An investigation of a mixed convection in a
⊔ shape channel moving with a
reciprocating motion
☆
Wu-Shung Fu
⁎
, Sin-Hong Lian, Yu-Chih Lai
Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, 30056, Taiwan, ROC
a b s t r a c t
a r t i c l e i n f o
Available online 1 July 2009 Keywords:
Mixed convection Moving boundary ALE
Reciprocating motion
This study focuses on utilizing numerical calculation to investigate the heat transfer mechanisms in a⊔ shape reciprocating channel system comprised of a horizontal channel at the bottom and vertical channels on both left and right sides. The issue is considered one kind of moving boundary problems and thefinite element and Arbitrary Lagrangian–Eulerian (ALE) kinematic methods can be applied to this study. Due to the high tempera-ture at the bottom surface of the horizontal channel and the direction of inlet coolingfluids in the same direc-tion of the gravity, the heat transfer mechanisms induced by the mixed convecdirec-tionflow become extremely complex. The results show that thermal layers near the heat surface are disturbed drastically and the effect of reciprocating motion upon the heat transfer mechanisms strongly depends on a relationship between Reynolds and Grashof numbers.
© 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Protecting a piston from heat damage could effectively enhance the thermal efficiency of the heat engine and economize the usage of energy[1]. Numerous studies were then to investigate similar objects. In order to simulate the heat transfer phenomena of pistons in a re-ciprocating motion more realistically, the heat dissipation phenomena in a⊔ shape reciprocating channel assumed as the piston action have been studied numerically by the authors and the related studies were reviewed in detail in[2].
In the previous study[2]the forced convection mechanisms were investigated exclusively. However, the temperature of pistons is usually very high and the effect of natural convection on the heat transfer mech-anisms of the reciprocating object needs to be considered also. Thus this study aims to investigate the numerical calculation of the mixed con-vection mechanisms in the⊔ shape reciprocating channel. Effects of Reynolds and Grashof numbers on the heat transfer mechanisms are examined in detail.
Usually when the problem of mixed convection is investigated, the relationship between the directions of inlet coolingfluids and gravity, and the positions of heat region relative to the direction of gravity should be examinedfirst. In this study the vertical channels on the left and right sides provide the coolingfluids to flow into and out of the ⊔ shape channel, respectively. A heat region is installed at the bottom of the horizontal channel in the⊔ shape channel system. The inlet
coolingfluids have the same direction as the gravity. Due to the posi-tion of heat region, the phenomena of opposite and aidingflows can be observed in the left and right channels, respectively. Additionally, because of the mutual counteractions caused by the buoyancy of up-ward direction and the impulse of coolingfluids in a horizontal direc-tion, thermal layers attaching to the heat region of horizontal channel will be disturbed drastically. As a result, the local Nusselt numbers distributed on the heat surface vary with time in a periodical duration. These interesting and complicated phenomena have not been inves-tigated yet.
2. Physical model
A physical model implemented in this study is shown inFig. 1. The total channel width and length are w0and h0, respectively, and the channel width is w. The horizontal channel means the region sur-rounded by BO0FGP0C. The bottom surface BC is heat surface and at
constant temperature TH.Besides, the temperature and velocity of inlet coolingfluids are T0and v0, respectively. Other surfaces of the channel are insulated. The original length between OP and MN is w and the maximum elongation length is 2w. A part of the channel circled by M0BCN0G0GFF0 is called as a reciprocating channel. The
ad-justable length w is the moving distance of the reciprocating channel. Therefore, computational grids in this region areflexible. As the chan-nel moves downward, MN isfixed and OP moves downward with a velocity of vc, the original region is then elongated. Afterward the OP moves upward and returns to the original position. The mesh velocity of the computational grids inside the horizontal channel is equal to that of OP . The right channel length hlis long enough for satisfying the convergent conditions of the temperature and velocity at the outlet of the channel. The reciprocating velocity of the horizontal channel is vc, International Communications in Heat and Mass Transfer 36 (2009) 921–924
☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author.
E-mail addresses:wsfu@mail.nctu.edu.tw(W.-S. Fu),
edenas.me94g@nctu.edu.tw(S.-H. Lian).
0735-1933/$– see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.06.006
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International Communications in Heat and Mass Transfer
and can be expressed as the equation, vc= vmsin(2πfct), where vmis the maximum reciprocating velocity of the piston and equals to 2πfclc. When the channel moves reciprocally, it will cause the motions of the coolingfluids to be time-dependent. The circumstance is regarded as a moving boundary problem and therefore the Arbitrary Lagrangian– Eulerian (ALE) method is properly applied to this study.
For facilitating the analysis, the following assumptions are made. (1) Thefluid is air and the flow field is two-dimensional,
incom-pressible and laminar.
(2) Except the density of thefluid, other properties of the fluid are assumed to be constant, and Boussinesq assumption is adopted. (3) Apply the no-slip condition to all boundaries. Thus thefluid velo-city on the moving boundaries is equal to the moving velovelo-city of the boundaries.
Based upon the characteristics scales of w, v0,ρv02, and T0, the dimensionless variables are defined as follows:
X =x w; Y = y w; U = u v0; V = v v0; ˆV = ˆv v0; Vm =vm v0; Vc =vc v0 Fc= fcw v0 ; P = p− p∞ ρv2 0 ; τ =tv0 w; θ = T− T0 Th− T0 ; Re =v0w m ; Pr = m α Gr =gβ Tðh− T0Þw 3 m2 ; Vc= Vmsin 2ð πFcτÞ ð1Þ and v̂ is defined as the mesh velocity.
According to the above assumptions and dimensionless variables, the dimensionless ALE governing equations are expressed as the fol-lowing equations: Continuity equation AU AX + AVAY = 0 ð2Þ Momentum equation AU Aτ + U AU AX + V− ˆV AU AY = − AP AX + 1 Re A2 U AX2 + A2 U AY2 ! ð3Þ AV Aτ + U AV AX + V− ˆV AV AY = − AP AY + 1 Re A2V AX2 + A2V AY2 ! + Gr Re2θ ð4Þ Energy equation Aθ Aτ + UAXAθ + V− ˆV Aθ AY = 1 Re Pr A2 θ AX2 + A2 θ AY2 ! ð5Þ
In this study, the cooling channel moves only in a vertical direc-tion and therefore the horizontal mesh velocity is absent in the above governing equations. According to ALE method, the mesh velocity V̂ is linearly distributed in the region between MN (fixed) and OP (movable). The mesh velocity Vη1at the positionη1is proportional to the distance between MN and OP , and is defined as the following equation, Vη 1 = η1 η0 · Vc ð6Þ
In the other regions, the mesh velocities are all set to be 0. The bound-ary conditions and solutions method used in this study are similar to those adopted in[2]except the term Gr/Re2taken into consideration in Eq.(4).
Nomenclature
Fc dimensionless reciprocating frequency of the piston
Gr Grashof number
h1 dimensional height of the inlet channel and outlet channel [m]
Lc dimensionless reciprocating amplitude of the piston NuX local Nusselt number
NuX average Nusselt number on the heat surface
Nuc time-average Nusselt number per cycle
p dimensional pressure [N m− 2] p∞ reference pressure [N m− 2] P dimensionless pressure Pr Prandtl number Re Reynolds number t dimensional time [s] T dimensional temperature [K]
u, v dimensional velocities of in x and y directions [m s− 1] v0 dimensional velocities of the inletfluid [m s− 1] U, V dimensionless velocities of in X and Y directions Vc dimensionless reciprocating velocity of the piston Vm dimensionless maximum reciprocating velocity of the
piston
V̂ dimensionless mesh velocity in y-direction x, y dimensional Cartesian coordinates [m] X, Y dimensionless Cartesian coordinates
Greek symbols
α thermal diffusivity [m2s− 1] v kinematics viscosity [m2s− 1]
η0 total length of the moving mesh region
η1 length counted from the bottom of the moving mesh region
θ dimensionless temperature
ρ density [kg m− 3]
τ dimensionless time
Fig. 1. Physical model of the⊔ shape channel. 922 W.-S. Fu et al. / International Communications in Heat and Mass Transfer 36 (2009) 921–924
3. Results and discussion
The workingfluid is air (Pr=0.7). Combinations of main param-eter are tabulated inTable 1. To obtain the optimal computational mesh, three models with different number of elements are used for mesh tests. According to the results, the computational mesh with 20,020 elements which correspond to 61,983 nodes is used through all analysis in this study. The local Nusselt number NuXof the heat surface at timeτ, the average Nusselt number NuXof the heat surface at time
τ, and the cyclical average Nusselt number Nucare calculated by the
following equations, respectively.
NuX= − AYAθ ð7Þ NuX= 1 w0 Z BC NuXdX ð8Þ Nuc= 1 τp Z τp NuXdτ ð9Þ
whereτpis a cyclical time.
For satisfying the boundary conditions at the outlet of the channel, the length from the horizontal channel to the outlet of the right chan-nel is about 70 times the width by numerical tests. In addition, an im-plicit scheme is implemented to deal with the time differential terms of the governing equations. The time stepΔτ = 1
60Fcis chosen for all
cases, i.e. totally 60 time steps are required to achieve one periodic cycle.
The dimensionless stream functionΨ is defined as
U = AW
AY; V = − AWAX: ð10Þ
In order to show the variations of theflow field more clearly, the streamlines in the vicinity of the horizontal channel are indicated ex-clusively. Besides, the sign‘▲’ in the following figures indicates the moving direction of the horizontal channel.
InFig. 2, the variations of streamlines in one period are indicated
for a situation of Re = 200, Gr/Re2= 10, F
c= 0.2 and Lc= 0.5. Shown
inFig. 2(a), the reciprocating channel completes an upward
move-ment and stops at the highest position. Since the directions of the buoy-ancy force and inlet coolingfluids are opposite, the cooling fluid streams are suppressed by ascending heatfluids and flow along the right side of the left vertical channel tightly in order to turn to the right andflow into the horizontal channel easily. The impingement region caused by the coolingfluid streams is then deviated slightly from the vertical posi-tion. Simultaneously part of heatfluids caused by the buoyancy force flow upwards along the left side of the left vertical channel that leads circulation zones to be observed in the left corner of the heat surface. An oppositeflow is apparently indicated in the left vertical channel.
Later on, the coolingfluid streams rush into the horizontal channel andflow along its heated bottom surface; meanwhile, the densities of
the coolingfluids neighboring to the heat surface become light. As a result, the coolingfluids intend to depart from the heat surface and separate at about X = 4.0. The coolingfluid streams departing from the heat surface tend to move in right and upward direction. A large circulation zone induced by the coolingfluids flow is observed in the space below the coolingfluid streams and near the lower right corner. In the right vertical channel, the directions of the buoyancy force and the coolingfluid streams are consistent, and the cooling fluid streams easilyflow out of the channel. An aiding flow is found in the right vertical channel.
InFig. 2(b), the reciprocating channel moves downwards and has
the largest magnitude of downward velocity at that instant. The in-fluence of the buoyancy force is weak. The impingement region is no longer deviated, and the impingement position occurs at X≈0.8. Due to the influences of the downward movement of reciprocating channel and the inlet coolingfluids, the directions of the cooling fluids in the lower and upper regions of the horizontal channel are rightward and upwards respectively. Near the positions of X = 3.0 and 4.5, two circulation zones are separately formed in the central region of the horizontal channel. Afterwards the coolingfluids flow into the right vertical channel.
Shown in Fig. 2(c), the reciprocating channel ends downward movement motion and stops at the lowest position. The buoyancy force changes into a larger magnitude relative toFig. 2(b). Due to the influence of the downward motion caused by the reciprocating chan-nel, the inlet coolingfluids impinge straightly on the heat surface, and the directions of the coolingfluids in the horizontal channel are changed from approximately vertical (Fig. 2(b)) to horizontal orientations. A reattachment position is found at X≈5.0.
InFig. 2(d), the reciprocating channel has the largest upward
velo-city which makes the buoyancy force in the reciprocating channel to be strong. Consequently, relative to the former ones, the impingement phenomenon weakens remarkably and the coolingfluids almost could notflow tightly along the bottom surface of the horizontal channel. Naturally the reattachment points are hard to be found out. The dual effects of the impulse of coolingfluids and buoyancy force in the right vertical channel cause the coolingfluids to flow straightly in the right vertical channel.
InFig. 3, the comparisons of local Nusselt numbers of stationary
and different periodical states in a cycle are shown for Re = 200, Gr/ Re2= 10, L
c= 0.5, Fc= 0.2 situation. Because of the occurrence of impingement at the inlet region of the heat surface, the largest mag-nitude of local Nusselt number is then obtained near the inlet region at any stage. According to the reasons mentioned before, the impinge-ment of the coolingfluids on the heat surface at 1 = 4τp and 2= 4τp
stages are stronger than those at 3= 4τpand 4= 4τpstages, and then
the local Nusselt numbers at 1= 4τpand 2= 4τpstages are larger than
those of 3= 4τpand 4= 4τp stages. Also at the stages of 1= 4τpand
2= 4τp, the reattachment points exist remarkably which result in
convex distributions of local Nusselt numbers near X = 4.0 and 5.0, respectively. In most right regions of the heat surface, the directions of the coolingfluid flows and buoyancy force are consistent; the larger local Nusselt numbers could be obtained doubtlessly. However at the stationary state, thefluids in the channel are not disturbed by the re-ciprocating motion, and thefluids in most right region of the horizon-tal channel are suppressed by the coolingfluid streams which form a mainflow field to lead the cooling fluids to flow out of the channel. Then the local Nusselt numbers near the most right region of the heat surface at the stationary state are smaller than those under the re-ciprocating situation.
Tabulated inTable 1, the cyclical average Nusselt number Nucof
the situation of Re = 200, Gr/Re2= 10, F
c= 0.2 and Lc= 0.5 is slightly smaller than that of the situation of Re = 200, Gr/Re2= 10, F
c= Lc= 0. The cyclical average Nusselt number Nucof the situation of Re = 300,
Gr/Re2= 10, F
c= 0.2 and Lc= 0.5 tabulated inTable 1is larger than that in the corresponding stationary state. Those phenomena imply Table 1
Computed parameters combinations.
Re ReGr2 Lc Fc Vm NuX Nuc Nuc; nð Þ NuX; nð Þ 200 1 0 0 0 4.607(0) – – 200 10 0 0 0 7.747(1) – – 300 10 0 0 0 9.881(2) – – 200 10 0.2 0.5 0.628 – 7.674(1) 0:991 Nuc; 1ð Þ NuX; 1ð Þ 300 10 0.2 0.5 0.628 – 10.010(2) 1:013 Nuc; 2ð Þ NuX; 2ð Þ 923 W.-S. Fu et al. / International Communications in Heat and Mass Transfer 36 (2009) 921–924
the enhancement of heat transfer to be affected deeply by the relation-ship between the buoyancy force and the impulse of coolingfluids.
4. Conclusions
Heat transfer phenomena of a mixed convection in a reciprocating channel are investigated numerically and the interesting results are obtained. Some conclusions could be summarized as follows.
(1) Opposite and aidingflows can be observed in the left and right vertical channels, respectively.
(2) Contributions of a reciprocating motion to heat transfer rates are strongly influenced by the relationship between the buoy-ancy force and impulse of coolingfluids.
Acknowledgement
The support of this study by the National Science Council of Taiwan, ROC, under contract NSC95-2212-E-009-002 is gratefully acknowledged. References
[1] S.W. Chang, L.M. Su, Heat transfer in a reciprocating ductfitted with transverse ribs, Experimental Heat Transfer 12 (1999) 95–115.
[2] W.S. Fu, S.H. Lian, Y.H. Liao, An investigation of heat transfer of a reciprocating piston, International Journal of Heat and Mass Transfer 49 (2006) 4360–4371. Fig. 3. The distributions of local Nusselt numbers of one cycle of Re = 200, Gr/Re2
= 10, Fc= 0.2, and Lc= 0.5 and stationary state.
Fig. 2. A history of development of distributions of streamlines of one cycle for Re = 200, Gr/Re2
= 10, Fc= 0.2, and Lc= 0.5 situation.
