In this work, we describe a dynamically adaptive wavelet method for solving the natural-convection flow a differentially heated cavity in three spatial dimensions. The adaptive method takes advantage of an interpolating wavelet for the adaptive approximation in the design of a simple refinement strategy that reflects the local demand of the physical solution. The derivative approximation is computed via consistent finite-difference approximation on an adaptive grid. To demonstrate the versatility of the algorithm, we simulate the 3-D differentially heated cavity with various values of Rayleigh number. The results are compared with those obtained by other computational approaches.

1.
Vasilyev
O. V.
, and
Paolucci
S.
,
1996
. “
Dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain
”.
Journal of Computational Physics
,
125
, pp.
498
512
.
2.
Holmstro¨m
M.
,
1999
. “
Solving hyperbolic PDEs using interpolating wavelets
”.
SIAM Journal of Scientific Computing
,
21
, pp.
405
420
.
3.
Barinka
A.
,
Barsch
T.
,
Charton
P.
,
Cohen
A.
,
Dahlke
S.
,
Dahmen
W.
, and
Urban
K.
,
2001
. “
Adaptive wavelet schemes for elliptic problems — implementation and numerical experiments
”.
SIAM Journal of Scientific Computing
,
23
, pp.
910
939
.
4.
Griebel, M., and Koster, F., 2000. “Adaptive wavelet solvers for the unsteady incompressible Navier-Stokes equations”. In Advances in Mathematical Fluid Mechanics, J. M. et al., ed. Springer Verlag, pp. 67–118.
5.
Rastigejev, Y., and Paolucci, S., 2006. “Wavelet based adaptive multiresolution computation of viscous reactive flows”. International Journal for Numerical Methods in Fluids, to appear.
6.
Schneider, K., and Farge, M., 2000. “Numerical simulation of a mixing layer in an adaptive basis”. C. R. Acad. Sci. Paris, t. 328, Se´rie II b, pp. 263–269.
7.
Rastigejev, Y., 2002. “Multiscale Computations with a Wavelet Adaptive Algorithm”. PhD Thesis, University of Notre Dame, Notre Dame, IN.
8.
Wirasaet
D.
, and
Paolucci
S.
,
2005
. “
An adaptive wavelet method for incompressible flows in complex domains
”.
Journal of Fluids Engineering
,
127
, pp.
656
665
.
9.
Wirasaet, D., and Paolucci, S., 2005. “Application of an adaptive wavelet method to natural-convection flow in a differentially heated cavity”. In Proceedings of 2005 ASME Summber Heat Transfer Conference, ASME.
10.
Paolucci
S.
, and
Chenoweth
D. R.
,
1989
. “
Transition to chaos in a differentailly heated veritical cavity
”.
Journal of Fluid Mechanics
,
201
, pp.
379
410
.
11.
Le Que´re´, P., and Behnia, M., 1998. “From onset of unsteadiness to chaos in a differentially heated square cavity”. Journal of Fluid Mechanics, pp. 81–107.
12.
Le Que´re´
P.
,
1991
. “
Accurate solutions to the square thermally driven cavity at high Rayleigh number
”.
Computers in Fluids
,
20
, pp.
29
41
.
13.
Suslov, S. A., and Paolucci, S., 1996. “A Petrov-Galerkin method for the direct simulation of fully enclosed flows”. In Proceedings of the ASME Heat Transfer Division, Volume 4, Vol. HTD-335.
14.
Fusegi
T.
,
Hyun
J. M.
,
Kuwahara
K.
, and
Farouk
B.
,
1991
. “
A numerical study of three-dimensional natural convection in a differentially heated cubical enclosure
”.
International Journal of Heat and Mass Transfer
,
34
, pp.
1543
1557
.
15.
Wakashima
S.
, and
Saitoh
T. S.
,
2004
. “
Benchmark solutions for natural convection in a cubic cavity using the high-order time-space method
”.
International Journal of Heat and Mass Transfer
,
47
, pp.
853
864
.
16.
Tric
E.
,
Labrosse
G.
, and
Bertrouni
M.
,
2000
. “
A first incursion into the 3d structure of natural convection of air in a differentially heated cubic cavity from accurate numerical solutions
”.
International Journal of Heat and Mass Transfer
,
43
, pp.
4043
4056
.
17.
Christon
M. A.
,
Gresho
P. M.
, and
Sutton
S. B.
,
2002
. “
Computational predictability of time-dependent natural convection flows in enclosures (including a benchmark solution)
”.
International Journal for Numerical Methods in Fluids
,
40
, pp.
953
980
.
18.
Deslauriers
G.
, and
Dubuc
S.
,
1989
. “
Symmetric iterative interpolation processes
”.
Constructive Approximation
,
5
, pp.
49
68
.
19.
Donoho, D., 1992. Interpolating wavelet transform. Tech. rep., Department of Statistics, Stanford University.
20.
Vasilyev
O. V.
,
2003
. “
Solving multi-dimensional evolution problems with localized structure using second generation wavelets
”.
International Journal of Computational Fluid Dynamics
,
17
, pp.
151
168
.
21.
Bertoluzza, S., 1995. “Adaptive wavelet collocation for the solution of steady-state equations”. In Wavelet Applications II: 17–21 April 1995, Orlando, Florida, Vol. 2491 of Proceedings of SPIE-the International Society for Optical Engineering, pp. 947–956.
22.
Jameson
L.
,
1998
. “
A wavelet-optimized, very high order numerical method
”.
SIAM Journal of Scientific Computing
,
19
, pp.
1980
2013
.
23.
Walde´n
J.
,
1999
. “
Filter bank methods for hyperbolic pdes
”.
SIAM Journal of Numerical Analysis
,
4
, pp.
1183
1233
.
24.
Wirasaet, D., 2006. “In preparation”. PhD thesis, University of Notre Dame.
25.
Kim
J.
, and
Moin
P.
,
1985
. “
Application of a fractionalstep method to incompressible Navier-Stokes equation
”.
Journal of Computational Physics
,
59
, pp.
308
323
.
26.
Saad, Y., 1994. SPARSKIT: A Basic Toolkit for Sparse Matrix Computations. (URL http://www-users.cs.umn.edu/~saad).
This content is only available via PDF.
You do not currently have access to this content.