Multiscaling in three-dimensional fluid turbulence

Abstract

Fluid turbulence remains one of the most fascinating unsolved problems in Physics. Different aspects of it have attracted physicists and fluid dynamicists for decades. This thesis studies multiscaling of velocity structure functions and related issues in deterministic and stochastic models for fluid turbulence. Turbulent velocity fluctuations are generated in a viscous fluid typically by external forcing at large spatial scales. When the viscous energy losses in the fluid are compensated by maintaining a continuous injection of energy from outside, these fluctuations are sustained and the system reaches a nonequilibrium statistical steady state. Far away from boundaries and at length scales smaller than those at which energy is supplied to the fluid, this steady state is homogeneous and isotropic. These small scales can be divided into the inertial range, in which dissipation is not significant, and the dissipation range, at smaller length scales. The inertial?range multiscaling of velocity structure functions has been studied extensively by several groups, both experimentally and by direct numerical simulations (DNS) of the three?dimensional Navier–Stokes equation (3D NSE). In this thesis, we study the inertial?range multiscaling of velocity structure functions in the randomly forced Navier–Stokes equation (RFNSE), along with issues relating to structures of high?vorticity regions and probability distributions for velocity differences. We also examine the feasibility of obtaining the RFNSE from the 3D NSE by a coarse?graining procedure, similar to that used to map the deterministic Kuramoto–Sivashinsky (KS) equation onto the stochastic Kardar–Parisi–Zhang (KPZ) equation. We then examine wave?vector (k?space) velocity structure functions, especially in the dissipation range; we study these both via DNS of the 3D NSE and via a field?theoretical closure approximation. Finally, we examine the crossover from inertial? to dissipation?range asymptotics by developing k?space versions of Extended Self?Similarity (ESS) and Generalised Extended Self?Similarity (GESS).

This thesis is organized as follows: • Chapter 1 We give a brief survey of the phenomenology of homogeneous and isotropic fluid turbulence. We describe various experimental observations and some phenomenological explanations, using dimensional analysis, etc. We then introduce the 3D NSE and the RFNSE; earlier numerical results for the former and analytical studies for the latter are summarised. Some studies of related partial differential equations—such as the KS, KPZ, and Burgers equations—are mentioned, especially in the context of the coarse?graining procedure that maps the KS equation onto the KPZ equation. • Chapter 2 We study the RFNSE in detail via a direct numerical simulation (DNS) using a pseudospectral method. The RFNSE is stirred by a stochastic force with zero mean and variance ? k^?y, with k the wave vector and dimension d = 3. We provide the first evidence for multiscaling of velocity structure functions for p > 4. We extract the multiscaling exponent ratios ??/?? using extended self similarity (ESS), examine their dependence on y, and show that for y = 4, they agree with those obtained for the Navier–Stokes equation forced at large spatial scales (3D NSE). We also show that well?defined vortex filaments, which appear clearly in studies of the 3D NSE, are absent in the RFNSE. • Chapter 3 Motivated by the success of the RFNSE in yielding multiscaling of velocity structure functions, we carry out a numerical coarse?graining of the 3D NSE to test whether the RFNSE can be obtained from it. We find that although the procedure works well for the mapping KS ? KPZ, it does not lead to the RFNSE in the Navier–Stokes case. We examine the reasons for this critically. • Chapter 4 We propose and verify k?space versions of ESS and GESS by conducting a low?Reynolds?number DNS of the 3D NSE. Using this study (along with results from the GOY shell model for turbulence), we uncover a universal crossover from inertial? to dissipation?range asymptotic behaviours. • Chapter 5 We present a field?theoretical one?loop calculation of a stochastic 3D NSE, which explains detailed features of ESS (Chapter 4). In particular, we propose a form for the order?k?space structure function S?(k) that interpolates between inertial? and dissipation?range behaviours. We also predict the far?dissipation?range form of S?(k).