Abstract
The transient convective flow adjacent to an inclined semi-infinite plate which is heated by a linear temperature gradient is investigated with scaling analysis and direct numerical simulation (DNS) in this study. Both Pr < 1 and Pr > 1 fluids are considered. The initial ambient fluid is quiescent and thermally homogeneous. Important parameters characterizing the thermal boundary layer flow are thickness, characteristic velocity, and time to reach the steady stage. Scaling analysis is carried out to obtain scales for these flow parameters. Compared to previous similar studies, the obtained scale relations are more generalized and they can be utilized for different inclination angles. The derived scales are compared against the DNS results for a variety of flow parameters, e.g., Rayleigh number Ra, Prandtl number Pr, stratification factor s (s = dθw(y)/dy, where θw(y) is the local temperature at a streamwise location of y), inclination angle of the heated plate α, evolutionary time τ, and streamwise location y. The scale relations and the DNS results compare well suggesting the proposed scale laws can provide a sound description for the dynamics of the convective flow subjected to a tilted surface and a linear heating condition.
Issue Section:
Natural and Mixed Convection
References
1.
Prandtl
,
L.
, 1904
, “
On Fluid Flow With Very Little Friction
,” Proceedings of the 3rd International Congress of Mathematicians
,
Germany, Heidelberg
, pp. 484
–491
.2.
Jacobi
,
A.
, 1993
, “
A Scale Analysis Approach to the Correlation of Continuous Moving Sheet (Backward Boundary Layer) Forced Convective Heat Transfer
,” ASME J. Heat Mass Transfer-Trans. ASME
,
115
(4
), pp. 1058
–1061
.10.1115/1.29113623.
Henkes
,
R.
, and
Hoogendoorn
,
C.
, 1994
, “
Scaling of the Turbulent Natural Convection Flow in a Heated Square Cavity
,” ASME J. Heat Mass Transfer-Trans. ASME
,
116
(2
), pp. 400
–408
.10.1115/1.29114124.
Wang
,
X.
,
Castillo
,
L.
, and
Araya
,
G.
, 2008
, “
Temperature Scalings and Profiles in Forced Convection Turbulent Boundary Layers
,” ASME J. Heat Mass Transfer-Trans. ASME
,
130
(2
), p. 021701
.10.1115/1.28137815.
Saha
,
S. C.
,
Patterson
,
J. C.
, and
Lei
,
C.
, 2011
, “
Scaling of Natural Convection of an Inclined Flat Plate: Sudden Cooling Condition
,” ASME J. Heat Mass Transfer-Trans. ASME
,
133
(4
), p. 041503
.10.1115/1.40029826.
Liu
,
Y.
,
Lei
,
C.
, and
Patterson
,
J. C.
, 2014
, “
Natural Convection in a Differentially Heated Cavity With Two Horizontal Adiabatic Fins on the Sidewalls
,” Int. J. Heat Mass Transfer
,
72
, pp. 23
–36
.10.1016/j.ijheatmasstransfer.2013.12.0837.
Liu
,
Y.
,
Lei
,
C.
, and
Patterson
,
J. C.
, 2015
, “
Plume Separation From an Adiabatic Horizontal Thin Fin Placed at Different Heights on the Sidewall of a Differentially Heated Cavity
,” Int. Commun. Heat Mass Transfer
,
61
, pp. 162
–169
.10.1016/j.icheatmasstransfer.2014.12.0078.
Liu
,
Y.
,
Zhang
,
S.
,
Huang
,
H.
,
Suo
,
Q.
,
Bian
,
Y.
, and
Zhao
,
Y.
, 2019
, “
Enhancing the Flow and Heat Transfer in a Convective Cavity Using Symmetrical and Adiabatic Twin Fins
,” Int. J. Heat Mass Transfer
,
142
, p. 118447
.10.1016/j.ijheatmasstransfer.2019.1184479.
Patterson
,
J. C.
, and
Imberger
,
J.
, 1980
, “
Unsteady Natural-Convection in a Rectangular Cavity
,” J. Fluid Mech.
,
100
(1
), pp. 65
–86
.10.1017/S002211208000101210.
Xu
,
F.
,
Patterson
,
J. C.
, and
Lei
,
C.
, 2010
, “
Temperature Oscillations in a Differentially Heated Cavity With and Without a Fin on the Sidewall
,” Int. Commun. Heat Mass Transfer
,
37
(4
), pp. 350
–359
.10.1016/j.icheatmasstransfer.2010.01.00411.
Liu
,
Y.
,
Bian
,
Y.
,
Zhao
,
Y.
,
Zhang
,
S.
, and
Suo
,
Q.
, 2019
, “
Scaling Laws for the Transient Convective Flow in a Differentially and Linearly Heated Rectangular Cavity at Pr > 1
,” Phys. Fluids
,
31
(4
), p. 043601
.10.1063/1.508790712.
Patterson
,
J. C.
,
Graham
,
T.
,
Schopf
,
W.
, and
Armfield
,
S. W.
, 2002
, “
Boundary Layer Development on a Semi-Infinite Suddenly Heated Vertical Plate
,” J. Fluid Mech.
,
453
, pp. 39
–55
.10.1017/S002211200100655313.
Lin
,
W.
,
Armfield
,
S. W.
, and
Patterson
,
J. C.
, 2007
, “
Cooling of a Pr < 1 Fluid in a Rectangular Container
,” J. Fluid Mech.
,
574
, pp. 85
–108
.10.1017/S002211200600370314.
Xu
,
F.
,
Patterson
,
J. C.
, and
Lei
,
C.
, 2010
, “
Pr < 1 Intrusion Flow Induced by a Vertical Heated Wall
,” Phys. Rev. E
,
82
, p. 026318
.10.1103/PhysRevE.82.02631815.
Xu
,
F.
,
Patterson
,
J. C.
, and
Lei
,
C. W.
, 2009
, “
Transient Natural Convection Flows Around a Thin Fin on the Sidewall of a Differentially Heated Cavity
,” J. Fluid Mech.
,
639
, pp. 261
–290
.10.1017/S002211200999099116.
Mahajan
,
R. L.
, and
Gebhart
,
B.
, 1978
, “
Higher Order Approximations to the Natural Convection Flow Over a Uniform Flux Vertical Surface
,” Int. J. Heat Mass Transfer
,
21
(5
), pp. 549
–556
.10.1016/0017-9310(78)90051-017.
Cianfrini
,
C.
,
Corcione
,
M.
, and
Fontana
,
D. M.
, 2002
, “
Laminar Free Convection From a Vertical Plate With Uniform Surface Heat Flux in Chemically Reacting Systems
,” Int. J. Heat Mass Transfer
,
45
(2
), pp. 319
–329
.10.1016/S0017-9310(01)00144-218.
Lin
,
W.
,
Armfield
,
S. W.
, and
Morgan
,
P. L.
, 2002
, “
Unsteady Natural Convection Boundary-Layer Flow Along a Vertical Isothermal Plate in a Linearly Stratified Fluid With Pr > 1
,” Int. J. Heat Mass Transfer
,
45
(2
), pp. 451
–459
.10.1016/S0017-9310(01)00154-519.
Liu
,
Y.
, and
Huang
,
H.
, 2020
, “
Effect of Three Modes of Linear Thermal Forcing on Convective Flow and Heat Transfer in Rectangular Cavities
,” Int. J. Heat Mass Transfer
,
147
, p. 118951
.10.1016/j.ijheatmasstransfer.2019.11895120.
Liu
,
Y.
, 2019
, “
Scaling of Convective Boundary Layer Flow Induced by Linear Thermal Forcing at Pr < 1 and Pr > 1
,” Phys. Rev. E
,
100
(4
), p. 043112
.10.1103/PhysRevE.100.04311221.
Lin
,
W.
,
Armfield
,
S. W.
,
Patterson
,
J. C.
, and
Lei
,
C.
, 2009
, “
Prandtl Number Scaling of Unsteady Natural Convection Boundary Layers for Pr > 1 Fluids Under Isothermal Heating
,” Phys. Rev. E
,
79
(6
), p. 066313
.10.1103/PhysRevE.79.06631322.
Armfield
,
S. W.
,
Patterson
,
J. C.
, and
Lin
,
W.
, 2007
, “
Scaling Investigation of the Natural Convection Boundary Layer on an Evenly Heated Plate
,” Int. J. Heat Mass Transfer
,
50
(7–8
), pp. 1592
–1602
.10.1016/j.ijheatmasstransfer.2006.08.02023.
Lin
,
W.
,
Armfield
,
S. W.
, and
Patterson
,
J. C.
, 2008
, “
Unsteady Natural Convection Boundary-Layer Flow of a Linearly-Stratified Fluid With Pr < 1 on an Evenly Heated Semi-Infinite Vertical Plate
,” Int. J. Heat Mass Transfer
,
51
(1–2
), pp. 327
–343
.10.1016/j.ijheatmasstransfer.2007.03.04024.
Lin
,
W.
, and
Armfield
,
S. W.
, 2013
, “
Scalings for Unsteady Natural Convection Boundary Layers on an Evenly Heated Plate With Time-Dependent Heating Flux
,” Phys. Rev. E
,
88
(6
), p. 063013
.10.1103/PhysRevE.88.06301325.
Ganesan
,
P.
, and
Palani
,
G.
, 2003
, “
Natural Convection Effects on Impulsively Started Inclined Plate With Heat and Mass Transfer
,” Heat Mass Transfer
,
39
(4
), pp. 277
–283
.10.1007/s00231-002-0380-126.
Ganesan
,
P.
, and
Palani
,
G.
, 2004
, “
Finite Difference Analysis of Unsteady Natural Convection MHD Flow Past an Inclined Plate With Variable Surface Heat and Mass Flux
,” Int. J. Heat Mass Transfer
,
47
(19–20
), pp. 4449
–4457
.10.1016/j.ijheatmasstransfer.2004.04.03427.
Jeschke
,
P.
, and
Beer
,
H.
, 2001
, “
Longitudinal Vortices in a Laminar Natural Convection Boundary Layer Flow on an Inclined Flat Plate and Their Influence on Heat Transfer
,” J. Fluid Mech.
,
432
, pp. 313
–339
.10.1017/S002211200000319028.
Saha
,
S. C.
,
Patterson
,
J. C.
, and
Lei
,
C. W.
, 2010
, “
Natural Convection Boundary-Layer Adjacent to an Inclined Flat Plate Subject to Sudden and Ramp Heating
,” Int. J. Therm. Sci.
,
49
(9
), pp. 1600
–1612
.10.1016/j.ijthermalsci.2010.03.01729.
Saha
,
S. C.
, and
Khan
,
M. M. K.
, 2012
, “
An Improved Boundary Layer Scaling With Ramp Heating on a Sloping Plate
,” Int. J. Heat Mass Transfer
,
55
(9–10
), pp. 2268
–2284
.10.1016/j.ijheatmasstransfer.2012.01.03830.
Saha
,
S. C.
,
Patterson
,
J. C.
, and
Lei
,
C.
, 2010
, “
Scaling of Natural Convection of an Inclined Flat Plate: Ramp Cooling Condition
,” Int. J. Heat Mass Transfer
,
53
(23–24
), pp. 5156
–5166
.10.1016/j.ijheatmasstransfer.2010.07.04731.
Saha
,
S. C.
,
Brown
,
R. J.
, and
Gu
,
Y. T.
, 2012
, “
Scaling for the Prandtl Number of the Natural Convection Boundary Layer of an Inclined Flat Plate Under Uniform Surface Heat Flux
,” Int. J. Heat Mass Transfer
,
55
(9–10
), pp. 2394
–2401
.10.1016/j.ijheatmasstransfer.2012.01.01932.
Liu
,
Y.
, 2020
, “
Dynamics and Scale Analysis of the Transient Convective Flow Induced by Cooling a Pr < 1 Fluid With Linear Thermal Forcing
,” Int. J. Heat Mass Transfer
,
154
, p. 119767
.10.1016/j.ijheatmasstransfer.2020.11976733.
Liu
,
Y.
, and
Ren
,
S.
, 2020
, “
Improved Scaling Analysis of the Transient Buoyancy-Driven Flow Induced by a Linear Temperature Gradient
,” Int. J. Heat Mass Transfer
,
162
, p. 120386
.10.1016/j.ijheatmasstransfer.2020.12038634.
Liu
,
Y.
, and
Zhang
,
S.
, 2022
, “
Convection in a Differentially Heated Cavity With a Conducting Fin Attached at the Bottom and Ceiling
,” Int. J. Heat Mass Transfer
,
186
, p. 122430
.10.1016/j.ijheatmasstransfer.2021.12243035.
Liu
,
Y.
, and
Ren
,
S.
, 2021
, “
Receptivity of Incompressible Convective Boundary Layers Induced by Linear Thermal Forcing
,” Phys. Fluids
,
33
(3
), p. 034127
.10.1063/5.004463836.
Liu
,
Y.
, and
Ren
,
S.
, 2021
, “
Scale Law Analysis of the Curved Boundary Layer Evolving Around a Horizontal Cylinder at Pr > 1
,” Phys. Fluids
,
33
(7
), p. 073614
.10.1063/5.006020237.
Liu
,
Y.
, and
Liu
,
C.
, 2022
, “
Unified Scale Laws for Transient Convective Boundary Layers: From Flat to Curved Boundary Layers
,” Phys. Rev. Fluids
,
7
(5
), p. 054101
.10.1103/PhysRevFluids.7.05410138.
Xu
,
F.
,
Patterson
,
J.
, and
Lei
,
C.
, 2005
, “
Shadowgraph Observations of the Transition of the Thermal Boundary Layer in a Side-Heated Cavity
,” Exp. Fluids
,
38
(6
), pp. 770
–779
.10.1007/s00348-005-0960-1Copyright © 2023 by ASME
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