Multiscale velocity correlations in turbulence and Burgers turbulence: Fusion rules, Markov processes in scale, and multifractal predictions

Emergence of universal scaling in isotropic turbulence

Universal properties of turbulence have been associated traditionally with very high Reynolds numbers, but recent work has shown that the onset of the power laws in derivative statistics occurs at modest microscale Reynolds numbers of the order of 10, with the corresponding exponents being consistent with those for the inertial range structure functions at very high Reynolds numbers. In this paper we use well-resolved direct numerical simulations of homogeneous and isotropic turbulence to establish this result for a range of initial conditions with different forcing mechanisms. We also show that the moments of transverse velocity gradients possess larger scaling exponents than those of the longitudinal moments, confirming past results that the former are more intermittent than the latter.

Spontaneous symmetry breaking for extreme vorticity and strain in the three-dimensional Navier-Stokes equations

We investigate the spatio-temporal structure of the most likely configurations realizing extremely high vorticity or strain in the stochastically forced three-dimensional incompressible Navier-Stokes equations. Most likely configurations are computed by numerically finding the highest probability velocity field realizing an extreme constraint as solution of a large optimization problem. High-vorticity configurations are identified as pinched vortex filaments with swirl, while high-strain configurations correspond to counter-rotating vortex rings. We additionally observe that the most likely configurations for vorticity and strain spontaneously break their rotational symmetry for extremely high observable values. Instanton calculus and large deviation theory allow us to show that these maximum likelihood realizations determine the tail probabilities of the observed quantities. In particular, we are able to demonstrate that artificially enforcing rotational symmetry for large strain configurations leads to a severe underestimate of their probability, as it is dominated in likelihood by an exponentially more likely symmetry-broken vortex-sheet configuration. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Hidden scale invariance in Navier-Stokes intermittency

We expose a hidden scaling symmetry of the Navier-Stokes equations in the limit of vanishing viscosity, which stems from dynamical space-time rescaling around suitably defined Lagrangian scaling centres. At a dynamical level, the hidden symmetry projects solutions which differ up to Galilean invariance and global temporal scaling onto the same representative flow. At a statistical level, this projection repairs the scale invariance, which is broken by intermittency in the original formulation. Following previous work by the first author, we here postulate and substantiate with numerics that hidden symmetry statistically holds in the inertial interval of fully developed turbulence. We show that this symmetry accounts for the scale-invariance of a certain class of observables, in particular, the Kolmogorov multipliers. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Generalized Description of Intermittency in Turbulence via Stochastic Methods

We present a generalized picture of intermittency in turbulence that is based on the theory of stochastic processes. To this end, we rely on the experimentally and numerically verified finding by R. Friedrich and J. Peinke [Phys. Rev. Lett. 78, 863 (1997)] that allows for an interpretation of the turbulent energy cascade as a Markov process of velocity increments in scale. It is explicitly shown that phenomenological models of turbulence, which are characterized by scaling exponents ζn of velocity increment structure functions, can be reproduced by the Kramers–Moyal expansion of the velocity increment probability density function that is associated with a Markov process. We compare the different sets of Kramers–Moyal coefficients of each phenomenology and deduce that an accurate description of intermittency should take into account an infinite number of coefficients. This is demonstrated in more detail for the case of Burgers turbulence that exhibits pronounced intermittency effects. Moreover, the influence of nonlocality on Kramers–Moyal coefficients is investigated by direct numerical simulations of a generalized Burgers equation. Depending on the balance between nonlinearity and nonlocality, we encounter different intermittency behavior that ranges from self-similarity (purely nonlocal case) to intermittent behavior (intermediate case that agrees with Yakhot’s mean field theory [Phys. Rev. E 63 026307 (2001)]) to shock-like behavior (purely nonlinear Burgers case).

Probability Density Functions in Homogeneous and Isotropic Magneto-Hydrodynamic Turbulence

We derive a hierarchy of evolution equations for multi-point probability density functions in magneto-hydrodynamic (MHD) turbulence. We discuss the relation to the moment hierarchy in MHD turbulence formulated by Chandrasekhar (S. Chandrasekhar, Proc. R. Soc. Lond. A 1951, 204, 435–449) and derive a functional equation for a joint characteristic functional, which can be considered as the analogon to the Hopf functional in hydrodynamic turbulence. Furthermore, we develop a closure method for the evolution equation of the single-point magnetic field probability density function, which is based on a joint Gaussian assumption for unclosed terms. It is explicitly shown that this closure, together with the assumptions of homogeneity and isotropy, leads to vanishing nonlinear terms. We discuss the implications of this finding for magnetic field generation and give a brief outlook on an axisymmetric theory, which includes a mean magnetic field.

Instanton based importance sampling for rare events in stochastic PDEs

We present a new method for sampling rare and large fluctuations in a nonequilibrium system governed by a stochastic partial differential equation (SPDE) with additive forcing. To this end, we deploy the so-called instanton formalism that corresponds to a saddle-point approximation of the action in the path integral formulation of the underlying SPDE. The crucial step in our approach is the formulation of an alternative SPDE that incorporates knowledge of the instanton solution such that we are able to constrain the dynamical evolutions around extreme flow configurations only. Finally, a reweighting procedure based on the Girsanov theorem is applied to recover the full distribution function of the original system. The entire procedure is demonstrated on the example of the one-dimensional Burgers equation. Furthermore, we compare our method to conventional direct numerical simulations as well as to Hybrid Monte Carlo methods. It will be shown that the instanton-based sampling method outperforms both approaches and allows for an accurate quantification of the whole probability density function of velocity gradients from the core to the very far tails.

Onset of intermittency in stochastic Burgers hydrodynamics

We study the onset of intermittency in stochastic Burgers hydrodynamics, as characterized by the statistical behavior of negative velocity gradient fluctuations. The analysis is based on the response functional formalism, where specific velocity configurations-the viscous instantons-are assumed to play a dominant role in modeling the left tails of velocity gradient probability distribution functions. We find, as expected on general grounds, that the field-theoretical approach becomes meaningful in practice only if the effects of fluctuations around instantons are taken into account. Working with a systematic cumulant expansion, it turns out that the integration of fluctuations yields, in leading perturbative order, to an effective description of the Burgers stochastic dynamics given by the renormalization of its associated heat kernel propagator and the external force-force correlation function.