We develop a High-Order Symmetric Interior Penalty (SIP) Discontinuous Galerkin (DG) Finite Element Method (FEM) to investigate two-dimensional in space natural convective flows in a vertical cavity. The physical problem is modeled by a coupled nonlinear system of partial differential equations and admits various solutions including stable and unstable modes in the form of traveling and/or standing waves, depending on the governing parameters. These flows are characterized by steep boundary and internal layers which evolve with time and can be well-resolved by high-order methods that also are adept to adaptive meshing. The standard no-slip boundary conditions which apply on the lateral walls, and the periodic conditions prescribed on the upper and lower boundaries, present additional challenges. The numerical scheme proposed herein is shown to successfully address these issues and furthermore, large Prandtl number values can be handled naturally. Discontinuous source terms and coefficients are an innate feature of multiphase flows involving heterogeneous fluids and will be a topic of subsequent work. Spatially adaptive Discontinuous Galerkin Finite Elements are especially suited to such problems.

1.
Y.
Shu
,
B. Q.
Li
, and
H. C.
de Groh
 III
(
2002
)
Numer. Heat Tr. A-Appl.
42
,
345
364
.
2.
A. C.
Aristotelous
,
I.
Klapper
,
Y.
Grabovsky
,
B.
Pabst
,
B.
Pitts
, and
P. S.
Stewart
(
2015
)
Phys. Rev. E
92
,
022703, 7
p.
3.
A. C.
Aristotelous
and
M. A.
Haider
(
2014
)
Int. J. Numer. Meth. Biomed. Eng.
30
,
767
780
.
4.
S.
Simic-Stefani
,
M.
Kawaji
, and
H. H.
Hu
(
2006
)
J. Cryst. Growth
294
,
373
384
.
5.
B.
Straughan
(
2009
)
Ricerche Mat.
58
,
157
162
.
6.
B.
Straughan
and
F.
Franchi
(
1984
)
Proc. Roy. Soc. Edinb.
96A
,
175
178
.
7.
C. I.
Christov
(
2009
)
Mech. Res. Comm.
36
,
481
486
.
8.
S.
Bargmann
,
P.
Steinmann
, and
P. M.
Jordan
(
2008
)
Phys. Lett. A
372
,
4418
4424
.
9.
M.
Ostoja-Starzewski
(
2009
)
Int. J. Eng. Sci.
47
,
807
810
.
10.
N.
Afrin
,
Y.
Zhang
, and
J. K.
Chen
(
2011
)
Int. J. Heat Mass Transfer
54
,
2419
2426
.
11.
J. W.
Elder
(
1965
)
J. Fluid Mech.
23
,
77
98
.
12.
A. E.
Gill
and
A.
Davey
(
1969
)
J. Fluid Mech.
35
,
775
798
.
13.
R. F.
Bergholz
(
1978
)
J. Fluid Mech.
84
,
743
768
.
14.
C. I.
Christov
and
G. M.
Homsy
(
2001
)
J. Fluid Mech.
430
,
335
360
.
15.
X. H.
Tang
and
C. I.
Christov
(
2007
)
Math. Comput. Simul.
74
,
203
213
.
16.
N. C.
Papanicolaou
,
C. I.
Christov
, and
G. M.
Homsy
(
2009
)
Int. J. Numer. Method Fluid
59
,
945
965
.
17.
N. C.
Papanicolaou
,
C. I.
Christov
, and
P. M.
Jordan
(
2011
)
Eur. J. Mech. B-Fluid
30
,
68
75
.
18.
N. C.
Papanicolaou
and
A. C.
Aristotelous
, “
High-order discontinuous Galerkin methods for coupled thermoconvective flows under gravity modulation
,” in
AMiTaNS’15
,
AIP Conference Proceedings
, Vol.
1684
, edited by
M.D.
Todorov
(
American Institute of Physics
,
Melville, NY
,
2015
), Article 090010,
9
p.
19.
T.
Gudi
,
N.
Nataraj
, and
A. K.
Pani
(
2008
)
J. Sci. Comput.
37
,
139
161
.
20.
K.
Vemaganti
(
2007
)
Num. Meth. PDE
23
,
587
596
.
21.
D. N.
Arnold
(
1982
)
SIAM J. Numer. Anal.
19
,
742
760
.
22.
D. N.
Arnold
,
F.
Brezzi
,
B.
Cockburn
, and
L. D.
Marini
(
2002
)
SIAM J. Numer. Anal.
39
,
1749
1779
.
23.
I.
Mozolevski
,
E.
Süli
, and
P. R.
Bösing
(
2007
)
J. Sci. Comput.
30
,
465
491
.
24.
E. H.
Georgoulis
and
P.
Houston
(
2009
)
IMA J. of Numer. Anal.
29
,
573
594
.
25.
G. A.
Baker
(
1977
)
Math. Comp.
31
,
44
59
.
26.
A. C.
Aristotelous
,
O. A.
Karakashian
, and
S. M.
Wise
(
2015
)
IMA J. Numer. Anal.
35
,
1167
1198
.
This content is only available via PDF.
You do not currently have access to this content.