We solve the advection-diffusion equation for a stochastically stationary passive scalar $\theta$, in conjunction with forced 3D Navier-Stokes equations, using direct numerical simulations in periodic domains of various sizes, the largest being ${8192}^3$. The Taylor-scale Reynolds number varies in the range 140–650 and the Schmidt number $\mathrm{Sc}\equiv \nu /D$ in the range 1–512, where $\nu$ is the kinematic viscosity of the fluid and $D$ is the molecular diffusivity of $\theta$. Our results show that turbulence becomes an ineffective mixer when Sc is large. First, the mean scalar dissipation rate $⟨\chi ⟩=2D⟨|\nabla \theta {|}^2⟩$, when suitably nondimensionalized, decreases as $1/\log \mathrm{Sc}$. Second, 1D cuts through the scalar field indicate increasing density of sharp fronts on larger scales, oscillating with large excursions leading to reduced mixing, and additionally suggesting weakening of scalar variance flux across the scales. The scaling exponents of the scalar structure functions in the inertial-convective range appear to saturate with respect to the moment order and the saturation exponent approaches unity as Sc increases, qualitatively consistent with 1D cuts of the scalar.

中文翻译：

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施密特数很大时，湍流是无效的混合器

我们求解随机平稳无源标量的对流扩散方程 $\theta$结合强制3D Navier-Stokes方程，在各种大小的周期域中使用直接数值模拟，最大的是 ${8192}^3$。泰勒尺度雷诺数在140–650之间变化，而施密特数在$\mathrm{SC}\equiv \nu /d$ 在1–512范围内，其中 $\nu$ 是流体的运动粘度， $d$ 是分子的扩散率 $\theta$。我们的结果表明，当Sc大时，湍流成为无效的混合器。一，平均标量耗散率$⟨\chi ⟩=2d⟨|\nabla \theta {|}^2⟩$，当适当地进行无量纲化时，减少为 $\mathrm{1个}/\mathrm{日志}\mathrm{SC}$。其次，通过标量场的一维切割表明较大尺度上锋利的锋面密度增加，随着大幅度偏移而振荡，导致混合减少，并且还提示了整个尺度上标量方差通量减弱。在惯性对流范围内，标量结构函数的标度指数似乎相对于矩阶饱和，并且随着Sc的增加，饱和指数趋于统一，在质量上与标量的一维切分一致。