KSME International Journal

, Volume 11, Issue 1, pp 106–114 | Cite as

A convection scheme sensitized to the convection direction of a scalar quantity

  • Ji Ryong Cho


It is generally believed that higher order differencing schemes for the convection transport term, e.g., the QUICK scheme and its variants, are superior to the first order simple upwind differencing scheme in the sense that the former produces less numerical diffusion than the latter. In this paper it is shown that this conclusion is no more correct when the flow changes its direction quite rapidly and the grid density is not sufficient. In this situation the simple upwind differencing returns much steeper change of convective variables than higher order schemes. The failure of usual higher order schemes for this flow condition is attributed to the ignorance of convection direction of variables, and a new convection scheme sensitized to the direction of convective transport of a scalar quantity is devised and applied to typical benchmark flows. Results show that the proposed scheme is sufficiently accurate for computation of scalar fields, and also show the optimum behavior for tested problems.

Key Words

Convection Scheme Simple Upwind Difference Scheme (SUDS) Quadratic Upstream Interpola on for Convective Kinematics (QUICK) Total Variation Diminishing (TVD) Constraint Convection Direction Constraint (CDC) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Atias, M., Wolfshtein, M. and Israel, M., 1977, “Efficiency of Navier-Stokes Solvers,”AIAA J. Vol. 15, pp. 263–266.MATHCrossRefGoogle Scholar
  2. de G. Allen, D. N. and Southwell, R. V., 1955. “Relaxation Methods Applied to Determine the Motion, in Two Dimensions, of a Viscous Fluid Past a Fixed Cylinder,”Q. J. Mech. Appl. Maths., Vol. 8, pp. 129–145.MATHCrossRefGoogle Scholar
  3. Gaskell, P. H. and Lau, A. K. C., 1988, “Curvature-Compensated Convective Transport: SMART, A New Boundedness-Preserving Transport Algorithm,”Int. J. Numer. Methods Fluids, Vol. 8, pp. 617–641.MATHCrossRefMathSciNetGoogle Scholar
  4. Harten, A., 1983, “High Resolution Schemes for Hyperbolic Conservation Laws,”J. Comput. Phys., Vol. 49, pp. 357–393.MATHCrossRefMathSciNetGoogle Scholar
  5. Khosla, P. K. and Rubin, S. G., 1974, “A Diagonally Dominant Second-Order Accurate Implicit Scheme,”Comput. Fluids, Vol. 2, pp. 207–218.MATHCrossRefGoogle Scholar
  6. Leonard, B. P., 1979, “A Stable and Accurate Convective Modelling Procedure Based on Quadratic Upstream Interpolation,”Comput. Methods Appl. Mech. Eng., Vol. 19, pp. 59–98.MATHCrossRefGoogle Scholar
  7. Leonard, B. P., 1988, “Simple High Accuracy Resolution Program for Convective Modelling of Discontinuities,”Int. J. Numer. Methods Fluids, Vol. 8, pp. 1291–1318.MATHCrossRefGoogle Scholar
  8. Leonard, B. P. and Drummond, J. E., 1995, “Why You should not Use ‘Hybrid’, ‘Power-Law’ or Related Exponential Schemes for Convective Modelling-There are much better Alternatives,”Int. J. Numer. Methods Fluids, Vol. 20, pp. 421–442.MATHCrossRefGoogle Scholar
  9. Lien, F. S. and Leschziner, M. A., 1993, “Approximation of Turbulence Convection in Complex Flows with a TVD-MUSCL Scheme,”Proc. of 5th Int. IAHR Symp. on Refined Flow Modelling and Turbulence Measurements, Paris.Google Scholar
  10. Lien, F. S. and Leschziner, M. A., 1994, “Upstream Monotonic Approximations for Turbulent Flows,”Proc. of 6th Biennial Colloquium on Compuatational Fluid Dynamics, paper 2.5, Manchester.Google Scholar
  11. Patankar, S. V., 1980,Numerical Heat Transfer and Fluid Flow, Hemisphere, New York.MATHGoogle Scholar
  12. Raithby, G. D., 1976, “Skew Upwind Differencing for Problem Involving Fluid Flow,”Comput. Methods Appl. Mech. Eng., Vol. 9, pp. 153–164.MATHCrossRefMathSciNetGoogle Scholar
  13. Shyy, W., 1985, “A Study of Finite Difference Approximations to Steady State, Convection-Dominated Flow Problems,”J. Comput. Phys., Vol. 57, pp. 415–438.MATHCrossRefMathSciNetGoogle Scholar
  14. Spalding, D. B., 1972, “A Novel Finite Difference Formulation for Differential Expressions Involving Both First and Second Derivatives,”Int. J. Numer. Methods Eng., Vol. 4, pp. 551–559.CrossRefGoogle Scholar
  15. Sweby, P. K., 1984, “High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws,”SIAM J. Numer. Anal., Vol. 21, pp. 995–1011.MATHCrossRefMathSciNetGoogle Scholar
  16. Vest, C. M. and Arpaci, V., 1969, “Stability of Natural Convection in a Vertical Slot,”J. Fluid Mech., Vol. 36, pp. 1–15.MATHCrossRefGoogle Scholar
  17. Zhu, J. and Rodi, W., 1991, “A Low Dispersion and Bounded Convection Scheme,”Computer Methods Applied Mech. Eng., Vol. 92, pp. 87–96.MATHCrossRefGoogle Scholar

Copyright information

© The Korean Society of Mechanical Engineers (KSME) 1997

Authors and Affiliations

  • Ji Ryong Cho
    • 1
  1. 1.Department of Mechanical Engineering, College of EngineeringInje Univ.Kimhae, KyongnamSouth Korea

Personalised recommendations