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Transient Flows around a Fin at Different Positions
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 126 (2015) 393 – 398
7th International Conference on Fluid Mechanics, ICFM7
Transient flows around a fin at different positions Jia Ma*, Feng Xu School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
Abstract Transient flows in a differently heated cavity with a fin at different positions on the sidewall are investigated using a scaling analysis and direct numerical simulations. The results show that the development of transient flows around the fin is dependent on the fin position and the Rayleigh number. The obtained scaling relations show that the thickness and velocity of the transient flows around the fin are determined by different dynamic and energy balances, and only depended on the fin position. Additionally, there is a good agreement between the scaling predictions and the corresponding numerical results. © 2015 2015Published The Authors. Published Elsevier Ltd.access article under the CC BY-NC-ND license © by Elsevier Ltd. by This is an open (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM) Keywords: natural convection; fin; scaling analysis; numerical simulation; Rayleigh number
1. Introduction Natural convection in a differentially heated cavity is one of the classical problems of fluid mechanics, and has wide industrial application. The study of natural convection in the cavity is therefore of fundamental interest and of practical significance. One of the earliest studies of natural convection in the differentially heated cavity was presented by Batchelor [1]. It has been demonstrated that conduction dominates heat transfer through the cavity for low Rayleigh numbers (e.g. < 103) but convective heat transfer is dominant for higher Rayleigh numbers. If the Rayleigh number is sufficiently large, distinct thermal boundary layers adjacent to the cooled and heated sidewalls are formed and the fluid in the core becomes stratified at the steady state [2,3]. As the Rayleigh number increases further (e.g. larger than a critical value), the flow may become time-periodic [4-6] and even turbulent [7,8]. Fundamental scaling relations quantifying transient natural convection were also obtained by Patterson and Imberger [9].
* Corresponding author. Tel.: +86 18811226827. E-mail address: [email protected]
1877-7058 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM)
doi:10.1016/j.proeng.2015.11.226
394
Jia Ma and Feng Xu / Procedia Engineering 126 (2015) 393 – 398
Recently, the study of natural convection in a differentially heated cavity with a fin on the sidewall has also been given considerable attention due to its relevance to industrial applications [10-12]. The development of natural convection induced by a fin on the sidewall of the differentially heated cavity following sudden heating and cooling may be classified into three stages: an early stage, a transitional stage, and a fully developed stage [13-18]. The fin influences on laminar natural convection flows in the cavity [10-12] and even may trigger the transition to unsteady natural convection flows in the cavity [14,15]. That is, the transition to a periodic flow around the fin may happen if the Rayleigh number is sufficiently large. The critical Rayleigh number is also dependent on the fin length (the fin thickness is often considered to be negligibly small in comparison with its length, i.e. so-called thin fin, refer to [1012] for details). The flow separation and oscillations of the thermal flow around the fin may in turn trigger traveling waves in the thermal boundary layer downstream of the fin [14-17]. The above-mentioned studies of natural convection in the cavity with a fin have been in the context of the fin length and the Rayleigh number and the Prandtl number. However, since the fin position may change the area dominated by the oscillations downstream of the fin and in turn heat transfer through the sidewall downstream of the fin, it is necessary to perform a further quantitative investigation in order to obtain insights into the correlation between the fin position and the transient flow around the fin. This motivates the present scaling analysis of the transient flow around the fin at different position. Nomenclature A g H, L Nu p Pr Q Ra Ray t 't T T0 Tc, Th 'T u, v U IQ VPg x, y yf Yf
E GI GP GT N Q U
τ τs τPv
aspect ratio of the cavity acceleration due to gravity height and length of the cavity Nusselt number dimensionless pressure Prandtl number dimensionless volumetric flow rate Rayleigh number local Rayleigh number calculated based on the fin position dimensionless time dimensionless time step dimensionless temperature initial temperature of the fluid in the cavity temperatures of the cold and hot sidewalls temperature difference between the hot and cold sidewalls dimensionless velocities in x- and y-directions velocity of the intrusion under the viscous regime velocity of the starting plume under the inertial regime dimensionless horizontal and vertical coordinates dimensionless position of the fin position of the fin coefficient of thermal expansion thickness of the intrusion thickness of the starting plume thickness of the thermal boundary layer thermal diffusivity kinematic viscosity density time time at which the convection term balances the conduction term time at which the viscous term balances the inertial term
395
Jia Ma and Feng Xu / Procedia Engineering 126 (2015) 393 – 398
2. Numerical procedures The numerical results consistent with the corresponding experiments in the previous studies (see e.g. [15,16]) show that two-dimensional (2D) numerical simulations may well characterize the transient features of natural convection flows in a differentially heated cavity with an adiabatic fin on the sidewall. Accordingly, a twodimensional numerical simulation was adopted for the present numerical simulation. The two-dimensional rectangular computational domain and boundary conditions are shown in Fig. 1. u
v
(-2.083, 1) u = v = 0, T = -1/2
0,
wT wy
0
. (2.083, 1)
P1 Fin
P2
y (-2.083, 0) u
v
x wT 0, wy
Fin
.
u = v = 0, T = 1/2 (2.083, 0)
0
Fig. 1. Schematic of the computation domain and boundary conditions. Here, the point P1 is at (x = 2.075, y = 0.917) and the point P2 is downstream of the fin (x = 2 with a distance of 0.05 to the fin).
The dimensionless forms of the governing equations (Naviere-Stokes and energy equations with the Boussinesq approximation) are written as wu wv wx wy
0,
(1)
wu wu wu u v wt wx wy
wp Pr w 2u w 2u 1/ 2 ( 2 2 ), wx Ra wx wy
(2)
wv wv wv u v wt wx wy
wp Pr w 2 v w 2 v 1/ 2 ( 2 2 ) PrT , wy Ra wx wy
(3)
1 w 2T w 2T ( ). Ra1/ 2 wx 2 wy 2
(4)
wT wT wT u v wt wx wy
The three dimensionless parameters which govern the flow are the Rayleigh number (Ra), the Prandtl number (Pr) and the aspect ratio (A), defined as g E Th Tc H 3 (5) Ra , QN Q Pr , (6) N H A . (7) L The governing equations were implicitly solved using a finite-volume SIMPLE algorithm. The advection terms were discretized by a QUICK scheme, and the time integration was by a second-order backward difference method. The discretized equations were iterated with under-relaxation factors [15,16]. In consideration of the accuracy of the simulation and saving the computing time, the grid system of 300u499 with grid inflation factors of 1.0258 in the xdirection and 1.0319 in the y-direction was adopted after the grid and time step dependence test.
396
Jia Ma and Feng Xu / Procedia Engineering 126 (2015) 393 – 398
3. Results and discussion
Fig. 2. Isotherms (contours from -0.4438 to 0.4313 with an interval of 0.0416) of the intrusion around the fin in the initial stage (t = 1.24) for Ra = 108 in the cases with different fin positions. (a) yf = 1/6. (b) yf = 1/3. (c) yf = 1/2. (d) yf = 2/3. (e) yf = 5/6.
In the initial stage, following sudden heating, heat conduction through the sidewall may result in a growing thermal boundary adjacent to the finned sidewall, which is separated into two sections by the fin, as shown in Fig. 2. It shows that the horizontal motion of the intrusion front is not synchronized compared with that under the ceiling (see dashed lines). Indeed, both intrusion flows under the fin and the ceiling are typical horizontal gravity currents. Accordingly, considering the variable flux of the intrusion (UIνGI ~ N5/2Rayτ3/2Yf -3) and the balance between the pressure gradient and inertia (gE'TGI/(UIντ) ~νUIν/δI2), a velocity scale may be obtained, N 17/10 Ra y 4/5W 7/10 N 17/10 Ra 4/5W 7/10 ~ . U Iv ~ (8) 12/5 12/5 Yf
H
0.4
y = 1/6 f
y = 1/3 f
y = 1/2 f
Iv
f
U κ-1Ra-9/20Y -7/20H27/20
The measurement results are shown in Fig. 3, which plots the normalized velocity UIvN-1Ra-9/20Yf -7/20H27/20 of the intrusion fronts against the time scale (τ/τs)7/10. The clear linear correlation between the normalized velocity and the time scale confirms the scaling Equation (8). Note that the balance between the pressure gradient and inertia does not work once the intrusion front approaches the fin tip due to the presence of the entrainment at the fin tip.
0.2
y = 2/3 f
y = 5/6 f
0 0
1
2 (τ/τs)7/10
3
4
Fig. 3. Dependence of the velocity of the intrusion front on the fin position and time for Ra = 108.
After the intrusion bypass the fin, a starting plume is formed on the downstream side of the fin. As the fin moves downstream, the plume head becomes larger. This is because the magnitude of the thermal fluid discharged from the thermal boundary layer to the plume head depends on the scales of the velocity and thickness of the thermal boundary layer upstream of the fin. That is, these scales are determined by the fin position. For the starting plume in the absence of entrainment, the ascent is governed by a balance between the inertial forces (~ VPg/τ) and the buoyancy forces (~ gE'T) firstly. Together with the imposed volume flux (VPgGP ~ NRay1/4), this balance yields a scale of the vertical velocity given by N 2 PrRa yW N 2 PrRaW ~ . VPg ~ (9) 3 3 Yf
H
397
Jia Ma and Feng Xu / Procedia Engineering 126 (2015) 393 – 398
Pg
V κ-1Ra-1/2Y -1/2H3/2
The measured velocity of the plume front, non-dimensionalised by VPgN -1Ra-1/2Yf -1/2H3/2, is plotted against τ/τPv in Fig. 4. The good linear relation between the normalized velocity and time demonstrates the correctness of the scaling Equation (9).
0.4 0.3 0.2 yf = 1/6
0.1 0 20
yf = 1/3
40
60 τ/τpν
80
100
Fig. 4. Dependence of the velocity of the starting plume on the fin position and time for Ra = 10 8.
4. Conclusions Scaling analysis and direct numerical simulations were performed for natural convection in a differently heated cavity with a fin at different positions on the sidewall. The features of the flow around the fin, including the intrusion and starting plume, may be dominated by different regimes in the initial and transitional stages. The simple scaling relations about the intrusion and the starting plume were obtained and validated by the present numerical results. It has been demonstrated that the velocity and thickness of the lower intrusion and the plume are dependent on the fin position.
Acknowledgements The authors would like to thank the National Natural Science Foundation of China (11272045) and the 111 Project (B13002) for their financial support.
References [1] G.K. Bachelor, Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quart. Appl. Math. 12 (1954) 209-233. [2] A.E. Gill, The boundary-layer regime for convection in a rectangular cavity. J. Fluid Mech. 26 (1966) 515-536. [3] G. De Vahl Davis, Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. Numer. Meth. Fluids 3 (1983) 249– 264. [4] A.E. Gill, A. Davey, Instabilities of a buoyancy-driven system. J. Fluid Mech. 35 (1969) 775-798. [5] S. Paolucci, D.R. Chenoweth, Transition to chaos in a differentially heated vertical cavity. J. Fluid Mech. 201 (1989) 379-410. [6] P. Le Quere, M. Behnia, From onset of unsteadiness to chaos in a differentially heated square cavity. J. Fluid Mech. 359 (1998) 81-107. [7] J.W. Elder, Turbulent free convection in a vertical slot. J. Fluid Mech. 23 (1965) 99-111. [8] S. Paolucci, Direct numerical simulation of two-dimensional turbulent natural convection in an enclosed cavity. J. Fluid Mech. 215 (1990) 229-262. [9] J.C. Patterson, J. Imberger, Unsteady natural convection in a rectangular cavity. J. Fluid Mech. 100 (1980) 65-86. [10] X. Shi, J.M. Khodadadi, Laminar natural convection heat transfer in a differentially heated square cavity due to a thin fin on the hot wall, J. Heat Transfer 125 (2003) 624-634. [11] S.H. Tasnim, M.R. Collins, Numerical analysis of heat transfer in a square cavity with a baffle on the hot wall, Int. Comm. Heat Mass Transfer 31 (2004) 639-650. [12] E. Bilgen, Natural convection in cavities with a thin fin on the hot wall, Int. J. Heat Mass Transfer 48 (2005) 3493-3505. [13] F. Xu, J.C. Patterson, C. Lei, Experimental observations of the thermal flow around a square obstruction on a vertical wall in a differentially heated cavity, Exp. Fluids 40 (2006) 363-371.
398
Jia Ma and Feng Xu / Procedia Engineering 126 (2015) 393 – 398
[14] F. Xu, J.C. Patterson, C. Lei, An experimental study of the unsteady thermal flow around a thin fin on a sidewall of a differentially heated cavity, Int. J. Heat Fluid Flow 29 (2008) 1139-1153. [15] F. Xu, J.C. Patterson, C. Lei, Transition to a periodic flow induced by a thin fin on the sidewall of a differentially heated cavity, Int. J. Heat Mass Transfer 52 (2009) 620-628. [16] F. Xu, J.C. Patterson, C. Lei, The effect of the fin length on the transition of the natural convection flows in a differentially heated cavity, Int. J. Therm. Sci. 70 (2013) 92-101. [17] Y. Liu, C. Lei, J.C. Patterson, Natural convection in a differentially heated cavity with two horizontal adiabatic fins on the sidewalls, Int. J. Heat Mass Transfer 72 (2014) 23-36. [18] F. Xu, J.C. Patterson, C. Lei, Transient natural convection around a thin fin on the sidewall of a differentially heated cavity, J. Fluid Mech. 639 (2009) 261-290.
ScienceDirect Procedia Engineering 126 (2015) 393 – 398
7th International Conference on Fluid Mechanics, ICFM7
Transient flows around a fin at different positions Jia Ma*, Feng Xu School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
Abstract Transient flows in a differently heated cavity with a fin at different positions on the sidewall are investigated using a scaling analysis and direct numerical simulations. The results show that the development of transient flows around the fin is dependent on the fin position and the Rayleigh number. The obtained scaling relations show that the thickness and velocity of the transient flows around the fin are determined by different dynamic and energy balances, and only depended on the fin position. Additionally, there is a good agreement between the scaling predictions and the corresponding numerical results. © 2015 2015Published The Authors. Published Elsevier Ltd.access article under the CC BY-NC-ND license © by Elsevier Ltd. by This is an open (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM) Keywords: natural convection; fin; scaling analysis; numerical simulation; Rayleigh number
1. Introduction Natural convection in a differentially heated cavity is one of the classical problems of fluid mechanics, and has wide industrial application. The study of natural convection in the cavity is therefore of fundamental interest and of practical significance. One of the earliest studies of natural convection in the differentially heated cavity was presented by Batchelor [1]. It has been demonstrated that conduction dominates heat transfer through the cavity for low Rayleigh numbers (e.g. < 103) but convective heat transfer is dominant for higher Rayleigh numbers. If the Rayleigh number is sufficiently large, distinct thermal boundary layers adjacent to the cooled and heated sidewalls are formed and the fluid in the core becomes stratified at the steady state [2,3]. As the Rayleigh number increases further (e.g. larger than a critical value), the flow may become time-periodic [4-6] and even turbulent [7,8]. Fundamental scaling relations quantifying transient natural convection were also obtained by Patterson and Imberger [9].
* Corresponding author. Tel.: +86 18811226827. E-mail address: [email protected]
1877-7058 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM)
doi:10.1016/j.proeng.2015.11.226
394
Jia Ma and Feng Xu / Procedia Engineering 126 (2015) 393 – 398
Recently, the study of natural convection in a differentially heated cavity with a fin on the sidewall has also been given considerable attention due to its relevance to industrial applications [10-12]. The development of natural convection induced by a fin on the sidewall of the differentially heated cavity following sudden heating and cooling may be classified into three stages: an early stage, a transitional stage, and a fully developed stage [13-18]. The fin influences on laminar natural convection flows in the cavity [10-12] and even may trigger the transition to unsteady natural convection flows in the cavity [14,15]. That is, the transition to a periodic flow around the fin may happen if the Rayleigh number is sufficiently large. The critical Rayleigh number is also dependent on the fin length (the fin thickness is often considered to be negligibly small in comparison with its length, i.e. so-called thin fin, refer to [1012] for details). The flow separation and oscillations of the thermal flow around the fin may in turn trigger traveling waves in the thermal boundary layer downstream of the fin [14-17]. The above-mentioned studies of natural convection in the cavity with a fin have been in the context of the fin length and the Rayleigh number and the Prandtl number. However, since the fin position may change the area dominated by the oscillations downstream of the fin and in turn heat transfer through the sidewall downstream of the fin, it is necessary to perform a further quantitative investigation in order to obtain insights into the correlation between the fin position and the transient flow around the fin. This motivates the present scaling analysis of the transient flow around the fin at different position. Nomenclature A g H, L Nu p Pr Q Ra Ray t 't T T0 Tc, Th 'T u, v U IQ VPg x, y yf Yf
E GI GP GT N Q U
τ τs τPv
aspect ratio of the cavity acceleration due to gravity height and length of the cavity Nusselt number dimensionless pressure Prandtl number dimensionless volumetric flow rate Rayleigh number local Rayleigh number calculated based on the fin position dimensionless time dimensionless time step dimensionless temperature initial temperature of the fluid in the cavity temperatures of the cold and hot sidewalls temperature difference between the hot and cold sidewalls dimensionless velocities in x- and y-directions velocity of the intrusion under the viscous regime velocity of the starting plume under the inertial regime dimensionless horizontal and vertical coordinates dimensionless position of the fin position of the fin coefficient of thermal expansion thickness of the intrusion thickness of the starting plume thickness of the thermal boundary layer thermal diffusivity kinematic viscosity density time time at which the convection term balances the conduction term time at which the viscous term balances the inertial term
395
Jia Ma and Feng Xu / Procedia Engineering 126 (2015) 393 – 398
2. Numerical procedures The numerical results consistent with the corresponding experiments in the previous studies (see e.g. [15,16]) show that two-dimensional (2D) numerical simulations may well characterize the transient features of natural convection flows in a differentially heated cavity with an adiabatic fin on the sidewall. Accordingly, a twodimensional numerical simulation was adopted for the present numerical simulation. The two-dimensional rectangular computational domain and boundary conditions are shown in Fig. 1. u
v
(-2.083, 1) u = v = 0, T = -1/2
0,
wT wy
0
. (2.083, 1)
P1 Fin
P2
y (-2.083, 0) u
v
x wT 0, wy
Fin
.
u = v = 0, T = 1/2 (2.083, 0)
0
Fig. 1. Schematic of the computation domain and boundary conditions. Here, the point P1 is at (x = 2.075, y = 0.917) and the point P2 is downstream of the fin (x = 2 with a distance of 0.05 to the fin).
The dimensionless forms of the governing equations (Naviere-Stokes and energy equations with the Boussinesq approximation) are written as wu wv wx wy
0,
(1)
wu wu wu u v wt wx wy
wp Pr w 2u w 2u 1/ 2 ( 2 2 ), wx Ra wx wy
(2)
wv wv wv u v wt wx wy
wp Pr w 2 v w 2 v 1/ 2 ( 2 2 ) PrT , wy Ra wx wy
(3)
1 w 2T w 2T ( ). Ra1/ 2 wx 2 wy 2
(4)
wT wT wT u v wt wx wy
The three dimensionless parameters which govern the flow are the Rayleigh number (Ra), the Prandtl number (Pr) and the aspect ratio (A), defined as g E Th Tc H 3 (5) Ra , QN Q Pr , (6) N H A . (7) L The governing equations were implicitly solved using a finite-volume SIMPLE algorithm. The advection terms were discretized by a QUICK scheme, and the time integration was by a second-order backward difference method. The discretized equations were iterated with under-relaxation factors [15,16]. In consideration of the accuracy of the simulation and saving the computing time, the grid system of 300u499 with grid inflation factors of 1.0258 in the xdirection and 1.0319 in the y-direction was adopted after the grid and time step dependence test.
396
Jia Ma and Feng Xu / Procedia Engineering 126 (2015) 393 – 398
3. Results and discussion
Fig. 2. Isotherms (contours from -0.4438 to 0.4313 with an interval of 0.0416) of the intrusion around the fin in the initial stage (t = 1.24) for Ra = 108 in the cases with different fin positions. (a) yf = 1/6. (b) yf = 1/3. (c) yf = 1/2. (d) yf = 2/3. (e) yf = 5/6.
In the initial stage, following sudden heating, heat conduction through the sidewall may result in a growing thermal boundary adjacent to the finned sidewall, which is separated into two sections by the fin, as shown in Fig. 2. It shows that the horizontal motion of the intrusion front is not synchronized compared with that under the ceiling (see dashed lines). Indeed, both intrusion flows under the fin and the ceiling are typical horizontal gravity currents. Accordingly, considering the variable flux of the intrusion (UIνGI ~ N5/2Rayτ3/2Yf -3) and the balance between the pressure gradient and inertia (gE'TGI/(UIντ) ~νUIν/δI2), a velocity scale may be obtained, N 17/10 Ra y 4/5W 7/10 N 17/10 Ra 4/5W 7/10 ~ . U Iv ~ (8) 12/5 12/5 Yf
H
0.4
y = 1/6 f
y = 1/3 f
y = 1/2 f
Iv
f
U κ-1Ra-9/20Y -7/20H27/20
The measurement results are shown in Fig. 3, which plots the normalized velocity UIvN-1Ra-9/20Yf -7/20H27/20 of the intrusion fronts against the time scale (τ/τs)7/10. The clear linear correlation between the normalized velocity and the time scale confirms the scaling Equation (8). Note that the balance between the pressure gradient and inertia does not work once the intrusion front approaches the fin tip due to the presence of the entrainment at the fin tip.
0.2
y = 2/3 f
y = 5/6 f
0 0
1
2 (τ/τs)7/10
3
4
Fig. 3. Dependence of the velocity of the intrusion front on the fin position and time for Ra = 108.
After the intrusion bypass the fin, a starting plume is formed on the downstream side of the fin. As the fin moves downstream, the plume head becomes larger. This is because the magnitude of the thermal fluid discharged from the thermal boundary layer to the plume head depends on the scales of the velocity and thickness of the thermal boundary layer upstream of the fin. That is, these scales are determined by the fin position. For the starting plume in the absence of entrainment, the ascent is governed by a balance between the inertial forces (~ VPg/τ) and the buoyancy forces (~ gE'T) firstly. Together with the imposed volume flux (VPgGP ~ NRay1/4), this balance yields a scale of the vertical velocity given by N 2 PrRa yW N 2 PrRaW ~ . VPg ~ (9) 3 3 Yf
H
397
Jia Ma and Feng Xu / Procedia Engineering 126 (2015) 393 – 398
Pg
V κ-1Ra-1/2Y -1/2H3/2
The measured velocity of the plume front, non-dimensionalised by VPgN -1Ra-1/2Yf -1/2H3/2, is plotted against τ/τPv in Fig. 4. The good linear relation between the normalized velocity and time demonstrates the correctness of the scaling Equation (9).
0.4 0.3 0.2 yf = 1/6
0.1 0 20
yf = 1/3
40
60 τ/τpν
80
100
Fig. 4. Dependence of the velocity of the starting plume on the fin position and time for Ra = 10 8.
4. Conclusions Scaling analysis and direct numerical simulations were performed for natural convection in a differently heated cavity with a fin at different positions on the sidewall. The features of the flow around the fin, including the intrusion and starting plume, may be dominated by different regimes in the initial and transitional stages. The simple scaling relations about the intrusion and the starting plume were obtained and validated by the present numerical results. It has been demonstrated that the velocity and thickness of the lower intrusion and the plume are dependent on the fin position.
Acknowledgements The authors would like to thank the National Natural Science Foundation of China (11272045) and the 111 Project (B13002) for their financial support.
References [1] G.K. Bachelor, Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quart. Appl. Math. 12 (1954) 209-233. [2] A.E. Gill, The boundary-layer regime for convection in a rectangular cavity. J. Fluid Mech. 26 (1966) 515-536. [3] G. De Vahl Davis, Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. Numer. Meth. Fluids 3 (1983) 249– 264. [4] A.E. Gill, A. Davey, Instabilities of a buoyancy-driven system. J. Fluid Mech. 35 (1969) 775-798. [5] S. Paolucci, D.R. Chenoweth, Transition to chaos in a differentially heated vertical cavity. J. Fluid Mech. 201 (1989) 379-410. [6] P. Le Quere, M. Behnia, From onset of unsteadiness to chaos in a differentially heated square cavity. J. Fluid Mech. 359 (1998) 81-107. [7] J.W. Elder, Turbulent free convection in a vertical slot. J. Fluid Mech. 23 (1965) 99-111. [8] S. Paolucci, Direct numerical simulation of two-dimensional turbulent natural convection in an enclosed cavity. J. Fluid Mech. 215 (1990) 229-262. [9] J.C. Patterson, J. Imberger, Unsteady natural convection in a rectangular cavity. J. Fluid Mech. 100 (1980) 65-86. [10] X. Shi, J.M. Khodadadi, Laminar natural convection heat transfer in a differentially heated square cavity due to a thin fin on the hot wall, J. Heat Transfer 125 (2003) 624-634. [11] S.H. Tasnim, M.R. Collins, Numerical analysis of heat transfer in a square cavity with a baffle on the hot wall, Int. Comm. Heat Mass Transfer 31 (2004) 639-650. [12] E. Bilgen, Natural convection in cavities with a thin fin on the hot wall, Int. J. Heat Mass Transfer 48 (2005) 3493-3505. [13] F. Xu, J.C. Patterson, C. Lei, Experimental observations of the thermal flow around a square obstruction on a vertical wall in a differentially heated cavity, Exp. Fluids 40 (2006) 363-371.
398
Jia Ma and Feng Xu / Procedia Engineering 126 (2015) 393 – 398
[14] F. Xu, J.C. Patterson, C. Lei, An experimental study of the unsteady thermal flow around a thin fin on a sidewall of a differentially heated cavity, Int. J. Heat Fluid Flow 29 (2008) 1139-1153. [15] F. Xu, J.C. Patterson, C. Lei, Transition to a periodic flow induced by a thin fin on the sidewall of a differentially heated cavity, Int. J. Heat Mass Transfer 52 (2009) 620-628. [16] F. Xu, J.C. Patterson, C. Lei, The effect of the fin length on the transition of the natural convection flows in a differentially heated cavity, Int. J. Therm. Sci. 70 (2013) 92-101. [17] Y. Liu, C. Lei, J.C. Patterson, Natural convection in a differentially heated cavity with two horizontal adiabatic fins on the sidewalls, Int. J. Heat Mass Transfer 72 (2014) 23-36. [18] F. Xu, J.C. Patterson, C. Lei, Transient natural convection around a thin fin on the sidewall of a differentially heated cavity, J. Fluid Mech. 639 (2009) 261-290.