Implicit Methods for Simulating Low Reynolds Number Free Surface Flows: Improvements on MAC-Type Methods
This paper is concerned with describing the main improvements introduced to the MAC (Marker-And-Cell) method for the numerical simulation of low Reynolds number free surface flows, namely: a stable implicit treatment of the pressure boundary condition for projection methods, a semi-implicit method based on the Crank–Nicolson (C–N) discretization of the momentum equations, a more accurate method for moving the massless particles representing the free surface and a viscoelastic model based on the Pom-Pom constitutive law, are discussed. Low Reynolds number free surface flows appear in a number of important industrial processes in the oil, food, cosmetic and medical industries and their simulation present a challenge for explicit MAC-type methods due to their parabolic time step constraint. The simulation of moving boundary problems presents a number of difficulties for a numerical method. For the semi-implicit (C–N) MAC method the main difficulty appears in applying the projection method to uncouple velocity and pressure, this is in addition to other difficulties of correctly imposing the boundary conditions on the free surface and the free surface representation itself.
KeywordsNavier–Stokes equations Viscoelastic fluid flows Free surface MAC scheme Implicit strategy Jet buckling Extrudate swell MAC method review Non-Newtonian fluids
The authors would like to acknowledge the financial support of FAPESP (projects nos. 2013/07375-0, 2011/09194-7, 2009/15892-9) and CNPq (projects nos. 305447/2010-6 , 473589/2013-3).
- 4.Bonito, A., Clément, P., Picasso, M.: Viscoelastic flows with complex free surfaces: numerical analysis and simulation. Glowinski, R., Xu, J. (eds.) Handbook of Numerical Analysis, Numerical Methods for Non-Newtonian Fluids vol. 16, pp. 305–369 (2011)Google Scholar
- 7.Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics, 3rd edn. Springer, New York (2000)Google Scholar
- 8.Ciarlet, P.G., Glowinsk, R., Lions, J.L.: Numerical methods for non-newtonian fluids. Handbook of Numerical Analysis, vol. 16, North-Holland, Amsterdam (2011)Google Scholar
- 14.Martins, F.P., Oishi, C.M., Sousa, F.S., Cuminato, J.A.: Numerical assessment of mass conservation on a MAC-type method for viscoelastic free surface flows. In: 6th European Congress on Computational Methods in Applied Sciences and Egineering (ECCOMAS 2012), vol. 1, pp. 6545–6562 (2012)Google Scholar
- 15.McKee, S., Tomé, M.F., Cuminato, J.A., Castelo, A., Ferreira, V.G.: Recent advances in the marker-and-cell method. Arch. Comput. Meth. Eng. 11, 107–142 (2004)Google Scholar
- 16.McKee, S., Tomé, M.F., Ferreira, V.G., Cuminato, J.A., Castelo, A., Sousa, F.S., Mangiavacchi, N.: MAC Method. Comput. Fluids 37, 907–930 (2008)Google Scholar
- 32.Yang, B., Ouyang, J., Wang, F.: Simulation of stress distribution near weld line in the viscoelastic melt mold filling process. J. Appl. Math. (2013)Google Scholar