EXAMPLE 4 PLANAR DIE SWELL FLOW OF A NEWTONIAN LIQUID
In this example, we simulate the planar extrudate swell (slit die) for a Newtonian liquid. In
Fig. 1, we display the finite element mesh and the flow boundary conditions. As five different types of boundary conditions are involved, we define five boundary sets. Such a simulation problem is characterized by an unknown boundary (the free surface of the jet), the shape of which is part of the problem. In this example, non-dimensional values are used.
2D extrusion, Newtonian fluid, remeshing technique : spines.
swell.msh, swell.dat, swell.cons, swell.lst, res, flum, flur
Together with the free surface boundary condition, we introduce the concept of fixed and moving subdomains. Indeed, as the shape of a boundary set is deformed, the finite element mesh in the neighbourhood of that boundary region is also deformed. As it is not necessary to remesh the whole domain, but the region adjacent to the free surface only, two subdomains are defined. The first one corresponds to the so-called fixed domain, whereas the second is called the moving domain. It is obvious that all free surfaces (in this example, there is only one free surface) must belong to the boundary of the moving domain. Let's note the i-th subdomain 'Si' (or 'SDi') and the j-th boundary set 'BSj'.
outflow (boundary 4) (b)
plane of symmetry (boundary 5)
(a) rigid wall (boundary 2)
free surface (boundary 3) I : fixed subdomain II : mobile subdomain
inflow (boundary 1)
Fig. 1. a) Finite element mesh, b) Flow geometry and boundary conditions. The initial mesh is included in the box x=[0, 1], y=[0, 8].
- Read a mesh : swell.msh - Create a new task : 2D planar, steady-state. - Create a sub-task: Isothermal Generalized Newtonian - Domain: whole mesh (S1+S2) - Material data Constant viscosity : fac = 1 Poise No density, no inertia and no gravity - Flow boundary conditions BS1 : inflow : Volumetric flow rate Q = 1 (automatic) BS2 : vn = 0, vs = 0 BS3 : free surface see note 1 June 2003 4.2 Version 3.10.0
Boundary conditions Free surface starts at intersection with BS2 BS4 : fn = 0, fs = 0 see note 2 BS5 : fs = 0, vn = 0 - Remeshing see note 3 Domain : S2 Method of spines. Inlet : intersection with S1 Outlet : intersection with BS4 - Assign stream function. PSI = 0 at the node closest to coordinates (1, 0) - Outputs Fluent Post Probe (optional) probe 1 : prefix : swell_1 location : (1, 8) probe 2 : prefix : swell_2 location : (0, 0) - Save and Exit Mesh file : swell.msh Data file : swell.dat Result file : res Fluent Output files : flum, flur Note 1 : free surface along boundary 3 The free surface is a streamline, along which we integrate the kinematic equation. That equation, when no surface tension is specified, is hyperbolic, and requires an initial condition at the starting point of the free surface. The boundary set 3 is the free surface, along which fluid particles move from point (a) to point (b), according to Fig. 1b. The initial point of the free surface (point a) is fixed. This point of the free surface is located at the common point between current boundary set 3 and boundary set 2. Note 2 : fn & fs imposed along boundary 4 We impose a zero traction along the exit. Note that in the case of a Newtonian fluid, an “outflow” condition could also be used along the exit, as “outflow” means zero tangential velocity and zero normal traction. However for a viscoelastic model, the “outflow” condition can no longer be used as, in that case,“outflow” means a fully developed profile as explained in Example III. Note 3 : Remeshing with the method of spines The flow boundary conditions involve a free surface, the shape of which is unknown. The corresponding boundary side is deformed while the finite element mesh should be deformed accordingly, in order to ensure that the elements maintain a proper shape.
In 2D extrusion, the method of spines is the default option. In many situations, it is also the most efficient method. This technique is applied along a set of slices, in the flow direction. The fluid enters the remeshing domain at its intersection with subdomain 1 : this line is the inlet of the system of spines. The fluid leaves the remeshing domain along boundary set 4 : this line is the outlet of the system of spines.
The two input files for POLYFLOW are SWELL.MSH and SWELL.DAT. The latter is taken as the standard input data file for POLYFLOW. As standard output file, the listing, we select the name SWELL.LST. POLYFLOW also generates a result file RES for a possible restart, together with files for graphic post-processing.
In Fig. 2 a-c, we display the deformed mesh, the streamlines, and the contour lines of the vertical velocity component. a) b) c)
Fig. 2. Die swell of a Newtonian liquid : a) Deformed mesh, b) Streamlines (init. val. = 0, incr. = 0.1, fin. val. = 1.), c) Contour lines of the vertical velocity component (init. val. = 0, incr. = 0.15, fin. val. = 1.5).