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.

References

References
1.
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 Method
s,”
Comput. Model. Eng. Sci.
,
3
, pp.
11
52
.10.3970/cmes.2002.003.011
2.
Chen
,
W.
,
2002
, “
New RBF Collocation Schemes and Kernel RBFs With Applications
,”
Lect. Notes Comput. Sci. Eng.
,
26
, pp.
75
86
.10.1007/978-3-642-56103-0
3.
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
.10.1016/0898-1221(90)90270-T
4.
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
.10.1016/S0955-7997(98)00098-8
5.
Sarler
,
B.
,
2002
, “
Towards a Mesh-Free Computation of Transport Phenomena
,”
Eng. Anal. Boundary Elem.
,
26
, pp.
731
738
.10.1016/S0955-7997(02)00044-9
6.
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
.10.3970/cmes.2000.001.205
7.
Buhman
,
M. D.
,
2000
,
Radial Basis Functions
,
Cambridge University Press
,
Cambridge
.
8.
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
.10.1002/num.20062
9.
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
.10.1016/0017-9310(72)90054-3
10.
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
.10.1080/104077802317418214
11.
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
.10.2514/1.22414
12.
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
.10.3970/cmes.2010.061.023
13.
Hardy
,
R. L.
,
1971
, “
Multiquadric Equations of Topography and Other Irregular Surfaces
,”
J. Geophys. Res.
,
76
, pp.
1905
1915
.10.1029/JB076i008p01905
14.
Franke
,
R.
,
1979
, “
A Critical Comparison of Some Methods for Interpolation of Scattered Data
,” Naval Postgraduate School, Report No. TR NPS-53-79-003.
15.
Franke
,
C.
, and
Schaback
,
R.
,
1998
, “
Solving Partial Differential Equations Using Radial Basis Functions
,”
Appl. Math. Comput.
,
3
, pp.
73
82
.10.1016/S0096-3003(97)10104-7
16.
Fasshauer
,
G. E.
,
2007
,
Meshfree Approximation Methods With matlab (Volume 6 of Interdisciplinary Mathematical Sciences)
,
World Scientific Pub. Co.
,
Singapore
.
17.
Atluri
,
S. N.
, and
Zhu
,
T.
,
1998
, “
A New Mesh-Less Local Petrov-Galerkin Approach in Computational Mechanics
,”
Comput. Mech.
,
22
, pp.
117
127
.10.1007/s004660050346
18.
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
.10.1007/s00466-008-0292-0
19.
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
.
20.
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
.
21.
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.
22.
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.
23.
Gewali
,
L.
, and
Pepper
,
D. W.
,
2010
, “
Adaptive Node Placement for Mesh-Free Methods
,” ICCES 10, Las Vegas, NV, Mar. 28–Apr. 1.
24.
Incropera
,
F. P.
, and
DeWitt
,
D. P.
,
2002
,
Fundamentals of Heat and Mass Transfer
,
5th ed.
,
J. Wiley & Sons
,
New York
.
25.
Pepper
,
D. W.
,
2010
, “
Meshless Methods for PDEs
,”
Scholarpedia
,
5
(
5
), p.
9838
.10.4249/scholarpedia.9838
26.
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
.
27.
De
Vahl 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
.10.1002/fld.1650030305
28.
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
.10.1002/fld.1650180705
29.
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
.10.1002/fld.1650110704
30.
Blackwell
,
B. F.
, and
Pepper
,
D. W.
,
1992
, “
Benchmark Problems for Heat Transfer Codes
,”
ASME Winter Annual Meeting (HTD)
,
21
, pp.
190
198
.
31.
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.
32.
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
.10.1016/j.enganabound.2009.03.006
You do not currently have access to this content.