Forced two-dimensional turbulence lives on the balance between the energy input and two dissipative mechanisms, viscosity and linear friction, resulting in a double cascade of energy and enstrophy. While it is known that the energy cascade is governed by the Reynolds number ${\mathrm{Re}}_{\alpha }=u_{\mathrm{rms}}/\left(\alpha L_{\mathrm{f}}\right)$, it has been more convenient to report $\mathrm{Re}=u_{\mathrm{rms}}{L}_{\mathrm{f}}/\nu$ (where $u_{\mathrm{rms}}$ is the fluctuating velocity, $L_{\mathrm{f}}$ the forcing scale, $\alpha$ the friction coefficient, and $\nu$ the kinematic viscosity). Therefore, it is unclear for which range of parameters the various hallmarks of fully developed turbulence will emerge. Here we use multiple laboratory setups in which a quasi-two-dimensional flow is generated in electromagnetic layers of fluids, over a wide range of $\mathrm{Re}$ and ${\mathrm{Re}}_{\alpha }$. The friction coefficient measured during turbulence decay is correctly estimated by a linear shear assumption, allowing us to readily estimate ${\mathrm{Re}}_{\alpha }$. We consider several observables characterizing the turbulence development: the fraction of energy input converted to fluctuating energy, the correlation scale of the flow, the velocity structure functions, the probability distribution of the velocity fluctuations and velocity differences, the single-particle diffusivity, and the separation time between particle pairs. All descriptors collapse on master curves against ${\mathrm{Re}}_{\alpha }$, providing a criterion for fully developed turbulence for this class of flows. Moreover, dedicated experiments in which the local $\mathrm{Re}$ and ${\mathrm{Re}}_{\alpha }$ are spatially decoupled show that only the latter is correlated with the growth of turbulent energy. Finally, a scaling relation is proposed that relates the amount of energy going to the large scales to the forcing scale-to-layer thickness ratio.

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• Transition to turbulence
• Turbulence
• Two-dimensional turbulence