An investigation of flow reversal of mixed convection in a three
dimensional rectangular channel with a finite length
Wu-Shung Fu
⇑, Yu-Chih Lai, Yun Huang, Kuan-Lan Liu
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, ROC
a r t i c l e
i n f o
Article history:
Received 20 August 2012
Received in revised form 15 April 2013 Accepted 15 April 2013
Available online 29 May 2013 Keywords:
Reversal flow Mixed convection Three dimensional channel
a b s t r a c t
An investigation of flow reversal of mixed flow in a three dimensional channel is studied numerically. At a high Richardson number, natural convection dominates the flow and thermal fields of mixed convection. Behaviors of the residual mass flow rate produced by the difference of the strength of natural and forced convections coexisting in mixed convection are worthy to be examined deeply for industrial applications. Due to the necessity of considering the fluid compressibility, methods of the Roe scheme, preconditioning and dual time stepping are adopted to solve governing equations. The results show that at a high Rich-ardson number situation, via the outlet in large quantities of fluids are sucked into the channel from the outside that causes the flow and thermal fields in the channel to be unsteady and vice versa.
Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Strictly speaking, except in situations of low temperature differ-ences between cooling fluids and heat sources, heat transfer phe-nomena of forced convection only in realistic situations are hardly observed; instead, the coexistence of forced and natural convections, also called mixed convection, usually appears. Heat transfer phenomena of mixed convection are strongly influenced by the ratio of natural convection to forced convection. And mixed convection is mainly divided into three parts of cross, opposite and aiding flows. Due to the same direction of the buoyancy force and a fluid flow in aiding flow mixed convection, the number of analyses of aiding flow mixed convection are relatively more than those of the other two mixing flows. From a viewpoint of enhancement of heat transfer, aiding flow mixed convection is usually regarded to be advantageous, so it attracts more investigations than those of the other two types.
Metais et al.[1]conducted an experimental work to draw a dia-gram indicating a flow relationship between Reynolds and Grashof numbers. In the diagram, regions of natural, forced and mixed con-vections divided into a laminar and a turbulent flows were clearly delimited. Behzadmehr et al.[2] investigated aiding flow mixed convection with the Boussinesq assumption under conditions of uniform heat flux and low Reynolds numbers in a vertical circular duct. Relationships of Grashof and Nusselt numbers were yielded to distinguish the laminar and turbulent regions. The results
showed that the regions of Re = 1000, 8 105< Gr < 5 107and
Re = 1500, 2 106< Gr < 108were included in the turbulent region.
Tanaka et al.[3]conducted an experimental work like[1]to draw a diagram of Reynolds via Grashof numbers, the domain of Reynolds number was between 1000 and 5000. Regions of natural, forced and mixed convections were divided into the laminar and turbu-lent flows. Celata et al. [4] showed the experimental results of the distribution of the buoyancy parameter of Bo in a figure of Rey-nolds number via Grashof number with water. In the range of Bo 6 1, a laminar phenomenon was observed, and the correspond-ing Nusselt number was slightly smaller than that of pure forced convection. Boulama and Galanis[5] studied aiding flow mixed convection in a two-dimensional vertical parallel plates with a condition of fully developed flow at the outlet, and the analytical solutions were dependent on the parameters which combined the effects of thermal and solutal buoyancy. The results revealed that buoyancy effects significantly improved heat and momentum transfer rates near the heated walls. Desrayaud and Lauriat[6]
investigated aiding flow mixed convection of a two-dimensional vertical duct with a high wall temperature. The results showed that when the magnitude of Gr/Re2 was larger than 1, phenomena of
flow reversal were observed. Under a condition of a constant Gras-hof number, the larger the Reynolds number was, the more diffi-cult the flow reversal was found. Zghal et al. [7] investigated aiding flow mixed convection in a two-dimensional vertical duct with the Boussinesq assumption. Effects of parameters of length, Reynolds number and Richardson number on Nusselt number were examined. Appearance of flow reversal was mainly determined between a relationship of Peclet and Richardson numbers. Ingham et al. [8] investigated phenomena of flow reversal of mixed 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.04.025
⇑Corresponding author. Address: 1001 Ta Hsueh Road, Hsinchu 30056, Taiwan, ROC. Tel.: +886 3 5712121x55110; fax: +886 3 5735065.
E-mail address:wsfu@mail.nctu.edu.tw(W.-S. Fu).
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International Journal of Heat and Mass Transfer
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m tconvection in a two-dimensional constant temperature vertical duct with the Boussinesq assumption. In a range of 300 6 Gr/ Re 6 70, the flow reversal easily appeared at the larger magnitude of jGr/Rej. Barletta [9,10]adopted an analytical method and the Boussinesq assumption to investigate mixed convection in a rect-angular cross-section duct with a fully developed flow condition in the z axis. Thermal conditions of walls were composed of differ-ent combinations of high and low temperatures and constant heat flux. Phenomena of flow reversal were examined under different shapes of rectangular cross section. Barletta[11]investigated vis-cous dissipation effect of mixed convection in a two-dimensional vertical duct. The phenomenon of flow reversal in opposing flow mixed convection was more apparent than that in aiding flow mixed convection under the Boussinesq assumption. Yang et al.
[12]adopted the Boussinesq assumption to study mixed convec-tion in a long two-dimensional vertical duct. Heat transfer mecha-nisms were investigated under positive and negative magnitudes of Richardson numbers. Nguyen et al.[13]investigated transient mixed convection in a high heat flux circular duct with the Bous-sinesq assumption numerically. Boundary conditions at the inlet and outlet were a uniform velocity and a fully developed flow, respectively. In an opposite situation, the flow reversal appeared near the outlet region at Gr = 3 105, and in an aiding situation
the flow reversal appeared near the center of the axis at Gr = 106.
In retrospect, the Boussinesq assumption which is only useful for the temperature differences smaller than 30 K[14]is still con-veniently used by most of the above literature. According to the limitation of the Boussinesq assumption, the analysis of mixed convection is then necessary to add an extra domain to the original domain that causes a fully developed condition to be adopted at the edge of the domain newly added. Regretfully, by using the Boussinesq assumption, some interesting and important
character-istics of mixed convection have the possibility to be omitted espe-cially in the range of the high magnitude of Richardson number. Doubtless, at a high Richardson number situation natural convec-tion dominates flow and heat transfer mechanisms. Then the amount of fluid which is induced by natural convection and flows through the domain is substantially larger than the amount of fluid which is provided by forced convection and flows through the do-main. Behaviors of the residue of the fluid caused by the difference between the amount of fluid provided by forced convection and the amount of fluid induced by natural convection are not deeply discussed yet. As well, the inlet is usually filled with the amount of fluid provided by forced convection. As a result, a problem of consideration of behaviors of the residue of the fluid mentioned above becomes important and is worthy of deep investigation.
The aim of the study investigates flow reversal and heat transfer mechanisms of mixed convection in a three-dimensional vertical channel with a finite length under larger Richardson numbers numerically. A non-reflecting boundary condition is assigned at the outlet of the domain to adjust fluids sucked into the channel or discharged to the outside of the channel according to the mag-nitude of the Richardson number. The compressibility of fluid has been taken into consideration for matching the usage of the non-reflecting boundary condition which also means that the Bous-sinesq assumption is no longer needed. Necessary methods of the Roe scheme[16], preconditioning and dual time stepping[17]for solving a low speed compressible flow are used. The results show that in high Richardson number situations natural convection is dominant and causes part of the amount of fluid from the outside via the outlet to be sucked into the channel. In a certain situation, the amount of fluid sucked into the channel is approximately equal to that flowing into the channel via the inlet provided by forced convection. That leads to drastic impingement between both the Nomenclature
a sound speed (m s1)
A area (m2)
Cp constant-pressure specific heat (J kg1k1)
Cv constant-volume specific heat (J kg1k1)
d width of the square channel (m) e internal energy (J kg1)
g acceleration of gravity (m s2)
Gr Grashof number defined in Eq.(12)
h enthalpy (J)
k thermal diffusivity (W m1k1)
k0 surrounding thermal diffusivity (W m1k1)
l length of the square channel (m) _
Minlet;n:c: dimensionless mass flow rate of natural convection de-fined in Eq.(36)
_
Minlet dimensionless mass flow rate at inlet defined in Eq.(33) (kg s1)
_
Moutlet dimensionless mass flow rate from outside at the outlet defined in Eq.(34)(kg s1)
Nu area averaged Nusselt number defined in Eq.(38)
Nux local Nusselt number defined in Eq.(35)
(Nux)t time averaged local Nusselt number defined in Eq.(37)
ðNuÞt time and area averaged Nussel number defined in Eq.
(39)
P pressure (Pa)
P0 surrounding pressure (Pa)
Pr Prandtl number R gas constant (J kg1k1)
Ra Rayleigh number defined in Eq.(12)
Re Reynolds number defined in Eq.(10)
Ri Richardson number defined in Eq.(11)
t time (s)
t⁄ dimensionless time defined in Eq.(9)
T temperature(K)
T⁄ dimensionless temperature defined in Eq.(9)
T0 temperature of surroundings (K)
Th temperature of heat surface (K)
DT time difference (K)
u,
v
, w velocities in x, y and z directions (m/s)U, V, W dimensionless velocities in x, y and z directions defined in Eq.(9)
x, y, z Cartesian coordinates (m)
X, Y, Z dimensionless Cartesian coordinates in Eq.(9)
Greek symbols
a
Thermal diffusivity rate (m2/s)b volumetric thermal expansion coefficient (K1)
q
density (kg m3)q
0 surrounding density (kg m3)t
kinematics viscosity (m2s1)l
absolute viscosity (N s/m2)l
0 viscosity of surrounding (N s/m2)c
specific heat ratioC preconditioning matrix[17]
amounts of fluid mentioned above to occur and the flow field to be unsteady. Oppositely, in low Richardson number situations the amount of fluid provided by forced convection is larger than the amount of fluid sucked into the channel induced by natural con-vection. As a result, the impingement mentioned above becomes peaceful and the flow in the channel displays a steady situation.
2. Physical model
A physical model investigated in this study is a three-dimen-sional vertical rectangular channel and shown inFig. 1. The cross section of the channel is square and the width is d. The length of the channel is finite and equal to l. The direction of gravity g is downward and parallel to the vertical channel. The temperature of four heat wall surfaces is constant and equal to Thwhich is
high-er than a temphigh-erature T0of surroundings. Boundary conditions of
the temperature and velocity and pressure at the outlet of the channel EFGH are non-reflecting which was developed by Fu et al.[15].
In the situation of natural convection, via the inlet the amount of fluid flowing into the channel has difficulty to be predicted in advance. Then the non-reflecting boundary conditions of the veloc-ity, temperature and pressure are held at the inlet to determine the amount of fluid induced by natural convection and flowing into the channel and shown inFig. 1(a).
The same channel is also used in mixed convection, the amount of fluid via the inlet flowing into the channel is provided by forced convection and evenly distributed on the inlet and shown in
Fig. 1(b). The velocity and temperature of the flowing fluid are equal to u0and T0, respectively. In this situation of mixed
convec-tion, the amount of fluid induced by natural convection via the inlet flows into the channel is no longer permitted because of the complete occupancy of the amount of fluid provided by forced convection at the inlet. Therefore, in a large Richardson number
situation the amount of fluid induced by natural convection is per-mitted only from the central region of the outlet to flow down-wards into the channel and compulsively impinges the amount of fluid which is provided by forced convection and flowing up-wards into the channel from the inlet. Both the amounts of fluid then mix together and newly compose upward streams along the heat walls via the outlet to flow out of the channel. Phenomena of flow reversal are then to be observed in the channel.
For facilitating the analysis, several assumptions are made and indicated as follows.
(1) A laminar flow.
(2) Properties of fluids follow the equation of state of an ideal gas.
(3) The pressure gradient in normal direction of surfaces is equal to zero.
The governing equation is expressed as follows
@U @t þ @F1 @x þ @F2 @yþ @F3 @z ¼ S ð1Þ
The quantities included in U, Fiand S are separately shown in the
following equations, respectively
U ¼
q
q
uq
v
q
wq
e 0 B B B B B B @ 1 C C C C C C A ð2Þ Fi¼q
uiq
uiu1þ Pdi1l
Ai1q
uiu2þ Pdi2l
Ai2q
uiu3þ Pdi3l
Ai3 ðq
e þ PÞuil
Aijuj k@x@Ti 0 B B B B B B @ 1 C C C C C C A ;8
i ¼ 1ðxÞ; 2ðyÞ; 3ðzÞ ð3Þ S ¼ 0 ðq
q
0Þg 0 0 ðq
q
0Þgu1 0 B B B B B B @ 1 C C C C C C A ð4Þ where Aij¼@u@xj iþ @ui@xjand the ideal gas equation is written by
P ¼
q
RT ð5ÞOn the surfaces of ABFE, BCGF, CDHG and DAEH:
T ¼ Th; for both convections ð6Þ
On the surface of ABCD:
u ¼ u0; T ¼ T0 ðforced convectionÞ ð7Þ
On the surfaces of ABCD (for natural convection) and EFGH (for both convections):
The non-reflecting conditions are held.
The Sutherland’s law is adopted to evaluate the viscosity and the thermal conductivity as follows, respectively
l
ðTÞ ¼l
0 T T0 2 3T 0þ 110 T þ 110 kðTÞ ¼l
ðTÞc
R ðc
1ÞPr ð8Þ whereq
0= 1.1842 kg/m3, g = 9.81 m/s2,l
0= 1.85 105N s/m2, T0= 298.0592 K,c
= 1.4, R = 287 J/kg/K and Pr = 0.7.To simplify the analysis, the following dimensionless variables are made X ¼x d; Y ¼ y d; Z ¼ z d U ¼ u u0;Re¼950 ; V ¼
v
u0;Re¼950 ; W ¼ w u0;Re¼950 l ¼ 3d t¼ tl
0q
0d 2 T ¼T Th ð9Þwhere u0,Re=950means a uniform velocity u0to be assigned at the
in-let under the situation of Re = 950.
To investigate the heat transfer in the cases of different Richard-son numbers, the compressibility and viscosity of the working fluid are considered. Definitions of Reynolds number, Re, Richardson number, Ri, and Rayleigh number, Ra, are represented as follows, respectively Re ¼
q
0u0dl
0 ð10Þ Ri ¼ Gr Re2¼ gq
2 0bðTh T0Þd 3l
2 0l
2 0q
2 0u20d 2¼ gbdðTh T0Þ u2 0 ð11Þ Ra ¼ Pr Gr ¼ ð0:7Þ gq
2 0bðTh T0Þd 3l
2 0 ð12Þ 3. Numerical methodMethods of the Roe scheme[16]and preconditioning[17]are adopted to resolve the governing equations of the compressible flow shown in Eq.(1). Besides, the dual time stepping method is added to calculate transient states. And Eq.(13)can be obtained
C
@Up @s
þ @U @t þ @F1 @x þ @F2 @yþ @F3 @z ¼ S ð13Þwhere
s
is an artificial time, t is a physical time.Cis a precondition-ing matrix proposed by Weiss and Smith[17]and UPis a primitiveform of [
q
,q
u,q
v,q
w,q
e]T. By the discretization of Eqs.(13) and (14)can be obtained. Terms of the@Up
@s and@U@tare differentiated by a
first-order forward and a second-first-order backward differences, respec-tively. Terms of the@F1
@x; @F2
@y and @F3
@z are differentiated by a central
difference
C
U kþ1 p U k pD
s
þ 3Unþ1 4Unþ Un1 2D
t þ 1D
x F k 1 iþ12;j;k F k 1 i12;j;k þ 1D
y F k 2 i;jþ12;k F k 2 i;j12;k þ1D
z F k 3 i;j;kþ12 F k 3 i;j;k12 ¼ S ð14ÞAfterward terms of the Uk+1and Fkþ1i in Eq.(14)are necessary to
be linearized and expressed as follows, respectively
Ukþ1¼ Ukþ M
D
Up ð15Þ where M ¼@U @UpandDUp¼ U kþ1 p U k p Fkþ11 ¼ F k 1þ ApD
Up ð16Þ where the Ap¼ @Fk 1@Upis the flux Jacobian and the same method is used for the Bp¼@F k 2 @Up and Cp¼ @Fk 3 @Up in linearization of the F kþ1 2 and F kþ1 3 , respectively.
To substitute Eqs. (15) and (16) into Eq. (14), the following equation is obtained
C
U kþ1 p U k pD
s
þ 3Ukþ1 4Unþ Un1 2D
t þ dx F k 1þ ApD
Up þ dy Fk2þ BpD
Up þ dz Fk3þ CpD
Up ¼ Sk ð17Þwhere dx, dy, and dzare central-difference operators.
Eq.(17)can be rearranged as the following form
I
D
s
þC
1M 3 2D
tþC
1 dxAkpþ dyBkpþ dzCkpD
Up¼C
1Rk ð18Þ where Rk ¼ S 3Uk4UnþUn1 2Dt ðdxFk1þ dyFk2þ dzFk3ÞTo solve a problem of the convergence of a low-speed com-pressible flow, the solver of.
Eq.(19)is newly derived from the LUSGS implicit method orig-inally proposed by Yoon and Jamesont[18]
Ap¼
C
1AkpBp¼
C
1BkpBp¼
C
1Ckpð19Þ
Ap, Bpand Cpcan be divided into two parts
Ap¼ Aþp þ A p Bp¼ Bþp þ B p Cp¼ Cþp þ C p ð20Þ where A p ¼ 1 2ðAp jkAjIÞ Bp ¼ 1 2ðBp jkBjIÞ C p ¼ 1 2ðCp jkCjIÞ ð21Þ
To substitute Eq. (20) into Eq. (18), the following equation is obtained I
D
s
þC
1M 3 2D
tþ dn A þ pþ A p þ dg Bþp þ B p þ df Cþp þ C pD
Up ¼C
1Rk ð22Þ dn Aþpþ A pcan be derived as following equation
dn ~Aþp þ ~Ap ¼ d nA~ þ p þ dþnA~ p ¼ ~ Aþ p;i ~Aþp;i1
D
n þ ~ A p;iþ1 ~Ap;iD
n ð23ÞTo substitute Eq.(23)into Eqs.(22) and (24)can be obtained
I
D
s
þC
1M 3 2D
tþ eAþ p;i eA þ p;i1D
n þ eA p;iþ1 eAp;iD
n þ eBþ p;j eB þ p;j1D
g
" þeB p;jþ1 eBp;jD
g
þ e Cþ p;k eCþp;k1D
f þ eC p;kþ1 eCp;kD
f #D
Up¼C
1Rk ð24Þ Eq.(24)can be rearranged as followsðL þ D þ UÞ
D
Up¼C
1Rk ð25ÞL ¼ 1
D
nðA þ pÞi1;j;kþ 1D
g
ðB þ pÞi;j1;kþ 1D
fðC þ pÞi;j;k1 D ¼ ID
s
þC
1M 3 2D
tþ 1D
nððA þ pÞi;j;k A p i;j;kÞ þ 1D
g
ððB þ pÞi;j;k ðB pÞi;j;kÞ þ 1D
fððC þ pÞi;j;k ðC pÞi;j;kÞ U ¼ 1D
n A p iþ1;j;kþ 1D
g
ðB pÞi;jþ1;kþ 1D
fðC pÞi;j;kþ1As for the computation of Rk
¼ S 3Uk 4UnþUn1 2Dt dxFk1þ dyFk2þ dzFk3
in the RHS (right hand side) of Eq.(18), terms of the Fiin Eq.(3)based on the Cartesian coordinate can be divided
into two parts. One is an inviscid term Finviscid
Finviscid¼
q
uiq
uiu þ Pdi1q
uiv
þ Pdi2q
uiw þ Pdi3 ðq
e þ PÞui 0 B B B B B B @ 1 C C C C C C A ð26ÞThe other is a viscous term Fviscous
Fviscous¼ 0
l
Ai1l
Ai2l
Ai3l
Aijujþ k@T@xi 0 B B B B B B @ 1 C C C C C C A ð27ÞA Roe upwind difference scheme[16]is employed in discretization of the terms of the Finviscidat the cells interface i þ12
and expressed as follows at a low Mach number situation
Finviscid;iþ1 2¼ 1 2ðFRþ FLÞ 1 2 j
C
1 ApjD
UP n o ð28ÞThe MUSCL scheme with a third order proposed by Abalakin et al.
[19]is used to compute the terms of the Finvicid, and the related
derivative terms of Aij¼ @uj
@xiþ
@ui
@xjin Eq.(27)are computed by a fourth order central difference
@u
@x¼
ui2 8ui1þ 8uiþ1 uiþ2
12
D
x þ oðD
x4Þ ð29Þ
The advantage of usage of the LUSGS implicit method is to improve efficiency.
On the heat surface, the boundary conditions are
Pði; 0; kÞ ¼ Pði; 1; kÞ uði; 0; kÞ ¼ uði; 1; kÞ
v
ði; 0; kÞ ¼v
ði; 1; kÞ wði; 0; kÞ ¼ wði; 1; kÞ Tði; 0; kÞ ¼ 2Th Tði; 1; kÞ ð30ÞWhere This the wall temperature.
0 indicates the ghost cell and 1 indicates the cell most near the wall.
As for the boundary conditions at the outlet, in order to avoid the flow in the channel polluted by the reflections of acoustic waves induced by the compressible flow, the non-reflecting boundary conditions are then necessarily used at the outlet of the channel.
In a high speed compressible flow condition, the method of LODI (local one-dimensional inviscid relations) proposed by Poinsot and Lele[20]was suitably adopted for determining the non-reflecting boundary conditions at the outlet. However, a preconditioning matrix is not necessary in the above method that causes the method to be not adopted appropriately for determin-ing the non-reflectdetermin-ing boundary conditions at the outlet under a low speed compressible flow. As a result, the method developed by Fu et al. [15] is necessary for resolving the non-reflection boundary conditions under a low speed compressible flow. Table 1
Comparisons for all situations (Ra = 6.78 105
, DT = 100K).
Flow M_inlet M_outlet U0 Umax ðNuÞi Ri t
⁄ Natural convection Re = 0 1 0 1 1.43 9.46 0.09 Mixed convection Re = 950 1 ffi0 1 1.46 8.87 1.0 0.07 Mixed convection Re = 400 0.43 0.02 0.46 1.27 7.16 5.9 0.07 Mixed convection Re = 200 0.21 0.12 0.22 1.13 7.71 23.6 0.09 Mixed convection Re = 100 0.11 0.11 0.11 0.95 7.64 94.2 0.09
Fig. 2. Grid distributions under natural convection (Ra = 6.78 105, DT = 100K).
Fig. 3. Variation of mass flow rate with time for natural convection at the inlet (Ra = 6.78 105
A procedure calculating the equations mentioned above is briefly described as follows.
(1) Assign the inlet conditions of pressure, velocity and temperature.
(2) Use the MUSCL method to calculate Eq.(18)to obtain the magnitude of theDUP.
(3) Substitute the magnitude of theDUPinto Eq.(28)and use
the Roe method to calculate the magnitudes of inviscid terms of the Finviscid.
(4) Calculate Eq.(29)to obtain the magnitudes of viscous terms and substitute in Eq.(27).
(5) Solve Ukþ1p Ukþ1 p ¼ U k pþ
D
U k p ð31Þ(6) Calculate Eq.(14)and examine the convergence of the iter-ational computation of Ukþ1
p . Repeat (2) (5) until until Ukþ1
p Ukp
Ds <
e
,e
= 103.
4. Results and discussion
In this study, there are five situations of which the ratio of the height to the width is 3, tabulated inTable 1to be performed. Ray-leigh number is 6.78 105in those five situations. The mass flow
rate of the situation of Re = 950 assigned in mixed convection is the same as that obtained by the situation of natural convection. The definition of the dimensionless mass flow rate at the inlet
_
Minletand the dimensionless mass flow rate sucked from the
out-side _Moutlet are presented as follows, respectively
_ mRe¼950¼
q
d 2 u0;Re¼950 ð32Þ _ Minlet¼q
u0d2= _mRe¼950 ð33Þ _ Moutlet¼q
juoutletjd 2 = _mRe¼950;uoutlet<0 ð34Þ The total mass flow rate which flows out of the channel and has advantage to heat transfer rates of heat walls is obtained by the addition of _Minlet and _Moutlet. In situations of mixed convection ofRe = 400, 200 and 100, the mass flow rates of _Moutlet are mainly
caused by the amount of fluid to be sucked from the outside. The reason is suggested as that the strength of the driving force of nat-ural convection is larger than those of forced convections assigned in mixed convections under situations of large magnitudes of the Richardson numbers. However, the viscous dissipation and impingement between the _Minlet and _Moutlet occurring in situations
of Re = 400, 200 and 100 cannot avoid. The total mass flow rates of situations of Re = 400, 200 and 100 have difficulty to be equal to that of the situation of natural convection. The dimensionless time t⁄are from t⁄= 0 to the steady state for situations of Re = 0,
950, 400, and the beginning of fully unsteady state for situations of Re = 100 and 200, respectively.
Shown in Fig. 2, the results of local Nusselt numbers with different grid distributions along x axis are indicated, and the grid
distributions of 120 50 50, 96 40 40 and 72 30 30 in x, y, z directions under natural convection are tested. According to the results of local Nusselt numbers, the grid distribution of 96 40 40 is adopted. The definition of the local Nusselt number Nuxis expressed as follows Nux¼ d k0ðTh T0Þ kðTÞ@T @z ð35Þ
Shown inFig. 3, the variation of the mass flow rate with time for natural convection at the inlet is indicated. The mass flow rate shar-ply increases at an initial stage, and reaches a plateau of develop-ment gradually. The negative magnitude at the initial stage indicates part of the amount of fluid originally staying in the chan-nel to be expanded by the heat wall and extruded to the outside. The definition of the dimensionless mass flow rate _Minlet;n:c:is
pre-sented as follows
_
Minlet;n:c:¼
q
uinletd 2= _mRe¼950 ð36Þ
Shown inFig. 4, the variations of the streamlines and thermal field with time for natural convection are indicated. At first the high tem-perature of the heat surface causes the densities of the fluids in the channel to become small, and then the volume of the fluid is ex-panded that causes the fluids to flow out of the channel shown in
Fig. 4(a1). Accordingly, the fluids near the inlet region gradually form flow reversal shown inFig. 4(a2–3) in order to supplement the lack of the fluids which are discharged mentioned above. Accompanying with the increment of time, because of the influence of natural convection the flow field becomes steady and all the flu-ids flow into the channel from the inlet shown inFig. 4a(d). In addi-tion, orderly thermal boundary layers are observed as the flow field reaches the steady state show inFig. 4(b).
Shown inFig. 5, the distributions of the streamlines and thermal field for the Reynolds number of 950 are indicated, respectively. The darker the color is, the higher the temperature is shown. Because of the same mass flow rate at the inlet of both situations of natural convection and Re = 950, the streamlines and thermal filed for the Reynolds number of 950 are similar to the situation of natural convection shown in Fig. 4(a4) and Fig. 4(b), respectively.
Shown inFig. 6, the distributions of streamlines and thermal field for the Reynolds number of 400 are indicated, respectively. InFig. 6(a), a part of the fluids via the outlet are sucked into the channel from the outside, and a region of the flow reversal is ob-served. The Rayleigh number of this situation is the same with that of natural convection shown inFig. 4, based upon the reason men-tioned above and then the mass flow rate at the inlet of the situa-tion of Re = 400 is smaller than that of natural convecsitua-tion. As a result, via the outlet the insufficiency of the mass flow rate is sup-plemented by the fluids from the outside. As well, inFig. 6(b) the flow reversal depresses the thermal field near the outlet that some-what increases the heat transfer rate of the region near the outlet. InFig. 7, the variations of streamlines with time for the Rey-nolds number of 200 are indicated. FromTable 1, at the inlet the mass flow rate of natural convection is much larger than that of the situation of Re = 200 that causes the insufficient mass flow rate to be supplemented from the outside of the channel. Then some fluids via the central region of the outlet flows into the channel and impinges the amount of fluid provided by forced convection flowing upwards from the inlet. Afterward, both the amounts of fluid newly coalesce and form a new stream flowing upward along the heat wall. Due to the occurrence of impingement, an unsteady phenomenon is apparently observed in Fig. 7(d). Naturally, the phenomenon is advantageous to heat transfer mechanisms of the heat wall.
Fig. 5. Distributions of streamlines and thermal field for the Reynolds number of 950.
Fig. 6. Distributions of streamlines and thermal field for the Reynolds number of 400.
The variations of streamlines with time for the Reynolds num-ber of Re = 100 are shown inFig. 8. Since the mass flow rate at the inlet of this situation is also much less than that of the situation of natural convection. The location of the impingement caused by
_
Minletand _Moutletis more close to the inlet than that shown inFig. 7.
This phenomenon leads more drastic impingement to occur that slightly decreases the mass flow rate of _Moutlet. Then the mass flow
rate of _Moutlet of Re = 100 is slightly smaller than that of Re = 200.
InFig. 9, the variations of thermal field with time corresponding to streamlines shown inFig. 8are indicated, respectively. In an ear-lier stage, a mixing effect of both the amounts of fluid mentioned above just begins, and then orderly thermal boundary layers are observed on heat walls. Gradually, the mixing effect becomes com-plex and drastic, and then orderly thermal boundary layers are no longer observed on heat walls and high temperature fluids are irregularly distributed in the channel.
Time averaged local Nusselt numbers distributed on the central line of the heat surface of all situations are indicated inFig. 10, respectively. The definition of the time averaged local Nusselt number (Nux)tis expressed as follows. The time interval t is
calcu-lated from t⁄ = 0 to 0.15 ðNuxÞt¼ 1 t Z d k0ðTh T0Þ kðTÞ@T @z dt ð37Þ
In the front region (x < 0.5), under the situation of natural convec-tion the amount of fluid begins to be sucked from the outside of the channel and flows into the channel. The buoyancy force driving the amount of fluid to flow upwards is gradually strengthened, and then the upward velocity is also accelerated little by little. Oppo-sitely, under the situation of forced convection a certain quantity of mass flow rate is evenly provided at the inlet to flow into the channel. As a result, in this region time averaged local Nusselt Fig. 7. Variations of streamlines with time for the Reynolds number of 200.
numbers of the situation of Re = 950 are slightly larger than those of the situation of natural convection. The other situations of forced
convection, mass flow rates at the inlet are smaller than the mass flow rate of the situation of natural convection at the inlet. Naturally, time averaged local Nusselt numbers of the other three Fig. 9. Variations of thermal field with time for the Reynolds number of 100.
Fig. 10. Time averaged local Nusselt numbers on the central line of the heat surface.
situations of forced convection are smaller than that of the situation of natural convection. Beyond the front region, under the situation of natural convection the upward fluid driven by the buoyancy force which is gradually strengthened is mainly concentrated on the heat surface. Then time averaged local Nusselt numbers of natural con-vection are larger than those of situations of forced concon-vection in
which the mass flow rate is evenly distributed on the cross section. According to the statements mentioned above, in situations of low-er Reynolds numblow-ers (Re = 100 and 200) via the outlet the fluids are sucked into the channel and impinges the upward fluid provided by forced convection. This phenomenon causes the drastic impinge-ment to occur in the channel. Consequently, time averaged local Nusselt numbers of the situations of lower Reynolds numbers (Re = 100 and 200) are larger than those of the situation of Re = 400. Near the outlet region, a part of time averaged local Nus-selt numbers of the situation of Re = 100 are even larger than those of the situation of Re = 950. However, the mass flow rate of the sit-uation of Re = 400 is medium and slightly smaller than that of the situation of natural convection. The amount of fluid sucked into the channel from the outside is small and has no ability to disturb the flow field provided by forced convection. The smallest distribu-tion of time averaged local Nusslet numbers is then indicated.
Fig. 11indicates the variations of area averaged Nusselt num-bers with time. The definition of area averaged Nusselt number Nu is defined as follows Nu ¼1 A Z Z A Nudxdy ¼1 A Z Z A d k0ðTh T0Þ kðTÞ@T @z dxdy ð38Þ
In an earlier time region, natural convection begins to develop that causes the magnitudes of area averaged Nusselt numbers of natural convection to be smaller than those of situations of forced convection. After a certain developing time, according to the rea-sons mentioned above the magnitudes of the area averaged Nus-selt numbers of natural convection are gradually larger than those of situations of forced convection. Among situations of forced convection, the magnitudes of the area averaged Nussslt numbers of the situation of Re = 950 situation are apparently larger than those of the other three situations of forced convection. According to the reason suggested above, the magnitudes of area averaged Nusselt numbers of the situation of Re = 400 are smaller than those of the other two situations of Re = 100 and 200. The similar phe-nomena of the flow reversal are found out in situations of Re = 100 and 200. The deviation of the magnitudes of situations of Re = 100 and 200 is small.
InFig. 12, the variation of time and area averaged Nusselt num-bers of all situations is indicated. The definition of the time and area averaged Nusselt number is expressed as follows
ðNuÞt¼ 1 t Z t Nudt ð39Þ
According to the reasons mentioned above, the maximum and min-imum magnitudes of the time and area averaged Nusselt numbers are situations of natural convection and Re = 400, respectively. The difference of the magnitudes of situations of Re = 100 and 200 is slight. The magnitude of the time and area averaged Nusselt num-ber of natural convection is slightly larger than that of the situation of Re = 950. The main reason is suggested as that the amount of Fig. 12. Variations of time and area averaged Nusselt numbers with different
Reynolds numbers.
Fig. 13. Comparisons of distributions of velocities and temperature of present results with those of the previous work[13].
Fig. 14. Comparison of the average Nusselt number of the present result with that of the previous work[3].
fluid provided by forced convection is evenly distributed in the cross section of the channel, and the amount of fluid in the channel induced by natural convection is mainly concentrated on heat walls. In Fig. 13, comparisons of distributions of temperature and velocity of present results with those of the previous work[13]
are indicated, respectively. The physical model of the previous work was a circular cylinder, then an equivalent hydraulic diame-ter regarded as the width of a square duct is used to calculate dis-tributions of temperature and velocity on the center line of the outlet cross section. The trend of both results is consistent. Since geometries of the two models are different, the existences of the slight deviations between both results are reasonable.
In Fig. 14, comparison of the present result with that of an experimental result of [3] is shown. For the Reynolds number and heat flux of both situations are 3000 and 57 Wm2,
respec-tively. The ratio of the height to the width is 10 in this work be-cause of the limitation of computation memory, and the ratio of
[3]was 25. The shorter the height is, the larger the average Nusselt number is achieved. Naturally, the result of this work ought to be larger than that of[3].
5. Conclusions
An investigation of flow reversal of mixed convection in a three dimensional rectangular channel with the consideration of the compressibility of the working fluid is studied numerically. The occurrence of flow reversal of mixed convection in the channel is mainly depended by the balance of mass flow rates induced by nat-ural convection and provided by forced convection. The mass flow rate induced by natural convection is larger than that provided by forced convection at the inlet that causes the phenomenon of flow reversal to occur easily. The phenomenon will decay accompany-ing with the decrement of the difference between both mass flow rates. In the range of this work, several related features of the re-sults are drawn as follows.
1. The magnitude of Richardson number dominates the mecha-nisms of the flow reversal of mixed convection.
2. Drastic phenomena of the flow reversal of mixed convection are mainly caused by the mutual impingement between the amount of downward fluid sucked from the outlet and the amount of upward fluid from the inlet provided by forced convection.
3. Under the same mass flow rate, the magnitude of time and area averaged Nusselt number of natural convection is slightly supe-rior to that of mixed convection
Acknowledgement
The authors gratefully acknowledge the support of the Natural Science Council, Taiwan, ROC under Contact NSC100-2221-E-009-086-MY2
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![Fig. 14. Comparison of the average Nusselt number of the present result with that of the previous work [3] .](https://thumb-ap.123doks.com/thumbv2/9libinfo/7657408.140002/10.892.89.414.451.953/fig-comparison-average-nusselt-number-present-result-previous.webp)
