THERMAL ENERGY TRANSPORTATION BY AN OSCILLATORY FLOW

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1 ISP-6, 5, PRAGUE 6 H INERNAIONAL SYMPOSIUM ON RANSPOR PHENOMENA HERMAL ENERGY RANSPORAION BY AN OSCILLAORY FLOW Akira Inukai, Makoto Hishida, Gaku anaka Chiba University, Department of Electronics and Mechanical Engineering -33, Yayoi-cho, Inage-ku, Chiba-city, JAPAN, Zip code: 63-85, Corresponding author: Makoto Hishida hishida@faculty.chiba-u.jp phone, fax Keywords: hermal Energy ransportation, Oscillatory Flow, Numerical Analysis, Cooling device Abstract his paper deals with the transportation of heat (or enthalpy) by an oscillatory flow that is a kind of reciprocating flow superimposed with a steady flow. he velocity and temperature fields, transportation rate of heat (or enthalpy), work rate and transportation efficiency of heat were investigated by numerical analysis. he results obtained are () the transportation rate of heat increased largely by superimposing a steady flow on a reciprocating flow, () on the other hand, work required by the present oscillatory flow increased only a little, (3)the transportation rate of heat by the present oscillatory flow was less than the arithmetic sum of those by the steady flow and by the reciprocating flow, (4) in the upstream heat was transported mainly by steady flow component and in the downstream mainly by oscillatory flow component, (5) the phase difference between the velocity and temperature variations of the oscillatory flow remained the same as that of the sinusoidal oscillatory flow (reciprocating flow). Introduction hermal energy transportation devices with small size and high transportation rate are highly required for achieving effective cooling of electronic devices and equipments. As pumping technologies are difficult to realize for the cooling of small size machines, the heat transportation device called dream pipe that utilizes reciprocating flow is considered to become one of important candidate technologies. Since the dream pipe was invented by Kurzweg [], many researchers have attempted to improve its performance. Kaviany [] pointed out that the heat transportation rate can be enhanced by utilizing heat transfer between pipe material and reciprocating flow. Nishio et al. [3,4] investigated the conditions for achieving the optimum performance of the dream pipe and they reported that water is the best working fluid because it can gain a high effective thermal conductivity in a relatively wide range of reciprocating frequency. In addition they pointed out that there exists an optimum size for the dream pipe. According to their study the optimum pipe radius should be a comparable magnitude to the thickness of thermal boundary layer of the reciprocating flow. Based on these investigations, Nishio et al. [5] proposed a new kind of dream pipe, inversed oscillation-phase heat transport tube, which can achieve higher heat transportation performance than the existed dream pipe. Ohno et al. [6] proposed an annular dream pipe that can improve the performance in a high frequency range. On the other hand, anaka et al. [7] investigated the oscillatory flow in a loop channel that included a -branch. hey found that a slow circulation flow superimposed on a sinusoidal reciprocating flow was established in a loop channel, even when a simple sinusoidal reciprocating flow was imposed at one end of the loop channel. In our preliminary

2 experiments, we also found that the similar oscillatory flow superimposed by a slow circulation flow (or steady flow) was established in a loop channel shown in Fig.. he flow rate of the superimposed circulation flow reached as much as 5% of the maximum flow rate of the simple sinusoidal reciprocating flow imposed at the left end of the preliminary experimental loop channel. We may expect a high performance of heat transportation by the new type of oscillatory flow involving the circulation flow component or steady flow component. Only a limited number of studies, however, have been reported so far on the heat transportation of this kind of oscillatory flow. he main purpose of the present study is to investigate the heat transportation performance of the new type of oscillatory flow in a single round tube. he computer code FLUEN was used for the numerical analysis [8]. In order to investigate the effects of the oscillation frequency and the flow rate of the steady flow component Womersley number was varied in a range from to 5 and steady flow parameter w st,i/ w osmax,i was varied in a rage from. to.5. In the present paper the oscillatory flow with no steady flow component is referred as the sinusoidal oscillatory flow and the one with steady flow component as the oscillatory flow. Sinusoidal oscillatory flow 5 mm 85 Fig. Preliminary experimental setup Nomenclature Sinusoidal oscillatory flow + Steady flow Reservoir 4 4 a : Radius C* : Specific heat Pr : Prandtl number q : Heat flux of enthalpy flux q ~ : Average heat (enthalpy) flux in a cycle q : Average heat (enthalpy) flux in a cycle and in cross sectional area of pipe r : Radial coordinate : emperature H : emperature of heating section C : emperature of cooling section t : ime t cy : Period of a cycle u : Velocity in radial direction Wo : Womersley number = a*(ω*ν*) / w : Velocity in axial direction W ~ : Average work rate in a cycle z : Axial coordinate η : Heat (enthalpy) transportation efficiency ν* : Kinetic viscosity ρ* : Density ω* : Angular frequency * : Dimensional symbols subscript os : Oscillatory flow component st : Steady state flow component osmax : Maximum of oscillatory component i : Left end (inlet) of heating section 3 Analysis As in any channel composing the loop channel shown in Fig. the oscillatory flow composed of steady flow component (hereafter steady component) and oscillatory flow component (hereafter oscillatory component) is established, in the present analysis we investigated the heat transportation performance of the oscillatory flow in a single round pipe. Figure shows the analytical model that has a heating section with a uniform temperature of H, heat transportation section (adiabatic section) and a cooling section with a uniform temperature of C. he basic equations are mass conservation equation (), momentum conservation equations ()(3) and energy conservation equation (4). hey are given in dimensionless form as: r = H Fig. Analytical model = C z

3 HERMAL ENERGY RANSPORAION BY AN OSCILLAORY FLOW u u + + = r () P w w Wo ( + u + w ) = t r z z r r r z () P u u u u Wo ( + u + w ) = (3) t r r r PrWo ( + u + w ) = + + (4) t r r Boundary conditions are written as: at z =, w = wosmax, i(cost + wst,i / wosmax, i ), (5a) u =, = ( for w ), = ( for w ) u at z = 75, =, =, P = const (5b) = ( for w ), = ( for w ) u at r =, =, =, = (5c) at r =, z = 5, w =,u =, = (5d) at r =, z = 5 5, w =,u =, = (5e) at r =, z = 5 75, w =,u =, = (5f) Initial condition is described as: at t =, u = w =, = (5g) Here, the reference length for dimensionless length is the pipe radius a*, time /ω*, velocity a*ω*, pressure ρ*ν*ω*. he dimensionless temperature is defined as =(*- C *)/( H *- C *). he working fluid is water whose Prandtl number is Pr=4.64. he dimensionless length of the heating section is z=~5, the adiabatic section z=5~5 and the cooling section z=5~75. he maximum velocity of the oscillatory component at the inlet of heating section was fixed at w osmax,i =.5, Womersley number Wo was varied from to 5 and the steady flow parameter defined as w st,i /w osmax,i was from to.5, where w st,i is the uniform velocity of the steady component at the left end of the heating section. According to Hino's study [9], the present flow regime is laminar. For the numerical analysis the computer code FLUEN [8] was used. he whole computational domain was divided into 3, cells ( times,5). he calculated velocity and temperature distributions change as computational time or cycles elapsed because of the transient calculation method employed by the present analysis. Eventually, however, identical velocity and temperature variations were reached. In the present study, we investigated the velocity and temperature fields, transportation heat rate (enthalpy rate) and the work of the oscillatory flow after the identical fields within cycles were established. In the preliminary calculation we compared the calculated velocity and temperature of a sinusoidal reciprocating flow with the theoretical ones given by Kaviany []. he calculated results agreed with Kaviany' theoretical ones within the differences of ±3% and ±5% respectively. 4 Analytical Results and Discussion 4. Velocity Field in Fully Developed Region Fully developed velocity region were established in the longitudinal domain of about 5 z 6. It is defined as the longitudinal domain where the velocity distributions over pipe cross sections are the same. Figure 3 shows the velocity distribution over the cross section of pipe in the fully w t/t cy r.6.8 Fig. 3 Variation of velocity profile (w st,i /w osmax,i =.5, Wo = ~5) Wo= Wo=6 Wo=5 3

4 developed region for the oscillatory flow with Wo=, 6, and 5 and w st,i /w osmax,i =.5. When Wo= there exists no core region where uniform velocity profiles are attained, whereas the oscillatory flows with Wo = 6 and 5 have core regions where the velocity distributions are flat. heir velocity boundary layers are restricted to the vicinity of the pipe wall. he boundary was temporary defined as the position where the velocity at t/t cy =.5 and t/t cy =.55 in one cycle are about ±3% less or larger than those of the uniform velocity in the core region. he moments of t/t cy =.5 and t/t cy =.55 are the time when the oscillatory velocity distributions become the maximum and minimum within one cycle. For the oscillatory flow of Wo=6, the core region spans from r= to r=.3 and the flow of Wo=5 it expands to r=.65. Inspection of Eqs. ()~(3) reveals that the velocity distribution in the fully developed region equals the sum of the velocity distributions of the sinusoidal oscillatory flow and Hagen-Poiseuille flow (steady flow). he present calculated velocity distribution agreed well with the sum of those velocities with an error of less than ±3%. As the present study is restricted to the flow in a very small range of the steady flow parameter, w st,i /w osmax,i.5, the oscillatory flow fields are almost characterized by the sinusoidal oscillatory flow. 4. emperature Field In Figs. 4 and 5 are plotted axial distributions of the maximum and minimum temperatures along the central line of pipe. he maximum and minimum temperatures are attained at the dimensionless time of t/t cy =.5 and.75 in a cycle. Figures 6 and 7 show the variations of the radial temperature distribution in a cycle in the cross sections of z=37.5 and emperature distribution of sinusoidal oscillatory flow he longitudinal temperature distributions of sinusoidal oscillatory flow have constant temperature gradient and identical amplitude of temperature oscillation over a wide axial region in the adiabatic section, except only narrow vicinities at the inlet and exit. As shown in Fig.6 the sinusoidal oscillatory flow with Wo= has no core region where uniform temperature distribution exists whereas the oscillatory flow with Wo=6 has the core region of uniform temperature and its thermal boundary layer is restricted to the vicinity of the pipe wall. he perpendicular chain lines in Fig.7 illustrate the boundary between the central core region and the boundary layer near the wall. he boundary was temporary defined as the position where the temperatures at t/t cy =.5 and t/t cy =.55 in one cycle are about ±3% less or larger than those of the uniform temperatures in core region. For the oscillatory flow of Wo=6, the core region spans from r= to r=.37 and the flow of Wo=5 it expands to r=.7. As Prandtl number is Pr=4.64 in the present analysis the temperature boundary layers are slightly thinner than the velocity ones Fig. 4 Axial distribution of the maximum and the minimum temperature along the center line of pipe (Wo =, W st,i /W osmax,i = ~.5) w st,i / w osmax,i..5 t/t cy =.5 t/t cy = z w st,i / w osmax,i..5 t/t cy =.5 t/t cy = z Fig. 5 Axial distribution of the maximum and the minimum temperature along the center line of pipe (Wo = 6, W st,i /W osmax,i = ~.5) 4

5 HERMAL ENERGY RANSPORAION BY AN OSCILLAORY FLOW.9.8 t/t cy w st,i /w osmax,i =.5 w st,i /w osmax,i =..9 t/t cy w st,i /w osmax,i = w st,i /w osmax,i = w st,i /w osmax,i = r.6.8 (a) Wo=, z=37.5, w st,i /w osmax,i = ~ r.6.8 (a) Wo=6, z=37.5, w st,i /w osmax,i = ~.5.9 t/t cy w st,i /w osmax,i = w st,i /w osmax,i = w st,i /w osmax,i = w st,i /w osmax,i = t/t cy w st,i /w osmax,i = w st,i /w osmax,i =..4. w st,i /w osmax,i =...4 r.6.8 (b) Wo=, z=45, w st,i /w osmax,i = ~.5 Fig.6 Variation of radial temperature distribution..4 r.6.8 (b) Wo=6, z=45, w st,i /w osmax,i = ~.5 Fig.7 Variation of radial temperature distribution 4.. emperature distribution of the oscillatory flow he oscillatory flows have no such axial region where the constant temperature gradient in z direction dominates. he following features become conspicuous in the temperature fields in the adiabatic section: as the steady flow parameter w st,i /w osmax,i increases, () the temperature gradient in z-direction and the amplitude of temperature oscillation become smaller in the upstream, and they become larger in the downstream, () the average temperature in one cycle becomes higher at any axial position in the adiabatic section. hese features 5

6 are resulted from the fact that the quantity of heat transported by the steady component increases with the steady flow parameter w st,i /w osmax,i. 4.3 ransportation of Heat Wo 6 5 he oscillating heat flux or enthalpy flux transported by the oscillatory flow from the heating section to the cooling section per unit cross sectional area of pipe is obtained from the velocity and temperature variations as: * * * * * * * * q(r,z,t) = [ ρ C (w os + w st )( c ) + λ ] * * * * * * * ρ C a ω (H C ) (6), where the first term of the right side is the enthalpy flux and the second term the heat flux by molecular conduction. As in the present calculations the second term was less than / of the first term, the enthalpy transportation dominates. hat is, the present analysis mainly concerns the enthalpy transportation. Neglecting the heat conduction term, the average enthalpy flux in one cycle is calculated by: tcy q ~ (r,z ) = wdt= ( wos + wst ) d(t / tcy ) (7) tcy In the present paper the term 'heat flux' is used instead of 'enthalpy flux'. he average heat flux in one cycle and in a unit cross section of pipe is written as: q = r q ~ ( r,z ) dr (8) Dimensionless heat flux is defined as: q = q* / a* ω * ρ * C*(H * C*) (9) 4.3. Average heat flux in one cycle of sinusoidal oscillatory flow Figure 8 shows the radial distribution of the average heat flux q in one cycle of the sinusoidal oscillatory flow in the fully developed region. he fully developed region was temporally defined as the axial region where exist identical longitudinal temperature gradient and identical amplitude of temperature oscillation along the length of pipe, at any instant in one cycle. In the flow of Wo=, heat Fig.8 Radial distribution of average heat flux q in one cycle of sinusoidal oscillatory flow (Wo=~5) [w(r,t) w st (r)] / w osmax (r), [(r,z,t) ave (r,z)] / (r,z) Fig.9 Velocity and temperature variations in the fully developed region of sinusoidal oscillatory flow (Wo=6) transportation from the heating section to the cooling section takes place in the central area of pipe where r is small. In the vicinity of pipe wall (large r), on the other hand, heat is transported in the reverse direction, from the cooling section to the heating section. As Wo is increasing, heat transportation becomes to occur in the vicinity of pipe wall, and in the central area small amount of heat is transported in the reverse direction. From the velocity and temperature variations shown in Fig. 9 we may understand that the velocity and temperature variations of the sinusoidal oscillatory flow can be approximated by the following equations: w'(r,t ) = w '(r ) cosπ [t / t φ '(r )] () '(r,z,t) r w st,i / w osmax,i.5. emperature Velocity r = r = t / t cy osmax, i cy w = '(r,z)cos π[t/tcy φ '(r)] + ave'(r,z) () 6

7 HERMAL ENERGY RANSPORAION BY AN OSCILLAORY FLOW Substituting Eqs.() and () into Eq.(7) we get the following equation: q ~ '(r,z) = wosmax'(r) '(r,z) cos π[t/tcy φw '(r)] () cos π[t/tcy φ '(r)]d(t/t cy) Here, (') represents the sinusoidal oscillatory flow. he phase difference between the temperature and velocity variations, Ф '(r)- Ф W '(r) is an important factor for the integration. For example, when Ф '(r)-ф W '(r) is, the integration has the maximum value of π and when it is.5 the minimum value of π. In the ranges of Ф '(r)-ф W '(r).5 and.75 Ф '(r)-ф W '(r). q'(r,z) has positive values and in the range of.5 Ф '(r)-ф W '(r).75 it has negative values. hus, the average heat flux in one cycle of the sinusoidal oscillatory flow is determined by the amplitudes of velocity variation w osmx '(r) and of temperature variation '(r,z) and the phase difference Ф '(r)-ф W '(r). he heat conduction in the radial direction causes this phase difference between the velocity and temperature Average heat flux in one cycle of the oscillatory flow Figures and show the radial distribution of the average heat flux q in one cycle in two cross sections of the pipe. Solid lines illustrate the total average heat flux q transported by the oscillatory flow and dotted lines illustrate average heat flux q st by the steady component, which is calculated with Eq. (7) by putting w os =. he difference between the solid and dotted lines is the average heat flux q õs transported by the oscillatory component, which is calculated with Eq. (7) by putting w st =. From Figs. and we see the average heat flux distributions over the cross section differ in longitudinal direction depending on axial position unlike the sinusoidal oscillatory flow. Both the total average heat flux q transported by the oscillatory flow and q st by the steady component increase with increasing the steady flow parameter w st,i /w osmax,i. hese will be explained by the following reasons..3.. w st,i / w osmax,i r (a) Wo=, z=37.5 (b) Wo=, z=45 Fig. Radial distribution of average heat flux (a) Wo=6, z=37.5 ~ q st r.3.. (b) Wo=6, z=45 Fig. Radial distribution of average heat flux w st,i / w osmax,i..5 ~ q qst st w st,i / w osmax,i..5 ~ q st r r.6.8 w st,i / w osmax,i..5 ~ q qst st 7

8 , As shown in Fig.9 the temperature variations of the present oscillatory flow can be roughly approximated by the following sinusoidal function although they gradually apart from the sinusoidal curve as the steady flow parameter w st,i /w osmax,i increases over w st,i /w osmax,i.. (r,z,t) = (r,z)cos π[t /tcy φ '(r)] + ave(r,z) (3) As explained before, the velocity variation is expressed by: w(r,t ) = wosmax, i(r ) cosπ [t / tcy φw' (r )] (4) It should be noted that the temperature oscillation phase Φ '(r) remained the same for a wide range of w st,i /w osmax,i. In another word, the phase difference Ф '(r)-ф W '(r) of the present oscillatory flow was the same as that of the sinusoidal oscillatory flow. Substituting Eqs. (3) and (4) into Eq. (7) we obtain: q ~ ( r,z ) = wos max' ( r ) ( r,z ) cosπ [t / tcy φw' ( r )] cosπ [t / tcy φ '( r )]d(t / tcy ) + wst( r )ave( r,z ) ( r,z ) = q ~ '( r,z ) + wst( r )ave( r,z ) ' ( r,z ) = q ~ ( r,z ) q ~ os + st( r,z ) (5) As Eq. (5) suggests, the total average heat flux q (r,z) transported by the oscillatory flow is expressed as the sum of q õs (r,z) by the oscillatory component and q st(r,z) by the steady component. he heat fluxes q õs (r,z) and q st(r,z) are given by: ( r,z ) q ~ os( r,z ) = q ~ '( r,z ) (6) ' ( r,z ) q ~ st (r,z) = wst(r) ave(r,z) (7) he velocity of the steady flow component w st (r) and ave (r,z) in the right term of Eq.(7) increase with the steady flow parameter w st,i /w osmax. he increase in ave (r,z) with the parameter w st,i /w osmax are illustrated in Figs. (4)~(7). hese account for the fact that the average heat flux q st(r,z) transported by the steady component increases with the steady component w st,i /w osmax,i. On the other hand, the average heat flux q õs (r,z) transported by the oscillatory component may increase or decrease with the steady flow parameter w st,i /w osmax depending on Wo, because as Figs. (6) and (7) illustrate (r,z) in Eq. (6) increases or decreases depending on Wo by adding the steady flow component. he total average heat flux q increases with w st,i /w osmax,i because the increase in q st(r,z) exceeds the decrease in q õs (r,z) in all the present parameter ranges Average heat flux in one cycle and in cross section of pipe Figure shows the ratio of the average heat flux q os ( z ) of the oscillatory component to the total average heat flux q( z ) of the oscillatory flow as well as the ratio of the average heat flux q st ( z ) of the steady component to the total one q ( z ). he average heat fluxes, q os ( z ), q st ( z ) and q ( z ) are the heat flux averaged in one cycle as well as averaged over the cross section of pipe. he average heat fluxes q os ( z ) and q st ( z ) are written as: qos(r,z) = dr r wosmax'(r) (r,z)cos π[t /tcy φw'(r)] (8) cos π[t /t φ '(r)]d(t /t ) cy qst( z ) = dr r wst(r ) ave(r,z )d(t / tcy ) (9) he average heat flux q st ( z ) of the steady component decreases in the axial direction, because the average temperature ave in Eq. (9) decreases in the axial direction. q os / q q st q q st q os.8.6 /.4. q os q st / q / q Fig. Heat fluxes transported by the steady and oscillatory conponents ( Wo = 6 ) cy w st,i / w osmax,i z

9 HERMAL ENERGY RANSPORAION BY AN OSCILLAORY FLOW While the average heat flux q os ( z ) of the oscillatory component increases in the axial direction, resulting from the increase in the amplitude of temperature variation along the axial length. Figure 3 illustrates the enhancement of heat transportation by adding the steady flow to the sinusoidal oscillatory flow. For example, the oscillatory flow of Wo= with w st,i /w osmax,i =. can transport about times as much as the heat of the sinusoidal oscillatory flow. he straight lines in Fig.3 show the total sum of the heat flux obtained by simply adding the heat flux of the sinusoidal oscillatory flow to the one of the steady flow, Hagen-Poiseuille flow with mean velocity of w st,i. he present numerically calculated results are about 3% less than those of the simply summed up total heat fluxes. his, however, implies that the heat transported by the oscillatory flow including steady flow component is roughly estimated by adding heat transported by the sinusoidal oscillatory flow to the one by steady flow. Figure 4 shows the relationship between the heat flux q * [W/m ] of the oscillatory flow and the average velocity w * st,i [m/s] of the steady flow component. he straight line indicates the heat flux of the steady flow component. In the low value range of w * st,i [m/s], heat flux transported by the oscillatory flow is much higher than the one by the steady flow component. he former, however, approaches to the latter as w * st,i [m/s] increases. 4.4 Work Required by the Oscillatory Flow Work rate required by the oscillatory flow is calculated by: W ~ = π d( t / tcy ) P( t ) r w( r )dr (), where P is the pressure difference between the inlet and exit of the adiabatic section. he reference pressure for the dimensionless pressure was chosen as a* ω* ρ*ν*. he work rate of the oscillatory flow was larger that of the sinusoidal oscillatory flow only by 3%. his is because work of the oscillatory flow is mainly consumed by the acceleration and deceleration of the flow and the one consumed by the flow friction is negligible. 4.5 Heat ransportation Efficiency Heat transportation efficiency is defined as: Q ~ W ~ η = * / * = π( a*) q * / W ~ * (), where Q * [W] is the average heat transfer rate in one cycle through the cross section of pipe. he heat transportation efficiency is shown in Fig.3. As the work rate of the oscillatory flow remained almost the same as that of the sinusoidal oscillatory flow, the heat transportation efficiency increased in the same manner as q, that is, it increased with increasing Wo as well as the steady flow parameter w st,i/ w osmax,i. he heat transportation efficiency of the present oscillatory flow was in the range of 9.7 ~ q* [W/m ] q / q', η / η' Wo q st st + q w st,i / w osmax,i Fig.3 Heat fux and heat transportation efficiency of the present oscillatory flow 3.E+5.E+5.E+5 Wo= 6 Wo=6 Wo= Wo 6 5 q st * 5 Wo=5.E w st,i * [m/s] Fig.4 Heat flux of the present oscillatory flow 9

10 5 Conclusion In the present analytical study we investigated the thermal energy transportation performance of the oscillatory flow that was established by superimposing a very slow steady flow to a sinusoidal reciprocating flow. he velocity and temperature fields, transportation rate of heat, work rate and transportation efficiency of heat were numerically analyzed. he results obtained are summarized in the followings. () he transportation rate of heat increased largely by superimposing a steady flow on a sinusoidal oscillatory flow. () On the other hand, work required by the oscillatory flow increased only a little. (3) he transportation rate of heat by the oscillatory flow was less than the arithmetic sum of those by the steady flow and by the sinusoidal reciprocating flow. (4) In the upstream heat was transported mainly by the steady flow component and in the downstream mainly by the oscillatory flow component. (5) he phase difference between the velocity and temperature variations of the oscillatory flow remained the same as that of the sinusoidal oscillatory flow. [4] Nishio S and Zhang W.M. Oscillation-controlled heat transport tube (nd report, Optimum condition), rans. JSME, Vol.6, No.57, pp , 994. [5] Nishio S, Shi X.H and Funatsu K. Study on oscillation-controlled heat transport tube (3nd report, Inverted oscillation-phase heat transport tube), rans. JSME, Vol.6, No.578, pp , 994. [6] Ohno Y, anaka G and Hishida M. Enhanced heat transfer during oscillatory flow in annular channel, rans. JSME, Vol.7, No.698, pp 6-69, 4. [7] anaka G, Sakai E and Hishida M. Flow distribution in a right angle branch during oscillatory flow, rans. JSME, Vol.68, No.67, pp 9-5,. [8] FLUEN 6. Users Guide, Fluent Inc., 3. [9] Hino M, et al. Experiments on transition to turbulence in an oscillatory tube flow, J. Fluids. Mech., Vol.56, pp 9-3, 985. Acknowledgement his research was supported in part by the Iwatani Naoji foundation. References [] Kurzweg U.H and Zhao L. Heat transfer by highfrequency oscillations, Phys. Fluids, vol.7, No., pp 64-67, 984. [] Kaviany M. Performance of a heat exchanger based on enhanced heat diffusion in fluids by oscillation, rans. ASME, J. Heat rans., Vol., pp 49-55, 99. [3] Nishio S, Honma M and Zhang W.M. Oscillationcontrolled heat transport tube (st report, Effect of liquid properties), rans. JSME, Vol.6, No.569, pp 33-39, 994.