A new model for heat transfer of fins swinging back and forth in a flow

A new model for heat transfer of ®ns swinging back and forth

in a ¯ow

Wu-Shung Fu

_{, Suh-Jenq Yang}

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30056, Taiwan, ROC Received 11 April 2000; received in revised form 22 June 2000

Abstract

In this paper, a new concept of an electronic device cooling method is proposed. In this method, extremely thin ®ns are used for swinging back and forth in a ¯owing ¯uid. The boundary layers attaching on the ®ns are then contracted and disturbed, and the heat transfer rate of the ®ns can be enhanced remarkably. The dynamic behavior between the ®ns and ¯uid is classi®ed into a class of the moving boundary problems. A Galerkin ®nite element formulation with an arbitrary Lagrangian±Eulerian kinematic description method is adopted to solve this problem. The parameters of velocities of the ¯uid and the swinging speed of the ®ns are employed to investigate the variations of the ¯ow and thermal ®elds. The results show that the velocity and thermal boundary layers may be contracted and disturbed, which results in a signi®cant heat transfer enhancement, being attained. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction

Accompanying with the progress of the semicon-ductor technology, the miniaturization of components becomes a trend of development of a new electronic device that results in large heat rate being generated per unit area and the temperature of the electronic device being high. The performance and reliability of the elec-tronic device are deeply a€ected by its temperature. Therefore, how to control the thermal dissipation and enhance the heat transfer rate from a small size elec-tronic device e€ectively becomes a very important sub-ject.

In general, the techniques for enhancing the heat transfer are divided into two parts: ``passive'' and ``ac-tive'' methods [1,2], and both these methods have been massively used to enhancing the heat transfer of high power electronic devices. Yeh [3] summarized and re-viewed the results of recent developments and researches of the heat transfer technologies in electronic equipment, such as air cooling, liquid cooling, jet impingement, heat

pipe, micro-channel cooling and phases change. Sathe and Sammakia [4] made a survey of recent developments in detail for air cooling method in electronic packages.

One of the above technologies of adding a ®nned heat sink on a hot component to enlarge the heat transfer area for enhancing the thermal performance is universally employed in the electronic device and heat exchangers, and several papers [5±8] had studied in this topic. Furthermore, vibrating a heated body surface also is an e€ective method to enhance the heat transfer rate and had been studied experimentally and theoretically [9±13], and the results indicated the heat transfer rate of the heated body to be increased remarkably.

However, at present it appears that the heat transfer eciency of the ®nned heat sink may fail to catch up with the increasing rate of the heat generation of a new electronic device. As for the method of vibrating of the heated body surface, it seriously reduces the reliability and stability of the electronic device.

Thus, a new cooling concept, in which the ®ns of the ®nned heat sink swing back and forth in ¯owing ¯uid, is proposed to enhance the heat transfer of the high power electronic device reliably and stably. For realizing this concept, the ®ns of the ®nned heat sink are needed to be made of extremely thin metal, then these thin ®ns could be easily swinging back and forth by the ¯owing ¯uid, or

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*_{Corresponding author. Tel.: +886-3-5712121/5510; fax:}

+886-3-5720634.

E-mail address: wsfu@cc.nctu.edu.tw (W.-S. Fu).

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these thin ®ns are forced to be oscillated by an oscilla-tion exciter installed at a proper place. As this apparatus is executed, the velocity and thermal boundary layers attached on the ®ns are contracted and disturbed dras-tically because of the swinging of the ®ns, which natu-rally results in the heat transfer rate being enhanced remarkably.

Due to the interaction between the ¯owing ¯uid and swinging ®ns, the variations of the ¯ow and thermal ®elds become time-dependent, and this behavior belongs to a class of the moving boundary problems, which is hardly analyzed by either the Lagrangian or Eulerian

kinematic description method solely. From the physical point of view, for analyzing the above phenomena val-idly, the moving interfaces between the ¯uid and ®ns have to be taken into consideration. An arbitrary La-grangian±Eulerian (ALE) kinematic description method [14±19], in which the computational meshes may move with the ¯uid, be held ®xed, or be moved in any other prescribed way, is an appropriate kinematic description method to analyze this new cooling concept mentioned above.

Consequently, the ALE kinematic description method is adopted to describe the variations of the ¯ow Nomenclature

d dimensional thickness of the ®ns (m)

D dimensionless thickness of the ®ns …D ˆ d=w2†

h dimensional width of the channel (m) H dimensionless width of the channel

…H ˆ h=w2†

h1 dimensional distance from the wall of the

channel to the ®ns (m)

H1 dimensionless distance from the wall of the

channel to the ®ns …H1ˆ h1=w2†

h2 dimensional pitch of the ®ns (m)

H2 dimensionless pitch of the ®ns …H2ˆ h2=w2†

Ni shape function

n normal vector of coordinates ne number of elements

Nu overall average Nusselt number of the ®ns NuX local Nusselt number on the top or bottom

surface of the ®ns

NuX average Nusselt number on the top or bottom

surface of the ®ns

p dimensional pressure …N mÿ2_†

p1 referential pressure …N mÿ2†

P dimensionless pressure …P ˆ …p ÿ p1†=qu20†

Pr Prandtl number …Pr ˆ a=m† Re Reynolds number …Re ˆ u0w2=m†

Rei Reynolds number for the Blasius solution

…Reiˆ u0i=m†

sb dimensional swinging speed of the ®ns …m sÿ1†

Sb dimensionless swinging speed of the ®ns

…Sbˆ sb=u0†

t dimensional time (s)

T dimensional temperature (°C)

Tf dimensional temperature of the ®ns (°C)

T0 dimensional temperature of the inlet ¯uid (°C)

u; v dimensional velocities in x- and y-directions …m sÿ1_†

U; V dimensionless velocities in X - and Y -directions …U ˆ u=u0; V ˆ v=u0†

u0 dimensional velocity of the inlet

¯uid …m sÿ1_†

vb dimensional swinging velocity of the ®ns

…m sÿ1_†

Vb dimensionless swinging velocity of the ®ns

…Vbˆ vb=u0†

^v dimensional mesh velocity in y-direction …m sÿ1_†

^

V dimensionless mesh velocity in Y -direction … ^V ˆ ^v=u0†

w dimensional length of the channel (m) W dimensionless length of the channel

…W ˆ w=w2†

w1 dimensional distance from the inlet to the

front side of the ®ns (m)

W1 dimensionless distance from the inlet to the

front side of the ®ns …W1ˆ w1=w2†

w2 dimensional length of the ®ns (m)

W2 dimensionless length of the ®ns …W2ˆ w2=w2†

x; y dimensional Cartesian coordinates (m) X ; Y dimensionless Cartesian coordinates

…X ˆ x=w2; Y ˆ y=w2†

Greek symbols

a thermal di€usivity …m2 _sÿ1_†

/ computational variables

i the length of a ¯at plate for the Blasius solution (m) k penalty parameter m kinematic viscosity …m2_sÿ1_† h dimensionless temperature …h ˆ …T ÿ T0†= …Tfÿ T0†† q density …kg mÿ3_† s dimensionless time …s ˆ tu0=w2† Superscripts (e) element m iteration number T transpose matrix Other ‰Š matrix fg column vector hi row vector

and thermal ®elds induced by the interaction between the swinging ®ns and ¯owing ¯uid and the enhancement of heat transfer of the ®ns numerically. A Galerkin ®nite element method and a backward di€erence scheme, dealing with the time terms, are used to solve the gov-erning equations. Several di€erent ¯ow rates and swinging speeds of the ®ns are considered in this study. 2. Physical model

A two-dimensional horizontal channel with width h and length w as sketched in Fig. 1 is used in this study. Three extremely thin ®ns with thickness d and length w2

are arranged with a pitch of h2in the channel. The ratio

of d to w2 is about 0.01. The distance from the inlet to

the front surface of the ®n is w1, and the distance from

the wall of the channel to the ®n is h1. The inlet velocity

and temperature of the ¯uid are uniform and equal to u0

and T0, respectively. These ®ns are made of high

con-ductivity material and maintain at a constant tempera-ture Tf, which is higher than T0. Initially …t ˆ 0†, these

thin ®ns are assumed to be stationary and the ¯uid is ¯owing steadily. As the time t > 0, these thin ®ns are swung back and forth induced by the ¯owing ¯uid or an oscillation exciter (the photograph of the swinging ®n is showed in Appendix A). Then, the variations of the ¯ow and thermal ®elds become time-dependent and catalog to a class of the moving boundary problems. As a result, the ALE method is properly utilized to analyze this subject. The detail of the ALE kinematic description method is delineated in Hughes et al. [15], Donea et al. [16], and Ramaswamy and Kawahara [17].

In order to facilitate the analysis, the following as-sumptions are made.

1. The ¯uid is air and the ¯ow ®eld is two-dimensional, incompressible and laminar.

2. The ¯uid properties are constant and the e€ect of the gravity is neglected.

3. The moving direction of the ®ns is in y-direction only, and the ®ns oscillate with a constant swinging speed sb.

4. The no-slip condition is held on the interfaces be-tween the ¯uid and ®ns.

Based upon the characteristic scales of w2; u0; qu20and

T0, the dimensionless variables are de®ned as follows:

X ˆ_wx 2; Y ˆ y w2; U ˆ u u0; V ˆ v u0; ^ V ˆ_u^v 0; Vbˆv_ub 0; P ˆ p ÿ p1 qu2 0 ; s ˆ tu0 w2; h ˆ T ÿ T0 Tfÿ T0; Re ˆu0_mw2; Pr ˆ_am; …1† where ^v is the mesh velocity and vb is the swinging

velocity of the ®ns.

According to the above assumptions and dimen-sionless variables, the dimendimen-sionless ALE governing equations [16±19] are expressed as the following equa-tions: continuity equation oU oX‡ oV oY ˆ 0 …2† momentum equations oU os‡ U oU oX ‡ …V ÿ ^V † oU oY ˆ ÿ oP oX‡ 1 Re o2_U oX2  ‡o_oY2U₂  …3† oV os‡ U oV oX‡ …V ÿ ^V † oV oY ˆ ÿ oP oY‡ 1 Re o2_V oX2  ‡o_oY2V₂  ; …4† energy equation oh os‡ U oh oX‡ …V ÿ ^V † oh oY ˆ 1 Pr Re o2_h oX2  ‡_oYo2h₂  : …5† As the time s > 0, the boundary conditions are as follows: on the ¯uid inlet surface AB (excluding the points A and B)

U ˆ 1; V ˆ 0; h ˆ 0 …6† on the wall surfaces of the channel BC and AD U ˆ V ˆ 0; oh=on ˆ 0; …7† on the ¯uid outlet surface CD (excluding the point C and D)

oU=on ˆ oV =on ˆ oh=on ˆ 0; …8† on the interfaces between the ¯uid and ®ns

U ˆ 0; V ˆ Vb; h ˆ 1: …9†

3. Numerical method

A Galerkin ®nite element method and a backward scheme, dealing with the time terms, are adopted to solve the governing equations (2)±(5). The Newton±

Raphson iteration algorithm and a penalty function model [20] are utilized to simplify the nonlinear and pressure terms in the momentum equations, respectively. The velocity and temperature terms are approximated by quadrilateral and nine-node quadratic isoparametric elements. The discretization processes of the governing equations are similar to the one used in Fu et al. [21]. Then, the momentum equation (3) and (4) can be ex-pressed as the following matrix form:

Xne 1

…‰AŠ…e†_{‡ ‰KŠ}…e†_{‡ k‰LŠ}…e†_†fqg…e† s‡Dsˆ Xne 1 ff g…e†_{; …10†} where …fqg…e†s‡Ds†Tˆ hU1; U2; . . . ; U9; V1; V2; . . . ; V9im‡1s‡Ds; …11†

‰AŠ…e†_{consists of the …m†th iteration values of U and V at}

time sDs, ‰KŠ…e†_{consists of the shape function N} i; ^V and

time di€erential terms, ‰LŠ…e† _{consists of the penalty}

function terms, ff g…e†_{consists of the known values of U}

and V at time s and …m†th iteration values of U and V at time s ‡ Ds.

The energy equation (5) can be expressed as the fol-lowing matrix form:

Xne 1

…‰MŠ…e†_{‡ ‰ZŠ}…e†_†gfcg…e† s‡Dsˆ Xne 1 frg…e†_{; …12†} where …fcg…e†s‡Ds†Tˆ hh1; h2; . . . x; h9is‡Ds; …13†

‰MŠ…e†_{consists of the values of U and V at time s ‡ Ds,}

‰ZŠ…e†_{consists of the shape function Ni; ^V and time}

dif-ferential terms, frg…e†_{consists of the known values of h}

at time s.

In Eqs. (10) and (12), Gaussian quadrature procedure are conveniently used to execute the numerical integra-tion. The terms with the penalty parameter k are inte-grated by 2  2 Gaussian quadrature, and the other terms are integrated by 3  3 Gaussian quadrature. The value of penalty parameter k used in this study is 106_.

The frontal method solver [22,23] is applied to solve Eqs. (10) and (12).

The mesh velocity ^V is linearly distributed and in-versely proportional to the distance between the nodes and ®ns. In addition, the boundary layer thickness on the ®ns surface are extremely thin and can be approxi-mately estimated by Reÿ1=2 _{[24]. To avoid the}

compu-tational nodes in the vicinity of the ®ns to slip away from the boundary layer, the mesh velocity adjacent to the ®ns are expediently assigned equal to the velocity of the ®ns.

A brief outline of the solution procedure is described as follows:

1. Determine the optimal mesh distribution and number of the elements and nodes.

2. Solve the values of the U; V and h at the steady state and regard them as the initial values.

3. Determine the time step Ds and the mesh velocities ^V of the computational meshes.

4. Update the coordinates of the nodes and examine the determinant of the Jacobian transformation matrix to ensure the one-to-one mapping to be satis®ed dur-ing the Gaussian quadrature numerical integration. 5. Solve Eq. (10), until the following criteria for

conver-gence are satis®ed: /m‡1_{ÿ /}m /m‡1    s‡Ds < 10ÿ3_{; where / ˆ U; V : …14†}

6. Substitute the U and V into Eq. (12) to obtain h. 7. Continue the next time step calculation until the

as-signed amplitude of the ®ns is reached. 4. Results and discussion

The dimensionless geometric parameters are listed in Table 1. The working ¯uid is air with Pr ˆ 0:71 and Reynolds number is varied from 500 to 1500. Several di€erent swinging speeds of the ®ns, Sb, are evaluated

and the maximum amplitude of the ®ns is assigned to be 0.05. Since the thickness of the ®ns D …ˆ 0:01† is very thin, the blockage e€ect and the heat transfer from the right and left surfaces of the ®ns can be neglected. The local Nusselt number NuX…X ; s† on the top or bottom

surface of the ®n is de®ned by the following equation:

Table 1

The dimensionless geometric parameters

H H1 H2 D W W1 W2 W3

7.0 3.085 0.4 0.01 15.0 4.0 1.0 10.0

Fig. 2. Comparison of the results of the average Nusselt numbers on the top and bottom surfaces of the middle ®n at steady state for di€erent Reynolds numbers of the present study with the Blasius solution.

NuX…X ; s† ˆ ÿ_oYoh: …15†

The average Nusselt number NuX…s† on the top or

bottom surface of the ®n is de®ned by NuX…s† ˆ_W1

2

Z W2

0 NuX dX ; …16†

where W2is the length of the ®n. In addition, the overall

average Nusselt number Nu…s† of the ®n is de®ned as Nu…s† ˆ_2W1 2 Z W2 0 NuXjtopdX  ‡ Z W2 0 NuXjbottom dX  : …17†

For obtaining an optimal computational meshes, a series of numerical tests for various meshes at the steady state are executed. The nonuniform distribution of 3872 elements corresponding to 15 942 nodes is chosen. Since the pitch of the ®ns …H2ˆ 0:4† is larger than the

thick-ness of the ®ns …D ˆ 0:01†, the ¯ow and thermal ®elds at the steady state are similar to the ¯uid ¯owing over a ¯at plate. The Blasius solution [25] for the average Nusselt number with laminar ¯ow over a ¯at plate of length i is NuX ˆ 0:664Re1=2i Pr1=3: …18†

In Fig. 2, the average Nusselt numbers on the top and bottom surfaces of the middle ®n at the steady state

Fig. 3. The transient developments of the velocity vectors and isothermal lines around the middle ®n for the swinging speed of the ®ns Sbˆ 0:025 and Re ˆ 1000 case: (a) s ˆ 0:0, (b) s ˆ 1:0, (c) s ˆ 4:0, (d) s ˆ 8:0, and (e) s ˆ 24:0.

for di€erent Reynolds numbers are compared with those of the Blasius solutions. Both the results are consistent well. As for the time step Ds, the time step Ds ˆ 1:0  10ÿ2_{is chosen for the swinging speed of the}

®ns Sbˆ 0:025 case, and Ds ˆ 5:0  10ÿ3; 2:5  10ÿ3

and 2:0  10ÿ3_{are chosen for the swinging speed of the}

®ns Sbˆ 0:5, 1.0 and 2.0 cases, respectively. Besides, the

residual of the continuity equation

Residual ˆoU_oX‡oV_oY …19† is checked for each element on each time to ensure the mass conservation law to be satis®ed. In the computing processes, the residual of the continuity equation for each element is smaller than 1:0  10ÿ6_.

For illustrating the variations of the ¯ow and thermal ®elds more detailed, the middle ®n is focused on and the velocity vectors and isothermal lines around the middle ®n are presented only. However, it should be noted the computational domain included three swinging ®ns, and a much larger region was calculated than what is displayed in the subsequent ®gures. In addition, the velocity vectors shown in the following ®gures are scaled relatively to the maximum velocity in the ¯ow ®eld.

Fig. 3 present the transient developments of the velocity vectors and isothermal lines around the middle ®n under the swinging speed of the ®ns Sbˆ 0:025 and

Re ˆ 1000 case. At the time s ˆ 0:0, the ®n is stationary and the ¯uid is ¯owing steadily, as shown in Fig. 3(a). As the time s > 0, the ®ns start in motion of swinging back and forth. As shown in Fig. 3(b), the ®n is on the way to move upward. The ¯uid close to the top surface of the ®n is pushed by the ®n and ¯ows upward. As a result, the heat transfer is enhanced. Conversely, the ¯uid near the bottom surface of the ®n simultaneously replenishes the vacant space induced by the movement of the ®n. Most of the ¯uid near the bottom surface of

the ®n are dicult to catch up to the bottom surface of the ®n simultaneously, this ¯ow is disadvantageous to the heat transfer. Afterwards, the ®n moves upward continuously until the amplitude of the ®n is equal to 0.05. The variations of the ¯ow ®elds are similar to those of the ¯ow ®elds mentioned above.

The motion of the ®n turns downward immediately as the ®n reaches the maximum upper amplitude. As shown in Fig. 3(c), the ®n is on the way to move downward and the position of the middle ®n is at the center of the channel. The ®n pushes the ¯uid near the bottom surface of the ®n, which is pro®table for the heat transfer. In the meantime, the ¯uid close to the top surface of the ®n continuously replenishes the vacant space near the top surface of the ®n as the ®n moves downward.

In Fig. 3(d), the ®n is on the way to move upward and the position of the middle ®n returns to the center of the channel. The variations of the ¯ow ®elds are similar to those as shown in Fig. 3(b).

As the time increases, the ®n swings back and forth as mentioned above. Since the ®ns swing with a small speed, the variations of the ¯ow ®elds are slight.

As for the thermal ®elds, the variations of the ther-mal ®elds usually correspond to the variations of the ¯ow ®elds. Since the swinging speed of the ®n is very slow, the ¯ow ®elds are similar to the ¯uid ¯owing over a ¯at plate. Thus, the variations of the thermal ®elds are very slight and the distributions of the isothermal lines are similar to those of the ¯uid ¯owing through a sta-tionary plate.

Fig. 4(a) shows the variations of the average Nusselt number NuX on the top and bottom surfaces of the

middle ®n with time at the same conditions as shown in Fig. 3. Based upon these reasons mentioned earlier, as the ®n moves upward, the average Nusselt number on the top surface of the ®n increases slightly, but the

Fig. 4. The variations of the Nusselt number on the middle ®n with time for the swinging speed of the ®ns Sbˆ 0:025 and Re ˆ 1000

average Nusselt number on the bottom surface de-creases. As the ®n moves downward, the results of the variations of the average Nusselt number on the ®n are opposite to those of the ®n moving upward. The varia-tions of the overall average Nusselt number Nu on the middle ®n with time are very slight compared with the steady state, as shown in Fig. 4(b).

Fig. 5 shows the transient developments of the velocity vectors and isothermal lines around the middle ®n under the swinging speed of the ®ns Sbˆ 0:5 and

Re ˆ 1000 case. Since the swinging speed of the ®n is greater than that of the case above, the variations of the

¯ow and thermal ®elds near the ®n are more drastic in this case. As shown in Fig. 5(b) and (c), the ®n is on the way to move upward. The ¯uid near the top surface on the ®n is pushed by the ®n and ¯ows upward. In the meantime, the ¯uid close to the left and bottom surfaces of the ®n ®lls the vacant space near the bottom surface of the ®n induced by the movement of the ®n. As a re-sult, a small recirculation zone, which is disadvan-tageous to the heat transfer, and a reattachement ¯ow, which is advantageous to the heat transfer, are observed around the left corner of the bottom surface of the ®n. The distributions of the isothermal lines near the top

(a)

(b)

(c)

(d)

(e)

Fig. 5. The transient developments of the velocity vectors and isothermal lines around the middle ®n for the swinging speed of the ®ns Sbˆ 0:5 and Re ˆ 1000 case: (a) s ˆ 0:0, (b) s ˆ 0:05, (c) s ˆ 0:1, (d) s ˆ 0:2, (e) s ˆ 0:4, (f) s ˆ 0:6, (g) s ˆ 0:8, (h) s ˆ 1:0, (i) s ˆ 1:2,

surface of the ®n are denser than those of the bottom surface of the ®n. Besides, the distributions of the iso-thermal lines become denser near the reattachement ¯ow.

In Fig. 5(d), the ®n is on the way to move downward and the position of the middle ®n is at the center of the channel. Since the ®n moves downward, the ®n pushes the ¯uid near the bottom surface of the ®n. Conversely, the ¯uid near the left and top surfaces of the ®n con-tinuously replenishes the vacant space near the top surface of the ®n induced by the movement of the ®n. As a result, a recirculation zone and a reattachement ¯ow are formed around the left corner of the top surface of the ®n. Similarly, the distributions of isothermal lines become denser near the reattachement ¯ow and sparser

near the recirculation zone around the left corner of the top surface of the ®n.

As shown in Fig. 5(e)±(j), since the ®n is in motion of swinging back and forth with a high swinging speed, the recirculation zones and reattachement ¯ows are formed around the ®n continuously and migrate to the down-stream gradually, which may cause the boundary layers of the ¯ow and thermal ®elds to be contracted and disturbed during the transient developments. Conse-quently, the heat transfer is enhanced remarkably.

The variations of the overall average Nusselt number Nu on the surfaces of the middle ®n with time at the same conditions as shown in Fig. 5 are indicated in Fig. 6. According to the variations of the ¯ow and thermal ®elds mentioned above, the overall average Nusselt

number during the transient developments is enhanced remarkably. Furthermore, as the time increases, the ¯ow and thermal ®elds may approach to a periodic state, and the mean increment of the overall average Nusselt number on the middle ®n is about 15% in the computing range.

In addition, Fig. 7 shows the variations of the overall average Nusselt number Nu on the middle ®n with time for various cases. Fig. 7(a) and (b) indicate the swinging speed of the ®ns Sbˆ 1:0 and 2.0 under Re ˆ 1000 cases,

respectively. The variations of the overall average Nusselt number with time are hardly found out to be periodic, this is suggested as that the swinging speed of the ®n is too fast and the ¯ow and thermal ®elds are unable to develop regular patterns in time. In the com-puting range, the mean increment of the overall average Nusselt number on the middle ®n are about 50% and 120% in these two cases, respectively, which are larger than those of the cases above. As expected, the

Fig. 6. The variations of the overall average Nusselt number Nu on the middle ®n with time for the swinging speed of the ®ns Sbˆ 0:5 and Re ˆ 1000 case.

Fig. 7. The variations of the overall average Nusselt number Nu on the middle ®n with time for various cases: (a) Re ˆ 1000; Sbˆ 1:0,

enhancement of the heat transfer increases signi®cantly with increased the swinging speed of the ®ns.

Fig. 7(c) and (d) present the swinging speed of the ®ns Sbˆ 1:0 under Re ˆ 500 and Re ˆ 1500 cases,

re-spectively. In the computing range, the mean increment of the overall average Nusselt number on the middle ®n for these two cases are about 18% and 80%, respectively. As expected, the heat transfer increases with increased Reynolds number.

5. Conclusions

A numerical simulation for the heat transfer of ex-tremely thin ®ns of a ®nned heat sink which swing back and forth induced by a ¯owing ¯uid or an oscillation exciter is presented. Some conclusions are summarized as follows:

1. As the ®ns swing with a small speed, the variations of the ¯ow and thermal ®elds are slight and similar to the ¯uid ¯owing over a ¯at plate.

2. As the ®ns swing with a large speed, the recirculation zones and reattachement ¯ows are formed around the ®ns continuously and migrate to the downstream gradually. This may cause the velocity and thermal boundary layers to be contracted and disturbed, which results in the enhancement of heat transfer re-markably.

3. As the ®ns swing with a relatively low speed, the vari-ations of the ¯ow and thermal ®elds may approach to regular patterns with time, which result in the varia-tions of the overall average Nusselt number being a periodic state. However, the variations of the ¯ow and thermal ®elds are unable to develop regular pat-terns with time as the ®ns swing with a large speed. Acknowledgements

The support of this work by the National Science Council of Taiwan, R.O.C., under contract NSC89-2212-E-009-010 is gratefully acknowledged.

Appendix A

The photographs of the swinging of the ®n induced by the ¯owing ¯uid are showed in Fig. 8. A ®nned heated sink with single extremely thin ®n is set on the test section of a small wind tunnel. The extremely thin ®n is made of Co-based amorphous ribbon and 25 lm in thickness. As shown in Fig. 8(a), both the ¯uid and ®n are stationary. As shown in Fig. 8(b) and (c), the ¯uid ¯ows through the tunnel. The extremely thin ®n is then swinging induced by the ¯owing ¯uid. As a result, the images of the ®n in the photographs are foggy.

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