Understanding the role of frictional drag in diffusive transport is an important problem in the field of active turbulence. Using a generic continuum model that applies well to living systems, we investigate the role of Ekman friction on the passive transport of Lagrangian tracers that go with the local flow. We find that the crossover from ballistic to diffusive regime happens at a time scale ${\tau }_c$ that attains a minimum at zero friction, meaning that both injection and dissipation of energy delay the relaxation of tracers. We explain this by proposing that ${\tau }_c\sim {\ell }^{*}/u_{\mathrm{\text{rms}}}$, where ${\ell }^{*}$ is a dominant length scale extracted from energy spectrum peak, and $u_{\mathrm{\text{rms}}}$ is a velocity scale that sets the kinetic energy at steady state, both scales monotonically decrease with friction. Finally, we predict friction scaling laws for ${\ell }^{*}$, $u_{\mathrm{\text{rms}}}$ and the diffusion coefficient $𝒟\sim {\ell }^{*}{u}_{\mathrm{\text{rms}}}/2$, that are valid over a wide range of fluid friction. The findings of our report should apply to transport phenomena in generic active systems such as dense bacterial suspensions, micro-tubule networks or even artificial swimmers, to name a few.

中文翻译：

---

主动湍流中的运输摩擦定律

理解摩擦阻力在扩散运输中的作用是主动湍流领域的重要问题。使用适用于生命系统的通用连续体模型，我们研究了埃克曼摩擦在拉格朗日示踪剂与局部水流一起被动传输中的作用。我们发现从弹道体制到扩散体制的转变发生在一个时间尺度上${\tau }_C$在零摩擦时达到最小值，这意味着能量的注入和耗散都会延迟示踪剂的松弛。我们通过提出以下解释${\tau }_C〜{\ell }^{*}/{ü}_{\mathrm{\text{均方根}}}$，在哪里 ${\ell }^{*}$ 是从能谱峰中提取的主要长度尺度，并且 ${ü}_{\mathrm{\text{均方根}}}$是一个将动能设定为稳态的速度标尺，两个标尺都随着摩擦而单调减小。最后，我们预测了${\ell }^{*}$， ${ü}_{\mathrm{\text{均方根}}}$ 和扩散系数 $𝒟〜{\ell }^{*}{ü}_{\mathrm{\text{均方根}}}/2$，适用于各种流体摩擦。我们报告的发现应适用于一般活动系统中的运输现象，例如致密的细菌悬浮液，微管网络甚至是人工游泳者，仅举几例。