A convection scheme sensitized to the convection direction of a scalar quantity
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 WordsConvection Scheme Simple Upwind Difference Scheme (SUDS) Quadratic Upstream Interpola on for Convective Kinematics (QUICK) Total Variation Diminishing (TVD) Constraint Convection Direction Constraint (CDC)
Atias, M., Wolfshtein, M. and Israel, M., 1977, “Efficiency of Navier-Stokes Solvers,”AIAA J.
Vol. 15, pp. 263–266.MATHCrossRefGoogle Scholar
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
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
Harten, A., 1983, “High Resolution Schemes for Hyperbolic Conservation Laws,”J. Comput. Phys.
, Vol. 49, pp. 357–393.MATHCrossRefMathSciNetGoogle Scholar
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
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
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
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
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
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
Patankar, S. V., 1980,Numerical Heat Transfer and Fluid Flow
, Hemisphere, New York.MATHGoogle Scholar
Raithby, G. D., 1976, “Skew Upwind Differencing for Problem Involving Fluid Flow,”Comput. Methods Appl. Mech. Eng.
, Vol. 9, pp. 153–164.MATHCrossRefMathSciNetGoogle Scholar
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
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
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
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
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
© The Korean Society of Mechanical Engineers (KSME) 1997