Turbulence at Low Reynolds Numbers

TL;DR

This work shows that turbulence-like energy transfer can persist at $\mathrm{Re} \sim \mathcal{O}(1)$ by reframing the inter-scale flux as mechanical work between scale-dependent stress and rate-of-strain in a 2D setting. By applying directionally biased perturbations to electromagnetically driven quasi-2D flows, the authors align the eigenframes of $\tau_{ij}^{(\rm L)}$ and $s_{ij}^{(\rm L)}$, greatly amplifying the spectral energy flux $\Pi^{(\rm L)}$ (up to about $800\times$) even when inertial forces are not dominant. The main contributions are: (i) a concrete tensor-alignment framework predicting flux magnitude via $\Pi^{(\rm L)} = -2\sigma\gamma\cos(2\theta^{(\rm L)})$, (ii) experimental demonstration of large low-Re energy transfer in both shear and cellular 2D flows, and (iii) identification of practical routes to engineer multiscale transport in microfluidic and biological systems lacking inertia. These findings broaden turbulence concepts to non-inertia-dominated regimes and suggest new strategies for enhanced mixing in low-Re environments, with a straightforward path to extending the approach to 3D flows.

Abstract

Turbulence -- ubiquitous in nature and engineering alike [1-5] -- is traditionally viewed as an intrinsically inertial phenomenon, emerging only when the Reynolds number (Re), which quantifies the ratio of inertial to dissipative forces [6], far exceeds unity [7, 8]. Here, we demonstrate that strong energy flux between different length scales of motion -- a defining hallmark of turbulence [9] -- can persist even at Re ~ 1, thereby extending the known regime of turbulent flows beyond the classical high-Re paradigm. We show that scale-to-scale energy transfer can be recast as a mechanical process between turbulent stress and large-scale flow deformation. In quasi-two-dimensional (quasi-2D) flows driven by electromagnetic forcing, we introduce directionally biased perturbations that enhance this interaction, amplifying the spectral energy flux by more than two orders of magnitude, even in the absence of dominant inertial forces. This study establishes a new regime of 2D Navier-Stokes (N-S) turbulence, challenging long-standing assumptions about the high Re conditions required for turbulent flows. Beyond revising classical belief, our results offer a generalizable strategy for engineering multiscale transport in flows that lack inertial dominance, such as those found in microfluidic [10, 11] and low-Re biological [12-15] systems.

Figures (7)

• Figure 1: Experimental setup and the characterization of the background flow and physical perturbations. (a) A schematic of the experimental setup for cellular flow. The vertical magnetic field from the permanent magnets interacts with the horizontal direct current to generate the Lorentz force on the fluid, which acts nearly within the plane. The tracer particle on the fluid’s surface represents the 2D space under study. A rod array is controlled by a linear actuator. The generation of shear flow only requires the rearrangement of magnets (see Supplementary Material). (b) The flow field and the extensional direction of $s_{ij}^{(\rm L)}$ ($\hat{\sigma}$) of the hydrodynamic shear. (c) The flow field and $\hat{\sigma}$ of cellular flow. Red double-headed arrows indicate $\hat{\sigma}$, and the black arrows are the velocity field (down-sampled for clarity). (d) Flow field generated by rods moving in a quiescent fluid. (e) Field of extensional direction of $\tau_{ij}^{(\rm L)}$ ($\hat{\gamma}$).
• Figure 2: Experimental results of energy flux enhancement. This figure illustrates how the added directionally biased physical perturbation can significantly increase the energy flux by enhancing the tensor alignment. (a and b) The $\Pi^{(\rm L)}$ for different cutoff lengths for shear and cellular flows, respectively. (c and e) The tensor orientation without physical perturbation for shear and cellular flows, respectively. (d and f) The tensor orientation with physical perturbation for the case with mechanical angle $\theta \approx 0$ for shear and cellular flows, respectively. Red double-headed arrows are the extensional direction of the local $s_{ij}^{(\rm L)}$ ($\hat{\sigma}$), and blue double-headed arrows are the extensional direction of the local $\tau_{ij}^{(\rm L)}$ ($\hat{\gamma}$). The background color represents the spatial distribution of instantaneous $\Pi^{(\rm L)}$. The results in c-f are based on a cutoff length scale of 0.7. The blue dashed box outlines the specific subdomain of the flow field used for all subsequent analysis presented in this paper. All lengths are normalized by half of the measurement domain size W. $\Pi^{(\rm L)}$ is normalized by the frictional dissipation $\alpha u^2$. For a, $Re$ = 1.23 (background), 0.15 (pure perturbation), 1.27 ($\theta \approx 0$), 1.45 ($\theta \approx \pi/4$), and 1.38 ($\theta \approx \pi/2$). For b, $Re$ = 1.43 (background), 0.04 (pure perturbation), 1.14 ($\theta \approx 0$), 1.26 ($\theta \approx \pi/4$), and 1.12 ($\theta \approx \pi/2$). For c-f, $Re$ = 1.23, 1.27, 1.43, and 1.14, respectively.
• Figure 3: Experimental results of stress-strain orientation statistics. (a and b) Probability density functions of $\theta^{(L)}$ for shear and cellular flows, respectively. (c and e) Efficiency, $\eta$, without physical perturbation for shear and cellular flows, respectively. (d and f) Efficiency, $\eta$, with physical perturbation for hydrodynamic shear and cellular flows. The results in c-f are based on a cutoff length scale of 0.7. The blue dashed box outlines the specific subdomain of the flow field used for all subsequent analysis presented in this paper. All lengths are normalized by half the domain size W. $\Pi^{(\rm L)}$ is normalized by the frictional dissipation $\alpha u^2$. Since the perturbed flows include additional energy input, the Re differs between perturbed and unperturbed cases. For a, $Re$ = 1.23 (background), 0.15 (pure perturbation), 1.27 ($\theta \approx 0$), 1.45 ($\theta \approx \pi/4$), and 1.38 ($\theta \approx \pi/2$). For b, $Re$ = 1.43 (background), 0.04 (pure perturbation), 1.14 ($\theta \approx 0$), 1.26 ($\theta \approx \pi/4$) and 1.12 ($\theta \approx \pi/2$). For c-f, $Re$ = 1.23, 1.27, 1.43 and 1.14, respectively. The legend is the same as that of Fig. \ref{['fig2']}.
• Figure 4: Energy flux enhancement and element-wise understanding. (a and b) $\Pi^{(\rm L)}$ for different Re with shear and cellular flows, respectively. (c and d) Energy flux ratio between enhanced and background flows at different Re with shear and cellular flow configurations. (e and g) Eigenvalue magnitudes of $s_{i,j}^{(L)}$ for shear and cellular flows, respectively. (f and h) Eigenvalue magnitudes of $\tau_{i,j}^{(L)}$ for shear and cellular flows, respectively. (i and k) Efficiency ($\eta$) for shear and cellular flows, respectively. (j) Magnitudes of $\Pi^{(\rm L)}$ versus kinetic energy contained in the system for background flow, perturbed flow and weak 2D turbulent flows in the literaturefang2017multiplefang2016advectionfang2021spectralliao2013spatialliao2014geometryliao2012forcingliao2015longrangeliao2015correlationskelley2011spatiotemporalni2014extracting. Flows with magnitudes of $\Pi^{(\rm L)}$ above the dashed line are typically considered as weak turbulence. Panels a-h, i, and k share the same legend as Fig. \ref{['fig2']}. All results are based on a cut-off length scale of 0.7. Shaded areas in a-d indicate 99% confidence interval.
• Figure S1: Top views of experimental setup. (a) Schematic of shear flow. Two lines of permanent magnets are arranged with opposite polarities to generate a shear flow in the middle of the domain. The grey walls on both sides reinforce the no-slip boundary condition at the edges of the shear flow. The magnetic field driving the shear flow in the central region is a secondary magnetic field arising from the magnet arrays. A 4×4 rod array (black grid) is positioned at the center to introduce controlled perturbations. The black arrow indicates the direction of the applied DC current, while the blue and red arrows (F) denote the Lorentz forces induced in opposite directions. (b) Schematic of cellular flow. Six groups of permanent magnets are arranged with alternating polarities to produce cellular flow patterns. As in the shear configuration, the black arrow indicates the direction of the DC current, and the blue and red arrows (F) represent the corresponding Lorentz forces.