浸没边界–简化热格子Boltzmann方法研究及其应用
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摘要
针对流固耦合传热问题,本文提出了一种基于浸没边界–简化热格子玻尔兹曼方法 (immersed boundary method–simplified thermal lattice Boltzmann method,IB-STLBM)的耦合模型.不同于传统的格子玻尔兹曼方法使用分布函数演化流场和温度场,简化热格子玻尔兹曼方法 (simplified thermal lattice Boltzmann method,STLBM)的演化过程不需要依赖分布函数,只涉及平衡态分布函数和非平衡态分布函数,能够直接演化宏观量,极大减小了计算过程中所占用的虚拟内存,简化了边界条件的实现方式,同时具有较高的稳定性.传统的浸没边界法对流场的计算采用欧拉网格,对固体边界采用拉格朗日网格,认为固体边界是对流场产生某种体积力.在应用浸没边界法时,汲取介观的思想,把固体的介入看作是对流场的干扰,打破了固体附近流体介观微团颗粒原始的平衡状态,这种干扰可以看作是在耦合边界上产生的一个非平衡项,可用非平衡态分布函数来表示.基于此,在模型中浸没边界法与简化热格子玻尔兹曼方法更紧密联系在一起,更大程度发挥二者的优点,整个计算过程更加简单直观,符合物理特性.通过对热圆柱绕流和内含热颗粒的封闭方腔自然对流问题的模拟以及对其结果的分析,验证了该算法在求解流固耦合传热问题的有效性和可行性.
An Immersed boundary-simplified thermal lattice Boltzmann method(IB-STLBM) for fluid-structure interaction problem with heat transfer is developed in this work. In the IB-STLBM, an effective simplified thermal lattice Boltzmann method without the evolution of distribution is used for the intermediate flow field. Different from the stander thermal lattice Boltzmann method, STLBM directly updates the macroscopic variables instead of the distribution functions, which offers several distinct benefits:lower cost in virtual memories, simpler implementation of physical boundary condition and higher numerical stability. In addition, from the mesoscopic view, the existence of solid boundary in the field is considered as an interference of system, which breaks the original equilibrium state of fluid particle, and a nonequilibrium state occurs on the fluid-structure interaction physics boundary. On this basis, in the present IB-STLBM,fluid-structural interaction duo to Immersed boundary appearance in the fluid can be expressed by the non-equilibrium distribution function, which is calculated by the popular non-equilibrium bounce-back boundary condition of the LBM.Hence, the solution procedure of present IB-STLBM can satisfy the non-slip boundary by a simpler way. Numerical experiments for the forced convection over a stationary heated circular cylinder and natural convection in a square cavity with a circle particle are presented to verify the stability, the capability and the flexibility of IB-STLBM for fluid-structure interaction problem with heat transfer. In the case of a stationary heated circular cylinder, quantitative and qualitative comparisons are carried out with previous study. The results of the drag coefficient and the avenge Nusselt numbers on the cylinder are in accordance with the results of previous study. From the case of natural convection in a square cavity with a circle particle, some interesting phenomena can be found. First, the temperature field is clearly stirred by the suspended particle. Second, the temporal trajectories of the particle exhibited regular changes. Third, the particle enhances heat transfer and the average Nusselt numbers periodically oscillate with time.
An Immersed boundary-simplified thermal lattice Boltzmann method(IB-STLBM) for fluid-structure interaction problem with heat transfer is developed in this work. In the IB-STLBM, an effective simplified thermal lattice Boltzmann method without the evolution of distribution is used for the intermediate flow field. Different from the stander thermal lattice Boltzmann method, STLBM directly updates the macroscopic variables instead of the distribution functions, which offers several distinct benefits:lower cost in virtual memories, simpler implementation of physical boundary condition and higher numerical stability. In addition, from the mesoscopic view, the existence of solid boundary in the field is considered as an interference of system, which breaks the original equilibrium state of fluid particle, and a nonequilibrium state occurs on the fluid-structure interaction physics boundary. On this basis, in the present IB-STLBM,fluid-structural interaction duo to Immersed boundary appearance in the fluid can be expressed by the non-equilibrium distribution function, which is calculated by the popular non-equilibrium bounce-back boundary condition of the LBM.Hence, the solution procedure of present IB-STLBM can satisfy the non-slip boundary by a simpler way. Numerical experiments for the forced convection over a stationary heated circular cylinder and natural convection in a square cavity with a circle particle are presented to verify the stability, the capability and the flexibility of IB-STLBM for fluid-structure interaction problem with heat transfer. In the case of a stationary heated circular cylinder, quantitative and qualitative comparisons are carried out with previous study. The results of the drag coefficient and the avenge Nusselt numbers on the cylinder are in accordance with the results of previous study. From the case of natural convection in a square cavity with a circle particle, some interesting phenomena can be found. First, the temperature field is clearly stirred by the suspended particle. Second, the temporal trajectories of the particle exhibited regular changes. Third, the particle enhances heat transfer and the average Nusselt numbers periodically oscillate with time.
引文
1邹勇,朱桂平,李来等.液桥内热质耦合对流不稳定性及旋转磁场法控制.力学学报,2017,49(6):1280-1289(Zou Yong,Zhu Guiping,Li Lai,et al.Instability of coupled thermo-solute capillary convection in liquid bridge and control by rotating magnetic field.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1280-1289(in Chinese))
2刘成,叶正寅,叶坤.转捩位置对全动舵面热气动弹性的影响.力学学报,2017,49(4):802-810(Liu Cheng,Ye Zhengyin,Ye Kun.The e ect of transiton location on aerothermoelasticity of a hypersonic all-movable centrol surface.Chinese Journal of Theoretical and Applied Mechanics,2017,49(4):802-810(in Chinese))
3徐飞彬,周全,卢志明.二维方腔热对流系统中纳米颗粒混合及凝并特性的数值模拟.力学学报,2015,47(5):740-750(Xu Feibin,Zhou Quan,Lu Zhiming.Numerical simulation of Brownian coagulation and mixing of Nanoparticles in 2-D Rayleigh-B’enard convection.Chinese Journal of Theoretical and Applied Mechanics,2015,47(5):740-750(in Chinese))
4 Mills ZG,Aziz B,Alexeev A.Beating synthetic cilia enhance heat transport in microfluidic channels.Soft Matter,2012,8(45):11508-11513
5 Peskin CS.Flow pat terns around heart valves:a numerical method.Journal of Computational Physics,1972,10(2):225-271
6 Peskin CS,Printz ABF.Improved volume conservation in the computation of flows with immersed elastic boundaries.Journal of Computational Physics,1993,105(1):33-46
7 Chen S,Doolen GD.Lattice boltzmann method for fluid flows.Annu.Rev.Fluid.Mech,1998,30:329-364
8 Feng Z,Michaelides E.The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems.Journal of Computational Physics,2004,195(2):602-628
9 Feng Z,Michaelides E.Proteus:A direct forcing method in the simulations of particulate flows.Journal of Computational Physics,2005,202(1):20-51
10 Cheng YG,Zhu LD,Zhang CZ.Numerical study of stability and accuracy of the immersed boundary method coupled to the Lattice Boltzmann BGK model.Communications in Computational Physics,2014,16(1):136-168
11 Kang SK,Hassan YA.A comparative study of direct-forcing immersed boundary-lattice Boltzmann methods for stationary complex boundaries.International Journal Numerical Methods Fluids,2011,66(9):1132-1158
12 Shan XW.Simulation of Rayleigh-B′enard convection using a lattice Boltzmann method.Physical Review E,1997,55(3):2780
13 He XY,Chen SY,Doolen GD.A novel thermal model for the lattice Boltzmann method in incompressible limit.Journal of Computational Physics,1998,146(1):282-300
14 Jeong HK,Yoon HS,Ha MY,et al.An immersed boundary-thermal lattice Boltzmann method using an equilibrium internal energy density approach for the simulation of flows with heat transfer.Journal of Computational Physics,2010,229(7):2526-2543
15 Kang SK,Hassan YA.A direct-forcing immersed boundary method for the thermal lattice Boltzmann method.Computational Fluids,2011,49(1):36-45
16 Seta T.Implicit temperature-correction-based immersed-boundary thermal lattice Boltzmann method for the simulation of natural convection.Physical Review E,2013,87(6):063304
17 Zhang H,Yuan HZ,Trias FX,et al.Particulate immersed boundary method for complex fluid-particle interaction problems with heat transfer.Computers&Mathematics with Applications,2016,71:391-407
18 Hu Y,Li DC,Shu S,et al.An efficient immersed boundary-lattice boltzmann method for the simulation of thermal flow problems.Communications in Computational Physics,2016,20:1210-1257
19 Chen Z,Shu C,Tan D.A simplified thermal lattice Boltzmann method without evolution of distribution functions.International Journal of Heat Mass Transfer,2017,105:741
20 Peng Y,Shu C,Chew Y.Simplified thermal lattice Boltzmann model for incompressible thermal flows.Physical Review E,2003,68:026701
21 Qian Y,Humieres D.Lallemand P.Lattice BGK models for NavierStokes equation.Europhysicas Letters,1992,17(6):479-484
22 Guo ZL,Shu C.Lattice Boltzmann Method and Its Applications in Engineering.World Scientific,2013
23 Wang Y,Shu C,Teo C.Thermal lattice Boltzmann flux solver and its application for simulation of incompressible thermal flows.Computer&Fluids,2014,94:98-111
24 Gray DD,Giorgini A.The validity of the Boussinesq approximation for liquids and gases.International Journal of Heat Mass Transfer,1976,19:545
25 Peskin CS.The immersed boundary method.Acta Numer,2002,11:479-517
26 Niu X,Shu C,Chew Y,et al.A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows.Physics Letters A,2006,354(3):173-182
27 Chen MF,Niu XD.An improved momentum-exchanged immersed boundary-based lattice Boltzmann method for incompressible viscous thermal flows.Int.J.Modern Phys,2016,42:1660161
28 Chen MF,Niu XD,Yamaguchi H,et al.A lattice Boltzmann modeling fluid-structure interaction problem and its applications in natural convections in a square cavity with particles suspended inside.Adv App Math Mech,2018,10(2):303-328
29陈木凤,李翔,牛小东等.两个非磁性颗粒在磁流体中的沉降现象研究.物理学报,2017,66(16):164703(Chen Mufeng,Li Xiang,Niu Xiaodong,et al.Sedimentation of two non-magnetic particles in magnetic fluid.Acta Phy.Sin,2018,66(16):164703(in Chinese))
30 Chorin AJ.Numerical solution of the Navier-Stokes equations.Math.Compo,1968,22:745-762
31 Chorin AJ.Numerical solution of incompressible flow problems.Stud.Numer.Anal,1968,2:64-71
32 Temam R.Sur l’approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires(I).Arch.Ration.Mech.Anal,1969,32:135-153
33 Temam R.Sur.l’approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires(II).Arch.Ration.Mech.Anal,1969,33:377-385
34 Kim J,Moin P.Application of a fractional-step method to incompressible Navier-Stokes equations.Journal of Computational Physics,1985,59:308-323
35 Ren W,Shu C,Yang W.An efficient immersed boundary method for thermal flow problems with heat flux boundary conditions.Heat&Mass Transfer,2013,64:694-705
36 Chen DJ,Lin KH,Lin CA.Immersed boundary method based lattice Boltzmann to simulate 2D and 3D complex geometry flows.International Journal of Modern Physics C,2007,18:585-594
37 Hu Y,Yuan H,Shu S,et al.An improved momentum exchangedbased immersed boundary-lattice Boltzmann method by using an iterative technique.Computers&Mathematics with Applications,2014,68(3):140-155
38 Dennis SCR,Chang GZ.Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100.Journal of Fluid Mechanics,1970,42(3):471-489
39 Ahmad RA,Qureshi ZH.Laminar mixed convection from a uniform heat flux horizontal cylinder in a crossflow.Journal of Thermophysics&Heat Transfer,1992,6(2):277-287
40 Bharti R,Chhabra RP,Eswaran V.A numerical study of the steady forced convection heat transfer from an unconfined circular cylinder.Heat&Mass Transfer,2007,43:639-648
2刘成,叶正寅,叶坤.转捩位置对全动舵面热气动弹性的影响.力学学报,2017,49(4):802-810(Liu Cheng,Ye Zhengyin,Ye Kun.The e ect of transiton location on aerothermoelasticity of a hypersonic all-movable centrol surface.Chinese Journal of Theoretical and Applied Mechanics,2017,49(4):802-810(in Chinese))
3徐飞彬,周全,卢志明.二维方腔热对流系统中纳米颗粒混合及凝并特性的数值模拟.力学学报,2015,47(5):740-750(Xu Feibin,Zhou Quan,Lu Zhiming.Numerical simulation of Brownian coagulation and mixing of Nanoparticles in 2-D Rayleigh-B’enard convection.Chinese Journal of Theoretical and Applied Mechanics,2015,47(5):740-750(in Chinese))
4 Mills ZG,Aziz B,Alexeev A.Beating synthetic cilia enhance heat transport in microfluidic channels.Soft Matter,2012,8(45):11508-11513
5 Peskin CS.Flow pat terns around heart valves:a numerical method.Journal of Computational Physics,1972,10(2):225-271
6 Peskin CS,Printz ABF.Improved volume conservation in the computation of flows with immersed elastic boundaries.Journal of Computational Physics,1993,105(1):33-46
7 Chen S,Doolen GD.Lattice boltzmann method for fluid flows.Annu.Rev.Fluid.Mech,1998,30:329-364
8 Feng Z,Michaelides E.The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems.Journal of Computational Physics,2004,195(2):602-628
9 Feng Z,Michaelides E.Proteus:A direct forcing method in the simulations of particulate flows.Journal of Computational Physics,2005,202(1):20-51
10 Cheng YG,Zhu LD,Zhang CZ.Numerical study of stability and accuracy of the immersed boundary method coupled to the Lattice Boltzmann BGK model.Communications in Computational Physics,2014,16(1):136-168
11 Kang SK,Hassan YA.A comparative study of direct-forcing immersed boundary-lattice Boltzmann methods for stationary complex boundaries.International Journal Numerical Methods Fluids,2011,66(9):1132-1158
12 Shan XW.Simulation of Rayleigh-B′enard convection using a lattice Boltzmann method.Physical Review E,1997,55(3):2780
13 He XY,Chen SY,Doolen GD.A novel thermal model for the lattice Boltzmann method in incompressible limit.Journal of Computational Physics,1998,146(1):282-300
14 Jeong HK,Yoon HS,Ha MY,et al.An immersed boundary-thermal lattice Boltzmann method using an equilibrium internal energy density approach for the simulation of flows with heat transfer.Journal of Computational Physics,2010,229(7):2526-2543
15 Kang SK,Hassan YA.A direct-forcing immersed boundary method for the thermal lattice Boltzmann method.Computational Fluids,2011,49(1):36-45
16 Seta T.Implicit temperature-correction-based immersed-boundary thermal lattice Boltzmann method for the simulation of natural convection.Physical Review E,2013,87(6):063304
17 Zhang H,Yuan HZ,Trias FX,et al.Particulate immersed boundary method for complex fluid-particle interaction problems with heat transfer.Computers&Mathematics with Applications,2016,71:391-407
18 Hu Y,Li DC,Shu S,et al.An efficient immersed boundary-lattice boltzmann method for the simulation of thermal flow problems.Communications in Computational Physics,2016,20:1210-1257
19 Chen Z,Shu C,Tan D.A simplified thermal lattice Boltzmann method without evolution of distribution functions.International Journal of Heat Mass Transfer,2017,105:741
20 Peng Y,Shu C,Chew Y.Simplified thermal lattice Boltzmann model for incompressible thermal flows.Physical Review E,2003,68:026701
21 Qian Y,Humieres D.Lallemand P.Lattice BGK models for NavierStokes equation.Europhysicas Letters,1992,17(6):479-484
22 Guo ZL,Shu C.Lattice Boltzmann Method and Its Applications in Engineering.World Scientific,2013
23 Wang Y,Shu C,Teo C.Thermal lattice Boltzmann flux solver and its application for simulation of incompressible thermal flows.Computer&Fluids,2014,94:98-111
24 Gray DD,Giorgini A.The validity of the Boussinesq approximation for liquids and gases.International Journal of Heat Mass Transfer,1976,19:545
25 Peskin CS.The immersed boundary method.Acta Numer,2002,11:479-517
26 Niu X,Shu C,Chew Y,et al.A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows.Physics Letters A,2006,354(3):173-182
27 Chen MF,Niu XD.An improved momentum-exchanged immersed boundary-based lattice Boltzmann method for incompressible viscous thermal flows.Int.J.Modern Phys,2016,42:1660161
28 Chen MF,Niu XD,Yamaguchi H,et al.A lattice Boltzmann modeling fluid-structure interaction problem and its applications in natural convections in a square cavity with particles suspended inside.Adv App Math Mech,2018,10(2):303-328
29陈木凤,李翔,牛小东等.两个非磁性颗粒在磁流体中的沉降现象研究.物理学报,2017,66(16):164703(Chen Mufeng,Li Xiang,Niu Xiaodong,et al.Sedimentation of two non-magnetic particles in magnetic fluid.Acta Phy.Sin,2018,66(16):164703(in Chinese))
30 Chorin AJ.Numerical solution of the Navier-Stokes equations.Math.Compo,1968,22:745-762
31 Chorin AJ.Numerical solution of incompressible flow problems.Stud.Numer.Anal,1968,2:64-71
32 Temam R.Sur l’approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires(I).Arch.Ration.Mech.Anal,1969,32:135-153
33 Temam R.Sur.l’approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires(II).Arch.Ration.Mech.Anal,1969,33:377-385
34 Kim J,Moin P.Application of a fractional-step method to incompressible Navier-Stokes equations.Journal of Computational Physics,1985,59:308-323
35 Ren W,Shu C,Yang W.An efficient immersed boundary method for thermal flow problems with heat flux boundary conditions.Heat&Mass Transfer,2013,64:694-705
36 Chen DJ,Lin KH,Lin CA.Immersed boundary method based lattice Boltzmann to simulate 2D and 3D complex geometry flows.International Journal of Modern Physics C,2007,18:585-594
37 Hu Y,Yuan H,Shu S,et al.An improved momentum exchangedbased immersed boundary-lattice Boltzmann method by using an iterative technique.Computers&Mathematics with Applications,2014,68(3):140-155
38 Dennis SCR,Chang GZ.Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100.Journal of Fluid Mechanics,1970,42(3):471-489
39 Ahmad RA,Qureshi ZH.Laminar mixed convection from a uniform heat flux horizontal cylinder in a crossflow.Journal of Thermophysics&Heat Transfer,1992,6(2):277-287
40 Bharti R,Chhabra RP,Eswaran V.A numerical study of the steady forced convection heat transfer from an unconfined circular cylinder.Heat&Mass Transfer,2007,43:639-648