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Flow downward penetration of vertical parallel plates natural convection with an asymmetrically heated wall
International Communications in Heat and Mass Transfer 74 (2016) 55–62
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Flow downward penetration of vertical parallel plates natural convection with an asymmetrically heated wall☆ Wu-Shung Fu a,⁎, Wei-Siang Chao a, Tzu-En Peng a, Chung-Gang Li b a b
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, ROC Complex Phenomena Unified Simulation Research Team, Advanced Institute for Computation Science, RIKEN, Kobe 650-0047, Japan
a r t i c l e
i n f o
Available online 17 March 2016 Keywords: Compressible flow Non-reflecting boundary Natural convection
a b s t r a c t Phenomena of the flow downward penetration in vertical parallel plates natural convection with an asymmetrically heated wall are investigated numerically. For clarifying the occurrence of flow downward penetration, the width of parallel plates is further broadened and broader than that investigated in the previous study in which the flow reversal was studied in detail. In order to simulate realistically, the compressibility and the viscosity of the working fluid are simultaneously considered, and the non-reflecting boundary is adopted at apertures of plates. The methods of the Roe scheme, preconditioning, and dual time stepping matching LUSGS are coordinated to solve governing equations of a low-speed compressible flow problem. The flow downward penetration, in which fluids are sucked from surroundings outside the outlet to flow into the parallel plates and through the inlet flow out of parallel plates, is composed of a few fluids and firstly observed. Also, two flow reversal caused by the buoyancy force appear in the upper left region, and one flow reversal induced by the flow downward penetration appears in the low right region. The usage of the existing results and present study, an equivalent ratio of the width to the length of parallel plates, of which the heat transfer rate is the same as the single vertical heated plate, can be expediently obtained. In this study, the ratio is about 0.6. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Until now, lots of numerical and experimental studies [1–26] still investigate natural convection in a channel or parallel plates because of its importance and applications in academic and industrial research. The flow reversal, which is one of unique and complex characteristics of natural convection, is often observed in some situations when a ratio of the width to the height of above situations exceeds a certain threshold. Since the flow field and heat transfer mechanism of natural convection are remarkably affected by the flow reversal. The investigation of the flow reversal becomes even firmer as time goes by. In the past, Sparrow et al. [2] conducted experimental and numerical studies to investigate the flow reversal in vertical parallel plates natural convection and flow reversal. The flow reversal was formed by a pocket of recirculating flow when Ra/A ratio exceeded a certain magnitude. Numerical solutions obtained by a parabolic finite difference scheme yielded Nusselt numbers in good agreement of the experimental results. An experimental work of natural convection in two vertical plates was conducted by Al-Azzawi [3]. One of the plates regarded as a heat wall was heated electrically, and the other was made of glass. The height of plates are 1 m and 2 m, respectively. The width between two plates ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. Tel.: +886 3 5712121x55173; fax: +886 3 5735065. E-mail address: [email protected] (W.-S. Fu).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2016.03.006 0735-1933/© 2016 Elsevier Ltd. All rights reserved.
varies from 25 to 150 mm. The thermal conditions of the heated wall were a uniform heat flux and a constant wall temperature, respectively. An experimental equation was derived to reveal a relationship between the Nusselt number and modified Rayleigh number. Chang and Lin [5] performed a numerical study to investigate the reversed flow and oscillating wake in an asymmetrically heated channel. The Boussinesq assumption and an extended domain were adopted. The phenomenon of flow reversal was indicated near the upper adiabatic wall. The larger the Rayleigh number was, the more remarkable phenomenon was revealed. Kihm et al. [8] performed a numerical study matching smoke visualization to investigate a problem of flow reversal of natural convection in isothermal vertical walls. The Boussinesq assumption and an extended boundary were adopted. The phenomenon of flow reversal was observed in the upper central region because of isothermal vertical walls, and variations of the entrance lengths with the Rayleigh numbers were indicated. Ospir et al. [20] conducted an experimental study to investigate an evolution of flow reversal from a single cell to a final eight-shaped structure. Results showed that the increase in the modified Rayleigh number resulted in an increment in the penetration of the flow reversal. Recently, Li et al. [21] firstly added an effect of radiation on investigating the flow reversal of natural convection in asymmetrically heated vertical channels numerically. Experimental results of Webb and Hill [4] were used as boundary conditions at the inlet and outlet of the channel, and then the governing equations of an elliptic nature were solved and an extended boundary adopted in the above
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Nomenclature A d e g h k k0 l M: y
area(m2) width of the plate (m) internal energy (J⋅ kg−1) acceleration of gravity (m⋅ s‐2) height of the plate (m) thermal conductivity (W⋅ mK−1) surrounding thermal conductivity (W⋅ mK−1) length of the plate (m) total mass flow rate defined in Eq. (13) (kg ⋅ s‐1) M : y ¼ ∫ ρudA
Nux
local Nusselt number Nux ¼ k0 ðT h−T c Þ ½kðTÞ ∂T ∂y
Nux
time-averaged local Nusselt number defined in Eq. (11) Nux ¼ 1t ∫ Nux dt
NuA
area-averaged local Nusselt number defined in Eq. (12) NuA ¼ A1 ∫ ∫ Nux dxdz
P P0 Pr R Ra
pressure (Pa) surrounding pressure (Pa) Prandtl number gas constant (J⋅ kg−1 ⋅ K−1) gρ2 ðT −T Þh3 Rayleigh number defined in Eq. (8) Ra ¼ Pr 0 h 2c
Ra⁎
modified Rayleigh number defined in Eq. (9) Ra ¼ Ra hb time (s) dimensionless time t ¼ t α2 h temperature (K) temperature of surroundings (K) temperature of heat surface (K) velocities in x, y, and z directions (m⋅ s−1) Cartesian coordinates (m) dimensionless Cartesian coordinates
A
defined
in
Eq.
(10)
h
t
t t⁎ T T0 Th u ,v , w x , y, z X , Y, Z
wh
T 0 μðTÞ
Greek symbols α Thermal diffusivity rate (m2 ⋅ s−1) ρ density (kg ⋅ m‐3) surrounding density (kg ⋅ m‐3) ρ0 μ viscosity (kg ⋅ m‐1 ⋅ s‐1) surrounding viscosity (kg⋅ m‐1 ⋅ s‐1) μ0 γ specific heat ratio
inlet flowing into parallel plates. In those situations the fluids, which were caused by the flow reversal through the outlet flowing into parallel plates, completely through the outlet flow out of parallel plates again. As the ratio of the width to the length is increased further and larger than those adopted in the previous study [26], in addition to the indication of the flow reversal, new phenomena are expected to be observed and able to encourage further investigation of this subject. Therefore, the aim of the study further enlarges the ratio of the width to the length of parallel plates adopted by Fu et al. [26] to investigate new phenomena of natural convection occurring in vertical parallel plates numerically. In order to simulate the situation more realistically, the compressibility and viscosity of the working fluid are considered, and the non-reflecting boundary condition [27], which causes the Boussinesq assumption to be no longer necessary, held at the inlet and outlet of the plates is adopted. For solving a low-speed compressible flow problem, the method of the Roe scheme [28], preconditioning [29], and dual time stepping [30] matching the LUSGS [31] are simultaneously used to solve the governing equations. Then the magnitudes of the velocity, temperature, pressure, and density at the inlet and outlet of the plate are simultaneously calculated. The results show that the total mass flow rate flowing into parallel plates is almost equivalent to the mass flow rate of the large modified Rayleigh number situations indicated by Fu et al. [26]. However, the width of this study is larger than that adopted by Fu et al. [26], and effects of the buoyancy force on flowing behaviors of the fluids, which are close to the right wall, naturally become weak. A few fluids, which are caused by the flow reversal appearing in the left upper region of parallel plates, unexpectedly form a downward flow and through the inlet flow out of parallel plates. This phenomenon is newly observed and named the flow downward penetration. Relative to occurrence of the flow reversal in the left upper region, some fluids close to the right low region are affected by the flow downward penetration to form the new flow reversal inside the inlet. With the concept of the modified Rayleigh number, an equivalent ratio of a single vertical heated plate can be expediently calculated under the same Nusselt number situation and the ratio is about 0.6 in this study. Also, both existing and present results are well consistent. 2. Physical backgrounds In this section, the physical model and the governing equations used are introduced, and the suitable initial and boundary conditions are derived. 2.1. Physical model
literature was not necessary. The flow reversal and distributions of streamlines and isotherms under a steady situation were indicated. The mass flow rate kept constant when the wall spacing exceeded a certain magnitude. The onset of flow reversal delayed upon the effect of surface radiation and enhancement of the cooling of the heated wall by radiation were shown. Desrayaud et al. [22] used eight different methods to study natural convection between parallel plates with an asymmetrically heated wall. The flow reversal was observed and results obtained by the above method were in good agreement. Because an elliptic natural of the problem was solved in the above literature, the results were mainly indicated by steady situations. Recently, Fu et al. [26] adopted the non-reflecting boundary, which was held at apertures because of consideration of the compressibility of fluid, to investigate the flow reversal of natural convection. Accompanied with increment of the ratio of the width to the length of parallel plates, the flow reversal became remarkable and drastic. The amount of fluid through the outlet flowing into parallel plates was gradually larger than that through the
From the results obtained by Fu et al. [26], except on the edge, phenomena on the cross sections of the XY plane in the direction of the Z axis are similar. Relative to Fu et al. [26], the width of this study is further enlarged for investigation of new phenomena of natural convection in vertical parallel plates with an asymmetrically heated wall. Accordingly, the depth in the Z axis of this study is shortened and shorter than that used by Fu et al. [26] for saving the total grids. The same periodical conditions used by Fu et al. [26] are still adopted in this study. Then the physical model of vertical parallel plates investigated in this study is indicated in Fig. 1. The left wall is heated and possesses a constant temperature Th, and the right wall is adiabatic. The width and length of parallel plates are d and l, respectively. The pressure and temperature of fluids in surroundings outside the inlet and outlet are Po and To, respectively. The direction of gravity is the negative y direction. In order to expand industrial and academic research, the compressibility and viscosity of working fluids are taken into consideration simultaneously. Thus, the Boussinesq assumption is no longer necessary and the non-reflecting boundary condition developed by [27] for a low-speed compressible flow can be suitably adopted on the inlet and outlet. For simplifying explanations, the upper and low
W.-S. Fu et al. / International Communications in Heat and Mass Transfer 74 (2016) 55–62
2
Non-reflecting boundary
2
outlet
Heated wall
T0
298 K
P0
1.013 105 Pa
Adiabatic wall
l Th
3 9 ρu > > > ρu þ P−τxx 7 > 6 > 7 > 6 > ρuv−τ 7 > 6 > xy F¼6 7 > > 7 > 6 ρuw−τxz > 5 > 4 > > ∂T > > > ρEu þ Pu−k −uτxx −vτ xy −wτxz > ∂x > 3 > 2 > > ρv > > > 7 > 6 ρuv−τyx > > 7 6 > 2 > 7 6 > ρv þ P−τ yy > 7 6 G¼6 > 7 > ρvw−τ yz > 7 > 6 > 5 > 4 > ∂T = ρEv þ Pv−k −uτ yx −vτyy −wτ yz ∂y 3> 2 > ρw > > > 7> 6 ρuw−τ zx > 7> 6 > > 7 6 ρvw−τ zy > 7> > H¼6 > 2 7 6 ρw þ P−τzz > 7> 6 > > 5 4 > ∂T > ρEw þ Pw−k −uτzx −vτzy −wτ zz > > > > 3 ∂z 2 > > > 0 > > > 6 −ðρ−ρ0 Þg 7 > > 7 6 > > > 7 6 S¼6 0 > > 7 > > 5 4 0 > > ; −ðρ−ρ0 Þgu
57
400 K
g
ð4Þ
x Sutherland's law is adopted to evaluate the viscosity and thermal conductivity shown as follows:
d
y
2 T 3 T 0 þ 110 T0 T þ 110 μ ðT ÞγR kðT Þ ¼ ðγ−1Þ Pr
inlet Non-reflecting boundary
μ ðT Þ ¼ μ 0
Fig. 1. Physical model.
apertures of parallel plates are conveniently called the outlet and inlet, respectively. 2.2. Governing equations For facilitating the analysis, the following assumptions are made. 1. The work fluid is ideal gas and follows the equation of state of an ideal gas. 2. Magnitudes of gradients of density and pressure on the whole surfaces in the normal direction are zero. 3. Radiation heat transfer is neglected.
ð1Þ
and the equation of state of an ideal gas is used. P ¼ ρRT
ð2Þ
The terms included in U, F, G, H, and S are separately shown in the following equations: 3 ρ 6 ρu 7 7 6 7 U¼6 6 ρv 7 4 ρw 5 ρE
in which
P 1 þ u2 þ v2 þ w2 ρðγ−1Þ 2 ρ0 ¼ 1:1842kg=m3 ; g ¼ 9:81m=s2 ; μ 0 ¼ 1:85 10−5 Ns=m2 ; T0 ¼ 298:05K; γ ¼ 1:4; R ¼ 287J=kg=K; Pr ¼ 0:72: E¼
ð6Þ
To simplify the analysis, the following dimensionless variables are made:
The governing equations are expressed as follows: ∂U ∂F ∂G ∂H þ þ þ ¼S ∂t ∂x ∂y ∂z
ð5Þ
2
ð3Þ
x y z ; Y¼ ; Z¼ ; h h h α t ¼ t 2 h X¼
ð7Þ
3. Numerical method Since the study is an extensive study of the previous study [26], the numerical method used is basically modified from the previous study [26]. The width is broadened in this study, and the density of grids in the width is then increased. The largest ratio (d/l) of the width to length by Fu et al. [26] is 0.1125. In this study, six different ratios are adopted, and the related magnitudes corresponding grid distributions and modified Rayleigh numbers are tabulated in Table 1. Except the grid distribution, the detailed finite differentiation of the governing equations and computing procedures are the same as those adopted by Fu et al. [26].
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Table 1 Related parameters Fu et al. [26] study
This study 2
3
4
5
6
7
d l
0.1125
0.100
0.150
0.175
0.200
0.250
0.300
0.400
Grid distribution
200 × 100 × 50 1.072 × 105
200 × 80 × 4 6.69 × 104
200 × 120 × 4 3.39 × 105
200 × 140 × 4 6.27 × 105
200 × 160 × 4 1.07 × 106
200 × 240 × 4 2.61 × 106
200 × 240 × 4 5.42 × 106
200 × 320 × 4 1.71 × 107
R a ¼ Ra dl
1
4. Results and discussion Basically, this work is an extensive study of the previous study [26]. For highlighting the relationship between the width and length, the Rayleigh number and modified Rayleigh number are defined as follows, respectively, 3
Ra ¼ Pr
gρ2o ðT h −T o Þd
Ra ¼ Ra
T o μT 2 d l
ð8Þ
ð9Þ
Due to complex variation of the flow field, several signals of exaggerated arrows are conveniently adopted to indicate variational flow fields. In Fig. 2(a–l), the history of development of flow fields is indicated under the situation of Ra⁎ = 2.61 × 106 and d l ¼ 0:25. Shown in initial stages of Fig. 2(a–c), fluids in parallel plates are heated by the left heated wall that causes the volume of the fluid to be expanded. Thus, the fluids start to flow out of parallel plates from both the inlet and outlet, respectively. Afterward, the fluids are affected by the buoyancy force produced by the left heated wall to flow upward. Since the width of parallel plates is wide. For supplementing the upward mass flow rate caused by the buoyancy force, fluids from surroundings outside the inlet and outlet are simultaneously sucked into parallel plates. The heat transfer mode varies from heat conduction to heat convection. Because of the higher modified Rayleigh number, the upward flow along the left heated wall becomes unsteady especially in the upper region of the heated wall. Several flow reversals, which are composed of fluids mainly sucked from surroundings outside the outlet, are casually formed on the upper heated wall and revealed in Fig. 2(d and e). These phenomena are similar to those indicated in the previous study [26]. In Fig. 2(f), the fluids are gradually accelerated to flow into parallel plates because of the strong strength of flow reversals. However, the width of this study is enlarged and wider than that of Fu et al. [26]. Effects of the buoyancy force on the fluids in the upper region of the right wall naturally become weak. Accordingly, the penetration length, which means the downward distance of movement of the fluid from the outlet, of the fluids close to the right wall is longer than that of the fluids close to the flow reversals. After piercing through the outlet, a few fluids close to the right upper corner are squeezed and reluctantly depart from the penetration path to flow back to the outlet. A small flow reversal is then formed in the right upper corner. In Fig. 2(g), due to the buoyancy force, three flow reversals on the heated wall shown in the above figure subsequently merge into one large flow reversal close to the left upper corner. Meanwhile some fluids, which are dragged from surroundings outside the outlet and accelerated by the large flow reversal, concentrate in the central region of the outlet to flow into parallel plates. As a result, the small flow reversal in the right upper corner becomes large. Since the width is large, the buoyancy force are difficult to affect behaviors of fluid close to the right wall. Then a few fluids, which are dragged by the large flow reversal and possess large downward impulse, along the right wall directly flow downward and penetrate the inlet to surroundings. Relative the flow reversal, the phenomenon is suitably named the flow downward
penetration. At this instant, except the region of the flow downward penetration, fluids in the other region are sucked from surroundings outside the inlet by the buoyancy force to flow into parallel plates. However, the buoyancy force is weak in the right region of the inlet, and then part of fluids close to the right low region of the inlet are immediately dragged by the flow downward penetration after piercing through the inlet and compulsively turned the direction to flow through the inlet. As a result, the small flow reversal neighbors on the flow downward penetration to be observed as shown in Fig. 2(h). As time passes revealed in Fig. 2(i–l), two upper flow reversals, one low flow reversal and the flow downward penetration become remarkable, respectively. In Fig. 3(a–l), history of development of thermal fields is indicated under the situation of Ra= 2.61× 106, d l ¼ 0:25. The darker the color, the higher the temperature indicated. Each instant shown in the figure is corresponding to that indicated in Fig. 2. In initial stages Fig. 2(a–c), the heat transfer mode varies from heat conduction to heat convection gradually. An orderly thermal boundary layer is clearly observed on the left heated wall, and the thickness of the thermal boundary becomes thicker from the inlet to outlet accompanied with increment of time. Afterward, in Fig. 3 (d–f), due to the higher modified Rayleigh number, the thermal field becomes unsteady on the upper heated wall, and several convex thermal lumps, of which the locations are corresponding to those of the flow reversals revealed in Fig. 2, are formed. The lump is almost filled by fluids with the dark color, and the thermal boundary region between the lumps becomes extremely thin. These phenomena firmly indicate the influence of behaviors of lumps even on the bottom of the thermal boundary. It implies that similar phenomena of the instability of the thermal boundary occur. Except the regions composed of the thermal boundary and lumps, the color of the other region is almost white that means the temperature of the other region not to be affected by the left heated wall. Flowing behaviors of fluids in the other region indicated in Fig. 2 are caused by the suction effect of the buoyancy force and not affected by the temperature difference, which plays an important role of the driving force in natural convection. As time passes, three thermal lumps merge to one thermal lump on the upper corner of the heated wall because of the buoyancy force as shown in Fig. 3(g). Afterward, the thermal boundary and thermal lump periodically unify [Fig. 3(h and k)] and divide [Fig. 3(i, j, and l)] to be observed. It implies that the heat transfer between the thermal boundary and lump is sometimes interrupted by the cool fluids, which are sucked by the flow reversal, to flow into parallel plates and flow out of parallel plates again after encircling the thermal lump. Since the cool fluids are indicated by the white color, the space between the thermal boundary and lump is periodically indicated by the white color, too. Variations of local Nusselt numbers of different locations with time for Ra⁎2.61 × 106 are indicated in Fig. 4. The definition of the local Nusselt number and the time-averaged local Nusselt number are defined as follows, respectively,
Nux ¼
h ∂T K ðT Þ k0 ðT h −T c Þ ∂y
ð10Þ
W.-S. Fu et al. / International Communications in Heat and Mass Transfer 74 (2016) 55–62
59
Fig. 2. History of development of flow fields under Ra⁎ =2.61×106,dl ¼ 0:25.
Nux ¼
1 t
Z Nux dt
ð11Þ
t
In the initial stage, the heat transfer mechanism is mainly dependent on heat conduction mode that causes the variations and the magnitudes of the local Nusselt number to be stable and similar. Due to occurrence of unsteady phenomena along the left upper wall, the variation of the local Nusselt number becomes drastic with increment of the distance from the inlet. Shown in Fig. 5, distributions of area-averaged local Nusselt numbers under different modified Rayleigh numbers are revealed.
The definition of the area-averaged Nusselt number is defined as follows:
NuA ¼
1 A
Z Z Nux dxdz w
ð12Þ
h
Since the width is wider, the modified Rayleigh number becomes larger. Behaviors of fluids in the wide space are able to be more active. As a result, the larger the modified Rayleigh number is, the larger the area-averaged Nusselt number is achieved.
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Fig. 3. History of development of thermal fields under Ra⁎ =2.61×106,dl ¼ 0:25.
In Fig. 6, distributions of mass flow rates flowing through the inlet and outlet are indicated, respectively. The definition of mass flow rate is defined as follows, _y¼ M
Z ρudA
ð13Þ
A
In the previous study [26], part of small modified Rayleigh number situations had the same mass flow rate ① through the inlet flowing into and through the outlet flowing out of parallel plates. In other part of large modified Rayleigh number situations, part of the fluids ②, which are sucked from surroundings outside the outlet, flow into and out of parallel plates through the outlet because of occurrence of the flow reversal. The other fluids ③, which are sucked from the outside of the inlet, through the inlet flow into and through the outlet flow out of parallel plates due to the buoyancy force.
However, in this study co-existence of the flow reversal and downward penetration are observed that causes flow phenomena to be more complex. Thus, the fluids, which are sucked from surroundings outside the outlet by the left upper flow reversal, through the outlet flow into parallel plates. Most of the fluids ④ through the outlet flow out of parallel plates because of the upper flow reversal, but a few other fluids ⑥ flow out of parallel plates through the inlet due to the flow downward penetration. Similarly, the other fluids, which are sucked from surroundings outside the inlet by the buoyancy force, through the inlet flow into parallel plates. Most of the fluids ⑤ through the outlet flow out of parallel plates due to the buoyancy force, but a few other fluids ⑥ through the inlet flow out of parallel plates because of the right low flow reversal. In Fig. 7, comparison of existing results and present results is revealed. Present results are consistent well with the existing results [3] in both investigated (solid line) and extended (dashed line) regions. The signal of X indicates the result of the situation of a single vertical
W.-S. Fu et al. / International Communications in Heat and Mass Transfer 74 (2016) 55–62
61
Fig. 4. Variations of local Nusselt number of different locations with dimensionless time for Ra⁎ =2.61×106.
heated plate. The time and area-averaged Nusselt number of X is calculated by the empirical equation derived from [32]. The width of the situation of the single vertical heated plate is close to infinite. For application, the corresponding modified Rayleigh number of the magnitude of X is expediently assumed to be equivalently located in the extended region of Al-Azzawi [3]. The usage of the empirical equation derived by Al-Azzawi [3], the corresponding modified Rayleigh number and equivalent ratio of the width to the length of the single vertical heated plate can be calculated. The magnitudes of the correspond ing modified Rayleigh number, Ra⁎, and ratio, d l , are about 1.46 × 108 and 0.6, respectively. It implies that the ratio of d l is larger than 0.6, and the effect of the heat transfer of the left heated wall caused by the right wall can be negligible.
Fig. 5. Variations of area-averaged Nusselt numbers of different modified Rayleigh numbers with dimensionless time.
5. Conclusions An investigation of the flow downward penetration of natural convection in vertical parallel plates with an asymmetrically heated wall is conducted numerically. In order to expand industrial and academic research of the previous study [26], the ratio of the width to the length is enlarged and larger than that adopted by Fu et al. [26]. Related methods of the Roe scheme [28], preconditioning [29], and dual time stepping [30] matching the method of LUSGS [31] are used simultaneously to solve low-speed compressible flow problems. Several conclusions are drawn and shown as follows: 1. Due to increment of the width, the buoyancy force caused by the left heated wall is difficult to affect flowing behaviors of the fluids close to the right wall, which leads the flow downward penetration to form along the right wall.
Fig. 6. Distributions of mass flow rates flowing through the inlet and outlet.
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Fig. 7. Comparison of existing results and present results.
2. The flow reversal indicated in the left upper region is caused by the buoyancy force. Oppositely, the flow reversal indicated in the right low region is induced by the flow downward penetration. 3. Both present numerical results and existing experimental result in good agreement. Acknowledgments The authors gratefully acknowledge the support of the Natural Science Council, Taiwan, ROC, under contact NSC102-2221-E-009-053MY2. References [1] W. Aung, Fully developed laminar free convection between vertical plates heated asymmetrically, Int. J. Heat Mass Transf. 15 (1972) 1577–1580. [2] E. Sparrow, G. Chrysler, L. Azevedo, Observed flow reversals and measured predicted Nusselt numbers for natural convection in a one-sided heated vertical channel, J. Heat Transf. 106 (1984) 325–332. [3] A.R.H. Al-Azzawi, Natural Convection in a Vertical Channel Related to Passive Solar Systems, Department of Mechanical Engineering, University of Strathclyde, Glasgow, UK, Ph. D thesis, 1987. [4] B. Webb, D. Hill, High Rayleigh number laminar natural convection in an asymmetrically heated vertical channel, J. Heat Transf. 111 (1989) 649–656. [5] T.S. Chang, T.F. Lin, On the reversed flow and oscillating wake in an asymmetrically heated channel, Int. J. Numer. Methods Fluids 10 (1990) 443–459. [6] S. Kim, N.K. Anand, W. Aung, Effect of wall conduction on free convection between asymmetrically heated vertical plates: uniform wall heat flux, Int. J. Heat Mass Transf. 33 (1990) 1013–1023. [7] A. La Pica, G. Rodono, R. Volpes, An experimental investigation on natural convection of air in a vertical channel, Int. J. Heat Mass Transf. 36 (1993) 611–616.
[8] K. Kihm, J. Kim, L. Fletcher, Onset of flow reversal and penetration length of natural convective flow between isothermal vertical walls, Trans. Am. Soc. Mech. Eng. J. Heat Transf. 117 (1995) 776–779. [9] A.G. Fedorov, R. Viskanta, Turbulent natural convection heat transfer in an asymmetrically heated, vertical parallel-plate channel, Int. J. Heat Mass Transf. 40 (1997) 3849–3860. [10] T.A.M. Versteegh, F.T.M. Nieuwstadt, Turbulent budgets of natural convection in an infinite, differentially heated, vertical channel, Int. J. Heat Fluid Flow 19 (1998) 135–149. [11] M.A. Habib, S.A.M. Said, S.A. Ahmed, A. Asghar, Velocity characteristics of turbulent natural convection in symmetrically and asymmetrically heated vertical channels, Exp. Thermal Fluid Sci. 36 (2002) 77–87. [12] T.F. Ayinde, S.A.M. Said, M.A. Habib, Experimental investigation of turbulent natural convection flow in a channel, Heat Mass Transf. 42 (2006) 169–177. [13] T. Yilmaz, S.M. Fraser, Turbulent natural convection in a vertical parallel-plate channel with asymmetric heating, Int. J. Heat Mass Transf. 50 (2007) 2612–2623. [14] T.F. Ayinde, S.A.M. Said, M.A. Habib, Turbulent natural convection flow in a vertical channel with antisymmetric heating, Heat Mass Transf. 44 (2008) 1207–1216. [15] A. Andreozzi, B. Buonomo, O. Manca, Transient natural convection in a symmetrically heated vertical channel at uniform wall heat flux, Numer. Heat Transf. sA 55 (2009) 409–431. [16] Q. Lu, S. Qiu, G. Su, W. Tian, Z. Ye, Experimental research on heat transfer of natural convection in vertical rectangular channels with large aspect ratio, Exp. Thermal Fluid Sci. 34 (2010) 73–80. [17] B. Buonomo, O. Manca, Natural convection slip flow in a vertical microchannel heated at uniform heat flux, Int. J. Therm. Sci. 49 (2010) 1333–1344. [18] V.I. Terekhov, A.L. Ekaid, Laminar natural convection between vertical isothermal heated plates with different temperatures, J. Eng. Thermophys. 20 (2011) 416–433. [19] B. Buonomo, O. Manca, Transient natural convection in a vertical microchannel heated at uniform heat flux, Int. J. Therm. Sci. 56 (2012) 35–47. [20] D. Ospir, C. Popa, C. Chereches, G. Polidori, S. Fohanno, Flow visualization of natural convection in a vertical channel with asymmetric heating, Int. Commun. Heat Mass Transf. 39 (2012) 486–493. [21] R. Li, M. Bousetta, E. Chénier, G. Lauriat, Effect of surface radiation on natural convective flows and onset of flow reversal in asymmetrically heated vertical channels, Int. J. Therm. Sci. 65 (2013) 9–27. [22] G. Desrayaud, E. Chénier, A. Joulin, A. Bastide, B. Brangeon, J.P. Caltagirone, Y. Cherif, R. Eymard, C. Garnier, S. Giroux-Julien, Y. Harnane, P. Joubert, N. Laaroussi, S. Lassue, P. Le Quéré, R. Li, D. Saury, A. Sergent, S. Xin, A. Zoubir, Benchmark solutions for natural convection flows in vertical channels submitted to different open boundary conditions, Int. J. Therm. Sci. 72 (2013) 18–33. [23] V.I. Terekhov, A.L. Ekaid, Turbulent free convection between vertical isothermal plates with asymmetrical heating, Thermophys. Aeromech. 20 (2013) 151–162. [24] W. Aung, G. Worku, Developing flow and flow reversal in a vertical channel with asymmetric wall temperatures, J. Heat Transf. 108 (1986) 229–304. [25] C. Cheng, H. Kou, W. Huang, Flow reversal and heat transfer of fully developed mixed convection in vertical channels, J. Thermophys. Heat Transf. 4 (1990) 375–383. [26] W.S. Fu, S.H. Huang, C.G. Li, An investigation of unsteady flow reversal of natural convection in vertical parallel plates by the modified local one- dimensional inviscid relations method, Int. J. Heat Mass Transf. 86 (2015) 124–138. [27] W.S. Fu, C.G. Li, C.P. Huang, J.C. Huang, An investigation of a high temperature difference natural convection in a finite length channel without Boussinesq assumption, Int. J. Heat Mass Transf. 52 (2009) 2571–2580. [28] P.L. Roe, Approximation Riemann solver, parameter vectors, and difference schemes, J. Comput. Phys. 43 (1981) 357–372. [29] J.M. Weiss, W.A. Simth, Preconditioning applied to variable and constants density flows, AIAA 33 (1995) 2050–2056. [30] S. Yoon, A. Jamesont, Lower-upper symmetric-Gauss–Seidel method for the Euler and Navier–Stokes equations, AIAA 26 (1988) 1025–1026. [31] I. Abalakin, A. Dervieux, T. Kozubskaya, A vertex centered high order MUSCL scheme applying to linearised Euler acoustics, INRIA 4459, 2002. [32] S.W. Churchill, H.H.S. Chu, Correlating equations for laminar and turbulent free convection from a vertical plate, Int. J. Heat Mass Transf. 18 (1975) 1323–1329.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Flow downward penetration of vertical parallel plates natural convection with an asymmetrically heated wall☆ Wu-Shung Fu a,⁎, Wei-Siang Chao a, Tzu-En Peng a, Chung-Gang Li b a b
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, ROC Complex Phenomena Unified Simulation Research Team, Advanced Institute for Computation Science, RIKEN, Kobe 650-0047, Japan
a r t i c l e
i n f o
Available online 17 March 2016 Keywords: Compressible flow Non-reflecting boundary Natural convection
a b s t r a c t Phenomena of the flow downward penetration in vertical parallel plates natural convection with an asymmetrically heated wall are investigated numerically. For clarifying the occurrence of flow downward penetration, the width of parallel plates is further broadened and broader than that investigated in the previous study in which the flow reversal was studied in detail. In order to simulate realistically, the compressibility and the viscosity of the working fluid are simultaneously considered, and the non-reflecting boundary is adopted at apertures of plates. The methods of the Roe scheme, preconditioning, and dual time stepping matching LUSGS are coordinated to solve governing equations of a low-speed compressible flow problem. The flow downward penetration, in which fluids are sucked from surroundings outside the outlet to flow into the parallel plates and through the inlet flow out of parallel plates, is composed of a few fluids and firstly observed. Also, two flow reversal caused by the buoyancy force appear in the upper left region, and one flow reversal induced by the flow downward penetration appears in the low right region. The usage of the existing results and present study, an equivalent ratio of the width to the length of parallel plates, of which the heat transfer rate is the same as the single vertical heated plate, can be expediently obtained. In this study, the ratio is about 0.6. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Until now, lots of numerical and experimental studies [1–26] still investigate natural convection in a channel or parallel plates because of its importance and applications in academic and industrial research. The flow reversal, which is one of unique and complex characteristics of natural convection, is often observed in some situations when a ratio of the width to the height of above situations exceeds a certain threshold. Since the flow field and heat transfer mechanism of natural convection are remarkably affected by the flow reversal. The investigation of the flow reversal becomes even firmer as time goes by. In the past, Sparrow et al. [2] conducted experimental and numerical studies to investigate the flow reversal in vertical parallel plates natural convection and flow reversal. The flow reversal was formed by a pocket of recirculating flow when Ra/A ratio exceeded a certain magnitude. Numerical solutions obtained by a parabolic finite difference scheme yielded Nusselt numbers in good agreement of the experimental results. An experimental work of natural convection in two vertical plates was conducted by Al-Azzawi [3]. One of the plates regarded as a heat wall was heated electrically, and the other was made of glass. The height of plates are 1 m and 2 m, respectively. The width between two plates ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. Tel.: +886 3 5712121x55173; fax: +886 3 5735065. E-mail address: [email protected] (W.-S. Fu).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2016.03.006 0735-1933/© 2016 Elsevier Ltd. All rights reserved.
varies from 25 to 150 mm. The thermal conditions of the heated wall were a uniform heat flux and a constant wall temperature, respectively. An experimental equation was derived to reveal a relationship between the Nusselt number and modified Rayleigh number. Chang and Lin [5] performed a numerical study to investigate the reversed flow and oscillating wake in an asymmetrically heated channel. The Boussinesq assumption and an extended domain were adopted. The phenomenon of flow reversal was indicated near the upper adiabatic wall. The larger the Rayleigh number was, the more remarkable phenomenon was revealed. Kihm et al. [8] performed a numerical study matching smoke visualization to investigate a problem of flow reversal of natural convection in isothermal vertical walls. The Boussinesq assumption and an extended boundary were adopted. The phenomenon of flow reversal was observed in the upper central region because of isothermal vertical walls, and variations of the entrance lengths with the Rayleigh numbers were indicated. Ospir et al. [20] conducted an experimental study to investigate an evolution of flow reversal from a single cell to a final eight-shaped structure. Results showed that the increase in the modified Rayleigh number resulted in an increment in the penetration of the flow reversal. Recently, Li et al. [21] firstly added an effect of radiation on investigating the flow reversal of natural convection in asymmetrically heated vertical channels numerically. Experimental results of Webb and Hill [4] were used as boundary conditions at the inlet and outlet of the channel, and then the governing equations of an elliptic nature were solved and an extended boundary adopted in the above
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Nomenclature A d e g h k k0 l M: y
area(m2) width of the plate (m) internal energy (J⋅ kg−1) acceleration of gravity (m⋅ s‐2) height of the plate (m) thermal conductivity (W⋅ mK−1) surrounding thermal conductivity (W⋅ mK−1) length of the plate (m) total mass flow rate defined in Eq. (13) (kg ⋅ s‐1) M : y ¼ ∫ ρudA
Nux
local Nusselt number Nux ¼ k0 ðT h−T c Þ ½kðTÞ ∂T ∂y
Nux
time-averaged local Nusselt number defined in Eq. (11) Nux ¼ 1t ∫ Nux dt
NuA
area-averaged local Nusselt number defined in Eq. (12) NuA ¼ A1 ∫ ∫ Nux dxdz
P P0 Pr R Ra
pressure (Pa) surrounding pressure (Pa) Prandtl number gas constant (J⋅ kg−1 ⋅ K−1) gρ2 ðT −T Þh3 Rayleigh number defined in Eq. (8) Ra ¼ Pr 0 h 2c
Ra⁎
modified Rayleigh number defined in Eq. (9) Ra ¼ Ra hb time (s) dimensionless time t ¼ t α2 h temperature (K) temperature of surroundings (K) temperature of heat surface (K) velocities in x, y, and z directions (m⋅ s−1) Cartesian coordinates (m) dimensionless Cartesian coordinates
A
defined
in
Eq.
(10)
h
t
t t⁎ T T0 Th u ,v , w x , y, z X , Y, Z
wh
T 0 μðTÞ
Greek symbols α Thermal diffusivity rate (m2 ⋅ s−1) ρ density (kg ⋅ m‐3) surrounding density (kg ⋅ m‐3) ρ0 μ viscosity (kg ⋅ m‐1 ⋅ s‐1) surrounding viscosity (kg⋅ m‐1 ⋅ s‐1) μ0 γ specific heat ratio
inlet flowing into parallel plates. In those situations the fluids, which were caused by the flow reversal through the outlet flowing into parallel plates, completely through the outlet flow out of parallel plates again. As the ratio of the width to the length is increased further and larger than those adopted in the previous study [26], in addition to the indication of the flow reversal, new phenomena are expected to be observed and able to encourage further investigation of this subject. Therefore, the aim of the study further enlarges the ratio of the width to the length of parallel plates adopted by Fu et al. [26] to investigate new phenomena of natural convection occurring in vertical parallel plates numerically. In order to simulate the situation more realistically, the compressibility and viscosity of the working fluid are considered, and the non-reflecting boundary condition [27], which causes the Boussinesq assumption to be no longer necessary, held at the inlet and outlet of the plates is adopted. For solving a low-speed compressible flow problem, the method of the Roe scheme [28], preconditioning [29], and dual time stepping [30] matching the LUSGS [31] are simultaneously used to solve the governing equations. Then the magnitudes of the velocity, temperature, pressure, and density at the inlet and outlet of the plate are simultaneously calculated. The results show that the total mass flow rate flowing into parallel plates is almost equivalent to the mass flow rate of the large modified Rayleigh number situations indicated by Fu et al. [26]. However, the width of this study is larger than that adopted by Fu et al. [26], and effects of the buoyancy force on flowing behaviors of the fluids, which are close to the right wall, naturally become weak. A few fluids, which are caused by the flow reversal appearing in the left upper region of parallel plates, unexpectedly form a downward flow and through the inlet flow out of parallel plates. This phenomenon is newly observed and named the flow downward penetration. Relative to occurrence of the flow reversal in the left upper region, some fluids close to the right low region are affected by the flow downward penetration to form the new flow reversal inside the inlet. With the concept of the modified Rayleigh number, an equivalent ratio of a single vertical heated plate can be expediently calculated under the same Nusselt number situation and the ratio is about 0.6 in this study. Also, both existing and present results are well consistent. 2. Physical backgrounds In this section, the physical model and the governing equations used are introduced, and the suitable initial and boundary conditions are derived. 2.1. Physical model
literature was not necessary. The flow reversal and distributions of streamlines and isotherms under a steady situation were indicated. The mass flow rate kept constant when the wall spacing exceeded a certain magnitude. The onset of flow reversal delayed upon the effect of surface radiation and enhancement of the cooling of the heated wall by radiation were shown. Desrayaud et al. [22] used eight different methods to study natural convection between parallel plates with an asymmetrically heated wall. The flow reversal was observed and results obtained by the above method were in good agreement. Because an elliptic natural of the problem was solved in the above literature, the results were mainly indicated by steady situations. Recently, Fu et al. [26] adopted the non-reflecting boundary, which was held at apertures because of consideration of the compressibility of fluid, to investigate the flow reversal of natural convection. Accompanied with increment of the ratio of the width to the length of parallel plates, the flow reversal became remarkable and drastic. The amount of fluid through the outlet flowing into parallel plates was gradually larger than that through the
From the results obtained by Fu et al. [26], except on the edge, phenomena on the cross sections of the XY plane in the direction of the Z axis are similar. Relative to Fu et al. [26], the width of this study is further enlarged for investigation of new phenomena of natural convection in vertical parallel plates with an asymmetrically heated wall. Accordingly, the depth in the Z axis of this study is shortened and shorter than that used by Fu et al. [26] for saving the total grids. The same periodical conditions used by Fu et al. [26] are still adopted in this study. Then the physical model of vertical parallel plates investigated in this study is indicated in Fig. 1. The left wall is heated and possesses a constant temperature Th, and the right wall is adiabatic. The width and length of parallel plates are d and l, respectively. The pressure and temperature of fluids in surroundings outside the inlet and outlet are Po and To, respectively. The direction of gravity is the negative y direction. In order to expand industrial and academic research, the compressibility and viscosity of working fluids are taken into consideration simultaneously. Thus, the Boussinesq assumption is no longer necessary and the non-reflecting boundary condition developed by [27] for a low-speed compressible flow can be suitably adopted on the inlet and outlet. For simplifying explanations, the upper and low
W.-S. Fu et al. / International Communications in Heat and Mass Transfer 74 (2016) 55–62
2
Non-reflecting boundary
2
outlet
Heated wall
T0
298 K
P0
1.013 105 Pa
Adiabatic wall
l Th
3 9 ρu > > > ρu þ P−τxx 7 > 6 > 7 > 6 > ρuv−τ 7 > 6 > xy F¼6 7 > > 7 > 6 ρuw−τxz > 5 > 4 > > ∂T > > > ρEu þ Pu−k −uτxx −vτ xy −wτxz > ∂x > 3 > 2 > > ρv > > > 7 > 6 ρuv−τyx > > 7 6 > 2 > 7 6 > ρv þ P−τ yy > 7 6 G¼6 > 7 > ρvw−τ yz > 7 > 6 > 5 > 4 > ∂T = ρEv þ Pv−k −uτ yx −vτyy −wτ yz ∂y 3> 2 > ρw > > > 7> 6 ρuw−τ zx > 7> 6 > > 7 6 ρvw−τ zy > 7> > H¼6 > 2 7 6 ρw þ P−τzz > 7> 6 > > 5 4 > ∂T > ρEw þ Pw−k −uτzx −vτzy −wτ zz > > > > 3 ∂z 2 > > > 0 > > > 6 −ðρ−ρ0 Þg 7 > > 7 6 > > > 7 6 S¼6 0 > > 7 > > 5 4 0 > > ; −ðρ−ρ0 Þgu
57
400 K
g
ð4Þ
x Sutherland's law is adopted to evaluate the viscosity and thermal conductivity shown as follows:
d
y
2 T 3 T 0 þ 110 T0 T þ 110 μ ðT ÞγR kðT Þ ¼ ðγ−1Þ Pr
inlet Non-reflecting boundary
μ ðT Þ ¼ μ 0
Fig. 1. Physical model.
apertures of parallel plates are conveniently called the outlet and inlet, respectively. 2.2. Governing equations For facilitating the analysis, the following assumptions are made. 1. The work fluid is ideal gas and follows the equation of state of an ideal gas. 2. Magnitudes of gradients of density and pressure on the whole surfaces in the normal direction are zero. 3. Radiation heat transfer is neglected.
ð1Þ
and the equation of state of an ideal gas is used. P ¼ ρRT
ð2Þ
The terms included in U, F, G, H, and S are separately shown in the following equations: 3 ρ 6 ρu 7 7 6 7 U¼6 6 ρv 7 4 ρw 5 ρE
in which
P 1 þ u2 þ v2 þ w2 ρðγ−1Þ 2 ρ0 ¼ 1:1842kg=m3 ; g ¼ 9:81m=s2 ; μ 0 ¼ 1:85 10−5 Ns=m2 ; T0 ¼ 298:05K; γ ¼ 1:4; R ¼ 287J=kg=K; Pr ¼ 0:72: E¼
ð6Þ
To simplify the analysis, the following dimensionless variables are made:
The governing equations are expressed as follows: ∂U ∂F ∂G ∂H þ þ þ ¼S ∂t ∂x ∂y ∂z
ð5Þ
2
ð3Þ
x y z ; Y¼ ; Z¼ ; h h h α t ¼ t 2 h X¼
ð7Þ
3. Numerical method Since the study is an extensive study of the previous study [26], the numerical method used is basically modified from the previous study [26]. The width is broadened in this study, and the density of grids in the width is then increased. The largest ratio (d/l) of the width to length by Fu et al. [26] is 0.1125. In this study, six different ratios are adopted, and the related magnitudes corresponding grid distributions and modified Rayleigh numbers are tabulated in Table 1. Except the grid distribution, the detailed finite differentiation of the governing equations and computing procedures are the same as those adopted by Fu et al. [26].
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Table 1 Related parameters Fu et al. [26] study
This study 2
3
4
5
6
7
d l
0.1125
0.100
0.150
0.175
0.200
0.250
0.300
0.400
Grid distribution
200 × 100 × 50 1.072 × 105
200 × 80 × 4 6.69 × 104
200 × 120 × 4 3.39 × 105
200 × 140 × 4 6.27 × 105
200 × 160 × 4 1.07 × 106
200 × 240 × 4 2.61 × 106
200 × 240 × 4 5.42 × 106
200 × 320 × 4 1.71 × 107
R a ¼ Ra dl
1
4. Results and discussion Basically, this work is an extensive study of the previous study [26]. For highlighting the relationship between the width and length, the Rayleigh number and modified Rayleigh number are defined as follows, respectively, 3
Ra ¼ Pr
gρ2o ðT h −T o Þd
Ra ¼ Ra
T o μT 2 d l
ð8Þ
ð9Þ
Due to complex variation of the flow field, several signals of exaggerated arrows are conveniently adopted to indicate variational flow fields. In Fig. 2(a–l), the history of development of flow fields is indicated under the situation of Ra⁎ = 2.61 × 106 and d l ¼ 0:25. Shown in initial stages of Fig. 2(a–c), fluids in parallel plates are heated by the left heated wall that causes the volume of the fluid to be expanded. Thus, the fluids start to flow out of parallel plates from both the inlet and outlet, respectively. Afterward, the fluids are affected by the buoyancy force produced by the left heated wall to flow upward. Since the width of parallel plates is wide. For supplementing the upward mass flow rate caused by the buoyancy force, fluids from surroundings outside the inlet and outlet are simultaneously sucked into parallel plates. The heat transfer mode varies from heat conduction to heat convection. Because of the higher modified Rayleigh number, the upward flow along the left heated wall becomes unsteady especially in the upper region of the heated wall. Several flow reversals, which are composed of fluids mainly sucked from surroundings outside the outlet, are casually formed on the upper heated wall and revealed in Fig. 2(d and e). These phenomena are similar to those indicated in the previous study [26]. In Fig. 2(f), the fluids are gradually accelerated to flow into parallel plates because of the strong strength of flow reversals. However, the width of this study is enlarged and wider than that of Fu et al. [26]. Effects of the buoyancy force on the fluids in the upper region of the right wall naturally become weak. Accordingly, the penetration length, which means the downward distance of movement of the fluid from the outlet, of the fluids close to the right wall is longer than that of the fluids close to the flow reversals. After piercing through the outlet, a few fluids close to the right upper corner are squeezed and reluctantly depart from the penetration path to flow back to the outlet. A small flow reversal is then formed in the right upper corner. In Fig. 2(g), due to the buoyancy force, three flow reversals on the heated wall shown in the above figure subsequently merge into one large flow reversal close to the left upper corner. Meanwhile some fluids, which are dragged from surroundings outside the outlet and accelerated by the large flow reversal, concentrate in the central region of the outlet to flow into parallel plates. As a result, the small flow reversal in the right upper corner becomes large. Since the width is large, the buoyancy force are difficult to affect behaviors of fluid close to the right wall. Then a few fluids, which are dragged by the large flow reversal and possess large downward impulse, along the right wall directly flow downward and penetrate the inlet to surroundings. Relative the flow reversal, the phenomenon is suitably named the flow downward
penetration. At this instant, except the region of the flow downward penetration, fluids in the other region are sucked from surroundings outside the inlet by the buoyancy force to flow into parallel plates. However, the buoyancy force is weak in the right region of the inlet, and then part of fluids close to the right low region of the inlet are immediately dragged by the flow downward penetration after piercing through the inlet and compulsively turned the direction to flow through the inlet. As a result, the small flow reversal neighbors on the flow downward penetration to be observed as shown in Fig. 2(h). As time passes revealed in Fig. 2(i–l), two upper flow reversals, one low flow reversal and the flow downward penetration become remarkable, respectively. In Fig. 3(a–l), history of development of thermal fields is indicated under the situation of Ra= 2.61× 106, d l ¼ 0:25. The darker the color, the higher the temperature indicated. Each instant shown in the figure is corresponding to that indicated in Fig. 2. In initial stages Fig. 2(a–c), the heat transfer mode varies from heat conduction to heat convection gradually. An orderly thermal boundary layer is clearly observed on the left heated wall, and the thickness of the thermal boundary becomes thicker from the inlet to outlet accompanied with increment of time. Afterward, in Fig. 3 (d–f), due to the higher modified Rayleigh number, the thermal field becomes unsteady on the upper heated wall, and several convex thermal lumps, of which the locations are corresponding to those of the flow reversals revealed in Fig. 2, are formed. The lump is almost filled by fluids with the dark color, and the thermal boundary region between the lumps becomes extremely thin. These phenomena firmly indicate the influence of behaviors of lumps even on the bottom of the thermal boundary. It implies that similar phenomena of the instability of the thermal boundary occur. Except the regions composed of the thermal boundary and lumps, the color of the other region is almost white that means the temperature of the other region not to be affected by the left heated wall. Flowing behaviors of fluids in the other region indicated in Fig. 2 are caused by the suction effect of the buoyancy force and not affected by the temperature difference, which plays an important role of the driving force in natural convection. As time passes, three thermal lumps merge to one thermal lump on the upper corner of the heated wall because of the buoyancy force as shown in Fig. 3(g). Afterward, the thermal boundary and thermal lump periodically unify [Fig. 3(h and k)] and divide [Fig. 3(i, j, and l)] to be observed. It implies that the heat transfer between the thermal boundary and lump is sometimes interrupted by the cool fluids, which are sucked by the flow reversal, to flow into parallel plates and flow out of parallel plates again after encircling the thermal lump. Since the cool fluids are indicated by the white color, the space between the thermal boundary and lump is periodically indicated by the white color, too. Variations of local Nusselt numbers of different locations with time for Ra⁎2.61 × 106 are indicated in Fig. 4. The definition of the local Nusselt number and the time-averaged local Nusselt number are defined as follows, respectively,
Nux ¼
h ∂T K ðT Þ k0 ðT h −T c Þ ∂y
ð10Þ
W.-S. Fu et al. / International Communications in Heat and Mass Transfer 74 (2016) 55–62
59
Fig. 2. History of development of flow fields under Ra⁎ =2.61×106,dl ¼ 0:25.
Nux ¼
1 t
Z Nux dt
ð11Þ
t
In the initial stage, the heat transfer mechanism is mainly dependent on heat conduction mode that causes the variations and the magnitudes of the local Nusselt number to be stable and similar. Due to occurrence of unsteady phenomena along the left upper wall, the variation of the local Nusselt number becomes drastic with increment of the distance from the inlet. Shown in Fig. 5, distributions of area-averaged local Nusselt numbers under different modified Rayleigh numbers are revealed.
The definition of the area-averaged Nusselt number is defined as follows:
NuA ¼
1 A
Z Z Nux dxdz w
ð12Þ
h
Since the width is wider, the modified Rayleigh number becomes larger. Behaviors of fluids in the wide space are able to be more active. As a result, the larger the modified Rayleigh number is, the larger the area-averaged Nusselt number is achieved.
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Fig. 3. History of development of thermal fields under Ra⁎ =2.61×106,dl ¼ 0:25.
In Fig. 6, distributions of mass flow rates flowing through the inlet and outlet are indicated, respectively. The definition of mass flow rate is defined as follows, _y¼ M
Z ρudA
ð13Þ
A
In the previous study [26], part of small modified Rayleigh number situations had the same mass flow rate ① through the inlet flowing into and through the outlet flowing out of parallel plates. In other part of large modified Rayleigh number situations, part of the fluids ②, which are sucked from surroundings outside the outlet, flow into and out of parallel plates through the outlet because of occurrence of the flow reversal. The other fluids ③, which are sucked from the outside of the inlet, through the inlet flow into and through the outlet flow out of parallel plates due to the buoyancy force.
However, in this study co-existence of the flow reversal and downward penetration are observed that causes flow phenomena to be more complex. Thus, the fluids, which are sucked from surroundings outside the outlet by the left upper flow reversal, through the outlet flow into parallel plates. Most of the fluids ④ through the outlet flow out of parallel plates because of the upper flow reversal, but a few other fluids ⑥ flow out of parallel plates through the inlet due to the flow downward penetration. Similarly, the other fluids, which are sucked from surroundings outside the inlet by the buoyancy force, through the inlet flow into parallel plates. Most of the fluids ⑤ through the outlet flow out of parallel plates due to the buoyancy force, but a few other fluids ⑥ through the inlet flow out of parallel plates because of the right low flow reversal. In Fig. 7, comparison of existing results and present results is revealed. Present results are consistent well with the existing results [3] in both investigated (solid line) and extended (dashed line) regions. The signal of X indicates the result of the situation of a single vertical
W.-S. Fu et al. / International Communications in Heat and Mass Transfer 74 (2016) 55–62
61
Fig. 4. Variations of local Nusselt number of different locations with dimensionless time for Ra⁎ =2.61×106.
heated plate. The time and area-averaged Nusselt number of X is calculated by the empirical equation derived from [32]. The width of the situation of the single vertical heated plate is close to infinite. For application, the corresponding modified Rayleigh number of the magnitude of X is expediently assumed to be equivalently located in the extended region of Al-Azzawi [3]. The usage of the empirical equation derived by Al-Azzawi [3], the corresponding modified Rayleigh number and equivalent ratio of the width to the length of the single vertical heated plate can be calculated. The magnitudes of the correspond ing modified Rayleigh number, Ra⁎, and ratio, d l , are about 1.46 × 108 and 0.6, respectively. It implies that the ratio of d l is larger than 0.6, and the effect of the heat transfer of the left heated wall caused by the right wall can be negligible.
Fig. 5. Variations of area-averaged Nusselt numbers of different modified Rayleigh numbers with dimensionless time.
5. Conclusions An investigation of the flow downward penetration of natural convection in vertical parallel plates with an asymmetrically heated wall is conducted numerically. In order to expand industrial and academic research of the previous study [26], the ratio of the width to the length is enlarged and larger than that adopted by Fu et al. [26]. Related methods of the Roe scheme [28], preconditioning [29], and dual time stepping [30] matching the method of LUSGS [31] are used simultaneously to solve low-speed compressible flow problems. Several conclusions are drawn and shown as follows: 1. Due to increment of the width, the buoyancy force caused by the left heated wall is difficult to affect flowing behaviors of the fluids close to the right wall, which leads the flow downward penetration to form along the right wall.
Fig. 6. Distributions of mass flow rates flowing through the inlet and outlet.
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Fig. 7. Comparison of existing results and present results.
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