Effect of transverse magnetic field on magnetogasdynamic flow and heat transfer characteristics in insulated / conductive circular tubes

Zhao Qi-Jin; Mao Bao-Quan; Bai Xiang-Hua; Yang Yu-Ying; Chen Chun-Lin

doi:10.7498/aps.71.20220051

Abstract

The regulating effect of magnetic field on magnetogasdynamic flow and heat transfer characteristics in circular tubes has important applications in many fields, but there is still a lack of relevant basic research. Considering the conductivity of the tube wall and the insufficient development of turbulence, the physical model and mathematical model of magnetogasdynamic flow in a circular tube under a given transverse magnetic field are constructed, and the numerical algorithm is designed within a theoretical framework of the finite volume method. The effect of factors including Hartman number (Ha) and wall conductivity ratio (C) on the flow and heat transfer characteristics are obtained through analyzing the distributions of velocity, turbulent kinetic energy, and temperature. Furthermore, the regulation mechanism of the transverse magnetic field is discussed by analyzing the spatial distribution of induced current, electromagnetic force and Joule heat. The results show that the distribution of velocity and the distribution of turbulent kinetic energy in the circular tube under a given transverse magnetic field are both anisotropic. The turbulent kinetic energy near the Hartmann boundary layer is much lower than that near the Roberts boundary layer, and the anisotropic distribution of velocity and turbulent kinetic energy become more and more evident with the increase of Ha and the extension of the flow. The transverse magnetic field has a suppression effect on the heat transfer in the tube. For different values of C, the average Nusselt number ($ \overline {Nu} $) shows a first-decreasing-and-then-increasing trend with Ha increasing, that is, there is a “saturation effect” in heat transfer suppression. When the wall conductivity is small (C $\leqslant $ 0.67), the change of $ \overline {Nu} $ under the condition of conductive wall is basically consistent with that of an insulating wall. However, when C exceeds a certain value (C $\geqslant $ 66.67), the $ \overline {Nu} $ under the condition of small Ha increases in comparison with that of the insulating wall, while the $ \overline {Nu} $ decreases under the condition of large Ha . The change of flow characteristics in the circular tube results from the variation of electromagnetic force under the coupling of magnetic field and fluid, while the change of heat transfer characteristics originates from the coupling effect of the suppression of turbulence and the Joule heating. When Ha is small, the suppression effect of the magnetic field on turbulence is dominant, and the $ \overline {Nu} $ decreases with the increase of Ha. When Ha exceeds a certain value (Ha $\geqslant $ 222), the large accumulation of Joule heat in the circular tube enhances the heat transfer, resulting in the increase of the $ \overline {Nu} $ with the continuous increase of Ha.

Keywords:

Authors and contacts

Weapons and Control Department, Army Academy of Armored Forces, Beijing 100072, China

Corresponding author: Mao Bao-Quan, mbq123321@126.com

Funds: Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 52006241).

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References

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[42]	Bradshaw P 1974 Nature 249 135

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图 1 物理模型示意图

Figure 1. Schematic diagram of physical model.

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图 2 不同Ha下圆管yz截面上沿z = 0方向的流速数值解与文献[40]中实验数据的对比

Figure 2. Comparison between the numerical solution of flow velocities along z = 0 direction on the yz cross-section of the circular tube under different Ha and the experimental results in reference [40].

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图 3 Re = 21375, Ha = 320, C = 0.0457时, 圆管yz截面上沿z = 0方向的流速数值解与文献[22]中实验数据的对比

Figure 3. Comparison between the numerical solution of flow velocities along z = 0 direction on the yz cross-section of the circular tube and the experimental results in Ref. [22] under the condition of Re = 21375, Ha = 320, C = 0.0457.

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图 4 不同Re下圆管壁面处沿流动方向的努塞尔数数值解与Gnielinski经验解的对比

Figure 4. Comparison between the numerical solution of Nusselt number along the flow direction at the wall of the circular tube under different Re and the Gnielinski empirical solution.

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图 5 不同Ha及不同C条件下圆管x = 200 mm截面上的速度幅值分布云图

Figure 5. The contours of velocity amplitude distribution of the x = 200 mm cross-section of the circular tube under the conditions of different Ha and different C.

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图 6 圆管x = 200 mm和x = 300 mm截面上y = 0沿线和z = 0沿线的无量纲速度分布曲线:　(a) x = 200 mm截面, y = 0; (b) x = 200 mm截面, z = 0; (c) x = 300 mm, y = 0; (d) x = 300 mm截面, z = 0

Figure 6. The profiles of the dimensionless velocity distribution on the x = 200 mm and x = 300 mm cross-sections of the circular tube: (a) x = 200 mm cross-section, y = 0; (b) x = 200 mm cross-section, z = 0; (c) x = 300 mm cross-section, y = 0; (d) x = 300 mm cross-section, z = 0.

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图 7 不同Ha及不同C条件下圆管x = 200 mm截面上的湍流动能分布云图

Figure 7. The contours of turbulent kinetic energy distribution of the x = 200 mm cross-section of the circular tube under the conditions of different Ha and different C.

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图 8 圆管x = 200 mm和x = 300 mm截面上y = 0沿线和z = 0沿线上的湍流动能分布曲线:　(a) x = 200 mm截面, y = 0 z; (b) x = 200 mm截面, z = 0; (c) x = 300 mm, y = 0; (d) x = 300 mm截面, z = 0

Figure 8. The profiles of the turbulent kinetic energy distribution on the x = 200 mm and x = 300 mm cross-sections of the circular tube: (a) x = 200 mm cross-section, y = 0; (b) x = 200 mm cross-section, z = 0; (c) x = 300 mm cross-section, y = 0; (b) x = 300 mm cross-section, z = 0.

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图 9 不同Ha及不同C条件下圆管x = 300 mm截面上的温度分布云图

Figure 9. The contours of temperature distribution of the x = 300 mm cross-section of the circular tube under the conditions of different Ha and different C.

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图 10 不同Ha下圆管y = 0, z = r₀壁面和z = 0, y = r₀壁面处沿x方向的局部努塞尔数分布曲线:　(a) y = 0, z = r_0, (b) z = 0, y = r₀

Figure 10. The profiles of the local Nusselt number along the x direction at the wall of y = 0, z = r₀ and the wall of z = 0, y = r₀ of the circular tube under different Ha: (a) y = 0, z = r_0, (b) z = 0, y = r₀.

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图 11 不同C下圆管壁面处的平均努塞尔数$ \overline {Nu} $随Ha的变化

Figure 11. Variation of the average Nusselt number $ \overline {Nu} $at the wall of circular tube with Ha under different C.

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图 12 Ha为74时不同C下圆管x = 200 mm截面上的感应电流分布:　(a) C = 0; (b) C = 66.67; (c) C = 6667

Figure 12. Induced current distributions on the x = 200 mm cross-sections at different C when Ha is 74: (a) C = 0; (b) C = 66.67; (c) C = 6667.

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图 13 Ha为74时不同C下圆管x = 200 mm截面上的电磁力矢量:　(a) C = 0; (b) C = 66.67; (c) C = 6667

Figure 13. Electromagnetic force vectors on the x = 200 mm cross-sections at different C when Ha is 74: (a) C = 0; (b) C = 66.67; (c) C = 6667.

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图 14 Ha为74时不同C下圆管y = 0截面和z = 0截面上的焦耳热分布:　(a) C = 0, y = 0截面; (b) C = 0, z = 0截面; (c) C = 66.67, y = 0截面; (d) C = 66.67, z = 0截面; (e) C = 6667, y = 0截面; (f) C = 6667, z = 0截面

Figure 14. Joule heat distributions on the y = 0 mm cross-sections and the z = 0 mm cross-sections at different C when Ha is 74: (a) C = 0, y = 0 mm cross-section; (b) C = 0, z = 0 mm cross-section; (c) C = 66.67, y = 0 mm cross-section; (d) C = 66.67, z = 0 mm cross-section; (e) C = 6667, y = 0 mm cross-section; (f) C = 6667, z = 0 mm cross-section.

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表 1 不同网格尺寸设置及与之对应的努塞尔数计算结果和误差

Table 1. Different mesh size settings and the corresponding Nusselt number calculation results and errors.

网格编号	网格单元总数	壁面平均努塞尔数 $ \overline {Nu} $	努塞尔数误差 $\varepsilon {}_{\overline {Nu} }{\text{ = } }\left\| {1 - \overline {Nu} ({M_i})/\overline {Nu} ({M_6})} \right\|$
M1	1021557	45.4185	4.65%
M2	1740325	46.3474	2.70%
M3	2883249	46.7952	1.76%
M4	4202281	47.4477	0.39%
M5	5613165	47.5001	0.28%
M6	7484076	47.6335	—

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map

[1]	Bedick C R, Woodside C R, Baylor R, Paul-Irudayaraj M 2020 Combust. Flame 213 140 Google Scholar
[2]	Zhao K, Lu Y, Li F, Hu J, Ming M, Wang X, Li S 2020 Acta Astronaut. 171 257 Google Scholar
[3]	Kim K S, Lee S H, Yun Y, Kwon S 2020 Acta Astronaut. 173 31 Google Scholar
[4]	Veefkind A 2004 IEEE T. Plasma Sci. 32 2197 Google Scholar
[5]	Wang Y, Ma T, Pei D, Chen C, Zhang D, Li C 2020 IEEE T. Plasma Sci. 48 1008 Google Scholar
[6]	李程, 毛保全, 白向华, 李晓刚 2018 兵工学报 39 851 Google Scholar Li C, Mao B Q, Bai X H, Li X G 2018 Acta Armam. 39 851 Google Scholar
[7]	成玉国, 夏广庆 2017 66 075204 Google Scholar Cheng Y G, Xia Y G 2017 Acta Phys. Sin. 66 075204 Google Scholar
[8]	Hedlund B, Houpt A, Gordeyev S, Leonov S 2018 AIAA J. 56 2699 Google Scholar
[9]	丁明松, 傅杨奥骁, 高铁锁, 董维中, 江涛, 刘庆宗 2020 69 214703 Google Scholar Ding M S, Fu Y A X, Gao T S, Dong W Z, Jiang T, Liu Q Z 2020 Acta Phys. Sin. 69 214703 Google Scholar
[10]	Mistrangelo C, Bühler L, Smolentsev S, Klüber V, Maione I, Aubert J 2021 Fusion Eng. Des. 173 112795 Google Scholar
[11]	Maurya A, Kumar R, Jha P K 2020 J. Manuf. Process. 60 596 Google Scholar
[12]	Yang R J, Hou H H, Wang Y N, Fu L M 2016 Sensor. Actuat. B-Chem. 224 1 Google Scholar
[13]	Moitoi A J, Shaw S 2022 Microvasc. Res. 139 104262 Google Scholar
[14]	Erdem M, Varol Y 2020 J. Therm. Anal. Calorim. 139 3897 Google Scholar
[15]	Krasnov D, Zikanov O, Boeck T 2012 J. Fluid Mech. 704 421 Google Scholar
[16]	Zikanov O, Krasnov D, Boeck T, Sukoriansky S 2019 J. Fluid Mech. 867 661 Google Scholar
[17]	Chatterjee D, Gupta S K J 2016 Appl. Fluid Mech. 9 2167 Google Scholar
[18]	Belyaev I, Sardov P, Melnikov I, Frick P 2021 Int. J. Therm. Sci. 161 106773 Google Scholar
[19]	Chaudhary R, Vanka S P, Thomas B G 2010 Phys. Fluids 22 075102 Google Scholar
[20]	Hunt J C R 1965 J. Fluid Mech. 21 577 Google Scholar
[21]	Tao Z, Ni M J 2015 Sci. China Phys. Mech. 58 024701
[22]	Zhang X, Pan C, Xu Z 2017 Fusion Eng. Des. 125 647 Google Scholar
[23]	Artemov V I, Makarov M V, Minko K B, Minko K B, Yankov G G 2020 Int. J. Heat Mass Tran. 146 118822 Google Scholar
[24]	Singh R J, Gohil T B 2019 Comput. Fluids 179 476 Google Scholar
[25]	Yarahmadi M, Goudarzi H M, Shafii M B 2015 Exp. Therm. Fluid Sci. 68 601 Google Scholar
[26]	Sha L, Ju Y, Zhang H, Wang J 2017 Appl. Therm. Eng 113 566 Google Scholar
[27]	Abadeh A, Sardarabadi M, Abedi M, Pourramezan M, Passandideh-Fard M, Maghrebi M 2020 J. Mol. Liq. 299 112206 Google Scholar
[28]	Shahsavar A, Saghafian M, Salimpour M R, Shafii M B 2016 Exp Therm. Fluid Sci. 76 1 Google Scholar
[29]	Afrand M, Karimipour A, Nadooshan A A, Akbari M 2016 Physica. E 84 474 Google Scholar
[30]	Khosravi A, Malekan M 2019 Eur. Phys. J. Plus 134 1 Google Scholar
[31]	Malekan M, Khosravi A 2018 Powder Techno. 333 364 Google Scholar
[32]	Hussam W K, Hamid A H A, Ng Z Y, Sheard G J 2018 Int. J. Therm. Sci. 134 453 Google Scholar
[33]	李开, 刘伟强 2016 65 064701 Google Scholar Li K, Liu W Q 2016 Acta Phys. Sin. 65 064701 Google Scholar
[34]	Liu Z, Li Y, Su Y 2018 Int. J. Adv. Manuf. Tech. 98 2015 Google Scholar
[35]	Tsilingiris P T 2008 Energ. Convers. Manage. 49 1098 Google Scholar
[36]	Stephan K, Laesecke A 1985 J. Phys. Chem. Ref. Data 14 227 Google Scholar
[37]	Menter F R 1994 AIAA J. 32 1598 Google Scholar
[38]	Widlund O, Zahrai S, Bark F H 1998 Phys. Fluids 10 1987 Google Scholar
[39]	Walker J S 1981 Journal de Mécanique 20 79
[40]	Takeuchi J, Satake S, Morley N B, Kunugi T, Yokomine T, Abdou, M A 2008 Fusion Eng. Des. 83 1082 Google Scholar
[41]	Gnielinski V 1976 Int. Chem. Eng 16 359
[42]	Bradshaw P 1974 Nature 249 135

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