Y. L. He and W. Q. Tao Xi an Jiaotong University, Xi an, China. T. S. Zhao Hong Kong University of Science and Technology, Kowloon, Hong Kong, China

Numerical Heat Transfer, Part A, 44: 399 431, 2003 Copyright # Taylor & Francis Inc. ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/10407780390206625 STEADY NATURAL CONVECTION IN A TILTED LONG CYLINDRICAL ENVELOPE WITH LATERAL ADIABATIC SURFACE, PART 2: HEAT TRANSFER RATE, FLOW PATTERNS AND TEMPERATURE DISTRIBUTIONS Y. L. He and W. Q. Tao Xi an Jiaotong University, Xi an, China T. S. Zhao Hong Kong University of Science and Technology, Kowloon, Hong Kong, China Z. Q. Chen Xi an Jiaotong University, Xi an, China Numerical results for the steady natural convection heat transfer in a tilted cylindrical envelope with constant but different end temperatures (300 versus 80 K) and an adiabatic lateral wall are presented. This envelope is supposed to be a simplified model for the pulse tube in a pulse tube cryocooler when the pulse tube is positioned at different orientations. Variation of the average heat transfer rate with the inclination angle, and details of velocity and temperature distributions, are provided for four typical inclination angles. The major numerical results are as follows. First, the flow pattern caused by the natural convection in the pulse tube is very complicated, characterized by the global circulation between the hot and cold end accompanied by several local recirculations, leading to a multicell structure of flow pattern. Second, from the temperature distribution, especially the section average temperature distribution, the heat transport mechanism of natural convection in the pulse tube may be classified into two classes, conduction-dominated and convection-dominated. The maximum heat transfer rate occurs at inclination angle of 120. Comparisons with available test results are made, and the agreement between the predicted and the test data is reasonably good. Third, even for the conduction-dominated case, there exists very weak flow within the pulse tube. Finally, to reduce the loss of cooling capacity a pulse tube in operation should be positioned in the range of inclination angle from 0 to 80. Received 15 July 2002; accepted 25 November 2002. This work was supported by the National Key Project of Fundamental R&D of China (Grant 2000026303) and the National Natural Science Foundation of China (Grant 50276046). The support of these foundations is greatly acknowledged. Thanks also go to Mr. Ding Wen-Jing for executing some additional computations. Address correspondence to Y. L. He, School of Energy and Power Engineering, Xi an Jiaotong University, Xi an, 710049 China. E-mail: yalinghe@mail.xjtu.edu.cn 399

400 Y. L. HE ET AL. NOMENCLATURE D diameter g gravitational acceleration L length of the pulse tube Q heat transfer rate r radius Ra Rayleigh number, ¼ Pr gbðth TcÞl3 : if n 2 l ¼ L, Ra¼ Ra L, if l ¼ R, Ra¼ Ra D T temperature u, v, w velocity components z axial coordinate y inclination angle j angle coordinate Subscripts c cold cond conduction h hot m mean 1. INTRODUCTION In recent years, it has been recognized that secondary flows exist in the pulse tube of a pulse-tube cryocooler [1, 2]. Such secondary flows result from the combined effect of forced convection of the pulsating streaming and the natural convection caused by the large temperature difference between the hot and cold ends. As a first approximation, the forced convection of the pulsating streaming and the natural convection due to temperature difference may be decoupled [3], and each kind of flow can be investigated in detail to reveal their inherent characteristics. The secondary flow and the mass streaming in a pulse tube can be a major heat loss mechanism. It carries heat from the hot heat exchanger (i.e, hot end) to the cold heat exchanger (cold end), thereby reducing the cooling power of a pulse-tube cryocooler. In a companion article [4], a three-dimensional physical and mathematical model with variable thermophysical properties, and numerical methods, were proposed. The geometry and the cylindrical coordinates are presented in Figure 2 of [4]. The diameter of the tube is 27.8 mm and its ratio of length to diameter is 9. This model serves to simulate the natural convection in a pulse tube of a pulse-tube cryocooler. A number of special features inherent to the problem studied were revealed, and a series of convergence criteria were proposed. In this article, the numerical results conducted for 19 inclination angles are presented in detail. First, the effect of the inclination angle on the average heat transfer rate is provided. Temperature contours and velocity vectors in longitudinal and cross sections are presented for four typical inclination angles, y ¼ 0, 90, 120, and 180, with focus being put on y ¼ 90 and 120. Flow field investigations show that the flow pattern in the pulse tube is very complicated, characterized by a multicell structure, and even for the inclination with cold end down, such complicated structure still exists. Then, the section average fluid temperature distribution along the axis is summarized to analyze the heat transfer mechanism of the 19 inclinations. Comparisons of the predicted results with available test data are conducted. According to the temperature distribution patterns, two kinds of heat transfer mechanisms are delineated, i.e., diffusion-dominated and convectiondominated. Finally, some conclusions are drawn. It is worth noting that the major purpose of this study is to reveal qualitatively the physical characteristics of the natural convection in the long cylindrical envelope with different inclined angles

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 401 and to predict the heat transfer rate, which may be compared with the available data. The major nondimensional quantities are Rayleigh number and Nusselt number. As for the flow and temperature fields, only their patterns are important; the absolute values are of less importance. Therefore all the results of velocities and temperatures will be presented in a dimensional way and no attempt is made to make them dimensionless. Although the specific values of velocity and temperature predicted in this article apply only to the case studied, the physical mechanism revealed can also be applied to the whole category of problems. For natural convection in enclosures, the numerical results are usually presented in such a way [5]. 2. AVERAGE HEAT TRANSFER RATE IN THE ENVELOPE UNDER DIFFERENT INCLINATIONS Computational results for the average heat transfer rate in the envelope are summarized in Table 1 for every 10 of inclination angle. The corresponding Nusselt numbers based on the cylinder length are also listed. In the table, Q cond represents the conduction heat transfer rate as if the heat transfer process in the envelope were pure heat conduction from the hot end to the cold end with the same end temperature difference; Ra L and Ra D are the Rayleigh numbers with envelope length and diameter as their characteristic length, respectively. All the thermophysical properties in the calculation of Q cond,ra L,Ra D, and Nu ml are volume-average ones. For the configuration studied, the geometric size and the temperature difference are all fixed, thus the differences of the first three quantities for different inclinations are Table 1 Computational results for y ¼ 0 180 y (deg) Ra L 610 77 Ra D 610 77 GM (kg=s) Q m (W) Q cond (W) Nu ml 0 1.559 2.139 5.890610 78 2.964610 72 2.943610 72 1.007 10 1.550 2.126 4.640610 77 3.683610 72 2.945610 72 1.251 20 1.545 2.118 1.976610 76 4.596610 72 2.946610 72 1.560 30 1.542 2.115 2.060610 76 6.188610 72 2.947610 72 2.100 40 1.549 2.125 2.936610 76 8.864610 72 2.945610 72 3.101 50 1.562 2.142 3.746610 76 1.311610 71 2.943610 72 4.456 60 1.587 2.176 4.337610 76 1.889610 71 2.938610 72 6.431 70 1.632 2.239 4.940610 76 3.067610 71 2.928610 72 10.47 80 1.694 2.324 5.687610 76 7.400610 71 2.912610 72 25.41 90 1.322 1.813 1.804610 75 4.563 2.940610 72 160.1 100 1.583 2.172 3.709610 75 6.965 2.850610 72 244.3 110 1.512 2.075 4.799610 75 7.057 2.797610 72 252.5 120 1.914 2.626 5.404610 75 7.221 2.782610 72 259.6 130 1.957 2.685 4.775610 75 6.613 2.770610 72 238.7 140 1.880 2.579 4.531610 75 6.168 2.768610 72 222.9 150 2.139 2.934 6.132610 75 5.834 2.728610 72 213.8 160 1.905 2.613 4.714610 75 5.423 2.774610 72 195.5 170 1.824 2.502 5.165610 75 5.047 2.788610 72 181.6 180 1.596 2.189 8.739610 75 4.388 2.838610 72 154.7

402 Y. L. HE ET AL. Figure 1. Variation of heat transfer rate with inclination angle. from the thermophysical properties, which come from the difference in temperature distribution. To show the effect of the inclination on the heat transfer rate more clearly, the variation of Q m versus y is presented in Figure 1. From Table 1 and Figure 1, the following features may be noted. (1) In the range of y ¼ 0 80, the heat transfer rate in the envelope is of the order of 10 72 to 10 71 W, while in the range of y ¼ 90 180 the heat transfer rate increases significantly, ranging from 4 to 8 W. In terms of dimensionless parameter, the Nusselt number based on the length L varies from 1(y ¼ 0, pure diffusion case) to 260 (y ¼ 120 ). Actually, according to the definitions, Nu L is the ratio of Q m over Q cond. Thus the value of the Nusselt number is actually an indication of how many times the heat transfer rate of natural convection is to that of pure conduction. The different heat transfer mechanisms will be discussed later. (2) The maximum heat transfer rate occurs at y ¼ 120, not at y ¼ 180, at which the vertical distance between the hot and cold ends is the largest. This situation may be caused by the following fact: at an appropriate inclination angle, the inclined surface provides a favorable path for fluid to move up. Detailed flow pattern inspection will be presented in subsequent paragraphs. Comparison with available experimental data is now conducted. For this purpose the picture presented in [3] is cited in Figure 2, where the black=white squares are the estimated convective heat loss of the pulse tube. As can be seen from Figures 1 and 2, qualitatively, our numerical results agree with [3] very well. For example, both our numerical prediction and the experimental result in [3] show that at y ¼ 120, natural-convection heat transfer rate is the maximum. However, quantitatively, our numerical results are about 35% lower than the estimated data of [3]. This deviation is estimated as follows. The test conditions of [3] are: inner diameter of the pulse tube ¼ 13.4 mm, length of pulse tube ¼ 250 mm (ratio of L=D ¼ 18.7). T h ¼ 300 K, 52.5 K. The maximum heat transfer rate equals 5.8 W.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 403 Figure 2. Estimated excess heat transfer rate as a function of inclination angle, from [3]. Neglecting the effect of L=D, for our case we should have a maximum heat transfer rate of Q expected max ¼ 5:8 27:7 2 300 80 ¼ 22:03 W 13:4 300 52:5 Thus our numerical result is 65.6% (¼7.2262=22.03, the factor 2 is multiplied here, since 7.22 W is the heat transfer rate of the half-cylinder) of the expected value. The deviation between the present numerical result and the expected one from [3] may be caused by following facts [6]. First, the present computation was conducted under the assumption of perfect adiabatic lateral surface of the pulse tube. Actually, this was not the case, and the heat loss through wall conduction of the pulse tube increases the total heat transfer rate, which was estimated in the name of convection in [3]. This is the major factor to account for the aforementioned deviation. Second, in our numerical model [4], the source terms in the momentum equations were simplified in that the thermophysical properties were assumed to be constant. This simplification may lead to some computation error. Finally, the data presented in [3] were estimated from cooling capacity of the pulse-tube refrigerator and the no-load temperature, rather than from direct measurement. This kind of estimation may include some imperfections. The white triangles in Figure 2 were obtained from the available heat transfer correlations by the authors of [3]. They differ from the estimated values significantly. As indicated in [4], the available heat transfer correlations for natural convection in an envelope are usually derived under the Boussinesq assumption, which certainly cannot be applied for the case studied. Further investigation is now underway in the authors group to reveal the reasons for the above deviation.

404 Y. L. HE ET AL. 3. VELOCITY AND TEMPERATURE DISTRIBUTIONS FOR u ¼ 0 (VERTICAL POSITION WITH COLD END DOWN) 3.1. Temperature Distribution In Figure 3a the isothermals in the section through j ¼ 0 and j ¼ 180, i.e., the longitudinal section, are presented. To have a better presentation the two coordinates (Z and r) in Figure 3 are not to scale, and the pulse tube is positioned horizontally to Figure 3. Temperature contour in longitudinal section for y ¼ 0 :(a) temperature contour; (b) temperature field; (c) axial distribution of section average temperature.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 405 save space. In Figure 3b temperature field is presented by black and white color. In the figure, different shades of black and white represent different temperatures. The isothermal are mainly a group of parallel lines with equal distance, and different shade regions are basically a series of rectangles parallel to the two ends, indicating the character of the conduction mechanism. This kind of feature may be clearly observed from the axial distribution of section average fluid temperature shown in Figure 3c. 3.2. Velocity Distribution Even though y ¼ 0 is a typical diffusion-dominated situation, fluid flow still exists, though very week. In Figure 4a velocity vectors in the longitudinal section are shown, and two large circulation streams (upward to z ¼ 0 and downward to z ¼ 0.25 m) can be observed. The upward circulation spans a space from z ¼ 0to about z ¼ 0.08 m, in which the fluid flows upward along the longitudinal section, turns around when approaching the hot end, flows downward along the circumferential surface, and then turns around again near the position z ¼ 0.08 m. To have a better look, the local velocity distributions in the r j cross section near the hot end and at z ¼ 0.0726 m are presented in Figures 4b and 4c, where the fluid flow comes from the center to the circumference (near z ¼ 0) and from the circumference to the center of the cylinder (z ¼ 0.0726 m). The same situation exists in the downward circulation stream. Such a special flow pattern results in a relative large axial velocity in the center of the cylinder compared to the circumferential region, as can be seen from Figure 4a. As far as the absolute value of the velocity components is concerned, it is very small, usually of the order of 10 75 to 10 77 m=s. Details of the cross-section velocity and temperature distributions for y ¼ 0 can be found in [6, 7]. 4. VELOCITY AND TEMPERATURE DISTRIBUTIONS FOR u ¼ 90 (HORIZONTAL POSITION) 4.1. Velocity Distribution Velocity distributions in the entire longitudinal section and three local regions are presented in Figures 5 and 6, respectively. In Figure 7 the cross-section velocity fields are illustrated. It can be seen that the fluid flow in the horizontal envelope constitutes a large circulation: in the upper half of the envelope the heated stream goes from the hot end to the cold end, while in the lower half the cooled stream moves from the cold end to the hot one. The two streams, hot and cold, meet each other near the center of the envelope, resulting in zero axial velocity. Figure 7 shows that at each cross section there is at least one flow circulation and the position of this circulation differs from section to section. This implies that in the cylindrical envelope, flow patterns are of multicell type in nature; i.e., apart from the global circulation that occurs in the entire region, there are a number of small vortices accompanying the big one, leading to a complicated flow structure. 4.2. Temperature Distribution Fluid temperature contours in the longitudinal section are presented in Figure 8a. It can be seen that near the two ends the isothermals are very crowded,

406 Y. L. HE ET AL. Figure 4. Velocity vector in longitudinal section for y ¼ 0 and two r y cross sections: (a) entire section; (b) velocity field in r y cross section near z ¼ 0 (hot end); (c) velocity field in r y cross section at z ¼ 0.0726 m.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 407 Figure 4. Continued. indicating a steep temperature gradient. In Figures 8b 8d, local magnified pictures for the isothermals near the hot end, in the center part of the envelope, and near the cold end are presented, where the crowdedness of the isothermal distribution near the two ends can be clearly observed. It also can be observed that in the major part of the section, isothermal are basically parallel lines, indicating diffusion-dominated mechanism. Fluid temperature distributions at nine cross sections are shown in Figure 5. Velocity vector in longitudinal section for y ¼ 90.

408 Y. L. HE ET AL. Figure 6. Details of flow pattern in longitudinal section for y ¼ 90 :(a) velocity field near hot end; (b) velocity field near center position; (c) velocity field near cold end. Figure 9. Once again we see that at each cross section, the isothermals are basically a series of horizontal parallel lines with temperature decreasing downward. This lamination of temperature is caused by the diffusion-dominated transport mechanism. Section average fluid temperature distributions along the axis are provided in Figure 10. Apart from the two ends, where very steep temperature gradients can be found, in the major part of the pulse tube, the average fluid temperature decreases almost linearly from the hot end to the cold end. All the above distribution characters show that for inclination angle y ¼ 90, the diffusion process plays an important role in the transport of the heat. More detail discussion of the heat transfer mechanism will be given later.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 409 Figure 7. Cross-section velocity distribution (y ¼ 90 ).

410 Y. L. HE ET AL. Figure 8. Temperature contours in longitudinal section (y ¼ 90 ): (a) entire section; (b) local isothermals in the vicinity of hot end; (c) local isothermals in the center part; (d) local isothermals in the vicinity of the cold end. 4.3. Comparison with Available Test Data Kimura and Bejan [8] performed an experimental study of the natural convection in a horizontal tube with the two ends at different but constant temperatures.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 411 Figure 8. Continued The Rayleigh number based on the tube diameter ranged from 10 8 to 10 9, with the end temperature difference around 70 C. Kimura and Bejan found that at each cross section normal to the axial direction, the temperature depth variation along the vertical diameter is almost linear. In addition, in the cross section through the z r plane, the measured velocity distributions show that the flow consists of two thin jets, one flowing toward the cold end along the top surface and the other flowing in the opposite direction along the bottom. Although the test temperature difference of [8] is much less than that of the present study, the Rayleigh number range of the two studies is quite close, hence some comparisons for the velocity and temperature profiles are made here. In Figure 11, the local temperature distributions along the vertical diameter at five cross sections (z ¼ 0.011, 0.059, 0.119, 0.18, and 0.239 m) are provided. It can seen that the temperature variation in the core region of the tube is almost linear in the vertical direction, and it is practically independent on the axial position. This variation pattern is very much similar to that found in the measurement results of [8]. The axial velocity distributions along the vertical diameters of the five cross sections are presented in Figure 12. It is quite clear that in the upper part of the tube there is quite strong streaming coming from the hot end to the cold end, while in the lower part of the tube an opposite streaming can be observed. Such variation of the axial velocity is similar to that observed by the measurement results of [8]. However, there are some qualitative discrepancies between the present velocity distribution and the measured one in [8]. First, since the fluid temperature in the upper portion is much higher than that of the lower portion (for this study the maximum temperature difference is 220 K, versus 70 K in [8]), the viscosity and density variation characteristics with temperature lead to a larger portion of the cross section where fluid goes from the hot to the cold end, and this phenomenon is especially significant near the cold end. Second, as seen in Figure 12c, the axial velocity distribution of hot-to-cold stream has two peaks, one in the upper part of the envelope and the other located at the centerline. This kind of fluid motion was not found in [8]. By referring to Figure 14b in the companion paper [4], where the velocity distribution for 70 C of the end temperature difference is provided, we can see that the velocity curve varies monotonically in the core region and does not show such a second peak. The authors consideration is as follows. In [8] water was used as

412 Y. L. HE ET AL. Figure 9. Temperature contours at nine cross sections for y ¼ 90.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 413 Figure 10. Axial distribution of section average temperature for y ¼ 90. the heat transfer medium and the absolute value of the axial velocity was appreciably lower than the one of this study, where gas was used at the same order of Rayleigh number. When the fluid impinges on the two end walls, some sort of injection effect may be caused, leading to the existence of a lower-pressure region outside the injection region. And the existence of the second peak in the hot-to-cold stream velocity distribution may be partially attributed to this lower-pressure region. By careful inspection of the flow pattern near the hot end, we can also observe such a second peak in the velocity distribution of the cold-to-hot steam, though it is not so significant as the one near the cold end. Thus it is believed that such differences are caused by the severe variation of the thermal physical properties associated with the present study. The good qualitative agreement in the total heat transfer variation curve with the inclination angle and the local temperature and velocity variation patterns with distance give us additional confidence in the reliability of the present numerical treatment. 5. VELOCITY AND TEMPERATURE DISTRIBUTIONS FOR u ¼ 120 (TILTED POSITION WITH HOT END DOWN) 5.1. Temperature Distribution In the longitudinal section the temperature contour (Figure 13a) seems abnormal in that near the cold end a contour with a temperature as high as 203 K occurs, while near the hot end a low-temperature contour of 135 K can be found. This complicated temperature distribution is caused by the convective flow of the fluid. If the temperature distributions in the local regions near the hot and cold ends are magnified, a very steep change of fluid temperature within 1 2 mm can be found (see Figures 13b and 13c), and within such thin a layer the fluid temperature may rise

414 Y. L. HE ET AL. Figure 11. Local temperature distributions along the vertical diameter at five cross sections for y ¼ 90 : (a) z ¼ 0.011 m; (b) z ¼ 0.059 m; (c) z ¼ 0.119 m; (d) z ¼ 0.18 m; (e) z ¼ 0.239 m. from 80 to 203 K or fall from 300 to 135 K. Such steep variation of gas temperature near the two ends can also be observed in the section average gas temperature curve presented in Figure 20m. As can be seen there, a very steep temperature gradient exists near the two ends such that near the cold end the section average fluid temperature is higher than that near the hot end. This is regarded as a major indication of convection-dominated heat transfer mechanism. More discussion will be provided later in this article.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 415 Figure 11. Continued. The temperature distributions at nine cross sections are presented in Figure 14. The major characteristics of these contours are as follows. First, for most of the cross sections the fluid temperature in the upper part of each section is higher than in the lower part, reflecting one of the basic features of natural convection in enclosures. Second, the isothermals in each section are of very different shape, from approximately parallel lines (Figure 14 f ) to S-type curves (Figures 14g 14i). The S-type isothermal is a very typical contour line in the convection-dominated region in natural convection in a horizontal annulus [9]. Third, in the center region, the

416 Y. L. HE ET AL. Figure 11. Continued. isothermals seem quite crowded, indicating a steep temperature gradient. Actually, this is a common feature of fluid temperature distribution in the envelope and is caused mainly by the global fluid circulation in the envelope. As can be seen from Figure 5 (for y ¼ 90 ) and Figure 15 (for y ¼ 120 ), the global flow pattern is that the Figure 12. Axial velocity distribution along the vertical diameter of the five cross sections for y ¼ 90 : (a) z ¼ 0.011 m; (b) z ¼ 0.059 m; (c) z ¼ 0.119 m; (d) z ¼ 0.18 m; (e) z ¼ 0.239 m.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 417 Figure 12. Continued. hot stream goes from the hot end to the cold end in the upper part of the envelope and the cold stream goes from the cold end to the hot one in the lower part of the envelope. The two streams meet near the center of the envelope, causing a sharp change in fluid temperature. The distribution is further complicated by many local fluid vortices, as can be seen from Figure 16, presented later. Therefore the complicated isothermals in each section simply imply that for this inclination a very complicated multicell structure exists.

418 Y. L. HE ET AL. Figure 12. Continued. 5.2. Velocity Distribution Velocity vectors in the longitudinal section for y ¼ 120 are shown in Figure 15, where the pulse tube is positioned as it should be. It can be observed that near the hot end along the bottom line of the rectangle, there is an appreciable flow stream moving upward. The inclined wall creates a favorable condition for fluid to move up, and under a certain inclination this favorable condition may dominate over other factors that prevent fluid flow, leading to the strongest natural convection in the envelope. Both our numerical results and experimental estimation in [3] found that

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 419 Figure 13. Temperature distribution in the longitudinal section for y ¼ 120 :(a) temperature contour in the entire longitudinal section; (b) local temperature contour near cold end; (c) local temperature contour near hot end.

420 Y. L. HE ET AL. Figure 14. Temperature contours of nine cross sections for y ¼ 120.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 421 Figure 15. Velocity vectors in longitudinal section for y ¼ 120. this inclination is around 120. Three magnified local velocity distribution are provided in Figure 16. Two features may be noted. First, in the middle part of the longitudinal section the velocity vector in both the upper half and the lower half of the section are very small, indicating that stream turns and becomes part of the local recirculation flow. This consideration can be further confirmed by carefully inspecting Figure 17. Second, near the cold end here are two circulations in the upper left corner and lower left corner, with the former being much stronger than the latter. These are truly three-dimensional vortices, as can be confirmed by the cross-section velocity distribution in Figure 17i. In Figure 17, velocity vectors for nine cross sections are presented. Recirculations with different strength can be found at each cross section, especially near the cold end. Figures 17e and 17f present the velocity fields in the middle part of the section. Careful inspection of Figure 17e can show that in the region of Y ¼ 0to 0.013, fluid leaves the Y axis and moves to the inner part of the section, while in the region of Y ¼ 0to 70.13, fluid leaves the Y axis and moves to the circumferential periphery. Such a cross-section velocity field is consistent with the velocity field in the middle part of the longitudinal section indicated above. Figure 17i presents two strong vortices which are the counterpart of the two longitudinal circulations shown in Figure 16c. Multicell structure flow pattern in the pulse tube can be well demonstrated from such cross-section velocity distributions.

422 Y. L. HE ET AL. Figure 16. Details of flow pattern in longitudinal section for y ¼ 120 :(a) near hot end; (b) near center point; (c) near cold end. 6. VELOCITY AND TEMPERATURE DISTRIBUTIONS FOR u ¼ 180 (VERTICAL POSITION WITH HOT END DOWN) In Figure 18, velocity vectors and temperature contours in the longitudinal section for vertical position with the hot end down are provided, with the pulse tube positioned horizontally to save the space. Appreciable flow circulation between the hot and cold ends can be found, while the fluid temperature seems more or less uniform in the major part of the longitudinal section (This characteristic will be

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 423 Figure 16. Continued. further enhanced in the later discussion). However, in the vicinity of the hot and cold ends a very steep temperature gradient can be found, and Figure 19 gives such a picture, where the isothermals near the two ends of the pulse tube are presented in detail. From the above presentation for the velocity and temperature distributions in a pulse tube with different inclinations, it can be concluded that the natural convection in the pulse tube possesses a very complicated flow pattern, characterized by a global recirculation between the hot and cold ends accompanied by several local vortices, leading to a multicell flow structure. This multicell structure varies from inclination to inclination. In Table 1 a parameter symbolized by GM is presented, which is the average axial flow rate for each inclination. It can be found that, basically, the value of GM increases with the increase in heat transfer rate Q m. But the variation trends of GM and Q m are not always consistent. For y ¼ 150 and 180 the values of GM are higher than for y ¼ 120, at which the heat transfer rate is the highest. The reason may be as follows. The definition of GM is GM ¼ 1 N X N k¼1 Z rði; j; kþ abs½wði; j; kþšrdrdj O k ð1þ where N is the number of sections in the axial direction (77 in total). The positions of the 77 cross sections are fixed for all inclination angles. But the multicell structure defers from one inclination to other. Thus the fixed 77 sections may not include all the sections where the multicell circulation is strongest for some inclinations, leading to some inconsistency between the heat transfer rate and the value of GM.

424 Y. L. HE ET AL. Figure 17. Velocity vectors at nine cross sections for y ¼ 120.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 425 Figure 18. Velocity vectors and temperature contours in longitudinal section for y ¼ 180 :(a) velocity vectors; (b) temperature contours. In addition, when the tube position changes from y ¼ 0 to y ¼ 180, the heat transfer mechanism changes from diffusion-dominated to convection-dominated. In the following we will discuss this point more comprehensively. 7. HEAT TRANSFER MECHANISM In the pulse tube the cooling capacity loss is equivalent to the heat transfer from the hot end to the cold end. From the above presentation we may delineate two mechanisms for the transfer of heat from the hot end to the cold end; (1) diffusiondominated and (2) convection-dominated. These two mechanisms may be best illustrated from the section average fluid temperature distribution along the axis. To have a better look at these axial temperature distributions, numerical results for all 19 inclinations are collected in Figure 20. It can been that all 19 curves for the section average axial temperature distribution can be classified into two classes. The first class is a conduction class, from y ¼ 0 to y ¼ 80. The major characteristic of this type is the linear-type temperature decease from the hot end and to the cold end, except in the vicinity of the hot and cold ends, where steep gradients may exist. As presented in Table 1, the total heat transfer rate of this class is of the order of 10 72 10 71 W, increasing with the increase in inclination angle. The second class is a convection class, from y ¼ 90 to y ¼ 180. The key feature of this class is that there exists an adverse temperature gradient in the major part between the hot and cold

426 Y. L. HE ET AL. Figure 19. Local isothermal near two ends for y ¼ 180 :(a) local isothermals near hot end; (b) local isothermals in the middle; (c) local isothermals near cold end.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 427 Figure 20. Summary of section average axial temperature distributions. ends, i.e., the section average temperature is increasing from the hot end to the cold end. If there is no convective contribution, it would not be possible to transfer heat from the hot end to the cold end. It is convective streaming of the fluid that carries heat from the hot end to the cold end. In the convection-dominated region, the heat transfer rate varies from 4.39 to 7.22 W, which is about two orders larger than that resulting from pure heat conduction (2.72610 72 W 2.95610 72 W).

428 Y. L. HE ET AL. Figure 20. Continued. 8. CONCLUSIONS In this article, numerical simulation results for three-dimensional steady natural convection in a tilted pulse tube were presented. The tube has a diameter of 27.8 mm and a length-to-diameter ratio of 9. The hot and cold ends were kept at 300 and 80 K, respectively. The inclination angle was varied from 0 (vertical with cold

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 429 Figure 20. Continued. end down) to 180 (vertical with hot end down). Computations were conducted for every 10 of inclination angle. Detailed temperature and velocity distributions in the longitudinal section and nine cross sections were presented for four typical inclination angles. Comparisons with available test data were conducted, and the agreement is reasonably good. The results presented here, seemingly first in the literature, provide useful information for further understanding the complicated secondary

430 Y. L. HE ET AL. Figure 20. Continued. flow structure in a pulse tube of low frequency, for which both the pulsating forced flow and the natural convection should be taken into account. The major findings of this study may be summarized as follows. (1) The flow pattern caused by the natural convection in the pulse tube is very complicated, characterized by the global circulation between the hot and cold ends accompanied by several local recirculations, leading to a multicell structure of flow pattern. (2) From the temperature distribution, especially the section average temperature distribution, the heat transport mechanism of natural convection in the pulse tube may be classified into two classes, conduction-dominated (y ¼ 0 80 ) and convectiondominated (y ¼ 90 180 ). The maximum heat transfer rate occurs at y ¼ 120. (3) Even for the conduction-dominated case, there exists weak flow within the pulse tube and the intensity of the flow circulation increases with the inclination angle. (4) For the Rayleigh number predicted, the temperature distribution along a vertical diameter in each cross section is almost linear, and the axial velocity distribution along the vertical diameter shows two opposite steamings in the upper and lower portions of the tube, representing the major trends of the fluid motion in the tube. (5) Thus, to reduce the cooling capacity loss in practical operation, the pulse tube should not be positioned in the range of y ¼ 90 180. REFERENCES 1. E. S. Jeong, Secondary Flow in Basic Pulse Tube Refrigerators, Cryogenics, vol. 36, pp. 317 323, 1996. 2. J. M. Lee, P. Kittel, K. D. Timmergaus, and R. Radebaugh, Flow Patterns Intrinsic to the Pulse Tube Refrigerator, in Proc. Seventh Int. Crycooler Conf., Phillips Laboratory, Kirtland AFB, NM, 1993, pp. 125 139.

CONVECTION IN A CYLINDRICAL ENVELOPE, PART 2 431 3. G. Thummes, M. Schreiber, R. Landgraf, and C. Heiden, Convective Heat Losses in Pulse Tube Coolers: Effect of Pulse Tube Inclination, Cryocoolers, vol. 9, pp. 393 402, 1997. 4. Y. L. He, W. O. Tao, T. S. Zhao, and Z. O. Chen, Natural Convection in a Tilted Long Cylindrical Envelope with Lateral Adiabatic Surface, Part 1: Theoretical Modeling and Numerical Treatments, Numer. Heat Transfer A, 5. G. Barakos and E. Mitsoulis, Natural Convection Flow in a Square Cavity Revisited: Laminar and Turbulent Models with Wall Functions, Int. J. Numer. Meth. Fluids, vol. 18, pp. 695 719, 1994. 6. Y. L. He, Theoretical and Experimental Investigations on the Performance Improvements of Split-Stirling Refrigerator and Pulse Tube Cryocooler, Ph.D. thesis, school of Energy and Power Engineering, Xi an Jiaotong University, Xi an, China, 2002. 7. Y. L. He, W. Q. Tao, and Z. Q. Chen, Natural Convection in a Vertical Cylindrical Envelope with Adiabatic Lateral Wall, submitted. 8. S. Kimura and A. Bejan, Experimental Study of Natural Convection in a Horizontal Cylinder with Different End Temperatures, Int. J. Heat Mass Transfer, vol. 23, pp. 1117 1126, 1980. 9. M. Yang and W. Q. Tao, Numerical Study of Natural Convection Heat Transfer in a Cylindrical Envelope with Internal Concentric Slotted Hollow Cylinder, Numer. Heat Transfer A, vol. 22, pp. 289 306, 1992.