is the most sensitive quantity, and such discrepancies are justified given the very dense mesh used by Leone [4]. The temperature distribution at the synthetic outlet shows a better agreement as evidenced in Figure 11. The overall pressure distribution along the perimeter of the domain (upper and lower walls, inflow and outflow) for this problem (not shown before) is given in Figure 12, as obtained using the Picard iteration (P) and meshes M1 and M2. Good agreement is obtained between the two meshes Extrudate-swell problem We present results for flows with the individual effects of inertia, gravity, compressibility, pressure dependence of viscosity, slip at the wall, and surface tension. The flow is from either a round tube or a two-dimensional slit. The results are given in terms of the dimensionless swell ratio,, and of the dimensionless change in pressure drop over and above the fully developed values, n ex.theseare defined as follows [13]:

13 FREE BOUNDARY CONDITION planar axisym planar (L =6, OBC) axisym (L =6, OBC) Exit Correction, n Reynolds Number, Re = UR/ Figure 14. Exit correction as a function of Re number. The results of the truncated domain (L 2 D 6H ) by applying the free boundary condition to the extrudate-swell problem coincide with those from the long domain (inertia flow). Note that results without the OBC would need a length at least equal to Re. OBC, open boundary condition. (i) The dimensionless extrudate-swell ratio,, definedby D h f R (ii) The dimensionless pressure or exit correction, n ex,definedby or D h f H. (21) n ex D P w P 0 2 w, (22) where P w is the overall pressure drop in the system calculated at the wall, P 0 is the pressure drop based on the fully developed flow in the tube (channel) without the extrudate region, and w is the shear stress for fully developed Poiseuille flow at the tube (channel) wall. Tanner [13] provided a selection of values from the literature and estimated the extrapolated values as D (planar) and D (axisymmetric) an infinite number of degrees of freedom (DOF). The converged results obtained by Georgiou and Boudouvis [25] are and The exit correction n ex is obtained from the wall pressure values given by the simulations at the entry of the domain upstream according to Equation (22). This is a very sensitive quantity and reflects the adequacy of the domain length, the imposed entry profile, and the finite element mesh used. The creeping values are (planar) and (axisymmetric) [26]. These results have been reproduced with either the long. 5, C40/ or the truncated flow domain. 5, C6/ Effect of Re. First, we show results for inertia flows and specifically for two Re numbers of 100 and 200 in the planar geometry. As pointed out in [19], the length of the extrudate domain L 2 to obtain correct results must be at least equal to Re. The imposition of the FBC at a length L 2 D 6 gives virtually identical results with those from the long domains, for all variables. As an example, Figure 13(a) shows the axial velocity distribution and Figure 13(b) shows the free surface development. Similar results are obtained for the axisymmetric case as well. A very interesting development concerns the exit correction, which can be accurately obtained even for highly elevated Re numbers (we have reached ReD10,000 with good convergence in the truncated domain), something that would be very hard to do with the full domain of 10,000R. This good agreement for the exit correction is shown in Figure 14 with the corresponding results from the long and truncated domains.

14 1312 E. MITSOULIS AND N. A. MALAMATARIS 6 centerline 5 Velocity, u x / U 4 3 free surface 2 1 planar, B=0.22 (L =40) axisym, B=0.11 (L =40) planar, B=0.22 (L =6, OBC) axisym, B=0.11 (L =6, OBC) (a) Free Surface, h / H planar, B=0.22 (L =40) axisym, B=0.11 (L =40) planar, B=0.22 (L =6, OBC) axisym, B=0.11 (L =6, OBC) (b) Figure 15. (a) Axial velocity profile and (b) free surface profile along the boundary of the truncated domain (L 2 D 6H ) by applying the free boundary condition to the extrudate-swell problem. The results coincide with those from the long domain (compressible flow). OBC, open boundary condition Effect of B. Then, the effect of compressibility is studied for the two geometries (planar and axisymmetric) at the upper limit of converged solutions, namely for B D 0.22 (planar) and B D 0.11 (axisymmetric). The imposition of the FBC at a length L 2 D 6 gives virtually identical results with those from the long domains (L 2 D 40), for all variables. As an example, Figure 15(a) shows the axial velocity distribution and Figure 15(b) shows the free surface development. All is well and the FBCisvalid Effect of B p. Then, the effect of pressure dependence of viscosity is studied for the two geometries (planar and axisymmetric) at the upper limit of converged solutions, namely for B p D 5 (planar) and B p D 2 (axisymmetric). The imposition of the FBC at a length L 2 D 6 gives virtually identical results with those from the long domains (L 2 D 40) for all variables. As an example, Figure 16(a) shows the axial velocity distribution and Figure 16(b) shows the free surface development. All is well and the FBC is valid Effect of B sl. Then, the effect of slip-at-the-wall is studied for the two geometries (planar and axisymmetric). The computations reached B sl D 100 for which a plug velocity profile is evident

15 FREE BOUNDARY CONDITION Velocity, u x / U centerline free surface planar, B =5 (L =40) axisym, B =2 (L =40) planar, B =5 (L =6, OBC) axisym, B =2 (L =6, OBC) (a) Free Surface, h / H planar, B =5 (L =40) axisym, B =2 (L =40) planar, B =5 (L =6, OBC) axisym, B =2 (L =6, OBC) (b) Figure 16. (a) Axial velocity profile and (b) free surface profile along the boundary of the truncated domain (L 2 D 6H ) by applying the free boundary condition to the extrudate-swell problem. The results coincide with those from the long domain (flow with a pressure dependence of viscosity). OBC, open boundary condition. throughout and no swelling is obtained ( D 1). Here, we present results for B sl D 0.1 for the two geometries. The imposition of the FBC at a length L 2 D 6 gives virtually identical results with those from the long domains (L 2 D 40), for all variables. As an example, Figure 17(a) shows the axial velocity distribution and Figure 17(b) shows the free surface development. All is well and the FBC is valid Effect of Ca. Then, the effect of surface tension (capillary effects) is studied for the two geometries (planar and axisymmetric). Whereas for the long domain (L 2 D 40) with no FBC present the computations reached down to Ca D 1, exhibiting always quadratic convergence with N-R, the results with the FBC (L 2 D 6) failedtoconvergeforca < 50. EvenatCa D 50, the results from the two domains did not match, as shown in Figure 18(a) for the axial velocity distribution and in Figure 18(b) for the free surface development. As a consequence, the finding here is that the FBC is not suitable for horizontal extrusion with surface-tension effects, as the integrals at the outflow interfere with the FBC and mess up the results.

16 1314 E. MITSOULIS AND N. A. MALAMATARIS planar, B =0.1 (L =40) axisym, B =0.1 (L =40) planar, B =0.1 (L =6, OBC) axisym, B =0.1 (L =6, OBC) Velocity, u x / U centerline 0.6 free surface (a) Free Surface, h / H planar, B =0.1 (L =40) axisym, B =0.1 (L =40) planar, B =0.1 (L =6, OBC) axisym, B =0.1 (L =6, OBC) (b) Figure 17. (a) Axial velocity profile and (b) free surface profile along the boundary of the truncated domain (L 2 D 6H ) by applying the free boundary condition to the extrudate-swell problem. The results coincide with those from the long domain (flow with slip at the wall). OBC, open boundary condition. This result is not surprising, because the boundary condition is known for this flow situation and it is the stress-free condition for the planar case or a force balance for the axisymmetric case (Figure 2). Thus, when the boundary condition is known at the synthetic outflow, then the concept of the FBC is not useful Effect of St. We continue our studies with gravity flows and specifically for St D 7. As pointed out in [27], the length of the extrudate domain L 2 plays an important role in gravity flows, and the results are usually independent of the exit boundary condition only in the first quarter of L 2. Gravity constantly applies an axial force on the fluid, and this drastically affects the results. Thus, the imposition of the FBC at a length L 2 D 6 does not give identical results with those from the long domains! As an example, Figure 19(a) shows the axial velocity distribution and Figure 19(b) shows the free surface development both for planar and axisymmetric geometries. Very different results are obtained in this case between the truncated and the long domains (L 2 D 100). Obviously, the same occurs for the exit correction (not shown here). Thus, a very interesting finding follows from this study, namely, that the FBC is not suitable for vertical extrusion under creeping flow conditions.

17 FREE BOUNDARY CONDITION centerline Velocity, u x / U 0.8 free surface (a) planar, Ca=50 (L =40) planar, Ca=50 (L =6, OBC) Free Surface, h / H planar, Ca=50 (L =40) planar, Ca=50 (L =6, OBC) (b) Figure 18. (a) Axial velocity profile and (b) free surface profile along the boundary of the truncated domain (L 2 D 6H ) by applying the free boundary condition to the extrudate-swell problem. The results do not coincide with those from the long domain (flow with surface-tension effects). OBC, open boundary condition. This is not surprising, as this is a situation where the flow phenomena are influenced from the events downstream the synthetic boundary. Depending on the length of the extrudate, the influence of gravity is proportional to this length, so that it is impossible for the FBC to give a correct result in truncated domains of this kind. The FBC is valid in cases where the flow phenomena are of a convective nature, like the BFS or the SBFS (or any other channel flow, like flow over an obstacle, or a cylinder, etc.), or in cases where gravity is balanced by viscous stresses, like vertical or inclined free surface flow (see Malamataris et al. [28, 29]), or in cases of vertical extrusion where the inertial forces are not negligible. All of these flow situations have the common attribute that the events upstream the synthetic outflow influence the flow phenomena downstream the outflow. The FBC has been conceived for such cases. It should be noted that there does not exist any suitable boundary condition for the vertical extrusion under creeping flow condition, because of the fact that gravity has a thinning effect in the extrudate that varies with its length. It should be also noted that the difference between the vertical and the horizontal extrusion is a term in the volume integral of the governing equation (the force of gravity). The surface integrals are the same, and although the stress-free boundary condition is valid in both flow cases, the imposition of this boundary condition is unsuitable in the vertical extrusion.

18 1316 E. MITSOULIS AND N. A. MALAMATARIS planar, St=7 (L =100) axisym, St=7 (L =100) planar, St=7 (L =6,OBC) axisym, St=7 (L =6, OBC) Velocity, u x / U centerline free surface (a) Free Surface, h / H planar, St=7 (L =100) axisym, St=7 (L =100) planar, St=7 (L =6, OBC) axisym, St=7 (L =6, OBC) 0.5 (b) Figure 19. (a) Axial velocity profile and (b) free surface profile along the boundary of the truncated domain (L 2 D 6H ) by applying the free boundary condition to the extrudate-swell problem. The results do not coincide with those from the long domain (gravity flow). OBC, open boundary condition Vertically falling-film problem The above results for the extrudate-swell problem may bring into question previous results with free surfaces where gravity and surface-tension effects were present. A typical such case is the vertically falling-film problem, where a Newtonian fluid flows initially in a slit and then exits and falls downwards on the flat wall, acquiring a free surface (Figure 20). This problem has been studied before by Malamataris and co-workers [5, 6, 28, 29] under various, mainly transient, conditions. Here, we consider a simple steady-state case where Re, St, andca effects are present, but the flow field and the free surface location are dominated by gravity. The results are given for a test case of Re D, St D 17.5, andca D 25 for a long domain ( 4, C40) and a truncated domain ( 4, C6) where the FBC is applied. The boundary conditions are given in Figure 20. The same meshes have been used as in the extrudate-swell case. The problem was solved with two different codes employing either the N-R or the P-iterative scheme with identical results. The N-R scheme was proven superior as it converged within five iterations at the conditions set. The results for the axial velocity distribution are shown in Figure 21(a) and for the free surface development in Figure 21(b). Here, all is well and the FBC is valid, as the results from

19 FREE BOUNDARY CONDITION 1317 A u z =f(r) u r =0 D L 1 H u x =u y =0 wall 0 x y S u x =u y =0 L 2 g h(y) free surface n =( 2R c Ca)n n u=0 B h f C' u y =0, xx=0 or OBC C p=free Figure 20. Schematic diagram of flow domain and boundary conditions for the vertically falling-film problem with free surface. OBC, open boundary condition. the two domains coincide. The same was true when each effect was studied individually, that is, only gravity is present or only surface tension is present. The main difference with the extrudate-swell problem is the presence of the flat wall, which acts as a guide for the falling fluid and provides a solid boundary instead of a symmetry line. Thus, the FBC works well for free-surface flows with both gravity and surface tension, provided that gravity is balanced by viscous stresses, so that the convective nature of the flow is dominant. This finding has implications in free-surface flows guided by walls underneath, like in coating flows or in falling-film flows. 5. CONCLUSIONS The FBC (OBC) has been revisited with the purpose of using it in programs not employing the Newton Raphson (N-R) iterative scheme but a direct substitution scheme (Picard iteration) and in finding possible problems where it might not be suitable. The FEM formulation for the Picard scheme is given in some detail, and its results compare favorably with the superior N-R scheme for the two standard benchmark problems for which FBC was tested, namely, the BFS and SBFS problems. For the benchmark problem of extrudate swell, a well-known free-surface problem in fluid mechanics, the FBC was found to work excellently for flows with inertia (Re), compressible flows, flows with a pressure dependence of viscosity, or flows with slip at the wall. It is not surprising that it is not suitable for vertical extrusion under creeping flow conditions, as gravity affects the flow phenomena depending on the length of the extrudate. It was also shown that the FBC is not suitable

20 1318 E. MITSOULIS AND N. A. MALAMATARIS free surface Velocity, u x / U Re=, St=17.5, Ca=25 (L =40) Re=, St=17.5, Ca=25 (L =6, OBC) (a) Re=, St=17.5, Ca=25 (L =40) Re=, St=17.5, Ca=25 (L =6, OBC) Free Surface, h / H (b) Figure 21. (a) Axial velocity profile and (b) free surface profile along the boundary of the truncated domain (L 2 D 6H ) by applying the free boundary condition to the vertically falling-film problem. The results coincide with those from the long domain (flow with inertia, gravity, and surface-tension effects in a vertically falling film). OBC, open boundary condition. for horizontal extrusion under creeping flow conditions with surface-tension effects, as the boundary condition there is already known. The study of extrudate swell reinforces once more the important issue that the FBC is primarily suitable for flows in cases where the boundary conditions are not known and the flow phenomena downstream are determined by the conditions upstream (convective flows). The present results are offered as extra reference solutions for researchers working with the numerical simulation of fluid flows. They are the prelude for a study of combined effects together with non-newtonian fluids, exhibiting viscoplasticity and mainly viscoelasticity, where very long domains are needed to fully relax the stresses [10, 11], especially for integral constitutive equations of the K-BK type [30, 31]. Such a study is currently under way by the authors. APPENDIX A: APPENDIX The FEM casts the differential equations into integral form according to the Galerkin principles [18, 19, 32, 33]. For the primitive variable formulation u v p T h, FEM approximates

21 FREE BOUNDARY CONDITION 1319 the field variables for the velocities u v, pressure p, temperature T, and free surface h, as follows: u D ' T U D T D ' T T D nx ' i U i id1 nx ' i T i id1 v D ' T V D h D ' T H D nx ' i V i id1 nx ' i H i id1 p D T P D mx id1 ip i (A.1) where U,V,P,T,andH are array columns of the nodal unknowns for each element, and ', are array rows of the basis (interpolation) functions, and the superscript T refers to the transpose of a vector. The pressure basis functions are of lower order than the other basis functions ', and interpolation for pressure is carried out over m nodes, whereas for the other variables it is carried over n nodes, with m<n. Nine-node Lagrangian isoparametric quadrilateral elements are used, and that choice fixes the basis functions.n D 9, m D 4/. A previous work [24] contains detailed derivations of the FEM formulation based on the stiffness matrix and load vector approach advocated by Huebner and Thornton [32]. This approach is better suited for Picard schemes and will be given in the succeeding equations with the appropriate modifications to incorporate the FBC. A.1. Mass and momentum discrete equations Combining the discrete forms of the conservation equations of mass and momentum (including compressibility) into one matrix equation leads (in two dimensions) to the following system of an element (stiffness) matrix, a vector of unknowns, and an RHS (load) vector for each element: S 11 S 12 L 1 U 4 S 21 S 22 L V 5 C L T 1, L T 2, S 33 P 2 4 A U/CA 2.V/ 0 0 U 0 A 1.U/CA 2.V/ V 5 D 4 F 3 (A.2) 1 F P 0 where S 11 D 2K 11 C K K K 22 compressible terms (A.3) S 22 D K 11 C 2K K K 22 compressible terms (A.4) S 12 D K K 12 compressible (A.5) S 21 D K K 21, with S 21 D S 12 (A.6) compressible

22 1320 E. MITSOULIS AND N. A. MALAMATARIS L 1, D K 21 D L K 11 D K 22 D K d (A.7) (A.8) d, with K 21 D K L 1 D L 2 S 33 @x d T d T d (A.11) (A.12) T d.d L1 for incompressible fluids/ (A.13) T d.d L2 for incompressible fluids/ (A.14).S 33 D 0 for incompressible fluids/ (A.15) F 1 D A 1.U/D A 2.V/D T x 'd T T C ˇsl tn W x 'd (A.16) (A.17) (A.18) tr surface tractions F 2 D T y 'd sl C slip ˇsl tn W y 'd. (A.19) tr surface tractions sl slip

23 FREE BOUNDARY CONDITION 1321 A.2. Contribution from the FBC With the FBC, the extra terms along the outflow boundary are F D n pi C 'd D nx. p C xx / C n y xy ' i d, i D 1, 3 (A.20) OBC open boundary condition OBC open boundary condition F q D OBC.n krt / 'd D open boundary condition k n C y ' i d, i D 1, 3. open boundary condition After the appropriate manipulations, the following matrix system is obtained: 2 SO11 S O12 S O13 4 S O21 S O22 S O23 U V P 3 5 D F O1 F O2, (A.22) where the components of the element (stiffness) matrix [S O ] of Equation (A.22) are S O11 D K 'i d, i D 1, 3 (A.23) S O12 D K 'i d, i D 1, 3 (A.24) S O13 D K O13 D 0 (A.25) S O21 D K O21 D 'i d, i D 1, 3.K O21 D 0 for incompressible fluids/ (A.26) S O22 D K O22 CK O22c 'i d 'i d, i D 1, 3.K O22c D 0 for incomp. fluids/ (A.27) S O23 D K O23 CK O23c D ' 'i d, i D 1, 3.K O23c D 0 for incomp. fluids/ (A.28) F O1 D F O2 D 0. (A.29) The above contributions of [S O ] and [F O ] must be added to the corresponding terms of Equation (A.2) for the elements having the FBC on one side.

24 1322 E. MITSOULIS AND N. A. MALAMATARIS For the contribution to the energy equation from the FBC (Equation (A.21)), the matrix equation (Equation (A.22)) has a fourth unknown, T, and the stiffness element corresponding to this is S O44 D K O44 D k n C y ' i d, i D 1, 3. It should be noted that when using the N-R iteration, Equations (A.20) and (A.21), as such, simply constitute the residuals, ¹Rº, from which the Jacobian ŒJ D Œ@R=@x is derived, and the system is solved for the vector of unknowns ¹ xº, according to ŒJ ¹ xºd ¹Rº. Thus, it is not necessary to derive stiffness matrices and load vectors, as in the above. ACKNOWLEDGEMENTS Financial support for one of the authors (E. M.) was provided by the National Technical University of Athens (NTUA), Greece, in the form of a basic-research project, code named PEBE This assistance is gratefully acknowledged. REFERENCES 1. Papanastasiou TC, Malamataris N, Ellwood K. A new outflow boundary condition. International Journal for Numerical Methods in Fluids 1992; 14: Sani RL, Gresho PM. Résumé and remarks on the open boundary condition minisymposium. International Journal for Numerical Methods in Fluids 1994; 18: Gartling DK. A test problem for outflow boundary conditions flow over a backward-facing step. International Journal for Numerical Methods in Fluids 1990; 11: Leone JM Jr. Open boundary condition symposium benchmark solution: stratified flow over a backward-facing step. International Journal for Numerical Methods in Fluids 1990; 11: Malamataris NG. Computer-aided analysis of flow on moving and unbounded domains: phase-change fronts and liquid leveling. Ph.D. Dissertation, The University of Michigan, Malamataris NT, Papanastasiou TC. Unsteady free surface flows on truncated domains. Industrial & Engineering Chemical Research 1991; 30: Griffiths DF. The no boundary condition outflow boundary condition. International Journal for Numerical Methods in Fluids 1997; 24: Renardy M. Imposing no boundary condition at outflow: why does it work? International Journal for Numerical Methods in Fluids 1997; 24: Wang MMT, Sheu TWH. Implementation of a free boundary condition to Navier Stokes equations. International Journal of Numerical Methods in Heat and Fluid Flow 1997; 7: Park SJ, Lee SJ. On the use of the open boundary condition method in the numerical simulation of nonisothermal viscoelastic flow. Journal of Non-Newtonian Fluid Mechanics 1999; 87: Sunwoo KB, Park SJ, Lee SJ, Ahn KH, Lee SJ. Numerical simulation of three-dimensional viscoelastic flow using the open boundary condition method in coextrusion process. Journal of Non-Newtonian Fluid Mechanics 2001; 99: Dimakopoulos Y, Tsamopoulos J. On the gas-penetration in straight tubes completely filled with a viscoelastic fluid. Journal of Non-Newtonian Fluid Mechanics 2004; 117: Tanner RI. Engineering Rheology (2nd edn). Oxford University Press: Oxford, Tanner RI. Die-swell reconsidered: some numerical solutions using a finite element program. Applied Polymer Symposia 1973; 20: Nickell RE, Tanner RI, Caswell B. The solution of viscous incompressible jet and free-surface flows using finite-element methods. Journal of Fluid Mechanics 1974; 65: Bird RB, Stewart WE, Lightfoot EN. Transport Phenomena. Wiley: New York, Cardinaels R, Van Puyvelde P, Moldenaers P. Evaluation and comparison of routes to obtain pressure coefficients from high-pressure capillary rheometry data. Rheologica Acta 2007; 46: Silliman WJ, Scriven LE. Separating flow near a static contact lineslip at a wall and shape of a free surface. Journal of Computational Physics 1980; 34: Georgiou GC, Papanastasiou TC, Wilkes JO. Laminar Newtonian jets at high Reynolds number and high surface tension. AIChE Journal 1988; 34: Omodei BJ. On the die-swell of an axisymmetric Newtonian jet. Computers and Fluids 1980; 8: Mitsoulis E, Georgiou GC. Extrudate swell revisited: the effect of various fluid mechanics parameters. AIChE Journal 2010; 56: Hannachi A, Mitsoulis E. Sheet coextrusion of polymer solutions and melts: comparison between simulation and experiments. Advances in Polymer Technology 1993; 12: