Exact scaling laws in isotropic binary fluid turbulence

Abstract

Binary fluid turbulence distinguishes itself from ordinary fluid turbulence by virtue of interfacial dynamics. Whether Kolmogorov-like scaling laws also exist for binary fluid turbulence is a fundamental question to explore. Starting from tensor formalism à la von Kármán and Howarth, here we derive exact scaling laws for isotropic Cahn-Hilliard-Navier-Stokes (CHNS) turbulence both in terms of two point correlators and increments. In particular, we derive the CHNS analogs for $1/3$, $4/3$, $2/15$ and $4/5$ laws known for isotropic hydrodynamic turbulence and show that the new scaling laws contain contributions both from the bulk flow and interface. The $2/15$ and $4/5$ laws of CHNS turbulence are found to be expressed purely in terms of two-point correlators and structure functions and their derivatives, respectively. However, unlike their hydrodynamic counterparts, these relations involve additional contributions from non-longitudinal directions. By means of direct numerical simulations with up to $1024^3$ grid points, all the derived exact laws are numerically verified and the scale dependence of the cascade rates obtained from different exact laws are thoroughly compared. As one moves from the homogeneous (but not necessarily isotropic) divergence form to the isotropic $4/5$ form, the inertial range is found to shift towards larger scales with a comparatively flatter cascade rate profile as a result of successive integrations over the small scales.

Figures (4)

• Figure 1: Mid-plane snapshot of three-dimensional $\phi$ field for $512^3$ and $1024^3$ runs.
• Figure 2: (a,c,e) and (b,d,f): Plots of $\mathcal{C}^{1,2,3}_{rrr}$, $\mathcal{C}^{1,2,3}_{rnn}$, $\mathcal{C}^{1,2,3}_{nnr}$ and $\mathcal{C}^{1,2,3}_{rrr}+2\mathcal{C}^{1,2,3}_{nnr}$ for Run1 and Run2, respectively.
• Figure 3: (a) and (b): Scaling of isotropic exact relations in the correlator form ($1/3$ and $2/15$ laws) and structure function forms ($4/3$ and $4/5$ exact laws), for Run1 and Run2, respectively. The left and right vertical lines in both the plots represent the Kolmogorov scale and the integral scales, respectively whereas the shaded region represent the regime where the linear scaling w.r.t. $r$ holds for the respective exact laws.
• Figure 4: (a) Plots of total energy cascade rates $\mathcal{A}^{\mathcal{B}}_{div}(r)$, $\mathcal{A}^{\mathcal{B}}_{iso_1}(r)$ and $\mathcal{A}^{\mathcal{B}}_{iso_2}(r)$ in log-log scale for Run2. For better comparison, in the inset we plot a zoomed in version of the three with arbitrary shifting. (b) Plots of the corresponding two-point average energy dissipation rates $D(r)$, $D_{iso_1}(r)$ and $D_{iso_2}(r)$, two-point average energy injection rates $F(r)$, $F_{iso_1}(r)$ and $F_{iso_2}(r)$ along with the energy cascade rates in semi-log scale for Run2. The horizontal (dashed gray) lines on both the plot represent the constant energy injection rate $\varepsilon_{in}$ whereas the vertical lines on the left plot represent the Kolomogorv scale $\eta$ (left) and the integral scale $L$ (right), respectively.