Numerical investigation of heat transfer characteristics of the heated blocks in the channel with a transversely oscillating cylinder

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Numerical investigation of heat transfer characteristics of

the heated blocks in the channel with a transversely

oscillating cylinder

Wu-Shung Fu

*

_{, Bao-Hong Tong}

Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30056, Taiwan, ROC Received 24 January 2003; received in revised form 21 May 2003

Abstract

A numerical simulation is performed to study the influence of an oscillating cylinder on the heat transfer from heated blocks in a channel flow. An arbitrary Lagrangian–Eulerian kinematics description method is adopted to describe the flow and thermal fields. A penalty consistent finite element formulation is applied to solve the governing equations. The effects of Reynolds number, oscillating amplitude and oscillating frequency on the heat transfer characteristics of the heated wall are examined. The results show that the heat transfer from heated blocks is enhanced remarkably as the oscillating frequency of the cylinder is in lock-in region.

1. Introduction

A problem of forced convection in a channel ﬂow with heated blocks is of practical importance and widely considered in the design of devices such as heat ex-changers, and arrays of electronic components. There-fore, there is an urgent need for improving heat transfer performance of the heated blocks set in the channel.

Up to now, forced convection in a channel with he-ated blocks has been a subject of active research. Several studies [1–5] had investigated the heat transfer and ﬂow characteristics in a channel with heated blocks. Fur-thermore, numerous methods including passive and ac-tive methods have been proposed to enhance the heat transfer from the heated blocks. One of these methods involves vortex generators or turbulence promoters in-stalled in the channel and often used to enhance the heat transfer of the above subject. Liou et al. [6–8] curried out a series of numerical and experimental studies on the

turbulent flows in a channel with turbulence promoters, and the results showed that the pitch ratio, Reynolds number and eccentric ratio affected the phenomena of separation, reattachment and heat transfer rate. Amon et al. [9,10] studied self-sustained oscillatory flows in communicating channels. The results indicated that self-sustained oscillations that resulted in very well mixed flows. On an equal pumping power basis, the heat transfer in communicating channels flows was higher than the one in a flat channel flow. Lin and Hung [11] studied transient forced convection heat transfer in a vertical rib-heated channel with a turbulence promoter, and found that the utilization of a turbulence promoter could effectively improve the heat transfer performance in the fully-developed region. Ghaddar et al. [12,13] in-vestigate modulatory heat transfer enhancement in grooved channels by direct numerical simulation. The results show that resonant oscillatory forcing at modu-latory amplitudes as small as 20% of the mean flow re-sults in a doubling of transport as measured by a time, space-averaged Nusselt number. Myrum et al. [14,15] studied the effects of position of a vortex generator above or just downstream of a rib on the heat transfer in a heated duct. It was found that the maximum en-hancement of heat transfer was about 30%. Garimella

*

Corresponding author. Tel.: 572-7925; fax: +886-3-572-0634/572-7925.

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

www.elsevier.com/locate/ijhmt

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and Eibeck [16] examined the effect of protrusion of a vortex generator on heat transfer from an array of dis-crete heated elements. The maximum heat transfer en-hancement was about 40%. Iyer and Kakac [17] investigated the instability and heat transfer in grooved channel flow. The results showed that beyond a critical Reynolds number, the heat transfer rate could be en-hanced. Ortega [18] studied asymmetric oscillator by numerical method, and try to understand the resonance phenomenon for the asymmetric oscillator. Chen and Wang [19] adopted experimental and numerical methods to study the forced convective flow in a channel with heated blocks in tandem. The results compared the variations of the Sherwood number along the heated

surfaces between the laminar and turbulent convection cases and discussed the eﬀect of the block spacing on heat transfer. Wu and Perng [20] studied a numerically investigated heat transfer enhancement in a horizontal block-heated channel an oblique plate installed. The maximum increase of average Nusselt number was 39.5% when the oblique angle was p=3. Herman and Kang [21] investigated the heat transfer enhancement in a grooved channel with curved vanes. The results showed that ﬂow oscillations contributed to heat transfer enhancement.

Most methods mentioned above are passive, and the improvement of heat transfer rate and the eﬀective range are limited. Therefore, an eﬀective method proposed in Nomenclature

d diameter of cylinder [m]

fc oscillating frequency of cylinder [s1]

Fc dimensionless oscillating frequency of

cyl-inder (Fc¼ fcd=uc)

hc distance from bottom side to center of

cyl-inder [m]

he height of the channel [m]

hd height of the block [m]

k thermal conductivity

lc oscillating amplitude of cylinder [m]

Lc dimensionless oscillating amplitude

(Lc¼ lc=d)

Nu periodic-averaged Nusselt number

Nuoverall average Nusselt number of overall heated

surface

Nusurface average Nusselt number of one surface of a

block

NuX local Nusselt number

p dimensional pressure [N m2_]

p1 reference pressure [N m2]

P dimensionless pressure (P¼ ðp p1Þ=qu20)

Pr Prandtl number (Pr¼ t=a) r radius of cylinder [m]

R dimensionless radial coordinate (R¼ r=d) Re Reynolds number (Re¼ u0d=t)

s distance between two heated blocks [m] t time [s]

T temperature [K]

TH temperature of high temperature region [K]

Tmx local mean temperature [K]

T0 temperature of inlet ﬂuid [K]

u, v velocities in x and y directions [m s1_]

U, V dimensionless velocities in X and Y direc-tions (U¼ u=u0, V ¼ v=u0)

u0 velocities of inlet ﬂuid [m s1]

vc oscillating velocity of cylinder [m s1]

Vc dimensionless oscillating velocity of cylinder

(Vc¼ vc=u0)

vm maximum oscillating velocity of cylinder

[m s1_]

Vm dimensionless maximum oscillating velocity

of cylinder (Vm¼ vm=u0)

^

vv mesh velocity in y-direction [m s1_]

^ V

V dimensionless mesh velocity in Y -direction ( ^VV ¼ ^vv=u0)

w length of channel [m]

w1 length from inlet region to the center of the

cylinder [m]

w2 length from outlet region to the center of the

cylinder [m]

w3 length from the center of the cylinder to the

ﬁrst block [m]

wb dimensional length of the heated block [m]

x, y dimensional Cartesian coordinates [m] X, Y dimensionless Cartesian coordinates

(X¼ x=he, Y ¼ y=he) Greek symbols a thermal diﬀusivity [m2_s1_] U computational variables k penalty parameter m kinematics viscosity [m2_s1_] h dimensionless temperature (h¼ ðT T0Þ= ðTH T0Þ)

hmX dimensionless local mean temperature

q density [kg m3_]

s dimensionless time (s¼ tu0=he)

sp dimensionless time of one oscillating cycle

W dimensionless stream function Others

j j absolute value

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this study is to utilize an oscillating cylinder suspended in the channel and in crossﬂow to cause the occurrence of ﬂow vibration and vortex shedding, which could en-hance the heat transfer rate of the heated blocks set in the channel walls.

The subject of the present work is therefore to in-vestigate the influence of the flow passing over the os-cillating cylinder on the heated blocks in the channel. The subject mentioned above belongs to the class of moving boundary problems, and the arbitrary Lagran-gian–Eulerian (ALE) method modified by Fu and Yang [22] is suitably adopted to solve this problem. The heat transfer characteristics along the heated blocks are pre-sented in detail. The mainly effects of Reynolds number, oscillating amplitude and oscillating frequency on the flow structures and heat transfer characteristics are in-vestigated.

2. Physical model

The physical model used in this study is shown in Fig. 1. A two-dimensional channel with height heand length

wis used to simulate this problem. An insulated cylinder of diameter d is set centrally within the channel. The distances from the inlet and outlet of the channel to the center of cylinder are w1 and w2, respectively. The

dis-tance from the center of the cylinder to the front surface of the ﬁrst block is w3. The height and length of the

heated blocks are hband wb, respectively. The distance

between the blocks is s. The numbers of I, II, III and IV of the blocks are arranged from front to rear position. In this study he=dis 4, hb=dis 0.8, wb=d is 2, s=d is 1. The

inlet velocity u0 and temperature T0 of the ﬂuid are

uniform. The temperature of the heated blocks is TH

which is higher than T0. The wall of the channel is

in-sulated. Initially, the cylinder is stationary at the posi-tion of the center of the channel and the ﬂuid ﬂows steadily. The distance from the wall of the channel to the center of the cylinder is hc. As the time t > 0, the cylinder

is in oscillating motion normal to the inlet ﬂow with amplitude lc. The oscillating velocity of the cylinder is

vc¼ 2plccosð2pfctÞ. The oscillating cylinder and the

flow then affects each other, and the variations of the flow field become time-dependent and are classified into a class of moving boundary problems. As a result, the ALE method is properly utilized to analyze this prob-lem.

For facilitating the analysis, the following assump-tions are made.

(1) The fluid is air and the flow field is two-dimensional, incompressible and laminar.

(2) The ﬂuid properties are constant and the eﬀect of the gravity is neglected.

(3) The no-slip condition is held on the interfaces be-tween the ﬂuid and cylinder.

Based upon the characteristics scales of d, u0, qu20and

T0, the dimensionless variables are deﬁned as follows:

X¼x d; Y ¼ y d; U¼ u u0 ; V ¼ v u0 ; VV^¼ ^vv u0 ; Vc¼ vc u0 ; Lc¼ lc d; Hc¼ hc d; Fc¼ fcd u0 ; P¼p p1 qu2 0 ; s¼tu0 d ; h¼ T T0 TH T0 ; Re¼u0d t ; Pr¼ t a; ð1Þ where ^vvis the mesh velocity, vc, fc, hc and lc are the

oscillating velocity, the oscillating frequency, the posi-tion and the oscillating amplitude of the cylinder, re-spectively.

According to the above assumptions and dimen-sionless variables, the dimendimen-sionless ALE governing equations are expressed as:

Continuity equation oU oXþ oV oY ¼ 0; ð2Þ d w c h A B C D c v c v 1 w 0 0 ∂ = ∂ ∂ = =v T y u T v u x , , 0 = Φ = ∂ 0 0 T u 0 = ∂ ∂T r x y 2 w b h s

block I block II block III block IV

b w 3 w H T T= T=TH T=TH T=TH e h Φ

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Momentum equation oU osþ U oU oXþ ðV ^VVÞ oU oY ¼ oP oXþ 1 Re o2_U oX2 þo 2_U oY2 ; ð3Þ oV osþ U oV oXþ ðV ^VVÞ oV oY ¼ oP oYþ 1 Re o2_V oX2 þo 2_V oY2 ; ð4Þ Energy equation oh osþ U oh oXþ ðV ^VVÞ oh oY¼ 1 Re Pr o2_h oX2 þo 2_h oY2 ; ð5Þ As the time s > 0, the boundary conditions are as fol-lows:

On the inlet surface AB

U¼ 1; V ¼ 0; h¼ 0; ð6Þ On the wall of channel

U¼ 0; V ¼ 0; oh

oY¼ 0; ð7Þ

On the surfaces of the block I, II, III, IV

U¼ 0; V ¼ 0; h¼ 1; ð8Þ On the outlet surface CD

oU oX ¼ 0; oV oX¼ 0; oh oX¼ 0; ð9Þ

On the interfaces between the ﬂuid and cylinder U¼ 0; V ¼ Vc;

oh

oR¼ 0: ð10Þ

3. Numerical method

The governing equations and boundary conditions are solved through the Galerkin ﬁnite element formu-lation and a backward scheme is adopted to deal with the time terms of the governing equations. The pressure is eliminated from the governing equations using the consistent penalty method. The velocity and tempera-ture terms are expressed as quadrilateral element and eight-node quadratic Lagrangian interpolation function. The Newton–Raphson iteration algorithm is utilized to simplify the nonlinear terms in the momentum tions. The discretion processes of the governing equa-tions are similar to the one used in Fu and Yang [22].

A brief outline of the solution procedures are de-scribed as follows:

(1) Determine the optimal mesh distribution and num-ber 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 velocity ^VV of the computational meshes.

(4) Update the coordinates of the nodes and examine the determinant of the Jacobian transformation ma-trix to ensure the one to one mapping to be satisﬁed during the Gaussian quadrature numerical integra-tion.

(5) Solve Eq. (11), until the following criteria for con-vergence are satisﬁed:

Umþ1 Um Umþ1 sþDs <103; where U¼ U ; V ; and h: ð11Þ (6) Continue the next time step calculation until

peri-odic solutions are attained.

4. Results and discussion

The working ﬂuid is air with Pr¼ 0:71. The main parameters of Reynolds number Re, oscillating ampli-tude Lc and oscillating frequency Fc are examined and

the combinations of these parameters are tabulated in Table 1.

In the channel, the thermal boundary layer grows in the downstream direction gradually. In order to describe the wall heat transfer in the channel region realistically, it is necessary to deﬁne the local mean temperature Tmx

of the stream as:

Tmx¼ 1 u u he Z h 0 uTdy; where uu¼ Z h 0 udy: ð12Þ

The dimensionless variable hmX is deﬁned as

hmX¼

Tmx T0

TH T0

: ð13Þ

The local Nusselt number is calculated by the following equation: Table 1 Parameter combinations Case Fc Lc Re 1 0.1 0.1 250 2 0.2 0.1 250 3 0.4 0.1 250 4 0.2 0.05 250 5 0.2 0.2 250 6 0.2 0.4 250 7 0.2 0.1 100 8 0.2 0.1 500

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NuX ¼ oh on block surface ₁ 1 hmX : ð14Þ

The time-averaged local Nusselt number per oscillatory cycle is deﬁned by

Nu¼ 1 sp

Z sp

0

NuXds where spis a period of the cycle:

ð15Þ The average Nusselt number along one surface of a heated block is found as follows:

Nusurface¼ 1 Xs Z surface NuXdX ;

where Xs is length of heated surface: ð16Þ

The time-average Nusselt number along one surface of a heated block per cycle period is expressed as follows: Nusurface¼ 1 sp Z sp 0 Nusurfaceds: ð17Þ

The overall Nusselt number of the four heated blocks is deﬁned by Nuoverall¼ 1 4ðwbþ 2hbÞ X block4 block1 Nusurface: ð18Þ

The time-average overall Nusselt number of the four heated blocks per cycle period is deﬁned by

Nuoverall¼ 1 sp Z sp 0 Nuoverallds: ð19Þ

For matching the boundary conditions at the inlet and outlet of the channel mentioned above, the dimen-sionless lengths from the inlet and outlet to the cylinder are determined by numerical tests and equal to 15.0 and 120.0, respectively. To obtain an optimal computational mesh, three different nonuniform distributed elements, which provide a finer element resolution near the cyl-inder and walls, are used for the mesh tests. Fig. 2(a) and (b), show the velocity profiles along the line through the center of the cylinder and parallel to the Y -axis at the steady state for Re¼ 500, respectively. Based upon the results, the computational mesh with 6700 elements, which is corresponding to 20216 nodes, is used for all cases in this study. In addition, an implicit scheme is employed to deal with the time differential terms of the governing equations. After the time step test, the time step Ds¼ 0:01 is chosen for all cases in this study.

The dimensionless stream function W is deﬁned as U¼oW

oY; V ¼ oW

oX: ð20Þ

Fig. 3 shows a comparison of the present results with those of Kim [22] under the same situation, both the results show good agreement.

For a better understanding of the phenomena around the oscillating cylinder and heated blocks, the flow and thermal fields close to the oscillating cylinder and heated blocks are illustrated in the following figures. The vari-ations of the streamlines under Re¼ 250, Lc¼ 0:2 and

Fc¼ 0:2 are indicated in Fig. 4. Fig. 4(a) shows the

streamlines in the empty channel with heated blocks at steady state. Due to the obstruction of the first block, the streamlines begin to deflect at upstream of the first block and the fluid is accelerated upward. Therefore, a small vortex is formed on the top surface of the first

-2 -1 0 1 2 Y -0.4 -0.2 0 0.2 0.4 0.6 5400 elements 6700 elements 7200 elements -2 -1 0 1 2 Y 0 0.4 0.8 1.2 1.6 2 U 5400 elements elements elements 6700 7200 (a) (b)

Fig. 2. Comparison of the velocity proﬁles along the line through the center of the cylinder and parallel the Y -axis for diﬀerent mesh.

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block. At the same time, other larger vortex down-stream of the last heated block is observed. Weak clockwise vortices are formed within the grooves, which are similar to the cavity ﬂow. These vortices are the main impediment for the heat transfer from the heated blocks.

At the time s¼ 0, shown in Fig. 4(b), the cylinder is stationary and the flow is steady. The fluid flows upward into the narrow space between the first block and cyl-inder, the fluid is accelerated more drastically due to the contraction effect. Besides, due to the effect of the cyl-inder, a vortex forming on the top surfaces of the blocks is larger than that of the channel without the cylinder installed shown above and reaches the fourth block. As a result, the heat transfer from the top surface of the heated blocks of Fig. 4(b) situation is worse than that of Fig. 4(a) situation.

As the time s > 0, the cylinder starts to oscillate with the oscillating velocity Vc¼ 2pFcLccosð2pFcsÞ, where the

oscillating frequency Fcis 0.2 and the oscillating

ampli-tude Lc is 0.2. Fig. 4(c) shows the cylinder that has

moved upward to the maximum amplitude. The cylinder squeezes the fluid near the upper region of the channel and the fluid near the bottom surface of the cylinder replenishes the vacant space induced by the oscillating cylinder. As a result, the original vortex behind the cylinder is shed from the cylinder and moves down-stream. Afterward, the cylinder moves downward di-rection immediately and the Fig. 4(d) shows the cylinder on the way to move downward. The fluid near the top surface of the cylinder simultaneously replenishes the vacant space created by the movement of the cylinder, and the cylinder presses the fluid near the bottom sur-face of the cylinder. Then a new vortex is formed gradually behind the cylinder. The oscillating cylinder

influences the formation of large vortex on the top surfaces of the blocks. As shown in the Fig. 4(e), because of the vortex shedding and the oscillating motion of the cylinder, the large vortex on the top surfaces of heated blocks is difficult to maintain its original state, which causes the large vortex to be split into small vortices and move downstream with the flow. The cylinder reaches the lowest amplitude. A new vortex forms on the top surface of the first block and pushes the existing vortices to the downstream as the cylinder is on the way to move upward shown in Fig. 4(f). When the cylinder returns to the center of the channel with the maximum upper ve-locity, the cylinder completes the first oscillation cycle, and the cycle time s is 5.

As the time increases, shown in Fig. 4(g), during the second oscillation cycle and the same situation as the Fig. 4(f), the vibrational ﬂow not only pushes the vor-tices generated on the top surfaces of the blocks to the downstream continually but also starts to aﬀect the

Fig. 4. The transient developments of streamlines for case 2. (a) Empty channel, (b) s¼ 0, (c) s ¼ 1:25, (d) s ¼ 2:5, (e) s ¼ 3:75, (f) s¼ 5, (g) s ¼ 10, (h) s ¼ 115, (i) s ¼ 116:25, (j) s ¼ 117:5, (k) s¼ 118:75, (l) s ¼ 120.

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vortices between the blocks and behind the last block gradually.

Fig. 4(h)–(l) show the variations of the streamlines during one steadily periodic cycle. As shown in these ﬁgures, the new vortices on the heated blocks are gen-erated periodically by the oscillating cylinder and push other vortices to the downstream continually. The vor-tices between the blocks form and are ejected alternately. Comparing Fig. 4(h) and (l), the streamlines are identi-cal which means that the variations of the ﬂow become a steadily periodic motion with time.

The distributions of local Nusselt number NuXon the

different surfaces of the blocks in a channel with a sta-tionary cylinder and with an oscillating cylinder during one period are shown in Fig. 5. The situations of the different times in Fig. 5 correspond to those shown in the Fig. 4(h)–(l). The shedding vortices affect the local heat transfer from blocks remarkably due to the movement of vortices. However, the influence of vortex shedding is not obvious on the first block.

The variations of the overall average Nusselt number Nuoverall with time of all blocks for the situation of

Re¼ 250, Fc¼ 0:2 and Lc¼ 0:2 (case A) are shown in

Fig. 6 and compare the Nusselt number with the same situations of an empty channel (case B) and a channel with a stationary cylinder as the flow in steady state (case C). In the beginning, the difference between the cases of A and C is small. As the time increases, the effect of the oscillating cylinder on the heat transfer of case A is apparent gradually. Finally, the variation of the heat transfer approaches a periodic state with time (s P 80). The increase of the heat transfer of case A compared with those of cases B and C is more than 60% and 120%, respectively.

When the oscillating frequency approaches the nat-ural shedding frequency, it would cause the lock-in phenomenon to happen [23]. For a ﬂow passing through a stationary cylinder, the natural frequency of vortex shedding Fc is about 0.2 over a range of Reynolds

number from 2· 102 _{to 10}4 _{[24]. The eﬀects of the}

os-cillating frequency on the heat transfer of each surface of the blocks are shown in Fig. 7 for the Lc¼ 0:1 and

Re¼ 250 situation. Comparing with the stationary cyl-inder, the average Nusselt numbers Nusurface along each

surface of the heated blocks are enhanced substantially. It can be observed that as the oscillating frequency is in the lock-in regime (Fc¼ 0:2), the heat transfer of each

surface, except for the front surface of the ﬁrst block, is greater than those of other frequencies situations obvi-ously. Conversely, when the values of Fcare equal to 0.1

and 0.4, because of the flow in the unlock-in flow, the enhancement for the heat transfer rate of these flows are not as significant as that in the lock-in regime.

Fig. 8 shows the eﬀects of the oscillating amplitude on the heat transfer of each surface of the blocks for the Fc¼ 0:2 and Re ¼ 250 situation. Comparing with the

stationary cylinder, the increase of the surface average Nusselt numbers Nusurface along each surface of the

he-ated blocks is signiﬁcant. Therefore, the eﬀect of the oscillating amplitude on the heat transfer from the he-ated block is remarkable as the oscillating amplitude is larger than 0.1.

The eﬀects of the Reynolds number on the heat transfer from each surface of the blocks for the Lc¼ 0:1

and Fc¼ 0:2 situation are shown in Fig. 9. The higher

the Reynolds number is, the velocity and disturbance of ﬂuid are quicker and more drastic, respectively. Thus the heat transfer rate is enhanced remarkably with the in-crement of the Reynolds number.

Table 2 shows the comparison of the time average overall Nusselt number Nuoverall of the heated blocks in

the channel installed with the oscillating cylinder with those of having no installation of the oscillating cylin-der. The results show the heat transfer could be im-proved by the oscillating cylinder just only as the oscillating frequency is in the lock-in region.

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Fig. 10 shows the pictures of flow visualization for the Re¼ 267 situation. The Ni–Cr wire on which par-affin was applied and evaporated parpar-affin

instanta-neously by resistive heating to generate white smoke is installed at the distance of 0.8D downstream from the center of the cylinder, and the oscillating cylinder is controlled by a stepper motor. The experimental study is intended to simulate as closely as possible to the above

0 40 80 120 t 1.5 2 2.5 3 3.5 4 4.5 Nu overall oscillating cylinder Stationary cylinder Without cylinder

Fig. 6. Vibration of the overall-average Nusselt number with time. F T B F T B F T B F T B 0 2 4 6 8 Nu surface Fc=0.0 Fc=0.1 Fc=0.2 Fc=0.4

Block I Block II Block III Block IV Fig. 7. The variations of average Nusselt number of each block for various frequencies under Lc¼ 0:1, Re ¼ 250 situation. Fig. 5. The distributions of the local Nusselt number on the diﬀerent surfaces of the blocks for case 2.

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numerical simulation model, however, the blocks are not equipped with the heat sources. Fig. 10(a) shows the streaklines in the channel under without cylinder situa-tion. As the streaklines develop over the blocks, they are

close to the top surfaces of the blocks. These phenomena are similar to the numerical results shown in Fig. 4(a). Fig. 10(b) shows the phenomena of the streaklines de-veloping over the blocks under the stationary cylinder installed in the channel situation. Since the space be-tween the cylinder and the ﬁrst block in contracted, the streaklines passing this space is like to pass expansion region. The streaklines are no longer close to the top surfaces of the blocks as shown in Fig. 10(a), which indicates a circulation forming on the top surface of the blocks and is similar to the numerical results shown in Fig. 4(b). Fig. 10(c)–(f) indicate the periodical phe-nomena of the cylinder oscillating under Re¼ 267 and Fc¼ 0:2 situation. A small vortex observed on the top

surface of second block (Fig. (c)) moves downstream continuously ((c)! (d) ! (e) ! (f)) which are similar to these results of Figs. 4(g)–(k) quantitatively.

5. Conclusions

The heat transfer characteristics of the heated blocks in the channel with a transversely oscillating cylinder in a cross ﬂow are investigated numerically. Some conclu-sions are summarized as follows:

1. The heat transfer rate could be improved substan-tially as the cylinder oscillating in the lock-in region. 2. The inﬂuence of oscillating amplitude on the heat transfer rate is not obvious under the lock-in region. 3. The heat transfer rate is increased when the Reynolds

number increases.

Acknowledgements

The support of this work by the National Science Council of Taiwan, ROC, under contract NSC89-2212-E009-019 is gratefully acknowledged.

F T B F T B F T B F T B 0 2 4 6 8 Nu surface Lc=0.05 Lc=0.1 Lc=0.2 Lc=0.4 Lc=0.0

Block I Block II Block III Block IV

Fig. 8. The variations of overall Nusselt number average Nusselt number of each blocks for various oscillating ampli-tudes under Fc¼ 0:2, Re ¼ 250 situation.

F T B F T B F T B F T B 0 2 4 6 8 10 Nu surface Re=100 Re=250 Re=500

Block I Block II Block III Block IV

Fig. 9. The variations of overall Nusselt number average Nusselt number of each blocks for various Reynolds number under Fc¼ 0:2, Lc¼ 0:1 situation.

Table 2

Compare the Nuoverallin the channel with the oscillating cylinder with those without oscillating cylinder

Case

ðNuoverallÞoscillating cylinder ðNuoverallÞempty channel ðNuoverallÞempty channel

ð%Þ 1 _)0.9 2 62 3 )5 4 33 5 59 6 60 7 )7 8 116.5

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