Lagrangian dispersion statistics of compressible turbulence

Abstract

Turbulent transport of mass, momentum and energy are ubiquitous across physical processes spanning a vast expanse of scales, from the formation of clouds, the transport of pollutants and nutrients across oceans, and the spread of airborne diseases to those leading to the birth of stars and planets. While the density of a fluid is not ideally constant, most terrestrial flows can be reasonably approximated as incompressible. The dispersion of Lagrangian particles or tracers in incompressible turbulence is a fascinating topic which has been subject to immense research, especially because turbulent velocity fluctuations are highly intermittent and non-universal. However, in compressible turbulence, which is widely prevalent in various astrophysical systems, there is a paucity of comprehensive understanding of tracer dispersion. There, shocks not only alter the intermittency of the flows but also induce the clustering of tracers, profoundly affecting their dynamics.

In this thesis, the statistics of the dispersion of tracers are investigated in different hydrodynamic models of compressible turbulence, ranging from the randomly-forced Burgers equation (also called Burgers turbulence) representing a pressure-less and irrotational flow, to the more complex compressible Navier-Stokes equations. Defining a Lagrangian interval-collapse-time as the time taken for the separation between a pair of tracers to become infinitesimal as they approach each other at a shock, a hitherto-unexplored, unique methodology to study dynamic scaling in compressible turbulence is proposed in Chapter 2. The distributions of these collapse-times obtained from the direct numerical simulation (DNS) of Burgers turbulence in one-spatial dimension (1D) unveil that 1D Burgers turbulence indeed exhibits dynamic multiscaling. We formulated a heuristic theoretical framework for the scaling of these collapse-time distributions whose predictions are in close agreement with the DNS results, thus providing analytic insights into the dynamic scaling properties of such flows.

Extension to higher-dimensional systems implies analyzing the collapse-times of the respective simplices of tracers – triangles in two and tetrahedra in three spatial dimensions. However, a more pertinent issue from both a fundamental and an application perspective is the effect of shocks on the statistical properties of relative or pair dispersion of tracers and its intermittency, the latter being a tell-tale sign of dynamic multiscaling. Seeking unprecedented insights into this, we analyze the characteristics of Lagrangian exit-times – the doubling and the halving times of inter-particle separations – as outlined in Chapter 3. Their distributions, evaluated from our DNS of 2D Burgers turbulence where shocks are filamentary structures, clearly show that the doubling and halving time statistics are different. This demonstrates that the pair dispersion of tracers in compressible turbulence is (a) indeed intermittent and (b) considerably different from that in incompressible turbulence. The moments of these exit-times accurately quantify the intermittency of pair dispersion. We again developed a heuristic theoretical framework – a generalisation of the multifractal model of pair dispersion in turbulence – whose prediction of the scaling of the exit-time statistics agrees excellently with our DNS results.

Next, we explored the pair dispersion of tracers in the turbulent flow of an isothermal gas in 2D, governed by the compressible Navier-Stokes equations, via high-resolution numerical simulations in Chapter 4. Turbulence is driven by both solenoidal and irrotational large-scale forces, and ranges from transonic to supersonic. Unlike 2D incompressible turbulence with a large-scale forcing, the mean-squared inter-particle separation grows as a power–law in time across spatial scales smaller than the wavelength of the external force. The intermittency of pair dispersion is accurately quantified by Lagrangian exit-times with the halving and doubling-time statistics being different from each other again. The halving-time statistics appear universal. However, the doubling-time statistics depend on the turbulent Mach number Ma when the external force is solenoidal, and a bridge relation is obtained that expresses them solely in terms of the statistics of the fluctuations of the solenoidal component of the velocity, indicating that the separation of the tracers is predominantly mediated by the vortices. Therefore, the Ma−dependence is a reflection of a change in the topology of the strongest vortices. In contrast, when the force is purely irrotational, the separation of tracers is also significantly influenced by the large flow divergences, and the doubling-time statistics are independent of the Mach number.

Finally, in Chapter 5, it is shown that the large-scale transport of tracers in different forms of isothermal compressible turbulence, governed by the Navier-Stokes equations in 2D, are qualitatively different. In particular, when the kinetic energy of the irrotational component of the flow exceeds that of the solenoidal component, tracers undergo heterogeneous diffusion which gives rise to their superdiffusive transport that persists up to very late times. However, when it is the other way around, they exhibit diffusion with a constant effective diffusivity at late times, akin to those in incompressible turbulence. Tracers also undergo superdiffusion in 2D Burgers turbulence although the underlying transport mechanism is different. The dynamics of shocks in Burgers turbulence offer useful insights, aided by which we surmise that the superdiffusion of tracers in compressible Navier-Stokes turbulence is driven by the persistent motion of the clusters of tracers that are entrained to the shocks.

This thesis provides new insights into the fundamental aspects of tracer dispersion in various models of compressible turbulence. These findings have far-reaching implications for astrophysical processes where compressible turbulence plays a crucial role, such as star and planetesimal formation in molecular clouds, accretion of matter onto stars, black holes, and protoplanets, as well as the transport of cosmic rays and heat in the interstellar medium.