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Thermo-dynamic irreversibility induced by natural convection in square enclosure with inner cylinder. Part-II: Effect of vertical position of inner cylinder
International Journal of Heat and Mass Transfer 97 (2016) 1120–1139
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Thermo-dynamic irreversibility induced by natural convection in square enclosure with inner cylinder. Part-II: Effect of vertical position of inner cylinder Jeong Hoon Doo b, Gi Su Mun a, Man Yeong Ha a,⇑, Seon Yoo Seong a a b
School of Mechanical Engineering, Pusan National University, Jang Jeon 2-Dong, Geum Jeong Gu, Busan 609-735, Republic of Korea Rolls-Royce and Pusan National University Technology Centre in Thermal Management, Jang Jeon 2-Dong, Geum Jeong Gu, Busan 609-735, Republic of Korea
a r t i c l e
i n f o
Article history: Received 24 June 2015 Received in revised form 15 February 2016 Accepted 18 February 2016 Available online 10 March 2016 Keywords: Natural convection Vertical position of inner cylinder Entropy generation
a b s t r a c t Two-dimensional numerical simulations were conducted to investigate the natural convection heat transfer induced by the temperature difference between cold walls of the square enclosure and a hot inner circular cylinder in the Prandtl number range of 102 6 Pr 6 102 and Rayleigh number range of 103 6 Ra 6 106. In this paper as a sequent research of Mun et al. (2015), the additional geometrical configuration of the system, which is the variation in the vertical position of the inner cylinder, was considered. The change in the structure of convection cells and corresponding heat transfer characteristics were analyzed, which are induced by the variation in the vertical position of the inner cylinder located in the enclosure. And the characteristics of the entropy generation due to heat transfer and fluid friction associated with the flow and thermal structures were addressed in this study. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Buoyancy-induced flow in an enclosure attracts academic attention due to a variety of its applications including solar collector receivers, heat exchangers, nuclear reactors and electronic packaging. Many researchers have investigated parameters that affect the natural convection characteristics in an enclosure and reported that the characteristics of the natural convection in an enclosure is influenced by the geometry and position of an inner body, the configuration of the enclosure and thermo-physical properties of working fluids [1–23]. The accompanying study presented in [24] undertook twodimensional numerical study on the natural convection characteristics in the square enclosure having an inner circular cylinder in the Rayleigh number range of 103 6 Ra 6 106 and the Prandtl number range of 102 6 Pr 6 102, and the natural convection flow is driven by the temperature difference between a hot cylinder surface and cold walls of the enclosure. In the accompanying study [24], authors have focused on the effect of the variation in the tilted angle of the enclosure (0° 6 c 6 45°) on the natural convection characteristics and the production of the thermo-dynamic irreversibility induced by the natural convection was addressed ⇑ Corresponding author. Tel.: +82 51 510 2440; fax: +82 51 515 3101. E-mail address: [email protected] (M.Y. Ha). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.02.054 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
through the analysis on the local entropy generation due to heat transfer and fluid friction. In addition, the heat transfer characteristics on the cylinder surface and walls of the enclosure associated with the structure of the convection cell formed in the enclosure were properly analyzed. As a summary of the accompanying study [24], the transition of the flow regime between the steady state to the unsteady one occurs from Ra = 105 and Pr = 0.1 except cases in the tilted angle range of 15° 6 c 6 30°. As reported previously, the state of the system is directly dependent on the thermal boundary condition and fluid property. Also, the state of the system is influenced by the geometrical configuration of the system, which means that the system at Ra = 105 and Pr = 0.1 shows the quasistate between the steady state and the unsteady one depending on the geometrical configuration of the system. In this paper as a sequent research of [24], therefore, authors focus on the additional geometrical configuration of the system, which is the variation in the vertical position of the inner cylinder, in the same ranges of the Rayleigh number and Prandtl number as those considered in [24]. The change in the structure of convection cells and corresponding heat transfer characteristics were carefully analyzed according to the variation in the vertical position of the inner cylinder located in the enclosure. And the characteristics of the entropy generation due to heat transfer and fluid friction associated with the flow and thermal structures were addressed in this study.
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Nomenclature Be fi g k L n Nu P Pr q R Ra S t T ui xi
c
Bejan number momentum forcing gravitational acceleration thermal conductivity length of the enclosure normal direction to the wall local Nusselt number dimensionless pressure Prandtl number mass source or sink radius of circular cylinder Rayleigh number distance along the enclosure dimensionless time temperature dimensionless velocity vector Cartesian coordinates system
q l m u h
Sub/superscripts dimensional value cyl cylinder en enclosure Abbreviations FFI fluid friction irreversibility HTI heat transfer irreversibility Mathematical symbol surface-averaged quantity hi volume-averaged quantity
Greek symbols a thermal diffusivity b thermal expansion coefficient
Here, the dimensionless governing equations (4)–(6) are obtained from the dimensional governing equations (1)–(3) using the dimensionless variables defined as follows:
2. Computational details 2.1. Numerical methods In this study, the numerical method is exactly the same as those used in the accompanying study [24]. The immersed boundary method was used to capture the virtual boundary of the inner circular cylinder in the Cartesian coordinate system. Fluid considered in this study is incompressible and Newtonian. Fluid flow is assumed to be laminar in the absence of heat generation, chemical reactions and thermal radiation. And also viscous dissipation in the energy equation has been neglected. Based on these assumptions, the dimensional governing equations of mass, momentum and energy for unsteady incompressible viscous flow and thermal fields are expressed as follows:
@ui ¼0 @xi
ð1Þ
"
# @ui @ 2 u @P @ui q þ uj ¼ þ l i þ qgbðT T c Þðsin udi1 þ cos udi2 Þ @t @xj @xi @xj @xj ð2Þ @T @T @2T þ uj ¼ a @t @xj @xj @xj
ð3Þ
The governing equations to which the immersed boundary method is applied are the continuity, momentum and energy equations in their non-dimensional forms, which are expressed as follows:
@ui q¼0 @xi
ð4Þ
t¼
t a 2
L
;
xi ¼
xi ; L
ui ¼
@ui @ui @P @ ui ¼ þ Pr þ RaPrhðsin udi1 þ cos udi2 Þ þ f i þ uj @xi @t @xj @xj @xj ð5Þ ð6Þ
ui L
a
;
P¼
P L2
qa2
;
h¼
T Tc Th Tc
ð7Þ
In the above equations, q, T, and a represent the density, dimensional temperature and thermal diffusivity, respectively. The superscript ⁄ in Eq. (7) represents the dimensional variables. xi is the dimensionless Cartesian coordinate, ui is the corresponding dimensionless velocity component, t is the dimensionless time, P is the dimensionless pressure and h is the dimensionless temperature. The above non-dimensionalization results in two dimensionless parameters are Pr ¼ m=a and Ra ¼ gbL3 ðT h T c Þ=ma, where m; g, and b are the kinematic viscosity, gravitational acceleration, and volume expansion coefficient, respectively. The terms of q; f i , and h in Eqs. (4)–(6) are related to the immersed boundary method. The mass source/sink q in Eq. (4) and momentum forcing f i in Eq. (5) are imposed on the body surface and inside the body to satisfy the no-slip condition and mass conservation in the cell containing the virtual boundary. In Eq. (6), the heat source/sink h is applied to satisfy the iso-thermal boundary condition at the virtual boundary. The momentum forcing term f i and heat source/sink term h in Eqs. (5) and (6) were obtained by
fi ¼
h¼
U i uni 3 1 @Pn Þþ þ NLðuni Þ NLðun1 i 2 2 Dt @xi rffiffiffiffiffiffi 1 Pr ½DIFðunþ1 Þ þ DIFðuni Þ hn di2 i 2 Ra
ð8Þ
H hn 3 1 1 þ NLðhn Þ NLðhn1 Þ pffiffiffiffiffiffiffiffiffiffi ½DIFðhnþ1 Þ 2 2 Dt 2 RaPr þ DIFðhn Þ hn di2
2
@h @h @2h þ uj ¼ þh @t @xj @xj @xj
tilted angle of the enclosure Kronecker delta density dynamic viscosity kinematic viscosity angle of circular cylinder dimensionless temperature
di2
ð9Þ
where U i and H represented the desired velocity and temperature to realize the viscous and thermal boundary conditions on the surface of the immersed object, and superscripts n and n þ 1 in Eq. (9) represent the time levels. The nonlinear term NLðuÞ and diffusion term DIFðuÞ in Eqs. (8) and (9) are defined as:
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NLðui Þ ¼
@uj ui ; @xj
DIFðui Þ ¼
@ 2 ui @xj @xj
ð10Þ
A second-order linear or bilinear interpolation scheme was applied to satisfy the no-slip and isothermal conditions on the immersed boundary. Further details for the immersed boundary method is described in Kim et al. [25], Kim and Choi [26] and Choi [27]. A second-order accurate finite volume method was used for the spatial discretization of governing equations (4)–(6). To simulate the time advancement of the flow field, the fractional step method proposed by Choi and Moin [28] was employed. In the discretization process, the advection terms were treated explicitly using the second-order Adams–Bashforth scheme, and the diffusion terms were treated implicitly using the second-order accurate Crank–Nicolson scheme. Once the velocity and temperature fields are obtained, the local and surface-averaged Nusselt numbers are calculated by the equations below:
Nu ¼
@h ; @nwall
Nu ¼
Z
1 S
S
Nu ds
ð11Þ
0
where, n represents the direction normal to the wall and S the length of surface. The local entropy generation rate was calculated to investigate the irreversibility caused by heat transfer and fluid friction in natural convection. The dimensional equations of local entropy generation rate are expressed as follow:
SHTI ¼
SFFI ¼
kf
"
T 2o
@T @x
2
þ
@T @y
2 #
( " 2 2 # 2 ) @u @v @u @v 2 þ þ þ To @x @y @y @x
l
ð12Þ
ð13Þ
where SHTI and SFFI are the dimensional local entropy generation rates due to heat transfer and fluid friction, respectively. By using dimensionless variables in Eq. (7), the dimensionless form of total entropy generation rate in two-dimensional Cartesian coordinate can be derived as
STotal ¼ SHTI þ SFFI SHTI
" 2 # 2 @h @h ¼ þ @x @y ( " 2 # 2 ) 2 @u @v @u @ v þ þ þ @x @y @x @y
SFFI ¼ / 2
ð14Þ ð15Þ
The Bejan number Be indicating a relative dominance of the irreversibility due to the heat transfer and fluid friction is defined as below:
Be ¼
hSHTI i hSHTI i þ hSFFI i
Be > 0:5 implies that the irreversibility due to heat transfer is dominant, and Be < 0:5 implies that the irreversibility due to the fluid friction is dominant. And Be ¼ 0:5 implies that the heat transfer irreversibility and fluid friction irreversibility has the equal contribution to the total irreversibility [29–36]. When we run the computational cases considered in this study in our computer system, the maximum ram usage required to run our computer code is about 0.15 GB and the maximum run time taken is about 10 days, including the time to obtain the timeaveraged flow and thermal fields. 2.2. Computational conditions A schematic of the system configuration considered in this study is shown in Fig. 1. The system consists of a square enclosure with an inner circular cylinder. The length of each side wall of the enclosure is L, and the radius of the inner cylinder is R = 0.2L, which are exactly the same geometries of the enclosure and inner cylinder as those considered in the accompanying study [24]. In this study, the vertical position of the inner cylinder is considered as a main simulation parameter. d represents the dimensionless distance between the center of the inner cylinder and center of the enclosure, and the position of the cylinder varies in the range of 0.2 6 d 6 0.2 along the vertical centerline of the enclosure as shown in Fig. 1. The non-dimensionless temperatures of hc ¼ 0 (cold) and hh ¼ 1 (hot) were imposed on walls of the enclosure and surface of the inner cylinder, respectively. No-slip and impermeability conditions were imposed on walls of the enclosure and the surface of the inner cylinder. The Boussinesq approximation was used to model the variation in the fluid density due to the change in the fluid temperature. The gravitational acceleration is set in the negative y-direction. The grid system used in this study is exactly the same as that used in the accompanying study [24] in terms of the number and resolution of grids. Extra numerical simulations were conducted to verify the accuracy of the present numerical method, and detailed results of the validation test in terms of the surfaceaveraged Nusselt number on the cylinder surface and local entropy generation rates are presented in the companion paper [24].
ð16Þ
where SHTI and SFFI are the dimensionless local entropy generation rates due to heat transfer and fluid friction, respectively. In above equation, / is called the irreversibility distribution ratio, which is defined as
/¼
lT o a 2 k
LDT
ð17Þ
In the present study, / is taken as 104, based on the result reported by Ilis et al. [31]. The volume-averaged entropy generation rates due to heat transfer and fluid friction and volumeaveraged total entropy generation rate are obtained by Eq. (18). Here, V represents the volume of fluid filled in the enclosure.
Z Z 1 1 SHTI dV; hSFFI i ¼ SFFI dV; V V V V hSTotal i ¼ hSHTI i þ hSFFI i
hSHTI i ¼
ð18Þ
ð19Þ
Fig. 1. Computational domain and its coordinate system.
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3. Results 3.1. Transition of the flow regime Table 1 shows the transition of the flow regime from the steady state to the unsteady one. In this study, numerical solutions show the time-independent characteristics in a relatively low Rayleigh
number range of 103 6 Ra 6 104 regardless of variations in the Prandtl number and location of the inner cylinder. When Ra = 105 and Ra = 106, all numerical solutions still show the timeindependent characteristics in a relatively high Prandtl number range of 0.7 6 Pr 6 100. However, when the Prandtl number decreases to Pr = 0.1 and Pr = 0.01 at Ra = 105 and Ra = 106, the flow experiences the transition from the steady state to the unsteady
Table 1 Flow transition from steady state to unsteady state for different Prandtl number and different location of the cylinder.
S : Steady, U : Unsteady Ra
105
106
Pr 0.01 0.1 0.7 7 100 0.01 0.1 0.7 7 100
Location of cylinder inside enclosure δ -0.2
-0.1
0
0.1
0.2
U S S S S U U S S S
U S S S S U U S S S
U U S S S U U S S S
U S S S S U S S S S
U S S S S U U S S S
Fig. 2. Distribution of the surface-averaged Nusselt numbers on the cylinder surface as a function of the Prandtl number at the Rayleigh numbers of Ra = 103, Ra = 104, Ra = 105 and Ra = 106; (a) d = 0.1, (b) d = 0.2, (c) d = 0.1 and (d) d = 0.2.
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one, depending on the location of the cylinder. When Pr = 0.1 at Ra = 105, the case only that the inner cylinder is located at the center of the enclosure (d = 0.0) shows the unsteady characteristics in the flow and thermal fields. When the Prandtl number is reduced to Pr = 0.01 at Ra = 105, all numerical solutions show the timedependent characteristics regardless of the variation in the location of the inner cylinder. When the Rayleigh number increases from Ra = 105 to Ra = 106, the flow instability in the enclosure is intensified further by a strong buoyancy effect. As a result, all numerical solutions at Pr = 0.1 and Ra = 106 show the timedependent characteristics regardless of the variation in the location of the inner cylinder, except the case that the inner cylinder is located at d = 0.1 showing the steady state characteristics. When the Prandtl number is reduced to Pr = 0.01 at Ra = 106, all numerical solutions show the unsteady state characteristics in the flow and thermal fields. As shown in this table, the ranges of the
Rayleigh number and Prandtl number in which the flow experiences the transition from the steady state to the unsteady one are consistent with to those reported in the companion paper [24]. In this study, the data are obtained from the time-averaged flow and thermal fields for cases showing the time-dependent characteristics in the numerical solution. Fig. 2 shows the distribution of the surface-averaged Nusselt number on the cylinder surface Nucyl as a function of the Prandtl number at four different Rayleigh numbers (Ra = 103, Ra = 104, Ra = 105 and Ra = 106) and four different locations of the cylinder (d = 0.1, d = 0.2, d = 0.1 and d = 0.2). As reported in the companion paper [24], the Prandtl number is an intrinsic thermo-physical property of the fluid, which means that the state of the flow in the system (i.e. flow unsteadiness and complexity of convection cells) is directly correlated with the Prandtl number at a certain Rayleigh number. In these figures, the values of Nucyl remain almost
Fig. 3. Distribution of isotherms, streamlines, entropy generation due to heat transfer and fluid friction at Ra = 104 and Pr = 0.1 (contour values range from 0 to 1 with 10 levels); (a) d = 0.1, (b) d = 0.2, (c) d = 0.1 and (d) d = 0.2.
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unchanged according to the variation in the Prandtl number regardless of the location of the cylinder at a relatively low Rayleigh numbers of Ra = 103 and Ra = 104. However, the Prandtl number dependency on the flow state is clearly visible at a relatively high Rayleigh numbers of Ra = 105 and Ra = 106 and shows a consistent trend of the variation in Nucyl regardless of the location of the inner cylinder. At Ra = 105 and Ra = 106, the values of Nucyl decreases very slightly as the Prandtl number decreases from Pr = 100 to Pr = 0.7 but steeply decreases by reducing the Prandtl number from Pr = 0.1 to Pr = 0.01, showing the change in the slope of Nucyl . In light of this feature, it can be expected that a critical Prandtl number at which the transition of the flow state initiates exists in 0.1 6 Pr 6 0.7 as reported in the companion paper [24]. Therefore, the change in the flow and thermal structures and consequent irreversibility production between Pr = 0.7 and Pr = 0.1 were highlighted in this paper.
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3.2. Rayleigh number of Ra = 103 and Ra = 104 Fig. 3(a)–(d) shows distributions of isotherms, streamlines, local entropy generation rates due to heat transfer and fluid friction, the local total entropy generation and the Bejan number at Pr = 0.1 and Ra = 104 for four different locations of the cylinder. In cases of Ra = 103 and Ra = 104, the flow and thermal structures at Ra = 103 are very similar to those at Ra = 104 at the same Prandtl number and location of the cylinder because the buoyancy effect is still low at Ra = 104. In addition, a significant change in the flow and thermal structures according the variation in the Prandtl number is not identified in this Rayleigh number range as aforementioned in Section 3.1. In Fig. 3(a), when the inner cylinder is located in the upper side of the enclosure at d = 0.1, a pair of two primary vortices is formed in the left and right parts of the enclosure. Each primary vortex has
Fig. 3 (continued)
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a bi-cellular structure with two vortex cores which are separately located in upper side and lower side of the enclosure. And the lower inner vortex is much larger in size than the lower one. In this case, isotherms near the upper side of the cylinder are very slightly denser than those near the lower side of the cylinder. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 38.576) occurs near the upper-center of the cylinder surface. High local entropy generation rates due to fluid friction (SFFImax = 9.270) occur near the lower-left and lower-right of the cylinder surface due to the flow circulating motion in two lower inner vortices rotating in the counter-clockwise (left one) and clockwise (right one) directions. The distribution of the local total entropy generation rate (STOTALmax = 38.57) is similar to that of the entropy generation rate due to the heat transfer because the conduction is a relatively dominant heat transfer mode rather than the convection due to a very weak convection velocity of primary vortices when Ra = 104. Therefore, the Bejan number has high values in the enclosure except for corner regions, the side and bottom walls of the enclosure. In Fig. 3(b), when the cylinder moves up to d = 0.2, a bi-cellular structure of each primary vortex is altered to a uni-cellular one having a single vortex core which is located in the lower part of the enclosure. The isotherms near the upper side of the cylinder become denser due to a shorter distance between the hot cylinder surface and the cold top wall of the enclosure as shown in Fig. 3(b). As a result, a high local entropy generation rate due to heat transfer (SHTImax = 129.24) occurs near the uppercenter of the cylinder surface, and high local entropy generation rates due to fluid friction (SFFImax = 8.981) occurs at the same locations as the case of d = 0.1. The high local total entropy generation rate (STOTALmax = 129.24) occurs at the same locations where the high entropy generation rate due to the heat transfer occurs. The overall distribution of the Bejan number is also similar to that of d = 0.1. In Fig. 3(c), when the cylinder is located at d = 0.1, a pair of two primary vortices shows a bi-cellular structure, and the upper inner vortex in each primary vortex becomes larger in size than the lower one. Therefore, the flow structure at d = 0.1 shows a pattern which is upside down about the horizontal centerline of the enclosure, compared to the case of d = 0.1. In this case, isotherms become dense near the lower side of the cylinder due to a shorter distance between the hot cylinder surface and the cold bottom wall but coarse near the upper side of the cylinder due the upward motion of the fluid in two primary vortices, showing a weak ascending plume. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 48.931) occurs near the lower-center of the cylinder surface. High local entropy generation rates due to fluid friction (SFFImax = 16.768) occur near the upperleft and upper-right of the cylinder surface due to the flow circulating motion in the upper inner vortices. The distribution of the local total entropy generation rate (STOTALmax = 48.934) is similar that of the local entropy generation rate due to the heat transfer. The Bejan number has a similar pattern which is upside down about the horizontal centerline of the enclosure, compared to the case of d = 0.1. In Fig. 3(d), when the cylinder moves down to d = 0.2, a bi-cellular structure of each primary vortex presented at d = 0.1 is altered to a uni-cellular one. In this case, an ascending plume near the upper surface of the cylinder is intensified further, compared to the case at d = 0.1. Following the plume intensified, the isotherms become coarser near the upper surface of the cylinder. Very dense isotherms are identified near the lower surface of the cylinder. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 131.561) occurs near the lower-center of the cylinder surface. High local entropy generation rate due to fluid friction (SFFImax = 21.069) occurs at the same locations as the case of d = 0.1. The maximum local total entropy generation rate (STOTALmax = 131.561) occurs near the lower-center of the cylinder surface. The irreversibility caused by fluid friction is intensified
Fig. 4. Distribution of the local Nusselt numbers on the cylinder surface for four different locations of the cylinder at Pr = 0.1 when Ra = 103 and Ra = 104.
near the upper side part of the cylinder surface due to the change in the location of the cylinder. As a result, the low Bejan number occurs near the upper part of the cylinder in contrast to the case at d = 0.1. Fig. 4 shows distribution of the local Nusselt number on the cylinder surface Nucyl at four different locations of the cylinder at Pr = 0.1 when Ra = 103 and Ra = 104. When d = 0.1 and d = 0.2 at Ra = 103, the maximum value of Nucyl occur at u = 0° (or u = 360°) and the minimum value at u = 180°. When the Rayleigh number increases to Ra = 104 from Ra = 103, the maximum value of Nucyl is still identified at u = 0° (or u = 360°) for both cases of d = 0.1 and d = 0.2 and is significantly augmented at both Ra = 103 and Ra = 104 as the cylinder moves up to d = 0.2 from d = 0.1. When d = 0.1 and Ra = 104, the values of Nucyl are slightly augmented in the lower part of the cylinder (120° 6 u 6 240°) showing a moderate high peak at u = 180° due to a slight increase in the buoyancy effect, compared to the case of d = 0.1 and Ra = 103. When d = 0.1 and d = 0.2, the distribution pattern of Nucyl at Ra = 104 is very similar to that at Ra = 103, showing the maximum value at u = 180° and the minimum value at u = 0° (or u = 360°). However, values of Nucyl at Ra = 104 are slightly augmented near u = 180° and slightly reduced near u = 0° (or u = 360°) due to the increase in the convection velocity, compared to the cases at Ra = 103. 3.3. Rayleigh number of Ra = 105 Fig. 5(a)–(d) shows distributions of isotherms, streamlines, local entropy generation rates due to heat transfer and fluid friction, the local total entropy generation and the Bejan number at Pr = 0.7 and Ra = 105 for four different locations of the cylinder. When the Rayleigh number increases from Ra = 104 and Ra = 105, the buoyancy effect is substantially intensified in the enclosure. Therefore, the core of each primary vortex keeps staying in the upper part of the cylinder regardless of the location of the cylinder, and the isotherms are distorted significantly, following the fluid motion of the convection cells formed in the enclosure as shown in Fig. 5(a)–(d). In Fig. 5(a), when the cylinder is located at d = 0.1, a pair of two primary vortices having a single vortex core is formed in the left and right parts of the enclosure, and a pair of two small scale secondary vortices is generated near the upper-center part of the cylinder, which is caused by the flow separation of the primary vortices near the upper part of the cylinder surface. Following the fluid motion of the primary and secondary vortices, two ascending plumes and a single descending plume are generated, which makes isotherms
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distorted in the upper part of the cylinder. The isotherms near the lower surface of the cylinder are attached to the cylinder surface due to a strong upward fluid motion in the primary vortices. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 127.706) occurs near the lower-center of the cylinder surface in contrast to the case at d = 0.1 and Ra = 104 presented in Fig. 3(a). High local entropy generation rates due to fluid friction (SFFImax = 748.74) occur near the middle-left and middle-right of the cylinder surface, and moderate high entropy generation rates due to fluid friction occur near the upper region of each side wall of the enclosure. The magnitude of the irreversibility caused by fluid friction is more intensified than that of the irreversibility caused by heat transfer in the enclosure with increasing Rayleigh number. Therefore, the overall distribution of the local total entropy generation rate is similar to that of local entropy generation due to fluid friction. As a result, high local total entropy
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generation rate (STOTALmax = 808.192) occurs near the middle-left and middle-right of the cylinder surface. Low Bejan number occurs near the side and lower region of the cylinder including the side surface of the cylinder and side walls of the enclosure due to the high convection velocity of primary vortices, and high Bejan number occurs near the upper region of the cylinder where a pair of secondary vortices having the weak convection velocity is located. In Fig. 5(b), when the cylinder moves up to d = 0.2, the space between the top wall of the enclosure and the cylinder surface is extremely confined. As a result, most of the convection flow with a high momentum which moves along the side surface of the cylinder keeps moving up to the top wall of the enclosure, and a very small portion of the convection flow is entrained to the space confined extremely, having a very low momentum. Therefore, a pair of two secondary vortices formed in the upper-center part of the cylinder at d = 0.1 vanishes as the cylinder is located at d = 0.2,
Fig. 5. Distribution of isotherms, streamlines, entropy generation due to heat transfer and fluid friction at Ra = 105 and Pr = 0.7 (contour values range from 0 to 1 with 10 levels); (a) d = 0.1, (b) d = 0.2, (c) d = 0.1and (d) d = 0.2.
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and a pair of two ascending plumes is generated on the upper-left and upper-right surfaces of the cylinder, following the convection flow with a high momentum. When the cylinder moves up to d = 0.2 from d = 0.1, the location at which high local entropy generation rate due to heat transfer (SHTImax = 125.792) occurs is still observed near the lower-center of the cylinder surface whereas the location at which high local entropy generation rates due to fluid friction (SFFImax = 609.250) occur moves to the lower-left and lower-right of the cylinder surface from middle-left and middle-right of the cylinder surface presented in Fig. 5(a). The high local total entropy generation rate (STOTALmax = 682.012) occurs at the same locations where the high entropy generation rate due to the heat transfer occurs. The overall pattern of the Bejan number distribution is also similar to that of d = 0.1. In Fig. 5(c), when the cylinder is located at d = 0.1, a single ascending plume induced by the fluid motion of two primary vortices is clearly visible in the upper part of the cylinder. Due to the ascending plume
intensified, the isotherms near the top wall of the enclosure become dense even though the cylinder is located in the lower part of the enclosure. As the space between the cylinder surface and bottom wall of the enclosure is confined, the magnitude of the convection velocity in the lower-center region of the cylinder is lower than that in the region off-centered slightly, which means that the thermal boundary layer near the off-centered region on the lower surface of the cylinder is thinner than that in the center region. As a result, high local entropy generation rate due to heat transfer (SHTImax = 118.490) occurs near the lower-left and lower-right of the cylinder surface rather than near the lower-center region. High local entropy generation rate due to fluid friction (SFFImax = 1177.91) occurs near the upper-left and upper-right of the cylinder surface due to the increase in the size of the upper inner vortices. The high local total entropy generation rate (STOTALmax = 1240.94) occurs near upper-left and upper-right of the cylinder which is the same location where the high local
Fig. 5 (continued)
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entropy generation rate due to fluid friction occurs. The Bejan number has low values near the corner region and upper part of the cylinder including the walls of the enclosure and the side surface of the cylinder in general due to the intensified viscous boundary layer generated by the high convection velocity, but high values are located away from the walls of the enclosure and cylinder surface. Especially, the high Bejan number distribution is intensified near the lower region of the cylinder due to the weak fluid motion. In Fig. 5(d), when the cylinder moves down to d = 0.2, the flow and thermal structures at d = 0.2 are very similar to those at d = 0.1. However, the space between the bottom wall of the enclosure and cylinder surface is extremely confined. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 144.837) occurs near the lower-center of the cylinder surface in contrast to the case at d = 0.1 whereas high local entropy generation rate due to fluid friction (SFFImax = 1239.9) and high local total entropy generation rate (STOTALmax = 1239.9) occurs at the same locations as the case at d = 0.1. And overall distribution of the Bejan number is also similar to that of the case at d = 0.1. Fig. 6 shows distribution of the local Nusselt number on the cylinder surface Nucyl at four different locations of the cylinder when Ra = 105 at Pr = 0.7. When the cylinder is located at d = 0.1, two low peaks of Nucyl occur at u = 30° and u = 330° due to the effect of a pair of two ascending plumes presented in Fig. 5(a). The maximum value of Nucyl occurs at u = 180°. When the cylinder moves up to d = 0.2, two ascending plumes are tilted further to each side wall of the enclosure. As a result, two low peaks of Nucyl occur at u = 60° and u = 300°. In this case, the value Nucyl is maximized at u = 180° in contrast to the case of d = 0.2, Ra = 104 and Pr = 0.7 showing the minimum value of Nucyl at u = 180° in Fig. 4, which indicates that the convection velocity in the lower part of the enclosure is significantly augmented. When the cylinder is located at d = 0.1, the minimum value of Nucyl occurs at u = 0° (or at u = 360°) due to the presence of a single ascending plume presented in Fig. 5(c), and two high peaks at u = 140° and u = 220°. When the cylinder moves down to d = 0.2, high values of Nucyl are almost uniformly distributed in 130° 6 u 6 230° due to a very short distance between the cylinder surface and the bottom wall of the enclosure, and the minimum value of Nucyl is still identified at u = 0° (or at u = 360°). Fig. 7(a)–(d) shows distributions of isotherms, streamlines, local entropy generation rates due to heat transfer and fluid friction, the local total entropy generation and the Bejan number at Pr = 0.1 and Ra = 105 for four different locations of the inner cylinder. According to findings in the accompanying study [24], the Grashof number Gr,
Fig. 6. Distribution of the local Nusselt numbers on the cylinder surface for four different locations of the cylinder at Pr = 0.7 when Ra = 105.
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which approximates the ratio of the buoyancy to the viscous force acting on a fluid, increases with decreasing the Prandtl number at the same Rayleigh number, which means that the buoyancy as a driving force in the natural convection becomes strong or the momentum dissipation of the vortex due to the viscous effect becomes weak. Therefore, the complexity of the convection cells in the enclosure increases by reducing the Prandtl number from Pr = 0.7 to Pr = 0.1 at a relatively high Rayleigh number of Ra = 105 as shown in Fig. 7. In Fig. 7(a), when the cylinder is located at d = 0.1, a pair of two primary vortices is formed in left and right parts of the enclosure, having a single vortex core located in the lower part of the cylinder. In addition, several small scale secondary vortices are formed near the bottom wall of the enclosure. In the primary vortex, the fluid cooled moves downward along each side wall of the enclosure. The fluid is re-heated in the lower part of the enclosure and directed to the lower-center region of the cylinder. And then the fluid re-heated moves upward along the cylinder surface. In this fluid motion, a small portion of the fluid reaches the vicinity of the bottom wall of the enclosure, and most of fluid is lifted up toward the lower surface of the cylinder before reaching the bottom wall. As a result, the isotherms in the off-centered region on the lower surface of the cylinder are denser than those in the center region. Therefore, high local entropy generation rates due to heat transfer (SHTImax = 122.481) occur near the lower-left and lower right of the cylinder surface. And high local entropy generation rates due to fluid friction (SFFImax = 831.503) also occur near the lower-left and lower-right of the cylinder surface due to the flow circulating motion in the primary vortices having a single vortex core located in the lower part of the cylinder. The high local total entropy generation rate (STOTALmax = 923.841) occurs near upper-left and upper-right of the cylinder as the same location where the high entropy generation rate due to fluid friction occurs. The Bejan number has high values near the bottom wall of the enclosure and the upper region of the cylinder due to the weak convection velocity because the large amount of circulating fluid is tied up to the near the side parts of the enclosure. In the side parts of the enclosure, the low Bejan number occurs due to the primary vortices having the high convection velocity. In Fig. 7(b), when the cylinder moves up to d = 0.2, the flow structure at d = 0.2 is similar to that at d = 0.1. However, in this case, the secondary vortices which are located at each bottom corner of the enclosure as well as the center region of the bottom wall of the enclosure act as a virtual blunt body. As a result, most of the fluid which is converged to the vertical centerline of the enclosure in the lower part of the enclosure is smoothly directed to the lower-center of the cylinder surface along the virtual wall boundary. Therefore, dense isotherms are distributed in the lower-center of the cylinder surface. Due to this, a high local entropy generation rate due to heat transfer (SHTImax = 136.436) occurs near the lower-center of the cylinder surface in contrast to the case at d = 0.1 whereas high local entropy generation rate due to fluid friction (SFFImax = 498.076) and high local total entropy generation rate (STOTALmax = 603.812) occur at the same locations as the case at d = 0.1. The overall distribution of the Bejan number is similar to that at d = 0.1 except for the expansion of the region having low Bejan number because the fluid is smoothly directed to the lower part of the cylinder due to the change in position of cylinder. In Fig. 7(c), when the cylinder is located at d = 0.1, the primary vortex having a uni-cellular structure at d = 0.1 and d = 0.2 is altered to that having a bi-cellular structure. And the upper inner vortices are larger in size than the lower one. In addition, very small scale secondary vortices are generated in each upper corner of the enclosure and the upper surface of the cylinder. In this case, small portion of the convection flow is entrained to a relatively narrow space between the cylinder surface and the bottom wall of the enclosure. As a result, dense
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Fig. 7. Distribution of isotherms, streamlines, entropy generation due to heat transfer and fluid friction at Ra = 105 and Pr = 0.1 (contour values range from 0 to 1 with 10 levels); (a) d = 0.1, (b) d = 0.2, (c) d = 0.1and (d) d = 0.2.
isotherms are distributed near the lower-left and lower-right of the cylinder surface rather than the lower-center of the cylinder surface. A single ascending plume formed above the upper surface of the cylinder is identified and is stretched to each side wall of the enclosure, following the flow circulating motion in the upper inner vortices. And a pair of two plumes formed on each side of the cylinder surface is stretched in the horizontal direction (x) due to the flow circulating motion in lower inner and upper inner vortices of each primary vortex. High local entropy generation rates due to heat transfer (SHTImax = 104.792) occur near the lower-left and lower-right of the cylinder surface because the thermal boundary layer is attached further to these regions by the flow circulating motion in the lower inner vortices whereas high local entropy generation rate due to fluid friction (SFFImax = 1846.77) and high local total entropy generation rate (STOTALmax = 1940.55) occur near upper-left and upper-right of the cylinder surface. The high Bejan number occurs near lower region of the cylinder,
upper-center of the cylinder surface and center region of the top wall of the enclosure due to the small portion of the convection flow entrained to a relatively narrow space between the cylinder surface and the bottom wall of the enclosure, secondary vortices having weak convection velocity and flow not reaching the center region of the top wall of the enclosure, respectively. The low Bejan number occurs near the upper-center and side region of the enclosure due to two upper inner vortices having strong ascending and descending flow motion. In Fig. 7(d), when the cylinder moves down to d = 0.2 from d = 0.1, the upper inner vortices increase in size while the primary vortex having a bi-cellular structure maintains its structure. And additional small scale secondary vortices are generated near the middle region of each side wall of the enclosure in company with several small scale vortices formed in each upper corner of the enclosure and upper surface of the cylinder. In this case, a single ascending plume formed above the upper surface of the cylinder as well as a pair of two ascending
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Fig. 7 (continued)
plumes near each side of the cylinder surface is identified, similar to the case at d = 0.1. A high local entropy generation rate due to heat transfer (SHTImax = 140.987) occurs near the lower-center of the cylinder surface whereas high local entropy generation rate due to fluid friction (SFFImax = 1373.23) and high local total entropy generation rate (STOTALmax = 1472.56) occur in the same locations as the case at d = 0.1. And overall distribution of the Bejan number is also similar to the case at d = 0.1. Fig. 8 shows distribution of the local Nusselt number on the cylinder surface Nucyl at four different locations of the cylinder when Ra = 105 at Pr = 0.1. When the cylinder is located at d = 0.1, two low peaks of Nucyl occur at u = 80° and u = 280°, which are the minimum values, and two high peaks of Nucyl occur at u = 130° and u = 230°, which are the maximum values. When the cylinder moves up to d = 0.2, two high peaks of Nucyl occur at u = 180° and u = 0° (or u = 360°), and the value at u = 180°, which is the maximum value, is very slightly higher than that at u = 0° (or
u = 360°). Two low peaks of Nucyl occur at u = 80° and u = 280°, which are the minimum values. When the cylinder is located at d = 0.1, two high peaks of Nucyl occur at u = 60° and u = 300°, and another two high peaks of Nucyl occur at u = 140° and u = 220°. The value of Nucyl are maximized at u = 140° and u = 220° and minimized at u = 0° (or u = 360°). When the cylinder moves down to d = 0.2, triple high peaks of Nucyl occur at u = 130°, u = 180° and u = 230°, and two low peaks of Nucyl occur at u = 50° and u = 310°. The value of Nucyl is maximized at u = 180° and minimized at u = 0° (or u = 360°). 3.4. Rayleigh number of Ra = 106 Fig. 9(a)–(d) shows distributions of isotherms, streamlines, local entropy generation rates due to heat transfer and fluid friction, the local total entropy generation and the Bejan number at Pr = 0.7 and Ra = 106 for four different locations of the inner cylinder. In Fig. 10,
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Fig. 8. Distribution of the local Nusselt numbers on the cylinder surface for four different locations of the cylinder at Pr = 0.1 when Ra = 105.
when the cylinder is located at d = 0.1, a pair of two primary vortices is formed in left and right sides of the enclosure, having a single vortex core located in the lower part of the cylinder. In addition, a pair of two small scale secondary vortices is formed in the center region of the bottom wall of the enclosure. When the Rayleigh number increases from Ra = 105 to Ra = 106, the strength of the convection flow is significantly intensified due to the increase in the buoyancy effect. Therefore, in contrast to the case at d = 0.1, Pr = 0.7 and Ra = 105 presented in Fig. 5(a), a pair of the secondary vortices formed in the upper part of the cylinder surface shown in Fig. 5(a) diminishes in this case. And hence a single ascending plume is generated above the upper surface of the cylinder, which is stretched to each side wall of the enclosure. As a result, isotherms near the center region of the top wall of the enclosure become dense by the upward motion of the single plume. The isotherms near the lower-center of the cylinder surface become denser than in other region of the lower surface due to the upward motion of the primary vortex in the lower part of the cylinder. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 329.3) occurs near the lower-center region of the cylinder surface. In addition, a moderate high local entropy generation rate due to heat transfer occurs in the center region of the top wall of the enclosure due to the effect of a single ascending plume intensified. High local entropy generation rate due to fluid friction (SFFImax = 28309.2) occurs near middle-left and middle-right of the cylinder surface, and moderate high local entropy generation rates due to fluid friction occur near the upper region of each side wall of the enclosure as the vortex core around which the high momentum circulating flow exists is located in the upper part of the cylinder. Due to the increasing up to Ra = 106, the convection velocity in an enclosure become more strong than that at Ra = 105. As a result, the portion of irreversibility due to the fluid friction is quite intensified rather than irreversibility due to the heat transfer in the enclosure. Therefore, the high local total entropy generation rate (STOTALmax = 28543.8) occurs at the same locations where the high local entropy generation rate due to the fluid friction occur. The low Bejan number is distributed evenly in an enclosure due to the intensified irreversibility caused by the fluid friction. The high Bejan number occurs near center of the bottom walls of the enclosure due to the secondary vortices having weak convection velocity and lower part of the cylinder surface due to the re-heated fluid before the ascending fluid reaches from lower region of the enclosure to the lower-center of the cylinder surface. In Fig. 9(b), when the cylinder moves up to d = 0.2, four small scale secondary vortices are generated in a very narrow space between the cylinder
surface and the top wall of the enclosure, and the upward, downward and upward motions of the fluid from vertical centerline of the enclosure to the side wall of the enclosure are produced by the interaction between neighboring vortices. However, the strength of the secondary vortices is very low due to the space narrowed extremely. As a result, significant distortion of the isotherms in the upper part of the cylinder is not identified, but a pair of two ascending plumes which is located near the upper-left and upperright of the cylinder surface is clearly visible with the distribution of isothermals distorted by the upward motion of the fluid in the primary vortex. The isotherms near the lower-center of the cylinder surface become dense due to the upward motion of the fluid in the primary vortex in the lower part of the cylinder. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 321.721) occurs near the lower-center of the cylinder surface, similar to the case at d = 0.1. In addition, moderate high local entropy generation rates due to heat transfer occur in two regions off-centered on the top wall of the enclosure due to the effect of two ascending plumes. High local entropy generation rate due to fluid friction (SFFImax = 22922.5) occurs near middle-left and middle-right of the cylinder surface, and moderate high local entropy generation rate due to fluid friction occur near the upper region of each side wall of the enclosure. And high local total entropy generation rate (STOTALmax = 23127.6) occurs at the same location where the high local entropy generation rate due to the fluid friction occurs, which shows almost the same trend as the case at d = 0.1. The distribution of low Bejan number near the cylinder surface and each side wall of the enclosure is similar to that at d = 0.1. The high Bejan number occurs near the center region of the bottom wall of the enclosure and upper part of the cylinder due to the weak convection velocity and secondary vortices, respectively. In Fig. 9(c) and (d), the overall flow and thermal structures at d = 0.1 are similar to those at d = 0.2. In these figures, a single ascending plume is intensified and stretched to the top wall of the enclosure. The ascending plume intensified significantly makes the isotherms near the top wall of the cylinder very dense. As a result, high local entropy generation rates due to heat transfer (SHTImax = 402.811 at d = 0.1 and SHTImax = 362.77 at d = 0.2) occur near the center region of the top wall of the enclosure. And moderate high local entropy generation rates due to heat transfer occur in the lower surface of the cylinder at d = 0.1 and d = 0.2. High local entropy generation rates due to fluid friction (SFFImax = 38691.7 at d = 0.1 and SFFImax = 54391.4 at d = 0.2) and high local total entropy generation rates (STOTALmax = 38970 at d = 0.1 and STOTALmax = 54678.2) occur near middle-left and middle-right of the cylinder surface, and moderate high local entropy generation rate due to fluid friction and moderate high local total entropy generation rate occur near the upper region of each side wall of the enclosure at d = 0.1 and d = 0.2. The overall distribution of the Bejan number at d = 0.1 is also are similar to that at d = 0.2 except for the lower region of the cylinder due to the change in the location of the cylinder. Fig. 10 shows distribution of the local Nusselt number on the cylinder surface Nucyl at four different locations of the cylinder when Ra = 106 at Pr = 0.7. When the cylinder is located at d = 0.1, a simple pattern of Nucyl is distributed, showing a single high peak at u = 180° and a single low peak at u = 0° (or u = 360°). When the cylinder moves up to d = 0.2, two low peaks of Nucyl occur at u = 30° and u = 330° due to small scale secondary vortices formed near the upper part of the cylinder. Two high peaks of Nucyl occurs at u = 180° and u = 0° (or u = 360°), and the maximum value occurs at u = 180°. When the cylinder is located at d = 0.1, high values of Nucyl are almost uniformly distributed in 120° 6 u 6 240°, and the minimum value of Nucyl occurs at u = 0° (or u = 360°). When the cylinder moves down to d = 0.2, two high peaks of Nucyl occur at u = 120° and u = 240° and a single moderate low peak at
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Fig. 9. Distribution of isotherms, streamlines, entropy generation due to heat transfer and fluid friction at Ra = 106 and Pr = 0.7 (contour values range from 0 to 1 with 10 levels); (a) d = 0.1, (b) d = 0.2, (c) d = 0.1and (d) d = 0.2.
u = 180° due to a very narrow space between the cylinder surface and the bottom wall of the enclosure. The minimum value of Nucyl occurs at u = 0° (or u = 360°). Fig. 11(a)–(d) shows distributions of isotherms, streamlines, local entropy generation rates due to heat transfer and fluid friction, the local total entropy generation and the Bejan number at Pr = 0.1 and Ra = 106 for four different locations of the inner cylinder. As aforementioned in Section 3.1, the flow instability is intensified further by reducing the Prandtl number from Pr = 0.7 to Pr = 0.1 at a relatively high Rayleigh number. As a result, the flow unsteadiness and complexity of the convection cells are augmented at Ra = 106 and Pr = 0.1. In Fig. 11(a), when the cylinder is located at d = 0.1, a pair of two primary vortices is formed in the left and right parts of the cylinder, having a single vortex core located in the lower part of the primary vortex. And four different secondary vortices are generated in
the upper part of the cylinder. In addition, several small scale secondary vortices are identified in each corner of the enclosure and the center region of the bottom wall of the enclosure. Due to this structure of complex convection cells, isotherms in the enclosure are significantly distorted, especially in the upper region of the cylinder. The upward and downward motions of the fluid are generated by the interaction between neighboring secondary vortices which are located in the upper part of the cylinder. As a result, a single ascending plume due to the upward motion of the fluid and a pair of two descending plumes due to downward motion of the fluid is formed in the upper part of the cylinder. In addition, a pair of two plumes stretched in the horizontal direction (x) is formed in each side of the cylinder surface, following the fluid motion of the primary vortex which is separated from the side surface of the cylinder and then directed to each side wall of the enclosure. As a result, high local entropy generation rate due to
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Fig. 9 (continued)
heat transfer (SHTImax = 461.058) occurs near the upper-left and upper-right of the cylinder surface, and moderate high local entropy generation rates due to heat transfer occur near the lower surface of the cylinder as well as the center region of the top wall of the enclosure. high local entropy generation rate due to fluid friction (SFFImax = 12243.2) occurs near middle region of each side wall of the enclosure, and moderate high local entropy generation rates due to fluid friction occur near the lower-left and lower-right of the cylinder surface. High local total entropy generation rate (STOTALmax = 12344.6) occurs at the same locations where the high entropy generation rate due to the fluid friction occur. The low Bejan number occurs on the walls of the enclosure and side part of the cylinder surface where the convection cells in an enclosure reach directly, and also occurs near the region which is away from the cylinder acting on the heat source. In Fig. 11(b), when the cylinder moves up to d = 0.2, four different secondary vortices in the upper part of the cylinder and a pair of two secondary vortices in each upper corner of the enclosure, which are presented in Fig. 12, shrink in size and are altered in
Fig. 10. Distribution of the local Nusselt numbers on the cylinder surface for four different locations of the cylinder at Pr = 0.7 when Ra = 106.
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Fig. 11. Distribution of isotherms, streamlines, entropy generation due to heat transfer and fluid friction at Ra = 106 and Pr = 0.1 (contour values range from 0 to 1 with 10 levels); (a) d = 0.1, (b) d = 0.2, (c) d = 0.1 and (d) d = 0.2.
shape as the space between the cylinder surface and top wall of the enclosure is extremely narrowed. As a result, a pair of two primary vortices having two vortex cores is widely formed in the left and right parts of the cylinder, and very small scale secondary vortices are formed near the top and bottom walls of the enclosure. As a pair of two inner vortices formed in upper part of the cylinder is very small in size, most of the convection flow in the enclosure flows following the circulating motion in the inner vortices having a vortex core located in the lower part of the cylinder. Due to this, the isotherms in the upper region of the cylinder are almost uniformly distributed in the horizontal direction (x) whereas the isotherms near the upper-left and upper-right of the cylinder surface are stretched to each side wall of the enclosure. As a result, high local entropy generation rates due to heat transfer (SHTImax = 388.271) occur in the regions slightly off-centered on the lower surface of the cylinder. The high local entropy generation
rate due to fluid friction (SFFImax = 12832.4) and high local total entropy generation rate (STOTALmax = 13120.3) occur near the lower-left and lower-right of the cylinder surface, and moderate high local entropy generation rates due to fluid friction occur near the upper-left and upper-right of each side wall of the enclosure. The Bejan number are altered in distribution, compared to that at d = 0.1. The low Bejan number occurs near the side and lower space in an enclosure due to the each primary vortex having high convection velocity, but high Bejan number occurs near the cylinder surface and upper region of the enclosure. In Fig. 11(c), when the cylinder is located at d = 0.1, a pair of two primary vortices showing a bi-cellular structure is formed in the left and right parts of the cylinder. In addition, very small scale secondary vortices are generated near the upper surface of the cylinder and walls of the enclosure. A single ascending plume is generated by the upward motion of two upper inner vortices,
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Fig. 11 (continued)
and a pair of two plumes stretched in the horizontal direction (x) is generated by the flow circulating motion in each lower inner vortex. In the upper inner vortex which is larger in size than the lower one, the fluid cooled by the cold top wall of the enclosure moves down and is directed to the upper-left and upper-right of the hot cylinder surface, following the flow circulating motion in each upper inner vortex. As a result, the thermal boundary layer near the upper-left and upper-right of the cylinder surface is attached further by the fluid motion. Therefore, high local entropy generation rates due to heat transfer (SHTImax = 406.307) occur near the upper-left and upper-right of the cylinder surface, moderate high local entropy generation rates due to heat transfer occur near the top wall of the enclosure due to the effect of a single ascending plume. High local entropy generation rate due to fluid friction (SFFImax = 71782.6) and high local total entropy generation rate (STOTALmax = 72188.9) occur near the upper-left and upper-right of the cylinder surface. The high Bejan number distributes near the
lower sides of the enclosure and the lower region of the cylinder surface due to the fluid motion with low inner vortices having relatively low convection velocity compared to the upper inner vortices and the narrow space between the cylinder surface and bottom wall of the enclosure, respectively. In Fig. 11(d), when the cylinder moves down to d = 0.2, the overall flow and thermal structures at d = 0.2 are very similar to those at d = 0.1. However, the upper inner vortices increases and the lower inner vortices decreases in size as the cylinder moves down to d = 0.2 from d = 0.1. Therefore, the locations at which high local entropy generation rates due to heat transfer (SHTImax = 431.983) and fluid friction (SFFImax = 53035.8) and high total entropy generation rate (STOTALmax = 53452.4) occurs are almost the same as the locations at d = 0.1. And overall distribution of the Bejan number is also almost the same as that at d = 0.1. Fig. 12 shows distribution of the local Nusselt number on the cylinder surface Nucyl at four different locations of the cylinder
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Fig. 12. Distribution of the local Nusselt numbers on the cylinder surface for four different locations of the cylinder at Pr = 0.1 when Ra = 106.
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when Ra = 106 at Pr = 0.1. When the cylinder is located at d = 0.1, four high peaks of Nucyl occur at u = 30°, u = 330° and u = 160°, u = 200°. Three low peaks of Nucyl occur at u = 0° (or u = 360°), u = 75° and u = 285° due to the downward and upward motions of four different secondary vortices formed in the upper part of the cylinder, and another single low peak occurs at u = 180°. The value of Nucyl is maximized at u = 30° and u = 330° and minimized at u = 75° and u = 285°. When the cylinder moves up to d = 0.2, the values of Nucyl remain almost unchanged in 0° 6 u 6 30° and 330° 6 u 6 360°. Two low peaks of Nucyl occur at u = 70° and u = 290°, which are the minimum values. High values of Nucyl are distributed in 150° 6 u 6 210° in which Nucyl is maximized. When the cylinder is located at d = 0.1, four high peaks of Nucyl occur at u = 55°, u = 305° and u = 150°, u = 210°, and the value is maximized at u = 55° and u = 305°. Four low peaks of Nucyl occur at u = 0° (or u = 360°), u = 90°, u = 180° and u = 270°, and the value is minimized at u = 0° (or u = 360°). When the cylinder moves down to d = 0.2, the distribution pattern and values of Nucyl at d = 0.2 are very similar to those at d = 0.1.
Fig. 13. Variation in the volume-averaged entropy generation rates as a function of the Grashof number; (a) hSHTIi and (b) hSFFIi.
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Fig. 14. Variation in Bejan numbers Be as a function of the Grashof number.
3.5. Volume-averaged entropy generation and Bejan number Fig. 13(a) and (b) shows the variation in the volume-averaged entropy generation rates due to heat transfer hSHTI i and fluid friction hSFFI i as a function of the Grashof number Gr, respectively, at four different Rayleigh numbers and four different locations of the cylinder. Here, the Grashof number is defined as Gr ¼ Ra=Pr. In Fig. 13(a), the values of hSHTI i at Ra=104 are almost the same as those of Ra = 103 because the convection velocity in the enclosure is very weak and hence the heat conduction is a dominant heat transfer process while the Rayleigh number increases from Ra = 103 to Ra = 104. The values of hSHTI i increase substantially with increasing Rayleigh number to Ra = 105 and Ra = 106 from Ra = 104. In Fig. 13(b), the values of hSFFI i at Ra = 103 are almost zero with the order of 103. When the Rayleigh number increases to Ra = 104 and Ra = 105, the values of hSFFI i at Ra = 104 and Ra = 105 increase up to the order of 101 and 101, respectively. When the Rayleigh number increases further to Ra = 106, the values of hSFFI i increase significantly up to the order of 102. In Fig. 13(a) and (b), hSHTI i and hSFFI i at relatively low Rayleigh numbers of Ra = 103 and Ra = 104 remain almost unchanged according to the variation in the Prandtl number regardless of the variation in the location of the cylinder. However, the Prandtl number dependency on the entropy generation is clearly identified at relatively high Rayleigh numbers of Ra = 105 and Ra = 106. In Fig. 13(a) and (b), hSHTI i and hSFFI i decreases slightly with decreasing Prandtl number from Pr = 100 to Pr = 0.7 and then steeply decreases by reducing the Prandtl number from Pr = 0.7 to Pr = 0.01 at Ra = 105 and Ra = 106. Fig. 14 shows the variation in the Bejan number Be as a function of the Grashof number Gr at four different Rayleigh numbers and four different locations of the cylinder. In this figure, the Bejan numbers are distributed in the vicinity of Be ¼ 1 at Ra = 103 and Ra = 104. Therefore, the irreversibility in the system is dominantly produced by the heat transfer at a relatively low Rayleigh numbers of Ra = 103 and Ra = 104. At a relatively high Rayleigh numbers of Ra = 105 and Ra = 106, the Bejan number is almost independent on the variation in the Prandtl number at each Rayleigh number and each location of the cylinder in the Prandtl number range of 0.7 6 Pr 6 100 but increases steeply by reducing the Prandtl number from Pr = 0.7 to Pr = 0.01. When the Rayleigh number increases to Ra = 105 from Ra = 104, the Bejan numbers are reduced and distributed below Be ¼ 0:5, except two cases of d = 0.1 and d = 0.2 at
Pr = 0.01. In these two cases, the heat transfer is still dominant mechanism to produce the irreversibility in the system rather than the fluid friction. When the Rayleigh number increases further to Ra = 106, all Bejan numbers are distributed below Be ¼ 0:5, which means that the fluid friction is a dominant mechanism to produce the irreversibility in the system.
4. Summary and conclusions In this study, two-dimensional numerical simulations were conducted to investigate the effect of the variation in the vertical position of the inner cylinder on the characteristics of the natural convection in the enclosure in the Rayleigh number range of 103 6 Ra 6 106 and Prandtl number range of 102 6 Pr 6 102. The flow instability is significantly intensified at a relatively high Rayleigh numbers of Ra = 105 and Ra = 106 and a relatively low Prandtl numbers of Pr = 0.1 and Pr = 0.01, which leads to the unsteadiness of the flow and thermal fields. The state of the flow in the system (i.e. flow unsteadiness and complexity of convection cells) is directly correlated with the Prandtl number at a certain Rayleigh number. And the configuration of the system including the geometry and position of the inner body and geometry of the enclosure is also one of crucial factors to affect the state of the flow. In this study, the ranges of the Rayleigh number and Prandtl number in which the flow experiences the transition from the steady state to the unsteady one are consistent with those reported in the companion paper [24]. In this study, the flow bifurcation problem showing an asymmetric flow structure in the enclosure is not identified for all simulation cases considered in this study, in contrast to the results reported in the companion paper [24]. At a relatively low Rayleigh numbers of Ra = 103 and Ra = 104, the degree of the irreversibility produced in the system is almost the same regardless of the variation in the location of the cylinder at the same Rayleigh number and Prandtl number. However, when the Rayleigh number increases to Ra = 105 and Ra = 106, the variation in the magnitude of the irreversibility according to the variation in the location of the cylinder becomes large at the same Rayleigh number and Prandtl number, especially at a relatively low Prandtl numbers of Pr = 0.1 and Pr = 0.01, due to the significant change in the structure of convection cells in the enclosure. How-
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ever their orders of the magnitudes are still similar to each other at the same Rayleigh number. Acknowledgement This research was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (No. NRF-2013R1A2A2A01067251). References [1] J.R. Lee, M.Y. Ha, A numerical study of natural convection in a horizontal enclosure with a conducting body, Int. J. Heat Mass Transf. 48 (2005) 3308– 3318. [2] C. Shu, Y.D. Zhu, Efficient computation of natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder, Int. J. Numer. Meth. Fluids 38 (2002) 429–445. [3] D. Angeli, P. Levoni, G. Barozzi, Numerical predictions for stable buoyant regimes within a square cavity containing a heated horizontal cylinder, Int. J. Heat Mass Transf. 51 (2008) 553–565. [4] F. Moukalled, S. Acharya, Natural convection in the annulus between concentric horizontal circular and square cylinders, J. Thermophys. Heat Transfer 10 (1996) 524–531. [5] X. Xu, Z. Yu, Y. Hu, L. Fan, K. Cen, A numerical study of laminar natural convective heat transfer around a horizontal cylinder inside a concentric airfilled triangular enclosure, Int. J. Heat Mass Transf. 53 (2010) 345–355. [6] A. Kumar De, A. Dalal, A numerical study of natural convection around a square, horizontal, heated cylinder placed in an enclosure, Int. J. Heat Mass Transf. 49 (2006) 4608–4623. [7] H.K. Park, M.Y. Ha, H.S. Yoon, Y.G. Park, Changmin Son, A numerical study on natural convection in an inclined square enclosure with a circular cylinder, Int. J. Heat Mass Transf. 66 (2013) 295–314. [8] B.S. Kim, D.S. Lee, M.Y. Ha, H.S. Yoon, A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations, Int. J. Heat Mass Transf. 51 (2008) 1888–1906. [9] D.H. Kang, M.Y. Ha, H.S. Yoon, C. Choi, Bifurcation to unsteady natural convection in square enclosure with a circular cylinder at Rayleigh number of d, Int. J. Heat Mass Transf. 64 (2013) 926–944. [10] J.M. Lee, M.Y. Ha, H.S. Yoon, Natural convection in a square enclosure with a circular cylinder at different horizontal and diagonal locations, Int. J. Heat Mass Transf. 53 (2010) 5905–5919. [11] Y.G. Park, M.Y. Ha, H.S. Yoon, Study on natural convection in a cold square enclosure with a pair of hot horizontal cylinders positioned at different vertical locations, Int. J. Heat Mass Transf. 65 (2013) 696–712. [12] H.S. Yoon, M.Y. Ha, B.S. Kim, D.H. Yu, Effect of the position of a circular cylinder in a square enclosure on natural convection at Rayleigh number of Ra ¼ 105 , Phys. Fluids 21 (2009) 047101. [13] H.J. Lee, J.H. Doo, M.Y. Ha, H.S. Yoon, Effects of thermal boundary conditions on natural convection in a square enclosure with an inner circular cylinder locally heated from the bottom wall, Int. J. Heat Mass Transf. 65 (2013) 435–450. [14] A.A. Mohamad, R. Viskanta, Transient natural convection of low-Prandtlnumber fluids in a differentially heated cavity, Int. J. Numer. Meth. Fluids 13 (1991) 61–81. [15] X. Xu, G. Sun, Z. Yu, Y. Hu, L. Fan, K.F. Cen, Numerical investigation of laminar natural convective heat transfer from a horizontal triangular cylinder to its concentric cylindrical enclosure, Int. J. Heat Mass Transf. 52 (2009) 3176– 3186.
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[16] X. Xu, Z. Yu, Y. Hu, L. Fan, K.F. Cen, Transient natural convective heat transfer of a low-Prandtl-number fluid from a heated horizontal circular cylinder to its coaxial triangular enclosure, Int. J. Heat Mass Transf. 55 (2012) 995–1003. [17] A. Koca, H.F. Oztop, Y. Varol, The effect of Prandtl number on natural convection in triangular enclosures with localized heating from below, Int. Commun. Heat Mass Transfer 34 (2007) 511–519. [18] Z. Yu, X. Xu, Y. Hu, L. Fan, K.F. Cen, Prandtl number dependence of laminar natural convection heat transfer in a horizontal cylindrical enclosure with an inner coaxial triangular cylinder, Int. J. Heat Mass Transf. 53 (2010) 1333– 1340. [19] J.S. Yoo, Prandtl number effect on bifurcation and dual solutions in natural convection in a horizontal annulus, Int. J. Heat Mass Transf. 42 (1999) 3279– 3290. [20] U. Projahn, H. Beer, Darmstadt, Prandtl number effects on natural convection heat transfer in concentric and eccentric horizontal cylindrical annuli, Wärmeund Stoffübertragung 19 (1985) 249–254. [21] Z. Yu, X. Xu, Y. Hu, L. Fan, K.F. Cen, Transient natural convective heat transfer of a low-Prandtl-number fluid inside a horizontal circular cylinder with an inner coaxial triangular cylinder, Int. J. Heat Mass Transf. 53 (2010) 5102–5110. [22] J.R. Lee, Numerical study for Prandtl number dependency on natural convection in an enclosure with square adiabatic body, Korean Soc. Comput. Fluids Eng. 16 (2011) 29–36. [23] A. Bouras, M. Djezzar, C. Ghernoug, Numerical simulation of natural convection between two elliptical cylinders: Influence of Rayleigh number and Prandtl number, Energy Proc. 36 (2013) 788–797. [24] G.S. Mun, J.H. Doo, M.Y. Ha, Thermo-dynamic irreversibility induced by natural convection in square enclosure with inner cylinder (Part-I: Effect of tilted angle of enclosure), Int. J. Heat Mass Transf. 97 (2016) 1102–1119. [25] J. Kim, D. Kim, H. Choi, An immersed-boundary finite volume method for simulations of flow in complex geometries, J. Comput. Phys. 171 (2001) 132– 150. [26] J. Kim, H. Choi, An immersed-boundary finite-volume method for simulation of heat transfer in complex geometries, KSME Int. J. 18 (2004) 1026–1035. [27] C. Choi, A numerical study of the effect of a circular cylinder on natural convection in rectangular channel heated from bottom wall (Ph.D thesis), Pusan National University, 2014, pp. 18–20. [28] H. Choi, P. Moin, Effects of the computational time step on numerical solutions of turbulent flow, J. Comput. Phys. 113 (1994) 1–4. [29] M.A. Magherbi, A. Abbassi, A.B. Brahim, Entropy generation at the onset of natural convection, Int. J. Heat Mass Transf. 46 (2003) 3441–3450. [30] T. Basak, R. Anandalakshmi, P. Kumar, S. Roy, Entropy generation vs energy flow due to natural convection in a trapezoidal cavity with isothermal and non-isothermal hot bottom wall, Energy 37 (2012) 514–532. [31] G.G. Ilis, M. Mobedi, B. Sunden, Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls, Int. Commun. Heat Mass Transfer 35 (2008) 696–703. [32] S. Bhardwaj, A. Dalal, S. Pati, Influence of wavy wall and non-uniform heating on natural convection heat transfer and entropy generation inside porous complex enclosure, Energy 79 (2015) 467–481. [33] A. Bejan, Entropy generation minimization: the new thermodynamics of finitesize devices and finite-time processes, J. Appl. Phys. 79 (1996) 1191–1218. [34] A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, FL, 1982. [35] GH.R. Kefayati, FDLBM simulation of entropy generation due to natural convection in an enclosure filled with non-Newtonian nanofluid, Powder Technol. 273 (2015) 176–190. [36] GH.R. Kefayati, FDLBM simulation of entropy generation in double diffusive natural convection of power-law fluids in an enclosure with Soret and Dufour effects, Int. J. Heat Mass Transf. 89 (2015) 267-290.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Thermo-dynamic irreversibility induced by natural convection in square enclosure with inner cylinder. Part-II: Effect of vertical position of inner cylinder Jeong Hoon Doo b, Gi Su Mun a, Man Yeong Ha a,⇑, Seon Yoo Seong a a b
School of Mechanical Engineering, Pusan National University, Jang Jeon 2-Dong, Geum Jeong Gu, Busan 609-735, Republic of Korea Rolls-Royce and Pusan National University Technology Centre in Thermal Management, Jang Jeon 2-Dong, Geum Jeong Gu, Busan 609-735, Republic of Korea
a r t i c l e
i n f o
Article history: Received 24 June 2015 Received in revised form 15 February 2016 Accepted 18 February 2016 Available online 10 March 2016 Keywords: Natural convection Vertical position of inner cylinder Entropy generation
a b s t r a c t Two-dimensional numerical simulations were conducted to investigate the natural convection heat transfer induced by the temperature difference between cold walls of the square enclosure and a hot inner circular cylinder in the Prandtl number range of 102 6 Pr 6 102 and Rayleigh number range of 103 6 Ra 6 106. In this paper as a sequent research of Mun et al. (2015), the additional geometrical configuration of the system, which is the variation in the vertical position of the inner cylinder, was considered. The change in the structure of convection cells and corresponding heat transfer characteristics were analyzed, which are induced by the variation in the vertical position of the inner cylinder located in the enclosure. And the characteristics of the entropy generation due to heat transfer and fluid friction associated with the flow and thermal structures were addressed in this study. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Buoyancy-induced flow in an enclosure attracts academic attention due to a variety of its applications including solar collector receivers, heat exchangers, nuclear reactors and electronic packaging. Many researchers have investigated parameters that affect the natural convection characteristics in an enclosure and reported that the characteristics of the natural convection in an enclosure is influenced by the geometry and position of an inner body, the configuration of the enclosure and thermo-physical properties of working fluids [1–23]. The accompanying study presented in [24] undertook twodimensional numerical study on the natural convection characteristics in the square enclosure having an inner circular cylinder in the Rayleigh number range of 103 6 Ra 6 106 and the Prandtl number range of 102 6 Pr 6 102, and the natural convection flow is driven by the temperature difference between a hot cylinder surface and cold walls of the enclosure. In the accompanying study [24], authors have focused on the effect of the variation in the tilted angle of the enclosure (0° 6 c 6 45°) on the natural convection characteristics and the production of the thermo-dynamic irreversibility induced by the natural convection was addressed ⇑ Corresponding author. Tel.: +82 51 510 2440; fax: +82 51 515 3101. E-mail address: [email protected] (M.Y. Ha). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.02.054 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
through the analysis on the local entropy generation due to heat transfer and fluid friction. In addition, the heat transfer characteristics on the cylinder surface and walls of the enclosure associated with the structure of the convection cell formed in the enclosure were properly analyzed. As a summary of the accompanying study [24], the transition of the flow regime between the steady state to the unsteady one occurs from Ra = 105 and Pr = 0.1 except cases in the tilted angle range of 15° 6 c 6 30°. As reported previously, the state of the system is directly dependent on the thermal boundary condition and fluid property. Also, the state of the system is influenced by the geometrical configuration of the system, which means that the system at Ra = 105 and Pr = 0.1 shows the quasistate between the steady state and the unsteady one depending on the geometrical configuration of the system. In this paper as a sequent research of [24], therefore, authors focus on the additional geometrical configuration of the system, which is the variation in the vertical position of the inner cylinder, in the same ranges of the Rayleigh number and Prandtl number as those considered in [24]. The change in the structure of convection cells and corresponding heat transfer characteristics were carefully analyzed according to the variation in the vertical position of the inner cylinder located in the enclosure. And the characteristics of the entropy generation due to heat transfer and fluid friction associated with the flow and thermal structures were addressed in this study.
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Nomenclature Be fi g k L n Nu P Pr q R Ra S t T ui xi
c
Bejan number momentum forcing gravitational acceleration thermal conductivity length of the enclosure normal direction to the wall local Nusselt number dimensionless pressure Prandtl number mass source or sink radius of circular cylinder Rayleigh number distance along the enclosure dimensionless time temperature dimensionless velocity vector Cartesian coordinates system
q l m u h
Sub/superscripts dimensional value cyl cylinder en enclosure Abbreviations FFI fluid friction irreversibility HTI heat transfer irreversibility Mathematical symbol surface-averaged quantity hi volume-averaged quantity
Greek symbols a thermal diffusivity b thermal expansion coefficient
Here, the dimensionless governing equations (4)–(6) are obtained from the dimensional governing equations (1)–(3) using the dimensionless variables defined as follows:
2. Computational details 2.1. Numerical methods In this study, the numerical method is exactly the same as those used in the accompanying study [24]. The immersed boundary method was used to capture the virtual boundary of the inner circular cylinder in the Cartesian coordinate system. Fluid considered in this study is incompressible and Newtonian. Fluid flow is assumed to be laminar in the absence of heat generation, chemical reactions and thermal radiation. And also viscous dissipation in the energy equation has been neglected. Based on these assumptions, the dimensional governing equations of mass, momentum and energy for unsteady incompressible viscous flow and thermal fields are expressed as follows:
@ui ¼0 @xi
ð1Þ
"
# @ui @ 2 u @P @ui q þ uj ¼ þ l i þ qgbðT T c Þðsin udi1 þ cos udi2 Þ @t @xj @xi @xj @xj ð2Þ @T @T @2T þ uj ¼ a @t @xj @xj @xj
ð3Þ
The governing equations to which the immersed boundary method is applied are the continuity, momentum and energy equations in their non-dimensional forms, which are expressed as follows:
@ui q¼0 @xi
ð4Þ
t¼
t a 2
L
;
xi ¼
xi ; L
ui ¼
@ui @ui @P @ ui ¼ þ Pr þ RaPrhðsin udi1 þ cos udi2 Þ þ f i þ uj @xi @t @xj @xj @xj ð5Þ ð6Þ
ui L
a
;
P¼
P L2
qa2
;
h¼
T Tc Th Tc
ð7Þ
In the above equations, q, T, and a represent the density, dimensional temperature and thermal diffusivity, respectively. The superscript ⁄ in Eq. (7) represents the dimensional variables. xi is the dimensionless Cartesian coordinate, ui is the corresponding dimensionless velocity component, t is the dimensionless time, P is the dimensionless pressure and h is the dimensionless temperature. The above non-dimensionalization results in two dimensionless parameters are Pr ¼ m=a and Ra ¼ gbL3 ðT h T c Þ=ma, where m; g, and b are the kinematic viscosity, gravitational acceleration, and volume expansion coefficient, respectively. The terms of q; f i , and h in Eqs. (4)–(6) are related to the immersed boundary method. The mass source/sink q in Eq. (4) and momentum forcing f i in Eq. (5) are imposed on the body surface and inside the body to satisfy the no-slip condition and mass conservation in the cell containing the virtual boundary. In Eq. (6), the heat source/sink h is applied to satisfy the iso-thermal boundary condition at the virtual boundary. The momentum forcing term f i and heat source/sink term h in Eqs. (5) and (6) were obtained by
fi ¼
h¼
U i uni 3 1 @Pn Þþ þ NLðuni Þ NLðun1 i 2 2 Dt @xi rffiffiffiffiffiffi 1 Pr ½DIFðunþ1 Þ þ DIFðuni Þ hn di2 i 2 Ra
ð8Þ
H hn 3 1 1 þ NLðhn Þ NLðhn1 Þ pffiffiffiffiffiffiffiffiffiffi ½DIFðhnþ1 Þ 2 2 Dt 2 RaPr þ DIFðhn Þ hn di2
2
@h @h @2h þ uj ¼ þh @t @xj @xj @xj
tilted angle of the enclosure Kronecker delta density dynamic viscosity kinematic viscosity angle of circular cylinder dimensionless temperature
di2
ð9Þ
where U i and H represented the desired velocity and temperature to realize the viscous and thermal boundary conditions on the surface of the immersed object, and superscripts n and n þ 1 in Eq. (9) represent the time levels. The nonlinear term NLðuÞ and diffusion term DIFðuÞ in Eqs. (8) and (9) are defined as:
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NLðui Þ ¼
@uj ui ; @xj
DIFðui Þ ¼
@ 2 ui @xj @xj
ð10Þ
A second-order linear or bilinear interpolation scheme was applied to satisfy the no-slip and isothermal conditions on the immersed boundary. Further details for the immersed boundary method is described in Kim et al. [25], Kim and Choi [26] and Choi [27]. A second-order accurate finite volume method was used for the spatial discretization of governing equations (4)–(6). To simulate the time advancement of the flow field, the fractional step method proposed by Choi and Moin [28] was employed. In the discretization process, the advection terms were treated explicitly using the second-order Adams–Bashforth scheme, and the diffusion terms were treated implicitly using the second-order accurate Crank–Nicolson scheme. Once the velocity and temperature fields are obtained, the local and surface-averaged Nusselt numbers are calculated by the equations below:
Nu ¼
@h ; @nwall
Nu ¼
Z
1 S
S
Nu ds
ð11Þ
0
where, n represents the direction normal to the wall and S the length of surface. The local entropy generation rate was calculated to investigate the irreversibility caused by heat transfer and fluid friction in natural convection. The dimensional equations of local entropy generation rate are expressed as follow:
SHTI ¼
SFFI ¼
kf
"
T 2o
@T @x
2
þ
@T @y
2 #
( " 2 2 # 2 ) @u @v @u @v 2 þ þ þ To @x @y @y @x
l
ð12Þ
ð13Þ
where SHTI and SFFI are the dimensional local entropy generation rates due to heat transfer and fluid friction, respectively. By using dimensionless variables in Eq. (7), the dimensionless form of total entropy generation rate in two-dimensional Cartesian coordinate can be derived as
STotal ¼ SHTI þ SFFI SHTI
" 2 # 2 @h @h ¼ þ @x @y ( " 2 # 2 ) 2 @u @v @u @ v þ þ þ @x @y @x @y
SFFI ¼ / 2
ð14Þ ð15Þ
The Bejan number Be indicating a relative dominance of the irreversibility due to the heat transfer and fluid friction is defined as below:
Be ¼
hSHTI i hSHTI i þ hSFFI i
Be > 0:5 implies that the irreversibility due to heat transfer is dominant, and Be < 0:5 implies that the irreversibility due to the fluid friction is dominant. And Be ¼ 0:5 implies that the heat transfer irreversibility and fluid friction irreversibility has the equal contribution to the total irreversibility [29–36]. When we run the computational cases considered in this study in our computer system, the maximum ram usage required to run our computer code is about 0.15 GB and the maximum run time taken is about 10 days, including the time to obtain the timeaveraged flow and thermal fields. 2.2. Computational conditions A schematic of the system configuration considered in this study is shown in Fig. 1. The system consists of a square enclosure with an inner circular cylinder. The length of each side wall of the enclosure is L, and the radius of the inner cylinder is R = 0.2L, which are exactly the same geometries of the enclosure and inner cylinder as those considered in the accompanying study [24]. In this study, the vertical position of the inner cylinder is considered as a main simulation parameter. d represents the dimensionless distance between the center of the inner cylinder and center of the enclosure, and the position of the cylinder varies in the range of 0.2 6 d 6 0.2 along the vertical centerline of the enclosure as shown in Fig. 1. The non-dimensionless temperatures of hc ¼ 0 (cold) and hh ¼ 1 (hot) were imposed on walls of the enclosure and surface of the inner cylinder, respectively. No-slip and impermeability conditions were imposed on walls of the enclosure and the surface of the inner cylinder. The Boussinesq approximation was used to model the variation in the fluid density due to the change in the fluid temperature. The gravitational acceleration is set in the negative y-direction. The grid system used in this study is exactly the same as that used in the accompanying study [24] in terms of the number and resolution of grids. Extra numerical simulations were conducted to verify the accuracy of the present numerical method, and detailed results of the validation test in terms of the surfaceaveraged Nusselt number on the cylinder surface and local entropy generation rates are presented in the companion paper [24].
ð16Þ
where SHTI and SFFI are the dimensionless local entropy generation rates due to heat transfer and fluid friction, respectively. In above equation, / is called the irreversibility distribution ratio, which is defined as
/¼
lT o a 2 k
LDT
ð17Þ
In the present study, / is taken as 104, based on the result reported by Ilis et al. [31]. The volume-averaged entropy generation rates due to heat transfer and fluid friction and volumeaveraged total entropy generation rate are obtained by Eq. (18). Here, V represents the volume of fluid filled in the enclosure.
Z Z 1 1 SHTI dV; hSFFI i ¼ SFFI dV; V V V V hSTotal i ¼ hSHTI i þ hSFFI i
hSHTI i ¼
ð18Þ
ð19Þ
Fig. 1. Computational domain and its coordinate system.
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3. Results 3.1. Transition of the flow regime Table 1 shows the transition of the flow regime from the steady state to the unsteady one. In this study, numerical solutions show the time-independent characteristics in a relatively low Rayleigh
number range of 103 6 Ra 6 104 regardless of variations in the Prandtl number and location of the inner cylinder. When Ra = 105 and Ra = 106, all numerical solutions still show the timeindependent characteristics in a relatively high Prandtl number range of 0.7 6 Pr 6 100. However, when the Prandtl number decreases to Pr = 0.1 and Pr = 0.01 at Ra = 105 and Ra = 106, the flow experiences the transition from the steady state to the unsteady
Table 1 Flow transition from steady state to unsteady state for different Prandtl number and different location of the cylinder.
S : Steady, U : Unsteady Ra
105
106
Pr 0.01 0.1 0.7 7 100 0.01 0.1 0.7 7 100
Location of cylinder inside enclosure δ -0.2
-0.1
0
0.1
0.2
U S S S S U U S S S
U S S S S U U S S S
U U S S S U U S S S
U S S S S U S S S S
U S S S S U U S S S
Fig. 2. Distribution of the surface-averaged Nusselt numbers on the cylinder surface as a function of the Prandtl number at the Rayleigh numbers of Ra = 103, Ra = 104, Ra = 105 and Ra = 106; (a) d = 0.1, (b) d = 0.2, (c) d = 0.1 and (d) d = 0.2.
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one, depending on the location of the cylinder. When Pr = 0.1 at Ra = 105, the case only that the inner cylinder is located at the center of the enclosure (d = 0.0) shows the unsteady characteristics in the flow and thermal fields. When the Prandtl number is reduced to Pr = 0.01 at Ra = 105, all numerical solutions show the timedependent characteristics regardless of the variation in the location of the inner cylinder. When the Rayleigh number increases from Ra = 105 to Ra = 106, the flow instability in the enclosure is intensified further by a strong buoyancy effect. As a result, all numerical solutions at Pr = 0.1 and Ra = 106 show the timedependent characteristics regardless of the variation in the location of the inner cylinder, except the case that the inner cylinder is located at d = 0.1 showing the steady state characteristics. When the Prandtl number is reduced to Pr = 0.01 at Ra = 106, all numerical solutions show the unsteady state characteristics in the flow and thermal fields. As shown in this table, the ranges of the
Rayleigh number and Prandtl number in which the flow experiences the transition from the steady state to the unsteady one are consistent with to those reported in the companion paper [24]. In this study, the data are obtained from the time-averaged flow and thermal fields for cases showing the time-dependent characteristics in the numerical solution. Fig. 2 shows the distribution of the surface-averaged Nusselt number on the cylinder surface Nucyl as a function of the Prandtl number at four different Rayleigh numbers (Ra = 103, Ra = 104, Ra = 105 and Ra = 106) and four different locations of the cylinder (d = 0.1, d = 0.2, d = 0.1 and d = 0.2). As reported in the companion paper [24], the Prandtl number is an intrinsic thermo-physical property of the fluid, which means that the state of the flow in the system (i.e. flow unsteadiness and complexity of convection cells) is directly correlated with the Prandtl number at a certain Rayleigh number. In these figures, the values of Nucyl remain almost
Fig. 3. Distribution of isotherms, streamlines, entropy generation due to heat transfer and fluid friction at Ra = 104 and Pr = 0.1 (contour values range from 0 to 1 with 10 levels); (a) d = 0.1, (b) d = 0.2, (c) d = 0.1 and (d) d = 0.2.
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unchanged according to the variation in the Prandtl number regardless of the location of the cylinder at a relatively low Rayleigh numbers of Ra = 103 and Ra = 104. However, the Prandtl number dependency on the flow state is clearly visible at a relatively high Rayleigh numbers of Ra = 105 and Ra = 106 and shows a consistent trend of the variation in Nucyl regardless of the location of the inner cylinder. At Ra = 105 and Ra = 106, the values of Nucyl decreases very slightly as the Prandtl number decreases from Pr = 100 to Pr = 0.7 but steeply decreases by reducing the Prandtl number from Pr = 0.1 to Pr = 0.01, showing the change in the slope of Nucyl . In light of this feature, it can be expected that a critical Prandtl number at which the transition of the flow state initiates exists in 0.1 6 Pr 6 0.7 as reported in the companion paper [24]. Therefore, the change in the flow and thermal structures and consequent irreversibility production between Pr = 0.7 and Pr = 0.1 were highlighted in this paper.
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3.2. Rayleigh number of Ra = 103 and Ra = 104 Fig. 3(a)–(d) shows distributions of isotherms, streamlines, local entropy generation rates due to heat transfer and fluid friction, the local total entropy generation and the Bejan number at Pr = 0.1 and Ra = 104 for four different locations of the cylinder. In cases of Ra = 103 and Ra = 104, the flow and thermal structures at Ra = 103 are very similar to those at Ra = 104 at the same Prandtl number and location of the cylinder because the buoyancy effect is still low at Ra = 104. In addition, a significant change in the flow and thermal structures according the variation in the Prandtl number is not identified in this Rayleigh number range as aforementioned in Section 3.1. In Fig. 3(a), when the inner cylinder is located in the upper side of the enclosure at d = 0.1, a pair of two primary vortices is formed in the left and right parts of the enclosure. Each primary vortex has
Fig. 3 (continued)
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a bi-cellular structure with two vortex cores which are separately located in upper side and lower side of the enclosure. And the lower inner vortex is much larger in size than the lower one. In this case, isotherms near the upper side of the cylinder are very slightly denser than those near the lower side of the cylinder. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 38.576) occurs near the upper-center of the cylinder surface. High local entropy generation rates due to fluid friction (SFFImax = 9.270) occur near the lower-left and lower-right of the cylinder surface due to the flow circulating motion in two lower inner vortices rotating in the counter-clockwise (left one) and clockwise (right one) directions. The distribution of the local total entropy generation rate (STOTALmax = 38.57) is similar to that of the entropy generation rate due to the heat transfer because the conduction is a relatively dominant heat transfer mode rather than the convection due to a very weak convection velocity of primary vortices when Ra = 104. Therefore, the Bejan number has high values in the enclosure except for corner regions, the side and bottom walls of the enclosure. In Fig. 3(b), when the cylinder moves up to d = 0.2, a bi-cellular structure of each primary vortex is altered to a uni-cellular one having a single vortex core which is located in the lower part of the enclosure. The isotherms near the upper side of the cylinder become denser due to a shorter distance between the hot cylinder surface and the cold top wall of the enclosure as shown in Fig. 3(b). As a result, a high local entropy generation rate due to heat transfer (SHTImax = 129.24) occurs near the uppercenter of the cylinder surface, and high local entropy generation rates due to fluid friction (SFFImax = 8.981) occurs at the same locations as the case of d = 0.1. The high local total entropy generation rate (STOTALmax = 129.24) occurs at the same locations where the high entropy generation rate due to the heat transfer occurs. The overall distribution of the Bejan number is also similar to that of d = 0.1. In Fig. 3(c), when the cylinder is located at d = 0.1, a pair of two primary vortices shows a bi-cellular structure, and the upper inner vortex in each primary vortex becomes larger in size than the lower one. Therefore, the flow structure at d = 0.1 shows a pattern which is upside down about the horizontal centerline of the enclosure, compared to the case of d = 0.1. In this case, isotherms become dense near the lower side of the cylinder due to a shorter distance between the hot cylinder surface and the cold bottom wall but coarse near the upper side of the cylinder due the upward motion of the fluid in two primary vortices, showing a weak ascending plume. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 48.931) occurs near the lower-center of the cylinder surface. High local entropy generation rates due to fluid friction (SFFImax = 16.768) occur near the upperleft and upper-right of the cylinder surface due to the flow circulating motion in the upper inner vortices. The distribution of the local total entropy generation rate (STOTALmax = 48.934) is similar that of the local entropy generation rate due to the heat transfer. The Bejan number has a similar pattern which is upside down about the horizontal centerline of the enclosure, compared to the case of d = 0.1. In Fig. 3(d), when the cylinder moves down to d = 0.2, a bi-cellular structure of each primary vortex presented at d = 0.1 is altered to a uni-cellular one. In this case, an ascending plume near the upper surface of the cylinder is intensified further, compared to the case at d = 0.1. Following the plume intensified, the isotherms become coarser near the upper surface of the cylinder. Very dense isotherms are identified near the lower surface of the cylinder. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 131.561) occurs near the lower-center of the cylinder surface. High local entropy generation rate due to fluid friction (SFFImax = 21.069) occurs at the same locations as the case of d = 0.1. The maximum local total entropy generation rate (STOTALmax = 131.561) occurs near the lower-center of the cylinder surface. The irreversibility caused by fluid friction is intensified
Fig. 4. Distribution of the local Nusselt numbers on the cylinder surface for four different locations of the cylinder at Pr = 0.1 when Ra = 103 and Ra = 104.
near the upper side part of the cylinder surface due to the change in the location of the cylinder. As a result, the low Bejan number occurs near the upper part of the cylinder in contrast to the case at d = 0.1. Fig. 4 shows distribution of the local Nusselt number on the cylinder surface Nucyl at four different locations of the cylinder at Pr = 0.1 when Ra = 103 and Ra = 104. When d = 0.1 and d = 0.2 at Ra = 103, the maximum value of Nucyl occur at u = 0° (or u = 360°) and the minimum value at u = 180°. When the Rayleigh number increases to Ra = 104 from Ra = 103, the maximum value of Nucyl is still identified at u = 0° (or u = 360°) for both cases of d = 0.1 and d = 0.2 and is significantly augmented at both Ra = 103 and Ra = 104 as the cylinder moves up to d = 0.2 from d = 0.1. When d = 0.1 and Ra = 104, the values of Nucyl are slightly augmented in the lower part of the cylinder (120° 6 u 6 240°) showing a moderate high peak at u = 180° due to a slight increase in the buoyancy effect, compared to the case of d = 0.1 and Ra = 103. When d = 0.1 and d = 0.2, the distribution pattern of Nucyl at Ra = 104 is very similar to that at Ra = 103, showing the maximum value at u = 180° and the minimum value at u = 0° (or u = 360°). However, values of Nucyl at Ra = 104 are slightly augmented near u = 180° and slightly reduced near u = 0° (or u = 360°) due to the increase in the convection velocity, compared to the cases at Ra = 103. 3.3. Rayleigh number of Ra = 105 Fig. 5(a)–(d) shows distributions of isotherms, streamlines, local entropy generation rates due to heat transfer and fluid friction, the local total entropy generation and the Bejan number at Pr = 0.7 and Ra = 105 for four different locations of the cylinder. When the Rayleigh number increases from Ra = 104 and Ra = 105, the buoyancy effect is substantially intensified in the enclosure. Therefore, the core of each primary vortex keeps staying in the upper part of the cylinder regardless of the location of the cylinder, and the isotherms are distorted significantly, following the fluid motion of the convection cells formed in the enclosure as shown in Fig. 5(a)–(d). In Fig. 5(a), when the cylinder is located at d = 0.1, a pair of two primary vortices having a single vortex core is formed in the left and right parts of the enclosure, and a pair of two small scale secondary vortices is generated near the upper-center part of the cylinder, which is caused by the flow separation of the primary vortices near the upper part of the cylinder surface. Following the fluid motion of the primary and secondary vortices, two ascending plumes and a single descending plume are generated, which makes isotherms
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distorted in the upper part of the cylinder. The isotherms near the lower surface of the cylinder are attached to the cylinder surface due to a strong upward fluid motion in the primary vortices. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 127.706) occurs near the lower-center of the cylinder surface in contrast to the case at d = 0.1 and Ra = 104 presented in Fig. 3(a). High local entropy generation rates due to fluid friction (SFFImax = 748.74) occur near the middle-left and middle-right of the cylinder surface, and moderate high entropy generation rates due to fluid friction occur near the upper region of each side wall of the enclosure. The magnitude of the irreversibility caused by fluid friction is more intensified than that of the irreversibility caused by heat transfer in the enclosure with increasing Rayleigh number. Therefore, the overall distribution of the local total entropy generation rate is similar to that of local entropy generation due to fluid friction. As a result, high local total entropy
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generation rate (STOTALmax = 808.192) occurs near the middle-left and middle-right of the cylinder surface. Low Bejan number occurs near the side and lower region of the cylinder including the side surface of the cylinder and side walls of the enclosure due to the high convection velocity of primary vortices, and high Bejan number occurs near the upper region of the cylinder where a pair of secondary vortices having the weak convection velocity is located. In Fig. 5(b), when the cylinder moves up to d = 0.2, the space between the top wall of the enclosure and the cylinder surface is extremely confined. As a result, most of the convection flow with a high momentum which moves along the side surface of the cylinder keeps moving up to the top wall of the enclosure, and a very small portion of the convection flow is entrained to the space confined extremely, having a very low momentum. Therefore, a pair of two secondary vortices formed in the upper-center part of the cylinder at d = 0.1 vanishes as the cylinder is located at d = 0.2,
Fig. 5. Distribution of isotherms, streamlines, entropy generation due to heat transfer and fluid friction at Ra = 105 and Pr = 0.7 (contour values range from 0 to 1 with 10 levels); (a) d = 0.1, (b) d = 0.2, (c) d = 0.1and (d) d = 0.2.
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and a pair of two ascending plumes is generated on the upper-left and upper-right surfaces of the cylinder, following the convection flow with a high momentum. When the cylinder moves up to d = 0.2 from d = 0.1, the location at which high local entropy generation rate due to heat transfer (SHTImax = 125.792) occurs is still observed near the lower-center of the cylinder surface whereas the location at which high local entropy generation rates due to fluid friction (SFFImax = 609.250) occur moves to the lower-left and lower-right of the cylinder surface from middle-left and middle-right of the cylinder surface presented in Fig. 5(a). The high local total entropy generation rate (STOTALmax = 682.012) occurs at the same locations where the high entropy generation rate due to the heat transfer occurs. The overall pattern of the Bejan number distribution is also similar to that of d = 0.1. In Fig. 5(c), when the cylinder is located at d = 0.1, a single ascending plume induced by the fluid motion of two primary vortices is clearly visible in the upper part of the cylinder. Due to the ascending plume
intensified, the isotherms near the top wall of the enclosure become dense even though the cylinder is located in the lower part of the enclosure. As the space between the cylinder surface and bottom wall of the enclosure is confined, the magnitude of the convection velocity in the lower-center region of the cylinder is lower than that in the region off-centered slightly, which means that the thermal boundary layer near the off-centered region on the lower surface of the cylinder is thinner than that in the center region. As a result, high local entropy generation rate due to heat transfer (SHTImax = 118.490) occurs near the lower-left and lower-right of the cylinder surface rather than near the lower-center region. High local entropy generation rate due to fluid friction (SFFImax = 1177.91) occurs near the upper-left and upper-right of the cylinder surface due to the increase in the size of the upper inner vortices. The high local total entropy generation rate (STOTALmax = 1240.94) occurs near upper-left and upper-right of the cylinder which is the same location where the high local
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entropy generation rate due to fluid friction occurs. The Bejan number has low values near the corner region and upper part of the cylinder including the walls of the enclosure and the side surface of the cylinder in general due to the intensified viscous boundary layer generated by the high convection velocity, but high values are located away from the walls of the enclosure and cylinder surface. Especially, the high Bejan number distribution is intensified near the lower region of the cylinder due to the weak fluid motion. In Fig. 5(d), when the cylinder moves down to d = 0.2, the flow and thermal structures at d = 0.2 are very similar to those at d = 0.1. However, the space between the bottom wall of the enclosure and cylinder surface is extremely confined. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 144.837) occurs near the lower-center of the cylinder surface in contrast to the case at d = 0.1 whereas high local entropy generation rate due to fluid friction (SFFImax = 1239.9) and high local total entropy generation rate (STOTALmax = 1239.9) occurs at the same locations as the case at d = 0.1. And overall distribution of the Bejan number is also similar to that of the case at d = 0.1. Fig. 6 shows distribution of the local Nusselt number on the cylinder surface Nucyl at four different locations of the cylinder when Ra = 105 at Pr = 0.7. When the cylinder is located at d = 0.1, two low peaks of Nucyl occur at u = 30° and u = 330° due to the effect of a pair of two ascending plumes presented in Fig. 5(a). The maximum value of Nucyl occurs at u = 180°. When the cylinder moves up to d = 0.2, two ascending plumes are tilted further to each side wall of the enclosure. As a result, two low peaks of Nucyl occur at u = 60° and u = 300°. In this case, the value Nucyl is maximized at u = 180° in contrast to the case of d = 0.2, Ra = 104 and Pr = 0.7 showing the minimum value of Nucyl at u = 180° in Fig. 4, which indicates that the convection velocity in the lower part of the enclosure is significantly augmented. When the cylinder is located at d = 0.1, the minimum value of Nucyl occurs at u = 0° (or at u = 360°) due to the presence of a single ascending plume presented in Fig. 5(c), and two high peaks at u = 140° and u = 220°. When the cylinder moves down to d = 0.2, high values of Nucyl are almost uniformly distributed in 130° 6 u 6 230° due to a very short distance between the cylinder surface and the bottom wall of the enclosure, and the minimum value of Nucyl is still identified at u = 0° (or at u = 360°). Fig. 7(a)–(d) shows distributions of isotherms, streamlines, local entropy generation rates due to heat transfer and fluid friction, the local total entropy generation and the Bejan number at Pr = 0.1 and Ra = 105 for four different locations of the inner cylinder. According to findings in the accompanying study [24], the Grashof number Gr,
Fig. 6. Distribution of the local Nusselt numbers on the cylinder surface for four different locations of the cylinder at Pr = 0.7 when Ra = 105.
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which approximates the ratio of the buoyancy to the viscous force acting on a fluid, increases with decreasing the Prandtl number at the same Rayleigh number, which means that the buoyancy as a driving force in the natural convection becomes strong or the momentum dissipation of the vortex due to the viscous effect becomes weak. Therefore, the complexity of the convection cells in the enclosure increases by reducing the Prandtl number from Pr = 0.7 to Pr = 0.1 at a relatively high Rayleigh number of Ra = 105 as shown in Fig. 7. In Fig. 7(a), when the cylinder is located at d = 0.1, a pair of two primary vortices is formed in left and right parts of the enclosure, having a single vortex core located in the lower part of the cylinder. In addition, several small scale secondary vortices are formed near the bottom wall of the enclosure. In the primary vortex, the fluid cooled moves downward along each side wall of the enclosure. The fluid is re-heated in the lower part of the enclosure and directed to the lower-center region of the cylinder. And then the fluid re-heated moves upward along the cylinder surface. In this fluid motion, a small portion of the fluid reaches the vicinity of the bottom wall of the enclosure, and most of fluid is lifted up toward the lower surface of the cylinder before reaching the bottom wall. As a result, the isotherms in the off-centered region on the lower surface of the cylinder are denser than those in the center region. Therefore, high local entropy generation rates due to heat transfer (SHTImax = 122.481) occur near the lower-left and lower right of the cylinder surface. And high local entropy generation rates due to fluid friction (SFFImax = 831.503) also occur near the lower-left and lower-right of the cylinder surface due to the flow circulating motion in the primary vortices having a single vortex core located in the lower part of the cylinder. The high local total entropy generation rate (STOTALmax = 923.841) occurs near upper-left and upper-right of the cylinder as the same location where the high entropy generation rate due to fluid friction occurs. The Bejan number has high values near the bottom wall of the enclosure and the upper region of the cylinder due to the weak convection velocity because the large amount of circulating fluid is tied up to the near the side parts of the enclosure. In the side parts of the enclosure, the low Bejan number occurs due to the primary vortices having the high convection velocity. In Fig. 7(b), when the cylinder moves up to d = 0.2, the flow structure at d = 0.2 is similar to that at d = 0.1. However, in this case, the secondary vortices which are located at each bottom corner of the enclosure as well as the center region of the bottom wall of the enclosure act as a virtual blunt body. As a result, most of the fluid which is converged to the vertical centerline of the enclosure in the lower part of the enclosure is smoothly directed to the lower-center of the cylinder surface along the virtual wall boundary. Therefore, dense isotherms are distributed in the lower-center of the cylinder surface. Due to this, a high local entropy generation rate due to heat transfer (SHTImax = 136.436) occurs near the lower-center of the cylinder surface in contrast to the case at d = 0.1 whereas high local entropy generation rate due to fluid friction (SFFImax = 498.076) and high local total entropy generation rate (STOTALmax = 603.812) occur at the same locations as the case at d = 0.1. The overall distribution of the Bejan number is similar to that at d = 0.1 except for the expansion of the region having low Bejan number because the fluid is smoothly directed to the lower part of the cylinder due to the change in position of cylinder. In Fig. 7(c), when the cylinder is located at d = 0.1, the primary vortex having a uni-cellular structure at d = 0.1 and d = 0.2 is altered to that having a bi-cellular structure. And the upper inner vortices are larger in size than the lower one. In addition, very small scale secondary vortices are generated in each upper corner of the enclosure and the upper surface of the cylinder. In this case, small portion of the convection flow is entrained to a relatively narrow space between the cylinder surface and the bottom wall of the enclosure. As a result, dense
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Fig. 7. Distribution of isotherms, streamlines, entropy generation due to heat transfer and fluid friction at Ra = 105 and Pr = 0.1 (contour values range from 0 to 1 with 10 levels); (a) d = 0.1, (b) d = 0.2, (c) d = 0.1and (d) d = 0.2.
isotherms are distributed near the lower-left and lower-right of the cylinder surface rather than the lower-center of the cylinder surface. A single ascending plume formed above the upper surface of the cylinder is identified and is stretched to each side wall of the enclosure, following the flow circulating motion in the upper inner vortices. And a pair of two plumes formed on each side of the cylinder surface is stretched in the horizontal direction (x) due to the flow circulating motion in lower inner and upper inner vortices of each primary vortex. High local entropy generation rates due to heat transfer (SHTImax = 104.792) occur near the lower-left and lower-right of the cylinder surface because the thermal boundary layer is attached further to these regions by the flow circulating motion in the lower inner vortices whereas high local entropy generation rate due to fluid friction (SFFImax = 1846.77) and high local total entropy generation rate (STOTALmax = 1940.55) occur near upper-left and upper-right of the cylinder surface. The high Bejan number occurs near lower region of the cylinder,
upper-center of the cylinder surface and center region of the top wall of the enclosure due to the small portion of the convection flow entrained to a relatively narrow space between the cylinder surface and the bottom wall of the enclosure, secondary vortices having weak convection velocity and flow not reaching the center region of the top wall of the enclosure, respectively. The low Bejan number occurs near the upper-center and side region of the enclosure due to two upper inner vortices having strong ascending and descending flow motion. In Fig. 7(d), when the cylinder moves down to d = 0.2 from d = 0.1, the upper inner vortices increase in size while the primary vortex having a bi-cellular structure maintains its structure. And additional small scale secondary vortices are generated near the middle region of each side wall of the enclosure in company with several small scale vortices formed in each upper corner of the enclosure and upper surface of the cylinder. In this case, a single ascending plume formed above the upper surface of the cylinder as well as a pair of two ascending
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Fig. 7 (continued)
plumes near each side of the cylinder surface is identified, similar to the case at d = 0.1. A high local entropy generation rate due to heat transfer (SHTImax = 140.987) occurs near the lower-center of the cylinder surface whereas high local entropy generation rate due to fluid friction (SFFImax = 1373.23) and high local total entropy generation rate (STOTALmax = 1472.56) occur in the same locations as the case at d = 0.1. And overall distribution of the Bejan number is also similar to the case at d = 0.1. Fig. 8 shows distribution of the local Nusselt number on the cylinder surface Nucyl at four different locations of the cylinder when Ra = 105 at Pr = 0.1. When the cylinder is located at d = 0.1, two low peaks of Nucyl occur at u = 80° and u = 280°, which are the minimum values, and two high peaks of Nucyl occur at u = 130° and u = 230°, which are the maximum values. When the cylinder moves up to d = 0.2, two high peaks of Nucyl occur at u = 180° and u = 0° (or u = 360°), and the value at u = 180°, which is the maximum value, is very slightly higher than that at u = 0° (or
u = 360°). Two low peaks of Nucyl occur at u = 80° and u = 280°, which are the minimum values. When the cylinder is located at d = 0.1, two high peaks of Nucyl occur at u = 60° and u = 300°, and another two high peaks of Nucyl occur at u = 140° and u = 220°. The value of Nucyl are maximized at u = 140° and u = 220° and minimized at u = 0° (or u = 360°). When the cylinder moves down to d = 0.2, triple high peaks of Nucyl occur at u = 130°, u = 180° and u = 230°, and two low peaks of Nucyl occur at u = 50° and u = 310°. The value of Nucyl is maximized at u = 180° and minimized at u = 0° (or u = 360°). 3.4. Rayleigh number of Ra = 106 Fig. 9(a)–(d) shows distributions of isotherms, streamlines, local entropy generation rates due to heat transfer and fluid friction, the local total entropy generation and the Bejan number at Pr = 0.7 and Ra = 106 for four different locations of the inner cylinder. In Fig. 10,
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Fig. 8. Distribution of the local Nusselt numbers on the cylinder surface for four different locations of the cylinder at Pr = 0.1 when Ra = 105.
when the cylinder is located at d = 0.1, a pair of two primary vortices is formed in left and right sides of the enclosure, having a single vortex core located in the lower part of the cylinder. In addition, a pair of two small scale secondary vortices is formed in the center region of the bottom wall of the enclosure. When the Rayleigh number increases from Ra = 105 to Ra = 106, the strength of the convection flow is significantly intensified due to the increase in the buoyancy effect. Therefore, in contrast to the case at d = 0.1, Pr = 0.7 and Ra = 105 presented in Fig. 5(a), a pair of the secondary vortices formed in the upper part of the cylinder surface shown in Fig. 5(a) diminishes in this case. And hence a single ascending plume is generated above the upper surface of the cylinder, which is stretched to each side wall of the enclosure. As a result, isotherms near the center region of the top wall of the enclosure become dense by the upward motion of the single plume. The isotherms near the lower-center of the cylinder surface become denser than in other region of the lower surface due to the upward motion of the primary vortex in the lower part of the cylinder. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 329.3) occurs near the lower-center region of the cylinder surface. In addition, a moderate high local entropy generation rate due to heat transfer occurs in the center region of the top wall of the enclosure due to the effect of a single ascending plume intensified. High local entropy generation rate due to fluid friction (SFFImax = 28309.2) occurs near middle-left and middle-right of the cylinder surface, and moderate high local entropy generation rates due to fluid friction occur near the upper region of each side wall of the enclosure as the vortex core around which the high momentum circulating flow exists is located in the upper part of the cylinder. Due to the increasing up to Ra = 106, the convection velocity in an enclosure become more strong than that at Ra = 105. As a result, the portion of irreversibility due to the fluid friction is quite intensified rather than irreversibility due to the heat transfer in the enclosure. Therefore, the high local total entropy generation rate (STOTALmax = 28543.8) occurs at the same locations where the high local entropy generation rate due to the fluid friction occur. The low Bejan number is distributed evenly in an enclosure due to the intensified irreversibility caused by the fluid friction. The high Bejan number occurs near center of the bottom walls of the enclosure due to the secondary vortices having weak convection velocity and lower part of the cylinder surface due to the re-heated fluid before the ascending fluid reaches from lower region of the enclosure to the lower-center of the cylinder surface. In Fig. 9(b), when the cylinder moves up to d = 0.2, four small scale secondary vortices are generated in a very narrow space between the cylinder
surface and the top wall of the enclosure, and the upward, downward and upward motions of the fluid from vertical centerline of the enclosure to the side wall of the enclosure are produced by the interaction between neighboring vortices. However, the strength of the secondary vortices is very low due to the space narrowed extremely. As a result, significant distortion of the isotherms in the upper part of the cylinder is not identified, but a pair of two ascending plumes which is located near the upper-left and upperright of the cylinder surface is clearly visible with the distribution of isothermals distorted by the upward motion of the fluid in the primary vortex. The isotherms near the lower-center of the cylinder surface become dense due to the upward motion of the fluid in the primary vortex in the lower part of the cylinder. As a result, a high local entropy generation rate due to heat transfer (SHTImax = 321.721) occurs near the lower-center of the cylinder surface, similar to the case at d = 0.1. In addition, moderate high local entropy generation rates due to heat transfer occur in two regions off-centered on the top wall of the enclosure due to the effect of two ascending plumes. High local entropy generation rate due to fluid friction (SFFImax = 22922.5) occurs near middle-left and middle-right of the cylinder surface, and moderate high local entropy generation rate due to fluid friction occur near the upper region of each side wall of the enclosure. And high local total entropy generation rate (STOTALmax = 23127.6) occurs at the same location where the high local entropy generation rate due to the fluid friction occurs, which shows almost the same trend as the case at d = 0.1. The distribution of low Bejan number near the cylinder surface and each side wall of the enclosure is similar to that at d = 0.1. The high Bejan number occurs near the center region of the bottom wall of the enclosure and upper part of the cylinder due to the weak convection velocity and secondary vortices, respectively. In Fig. 9(c) and (d), the overall flow and thermal structures at d = 0.1 are similar to those at d = 0.2. In these figures, a single ascending plume is intensified and stretched to the top wall of the enclosure. The ascending plume intensified significantly makes the isotherms near the top wall of the cylinder very dense. As a result, high local entropy generation rates due to heat transfer (SHTImax = 402.811 at d = 0.1 and SHTImax = 362.77 at d = 0.2) occur near the center region of the top wall of the enclosure. And moderate high local entropy generation rates due to heat transfer occur in the lower surface of the cylinder at d = 0.1 and d = 0.2. High local entropy generation rates due to fluid friction (SFFImax = 38691.7 at d = 0.1 and SFFImax = 54391.4 at d = 0.2) and high local total entropy generation rates (STOTALmax = 38970 at d = 0.1 and STOTALmax = 54678.2) occur near middle-left and middle-right of the cylinder surface, and moderate high local entropy generation rate due to fluid friction and moderate high local total entropy generation rate occur near the upper region of each side wall of the enclosure at d = 0.1 and d = 0.2. The overall distribution of the Bejan number at d = 0.1 is also are similar to that at d = 0.2 except for the lower region of the cylinder due to the change in the location of the cylinder. Fig. 10 shows distribution of the local Nusselt number on the cylinder surface Nucyl at four different locations of the cylinder when Ra = 106 at Pr = 0.7. When the cylinder is located at d = 0.1, a simple pattern of Nucyl is distributed, showing a single high peak at u = 180° and a single low peak at u = 0° (or u = 360°). When the cylinder moves up to d = 0.2, two low peaks of Nucyl occur at u = 30° and u = 330° due to small scale secondary vortices formed near the upper part of the cylinder. Two high peaks of Nucyl occurs at u = 180° and u = 0° (or u = 360°), and the maximum value occurs at u = 180°. When the cylinder is located at d = 0.1, high values of Nucyl are almost uniformly distributed in 120° 6 u 6 240°, and the minimum value of Nucyl occurs at u = 0° (or u = 360°). When the cylinder moves down to d = 0.2, two high peaks of Nucyl occur at u = 120° and u = 240° and a single moderate low peak at
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Fig. 9. Distribution of isotherms, streamlines, entropy generation due to heat transfer and fluid friction at Ra = 106 and Pr = 0.7 (contour values range from 0 to 1 with 10 levels); (a) d = 0.1, (b) d = 0.2, (c) d = 0.1and (d) d = 0.2.
u = 180° due to a very narrow space between the cylinder surface and the bottom wall of the enclosure. The minimum value of Nucyl occurs at u = 0° (or u = 360°). Fig. 11(a)–(d) shows distributions of isotherms, streamlines, local entropy generation rates due to heat transfer and fluid friction, the local total entropy generation and the Bejan number at Pr = 0.1 and Ra = 106 for four different locations of the inner cylinder. As aforementioned in Section 3.1, the flow instability is intensified further by reducing the Prandtl number from Pr = 0.7 to Pr = 0.1 at a relatively high Rayleigh number. As a result, the flow unsteadiness and complexity of the convection cells are augmented at Ra = 106 and Pr = 0.1. In Fig. 11(a), when the cylinder is located at d = 0.1, a pair of two primary vortices is formed in the left and right parts of the cylinder, having a single vortex core located in the lower part of the primary vortex. And four different secondary vortices are generated in
the upper part of the cylinder. In addition, several small scale secondary vortices are identified in each corner of the enclosure and the center region of the bottom wall of the enclosure. Due to this structure of complex convection cells, isotherms in the enclosure are significantly distorted, especially in the upper region of the cylinder. The upward and downward motions of the fluid are generated by the interaction between neighboring secondary vortices which are located in the upper part of the cylinder. As a result, a single ascending plume due to the upward motion of the fluid and a pair of two descending plumes due to downward motion of the fluid is formed in the upper part of the cylinder. In addition, a pair of two plumes stretched in the horizontal direction (x) is formed in each side of the cylinder surface, following the fluid motion of the primary vortex which is separated from the side surface of the cylinder and then directed to each side wall of the enclosure. As a result, high local entropy generation rate due to
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Fig. 9 (continued)
heat transfer (SHTImax = 461.058) occurs near the upper-left and upper-right of the cylinder surface, and moderate high local entropy generation rates due to heat transfer occur near the lower surface of the cylinder as well as the center region of the top wall of the enclosure. high local entropy generation rate due to fluid friction (SFFImax = 12243.2) occurs near middle region of each side wall of the enclosure, and moderate high local entropy generation rates due to fluid friction occur near the lower-left and lower-right of the cylinder surface. High local total entropy generation rate (STOTALmax = 12344.6) occurs at the same locations where the high entropy generation rate due to the fluid friction occur. The low Bejan number occurs on the walls of the enclosure and side part of the cylinder surface where the convection cells in an enclosure reach directly, and also occurs near the region which is away from the cylinder acting on the heat source. In Fig. 11(b), when the cylinder moves up to d = 0.2, four different secondary vortices in the upper part of the cylinder and a pair of two secondary vortices in each upper corner of the enclosure, which are presented in Fig. 12, shrink in size and are altered in
Fig. 10. Distribution of the local Nusselt numbers on the cylinder surface for four different locations of the cylinder at Pr = 0.7 when Ra = 106.
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Fig. 11. Distribution of isotherms, streamlines, entropy generation due to heat transfer and fluid friction at Ra = 106 and Pr = 0.1 (contour values range from 0 to 1 with 10 levels); (a) d = 0.1, (b) d = 0.2, (c) d = 0.1 and (d) d = 0.2.
shape as the space between the cylinder surface and top wall of the enclosure is extremely narrowed. As a result, a pair of two primary vortices having two vortex cores is widely formed in the left and right parts of the cylinder, and very small scale secondary vortices are formed near the top and bottom walls of the enclosure. As a pair of two inner vortices formed in upper part of the cylinder is very small in size, most of the convection flow in the enclosure flows following the circulating motion in the inner vortices having a vortex core located in the lower part of the cylinder. Due to this, the isotherms in the upper region of the cylinder are almost uniformly distributed in the horizontal direction (x) whereas the isotherms near the upper-left and upper-right of the cylinder surface are stretched to each side wall of the enclosure. As a result, high local entropy generation rates due to heat transfer (SHTImax = 388.271) occur in the regions slightly off-centered on the lower surface of the cylinder. The high local entropy generation
rate due to fluid friction (SFFImax = 12832.4) and high local total entropy generation rate (STOTALmax = 13120.3) occur near the lower-left and lower-right of the cylinder surface, and moderate high local entropy generation rates due to fluid friction occur near the upper-left and upper-right of each side wall of the enclosure. The Bejan number are altered in distribution, compared to that at d = 0.1. The low Bejan number occurs near the side and lower space in an enclosure due to the each primary vortex having high convection velocity, but high Bejan number occurs near the cylinder surface and upper region of the enclosure. In Fig. 11(c), when the cylinder is located at d = 0.1, a pair of two primary vortices showing a bi-cellular structure is formed in the left and right parts of the cylinder. In addition, very small scale secondary vortices are generated near the upper surface of the cylinder and walls of the enclosure. A single ascending plume is generated by the upward motion of two upper inner vortices,
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Fig. 11 (continued)
and a pair of two plumes stretched in the horizontal direction (x) is generated by the flow circulating motion in each lower inner vortex. In the upper inner vortex which is larger in size than the lower one, the fluid cooled by the cold top wall of the enclosure moves down and is directed to the upper-left and upper-right of the hot cylinder surface, following the flow circulating motion in each upper inner vortex. As a result, the thermal boundary layer near the upper-left and upper-right of the cylinder surface is attached further by the fluid motion. Therefore, high local entropy generation rates due to heat transfer (SHTImax = 406.307) occur near the upper-left and upper-right of the cylinder surface, moderate high local entropy generation rates due to heat transfer occur near the top wall of the enclosure due to the effect of a single ascending plume. High local entropy generation rate due to fluid friction (SFFImax = 71782.6) and high local total entropy generation rate (STOTALmax = 72188.9) occur near the upper-left and upper-right of the cylinder surface. The high Bejan number distributes near the
lower sides of the enclosure and the lower region of the cylinder surface due to the fluid motion with low inner vortices having relatively low convection velocity compared to the upper inner vortices and the narrow space between the cylinder surface and bottom wall of the enclosure, respectively. In Fig. 11(d), when the cylinder moves down to d = 0.2, the overall flow and thermal structures at d = 0.2 are very similar to those at d = 0.1. However, the upper inner vortices increases and the lower inner vortices decreases in size as the cylinder moves down to d = 0.2 from d = 0.1. Therefore, the locations at which high local entropy generation rates due to heat transfer (SHTImax = 431.983) and fluid friction (SFFImax = 53035.8) and high total entropy generation rate (STOTALmax = 53452.4) occurs are almost the same as the locations at d = 0.1. And overall distribution of the Bejan number is also almost the same as that at d = 0.1. Fig. 12 shows distribution of the local Nusselt number on the cylinder surface Nucyl at four different locations of the cylinder
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Fig. 12. Distribution of the local Nusselt numbers on the cylinder surface for four different locations of the cylinder at Pr = 0.1 when Ra = 106.
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when Ra = 106 at Pr = 0.1. When the cylinder is located at d = 0.1, four high peaks of Nucyl occur at u = 30°, u = 330° and u = 160°, u = 200°. Three low peaks of Nucyl occur at u = 0° (or u = 360°), u = 75° and u = 285° due to the downward and upward motions of four different secondary vortices formed in the upper part of the cylinder, and another single low peak occurs at u = 180°. The value of Nucyl is maximized at u = 30° and u = 330° and minimized at u = 75° and u = 285°. When the cylinder moves up to d = 0.2, the values of Nucyl remain almost unchanged in 0° 6 u 6 30° and 330° 6 u 6 360°. Two low peaks of Nucyl occur at u = 70° and u = 290°, which are the minimum values. High values of Nucyl are distributed in 150° 6 u 6 210° in which Nucyl is maximized. When the cylinder is located at d = 0.1, four high peaks of Nucyl occur at u = 55°, u = 305° and u = 150°, u = 210°, and the value is maximized at u = 55° and u = 305°. Four low peaks of Nucyl occur at u = 0° (or u = 360°), u = 90°, u = 180° and u = 270°, and the value is minimized at u = 0° (or u = 360°). When the cylinder moves down to d = 0.2, the distribution pattern and values of Nucyl at d = 0.2 are very similar to those at d = 0.1.
Fig. 13. Variation in the volume-averaged entropy generation rates as a function of the Grashof number; (a) hSHTIi and (b) hSFFIi.
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Fig. 14. Variation in Bejan numbers Be as a function of the Grashof number.
3.5. Volume-averaged entropy generation and Bejan number Fig. 13(a) and (b) shows the variation in the volume-averaged entropy generation rates due to heat transfer hSHTI i and fluid friction hSFFI i as a function of the Grashof number Gr, respectively, at four different Rayleigh numbers and four different locations of the cylinder. Here, the Grashof number is defined as Gr ¼ Ra=Pr. In Fig. 13(a), the values of hSHTI i at Ra=104 are almost the same as those of Ra = 103 because the convection velocity in the enclosure is very weak and hence the heat conduction is a dominant heat transfer process while the Rayleigh number increases from Ra = 103 to Ra = 104. The values of hSHTI i increase substantially with increasing Rayleigh number to Ra = 105 and Ra = 106 from Ra = 104. In Fig. 13(b), the values of hSFFI i at Ra = 103 are almost zero with the order of 103. When the Rayleigh number increases to Ra = 104 and Ra = 105, the values of hSFFI i at Ra = 104 and Ra = 105 increase up to the order of 101 and 101, respectively. When the Rayleigh number increases further to Ra = 106, the values of hSFFI i increase significantly up to the order of 102. In Fig. 13(a) and (b), hSHTI i and hSFFI i at relatively low Rayleigh numbers of Ra = 103 and Ra = 104 remain almost unchanged according to the variation in the Prandtl number regardless of the variation in the location of the cylinder. However, the Prandtl number dependency on the entropy generation is clearly identified at relatively high Rayleigh numbers of Ra = 105 and Ra = 106. In Fig. 13(a) and (b), hSHTI i and hSFFI i decreases slightly with decreasing Prandtl number from Pr = 100 to Pr = 0.7 and then steeply decreases by reducing the Prandtl number from Pr = 0.7 to Pr = 0.01 at Ra = 105 and Ra = 106. Fig. 14 shows the variation in the Bejan number Be as a function of the Grashof number Gr at four different Rayleigh numbers and four different locations of the cylinder. In this figure, the Bejan numbers are distributed in the vicinity of Be ¼ 1 at Ra = 103 and Ra = 104. Therefore, the irreversibility in the system is dominantly produced by the heat transfer at a relatively low Rayleigh numbers of Ra = 103 and Ra = 104. At a relatively high Rayleigh numbers of Ra = 105 and Ra = 106, the Bejan number is almost independent on the variation in the Prandtl number at each Rayleigh number and each location of the cylinder in the Prandtl number range of 0.7 6 Pr 6 100 but increases steeply by reducing the Prandtl number from Pr = 0.7 to Pr = 0.01. When the Rayleigh number increases to Ra = 105 from Ra = 104, the Bejan numbers are reduced and distributed below Be ¼ 0:5, except two cases of d = 0.1 and d = 0.2 at
Pr = 0.01. In these two cases, the heat transfer is still dominant mechanism to produce the irreversibility in the system rather than the fluid friction. When the Rayleigh number increases further to Ra = 106, all Bejan numbers are distributed below Be ¼ 0:5, which means that the fluid friction is a dominant mechanism to produce the irreversibility in the system.
4. Summary and conclusions In this study, two-dimensional numerical simulations were conducted to investigate the effect of the variation in the vertical position of the inner cylinder on the characteristics of the natural convection in the enclosure in the Rayleigh number range of 103 6 Ra 6 106 and Prandtl number range of 102 6 Pr 6 102. The flow instability is significantly intensified at a relatively high Rayleigh numbers of Ra = 105 and Ra = 106 and a relatively low Prandtl numbers of Pr = 0.1 and Pr = 0.01, which leads to the unsteadiness of the flow and thermal fields. The state of the flow in the system (i.e. flow unsteadiness and complexity of convection cells) is directly correlated with the Prandtl number at a certain Rayleigh number. And the configuration of the system including the geometry and position of the inner body and geometry of the enclosure is also one of crucial factors to affect the state of the flow. In this study, the ranges of the Rayleigh number and Prandtl number in which the flow experiences the transition from the steady state to the unsteady one are consistent with those reported in the companion paper [24]. In this study, the flow bifurcation problem showing an asymmetric flow structure in the enclosure is not identified for all simulation cases considered in this study, in contrast to the results reported in the companion paper [24]. At a relatively low Rayleigh numbers of Ra = 103 and Ra = 104, the degree of the irreversibility produced in the system is almost the same regardless of the variation in the location of the cylinder at the same Rayleigh number and Prandtl number. However, when the Rayleigh number increases to Ra = 105 and Ra = 106, the variation in the magnitude of the irreversibility according to the variation in the location of the cylinder becomes large at the same Rayleigh number and Prandtl number, especially at a relatively low Prandtl numbers of Pr = 0.1 and Pr = 0.01, due to the significant change in the structure of convection cells in the enclosure. How-
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