Intermittency and non-universality of pair dispersion in isothermal compressible turbulence

Abstract

Statistical properties of the pair dispersion of Lagrangian particles (tracers) in incompressible turbulent flows provide insights into transport and mixing. We explore the same in transonic to supersonic compressible turbulence of an isothermal ideal gas in two dimensions, driven by large-scale solenoidal and irrotational stirring forces, via direct numerical simulations. We find that the scaling exponents of the order-$p$ negative moments of the distribution of exit times -- in particular, the doubling and halving times of pair separations -- are nonlinear functions of $p$. Furthermore, the doubling and halving time statistics are different. The halving-time exponents are universal -- they satisfy their multifractal model-based prediction, irrespective of the nature of the stirring. However, the doubling-time exponents are not. In the solenoidally-stirred flows, the doubling time exponents can be expressed solely in terms of the multifractal scaling exponents obtained from the structure functions of the solenoidal component of the velocity. Moreover, they depend strongly on the Mach number, Ma, as elongated patches of high vorticity emerge along shock fronts at high Ma. In contrast, in the irrotationally-stirred flows, the doubling-time exponents do not satisfy any known multifractal model-based relation, and are independent of Ma. Our findings are of potential relevance to astrophysical disks and molecular clouds wherein turbulent transport and mixing of gases often govern chemical kinetics and the rates of formation of stars and planetesimals.

Figures (11)

• Figure 1: Typical snapshots of $\nabla\cdot{\bf u}$, the vorticity ${\bm \omega}={\bm \nabla} \times\bm{u}$, and $\log_{10}\rho$, for solenoidally-driven supersonic turbulence, run S2 (${\bm \nabla}\cdot\bm{u}$ and ${\bm \omega}$ are normalized by their respective root-mean-square values). Thin filaments in the ${\bm \nabla}\cdot\bm{u}$ profile are the shocks. The vorticity, ${\bm \omega}$, is large in the vicinity of curved shocks. There are strong density gradients across shocks.
• Figure 2: Dynamic exponents for doubling and halving times (see text) $\chi^{\rm D}_{\rm p}$ and $\chi^{\rm H}_{\rm p}$, respectively, as functions of the order $p$ for the two supersonic runs [S2 (left panel) and C2 (right panel)]. The blue shaded region corresponds to the bridge relation, $p-\zeta_p$, derived from the multifractal model. The pink shaded region corresponds to the expression $p-\zeta_p^{\rm s}$. In both S2 and C2, the multifractal prediction for $\chi^{\rm H}_{\rm p}$ is correct within errorbars, but not for $\chi^{\rm D}_{\rm p}$. For S2 $\chi^{\rm D}_{\rm p} = p -\zeta_p^{\rm s}$ is correct within errorbars. For C2 $\chi^{\rm D}_{\rm p}$ does not satisfy any known bridge relation.
• Figure 3: Typical snapshots of $\nabla\cdot{\bf u}$, vorticity $\omega=\hat{z}\cdot{\bm \nabla} \times\bm{u}$, and $\log_{10}\rho$, in the cases (a) S1, (b) S2, (c) C1 and (d) C2. In each plot, ${\bm \nabla}\cdot\bm{u}$ and $\omega$ are normalized by their respective root-mean-square (rms) values. Shocks are clearly visible in the ${\bm \nabla}\cdot\bm{u}$ profiles as the blue filament--like structures. The vorticity field, $\omega$, is smoother in S1 than in S2 which contains nearly one-dimensional structures of intense vorticity bordering the shocks; in all figures, the areas of high vorticity lie in the vicinity of shocks Similarly the spatial distribution of $\omega$ is more homogeneous in C1 than in C2. The values of $\rho$ are clearly enhanced near the shocks; for a given type of external forcing, $\rho$ tends to take much higher values at higher Mach numbers, $\hbox{Ma}$.
• Figure 4: Kinetic energy flux for the four DNS runs. We obtain a range of direct cascade.
• Figure 5: Compensated velocity spectra Log-log plots of the compensated (by $k^2$) spectra of the total velocity (blue), $E(k)$, its compressive component (red), $E_{\rm c}(k)$, and its solenoidal components (violet), $E_{\rm s}(k)$ for the different runs. The spectra are normalized by $\sum_k E(k)$.