50 Years of CFD in Engineering Sciences pp 75-118 | Cite as

# The SUPER Numerical Scheme for the Discretization of the Convection Terms in Computational Fluid Dynamics Computations

- 127 Downloads

## Abstract

The numerical simulation of fluid flow and heat/mass transfer phenomena requires the numerical solution of the Navier–Stokes and energy conservation equations coupled with the continuity equation. Numerical or false diffusion is the phenomenon of producing errors in the calculations that compromise the accuracy of the computational solution. According to Spalding, the Taylor series analysis that reveals the truncation/discretization errors of the differential equations terms should not be classified as false diffusion. Numerical diffusion, in the strict sense of definition, appears in multidimensional flows when the differencing scheme fails to account for the true direction of the flow. Numerical errors associated with false diffusion are investigated via two- and three-dimensional problems. A numerical scheme must satisfy some necessary criteria for the successful solution of the convection–diffusion formulations. The common practice of approximating the diffusion terms via the central-difference approximation is satisfactory. Attention is directed to the convection terms since their approximations induce false diffusion. The conservation equations of all the dependent variables in this study are discretized by the finite volume method. The performance of different numerical schemes (e.g. hybrid, van Leer, SUCCA and the novel SUPER version) is studied in this chapter by the numerical simulation of the transport of a scalar quantity in an inclined and tubular airflow, heat conduction in a cylindrical heat exchanger, and the water vapour condensation in an enclosed space. The numerical accuracy of the predictions obtained when using the various schemes was also studied in the classical cases of the backward-facing step and of an inclined inflow. The study focused on the transport of a contaminant concentration by means of an airflow, the diffusion of temperature in an water flow and at the solid surface of a triple tube, the mass transfer interaction between liquid droplets and humid air and on circular airflow predictions. An Eulerian one- and two-phase flow model is developed within the CFD general-purpose computer program PHOENICS, which considers the phases as interpenetrating continua. The phases may move at different velocities (slip velocity) in a manner that is dictated by the interphase friction. According to the numerical results obtained, it is concluded that the predictions improve when using SUPER in all cases of inclined flow and of humid air precipitation while they are similar to the predictions of the other schemes in the case of heat conduction in the tubular flow.

## Keywords

False diffusion Numerical dispersion Computational errors Finite volume method Discretization schemes The SUPER scheme## Notes

### Acknowledgements

The second author (D. P. Karadimou) gratefully acknowledges the financial support from the State Scholarships Foundation of Greece through the ‘‘IKY Fellowships of Excellence for Postgraduate studies in Greece-SIEMENS’’ Program.

## References

- 1.Patel, M. K., Markatos, N. C., & Cross, M. (1985). Technical note-method of reducing false diffusion errors in convection diffusion problems.
*Applied Mathematical Modelling,**9,*302–306.MathSciNetCrossRefGoogle Scholar - 2.Patel, M. K., & Markatos, N. C. (1986). An evaluation of eight discretization schemes for two-dimensional convection-diffusion equations.
*International Journal for Numerical Methods,**6,*129–154.MathSciNetCrossRefGoogle Scholar - 3.Darwish, M., & Moukalled, F. (1996). A new approach for building bounded skew-upwind schemes.
*Computer Methods in Applied Mechanics and Engineering,**129,*221–233.MathSciNetCrossRefGoogle Scholar - 4.De Vahl Davis, G., & Mallinson, G. D. (1976). An evaluation of upwind and central difference approximations by a study of recirculating flows.
*Computers & Fluids,**4*(1), 29–43.CrossRefGoogle Scholar - 5.Fromm, J. E. (1968). A method for reducing dispersion in convective difference schemes.
*Journal of Computational Physics,**3,*176–189.CrossRefGoogle Scholar - 6.Spalding, D. B. (1972). A novel finite-difference formulation for different expressions involving both first and second derivatives.
*International Journal for Numerical Methods in Engineering,**4,*551–559.CrossRefGoogle Scholar - 7.Leonard, B. P. (1979). A stable and accurate convective modelling procedure based on quadratic upstream interpolating.
*Computational Mechanics and Applied Mechanical Engineering,**4,*557–559.Google Scholar - 8.Van Leer, B. (1985). Upwind-difference methods for aerodynamics problems governed by the Euler equations.
*Lectures in Applications of Mathematics,**22,*327–336.MathSciNetGoogle Scholar - 9.Raithby, G. D. (1976). Skew upstream differencing schemes for problems involving fluid flow.
*Computer Methods in Applied Mechanics and Engineering,**9,*151–156.MathSciNetzbMATHGoogle Scholar - 10.Patel, M. K., Markatos, N. C., & Cross, M. (1988). An assessment of flow oriented schemes for reducing ‘false diffusion’.
*International Journal for Numerical Methods in Engineering,**26,*2279–2304.CrossRefGoogle Scholar - 11.Carey, C., Scanlon, T. J., & Fraser, S. M. (1993). SUCCA-an alternative scheme to reduce the effects of multidimensional false diffusion.
*Applied Mathematical Modelling,**17*(5), 263–270.CrossRefGoogle Scholar - 12.Karadimou, D. P., & Markatos, N. C. (2012). A novel flow oriented discretization scheme for reducing false diffusion in three-dimensional (3D) flows: An application in the indoor environment.
*Atmospheric Environment,**61,*327–339.CrossRefGoogle Scholar - 13.Versteeg, H. K., & Malalasekera, W. (1995).
*An introduction to computational fluid dynamics—The finite volume method*. Essex, England: Longman Group Ltd.Google Scholar - 14.Spalding, D. B. (1978). Numerical computation of multiphase flow and heat-transfer. In C. Taylor, & K. Morgan (Eds.),
*Contribution to recent advances in numerical methods in fluids*(pp. 139–167). Pineridge Press.Google Scholar - 15.Markatos, N. C. (1983). Modelling of two-phase transient flow and combustion of granular propellants.
*International Journal of Multiphase Flow,**12*(6), 913–933.CrossRefGoogle Scholar - 16.Spalding, D. B. (1981). A general-purpose computer program for multi-dimensional one or two-phase flow.
*Mathematics and Computers in Simulation, XII*, 267–276.CrossRefGoogle Scholar - 17.Huang, P. G., Launder, B. E., & Leschziner, M. A. (1985). Discretization of nonlinear convection processes: A broad-range comparison of four schemes.
*Computer Methods in Applied Mechanics and Engineering,**48,*1–24.CrossRefGoogle Scholar - 18.Scanlon, T. J., Carey, C., & Fraser, S. M. (1993). SUCCA3D—an alternative scheme to false-diffusion in 3D flows.
*Proceedings of the Institution of Mechanical Engineers, Journal of Mechanical Engineering Science,**207,*307–313.CrossRefGoogle Scholar - 19.Karadimou, D. P., & Markatos, N. C. (2018).
*Study of the numerical diffusion in computational calculations*(pp. 65–78). London, UK: Intechopen Publisher.Google Scholar - 20.Gomma, A., Halim, M. A., & Elsaid, A. M. (2016). Experimental and numerical investigations of a triple-concentric tube heat exchanger.
*Applied Thermal Engineering,**99,*1303–1315.CrossRefGoogle Scholar - 21.Benavides, A., & Van Wachem, B. (2009). Eulerian-Eulerian prediction of dilute turbulent gas particle flow in a backward facing step.
*International Journal of Heat and Fluid Flow,**30,*452–461.CrossRefGoogle Scholar - 22.Chen, F., Yu, S., & Lai, A. (2006). Modeling particle distribution and deposition in indoor environments with a new drift-flux model.
*Atmospheric Environment,**40,*357–367.CrossRefGoogle Scholar - 23.Yakhot, V., Orszag, S. A., Thangam, S., Gatski, T. B., & Speziale, C. G. (1992). Development of turbulence models for shear flows by a double expansion technique.
*Physics of Fluids A,**4*(7), 1510–1520.MathSciNetCrossRefGoogle Scholar - 24.Markatos, N. C. (1986). The mathematical modelling of turbulent flows.
*Applied Mathematical Modelling,**10,*190–220.CrossRefGoogle Scholar - 25.Argyropoulos & Markatos. (2015). Recent advances on the numerical modelling of turbulent flows.
*Applied Mathematical Modelling,**39*(2), 693–732.MathSciNetCrossRefGoogle Scholar - 26.Padfield, T.
*Conservation physics. An online textbook in serial form.*http://www.conservationphysics.org/atmcalc/atmoclc2.pdf. - 27.Wexler, A. (1983).
*ASHRAE Handbook: Thermodynamic properties of dry air, moist air and water and SI psychrometric charts*(p. 360). New York.Google Scholar - 28.Smagorinsky, J. (1963). General circulation experimental with the primitive equations.
*Monthly Weather Review,**93*(3), 99.CrossRefGoogle Scholar - 29.Karadimou, D. P., & Markatos, N. C. (2013). Two-phase transient mathematical modelling of indoor aerosol by means of a flow-oriented discretization scheme. In
*CD Proceedings of 10th HSTAM International Congress on Mechanics*, Chania, Crete, May 2013.Google Scholar - 30.El Diasty, R., Fazio, P., & Budaiwi, I. (1992). Modelling of indoor air humidity: The dynamic behavior within an enclosure.
*Energy and Buildings,**19,*61–73.CrossRefGoogle Scholar - 31.Lu, X. (2002). Modelling of heat and moisture transfer in buildings II, Applications to indoor thermal and moisture control.
*Energy and Buildings,**34,*1045–1054.CrossRefGoogle Scholar - 32.Lu, X. (2003). Estimation of indoor moisture generation rate from measurement in buildings.
*Building and Environment,**38,*665–675.CrossRefGoogle Scholar - 33.Ma, X., Li, X., Shao, X., & Jiang, X. (2013). An algorithm to predict the transient moisture distribution for wall condensation under a steady flow field.
*Building and Environment,**67,*56–68.CrossRefGoogle Scholar - 34.Mimouni, S., Foissac, A., & Lavieville, J. (2011). CFD modeling of wall steam condensation by a two-phase flow approach.
*Nuclear Engineering and Design,**241,*4445–4455.CrossRefGoogle Scholar - 35.Argyropoulos, et al. (2017). Mathematical modelling and computing simulation of toxic gas building infiltration.
*Process Safety and Environmental Protection, 111*, 687–700.CrossRefGoogle Scholar - 36.Isetti, C., Laurenti, L., & Ponticiello, A. (1988). Predicting vapour content of the indoor air and latent loads for air-conditioned environments: Effect of moisture storage capacity of the walls.
*Energy and Buildings,**12,*141–148.CrossRefGoogle Scholar - 37.Teodosiu, C., Hohota, R., Rusaouen, G., & Woloszyn, M. (2003). Numerical prediction of indoor air humidity and its effects on indoor environment.
*Building and Environment,**38,*655–664.CrossRefGoogle Scholar - 38.Stavrakakis, G. M., Karadimou, D. P., Zervas, P. L., Sarimveis, H., & Markatos, N. C. (2011). Selection of window sizes for optimizing occupational comfort and hygiene based on computational fluid dynamics and neural networks.
*Building and Environmnet,**46,*298–314.CrossRefGoogle Scholar - 39.Pu, L., Xiao, F., Li, Y., & Ma, Z. (2012). Effects of initial mist conditions on simulation accuracy of humidity distribution in an environmental chamber.
*Building and Environment,**47,*217–222.CrossRefGoogle Scholar - 40.Liu, J., Aizawa, H., & Yoshino, H. (2004). CFD prediction of surface condensation on walls and its experimental validation.
*Building and Environment,**39,*905–911.CrossRefGoogle Scholar - 41.Karabelas, S. J., & Markatos, N. C. (2008). Water vapor condensation in forced convection flow over an airfoil.
*Aerospace and Technology,**12,*150–158.CrossRefGoogle Scholar - 42.Ishii, M., & Mishima, K. (1984). Two-fluid model and hydrodynamic constitutive relations.
*Nuclear Engineering and Design*, (82), 107–126.Google Scholar - 43.Lee, S. L. (1987). Particle drag in a dilute turbulent two-phase suspension flow.
*International Journal of Multiphase Flow,**2*(13), 247–256.CrossRefGoogle Scholar - 44.Hetsroni, G. (1989). Particles-turbulence interaction.
*International Journal of Multiphase Flow,**5*(15), 735–746.CrossRefGoogle Scholar - 45.ASHRAE handbook-Fundamentals, 2013.Google Scholar