A semi-staggered dilation-free finite-volume method for the numerical solution of viscoelastic fluid flows on all-hexahedral elements Mehmet SAHIN & Helen J. WILSON Department of Mathematics, University College London Gower Street, London, WC1E 6BT, UK 29 April 2006 Crete/GREECE
Why semi-staggered? • The semi-staggered grid arrangement has better pressure coupling compare to the non-staggered (collocated) approach while being capable of handling non- cartesian grids. • It eliminates the need for a pressure boundary condition since it is defined at interior points. • The summation of the continuity equation within each element can be exactly reduced to the domain boundary, which is important for the global mass conservation. • The most appealing feature of the method is leading to very simple algorithm consistent with the boundary and initial conditions required by the Navier-Stokes equations. • Recent works on semi-staggered methods: Rida et al. (1997), Kobayashi et al. (1999), Wright and Smith (2001, 2003), etc.
Why all-hexahedral mesh? • First, hexahedral meshes perform quite well when local mesh is strongly unisotropic such as boundary layers. • Second, they are more accurate and efficient for a fixed number of grid points. • In addition, when cell-centered discretization is used other elements cause the number of unknows increase significantly for a same number of mesh points which is case for the pressure term in our semi-staggered discretization. • All-hexahedral mesh generation for a general geometry is still difficult task compare to other elements such as tetrahedral, pyramid, etc. However, recently several all-hexahedral commercial grid generation programs become available (HEXPRESS, KUBRIX, etc.)
Why dilation-free? • The problem with non-conservative approaches is that high pressure gradients result in high values of the fourth-order pressure derivatives and thus a significant mass source is generated (such as at high blockage ratios, etc.). • Unequal-order approximation is dilation-free but it leads to very coarse mesh for the pressure term and it may not be very accurate. • Most of the block preconditioners for saddle point problems are designed for dilation free methods. These preconditioners lead to more robust solution techniques compare to SIMPLE, SIMPLER, etc. type decoupled solution techniques.
Mathematical and Numerical Formulation for Navier-Stokes Equations The incompressible Navier-Stokes equations may be written in dimensionless form as: and the continuity equation Integrating the differential equations over an arbitrary irregular control volume gives
Numerical Discretization The discrete contribution from cell c 1 to cell c 2 for the momentum equation is given The gradient of velocity components are calculated from the Green’s Theorem. The discretization of above equations leads to a saddle point problem of the form:
Mathematical and Numerical Formulation for Viscoelastic Fluid Flows The governing equations of incompressible Oldroyd-B fluid can be written in dimensionless form as follows: Integrating the differential equations over an arbitrary irregular control volume gives
Numerical Discretization (Continued…) Any component of the extra stress tensor can be extrapolated to control volume boundaries using a Taylor series expansion The neighbouring cell center values may be expressed as This overdetermined system may be solved in a least square sense. The discretization of the above equations leads to
Parallelization and Efficiency Mesh Generation (GAMBIT, CUBIT, ...) Mesh Partition (METIS library) Parallel Unstructured Finite Volume Code Linear Solver with Block Preconditioners Kroylov subspace methods (PETSc library) Preconditioning (HYPRE library) Post Processing
A 3D unstructured mesh with 184008 hexahedral elements (R/H=0.5 and W/H=2.5)
Computed Txx contours at We=0.7 for an Oldroyd-B fluid past a confined cylinder
Computed Txx contours at We=0.7 for an Oldroyd-B fluid past a confined cylinder
Computed isobaric surfaces at We=0.7 for an Oldroyd-B fluid past a confined cylinder
Emerge of corner vortices at We=2.0 for an Oldroyd- B fluid past a confined cylinder (z=+4.99R) Corner vortex (Shiang et al. 2000)
Comparison of u-velocity between 2D and 3D results at x= 4R, ±2R and 0 at z=0 at We=0.7 2D 3D u_max 2.98 2.56 Txx_max 107.76 81.26 P( 10R) 55.30 49.42
Mesh Adaptation Mesh adaptation can be classified in general into three 1. r-refinement 2. h-refinement 3. p-refinement The mesh size control variable is chosen to be von Mises stress And the mesh size is computed from the following equation CUBIT mesh generator developed at Sandia National Laboratories is used to produce all-quadrilateral elements for a given size function.
An adaptive mesh for an Oldroyd-B fluid past a confined circular cylinder at We=0.7 Node # = 90292 Element #= 88870 Txx contours
An impulsively started flow for an Oldroyd-B fluid past a confined circular cylinder at We=0.7
Linear Stability (Normal Mode) Analysis We consider the 3D perturbations to the 2D base flow Where k is the wave number and ω is the growth rate in time. Discretizing the dimensionless perturbation equations leads to an algebraic system of equations Arnoldi method is used to compute the leading eigenvalues.
Normal mode analysis: Newtonian lid-driven cavity flow at Re=200.
Normal mode analysis: Oldroyd-B fluid past periodic array of cylinders at We=1.50. u-velocity Txx Leading eigenfunction (Txx) Exp. results of Liu We_crit=1.53 k_crit=3.1 Num. results of Smith et al. We_crit=1.475 k_crit=3.125
Normal mode analysis: Oldroyd-B fluid past a confined cylinder at We=0.7
Conclusions A new adaptive unstructured semi-staggered finite volume method is presented for viscoelastic fluid flow calculations with exact mass conservation within each element. A normal mode analysis code has been developed to investigate flow instabilities for both Newtonian and Oldroyd-B fluids. The use of highly efficient MUMPS, PETSc and HYPRE parallel libraries allows us to solve very large systems of algebraic equations on parallel machines. The accuracy of the proposed method is verified for the given test cases. Oldroyd-B model is capable of capturing corner vortices and merging of streamtraces as in the experimental works of Shiang et al. and McKinley. Oldroyd-B model can also capture the critical Weissenberg number for periodic array of cylinders in a channel. More work is needed for the efficient solution of Stokes problem in 3D.
CUBIT (Geometry and Mesh Generation Toolkit) http://sass1693.sandia.gov/cubit/ MUMPS (MUltifrontal Massively Parallel Sparse Direct Solver) http://graal.ens-lyon.fr/MUMPS/ PETSc (Portable, Extensible Toolkit for Scientific Computing) http://www-unix.mcs.anl.gov/petsc/petsc-2/ HYPRE (High Performance Preconditioners) http://www.llnl.gov/CASC/linear_solvers/ http://www.homepages.ucl.ac.uk/~ucahmsa/
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