Research Papers

A Meshless Method for Modeling Convective Heat Transfer

[+] Author and Article Information
Darrell W. Pepper

Professor and Director
Las Vegas, NV 89154
e-mail: darrell.pepper@unlv.edu

Xiuling Wang

Assistant Professor
Department of Mechanical Engineering,
Purdue University Calumet,
Hammond, IN 46323
e-mail: wangx@purduecal.edu

D. B. Carrington

Research Scientist
T-3 Fluid Dynamics and Solid Mechanics Group, LANL,
Los Alamos, NM 87545
e-mail: dcarring@lanl.gov

1Currently Distinguished Visiting Professor, U.S. Air Force Academy, CO 80840.

Manuscript received October 19, 2010; final manuscript received July 7, 2012; published online December 6, 2012. Assoc. Editor: Akshai Runchal.

J. Heat Transfer 135(1), 011003 (Dec 06, 2012) (9 pages) Paper No: HT-10-1486; doi: 10.1115/1.4007650 History: Received October 19, 2010; Revised July 07, 2012

A meshless method is used in a projection-based approach to solve the primitive equations for fluid flow with heat transfer. The method is easy to implement in a matlab format. Radial basis functions are used to solve two benchmark test cases: natural convection in a square enclosure and flow with forced convection over a backward facing step. The results are compared with two popular and widely used commercial codes: comsol, a finite element-based model, and fluent, a finite volume-based model.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Atluri, S. N., and Shen, S., 2002, “The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple & Less-Costly Alternative to the Finite Element and Boundary Element Methods,” Comput. Model. Eng. Sci., 3, pp. 11–52. [CrossRef]
Chen, W., 2002, “New RBF Collocation Schemes and Kernel RBFs With Applications,” Lect. Notes Comput. Sci. Eng., 26, pp. 75–86. [CrossRef]
Kansa, E. J., 1990, “Multiquadrics—A Scattered Data Approximation Scheme With Application to Computational Fluid Dynamics, Part I,” Comput. Math. With Appl., 19, pp. 127–145. [CrossRef]
Sarler, B., and Kuhn, G., 1999, “Primitive Variables Dual Reciprocity Boundary Element Method Solution of Incompressible Navier-Stokes Equations,” Eng. Anal. Boundary Elem., 23, pp. 443–455. [CrossRef]
Sarler, B., 2002, “Towards a Mesh-Free Computation of Transport Phenomena,” Eng. Anal. Boundary Elem., 26, pp. 731–738. [CrossRef]
Lin, H., and Atluri, S. N., 2001, “Meshless Local Petrov Galerkin Method (MLPG) for Convection-Diffusion Problems,” Comput. Model. Eng. Sci., 1, pp. 45–60. [CrossRef]
Buhman, M. D., 2000, Radial Basis Functions, Cambridge University Press, Cambridge.
Mai-Duy, N., and Tran-Cong, T., 2005, “An Efficient Indirect RBFN-Based Method for Numerical Solution of PDEs,” Numer. Methods Partial Differ. Equ., 21, pp. 770–790. [CrossRef]
Patankar, S. V., and Spalding, D. B., 1972, “Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic Flows,” Int. J. Heat Mass Transfer, 15(10), pp.1787–8061. [CrossRef]
Carrington, D. B., and Pepper, D. W., 2002, “Convective Heat Transfer Downstream of a 3-D Backward-Facing Step,” Numer. Heat Transfer, Part A, 41(6–7), pp. 555–578. [CrossRef]
Wang, X., and Pepper, D. W., 2007, “Application of an hp-Adaptive FEM for Solving Thermal Flow Problems,” J. Thermophys. Heat Transfer, 21(1), pp. 190–198. [CrossRef]
Carrington, D. B., Wang, X., and Pepper, D. W., 2010, “An h-Adaptive Finite Element Method for Turbulent Heat Transfer,” Comput. Model. Eng. Sci., 61(1), pp. 23–44. [CrossRef]
Hardy, R. L., 1971, “Multiquadric Equations of Topography and Other Irregular Surfaces,” J. Geophys. Res., 76, pp. 1905–1915. [CrossRef]
Franke, R., 1979, “A Critical Comparison of Some Methods for Interpolation of Scattered Data,” Naval Postgraduate School, Report No. TR NPS-53-79-003.
Franke, C., and Schaback, R., 1998, “Solving Partial Differential Equations Using Radial Basis Functions,” Appl. Math. Comput., 3, pp. 73–82. [CrossRef]
Fasshauer, G. E., 2007, Meshfree Approximation Methods With matlab (Volume 6 of Interdisciplinary Mathematical Sciences), World Scientific Pub. Co., Singapore.
Atluri, S. N., and Zhu, T., 1998, “A New Mesh-Less Local Petrov-Galerkin Approach in Computational Mechanics,” Comput. Mech., 22, pp. 117–127. [CrossRef]
Balachandran, G. R., Rajagopal, A., and Sivakumar, S. M., 2008, “Mesh Free Galerkin Method Based on Natural Neighbors and Conformal Mapping,” Comput. Mech., 42(6), pp. 885–905. [CrossRef]
Bern, M., and Eppstein, D., 1992, “Mesh Generation and Optimal Triangulation,” Computing in Euclidean Geometry, Vol. 1, D. Z.Du and F. K.Hwang, eds., World Scientific Publishing Co., Singapore, pp. 23–90.
Choi, Y., and Kim, S. J., 1999, “Node Generation Scheme for the Mesh-Less Method by Voronoi Diagram and Weighted Bubble Packing,” 5th US National Congress on Computational Mechanics, Boulder, CO.
Li, X. Y., Teng, S. H., and Ungor, A., 2000, “Point Placement for Meshless Methods Using Sphere Packing and Advancing Front Methods,” ICCES’00, Los Angeles, CA.
Sarler, B., Lorbiecka, A. Z., and Vertnik, R., 2010, “Heat and Mass Transfer Problems in Continuous Casting of Steel: Multiscale Solution by Meshless Method,” International Conference on Fluid Dynamics, Cairo, Egypt, Dec. 16–19, Vol. 10.
Gewali, L., and Pepper, D. W., 2010, “Adaptive Node Placement for Mesh-Free Methods,” ICCES 10, Las Vegas, NV, Mar. 28–Apr. 1.
Incropera, F. P., and DeWitt, D. P., 2002, Fundamentals of Heat and Mass Transfer, 5th ed., J. Wiley & Sons, New York.
Pepper, D. W., 2010, “Meshless Methods for PDEs,” Scholarpedia, 5(5), p. 9838. [CrossRef]
Spalding, D. B., 1985, “Numerical Simulation of Natural Convection in Porous Media,” Conference on Natural Convection: Fundamentals and Applications, Hemisphere, Washington, DC, pp. 655–673.
DeVahl Davis, G., 1983, “Natural Convection of Air in a Square Cavity: A Bench Mark Numerical Solution,” Int. J. Numer. Methods Fluids, 3, pp. 249–264. [CrossRef]
Barakos, G., Mitsoulis, E., and Assimacopoulos, D., 1994, “Natural Convection Flow in a Square Cavity Revisited: Laminar and Turbulent Models With Wall Functions,” Int. J. Numer. Methods Fluids, 18, pp. 695–719. [CrossRef]
Gartling, D. K., 1990, “A Test Problem for Outflow Boundary Conditions—Flow Over a Backward-Facing Step,” Int. J. Numer. Methods Fluids, 11, pp. 953–967. [CrossRef]
Blackwell, B. F., and Pepper, D. W., 1992, “Benchmark Problems for Heat Transfer Codes,” ASME Winter Annual Meeting (HTD), 21, pp. 190–198.
Kalla, N. K., 2007, “Solution of Heat Transfer and Fluid Flow Problems Using Meshless Radial Basis Function Method,” M.S. thesis, UNLV, Las Vegas, NV.
Zahab, Z. E., Divo, E., and Kassab, A. J., 2009, “A Localized Collocation Meshless Method (LCMM) for Incompressible Flows CFD Modeling With Applications to Transient Hemodynamics,” Eng. Anal. Boundary Elem., 33, pp. 1045–1061. [CrossRef]


Grahic Jump Location
Fig. 1

Irregular domain discretized using (a) three-noded triangular finite elements, (b) boundary element, and (c) arbitrary interior and boundary points using a meshless method

Grahic Jump Location
Fig. 2

Nodal placement within (a) patches and (b) clipping circles

Grahic Jump Location
Fig. 3

Steady-state conduction in a two-dimensional plate

Grahic Jump Location
Fig. 4

Natural convection within an enclosed cavity

Grahic Jump Location
Fig. 5

31 × 31 point distribution for natural convection in a square cavity (a) comsol mesh, (b) fluent mesh, and (c) meshless node distribution

Grahic Jump Location
Fig. 6

Velocity vectors for natural convection for Ra = 103 in a square cavity using (a) comsol (b) fluent, and (c) meshless

Grahic Jump Location
Fig. 7

Velocity profiles for Ra = 104 along (a) vertical and (b) horizontal central lines

Grahic Jump Location
Fig. 8

Isotherms for natural convection in a square cavity for Ra = 104 using (a) meshless, (b) comsol, and (c) fluent (right vertical wall heated)

Grahic Jump Location
Fig. 9

Problem configuration for forced convection in a backward facing step

Grahic Jump Location
Fig. 10

Typical mesh for backward facing step solution: (a) COMSOL mesh of 388 elements, (b) FLUENT mesh of 284, and (c) 284 nodes for the meshless method

Grahic Jump Location
Fig. 11

Velocity profiles for Re = 800

Grahic Jump Location
Fig. 12

Temperature profiles for Re = 800

Grahic Jump Location
Fig. 13

Isotherms for backward step flow using (a) meshless, (b) comsol, and (c) fluent




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In