Enhancing numerical stability in simulation of viscoelastic fluid flows at high weissenberg number problem

Authors

1 MSc. Student, Department Of Mechanical. Engineering., University Of Tehran, Tehran, Iran.

2 Asst. Prof., Department Of Mechanical. Engineering., University Of Tehran, Tehran, Iran.

Abstract

Now days, simulation of viscoelastic flows at high Weissenberg numbers is one of the most obstacles and important issues for rheologists to observe the rheological properties at sufficiently high weissenberg number. It is well known that the conformation tensor should, in principle, remain symmetric positive definite (SPD) as it evolves in time. In fact, this property is crucial for the well-posedness of its evolution equation . In practice this property is violated in many numerical simulations. Most likely, this is caused by the accumulation of spatial discretization errors that arises from numerical integration of the governing equations. In this research we apply a mathematical transformation, the so-called hyperbolic tangent, on the conformation tensor to bound the eigenvalues and prevent the generation of negative spurious eigenvalues during simulations . The flow of FENE-P fluid through a 2D channel is selected as the test case. Discrete solutions are obtained by spectral/hp element methods which based on the high orders polynomials and have high accuracy for physical instability problems. This enhanced formulation, hyperbolic tangent, prevails the previous numerical failure by bounding the magnitude of eigenvalues in a manner that positive definite is always satisfied. Under this new transformation, the maximum accessible Weissenberg number increases 100% comparing the classical constitutive equation(FENE-P classic).

Keywords

Main Subjects

References

[1] Hulsen MA (1990) A sufficient condition              for a positive definite configuration tensor in differential models. J Non-Newton Fluid 38(1): 93-100.
[2] Kwon Y, Leonov AI (1995) Stability constraints in the formulation of vis coelastic constitutive equations. J Non-Newton Fluid 58(1): 25-46.
[3] Dupret F, Marchal JM () Loss of evolution in the flow of viscoelastic fluids. J Non-Newton Fluid 20(C): 143-171.
[4] Owens RG, Phillips TN (2002) Computational rheology. Imperial College Press, London.
[5] Fattal R, Kupferman R (2004) Constitutive laws for the   matrix-logarithm of the conformation tensor. J Non-Newton Fluid 123: 281-285.
[6] Fattal R, Kupferman R (2005) Time-          dependent simulation of viscoelastic flow at high Weiss-enberg number using the log-     conformation representation. J Non-Newton Fluid 126: 23-37.
[7] Hulsen MA, Fattal R, Kupferman R (2005) Flow of viscoelastic fluid past a cylinder at high Weissenberg number: stabilized simulation using matrixlogarithms. J Comput Phys 127: 27-39.
[8] Kwon Y (2004) Finite element analysis of planar 4:1 contraction flow with the tensor-logarithmic formulation of differential constitutive equations. Korea-Aust Rheol J 4: 183-191.
[9] Leonov AI (1995) Viscoelastic constitutive equations and Rheology for high speed polymer processing. Polym Int 36: 187-193.
[10] Vaithianathan T, Robert A, Brasseur JG, Collins LR (2006) An improved algorithm for simulating three-dimensional, viscoelastic turbulence. J Non-Newton Fluid 140(1-3): 3-22.
[11] Coronado OM, Arora D, Behr M, Pasquali M (2007) A simple method for simulating general viscoelastic fluid flow with an alternate log conformation formulation. J Non-Newton Fluid 147: 189-199.
[12] Jafari A, Fiétier N, Deville MO (2010) A           new extended matrix logarithm formulation          for the simulation of viscoelastic fluids by     spectral elements. Comput Fluids 39(9): 1425-1438.
[13] Tome M, Castelo A, Afonso A, Alves M, Pinho    F (2012) Application of the log-conformation tensor to three-dimensional time-dependent        free surface flows. J Non-Newton Fluid 175176: 44-54.
[14] Saramito P (2014) On a modified non-singular log-conformation formulation for Johnson- segalman viscoelastic fluids. J Non-Newton Fluid 211: 16-30.
[15] Comminal R, Spangenberg J, Hattel JH (2015) Robust simulations of viscoelastic flows at high weissenberg numbers with the stream function/log-conformation formulation. J Non-Newton Fluid 223: 37-61.
[16] پارسایی م، دهقان س، جعفری آ، ایزدپناه ا (2018) حل عددی جریان چندلایه هسته-حلقه دو سیال با لزجت­های متفاوت به روش المان طیفی. مجله مکانیک سازه­ها و شاره­ها 260-249 :(4)8.
 [17]  Jafari A, Chitsaz A, Nouri R, Timothy NP (2018) Property preserving reformulation of constitutive laws for the conformation tensor. Theor Comp Fluid Dyn 32(6): 789-803.
[18] Cai W, Gottlieb D, Harten A (1990) Cell averaging Chebyshev methods for hyperbolic problems. Report No. 90-72, ICASE.
[19] McDonald BE (1989) Flux-corrected pseudo spectral method for scalar hyperbolic conservation laws. J Comput Phys 82(2): 413-428
[20] SJ Sherwin (1995) Triangular and tetrahedral spectral/hp element methods for fluid dynamics. PhD thesis, Princeton University.
[21] Bolis A (2013) fourier spectral/hp element method: Investigation of time-stepping and parallelisation strategies. PhD Thesis, Imperial College London.
[22] Karniadakis GE, Sherwin SJ (1999) Spectral/hp element methods for CFD, numerical mathematics and scientific computation. Oxford University Press, New York.
[23] Gear CW (1971) Numerical initial value problems in ordinary differential equations. Prentice Hall PTR.
[24] Cruz DOA, Pinho FT, Oliveira PJ (2005) Analytical solutions for fully developed laminar flow of some viscoelastic liquids with a Newtonian solvent contribution. J Non-Newton Fluid 132(1-3): 28-35.
Journal of Solid and Fluid Mechanics
Volume 9, Issue 3
October 2019
Pages 217-230

Files

Share

How to cite

Statistics