A finite volume approach in the simulation of viscoelastic expansion flows

Abstract A finite volume technique is presented for the numerical solution of viscoelastic flows. The flow of a differential upper-convected Maxwell (UCM) model fluid through an abrupt expansion has been chosen as a prototype example due to the existence of previous simulations in the literature. The conservation and constitutive equations are solved using the finite volume method (FVM) in a non-staggered grid with an upwind scheme for the viscoelastic stresses and a hybrid scheme for the velocities. An enhanced-in-speed pressure-correction algorithm is used and a new method for handling the source term of the momentum equations is introduced. Improved accuracy is achieved by a special discretization of the boundary conditions. Stable solutions are found for high Deborah numbers, further extending the range of previous similar simulations with the FVM. The solutions have been verified with grid refinement and show that at high elasticity levels, the domain length must be long enough to accommodate the slow relaxation of high viscoelastic stresses. The FVM is proven quite capable for numerically handling viscoelastic models with low computational cost and its use is recommended as a viable alternative to the solution of viscoelastic problems using a variety of constitutive models.

[1]  R. Tanner.  From A to (BK)Z in Constitutive Relations , 1988 .

[2]  R. A. Mashelkar,et al.  Advances in Transport Processes , 1982 .

[3]  K. Walters,et al.  Long-range memory effects in flows involving abrupt changes in geometry: Part I: flows associated with I-shaped and T-shaped geometries , 1977 .

[4]  K. Walters,et al.  Long range memory effects in flows involving abrupt changes in geometry: Part 2: the expansion/contraction/expansion problem , 1977 .

[5]  Roland Keunings,et al.  SIMULATION OF VISCOELASTIC FLUID FLOW , 1987 .

[6]  E. Mitsoulis,et al.  Numerical simulation of viscoelastic flow around a cylinder using an integral constitutive equation , 1995 .

[7]  Xiaolin Luo.  A control volume approach for integral viscoelastic models and its application to contraction flow of polymer melts , 1996 .

[8]  F. Baaijens,et al.  Viscoelastic flow past a confined cylinder of a polyisobutylene solution , 1995 .

[9]  F. W. Schmidt,et al.  USE OF A PRESSURE-WEIGHTED INTERPOLATION METHOD FOR THE SOLUTION OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON A NONSTAGGERED GRID SYSTEM , 1988 .

[10]  A. Beris,et al.  Spectral/finite-element calculations of the flow of a Maxwell fluid between eccentric rotating cylinders , 1987 .

[11]  C. Fletcher.  Computational techniques for fluid dynamics , 1992 .

[12]  L. Back,et al.  Shear-Layer Flow Regimes and Wave Instabilities and Reattachment Lengths Downstream of an Abrupt Circular Channel Expansion , 1972 .

[13]  J. P. V. Doormaal,et al.  ENHANCEMENTS OF THE SIMPLE METHOD FOR PREDICTING INCOMPRESSIBLE FLUID FLOWS , 1984 .

[14]  N. Phan-Thien,et al.  A boundary element investigation of extrudate swell , 1985 .

[15]  G. P Sasmal,et al.  A finite volume approach for calculation of viscoelastic flow through an abrupt axisymmetric contraction , 1995 .

[16]  Marcel Crochet,et al.  Plane flow of a fluid of second grade through a contraction , 1976 .

[17]  S. Majumdar.  Role of underrelaxation in momentum interpolation for calculation of flow with nonstaggered grids , 1988 .

[18]  Nhan Phan-Thien,et al.  VISCOELASTIC FLOW BETWEEN ECCENTRIC ROTATING CYLINDERS: UNSTRUCTURED CONTROL VOLUME METHOD , 1996 .

[19]  N. Phan-Thien,et al.  Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite volume method , 1995 .

[20]  R. Tanner,et al.  Numerical Simulation of Non-Newtonian Flow , 1984 .

[21]  Chia-Jung Hsu.  Numerical Heat Transfer and Fluid Flow , 1981 .

[22]  A. R. Davies,et al.  Long-range memory effects in flows involving abrupt changes in geometry Part 4: Numerical simulation using integral rheological models , 1981 .

[23]  R. Bird.  Dynamics of Polymeric Liquids , 1977 .

[24]  R. Tanner,et al.  Finite element and boundary element methods for extrusion computations , 1984 .

[25]  Daniel D. Joseph,et al.  Numerical simulation of viscoelastic flow past a cylinder , 1990 .

[26]  J. Yoo,et al.  A NUMERICAL STUDY OF THE PLANAR CONTRACTION FLOW OF A VISCOELASTIC FLUID USING THE SIMPLER ALGORITHM , 1991 .

[27]  John R. Whiteman,et al.  Numerical modelling of viscoelastic liquids using a finite-volume method , 1992 .