Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field
Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field
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DOI：
10.1017/s0022112076001675
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发表时间：
1976-03
影响因子：
3.7
通讯作者：
U. Schumann
中科院分区：
文献类型：
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作者：
U. Schumann
对于均匀磁场B0作用下的导电不可压缩流体，数值模拟了均匀湍流从初始各向同性三维态向准二维态的转变。假设磁雷诺数很小，因此与外加磁场B0相比，磁场的感应起伏很小，并且可以从准静态近似计算。如果外加磁场足够强，流场在B0方向的所有变化都会被抑制。例如，在聚变反应堆的液态金属冷却系统的设计中，这种效应是很重要的，而最终状态的性质与大气湍流有关。用Orszag&Patterson(1972)程序的扩展版本对不可压缩流体的Navier-Stokes方程进行了积分。初始流体动力学雷诺数为60。磁相互作用数N在0到50之间变化。采用周期边界条件。分辨率相当于实空间中的323个点。将完全非线性模拟与其他相同的线性模拟进行了比较；在大约五分之一的大规模周转时间内，线性模拟与非线性模拟的结果在3%以内是一致的。这种偏离是回归平衡趋势的结果，这种趋势主要是由于能量转移到高波数造成的。角能量转移和各组元之间的能量交换较小，当N较大时，它们几乎为零。对于N≈50，我们达到准二维态。对于垂直于B0的速度分量，向高波数传递的能量减少，而对于平行于B0的分量，能量传递相对增加。整体行为更类似于三维湍流，而不是纯粹的二维湍流。这一发现对大气湍流模型具有重要意义。对于衰变的湍流来说，实现纯二维态似乎是不可能的。磁场引起高度增强的压力波动，这有助于各向异性洛伦兹强迫的重新分布。
The transition of homogeneous turbulence from an initially isotropic three-dimensional to a quasi-two-dimensional state is simulated numerically for a conducting, incompressible fluid under a uniform magnetic field B0. The magnetic Reynolds number is assumed to be small, so that the induced fluctuations of the magnetic field are small compared with the imposed magnetic field B0, and can be computed from a quasi-static approximation. If the imposed magnetic field is strong enough, all variations of the flow field in the direction of B0 are damped out. This effect is important e.g. in the design of liquid-metal cooling systems for fusion reactors, and the properties of the final state are relevant to atmospheric turbulence. An extended version of the code of Orszag & Patterson (1972) is used to integrate the Navier-Stokes equations for an incompressible fluid. The initial hydrodynamic Reynolds number is 60. The magnetic interaction number N is varied between zero and 50. Periodic boundary conditions are used. The resolution corresponds to 323 points in real space. The full nonlinear simulations are compared with otherwise identical linear simulations; the linear results agree with the nonlinear ones within 3% for about one-fifth of the large-scale turnover time. This departure is a consequence of the return-to-equilibrium tendencies caused mainly by energy transfer towards high wavenumbers. The angular energy transfer and the energy exchange between different components are smaller, and become virtually zero for large values of N. For N ≈ 50 we reach a quasi-two-dimensional state. Here, the energy transfer towards high wavenumbers is reduced for the velocity components perpendicular to B0 but relatively increased for the component parallel to B0. The overall behaviour is more similar to three-than to purely two-dimensional turbulence. This finding is of great importance for turbulence models of the atmosphere. The realization of a purely two-dimensional state does not seem to be possible for decaying turbulence. The magnetic field causes highly intensified pressure fluctuations, which contribute to the redistribution of the anisotropic Lorentz forcing.