We study the mixing of a passive scalar in homogeneous turbulent flow with and without anisotropic external force. It is assumed that no scalar source exists and the scalar spectrum at an initial time instant $t_0$ is given by the form $C{k}^2+o(k^2)$ at the wavenumber $k\to 0$, where $C$ is independent of $k$. We have performed direct numerical simulations (DNSs) of the mixing, in which the initial integral length scales of the scalar field are comparable to those of the velocity field and the Schmidt number is unity. The DNSs show that even though the large-scale anisotropy of the velocity field grows with time owing to the external force, its scalar field counterpart remains almost unchanged, i.e., frozen with respect to time, in a state where the scalar field evolves in a self-similar manner. The degree of the anisotropy of each of the velocity and scalar fields is measured by the ratios of the integral length scales in different Cartesian directions. The DNSs also suggest that the scalar spectrum keeps the form $C{k}^2+o(k^2)$ at small $k$ for time $t(\ge t_0)$, and that $C$ is time-independent.

中文翻译：

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各向异性均质湍流中被动标量场的各向异性冻结

我们研究在有和没有各向异性外力的情况下，均质湍流中被动标量的混合。假设不存在标量源，并且在初始时刻标量频谱${Ť}_0$ 由以下形式给出 $C{ķ}^2+Ø（{ķ}^2）$ 在波数 $ķ\to 0$，在哪里 $C$ 独立于 $ķ$。我们已经进行了混合的直接数值模拟（DNS），其中标量场的初始积分长度比例与速度场的初始积分长度比例可比，而Schmidt数为1。DNS显示，即使速度场的大规模各向异性由于外力而随时间增长，但其标量场对应物几乎保持不变，即相对于时间处于冻结状态，而标量场则以自相似的方式。每个速度场和标量场的各向异性程度是通过不同笛卡尔方向上的积分长度比例之比来衡量的。DNS还建议标量频谱保持形式$C{ķ}^2+Ø（{ķ}^2）$ 在小 $ķ$ 时间 $Ť（\ge {Ť}_0）$， 然后 $C$ 与时间无关。