Multiple steady states are investigated for natural convection of fluids in a square enclosure with non-isothermally hot bottom wall, isothermally cold side walls, and thermally insulated top wall. A robust computation scheme involving steady-state governing equations has been developed to compute the steady states as a function of Rayleigh number () for two different Prandtl numbers (Pr = 0.026 and 0.1). Penalty Galerkin finite element method with Newton–Raphson solver is employed for the solution of the governing equations, while the solution branches are initiated by varying initial guess to the Newton–Raphson solver. In this context, a dual-perturbation scheme involving perturbations of the boundary conditions and various process parameters has been designed leading to the rich spectrum of the symmetric and asymmetric solution branches for the current symmetric problem. It is found that multiple steady states occur beyond a critical value of Ra, which depends on the magnitude of Pr. In addition to the basic solution branch (corresponding to the solutions obtained via uniform initial guesses), nineteen additional solution branches (six symmetric and thirteen asymmetric) are obtained for Pr = 0.026, while four additional solution branches (two symmetric and two asymmetric) are obtained for Pr = 0.1. The solution branches are associated with a wide spectrum of flow structures (24 distinct types for Pr = 0.026 excluding the reflection symmetric mirror images of the asymmetric solutions), which are reported for the first time. The flow structures lead to various heating scenarios within the enclosure resulting in a significant variation of heat transfer rates (more than 50%). The current results are important for the practical applications. The spectrum of the possible scenarios revealed in this work can be pivotal to design the optimal processes based on the process requirement (targeted heating or enhanced heating rates).
References
1.
A. A.
Hussien
, W.
Al-Kouz
, M.
El Hassan
, A. A.
Janvekar
, and A. J.
Chamkha
, “A review of flow and heat transfer in cavities and their applications
,” Eur. Phys. J. Plus
136
, 353
(2021
).2.
S.
Pandey
, Y. G.
Park
, and M. Y.
Ha
, “An exhaustive review of studies on natural convection in enclosures with and without internal bodies of various shapes
,” Int. J. Heat Mass Transfer
138
, 762
–795
(2019
).3.
A.
Rahimi
, A. D.
Saee
, A.
Kasaeipoor
, and E. H.
Malekshah
, “A comprehensive review on natural convection flow and heat transfer: The most practical geometries for engineering applications
,” Int. J. Numer. Methods Heat Fluid Flow
29
, 834
–877
(2019
).4.
D.
Das
, M.
Roy
, and T.
Basak
, “Studies on natural convection within enclosures of various (non-square) shapes—A review
,” Int. J. Heat Mass Transfer
106
, 356
–406
(2017
).5.
A.
Bairi
, E.
Zarco-Pernia
, and J. M. G.
de Maria
, “A review on natural convection in enclosures for engineering applications. The particular case of the parallelogrammic diode cavity
,” Appl. Therm. Eng.
63
, 304
–322
(2014
).6.
S. C.
Saha
and M. K. K.
Khan
, “A review of natural convection and heat transfer in attic-shaped space
,” Energy Build.
43
, 2564
–2571
(2011
).7.
B.
Wang
, T. M.
Shih
, X. W.
Chen
, R. R. G.
Chang
, and C. X.
Wu
, “Cascade-like and cyclic heat transfer characteristics affected by enclosure aspect ratios for low Prandtl numbers
,” Int. J. Heat Mass Transfer
124
, 131
–140
(2018
).8.
L.
El Moutaouakil
, Z.
Zrikem
, and A.
Abdelbaki
, “Interaction of surface radiation with natural convection in tall vertical cavities heated by a linear heat flux
,” Int. J. Numer. Methods Heat Fluid Flow
26
, 1975
–1996
(2016
).9.
J. C. K.
Tong
, E. M.
Sparrow
, and J. P.
Abraham
, “Unified treatment of natural convection in tall narrow and flat wide rectangular enclosures
,” Numer. Heat Transfer, Part A
54
, 763
–776
(2008
).10.
A.
Bahlaoui
, A.
Raji
, R.
El Ayachi
, M.
Hasnaoui
, M.
Lamsaadi
, and M.
Naimi
, “Coupled natural convection and radiation in a horizontal rectangular enclosure discretely heated from below
,” Numer. Heat Transfer, Part A
52
, 1027
–1042
(2007
).11.
T.
Basak
, S.
Roy
, and A. R.
Balakrishnan
, “Effects of thermal boundary conditions on natural convection flows within a square cavity
,” Int. J. Heat Mass Transfer
49
, 4525
–4535
(2006
).12.
C. J.
Ho
and F. J.
Tu
, “Visualization and prediction of natural convection of water near its density maximum in a tall rectangular enclosure at high Rayleigh numbers
,” ASME J. Heat Transfer
123
, 84
–95
(2001
).13.
G.
Ramirez
, J.
Nunez
, G.
Hernandez-Cruz
, and E.
Ramos
, “Natural convective three-dimensional flow structure in a cylindrical container
,” Int. Commun. Heat Mass Transfer
116
, 104616
(2020
).14.
H.
Welhezi
, N.
Ben-Cheikh
, and B.
Ben-Beya
, “Numerical analysis of natural convection between a heated cube and its spherical enclosure
,” Int. J. Therm. Sci.
150
, 105828
(2020
).15.
S. I.
Moldovan
, A. M.
Balasoiu
, and M.
Braun
, “Experimental investigation of natural convection flow in a laterally heated vertical cylindrical enclosure
,” Int. J. Heat Mass Transfer
139
, 205
–212
(2019
).16.
Y. P.
Hu
, Y. R.
Li
, and C. M.
Wu
, “Aspect ratio dependence of Rayleigh-Benard convection of cold water near its density maximum in vertical cylindrical containers
,” Int. J. Heat Mass Transfer
97
, 932
–942
(2016
).17.
N.
Hasan
, S. F.
Anwer
, and S.
Sanghi
, “Natural convection in a bottom heated horizontal cylinder
,” Phys. Fluids
17
, 064105
(2005
).18.
Q. G.
Wang
, W.
Li
, Z. C.
Chen
, and D. J.
Kukulka
, “Numerical analysis on natural convection in various enclosures
,” Numer. Heat Transfer, Part A
77
, 391
–408
(2020
).19.
S.
Bhowmick
, F.
Xu
, X.
Zhang
, and S. C.
Saha
, “Natural convection and heat transfer in a valley shaped cavity filled with initially stratified water
,” Int. J. Therm. Sci.
128
, 59
–69
(2018
).20.
T.
Basak
, R.
Anandalakshmi
, and M.
Roy
, “Heatlines based natural convection analysis in tilted isosceles triangular enclosures with linearly heated inclined walls: Effect of various orientations
,” Int. Commun. Heat Mass Transfer
43
, 39
–45
(2013
).21.
R. S.
Kaluri
, R.
Anandalakshmi
, and T.
Basak
, “Bejan's heatline analysis of natural convection in right-angled triangular enclosures: Effects of aspect-ratio and thermal boundary conditions
,” Int. J. Therm. Sci.
49
, 1576
–1592
(2010
).22.
E. H.
Ridouane
, A.
Campo
, and J. Y.
Chang
, “Natural convection patterns in right-angled triangular cavities with heated vertical sides and cooled hypotenuses
,” J. Heat Transfer -Trans. ASME
127
, 1181
–1186
(2005
).23.
K.
Lasfer
, M.
Bouzaiane
, and T.
Lili
, “Numerical study of laminar natural convection in a side-heated trapezoidal cavity at various inclined heated sidewalls
,” Heat Transfer Eng.
31
, 362
–373
(2010
).24.
N. I.
Tracy
and D. W.
Crunkleton
, “Oscillatory natural convection in trapezoidal enclosures
,” Int. J. Heat Mass Transfer
55
, 4498
–4510
(2012
).25.
R.
Anandalakshmi
and T.
Basak
, “Heat flow visualization analysis on natural convection in rhombic enclosures with isothermal hot side or bottom wall
,” Eur. J. Mech. B-Fluids
41
, 29
–45
(2013
).26.
M.
Ghalambaz
, S. A. M.
Mehryan
, A. I.
Alsabery
, A.
Hajjar
, M.
Izadi
, and A.
Chamkha
, “Controlling the natural convection flow through a flexible baffle in an L-shaped enclosure
,” Meccanica
55
, 1561
–1584
(2020
).27.
A. I.
Alsabery
, T.
Tayebi
, R.
Roslan
, A. J.
Chamkha
, and I.
Hashim
, “Entropy generation and mixed convection flow inside a wavy-walled enclosure containing a rotating solid cylinder and a heat source
,” Entropy
22
, 606
(2020
).28.
A.
Kadari
, N. E. S.
Chemloul
, and S.
Mekroussi
, “Numerical investigation of laminar natural convection in a square cavity with wavy wall and horizontal fin attached to the hot wall
,” J. Heat Transfer -Trans. ASME
140
, 072503
(2018
).29.
A. I.
Alsabery
, M. A.
Sheremet
, M.
Ghalambaz
, A. J.
Chamkha
, and I.
Hashim
, “Fluid-structure interaction in natural convection heat transfer in an oblique cavity with a flexible oscillating fin and partial heating
,” Appl. Therm. Eng.
145
, 80
–97
(2018
).30.
A.
da Silva
, E.
Fontana
, V. C.
Mariani
, and F.
Marcondes
, “Numerical investigation of several physical and geometric parameters in the natural convection into trapezoidal cavities
,” Int. J. Heat Mass Transfer
55
, 6808
–6818
(2012
).31.
P.
Biswal
and T.
Basak
, “Bejan's heatlines and numerical visualization of convective heat flow in differentially heated enclosures with concave/convex side walls
,” Energy
64
, 69
–94
(2014
).32.
P.
Biswal
and T.
Basak
, “Analysis of differential versus Rayleigh-Benard heating via heat flow visualization for thermal convection due to heating at enclosures with concave/convex walls
,” Numer. Heat Transfer, Part A
73
, 823
–848
(2018
).33.
E.
Mahravan
, A.
Abbassi
, and E. P.
Afsharian
, “Effects of sinusoidal hot wavy walls on natural convection heat transfer and entropy generation inside a Gamma-shaped enclosure
,” Int. J. Exergy
15
, 24
–45
(2014
).34.
M. N.
Hasan
, S.
Saha
, and S. C.
Saha
, “Effects of corrugation frequency and aspect ratio on natural convection within an enclosure having sinusoidal corrugation over a heated top surface
,” Int. Commun. Heat Mass Transfer
39
, 368
–377
(2012
).35.
T. N.
Anderson
, M.
Duke
, and J. K.
Carson
, “Suppression of natural convection heat transfer coefficients in an attic shaped enclosure
,” Int. Commun. Heat Mass Transfer
37
, 984
–986
(2010
).36.
M.
Bakkas
, M.
Hasnaoui
, and A.
Amahmid
, “Natural convective flows in a horizontal channel provided with heating isothermal blocks: Effect of the inter blocks spacing
,” Energy Convers. Manage.
51
, 296
–304
(2010
).37.
L.
Lukose
and T.
Basak
, “Numerical heat flow visualization analysis on enhanced thermal processing for various shapes of containers during thermal convection
,” Int. J. Numer. Methods Heat Fluid Flow
30
, 3535
–3583
(2020
).38.
R. S.
Kaluri
and T.
Basak
, “Analysis of entropy generation for distributed heating in processing of materials by thermal convection
,” Int. J. Heat Mass Transfer
54
, 2578
–2594
(2011
).39.
M.
Corcione
, “Effects of the thermal boundary conditions at the sidewalls upon natural convection in rectangular enclosures heated from below and cooled from above
,” Int. J. Therm. Sci.
42
, 199
–208
(2003
).40.
I.
Sezai
and A. A.
Mohamad
, “Natural convection in a rectangular cavity heated from below and cooled from top as well as the sides
,” Phys. Fluids
12
, 432
–443
(2000
).41.
M.
Bhattacharya
and T.
Basak
, “Critical role of Prandtl number on multiple steady states during natural convection in square enclosures: Analysis of heat transfer rates, flow and thermal maps
,” Int. J. Heat Mass Transfer
170
, 120900
(2021
).42.
D.
Venturi
, X. L.
Wan
, and G. E.
Karniadakis
, “Stochastic bifurcation analysis of Rayleigh-Benard convection
,” J. Fluid Mech.
650
, 391
–413
(2010
).43.
E. H.
Ridouane
, M.
Hasnaoui
, A.
Amahmid
, and A.
Raji
, “Interaction between natural convection and radiation in a square cavity heated from below
,” Numer. Heat Transfer, Part A
45
, 289
–311
(2004
).44.
S. H.
Xin
and P.
Le Quere
, “Natural-convection flows in air-filled, differentially heated cavities with adiabatic horizontal walls
,” Numer. Heat Transfer, Part A
50
, 437
–466
(2006
).45.
J.
Mizushima
and H.
Tanaka
, “Transitions of natural convection in a vertical fluid layer
,” Phys. Fluids
14
, L21
–L24
(2002
).46.
M.
Hasnaoui
, E.
Bilgen
, and P.
Vasseur
, “Natural-convection heat-transfer in rectangular cavities partially heated from below
,” J. Thermophys. Heat Transfer
6
, 255
–264
(1992
).47.
A. Y.
Gelfgat
, P. Z.
Bar-Yoseph
, and A. L.
Yarin
, “Stability of multiple steady states of convection in laterally heated cavities
,” J. Fluid Mech.
388
, 315
–334
(1999
).48.
V.
Erenburg
, A. Y.
Gelfgat
, E.
Kit
, P. Z.
Bar-Yoseph
, and A.
Solan
, “Multiple states, stability and bifurcations of natural convection in a rectangular cavity with partially heated vertical walls
,” J. Fluid Mech.
492
, 63
–89
(2003
).49.
E. H.
Ridouane
and A.
Campo
, “Formation of a pitchfork bifurcation in thermal convection flow inside an isosceles triangular cavity
,” Phys. Fluids
18
, 074102
(2006
).50.
G. A.
Holtzman
, R. W.
Hill
, and K. S.
Ball
, “Laminar natural convection in isosceles triangular enclosures heated from below and symmetrically cooled from above
,” J. Heat Transfer -Trans. ASME
122
, 485
–491
(2000
).51.
A.
Omri
, M.
Najjari
, and S.
Ben Nasrallah
, “Numerical analysis of natural buoyancy-induced regimes in isosceles triangular cavities
,” Numer. Heat Transfer, Part A
52
, 661
–678
(2007
).52.
X. Y.
Wang
, S.
Bhowmick
, Z. F.
Tian
, S. C.
Saha
, and F.
Xu
, “Experimental study of natural convection in a V-shape-section cavity
,” Phys. Fluids
33
, 014104
(2021
).53.
S.
Bhowmick
, S. C.
Saha
, M. M.
Qiao
, and F.
Xu
, “Transition to a chaotic flow in a V-shaped triangular cavity heated from below
,” Int. J. Heat Mass Transfer
128
, 76
–86
(2019
).54.
W.
Aich
, I.
Hajri
, and A.
Omri
, “Numerical analysis of natural convection in a prismatic enclosure
,” Therm. Sci.
15
, 437
–446
(2011
).55.
E. H.
Ridouane
, A.
Campo
, and J. Y.
Chang
, “Role of the inclination of an inverted-V upper plate on the heat and flow behavior of trapped gases in a modified Rayleigh-Benard cavity
,” Int. J. Numer. Methods Fluids
56
, 453
–465
(2008
).56.
J. J.
Serrano-Aguilera
, F. J.
Blanco-Rodriguez
, and L.
Parras
, “Global stability analysis of the natural convection between two horizontal concentric cylinders
,” Int. J. Heat Mass Transfer
172
, 121151
(2021
).57.
Y.
Hu
, D. C.
Li
, S.
Shu
, and X. D.
Niu
, “Study of multiple steady solutions for the 2D natural convection in a concentric horizontal annulus with a constant heat flux wall using immersed boundary-lattice Boltzmann method
,” Int. J. Heat Mass Transfer
81
, 591
–601
(2015
).58.
K.
Luo
, H. L.
Yi
, and H. P.
Tan
, “Eccentricity effect on bifurcation and dual solutions in transient natural convection in a horizontal annulus
,” Int. J. Therm. Sci.
89
, 283
–293
(2015
).59.
G.
Petrone
, E.
Chenier
, and G.
Lauriat
, “Stability of free convection in air-filled horizontal annuli: Influence of the radius ratio
,” Int. J. Heat Mass Transfer
47
, 3889
–3907
(2004
).60.
J.
Mizushima
and S.
Hayashi
, “Exchange of instability modes for natural convection in a narrow horizontal annulus
,” Phys. Fluids
13
, 99
–106
(2001
).61.
J. S.
Yoo
, “Prandtl number effect on bifurcation and dual solutions in natural convection in a horizontal annulus
,” Int. J. Heat Mass Transfer
42
, 3279
–3290
(1999
).62.
U. K.
Sarkar
, N.
Biswas
, and H. F.
Oztop
, “Multiplicity of solution for natural convective heat transfer and entropy generation in a semi-elliptical enclosure
,” Phys. Fluids
33
, 013606
(2021
).63.
Y. P.
Hu
, Y. R.
Li
, and C. M.
Wu
, “Rayleigh-Benard convection of cold water near its density maximum in a cubical cavity
,” Phys. Fluids
27
, 034102
(2015
).64.
J. F.
Torres
, D.
Henry
, A.
Komiya
, and S.
Maruyama
, “Bifurcation analysis of steady natural convection in a tilted cubical cavity with adiabatic sidewalls
,” J. Fluid Mech.
756
, 650
–688
(2014
).65.
D.
Puigjaner
, J.
Herrero
, C.
Simo
, and F.
Giralt
, “Bifurcation analysis of steady Rayleigh-Benard convection in a cubical cavity with conducting sidewalls
,” J. Fluid Mech.
598
, 393
–427
(2008
).66.
D.
Puigjaner
, J.
Herrero
, F.
Giralt
, and C.
Simo
, “Stability analysis of the flow in a cubical cavity heated from below
,” Phys. Fluids
16
, 3639
–3655
(2004
).67.
B. F.
Wang
, L.
Zhou
, J.
Jiang
, and D. J.
Sun
, “Numerical investigation of Rayleigh-Benard convection in a cylinder of unit aspect ratio
,” Fluid Dyn. Res.
48
, 015503
(2016
).68.
Y. R.
Li
, Y. P.
Hu
, Y. Q.
Ouyang
, and C. M.
Wu
, “Flow state multiplicity in Rayleigh-Benard convection of cold water with density maximum in a cylinder of aspect ratio 2
,” Int. J. Heat Mass Transfer
86
, 244
–257
(2015
).69.
D. J.
Ma
, D.
Henry
, and B. H.
Hadid
, “Three-dimensional numerical study of natural convection in vertical cylinders partially heated from the side
,” Phys. Fluids
17
, 124101
(2005
).70.
J. M.
Lopez
and F.
Marques
, “Three-dimensional instabilities in a discretely heated annular flow: Onset of spatio-temporal complexity via defect dynamics
,” Phys. Fluids
26
, 064102
(2014
).71.
V.
Travnikov
, K.
Eckert
, and S.
Odenbach
, “Influence of the Prandtl number on the stability of convective flows between non-isothermal concentric spheres
,” Int. J. Heat Mass Transfer
66
, 154
–163
(2013
).72.
A.
Kondrashov
and E.
Burkova
, “Stationary convective regimes in a thin vertical layer under the local heating from below
,” Int. J. Heat Mass Transfer
118
, 58
–65
(2018
).73.
E.
Ridouane
, A.
Campo
, and M.
Hasnaoui
, “Benefits derivable from connecting the bottom and top walls of attic enclosures with insulated vertical side walls
,” Numer. Heat Transfer, Part A
49
, 175
–193
(2006
).74.
H.
Liu
, Z.
Zeng
, Z. H.
Qiu
, and L. M.
Yin
, “Linear stability analysis of Rayleigh-Benard convection for cold water near its density maximum in a cylindrical container
,” Int. J. Heat Mass Transfer
173
, 121240
(2021
).75.
A. Y.
Gelfgat
, “Instability of natural convection of air in a laterally heated cube with perfectly insulated horizontal boundaries and perfectly conducting spanwise boundaries
,” Phys. Rev. Fluids
5
, 093901
(2020
).76.
J. N.
Reddy
, An Introduction to the Finite Element Method
(McGraw-Hill
, New York
, 1993
).77.
G.
De Vahl Davis
, “Natural convection of air in a square cavity—A bench mark numerical solution
,” Int. J. Numer. Methods Fluids
3
, 249
–264
(1983
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
2021
Author(s)
You do not currently have access to this content.