2.1. Theoretical framework of tensor alignment

Studying the details of the energy flux between scales in turbulence has historically been challenging. Filtering provides a foundational framework for analysing scale interactions in nonlinear systems (Germano Reference Germano1992; Liu et al. Reference Liu, Meneveau and Katz1994) and has been applied across a wide range of physical systems (Rai et al. Reference Rai, Hecht, Maltrud and Aluie2021; Storer et al. 2022, Reference Storer, Buzzicotti, Khatri, Griffies and Aluie2023; Yang et al. Reference Yang2023; Zhang et al. Reference Zhang2023; Li et al. Reference Li, Wang, Danilov, Koldunov, Liu, Müller, Sidorenko and Jung2024). When a nonlinear equation is filtered at a characteristic length scale, the filtering process introduces additional terms that encapsulate the interactions between the retained and eliminated degrees of freedom. These terms effectively serve as sources or sinks for the dynamics of the retained scales. For example, applying a low-pass filter – which removes flow structures smaller than a cutoff scale $ L$ – to the N–S equations results in a subgrid-scale stress tensor, $\tau _{\textit{ij}}^{({L})} = (u_i u_j)^{({L})} - u_i^{({L})}u_j^{({L})}$ , that captures the momentum transfer across the scale $ L$ (Pope Reference Pope2000).

A similar approach to the filtered kinetic energy, defined as $\textrm {E}^{({L})} = ( {1}/{2})u_i^{({L})}u_i^{({L})}$ , yields an expression for the spectral energy flux: $\varPi ^{({L})} = - \tau _{\textit{ij}}^{({L})} s_{\textit{ij}}^{({L})}$ , which quantifies the energy transfer between unresolved and resolved scales under a low-pass filter. Essentially, $\varPi ^{({L})}$ is the fundamental generator of turbulent energy flux, and significantly increasing its magnitude without substantially increasing Re is the goal of the paper. Here, $u_i$ denotes the $i{\rm th}$ component of the fluctuating velocity field and $s_{\textit{ij}}^{({L})} = (1/2)(\partial u_i^{({L})}/\partial x_j + \partial u_j^{({L})}/\partial x_i)$ represents the filtered rate of strain tensor. In this framework, $\tau _{\textit{ij}}^{({L})}$ and $s_{\textit{ij}}^{({L})}$ can be viewed analogously to force and displacement, respectively, such that their inner product reflects the work performed by the unresolved scales on the resolved ones through scale $L$ .

This perspective underscores the critical role of the geometric alignment between the stress tensor $\tau _{\textit{ij}}^{({L})}$ and the rate of strain tensor $s_{\textit{ij}}^{({L})}$ . When the two tensors are aligned, $\varPi ^{({L})} \lt 0$ , indicating an inverse energy flux towards larger scales. In contrast, if the tensors are anti-aligned, $\varPi ^{({L})} \gt 0$ , signifying a forward cascade towards smaller scales. Furthermore, $\varPi ^{({L})}$ can be recast as a function of the angular alignment between the eigenframes of $\tau _{\textit{ij}}^{({L})}$ and $s_{\textit{ij}}^{({L})}$ . In 2-D flows, this relationship takes the form (Liao & Ouellette Reference Liao and Ouellette2014; Fang & Ouellette Reference Fang and Ouellette2016)

(2.1) \begin{align} \varPi ^{({L})} = -2 \sigma ^{({L})} \gamma ^{({L})} \cos \big(2\theta ^{({L})} \big), \end{align}

where $ \sigma ^{({L})}$ and $ \gamma ^{({L})}$ are the principal extensional eigenvalues of the rate of strain tensor and the deviatoric part of the stress tensor, respectively, and $\theta ^{({L})}$ is the angle between their corresponding extensional eigenvectors $\hat {\sigma }^{({L})}$ and $\hat {\gamma }^{({L})}$ (see supplementary material).

This formulation clearly shows that the orientation between $\tau _{\textit{ij}}^{({L})}$ and $s_{\textit{ij}}^{({L})}$ not only determines the direction of the energy flux but also modulates its magnitude. When $\theta ^{({L})} \lt \pi /4$ , the energy flux is directed towards larger scales, corresponding to an inverse energy transfer. Conversely, when $\theta ^{({L})} \gt \pi /4$ , energy is transferred towards smaller scales, indicating a forward cascade. No net energy flux occurs when $\theta ^{({L})} = \pi /4$ . In conventional isotropic 2-D turbulence, the alignment between these tensors is governed by self-organisation, typically resulting in a net inverse energy flux. Fang & Ouellette (Reference Fang and Ouellette2016) have proposed using $\eta ^{({L})} = \cos (2\theta ^{({L})})$ as a quantitative measure of spectral flux efficiency, with experiments reporting a relatively low efficiency of about 27 % in isotropic 2-D turbulence. As Re decreases, both $ \gamma ^{({L})}$ and $\theta ^{({L})}$ in (2.1) decay rapidly to 0 and $\pi /4$ , respectively, leading to laminar flow with only large-scale features and negligible spectral energy flux.

In principle, a laminar flow with spatially varying velocity naturally exhibits a rate of strain tensor with non-negligible extensional eigenvalues. If a directionally biased perturbation, which can be represented spectrally as a stress tensor $\tau _{\textit{ij}}^{({L})}$ , is applied, then this externally imposed stress can couple with the existing rate of strain field in the low-Re regime to produce a non-negligible spectral energy flux. Moreover, the geometric alignment between the stress and rate of strain tensors can also be enhanced by a deliberately imposed perturbation direction. Importantly, the flow can still be kept in a low-Re regime with the perturbation applied. As a result, it becomes possible to generate a much stronger inter-scale energy flux at low Re, despite remaining in a nominally non-inertia-dominated regime.

Here, we define the Re as $\textit{Re}_\alpha = u / (L_i \alpha )$ instead of $\textit{Re}_\nu = u L_i/ \nu$ , where u is the root-mean-square velocity, $L_i$ is the characteristic length scale – chosen to be the spacing between rods (1.5 cm) – $\alpha$ is the linear friction coefficient and $\nu$ is the kinematic viscosity (Boffetta & Ecke Reference Boffetta and Ecke2012). The rod spacing is used because it represents the only relevant energy injection length scale in our experimental set-up. In the quasi-2-D flow studied here, energy dissipation is dominated not by viscosity but by linear friction with the bottom boundary. As a result, the Re defined in this context characterises the ratio of inertial forcing to frictional dissipation. From a length-scale perspective, the classical Reynolds number can be interpreted as $\textit{Re}_\nu \sim (L_f/L_d)^{4/3}$ , where $L_f$ and $L_d$ are the forcing and Kolmogorov scales. In direct analogy, $\textit{Re}_\alpha \sim L_\alpha / L_i$ is the ratio between the friction and forcing scales (Boffetta & Ecke Reference Boffetta and Ecke2012). Thus, like $\textit{Re}_\nu $ , $\textit{Re}_\alpha $ is also a ratio of dynamically relevant length scales, but with a different scaling exponent reflecting the dissipation mechanism. This formulation provides a more physically appropriate measure of momentum–dissipation balance in flows where conventional viscous effects are subdominant. Previous work on 2-D electromagnetic turbulence has shown that $\textit{Re}_\alpha $ is the relevant control parameter for characterising the onset and degree of turbulence in such friction-dominated systems (Shin et al. Reference Shin, Coletti and Conlin2023). In particular, key statistical descriptors of turbulent fluctuations collapse onto master curves when plotted against $\textit{Re}_\alpha $ . These results justify the use of $\textit{Re}_\alpha $ as the appropriate definition for 2-D flows. For simplicity of notation, $\textit{Re}_\alpha $ will be denoted as $\textit{Re}$ for the remainder of this paper.

Furthermore, as we show in figure 4(j), this ability to directly access the same stress–strain-rate energy-transfer pathway that underlies classical turbulence – without the need for inertial dominance – does not depend on how the Reynolds number is defined.

Figure 1.

Experimental set-up and the characterisation of the background flow and physical perturbations. (a) A schematic of the experimental set-up 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 2-D 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 $s_{\textit{ij}}^{({L})}$ ( $\hat {\sigma }^{({L})}$ ) of the hydrodynamic shear. (c) The flow field and $\hat {\sigma }^{({L})}$ of cellular flow. Red double-headed arrows indicate $\hat {\sigma }^{({L})}$ , 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 $\tau _{\textit{ij}}^{({L})}$ ( $\hat {\gamma }^{({L})}$ ).

Figure 1. Long description

Panel A: A schematic of the experimental setup for cellular flow. The setup includes electrodes, an electrolyte solution, a blackout film, magnets, tracer particles, a rod array, and a glass floor. The magnets generate a vertical magnetic field that interacts with a horizontal direct current to create a Lorentz force on the fluid. The tracer particles on the fluid’s surface represent the two-dimensional space under study. The rod array is controlled by a linear actuator. Panel B: A flow field diagram showing the extensional direction of hydrodynamic shear. Panel C: A flow field diagram of cellular flow with red double-headed arrows indicating the extensional direction and black arrows representing the velocity field. Panel D: A flow field generated by rods moving in a quiescent fluid. Panel E: A field diagram of the extensional direction of the flow.

2.2. Experimental methods

To apply this theoretical framework for significantly enhancing spectral energy flux at Re of the order of unity, we conducted a series of controlled experiments using an electromagnetically driven thin-layer flow system (Fang & Ouellette 2017, Reference Fang and Ouellette2018; Fang et al. Reference Fang, Balasuriya and Ouellette2019; Si & Fang Reference Si and Fang2024; Si et al. Reference Si, De Lillo, Boffetta and Fang2025). The working fluid is a uniform layer of saltwater (14 % NaCl by mass in deionised water) measuring 96.5 cm $\times$ 83.8 cm $\times$ 0.5 cm. This fluid sits on a glass substrate above permanent magnets. The magnets can be organised to generate a variety of flow patterns (see supplementary material). Two distinct base flow configurations were established – shear and cellular flow – each generated via Lorentz body forces arising from the interaction between a magnetic field produced by permanent magnets and a direct current across a thin electrolyte layer. Both flow types exhibited well-organised, large-scale rate of strain tensor structures.

Controlled perturbations were introduced through periodic oscillation of a $4 \times 4$ rod array actuated by a programmable linear actuator. For all the experiments, we make the rod move at around 2.5 times the background root-mean-square velocities, where the perturbation is strong enough to activate stress–strain interactions, yet does not dominate the background flow. The extensional direction of $\tau _{i,j}^{(L)}$ is aligned with the moving direction (figure 1 d,e), which is predicted by Si et al. (Reference Si, De Lillo, Boffetta and Fang2025). By precisely tuning the mechanical angle $\theta$ between the strain eigenvector $\hat {\sigma }^{({L})}$ of the background flow and the direction of the monopolar force a priori, we can achieve the manipulation of energy flux magnitude. The flow field, spanning $20\,{\rm cm} \times 20\,{\rm cm}$ , was recorded using an industrial camera and analysed via a particle tracking velocimetry algorithm (Ouellette et al. Reference Ouellette, Xu and Bodenschatz2006).

2.3. Enhanced scale-to-scale energy flux at Reynolds numbers of order one

In figure 2, we observed a significant energy flux enhancement in both shear and cellular flows. We had two control cases. The first is the background flow. We will contrast the energy flux magnitude of the background flow with that of the enhanced cases. In the second case, the rod array was moved in a quiescent fluid to rule out the possibility that the enhanced energy flux was due to its motion alone. Crucially, these two controls do more than rule out either ingredient in isolation. To test directly whether the enhancement requires a dynamical interaction between the two ingredients, we constructed the linearly superposed field from the independently measured background and rod-only experiments and evaluated its flux with an identical pipeline (supplementary material, figures S8 and S9). This superposed field contains both ingredients but no dynamical interaction between them, and it yields a mean flux more than an order of magnitude smaller than the measured coupled flow. The enhancement therefore cannot be obtained by superposing the two flows: it requires a nonlinear, dynamical interaction between the background strain field and the perturbation-induced stress.

Figure 2.

Experimental results of energy flux enhancement. (a,b) Scale-dependent energy flux, $\varPi ^{({L})}$ , in shear flow and cellular flow. The panels outlined in orange (left) correspond to shear flow, whereas the panels outlined in green (right) correspond to cellular flow. (c,e) The tensor orientation without perturbation. (d,f) The tensor orientation with perturbation for $\theta \approx 0$ . Red and blue arrows mark the extensional directions of $s_{\textit{ij}}^{({L})}$ and $\tau _{\textit{ij}}^{({L})}$ , and colour indicates instantaneous $\varPi ^{({L})}$ . The results in (c–f) are based on a cutoff length scale of 0.7. Dashed boxes show the analysed subdomain. All lengths are normalised by half of the measurement domain size W. Energy flux $\varPi ^{({L})}$ is normalised 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.

Experiments were grouped based on the mechanical angle $\theta$ . One obvious observation one can make is that the direction of energy flux can be manipulated based on $\theta$ a priori (Si et al. Reference Si, De Lillo, Boffetta and Fang2025). In this paper, we focus on the significant enhancement of the energy flux magnitude at low Re. At Re of the order of one, we observe that the energy flux magnitude is significantly enhanced in $\theta \approx 0$ and $\theta \approx \pi /2$ cases (figures 2 a and 2 b).

In addition, figure 2(a,b) shows a broad region ( $0.2W \lesssim L \lesssim 1.2W$ ) in which the spectral energy flux remains relatively constant, indicating that the flux enhancement is robust across the resolved filter scales rather than an artefact tied to a single cutoff choice. Although the rods introduce small-scale disturbances, the resulting stress–strain-rate alignment is not confined to the perturbation scale; instead, it remains coherent across all resolved filter lengths. To identify which scales dominate the energy flux, we examine a band-resolved decomposition of the stress following Xiao et al. (Reference Xiao, Wan, Chen and Eyink2009) (supplementary material, section ‘Scale locality of the enhanced energy flux’, figure S7). The flux is concentrated in the bands containing the rod spacing and is transferred directly between the rod scale and the filter scale, rather than through a progressive, scale-by-scale cascade. This is not surprising because, at low Re, stress is consistently added at the rod scale, and this rod-scale stress nonlinearly couples with the large-scale strain-rate fields to initiate energy flux according to (2.1). The relatively narrow scale separation is inherent to the experimental apparatus, as electromagnetically driven thin-layer 2-D turbulent flows typically exhibit less than one decade of scale separation in the literature (Kelley & Ouellette Reference Kelley and Ouellette2011; Liao et al. Reference Liao, Kelley and Ouellette2012; Liao & Ouellette 2013, Reference Liao and Ouellette2014; Fang & Ouellette 2016, Reference Fang and Ouellette2017; Si et al. Reference Si, De Lillo, Boffetta and Fang2025).

Tensor geometry provides a clearer understanding of the energy flux magnitude enhancement. In figure 2(c–f), we show the tensor alignment ( $\theta ^{(L)}$ ) between the extensional direction of stress and the rate of strain tensors. Without manipulation, shear (figure 2 c) and cellular (figure 2 e) flows have $\theta ^{(L)}$ mostly close to $\pi /4$ leading to very small energy flux magnitudes. With directionally biased physical perturbation at $\theta \approx 0$ , the perturbed shear (figure 2 d) and cellular (figure 2 f) flows have stress and rate of strain eigenframe well aligned ( $\theta ^{(L)} = 0$ ), causing an enhanced inverse energy flux and close-to-optimal $ \eta ^{({L})}$ .

Figure 3.

Experimental results of stress–strain orientation statistics. (a,f) Probability density functions (PDFs) of $\theta ^{(L)}$ . The panels outlined in orange correspond to shear flow, whereas the panels outlined in green correspond to cellular flow. (b,g) Efficiency $\eta ^{({L})}$ without perturbation and (d,i) efficiency with perturbation for shear and cellular flows. The results in (b,d,g,i) use a cutoff length of 0.7. (c,e,h,j) Corresponding particle trajectories over the past 1.5 s, with background colour indicating the vorticity field based on fluctuation velocity at the final time. (k) Quiescent perturbation case showing trajectories and vorticity field based on fluctuation velocity. The dashed box marks the subdomain used for subsequent analysis. All lengths are normalised by $W$ . Representative Reynolds numbers are the same as in figure 2.

In figures 3(a) and 3(f), we observe a clear correlation between $\theta$ and $\theta ^{(L)}$ even at Re of the order of unity. For background flows, pure perturbation and $\theta \approx \pi /4$ cases, we only observe a weak asymmetry around $\pi /4$ as compared with the $\theta \approx 0$ and $\theta \approx \pi /2$ cases. As a consequence, we see that the spatial distribution of $ \eta ^{({L})}$ is significantly enhanced for $\theta \approx 0$ cases for both shear and cellular flows. This enhancement is directly visible in the spatial distribution of $ \eta ^{({L})}$ (figure 3 b,d,g,i). In physical space, we observe that unperturbed shear (figure 3 c) and cellular (figure 3 h) flows show smooth, laminar fluid trajectories. On the other hand, perturbed shear (figure 3 e) and cellular (figure 3 j) flows show chaotic trajectories with complicated vorticity fields, even though their Re remains similar to that of the unperturbed cases in figures 3(c) and 3(h).

Although the largest instantaneous vorticity based on fluctuation velocity is seemingly around the rods, the high-vorticity structures extend over a region nearly an order of magnitude larger than a single rod diameter ( $0.02W$ ). Moreover, their geometry is also markedly different from the classical symmetric wake patterns produced by uniform flow past a cylinder. This indicates that the observed vorticity is not a trivial local perturbation generated by the rods, but instead results from multiscale interactions within the flow. Figure 3(k) shows the vorticity of the fluctuation velocity field from the rod perturbation in the quiescent fluid. We observe a very small amount of vorticity, which is consistent with the interpretation that the vorticity observed in figures 3(e) and 3(j) arises from the nonlinear, cross-scale coupling according to (2.1), consistent with the linear-superposition control reported above. In addition, our rod diameter is more than an order of magnitude smaller than the filter length (rod diameter $\approx$ 0.02W, filter length $=$ 0.7W) used for computing spectral energy flux. Thus, all flux-related diagnostics probe flow structures far above the immediate perturbation scale. We verified that excluding a narrow neighbourhood around the rods does not materially change the $ \eta ^{({L})}$ or flux statistics.

Notably, the tensor geometry framework remains effective even at Re of the order of unity, where inertial forces are non-dominant. In such regimes, flows are conventionally classified as laminar, and persistent spectral energy flux is generally considered absent in such regimes. However, our results challenge this assumption: despite the low Re, we observe clear and non-negligible scale-to-scale energy transfer, driven by the alignment between $s_{\textit{ij}}^{({L})}$ and $\tau _{\textit{ij}}^{({L})}$ . This behaviour is rooted in the N–S equations. The Re characterises the relative strength of advection to dissipation – viscosity in three-dimensional flows, or linear friction in quasi-2-D systems. When this ratio is finite, even if small, advection remains dynamically important. Consequently, the tensorial interactions responsible for energy flux – governed by stress–strain alignment – persist and remain meaningful. These observations indicate that the mechanisms underpinning spectral energy transport are not exclusive to high-Re turbulence but can extend into regimes previously deemed dynamically trivial.

Under generic forcing conditions at low Re, where advection and dissipation are comparable, spectral energy transfer between scales remains negligible and, as a spatial manifestation, flows exhibit a laminar behaviour. This limited flux arises from the geometric misalignment between $s_{\textit{ij}}^{({L})}$ and $\tau _{\textit{ij}}^{({L})}$ and negligible $\tau _{\textit{ij}}^{({L})}$ . However, our results demonstrate that if perturbations are engineered to align these two tensors, appreciable, force-sustained energy transfer between scales can be induced, even when $\textit{Re} \sim \mathcal{O}(1)$ . This finding suggests that a turbulent-like energy flux, traditionally associated with high-Re regimes, can be actively generated in nominally laminar systems via geometric control of tensor alignment.

Figure 4.

Energy flux enhancement and tensor-level contributions. (a,c) Energy flux $\varPi ^{({L})}$ as a function of $Re$ . The panels outlined in orange (left) correspond to shear flow, whereas the panels outlined in green (right) correspond to cellular flow. (b,d) Energy flux ratio between enhanced and background flows at different Re. (a–d) Obtained using block-averaged statistics, where error bars denote the 95 % confidence interval based on independent forcing cycles. (e,g) Eigenvalue magnitudes of $s_{\textit{ij}}^{(L)}$ . (f,h) Eigenvalue magnitudes of $\tau _{\textit{ij}}^{(L)}$ . (i,k) Efficiency $\eta ^{({L})}$ . (j) Energy flux $\varPi ^{({L})}$ versus system kinetic energy for background, perturbed and previously reported weak 2-D turbulent flows. Flows with magnitudes of $\varPi ^{({L})}$ above the dashed line are typically considered as in a weak turbulent regime (Liao et al. Reference Liao, Kelley and Ouellette2012; Liao & Ouellette 2013, Reference Liao and Ouellette2014, Reference Liao and Ouellette2015a , Reference Liao and Ouelletteb ; Fang & Ouellette 2016, 2017, Reference Fang and Ouellette2021; Kelley & Ouellette Reference Kelley and Ouellette2011; Ni et al. Reference Ni, Voth and Ouellette2014). Panels (a–h,i,k) share the same legend as figure 2. All results use a cutoff length of 0.7.

In the following, we compare the magnitude of energy flux enhancement observed near Re of unity, and systematically examine each constituent quantity in the energy flux expression (2.1) – $ \gamma ^{({L})}$ , $ \sigma ^{({L})}$ and $ \eta ^{({L})}$ – for both manipulated and background flows. Figures 4(a) and 4(c) illustrate the enhancement of spectral energy flux across a range of Re for both shear and cellular background flows. Because the physical perturbation is an additional forcing that itself changes Re, it is not feasible to generate perturbed and unperturbed flows with exactly matching Re. Thus, we perform a piecewise fit of the energy flux as a function of Re to capture the trend. To conservatively quantify enhancement, we compute the ratio of energy flux in the manipulated flow to that of the unperturbed background flow at the highest available Re (figures 4 b and 4 d). When Re is near unity, we observe a spectral energy flux strength enhancement up to 300 times. This enhancement is evident in both the $\theta \approx \pi /2$ and $\theta \approx 0$ configurations.

The magnitude of the spectral energy flux is governed by (2.1). To understand the origin of flux enhancement, we analyse each constituent quantity for $\varPi ^{(L)}$ . For both shear and cellular background flows, we observe that $ \sigma ^{({L})}$ increases approximately linearly with Re (figures 4 e and 4 g). In all manipulated and purely perturbed cases, $ \sigma ^{({L})}$ is consistently smaller than that of the background flow. This observation is consistent with physical intuition: the rate of strain tensor is a large-scale quantity, and unperturbed background flow should have the strongest rate of strain. In contrast, all perturbed cases exhibit a significant increase in $ \gamma ^{({L})}$ relative to the background flow (figures 4 f and 4 h). This is because the stress tensor is a small-scale quantity. Since the background flow is laminar, even modest additions of small-scale perturbations can substantially amplify $ \gamma ^{({L})}$ . Furthermore, we find that the efficiency of energy flux ( $ \eta ^{({L})}$ ), as governed by the tensor alignment angle $\theta ^{(L)}$ , is significantly enhanced in both $\theta \approx 0$ and $\theta \approx \pi /2$ configurations (figures 4 i and 4 k). These results are consistent with our theoretical framework and confirm that appreciable inter-scale energy transfer can be actively generated in nominally laminar systems via geometric control of tensor alignment.

Based on these observations, we conclude that the enhancement of spectral energy flux at $\textit{Re} \sim \mathcal{O}(1)$ arises from two primary mechanisms. First, the introduction of small-scale directional perturbations significantly increases $ \gamma ^{({L})}$ . Second, once the directional biased perturbations favour mechanical angles that promote energy flux ( $\theta \approx 0$ and $\theta \approx \pi /2$ ), they enhance $ \eta ^{({L})}$ by aligning the stress and rate of strain tensors thereby improving energy flux efficiency ( $\theta ^{(L)} \approx 0$ and $\theta ^{(L)} \approx \pi /2$ ). As a consequence, $ \eta ^{({L})}$ increases from near zero to values approaching 1 and $-1$ , respectively. Together, these two mechanisms enable a remarkable amplification of spectral energy flux, even in nominally laminar flows.

Figure 4(j) presents the magnitude of spectral energy flux $\varPi ^{({L})}$ against the total kinetic energy in the system. While the precise value of the friction coefficient $\alpha$ is often not reported in prior studies, precluding direct calculation of Re, the experimental apparatus used in the literature closely resembles our set-up. Assuming a comparable $\alpha$ and adopting a consistent length scale $L_i$ , Re becomes effectively proportional to the root-mean-square velocity. We therefore use the kinetic energy per unit mass, E, as a practical proxy for Re. Our results show that in both shear and cellular configurations, the background flow follows a similar trend as literature data – namely, spectral energy flux increases with growing kinetic energy. Strikingly, in the perturbed cases, we observe a substantial enhancement of energy flux even at very low kinetic energy levels. The horizontal dashed line marks a commonly accepted threshold above which 2-D flows are considered to be in a weak turbulent regime. The weak-turbulence threshold is defined based on the energy-flux magnitude and is chosen as the lower bound of the values reported in the literature. Our perturbed flows surpass this threshold despite their low kinetic energy content ( $\textit{Re} \sim \mathcal{O}(1)$ ), indicating that they already manifest a hallmark of turbulence – substantial energy transfer between different length scales – well before the inertial forces dominate.

The fact that our perturbed flows cross the weak-turbulence threshold (dashed line) at dramatically lower E than both the unperturbed flow and existing literature data demonstrates that the enhancement we achieve is independent of how Re is defined. Even when inertia is minimal, our perturbation method drives the system to a turbulent-like energy flux that normally requires substantially higher inertial forcing in conventionally driven flows.