Non-reciprocal Binary-fluid Turbulence

TL;DR

This work introduces a minimal 2D hydrodynamic model, NRCHNS, in which non-reciprocal interactions are encoded through off-diagonal diffusion between two Cahn–Hilliard fields coupled to Navier–Stokes flow. By tuning the non-reciprocal strength $\\mathcal{D}_{12}$ or lowering the viscosity, the system exhibits a novel turbulence regime with an inverse energy cascade and an energy spectrum $E(k) \\\sim k^{-5/3}$, accompanied by a non-reciprocal flux $\\mathbf{J}$ that becomes suppressed at high Reynolds numbers. The authors validate these findings with extensive pseudospectral DNS, revealing spectral plateaus, inverse-cascade flux balance, characteristic flow topology via Okubo–Weiss statistics, and multifractality in both spatial and temporal fluctuations. This non-reciprocal turbulence is qualitatively distinct from conventional 2D turbulence and active turbulence, offering new insights into energy transfer mechanisms and potential experimental realizations in non-reciprocal active fluids.

Abstract

Although effective non-reciprocal interactions have been investigated in a variety of fields, their consequences have not been explored in hydrodynamical turbulence. We initiate such an exploration by introducing non-reciprocal binary-fluid tubulence and uncover its properties by developing a two-dimensional (2D) Non-Reciprocal Cahn-Hilliard-Navier-Stokes (NRCHNS) model. We show that, as we increase the strength of the non-reciprocal terms, this model displays a hitherto unanticipated type of turbulence, with an inverse cascade of energy and an energy spectrum $E(k)\sim k^{-5/3}$, reminiscent of the well-known inverse cascade in forced, 2D fluid turbulence, but distinct from it, in so far as it develops a non-reciprocal flux $\mathbf J$. We demonstrate how NRCHNS turbulence suppresses $J(t) = |\mathbf J|$, as the Reynolds number increases. We compare and contrast 2D NRCHNS turbulence with its fluid-turbulence counterpart by examining spectra, fluxes, spectral balances, flow topologies, and signatures of multifractality.

Figures (6)

• Figure 1: Pseudocolor plots of fields in 2D NRCHNS turbulence at a representative time and with $\nu=0.0001$ and $\sigma_1=\sigma_2=3$: The non-reciprocity parameter $\mathcal{D}_{12}=0.12$ (column 1), $\mathcal{D}_{12}=0.16$ (column 2), $\mathcal{D}_{12}=0.24$ (column 3), and $\mathcal{D}_{12}=0.33$ (column 4): (a)-(d) $\omega$, with corresponding pseudocolor plots for (e)-(h) $\phi$ and (i)-(l) $\psi$, respectively. For the full spatiotemporal evolution of these fields see the Supplementary Movies V1-V4.
• Figure 2: Pseudocolor plots for 2D NRCHNS turbulence at a representative time with the non-reciprocity parameter $\mathcal{D}_{12}=0.33$ and $\sigma_1=\sigma_2=3$: The kinematic viscosity $\nu=12$ (column 1), $\nu=0.01$ (column 2), $\nu=0.001$ (column 3), and $\nu=0.0001$ (column 4): (a)-(d) $\omega$, with corresponding pseudocolor plots for (e)-(h) $\phi$ and (i)-(l) $\psi$, respectively. For the full spatiotemporal evolution of these fields see the Supplementary Movies V5-V7 and V4.
• Figure 3: Temporal evolution of NRCHNS turbulence: Plots versus the scaled time $t\Omega_L$ showing (a) the normalised kinetic energy $\mathfrak{E}(t)/\mathfrak{E}_0$, where $\mathfrak{E}_0 \equiv \mathfrak{E}(t=\Omega_L^{-1})$; (b) the coarsening length scale of the composition field, $L_\phi(t)$; and (c) the corresponding length scale of the $\psi$ field, $L_\psi(t)$, for $\mathcal{D}_{12} = 0.12, 0.16, 0.24,$ and $0.33$. (d) Dependence on $\mathcal{D}_{12}$ of the time-averaged lengths $\langle L_\phi \rangle$ (red) and $\langle L_\psi \rangle$ (blue), together with the integral-scale Reynolds number $Re$ (purple curve and right vertical axis). The magnitude $J(t)$ of the nonreciprocal flux \ref{['eq:fluxJ']}, for (e) $\mathcal{D}_{12} = 0.12, 0.16, 0.24,$ and $0.33$, with fixed $\nu=0.0001$, and (f) for $\nu = 12, 0.01, 0.001,$ and $0.0001$, with fixed $\mathcal{D}_{12}=0.33$.
• Figure 4: The dependence on the wavenumber $k$ of energy and concentration spectra, spectral transfers, and fluxes for $\nu = 0.0001$: Log-log plots of compensated spectra versus the scaled wavenumber $k$, for $\mathcal{D}_{12}=0.16,\,0.24,\,0.33$, with power-law scaling regions shaded in gray: (a) $k^{5/3}E(k)$ [scaling region $3 \lesssim k \lesssim 13$]; (b) $k^{-1}S^{\phi}(k)$ [scaling region $1 \lesssim k \lesssim 4$]; (c) $k^{-1}S^{\psi}(k)$ [scaling region $1 \lesssim k \lesssim 4$]. For plots of the energy flux, $\Pi(k)$ , see Figs.(d) for different $\mathcal{D}_{12}=0.16, 0.24$, and $0.33$. Log-lin plots of $T^u(k)$(red), $T^{\phi}(k)$(green), $T^{\psi}(k)$(orange), and $2\nu k^2 E(k)$(blue) [see Eq. \ref{['eq:spectralbalance']}] versus $k$, in the NESS, for $\alpha=0.0$ and (e) $\mathcal{D}_{12}=0.16$ and (f) $\mathcal{D}_{12}=0.33$.
• Figure 5: Flow topology in NRCHNS turbulence for $\mathcal{D}_{12}=0.33$: (a) Normalised probability distribution functions (PDFs) of the Okubo–Weiss parameter $\Lambda$ [see Eq. \ref{['eq:Okubo']}] for $\nu=0.01, 0.001$, and $0.0001$ (the top two curves have been translated upwards for clarity). Pseudocolor plots of the normalised Okubo-Weiss paramemter $\Lambda/\Lambda_{max}$ for (b) $\nu=0.01$, (c) $\nu=0.001$, and (d) $\nu=0.0001$; contours of the normalized vorticity $\omega/\omega_{max}=-0.6$ (dashed-brown contours), $\omega/\omega_{max}=0$ (white contours), and $\omega/\omega_{max}=0.6$ (black contours) are superimposed.