1. 08/05/2007 Cardiff, End of Year Workshop 1 Spectral Elements Method for free surface and viscoelastic flows Giancarlo Russo, supervised by Prof. Tim Phillips

2. 208/05/2007Cardiff, End of Year Workshop Outline The code in details The code in details - The discrete problem and the operators involved - Time and Spatial discretization’s techniques - Preconditioners Used - Dealing with the geometry: meshes and boundary conditions - Upwinding Techniques: SUPG and LUST - Results for the flow past a cylinder, Oldroyd-B model Matrix-Logarithm approach and free surface’s tracking - The log-conformation representation - First results from the channel flow - Free surface tracking for die swell and filament stretching. Models’ Overview: features and drawbacks Models’ Overview: features and drawbacks

3. 308/05/2007Cardiff, End of Year Workshop The Problem in in on on Domain change and set of admissible functions for the free surface

4. 408/05/2007Cardiff, End of Year Workshop New Variables New Differential Operators New Equations New Operators The modified problem Compatibility conditions for existence and uniqueness of a solution

5. 508/05/2007Cardiff, End of Year Workshop Key steps for conditions (1) and (2) to be satisfied The proof of condition (1) in the case of the usual divergence lies on the application that the operator is an isomorphism, and the whole point is the application of the divergence theorem. To prove (2), which involves the gradient, we have to apply the Poincare’ inequality instead. What we need then for these conditions to hold for our operators defined in (3) and (4) is for G, as defined in (5), to be bounded. Using the hypotesis on the domain and the free surface, we can bound the following quantities:

6. 608/05/2007Cardiff, End of Year Workshop So far we have proved that, picked a free surface map, the corresponding problem (with that surface) has got a unique solution; now we have to prove that such a map always exists, which means that the Cauchy problem (C1)-(C2) has a unique solution. Applying the Schauder’s fixed point theorem we have to prove that the following operator Existence and uniqueness of a solution for the free boundary problem This operator has to be: 1) CONTINOUS 2)COMPACT 3)A CONTRACTION 1) CONTINUITY: consider the following sequence, supposing it converges strongly in the let’s say. There will be a corresponding sequence u N satisfying the intial (weak) problem, and which will have the following properties: From the definition of the operator E, and taking the limit of the sequence of weak problems, we can deduce that namely E is continuous.

7. 708/05/2007Cardiff, End of Year Workshop 2)COMPACTNESS : the operator E is the composition of a continuous function and a compact embedding, therefore is compact. More precisely: Existence and uniqueness of a solution for the free boundary problem II 3)CONTRACTION : We have to prove that the operator E is a contraction, namely: If we write and combine with (OpE) we obtain Finally, expanding the continuity equation and since G is invertible, we write We remark that the y-component of the velocity field eventually vanishes when we approach the total relaxation stress configuration.

8. 808/05/2007Cardiff, End of Year Workshop The problem for the Oldroyd-B model Different Models for different viscoelastic fluids Model Fluids described FeaturesDrawbacksViscosityOldroyd-B Boger fluids; Polymer Solutions (molecules are far enough from each other so that their interactions can be neglected) Analytical Solutions Available for both transient and steady flow (Channel); direct derivation from F-P eq.; 1 st NSDiff predicted Infinite extensibility of dumbells allowed (unphysical); no shear thinning predicted; no 2 nd NSD Shear : constant over a wide range of shear rates; Ext.: possibly unbounded PTT (Phan Thien-Tanner) Polymer melts (molecules are far enough from each other so that their interactions can be neglected) Extension of dumbells bounded; shear thinning predicted; 1 st NSDiff predicted. No 2 nd NSD predicted Shear thinning, Extensional bounded. Pom Pom Polymer Melts (molecules are not so dispersed anymore) Vortex prediction in contraction flows due to branches effect; shear thinning and strain hardening predicted. Derivatives of ext. viscosity discontinuous; no 2 nd NSD. Shear thinning, constant plateau without Blackwell’s mod.; Tension Thickening at low strain rates. XPP (eXtended Pom Pom) Polymer Melts (molecules are not so dispersed anymore) Derivatives of ext. viscosity are smooth; 2 nd NSD also predicted by Giesekus term. Ext. unbounded and shear approaching constant plateau if ETA is switched off; possible unbounded stress Shear thinning. Tension Thickening at low strain rates.

9. 908/05/2007Cardiff, End of Year Workshop Setting up the spectral approximation: the weak formulation for the Oldroyd-B model We look forsuch that for all the following equations are satisfied : where f includes the UCD terms and b, c, d, and l are defined as follows : Remark:it simply means the velocity fields has to be chosen according to the boundary conditions. (5) (6)

10. 1008/05/2007Cardiff, End of Year Workshop The 1-D discretization process (note: all the results are obtained for N=5 and an error tolerance of 10°-05 in the CG routine) The spectral (Lagrange) basis : The spectral (Lagrange) basis : Approximating the solution: replacing velocity, pressure and stress by the following expansions and the integral by a Gaussian quadrature on the Gauss- Lobatto-Legendre nodes, namely the roots of L’(x), (5) becomes a linear system:Approximating the solution: replacing velocity, pressure and stress by the following expansions and the integral by a Gaussian quadrature on the Gauss- Lobatto-Legendre nodes, namely the roots of L’(x), (5) becomes a linear system: (5)

11. 1108/05/2007Cardiff, End of Year Workshop The 2-D discretization process I The 2-D spectral (tensorial) expansion : The 2-D spectral (tensorial) expansion :

12. 1208/05/2007Cardiff, End of Year Workshop The Discrete Problem The operators involved (discrete in a weak sense) Equation (3) is the first to be solved; then (1) and (2) are solved simultaneously, updating velocity and pressure. N and n+1 they respectively means discretized spatially or in time -D is the divergence operator acting on the velocity field; - D transposed is the gradient acting on the pressure; - B is the divergence operator acting on the stress; - B transposed is the gradient acting on the velocity; - C is the mass matrix for the velocity; - E is the stiffness matrix for the velocity; - g and h take into account all the RHS terms; - S is the non symmetric operator arising from the LHS of the constitutive equation.

13. 1308/05/2007Cardiff, End of Year Workshop How it is solved (uncoupled and explicitly) (I) In the system in the previous slide N and n+1 mean spatial and time discretization, since we know the spatial is done by SEM, let’s talk about the time discretization, which is performed through a fully explicit OIFS1 / Euler 1 scheme. Uncoupled simply means what highlighted in the previous slide, namely the constitutive equation is solved a timestep before the fields equations. OIFS 1 (Operator Integration Factor Splitting, 1 st order) is used to discretize the time derivative as follows: The convective problem is then solved used a fully 4 th order explicit RK scheme. Euler 1 : simply treats all the nonlinear terms following the EE scheme:

14. 1408/05/2007Cardiff, End of Year Workshop How it is solved (uncoupled and explicitly) (II) In order to use (1) to eliminate the velocity, we have to premultiply by the inverse of H, and then by D; so we’ll finally obtain the following equation to recover the pressure. The matrix acting on the pressure now is known as the UZAWA (U) operator. - To invert H, a Schur complement method is used to reduce the size of the problem, then a direct LU factorization is performed. - To invert U, since is symmetric, Preconditioned Conjugate Gradient methods is used. - Finally, to solve the constitutive equation where the matrix is not symmetric, BiConjugate Gradient Stabilized is used. U

15. 1508/05/2007Cardiff, End of Year Workshop The Preconditioners Used Since we have to deal with inverting 3 different operators, a combination of preconditioners is required; the application of different preconditioners suggested that the best combination is as follows: 1) No prec to invert the Helmoltz operator, but a direct LU factorization of its Schur complement; 2) A finite element matrices preconditioner for the inversion of the Uzawa operator, which looks like 1) No prec to invert the Helmoltz operator, but a direct LU factorization of its Schur complement; 2) A finite element matrices preconditioner for the inversion of the Uzawa operator, which looks like Where E and M are respectively the stiffness and the mass matrix of the FE problems (arising after a triangulation of the domain), R is a prolongation/restriction operator to match the FE/SE meshes and K is the number of the Spectral Elements. 3) A simple mass matrix is used as preconditioner for the BiCGStab. The FE preconditioner in 2) is a non overlapping version of the overlapping Schwarz FE preconditioner; the latter has been used as a preconditioner for the Helmoltz inversion, but it doesn’t perform as well as the direct LU factorization of the Schur complement does.

16. 1608/05/2007Cardiff, End of Year Workshop Meshes and Boundary Conditions The complete geometry of the problem lies in the mesh and the types of boundary conditions. The mesh is read as a TXT file which incorporates all the coordinates of the nodes (and of the element vertices, of course), and the relations between the elements’ edges and the boundary types. There are 5 types of boundary associated (as LOGICAL VARIABLES) to each of the nodes on the elements’ edges ( just 4 for the channel flow, no circular walls are present) : 1) Wall (BLUE) 2) Symmetry axis (RED) 3) Inflow (PURPLE) 4) Outflow (BROWN) 5) Circular Walls (flow past a cylinder or a sphere) (GREEN) Everywhere but in case 2) the boundary conditions are Dirichelet-like. On the symmetry axis, informations on the normal derivatives are provided (Neumann). For condition 5) to hold a control on the elements around the obstacle is performed, to provide that the nodes on the obstacle actually represent a circular segment.

17. 1708/05/2007Cardiff, End of Year Workshop Upwinding Techniques: SUPG and LUST Since the system is solved uncoupled, no particular care has to be taken about the compatibility condition between stress and velocity, which means the stress test functions can be freely picked. Upwinding is then used, choosing a modified stress test function in the form The difference between Streamlines Upwninding Petrov Galerkin (SUPG) and Local Upwinding Spectral Technique (LUST) is the computation of the factor h : SUPG: global quantity usually given by LUST: local factors h ij given by the positive roots of the equation, which comes from the Taylor’s expansion of the interpolant in around an element’s corner. Testing both on CHANNEL FLOW (8 elements, N=4, Re=1, beta = 0.111), LUST has been found slightly more reliable with respect to the maximum We number achievable:

18. 1808/05/2007Cardiff, End of Year Workshop Flow past a cylinder(1:1): Re =0. 23, We =0.43, Parabolic Inflow/Outflow, 20 Elem, N=6, beta=0.167, deltaT=10d-3, eps=10d-6 (10d-4 p); OLDROYD-B,(I, velocity)

19. 1908/05/2007Cardiff, End of Year Workshop Flow past a cylinder (1:1): Re =0. 23, We =0.43, Parabolic Inflow/Outflow, 20 Elem, N=6, beta=0.167, deltaT=10d-3, eps=10d-6 (10d-4 p); OLDROYD-B (II, stress)

20. 2008/05/2007Cardiff, End of Year Workshop Flow past a cylinder(1:1): Re =0. 23, We =0.43, Parabolic Inflow/Outflow, 20 Elem, N=6, beta=0.167, deltaT=10d-3, eps=10d-6 (10d-4 p); OLDROYD-B (III, pressure) TO BE FIXED: -Transient Flow - Drag calculation

21. 2108/05/2007Cardiff, End of Year Workshop To model the dynamics of polymer solutions the Oldroyd-B model is often used as constitutive equation: The Oldroyd-B model and the log-conformation representation A new equivalent constitutive equation is proposed by Fattal and Kupferman: Where the relative quantities are defined as follows: The main aim of this new approach is the chance of modelling flows with much higher Weissenberg number, because it looks like the oscillations due to the use of polynomials to approximate exponential behaviours are deeply reduced.

22. 2208/05/2007Cardiff, End of Year Workshop First results from the log-conformation channel flow: Re =1, beta=0.167, We =5, Parabolic Inflow/Outflow, 2 Elements, N=6

23. 2308/05/2007Cardiff, End of Year Workshop In the figure we remark the approach we are going to follow to track the free surfaces in the die swell and filament stretching problems: this is a completely “wet” approach, it means the values of the fields in blue nodes, the ones on the free surfaces, are extrapolated from the values we have in the interior nodes (the black ones) at each time step. After a certain number of timesteps we then redistribute the nodes to avoid big gaps between the free surface nodes an the neighbours. This approach has been proposed by Webster & al. in a finite differences context. Free surface problems: a complete “wet” approach

24. 2408/05/2007Cardiff, End of Year Workshop Future Work - Analyze the unsteady free surface die swell problem - Fix the code for calculating drag and transient flow; - Testing the log-conformation method for higher We and different geometries - Implementing the free surface “wet” approach in a SEM framework for the die swell and the filament stretching - Eventually join the latter with the log-conformation method for the constitutive equation

25. 2508/05/2007Cardiff, End of Year Workshop [1] PHILLIPS T.N., OWENS. R., Computational Rheology, Imperial College Press, 2002. [2] OWENS R.G., CHAUVIERE C., PHILLIPS T.N., A locally upwinded spectral technique for viscoelastic flows, Journal of Non-Newtonian Fluid Mechanics, 108:49-71, 2002. [3] GERRITSMA M.I., PHILLIPS T.N., Compatible spectral approximation, for the velocity-pressure-stress formulation of the Stokes problem, SIAM Journal of Scientific Computing, 1999, 20 (4) : 1530-1550. [4] FATTAL R.,KUPFERMAN R. Constitutive laws for the matrix logarithm of the conformation tensor, Journal of Non-Newtonian Fluid Mechanics,2004, 123: 281- 285. [5] FATTAL R.,KUPFERMAN R. Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation, Journal of Non-Newtonian Fluid Mechanics,2005, 126: 23-37, [6] HULSEN M.A.,FATTAL R.,KUPFERMAN R. Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms, Journal of Non-Newtonian Fluid Mechanics,2005, 127: 27-39. [7] VAN OS R. Spectral Element Methods for predicting the flow of polymer solutions and melts, Ph.D. thesis, The University of Wales, Aberystwyth, 2004. [8] WEBSTER M., MATALLAH H., BANAAI M.J., SUJATHA K.S., Computational predictions for viscoelastic filament stretching flows: ALE methods and free- surface techniques (CM and VOF), J. Non-Newtonian Fluid Mechanics, 137 (2006): 81-102. References