不同温度和浓度梯度下磁流体双扩散对流的格子Boltzmann方法模拟
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摘要
格子Boltzmann方法(Lattice Boltzmann Method, LBM)是基于微观层面上的一种新的数值模拟方法。LBM是把研究对象抽象为在微观上比分子大,在宏观上又无限小的微观粒子来研究。与传统数值模拟方法相比,LBM具有物理图像清晰、边界处理容易、计算效率高、本质并行性等独特优点。目前,LBM己成功应用于微尺度流动与换热、多孔介质、磁流体力学、晶体生长等许多领域。
本论文是用格子Boltzmann方法来模拟不同温度浓度梯度下矩形腔内二元混合导电气体的双扩散对流。我们采用常用的温度浓度格子Bhatnagar-Gross-Krook(TCLBGK)模型,并用D2G9模型来模拟速度场,用D2Q4模型模拟温度场和浓度场。并且认为矩形腔的上下壁是绝热的,左右壁温度和浓度是变化的,磁场沿x轴方向均匀加在矩形腔内。在模拟计算中,我们取Prandtl数Pr=1,Lewis数Le=2,热Raleigh数RaT=105,Hartmann数Hα=0,25,50,无量纲热产生和热吸收量Φ=0,方腔长宽比A=2,浓度与温度的效应比N=0.8,1.3。为了更详细的反映矩形腔左右壁温度浓度对双扩散对流的影响,我们分别模拟了左右壁温度梯度从下往上由0线性增加到2.0而浓度梯度不变为常数1.0、浓度梯度从下往上由0线性增大到2.0而温度梯度不变为常数1.0和温度浓度梯度都是由0线性增大到2.0时三种情况下的双扩散对流。
数值结果表明：对于第一种情况,在磁场强度Ha的取值范围内时,N=0.8和N=1.3对流都出现了明显的分层现象,且分为上层由热浮力效应驱动的沿顺时针方向旋转的热流动和下层由浓度浮力效应驱动的沿逆时针方向旋转的传质流动。在矩形腔的中上部温度梯度大于常浓度梯度的区域,热对流占主要地位；然而,在矩形腔的下部温度梯度小于常浓度梯度的区域,浓度梯度驱动的传质对流占主要地位。这种现象表明温度梯度越大热对流越强。
对于第二种情况,在磁场强度Ha的取值范围内时,N=0.8和N=1.3对流也分方向相反的上下两层,但浓度浮力效应驱动的逆时针旋转的传质对流在上部,且占据了空腔的大部分区域,而热浮力效应驱动的顺时针热流动占据下部的小部分区域。此现象表明浓度梯度对溶质扩散对流有重要的影响。且当Ha=0时,无论N=0.8还是N=1.3,在空腔下部区域都出现了溶质扩散停滞现象,但随着Ha的增大,传质区域增大。这表明此情况下磁场可以加强传质对流。
对于第三种情况,浓度梯度与温度梯度相互影响,对于N=0.8,热浮力效应占主导地位,流线基本没有出现分层现象,整个流场区域几乎只有热流动在沿顺时针方向旋转；对于N=1.3,浓度效应占主导地位,对流出现了明显的分层,但沿逆时针方向旋转的溶质扩散对流占据了空腔的大部分区域。此种情况表明浓度梯度和温度梯度相同时,传质对流和传热对流的强弱分别取决于各自的浮力效应。
The lattice Boltzmann method is a new numerical scheme based on microscopic model for fluids flow. According to LBM, fluid particles are much larger than the molecules on the microscopic level. However, on the macroscopic level, fluid particles are' infinitely small. Compared with the traditional Computational Fluid Dynamics (CFD) methods, LBM has many unique advantages, such as simple codes, easy implementation of boundary conditions, and fully parallelism. These features make LBM apply successfully to many fields, such as microscale flow and Transfer, the porous media, magnetohydrodynamics, the crystal growth etc.
In this paper, the two-dimensional, hydromagnetic double-diffusive convection of a binary gas mixture is simulated by a temperature-concentration lattice Bhatnagar-Gross-Krook (TCLBGK) model in a rectangular enclosure with the top and bottom walls being insulated, while linearly variable temperature or concentration gradient or both are imposed along the left and right walls from the bottom to the top and a uniform magnetic field is applied in x-direction. We take the Prandtl number Pr=1, the Lewis Le =2, the thermal Raleigh numberRaT=105, the Hartmann number Ha=0,25,50, the dimensionless heat generation or absorption (?)=0, the aspect ration A=2 for the enclosure and the ratio of buoyancy forces N=0.8,1.3. For more details about temperature and concentration gradients on the impact of double-diffusive convection, there are three possible circumstances to consider about the left and right walls of the cavity. One is that linearly variable temperature and constant concentration are imposed along the left and right walls, another is that constant temperature and linearly variable concentration are imposed along the two walls, the other is linearly variable temperature and linearly variable concentration.
In the first case, the calculation was firstly carried out at various values of the Hartmann number Ha for Le=2.0, N=0.8 and 1.3. The streamlines at N=0.8 and 1.3 are obviously stratified throughout the Ha range, and each recirculation cell splits into two smaller circulating cells of opposite directions situated close to each of the insulated upper and lower walls. The temperature gradient is larger than the constant concentration that in the upper part of the rectangular cavity, a large central clockwise thermal recirculation closes to the insulated upper wall. It is shown that thermal convection is stronger when temperature gradient is larger. For the second case, compared with the first case, however, there are some differences between them. For N=0.8, the flow is still primarily dominated by thermal buoyancy effects, but a large central clockwise thermal recirculation closes to the insulated lower wall, for N=1.3, the flow is also mainly dominated by compositional buoyancy effects, whereas, a big counterclockwise compositional recirculation exists in the insulated upper wall. It is found that variable concentration gradient has influence on the mass transfer. In the last case, For N=0.8, the zone of heat transfer ocuppies the cavity almost exclusively. For N=1.3, the diffusion of solute ocuppies the cavity almost exclusively. This situation shows that the strength of convection of the mass transfer and heat transfer depends on the strength of their buoyancy, respectively, when the concentration and temperature gradients are the same.
本论文是用格子Boltzmann方法来模拟不同温度浓度梯度下矩形腔内二元混合导电气体的双扩散对流。我们采用常用的温度浓度格子Bhatnagar-Gross-Krook(TCLBGK)模型,并用D2G9模型来模拟速度场,用D2Q4模型模拟温度场和浓度场。并且认为矩形腔的上下壁是绝热的,左右壁温度和浓度是变化的,磁场沿x轴方向均匀加在矩形腔内。在模拟计算中,我们取Prandtl数Pr=1,Lewis数Le=2,热Raleigh数RaT=105,Hartmann数Hα=0,25,50,无量纲热产生和热吸收量Φ=0,方腔长宽比A=2,浓度与温度的效应比N=0.8,1.3。为了更详细的反映矩形腔左右壁温度浓度对双扩散对流的影响,我们分别模拟了左右壁温度梯度从下往上由0线性增加到2.0而浓度梯度不变为常数1.0、浓度梯度从下往上由0线性增大到2.0而温度梯度不变为常数1.0和温度浓度梯度都是由0线性增大到2.0时三种情况下的双扩散对流。
数值结果表明：对于第一种情况,在磁场强度Ha的取值范围内时,N=0.8和N=1.3对流都出现了明显的分层现象,且分为上层由热浮力效应驱动的沿顺时针方向旋转的热流动和下层由浓度浮力效应驱动的沿逆时针方向旋转的传质流动。在矩形腔的中上部温度梯度大于常浓度梯度的区域,热对流占主要地位；然而,在矩形腔的下部温度梯度小于常浓度梯度的区域,浓度梯度驱动的传质对流占主要地位。这种现象表明温度梯度越大热对流越强。
对于第二种情况,在磁场强度Ha的取值范围内时,N=0.8和N=1.3对流也分方向相反的上下两层,但浓度浮力效应驱动的逆时针旋转的传质对流在上部,且占据了空腔的大部分区域,而热浮力效应驱动的顺时针热流动占据下部的小部分区域。此现象表明浓度梯度对溶质扩散对流有重要的影响。且当Ha=0时,无论N=0.8还是N=1.3,在空腔下部区域都出现了溶质扩散停滞现象,但随着Ha的增大,传质区域增大。这表明此情况下磁场可以加强传质对流。
对于第三种情况,浓度梯度与温度梯度相互影响,对于N=0.8,热浮力效应占主导地位,流线基本没有出现分层现象,整个流场区域几乎只有热流动在沿顺时针方向旋转；对于N=1.3,浓度效应占主导地位,对流出现了明显的分层,但沿逆时针方向旋转的溶质扩散对流占据了空腔的大部分区域。此种情况表明浓度梯度和温度梯度相同时,传质对流和传热对流的强弱分别取决于各自的浮力效应。
The lattice Boltzmann method is a new numerical scheme based on microscopic model for fluids flow. According to LBM, fluid particles are much larger than the molecules on the microscopic level. However, on the macroscopic level, fluid particles are' infinitely small. Compared with the traditional Computational Fluid Dynamics (CFD) methods, LBM has many unique advantages, such as simple codes, easy implementation of boundary conditions, and fully parallelism. These features make LBM apply successfully to many fields, such as microscale flow and Transfer, the porous media, magnetohydrodynamics, the crystal growth etc.
In this paper, the two-dimensional, hydromagnetic double-diffusive convection of a binary gas mixture is simulated by a temperature-concentration lattice Bhatnagar-Gross-Krook (TCLBGK) model in a rectangular enclosure with the top and bottom walls being insulated, while linearly variable temperature or concentration gradient or both are imposed along the left and right walls from the bottom to the top and a uniform magnetic field is applied in x-direction. We take the Prandtl number Pr=1, the Lewis Le =2, the thermal Raleigh numberRaT=105, the Hartmann number Ha=0,25,50, the dimensionless heat generation or absorption (?)=0, the aspect ration A=2 for the enclosure and the ratio of buoyancy forces N=0.8,1.3. For more details about temperature and concentration gradients on the impact of double-diffusive convection, there are three possible circumstances to consider about the left and right walls of the cavity. One is that linearly variable temperature and constant concentration are imposed along the left and right walls, another is that constant temperature and linearly variable concentration are imposed along the two walls, the other is linearly variable temperature and linearly variable concentration.
In the first case, the calculation was firstly carried out at various values of the Hartmann number Ha for Le=2.0, N=0.8 and 1.3. The streamlines at N=0.8 and 1.3 are obviously stratified throughout the Ha range, and each recirculation cell splits into two smaller circulating cells of opposite directions situated close to each of the insulated upper and lower walls. The temperature gradient is larger than the constant concentration that in the upper part of the rectangular cavity, a large central clockwise thermal recirculation closes to the insulated upper wall. It is shown that thermal convection is stronger when temperature gradient is larger. For the second case, compared with the first case, however, there are some differences between them. For N=0.8, the flow is still primarily dominated by thermal buoyancy effects, but a large central clockwise thermal recirculation closes to the insulated lower wall, for N=1.3, the flow is also mainly dominated by compositional buoyancy effects, whereas, a big counterclockwise compositional recirculation exists in the insulated upper wall. It is found that variable concentration gradient has influence on the mass transfer. In the last case, For N=0.8, the zone of heat transfer ocuppies the cavity almost exclusively. For N=1.3, the diffusion of solute ocuppies the cavity almost exclusively. This situation shows that the strength of convection of the mass transfer and heat transfer depends on the strength of their buoyancy, respectively, when the concentration and temperature gradients are the same.
引文
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[3]朱立明,柯葵.流体力学[M].上海：同济大学出版社,2009
[4]赵秀艳.金属射流的MHD特性分析[D].山东：山东农业大学,2009
[5]郭照立,郑楚光,李青.流体动力学的格子Boltzmann方法[M].湖北：湖北科学技术出版社,2002
[6]彭伟.格子Boltzmann方法基本模型比较及应用[D].湖北：华中科技大学,2004
[7]蒋荣勤.高雷诺数下格子Boltzmann方法的应用研究[D].哈尔滨：哈尔滨工程大学,2008
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