Numerical simulation of viscoelastic free‐surface flows using a streamfunction/log‐conformation formulation and the volume‐of‐fluid method

Abstract

Denne afhandling præsenterer en ny numerisk algoritme der kan anvendes til a simulere hvordan multifase viskoelastiske materialer flyder i to dimensioner. Simuleringer af viskolelastiske materialers flydeevne har både en videnskabelig relevans og en praktisk betydning for processer der involvere polymere. Den viskoelastiske opførelse giver anledning til dynamisk ustabilitet specielt i ekstrudering. De populære konstitutive modeller for fluider bestående af polymere er opsummeret. De numeriske problemstillinger der relaterer sig til det høje Weissenberg nummer problem er diskuteret og løsningen er fundet ved hjælp af log‐tilpasningsformuleringen. Den foreslåede numeriske algoritme anvender også den trykløse strømfunktion formulering til at finde strømningsprofilerne. Denne formulering er mere stabil og præcis end de klassiske metoder der adskiller trykog hastighedsberegningen. Den nye strømfunktion/log‐tilpasningsalgoritme giver sammen med en selvjusterende dæmpningsfaktor og tidskridt en robust og effektiv viskoelastisk løser. To faser/den frie overflade er modelleret med Volumen‐Af‐Fluid metoden og en ny cellevis konservativ ikke splittet advektions rutine, der bevarer væskens volumen andel og er mere præcis end andre nye procedurer. Simuleringerne af viskoelastiske sammenligningstests er i god overensstemmelse med tidligere udgivet resultater. Forbedringen af den numeriske stabilitet vha. strømfunktionen/log‐tilpasningsalgoritmen gør det muligt at simulere elastisk ustabilitet. Dette er den fysiske mekanisme der opløser den lagrede elastiske energi.This thesis presents a new numerical algorithm for the simulation of two‐dimensional multiphase viscoelastic flows. The simulation of viscoelastic flows has both a scientific importance and practical implications in polymer processing. This work has put the emphasis on the extrusion of polymeric materials, where viscoelastic effects cause dynamical instabilities, despite the very simple geometry. This thesis reviews the popular differential constitutive models derived from molecular theories of dilute polymer solutions, polymer networks, and entangled polymer melts, as well as the inelastic phenomenological models describing shear–thinning and viscoplastic (yield stress) fluids, based on the generalized Newtonian fluid model. In addition, the numerical issues related to the high Weissenberg number problem, and its remedy with the log–conformation representation, are discussed. The proposed algorithm utilizes a new streamfunction/log–conformation scheme. The drawbacks of the classical velocity–pressure decoupled method, which is by far the most popular approach, are remedied with the pure streamfunction formulation, which is derived from the pressureless vorticity‐based methods. The implicit pure streamfunction formulation is formally more accurate than the velocity–pressure decoupled method, because it is immune of decoupling errors. Moreover, the absence of decoupling enhances the stability of the calculation. The governing equations (conservation laws and constitutive models) are discretized with the finite– volume method, on a Cartesian grid. Discrete curl operators are applied to the discretized momentum equations in order to obtain the matrix system of the discrete streamfunction variables. The coupling of the streamfunction/log–conformation scheme with adaptive under‐relaxation and adaptive time‐stepping yield a robust and efficient viscoelastic flow solver algorithm. The potential extension of the method to threedimensional simulations is also discussed in this thesis. Bi‐phasic/free‐surface flows are modelled with the Volume–of–Fluid (VOF) method, and the standard piecewise–linear–interface–construction technique. In addition, a new Cellwise Conservative Unsplit (CCU) advection scheme is presented. The CCU scheme updates the liquid volume fractions based on cellwise backward‐tracking of the liquid volumes. The algorithm calculates non‐overlapping and conforming adjacent donating regions, which ensures the boundedness and conservativeness of the liquid volume. As a result, the CCU advection scheme is overall more accurate in classical benchmark tests, than the other state‐of‐the‐art multidimensional VOF–advection schemes. In complex flows, the convergence rate of the CCU scheme with mesh refinements is between 2 and 3. Moreover, the remaining geometrical errors are mostly due to the inability of the standard piecewise linear interface to represent subgrid material topologies (i.e. high curvatures and thin material filaments), rather than the proposed CCU advection scheme. This thesis reports examples of numerical simulations of the Oldroyd–B liquid, calculated with the proposed streamfunction/log–conformation/VOF–CCU methodology, implemented in Matlab. A thorough investigation of the viscoelastic flow in the lid‐driven cavity is conducted. The streamfunction/log–conformation shows secondorder accuracy and numerical stability at very large time‐step increments, which demonstrates the robustness of vi the scheme. The numerical results at moderate Weissenberg numbers are in good agreement with the literature. Moreover, the enhancement of numerical stability, with the streamfunction/log–conformation scheme, makes it possible to simulate elastic instabilities at high Weissenberg numbers. Quasi‐periodic elastic instabilities at the upstream corner appear to be a mechanism that dissipates the stored elastic energy. The simulations of viscoelastic flows in the planar 4:1 contraction are also in good agreement with data in the literature. Finally, preliminary simulations of extrudate swelling show that the fracture melt extrusion defect could be caused by instabilities in the stress layer at the surface of the die, triggered at moderate Weissenberg numbers