Development of an Implicit Physical Influence Upwind Scheme for Cell-Centered Finite Volume Method

Authors: Shidvash Vakilipour, Masoud Mohammadi, Rouzbeh Riazi, Scott Ormiston, Kimia Amiri, Sahar Barati

Abstract:

An essential component of a finite volume method (FVM) is the advection scheme that estimates values on the cell faces based on the calculated values on the nodes or cell centers. The most widely used advection schemes are upwind schemes. These schemes have been developed in FVM on different kinds of structured and unstructured grids. In this research, the physical influence scheme (PIS) is developed for a cell-centered FVM that uses an implicit coupled solver. Results are compared with the exponential differencing scheme (EDS) and the skew upwind differencing scheme (SUDS). Accuracy of these schemes is evaluated for a lid-driven cavity flow at Re = 1000, 3200, and 5000 and a backward-facing step flow at Re = 800. Simulations show considerable differences between the results of EDS scheme with benchmarks, especially for the lid-driven cavity flow at high Reynolds numbers. These differences occur due to false diffusion. Comparing SUDS and PIS schemes shows relatively close results for the backward-facing step flow and different results in lid-driven cavity flow. The poor results of SUDS in the lid-driven cavity flow can be related to its lack of sensitivity to the pressure difference between cell face and upwind points, which is critical for the prediction of such vortex dominant flows.

Keywords: Cell-centered finite volume method, physical influence scheme, exponential differencing scheme, skew upwind differencing scheme, false diffusion.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1130421

References:

[1] S. V. Patankar, Numerical Heat Transfer and Fluid Flow. Series in Computational Methods in Mechanics and Thermal Sciences. New York: McGraw-Hill, 1980.
[2] D. B. Spalding, “A novel finite difference formulation for differential expressions involving both first and second derivatives,” Int. J. Numer. Method Eng., vol. 4, no. 4, pp. 551-559, 1972.
[3] S. V. Patankar and D.B. Spalding, “A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows,” Int. J. Heat Mass Tran., vol. 15, p. 1787, 1972.
[4] G. D. Raithby, K. E. Torrance, “Upstream weighted differencing schemes and their application to elliptic problems involving fluid flow,” Comput. Fluids, vol. 8, no.12, pp. 191–206, 1974.
[5] G. D. Raithby, P. F. Galpin, J. P. Van Doormaal, “Prediction of heat and fluid flow in complex geometries using general orthogonal coordinates,” Numer. Heat Tr. A-Appl., vol. 9, no. 2, pp. 125-142, 1986.
[6] B. P. Leonard, “A stable and accurate convective modeling procedure based on quadratic upstream interpolation,” Comput. Methods Appl. M., vol. 19, no. 1, pp. 59–98, 1979.
[7] T. Han, J. A. C. Humphrey, B. E. Launder, “A comparison of hybrid and quadratic-upstream differencing in high Reynolds number elliptic flows,” Comput. Methods Appl. M., vol. 29, no. 1, pp. 81-95, 1981.
[8] Pollard, A. L. W. Siu, “The calculation of some laminar flows using various discretisation schemes,” Comput. Methods Appl. M., vol. 35, no. 3, pp. 293-313, 1982.
[9] T. Hayase, J. A. C. Humphrey, R. Greif, “A consistently formulated QUICK scheme for fast and stable convergence using finite-volume iterative calculation procedures,” J. Comput. Phys., vol. 98, No. 1, pp. 108-118, 1992.
[10] G. D. Raithby, “A critical evaluation of upstream differencing applied to problems involving fluid flow,” Comput. Methods Appl. M., vol. 9, No. 1, pp. 75-103, 1976.
[11] G. D. Raithby, “Skew upstream differencing schemes for problems involving fluid flow,” Comput. Methods Appl. M., vol. 9, no. 2, pp. 153-164, 1976.
[12] M. K. Patel, M. Cross, N. C. Markatos, “An assessment of flow oriented schemes for reducing ‘false diffusion’,” Int. J. Numer. Meth.Eng., vol. 26, no.10, pp.2279-2304, 1988.
[13] M. K. Patel, N. C. Markatos, M. Cross, “Method of reducing false-diffusion errors in convection—diffusion problems,” Appl. Math. Model., vol. 9, no. 4, pp. 302-306, 1985.
[14] J. P. Van Doormaal, A. Turan, G. D. Raithby, “Evaluation of new techniques for the calculation of internal recirculating flows,” Paper No. AIAA-87-0059, AIAA 25th Aerosp. Sci. Meet., Reno, NV, Jan. 12-15, 1987.
[15] A.A. Busnaina, X. Zheng, M.A.R. Sharif, “A modified skew upwind scheme for fluid flow and heat transfer computations,” Appl. Math. Model., vol. 15, No. 8, pp. 425-432, 1991.
[16] G. E. Schneider, M. J. Raw, “A skewed, positive influence coefficient upwinding procedure for control-volume-based finite-element convection-diffusion computation,” Numer. Heat Tr. A-Appl., vol. 9, no. 1, pp. 1-26, 1986.
[17] H. J. Saabas, B. R. Baliga, “Co-located equal-order control-volume finite-element method for multidimensional, incompressible, fluid flow—Part II: verification,” Numer. Heat Transfer, vol. 26, no. 4, pp. 409-424, 1994.
[18] Masson, C., H. J. Saabas, B. R. Baliga, “Co‐located equal‐order control‐volume finite element method for two‐dimensional axisymmetric incompressible fluid flow,” Int. J. Numer. Meth. Fl., vol. 18, no. 1, pp. 1-26, 1994.
[19] L. Dung Tran, M. Christian, S. Arezki, “A stable second‐order mass‐ weighted upwind scheme for unstructured meshes,” Int. J. Numer. Meth. Fl., vol. 51, no.7, pp.749-771, 2006.
[20] G. E. Schneider, M. J. Raw, “Control volume finite element method for heat transfer and fluid flow using collocated variables. 1. Computational procedure; 2. Application and validation,” Numer. Heat Transfer, vol. 11, no. 4, pp. 363–400, 1987.
[21] M. Darbandi, S. Vakilipour, “Developing implicit pressure‐weighted upwinding scheme to calculate steady and unsteady flows on unstructured grids,” Int. J. Numer. Meth. Fl., vol. 56, no. 2, pp. 115-141, 2008.
[22] M. Darbandi, S. Vakilipour, “Using fully implicit conservative statements to close open boundaries passing through recirculations,” Int. J. Numer. Meth. Fl., vol. 53, no.3, pp. 371-389, 2007.
[23] H. Alisadeghi, S. M. H. Karimian, “Different modelings of cell‐face velocities and their effects on the pressure–velocity coupling, accuracy and convergence of solution,” Int. J. Numer. Meth. Fl., vol. 65, no. 8, pp. 969-988, 2011.
[24] M. Rhie, W. L Chow, “Numerical study of the turbulent flow past an airfoil with trailing edge separation,” AIAA J., vol. 21, no. 11, pp. 1525–1532, 1983.
[25] S. Vakilipour, S. J. Ormiston, “A coupled pressure-based co-located finite-volume solution method for natural-convection flows,” Numer. Heat Tr. B-Fund., vol. 61, no. 2, pp. 91-115, 2012.
[26] G. D. Raithby and G. E. Schneider, “Elliptic Systems: Finite-Difference Methods II,” in Handbook of Numerical Heat Transfer, W. J. Minkowycz, E. M. Sparrow, G.E. Schneider, and R. H. Pletcher (Eds.), New York: Wiley, 1988, pp. 241-292.
[27] U. K. N. G. Ghia, K. N. Ghia, C. T. Shin, “High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method,” J. comput. Phys., vol. 48, no. 3, pp. 387-411, 1982.
[28] K. Gartling, “A test problem for outflow boundary conditions—flow over a backward‐facing step,” Int. J. Numer. Meth. Fl., vol. 11, no. 7, pp. 953-967, 1990.