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The Modified LS-STAG Method Application for Planar Viscoelastic Flow Computation in a 4:1 Contraction Channel

Authors: Marchevsky I.K., Puzikova V.V. Published: 23.06.2021
Published in issue: #3(96)/2021  
DOI: 10.18698/1812-3368-2021-3-46-63

 
Category: Mathematics and Mechanics | Chapter: Computational Mathematics  
Keywords: immersed boundary method, the LS-STAG method, rate type viscoelastic flow, contraction channel, convective derivative, incompressible flow

In this study we present the modification of the LS-STAG immersed boundary cut-cell method. This modification is designed for viscoelastic fluids. Linear and quasilinear viscoelastic fluid models of a rate type are considered. The obtained numerical method is implemented in the LS-STAG software package developed by the author. This software is created for viscous incompressible flows simulation both by the LS-STAG method and by it developed modifications. Besides of this, the software package is designed to compute extra-stresses for viscoelastic Maxwell, Jeffreys, upper-convected Maxwell, Maxwell-A, Oldroyd-B, Oldroyd-A, Johnson --- Segalman fluids on the LS-STAG mesh. The construction of convective derivatives discrete analogues is described for Oldroyd, Cotter --- Rivlin, Jaumann --- Zaremba --- Noll derivatives. The centers of base LS-STAG mesh cells are the locations for shear non-Newtonian stresses computation. The corners of these cells are the positions for normal non-Newtonian stresses computation. The first order predictor--corrector scheme is the basis for time-stepping numerical algorithm. Benchmark solutions for the planar flow of Oldroyd-B fluid in a 4:1 contraction channel are presented. A critical value of Weissenberg number is defined. Computational results are in good agreement with the data known in the literature

This work was supported by the Russian Science Foundation (RSF project no. 17-79-20445)

References

[1] Owens R.G., Phillips T.N. Computational rheology. Imperial College Press, 2002.

[2] Galdi G.P., Robertson A.M., Rannacher R., et al. Hemodynamical flows: modeling, analysis and simulation. Oberwolfach Seminars, vol. 37. Birkhauser Basel, Springer, 2008. DOI: https://doi.org/10.1007/978-3-7643-7806-6

[3] Kim J.M., Kim C., Kim J.H., et al. High-resolution finite element simulation of 4:1 planar contraction flow of viscoelastic fluid. J. Nonnewton. Fluid Mech., 2005, no. 129, iss. 1, pp. 23--37. DOI: https://doi.org/10.1016/j.jnnfm.2005.04.007

[4] Mittal R., Iaccarino G. Immersed boundary methods. Annu. Rev. Fluid Mech., 2005, vol. 37, pp. 239--261. DOI: https://doi.org/10.1146/annurev.fluid.37.061903.175743

[5] Cheny Y., Botella O. The LS-STAG method: a new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties. J. Comput. Phys., 2010, vol. 229, iss. 4, pp. 1043--1076. DOI: https://doi.org/10.1016/j.jcp.2009.10.007

[6] Osher S., Fedkiw R.P. Level set methods and dynamic implicit surfaces. Applied Mathematical Sciences, vol. 153. New York, Springer, 2002. DOI: https://doi.org/10.1007/b98879

[7] Marchevsky I.K., Puzikova V.V. Numerical simulation of wind turbine rotors autorotation by using the modified LS-STAG immersed boundary method. Int. J. Rotating Mach., 2017, vol. 2017, art. 6418108. DOI: https://doi.org/10.1155/2017/6418108

[8] Puzikova V.V. The LS-STAG immersed boundary method modification for viscoelastic flow computations. Proceedings of ISP RAS, 2017, vol. 29, no. 1, pp. 71--84 (in Russ.). DOI: https://doi.org/10.15514/ISPRAS-2017-29(1)-5

[9] Botella O., Cheny Y., Nikfarjam F., et al. Application of the LS-STAG immersed boundary/cut-cell method to viscoelastic flow computations. Commun. Comput. Phys., 2016, vol. 20, iss. 4, pp. 870--901. DOI: https://doi.org/10.4208/cicp.080615.010216a

[10] Wilkinson W.L. Non-Newtonian fluids. Pergamon Press, 1960.

[11] Maxwell J.C. On the dynamical theory of gases. Philos. Trans. R. Soc. Lond., 1867, vol. 157, pp. 49--88. DOI: https://doi.org/10.1098/rstl.1867.0004

[12] Jeffreys H. The Earth. Its origin, history and physical constitution. Cambridge Univ. Press, 1929.

[13] Oldroyd J.G. On the formulation of rheological equations of state. Proc. Roy. Soc. London, 1950, vol. 200, iss. 2063, pp. 523--541. DOI: https://doi.org/10.1098/rspa.1950.0035

[14] Johnson M.W. Jr., Segalman D. A model for viscoelastic fluid behavior which allows non-affine deformation. J. Nonnewton. Fluid Mech., 1977, vol. 2, iss. 3, pp. 255--270. DOI: https://doi.org/10.1016/0377-0257(77)80003-7

[15] Roache P.J. Computational fluid dynamics. Hermosa, 1998.

[16] Li X., Han X., Wang X. Numerical modeling of viscoelastic flows using equal low-order finite elements. Comput. Methods Appl. Mech. Eng., 2010, vol. 199, iss. 9--12, pp. 570--581. DOI: https://doi.org/10.1016/j.cma.2009.10.010