Natural convection on a convectively heated vertical wall, one of the fundamental issues of heat and mass transfer in many engineering applications, is investigated in this work. The configuration is governed by the Rayleigh number (RaL or Ra), the Prandtl number (Pr), and the non-dimensional convective heat transfer coefficient (CiL or Ci). A scaling analysis for the dynamics of the boundary layer flow and heat transfer is carried out. The scales of the velocity/thickness of the boundary layer flow and the temperature/thickness of the thermal boundary layer related to the non-dimensional governing parameters are obtained. The scales are validated using the numerical results by large eddy simulation. The results show that the non-dimensional velocity of the boundary layer flow is proportional to CiL2/5RaL2/5; the thickness from the wall to the layer of the maximal velocity is inversely proportional CiL1/5RaL1/5; the non-dimensional thickness of the thermal boundary layer is inversely proportional CiL1/5RaL1/5; the non-dimensional temperature in the thermal boundary layer is proportional to CiL4/5RaL−1/5. The reduction factor describing the thermal resistance of the thermal boundary layer is further discussed, which is proportional to Ci4/5Ra−1/5.
REFERENCES
1.
S.
Ostrach
, “An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force
,” Report NACA-TR-1111, 1953
.2.
E. M.
Sparrow
and J. L.
Gregg
, “Laminar free convection from a vertical plate with uniform surface heat flux
,” Trans. ASME, J. Heat Transfer
78
, 435
(1956
).3.
C. E.
Polymeropoulos
and B.
Gebhart
, “Incipient instability in free convection laminar boundary layers
,” J. Fluid Mech.
30
, 225
(1967
).4.
C. P.
Knowles
and B.
Gebhart
, “The stability of the laminar natural convection boundary layer
,” J. Fluid Mech.
34
, 657
(1968
).5.
R. P.
Dring
and B.
Gebhart
, “A theoretical investigation of disturbance amplification in external laminar natural convection
,” J. Fluid Mech.
34
, 551
(1968
).6.
B.
Gebhart
and R.
Mahajan
, “Characteristic disturbance frequency in vertical natural convection flow
,” Int. J. Heat Mass Transfer
18
, 1143
(1975
).7.
S. W.
Armfield
and J. C.
Patterson
, “Wave properties of natural-convection boundary-layers
,” J. Fluid Mech.
239
, 195
(1992
).8.
M. C.
Paul
, M.
Wilson
, and A.
Rees
, “Thermal receptivity of free convective flow from a heated vertical surface: Nonlinear waves
,” Comput. Therm. Sci.
2
, 421
(2010
).9.
Y.
Zhao
, C.
Lei
, and J. C.
Patterson
, “Natural transition in natural convection boundary layers
,” Int. Commun. Heat Mass Transfer
76
, 366
(2016
).10.
J. C.
Patterson
and J.
Imberger
, “Unsteady natural convection in a rectangular cavity
,” J. Fluid Mech.
100
, 65
(1980
).11.
W.
Lin
and S. W.
Armfield
, “Scaling laws for unsteady natural convection cooling of fluid with Prandtl number less than one in a vertical cylinder
,” Phys. Rev. E
72
, 016306
(2005
).12.
W.
Lin
, S. W.
Armfield
, and J. C.
Patterson
, “Cooling of a Pr <1 fluid in a rectangular container
,” J. Fluid Mech.
574
, 85
(2007
).13.
W.
Lin
and S. W.
Armfield
, “Unified Prandtl number scaling for start-up and fully developed natural-convection boundary layers for both Pr ≥ 1 and Pr ≤ 1 fluids with isothermal heating
,” Phys. Rev. E
86
, 066312
(2012
).14.
F.
Xu
, J. C.
Patterson
, and C.
Lei
, “Transient natural convection flows around a thin fin on the sidewall of a differentially heated cavity
,” J. Fluid Mech.
639
, 261
(2009
).15.
J.
Ma
, B. C.
Nie
, and F.
Xu
, “Transient flows on an evenly heated wall with a fin
,” Int. J. Heat Mass Transfer
118
, 235
(2018
).16.
J.
Ma
, B. C.
Nie
, and F.
Xu
, “Pr>1 unsteady thermal flows and heat transfer in a finned cavity with a uniform heat flux
,” Int. J. Therm. Sci.
129
, 83
(2018
).17.
S.
Saravanan
and C.
Sivaraj
, “Combined natural convection and thermal radiation in a square cavity with a nonuniformly heated plate
,” Comput. Fluids
117
, 125
(2015
).18.
C. D.
McConnochie
and R. C.
Kerr
, “The turbulent wall plume from a vertically distributed source of buoyancy
,” J. Fluid Mech.
787
, 237
(2016
).19.
S. K.
Kim
, S. Y.
Kim
, and Y. D.
Choi
, “Amplification of boundary layer instability by hot wall thermal oscillation in a side heated cavity
,” Phys. Fluids
17
, 014103
(2005
).20.
N.
Ahmed
, D.
Vieru
, C.
Fetecau
, and N. A.
Shah
, “Convective flows of generalized time-nonlocal nanofluids through a vertical rectangular channel
,” Phys. Fluids
30
, 052002
(2018
).21.
W. A.
Azhar
, D.
Vieru
, and C.
Fetecau
, “Free convection flow of some fractional nanofluids over a moving vertical plate with uniform heat flux and heat source
,” Phys. Fluids
29
, 082001
(2017
).22.
A.
Saxena
, V.
Kishor
, S.
Singh
, and A.
Srivastava
, “Experimental and numerical study on the onset of natural convection in a cavity open at the top
,” Phys. Fluids
30
, 057102
(2018
).23.
M.
Qiao
, Z. F.
Tian
, B. C.
Nie
, and F.
Xu
, “The route to chaos for plumes from a top-open cylinder heated from underneath
,” Phys. Fluids
30
, 124102
(2018
).24.
A.
Saxena
, S.
Singh
, and A.
Srivastava
, “Flow and heat transfer characteristics of an open cubic cavity with different inclinations
,” Phys. Fluids
30
, 087101
(2018
).25.
J. H.
Merkin
and I.
Pop
, “The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition
,” Commun. Nonlinear Sci.
16
, 3602
(2011
).26.
A.
Aziz
, “A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition
,” Commun. Nonlinear Sci.
14
, 1064
(2009
).27.
A.
Ishak
, “Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition
,” Appl. Math. Comput.
217
, 837
(2010
).28.
S. V.
Subhashini
, N.
Samuel
, and I.
Pop
, “Effects of buoyancy assisting and opposing flows on mixed convection boundary layer flow over a permeable vertical surface
,” Int. Commun. Heat Mass Transfer
38
, 499
(2011
).29.
M. M.
Rahman
, J. H.
Merkin
, and I.
Pop
, “Mixed convection boundary-layer flow past a vertical flat plate with a convective boundary condition
,” Acta Mech.
226
, 2441
(2015
).30.
J. M.
Shi
, D.
Gerlach
, M.
Breuer
, F.
Durst
, and C. F.
Lange
, “Analysis of heat transfer from single wires close to walls
,” Phys. Fluids
15
, 908
(2003
).31.
C.
Cintolesi
, A.
Petronio
, and V.
Armenio
, “Large eddy simulation of turbulent buoyant flow in a confined cavity with conjugate heat transfer
,” Phys. Fluids
27
, 095109
(2015
).32.
C.
Lei
and J. C.
Patterson
, “A direct three-dimensional simulation of radiation-induced natural convection in a shallow wedge
,” Int. J. Heat Mass Transfer
46
, 1183
(2003
).33.
G.
Barhaghi
and L.
Davidson
, “Natural convection boundary layer in a 5:1 cavity
,” Phys. Fluids
19
, 125106
(2007
).34.
D.
Li
, “Revisiting the subgrid-scale Prandtl number for large-eddy simulation
,” J. Fluid Mech.
802
, R2
(2016
).35.
P.
Moin
, K.
Squires
, W.
Cabot
, and S.
Lee
, “A dynamic subgrid-scale model for compressible turbulence and scalar transport
,” Phys. Fluids
3
, 2746
(1991
).36.
F.
Nicoud
, H. B.
Toda
, O.
Cabrit
, S.
Bose
, and J.
Lee
, “Using singular values to build a subgrid-scale model for large eddy simulations
,” Phys. Fluids
23
, 085106
(2011
).37.
E.
Lau
, G. H.
Yeoh
, V.
Timchenko
, and J. A.
Reizes
, “Large-eddy simulation of natural convection in an asymmetrically-heated vertical parallel-plate channel: Assessment of subgrid-scale models
,” Comput. Fluids
59
, 101
(2012
).38.
M.
Liu
, X. P.
Chen
, and K. N.
Premnath
, “Comparative study of the large eddy simulations with the lattice Boltzmann method using the wall-adapting local eddy-viscosity and Vreman subgrid scale models
,” Chin. Phys. Lett.
29
, 104706
(2012
).39.
F.
Nicoud
and F.
Ducros
, “Subgrid-scale stress modelling based on the square of the velocity gradient tensor
,” Flow, Turbul. Combust.
62
, 183
(1999
).40.
See https://cfd.direct/openfoam/user-guide/ for OpenFOAM User Guide.
41.
Y. S.
Tian
and T. G.
Karayiannis
, “Low turbulence natural convection in an air filled square cavity. Part I: The thermal and fluid flow fields
,” Int. J. Heat Mass Transfer
43
, 867
(2000
).42.
F.
Ampofo
and T. G.
Karayiannis
, “Experimental benchmark data for turbulent natural convection in an air filled square cavity
,” Int. J. Heat Mass Transfer
46
, 3551
(2003
).43.
T.
Kogawa
, J.
Okajima
, A.
Komiya
, S.
Armfield
, and S.
Maruyama
, “Large eddy simulation of turbulent natural convection between symmetrically heated vertical parallel plates for water
,” Int. J. Heat Mass Transfer
101
, 870
(2016
).© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.